LMFDB - Embedded newform 1350.2.q.e.1007.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1350,2,Mod(143,1350)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1350, base_ring=CyclotomicField(12))

chi = DirichletCharacter(H, H._module([2, 9]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1350.143");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1350 = 2 \cdot 3^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1350.q (of order $12$, degree $4$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$10.7798042729$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$2$ over $\Q(\zeta_{12})$
Coefficient field:	$\Q(\zeta_{24})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 450)
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			1007.1
Root			$0.965926 - 0.258819i$ of defining polynomial
Character	$\chi$	$=$	1350.1007
Dual form			1350.2.q.e.1043.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.258819 - 0.965926i) q^{2} +(-0.866025 + 0.500000i) q^{4} +(0.707107 + 0.707107i) q^{8} +O(q^{10})$	\(q+(-0.258819 - 0.965926i) q^{2} +(-0.866025 + 0.500000i) q^{4} +(0.707107 + 0.707107i) q^{8} +(1.50000 + 0.866025i) q^{11} +(-6.69213 - 1.79315i) q^{13} +(0.500000 - 0.866025i) q^{16} +(2.12132 - 2.12132i) q^{17} -4.00000i q^{19} +(0.448288 - 1.67303i) q^{22} +(-1.55291 + 5.79555i) q^{23} +6.92820i q^{26} +(-1.73205 + 3.00000i) q^{29} +(-2.00000 - 3.46410i) q^{31} +(-0.965926 - 0.258819i) q^{32} +(-2.59808 - 1.50000i) q^{34} +(-4.89898 - 4.89898i) q^{37} +(-3.86370 + 1.03528i) q^{38} +(6.00000 - 3.46410i) q^{41} +(-2.24144 - 8.36516i) q^{43} -1.73205 q^{44} +6.00000 q^{46} +(-3.10583 - 11.5911i) q^{47} +(-6.06218 + 3.50000i) q^{49} +(6.69213 - 1.79315i) q^{52} +(-8.48528 - 8.48528i) q^{53} +(3.34607 + 0.896575i) q^{58} +(4.33013 + 7.50000i) q^{59} +(-4.00000 + 6.92820i) q^{61} +(-2.82843 + 2.82843i) q^{62} +1.00000i q^{64} +(-0.896575 + 3.34607i) q^{67} +(-0.776457 + 2.89778i) q^{68} -3.46410i q^{71} +(-4.89898 + 4.89898i) q^{73} +(-3.46410 + 6.00000i) q^{74} +(2.00000 + 3.46410i) q^{76} +(-3.46410 - 2.00000i) q^{79} +(-4.89898 - 4.89898i) q^{82} +(-8.69333 + 2.32937i) q^{83} +(-7.50000 + 4.33013i) q^{86} +(0.448288 + 1.67303i) q^{88} +1.73205 q^{89} +(-1.55291 - 5.79555i) q^{92} +(-10.3923 + 6.00000i) q^{94} +(15.0573 - 4.03459i) q^{97} +(4.94975 + 4.94975i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q + 12 q^{11} + 4 q^{16} - 16 q^{31} + 48 q^{41} + 48 q^{46} - 32 q^{61} + 16 q^{76} - 60 q^{86}+O(q^{100}) $ 8 * q + 12 * q^11 + 4 * q^16 - 16 * q^31 + 48 * q^41 + 48 * q^46 - 32 * q^61 + 16 * q^76 - 60 * q^86

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1350\mathbb{Z}\right)^\times$.

$n$	$1001$	$1027$
$\chi(n)$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{1}{4}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	−0.258819	−	0.965926i	−0.183013	−	0.683013i
$2$	−0.258819	−	0.965926i	−0.183013	−	0.683013i
$3$		0			0
$3$		0			0
$4$	−0.866025	+	0.500000i	−0.433013	+	0.250000i
$5$		0			0
$5$		0			0
$6$		0			0
$7$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$7$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$8$	0.707107	+	0.707107i	0.250000	+	0.250000i
$9$		0			0
$10$		0			0
$11$	1.50000	+	0.866025i	0.452267	+	0.261116i	0.708787	−	0.705422i	$-0.249243\pi$
$11$	1.50000	+	0.866025i	0.452267	+	0.261116i	−0.256520	+	0.966539i	$0.582576\pi$
$12$		0			0
$13$	−6.69213	−	1.79315i	−1.85606	−	0.497331i	−0.856248	−	0.516565i	$-0.827210\pi$
$13$	−6.69213	−	1.79315i	−1.85606	−	0.497331i	−0.999815	+	0.0192343i	$0.993877\pi$
$14$		0			0
$15$		0			0
$16$	0.500000	−	0.866025i	0.125000	−	0.216506i
$17$	2.12132	−	2.12132i	0.514496	−	0.514496i	−0.401405	−	0.915901i	$-0.631478\pi$
$17$	2.12132	−	2.12132i	0.514496	−	0.514496i	0.915901	+	0.401405i	$0.131478\pi$
$18$		0			0
$19$		−	4.00000i		−	0.917663i	−0.888523	−	0.458831i	$-0.848268\pi$
$19$		−	4.00000i		−	0.917663i	0.888523	−	0.458831i	$-0.151732\pi$
$20$		0			0
$21$		0			0
$22$	0.448288	−	1.67303i	0.0955753	−	0.356692i
$23$	−1.55291	+	5.79555i	−0.323805	+	1.20846i	0.591703	+	0.806156i	$0.298456\pi$
$23$	−1.55291	+	5.79555i	−0.323805	+	1.20846i	−0.915508	+	0.402300i	$0.868211\pi$
$24$		0			0
$25$		0			0
$26$			6.92820i			1.35873i
$27$		0			0
$28$		0			0
$29$	−1.73205	+	3.00000i	−0.321634	+	0.557086i	−0.980825	−	0.194889i	$-0.937565\pi$
$29$	−1.73205	+	3.00000i	−0.321634	+	0.557086i	0.659192	+	0.751975i	$0.270899\pi$
$30$		0			0
$31$	−2.00000	−	3.46410i	−0.359211	−	0.622171i	0.628619	−	0.777714i	$-0.283621\pi$
$31$	−2.00000	−	3.46410i	−0.359211	−	0.622171i	−0.987829	+	0.155543i	$0.950287\pi$
$32$	−0.965926	−	0.258819i	−0.170753	−	0.0457532i
$33$		0			0
$34$	−2.59808	−	1.50000i	−0.445566	−	0.257248i
$35$		0			0
$36$		0			0
$37$	−4.89898	−	4.89898i	−0.805387	−	0.805387i	0.178545	−	0.983932i	$-0.442861\pi$
$37$	−4.89898	−	4.89898i	−0.805387	−	0.805387i	−0.983932	+	0.178545i	$0.942861\pi$
$38$	−3.86370	+	1.03528i	−0.626775	+	0.167944i
$39$		0			0
$40$		0			0
$41$	6.00000	−	3.46410i	0.937043	−	0.541002i	0.0480106	−	0.998847i	$-0.484712\pi$
$41$	6.00000	−	3.46410i	0.937043	−	0.541002i	0.889032	+	0.457845i	$0.151379\pi$
$42$		0			0
$43$	−2.24144	−	8.36516i	−0.341816	−	1.27568i	−0.896288	−	0.443473i	$-0.853746\pi$
$43$	−2.24144	−	8.36516i	−0.341816	−	1.27568i	0.554472	−	0.832203i	$-0.312920\pi$
$44$	−1.73205			−0.261116
$45$		0			0
$46$	6.00000			0.884652
$47$	−3.10583	−	11.5911i	−0.453032	−	1.69074i	−0.693811	−	0.720157i	$-0.744070\pi$
$47$	−3.10583	−	11.5911i	−0.453032	−	1.69074i	0.240779	−	0.970580i	$-0.422597\pi$
$48$		0			0
$49$	−6.06218	+	3.50000i	−0.866025	+	0.500000i
$50$		0			0
$51$		0			0
$52$	6.69213	−	1.79315i	0.928032	−	0.248665i
$53$	−8.48528	−	8.48528i	−1.16554	−	1.16554i	−0.983243	−	0.182300i	$-0.941646\pi$
$53$	−8.48528	−	8.48528i	−1.16554	−	1.16554i	−0.182300	−	0.983243i	$-0.558354\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$	3.34607	+	0.896575i	0.439360	+	0.117726i
$59$	4.33013	+	7.50000i	0.563735	+	0.976417i	0.997166	+	0.0752304i	$0.0239692\pi$
$59$	4.33013	+	7.50000i	0.563735	+	0.976417i	−0.433432	+	0.901186i	$0.642697\pi$
$60$		0			0
$61$	−4.00000	+	6.92820i	−0.512148	+	0.887066i	0.487753	+	0.872982i	$0.337817\pi$
$61$	−4.00000	+	6.92820i	−0.512148	+	0.887066i	−0.999901	+	0.0140840i	$0.995517\pi$
$62$	−2.82843	+	2.82843i	−0.359211	+	0.359211i
$63$		0			0
$64$			1.00000i			0.125000i
$65$		0			0
$66$		0			0
$67$	−0.896575	+	3.34607i	−0.109534	+	0.408787i	−0.998820	−	0.0485648i	$-0.984535\pi$
$67$	−0.896575	+	3.34607i	−0.109534	+	0.408787i	0.889286	+	0.457352i	$0.151202\pi$
$68$	−0.776457	+	2.89778i	−0.0941593	+	0.351407i
$69$		0			0
$70$		0			0
$71$		−	3.46410i		−	0.411113i	−0.978645	−	0.205557i	$-0.934100\pi$
$71$		−	3.46410i		−	0.411113i	0.978645	−	0.205557i	$-0.0659005\pi$
$72$		0			0
$73$	−4.89898	+	4.89898i	−0.573382	+	0.573382i	−0.933072	−	0.359690i	$-0.882883\pi$
$73$	−4.89898	+	4.89898i	−0.573382	+	0.573382i	0.359690	+	0.933072i	$0.382883\pi$
$74$	−3.46410	+	6.00000i	−0.402694	+	0.697486i
$75$		0			0
$76$	2.00000	+	3.46410i	0.229416	+	0.397360i
$77$		0			0
$78$		0			0
$79$	−3.46410	−	2.00000i	−0.389742	−	0.225018i	0.292306	−	0.956325i	$-0.405577\pi$
$79$	−3.46410	−	2.00000i	−0.389742	−	0.225018i	−0.682048	+	0.731307i	$0.738911\pi$
$80$		0			0
$81$		0			0
$82$	−4.89898	−	4.89898i	−0.541002	−	0.541002i
$83$	−8.69333	+	2.32937i	−0.954217	+	0.255682i	−0.702151	−	0.712028i	$-0.747777\pi$
$83$	−8.69333	+	2.32937i	−0.954217	+	0.255682i	−0.252066	+	0.967710i	$0.581110\pi$
$84$		0			0
$85$		0			0
$86$	−7.50000	+	4.33013i	−0.808746	+	0.466930i
$87$		0			0
$88$	0.448288	+	1.67303i	0.0477876	+	0.178346i
$89$	1.73205			0.183597			0.0917985	−	0.995778i	$-0.470738\pi$
$89$	1.73205			0.183597			0.0917985	+	0.995778i	$0.470738\pi$
$90$		0			0
$91$		0			0
$92$	−1.55291	−	5.79555i	−0.161903	−	0.604228i
$93$		0			0
$94$	−10.3923	+	6.00000i	−1.07188	+	0.618853i
$95$		0			0
$96$		0			0
$97$	15.0573	−	4.03459i	1.52884	−	0.409651i	0.606197	−	0.795314i	$-0.292694\pi$
$97$	15.0573	−	4.03459i	1.52884	−	0.409651i	0.922639	+	0.385664i	$0.126028\pi$
$98$	4.94975	+	4.94975i	0.500000	+	0.500000i
$99$		0			0
$100$		0			0
$101$	−12.0000	−	6.92820i	−1.19404	−	0.689382i	−0.234823	−	0.972038i	$-0.575451\pi$
$101$	−12.0000	−	6.92820i	−1.19404	−	0.689382i	−0.959221	+	0.282656i	$0.908784\pi$
$102$		0			0
$103$	−6.69213	−	1.79315i	−0.659395	−	0.176684i	−0.0864221	−	0.996259i	$-0.527543\pi$
$103$	−6.69213	−	1.79315i	−0.659395	−	0.176684i	−0.572973	+	0.819574i	$0.694210\pi$
$104$	−3.46410	−	6.00000i	−0.339683	−	0.588348i
$105$		0			0
$106$	−6.00000	+	10.3923i	−0.582772	+	1.00939i
$107$	−8.48528	+	8.48528i	−0.820303	+	0.820303i	−0.986151	−	0.165848i	$-0.946964\pi$
$107$	−8.48528	+	8.48528i	−0.820303	+	0.820303i	0.165848	+	0.986151i	$0.446964\pi$
$108$		0			0
$109$		−	2.00000i		−	0.191565i	−0.995402	−	0.0957826i	$-0.969465\pi$
$109$		−	2.00000i		−	0.191565i	0.995402	−	0.0957826i	$-0.0305354\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	0.776457	−	2.89778i	0.0730429	−	0.272600i	−0.919739	−	0.392529i	$-0.871600\pi$
$113$	0.776457	−	2.89778i	0.0730429	−	0.272600i	0.992782	+	0.119929i	$0.0382668\pi$
$114$		0			0
$115$		0			0
$116$		−	3.46410i		−	0.321634i
$117$		0			0
$118$	6.12372	−	6.12372i	0.563735	−	0.563735i
$119$		0			0
$120$		0			0
$121$	−4.00000	−	6.92820i	−0.363636	−	0.629837i
$122$	7.72741	+	2.07055i	0.699607	+	0.187459i
$123$		0			0
$124$	3.46410	+	2.00000i	0.311086	+	0.179605i
$125$		0			0
$126$		0			0
$127$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$127$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$128$	0.965926	−	0.258819i	0.0853766	−	0.0228766i
$129$		0			0
$130$		0			0
$131$	15.0000	−	8.66025i	1.31056	−	0.756650i	0.328368	−	0.944550i	$-0.393501\pi$
$131$	15.0000	−	8.66025i	1.31056	−	0.756650i	0.982188	+	0.187900i	$0.0601681\pi$
$132$		0			0
$133$		0			0
$134$	3.46410			0.299253
$135$		0			0
$136$	3.00000			0.257248
$137$	4.65874	+	17.3867i	0.398023	+	1.48544i	0.816569	+	0.577247i	$0.195873\pi$
$137$	4.65874	+	17.3867i	0.398023	+	1.48544i	−0.418546	+	0.908196i	$0.637460\pi$
$138$		0			0
$139$	−11.2583	+	6.50000i	−0.954919	+	0.551323i	−0.894606	−	0.446857i	$-0.852543\pi$
$139$	−11.2583	+	6.50000i	−0.954919	+	0.551323i	−0.0603135	+	0.998179i	$0.519210\pi$
$140$		0			0
$141$		0			0
$142$	−3.34607	+	0.896575i	−0.280796	+	0.0752389i
$143$	−8.48528	−	8.48528i	−0.709575	−	0.709575i
$144$		0			0
$145$		0			0
$146$	6.00000	+	3.46410i	0.496564	+	0.286691i
$147$		0			0
$148$	6.69213	+	1.79315i	0.550090	+	0.147396i
$149$	5.19615	+	9.00000i	0.425685	+	0.737309i	0.996484	−	0.0837813i	$-0.0266997\pi$
$149$	5.19615	+	9.00000i	0.425685	+	0.737309i	−0.570799	+	0.821090i	$0.693366\pi$
$150$		0			0
$151$	1.00000	−	1.73205i	0.0813788	−	0.140952i	−0.822464	−	0.568818i	$-0.807401\pi$
$151$	1.00000	−	1.73205i	0.0813788	−	0.140952i	0.903842	+	0.427865i	$0.140734\pi$
$152$	2.82843	−	2.82843i	0.229416	−	0.229416i
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−1.79315	+	6.69213i	−0.143109	+	0.534090i	0.856723	+	0.515776i	$0.172496\pi$
$157$	−1.79315	+	6.69213i	−0.143109	+	0.534090i	−0.999832	+	0.0183138i	$0.994170\pi$
$158$	−1.03528	+	3.86370i	−0.0823622	+	0.307380i
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−1.22474	+	1.22474i	−0.0959294	+	0.0959294i	−0.753443	−	0.657513i	$-0.771608\pi$
$163$	−1.22474	+	1.22474i	−0.0959294	+	0.0959294i	0.657513	+	0.753443i	$0.271608\pi$
$164$	−3.46410	+	6.00000i	−0.270501	+	0.468521i
$165$		0			0
$166$	4.50000	+	7.79423i	0.349268	+	0.604949i
$167$	−5.79555	−	1.55291i	−0.448474	−	0.120168i	0.0275115	−	0.999621i	$-0.491242\pi$
$167$	−5.79555	−	1.55291i	−0.448474	−	0.120168i	−0.475985	+	0.879453i	$0.657908\pi$
$168$		0			0
$169$	30.3109	+	17.5000i	2.33161	+	1.34615i
$170$		0			0
$171$		0			0
$172$	6.12372	+	6.12372i	0.466930	+	0.466930i
$173$	17.3867	−	4.65874i	1.32188	−	0.354198i	0.472200	−	0.881491i	$-0.343460\pi$
$173$	17.3867	−	4.65874i	1.32188	−	0.354198i	0.849683	+	0.527294i	$0.176793\pi$
$174$		0			0
$175$		0			0
$176$	1.50000	−	0.866025i	0.113067	−	0.0652791i
$177$		0			0
$178$	−0.448288	−	1.67303i	−0.0336006	−	0.125399i
$179$	−12.1244			−0.906217			−0.453108	−	0.891455i	$-0.649685\pi$
$179$	−12.1244			−0.906217			−0.453108	+	0.891455i	$0.649685\pi$
$180$		0			0
$181$	−2.00000			−0.148659			−0.0743294	−	0.997234i	$-0.523682\pi$
$181$	−2.00000			−0.148659			−0.0743294	+	0.997234i	$0.523682\pi$
$182$		0			0
$183$		0			0
$184$	−5.19615	+	3.00000i	−0.383065	+	0.221163i
$185$		0			0
$186$		0			0
$187$	5.01910	−	1.34486i	0.367033	−	0.0983461i
$188$	8.48528	+	8.48528i	0.618853	+	0.618853i
$189$		0			0
$190$		0			0
$191$	−3.00000	−	1.73205i	−0.217072	−	0.125327i	0.387522	−	0.921861i	$-0.373331\pi$
$191$	−3.00000	−	1.73205i	−0.217072	−	0.125327i	−0.604594	+	0.796534i	$0.706665\pi$
$192$		0			0
$193$	5.01910	+	1.34486i	0.361283	+	0.0968054i	0.434894	−	0.900482i	$-0.356786\pi$
$193$	5.01910	+	1.34486i	0.361283	+	0.0968054i	−0.0736115	+	0.997287i	$0.523452\pi$
$194$	−7.79423	−	13.5000i	−0.559593	−	0.969244i
$195$		0			0
$196$	3.50000	−	6.06218i	0.250000	−	0.433013i
$197$	4.24264	−	4.24264i	0.302276	−	0.302276i	−0.539628	−	0.841904i	$-0.681435\pi$
$197$	4.24264	−	4.24264i	0.302276	−	0.302276i	0.841904	+	0.539628i	$0.181435\pi$
$198$		0			0
$199$		−	26.0000i		−	1.84309i	−0.388270	−	0.921546i	$-0.626927\pi$
$199$		−	26.0000i		−	1.84309i	0.388270	−	0.921546i	$-0.373073\pi$
$200$		0			0
$201$		0			0
$202$	−3.58630	+	13.3843i	−0.252331	+	0.941713i
$203$		0			0
$204$		0			0
$205$		0			0
$206$			6.92820i			0.482711i
$207$		0			0
$208$	−4.89898	+	4.89898i	−0.339683	+	0.339683i
$209$	3.46410	−	6.00000i	0.239617	−	0.415029i
$210$		0			0
$211$	−11.5000	−	19.9186i	−0.791693	−	1.37125i	−0.924918	−	0.380166i	$-0.875867\pi$
$211$	−11.5000	−	19.9186i	−0.791693	−	1.37125i	0.133226	−	0.991086i	$-0.457467\pi$
$212$	11.5911	+	3.10583i	0.796081	+	0.213309i
$213$		0			0
$214$	10.3923	+	6.00000i	0.710403	+	0.410152i
$215$		0			0
$216$		0			0
$217$		0			0
$218$	−1.93185	+	0.517638i	−0.130842	+	0.0350589i
$219$		0			0
$220$		0			0
$221$	−18.0000	+	10.3923i	−1.21081	+	0.699062i
$222$		0			0
$223$	0.896575	+	3.34607i	0.0600391	+	0.224069i	0.989426	−	0.145038i	$-0.0463303\pi$
$223$	0.896575	+	3.34607i	0.0600391	+	0.224069i	−0.929387	+	0.369107i	$0.879664\pi$
$224$		0			0
$225$		0			0
$226$	−3.00000			−0.199557
$227$	−0.776457	−	2.89778i	−0.0515353	−	0.192332i	0.935359	−	0.353699i	$-0.115076\pi$
$227$	−0.776457	−	2.89778i	−0.0515353	−	0.192332i	−0.986894	+	0.161367i	$0.948410\pi$
$228$		0			0
$229$	17.3205	−	10.0000i	1.14457	−	0.660819i	0.197013	−	0.980401i	$-0.436876\pi$
$229$	17.3205	−	10.0000i	1.14457	−	0.660819i	0.947559	+	0.319582i	$0.103543\pi$
$230$		0			0
$231$		0			0
$232$	−3.34607	+	0.896575i	−0.219680	+	0.0588631i
$233$	6.36396	+	6.36396i	0.416917	+	0.416917i	0.884140	−	0.467223i	$-0.154745\pi$
$233$	6.36396	+	6.36396i	0.416917	+	0.416917i	−0.467223	+	0.884140i	$0.654745\pi$
$234$		0			0
$235$		0			0
$236$	−7.50000	−	4.33013i	−0.488208	−	0.281867i
$237$		0			0
$238$		0			0
$239$	−5.19615	−	9.00000i	−0.336111	−	0.582162i	0.647586	−	0.761992i	$-0.275778\pi$
$239$	−5.19615	−	9.00000i	−0.336111	−	0.582162i	−0.983698	+	0.179830i	$0.942445\pi$
$240$		0			0
$241$	5.00000	−	8.66025i	0.322078	−	0.557856i	−0.658838	−	0.752285i	$-0.728952\pi$
$241$	5.00000	−	8.66025i	0.322078	−	0.557856i	0.980917	+	0.194429i	$0.0622852\pi$
$242$	−5.65685	+	5.65685i	−0.363636	+	0.363636i
$243$		0			0
$244$		−	8.00000i		−	0.512148i
$245$		0			0
$246$		0			0
$247$	−7.17260	+	26.7685i	−0.456382	+	1.70324i
$248$	1.03528	−	3.86370i	0.0657401	−	0.245345i
$249$		0			0
$250$		0			0
$251$			10.3923i			0.655956i	0.944685	+	0.327978i	$0.106367\pi$
$251$			10.3923i			0.655956i	−0.944685	+	0.327978i	$0.893633\pi$
$252$		0			0
$253$	−7.34847	+	7.34847i	−0.461994	+	0.461994i
$254$		0			0
$255$		0			0
$256$	−0.500000	−	0.866025i	−0.0312500	−	0.0541266i
$257$	2.89778	+	0.776457i	0.180758	+	0.0484341i	0.348063	−	0.937471i	$-0.386840\pi$
$257$	2.89778	+	0.776457i	0.180758	+	0.0484341i	−0.167304	+	0.985905i	$0.553506\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$	−12.2474	−	12.2474i	−0.756650	−	0.756650i
$263$	23.1822	−	6.21166i	1.42948	−	0.383027i	0.540641	−	0.841253i	$-0.318182\pi$
$263$	23.1822	−	6.21166i	1.42948	−	0.383027i	0.888836	+	0.458226i	$0.151515\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	−0.896575	−	3.34607i	−0.0547671	−	0.204393i
$269$	6.92820			0.422420			0.211210	−	0.977441i	$-0.432260\pi$
$269$	6.92820			0.422420			0.211210	+	0.977441i	$0.432260\pi$
$270$		0			0
$271$	8.00000			0.485965			0.242983	−	0.970031i	$-0.421874\pi$
$271$	8.00000			0.485965			0.242983	+	0.970031i	$0.421874\pi$
$272$	−0.776457	−	2.89778i	−0.0470796	−	0.175704i
$273$		0			0
$274$	15.5885	−	9.00000i	0.941733	−	0.543710i
$275$		0			0
$276$		0			0
$277$	3.34607	−	0.896575i	0.201046	−	0.0538700i	−0.156891	−	0.987616i	$-0.550147\pi$
$277$	3.34607	−	0.896575i	0.201046	−	0.0538700i	0.357936	+	0.933746i	$0.383480\pi$
$278$	9.19239	+	9.19239i	0.551323	+	0.551323i
$279$		0			0
$280$		0			0
$281$	18.0000	+	10.3923i	1.07379	+	0.619953i	0.929214	−	0.369541i	$-0.120485\pi$
$281$	18.0000	+	10.3923i	1.07379	+	0.619953i	0.144575	+	0.989494i	$0.453818\pi$
$282$		0			0
$283$	31.7876	+	8.51747i	1.88958	+	0.506311i	0.998637	+	0.0521913i	$0.0166205\pi$
$283$	31.7876	+	8.51747i	1.88958	+	0.506311i	0.890941	+	0.454120i	$0.150046\pi$
$284$	1.73205	+	3.00000i	0.102778	+	0.178017i
$285$		0			0
$286$	−6.00000	+	10.3923i	−0.354787	+	0.614510i
$287$		0			0
$288$		0			0
$289$			8.00000i			0.470588i
$290$		0			0
$291$		0			0
$292$	1.79315	−	6.69213i	0.104936	−	0.391627i
$293$	3.10583	−	11.5911i	0.181444	−	0.677160i	−0.813919	−	0.580978i	$-0.802670\pi$
$293$	3.10583	−	11.5911i	0.181444	−	0.677160i	0.995364	−	0.0961820i	$-0.0306631\pi$
$294$		0			0
$295$		0			0
$296$		−	6.92820i		−	0.402694i
$297$		0			0
$298$	7.34847	−	7.34847i	0.425685	−	0.425685i
$299$	20.7846	−	36.0000i	1.20201	−	2.08193i
$300$		0			0
$301$		0			0
$302$	−1.93185	−	0.517638i	−0.111166	−	0.0297867i
$303$		0			0
$304$	−3.46410	−	2.00000i	−0.198680	−	0.114708i
$305$		0			0
$306$		0			0
$307$	−15.9217	−	15.9217i	−0.908698	−	0.908698i	0.0874688	−	0.996167i	$-0.472122\pi$
$307$	−15.9217	−	15.9217i	−0.908698	−	0.908698i	−0.996167	+	0.0874688i	$0.972122\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	3.00000	−	1.73205i	0.170114	−	0.0982156i	−0.412525	−	0.910946i	$-0.635353\pi$
$311$	3.00000	−	1.73205i	0.170114	−	0.0982156i	0.582640	+	0.812731i	$0.302020\pi$
$312$		0			0
$313$	−0.448288	−	1.67303i	−0.0253387	−	0.0945654i	0.952099	−	0.305791i	$-0.0989210\pi$
$313$	−0.448288	−	1.67303i	−0.0253387	−	0.0945654i	−0.977437	+	0.211226i	$0.932254\pi$
$314$	6.92820			0.390981
$315$		0			0
$316$	4.00000			0.225018
$317$	7.76457	+	28.9778i	0.436102	+	1.62755i	0.738416	+	0.674346i	$0.235574\pi$
$317$	7.76457	+	28.9778i	0.436102	+	1.62755i	−0.302314	+	0.953208i	$0.597759\pi$
$318$		0			0
$319$	−5.19615	+	3.00000i	−0.290929	+	0.167968i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−8.48528	−	8.48528i	−0.472134	−	0.472134i
$324$		0			0
$325$		0			0
$326$	1.50000	+	0.866025i	0.0830773	+	0.0479647i
$327$		0			0
$328$	6.69213	+	1.79315i	0.369511	+	0.0990102i
$329$		0			0
$330$		0			0
$331$	0.500000	−	0.866025i	0.0274825	−	0.0476011i	−0.851957	−	0.523612i	$-0.824584\pi$
$331$	0.500000	−	0.866025i	0.0274825	−	0.0476011i	0.879440	+	0.476011i	$0.157918\pi$
$332$	6.36396	−	6.36396i	0.349268	−	0.349268i
$333$		0			0
$334$			6.00000i			0.328305i
$335$		0			0
$336$		0			0
$337$	2.24144	−	8.36516i	0.122099	−	0.455679i	−0.877621	−	0.479356i	$-0.840870\pi$
$337$	2.24144	−	8.36516i	0.122099	−	0.455679i	0.999720	+	0.0236762i	$0.00753707\pi$
$338$	9.05867	−	33.8074i	0.492727	−	1.83888i
$339$		0			0
$340$		0			0
$341$		−	6.92820i		−	0.375183i
$342$		0			0
$343$		0			0
$344$	4.33013	−	7.50000i	0.233465	−	0.404373i
$345$		0			0
$346$	−9.00000	−	15.5885i	−0.483843	−	0.838041i
$347$	−20.2844	−	5.43520i	−1.08893	−	0.291777i	−0.330678	−	0.943744i	$-0.607278\pi$
$347$	−20.2844	−	5.43520i	−1.08893	−	0.291777i	−0.758248	+	0.651967i	$0.773944\pi$
$348$		0			0
$349$	8.66025	+	5.00000i	0.463573	+	0.267644i	0.713545	−	0.700609i	$-0.247088\pi$
$349$	8.66025	+	5.00000i	0.463573	+	0.267644i	−0.249973	+	0.968253i	$0.580422\pi$
$350$		0			0
$351$		0			0
$352$	−1.22474	−	1.22474i	−0.0652791	−	0.0652791i
$353$	8.69333	−	2.32937i	0.462699	−	0.123980i	−0.0199361	−	0.999801i	$-0.506346\pi$
$353$	8.69333	−	2.32937i	0.462699	−	0.123980i	0.482635	+	0.875821i	$0.339680\pi$
$354$		0			0
$355$		0			0
$356$	−1.50000	+	0.866025i	−0.0794998	+	0.0458993i
$357$		0			0
$358$	3.13801	+	11.7112i	0.165849	+	0.618958i
$359$	13.8564			0.731313			0.365657	−	0.930750i	$-0.380844\pi$
$359$	13.8564			0.731313			0.365657	+	0.930750i	$0.380844\pi$
$360$		0			0
$361$	3.00000			0.157895
$362$	0.517638	+	1.93185i	0.0272065	+	0.101536i
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−23.4225	+	6.27603i	−1.22264	+	0.327606i	−0.811710	−	0.584060i	$-0.801463\pi$
$367$	−23.4225	+	6.27603i	−1.22264	+	0.327606i	−0.410932	+	0.911666i	$0.634797\pi$
$368$	4.24264	+	4.24264i	0.221163	+	0.221163i
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$373$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$374$	−2.59808	−	4.50000i	−0.134343	−	0.232689i
$375$		0			0
$376$	6.00000	−	10.3923i	0.309426	−	0.535942i
$377$	16.9706	−	16.9706i	0.874028	−	0.874028i
$378$		0			0
$379$		−	19.0000i		−	0.975964i	−0.872854	−	0.487982i	$-0.837733\pi$
$379$		−	19.0000i		−	0.975964i	0.872854	−	0.487982i	$-0.162267\pi$
$380$		0			0
$381$		0			0
$382$	−0.896575	+	3.34607i	−0.0458728	+	0.171200i
$383$	−7.76457	+	28.9778i	−0.396751	+	1.48070i	0.422026	+	0.906584i	$0.361319\pi$
$383$	−7.76457	+	28.9778i	−0.396751	+	1.48070i	−0.818777	+	0.574111i	$0.805348\pi$
$384$		0			0
$385$		0			0
$386$		−	5.19615i		−	0.264477i
$387$		0			0
$388$	−11.0227	+	11.0227i	−0.559593	+	0.559593i
$389$	5.19615	−	9.00000i	0.263455	−	0.456318i	−0.703702	−	0.710495i	$-0.748471\pi$
$389$	5.19615	−	9.00000i	0.263455	−	0.456318i	0.967158	+	0.254177i	$0.0818045\pi$
$390$		0			0
$391$	9.00000	+	15.5885i	0.455150	+	0.788342i
$392$	−6.76148	−	1.81173i	−0.341506	−	0.0915064i
$393$		0			0
$394$	−5.19615	−	3.00000i	−0.261778	−	0.151138i
$395$		0			0
$396$		0			0
$397$	−2.44949	−	2.44949i	−0.122936	−	0.122936i	0.642962	−	0.765898i	$-0.277705\pi$
$397$	−2.44949	−	2.44949i	−0.122936	−	0.122936i	−0.765898	+	0.642962i	$0.777705\pi$
$398$	−25.1141	+	6.72930i	−1.25885	+	0.337309i
$399$		0			0
$400$		0			0
$401$	−22.5000	+	12.9904i	−1.12360	+	0.648709i	−0.942317	−	0.334723i	$-0.891357\pi$
$401$	−22.5000	+	12.9904i	−1.12360	+	0.648709i	−0.181280	+	0.983432i	$0.558024\pi$
$402$		0			0
$403$	7.17260	+	26.7685i	0.357293	+	1.33344i
$404$	13.8564			0.689382
$405$		0			0
$406$		0			0
$407$	−3.10583	−	11.5911i	−0.153950	−	0.574550i
$408$		0			0
$409$	1.73205	−	1.00000i	0.0856444	−	0.0494468i	−0.456566	−	0.889689i	$-0.650921\pi$
$409$	1.73205	−	1.00000i	0.0856444	−	0.0494468i	0.542211	+	0.840243i	$0.317588\pi$
$410$		0			0
$411$		0			0
$412$	6.69213	−	1.79315i	0.329698	−	0.0883422i
$413$		0			0
$414$		0			0
$415$		0			0
$416$	6.00000	+	3.46410i	0.294174	+	0.169842i
$417$		0			0
$418$	−6.69213	−	1.79315i	−0.327323	−	0.0877059i
$419$	−2.59808	−	4.50000i	−0.126924	−	0.219839i	0.795559	−	0.605876i	$-0.207177\pi$
$419$	−2.59808	−	4.50000i	−0.126924	−	0.219839i	−0.922484	+	0.386037i	$0.873844\pi$
$420$		0			0
$421$	−14.0000	+	24.2487i	−0.682318	+	1.18181i	0.291953	+	0.956433i	$0.405695\pi$
$421$	−14.0000	+	24.2487i	−0.682318	+	1.18181i	−0.974272	+	0.225377i	$0.927639\pi$
$422$	−16.2635	+	16.2635i	−0.791693	+	0.791693i
$423$		0			0
$424$		−	12.0000i		−	0.582772i
$425$		0			0
$426$		0			0
$427$		0			0
$428$	3.10583	−	11.5911i	0.150126	−	0.560277i
$429$		0			0
$430$		0			0
$431$		−	24.2487i		−	1.16802i	−0.811747	−	0.584010i	$-0.801483\pi$
$431$		−	24.2487i		−	1.16802i	0.811747	−	0.584010i	$-0.198517\pi$
$432$		0			0
$433$	20.8207	−	20.8207i	1.00058	−	1.00058i	0.000577367	−	1.00000i	$-0.499816\pi$
$433$	20.8207	−	20.8207i	1.00058	−	1.00058i	1.00000		0.000577367i	$-0.000183782\pi$
$434$		0			0
$435$		0			0
$436$	1.00000	+	1.73205i	0.0478913	+	0.0829502i
$437$	23.1822	+	6.21166i	1.10896	+	0.297144i
$438$		0			0
$439$	−8.66025	−	5.00000i	−0.413331	−	0.238637i	0.278889	−	0.960323i	$-0.410034\pi$
$439$	−8.66025	−	5.00000i	−0.413331	−	0.238637i	−0.692220	+	0.721686i	$0.743367\pi$
$440$		0			0
$441$		0			0
$442$	14.6969	+	14.6969i	0.699062	+	0.699062i
$443$	8.69333	−	2.32937i	0.413033	−	0.110672i	−0.0463181	−	0.998927i	$-0.514749\pi$
$443$	8.69333	−	2.32937i	0.413033	−	0.110672i	0.459351	+	0.888255i	$0.348082\pi$
$444$		0			0
$445$		0			0
$446$	3.00000	−	1.73205i	0.142054	−	0.0820150i
$447$		0			0
$448$		0			0
$449$	−34.6410			−1.63481			−0.817405	−	0.576063i	$-0.804588\pi$
$449$	−34.6410			−1.63481			−0.817405	+	0.576063i	$0.804588\pi$
$450$		0			0
$451$	12.0000			0.565058
$452$	0.776457	+	2.89778i	0.0365215	+	0.136300i
$453$		0			0
$454$	−2.59808	+	1.50000i	−0.121934	+	0.0703985i
$455$		0			0
$456$		0			0
$457$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$457$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$458$	−14.1421	−	14.1421i	−0.660819	−	0.660819i
$459$		0			0
$460$		0			0
$461$	−15.0000	−	8.66025i	−0.698620	−	0.403348i	0.108213	−	0.994128i	$-0.465487\pi$
$461$	−15.0000	−	8.66025i	−0.698620	−	0.403348i	−0.806833	+	0.590779i	$0.798820\pi$
$462$		0			0
$463$	−3.34607	−	0.896575i	−0.155505	−	0.0416674i	0.180227	−	0.983625i	$-0.442317\pi$
$463$	−3.34607	−	0.896575i	−0.155505	−	0.0416674i	−0.335732	+	0.941958i	$0.608984\pi$
$464$	1.73205	+	3.00000i	0.0804084	+	0.139272i
$465$		0			0
$466$	4.50000	−	7.79423i	0.208458	−	0.361061i
$467$	10.6066	−	10.6066i	0.490815	−	0.490815i	−0.417748	−	0.908563i	$-0.637180\pi$
$467$	10.6066	−	10.6066i	0.490815	−	0.490815i	0.908563	+	0.417748i	$0.137180\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$	−2.24144	+	8.36516i	−0.103171	+	0.385038i
$473$	3.88229	−	14.4889i	0.178508	−	0.666200i
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$	−7.34847	+	7.34847i	−0.336111	+	0.336111i
$479$	−3.46410	+	6.00000i	−0.158279	+	0.274147i	−0.934248	−	0.356624i	$-0.883928\pi$
$479$	−3.46410	+	6.00000i	−0.158279	+	0.274147i	0.775969	+	0.630771i	$0.217261\pi$
$480$		0			0
$481$	24.0000	+	41.5692i	1.09431	+	1.89539i
$482$	−9.65926	−	2.58819i	−0.439967	−	0.117889i
$483$		0			0
$484$	6.92820	+	4.00000i	0.314918	+	0.181818i
$485$		0			0
$486$		0			0
$487$	12.2474	+	12.2474i	0.554985	+	0.554985i	0.927875	−	0.372890i	$-0.121633\pi$
$487$	12.2474	+	12.2474i	0.554985	+	0.554985i	−0.372890	+	0.927875i	$0.621633\pi$
$488$	−7.72741	+	2.07055i	−0.349803	+	0.0937295i
$489$		0			0
$490$		0			0
$491$	19.5000	−	11.2583i	0.880023	−	0.508081i	0.00935679	−	0.999956i	$-0.497022\pi$
$491$	19.5000	−	11.2583i	0.880023	−	0.508081i	0.870666	+	0.491875i	$0.163688\pi$
$492$		0			0
$493$	2.68973	+	10.0382i	0.121139	+	0.452098i
$494$	27.7128			1.24686
$495$		0			0
$496$	−4.00000			−0.179605
$497$		0			0
$498$		0			0
$499$	−4.33013	+	2.50000i	−0.193843	+	0.111915i	−0.593780	−	0.804627i	$-0.702365\pi$
$499$	−4.33013	+	2.50000i	−0.193843	+	0.111915i	0.399937	+	0.916542i	$0.369032\pi$
$500$		0			0
$501$		0			0
$502$	10.0382	−	2.68973i	0.448027	−	0.120048i
$503$	16.9706	+	16.9706i	0.756680	+	0.756680i	0.975717	−	0.219037i	$-0.0702914\pi$
$503$	16.9706	+	16.9706i	0.756680	+	0.756680i	−0.219037	+	0.975717i	$0.570291\pi$
$504$		0			0
$505$		0			0
$506$	9.00000	+	5.19615i	0.400099	+	0.230997i
$507$		0			0
$508$		0			0
$509$	15.5885	+	27.0000i	0.690946	+	1.19675i	0.971528	+	0.236924i	$0.0761392\pi$
$509$	15.5885	+	27.0000i	0.690946	+	1.19675i	−0.280582	+	0.959830i	$0.590527\pi$
$510$		0			0
$511$		0			0
$512$	−0.707107	+	0.707107i	−0.0312500	+	0.0312500i
$513$		0			0
$514$		−	3.00000i		−	0.132324i
$515$		0			0
$516$		0			0
$517$	5.37945	−	20.0764i	0.236588	−	0.882959i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	5.19615i		−	0.227648i	−0.993501	−	0.113824i	$-0.963690\pi$
$521$		−	5.19615i		−	0.227648i	0.993501	−	0.113824i	$-0.0363099\pi$
$522$		0			0
$523$	18.3712	−	18.3712i	0.803315	−	0.803315i	−0.180297	−	0.983612i	$-0.557706\pi$
$523$	18.3712	−	18.3712i	0.803315	−	0.803315i	0.983612	+	0.180297i	$0.0577059\pi$
$524$	−8.66025	+	15.0000i	−0.378325	+	0.655278i
$525$		0			0
$526$	−12.0000	−	20.7846i	−0.523225	−	0.906252i
$527$	−11.5911	−	3.10583i	−0.504917	−	0.135292i
$528$		0			0
$529$	−11.2583	−	6.50000i	−0.489493	−	0.282609i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−46.3644	+	12.4233i	−2.00827	+	0.538113i
$534$		0			0
$535$		0			0
$536$	−3.00000	+	1.73205i	−0.129580	+	0.0748132i
$537$		0			0
$538$	−1.79315	−	6.69213i	−0.0773082	−	0.288518i
$539$	−12.1244			−0.522233
$540$		0			0
$541$	−4.00000			−0.171973			−0.0859867	−	0.996296i	$-0.527404\pi$
$541$	−4.00000			−0.171973			−0.0859867	+	0.996296i	$0.527404\pi$
$542$	−2.07055	−	7.72741i	−0.0889378	−	0.331921i
$543$		0			0
$544$	−2.59808	+	1.50000i	−0.111392	+	0.0643120i
$545$		0			0
$546$		0			0
$547$	−16.7303	+	4.48288i	−0.715337	+	0.191674i	−0.598090	−	0.801429i	$-0.704074\pi$
$547$	−16.7303	+	4.48288i	−0.715337	+	0.191674i	−0.117247	+	0.993103i	$0.537407\pi$
$548$	−12.7279	−	12.7279i	−0.543710	−	0.543710i
$549$		0			0
$550$		0			0
$551$	12.0000	+	6.92820i	0.511217	+	0.295151i
$552$		0			0
$553$		0			0
$554$	−1.73205	−	3.00000i	−0.0735878	−	0.127458i
$555$		0			0
$556$	6.50000	−	11.2583i	0.275661	−	0.477460i
$557$	4.24264	−	4.24264i	0.179766	−	0.179766i	−0.611488	−	0.791254i	$-0.709429\pi$
$557$	4.24264	−	4.24264i	0.179766	−	0.179766i	0.791254	+	0.611488i	$0.209429\pi$
$558$		0			0
$559$			60.0000i			2.53773i
$560$		0			0
$561$		0			0
$562$	5.37945	−	20.0764i	0.226919	−	0.846871i
$563$	−5.43520	+	20.2844i	−0.229066	+	0.854887i	0.751668	+	0.659542i	$0.229250\pi$
$563$	−5.43520	+	20.2844i	−0.229066	+	0.854887i	−0.980734	+	0.195346i	$0.937417\pi$
$564$		0			0
$565$		0			0
$566$		−	32.9090i		−	1.38327i
$567$		0			0
$568$	2.44949	−	2.44949i	0.102778	−	0.102778i
$569$	−16.4545	+	28.5000i	−0.689808	+	1.19478i	0.282092	+	0.959387i	$0.408972\pi$
$569$	−16.4545	+	28.5000i	−0.689808	+	1.19478i	−0.971900	+	0.235395i	$0.924362\pi$
$570$		0			0
$571$	−8.50000	−	14.7224i	−0.355714	−	0.616115i	0.631526	−	0.775355i	$-0.282429\pi$
$571$	−8.50000	−	14.7224i	−0.355714	−	0.616115i	−0.987240	+	0.159240i	$0.949096\pi$
$572$	11.5911	+	3.10583i	0.484649	+	0.129861i
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	14.6969	+	14.6969i	0.611842	+	0.611842i	0.943426	−	0.331584i	$-0.107583\pi$
$577$	14.6969	+	14.6969i	0.611842	+	0.611842i	−0.331584	+	0.943426i	$0.607583\pi$
$578$	7.72741	−	2.07055i	0.321418	−	0.0861236i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−5.37945	−	20.0764i	−0.222794	−	0.831479i
$584$	−6.92820			−0.286691
$585$		0			0
$586$	−12.0000			−0.495715
$587$	6.98811	+	26.0800i	0.288430	+	1.07644i	0.946296	+	0.323301i	$0.104793\pi$
$587$	6.98811	+	26.0800i	0.288430	+	1.07644i	−0.657866	+	0.753135i	$0.728541\pi$
$588$		0			0
$589$	−13.8564	+	8.00000i	−0.570943	+	0.329634i
$590$		0			0
$591$		0			0
$592$	−6.69213	+	1.79315i	−0.275045	+	0.0736980i
$593$	14.8492	+	14.8492i	0.609785	+	0.609785i	0.942890	−	0.333105i	$-0.108096\pi$
$593$	14.8492	+	14.8492i	0.609785	+	0.609785i	−0.333105	+	0.942890i	$0.608096\pi$
$594$		0			0
$595$		0			0
$596$	−9.00000	−	5.19615i	−0.368654	−	0.212843i
$597$		0			0
$598$	−40.1528	−	10.7589i	−1.64197	−	0.439964i
$599$	−22.5167	−	39.0000i	−0.920006	−	1.59350i	−0.799402	−	0.600796i	$-0.794850\pi$
$599$	−22.5167	−	39.0000i	−0.920006	−	1.59350i	−0.120603	−	0.992701i	$-0.538483\pi$
$600$		0			0
$601$	−9.50000	+	16.4545i	−0.387513	+	0.671192i	−0.992114	−	0.125336i	$-0.959999\pi$
$601$	−9.50000	+	16.4545i	−0.387513	+	0.671192i	0.604601	+	0.796528i	$0.293332\pi$
$602$		0			0
$603$		0			0
$604$			2.00000i			0.0813788i
$605$		0			0
$606$		0			0
$607$	−1.79315	+	6.69213i	−0.0727818	+	0.271625i	−0.992721	−	0.120435i	$-0.961571\pi$
$607$	−1.79315	+	6.69213i	−0.0727818	+	0.271625i	0.919939	+	0.392061i	$0.128238\pi$
$608$	−1.03528	+	3.86370i	−0.0419860	+	0.156694i
$609$		0			0
$610$		0			0
$611$			83.1384i			3.36342i
$612$		0			0
$613$	−17.1464	+	17.1464i	−0.692538	+	0.692538i	−0.962790	−	0.270252i	$-0.912893\pi$
$613$	−17.1464	+	17.1464i	−0.692538	+	0.692538i	0.270252	+	0.962790i	$0.412893\pi$
$614$	−11.2583	+	19.5000i	−0.454349	+	0.786956i
$615$		0			0
$616$		0			0
$617$	−26.0800	−	6.98811i	−1.04994	−	0.281331i	−0.307714	−	0.951479i	$-0.599564\pi$
$617$	−26.0800	−	6.98811i	−1.04994	−	0.281331i	−0.742227	+	0.670148i	$0.766231\pi$
$618$		0			0
$619$	−32.0429	−	18.5000i	−1.28791	−	0.743578i	−0.309633	−	0.950856i	$-0.600206\pi$
$619$	−32.0429	−	18.5000i	−1.28791	−	0.743578i	−0.978282	+	0.207279i	$0.933539\pi$
$620$		0			0
$621$		0			0
$622$	−2.44949	−	2.44949i	−0.0982156	−	0.0982156i
$623$		0			0
$624$		0			0
$625$		0			0
$626$	−1.50000	+	0.866025i	−0.0599521	+	0.0346133i
$627$		0			0
$628$	−1.79315	−	6.69213i	−0.0715545	−	0.267045i
$629$	−20.7846			−0.828737
$630$		0			0
$631$	2.00000			0.0796187			0.0398094	−	0.999207i	$-0.487325\pi$
$631$	2.00000			0.0796187			0.0398094	+	0.999207i	$0.487325\pi$
$632$	−1.03528	−	3.86370i	−0.0411811	−	0.153690i
$633$		0			0
$634$	25.9808	−	15.0000i	1.03183	−	0.595726i
$635$		0			0
$636$		0			0
$637$	46.8449	−	12.5521i	1.85606	−	0.497331i
$638$	4.24264	+	4.24264i	0.167968	+	0.167968i
$639$		0			0
$640$		0			0
$641$	16.5000	+	9.52628i	0.651711	+	0.376265i	0.789111	−	0.614250i	$-0.210541\pi$
$641$	16.5000	+	9.52628i	0.651711	+	0.376265i	−0.137401	+	0.990516i	$0.543875\pi$
$642$		0			0
$643$	−45.1719	−	12.1038i	−1.78141	−	0.477326i	−0.790566	−	0.612376i	$-0.790214\pi$
$643$	−45.1719	−	12.1038i	−1.78141	−	0.477326i	−0.990839	+	0.135050i	$0.956880\pi$
$644$		0			0
$645$		0			0
$646$	−6.00000	+	10.3923i	−0.236067	+	0.408880i
$647$	8.48528	−	8.48528i	0.333591	−	0.333591i	−0.520358	−	0.853948i	$-0.674201\pi$
$647$	8.48528	−	8.48528i	0.333591	−	0.333591i	0.853948	+	0.520358i	$0.174201\pi$
$648$		0			0
$649$			15.0000i			0.588802i
$650$		0			0
$651$		0			0
$652$	0.448288	−	1.67303i	0.0175563	−	0.0655210i
$653$	−12.4233	+	46.3644i	−0.486162	+	1.81438i	0.0886092	+	0.996066i	$0.471758\pi$
$653$	−12.4233	+	46.3644i	−0.486162	+	1.81438i	−0.574771	+	0.818314i	$0.694909\pi$
$654$		0			0
$655$		0			0
$656$		−	6.92820i		−	0.270501i
$657$		0			0
$658$		0			0
$659$	−7.79423	+	13.5000i	−0.303620	+	0.525885i	−0.976953	−	0.213454i	$-0.931529\pi$
$659$	−7.79423	+	13.5000i	−0.303620	+	0.525885i	0.673333	+	0.739339i	$0.264862\pi$
$660$		0			0
$661$	−11.0000	−	19.0526i	−0.427850	−	0.741059i	0.568831	−	0.822454i	$-0.307396\pi$
$661$	−11.0000	−	19.0526i	−0.427850	−	0.741059i	−0.996682	+	0.0813955i	$0.974062\pi$
$662$	−0.965926	−	0.258819i	−0.0375418	−	0.0100593i
$663$		0			0
$664$	−7.79423	−	4.50000i	−0.302475	−	0.174634i
$665$		0			0
$666$		0			0
$667$	−14.6969	−	14.6969i	−0.569068	−	0.569068i
$668$	5.79555	−	1.55291i	0.224237	−	0.0600841i
$669$		0			0
$670$		0			0
$671$	−12.0000	+	6.92820i	−0.463255	+	0.267460i
$672$		0			0
$673$	−12.5521	−	46.8449i	−0.483846	−	1.80574i	−0.585201	−	0.810888i	$-0.698984\pi$
$673$	−12.5521	−	46.8449i	−0.483846	−	1.80574i	0.101355	−	0.994850i	$-0.467682\pi$
$674$	−8.66025			−0.333581
$675$		0			0
$676$	−35.0000			−1.34615
$677$	1.55291	+	5.79555i	0.0596833	+	0.222741i	0.989326	−	0.145722i	$-0.0465506\pi$
$677$	1.55291	+	5.79555i	0.0596833	+	0.222741i	−0.929642	+	0.368464i	$0.879884\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$	−6.69213	+	1.79315i	−0.256255	+	0.0686633i
$683$	−8.48528	−	8.48528i	−0.324680	−	0.324680i	0.525879	−	0.850559i	$-0.323736\pi$
$683$	−8.48528	−	8.48528i	−0.324680	−	0.324680i	−0.850559	+	0.525879i	$0.823736\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$	−8.36516	−	2.24144i	−0.318919	−	0.0854540i
$689$	41.5692	+	72.0000i	1.58366	+	2.74298i
$690$		0			0
$691$	17.5000	−	30.3109i	0.665731	−	1.15308i	−0.313355	−	0.949636i	$-0.601453\pi$
$691$	17.5000	−	30.3109i	0.665731	−	1.15308i	0.979086	−	0.203445i	$-0.0652137\pi$
$692$	−12.7279	+	12.7279i	−0.483843	+	0.483843i
$693$		0			0
$694$			21.0000i			0.797149i
$695$		0			0
$696$		0			0
$697$	5.37945	−	20.0764i	0.203761	−	0.760448i
$698$	2.58819	−	9.65926i	0.0979645	−	0.365608i
$699$		0			0
$700$		0			0
$701$		−	27.7128i		−	1.04670i	−0.852118	−	0.523349i	$-0.824682\pi$
$701$		−	27.7128i		−	1.04670i	0.852118	−	0.523349i	$-0.175318\pi$
$702$		0			0
$703$	−19.5959	+	19.5959i	−0.739074	+	0.739074i
$704$	−0.866025	+	1.50000i	−0.0326396	+	0.0565334i
$705$		0			0
$706$	−4.50000	−	7.79423i	−0.169360	−	0.293340i
$707$		0			0
$708$		0			0
$709$	−6.92820	−	4.00000i	−0.260194	−	0.150223i	0.364229	−	0.931309i	$-0.381333\pi$
$709$	−6.92820	−	4.00000i	−0.260194	−	0.150223i	−0.624423	+	0.781086i	$0.714666\pi$
$710$		0			0
$711$		0			0
$712$	1.22474	+	1.22474i	0.0458993	+	0.0458993i
$713$	23.1822	−	6.21166i	0.868181	−	0.232628i
$714$		0			0
$715$		0			0
$716$	10.5000	−	6.06218i	0.392403	−	0.226554i
$717$		0			0
$718$	−3.58630	−	13.3843i	−0.133840	−	0.499496i
$719$	−34.6410			−1.29189			−0.645946	−	0.763383i	$-0.723537\pi$
$719$	−34.6410			−1.29189			−0.645946	+	0.763383i	$0.723537\pi$
$720$		0			0
$721$		0			0
$722$	−0.776457	−	2.89778i	−0.0288967	−	0.107844i
$723$		0			0
$724$	1.73205	−	1.00000i	0.0643712	−	0.0371647i
$725$		0			0
$726$		0			0
$727$	−3.34607	+	0.896575i	−0.124099	+	0.0332521i	−0.320334	−	0.947305i	$-0.603795\pi$
$727$	−3.34607	+	0.896575i	−0.124099	+	0.0332521i	0.196235	+	0.980557i	$0.437128\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−22.5000	−	12.9904i	−0.832193	−	0.480467i
$732$		0			0
$733$	20.0764	+	5.37945i	0.741538	+	0.198695i	0.609762	−	0.792585i	$-0.291265\pi$
$733$	20.0764	+	5.37945i	0.741538	+	0.198695i	0.131777	+	0.991279i	$0.457932\pi$
$734$	12.1244	+	21.0000i	0.447518	+	0.775124i
$735$		0			0
$736$	3.00000	−	5.19615i	0.110581	−	0.191533i
$737$	−4.24264	+	4.24264i	−0.156280	+	0.156280i
$738$		0			0
$739$			41.0000i			1.50821i	0.656754	+	0.754105i	$0.271929\pi$
$739$			41.0000i			1.50821i	−0.656754	+	0.754105i	$0.728071\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	6.21166	−	23.1822i	0.227884	−	0.850473i	−0.753345	−	0.657625i	$-0.771561\pi$
$743$	6.21166	−	23.1822i	0.227884	−	0.850473i	0.981229	−	0.192848i	$-0.0617723\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$	−3.67423	+	3.67423i	−0.134343	+	0.134343i
$749$		0			0
$750$		0			0
$751$	19.0000	+	32.9090i	0.693320	+	1.20087i	0.970744	+	0.240118i	$0.0771860\pi$
$751$	19.0000	+	32.9090i	0.693320	+	1.20087i	−0.277424	+	0.960748i	$0.589481\pi$
$752$	−11.5911	−	3.10583i	−0.422684	−	0.113258i
$753$		0			0
$754$	−20.7846	−	12.0000i	−0.756931	−	0.437014i
$755$		0			0
$756$		0			0
$757$	29.3939	+	29.3939i	1.06834	+	1.06834i	0.997487	+	0.0708518i	$0.0225717\pi$
$757$	29.3939	+	29.3939i	1.06834	+	1.06834i	0.0708518	+	0.997487i	$0.477428\pi$
$758$	−18.3526	+	4.91756i	−0.666596	+	0.178614i
$759$		0			0
$760$		0			0
$761$	−46.5000	+	26.8468i	−1.68562	+	0.973195i	−0.727822	+	0.685766i	$0.759467\pi$
$761$	−46.5000	+	26.8468i	−1.68562	+	0.973195i	−0.957802	+	0.287429i	$0.907200\pi$
$762$		0			0
$763$		0			0
$764$	3.46410			0.125327
$765$		0			0
$766$	30.0000			1.08394
$767$	−15.5291	−	57.9555i	−0.560725	−	2.09265i
$768$		0			0
$769$	11.2583	−	6.50000i	0.405986	−	0.234396i	−0.283078	−	0.959097i	$-0.591355\pi$
$769$	11.2583	−	6.50000i	0.405986	−	0.234396i	0.689063	+	0.724701i	$0.258022\pi$
$770$		0			0
$771$		0			0
$772$	−5.01910	+	1.34486i	−0.180641	+	0.0484027i
$773$	4.24264	+	4.24264i	0.152597	+	0.152597i	0.779277	−	0.626680i	$-0.215587\pi$
$773$	4.24264	+	4.24264i	0.152597	+	0.152597i	−0.626680	+	0.779277i	$0.715587\pi$
$774$		0			0
$775$		0			0
$776$	13.5000	+	7.79423i	0.484622	+	0.279797i
$777$		0			0
$778$	−10.0382	−	2.68973i	−0.359887	−	0.0964314i
$779$	−13.8564	−	24.0000i	−0.496457	−	0.859889i
$780$		0			0
$781$	3.00000	−	5.19615i	0.107348	−	0.185933i
$782$	12.7279	−	12.7279i	0.455150	−	0.455150i
$783$		0			0
$784$			7.00000i			0.250000i
$785$		0			0
$786$		0			0
$787$	13.4486	−	50.1910i	0.479392	−	1.78912i	−0.124693	−	0.992195i	$-0.539795\pi$
$787$	13.4486	−	50.1910i	0.479392	−	1.78912i	0.604085	−	0.796920i	$-0.293539\pi$
$788$	−1.55291	+	5.79555i	−0.0553203	+	0.206458i
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	39.1918	−	39.1918i	1.39174	−	1.39174i
$794$	−1.73205	+	3.00000i	−0.0614682	+	0.106466i
$795$		0			0
$796$	13.0000	+	22.5167i	0.460773	+	0.798082i
$797$	−23.1822	−	6.21166i	−0.821156	−	0.220028i	−0.176304	−	0.984336i	$-0.556414\pi$
$797$	−23.1822	−	6.21166i	−0.821156	−	0.220028i	−0.644852	+	0.764308i	$0.723081\pi$
$798$		0			0
$799$	−31.1769	−	18.0000i	−1.10296	−	0.636794i
$800$		0			0
$801$		0			0
$802$	18.3712	+	18.3712i	0.648709	+	0.648709i
$803$	−11.5911	+	3.10583i	−0.409041	+	0.109602i
$804$		0			0
$805$		0			0
$806$	24.0000	−	13.8564i	0.845364	−	0.488071i
$807$		0			0
$808$	−3.58630	−	13.3843i	−0.126166	−	0.470857i
$809$	8.66025			0.304478			0.152239	−	0.988344i	$-0.451352\pi$
$809$	8.66025			0.304478			0.152239	+	0.988344i	$0.451352\pi$
$810$		0			0
$811$	−29.0000			−1.01833			−0.509164	−	0.860670i	$-0.670045\pi$
$811$	−29.0000			−1.01833			−0.509164	+	0.860670i	$0.670045\pi$
$812$		0			0
$813$		0			0
$814$	−10.3923	+	6.00000i	−0.364250	+	0.210300i
$815$		0			0
$816$		0			0
$817$	−33.4607	+	8.96575i	−1.17064	+	0.313672i
$818$	−1.41421	−	1.41421i	−0.0494468	−	0.0494468i
$819$		0			0
$820$		0			0
$821$	30.0000	+	17.3205i	1.04701	+	0.604490i	0.921810	−	0.387642i	$-0.126710\pi$
$821$	30.0000	+	17.3205i	1.04701	+	0.604490i	0.125197	+	0.992132i	$0.460044\pi$
$822$		0			0
$823$	6.69213	+	1.79315i	0.233273	+	0.0625053i	0.373562	−	0.927605i	$-0.378136\pi$
$823$	6.69213	+	1.79315i	0.233273	+	0.0625053i	−0.140289	+	0.990111i	$0.544803\pi$
$824$	−3.46410	−	6.00000i	−0.120678	−	0.209020i
$825$		0			0
$826$		0			0
$827$	−23.3345	+	23.3345i	−0.811421	+	0.811421i	−0.984847	−	0.173426i	$-0.944516\pi$
$827$	−23.3345	+	23.3345i	−0.811421	+	0.811421i	0.173426	+	0.984847i	$0.444516\pi$
$828$		0			0
$829$		−	16.0000i		−	0.555703i	−0.960624	−	0.277851i	$-0.910378\pi$
$829$		−	16.0000i		−	0.555703i	0.960624	−	0.277851i	$-0.0896223\pi$
$830$		0			0
$831$		0			0
$832$	1.79315	−	6.69213i	0.0621663	−	0.232008i
$833$	−5.43520	+	20.2844i	−0.188319	+	0.702814i
$834$		0			0
$835$		0			0
$836$			6.92820i			0.239617i
$837$		0			0
$838$	−3.67423	+	3.67423i	−0.126924	+	0.126924i
$839$	−3.46410	+	6.00000i	−0.119594	+	0.207143i	−0.919607	−	0.392840i	$-0.871493\pi$
$839$	−3.46410	+	6.00000i	−0.119594	+	0.207143i	0.800013	+	0.599983i	$0.204826\pi$
$840$		0			0
$841$	8.50000	+	14.7224i	0.293103	+	0.507670i
$842$	27.0459	+	7.24693i	0.932064	+	0.249746i
$843$		0			0
$844$	19.9186	+	11.5000i	0.685626	+	0.395846i
$845$		0			0
$846$		0			0
$847$		0			0
$848$	−11.5911	+	3.10583i	−0.398040	+	0.106655i
$849$		0			0
$850$		0			0
$851$	36.0000	−	20.7846i	1.23406	−	0.712487i
$852$		0			0
$853$	4.48288	+	16.7303i	0.153491	+	0.572835i	0.999230	+	0.0392388i	$0.0124933\pi$
$853$	4.48288	+	16.7303i	0.153491	+	0.572835i	−0.845739	+	0.533597i	$0.820840\pi$
$854$		0			0
$855$		0			0
$856$	−12.0000			−0.410152
$857$	−0.776457	−	2.89778i	−0.0265233	−	0.0989862i	0.951395	−	0.307972i	$-0.0996504\pi$
$857$	−0.776457	−	2.89778i	−0.0265233	−	0.0989862i	−0.977919	+	0.208986i	$0.932984\pi$
$858$		0			0
$859$	−27.7128	+	16.0000i	−0.945549	+	0.545913i	−0.891695	−	0.452636i	$-0.850484\pi$
$859$	−27.7128	+	16.0000i	−0.945549	+	0.545913i	−0.0538535	+	0.998549i	$0.517150\pi$
$860$		0			0
$861$		0			0
$862$	−23.4225	+	6.27603i	−0.797772	+	0.213762i
$863$	−33.9411	−	33.9411i	−1.15537	−	1.15537i	−0.985460	−	0.169910i	$-0.945652\pi$
$863$	−33.9411	−	33.9411i	−1.15537	−	1.15537i	−0.169910	−	0.985460i	$-0.554348\pi$
$864$		0			0
$865$		0			0
$866$	−25.5000	−	14.7224i	−0.866525	−	0.500289i
$867$		0			0
$868$		0			0
$869$	−3.46410	−	6.00000i	−0.117512	−	0.203536i
$870$		0			0
$871$	12.0000	−	20.7846i	0.406604	−	0.704260i
$872$	1.41421	−	1.41421i	0.0478913	−	0.0478913i
$873$		0			0
$874$		−	24.0000i		−	0.811812i
$875$		0			0
$876$		0			0
$877$	5.37945	−	20.0764i	0.181651	−	0.677932i	−0.813671	−	0.581325i	$-0.802534\pi$
$877$	5.37945	−	20.0764i	0.181651	−	0.677932i	0.995323	−	0.0966065i	$-0.0307988\pi$
$878$	−2.58819	+	9.65926i	−0.0873472	+	0.325984i
$879$		0			0
$880$		0			0
$881$			34.6410i			1.16709i	0.812082	+	0.583543i	$0.198334\pi$
$881$			34.6410i			1.16709i	−0.812082	+	0.583543i	$0.801666\pi$
$882$		0			0
$883$	−18.3712	+	18.3712i	−0.618239	+	0.618239i	−0.945080	−	0.326840i	$-0.894016\pi$
$883$	−18.3712	+	18.3712i	−0.618239	+	0.618239i	0.326840	+	0.945080i	$0.394016\pi$
$884$	10.3923	−	18.0000i	0.349531	−	0.605406i
$885$		0			0
$886$	−4.50000	−	7.79423i	−0.151180	−	0.261852i
$887$	40.5689	+	10.8704i	1.36217	+	0.364992i	0.864613	−	0.502439i	$-0.167564\pi$
$887$	40.5689	+	10.8704i	1.36217	+	0.364992i	0.497557	+	0.867431i	$0.334230\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$	−2.44949	−	2.44949i	−0.0820150	−	0.0820150i
$893$	−46.3644	+	12.4233i	−1.55153	+	0.415730i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	8.96575	+	33.4607i	0.299191	+	1.11660i
$899$	13.8564			0.462137
$900$		0			0
$901$	−36.0000			−1.19933
$902$	−3.10583	−	11.5911i	−0.103413	−	0.385942i
$903$		0			0
$904$	2.59808	−	1.50000i	0.0864107	−	0.0498893i
$905$		0			0
$906$		0			0
$907$	−8.36516	+	2.24144i	−0.277761	+	0.0744257i	−0.395010	−	0.918677i	$-0.629259\pi$
$907$	−8.36516	+	2.24144i	−0.277761	+	0.0744257i	0.117250	+	0.993102i	$0.462592\pi$
$908$	2.12132	+	2.12132i	0.0703985	+	0.0703985i
$909$		0			0
$910$		0			0
$911$	−33.0000	−	19.0526i	−1.09334	−	0.631239i	−0.158875	−	0.987299i	$-0.550787\pi$
$911$	−33.0000	−	19.0526i	−1.09334	−	0.631239i	−0.934463	+	0.356059i	$0.884120\pi$
$912$		0			0
$913$	−15.0573	−	4.03459i	−0.498324	−	0.133525i
$914$		0			0
$915$		0			0
$916$	−10.0000	+	17.3205i	−0.330409	+	0.572286i
$917$		0			0
$918$		0			0
$919$		−	34.0000i		−	1.12156i	−0.827966	−	0.560778i	$-0.810502\pi$
$919$		−	34.0000i		−	1.12156i	0.827966	−	0.560778i	$-0.189498\pi$
$920$		0			0
$921$		0			0
$922$	−4.48288	+	16.7303i	−0.147636	+	0.550984i
$923$	−6.21166	+	23.1822i	−0.204459	+	0.763052i
$924$		0			0
$925$		0			0
$926$			3.46410i			0.113837i
$927$		0			0
$928$	2.44949	−	2.44949i	0.0804084	−	0.0804084i
$929$	−17.3205	+	30.0000i	−0.568267	+	0.984268i	0.428470	+	0.903556i	$0.359053\pi$
$929$	−17.3205	+	30.0000i	−0.568267	+	0.984268i	−0.996737	+	0.0807121i	$0.974281\pi$
$930$		0			0
$931$	14.0000	+	24.2487i	0.458831	+	0.794719i
$932$	−8.69333	−	2.32937i	−0.284760	−	0.0763011i
$933$		0			0
$934$	−12.9904	−	7.50000i	−0.425058	−	0.245407i
$935$		0			0
$936$		0			0
$937$	−3.67423	−	3.67423i	−0.120032	−	0.120032i	0.644539	−	0.764571i	$-0.277049\pi$
$937$	−3.67423	−	3.67423i	−0.120032	−	0.120032i	−0.764571	+	0.644539i	$0.777049\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	18.0000	−	10.3923i	0.586783	−	0.338779i	−0.177041	−	0.984203i	$-0.556653\pi$
$941$	18.0000	−	10.3923i	0.586783	−	0.338779i	0.763825	+	0.645424i	$0.223319\pi$
$942$		0			0
$943$	10.7589	+	40.1528i	0.350358	+	1.30755i
$944$	8.66025			0.281867
$945$		0			0
$946$	−15.0000			−0.487692
$947$	−3.88229	−	14.4889i	−0.126157	−	0.470826i	0.873721	−	0.486427i	$-0.161700\pi$
$947$	−3.88229	−	14.4889i	−0.126157	−	0.470826i	−0.999878	+	0.0156019i	$0.995034\pi$
$948$		0			0
$949$	41.5692	−	24.0000i	1.34939	−	0.779073i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−14.8492	−	14.8492i	−0.481014	−	0.481014i	0.424441	−	0.905455i	$-0.360471\pi$
$953$	−14.8492	−	14.8492i	−0.481014	−	0.481014i	−0.905455	+	0.424441i	$0.860471\pi$
$954$		0			0
$955$		0			0
$956$	9.00000	+	5.19615i	0.291081	+	0.168056i
$957$		0			0
$958$	6.69213	+	1.79315i	0.216213	+	0.0579341i
$959$		0			0
$960$		0			0
$961$	7.50000	−	12.9904i	0.241935	−	0.419045i
$962$	33.9411	−	33.9411i	1.09431	−	1.09431i
$963$		0			0
$964$			10.0000i			0.322078i
$965$		0			0
$966$		0			0
$967$	3.58630	−	13.3843i	0.115328	−	0.430409i	−0.883984	−	0.467518i	$-0.845148\pi$
$967$	3.58630	−	13.3843i	0.115328	−	0.430409i	0.999311	+	0.0371092i	$0.0118150\pi$
$968$	2.07055	−	7.72741i	0.0665501	−	0.248368i
$969$		0			0
$970$		0			0
$971$		−	12.1244i		−	0.389089i	−0.980894	−	0.194545i	$-0.937677\pi$
$971$		−	12.1244i		−	0.389089i	0.980894	−	0.194545i	$-0.0623229\pi$
$972$		0			0
$973$		0			0
$974$	8.66025	−	15.0000i	0.277492	−	0.480631i
$975$		0			0
$976$	4.00000	+	6.92820i	0.128037	+	0.221766i
$977$	5.79555	+	1.55291i	0.185416	+	0.0496821i	0.350332	−	0.936625i	$-0.386069\pi$
$977$	5.79555	+	1.55291i	0.185416	+	0.0496821i	−0.164916	+	0.986308i	$0.552735\pi$
$978$		0			0
$979$	2.59808	+	1.50000i	0.0830349	+	0.0479402i
$980$		0			0
$981$		0			0
$982$	−15.9217	−	15.9217i	−0.508081	−	0.508081i
$983$	40.5689	−	10.8704i	1.29395	−	0.346712i	0.454788	−	0.890600i	$-0.349715\pi$
$983$	40.5689	−	10.8704i	1.29395	−	0.346712i	0.839158	+	0.543888i	$0.183048\pi$
$984$		0			0
$985$		0			0
$986$	9.00000	−	5.19615i	0.286618	−	0.165479i
$987$		0			0
$988$	−7.17260	−	26.7685i	−0.228191	−	0.851620i
$989$	51.9615			1.65228
$990$		0			0
$991$	32.0000			1.01651			0.508257	−	0.861206i	$-0.330290\pi$
$991$	32.0000			1.01651			0.508257	+	0.861206i	$0.330290\pi$
$992$	1.03528	+	3.86370i	0.0328701	+	0.122673i
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	10.0382	−	2.68973i	0.317913	−	0.0851845i	−0.0963340	−	0.995349i	$-0.530712\pi$
$997$	10.0382	−	2.68973i	0.317913	−	0.0851845i	0.414247	+	0.910165i	$0.364045\pi$
$998$	3.53553	+	3.53553i	0.111915	+	0.111915i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1350.2.q.e.1007.1		8
3.2	odd	2		450.2.p.b.407.2	yes	8
5.2	odd	4	inner	1350.2.q.e.143.2		8
5.3	odd	4	inner	1350.2.q.e.143.1		8
5.4	even	2	inner	1350.2.q.e.1007.2		8
9.4	even	3		450.2.p.b.257.2	yes	8
9.5	odd	6	inner	1350.2.q.e.557.1		8
15.2	even	4		450.2.p.b.443.1	yes	8
15.8	even	4		450.2.p.b.443.2	yes	8
15.14	odd	2		450.2.p.b.407.1	yes	8
45.4	even	6		450.2.p.b.257.1	✓	8
45.13	odd	12		450.2.p.b.293.2	yes	8
45.14	odd	6	inner	1350.2.q.e.557.2		8
45.22	odd	12		450.2.p.b.293.1	yes	8
45.23	even	12	inner	1350.2.q.e.1043.1		8
45.32	even	12	inner	1350.2.q.e.1043.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
450.2.p.b.257.1	✓	8	45.4	even	6
450.2.p.b.257.2	yes	8	9.4	even	3
450.2.p.b.293.1	yes	8	45.22	odd	12
450.2.p.b.293.2	yes	8	45.13	odd	12
450.2.p.b.407.1	yes	8	15.14	odd	2
450.2.p.b.407.2	yes	8	3.2	odd	2
450.2.p.b.443.1	yes	8	15.2	even	4
450.2.p.b.443.2	yes	8	15.8	even	4
1350.2.q.e.143.1		8	5.3	odd	4	inner
1350.2.q.e.143.2		8	5.2	odd	4	inner
1350.2.q.e.557.1		8	9.5	odd	6	inner
1350.2.q.e.557.2		8	45.14	odd	6	inner
1350.2.q.e.1007.1		8	1.1	even	1	trivial
1350.2.q.e.1007.2		8	5.4	even	2	inner
1350.2.q.e.1043.1		8	45.23	even	12	inner
1350.2.q.e.1043.2		8	45.32	even	12	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1350 = 2 \cdot 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1350.q (of order \(12\), degree \(4\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.258819 - 0.965926i) q^{2} +(-0.866025 + 0.500000i) q^{4} +(0.707107 + 0.707107i) q^{8} +O(q^{10})\)	\(q+(-0.258819 - 0.965926i) q^{2} +(-0.866025 + 0.500000i) q^{4} +(0.707107 + 0.707107i) q^{8} +(1.50000 + 0.866025i) q^{11} +(-6.69213 - 1.79315i) q^{13} +(0.500000 - 0.866025i) q^{16} +(2.12132 - 2.12132i) q^{17} -4.00000i q^{19} +(0.448288 - 1.67303i) q^{22} +(-1.55291 + 5.79555i) q^{23} +6.92820i q^{26} +(-1.73205 + 3.00000i) q^{29} +(-2.00000 - 3.46410i) q^{31} +(-0.965926 - 0.258819i) q^{32} +(-2.59808 - 1.50000i) q^{34} +(-4.89898 - 4.89898i) q^{37} +(-3.86370 + 1.03528i) q^{38} +(6.00000 - 3.46410i) q^{41} +(-2.24144 - 8.36516i) q^{43} -1.73205 q^{44} +6.00000 q^{46} +(-3.10583 - 11.5911i) q^{47} +(-6.06218 + 3.50000i) q^{49} +(6.69213 - 1.79315i) q^{52} +(-8.48528 - 8.48528i) q^{53} +(3.34607 + 0.896575i) q^{58} +(4.33013 + 7.50000i) q^{59} +(-4.00000 + 6.92820i) q^{61} +(-2.82843 + 2.82843i) q^{62} +1.00000i q^{64} +(-0.896575 + 3.34607i) q^{67} +(-0.776457 + 2.89778i) q^{68} -3.46410i q^{71} +(-4.89898 + 4.89898i) q^{73} +(-3.46410 + 6.00000i) q^{74} +(2.00000 + 3.46410i) q^{76} +(-3.46410 - 2.00000i) q^{79} +(-4.89898 - 4.89898i) q^{82} +(-8.69333 + 2.32937i) q^{83} +(-7.50000 + 4.33013i) q^{86} +(0.448288 + 1.67303i) q^{88} +1.73205 q^{89} +(-1.55291 - 5.79555i) q^{92} +(-10.3923 + 6.00000i) q^{94} +(15.0573 - 4.03459i) q^{97} +(4.94975 + 4.94975i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q + 12 q^{11} + 4 q^{16} - 16 q^{31} + 48 q^{41} + 48 q^{46} - 32 q^{61} + 16 q^{76} - 60 q^{86}+O(q^{100}) \) 8 * q + 12 * q^11 + 4 * q^16 - 16 * q^31 + 48 * q^41 + 48 * q^46 - 32 * q^61 + 16 * q^76 - 60 * q^86

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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