LMFDB - Embedded newform 2700.2.s.a.1549.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2700,2,Mod(1549,2700)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2700, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 2, 3]))

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

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

chi := DirichletCharacter("2700.1549");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2700 = 2^{2} \cdot 3^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2700.s (of order $6$, degree $2$, 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:	$21.5596085457$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 180)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			1549.1
Root			$-0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	2700.1549
Dual form			2700.2.s.a.2449.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.866025 + 0.500000i) q^{7} +O(q^{10})$	$q+(-0.866025 + 0.500000i) q^{7} +(-3.46410 - 2.00000i) q^{13} +6.00000i q^{17} -2.00000 q^{19} +(2.59808 + 1.50000i) q^{23} +(-1.50000 - 2.59808i) q^{29} +(5.00000 - 8.66025i) q^{31} -10.0000i q^{37} +(4.50000 - 7.79423i) q^{41} +(3.46410 - 2.00000i) q^{43} +(-7.79423 + 4.50000i) q^{47} +(-3.00000 + 5.19615i) q^{49} -6.00000i q^{53} +(3.00000 - 5.19615i) q^{59} +(0.500000 + 0.866025i) q^{61} +(-9.52628 - 5.50000i) q^{67} -12.0000 q^{71} +4.00000i q^{73} +(-5.00000 - 8.66025i) q^{79} +(-7.79423 + 4.50000i) q^{83} +9.00000 q^{89} +4.00000 q^{91} +(-8.66025 + 5.00000i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q - 8 q^{19} - 6 q^{29} + 20 q^{31} + 18 q^{41} - 12 q^{49} + 12 q^{59} + 2 q^{61} - 48 q^{71} - 20 q^{79} + 36 q^{89} + 16 q^{91}+O(q^{100}) $ 4 * q - 8 * q^19 - 6 * q^29 + 20 * q^31 + 18 * q^41 - 12 * q^49 + 12 * q^59 + 2 * q^61 - 48 * q^71 - 20 * q^79 + 36 * q^89 + 16 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2700\mathbb{Z}\right)^\times$.

$n$	$1001$	$1351$	$2377$
$\chi(n)$	$e\left(\frac{1}{3}\right)$	$1$	$-1$

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			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−0.866025	+	0.500000i	−0.327327	+	0.188982i	−0.654654	−	0.755929i	$-0.727186\pi$
$7$	−0.866025	+	0.500000i	−0.327327	+	0.188982i	0.327327	+	0.944911i	$0.393852\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$11$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$12$		0			0
$13$	−3.46410	−	2.00000i	−0.960769	−	0.554700i	−0.0643593	−	0.997927i	$-0.520500\pi$
$13$	−3.46410	−	2.00000i	−0.960769	−	0.554700i	−0.896410	+	0.443227i	$0.853834\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			6.00000i			1.45521i	0.685994	+	0.727607i	$0.259367\pi$
$17$			6.00000i			1.45521i	−0.685994	+	0.727607i	$0.740633\pi$
$18$		0			0
$19$	−2.00000			−0.458831			−0.229416	−	0.973329i	$-0.573682\pi$
$19$	−2.00000			−0.458831			−0.229416	+	0.973329i	$0.573682\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	2.59808	+	1.50000i	0.541736	+	0.312772i	0.745782	−	0.666190i	$-0.232076\pi$
$23$	2.59808	+	1.50000i	0.541736	+	0.312772i	−0.204046	+	0.978961i	$0.565409\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−1.50000	−	2.59808i	−0.278543	−	0.482451i	0.692480	−	0.721437i	$-0.256518\pi$
$29$	−1.50000	−	2.59808i	−0.278543	−	0.482451i	−0.971023	+	0.238987i	$0.923185\pi$
$30$		0			0
$31$	5.00000	−	8.66025i	0.898027	−	1.55543i	0.0680129	−	0.997684i	$-0.478334\pi$
$31$	5.00000	−	8.66025i	0.898027	−	1.55543i	0.830014	−	0.557743i	$-0.188333\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$		−	10.0000i		−	1.64399i	−0.569495	−	0.821995i	$-0.692861\pi$
$37$		−	10.0000i		−	1.64399i	0.569495	−	0.821995i	$-0.307139\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	4.50000	−	7.79423i	0.702782	−	1.21725i	−0.264704	−	0.964330i	$-0.585274\pi$
$41$	4.50000	−	7.79423i	0.702782	−	1.21725i	0.967486	−	0.252924i	$-0.0813924\pi$
$42$		0			0
$43$	3.46410	−	2.00000i	0.528271	−	0.304997i	−0.212041	−	0.977261i	$-0.568011\pi$
$43$	3.46410	−	2.00000i	0.528271	−	0.304997i	0.740312	+	0.672264i	$0.234678\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−7.79423	+	4.50000i	−1.13691	+	0.656392i	−0.945662	−	0.325150i	$-0.894585\pi$
$47$	−7.79423	+	4.50000i	−1.13691	+	0.656392i	−0.191243	+	0.981543i	$0.561252\pi$
$48$		0			0
$49$	−3.00000	+	5.19615i	−0.428571	+	0.742307i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	6.00000i		−	0.824163i	−0.911147	−	0.412082i	$-0.864802\pi$
$53$		−	6.00000i		−	0.824163i	0.911147	−	0.412082i	$-0.135198\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$	3.00000	−	5.19615i	0.390567	−	0.676481i	−0.601958	−	0.798528i	$-0.705612\pi$
$59$	3.00000	−	5.19615i	0.390567	−	0.676481i	0.992524	+	0.122047i	$0.0389457\pi$
$60$		0			0
$61$	0.500000	+	0.866025i	0.0640184	+	0.110883i	0.896258	−	0.443533i	$-0.146275\pi$
$61$	0.500000	+	0.866025i	0.0640184	+	0.110883i	−0.832240	+	0.554416i	$0.812942\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−9.52628	−	5.50000i	−1.16382	−	0.671932i	−0.211604	−	0.977356i	$-0.567869\pi$
$67$	−9.52628	−	5.50000i	−1.16382	−	0.671932i	−0.952217	+	0.305424i	$0.901202\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−12.0000			−1.42414			−0.712069	−	0.702109i	$-0.752242\pi$
$71$	−12.0000			−1.42414			−0.712069	+	0.702109i	$0.752242\pi$
$72$		0			0
$73$			4.00000i			0.468165i	0.972217	+	0.234082i	$0.0752085\pi$
$73$			4.00000i			0.468165i	−0.972217	+	0.234082i	$0.924791\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−5.00000	−	8.66025i	−0.562544	−	0.974355i	−0.997274	−	0.0737937i	$-0.976489\pi$
$79$	−5.00000	−	8.66025i	−0.562544	−	0.974355i	0.434730	−	0.900561i	$-0.356844\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−7.79423	+	4.50000i	−0.855528	+	0.493939i	−0.862512	−	0.506036i	$-0.831110\pi$
$83$	−7.79423	+	4.50000i	−0.855528	+	0.493939i	0.00698436	+	0.999976i	$0.497777\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	9.00000			0.953998			0.476999	−	0.878904i	$-0.341725\pi$
$89$	9.00000			0.953998			0.476999	+	0.878904i	$0.341725\pi$
$90$		0			0
$91$	4.00000			0.419314
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−8.66025	+	5.00000i	−0.879316	+	0.507673i	−0.870433	−	0.492287i	$-0.836161\pi$
$97$	−8.66025	+	5.00000i	−0.879316	+	0.507673i	−0.00888289	+	0.999961i	$0.502828\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−9.00000	−	15.5885i	−0.895533	−	1.55111i	−0.833143	−	0.553058i	$-0.813461\pi$
$101$	−9.00000	−	15.5885i	−0.895533	−	1.55111i	−0.0623905	−	0.998052i	$-0.519872\pi$
$102$		0			0
$103$	6.92820	+	4.00000i	0.682656	+	0.394132i	0.800855	−	0.598858i	$-0.204379\pi$
$103$	6.92820	+	4.00000i	0.682656	+	0.394132i	−0.118199	+	0.992990i	$0.537712\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	9.00000i		−	0.870063i	−0.900415	−	0.435031i	$-0.856737\pi$
$107$		−	9.00000i		−	0.870063i	0.900415	−	0.435031i	$-0.143263\pi$
$108$		0			0
$109$	19.0000			1.81987			0.909935	−	0.414751i	$-0.136131\pi$
$109$	19.0000			1.81987			0.909935	+	0.414751i	$0.136131\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	10.3923	+	6.00000i	0.977626	+	0.564433i	0.901553	−	0.432670i	$-0.142428\pi$
$113$	10.3923	+	6.00000i	0.977626	+	0.564433i	0.0760733	+	0.997102i	$0.475762\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−3.00000	−	5.19615i	−0.275010	−	0.476331i
$120$		0			0
$121$	5.50000	−	9.52628i	0.500000	−	0.866025i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$		−	7.00000i		−	0.621150i	−0.950549	−	0.310575i	$-0.899478\pi$
$127$		−	7.00000i		−	0.621150i	0.950549	−	0.310575i	$-0.100522\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	3.00000	−	5.19615i	0.262111	−	0.453990i	−0.704692	−	0.709514i	$-0.748915\pi$
$131$	3.00000	−	5.19615i	0.262111	−	0.453990i	0.966803	+	0.255524i	$0.0822479\pi$
$132$		0			0
$133$	1.73205	−	1.00000i	0.150188	−	0.0867110i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−10.3923	+	6.00000i	−0.887875	+	0.512615i	−0.873247	−	0.487278i	$-0.837990\pi$
$137$	−10.3923	+	6.00000i	−0.887875	+	0.512615i	−0.0146279	+	0.999893i	$0.504656\pi$
$138$		0			0
$139$	10.0000	−	17.3205i	0.848189	−	1.46911i	−0.0346338	−	0.999400i	$-0.511026\pi$
$139$	10.0000	−	17.3205i	0.848189	−	1.46911i	0.882823	−	0.469706i	$-0.155640\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−4.50000	+	7.79423i	−0.368654	+	0.638528i	−0.989355	−	0.145519i	$-0.953515\pi$
$149$	−4.50000	+	7.79423i	−0.368654	+	0.638528i	0.620701	+	0.784047i	$0.286848\pi$
$150$		0			0
$151$	−4.00000	−	6.92820i	−0.325515	−	0.563809i	0.656101	−	0.754673i	$-0.272204\pi$
$151$	−4.00000	−	6.92820i	−0.325515	−	0.563809i	−0.981617	+	0.190864i	$0.938871\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−1.73205	−	1.00000i	−0.138233	−	0.0798087i	0.429289	−	0.903167i	$-0.358764\pi$
$157$	−1.73205	−	1.00000i	−0.138233	−	0.0798087i	−0.567521	+	0.823359i	$0.692098\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−3.00000			−0.236433
$162$		0			0
$163$			4.00000i			0.313304i	0.987654	+	0.156652i	$0.0500701\pi$
$163$			4.00000i			0.313304i	−0.987654	+	0.156652i	$0.949930\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−12.9904	−	7.50000i	−1.00523	−	0.580367i	−0.0954356	−	0.995436i	$-0.530424\pi$
$167$	−12.9904	−	7.50000i	−1.00523	−	0.580367i	−0.909790	+	0.415068i	$0.863758\pi$
$168$		0			0
$169$	1.50000	+	2.59808i	0.115385	+	0.199852i
$170$		0			0
$171$		0			0
$172$		0			0
$173$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$173$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−24.0000			−1.79384			−0.896922	−	0.442189i	$-0.854202\pi$
$179$	−24.0000			−1.79384			−0.896922	+	0.442189i	$0.854202\pi$
$180$		0			0
$181$	5.00000			0.371647			0.185824	−	0.982583i	$-0.440505\pi$
$181$	5.00000			0.371647			0.185824	+	0.982583i	$0.440505\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$	3.00000	+	5.19615i	0.217072	+	0.375980i	0.953912	−	0.300088i	$-0.0970159\pi$
$191$	3.00000	+	5.19615i	0.217072	+	0.375980i	−0.736839	+	0.676068i	$0.763683\pi$
$192$		0			0
$193$	−3.46410	−	2.00000i	−0.249351	−	0.143963i	0.370116	−	0.928986i	$-0.379318\pi$
$193$	−3.46410	−	2.00000i	−0.249351	−	0.143963i	−0.619467	+	0.785022i	$0.712651\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	12.0000i		−	0.854965i	−0.904024	−	0.427482i	$-0.859401\pi$
$197$		−	12.0000i		−	0.854965i	0.904024	−	0.427482i	$-0.140599\pi$
$198$		0			0
$199$	−2.00000			−0.141776			−0.0708881	−	0.997484i	$-0.522583\pi$
$199$	−2.00000			−0.141776			−0.0708881	+	0.997484i	$0.522583\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	2.59808	+	1.50000i	0.182349	+	0.105279i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$	2.00000	−	3.46410i	0.137686	−	0.238479i	−0.788935	−	0.614477i	$-0.789367\pi$
$211$	2.00000	−	3.46410i	0.137686	−	0.238479i	0.926620	+	0.375999i	$0.122700\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$			10.0000i			0.678844i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	12.0000	−	20.7846i	0.807207	−	1.39812i
$222$		0			0
$223$	−9.52628	+	5.50000i	−0.637927	+	0.368307i	−0.783815	−	0.620994i	$-0.786729\pi$
$223$	−9.52628	+	5.50000i	−0.637927	+	0.368307i	0.145889	+	0.989301i	$0.453396\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−10.3923	+	6.00000i	−0.689761	+	0.398234i	−0.803523	−	0.595274i	$-0.797043\pi$
$227$	−10.3923	+	6.00000i	−0.689761	+	0.398234i	0.113761	+	0.993508i	$0.463710\pi$
$228$		0			0
$229$	−0.500000	+	0.866025i	−0.0330409	+	0.0572286i	−0.882073	−	0.471113i	$-0.843853\pi$
$229$	−0.500000	+	0.866025i	−0.0330409	+	0.0572286i	0.849032	+	0.528341i	$0.177186\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	30.0000i		−	1.96537i	−0.185296	−	0.982683i	$-0.559325\pi$
$233$		−	30.0000i		−	1.96537i	0.185296	−	0.982683i	$-0.440675\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$239$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$240$		0			0
$241$	3.50000	+	6.06218i	0.225455	+	0.390499i	0.956456	−	0.291877i	$-0.0942799\pi$
$241$	3.50000	+	6.06218i	0.225455	+	0.390499i	−0.731001	+	0.682376i	$0.760947\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	6.92820	+	4.00000i	0.440831	+	0.254514i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−18.0000			−1.13615			−0.568075	−	0.822977i	$-0.692312\pi$
$251$	−18.0000			−1.13615			−0.568075	+	0.822977i	$0.692312\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$	5.19615	+	3.00000i	0.324127	+	0.187135i	0.653231	−	0.757159i	$-0.273413\pi$
$257$	5.19615	+	3.00000i	0.324127	+	0.187135i	−0.329104	+	0.944294i	$0.606747\pi$
$258$		0			0
$259$	5.00000	+	8.66025i	0.310685	+	0.538122i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−20.7846	+	12.0000i	−1.28163	+	0.739952i	−0.977147	−	0.212565i	$-0.931818\pi$
$263$	−20.7846	+	12.0000i	−1.28163	+	0.739952i	−0.304487	+	0.952517i	$0.598485\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$	9.00000			0.548740			0.274370	−	0.961624i	$-0.411531\pi$
$269$	9.00000			0.548740			0.274370	+	0.961624i	$0.411531\pi$
$270$		0			0
$271$	14.0000			0.850439			0.425220	−	0.905090i	$-0.360197\pi$
$271$	14.0000			0.850439			0.425220	+	0.905090i	$0.360197\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	1.73205	−	1.00000i	0.104069	−	0.0600842i	−0.447062	−	0.894503i	$-0.647530\pi$
$277$	1.73205	−	1.00000i	0.104069	−	0.0600842i	0.551131	+	0.834419i	$0.314196\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	16.5000	+	28.5788i	0.984307	+	1.70487i	0.644974	+	0.764204i	$0.276868\pi$
$281$	16.5000	+	28.5788i	0.984307	+	1.70487i	0.339333	+	0.940666i	$0.389799\pi$
$282$		0			0
$283$	−0.866025	−	0.500000i	−0.0514799	−	0.0297219i	0.474039	−	0.880504i	$-0.342796\pi$
$283$	−0.866025	−	0.500000i	−0.0514799	−	0.0297219i	−0.525519	+	0.850782i	$0.676129\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			9.00000i			0.531253i
$288$		0			0
$289$	−19.0000			−1.11765
$290$		0			0
$291$		0			0
$292$		0			0
$293$	5.19615	+	3.00000i	0.303562	+	0.175262i	0.644042	−	0.764990i	$-0.277256\pi$
$293$	5.19615	+	3.00000i	0.303562	+	0.175262i	−0.340480	+	0.940252i	$0.610589\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−6.00000	−	10.3923i	−0.346989	−	0.601003i
$300$		0			0
$301$	−2.00000	+	3.46410i	−0.115278	+	0.199667i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$			29.0000i			1.65512i	0.561379	+	0.827559i	$0.310271\pi$
$307$			29.0000i			1.65512i	−0.561379	+	0.827559i	$0.689729\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	9.00000	−	15.5885i	0.510343	−	0.883940i	−0.489585	−	0.871956i	$-0.662852\pi$
$311$	9.00000	−	15.5885i	0.510343	−	0.883940i	0.999928	−	0.0119847i	$-0.00381495\pi$
$312$		0			0
$313$	13.8564	−	8.00000i	0.783210	−	0.452187i	−0.0543564	−	0.998522i	$-0.517311\pi$
$313$	13.8564	−	8.00000i	0.783210	−	0.452187i	0.837567	+	0.546335i	$0.183977\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	10.3923	−	6.00000i	0.583690	−	0.336994i	−0.178908	−	0.983866i	$-0.557257\pi$
$317$	10.3923	−	6.00000i	0.583690	−	0.336994i	0.762598	+	0.646872i	$0.223923\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	12.0000i		−	0.667698i
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$	4.50000	−	7.79423i	0.248093	−	0.429710i
$330$		0			0
$331$	−13.0000	−	22.5167i	−0.714545	−	1.23763i	−0.963135	−	0.269019i	$-0.913301\pi$
$331$	−13.0000	−	22.5167i	−0.714545	−	1.23763i	0.248590	−	0.968609i	$-0.420033\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−6.92820	−	4.00000i	−0.377403	−	0.217894i	0.299285	−	0.954164i	$-0.403252\pi$
$337$	−6.92820	−	4.00000i	−0.377403	−	0.217894i	−0.676688	+	0.736270i	$0.736585\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		−	13.0000i		−	0.701934i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	20.7846	+	12.0000i	1.11578	+	0.644194i	0.940319	−	0.340293i	$-0.110526\pi$
$347$	20.7846	+	12.0000i	1.11578	+	0.644194i	0.175457	+	0.984487i	$0.443860\pi$
$348$		0			0
$349$	11.5000	+	19.9186i	0.615581	+	1.06622i	0.990282	+	0.139072i	$0.0444119\pi$
$349$	11.5000	+	19.9186i	0.615581	+	1.06622i	−0.374701	+	0.927146i	$0.622255\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	10.3923	−	6.00000i	0.553127	−	0.319348i	−0.197256	−	0.980352i	$-0.563203\pi$
$353$	10.3923	−	6.00000i	0.553127	−	0.319348i	0.750382	+	0.661004i	$0.229870\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−6.00000			−0.316668			−0.158334	−	0.987386i	$-0.550612\pi$
$359$	−6.00000			−0.316668			−0.158334	+	0.987386i	$0.550612\pi$
$360$		0			0
$361$	−15.0000			−0.789474
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	27.7128	−	16.0000i	1.44660	−	0.835193i	0.448320	−	0.893873i	$-0.352022\pi$
$367$	27.7128	−	16.0000i	1.44660	−	0.835193i	0.998277	+	0.0586798i	$0.0186891\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	3.00000	+	5.19615i	0.155752	+	0.269771i
$372$		0			0
$373$	12.1244	+	7.00000i	0.627775	+	0.362446i	0.779890	−	0.625917i	$-0.215275\pi$
$373$	12.1244	+	7.00000i	0.627775	+	0.362446i	−0.152115	+	0.988363i	$0.548608\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			12.0000i			0.618031i
$378$		0			0
$379$	−14.0000			−0.719132			−0.359566	−	0.933120i	$-0.617075\pi$
$379$	−14.0000			−0.719132			−0.359566	+	0.933120i	$0.617075\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	20.7846	+	12.0000i	1.06204	+	0.613171i	0.925997	−	0.377531i	$-0.123227\pi$
$383$	20.7846	+	12.0000i	1.06204	+	0.613171i	0.136047	+	0.990702i	$0.456560\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	7.50000	+	12.9904i	0.380265	+	0.658638i	0.991100	−	0.133120i	$-0.0424994\pi$
$389$	7.50000	+	12.9904i	0.380265	+	0.658638i	−0.610835	+	0.791758i	$0.709166\pi$
$390$		0			0
$391$	−9.00000	+	15.5885i	−0.455150	+	0.788342i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$		−	22.0000i		−	1.10415i	−0.833795	−	0.552074i	$-0.813837\pi$
$397$		−	22.0000i		−	1.10415i	0.833795	−	0.552074i	$-0.186163\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−9.00000	+	15.5885i	−0.449439	+	0.778450i	−0.998350	−	0.0574304i	$-0.981709\pi$
$401$	−9.00000	+	15.5885i	−0.449439	+	0.778450i	0.548911	+	0.835881i	$0.315043\pi$
$402$		0			0
$403$	−34.6410	+	20.0000i	−1.72559	+	0.996271i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−5.00000	+	8.66025i	−0.247234	+	0.428222i	−0.962757	−	0.270367i	$-0.912855\pi$
$409$	−5.00000	+	8.66025i	−0.247234	+	0.428222i	0.715523	+	0.698589i	$0.246188\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			6.00000i			0.295241i
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−6.00000	+	10.3923i	−0.293119	+	0.507697i	−0.974546	−	0.224189i	$-0.928027\pi$
$419$	−6.00000	+	10.3923i	−0.293119	+	0.507697i	0.681426	+	0.731887i	$0.261360\pi$
$420$		0			0
$421$	11.0000	+	19.0526i	0.536107	+	0.928565i	0.999109	+	0.0422075i	$0.0134391\pi$
$421$	11.0000	+	19.0526i	0.536107	+	0.928565i	−0.463002	+	0.886357i	$0.653228\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−0.866025	−	0.500000i	−0.0419099	−	0.0241967i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−6.00000			−0.289010			−0.144505	−	0.989504i	$-0.546159\pi$
$431$	−6.00000			−0.289010			−0.144505	+	0.989504i	$0.546159\pi$
$432$		0			0
$433$		−	26.0000i		−	1.24948i	−0.780833	−	0.624740i	$-0.785205\pi$
$433$		−	26.0000i		−	1.24948i	0.780833	−	0.624740i	$-0.214795\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−5.19615	−	3.00000i	−0.248566	−	0.143509i
$438$		0			0
$439$	−14.0000	−	24.2487i	−0.668184	−	1.15733i	−0.978412	−	0.206666i	$-0.933739\pi$
$439$	−14.0000	−	24.2487i	−0.668184	−	1.15733i	0.310228	−	0.950662i	$-0.399595\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−23.3827	+	13.5000i	−1.11094	+	0.641404i	−0.939074	−	0.343715i	$-0.888315\pi$
$443$	−23.3827	+	13.5000i	−1.11094	+	0.641404i	−0.171871	+	0.985119i	$0.554981\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	30.0000			1.41579			0.707894	−	0.706319i	$-0.249646\pi$
$449$	30.0000			1.41579			0.707894	+	0.706319i	$0.249646\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−8.66025	+	5.00000i	−0.405110	+	0.233890i	−0.688686	−	0.725059i	$-0.741812\pi$
$457$	−8.66025	+	5.00000i	−0.405110	+	0.233890i	0.283577	+	0.958950i	$0.408479\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	16.5000	+	28.5788i	0.768482	+	1.33105i	0.938386	+	0.345589i	$0.112321\pi$
$461$	16.5000	+	28.5788i	0.768482	+	1.33105i	−0.169904	+	0.985461i	$0.554346\pi$
$462$		0			0
$463$	−3.46410	−	2.00000i	−0.160990	−	0.0929479i	0.417340	−	0.908750i	$-0.362962\pi$
$463$	−3.46410	−	2.00000i	−0.160990	−	0.0929479i	−0.578331	+	0.815802i	$0.696296\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	36.0000i		−	1.66588i	−0.553362	−	0.832941i	$-0.686655\pi$
$467$		−	36.0000i		−	1.66588i	0.553362	−	0.832941i	$-0.313345\pi$
$468$		0			0
$469$	11.0000			0.507933
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$	15.0000	+	25.9808i	0.685367	+	1.18709i	0.973321	+	0.229447i	$0.0736918\pi$
$479$	15.0000	+	25.9808i	0.685367	+	1.18709i	−0.287954	+	0.957644i	$0.592975\pi$
$480$		0			0
$481$	−20.0000	+	34.6410i	−0.911922	+	1.57949i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$		−	16.0000i		−	0.725029i	−0.931978	−	0.362515i	$-0.881918\pi$
$487$		−	16.0000i		−	0.725029i	0.931978	−	0.362515i	$-0.118082\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−12.0000	+	20.7846i	−0.541552	+	0.937996i	0.457263	+	0.889332i	$0.348830\pi$
$491$	−12.0000	+	20.7846i	−0.541552	+	0.937996i	−0.998815	+	0.0486647i	$0.984503\pi$
$492$		0			0
$493$	15.5885	−	9.00000i	0.702069	−	0.405340i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	10.3923	−	6.00000i	0.466159	−	0.269137i
$498$		0			0
$499$	−2.00000	+	3.46410i	−0.0895323	+	0.155074i	−0.907314	−	0.420455i	$-0.861871\pi$
$499$	−2.00000	+	3.46410i	−0.0895323	+	0.155074i	0.817781	+	0.575529i	$0.195204\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			3.00000i			0.133763i	0.997761	+	0.0668817i	$0.0213050\pi$
$503$			3.00000i			0.133763i	−0.997761	+	0.0668817i	$0.978695\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$	4.50000	−	7.79423i	0.199459	−	0.345473i	−0.748894	−	0.662690i	$-0.769415\pi$
$509$	4.50000	−	7.79423i	0.199459	−	0.345473i	0.948353	+	0.317217i	$0.102748\pi$
$510$		0			0
$511$	−2.00000	−	3.46410i	−0.0884748	−	0.153243i
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−27.0000			−1.18289			−0.591446	−	0.806345i	$-0.701443\pi$
$521$	−27.0000			−1.18289			−0.591446	+	0.806345i	$0.701443\pi$
$522$		0			0
$523$			7.00000i			0.306089i	0.988219	+	0.153044i	$0.0489077\pi$
$523$			7.00000i			0.306089i	−0.988219	+	0.153044i	$0.951092\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	51.9615	+	30.0000i	2.26348	+	1.30682i
$528$		0			0
$529$	−7.00000	−	12.1244i	−0.304348	−	0.527146i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−31.1769	+	18.0000i	−1.35042	+	0.779667i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−7.00000			−0.300954			−0.150477	−	0.988614i	$-0.548081\pi$
$541$	−7.00000			−0.300954			−0.150477	+	0.988614i	$0.548081\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−6.06218	+	3.50000i	−0.259200	+	0.149649i	−0.623970	−	0.781449i	$-0.714481\pi$
$547$	−6.06218	+	3.50000i	−0.259200	+	0.149649i	0.364770	+	0.931098i	$0.381148\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	3.00000	+	5.19615i	0.127804	+	0.221364i
$552$		0			0
$553$	8.66025	+	5.00000i	0.368271	+	0.212622i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			6.00000i			0.254228i	0.991888	+	0.127114i	$0.0405714\pi$
$557$			6.00000i			0.254228i	−0.991888	+	0.127114i	$0.959429\pi$
$558$		0			0
$559$	−16.0000			−0.676728
$560$		0			0
$561$		0			0
$562$		0			0
$563$	33.7750	+	19.5000i	1.42345	+	0.821827i	0.996592	−	0.0824933i	$-0.0262883\pi$
$563$	33.7750	+	19.5000i	1.42345	+	0.821827i	0.426855	+	0.904320i	$0.359622\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−3.00000	−	5.19615i	−0.125767	−	0.217834i	0.796266	−	0.604947i	$-0.206806\pi$
$569$	−3.00000	−	5.19615i	−0.125767	−	0.217834i	−0.922032	+	0.387113i	$0.873472\pi$
$570$		0			0
$571$	−16.0000	+	27.7128i	−0.669579	+	1.15975i	0.308443	+	0.951243i	$0.400192\pi$
$571$	−16.0000	+	27.7128i	−0.669579	+	1.15975i	−0.978022	+	0.208502i	$0.933141\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$			38.0000i			1.58196i	0.611842	+	0.790980i	$0.290429\pi$
$577$			38.0000i			1.58196i	−0.611842	+	0.790980i	$0.709571\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	4.50000	−	7.79423i	0.186691	−	0.323359i
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$	23.3827	−	13.5000i	0.965107	−	0.557205i	0.0673658	−	0.997728i	$-0.478541\pi$
$587$	23.3827	−	13.5000i	0.965107	−	0.557205i	0.897741	+	0.440524i	$0.145207\pi$
$588$		0			0
$589$	−10.0000	+	17.3205i	−0.412043	+	0.713679i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	6.00000i		−	0.246390i	−0.992382	−	0.123195i	$-0.960686\pi$
$593$		−	6.00000i		−	0.246390i	0.992382	−	0.123195i	$-0.0393141\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−3.00000	+	5.19615i	−0.122577	+	0.212309i	−0.920783	−	0.390075i	$-0.872449\pi$
$599$	−3.00000	+	5.19615i	−0.122577	+	0.212309i	0.798206	+	0.602384i	$0.205782\pi$
$600$		0			0
$601$	11.0000	+	19.0526i	0.448699	+	0.777170i	0.998302	−	0.0582563i	$-0.0185541\pi$
$601$	11.0000	+	19.0526i	0.448699	+	0.777170i	−0.549602	+	0.835426i	$0.685221\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$	6.06218	+	3.50000i	0.246056	+	0.142061i	0.617957	−	0.786212i	$-0.287961\pi$
$607$	6.06218	+	3.50000i	0.246056	+	0.142061i	−0.371901	+	0.928272i	$0.621294\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	36.0000			1.45640
$612$		0			0
$613$			22.0000i			0.888572i	0.895885	+	0.444286i	$0.146543\pi$
$613$			22.0000i			0.888572i	−0.895885	+	0.444286i	$0.853457\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−10.3923	−	6.00000i	−0.418378	−	0.241551i	0.276005	−	0.961156i	$-0.410989\pi$
$617$	−10.3923	−	6.00000i	−0.418378	−	0.241551i	−0.694383	+	0.719605i	$0.744323\pi$
$618$		0			0
$619$	13.0000	+	22.5167i	0.522514	+	0.905021i	0.999657	+	0.0261952i	$0.00833914\pi$
$619$	13.0000	+	22.5167i	0.522514	+	0.905021i	−0.477143	+	0.878826i	$0.658328\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−7.79423	+	4.50000i	−0.312269	+	0.180289i
$624$		0			0
$625$		0			0
$626$		0			0
$627$		0			0
$628$		0			0
$629$	60.0000			2.39236
$630$		0			0
$631$	−40.0000			−1.59237			−0.796187	−	0.605050i	$-0.793153\pi$
$631$	−40.0000			−1.59237			−0.796187	+	0.605050i	$0.793153\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	20.7846	−	12.0000i	0.823516	−	0.475457i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	10.5000	+	18.1865i	0.414725	+	0.718325i	0.995400	−	0.0958109i	$-0.0305444\pi$
$641$	10.5000	+	18.1865i	0.414725	+	0.718325i	−0.580674	+	0.814136i	$0.697211\pi$
$642$		0			0
$643$	−26.8468	−	15.5000i	−1.05873	−	0.611260i	−0.133652	−	0.991028i	$-0.542670\pi$
$643$	−26.8468	−	15.5000i	−1.05873	−	0.611260i	−0.925082	+	0.379768i	$0.876004\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	9.00000i		−	0.353827i	−0.984226	−	0.176913i	$-0.943389\pi$
$647$		−	9.00000i		−	0.353827i	0.984226	−	0.176913i	$-0.0566112\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−5.19615	−	3.00000i	−0.203341	−	0.117399i	0.394872	−	0.918736i	$-0.370789\pi$
$653$	−5.19615	−	3.00000i	−0.203341	−	0.117399i	−0.598213	+	0.801337i	$0.704122\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−18.0000	−	31.1769i	−0.701180	−	1.21448i	−0.968052	−	0.250748i	$-0.919323\pi$
$659$	−18.0000	−	31.1769i	−0.701180	−	1.21448i	0.266872	−	0.963732i	$-0.414010\pi$
$660$		0			0
$661$	5.00000	−	8.66025i	0.194477	−	0.336845i	−0.752252	−	0.658876i	$-0.771032\pi$
$661$	5.00000	−	8.66025i	0.194477	−	0.336845i	0.946729	+	0.322031i	$0.104366\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		−	9.00000i		−	0.348481i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	24.2487	−	14.0000i	0.934719	−	0.539660i	0.0464181	−	0.998922i	$-0.485219\pi$
$673$	24.2487	−	14.0000i	0.934719	−	0.539660i	0.888301	+	0.459262i	$0.151886\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$677$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$678$		0			0
$679$	5.00000	−	8.66025i	0.191882	−	0.332350i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	12.0000i		−	0.459167i	−0.973289	−	0.229584i	$-0.926264\pi$
$683$		−	12.0000i		−	0.459167i	0.973289	−	0.229584i	$-0.0737364\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−12.0000	+	20.7846i	−0.457164	+	0.791831i
$690$		0			0
$691$	5.00000	+	8.66025i	0.190209	+	0.329452i	0.945319	−	0.326146i	$-0.105750\pi$
$691$	5.00000	+	8.66025i	0.190209	+	0.329452i	−0.755110	+	0.655598i	$0.772417\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	46.7654	+	27.0000i	1.77136	+	1.02270i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	21.0000			0.793159			0.396580	−	0.918000i	$-0.370197\pi$
$701$	21.0000			0.793159			0.396580	+	0.918000i	$0.370197\pi$
$702$		0			0
$703$			20.0000i			0.754314i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	15.5885	+	9.00000i	0.586264	+	0.338480i
$708$		0			0
$709$	17.5000	+	30.3109i	0.657226	+	1.13835i	0.981331	+	0.192328i	$0.0616038\pi$
$709$	17.5000	+	30.3109i	0.657226	+	1.13835i	−0.324104	+	0.946021i	$0.605063\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	25.9808	−	15.0000i	0.972987	−	0.561754i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−42.0000			−1.56634			−0.783168	−	0.621810i	$-0.786397\pi$
$719$	−42.0000			−1.56634			−0.783168	+	0.621810i	$0.786397\pi$
$720$		0			0
$721$	−8.00000			−0.297936
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−16.4545	+	9.50000i	−0.610263	+	0.352335i	−0.773068	−	0.634323i	$-0.781279\pi$
$727$	−16.4545	+	9.50000i	−0.610263	+	0.352335i	0.162805	+	0.986658i	$0.447946\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	12.0000	+	20.7846i	0.443836	+	0.768747i
$732$		0			0
$733$	12.1244	+	7.00000i	0.447823	+	0.258551i	0.706910	−	0.707303i	$-0.250088\pi$
$733$	12.1244	+	7.00000i	0.447823	+	0.258551i	−0.259087	+	0.965854i	$0.583422\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−2.00000			−0.0735712			−0.0367856	−	0.999323i	$-0.511712\pi$
$739$	−2.00000			−0.0735712			−0.0367856	+	0.999323i	$0.511712\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−28.5788	−	16.5000i	−1.04846	−	0.605326i	−0.126239	−	0.992000i	$-0.540291\pi$
$743$	−28.5788	−	16.5000i	−1.04846	−	0.605326i	−0.922217	+	0.386674i	$0.873624\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$	4.50000	+	7.79423i	0.164426	+	0.284795i
$750$		0			0
$751$	8.00000	−	13.8564i	0.291924	−	0.505627i	−0.682341	−	0.731034i	$-0.739038\pi$
$751$	8.00000	−	13.8564i	0.291924	−	0.505627i	0.974265	+	0.225407i	$0.0723712\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$			20.0000i			0.726912i	0.931611	+	0.363456i	$0.118403\pi$
$757$			20.0000i			0.726912i	−0.931611	+	0.363456i	$0.881597\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−22.5000	+	38.9711i	−0.815624	+	1.41270i	0.0932544	+	0.995642i	$0.470273\pi$
$761$	−22.5000	+	38.9711i	−0.815624	+	1.41270i	−0.908879	+	0.417061i	$0.863060\pi$
$762$		0			0
$763$	−16.4545	+	9.50000i	−0.595692	+	0.343923i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−20.7846	+	12.0000i	−0.750489	+	0.433295i
$768$		0			0
$769$	−15.5000	+	26.8468i	−0.558944	+	0.968120i	0.438641	+	0.898663i	$0.355460\pi$
$769$	−15.5000	+	26.8468i	−0.558944	+	0.968120i	−0.997585	+	0.0694574i	$0.977873\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	36.0000i		−	1.29483i	−0.762138	−	0.647415i	$-0.775850\pi$
$773$		−	36.0000i		−	1.29483i	0.762138	−	0.647415i	$-0.224150\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−9.00000	+	15.5885i	−0.322458	+	0.558514i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	34.6410	+	20.0000i	1.23482	+	0.712923i	0.968031	−	0.250832i	$-0.0807042\pi$
$787$	34.6410	+	20.0000i	1.23482	+	0.712923i	0.266788	+	0.963755i	$0.414038\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−12.0000			−0.426671
$792$		0			0
$793$		−	4.00000i		−	0.142044i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−20.7846	−	12.0000i	−0.736229	−	0.425062i	0.0844678	−	0.996426i	$-0.473081\pi$
$797$	−20.7846	−	12.0000i	−0.736229	−	0.425062i	−0.820696	+	0.571364i	$0.806414\pi$
$798$		0			0
$799$	−27.0000	−	46.7654i	−0.955191	−	1.65444i
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−54.0000			−1.89854			−0.949269	−	0.314464i	$-0.898175\pi$
$809$	−54.0000			−1.89854			−0.949269	+	0.314464i	$0.898175\pi$
$810$		0			0
$811$	−22.0000			−0.772524			−0.386262	−	0.922389i	$-0.626234\pi$
$811$	−22.0000			−0.772524			−0.386262	+	0.922389i	$0.626234\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−6.92820	+	4.00000i	−0.242387	+	0.139942i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	1.50000	+	2.59808i	0.0523504	+	0.0906735i	0.891013	−	0.453978i	$-0.149995\pi$
$821$	1.50000	+	2.59808i	0.0523504	+	0.0906735i	−0.838663	+	0.544651i	$0.816662\pi$
$822$		0			0
$823$	−42.4352	−	24.5000i	−1.47920	−	0.854016i	−0.479477	−	0.877555i	$-0.659174\pi$
$823$	−42.4352	−	24.5000i	−1.47920	−	0.854016i	−0.999723	+	0.0235383i	$0.992507\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	27.0000i		−	0.938882i	−0.882964	−	0.469441i	$-0.844455\pi$
$827$		−	27.0000i		−	0.938882i	0.882964	−	0.469441i	$-0.155545\pi$
$828$		0			0
$829$	7.00000			0.243120			0.121560	−	0.992584i	$-0.461210\pi$
$829$	7.00000			0.243120			0.121560	+	0.992584i	$0.461210\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−31.1769	−	18.0000i	−1.08022	−	0.623663i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$	15.0000	+	25.9808i	0.517858	+	0.896956i	0.999785	+	0.0207443i	$0.00660359\pi$
$839$	15.0000	+	25.9808i	0.517858	+	0.896956i	−0.481927	+	0.876211i	$0.660063\pi$
$840$		0			0
$841$	10.0000	−	17.3205i	0.344828	−	0.597259i
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$			11.0000i			0.377964i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	15.0000	−	25.9808i	0.514193	−	0.890609i
$852$		0			0
$853$	3.46410	−	2.00000i	0.118609	−	0.0684787i	−0.439522	−	0.898232i	$-0.644852\pi$
$853$	3.46410	−	2.00000i	0.118609	−	0.0684787i	0.558131	+	0.829753i	$0.311519\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	41.5692	−	24.0000i	1.41998	−	0.819824i	0.423681	−	0.905811i	$-0.360738\pi$
$857$	41.5692	−	24.0000i	1.41998	−	0.819824i	0.996296	+	0.0859870i	$0.0274043\pi$
$858$		0			0
$859$	10.0000	−	17.3205i	0.341196	−	0.590968i	−0.643459	−	0.765480i	$-0.722501\pi$
$859$	10.0000	−	17.3205i	0.341196	−	0.590968i	0.984655	+	0.174512i	$0.0558348\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			15.0000i			0.510606i	0.966861	+	0.255303i	$0.0821752\pi$
$863$			15.0000i			0.510606i	−0.966861	+	0.255303i	$0.917825\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$	22.0000	+	38.1051i	0.745442	+	1.29114i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	3.46410	+	2.00000i	0.116974	+	0.0675352i	0.557346	−	0.830281i	$-0.311820\pi$
$877$	3.46410	+	2.00000i	0.116974	+	0.0675352i	−0.440371	+	0.897816i	$0.645153\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	15.0000			0.505363			0.252681	−	0.967550i	$-0.418688\pi$
$881$	15.0000			0.505363			0.252681	+	0.967550i	$0.418688\pi$
$882$		0			0
$883$		−	29.0000i		−	0.975928i	−0.872864	−	0.487964i	$-0.837740\pi$
$883$		−	29.0000i		−	0.975928i	0.872864	−	0.487964i	$-0.162260\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−41.5692	−	24.0000i	−1.39576	−	0.805841i	−0.401813	−	0.915722i	$-0.631620\pi$
$887$	−41.5692	−	24.0000i	−1.39576	−	0.805841i	−0.993945	+	0.109881i	$0.964953\pi$
$888$		0			0
$889$	3.50000	+	6.06218i	0.117386	+	0.203319i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	15.5885	−	9.00000i	0.521648	−	0.301174i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−30.0000			−1.00056
$900$		0			0
$901$	36.0000			1.19933
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−0.866025	+	0.500000i	−0.0287559	+	0.0166022i	−0.514309	−	0.857605i	$-0.671952\pi$
$907$	−0.866025	+	0.500000i	−0.0287559	+	0.0166022i	0.485553	+	0.874207i	$0.338618\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−3.00000	−	5.19615i	−0.0993944	−	0.172156i	0.812040	−	0.583602i	$-0.198357\pi$
$911$	−3.00000	−	5.19615i	−0.0993944	−	0.172156i	−0.911434	+	0.411446i	$0.865024\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$			6.00000i			0.198137i
$918$		0			0
$919$	34.0000			1.12156			0.560778	−	0.827966i	$-0.310502\pi$
$919$	34.0000			1.12156			0.560778	+	0.827966i	$0.310502\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	41.5692	+	24.0000i	1.36827	+	0.789970i
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−3.00000	−	5.19615i	−0.0984268	−	0.170480i	0.812607	−	0.582812i	$-0.198048\pi$
$929$	−3.00000	−	5.19615i	−0.0984268	−	0.170480i	−0.911034	+	0.412332i	$0.864714\pi$
$930$		0			0
$931$	6.00000	−	10.3923i	0.196642	−	0.340594i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		−	16.0000i		−	0.522697i	−0.965244	−	0.261349i	$-0.915833\pi$
$937$		−	16.0000i		−	0.522697i	0.965244	−	0.261349i	$-0.0841672\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	22.5000	−	38.9711i	0.733479	−	1.27042i	−0.221908	−	0.975068i	$-0.571229\pi$
$941$	22.5000	−	38.9711i	0.733479	−	1.27042i	0.955387	−	0.295355i	$-0.0954381\pi$
$942$		0			0
$943$	23.3827	−	13.5000i	0.761445	−	0.439620i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−12.9904	+	7.50000i	−0.422131	+	0.243717i	−0.695988	−	0.718053i	$-0.745034\pi$
$947$	−12.9904	+	7.50000i	−0.422131	+	0.243717i	0.273858	+	0.961770i	$0.411700\pi$
$948$		0			0
$949$	8.00000	−	13.8564i	0.259691	−	0.449798i
$950$		0			0
$951$		0			0
$952$		0			0
$953$			24.0000i			0.777436i	0.921357	+	0.388718i	$0.127082\pi$
$953$			24.0000i			0.777436i	−0.921357	+	0.388718i	$0.872918\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$	6.00000	−	10.3923i	0.193750	−	0.335585i
$960$		0			0
$961$	−34.5000	−	59.7558i	−1.11290	−	1.92760i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	0.866025	+	0.500000i	0.0278495	+	0.0160789i	0.513860	−	0.857874i	$-0.328215\pi$
$967$	0.866025	+	0.500000i	0.0278495	+	0.0160789i	−0.486011	+	0.873953i	$0.661548\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−42.0000			−1.34784			−0.673922	−	0.738802i	$-0.735392\pi$
$971$	−42.0000			−1.34784			−0.673922	+	0.738802i	$0.735392\pi$
$972$		0			0
$973$			20.0000i			0.641171i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	20.7846	+	12.0000i	0.664959	+	0.383914i	0.794164	−	0.607704i	$-0.207909\pi$
$977$	20.7846	+	12.0000i	0.664959	+	0.383914i	−0.129205	+	0.991618i	$0.541243\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$		0			0
$983$	7.79423	−	4.50000i	0.248597	−	0.143528i	−0.370525	−	0.928823i	$-0.620822\pi$
$983$	7.79423	−	4.50000i	0.248597	−	0.143528i	0.619122	+	0.785295i	$0.287489\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	12.0000			0.381578
$990$		0			0
$991$	2.00000			0.0635321			0.0317660	−	0.999495i	$-0.489887\pi$
$991$	2.00000			0.0635321			0.0317660	+	0.999495i	$0.489887\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−8.66025	+	5.00000i	−0.274273	+	0.158352i	−0.630828	−	0.775923i	$-0.717285\pi$
$997$	−8.66025	+	5.00000i	−0.274273	+	0.158352i	0.356555	+	0.934274i	$0.383951\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2700.2.s.a.1549.1		4
3.2	odd	2		900.2.s.a.49.2		4
5.2	odd	4		2700.2.i.a.901.1		2
5.3	odd	4		540.2.i.a.361.1		2
5.4	even	2	inner	2700.2.s.a.1549.2		4
9.2	odd	6		900.2.s.a.349.1		4
9.4	even	3		8100.2.d.f.649.1		2
9.5	odd	6		8100.2.d.e.649.1		2
9.7	even	3	inner	2700.2.s.a.2449.2		4
15.2	even	4		900.2.i.a.301.1		2
15.8	even	4		180.2.i.a.121.1	yes	2
15.14	odd	2		900.2.s.a.49.1		4
20.3	even	4		2160.2.q.e.1441.1		2
45.2	even	12		900.2.i.a.601.1		2
45.4	even	6		8100.2.d.f.649.2		2
45.7	odd	12		2700.2.i.a.1801.1		2
45.13	odd	12		1620.2.a.b.1.1		1
45.14	odd	6		8100.2.d.e.649.2		2
45.22	odd	12		8100.2.a.h.1.1		1
45.23	even	12		1620.2.a.e.1.1		1
45.29	odd	6		900.2.s.a.349.2		4
45.32	even	12		8100.2.a.i.1.1		1
45.34	even	6	inner	2700.2.s.a.2449.1		4
45.38	even	12		180.2.i.a.61.1	✓	2
45.43	odd	12		540.2.i.a.181.1		2
60.23	odd	4		720.2.q.a.481.1		2
180.23	odd	12		6480.2.a.t.1.1		1
180.43	even	12		2160.2.q.e.721.1		2
180.83	odd	12		720.2.q.a.241.1		2
180.103	even	12		6480.2.a.h.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
180.2.i.a.61.1	✓	2	45.38	even	12
180.2.i.a.121.1	yes	2	15.8	even	4
540.2.i.a.181.1		2	45.43	odd	12
540.2.i.a.361.1		2	5.3	odd	4
720.2.q.a.241.1		2	180.83	odd	12
720.2.q.a.481.1		2	60.23	odd	4
900.2.i.a.301.1		2	15.2	even	4
900.2.i.a.601.1		2	45.2	even	12
900.2.s.a.49.1		4	15.14	odd	2
900.2.s.a.49.2		4	3.2	odd	2
900.2.s.a.349.1		4	9.2	odd	6
900.2.s.a.349.2		4	45.29	odd	6
1620.2.a.b.1.1		1	45.13	odd	12
1620.2.a.e.1.1		1	45.23	even	12
2160.2.q.e.721.1		2	180.43	even	12
2160.2.q.e.1441.1		2	20.3	even	4
2700.2.i.a.901.1		2	5.2	odd	4
2700.2.i.a.1801.1		2	45.7	odd	12
2700.2.s.a.1549.1		4	1.1	even	1	trivial
2700.2.s.a.1549.2		4	5.4	even	2	inner
2700.2.s.a.2449.1		4	45.34	even	6	inner
2700.2.s.a.2449.2		4	9.7	even	3	inner
6480.2.a.h.1.1		1	180.103	even	12
6480.2.a.t.1.1		1	180.23	odd	12
8100.2.a.h.1.1		1	45.22	odd	12
8100.2.a.i.1.1		1	45.32	even	12
8100.2.d.e.649.1		2	9.5	odd	6
8100.2.d.e.649.2		2	45.14	odd	6
8100.2.d.f.649.1		2	9.4	even	3
8100.2.d.f.649.2		2	45.4	even	6

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:	\( N \)	\(=\)	\( 2700 = 2^{2} \cdot 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2700.s (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.866025 + 0.500000i) q^{7} +O(q^{10})\)	\(q+(-0.866025 + 0.500000i) q^{7} +(-3.46410 - 2.00000i) q^{13} +6.00000i q^{17} -2.00000 q^{19} +(2.59808 + 1.50000i) q^{23} +(-1.50000 - 2.59808i) q^{29} +(5.00000 - 8.66025i) q^{31} -10.0000i q^{37} +(4.50000 - 7.79423i) q^{41} +(3.46410 - 2.00000i) q^{43} +(-7.79423 + 4.50000i) q^{47} +(-3.00000 + 5.19615i) q^{49} -6.00000i q^{53} +(3.00000 - 5.19615i) q^{59} +(0.500000 + 0.866025i) q^{61} +(-9.52628 - 5.50000i) q^{67} -12.0000 q^{71} +4.00000i q^{73} +(-5.00000 - 8.66025i) q^{79} +(-7.79423 + 4.50000i) q^{83} +9.00000 q^{89} +4.00000 q^{91} +(-8.66025 + 5.00000i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q - 8 q^{19} - 6 q^{29} + 20 q^{31} + 18 q^{41} - 12 q^{49} + 12 q^{59} + 2 q^{61} - 48 q^{71} - 20 q^{79} + 36 q^{89} + 16 q^{91}+O(q^{100}) \) 4 * q - 8 * q^19 - 6 * q^29 + 20 * q^31 + 18 * q^41 - 12 * q^49 + 12 * q^59 + 2 * q^61 - 48 * q^71 - 20 * q^79 + 36 * q^89 + 16 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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