LMFDB - Embedded newform 1600.2.o.g.543.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1600,2,Mod(543,1600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1600, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 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("1600.543");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1600 = 2^{6} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1600.o (of order $4$, degree $2$, 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:	$12.7760643234$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(i)$
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:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{10} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			543.3
Root			$-0.258819 - 0.965926i$ of defining polynomial
Character	$\chi$	$=$	1600.543
Dual form			1600.2.o.g.607.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.41421 + 1.41421i) q^{3} +(-2.44949 - 2.44949i) q^{7} +1.00000i q^{9} +O(q^{10})$	$q+(1.41421 + 1.41421i) q^{3} +(-2.44949 - 2.44949i) q^{7} +1.00000i q^{9} +3.46410 q^{11} +(-2.82843 + 2.82843i) q^{13} +(4.89898 - 4.89898i) q^{17} +3.46410i q^{19} -6.92820i q^{21} +(2.44949 - 2.44949i) q^{23} +(2.82843 - 2.82843i) q^{27} +6.92820 q^{29} +8.00000i q^{31} +(4.89898 + 4.89898i) q^{33} +(5.65685 + 5.65685i) q^{37} -8.00000 q^{39} +6.00000 q^{41} +(-1.41421 - 1.41421i) q^{43} +(-2.44949 - 2.44949i) q^{47} +5.00000i q^{49} +13.8564 q^{51} +(8.48528 - 8.48528i) q^{53} +(-4.89898 + 4.89898i) q^{57} +10.3923i q^{59} -13.8564i q^{61} +(2.44949 - 2.44949i) q^{63} +(1.41421 - 1.41421i) q^{67} +6.92820 q^{69} +(-4.89898 - 4.89898i) q^{73} +(-8.48528 - 8.48528i) q^{77} +8.00000 q^{79} +11.0000 q^{81} +(-4.24264 - 4.24264i) q^{83} +(9.79796 + 9.79796i) q^{87} +6.00000i q^{89} +13.8564 q^{91} +(-11.3137 + 11.3137i) q^{93} +(-4.89898 + 4.89898i) q^{97} +3.46410i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q - 64 q^{39} + 48 q^{41} + 64 q^{79} + 88 q^{81}+O(q^{100}) $ 8 * q - 64 * q^39 + 48 * q^41 + 64 * q^79 + 88 * q^81

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1600\mathbb{Z}\right)^\times$.

$n$	$577$	$901$	$1151$
$\chi(n)$	$e\left(\frac{3}{4}\right)$	$-1$	$-1$

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
$3$	1.41421	+	1.41421i	0.816497	+	0.816497i	0.985599	−	0.169102i	$-0.0540867\pi$
$3$	1.41421	+	1.41421i	0.816497	+	0.816497i	−0.169102	+	0.985599i	$0.554087\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−2.44949	−	2.44949i	−0.925820	−	0.925820i	0.0716124	−	0.997433i	$-0.477186\pi$
$7$	−2.44949	−	2.44949i	−0.925820	−	0.925820i	−0.997433	+	0.0716124i	$0.977186\pi$
$8$		0			0
$9$			1.00000i			0.333333i
$10$		0			0
$11$	3.46410			1.04447			0.522233	−	0.852803i	$-0.325099\pi$
$11$	3.46410			1.04447			0.522233	+	0.852803i	$0.325099\pi$
$12$		0			0
$13$	−2.82843	+	2.82843i	−0.784465	+	0.784465i	−0.980581	−	0.196116i	$-0.937167\pi$
$13$	−2.82843	+	2.82843i	−0.784465	+	0.784465i	0.196116	+	0.980581i	$0.437167\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	4.89898	−	4.89898i	1.18818	−	1.18818i	0.210606	−	0.977571i	$-0.432456\pi$
$17$	4.89898	−	4.89898i	1.18818	−	1.18818i	0.977571	−	0.210606i	$-0.0675437\pi$
$18$		0			0
$19$			3.46410i			0.794719i	0.917663	+	0.397360i	$0.130073\pi$
$19$			3.46410i			0.794719i	−0.917663	+	0.397360i	$0.869927\pi$
$20$		0			0
$21$		−	6.92820i		−	1.51186i
$22$		0			0
$23$	2.44949	−	2.44949i	0.510754	−	0.510754i	−0.404004	−	0.914757i	$-0.632382\pi$
$23$	2.44949	−	2.44949i	0.510754	−	0.510754i	0.914757	+	0.404004i	$0.132382\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	2.82843	−	2.82843i	0.544331	−	0.544331i
$28$		0			0
$29$	6.92820			1.28654			0.643268	−	0.765641i	$-0.277578\pi$
$29$	6.92820			1.28654			0.643268	+	0.765641i	$0.277578\pi$
$30$		0			0
$31$			8.00000i			1.43684i	0.695608	+	0.718421i	$0.255135\pi$
$31$			8.00000i			1.43684i	−0.695608	+	0.718421i	$0.744865\pi$
$32$		0			0
$33$	4.89898	+	4.89898i	0.852803	+	0.852803i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	5.65685	+	5.65685i	0.929981	+	0.929981i	0.997704	−	0.0677230i	$-0.0215734\pi$
$37$	5.65685	+	5.65685i	0.929981	+	0.929981i	−0.0677230	+	0.997704i	$0.521573\pi$
$38$		0			0
$39$	−8.00000			−1.28103
$40$		0			0
$41$	6.00000			0.937043			0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000			0.937043			0.468521	+	0.883452i	$0.344787\pi$
$42$		0			0
$43$	−1.41421	−	1.41421i	−0.215666	−	0.215666i	0.591003	−	0.806669i	$-0.298732\pi$
$43$	−1.41421	−	1.41421i	−0.215666	−	0.215666i	−0.806669	+	0.591003i	$0.798732\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−2.44949	−	2.44949i	−0.357295	−	0.357295i	0.505520	−	0.862815i	$-0.331301\pi$
$47$	−2.44949	−	2.44949i	−0.357295	−	0.357295i	−0.862815	+	0.505520i	$0.831301\pi$
$48$		0			0
$49$			5.00000i			0.714286i
$50$		0			0
$51$	13.8564			1.94029
$52$		0			0
$53$	8.48528	−	8.48528i	1.16554	−	1.16554i	0.182300	−	0.983243i	$-0.441646\pi$
$53$	8.48528	−	8.48528i	1.16554	−	1.16554i	0.983243	−	0.182300i	$-0.0583542\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−4.89898	+	4.89898i	−0.648886	+	0.648886i
$58$		0			0
$59$			10.3923i			1.35296i	0.736460	+	0.676481i	$0.236496\pi$
$59$			10.3923i			1.35296i	−0.736460	+	0.676481i	$0.763504\pi$
$60$		0			0
$61$		−	13.8564i		−	1.77413i	−0.461644	−	0.887066i	$-0.652740\pi$
$61$		−	13.8564i		−	1.77413i	0.461644	−	0.887066i	$-0.347260\pi$
$62$		0			0
$63$	2.44949	−	2.44949i	0.308607	−	0.308607i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	1.41421	−	1.41421i	0.172774	−	0.172774i	−0.615423	−	0.788197i	$-0.711015\pi$
$67$	1.41421	−	1.41421i	0.172774	−	0.172774i	0.788197	+	0.615423i	$0.211015\pi$
$68$		0			0
$69$	6.92820			0.834058
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		0			0
$73$	−4.89898	−	4.89898i	−0.573382	−	0.573382i	0.359690	−	0.933072i	$-0.382883\pi$
$73$	−4.89898	−	4.89898i	−0.573382	−	0.573382i	−0.933072	+	0.359690i	$0.882883\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−8.48528	−	8.48528i	−0.966988	−	0.966988i
$78$		0			0
$79$	8.00000			0.900070			0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000			0.900070			0.450035	+	0.893011i	$0.351411\pi$
$80$		0			0
$81$	11.0000			1.22222
$82$		0			0
$83$	−4.24264	−	4.24264i	−0.465690	−	0.465690i	0.434825	−	0.900515i	$-0.356810\pi$
$83$	−4.24264	−	4.24264i	−0.465690	−	0.465690i	−0.900515	+	0.434825i	$0.856810\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	9.79796	+	9.79796i	1.05045	+	1.05045i
$88$		0			0
$89$			6.00000i			0.635999i	0.948091	+	0.317999i	$0.103011\pi$
$89$			6.00000i			0.635999i	−0.948091	+	0.317999i	$0.896989\pi$
$90$		0			0
$91$	13.8564			1.45255
$92$		0			0
$93$	−11.3137	+	11.3137i	−1.17318	+	1.17318i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−4.89898	+	4.89898i	−0.497416	+	0.497416i	−0.910633	−	0.413217i	$-0.864405\pi$
$97$	−4.89898	+	4.89898i	−0.497416	+	0.497416i	0.413217	+	0.910633i	$0.364405\pi$
$98$		0			0
$99$			3.46410i			0.348155i
$100$		0			0
$101$		−	6.92820i		−	0.689382i	−0.938716	−	0.344691i	$-0.887984\pi$
$101$		−	6.92820i		−	0.689382i	0.938716	−	0.344691i	$-0.112016\pi$
$102$		0			0
$103$	−12.2474	+	12.2474i	−1.20678	+	1.20678i	−0.234712	+	0.972065i	$0.575415\pi$
$103$	−12.2474	+	12.2474i	−1.20678	+	1.20678i	−0.972065	+	0.234712i	$0.924585\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−12.7279	+	12.7279i	−1.23045	+	1.23045i	−0.266666	+	0.963789i	$0.585922\pi$
$107$	−12.7279	+	12.7279i	−1.23045	+	1.23045i	−0.963789	+	0.266666i	$0.914078\pi$
$108$		0			0
$109$		0			0			−	1.00000i	$-0.5\pi$
$109$		0			0				1.00000i	$0.5\pi$
$110$		0			0
$111$			16.0000i			1.51865i
$112$		0			0
$113$	−9.79796	−	9.79796i	−0.921714	−	0.921714i	0.0754362	−	0.997151i	$-0.475965\pi$
$113$	−9.79796	−	9.79796i	−0.921714	−	0.921714i	−0.997151	+	0.0754362i	$0.975965\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−2.82843	−	2.82843i	−0.261488	−	0.261488i
$118$		0			0
$119$	−24.0000			−2.20008
$120$		0			0
$121$	1.00000			0.0909091
$122$		0			0
$123$	8.48528	+	8.48528i	0.765092	+	0.765092i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	7.34847	+	7.34847i	0.652071	+	0.652071i	0.953491	−	0.301420i	$-0.0974607\pi$
$127$	7.34847	+	7.34847i	0.652071	+	0.652071i	−0.301420	+	0.953491i	$0.597461\pi$
$128$		0			0
$129$		−	4.00000i		−	0.352180i
$130$		0			0
$131$	−3.46410			−0.302660			−0.151330	−	0.988483i	$-0.548356\pi$
$131$	−3.46410			−0.302660			−0.151330	+	0.988483i	$0.548356\pi$
$132$		0			0
$133$	8.48528	−	8.48528i	0.735767	−	0.735767i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$137$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$138$		0			0
$139$		−	3.46410i		−	0.293821i	−0.989150	−	0.146911i	$-0.953067\pi$
$139$		−	3.46410i		−	0.293821i	0.989150	−	0.146911i	$-0.0469330\pi$
$140$		0			0
$141$		−	6.92820i		−	0.583460i
$142$		0			0
$143$	−9.79796	+	9.79796i	−0.819346	+	0.819346i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−7.07107	+	7.07107i	−0.583212	+	0.583212i
$148$		0			0
$149$	13.8564			1.13516			0.567581	−	0.823318i	$-0.307880\pi$
$149$	13.8564			1.13516			0.567581	+	0.823318i	$0.307880\pi$
$150$		0			0
$151$		−	8.00000i		−	0.651031i	−0.945537	−	0.325515i	$-0.894462\pi$
$151$		−	8.00000i		−	0.651031i	0.945537	−	0.325515i	$-0.105538\pi$
$152$		0			0
$153$	4.89898	+	4.89898i	0.396059	+	0.396059i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−5.65685	−	5.65685i	−0.451466	−	0.451466i	0.444375	−	0.895841i	$-0.353426\pi$
$157$	−5.65685	−	5.65685i	−0.451466	−	0.451466i	−0.895841	+	0.444375i	$0.853426\pi$
$158$		0			0
$159$	24.0000			1.90332
$160$		0			0
$161$	−12.0000			−0.945732
$162$		0			0
$163$	−1.41421	−	1.41421i	−0.110770	−	0.110770i	0.649550	−	0.760319i	$-0.274958\pi$
$163$	−1.41421	−	1.41421i	−0.110770	−	0.110770i	−0.760319	+	0.649550i	$0.774958\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−7.34847	−	7.34847i	−0.568642	−	0.568642i	0.363106	−	0.931748i	$-0.381716\pi$
$167$	−7.34847	−	7.34847i	−0.568642	−	0.568642i	−0.931748	+	0.363106i	$0.881716\pi$
$168$		0			0
$169$		−	3.00000i		−	0.230769i
$170$		0			0
$171$	−3.46410			−0.264906
$172$		0			0
$173$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$173$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−14.6969	+	14.6969i	−1.10469	+	1.10469i
$178$		0			0
$179$			10.3923i			0.776757i	0.921500	+	0.388379i	$0.126965\pi$
$179$			10.3923i			0.776757i	−0.921500	+	0.388379i	$0.873035\pi$
$180$		0			0
$181$			6.92820i			0.514969i	0.966282	+	0.257485i	$0.0828937\pi$
$181$			6.92820i			0.514969i	−0.966282	+	0.257485i	$0.917106\pi$
$182$		0			0
$183$	19.5959	−	19.5959i	1.44857	−	1.44857i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	16.9706	−	16.9706i	1.24101	−	1.24101i
$188$		0			0
$189$	−13.8564			−1.00791
$190$		0			0
$191$		0			0		1.00000			$0$
$191$		0			0		−1.00000			$\pi$
$192$		0			0
$193$	−4.89898	−	4.89898i	−0.352636	−	0.352636i	0.508453	−	0.861090i	$-0.330217\pi$
$193$	−4.89898	−	4.89898i	−0.352636	−	0.352636i	−0.861090	+	0.508453i	$0.830217\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	8.48528	+	8.48528i	0.604551	+	0.604551i	0.941517	−	0.336966i	$-0.109401\pi$
$197$	8.48528	+	8.48528i	0.604551	+	0.604551i	−0.336966	+	0.941517i	$0.609401\pi$
$198$		0			0
$199$	−16.0000			−1.13421			−0.567105	−	0.823646i	$-0.691937\pi$
$199$	−16.0000			−1.13421			−0.567105	+	0.823646i	$0.691937\pi$
$200$		0			0
$201$	4.00000			0.282138
$202$		0			0
$203$	−16.9706	−	16.9706i	−1.19110	−	1.19110i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	2.44949	+	2.44949i	0.170251	+	0.170251i
$208$		0			0
$209$			12.0000i			0.830057i
$210$		0			0
$211$	−3.46410			−0.238479			−0.119239	−	0.992866i	$-0.538046\pi$
$211$	−3.46410			−0.238479			−0.119239	+	0.992866i	$0.538046\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	19.5959	−	19.5959i	1.33026	−	1.33026i
$218$		0			0
$219$		−	13.8564i		−	0.936329i
$220$		0			0
$221$			27.7128i			1.86417i
$222$		0			0
$223$	7.34847	−	7.34847i	0.492090	−	0.492090i	−0.416874	−	0.908964i	$-0.636874\pi$
$223$	7.34847	−	7.34847i	0.492090	−	0.492090i	0.908964	+	0.416874i	$0.136874\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	12.7279	−	12.7279i	0.844782	−	0.844782i	−0.144695	−	0.989476i	$-0.546220\pi$
$227$	12.7279	−	12.7279i	0.844782	−	0.844782i	0.989476	+	0.144695i	$0.0462199\pi$
$228$		0			0
$229$	−20.7846			−1.37349			−0.686743	−	0.726900i	$-0.740960\pi$
$229$	−20.7846			−1.37349			−0.686743	+	0.726900i	$0.740960\pi$
$230$		0			0
$231$		−	24.0000i		−	1.57908i
$232$		0			0
$233$	14.6969	+	14.6969i	0.962828	+	0.962828i	0.999333	−	0.0365050i	$-0.0116225\pi$
$233$	14.6969	+	14.6969i	0.962828	+	0.962828i	−0.0365050	+	0.999333i	$0.511622\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	11.3137	+	11.3137i	0.734904	+	0.734904i
$238$		0			0
$239$	24.0000			1.55243			0.776215	−	0.630468i	$-0.217137\pi$
$239$	24.0000			1.55243			0.776215	+	0.630468i	$0.217137\pi$
$240$		0			0
$241$	10.0000			0.644157			0.322078	−	0.946713i	$-0.395619\pi$
$241$	10.0000			0.644157			0.322078	+	0.946713i	$0.395619\pi$
$242$		0			0
$243$	7.07107	+	7.07107i	0.453609	+	0.453609i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−9.79796	−	9.79796i	−0.623429	−	0.623429i
$248$		0			0
$249$		−	12.0000i		−	0.760469i
$250$		0			0
$251$	−24.2487			−1.53057			−0.765283	−	0.643695i	$-0.777401\pi$
$251$	−24.2487			−1.53057			−0.765283	+	0.643695i	$0.777401\pi$
$252$		0			0
$253$	8.48528	−	8.48528i	0.533465	−	0.533465i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	9.79796	−	9.79796i	0.611180	−	0.611180i	−0.332074	−	0.943253i	$-0.607748\pi$
$257$	9.79796	−	9.79796i	0.611180	−	0.611180i	0.943253	+	0.332074i	$0.107748\pi$
$258$		0			0
$259$		−	27.7128i		−	1.72199i
$260$		0			0
$261$			6.92820i			0.428845i
$262$		0			0
$263$	−7.34847	+	7.34847i	−0.453126	+	0.453126i	−0.896391	−	0.443265i	$-0.853820\pi$
$263$	−7.34847	+	7.34847i	−0.453126	+	0.453126i	0.443265	+	0.896391i	$0.353820\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−8.48528	+	8.48528i	−0.519291	+	0.519291i
$268$		0			0
$269$	13.8564			0.844840			0.422420	−	0.906400i	$-0.361181\pi$
$269$	13.8564			0.844840			0.422420	+	0.906400i	$0.361181\pi$
$270$		0			0
$271$			8.00000i			0.485965i	0.970031	+	0.242983i	$0.0781258\pi$
$271$			8.00000i			0.485965i	−0.970031	+	0.242983i	$0.921874\pi$
$272$		0			0
$273$	19.5959	+	19.5959i	1.18600	+	1.18600i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	5.65685	+	5.65685i	0.339887	+	0.339887i	0.856325	−	0.516437i	$-0.172742\pi$
$277$	5.65685	+	5.65685i	0.339887	+	0.339887i	−0.516437	+	0.856325i	$0.672742\pi$
$278$		0			0
$279$	−8.00000			−0.478947
$280$		0			0
$281$	−30.0000			−1.78965			−0.894825	−	0.446417i	$-0.852700\pi$
$281$	−30.0000			−1.78965			−0.894825	+	0.446417i	$0.852700\pi$
$282$		0			0
$283$	9.89949	+	9.89949i	0.588464	+	0.588464i	0.937215	−	0.348751i	$-0.113394\pi$
$283$	9.89949	+	9.89949i	0.588464	+	0.588464i	−0.348751	+	0.937215i	$0.613394\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−14.6969	−	14.6969i	−0.867533	−	0.867533i
$288$		0			0
$289$		−	31.0000i		−	1.82353i
$290$		0			0
$291$	−13.8564			−0.812277
$292$		0			0
$293$	−16.9706	+	16.9706i	−0.991431	+	0.991431i	−0.999964	−	0.00853273i	$-0.997284\pi$
$293$	−16.9706	+	16.9706i	−0.991431	+	0.991431i	0.00853273	+	0.999964i	$0.497284\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	9.79796	−	9.79796i	0.568535	−	0.568535i
$298$		0			0
$299$			13.8564i			0.801337i
$300$		0			0
$301$			6.92820i			0.399335i
$302$		0			0
$303$	9.79796	−	9.79796i	0.562878	−	0.562878i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−9.89949	+	9.89949i	−0.564994	+	0.564994i	−0.930722	−	0.365728i	$-0.880820\pi$
$307$	−9.89949	+	9.89949i	−0.564994	+	0.564994i	0.365728	+	0.930722i	$0.380820\pi$
$308$		0			0
$309$	−34.6410			−1.97066
$310$		0			0
$311$		−	24.0000i		−	1.36092i	−0.732787	−	0.680458i	$-0.761781\pi$
$311$		−	24.0000i		−	1.36092i	0.732787	−	0.680458i	$-0.238219\pi$
$312$		0			0
$313$	−9.79796	−	9.79796i	−0.553813	−	0.553813i	0.373726	−	0.927539i	$-0.378080\pi$
$313$	−9.79796	−	9.79796i	−0.553813	−	0.553813i	−0.927539	+	0.373726i	$0.878080\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	8.48528	+	8.48528i	0.476581	+	0.476581i	0.904036	−	0.427456i	$-0.140590\pi$
$317$	8.48528	+	8.48528i	0.476581	+	0.476581i	−0.427456	+	0.904036i	$0.640590\pi$
$318$		0			0
$319$	24.0000			1.34374
$320$		0			0
$321$	−36.0000			−2.00932
$322$		0			0
$323$	16.9706	+	16.9706i	0.944267	+	0.944267i
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$			12.0000i			0.661581i
$330$		0			0
$331$	−17.3205			−0.952021			−0.476011	−	0.879440i	$-0.657918\pi$
$331$	−17.3205			−0.952021			−0.476011	+	0.879440i	$0.657918\pi$
$332$		0			0
$333$	−5.65685	+	5.65685i	−0.309994	+	0.309994i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	9.79796	−	9.79796i	0.533729	−	0.533729i	−0.387951	−	0.921680i	$-0.626817\pi$
$337$	9.79796	−	9.79796i	0.533729	−	0.533729i	0.921680	+	0.387951i	$0.126817\pi$
$338$		0			0
$339$		−	27.7128i		−	1.50515i
$340$		0			0
$341$			27.7128i			1.50073i
$342$		0			0
$343$	−4.89898	+	4.89898i	−0.264520	+	0.264520i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	21.2132	−	21.2132i	1.13878	−	1.13878i	0.150116	−	0.988668i	$-0.452035\pi$
$347$	21.2132	−	21.2132i	1.13878	−	1.13878i	0.988668	−	0.150116i	$-0.0479647\pi$
$348$		0			0
$349$	−6.92820			−0.370858			−0.185429	−	0.982658i	$-0.559368\pi$
$349$	−6.92820			−0.370858			−0.185429	+	0.982658i	$0.559368\pi$
$350$		0			0
$351$			16.0000i			0.854017i
$352$		0			0
$353$	−19.5959	−	19.5959i	−1.04299	−	1.04299i	−0.999034	−	0.0439518i	$-0.986005\pi$
$353$	−19.5959	−	19.5959i	−1.04299	−	1.04299i	−0.0439518	−	0.999034i	$-0.513995\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	−33.9411	−	33.9411i	−1.79635	−	1.79635i
$358$		0			0
$359$	−24.0000			−1.26667			−0.633336	−	0.773877i	$-0.718315\pi$
$359$	−24.0000			−1.26667			−0.633336	+	0.773877i	$0.718315\pi$
$360$		0			0
$361$	7.00000			0.368421
$362$		0			0
$363$	1.41421	+	1.41421i	0.0742270	+	0.0742270i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−17.1464	−	17.1464i	−0.895036	−	0.895036i	0.0999556	−	0.994992i	$-0.468130\pi$
$367$	−17.1464	−	17.1464i	−0.895036	−	0.895036i	−0.994992	+	0.0999556i	$0.968130\pi$
$368$		0			0
$369$			6.00000i			0.312348i
$370$		0			0
$371$	−41.5692			−2.15817
$372$		0			0
$373$	−22.6274	+	22.6274i	−1.17160	+	1.17160i	−0.189776	+	0.981827i	$0.560776\pi$
$373$	−22.6274	+	22.6274i	−1.17160	+	1.17160i	−0.981827	+	0.189776i	$0.939224\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−19.5959	+	19.5959i	−1.00924	+	1.00924i
$378$		0			0
$379$		−	24.2487i		−	1.24557i	−0.782392	−	0.622786i	$-0.786001\pi$
$379$		−	24.2487i		−	1.24557i	0.782392	−	0.622786i	$-0.213999\pi$
$380$		0			0
$381$			20.7846i			1.06483i
$382$		0			0
$383$	−2.44949	+	2.44949i	−0.125163	+	0.125163i	−0.766914	−	0.641750i	$-0.778208\pi$
$383$	−2.44949	+	2.44949i	−0.125163	+	0.125163i	0.641750	+	0.766914i	$0.278208\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	1.41421	−	1.41421i	0.0718885	−	0.0718885i
$388$		0			0
$389$	13.8564			0.702548			0.351274	−	0.936273i	$-0.385749\pi$
$389$	13.8564			0.702548			0.351274	+	0.936273i	$0.385749\pi$
$390$		0			0
$391$		−	24.0000i		−	1.21373i
$392$		0			0
$393$	−4.89898	−	4.89898i	−0.247121	−	0.247121i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	2.82843	+	2.82843i	0.141955	+	0.141955i	0.774513	−	0.632558i	$-0.217995\pi$
$397$	2.82843	+	2.82843i	0.141955	+	0.141955i	−0.632558	+	0.774513i	$0.717995\pi$
$398$		0			0
$399$	24.0000			1.20150
$400$		0			0
$401$	18.0000			0.898877			0.449439	−	0.893311i	$-0.351624\pi$
$401$	18.0000			0.898877			0.449439	+	0.893311i	$0.351624\pi$
$402$		0			0
$403$	−22.6274	−	22.6274i	−1.12715	−	1.12715i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	19.5959	+	19.5959i	0.971334	+	0.971334i
$408$		0			0
$409$		−	22.0000i		−	1.08783i	−0.839140	−	0.543915i	$-0.816941\pi$
$409$		−	22.0000i		−	1.08783i	0.839140	−	0.543915i	$-0.183059\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	25.4558	−	25.4558i	1.25260	−	1.25260i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	4.89898	−	4.89898i	0.239904	−	0.239904i
$418$		0			0
$419$			38.1051i			1.86156i	0.365584	+	0.930778i	$0.380869\pi$
$419$			38.1051i			1.86156i	−0.365584	+	0.930778i	$0.619131\pi$
$420$		0			0
$421$			13.8564i			0.675320i	0.941268	+	0.337660i	$0.109635\pi$
$421$			13.8564i			0.675320i	−0.941268	+	0.337660i	$0.890365\pi$
$422$		0			0
$423$	2.44949	−	2.44949i	0.119098	−	0.119098i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−33.9411	+	33.9411i	−1.64253	+	1.64253i
$428$		0			0
$429$	−27.7128			−1.33799
$430$		0			0
$431$		−	24.0000i		−	1.15604i	−0.816023	−	0.578020i	$-0.803826\pi$
$431$		−	24.0000i		−	1.15604i	0.816023	−	0.578020i	$-0.196174\pi$
$432$		0			0
$433$	14.6969	+	14.6969i	0.706290	+	0.706290i	0.965753	−	0.259463i	$-0.0835456\pi$
$433$	14.6969	+	14.6969i	0.706290	+	0.706290i	−0.259463	+	0.965753i	$0.583546\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	8.48528	+	8.48528i	0.405906	+	0.405906i
$438$		0			0
$439$	8.00000			0.381819			0.190910	−	0.981608i	$-0.438856\pi$
$439$	8.00000			0.381819			0.190910	+	0.981608i	$0.438856\pi$
$440$		0			0
$441$	−5.00000			−0.238095
$442$		0			0
$443$	12.7279	+	12.7279i	0.604722	+	0.604722i	0.941562	−	0.336840i	$-0.109358\pi$
$443$	12.7279	+	12.7279i	0.604722	+	0.604722i	−0.336840	+	0.941562i	$0.609358\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	19.5959	+	19.5959i	0.926855	+	0.926855i
$448$		0			0
$449$			6.00000i			0.283158i	0.989927	+	0.141579i	$0.0452178\pi$
$449$			6.00000i			0.283158i	−0.989927	+	0.141579i	$0.954782\pi$
$450$		0			0
$451$	20.7846			0.978709
$452$		0			0
$453$	11.3137	−	11.3137i	0.531564	−	0.531564i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−29.3939	+	29.3939i	−1.37499	+	1.37499i	−0.522108	+	0.852879i	$0.674854\pi$
$457$	−29.3939	+	29.3939i	−1.37499	+	1.37499i	−0.852879	+	0.522108i	$0.825146\pi$
$458$		0			0
$459$		−	27.7128i		−	1.29352i
$460$		0			0
$461$			20.7846i			0.968036i	0.875058	+	0.484018i	$0.160823\pi$
$461$			20.7846i			0.968036i	−0.875058	+	0.484018i	$0.839177\pi$
$462$		0			0
$463$	−12.2474	+	12.2474i	−0.569187	+	0.569187i	−0.931901	−	0.362713i	$-0.881850\pi$
$463$	−12.2474	+	12.2474i	−0.569187	+	0.569187i	0.362713	+	0.931901i	$0.381850\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	4.24264	−	4.24264i	0.196326	−	0.196326i	−0.602097	−	0.798423i	$-0.705668\pi$
$467$	4.24264	−	4.24264i	0.196326	−	0.196326i	0.798423	+	0.602097i	$0.205668\pi$
$468$		0			0
$469$	−6.92820			−0.319915
$470$		0			0
$471$		−	16.0000i		−	0.737241i
$472$		0			0
$473$	−4.89898	−	4.89898i	−0.225255	−	0.225255i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	8.48528	+	8.48528i	0.388514	+	0.388514i
$478$		0			0
$479$		0			0			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$		0			0
$481$	−32.0000			−1.45907
$482$		0			0
$483$	−16.9706	−	16.9706i	−0.772187	−	0.772187i
$484$		0			0
$485$		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	$0.621633\pi$
$488$		0			0
$489$		−	4.00000i		−	0.180886i
$490$		0			0
$491$	−10.3923			−0.468998			−0.234499	−	0.972116i	$-0.575345\pi$
$491$	−10.3923			−0.468998			−0.234499	+	0.972116i	$0.575345\pi$
$492$		0			0
$493$	33.9411	−	33.9411i	1.52863	−	1.52863i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$		−	10.3923i		−	0.465223i	−0.972570	−	0.232612i	$-0.925273\pi$
$499$		−	10.3923i		−	0.465223i	0.972570	−	0.232612i	$-0.0747271\pi$
$500$		0			0
$501$		−	20.7846i		−	0.928588i
$502$		0			0
$503$	−12.2474	+	12.2474i	−0.546087	+	0.546087i	−0.925307	−	0.379220i	$-0.876192\pi$
$503$	−12.2474	+	12.2474i	−0.546087	+	0.546087i	0.379220	+	0.925307i	$0.376192\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	4.24264	−	4.24264i	0.188422	−	0.188422i
$508$		0			0
$509$	6.92820			0.307087			0.153544	−	0.988142i	$-0.450931\pi$
$509$	6.92820			0.307087			0.153544	+	0.988142i	$0.450931\pi$
$510$		0			0
$511$			24.0000i			1.06170i
$512$		0			0
$513$	9.79796	+	9.79796i	0.432590	+	0.432590i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−8.48528	−	8.48528i	−0.373182	−	0.373182i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−6.00000			−0.262865			−0.131432	−	0.991325i	$-0.541958\pi$
$521$	−6.00000			−0.262865			−0.131432	+	0.991325i	$0.541958\pi$
$522$		0			0
$523$	−24.0416	−	24.0416i	−1.05127	−	1.05127i	−0.998613	−	0.0526543i	$-0.983232\pi$
$523$	−24.0416	−	24.0416i	−1.05127	−	1.05127i	−0.0526543	−	0.998613i	$-0.516768\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	39.1918	+	39.1918i	1.70722	+	1.70722i
$528$		0			0
$529$			11.0000i			0.478261i
$530$		0			0
$531$	−10.3923			−0.450988
$532$		0			0
$533$	−16.9706	+	16.9706i	−0.735077	+	0.735077i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−14.6969	+	14.6969i	−0.634220	+	0.634220i
$538$		0			0
$539$			17.3205i			0.746047i
$540$		0			0
$541$		−	34.6410i		−	1.48933i	−0.667436	−	0.744667i	$-0.732608\pi$
$541$		−	34.6410i		−	1.48933i	0.667436	−	0.744667i	$-0.267392\pi$
$542$		0			0
$543$	−9.79796	+	9.79796i	−0.420471	+	0.420471i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	7.07107	−	7.07107i	0.302337	−	0.302337i	−0.539591	−	0.841928i	$-0.681421\pi$
$547$	7.07107	−	7.07107i	0.302337	−	0.302337i	0.841928	+	0.539591i	$0.181421\pi$
$548$		0			0
$549$	13.8564			0.591377
$550$		0			0
$551$			24.0000i			1.02243i
$552$		0			0
$553$	−19.5959	−	19.5959i	−0.833303	−	0.833303i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$557$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$558$		0			0
$559$	8.00000			0.338364
$560$		0			0
$561$	48.0000			2.02656
$562$		0			0
$563$	−4.24264	−	4.24264i	−0.178806	−	0.178806i	0.612029	−	0.790835i	$-0.290353\pi$
$563$	−4.24264	−	4.24264i	−0.178806	−	0.178806i	−0.790835	+	0.612029i	$0.790353\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−26.9444	−	26.9444i	−1.13156	−	1.13156i
$568$		0			0
$569$			6.00000i			0.251533i	0.992060	+	0.125767i	$0.0401390\pi$
$569$			6.00000i			0.251533i	−0.992060	+	0.125767i	$0.959861\pi$
$570$		0			0
$571$	−3.46410			−0.144968			−0.0724841	−	0.997370i	$-0.523093\pi$
$571$	−3.46410			−0.144968			−0.0724841	+	0.997370i	$0.523093\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−9.79796	+	9.79796i	−0.407894	+	0.407894i	−0.881004	−	0.473109i	$-0.843132\pi$
$577$	−9.79796	+	9.79796i	−0.407894	+	0.407894i	0.473109	+	0.881004i	$0.343132\pi$
$578$		0			0
$579$		−	13.8564i		−	0.575853i
$580$		0			0
$581$			20.7846i			0.862291i
$582$		0			0
$583$	29.3939	−	29.3939i	1.21737	−	1.21737i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−4.24264	+	4.24264i	−0.175113	+	0.175113i	−0.789221	−	0.614109i	$-0.789516\pi$
$587$	−4.24264	+	4.24264i	−0.175113	+	0.175113i	0.614109	+	0.789221i	$0.289516\pi$
$588$		0			0
$589$	−27.7128			−1.14189
$590$		0			0
$591$			24.0000i			0.987228i
$592$		0			0
$593$	−19.5959	−	19.5959i	−0.804708	−	0.804708i	0.179119	−	0.983827i	$-0.442675\pi$
$593$	−19.5959	−	19.5959i	−0.804708	−	0.804708i	−0.983827	+	0.179119i	$0.942675\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−22.6274	−	22.6274i	−0.926079	−	0.926079i
$598$		0			0
$599$	−48.0000			−1.96123			−0.980613	−	0.195952i	$-0.937220\pi$
$599$	−48.0000			−1.96123			−0.980613	+	0.195952i	$0.937220\pi$
$600$		0			0
$601$	−2.00000			−0.0815817			−0.0407909	−	0.999168i	$-0.512988\pi$
$601$	−2.00000			−0.0815817			−0.0407909	+	0.999168i	$0.512988\pi$
$602$		0			0
$603$	1.41421	+	1.41421i	0.0575912	+	0.0575912i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−2.44949	−	2.44949i	−0.0994217	−	0.0994217i	0.655646	−	0.755068i	$-0.272396\pi$
$607$	−2.44949	−	2.44949i	−0.0994217	−	0.0994217i	−0.755068	+	0.655646i	$0.772396\pi$
$608$		0			0
$609$		−	48.0000i		−	1.94506i
$610$		0			0
$611$	13.8564			0.560570
$612$		0			0
$613$	−14.1421	+	14.1421i	−0.571195	+	0.571195i	−0.932462	−	0.361267i	$-0.882344\pi$
$613$	−14.1421	+	14.1421i	−0.571195	+	0.571195i	0.361267	+	0.932462i	$0.382344\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	4.89898	−	4.89898i	0.197225	−	0.197225i	−0.601584	−	0.798810i	$-0.705464\pi$
$617$	4.89898	−	4.89898i	0.197225	−	0.197225i	0.798810	+	0.601584i	$0.205464\pi$
$618$		0			0
$619$			17.3205i			0.696170i	0.937463	+	0.348085i	$0.113168\pi$
$619$			17.3205i			0.696170i	−0.937463	+	0.348085i	$0.886832\pi$
$620$		0			0
$621$		−	13.8564i		−	0.556038i
$622$		0			0
$623$	14.6969	−	14.6969i	0.588820	−	0.588820i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	−16.9706	+	16.9706i	−0.677739	+	0.677739i
$628$		0			0
$629$	55.4256			2.20996
$630$		0			0
$631$			8.00000i			0.318475i	0.987240	+	0.159237i	$0.0509036\pi$
$631$			8.00000i			0.318475i	−0.987240	+	0.159237i	$0.949096\pi$
$632$		0			0
$633$	−4.89898	−	4.89898i	−0.194717	−	0.194717i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−14.1421	−	14.1421i	−0.560332	−	0.560332i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−42.0000			−1.65890			−0.829450	−	0.558581i	$-0.811346\pi$
$641$	−42.0000			−1.65890			−0.829450	+	0.558581i	$0.811346\pi$
$642$		0			0
$643$	32.5269	+	32.5269i	1.28274	+	1.28274i	0.939105	+	0.343632i	$0.111657\pi$
$643$	32.5269	+	32.5269i	1.28274	+	1.28274i	0.343632	+	0.939105i	$0.388343\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−7.34847	−	7.34847i	−0.288898	−	0.288898i	0.547746	−	0.836644i	$-0.315486\pi$
$647$	−7.34847	−	7.34847i	−0.288898	−	0.288898i	−0.836644	+	0.547746i	$0.815486\pi$
$648$		0			0
$649$			36.0000i			1.41312i
$650$		0			0
$651$	55.4256			2.17230
$652$		0			0
$653$	−8.48528	+	8.48528i	−0.332055	+	0.332055i	−0.853366	−	0.521312i	$-0.825443\pi$
$653$	−8.48528	+	8.48528i	−0.332055	+	0.332055i	0.521312	+	0.853366i	$0.325443\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	4.89898	−	4.89898i	0.191127	−	0.191127i
$658$		0			0
$659$			10.3923i			0.404827i	0.979300	+	0.202413i	$0.0648785\pi$
$659$			10.3923i			0.404827i	−0.979300	+	0.202413i	$0.935122\pi$
$660$		0			0
$661$		−	27.7128i		−	1.07790i	−0.842337	−	0.538952i	$-0.818821\pi$
$661$		−	27.7128i		−	1.07790i	0.842337	−	0.538952i	$-0.181179\pi$
$662$		0			0
$663$	−39.1918	+	39.1918i	−1.52208	+	1.52208i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	16.9706	−	16.9706i	0.657103	−	0.657103i
$668$		0			0
$669$	20.7846			0.803579
$670$		0			0
$671$		−	48.0000i		−	1.85302i
$672$		0			0
$673$	24.4949	+	24.4949i	0.944209	+	0.944209i	0.998524	−	0.0543149i	$-0.0172975\pi$
$673$	24.4949	+	24.4949i	0.944209	+	0.944209i	−0.0543149	+	0.998524i	$0.517297\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	8.48528	+	8.48528i	0.326116	+	0.326116i	0.851107	−	0.524992i	$-0.175932\pi$
$677$	8.48528	+	8.48528i	0.326116	+	0.326116i	−0.524992	+	0.851107i	$0.675932\pi$
$678$		0			0
$679$	24.0000			0.921035
$680$		0			0
$681$	36.0000			1.37952
$682$		0			0
$683$	21.2132	+	21.2132i	0.811701	+	0.811701i	0.984889	−	0.173188i	$-0.0554069\pi$
$683$	21.2132	+	21.2132i	0.811701	+	0.811701i	−0.173188	+	0.984889i	$0.555407\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−29.3939	−	29.3939i	−1.12145	−	1.12145i
$688$		0			0
$689$			48.0000i			1.82865i
$690$		0			0
$691$	31.1769			1.18603			0.593013	−	0.805193i	$-0.297938\pi$
$691$	31.1769			1.18603			0.593013	+	0.805193i	$0.297938\pi$
$692$		0			0
$693$	8.48528	−	8.48528i	0.322329	−	0.322329i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	29.3939	−	29.3939i	1.11337	−	1.11337i
$698$		0			0
$699$			41.5692i			1.57229i
$700$		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			0
$705$		0			0
$706$		0			0
$707$	−16.9706	+	16.9706i	−0.638244	+	0.638244i
$708$		0			0
$709$	−34.6410			−1.30097			−0.650485	−	0.759519i	$-0.725434\pi$
$709$	−34.6410			−1.30097			−0.650485	+	0.759519i	$0.725434\pi$
$710$		0			0
$711$			8.00000i			0.300023i
$712$		0			0
$713$	19.5959	+	19.5959i	0.733873	+	0.733873i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	33.9411	+	33.9411i	1.26755	+	1.26755i
$718$		0			0
$719$	−24.0000			−0.895049			−0.447524	−	0.894272i	$-0.647694\pi$
$719$	−24.0000			−0.895049			−0.447524	+	0.894272i	$0.647694\pi$
$720$		0			0
$721$	60.0000			2.23452
$722$		0			0
$723$	14.1421	+	14.1421i	0.525952	+	0.525952i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	7.34847	+	7.34847i	0.272540	+	0.272540i	0.830122	−	0.557582i	$-0.188271\pi$
$727$	7.34847	+	7.34847i	0.272540	+	0.272540i	−0.557582	+	0.830122i	$0.688271\pi$
$728$		0			0
$729$		−	13.0000i		−	0.481481i
$730$		0			0
$731$	−13.8564			−0.512498
$732$		0			0
$733$	−22.6274	+	22.6274i	−0.835763	+	0.835763i	−0.988298	−	0.152535i	$-0.951256\pi$
$733$	−22.6274	+	22.6274i	−0.835763	+	0.835763i	0.152535	+	0.988298i	$0.451256\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	4.89898	−	4.89898i	0.180456	−	0.180456i
$738$		0			0
$739$			10.3923i			0.382287i	0.981562	+	0.191144i	$0.0612196\pi$
$739$			10.3923i			0.382287i	−0.981562	+	0.191144i	$0.938780\pi$
$740$		0			0
$741$		−	27.7128i		−	1.01806i
$742$		0			0
$743$	26.9444	−	26.9444i	0.988494	−	0.988494i	−0.0114409	−	0.999935i	$-0.503642\pi$
$743$	26.9444	−	26.9444i	0.988494	−	0.988494i	0.999935	+	0.0114409i	$0.00364182\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	4.24264	−	4.24264i	0.155230	−	0.155230i
$748$		0			0
$749$	62.3538			2.27836
$750$		0			0
$751$			32.0000i			1.16770i	0.811863	+	0.583848i	$0.198454\pi$
$751$			32.0000i			1.16770i	−0.811863	+	0.583848i	$0.801546\pi$
$752$		0			0
$753$	−34.2929	−	34.2929i	−1.24970	−	1.24970i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	11.3137	+	11.3137i	0.411204	+	0.411204i	0.882158	−	0.470954i	$-0.156090\pi$
$757$	11.3137	+	11.3137i	0.411204	+	0.411204i	−0.470954	+	0.882158i	$0.656090\pi$
$758$		0			0
$759$	24.0000			0.871145
$760$		0			0
$761$	6.00000			0.217500			0.108750	−	0.994069i	$-0.465315\pi$
$761$	6.00000			0.217500			0.108750	+	0.994069i	$0.465315\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−29.3939	−	29.3939i	−1.06135	−	1.06135i
$768$		0			0
$769$			14.0000i			0.504853i	0.967616	+	0.252426i	$0.0812286\pi$
$769$			14.0000i			0.504853i	−0.967616	+	0.252426i	$0.918771\pi$
$770$		0			0
$771$	27.7128			0.998053
$772$		0			0
$773$	−8.48528	+	8.48528i	−0.305194	+	0.305194i	−0.843042	−	0.537848i	$-0.819238\pi$
$773$	−8.48528	+	8.48528i	−0.305194	+	0.305194i	0.537848	+	0.843042i	$0.319238\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	39.1918	−	39.1918i	1.40600	−	1.40600i
$778$		0			0
$779$			20.7846i			0.744686i
$780$		0			0
$781$		0			0
$782$		0			0
$783$	19.5959	−	19.5959i	0.700301	−	0.700301i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	15.5563	−	15.5563i	0.554524	−	0.554524i	−0.373219	−	0.927743i	$-0.621746\pi$
$787$	15.5563	−	15.5563i	0.554524	−	0.554524i	0.927743	+	0.373219i	$0.121746\pi$
$788$		0			0
$789$	−20.7846			−0.739952
$790$		0			0
$791$			48.0000i			1.70668i
$792$		0			0
$793$	39.1918	+	39.1918i	1.39174	+	1.39174i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−8.48528	−	8.48528i	−0.300564	−	0.300564i	0.540670	−	0.841235i	$-0.318171\pi$
$797$	−8.48528	−	8.48528i	−0.300564	−	0.300564i	−0.841235	+	0.540670i	$0.818171\pi$
$798$		0			0
$799$	−24.0000			−0.849059
$800$		0			0
$801$	−6.00000			−0.212000
$802$		0			0
$803$	−16.9706	−	16.9706i	−0.598878	−	0.598878i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	19.5959	+	19.5959i	0.689809	+	0.689809i
$808$		0			0
$809$			6.00000i			0.210949i	0.994422	+	0.105474i	$0.0336361\pi$
$809$			6.00000i			0.210949i	−0.994422	+	0.105474i	$0.966364\pi$
$810$		0			0
$811$	31.1769			1.09477			0.547385	−	0.836881i	$-0.315623\pi$
$811$	31.1769			1.09477			0.547385	+	0.836881i	$0.315623\pi$
$812$		0			0
$813$	−11.3137	+	11.3137i	−0.396789	+	0.396789i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	4.89898	−	4.89898i	0.171394	−	0.171394i
$818$		0			0
$819$			13.8564i			0.484182i
$820$		0			0
$821$		0			0		1.00000			$0$
$821$		0			0		−1.00000			$\pi$
$822$		0			0
$823$	12.2474	−	12.2474i	0.426919	−	0.426919i	−0.460658	−	0.887578i	$-0.652387\pi$
$823$	12.2474	−	12.2474i	0.426919	−	0.426919i	0.887578	+	0.460658i	$0.152387\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	4.24264	−	4.24264i	0.147531	−	0.147531i	−0.629483	−	0.777014i	$-0.716733\pi$
$827$	4.24264	−	4.24264i	0.147531	−	0.147531i	0.777014	+	0.629483i	$0.216733\pi$
$828$		0			0
$829$	13.8564			0.481253			0.240626	−	0.970618i	$-0.422647\pi$
$829$	13.8564			0.481253			0.240626	+	0.970618i	$0.422647\pi$
$830$		0			0
$831$			16.0000i			0.555034i
$832$		0			0
$833$	24.4949	+	24.4949i	0.848698	+	0.848698i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	22.6274	+	22.6274i	0.782118	+	0.782118i
$838$		0			0
$839$	−24.0000			−0.828572			−0.414286	−	0.910147i	$-0.635969\pi$
$839$	−24.0000			−0.828572			−0.414286	+	0.910147i	$0.635969\pi$
$840$		0			0
$841$	19.0000			0.655172
$842$		0			0
$843$	−42.4264	−	42.4264i	−1.46124	−	1.46124i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−2.44949	−	2.44949i	−0.0841655	−	0.0841655i
$848$		0			0
$849$			28.0000i			0.960958i
$850$		0			0
$851$	27.7128			0.949983
$852$		0			0
$853$	19.7990	−	19.7990i	0.677905	−	0.677905i	−0.281621	−	0.959526i	$-0.590872\pi$
$853$	19.7990	−	19.7990i	0.677905	−	0.677905i	0.959526	+	0.281621i	$0.0908721\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$857$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$858$		0			0
$859$		−	45.0333i		−	1.53652i	−0.640140	−	0.768259i	$-0.721124\pi$
$859$		−	45.0333i		−	1.53652i	0.640140	−	0.768259i	$-0.278876\pi$
$860$		0			0
$861$		−	41.5692i		−	1.41668i
$862$		0			0
$863$	31.8434	−	31.8434i	1.08396	−	1.08396i	0.0878249	−	0.996136i	$-0.472008\pi$
$863$	31.8434	−	31.8434i	1.08396	−	1.08396i	0.996136	−	0.0878249i	$-0.0279916\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	43.8406	−	43.8406i	1.48891	−	1.48891i
$868$		0			0
$869$	27.7128			0.940093
$870$		0			0
$871$			8.00000i			0.271070i
$872$		0			0
$873$	−4.89898	−	4.89898i	−0.165805	−	0.165805i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−28.2843	−	28.2843i	−0.955092	−	0.955092i	0.0439421	−	0.999034i	$-0.486008\pi$
$877$	−28.2843	−	28.2843i	−0.955092	−	0.955092i	−0.999034	+	0.0439421i	$0.986008\pi$
$878$		0			0
$879$	−48.0000			−1.61900
$880$		0			0
$881$	18.0000			0.606435			0.303218	−	0.952921i	$-0.401939\pi$
$881$	18.0000			0.606435			0.303218	+	0.952921i	$0.401939\pi$
$882$		0			0
$883$	18.3848	+	18.3848i	0.618697	+	0.618697i	0.945197	−	0.326500i	$-0.105869\pi$
$883$	18.3848	+	18.3848i	0.618697	+	0.618697i	−0.326500	+	0.945197i	$0.605869\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−31.8434	−	31.8434i	−1.06920	−	1.06920i	−0.997421	−	0.0717747i	$-0.977134\pi$
$887$	−31.8434	−	31.8434i	−1.06920	−	1.06920i	−0.0717747	−	0.997421i	$-0.522866\pi$
$888$		0			0
$889$		−	36.0000i		−	1.20740i
$890$		0			0
$891$	38.1051			1.27657
$892$		0			0
$893$	8.48528	−	8.48528i	0.283949	−	0.283949i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−19.5959	+	19.5959i	−0.654289	+	0.654289i
$898$		0			0
$899$			55.4256i			1.84855i
$900$		0			0
$901$		−	83.1384i		−	2.76974i
$902$		0			0
$903$	−9.79796	+	9.79796i	−0.326056	+	0.326056i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	18.3848	−	18.3848i	0.610456	−	0.610456i	−0.332609	−	0.943065i	$-0.607929\pi$
$907$	18.3848	−	18.3848i	0.610456	−	0.610456i	0.943065	+	0.332609i	$0.107929\pi$
$908$		0			0
$909$	6.92820			0.229794
$910$		0			0
$911$		−	24.0000i		−	0.795155i	−0.917568	−	0.397578i	$-0.869851\pi$
$911$		−	24.0000i		−	0.795155i	0.917568	−	0.397578i	$-0.130149\pi$
$912$		0			0
$913$	−14.6969	−	14.6969i	−0.486398	−	0.486398i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	8.48528	+	8.48528i	0.280209	+	0.280209i
$918$		0			0
$919$	32.0000			1.05558			0.527791	−	0.849374i	$-0.323020\pi$
$919$	32.0000			1.05558			0.527791	+	0.849374i	$0.323020\pi$
$920$		0			0
$921$	−28.0000			−0.922631
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$	−12.2474	−	12.2474i	−0.402259	−	0.402259i
$928$		0			0
$929$		−	42.0000i		−	1.37798i	−0.724773	−	0.688988i	$-0.758055\pi$
$929$		−	42.0000i		−	1.37798i	0.724773	−	0.688988i	$-0.241945\pi$
$930$		0			0
$931$	−17.3205			−0.567657
$932$		0			0
$933$	33.9411	−	33.9411i	1.11118	−	1.11118i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−4.89898	+	4.89898i	−0.160043	+	0.160043i	−0.782586	−	0.622543i	$-0.786100\pi$
$937$	−4.89898	+	4.89898i	−0.160043	+	0.160043i	0.622543	+	0.782586i	$0.286100\pi$
$938$		0			0
$939$		−	27.7128i		−	0.904373i
$940$		0			0
$941$		−	6.92820i		−	0.225853i	−0.993603	−	0.112926i	$-0.963978\pi$
$941$		−	6.92820i		−	0.225853i	0.993603	−	0.112926i	$-0.0360225\pi$
$942$		0			0
$943$	14.6969	−	14.6969i	0.478598	−	0.478598i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−21.2132	+	21.2132i	−0.689336	+	0.689336i	−0.962085	−	0.272749i	$-0.912067\pi$
$947$	−21.2132	+	21.2132i	−0.689336	+	0.689336i	0.272749	+	0.962085i	$0.412067\pi$
$948$		0			0
$949$	27.7128			0.899596
$950$		0			0
$951$			24.0000i			0.778253i
$952$		0			0
$953$	19.5959	+	19.5959i	0.634774	+	0.634774i	0.949262	−	0.314488i	$-0.101833\pi$
$953$	19.5959	+	19.5959i	0.634774	+	0.634774i	−0.314488	+	0.949262i	$0.601833\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	33.9411	+	33.9411i	1.09716	+	1.09716i
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−33.0000			−1.06452
$962$		0			0
$963$	−12.7279	−	12.7279i	−0.410152	−	0.410152i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	17.1464	+	17.1464i	0.551392	+	0.551392i	0.926842	−	0.375450i	$-0.122512\pi$
$967$	17.1464	+	17.1464i	0.551392	+	0.551392i	−0.375450	+	0.926842i	$0.622512\pi$
$968$		0			0
$969$			48.0000i			1.54198i
$970$		0			0
$971$	−17.3205			−0.555842			−0.277921	−	0.960604i	$-0.589645\pi$
$971$	−17.3205			−0.555842			−0.277921	+	0.960604i	$0.589645\pi$
$972$		0			0
$973$	−8.48528	+	8.48528i	−0.272026	+	0.272026i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−24.4949	+	24.4949i	−0.783661	+	0.783661i	−0.980447	−	0.196785i	$-0.936950\pi$
$977$	−24.4949	+	24.4949i	−0.783661	+	0.783661i	0.196785	+	0.980447i	$0.436950\pi$
$978$		0			0
$979$			20.7846i			0.664279i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−17.1464	+	17.1464i	−0.546886	+	0.546886i	−0.925539	−	0.378653i	$-0.876387\pi$
$983$	−17.1464	+	17.1464i	−0.546886	+	0.546886i	0.378653	+	0.925539i	$0.376387\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	−16.9706	+	16.9706i	−0.540179	+	0.540179i
$988$		0			0
$989$	−6.92820			−0.220304
$990$		0			0
$991$			16.0000i			0.508257i	0.967170	+	0.254128i	$0.0817886\pi$
$991$			16.0000i			0.508257i	−0.967170	+	0.254128i	$0.918211\pi$
$992$		0			0
$993$	−24.4949	−	24.4949i	−0.777322	−	0.777322i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	36.7696	+	36.7696i	1.16450	+	1.16450i	0.983479	+	0.181025i	$0.0579415\pi$
$997$	36.7696	+	36.7696i	1.16450	+	1.16450i	0.181025	+	0.983479i	$0.442059\pi$
$998$		0			0
$999$	32.0000			1.01244

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1600.2.o.g.543.3	yes	8
4.3	odd	2		1600.2.o.j.543.2	yes	8
5.2	odd	4		1600.2.o.j.607.4	yes	8
5.3	odd	4		1600.2.o.j.607.1	yes	8
5.4	even	2	inner	1600.2.o.g.543.2	yes	8
8.3	odd	2		1600.2.o.j.543.4	yes	8
8.5	even	2	inner	1600.2.o.g.543.1	✓	8
20.3	even	4	inner	1600.2.o.g.607.4	yes	8
20.7	even	4	inner	1600.2.o.g.607.1	yes	8
20.19	odd	2		1600.2.o.j.543.3	yes	8
40.3	even	4	inner	1600.2.o.g.607.2	yes	8
40.13	odd	4		1600.2.o.j.607.3	yes	8
40.19	odd	2		1600.2.o.j.543.1	yes	8
40.27	even	4	inner	1600.2.o.g.607.3	yes	8
40.29	even	2	inner	1600.2.o.g.543.4	yes	8
40.37	odd	4		1600.2.o.j.607.2	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1600.2.o.g.543.1	✓	8	8.5	even	2	inner
1600.2.o.g.543.2	yes	8	5.4	even	2	inner
1600.2.o.g.543.3	yes	8	1.1	even	1	trivial
1600.2.o.g.543.4	yes	8	40.29	even	2	inner
1600.2.o.g.607.1	yes	8	20.7	even	4	inner
1600.2.o.g.607.2	yes	8	40.3	even	4	inner
1600.2.o.g.607.3	yes	8	40.27	even	4	inner
1600.2.o.g.607.4	yes	8	20.3	even	4	inner
1600.2.o.j.543.1	yes	8	40.19	odd	2
1600.2.o.j.543.2	yes	8	4.3	odd	2
1600.2.o.j.543.3	yes	8	20.19	odd	2
1600.2.o.j.543.4	yes	8	8.3	odd	2
1600.2.o.j.607.1	yes	8	5.3	odd	4
1600.2.o.j.607.2	yes	8	40.37	odd	4
1600.2.o.j.607.3	yes	8	40.13	odd	4
1600.2.o.j.607.4	yes	8	5.2	odd	4

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:	\( N \)	\(=\)	\( 1600 = 2^{6} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1600.o (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(1.41421 + 1.41421i) q^{3} +(-2.44949 - 2.44949i) q^{7} +1.00000i q^{9} +O(q^{10})\)	\(q+(1.41421 + 1.41421i) q^{3} +(-2.44949 - 2.44949i) q^{7} +1.00000i q^{9} +3.46410 q^{11} +(-2.82843 + 2.82843i) q^{13} +(4.89898 - 4.89898i) q^{17} +3.46410i q^{19} -6.92820i q^{21} +(2.44949 - 2.44949i) q^{23} +(2.82843 - 2.82843i) q^{27} +6.92820 q^{29} +8.00000i q^{31} +(4.89898 + 4.89898i) q^{33} +(5.65685 + 5.65685i) q^{37} -8.00000 q^{39} +6.00000 q^{41} +(-1.41421 - 1.41421i) q^{43} +(-2.44949 - 2.44949i) q^{47} +5.00000i q^{49} +13.8564 q^{51} +(8.48528 - 8.48528i) q^{53} +(-4.89898 + 4.89898i) q^{57} +10.3923i q^{59} -13.8564i q^{61} +(2.44949 - 2.44949i) q^{63} +(1.41421 - 1.41421i) q^{67} +6.92820 q^{69} +(-4.89898 - 4.89898i) q^{73} +(-8.48528 - 8.48528i) q^{77} +8.00000 q^{79} +11.0000 q^{81} +(-4.24264 - 4.24264i) q^{83} +(9.79796 + 9.79796i) q^{87} +6.00000i q^{89} +13.8564 q^{91} +(-11.3137 + 11.3137i) q^{93} +(-4.89898 + 4.89898i) q^{97} +3.46410i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q - 64 q^{39} + 48 q^{41} + 64 q^{79} + 88 q^{81}+O(q^{100}) \) 8 * q - 64 * q^39 + 48 * q^41 + 64 * q^79 + 88 * q^81

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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