LMFDB - Embedded newform 900.3.l.g.757.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [900,3,Mod(757,900)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(900, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("900.757");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	900.l (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:	$24.5232237924$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(i)$
Coefficient field:	$\Q(i, \sqrt{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9 $ x^4 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 100)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			757.1
Root			$-1.22474 - 1.22474i$ of defining polynomial
Character	$\chi$	$=$	900.757
Dual form			900.3.l.g.793.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-4.89898 - 4.89898i) q^{7} +O(q^{10})$	$q+(-4.89898 - 4.89898i) q^{7} +15.0000 q^{11} +(-2.44949 + 2.44949i) q^{13} +(3.67423 + 3.67423i) q^{17} +17.0000i q^{19} +(7.34847 - 7.34847i) q^{23} -42.0000i q^{29} -14.0000 q^{31} +(-29.3939 - 29.3939i) q^{37} +39.0000 q^{41} +(24.4949 - 24.4949i) q^{43} +(-51.4393 - 51.4393i) q^{47} -1.00000i q^{49} +(66.1362 - 66.1362i) q^{53} +6.00000i q^{59} +92.0000 q^{61} +(-35.5176 - 35.5176i) q^{67} +102.000 q^{71} +(3.67423 - 3.67423i) q^{73} +(-73.4847 - 73.4847i) q^{77} -104.000i q^{79} +(-47.7650 + 47.7650i) q^{83} +87.0000i q^{89} +24.0000 q^{91} +(93.0806 + 93.0806i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q + 60 q^{11} - 56 q^{31} + 156 q^{41} + 368 q^{61} + 408 q^{71} + 96 q^{91}+O(q^{100}) $ 4 * q + 60 * q^11 - 56 * q^31 + 156 * q^41 + 368 * q^61 + 408 * q^71 + 96 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/900\mathbb{Z}\right)^\times$.

$n$	$101$	$451$	$577$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{4}\right)$

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			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−4.89898	−	4.89898i	−0.699854	−	0.699854i	0.264525	−	0.964379i	$-0.414785\pi$
$7$	−4.89898	−	4.89898i	−0.699854	−	0.699854i	−0.964379	+	0.264525i	$0.914785\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	15.0000			1.36364			0.681818	−	0.731522i	$-0.261190\pi$
$11$	15.0000			1.36364			0.681818	+	0.731522i	$0.261190\pi$
$12$		0			0
$13$	−2.44949	+	2.44949i	−0.188422	+	0.188422i	−0.795014	−	0.606591i	$-0.792536\pi$
$13$	−2.44949	+	2.44949i	−0.188422	+	0.188422i	0.606591	+	0.795014i	$0.292536\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	3.67423	+	3.67423i	0.216131	+	0.216131i	0.806866	−	0.590735i	$-0.201162\pi$
$17$	3.67423	+	3.67423i	0.216131	+	0.216131i	−0.590735	+	0.806866i	$0.701162\pi$
$18$		0			0
$19$			17.0000i			0.894737i	0.894350	+	0.447368i	$0.147639\pi$
$19$			17.0000i			0.894737i	−0.894350	+	0.447368i	$0.852361\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	7.34847	−	7.34847i	0.319499	−	0.319499i	−0.529076	−	0.848575i	$-0.677461\pi$
$23$	7.34847	−	7.34847i	0.319499	−	0.319499i	0.848575	+	0.529076i	$0.177461\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	42.0000i		−	1.44828i	−0.689655	−	0.724138i	$-0.742238\pi$
$29$		−	42.0000i		−	1.44828i	0.689655	−	0.724138i	$-0.257762\pi$
$30$		0			0
$31$	−14.0000			−0.451613			−0.225806	−	0.974172i	$-0.572502\pi$
$31$	−14.0000			−0.451613			−0.225806	+	0.974172i	$0.572502\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−29.3939	−	29.3939i	−0.794429	−	0.794429i	0.187782	−	0.982211i	$-0.439870\pi$
$37$	−29.3939	−	29.3939i	−0.794429	−	0.794429i	−0.982211	+	0.187782i	$0.939870\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	39.0000			0.951220			0.475610	−	0.879656i	$-0.342227\pi$
$41$	39.0000			0.951220			0.475610	+	0.879656i	$0.342227\pi$
$42$		0			0
$43$	24.4949	−	24.4949i	0.569649	−	0.569649i	−0.362381	−	0.932030i	$-0.618036\pi$
$43$	24.4949	−	24.4949i	0.569649	−	0.569649i	0.932030	+	0.362381i	$0.118036\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−51.4393	−	51.4393i	−1.09445	−	1.09445i	−0.995047	−	0.0994059i	$-0.968306\pi$
$47$	−51.4393	−	51.4393i	−1.09445	−	1.09445i	−0.0994059	−	0.995047i	$-0.531694\pi$
$48$		0			0
$49$		−	1.00000i		−	0.0204082i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	66.1362	−	66.1362i	1.24785	−	1.24785i	0.291187	−	0.956666i	$-0.405950\pi$
$53$	66.1362	−	66.1362i	1.24785	−	1.24785i	0.956666	−	0.291187i	$-0.0940502\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$			6.00000i			0.101695i	0.998706	+	0.0508475i	$0.0161922\pi$
$59$			6.00000i			0.101695i	−0.998706	+	0.0508475i	$0.983808\pi$
$60$		0			0
$61$	92.0000			1.50820			0.754098	−	0.656761i	$-0.228074\pi$
$61$	92.0000			1.50820			0.754098	+	0.656761i	$0.228074\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−35.5176	−	35.5176i	−0.530113	−	0.530113i	0.390493	−	0.920606i	$-0.372305\pi$
$67$	−35.5176	−	35.5176i	−0.530113	−	0.530113i	−0.920606	+	0.390493i	$0.872305\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	102.000			1.43662			0.718310	−	0.695723i	$-0.244916\pi$
$71$	102.000			1.43662			0.718310	+	0.695723i	$0.244916\pi$
$72$		0			0
$73$	3.67423	−	3.67423i	0.0503320	−	0.0503320i	−0.681493	−	0.731825i	$-0.738669\pi$
$73$	3.67423	−	3.67423i	0.0503320	−	0.0503320i	0.731825	+	0.681493i	$0.238669\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−73.4847	−	73.4847i	−0.954347	−	0.954347i
$78$		0			0
$79$		−	104.000i		−	1.31646i	−0.752819	−	0.658228i	$-0.771306\pi$
$79$		−	104.000i		−	1.31646i	0.752819	−	0.658228i	$-0.228694\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−47.7650	+	47.7650i	−0.575483	+	0.575483i	−0.933655	−	0.358173i	$-0.883400\pi$
$83$	−47.7650	+	47.7650i	−0.575483	+	0.575483i	0.358173	+	0.933655i	$0.383400\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$			87.0000i			0.977528i	0.872416	+	0.488764i	$0.162552\pi$
$89$			87.0000i			0.977528i	−0.872416	+	0.488764i	$0.837448\pi$
$90$		0			0
$91$	24.0000			0.263736
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	93.0806	+	93.0806i	0.959594	+	0.959594i	0.999215	−	0.0396209i	$-0.0126150\pi$
$97$	93.0806	+	93.0806i	0.959594	+	0.959594i	−0.0396209	+	0.999215i	$0.512615\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	24.0000			0.237624			0.118812	−	0.992917i	$-0.462091\pi$
$101$	24.0000			0.237624			0.118812	+	0.992917i	$0.462091\pi$
$102$		0			0
$103$	7.34847	−	7.34847i	0.0713444	−	0.0713444i	−0.670534	−	0.741879i	$-0.733935\pi$
$103$	7.34847	−	7.34847i	0.0713444	−	0.0713444i	0.741879	+	0.670534i	$0.233935\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−106.553	−	106.553i	−0.995821	−	0.995821i	0.00417071	−	0.999991i	$-0.498672\pi$
$107$	−106.553	−	106.553i	−0.995821	−	0.995821i	−0.999991	+	0.00417071i	$0.998672\pi$
$108$		0			0
$109$			64.0000i			0.587156i	0.955935	+	0.293578i	$0.0948460\pi$
$109$			64.0000i			0.587156i	−0.955935	+	0.293578i	$0.905154\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	84.5074	−	84.5074i	0.747853	−	0.747853i	−0.226223	−	0.974076i	$-0.572638\pi$
$113$	84.5074	−	84.5074i	0.747853	−	0.747853i	0.974076	+	0.226223i	$0.0726377\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	36.0000i		−	0.302521i
$120$		0			0
$121$	104.000			0.859504
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−66.1362	−	66.1362i	−0.520758	−	0.520758i	0.397043	−	0.917800i	$-0.370037\pi$
$127$	−66.1362	−	66.1362i	−0.520758	−	0.520758i	−0.917800	+	0.397043i	$0.870037\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	30.0000			0.229008			0.114504	−	0.993423i	$-0.463472\pi$
$131$	30.0000			0.229008			0.114504	+	0.993423i	$0.463472\pi$
$132$		0			0
$133$	83.2827	−	83.2827i	0.626185	−	0.626185i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−47.7650	−	47.7650i	−0.348650	−	0.348650i	0.510957	−	0.859607i	$-0.329291\pi$
$137$	−47.7650	−	47.7650i	−0.348650	−	0.348650i	−0.859607	+	0.510957i	$0.829291\pi$
$138$		0			0
$139$		−	163.000i		−	1.17266i	−0.810072	−	0.586331i	$-0.800572\pi$
$139$		−	163.000i		−	1.17266i	0.810072	−	0.586331i	$-0.199428\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−36.7423	+	36.7423i	−0.256939	+	0.256939i
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$		−	18.0000i		−	0.120805i	−0.998174	−	0.0604027i	$-0.980762\pi$
$149$		−	18.0000i		−	0.120805i	0.998174	−	0.0604027i	$-0.0192385\pi$
$150$		0			0
$151$	−20.0000			−0.132450			−0.0662252	−	0.997805i	$-0.521096\pi$
$151$	−20.0000			−0.132450			−0.0662252	+	0.997805i	$0.521096\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−105.328	−	105.328i	−0.670879	−	0.670879i	0.287039	−	0.957919i	$-0.407329\pi$
$157$	−105.328	−	105.328i	−0.670879	−	0.670879i	−0.957919	+	0.287039i	$0.907329\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−72.0000			−0.447205
$162$		0			0
$163$	6.12372	−	6.12372i	0.0375689	−	0.0375689i	−0.688073	−	0.725642i	$-0.741543\pi$
$163$	6.12372	−	6.12372i	0.0375689	−	0.0375689i	0.725642	+	0.688073i	$0.241543\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	80.8332	+	80.8332i	0.484031	+	0.484031i	0.906416	−	0.422385i	$-0.138807\pi$
$167$	80.8332	+	80.8332i	0.484031	+	0.484031i	−0.422385	+	0.906416i	$0.638807\pi$
$168$		0			0
$169$			157.000i			0.928994i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	176.363	−	176.363i	1.01944	−	1.01944i	0.0196336	−	0.999807i	$-0.493750\pi$
$173$	176.363	−	176.363i	1.01944	−	1.01944i	0.999807	−	0.0196336i	$-0.00624996\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	231.000i		−	1.29050i	−0.763970	−	0.645251i	$-0.776753\pi$
$179$		−	231.000i		−	1.29050i	0.763970	−	0.645251i	$-0.223247\pi$
$180$		0			0
$181$	−170.000			−0.939227			−0.469613	−	0.882872i	$-0.655607\pi$
$181$	−170.000			−0.939227			−0.469613	+	0.882872i	$0.655607\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	55.1135	+	55.1135i	0.294725	+	0.294725i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	42.0000			0.219895			0.109948	−	0.993937i	$-0.464932\pi$
$191$	42.0000			0.219895			0.109948	+	0.993937i	$0.464932\pi$
$192$		0			0
$193$	−197.184	+	197.184i	−1.02168	+	1.02168i	−0.0219186	+	0.999760i	$0.506977\pi$
$193$	−197.184	+	197.184i	−1.02168	+	1.02168i	−0.999760	+	0.0219186i	$0.993023\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	227.803	+	227.803i	1.15636	+	1.15636i	0.985253	+	0.171105i	$0.0547339\pi$
$197$	227.803	+	227.803i	1.15636	+	1.15636i	0.171105	+	0.985253i	$0.445266\pi$
$198$		0			0
$199$			136.000i			0.683417i	0.939806	+	0.341709i	$0.111006\pi$
$199$			136.000i			0.683417i	−0.939806	+	0.341709i	$0.888994\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−205.757	+	205.757i	−1.01358	+	1.01358i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$			255.000i			1.22010i
$210$		0			0
$211$	−211.000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$211$	−211.000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	68.5857	+	68.5857i	0.316063	+	0.316063i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−18.0000			−0.0814480
$222$		0			0
$223$	−24.4949	+	24.4949i	−0.109843	+	0.109843i	−0.759892	−	0.650049i	$-0.774748\pi$
$223$	−24.4949	+	24.4949i	−0.109843	+	0.109843i	0.650049	+	0.759892i	$0.274748\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	205.757	+	205.757i	0.906419	+	0.906419i	0.995981	−	0.0895621i	$-0.0285468\pi$
$227$	205.757	+	205.757i	0.906419	+	0.906419i	−0.0895621	+	0.995981i	$0.528547\pi$
$228$		0			0
$229$			68.0000i			0.296943i	0.988917	+	0.148472i	$0.0474354\pi$
$229$			68.0000i			0.296943i	−0.988917	+	0.148472i	$0.952565\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−191.060	+	191.060i	−0.820001	+	0.820001i	−0.986108	−	0.166107i	$-0.946880\pi$
$233$	−191.060	+	191.060i	−0.820001	+	0.820001i	0.166107	+	0.986108i	$0.446880\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	162.000i		−	0.677824i	−0.940818	−	0.338912i	$-0.889941\pi$
$239$		−	162.000i		−	0.677824i	0.940818	−	0.338912i	$-0.110059\pi$
$240$		0			0
$241$	29.0000			0.120332			0.0601660	−	0.998188i	$-0.480837\pi$
$241$	29.0000			0.120332			0.0601660	+	0.998188i	$0.480837\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−41.6413	−	41.6413i	−0.168588	−	0.168588i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−225.000			−0.896414			−0.448207	−	0.893930i	$-0.647937\pi$
$251$	−225.000			−0.896414			−0.448207	+	0.893930i	$0.647937\pi$
$252$		0			0
$253$	110.227	−	110.227i	0.435680	−	0.435680i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	132.272	+	132.272i	0.514679	+	0.514679i	0.915956	−	0.401278i	$-0.131434\pi$
$257$	132.272	+	132.272i	0.514679	+	0.514679i	−0.401278	+	0.915956i	$0.631434\pi$
$258$		0			0
$259$			288.000i			1.11197i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−95.5301	+	95.5301i	−0.363232	+	0.363232i	−0.865002	−	0.501769i	$-0.832683\pi$
$263$	−95.5301	+	95.5301i	−0.363232	+	0.363232i	0.501769	+	0.865002i	$0.332683\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$			24.0000i			0.0892193i	0.999004	+	0.0446097i	$0.0142044\pi$
$269$			24.0000i			0.0892193i	−0.999004	+	0.0446097i	$0.985796\pi$
$270$		0			0
$271$	110.000			0.405904			0.202952	−	0.979189i	$-0.434946\pi$
$271$	110.000			0.405904			0.202952	+	0.979189i	$0.434946\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	46.5403	+	46.5403i	0.168016	+	0.168016i	0.786107	−	0.618091i	$-0.212094\pi$
$277$	46.5403	+	46.5403i	0.168016	+	0.168016i	−0.618091	+	0.786107i	$0.712094\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−126.000			−0.448399			−0.224199	−	0.974543i	$-0.571977\pi$
$281$	−126.000			−0.448399			−0.224199	+	0.974543i	$0.571977\pi$
$282$		0			0
$283$	−292.714	+	292.714i	−1.03433	+	1.03433i	−0.0349356	+	0.999390i	$0.511123\pi$
$283$	−292.714	+	292.714i	−1.03433	+	1.03433i	−0.999390	+	0.0349356i	$0.988877\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−191.060	−	191.060i	−0.665715	−	0.665715i
$288$		0			0
$289$		−	262.000i		−	0.906574i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	73.4847	−	73.4847i	0.250801	−	0.250801i	−0.570498	−	0.821299i	$-0.693250\pi$
$293$	73.4847	−	73.4847i	0.250801	−	0.250801i	0.821299	+	0.570498i	$0.193250\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$			36.0000i			0.120401i
$300$		0			0
$301$	−240.000			−0.797342
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	221.679	+	221.679i	0.722081	+	0.722081i	0.969029	−	0.246948i	$-0.0794277\pi$
$307$	221.679	+	221.679i	0.722081	+	0.722081i	−0.246948	+	0.969029i	$0.579428\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−552.000			−1.77492			−0.887460	−	0.460885i	$-0.847532\pi$
$311$	−552.000			−1.77492			−0.887460	+	0.460885i	$0.847532\pi$
$312$		0			0
$313$	406.615	−	406.615i	1.29909	−	1.29909i	0.370097	−	0.928993i	$-0.379324\pi$
$313$	406.615	−	406.615i	1.29909	−	1.29909i	0.928993	−	0.370097i	$-0.120676\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	426.211	+	426.211i	1.34451	+	1.34451i	0.891506	+	0.453009i	$0.149649\pi$
$317$	426.211	+	426.211i	1.34451	+	1.34451i	0.453009	+	0.891506i	$0.350351\pi$
$318$		0			0
$319$		−	630.000i		−	1.97492i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−62.4620	+	62.4620i	−0.193381	+	0.193381i
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$			504.000i			1.53191i
$330$		0			0
$331$	−157.000			−0.474320			−0.237160	−	0.971471i	$-0.576217\pi$
$331$	−157.000			−0.474320			−0.237160	+	0.971471i	$0.576217\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	233.926	+	233.926i	0.694143	+	0.694143i	0.963141	−	0.268998i	$-0.0866924\pi$
$337$	233.926	+	233.926i	0.694143	+	0.694143i	−0.268998	+	0.963141i	$0.586692\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−210.000			−0.615836
$342$		0			0
$343$	−244.949	+	244.949i	−0.714137	+	0.714137i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−91.8559	−	91.8559i	−0.264714	−	0.264714i	0.562252	−	0.826966i	$-0.309935\pi$
$347$	−91.8559	−	91.8559i	−0.264714	−	0.264714i	−0.826966	+	0.562252i	$0.809935\pi$
$348$		0			0
$349$			500.000i			1.43266i	0.697759	+	0.716332i	$0.254181\pi$
$349$			500.000i			1.43266i	−0.697759	+	0.716332i	$0.745819\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−14.6969	+	14.6969i	−0.0416344	+	0.0416344i	−0.727617	−	0.685983i	$-0.759372\pi$
$353$	−14.6969	+	14.6969i	−0.0416344	+	0.0416344i	0.685983	+	0.727617i	$0.259372\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$			12.0000i			0.0334262i	0.999860	+	0.0167131i	$0.00532019\pi$
$359$			12.0000i			0.0334262i	−0.999860	+	0.0167131i	$0.994680\pi$
$360$		0			0
$361$	72.0000			0.199446
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	53.8888	+	53.8888i	0.146836	+	0.146836i	0.776703	−	0.629867i	$-0.216891\pi$
$367$	53.8888	+	53.8888i	0.146836	+	0.146836i	−0.629867	+	0.776703i	$0.716891\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−648.000			−1.74663
$372$		0			0
$373$	−276.792	+	276.792i	−0.742071	+	0.742071i	−0.972976	−	0.230906i	$-0.925831\pi$
$373$	−276.792	+	276.792i	−0.742071	+	0.742071i	0.230906	+	0.972976i	$0.425831\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	102.879	+	102.879i	0.272887	+	0.272887i
$378$		0			0
$379$		−	181.000i		−	0.477573i	−0.971072	−	0.238786i	$-0.923250\pi$
$379$		−	181.000i		−	0.477573i	0.971072	−	0.238786i	$-0.0767495\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−249.848	+	249.848i	−0.652345	+	0.652345i	−0.953557	−	0.301213i	$-0.902609\pi$
$383$	−249.848	+	249.848i	−0.652345	+	0.652345i	0.301213	+	0.953557i	$0.402609\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$			162.000i			0.416452i	0.978081	+	0.208226i	$0.0667690\pi$
$389$			162.000i			0.416452i	−0.978081	+	0.208226i	$0.933231\pi$
$390$		0			0
$391$	54.0000			0.138107
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	362.524	+	362.524i	0.913160	+	0.913160i	0.996520	−	0.0833596i	$-0.0265650\pi$
$397$	362.524	+	362.524i	0.913160	+	0.913160i	−0.0833596	+	0.996520i	$0.526565\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	405.000			1.00998			0.504988	−	0.863127i	$-0.331497\pi$
$401$	405.000			1.00998			0.504988	+	0.863127i	$0.331497\pi$
$402$		0			0
$403$	34.2929	−	34.2929i	0.0850939	−	0.0850939i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−440.908	−	440.908i	−1.08331	−	1.08331i
$408$		0			0
$409$		−	643.000i		−	1.57213i	−0.618146	−	0.786064i	$-0.712116\pi$
$409$		−	643.000i		−	1.57213i	0.618146	−	0.786064i	$-0.287884\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	29.3939	−	29.3939i	0.0711716	−	0.0711716i
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	207.000i		−	0.494033i	−0.969011	−	0.247017i	$-0.920550\pi$
$419$		−	207.000i		−	0.494033i	0.969011	−	0.247017i	$-0.0794503\pi$
$420$		0			0
$421$	−104.000			−0.247031			−0.123515	−	0.992343i	$-0.539417\pi$
$421$	−104.000			−0.247031			−0.123515	+	0.992343i	$0.539417\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−450.706	−	450.706i	−1.05552	−	1.05552i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	444.000			1.03016			0.515081	−	0.857141i	$-0.327762\pi$
$431$	444.000			1.03016			0.515081	+	0.857141i	$0.327762\pi$
$432$		0			0
$433$	530.315	−	530.315i	1.22474	−	1.22474i	0.258819	−	0.965926i	$-0.416667\pi$
$433$	530.315	−	530.315i	1.22474	−	1.22474i	0.965926	−	0.258819i	$-0.0833333\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	124.924	+	124.924i	0.285867	+	0.285867i
$438$		0			0
$439$			826.000i			1.88155i	0.339033	+	0.940774i	$0.389900\pi$
$439$			826.000i			1.88155i	−0.339033	+	0.940774i	$0.610100\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−268.219	+	268.219i	−0.605461	+	0.605461i	−0.941756	−	0.336296i	$-0.890826\pi$
$443$	−268.219	+	268.219i	−0.605461	+	0.605461i	0.336296	+	0.941756i	$0.390826\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$			483.000i			1.07572i	0.843033	+	0.537862i	$0.180768\pi$
$449$			483.000i			1.07572i	−0.843033	+	0.537862i	$0.819232\pi$
$450$		0			0
$451$	585.000			1.29712
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−33.0681	−	33.0681i	−0.0723591	−	0.0723591i	0.670001	−	0.742360i	$-0.266294\pi$
$457$	−33.0681	−	33.0681i	−0.0723591	−	0.0723591i	−0.742360	+	0.670001i	$0.766294\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	660.000			1.43167			0.715835	−	0.698269i	$-0.246046\pi$
$461$	660.000			1.43167			0.715835	+	0.698269i	$0.246046\pi$
$462$		0			0
$463$	−249.848	+	249.848i	−0.539628	+	0.539628i	−0.923420	−	0.383791i	$-0.874618\pi$
$463$	−249.848	+	249.848i	−0.539628	+	0.539628i	0.383791	+	0.923420i	$0.374618\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	367.423	+	367.423i	0.786774	+	0.786774i	0.980964	−	0.194190i	$-0.0622078\pi$
$467$	367.423	+	367.423i	0.786774	+	0.786774i	−0.194190	+	0.980964i	$0.562208\pi$
$468$		0			0
$469$			348.000i			0.742004i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	367.423	−	367.423i	0.776794	−	0.776794i
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$		−	822.000i		−	1.71608i	−0.513587	−	0.858038i	$-0.671684\pi$
$479$		−	822.000i		−	1.71608i	0.513587	−	0.858038i	$-0.328316\pi$
$480$		0			0
$481$	144.000			0.299376
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−149.419	−	149.419i	−0.306815	−	0.306815i	0.536858	−	0.843673i	$-0.319611\pi$
$487$	−149.419	−	149.419i	−0.306815	−	0.306815i	−0.843673	+	0.536858i	$0.819611\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−150.000			−0.305499			−0.152749	−	0.988265i	$-0.548813\pi$
$491$	−150.000			−0.305499			−0.152749	+	0.988265i	$0.548813\pi$
$492$		0			0
$493$	154.318	−	154.318i	0.313018	−	0.313018i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−499.696	−	499.696i	−1.00542	−	1.00542i
$498$		0			0
$499$		−	518.000i		−	1.03808i	−0.854751	−	0.519038i	$-0.826290\pi$
$499$		−	518.000i		−	1.03808i	0.854751	−	0.519038i	$-0.173710\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	382.120	−	382.120i	0.759683	−	0.759683i	−0.216582	−	0.976264i	$-0.569491\pi$
$503$	382.120	−	382.120i	0.759683	−	0.759683i	0.976264	+	0.216582i	$0.0694908\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$		−	342.000i		−	0.671906i	−0.941879	−	0.335953i	$-0.890942\pi$
$509$		−	342.000i		−	0.671906i	0.941879	−	0.335953i	$-0.109058\pi$
$510$		0			0
$511$	−36.0000			−0.0704501
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−771.589	−	771.589i	−1.49244	−	1.49244i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−9.00000			−0.0172745			−0.00863724	−	0.999963i	$-0.502749\pi$
$521$	−9.00000			−0.0172745			−0.00863724	+	0.999963i	$0.502749\pi$
$522$		0			0
$523$	−206.982	+	206.982i	−0.395759	+	0.395759i	−0.876734	−	0.480975i	$-0.840283\pi$
$523$	−206.982	+	206.982i	−0.395759	+	0.395759i	0.480975	+	0.876734i	$0.340283\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−51.4393	−	51.4393i	−0.0976078	−	0.0976078i
$528$		0			0
$529$			421.000i			0.795841i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−95.5301	+	95.5301i	−0.179231	+	0.179231i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	15.0000i		−	0.0278293i
$540$		0			0
$541$	−856.000			−1.58226			−0.791128	−	0.611651i	$-0.790506\pi$
$541$	−856.000			−1.58226			−0.791128	+	0.611651i	$0.790506\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	481.325	+	481.325i	0.879936	+	0.879936i	0.993527	−	0.113592i	$-0.0362357\pi$
$547$	481.325	+	481.325i	0.879936	+	0.879936i	−0.113592	+	0.993527i	$0.536236\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	714.000			1.29583
$552$		0			0
$553$	−509.494	+	509.494i	−0.921327	+	0.921327i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−36.7423	−	36.7423i	−0.0659647	−	0.0659647i	0.673355	−	0.739320i	$-0.264853\pi$
$557$	−36.7423	−	36.7423i	−0.0659647	−	0.0659647i	−0.739320	+	0.673355i	$0.764853\pi$
$558$		0			0
$559$			120.000i			0.214669i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	308.636	−	308.636i	0.548198	−	0.548198i	−0.377721	−	0.925919i	$-0.623292\pi$
$563$	308.636	−	308.636i	0.548198	−	0.548198i	0.925919	+	0.377721i	$0.123292\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	435.000i		−	0.764499i	−0.924059	−	0.382250i	$-0.875149\pi$
$569$		−	435.000i		−	0.764499i	0.924059	−	0.382250i	$-0.124851\pi$
$570$		0			0
$571$	−562.000			−0.984238			−0.492119	−	0.870528i	$-0.663778\pi$
$571$	−562.000			−0.984238			−0.492119	+	0.870528i	$0.663778\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−233.926	−	233.926i	−0.405418	−	0.405418i	0.474719	−	0.880137i	$-0.342550\pi$
$577$	−233.926	−	233.926i	−0.405418	−	0.405418i	−0.880137	+	0.474719i	$0.842550\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	468.000			0.805508
$582$		0			0
$583$	992.043	−	992.043i	1.70162	−	1.70162i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	40.4166	+	40.4166i	0.0688528	+	0.0688528i	0.740695	−	0.671842i	$-0.234497\pi$
$587$	40.4166	+	40.4166i	0.0688528	+	0.0688528i	−0.671842	+	0.740695i	$0.734497\pi$
$588$		0			0
$589$		−	238.000i		−	0.404075i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	628.294	−	628.294i	1.05952	−	1.05952i	0.0614050	−	0.998113i	$-0.480442\pi$
$593$	628.294	−	628.294i	1.05952	−	1.05952i	0.998113	−	0.0614050i	$-0.0195581\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	60.0000i		−	0.100167i	−0.998745	−	0.0500835i	$-0.984051\pi$
$599$		−	60.0000i		−	0.100167i	0.998745	−	0.0500835i	$-0.0159487\pi$
$600$		0			0
$601$	745.000			1.23960			0.619800	−	0.784760i	$-0.287214\pi$
$601$	745.000			1.23960			0.619800	+	0.784760i	$0.287214\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−558.484	−	558.484i	−0.920072	−	0.920072i	0.0769621	−	0.997034i	$-0.475478\pi$
$607$	−558.484	−	558.484i	−0.920072	−	0.920072i	−0.997034	+	0.0769621i	$0.975478\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	252.000			0.412439
$612$		0			0
$613$	95.5301	−	95.5301i	0.155840	−	0.155840i	−0.624880	−	0.780721i	$-0.714852\pi$
$613$	95.5301	−	95.5301i	0.155840	−	0.155840i	0.780721	+	0.624880i	$0.214852\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−382.120	−	382.120i	−0.619320	−	0.619320i	0.326037	−	0.945357i	$-0.394287\pi$
$617$	−382.120	−	382.120i	−0.619320	−	0.619320i	−0.945357	+	0.326037i	$0.894287\pi$
$618$		0			0
$619$			250.000i			0.403877i	0.979398	+	0.201939i	$0.0647241\pi$
$619$			250.000i			0.403877i	−0.979398	+	0.201939i	$0.935276\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	426.211	−	426.211i	0.684127	−	0.684127i
$624$		0			0
$625$		0			0
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	216.000i		−	0.343402i
$630$		0			0
$631$	962.000			1.52456			0.762282	−	0.647245i	$-0.224079\pi$
$631$	962.000			1.52456			0.762282	+	0.647245i	$0.224079\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	2.44949	+	2.44949i	0.00384535	+	0.00384535i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−582.000			−0.907956			−0.453978	−	0.891013i	$-0.649996\pi$
$641$	−582.000			−0.907956			−0.453978	+	0.891013i	$0.649996\pi$
$642$		0			0
$643$	−396.817	+	396.817i	−0.617134	+	0.617134i	−0.944795	−	0.327661i	$-0.893740\pi$
$643$	−396.817	+	396.817i	−0.617134	+	0.617134i	0.327661	+	0.944795i	$0.393740\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−852.422	−	852.422i	−1.31750	−	1.31750i	−0.915749	−	0.401751i	$-0.868402\pi$
$647$	−852.422	−	852.422i	−1.31750	−	1.31750i	−0.401751	−	0.915749i	$-0.631598\pi$
$648$		0			0
$649$			90.0000i			0.138675i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−352.727	+	352.727i	−0.540163	+	0.540163i	−0.923577	−	0.383414i	$-0.874748\pi$
$653$	−352.727	+	352.727i	−0.540163	+	0.540163i	0.383414	+	0.923577i	$0.374748\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$			153.000i			0.232170i	0.993239	+	0.116085i	$0.0370345\pi$
$659$			153.000i			0.232170i	−0.993239	+	0.116085i	$0.962966\pi$
$660$		0			0
$661$	658.000			0.995461			0.497731	−	0.867332i	$-0.334167\pi$
$661$	658.000			0.995461			0.497731	+	0.867332i	$0.334167\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−308.636	−	308.636i	−0.462722	−	0.462722i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	1380.00			2.05663
$672$		0			0
$673$	411.514	−	411.514i	0.611463	−	0.611463i	−0.331865	−	0.943327i	$-0.607678\pi$
$673$	411.514	−	411.514i	0.611463	−	0.611463i	0.943327	+	0.331865i	$0.107678\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−176.363	−	176.363i	−0.260507	−	0.260507i	0.564753	−	0.825260i	$-0.308971\pi$
$677$	−176.363	−	176.363i	−0.260507	−	0.260507i	−0.825260	+	0.564753i	$0.808971\pi$
$678$		0			0
$679$		−	912.000i		−	1.34315i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−547.461	+	547.461i	−0.801553	+	0.801553i	−0.983338	−	0.181785i	$-0.941813\pi$
$683$	−547.461	+	547.461i	−0.801553	+	0.801553i	0.181785	+	0.983338i	$0.441813\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$			324.000i			0.470247i
$690$		0			0
$691$	103.000			0.149059			0.0745297	−	0.997219i	$-0.476254\pi$
$691$	103.000			0.149059			0.0745297	+	0.997219i	$0.476254\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	143.295	+	143.295i	0.205588	+	0.205588i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	870.000			1.24108			0.620542	−	0.784173i	$-0.286913\pi$
$701$	870.000			1.24108			0.620542	+	0.784173i	$0.286913\pi$
$702$		0			0
$703$	499.696	−	499.696i	0.710805	−	0.710805i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−117.576	−	117.576i	−0.166302	−	0.166302i
$708$		0			0
$709$			92.0000i			0.129760i	0.997893	+	0.0648801i	$0.0206665\pi$
$709$			92.0000i			0.129760i	−0.997893	+	0.0648801i	$0.979334\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−102.879	+	102.879i	−0.144290	+	0.144290i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		−	192.000i		−	0.267038i	−0.991046	−	0.133519i	$-0.957372\pi$
$719$		−	192.000i		−	0.267038i	0.991046	−	0.133519i	$-0.0426276\pi$
$720$		0			0
$721$	−72.0000			−0.0998613
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	614.822	+	614.822i	0.845697	+	0.845697i	0.989593	−	0.143896i	$-0.0459629\pi$
$727$	614.822	+	614.822i	0.845697	+	0.845697i	−0.143896	+	0.989593i	$0.545963\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	180.000			0.246238
$732$		0			0
$733$	−683.408	+	683.408i	−0.932343	+	0.932343i	−0.997852	−	0.0655087i	$-0.979133\pi$
$733$	−683.408	+	683.408i	−0.932343	+	0.932343i	0.0655087	+	0.997852i	$0.479133\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−532.764	−	532.764i	−0.722882	−	0.722882i
$738$		0			0
$739$			470.000i			0.635995i	0.948092	+	0.317997i	$0.103010\pi$
$739$			470.000i			0.635995i	−0.948092	+	0.317997i	$0.896990\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	301.287	−	301.287i	0.405501	−	0.405501i	−0.474665	−	0.880166i	$-0.657431\pi$
$743$	301.287	−	301.287i	0.405501	−	0.405501i	0.880166	+	0.474665i	$0.157431\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$			1044.00i			1.39386i
$750$		0			0
$751$	−314.000			−0.418109			−0.209055	−	0.977904i	$-0.567039\pi$
$751$	−314.000			−0.418109			−0.209055	+	0.977904i	$0.567039\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	200.858	+	200.858i	0.265334	+	0.265334i	0.827217	−	0.561883i	$-0.189923\pi$
$757$	200.858	+	200.858i	0.265334	+	0.265334i	−0.561883	+	0.827217i	$0.689923\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−1227.00			−1.61235			−0.806176	−	0.591676i	$-0.798467\pi$
$761$	−1227.00			−1.61235			−0.806176	+	0.591676i	$0.798467\pi$
$762$		0			0
$763$	313.535	−	313.535i	0.410924	−	0.410924i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−14.6969	−	14.6969i	−0.0191616	−	0.0191616i
$768$		0			0
$769$		−	1121.00i		−	1.45774i	−0.684654	−	0.728869i	$-0.740046\pi$
$769$		−	1121.00i		−	1.45774i	0.684654	−	0.728869i	$-0.259954\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	183.712	−	183.712i	0.237661	−	0.237661i	−0.578220	−	0.815881i	$-0.696253\pi$
$773$	183.712	−	183.712i	0.237661	−	0.237661i	0.815881	+	0.578220i	$0.196253\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$			663.000i			0.851091i
$780$		0			0
$781$	1530.00			1.95903
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	230.252	+	230.252i	0.292569	+	0.292569i	0.838094	−	0.545525i	$-0.183670\pi$
$787$	230.252	+	230.252i	0.292569	+	0.292569i	−0.545525	+	0.838094i	$0.683670\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−828.000			−1.04678
$792$		0			0
$793$	−225.353	+	225.353i	−0.284178	+	0.284178i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−154.318	−	154.318i	−0.193623	−	0.193623i	0.603636	−	0.797260i	$-0.293718\pi$
$797$	−154.318	−	154.318i	−0.193623	−	0.193623i	−0.797260	+	0.603636i	$0.793718\pi$
$798$		0			0
$799$		−	378.000i		−	0.473091i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	55.1135	−	55.1135i	0.0686345	−	0.0686345i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	618.000i		−	0.763906i	−0.924182	−	0.381953i	$-0.875252\pi$
$809$		−	618.000i		−	0.763906i	0.924182	−	0.381953i	$-0.124748\pi$
$810$		0			0
$811$	−1286.00			−1.58570			−0.792848	−	0.609419i	$-0.791403\pi$
$811$	−1286.00			−1.58570			−0.792848	+	0.609419i	$0.791403\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	416.413	+	416.413i	0.509686	+	0.509686i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−186.000			−0.226553			−0.113276	−	0.993564i	$-0.536135\pi$
$821$	−186.000			−0.226553			−0.113276	+	0.993564i	$0.536135\pi$
$822$		0			0
$823$	−769.140	+	769.140i	−0.934556	+	0.934556i	−0.997986	−	0.0634301i	$-0.979796\pi$
$823$	−769.140	+	769.140i	−0.934556	+	0.934556i	0.0634301	+	0.997986i	$0.479796\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	679.733	+	679.733i	0.821927	+	0.821927i	0.986384	−	0.164457i	$-0.0525873\pi$
$827$	679.733	+	679.733i	0.821927	+	0.821927i	−0.164457	+	0.986384i	$0.552587\pi$
$828$		0			0
$829$			434.000i			0.523522i	0.965133	+	0.261761i	$0.0843033\pi$
$829$			434.000i			0.523522i	−0.965133	+	0.261761i	$0.915697\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	3.67423	−	3.67423i	0.00441085	−	0.00441085i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$			1176.00i			1.40167i	0.713324	+	0.700834i	$0.247189\pi$
$839$			1176.00i			1.40167i	−0.713324	+	0.700834i	$0.752811\pi$
$840$		0			0
$841$	−923.000			−1.09750
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−509.494	−	509.494i	−0.601528	−	0.601528i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−432.000			−0.507638
$852$		0			0
$853$	894.064	−	894.064i	1.04814	−	1.04814i	0.0493593	−	0.998781i	$-0.484282\pi$
$853$	894.064	−	894.064i	1.04814	−	1.04814i	0.998781	−	0.0493593i	$-0.0157179\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	503.370	+	503.370i	0.587363	+	0.587363i	0.936916	−	0.349553i	$-0.113667\pi$
$857$	503.370	+	503.370i	0.587363	+	0.587363i	−0.349553	+	0.936916i	$0.613667\pi$
$858$		0			0
$859$		−	683.000i		−	0.795111i	−0.917578	−	0.397555i	$-0.869859\pi$
$859$		−	683.000i		−	0.795111i	0.917578	−	0.397555i	$-0.130141\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−580.529	+	580.529i	−0.672687	+	0.672687i	−0.958335	−	0.285648i	$-0.907791\pi$
$863$	−580.529	+	580.529i	−0.672687	+	0.672687i	0.285648	+	0.958335i	$0.407791\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	1560.00i		−	1.79517i
$870$		0			0
$871$	174.000			0.199770
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−651.564	−	651.564i	−0.742947	−	0.742947i	0.230197	−	0.973144i	$-0.426063\pi$
$877$	−651.564	−	651.564i	−0.742947	−	0.742947i	−0.973144	+	0.230197i	$0.926063\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	78.0000			0.0885358			0.0442679	−	0.999020i	$-0.485904\pi$
$881$	78.0000			0.0885358			0.0442679	+	0.999020i	$0.485904\pi$
$882$		0			0
$883$	564.607	−	564.607i	0.639419	−	0.639419i	−0.310993	−	0.950412i	$-0.600662\pi$
$883$	564.607	−	564.607i	0.639419	−	0.639419i	0.950412	+	0.310993i	$0.100662\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−470.302	−	470.302i	−0.530216	−	0.530216i	0.390420	−	0.920637i	$-0.372330\pi$
$887$	−470.302	−	470.302i	−0.530216	−	0.530216i	−0.920637	+	0.390420i	$0.872330\pi$
$888$		0			0
$889$			648.000i			0.728909i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	874.468	−	874.468i	0.979247	−	0.979247i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$			588.000i			0.654060i
$900$		0			0
$901$	486.000			0.539401
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	896.513	+	896.513i	0.988438	+	0.988438i	0.999934	−	0.0114959i	$-0.00365935\pi$
$907$	896.513	+	896.513i	0.988438	+	0.988438i	−0.0114959	+	0.999934i	$0.503659\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	588.000			0.645445			0.322722	−	0.946494i	$-0.395402\pi$
$911$	588.000			0.645445			0.322722	+	0.946494i	$0.395402\pi$
$912$		0			0
$913$	−716.476	+	716.476i	−0.784749	+	0.784749i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−146.969	−	146.969i	−0.160272	−	0.160272i
$918$		0			0
$919$			98.0000i			0.106638i	0.998578	+	0.0533188i	$0.0169800\pi$
$919$			98.0000i			0.106638i	−0.998578	+	0.0533188i	$0.983020\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−249.848	+	249.848i	−0.270691	+	0.270691i
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	1626.00i		−	1.75027i	−0.483880	−	0.875135i	$-0.660773\pi$
$929$		−	1626.00i		−	1.75027i	0.483880	−	0.875135i	$-0.339227\pi$
$930$		0			0
$931$	17.0000			0.0182599
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	366.199	+	366.199i	0.390820	+	0.390820i	0.874980	−	0.484159i	$-0.160875\pi$
$937$	366.199	+	366.199i	0.390820	+	0.390820i	−0.484159	+	0.874980i	$0.660875\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−1242.00			−1.31987			−0.659936	−	0.751322i	$-0.729417\pi$
$941$	−1242.00			−1.31987			−0.659936	+	0.751322i	$0.729417\pi$
$942$		0			0
$943$	286.590	−	286.590i	0.303913	−	0.303913i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	778.938	+	778.938i	0.822532	+	0.822532i	0.986471	−	0.163939i	$-0.0524199\pi$
$947$	778.938	+	778.938i	0.822532	+	0.822532i	−0.163939	+	0.986471i	$0.552420\pi$
$948$		0			0
$949$			18.0000i			0.0189673i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−1017.76	+	1017.76i	−1.06796	+	1.06796i	−0.0704410	+	0.997516i	$0.522441\pi$
$953$	−1017.76	+	1017.76i	−1.06796	+	1.06796i	−0.997516	+	0.0704410i	$0.977559\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$			468.000i			0.488008i
$960$		0			0
$961$	−765.000			−0.796046
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−1219.85	−	1219.85i	−1.26147	−	1.26147i	−0.950378	−	0.311096i	$-0.899304\pi$
$967$	−1219.85	−	1219.85i	−1.26147	−	1.26147i	−0.311096	−	0.950378i	$-0.600696\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−1191.00			−1.22657			−0.613285	−	0.789861i	$-0.710152\pi$
$971$	−1191.00			−1.22657			−0.613285	+	0.789861i	$0.710152\pi$
$972$		0			0
$973$	−798.534	+	798.534i	−0.820692	+	0.820692i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	1164.73	+	1164.73i	1.19215	+	1.19215i	0.976462	+	0.215690i	$0.0692001\pi$
$977$	1164.73	+	1164.73i	1.19215	+	1.19215i	0.215690	+	0.976462i	$0.430800\pi$
$978$		0			0
$979$			1305.00i			1.33299i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	529.090	−	529.090i	0.538240	−	0.538240i	−0.384772	−	0.923012i	$-0.625720\pi$
$983$	529.090	−	529.090i	0.538240	−	0.538240i	0.923012	+	0.384772i	$0.125720\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	360.000i		−	0.364004i
$990$		0			0
$991$	−64.0000			−0.0645812			−0.0322906	−	0.999479i	$-0.510280\pi$
$991$	−64.0000			−0.0645812			−0.0322906	+	0.999479i	$0.510280\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	352.727	+	352.727i	0.353788	+	0.353788i	0.861517	−	0.507729i	$-0.169515\pi$
$997$	352.727	+	352.727i	0.353788	+	0.353788i	−0.507729	+	0.861517i	$0.669515\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	900.3.l.g.757.1		4
3.2	odd	2		100.3.f.b.57.1	✓	4
5.2	odd	4	inner	900.3.l.g.793.2		4
5.3	odd	4	inner	900.3.l.g.793.1		4
5.4	even	2	inner	900.3.l.g.757.2		4
12.11	even	2		400.3.p.k.257.2		4
15.2	even	4		100.3.f.b.93.2	yes	4
15.8	even	4		100.3.f.b.93.1	yes	4
15.14	odd	2		100.3.f.b.57.2	yes	4
60.23	odd	4		400.3.p.k.193.2		4
60.47	odd	4		400.3.p.k.193.1		4
60.59	even	2		400.3.p.k.257.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
100.3.f.b.57.1	✓	4	3.2	odd	2
100.3.f.b.57.2	yes	4	15.14	odd	2
100.3.f.b.93.1	yes	4	15.8	even	4
100.3.f.b.93.2	yes	4	15.2	even	4
400.3.p.k.193.1		4	60.47	odd	4
400.3.p.k.193.2		4	60.23	odd	4
400.3.p.k.257.1		4	60.59	even	2
400.3.p.k.257.2		4	12.11	even	2
900.3.l.g.757.1		4	1.1	even	1	trivial
900.3.l.g.757.2		4	5.4	even	2	inner
900.3.l.g.793.1		4	5.3	odd	4	inner
900.3.l.g.793.2		4	5.2	odd	4	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:	\( N \)	\(=\)	\( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	900.l (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-4.89898 - 4.89898i) q^{7} +O(q^{10})\)	\(q+(-4.89898 - 4.89898i) q^{7} +15.0000 q^{11} +(-2.44949 + 2.44949i) q^{13} +(3.67423 + 3.67423i) q^{17} +17.0000i q^{19} +(7.34847 - 7.34847i) q^{23} -42.0000i q^{29} -14.0000 q^{31} +(-29.3939 - 29.3939i) q^{37} +39.0000 q^{41} +(24.4949 - 24.4949i) q^{43} +(-51.4393 - 51.4393i) q^{47} -1.00000i q^{49} +(66.1362 - 66.1362i) q^{53} +6.00000i q^{59} +92.0000 q^{61} +(-35.5176 - 35.5176i) q^{67} +102.000 q^{71} +(3.67423 - 3.67423i) q^{73} +(-73.4847 - 73.4847i) q^{77} -104.000i q^{79} +(-47.7650 + 47.7650i) q^{83} +87.0000i q^{89} +24.0000 q^{91} +(93.0806 + 93.0806i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q + 60 q^{11} - 56 q^{31} + 156 q^{41} + 368 q^{61} + 408 q^{71} + 96 q^{91}+O(q^{100}) \) 4 * q + 60 * q^11 - 56 * q^31 + 156 * q^41 + 368 * q^61 + 408 * q^71 + 96 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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