LMFDB - Embedded newform 4050.2.c.r.649.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4050,2,Mod(649,4050)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4050, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4050.649");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4050 = 2 \cdot 3^{4} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4050.c (of order $2$, degree $1$, 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:	$32.3394128186$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 18)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			649.2
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	4050.649
Dual form			4050.2.c.r.649.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.00000 q^{4} -2.00000i q^{7} -1.00000i q^{8} +O(q^{10})$	$q+1.00000i q^{2} -1.00000 q^{4} -2.00000i q^{7} -1.00000i q^{8} +3.00000 q^{11} +2.00000i q^{13} +2.00000 q^{14} +1.00000 q^{16} -3.00000i q^{17} +1.00000 q^{19} +3.00000i q^{22} +6.00000i q^{23} -2.00000 q^{26} +2.00000i q^{28} +6.00000 q^{29} -4.00000 q^{31} +1.00000i q^{32} +3.00000 q^{34} +4.00000i q^{37} +1.00000i q^{38} -9.00000 q^{41} -1.00000i q^{43} -3.00000 q^{44} -6.00000 q^{46} -6.00000i q^{47} +3.00000 q^{49} -2.00000i q^{52} -12.0000i q^{53} -2.00000 q^{56} +6.00000i q^{58} +3.00000 q^{59} +8.00000 q^{61} -4.00000i q^{62} -1.00000 q^{64} -5.00000i q^{67} +3.00000i q^{68} +12.0000 q^{71} +11.0000i q^{73} -4.00000 q^{74} -1.00000 q^{76} -6.00000i q^{77} +4.00000 q^{79} -9.00000i q^{82} -12.0000i q^{83} +1.00000 q^{86} -3.00000i q^{88} +6.00000 q^{89} +4.00000 q^{91} -6.00000i q^{92} +6.00000 q^{94} -5.00000i q^{97} +3.00000i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4}+O(q^{10}) $ 2 * q - 2 * q^4	$ 2 q - 2 q^{4} + 6 q^{11} + 4 q^{14} + 2 q^{16} + 2 q^{19} - 4 q^{26} + 12 q^{29} - 8 q^{31} + 6 q^{34} - 18 q^{41} - 6 q^{44} - 12 q^{46} + 6 q^{49} - 4 q^{56} + 6 q^{59} + 16 q^{61} - 2 q^{64} + 24 q^{71} - 8 q^{74} - 2 q^{76} + 8 q^{79} + 2 q^{86} + 12 q^{89} + 8 q^{91} + 12 q^{94}+O(q^{100}) $ 2 * q - 2 * q^4 + 6 * q^11 + 4 * q^14 + 2 * q^16 + 2 * q^19 - 4 * q^26 + 12 * q^29 - 8 * q^31 + 6 * q^34 - 18 * q^41 - 6 * q^44 - 12 * q^46 + 6 * q^49 - 4 * q^56 + 6 * q^59 + 16 * q^61 - 2 * q^64 + 24 * q^71 - 8 * q^74 - 2 * q^76 + 8 * q^79 + 2 * q^86 + 12 * q^89 + 8 * q^91 + 12 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4050\mathbb{Z}\right)^\times$.

$n$	$2351$	$3727$
$\chi(n)$	$1$	$-1$

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 $

$2$	1.00000i	0.707107i
$2$	1.00000i	0.707107i
$3$	0	0
$3$	0	0
$4$	−1.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 2.00000i	− 0.755929i	−0.925820	−	0.377964i	$-0.876624\pi$
$7$	− 2.00000i	− 0.755929i	0.925820	−	0.377964i	$-0.123376\pi$
$8$	− 1.00000i	− 0.353553i
$9$	0	0
$10$	0	0
$11$	3.00000	0.904534	0.452267	−	0.891883i	$-0.350615\pi$
$11$	3.00000	0.904534	0.452267	+	0.891883i	$0.350615\pi$
$12$	0	0
$13$	2.00000i	0.554700i	0.960769	+	0.277350i	$0.0894562\pi$
$13$	2.00000i	0.554700i	−0.960769	+	0.277350i	$0.910544\pi$
$14$	2.00000	0.534522
$15$	0	0
$16$	1.00000	0.250000
$17$	− 3.00000i	− 0.727607i	−0.931476	−	0.363803i	$-0.881478\pi$
$17$	− 3.00000i	− 0.727607i	0.931476	−	0.363803i	$-0.118522\pi$
$18$	0	0
$19$	1.00000	0.229416	0.114708	−	0.993399i	$-0.463407\pi$
$19$	1.00000	0.229416	0.114708	+	0.993399i	$0.463407\pi$
$20$	0	0
$21$	0	0
$22$	3.00000i	0.639602i
$23$	6.00000i	1.25109i	0.780189	+	0.625543i	$0.215123\pi$
$23$	6.00000i	1.25109i	−0.780189	+	0.625543i	$0.784877\pi$
$24$	0	0
$25$	0	0
$26$	−2.00000	−0.392232
$27$	0	0
$28$	2.00000i	0.377964i
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	0	0
$31$	−4.00000	−0.718421	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	1.00000i	0.176777i
$33$	0	0
$34$	3.00000	0.514496
$35$	0	0
$36$	0	0
$37$	4.00000i	0.657596i	0.944400	+	0.328798i	$0.106644\pi$
$37$	4.00000i	0.657596i	−0.944400	+	0.328798i	$0.893356\pi$
$38$	1.00000i	0.162221i
$39$	0	0
$40$	0	0
$41$	−9.00000	−1.40556	−0.702782	−	0.711405i	$-0.748059\pi$
$41$	−9.00000	−1.40556	−0.702782	+	0.711405i	$0.748059\pi$
$42$	0	0
$43$	− 1.00000i	− 0.152499i	−0.997089	−	0.0762493i	$-0.975706\pi$
$43$	− 1.00000i	− 0.152499i	0.997089	−	0.0762493i	$-0.0242945\pi$
$44$	−3.00000	−0.452267
$45$	0	0
$46$	−6.00000	−0.884652
$47$	− 6.00000i	− 0.875190i	−0.899172	−	0.437595i	$-0.855830\pi$
$47$	− 6.00000i	− 0.875190i	0.899172	−	0.437595i	$-0.144170\pi$
$48$	0	0
$49$	3.00000	0.428571
$50$	0	0
$51$	0	0
$52$	− 2.00000i	− 0.277350i
$53$	− 12.0000i	− 1.64833i	−0.566352	−	0.824163i	$-0.691646\pi$
$53$	− 12.0000i	− 1.64833i	0.566352	−	0.824163i	$-0.308354\pi$
$54$	0	0
$55$	0	0
$56$	−2.00000	−0.267261
$57$	0	0
$58$	6.00000i	0.787839i
$59$	3.00000	0.390567	0.195283	−	0.980747i	$-0.437437\pi$
$59$	3.00000	0.390567	0.195283	+	0.980747i	$0.437437\pi$
$60$	0	0
$61$	8.00000	1.02430	0.512148	−	0.858898i	$-0.328850\pi$
$61$	8.00000	1.02430	0.512148	+	0.858898i	$0.328850\pi$
$62$	− 4.00000i	− 0.508001i
$63$	0	0
$64$	−1.00000	−0.125000
$65$	0	0
$66$	0	0
$67$	− 5.00000i	− 0.610847i	−0.952217	−	0.305424i	$-0.901202\pi$
$67$	− 5.00000i	− 0.610847i	0.952217	−	0.305424i	$-0.0987981\pi$
$68$	3.00000i	0.363803i
$69$	0	0
$70$	0	0
$71$	12.0000	1.42414	0.712069	−	0.702109i	$-0.247758\pi$
$71$	12.0000	1.42414	0.712069	+	0.702109i	$0.247758\pi$
$72$	0	0
$73$	11.0000i	1.28745i	0.765256	+	0.643726i	$0.222612\pi$
$73$	11.0000i	1.28745i	−0.765256	+	0.643726i	$0.777388\pi$
$74$	−4.00000	−0.464991
$75$	0	0
$76$	−1.00000	−0.114708
$77$	− 6.00000i	− 0.683763i
$78$	0	0
$79$	4.00000	0.450035	0.225018	−	0.974355i	$-0.427756\pi$
$79$	4.00000	0.450035	0.225018	+	0.974355i	$0.427756\pi$
$80$	0	0
$81$	0	0
$82$	− 9.00000i	− 0.993884i
$83$	− 12.0000i	− 1.31717i	−0.752506	−	0.658586i	$-0.771155\pi$
$83$	− 12.0000i	− 1.31717i	0.752506	−	0.658586i	$-0.228845\pi$
$84$	0	0
$85$	0	0
$86$	1.00000	0.107833
$87$	0	0
$88$	− 3.00000i	− 0.319801i
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	0	0
$91$	4.00000	0.419314
$92$	− 6.00000i	− 0.625543i
$93$	0	0
$94$	6.00000	0.618853
$95$	0	0
$96$	0	0
$97$	− 5.00000i	− 0.507673i	−0.967247	−	0.253837i	$-0.918307\pi$
$97$	− 5.00000i	− 0.507673i	0.967247	−	0.253837i	$-0.0816925\pi$
$98$	3.00000i	0.303046i
$99$	0	0
$100$	0	0
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	0	0
$103$	14.0000i	1.37946i	0.724066	+	0.689730i	$0.242271\pi$
$103$	14.0000i	1.37946i	−0.724066	+	0.689730i	$0.757729\pi$
$104$	2.00000	0.196116
$105$	0	0
$106$	12.0000	1.16554
$107$	3.00000i	0.290021i	0.989430	+	0.145010i	$0.0463216\pi$
$107$	3.00000i	0.290021i	−0.989430	+	0.145010i	$0.953678\pi$
$108$	0	0
$109$	16.0000	1.53252	0.766261	−	0.642529i	$-0.222115\pi$
$109$	16.0000	1.53252	0.766261	+	0.642529i	$0.222115\pi$
$110$	0	0
$111$	0	0
$112$	− 2.00000i	− 0.188982i
$113$	− 6.00000i	− 0.564433i	−0.959351	−	0.282216i	$-0.908930\pi$
$113$	− 6.00000i	− 0.564433i	0.959351	−	0.282216i	$-0.0910696\pi$
$114$	0	0
$115$	0	0
$116$	−6.00000	−0.557086
$117$	0	0
$118$	3.00000i	0.276172i
$119$	−6.00000	−0.550019
$120$	0	0
$121$	−2.00000	−0.181818
$122$	8.00000i	0.724286i
$123$	0	0
$124$	4.00000	0.359211
$125$	0	0
$126$	0	0
$127$	− 2.00000i	− 0.177471i	−0.996055	−	0.0887357i	$-0.971717\pi$
$127$	− 2.00000i	− 0.177471i	0.996055	−	0.0887357i	$-0.0282826\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	0	0
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	− 2.00000i	− 0.173422i
$134$	5.00000	0.431934
$135$	0	0
$136$	−3.00000	−0.257248
$137$	− 3.00000i	− 0.256307i	−0.991754	−	0.128154i	$-0.959095\pi$
$137$	− 3.00000i	− 0.256307i	0.991754	−	0.128154i	$-0.0409051\pi$
$138$	0	0
$139$	19.0000	1.61156	0.805779	−	0.592216i	$-0.201747\pi$
$139$	19.0000	1.61156	0.805779	+	0.592216i	$0.201747\pi$
$140$	0	0
$141$	0	0
$142$	12.0000i	1.00702i
$143$	6.00000i	0.501745i
$144$	0	0
$145$	0	0
$146$	−11.0000	−0.910366
$147$	0	0
$148$	− 4.00000i	− 0.328798i
$149$	−6.00000	−0.491539	−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000	−0.491539	−0.245770	+	0.969328i	$0.579041\pi$
$150$	0	0
$151$	−10.0000	−0.813788	−0.406894	−	0.913475i	$-0.633388\pi$
$151$	−10.0000	−0.813788	−0.406894	+	0.913475i	$0.633388\pi$
$152$	− 1.00000i	− 0.0811107i
$153$	0	0
$154$	6.00000	0.483494
$155$	0	0
$156$	0	0
$157$	4.00000i	0.319235i	0.987179	+	0.159617i	$0.0510260\pi$
$157$	4.00000i	0.319235i	−0.987179	+	0.159617i	$0.948974\pi$
$158$	4.00000i	0.318223i
$159$	0	0
$160$	0	0
$161$	12.0000	0.945732
$162$	0	0
$163$	− 4.00000i	− 0.313304i	−0.987654	−	0.156652i	$-0.949930\pi$
$163$	− 4.00000i	− 0.313304i	0.987654	−	0.156652i	$-0.0500701\pi$
$164$	9.00000	0.702782
$165$	0	0
$166$	12.0000	0.931381
$167$	− 12.0000i	− 0.928588i	−0.885681	−	0.464294i	$-0.846308\pi$
$167$	− 12.0000i	− 0.928588i	0.885681	−	0.464294i	$-0.153692\pi$
$168$	0	0
$169$	9.00000	0.692308
$170$	0	0
$171$	0	0
$172$	1.00000i	0.0762493i
$173$	6.00000i	0.456172i	0.973641	+	0.228086i	$0.0732467\pi$
$173$	6.00000i	0.456172i	−0.973641	+	0.228086i	$0.926753\pi$
$174$	0	0
$175$	0	0
$176$	3.00000	0.226134
$177$	0	0
$178$	6.00000i	0.449719i
$179$	12.0000	0.896922	0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000	0.896922	0.448461	+	0.893802i	$0.351972\pi$
$180$	0	0
$181$	14.0000	1.04061	0.520306	−	0.853980i	$-0.325818\pi$
$181$	14.0000	1.04061	0.520306	+	0.853980i	$0.325818\pi$
$182$	4.00000i	0.296500i
$183$	0	0
$184$	6.00000	0.442326
$185$	0	0
$186$	0	0
$187$	− 9.00000i	− 0.658145i
$188$	6.00000i	0.437595i
$189$	0	0
$190$	0	0
$191$	18.0000	1.30243	0.651217	−	0.758891i	$-0.274259\pi$
$191$	18.0000	1.30243	0.651217	+	0.758891i	$0.274259\pi$
$192$	0	0
$193$	5.00000i	0.359908i	0.983675	+	0.179954i	$0.0575949\pi$
$193$	5.00000i	0.359908i	−0.983675	+	0.179954i	$0.942405\pi$
$194$	5.00000	0.358979
$195$	0	0
$196$	−3.00000	−0.214286
$197$	− 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
$199$	10.0000	0.708881	0.354441	−	0.935079i	$-0.384671\pi$
$199$	10.0000	0.708881	0.354441	+	0.935079i	$0.384671\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 12.0000i	− 0.842235i
$204$	0	0
$205$	0	0
$206$	−14.0000	−0.975426
$207$	0	0
$208$	2.00000i	0.138675i
$209$	3.00000	0.207514
$210$	0	0
$211$	20.0000	1.37686	0.688428	−	0.725304i	$-0.258301\pi$
$211$	20.0000	1.37686	0.688428	+	0.725304i	$0.258301\pi$
$212$	12.0000i	0.824163i
$213$	0	0
$214$	−3.00000	−0.205076
$215$	0	0
$216$	0	0
$217$	8.00000i	0.543075i
$218$	16.0000i	1.08366i
$219$	0	0
$220$	0	0
$221$	6.00000	0.403604
$222$	0	0
$223$	26.0000i	1.74109i	0.492090	+	0.870544i	$0.336233\pi$
$223$	26.0000i	1.74109i	−0.492090	+	0.870544i	$0.663767\pi$
$224$	2.00000	0.133631
$225$	0	0
$226$	6.00000	0.399114
$227$	21.0000i	1.39382i	0.717159	+	0.696909i	$0.245442\pi$
$227$	21.0000i	1.39382i	−0.717159	+	0.696909i	$0.754558\pi$
$228$	0	0
$229$	−14.0000	−0.925146	−0.462573	−	0.886581i	$-0.653074\pi$
$229$	−14.0000	−0.925146	−0.462573	+	0.886581i	$0.653074\pi$
$230$	0	0
$231$	0	0
$232$	− 6.00000i	− 0.393919i
$233$	− 3.00000i	− 0.196537i	−0.995160	−	0.0982683i	$-0.968670\pi$
$233$	− 3.00000i	− 0.196537i	0.995160	−	0.0982683i	$-0.0313303\pi$
$234$	0	0
$235$	0	0
$236$	−3.00000	−0.195283
$237$	0	0
$238$	− 6.00000i	− 0.388922i
$239$	6.00000	0.388108	0.194054	−	0.980991i	$-0.437836\pi$
$239$	6.00000	0.388108	0.194054	+	0.980991i	$0.437836\pi$
$240$	0	0
$241$	−7.00000	−0.450910	−0.225455	−	0.974254i	$-0.572387\pi$
$241$	−7.00000	−0.450910	−0.225455	+	0.974254i	$0.572387\pi$
$242$	− 2.00000i	− 0.128565i
$243$	0	0
$244$	−8.00000	−0.512148
$245$	0	0
$246$	0	0
$247$	2.00000i	0.127257i
$248$	4.00000i	0.254000i
$249$	0	0
$250$	0	0
$251$	−21.0000	−1.32551	−0.662754	−	0.748837i	$-0.730613\pi$
$251$	−21.0000	−1.32551	−0.662754	+	0.748837i	$0.730613\pi$
$252$	0	0
$253$	18.0000i	1.13165i
$254$	2.00000	0.125491
$255$	0	0
$256$	1.00000	0.0625000
$257$	− 21.0000i	− 1.30994i	−0.755653	−	0.654972i	$-0.772680\pi$
$257$	− 21.0000i	− 1.30994i	0.755653	−	0.654972i	$-0.227320\pi$
$258$	0	0
$259$	8.00000	0.497096
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 18.0000i	− 1.10993i	−0.831875	−	0.554964i	$-0.812732\pi$
$263$	− 18.0000i	− 1.10993i	0.831875	−	0.554964i	$-0.187268\pi$
$264$	0	0
$265$	0	0
$266$	2.00000	0.122628
$267$	0	0
$268$	5.00000i	0.305424i
$269$	−24.0000	−1.46331	−0.731653	−	0.681677i	$-0.761251\pi$
$269$	−24.0000	−1.46331	−0.731653	+	0.681677i	$0.761251\pi$
$270$	0	0
$271$	20.0000	1.21491	0.607457	−	0.794353i	$-0.292190\pi$
$271$	20.0000	1.21491	0.607457	+	0.794353i	$0.292190\pi$
$272$	− 3.00000i	− 0.181902i
$273$	0	0
$274$	3.00000	0.181237
$275$	0	0
$276$	0	0
$277$	10.0000i	0.600842i	0.953807	+	0.300421i	$0.0971271\pi$
$277$	10.0000i	0.600842i	−0.953807	+	0.300421i	$0.902873\pi$
$278$	19.0000i	1.13954i
$279$	0	0
$280$	0	0
$281$	−6.00000	−0.357930	−0.178965	−	0.983855i	$-0.557275\pi$
$281$	−6.00000	−0.357930	−0.178965	+	0.983855i	$0.557275\pi$
$282$	0	0
$283$	− 4.00000i	− 0.237775i	−0.992908	−	0.118888i	$-0.962067\pi$
$283$	− 4.00000i	− 0.237775i	0.992908	−	0.118888i	$-0.0379328\pi$
$284$	−12.0000	−0.712069
$285$	0	0
$286$	−6.00000	−0.354787
$287$	18.0000i	1.06251i
$288$	0	0
$289$	8.00000	0.470588
$290$	0	0
$291$	0	0
$292$	− 11.0000i	− 0.643726i
$293$	− 30.0000i	− 1.75262i	−0.481749	−	0.876309i	$-0.659998\pi$
$293$	− 30.0000i	− 1.75262i	0.481749	−	0.876309i	$-0.340002\pi$
$294$	0	0
$295$	0	0
$296$	4.00000	0.232495
$297$	0	0
$298$	− 6.00000i	− 0.347571i
$299$	−12.0000	−0.693978
$300$	0	0
$301$	−2.00000	−0.115278
$302$	− 10.0000i	− 0.575435i
$303$	0	0
$304$	1.00000	0.0573539
$305$	0	0
$306$	0	0
$307$	7.00000i	0.399511i	0.979846	+	0.199756i	$0.0640148\pi$
$307$	7.00000i	0.399511i	−0.979846	+	0.199756i	$0.935985\pi$
$308$	6.00000i	0.341882i
$309$	0	0
$310$	0	0
$311$	18.0000	1.02069	0.510343	−	0.859971i	$-0.329518\pi$
$311$	18.0000	1.02069	0.510343	+	0.859971i	$0.329518\pi$
$312$	0	0
$313$	29.0000i	1.63918i	0.572953	+	0.819588i	$0.305798\pi$
$313$	29.0000i	1.63918i	−0.572953	+	0.819588i	$0.694202\pi$
$314$	−4.00000	−0.225733
$315$	0	0
$316$	−4.00000	−0.225018
$317$	− 18.0000i	− 1.01098i	−0.862832	−	0.505490i	$-0.831312\pi$
$317$	− 18.0000i	− 1.01098i	0.862832	−	0.505490i	$-0.168688\pi$
$318$	0	0
$319$	18.0000	1.00781
$320$	0	0
$321$	0	0
$322$	12.0000i	0.668734i
$323$	− 3.00000i	− 0.166924i
$324$	0	0
$325$	0	0
$326$	4.00000	0.221540
$327$	0	0
$328$	9.00000i	0.496942i
$329$	−12.0000	−0.661581
$330$	0	0
$331$	−4.00000	−0.219860	−0.109930	−	0.993939i	$-0.535063\pi$
$331$	−4.00000	−0.219860	−0.109930	+	0.993939i	$0.535063\pi$
$332$	12.0000i	0.658586i
$333$	0	0
$334$	12.0000	0.656611
$335$	0	0
$336$	0	0
$337$	1.00000i	0.0544735i	0.999629	+	0.0272367i	$0.00867079\pi$
$337$	1.00000i	0.0544735i	−0.999629	+	0.0272367i	$0.991329\pi$
$338$	9.00000i	0.489535i
$339$	0	0
$340$	0	0
$341$	−12.0000	−0.649836
$342$	0	0
$343$	− 20.0000i	− 1.07990i
$344$	−1.00000	−0.0539164
$345$	0	0
$346$	−6.00000	−0.322562
$347$	33.0000i	1.77153i	0.464131	+	0.885766i	$0.346367\pi$
$347$	33.0000i	1.77153i	−0.464131	+	0.885766i	$0.653633\pi$
$348$	0	0
$349$	16.0000	0.856460	0.428230	−	0.903670i	$-0.359137\pi$
$349$	16.0000	0.856460	0.428230	+	0.903670i	$0.359137\pi$
$350$	0	0
$351$	0	0
$352$	3.00000i	0.159901i
$353$	21.0000i	1.11772i	0.829263	+	0.558859i	$0.188761\pi$
$353$	21.0000i	1.11772i	−0.829263	+	0.558859i	$0.811239\pi$
$354$	0	0
$355$	0	0
$356$	−6.00000	−0.317999
$357$	0	0
$358$	12.0000i	0.634220i
$359$	−18.0000	−0.950004	−0.475002	−	0.879985i	$-0.657553\pi$
$359$	−18.0000	−0.950004	−0.475002	+	0.879985i	$0.657553\pi$
$360$	0	0
$361$	−18.0000	−0.947368
$362$	14.0000i	0.735824i
$363$	0	0
$364$	−4.00000	−0.209657
$365$	0	0
$366$	0	0
$367$	28.0000i	1.46159i	0.682598	+	0.730794i	$0.260850\pi$
$367$	28.0000i	1.46159i	−0.682598	+	0.730794i	$0.739150\pi$
$368$	6.00000i	0.312772i
$369$	0	0
$370$	0	0
$371$	−24.0000	−1.24602
$372$	0	0
$373$	− 34.0000i	− 1.76045i	−0.474554	−	0.880227i	$-0.657390\pi$
$373$	− 34.0000i	− 1.76045i	0.474554	−	0.880227i	$-0.342610\pi$
$374$	9.00000	0.465379
$375$	0	0
$376$	−6.00000	−0.309426
$377$	12.0000i	0.618031i
$378$	0	0
$379$	−23.0000	−1.18143	−0.590715	−	0.806880i	$-0.701154\pi$
$379$	−23.0000	−1.18143	−0.590715	+	0.806880i	$0.701154\pi$
$380$	0	0
$381$	0	0
$382$	18.0000i	0.920960i
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	0	0
$385$	0	0
$386$	−5.00000	−0.254493
$387$	0	0
$388$	5.00000i	0.253837i
$389$	18.0000	0.912636	0.456318	−	0.889817i	$-0.349168\pi$
$389$	18.0000	0.912636	0.456318	+	0.889817i	$0.349168\pi$
$390$	0	0
$391$	18.0000	0.910299
$392$	− 3.00000i	− 0.151523i
$393$	0	0
$394$	12.0000	0.604551
$395$	0	0
$396$	0	0
$397$	− 20.0000i	− 1.00377i	−0.864934	−	0.501886i	$-0.832640\pi$
$397$	− 20.0000i	− 1.00377i	0.864934	−	0.501886i	$-0.167360\pi$
$398$	10.0000i	0.501255i
$399$	0	0
$400$	0	0
$401$	27.0000	1.34832	0.674158	−	0.738587i	$-0.264507\pi$
$401$	27.0000	1.34832	0.674158	+	0.738587i	$0.264507\pi$
$402$	0	0
$403$	− 8.00000i	− 0.398508i
$404$	0	0
$405$	0	0
$406$	12.0000	0.595550
$407$	12.0000i	0.594818i
$408$	0	0
$409$	−17.0000	−0.840596	−0.420298	−	0.907386i	$-0.638074\pi$
$409$	−17.0000	−0.840596	−0.420298	+	0.907386i	$0.638074\pi$
$410$	0	0
$411$	0	0
$412$	− 14.0000i	− 0.689730i
$413$	− 6.00000i	− 0.295241i
$414$	0	0
$415$	0	0
$416$	−2.00000	−0.0980581
$417$	0	0
$418$	3.00000i	0.146735i
$419$	−12.0000	−0.586238	−0.293119	−	0.956076i	$-0.594693\pi$
$419$	−12.0000	−0.586238	−0.293119	+	0.956076i	$0.594693\pi$
$420$	0	0
$421$	20.0000	0.974740	0.487370	−	0.873195i	$-0.337956\pi$
$421$	20.0000	0.974740	0.487370	+	0.873195i	$0.337956\pi$
$422$	20.0000i	0.973585i
$423$	0	0
$424$	−12.0000	−0.582772
$425$	0	0
$426$	0	0
$427$	− 16.0000i	− 0.774294i
$428$	− 3.00000i	− 0.145010i
$429$	0	0
$430$	0	0
$431$	30.0000	1.44505	0.722525	−	0.691345i	$-0.242982\pi$
$431$	30.0000	1.44505	0.722525	+	0.691345i	$0.242982\pi$
$432$	0	0
$433$	− 7.00000i	− 0.336399i	−0.985753	−	0.168199i	$-0.946205\pi$
$433$	− 7.00000i	− 0.336399i	0.985753	−	0.168199i	$-0.0537952\pi$
$434$	−8.00000	−0.384012
$435$	0	0
$436$	−16.0000	−0.766261
$437$	6.00000i	0.287019i
$438$	0	0
$439$	−8.00000	−0.381819	−0.190910	−	0.981608i	$-0.561144\pi$
$439$	−8.00000	−0.381819	−0.190910	+	0.981608i	$0.561144\pi$
$440$	0	0
$441$	0	0
$442$	6.00000i	0.285391i
$443$	− 3.00000i	− 0.142534i	−0.997457	−	0.0712672i	$-0.977296\pi$
$443$	− 3.00000i	− 0.142534i	0.997457	−	0.0712672i	$-0.0227043\pi$
$444$	0	0
$445$	0	0
$446$	−26.0000	−1.23114
$447$	0	0
$448$	2.00000i	0.0944911i
$449$	9.00000	0.424736	0.212368	−	0.977190i	$-0.431882\pi$
$449$	9.00000	0.424736	0.212368	+	0.977190i	$0.431882\pi$
$450$	0	0
$451$	−27.0000	−1.27138
$452$	6.00000i	0.282216i
$453$	0	0
$454$	−21.0000	−0.985579
$455$	0	0
$456$	0	0
$457$	− 17.0000i	− 0.795226i	−0.917553	−	0.397613i	$-0.869839\pi$
$457$	− 17.0000i	− 0.795226i	0.917553	−	0.397613i	$-0.130161\pi$
$458$	− 14.0000i	− 0.654177i
$459$	0	0
$460$	0	0
$461$	30.0000	1.39724	0.698620	−	0.715493i	$-0.253798\pi$
$461$	30.0000	1.39724	0.698620	+	0.715493i	$0.253798\pi$
$462$	0	0
$463$	20.0000i	0.929479i	0.885448	+	0.464739i	$0.153852\pi$
$463$	20.0000i	0.929479i	−0.885448	+	0.464739i	$0.846148\pi$
$464$	6.00000	0.278543
$465$	0	0
$466$	3.00000	0.138972
$467$	− 15.0000i	− 0.694117i	−0.937843	−	0.347059i	$-0.887180\pi$
$467$	− 15.0000i	− 0.694117i	0.937843	−	0.347059i	$-0.112820\pi$
$468$	0	0
$469$	−10.0000	−0.461757
$470$	0	0
$471$	0	0
$472$	− 3.00000i	− 0.138086i
$473$	− 3.00000i	− 0.137940i
$474$	0	0
$475$	0	0
$476$	6.00000	0.275010
$477$	0	0
$478$	6.00000i	0.274434i
$479$	−42.0000	−1.91903	−0.959514	−	0.281659i	$-0.909115\pi$
$479$	−42.0000	−1.91903	−0.959514	+	0.281659i	$0.909115\pi$
$480$	0	0
$481$	−8.00000	−0.364769
$482$	− 7.00000i	− 0.318841i
$483$	0	0
$484$	2.00000	0.0909091
$485$	0	0
$486$	0	0
$487$	− 26.0000i	− 1.17817i	−0.808070	−	0.589086i	$-0.799488\pi$
$487$	− 26.0000i	− 1.17817i	0.808070	−	0.589086i	$-0.200512\pi$
$488$	− 8.00000i	− 0.362143i
$489$	0	0
$490$	0	0
$491$	15.0000	0.676941	0.338470	−	0.940977i	$-0.390091\pi$
$491$	15.0000	0.676941	0.338470	+	0.940977i	$0.390091\pi$
$492$	0	0
$493$	− 18.0000i	− 0.810679i
$494$	−2.00000	−0.0899843
$495$	0	0
$496$	−4.00000	−0.179605
$497$	− 24.0000i	− 1.07655i
$498$	0	0
$499$	13.0000	0.581960	0.290980	−	0.956729i	$-0.406019\pi$
$499$	13.0000	0.581960	0.290980	+	0.956729i	$0.406019\pi$
$500$	0	0
$501$	0	0
$502$	− 21.0000i	− 0.937276i
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	0	0
$505$	0	0
$506$	−18.0000	−0.800198
$507$	0	0
$508$	2.00000i	0.0887357i
$509$	0	0		−	1.00000i	$-0.5\pi$
$509$	0	0			1.00000i	$0.5\pi$
$510$	0	0
$511$	22.0000	0.973223
$512$	1.00000i	0.0441942i
$513$	0	0
$514$	21.0000	0.926270
$515$	0	0
$516$	0	0
$517$	− 18.0000i	− 0.791639i
$518$	8.00000i	0.351500i
$519$	0	0
$520$	0	0
$521$	3.00000	0.131432	0.0657162	−	0.997838i	$-0.479067\pi$
$521$	3.00000	0.131432	0.0657162	+	0.997838i	$0.479067\pi$
$522$	0	0
$523$	20.0000i	0.874539i	0.899331	+	0.437269i	$0.144054\pi$
$523$	20.0000i	0.874539i	−0.899331	+	0.437269i	$0.855946\pi$
$524$	0	0
$525$	0	0
$526$	18.0000	0.784837
$527$	12.0000i	0.522728i
$528$	0	0
$529$	−13.0000	−0.565217
$530$	0	0
$531$	0	0
$532$	2.00000i	0.0867110i
$533$	− 18.0000i	− 0.779667i
$534$	0	0
$535$	0	0
$536$	−5.00000	−0.215967
$537$	0	0
$538$	− 24.0000i	− 1.03471i
$539$	9.00000	0.387657
$540$	0	0
$541$	−4.00000	−0.171973	−0.0859867	−	0.996296i	$-0.527404\pi$
$541$	−4.00000	−0.171973	−0.0859867	+	0.996296i	$0.527404\pi$
$542$	20.0000i	0.859074i
$543$	0	0
$544$	3.00000	0.128624
$545$	0	0
$546$	0	0
$547$	1.00000i	0.0427569i	0.999771	+	0.0213785i	$0.00680549\pi$
$547$	1.00000i	0.0427569i	−0.999771	+	0.0213785i	$0.993195\pi$
$548$	3.00000i	0.128154i
$549$	0	0
$550$	0	0
$551$	6.00000	0.255609
$552$	0	0
$553$	− 8.00000i	− 0.340195i
$554$	−10.0000	−0.424859
$555$	0	0
$556$	−19.0000	−0.805779
$557$	− 30.0000i	− 1.27114i	−0.772043	−	0.635570i	$-0.780765\pi$
$557$	− 30.0000i	− 1.27114i	0.772043	−	0.635570i	$-0.219235\pi$
$558$	0	0
$559$	2.00000	0.0845910
$560$	0	0
$561$	0	0
$562$	− 6.00000i	− 0.253095i
$563$	39.0000i	1.64365i	0.569737	+	0.821827i	$0.307045\pi$
$563$	39.0000i	1.64365i	−0.569737	+	0.821827i	$0.692955\pi$
$564$	0	0
$565$	0	0
$566$	4.00000	0.168133
$567$	0	0
$568$	− 12.0000i	− 0.503509i
$569$	45.0000	1.88650	0.943249	−	0.332086i	$-0.107752\pi$
$569$	45.0000	1.88650	0.943249	+	0.332086i	$0.107752\pi$
$570$	0	0
$571$	−37.0000	−1.54840	−0.774201	−	0.632940i	$-0.781848\pi$
$571$	−37.0000	−1.54840	−0.774201	+	0.632940i	$0.781848\pi$
$572$	− 6.00000i	− 0.250873i
$573$	0	0
$574$	−18.0000	−0.751305
$575$	0	0
$576$	0	0
$577$	− 11.0000i	− 0.457936i	−0.973434	−	0.228968i	$-0.926465\pi$
$577$	− 11.0000i	− 0.457936i	0.973434	−	0.228968i	$-0.0735351\pi$
$578$	8.00000i	0.332756i
$579$	0	0
$580$	0	0
$581$	−24.0000	−0.995688
$582$	0	0
$583$	− 36.0000i	− 1.49097i
$584$	11.0000	0.455183
$585$	0	0
$586$	30.0000	1.23929
$587$	− 9.00000i	− 0.371470i	−0.982600	−	0.185735i	$-0.940533\pi$
$587$	− 9.00000i	− 0.371470i	0.982600	−	0.185735i	$-0.0594666\pi$
$588$	0	0
$589$	−4.00000	−0.164817
$590$	0	0
$591$	0	0
$592$	4.00000i	0.164399i
$593$	− 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
$596$	6.00000	0.245770
$597$	0	0
$598$	− 12.0000i	− 0.490716i
$599$	12.0000	0.490307	0.245153	−	0.969484i	$-0.421162\pi$
$599$	12.0000	0.490307	0.245153	+	0.969484i	$0.421162\pi$
$600$	0	0
$601$	−37.0000	−1.50926	−0.754631	−	0.656150i	$-0.772184\pi$
$601$	−37.0000	−1.50926	−0.754631	+	0.656150i	$0.772184\pi$
$602$	− 2.00000i	− 0.0815139i
$603$	0	0
$604$	10.0000	0.406894
$605$	0	0
$606$	0	0
$607$	28.0000i	1.13648i	0.822861	+	0.568242i	$0.192376\pi$
$607$	28.0000i	1.13648i	−0.822861	+	0.568242i	$0.807624\pi$
$608$	1.00000i	0.0405554i
$609$	0	0
$610$	0	0
$611$	12.0000	0.485468
$612$	0	0
$613$	− 16.0000i	− 0.646234i	−0.946359	−	0.323117i	$-0.895269\pi$
$613$	− 16.0000i	− 0.646234i	0.946359	−	0.323117i	$-0.104731\pi$
$614$	−7.00000	−0.282497
$615$	0	0
$616$	−6.00000	−0.241747
$617$	27.0000i	1.08698i	0.839416	+	0.543490i	$0.182897\pi$
$617$	27.0000i	1.08698i	−0.839416	+	0.543490i	$0.817103\pi$
$618$	0	0
$619$	−35.0000	−1.40677	−0.703384	−	0.710810i	$-0.748329\pi$
$619$	−35.0000	−1.40677	−0.703384	+	0.710810i	$0.748329\pi$
$620$	0	0
$621$	0	0
$622$	18.0000i	0.721734i
$623$	− 12.0000i	− 0.480770i
$624$	0	0
$625$	0	0
$626$	−29.0000	−1.15907
$627$	0	0
$628$	− 4.00000i	− 0.159617i
$629$	12.0000	0.478471
$630$	0	0
$631$	−40.0000	−1.59237	−0.796187	−	0.605050i	$-0.793153\pi$
$631$	−40.0000	−1.59237	−0.796187	+	0.605050i	$0.793153\pi$
$632$	− 4.00000i	− 0.159111i
$633$	0	0
$634$	18.0000	0.714871
$635$	0	0
$636$	0	0
$637$	6.00000i	0.237729i
$638$	18.0000i	0.712627i
$639$	0	0
$640$	0	0
$641$	3.00000	0.118493	0.0592464	−	0.998243i	$-0.481130\pi$
$641$	3.00000	0.118493	0.0592464	+	0.998243i	$0.481130\pi$
$642$	0	0
$643$	23.0000i	0.907031i	0.891248	+	0.453516i	$0.149830\pi$
$643$	23.0000i	0.907031i	−0.891248	+	0.453516i	$0.850170\pi$
$644$	−12.0000	−0.472866
$645$	0	0
$646$	3.00000	0.118033
$647$	18.0000i	0.707653i	0.935311	+	0.353827i	$0.115120\pi$
$647$	18.0000i	0.707653i	−0.935311	+	0.353827i	$0.884880\pi$
$648$	0	0
$649$	9.00000	0.353281
$650$	0	0
$651$	0	0
$652$	4.00000i	0.156652i
$653$	6.00000i	0.234798i	0.993085	+	0.117399i	$0.0374557\pi$
$653$	6.00000i	0.234798i	−0.993085	+	0.117399i	$0.962544\pi$
$654$	0	0
$655$	0	0
$656$	−9.00000	−0.351391
$657$	0	0
$658$	− 12.0000i	− 0.467809i
$659$	−36.0000	−1.40236	−0.701180	−	0.712984i	$-0.747343\pi$
$659$	−36.0000	−1.40236	−0.701180	+	0.712984i	$0.747343\pi$
$660$	0	0
$661$	−4.00000	−0.155582	−0.0777910	−	0.996970i	$-0.524787\pi$
$661$	−4.00000	−0.155582	−0.0777910	+	0.996970i	$0.524787\pi$
$662$	− 4.00000i	− 0.155464i
$663$	0	0
$664$	−12.0000	−0.465690
$665$	0	0
$666$	0	0
$667$	36.0000i	1.39393i
$668$	12.0000i	0.464294i
$669$	0	0
$670$	0	0
$671$	24.0000	0.926510
$672$	0	0
$673$	− 22.0000i	− 0.848038i	−0.905653	−	0.424019i	$-0.860619\pi$
$673$	− 22.0000i	− 0.848038i	0.905653	−	0.424019i	$-0.139381\pi$
$674$	−1.00000	−0.0385186
$675$	0	0
$676$	−9.00000	−0.346154
$677$	36.0000i	1.38359i	0.722093	+	0.691796i	$0.243180\pi$
$677$	36.0000i	1.38359i	−0.722093	+	0.691796i	$0.756820\pi$
$678$	0	0
$679$	−10.0000	−0.383765
$680$	0	0
$681$	0	0
$682$	− 12.0000i	− 0.459504i
$683$	9.00000i	0.344375i	0.985064	+	0.172188i	$0.0550836\pi$
$683$	9.00000i	0.344375i	−0.985064	+	0.172188i	$0.944916\pi$
$684$	0	0
$685$	0	0
$686$	20.0000	0.763604
$687$	0	0
$688$	− 1.00000i	− 0.0381246i
$689$	24.0000	0.914327
$690$	0	0
$691$	8.00000	0.304334	0.152167	−	0.988355i	$-0.451375\pi$
$691$	8.00000	0.304334	0.152167	+	0.988355i	$0.451375\pi$
$692$	− 6.00000i	− 0.228086i
$693$	0	0
$694$	−33.0000	−1.25266
$695$	0	0
$696$	0	0
$697$	27.0000i	1.02270i
$698$	16.0000i	0.605609i
$699$	0	0
$700$	0	0
$701$	−30.0000	−1.13308	−0.566542	−	0.824033i	$-0.691719\pi$
$701$	−30.0000	−1.13308	−0.566542	+	0.824033i	$0.691719\pi$
$702$	0	0
$703$	4.00000i	0.150863i
$704$	−3.00000	−0.113067
$705$	0	0
$706$	−21.0000	−0.790345
$707$	0	0
$708$	0	0
$709$	4.00000	0.150223	0.0751116	−	0.997175i	$-0.476069\pi$
$709$	4.00000	0.150223	0.0751116	+	0.997175i	$0.476069\pi$
$710$	0	0
$711$	0	0
$712$	− 6.00000i	− 0.224860i
$713$	− 24.0000i	− 0.898807i
$714$	0	0
$715$	0	0
$716$	−12.0000	−0.448461
$717$	0	0
$718$	− 18.0000i	− 0.671754i
$719$	−36.0000	−1.34257	−0.671287	−	0.741198i	$-0.734258\pi$
$719$	−36.0000	−1.34257	−0.671287	+	0.741198i	$0.734258\pi$
$720$	0	0
$721$	28.0000	1.04277
$722$	− 18.0000i	− 0.669891i
$723$	0	0
$724$	−14.0000	−0.520306
$725$	0	0
$726$	0	0
$727$	− 26.0000i	− 0.964287i	−0.876092	−	0.482143i	$-0.839858\pi$
$727$	− 26.0000i	− 0.964287i	0.876092	−	0.482143i	$-0.160142\pi$
$728$	− 4.00000i	− 0.148250i
$729$	0	0
$730$	0	0
$731$	−3.00000	−0.110959
$732$	0	0
$733$	14.0000i	0.517102i	0.965998	+	0.258551i	$0.0832450\pi$
$733$	14.0000i	0.517102i	−0.965998	+	0.258551i	$0.916755\pi$
$734$	−28.0000	−1.03350
$735$	0	0
$736$	−6.00000	−0.221163
$737$	− 15.0000i	− 0.552532i
$738$	0	0
$739$	−47.0000	−1.72892	−0.864461	−	0.502699i	$-0.832340\pi$
$739$	−47.0000	−1.72892	−0.864461	+	0.502699i	$0.832340\pi$
$740$	0	0
$741$	0	0
$742$	− 24.0000i	− 0.881068i
$743$	− 6.00000i	− 0.220119i	−0.993925	−	0.110059i	$-0.964896\pi$
$743$	− 6.00000i	− 0.220119i	0.993925	−	0.110059i	$-0.0351041\pi$
$744$	0	0
$745$	0	0
$746$	34.0000	1.24483
$747$	0	0
$748$	9.00000i	0.329073i
$749$	6.00000	0.219235
$750$	0	0
$751$	8.00000	0.291924	0.145962	−	0.989290i	$-0.453372\pi$
$751$	8.00000	0.291924	0.145962	+	0.989290i	$0.453372\pi$
$752$	− 6.00000i	− 0.218797i
$753$	0	0
$754$	−12.0000	−0.437014
$755$	0	0
$756$	0	0
$757$	− 2.00000i	− 0.0726912i	−0.999339	−	0.0363456i	$-0.988428\pi$
$757$	− 2.00000i	− 0.0726912i	0.999339	−	0.0363456i	$-0.0115717\pi$
$758$	− 23.0000i	− 0.835398i
$759$	0	0
$760$	0	0
$761$	−42.0000	−1.52250	−0.761249	−	0.648459i	$-0.775414\pi$
$761$	−42.0000	−1.52250	−0.761249	+	0.648459i	$0.775414\pi$
$762$	0	0
$763$	− 32.0000i	− 1.15848i
$764$	−18.0000	−0.651217
$765$	0	0
$766$	0	0
$767$	6.00000i	0.216647i
$768$	0	0
$769$	−2.00000	−0.0721218	−0.0360609	−	0.999350i	$-0.511481\pi$
$769$	−2.00000	−0.0721218	−0.0360609	+	0.999350i	$0.511481\pi$
$770$	0	0
$771$	0	0
$772$	− 5.00000i	− 0.179954i
$773$	− 18.0000i	− 0.647415i	−0.946157	−	0.323708i	$-0.895071\pi$
$773$	− 18.0000i	− 0.647415i	0.946157	−	0.323708i	$-0.104929\pi$
$774$	0	0
$775$	0	0
$776$	−5.00000	−0.179490
$777$	0	0
$778$	18.0000i	0.645331i
$779$	−9.00000	−0.322458
$780$	0	0
$781$	36.0000	1.28818
$782$	18.0000i	0.643679i
$783$	0	0
$784$	3.00000	0.107143
$785$	0	0
$786$	0	0
$787$	4.00000i	0.142585i	0.997455	+	0.0712923i	$0.0227123\pi$
$787$	4.00000i	0.142585i	−0.997455	+	0.0712923i	$0.977288\pi$
$788$	12.0000i	0.427482i
$789$	0	0
$790$	0	0
$791$	−12.0000	−0.426671
$792$	0	0
$793$	16.0000i	0.568177i
$794$	20.0000	0.709773
$795$	0	0
$796$	−10.0000	−0.354441
$797$	− 12.0000i	− 0.425062i	−0.977154	−	0.212531i	$-0.931829\pi$
$797$	− 12.0000i	− 0.425062i	0.977154	−	0.212531i	$-0.0681706\pi$
$798$	0	0
$799$	−18.0000	−0.636794
$800$	0	0
$801$	0	0
$802$	27.0000i	0.953403i
$803$	33.0000i	1.16454i
$804$	0	0
$805$	0	0
$806$	8.00000	0.281788
$807$	0	0
$808$	0	0
$809$	33.0000	1.16022	0.580109	−	0.814539i	$-0.303010\pi$
$809$	33.0000	1.16022	0.580109	+	0.814539i	$0.303010\pi$
$810$	0	0
$811$	−7.00000	−0.245803	−0.122902	−	0.992419i	$-0.539220\pi$
$811$	−7.00000	−0.245803	−0.122902	+	0.992419i	$0.539220\pi$
$812$	12.0000i	0.421117i
$813$	0	0
$814$	−12.0000	−0.420600
$815$	0	0
$816$	0	0
$817$	− 1.00000i	− 0.0349856i
$818$	− 17.0000i	− 0.594391i
$819$	0	0
$820$	0	0
$821$	18.0000	0.628204	0.314102	−	0.949389i	$-0.398297\pi$
$821$	18.0000	0.628204	0.314102	+	0.949389i	$0.398297\pi$
$822$	0	0
$823$	− 16.0000i	− 0.557725i	−0.960331	−	0.278862i	$-0.910043\pi$
$823$	− 16.0000i	− 0.557725i	0.960331	−	0.278862i	$-0.0899574\pi$
$824$	14.0000	0.487713
$825$	0	0
$826$	6.00000	0.208767
$827$	− 12.0000i	− 0.417281i	−0.977992	−	0.208640i	$-0.933096\pi$
$827$	− 12.0000i	− 0.417281i	0.977992	−	0.208640i	$-0.0669038\pi$
$828$	0	0
$829$	10.0000	0.347314	0.173657	−	0.984806i	$-0.444442\pi$
$829$	10.0000	0.347314	0.173657	+	0.984806i	$0.444442\pi$
$830$	0	0
$831$	0	0
$832$	− 2.00000i	− 0.0693375i
$833$	− 9.00000i	− 0.311832i
$834$	0	0
$835$	0	0
$836$	−3.00000	−0.103757
$837$	0	0
$838$	− 12.0000i	− 0.414533i
$839$	12.0000	0.414286	0.207143	−	0.978311i	$-0.433583\pi$
$839$	12.0000	0.414286	0.207143	+	0.978311i	$0.433583\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	20.0000i	0.689246i
$843$	0	0
$844$	−20.0000	−0.688428
$845$	0	0
$846$	0	0
$847$	4.00000i	0.137442i
$848$	− 12.0000i	− 0.412082i
$849$	0	0
$850$	0	0
$851$	−24.0000	−0.822709
$852$	0	0
$853$	26.0000i	0.890223i	0.895475	+	0.445112i	$0.146836\pi$
$853$	26.0000i	0.890223i	−0.895475	+	0.445112i	$0.853164\pi$
$854$	16.0000	0.547509
$855$	0	0
$856$	3.00000	0.102538
$857$	− 42.0000i	− 1.43469i	−0.696717	−	0.717346i	$-0.745357\pi$
$857$	− 42.0000i	− 1.43469i	0.696717	−	0.717346i	$-0.254643\pi$
$858$	0	0
$859$	−35.0000	−1.19418	−0.597092	−	0.802173i	$-0.703677\pi$
$859$	−35.0000	−1.19418	−0.597092	+	0.802173i	$0.703677\pi$
$860$	0	0
$861$	0	0
$862$	30.0000i	1.02180i
$863$	− 24.0000i	− 0.816970i	−0.912765	−	0.408485i	$-0.866057\pi$
$863$	− 24.0000i	− 0.816970i	0.912765	−	0.408485i	$-0.133943\pi$
$864$	0	0
$865$	0	0
$866$	7.00000	0.237870
$867$	0	0
$868$	− 8.00000i	− 0.271538i
$869$	12.0000	0.407072
$870$	0	0
$871$	10.0000	0.338837
$872$	− 16.0000i	− 0.541828i
$873$	0	0
$874$	−6.00000	−0.202953
$875$	0	0
$876$	0	0
$877$	− 8.00000i	− 0.270141i	−0.990836	−	0.135070i	$-0.956874\pi$
$877$	− 8.00000i	− 0.270141i	0.990836	−	0.135070i	$-0.0431261\pi$
$878$	− 8.00000i	− 0.269987i
$879$	0	0
$880$	0	0
$881$	42.0000	1.41502	0.707508	−	0.706705i	$-0.249819\pi$
$881$	42.0000	1.41502	0.707508	+	0.706705i	$0.249819\pi$
$882$	0	0
$883$	− 19.0000i	− 0.639401i	−0.947519	−	0.319700i	$-0.896418\pi$
$883$	− 19.0000i	− 0.639401i	0.947519	−	0.319700i	$-0.103582\pi$
$884$	−6.00000	−0.201802
$885$	0	0
$886$	3.00000	0.100787
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	0	0
$889$	−4.00000	−0.134156
$890$	0	0
$891$	0	0
$892$	− 26.0000i	− 0.870544i
$893$	− 6.00000i	− 0.200782i
$894$	0	0
$895$	0	0
$896$	−2.00000	−0.0668153
$897$	0	0
$898$	9.00000i	0.300334i
$899$	−24.0000	−0.800445
$900$	0	0
$901$	−36.0000	−1.19933
$902$	− 27.0000i	− 0.899002i
$903$	0	0
$904$	−6.00000	−0.199557
$905$	0	0
$906$	0	0
$907$	31.0000i	1.02934i	0.857389	+	0.514669i	$0.172085\pi$
$907$	31.0000i	1.02934i	−0.857389	+	0.514669i	$0.827915\pi$
$908$	− 21.0000i	− 0.696909i
$909$	0	0
$910$	0	0
$911$	−48.0000	−1.59031	−0.795155	−	0.606406i	$-0.792611\pi$
$911$	−48.0000	−1.59031	−0.795155	+	0.606406i	$0.792611\pi$
$912$	0	0
$913$	− 36.0000i	− 1.19143i
$914$	17.0000	0.562310
$915$	0	0
$916$	14.0000	0.462573
$917$	0	0
$918$	0	0
$919$	−38.0000	−1.25350	−0.626752	−	0.779219i	$-0.715616\pi$
$919$	−38.0000	−1.25350	−0.626752	+	0.779219i	$0.715616\pi$
$920$	0	0
$921$	0	0
$922$	30.0000i	0.987997i
$923$	24.0000i	0.789970i
$924$	0	0
$925$	0	0
$926$	−20.0000	−0.657241
$927$	0	0
$928$	6.00000i	0.196960i
$929$	−6.00000	−0.196854	−0.0984268	−	0.995144i	$-0.531381\pi$
$929$	−6.00000	−0.196854	−0.0984268	+	0.995144i	$0.531381\pi$
$930$	0	0
$931$	3.00000	0.0983210
$932$	3.00000i	0.0982683i
$933$	0	0
$934$	15.0000	0.490815
$935$	0	0
$936$	0	0
$937$	− 14.0000i	− 0.457360i	−0.973502	−	0.228680i	$-0.926559\pi$
$937$	− 14.0000i	− 0.457360i	0.973502	−	0.228680i	$-0.0734410\pi$
$938$	− 10.0000i	− 0.326512i
$939$	0	0
$940$	0	0
$941$	60.0000	1.95594	0.977972	−	0.208736i	$-0.0669349\pi$
$941$	60.0000	1.95594	0.977972	+	0.208736i	$0.0669349\pi$
$942$	0	0
$943$	− 54.0000i	− 1.75848i
$944$	3.00000	0.0976417
$945$	0	0
$946$	3.00000	0.0975384
$947$	3.00000i	0.0974869i	0.998811	+	0.0487435i	$0.0155217\pi$
$947$	3.00000i	0.0974869i	−0.998811	+	0.0487435i	$0.984478\pi$
$948$	0	0
$949$	−22.0000	−0.714150
$950$	0	0
$951$	0	0
$952$	6.00000i	0.194461i
$953$	− 9.00000i	− 0.291539i	−0.989319	−	0.145769i	$-0.953434\pi$
$953$	− 9.00000i	− 0.291539i	0.989319	−	0.145769i	$-0.0465657\pi$
$954$	0	0
$955$	0	0
$956$	−6.00000	−0.194054
$957$	0	0
$958$	− 42.0000i	− 1.35696i
$959$	−6.00000	−0.193750
$960$	0	0
$961$	−15.0000	−0.483871
$962$	− 8.00000i	− 0.257930i
$963$	0	0
$964$	7.00000	0.225455
$965$	0	0
$966$	0	0
$967$	22.0000i	0.707472i	0.935345	+	0.353736i	$0.115089\pi$
$967$	22.0000i	0.707472i	−0.935345	+	0.353736i	$0.884911\pi$
$968$	2.00000i	0.0642824i
$969$	0	0
$970$	0	0
$971$	36.0000	1.15529	0.577647	−	0.816286i	$-0.303971\pi$
$971$	36.0000	1.15529	0.577647	+	0.816286i	$0.303971\pi$
$972$	0	0
$973$	− 38.0000i	− 1.21822i
$974$	26.0000	0.833094
$975$	0	0
$976$	8.00000	0.256074
$977$	− 51.0000i	− 1.63163i	−0.578310	−	0.815817i	$-0.696287\pi$
$977$	− 51.0000i	− 1.63163i	0.578310	−	0.815817i	$-0.303713\pi$
$978$	0	0
$979$	18.0000	0.575282
$980$	0	0
$981$	0	0
$982$	15.0000i	0.478669i
$983$	0	0	1.00000			$0$
$983$	0	0	−1.00000			$\pi$
$984$	0	0
$985$	0	0
$986$	18.0000	0.573237
$987$	0	0
$988$	− 2.00000i	− 0.0636285i
$989$	6.00000	0.190789
$990$	0	0
$991$	−16.0000	−0.508257	−0.254128	−	0.967170i	$-0.581789\pi$
$991$	−16.0000	−0.508257	−0.254128	+	0.967170i	$0.581789\pi$
$992$	− 4.00000i	− 0.127000i
$993$	0	0
$994$	24.0000	0.761234
$995$	0	0
$996$	0	0
$997$	28.0000i	0.886769i	0.896332	+	0.443384i	$0.146222\pi$
$997$	28.0000i	0.886769i	−0.896332	+	0.443384i	$0.853778\pi$
$998$	13.0000i	0.411508i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4050.2.c.r.649.2		2
3.2	odd	2		4050.2.c.c.649.1		2
5.2	odd	4		162.2.a.b.1.1		1
5.3	odd	4		4050.2.a.v.1.1		1
5.4	even	2	inner	4050.2.c.r.649.1		2
9.2	odd	6		450.2.j.e.49.1		4
9.4	even	3		1350.2.j.a.1099.1		4
9.5	odd	6		450.2.j.e.349.2		4
9.7	even	3		1350.2.j.a.199.2		4
15.2	even	4		162.2.a.c.1.1		1
15.8	even	4		4050.2.a.c.1.1		1
15.14	odd	2		4050.2.c.c.649.2		2
20.7	even	4		1296.2.a.f.1.1		1
35.27	even	4		7938.2.a.i.1.1		1
40.27	even	4		5184.2.a.p.1.1		1
40.37	odd	4		5184.2.a.q.1.1		1
45.2	even	12		18.2.c.a.13.1	yes	2
45.4	even	6		1350.2.j.a.1099.2		4
45.7	odd	12		54.2.c.a.37.1		2
45.13	odd	12		1350.2.e.c.451.1		2
45.14	odd	6		450.2.j.e.349.1		4
45.22	odd	12		54.2.c.a.19.1		2
45.23	even	12		450.2.e.i.151.1		2
45.29	odd	6		450.2.j.e.49.2		4
45.32	even	12		18.2.c.a.7.1	✓	2
45.34	even	6		1350.2.j.a.199.1		4
45.38	even	12		450.2.e.i.301.1		2
45.43	odd	12		1350.2.e.c.901.1		2
60.47	odd	4		1296.2.a.g.1.1		1
105.62	odd	4		7938.2.a.x.1.1		1
120.77	even	4		5184.2.a.r.1.1		1
120.107	odd	4		5184.2.a.o.1.1		1
180.7	even	12		432.2.i.b.145.1		2
180.47	odd	12		144.2.i.c.49.1		2
180.67	even	12		432.2.i.b.289.1		2
180.167	odd	12		144.2.i.c.97.1		2
315.2	even	12		882.2.h.c.67.1		2
315.32	even	12		882.2.h.c.79.1		2
315.47	odd	12		882.2.h.b.67.1		2
315.52	even	12		2646.2.e.c.1549.1		2
315.67	odd	12		2646.2.h.h.667.1		2
315.97	even	12		2646.2.f.g.1765.1		2
315.122	odd	12		882.2.h.b.79.1		2
315.137	even	12		882.2.e.i.373.1		2
315.142	odd	12		2646.2.h.h.361.1		2
315.157	even	12		2646.2.h.i.667.1		2
315.167	odd	12		882.2.f.d.295.1		2
315.187	even	12		2646.2.h.i.361.1		2
315.202	even	12		2646.2.f.g.883.1		2
315.212	even	12		882.2.e.i.655.1		2
315.227	odd	12		882.2.e.g.373.1		2
315.247	odd	12		2646.2.e.b.2125.1		2
315.257	odd	12		882.2.e.g.655.1		2
315.272	odd	12		882.2.f.d.589.1		2
315.277	odd	12		2646.2.e.b.1549.1		2
315.292	even	12		2646.2.e.c.2125.1		2
360.67	even	12		1728.2.i.f.1153.1		2
360.77	even	12		576.2.i.g.385.1		2
360.157	odd	12		1728.2.i.e.1153.1		2
360.187	even	12		1728.2.i.f.577.1		2
360.227	odd	12		576.2.i.a.193.1		2
360.277	odd	12		1728.2.i.e.577.1		2
360.317	even	12		576.2.i.g.193.1		2
360.347	odd	12		576.2.i.a.385.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
18.2.c.a.7.1	✓	2	45.32	even	12
18.2.c.a.13.1	yes	2	45.2	even	12
54.2.c.a.19.1		2	45.22	odd	12
54.2.c.a.37.1		2	45.7	odd	12
144.2.i.c.49.1		2	180.47	odd	12
144.2.i.c.97.1		2	180.167	odd	12
162.2.a.b.1.1		1	5.2	odd	4
162.2.a.c.1.1		1	15.2	even	4
432.2.i.b.145.1		2	180.7	even	12
432.2.i.b.289.1		2	180.67	even	12
450.2.e.i.151.1		2	45.23	even	12
450.2.e.i.301.1		2	45.38	even	12
450.2.j.e.49.1		4	9.2	odd	6
450.2.j.e.49.2		4	45.29	odd	6
450.2.j.e.349.1		4	45.14	odd	6
450.2.j.e.349.2		4	9.5	odd	6
576.2.i.a.193.1		2	360.227	odd	12
576.2.i.a.385.1		2	360.347	odd	12
576.2.i.g.193.1		2	360.317	even	12
576.2.i.g.385.1		2	360.77	even	12
882.2.e.g.373.1		2	315.227	odd	12
882.2.e.g.655.1		2	315.257	odd	12
882.2.e.i.373.1		2	315.137	even	12
882.2.e.i.655.1		2	315.212	even	12
882.2.f.d.295.1		2	315.167	odd	12
882.2.f.d.589.1		2	315.272	odd	12
882.2.h.b.67.1		2	315.47	odd	12
882.2.h.b.79.1		2	315.122	odd	12
882.2.h.c.67.1		2	315.2	even	12
882.2.h.c.79.1		2	315.32	even	12
1296.2.a.f.1.1		1	20.7	even	4
1296.2.a.g.1.1		1	60.47	odd	4
1350.2.e.c.451.1		2	45.13	odd	12
1350.2.e.c.901.1		2	45.43	odd	12
1350.2.j.a.199.1		4	45.34	even	6
1350.2.j.a.199.2		4	9.7	even	3
1350.2.j.a.1099.1		4	9.4	even	3
1350.2.j.a.1099.2		4	45.4	even	6
1728.2.i.e.577.1		2	360.277	odd	12
1728.2.i.e.1153.1		2	360.157	odd	12
1728.2.i.f.577.1		2	360.187	even	12
1728.2.i.f.1153.1		2	360.67	even	12
2646.2.e.b.1549.1		2	315.277	odd	12
2646.2.e.b.2125.1		2	315.247	odd	12
2646.2.e.c.1549.1		2	315.52	even	12
2646.2.e.c.2125.1		2	315.292	even	12
2646.2.f.g.883.1		2	315.202	even	12
2646.2.f.g.1765.1		2	315.97	even	12
2646.2.h.h.361.1		2	315.142	odd	12
2646.2.h.h.667.1		2	315.67	odd	12
2646.2.h.i.361.1		2	315.187	even	12
2646.2.h.i.667.1		2	315.157	even	12
4050.2.a.c.1.1		1	15.8	even	4
4050.2.a.v.1.1		1	5.3	odd	4
4050.2.c.c.649.1		2	3.2	odd	2
4050.2.c.c.649.2		2	15.14	odd	2
4050.2.c.r.649.1		2	5.4	even	2	inner
4050.2.c.r.649.2		2	1.1	even	1	trivial
5184.2.a.o.1.1		1	120.107	odd	4
5184.2.a.p.1.1		1	40.27	even	4
5184.2.a.q.1.1		1	40.37	odd	4
5184.2.a.r.1.1		1	120.77	even	4
7938.2.a.i.1.1		1	35.27	even	4
7938.2.a.x.1.1		1	105.62	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:	\( N \)	\(=\)	\( 4050 = 2 \cdot 3^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4050.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000 q^{4} -2.00000i q^{7} -1.00000i q^{8} +O(q^{10})\)	\(q+1.00000i q^{2} -1.00000 q^{4} -2.00000i q^{7} -1.00000i q^{8} +3.00000 q^{11} +2.00000i q^{13} +2.00000 q^{14} +1.00000 q^{16} -3.00000i q^{17} +1.00000 q^{19} +3.00000i q^{22} +6.00000i q^{23} -2.00000 q^{26} +2.00000i q^{28} +6.00000 q^{29} -4.00000 q^{31} +1.00000i q^{32} +3.00000 q^{34} +4.00000i q^{37} +1.00000i q^{38} -9.00000 q^{41} -1.00000i q^{43} -3.00000 q^{44} -6.00000 q^{46} -6.00000i q^{47} +3.00000 q^{49} -2.00000i q^{52} -12.0000i q^{53} -2.00000 q^{56} +6.00000i q^{58} +3.00000 q^{59} +8.00000 q^{61} -4.00000i q^{62} -1.00000 q^{64} -5.00000i q^{67} +3.00000i q^{68} +12.0000 q^{71} +11.0000i q^{73} -4.00000 q^{74} -1.00000 q^{76} -6.00000i q^{77} +4.00000 q^{79} -9.00000i q^{82} -12.0000i q^{83} +1.00000 q^{86} -3.00000i q^{88} +6.00000 q^{89} +4.00000 q^{91} -6.00000i q^{92} +6.00000 q^{94} -5.00000i q^{97} +3.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4}+O(q^{10}) \) 2 * q - 2 * q^4	\( 2 q - 2 q^{4} + 6 q^{11} + 4 q^{14} + 2 q^{16} + 2 q^{19} - 4 q^{26} + 12 q^{29} - 8 q^{31} + 6 q^{34} - 18 q^{41} - 6 q^{44} - 12 q^{46} + 6 q^{49} - 4 q^{56} + 6 q^{59} + 16 q^{61} - 2 q^{64} + 24 q^{71} - 8 q^{74} - 2 q^{76} + 8 q^{79} + 2 q^{86} + 12 q^{89} + 8 q^{91} + 12 q^{94}+O(q^{100}) \) 2 * q - 2 * q^4 + 6 * q^11 + 4 * q^14 + 2 * q^16 + 2 * q^19 - 4 * q^26 + 12 * q^29 - 8 * q^31 + 6 * q^34 - 18 * q^41 - 6 * q^44 - 12 * q^46 + 6 * q^49 - 4 * q^56 + 6 * q^59 + 16 * q^61 - 2 * q^64 + 24 * q^71 - 8 * q^74 - 2 * q^76 + 8 * q^79 + 2 * q^86 + 12 * q^89 + 8 * q^91 + 12 * q^94

Introduction

L-functions

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Representations

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