LMFDB - Embedded newform 4900.2.e.i.2549.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4900,2,Mod(2549,4900)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4900, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4900.2549");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4900 = 2^{2} \cdot 5^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4900.e (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:	$39.1266969904$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 28)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2549.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	4900.2549
Dual form			4900.2.e.i.2549.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} +2.00000 q^{9} +O(q^{10})$	$q+1.00000i q^{3} +2.00000 q^{9} -3.00000 q^{11} +2.00000i q^{13} -3.00000i q^{17} +1.00000 q^{19} +3.00000i q^{23} +5.00000i q^{27} +6.00000 q^{29} -7.00000 q^{31} -3.00000i q^{33} +1.00000i q^{37} -2.00000 q^{39} +6.00000 q^{41} -4.00000i q^{43} +9.00000i q^{47} +3.00000 q^{51} +3.00000i q^{53} +1.00000i q^{57} -9.00000 q^{59} -1.00000 q^{61} +7.00000i q^{67} -3.00000 q^{69} -1.00000i q^{73} +13.0000 q^{79} +1.00000 q^{81} +12.0000i q^{83} +6.00000i q^{87} -15.0000 q^{89} -7.00000i q^{93} +10.0000i q^{97} -6.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{9}+O(q^{10}) $ 2 * q + 4 * q^9	$ 2 q + 4 q^{9} - 6 q^{11} + 2 q^{19} + 12 q^{29} - 14 q^{31} - 4 q^{39} + 12 q^{41} + 6 q^{51} - 18 q^{59} - 2 q^{61} - 6 q^{69} + 26 q^{79} + 2 q^{81} - 30 q^{89} - 12 q^{99}+O(q^{100}) $ 2 * q + 4 * q^9 - 6 * q^11 + 2 * q^19 + 12 * q^29 - 14 * q^31 - 4 * q^39 + 12 * q^41 + 6 * q^51 - 18 * q^59 - 2 * q^61 - 6 * q^69 + 26 * q^79 + 2 * q^81 - 30 * q^89 - 12 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4900\mathbb{Z}\right)^\times$.

$n$	$101$	$1177$	$2451$
$\chi(n)$	$1$	$-1$	$1$

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
$3$	1.00000i	0.577350i	0.957427	+	0.288675i	$0.0932147\pi$
$3$	1.00000i	0.577350i	−0.957427	+	0.288675i	$0.906785\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	2.00000	0.666667
$10$	0	0
$11$	−3.00000	−0.904534	−0.452267	−	0.891883i	$-0.649385\pi$
$11$	−3.00000	−0.904534	−0.452267	+	0.891883i	$0.649385\pi$
$12$	0	0
$13$	2.00000i	0.554700i	0.960769	+	0.277350i	$0.0894562\pi$
$13$	2.00000i	0.554700i	−0.960769	+	0.277350i	$0.910544\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 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
$22$	0	0
$23$	3.00000i	0.625543i	0.949828	+	0.312772i	$0.101257\pi$
$23$	3.00000i	0.625543i	−0.949828	+	0.312772i	$0.898743\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	5.00000i	0.962250i
$28$	0	0
$29$	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
$31$	−7.00000	−1.25724	−0.628619	−	0.777714i	$-0.716379\pi$
$31$	−7.00000	−1.25724	−0.628619	+	0.777714i	$0.716379\pi$
$32$	0	0
$33$	− 3.00000i	− 0.522233i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	1.00000i	0.164399i	0.996616	+	0.0821995i	$0.0261945\pi$
$37$	1.00000i	0.164399i	−0.996616	+	0.0821995i	$0.973806\pi$
$38$	0	0
$39$	−2.00000	−0.320256
$40$	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
$43$	− 4.00000i	− 0.609994i	−0.952353	−	0.304997i	$-0.901344\pi$
$43$	− 4.00000i	− 0.609994i	0.952353	−	0.304997i	$-0.0986555\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	9.00000i	1.31278i	0.754420	+	0.656392i	$0.227918\pi$
$47$	9.00000i	1.31278i	−0.754420	+	0.656392i	$0.772082\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	3.00000	0.420084
$52$	0	0
$53$	3.00000i	0.412082i	0.978543	+	0.206041i	$0.0660580\pi$
$53$	3.00000i	0.412082i	−0.978543	+	0.206041i	$0.933942\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	1.00000i	0.132453i
$58$	0	0
$59$	−9.00000	−1.17170	−0.585850	−	0.810419i	$-0.699239\pi$
$59$	−9.00000	−1.17170	−0.585850	+	0.810419i	$0.699239\pi$
$60$	0	0
$61$	−1.00000	−0.128037	−0.0640184	−	0.997949i	$-0.520392\pi$
$61$	−1.00000	−0.128037	−0.0640184	+	0.997949i	$0.520392\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	7.00000i	0.855186i	0.903971	+	0.427593i	$0.140638\pi$
$67$	7.00000i	0.855186i	−0.903971	+	0.427593i	$0.859362\pi$
$68$	0	0
$69$	−3.00000	−0.361158
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	− 1.00000i	− 0.117041i	−0.998286	−	0.0585206i	$-0.981362\pi$
$73$	− 1.00000i	− 0.117041i	0.998286	−	0.0585206i	$-0.0186383\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	13.0000	1.46261	0.731307	−	0.682048i	$-0.238911\pi$
$79$	13.0000	1.46261	0.731307	+	0.682048i	$0.238911\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	12.0000i	1.31717i	0.752506	+	0.658586i	$0.228845\pi$
$83$	12.0000i	1.31717i	−0.752506	+	0.658586i	$0.771155\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	6.00000i	0.643268i
$88$	0	0
$89$	−15.0000	−1.59000	−0.794998	−	0.606612i	$-0.792528\pi$
$89$	−15.0000	−1.59000	−0.794998	+	0.606612i	$0.792528\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	− 7.00000i	− 0.725866i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	10.0000i	1.01535i	0.861550	+	0.507673i	$0.169494\pi$
$97$	10.0000i	1.01535i	−0.861550	+	0.507673i	$0.830506\pi$
$98$	0	0
$99$	−6.00000	−0.603023
$100$	0	0
$101$	15.0000	1.49256	0.746278	−	0.665635i	$-0.231839\pi$
$101$	15.0000	1.49256	0.746278	+	0.665635i	$0.231839\pi$
$102$	0	0
$103$	11.0000i	1.08386i	0.840423	+	0.541931i	$0.182307\pi$
$103$	11.0000i	1.08386i	−0.840423	+	0.541931i	$0.817693\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 15.0000i	− 1.45010i	−0.688694	−	0.725052i	$-0.741816\pi$
$107$	− 15.0000i	− 1.45010i	0.688694	−	0.725052i	$-0.258184\pi$
$108$	0	0
$109$	1.00000	0.0957826	0.0478913	−	0.998853i	$-0.484750\pi$
$109$	1.00000	0.0957826	0.0478913	+	0.998853i	$0.484750\pi$
$110$	0	0
$111$	−1.00000	−0.0949158
$112$	0	0
$113$	6.00000i	0.564433i	0.959351	+	0.282216i	$0.0910696\pi$
$113$	6.00000i	0.564433i	−0.959351	+	0.282216i	$0.908930\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	4.00000i	0.369800i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−2.00000	−0.181818
$122$	0	0
$123$	6.00000i	0.541002i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 8.00000i	− 0.709885i	−0.934888	−	0.354943i	$-0.884500\pi$
$127$	− 8.00000i	− 0.709885i	0.934888	−	0.354943i	$-0.115500\pi$
$128$	0	0
$129$	4.00000	0.352180
$130$	0	0
$131$	3.00000	0.262111	0.131056	−	0.991375i	$-0.458163\pi$
$131$	3.00000	0.262111	0.131056	+	0.991375i	$0.458163\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	21.0000i	1.79415i	0.441877	+	0.897076i	$0.354313\pi$
$137$	21.0000i	1.79415i	−0.441877	+	0.897076i	$0.645687\pi$
$138$	0	0
$139$	−20.0000	−1.69638	−0.848189	−	0.529694i	$-0.822307\pi$
$139$	−20.0000	−1.69638	−0.848189	+	0.529694i	$0.822307\pi$
$140$	0	0
$141$	−9.00000	−0.757937
$142$	0	0
$143$	− 6.00000i	− 0.501745i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−3.00000	−0.245770	−0.122885	−	0.992421i	$-0.539215\pi$
$149$	−3.00000	−0.245770	−0.122885	+	0.992421i	$0.539215\pi$
$150$	0	0
$151$	17.0000	1.38344	0.691720	−	0.722166i	$-0.256853\pi$
$151$	17.0000	1.38344	0.691720	+	0.722166i	$0.256853\pi$
$152$	0	0
$153$	− 6.00000i	− 0.485071i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	13.0000i	1.03751i	0.854922	+	0.518756i	$0.173605\pi$
$157$	13.0000i	1.03751i	−0.854922	+	0.518756i	$0.826395\pi$
$158$	0	0
$159$	−3.00000	−0.237915
$160$	0	0
$161$	0	0
$162$	0	0
$163$	11.0000i	0.861586i	0.902451	+	0.430793i	$0.141766\pi$
$163$	11.0000i	0.861586i	−0.902451	+	0.430793i	$0.858234\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	12.0000i	0.928588i	0.885681	+	0.464294i	$0.153692\pi$
$167$	12.0000i	0.928588i	−0.885681	+	0.464294i	$0.846308\pi$
$168$	0	0
$169$	9.00000	0.692308
$170$	0	0
$171$	2.00000	0.152944
$172$	0	0
$173$	− 9.00000i	− 0.684257i	−0.939653	−	0.342129i	$-0.888852\pi$
$173$	− 9.00000i	− 0.684257i	0.939653	−	0.342129i	$-0.111148\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 9.00000i	− 0.676481i
$178$	0	0
$179$	−21.0000	−1.56961	−0.784807	−	0.619740i	$-0.787238\pi$
$179$	−21.0000	−1.56961	−0.784807	+	0.619740i	$0.787238\pi$
$180$	0	0
$181$	−10.0000	−0.743294	−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000	−0.743294	−0.371647	+	0.928374i	$0.621207\pi$
$182$	0	0
$183$	− 1.00000i	− 0.0739221i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	9.00000i	0.658145i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−9.00000	−0.651217	−0.325609	−	0.945505i	$-0.605569\pi$
$191$	−9.00000	−0.651217	−0.325609	+	0.945505i	$0.605569\pi$
$192$	0	0
$193$	11.0000i	0.791797i	0.918294	+	0.395899i	$0.129567\pi$
$193$	11.0000i	0.791797i	−0.918294	+	0.395899i	$0.870433\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 18.0000i	− 1.28245i	−0.767354	−	0.641223i	$-0.778427\pi$
$197$	− 18.0000i	− 1.28245i	0.767354	−	0.641223i	$-0.221573\pi$
$198$	0	0
$199$	7.00000	0.496217	0.248108	−	0.968732i	$-0.420191\pi$
$199$	7.00000	0.496217	0.248108	+	0.968732i	$0.420191\pi$
$200$	0	0
$201$	−7.00000	−0.493742
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	6.00000i	0.417029i
$208$	0	0
$209$	−3.00000	−0.207514
$210$	0	0
$211$	−4.00000	−0.275371	−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000	−0.275371	−0.137686	+	0.990476i	$0.543966\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	1.00000	0.0675737
$220$	0	0
$221$	6.00000	0.403604
$222$	0	0
$223$	8.00000i	0.535720i	0.963458	+	0.267860i	$0.0863164\pi$
$223$	8.00000i	0.535720i	−0.963458	+	0.267860i	$0.913684\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	3.00000i	0.199117i	0.995032	+	0.0995585i	$0.0317430\pi$
$227$	3.00000i	0.199117i	−0.995032	+	0.0995585i	$0.968257\pi$
$228$	0	0
$229$	−11.0000	−0.726900	−0.363450	−	0.931614i	$-0.618401\pi$
$229$	−11.0000	−0.726900	−0.363450	+	0.931614i	$0.618401\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 21.0000i	− 1.37576i	−0.725826	−	0.687878i	$-0.758542\pi$
$233$	− 21.0000i	− 1.37576i	0.725826	−	0.687878i	$-0.241458\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	13.0000i	0.844441i
$238$	0	0
$239$	12.0000	0.776215	0.388108	−	0.921614i	$-0.373129\pi$
$239$	12.0000	0.776215	0.388108	+	0.921614i	$0.373129\pi$
$240$	0	0
$241$	−1.00000	−0.0644157	−0.0322078	−	0.999481i	$-0.510254\pi$
$241$	−1.00000	−0.0644157	−0.0322078	+	0.999481i	$0.510254\pi$
$242$	0	0
$243$	16.0000i	1.02640i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	2.00000i	0.127257i
$248$	0	0
$249$	−12.0000	−0.760469
$250$	0	0
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	0	0
$253$	− 9.00000i	− 0.565825i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 3.00000i	− 0.187135i	−0.995613	−	0.0935674i	$-0.970173\pi$
$257$	− 3.00000i	− 0.187135i	0.995613	−	0.0935674i	$-0.0298271\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	12.0000	0.742781
$262$	0	0
$263$	− 3.00000i	− 0.184988i	−0.995713	−	0.0924940i	$-0.970516\pi$
$263$	− 3.00000i	− 0.184988i	0.995713	−	0.0924940i	$-0.0294839\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 15.0000i	− 0.917985i
$268$	0	0
$269$	−3.00000	−0.182913	−0.0914566	−	0.995809i	$-0.529152\pi$
$269$	−3.00000	−0.182913	−0.0914566	+	0.995809i	$0.529152\pi$
$270$	0	0
$271$	11.0000	0.668202	0.334101	−	0.942537i	$-0.391567\pi$
$271$	11.0000	0.668202	0.334101	+	0.942537i	$0.391567\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	13.0000i	0.781094i	0.920583	+	0.390547i	$0.127714\pi$
$277$	13.0000i	0.781094i	−0.920583	+	0.390547i	$0.872286\pi$
$278$	0	0
$279$	−14.0000	−0.838158
$280$	0	0
$281$	30.0000	1.78965	0.894825	−	0.446417i	$-0.147300\pi$
$281$	30.0000	1.78965	0.894825	+	0.446417i	$0.147300\pi$
$282$	0	0
$283$	29.0000i	1.72387i	0.507018	+	0.861936i	$0.330748\pi$
$283$	29.0000i	1.72387i	−0.507018	+	0.861936i	$0.669252\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	8.00000	0.470588
$290$	0	0
$291$	−10.0000	−0.586210
$292$	0	0
$293$	6.00000i	0.350524i	0.984522	+	0.175262i	$0.0560772\pi$
$293$	6.00000i	0.350524i	−0.984522	+	0.175262i	$0.943923\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 15.0000i	− 0.870388i
$298$	0	0
$299$	−6.00000	−0.346989
$300$	0	0
$301$	0	0
$302$	0	0
$303$	15.0000i	0.861727i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	28.0000i	1.59804i	0.601302	+	0.799022i	$0.294649\pi$
$307$	28.0000i	1.59804i	−0.601302	+	0.799022i	$0.705351\pi$
$308$	0	0
$309$	−11.0000	−0.625768
$310$	0	0
$311$	−27.0000	−1.53103	−0.765515	−	0.643418i	$-0.777516\pi$
$311$	−27.0000	−1.53103	−0.765515	+	0.643418i	$0.777516\pi$
$312$	0	0
$313$	23.0000i	1.30004i	0.759918	+	0.650018i	$0.225239\pi$
$313$	23.0000i	1.30004i	−0.759918	+	0.650018i	$0.774761\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	9.00000i	0.505490i	0.967533	+	0.252745i	$0.0813334\pi$
$317$	9.00000i	0.505490i	−0.967533	+	0.252745i	$0.918667\pi$
$318$	0	0
$319$	−18.0000	−1.00781
$320$	0	0
$321$	15.0000	0.837218
$322$	0	0
$323$	− 3.00000i	− 0.166924i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	1.00000i	0.0553001i
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−13.0000	−0.714545	−0.357272	−	0.934000i	$-0.616293\pi$
$331$	−13.0000	−0.714545	−0.357272	+	0.934000i	$0.616293\pi$
$332$	0	0
$333$	2.00000i	0.109599i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	34.0000i	1.85210i	0.377403	+	0.926049i	$0.376817\pi$
$337$	34.0000i	1.85210i	−0.377403	+	0.926049i	$0.623183\pi$
$338$	0	0
$339$	−6.00000	−0.325875
$340$	0	0
$341$	21.0000	1.13721
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 9.00000i	− 0.483145i	−0.970383	−	0.241573i	$-0.922337\pi$
$347$	− 9.00000i	− 0.483145i	0.970383	−	0.241573i	$-0.0776632\pi$
$348$	0	0
$349$	−26.0000	−1.39175	−0.695874	−	0.718164i	$-0.744983\pi$
$349$	−26.0000	−1.39175	−0.695874	+	0.718164i	$0.744983\pi$
$350$	0	0
$351$	−10.0000	−0.533761
$352$	0	0
$353$	− 21.0000i	− 1.11772i	−0.829263	−	0.558859i	$-0.811239\pi$
$353$	− 21.0000i	− 1.11772i	0.829263	−	0.558859i	$-0.188761\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−15.0000	−0.791670	−0.395835	−	0.918322i	$-0.629545\pi$
$359$	−15.0000	−0.791670	−0.395835	+	0.918322i	$0.629545\pi$
$360$	0	0
$361$	−18.0000	−0.947368
$362$	0	0
$363$	− 2.00000i	− 0.104973i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 5.00000i	− 0.260998i	−0.991448	−	0.130499i	$-0.958342\pi$
$367$	− 5.00000i	− 0.260998i	0.991448	−	0.130499i	$-0.0416579\pi$
$368$	0	0
$369$	12.0000	0.624695
$370$	0	0
$371$	0	0
$372$	0	0
$373$	− 25.0000i	− 1.29445i	−0.762299	−	0.647225i	$-0.775929\pi$
$373$	− 25.0000i	− 1.29445i	0.762299	−	0.647225i	$-0.224071\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	12.0000i	0.618031i
$378$	0	0
$379$	−8.00000	−0.410932	−0.205466	−	0.978664i	$-0.565871\pi$
$379$	−8.00000	−0.410932	−0.205466	+	0.978664i	$0.565871\pi$
$380$	0	0
$381$	8.00000	0.409852
$382$	0	0
$383$	− 33.0000i	− 1.68622i	−0.537740	−	0.843111i	$-0.680722\pi$
$383$	− 33.0000i	− 1.68622i	0.537740	−	0.843111i	$-0.319278\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 8.00000i	− 0.406663i
$388$	0	0
$389$	−15.0000	−0.760530	−0.380265	−	0.924878i	$-0.624167\pi$
$389$	−15.0000	−0.760530	−0.380265	+	0.924878i	$0.624167\pi$
$390$	0	0
$391$	9.00000	0.455150
$392$	0	0
$393$	3.00000i	0.151330i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	37.0000i	1.85698i	0.371361	+	0.928488i	$0.378891\pi$
$397$	37.0000i	1.85698i	−0.371361	+	0.928488i	$0.621109\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	3.00000	0.149813	0.0749064	−	0.997191i	$-0.476134\pi$
$401$	3.00000	0.149813	0.0749064	+	0.997191i	$0.476134\pi$
$402$	0	0
$403$	− 14.0000i	− 0.697390i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 3.00000i	− 0.148704i
$408$	0	0
$409$	−11.0000	−0.543915	−0.271957	−	0.962309i	$-0.587671\pi$
$409$	−11.0000	−0.543915	−0.271957	+	0.962309i	$0.587671\pi$
$410$	0	0
$411$	−21.0000	−1.03585
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 20.0000i	− 0.979404i
$418$	0	0
$419$	12.0000	0.586238	0.293119	−	0.956076i	$-0.405307\pi$
$419$	12.0000	0.586238	0.293119	+	0.956076i	$0.405307\pi$
$420$	0	0
$421$	−22.0000	−1.07221	−0.536107	−	0.844150i	$-0.680106\pi$
$421$	−22.0000	−1.07221	−0.536107	+	0.844150i	$0.680106\pi$
$422$	0	0
$423$	18.0000i	0.875190i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0	0
$428$	0	0
$429$	6.00000	0.289683
$430$	0	0
$431$	−15.0000	−0.722525	−0.361262	−	0.932464i	$-0.617654\pi$
$431$	−15.0000	−0.722525	−0.361262	+	0.932464i	$0.617654\pi$
$432$	0	0
$433$	− 10.0000i	− 0.480569i	−0.970702	−	0.240285i	$-0.922759\pi$
$433$	− 10.0000i	− 0.480569i	0.970702	−	0.240285i	$-0.0772408\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	3.00000i	0.143509i
$438$	0	0
$439$	1.00000	0.0477274	0.0238637	−	0.999715i	$-0.492403\pi$
$439$	1.00000	0.0477274	0.0238637	+	0.999715i	$0.492403\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 9.00000i	− 0.427603i	−0.976877	−	0.213801i	$-0.931415\pi$
$443$	− 9.00000i	− 0.427603i	0.976877	−	0.213801i	$-0.0685846\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 3.00000i	− 0.141895i
$448$	0	0
$449$	18.0000	0.849473	0.424736	−	0.905317i	$-0.360367\pi$
$449$	18.0000	0.849473	0.424736	+	0.905317i	$0.360367\pi$
$450$	0	0
$451$	−18.0000	−0.847587
$452$	0	0
$453$	17.0000i	0.798730i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 23.0000i	− 1.07589i	−0.842978	−	0.537947i	$-0.819200\pi$
$457$	− 23.0000i	− 1.07589i	0.842978	−	0.537947i	$-0.180800\pi$
$458$	0	0
$459$	15.0000	0.700140
$460$	0	0
$461$	−6.00000	−0.279448	−0.139724	−	0.990190i	$-0.544622\pi$
$461$	−6.00000	−0.279448	−0.139724	+	0.990190i	$0.544622\pi$
$462$	0	0
$463$	− 16.0000i	− 0.743583i	−0.928316	−	0.371792i	$-0.878744\pi$
$463$	− 16.0000i	− 0.743583i	0.928316	−	0.371792i	$-0.121256\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	21.0000i	0.971764i	0.874024	+	0.485882i	$0.161502\pi$
$467$	21.0000i	0.971764i	−0.874024	+	0.485882i	$0.838498\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−13.0000	−0.599008
$472$	0	0
$473$	12.0000i	0.551761i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	6.00000i	0.274721i
$478$	0	0
$479$	3.00000	0.137073	0.0685367	−	0.997649i	$-0.478167\pi$
$479$	3.00000	0.137073	0.0685367	+	0.997649i	$0.478167\pi$
$480$	0	0
$481$	−2.00000	−0.0911922
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	19.0000i	0.860972i	0.902597	+	0.430486i	$0.141658\pi$
$487$	19.0000i	0.860972i	−0.902597	+	0.430486i	$0.858342\pi$
$488$	0	0
$489$	−11.0000	−0.497437
$490$	0	0
$491$	24.0000	1.08310	0.541552	−	0.840667i	$-0.317837\pi$
$491$	24.0000	1.08310	0.541552	+	0.840667i	$0.317837\pi$
$492$	0	0
$493$	− 18.0000i	− 0.810679i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−11.0000	−0.492428	−0.246214	−	0.969216i	$-0.579187\pi$
$499$	−11.0000	−0.492428	−0.246214	+	0.969216i	$0.579187\pi$
$500$	0	0
$501$	−12.0000	−0.536120
$502$	0	0
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	9.00000i	0.399704i
$508$	0	0
$509$	−3.00000	−0.132973	−0.0664863	−	0.997787i	$-0.521179\pi$
$509$	−3.00000	−0.132973	−0.0664863	+	0.997787i	$0.521179\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	5.00000i	0.220755i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 27.0000i	− 1.18746i
$518$	0	0
$519$	9.00000	0.395056
$520$	0	0
$521$	39.0000	1.70862	0.854311	−	0.519763i	$-0.173980\pi$
$521$	39.0000	1.70862	0.854311	+	0.519763i	$0.173980\pi$
$522$	0	0
$523$	− 1.00000i	− 0.0437269i	−0.999761	−	0.0218635i	$-0.993040\pi$
$523$	− 1.00000i	− 0.0437269i	0.999761	−	0.0218635i	$-0.00695991\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	21.0000i	0.914774i
$528$	0	0
$529$	14.0000	0.608696
$530$	0	0
$531$	−18.0000	−0.781133
$532$	0	0
$533$	12.0000i	0.519778i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 21.0000i	− 0.906217i
$538$	0	0
$539$	0	0
$540$	0	0
$541$	35.0000	1.50477	0.752384	−	0.658725i	$-0.228904\pi$
$541$	35.0000	1.50477	0.752384	+	0.658725i	$0.228904\pi$
$542$	0	0
$543$	− 10.0000i	− 0.429141i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 8.00000i	− 0.342055i	−0.985266	−	0.171028i	$-0.945291\pi$
$547$	− 8.00000i	− 0.342055i	0.985266	−	0.171028i	$-0.0547087\pi$
$548$	0	0
$549$	−2.00000	−0.0853579
$550$	0	0
$551$	6.00000	0.255609
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	33.0000i	1.39825i	0.714997	+	0.699127i	$0.246428\pi$
$557$	33.0000i	1.39825i	−0.714997	+	0.699127i	$0.753572\pi$
$558$	0	0
$559$	8.00000	0.338364
$560$	0	0
$561$	−9.00000	−0.379980
$562$	0	0
$563$	9.00000i	0.379305i	0.981851	+	0.189652i	$0.0607361\pi$
$563$	9.00000i	0.379305i	−0.981851	+	0.189652i	$0.939264\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	9.00000	0.377300	0.188650	−	0.982044i	$-0.439589\pi$
$569$	9.00000	0.377300	0.188650	+	0.982044i	$0.439589\pi$
$570$	0	0
$571$	29.0000	1.21361	0.606806	−	0.794850i	$-0.292450\pi$
$571$	29.0000	1.21361	0.606806	+	0.794850i	$0.292450\pi$
$572$	0	0
$573$	− 9.00000i	− 0.375980i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	1.00000i	0.0416305i	0.999783	+	0.0208153i	$0.00662619\pi$
$577$	1.00000i	0.0416305i	−0.999783	+	0.0208153i	$0.993374\pi$
$578$	0	0
$579$	−11.0000	−0.457144
$580$	0	0
$581$	0	0
$582$	0	0
$583$	− 9.00000i	− 0.372742i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 12.0000i	− 0.495293i	−0.968850	−	0.247647i	$-0.920343\pi$
$587$	− 12.0000i	− 0.495293i	0.968850	−	0.247647i	$-0.0796572\pi$
$588$	0	0
$589$	−7.00000	−0.288430
$590$	0	0
$591$	18.0000	0.740421
$592$	0	0
$593$	− 21.0000i	− 0.862367i	−0.902264	−	0.431183i	$-0.858096\pi$
$593$	− 21.0000i	− 0.862367i	0.902264	−	0.431183i	$-0.141904\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	7.00000i	0.286491i
$598$	0	0
$599$	27.0000	1.10319	0.551595	−	0.834112i	$-0.314019\pi$
$599$	27.0000	1.10319	0.551595	+	0.834112i	$0.314019\pi$
$600$	0	0
$601$	14.0000	0.571072	0.285536	−	0.958368i	$-0.407828\pi$
$601$	14.0000	0.571072	0.285536	+	0.958368i	$0.407828\pi$
$602$	0	0
$603$	14.0000i	0.570124i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 47.0000i	− 1.90767i	−0.300329	−	0.953836i	$-0.597097\pi$
$607$	− 47.0000i	− 1.90767i	0.300329	−	0.953836i	$-0.402903\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−18.0000	−0.728202
$612$	0	0
$613$	− 25.0000i	− 1.00974i	−0.863195	−	0.504870i	$-0.831540\pi$
$613$	− 25.0000i	− 1.00974i	0.863195	−	0.504870i	$-0.168460\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 6.00000i	− 0.241551i	−0.992680	−	0.120775i	$-0.961462\pi$
$617$	− 6.00000i	− 0.241551i	0.992680	−	0.120775i	$-0.0385381\pi$
$618$	0	0
$619$	31.0000	1.24600	0.622998	−	0.782224i	$-0.285915\pi$
$619$	31.0000	1.24600	0.622998	+	0.782224i	$0.285915\pi$
$620$	0	0
$621$	−15.0000	−0.601929
$622$	0	0
$623$	0	0
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 3.00000i	− 0.119808i
$628$	0	0
$629$	3.00000	0.119618
$630$	0	0
$631$	−16.0000	−0.636950	−0.318475	−	0.947931i	$-0.603171\pi$
$631$	−16.0000	−0.636950	−0.318475	+	0.947931i	$0.603171\pi$
$632$	0	0
$633$	− 4.00000i	− 0.158986i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	15.0000	0.592464	0.296232	−	0.955116i	$-0.404270\pi$
$641$	15.0000	0.592464	0.296232	+	0.955116i	$0.404270\pi$
$642$	0	0
$643$	20.0000i	0.788723i	0.918955	+	0.394362i	$0.129034\pi$
$643$	20.0000i	0.788723i	−0.918955	+	0.394362i	$0.870966\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 21.0000i	− 0.825595i	−0.910823	−	0.412798i	$-0.864552\pi$
$647$	− 21.0000i	− 0.825595i	0.910823	−	0.412798i	$-0.135448\pi$
$648$	0	0
$649$	27.0000	1.05984
$650$	0	0
$651$	0	0
$652$	0	0
$653$	39.0000i	1.52619i	0.646288	+	0.763094i	$0.276321\pi$
$653$	39.0000i	1.52619i	−0.646288	+	0.763094i	$0.723679\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 2.00000i	− 0.0780274i
$658$	0	0
$659$	−12.0000	−0.467454	−0.233727	−	0.972302i	$-0.575092\pi$
$659$	−12.0000	−0.467454	−0.233727	+	0.972302i	$0.575092\pi$
$660$	0	0
$661$	11.0000	0.427850	0.213925	−	0.976850i	$-0.431375\pi$
$661$	11.0000	0.427850	0.213925	+	0.976850i	$0.431375\pi$
$662$	0	0
$663$	6.00000i	0.233021i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	18.0000i	0.696963i
$668$	0	0
$669$	−8.00000	−0.309298
$670$	0	0
$671$	3.00000	0.115814
$672$	0	0
$673$	14.0000i	0.539660i	0.962908	+	0.269830i	$0.0869676\pi$
$673$	14.0000i	0.539660i	−0.962908	+	0.269830i	$0.913032\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 27.0000i	− 1.03769i	−0.854867	−	0.518847i	$-0.826361\pi$
$677$	− 27.0000i	− 1.03769i	0.854867	−	0.518847i	$-0.173639\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−3.00000	−0.114960
$682$	0	0
$683$	21.0000i	0.803543i	0.915740	+	0.401771i	$0.131605\pi$
$683$	21.0000i	0.803543i	−0.915740	+	0.401771i	$0.868395\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 11.0000i	− 0.419676i
$688$	0	0
$689$	−6.00000	−0.228582
$690$	0	0
$691$	−13.0000	−0.494543	−0.247272	−	0.968946i	$-0.579534\pi$
$691$	−13.0000	−0.494543	−0.247272	+	0.968946i	$0.579534\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 18.0000i	− 0.681799i
$698$	0	0
$699$	21.0000	0.794293
$700$	0	0
$701$	−18.0000	−0.679851	−0.339925	−	0.940452i	$-0.610402\pi$
$701$	−18.0000	−0.679851	−0.339925	+	0.940452i	$0.610402\pi$
$702$	0	0
$703$	1.00000i	0.0377157i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	1.00000	0.0375558	0.0187779	−	0.999824i	$-0.494022\pi$
$709$	1.00000	0.0375558	0.0187779	+	0.999824i	$0.494022\pi$
$710$	0	0
$711$	26.0000	0.975076
$712$	0	0
$713$	− 21.0000i	− 0.786456i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	12.0000i	0.448148i
$718$	0	0
$719$	21.0000	0.783168	0.391584	−	0.920142i	$-0.371927\pi$
$719$	21.0000	0.783168	0.391584	+	0.920142i	$0.371927\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	− 1.00000i	− 0.0371904i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 32.0000i	− 1.18681i	−0.804902	−	0.593407i	$-0.797782\pi$
$727$	− 32.0000i	− 1.18681i	0.804902	−	0.593407i	$-0.202218\pi$
$728$	0	0
$729$	−13.0000	−0.481481
$730$	0	0
$731$	−12.0000	−0.443836
$732$	0	0
$733$	− 25.0000i	− 0.923396i	−0.887037	−	0.461698i	$-0.847240\pi$
$733$	− 25.0000i	− 0.923396i	0.887037	−	0.461698i	$-0.152760\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 21.0000i	− 0.773545i
$738$	0	0
$739$	19.0000	0.698926	0.349463	−	0.936950i	$-0.386364\pi$
$739$	19.0000	0.698926	0.349463	+	0.936950i	$0.386364\pi$
$740$	0	0
$741$	−2.00000	−0.0734718
$742$	0	0
$743$	− 48.0000i	− 1.76095i	−0.474093	−	0.880475i	$-0.657224\pi$
$743$	− 48.0000i	− 1.76095i	0.474093	−	0.880475i	$-0.342776\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	24.0000i	0.878114i
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−25.0000	−0.912263	−0.456131	−	0.889912i	$-0.650765\pi$
$751$	−25.0000	−0.912263	−0.456131	+	0.889912i	$0.650765\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	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	$-0.0115717\pi$
$758$	0	0
$759$	9.00000	0.326679
$760$	0	0
$761$	3.00000	0.108750	0.0543750	−	0.998521i	$-0.482683\pi$
$761$	3.00000	0.108750	0.0543750	+	0.998521i	$0.482683\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 18.0000i	− 0.649942i
$768$	0	0
$769$	34.0000	1.22607	0.613036	−	0.790055i	$-0.289948\pi$
$769$	34.0000	1.22607	0.613036	+	0.790055i	$0.289948\pi$
$770$	0	0
$771$	3.00000	0.108042
$772$	0	0
$773$	− 33.0000i	− 1.18693i	−0.804861	−	0.593464i	$-0.797760\pi$
$773$	− 33.0000i	− 1.18693i	0.804861	−	0.593464i	$-0.202240\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	6.00000	0.214972
$780$	0	0
$781$	0	0
$782$	0	0
$783$	30.0000i	1.07211i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	31.0000i	1.10503i	0.833503	+	0.552515i	$0.186332\pi$
$787$	31.0000i	1.10503i	−0.833503	+	0.552515i	$0.813668\pi$
$788$	0	0
$789$	3.00000	0.106803
$790$	0	0
$791$	0	0
$792$	0	0
$793$	− 2.00000i	− 0.0710221i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 42.0000i	− 1.48772i	−0.668338	−	0.743858i	$-0.732994\pi$
$797$	− 42.0000i	− 1.48772i	0.668338	−	0.743858i	$-0.267006\pi$
$798$	0	0
$799$	27.0000	0.955191
$800$	0	0
$801$	−30.0000	−1.06000
$802$	0	0
$803$	3.00000i	0.105868i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 3.00000i	− 0.105605i
$808$	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
$811$	−52.0000	−1.82597	−0.912983	−	0.407997i	$-0.866228\pi$
$811$	−52.0000	−1.82597	−0.912983	+	0.407997i	$0.866228\pi$
$812$	0	0
$813$	11.0000i	0.385787i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 4.00000i	− 0.139942i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	27.0000	0.942306	0.471153	−	0.882051i	$-0.343838\pi$
$821$	27.0000	0.942306	0.471153	+	0.882051i	$0.343838\pi$
$822$	0	0
$823$	5.00000i	0.174289i	0.996196	+	0.0871445i	$0.0277742\pi$
$823$	5.00000i	0.174289i	−0.996196	+	0.0871445i	$0.972226\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	36.0000i	1.25184i	0.779886	+	0.625921i	$0.215277\pi$
$827$	36.0000i	1.25184i	−0.779886	+	0.625921i	$0.784723\pi$
$828$	0	0
$829$	1.00000	0.0347314	0.0173657	−	0.999849i	$-0.494472\pi$
$829$	1.00000	0.0347314	0.0173657	+	0.999849i	$0.494472\pi$
$830$	0	0
$831$	−13.0000	−0.450965
$832$	0	0
$833$	0	0
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 35.0000i	− 1.20978i
$838$	0	0
$839$	24.0000	0.828572	0.414286	−	0.910147i	$-0.364031\pi$
$839$	24.0000	0.828572	0.414286	+	0.910147i	$0.364031\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	0	0
$843$	30.0000i	1.03325i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−29.0000	−0.995277
$850$	0	0
$851$	−3.00000	−0.102839
$852$	0	0
$853$	− 22.0000i	− 0.753266i	−0.926363	−	0.376633i	$-0.877082\pi$
$853$	− 22.0000i	− 0.753266i	0.926363	−	0.376633i	$-0.122918\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 15.0000i	− 0.512390i	−0.966625	−	0.256195i	$-0.917531\pi$
$857$	− 15.0000i	− 0.512390i	0.966625	−	0.256195i	$-0.0824690\pi$
$858$	0	0
$859$	−23.0000	−0.784750	−0.392375	−	0.919805i	$-0.628346\pi$
$859$	−23.0000	−0.784750	−0.392375	+	0.919805i	$0.628346\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	51.0000i	1.73606i	0.496512	+	0.868030i	$0.334614\pi$
$863$	51.0000i	1.73606i	−0.496512	+	0.868030i	$0.665386\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	8.00000i	0.271694i
$868$	0	0
$869$	−39.0000	−1.32298
$870$	0	0
$871$	−14.0000	−0.474372
$872$	0	0
$873$	20.0000i	0.676897i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	13.0000i	0.438979i	0.975615	+	0.219489i	$0.0704391\pi$
$877$	13.0000i	0.438979i	−0.975615	+	0.219489i	$0.929561\pi$
$878$	0	0
$879$	−6.00000	−0.202375
$880$	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
$883$	44.0000i	1.48072i	0.672212	+	0.740359i	$0.265344\pi$
$883$	44.0000i	1.48072i	−0.672212	+	0.740359i	$0.734656\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 39.0000i	− 1.30949i	−0.755849	−	0.654746i	$-0.772776\pi$
$887$	− 39.0000i	− 1.30949i	0.755849	−	0.654746i	$-0.227224\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−3.00000	−0.100504
$892$	0	0
$893$	9.00000i	0.301174i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 6.00000i	− 0.200334i
$898$	0	0
$899$	−42.0000	−1.40078
$900$	0	0
$901$	9.00000	0.299833
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 41.0000i	− 1.36138i	−0.732570	−	0.680691i	$-0.761680\pi$
$907$	− 41.0000i	− 1.36138i	0.732570	−	0.680691i	$-0.238320\pi$
$908$	0	0
$909$	30.0000	0.995037
$910$	0	0
$911$	48.0000	1.59031	0.795155	−	0.606406i	$-0.207389\pi$
$911$	48.0000	1.59031	0.795155	+	0.606406i	$0.207389\pi$
$912$	0	0
$913$	− 36.0000i	− 1.19143i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	1.00000	0.0329870	0.0164935	−	0.999864i	$-0.494750\pi$
$919$	1.00000	0.0329870	0.0164935	+	0.999864i	$0.494750\pi$
$920$	0	0
$921$	−28.0000	−0.922631
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	22.0000i	0.722575i
$928$	0	0
$929$	9.00000	0.295280	0.147640	−	0.989041i	$-0.452832\pi$
$929$	9.00000	0.295280	0.147640	+	0.989041i	$0.452832\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	− 27.0000i	− 0.883940i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 26.0000i	− 0.849383i	−0.905338	−	0.424691i	$-0.860383\pi$
$937$	− 26.0000i	− 0.849383i	0.905338	−	0.424691i	$-0.139617\pi$
$938$	0	0
$939$	−23.0000	−0.750577
$940$	0	0
$941$	27.0000	0.880175	0.440087	−	0.897955i	$-0.354947\pi$
$941$	27.0000	0.880175	0.440087	+	0.897955i	$0.354947\pi$
$942$	0	0
$943$	18.0000i	0.586161i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	45.0000i	1.46230i	0.682215	+	0.731152i	$0.261017\pi$
$947$	45.0000i	1.46230i	−0.682215	+	0.731152i	$0.738983\pi$
$948$	0	0
$949$	2.00000	0.0649227
$950$	0	0
$951$	−9.00000	−0.291845
$952$	0	0
$953$	− 18.0000i	− 0.583077i	−0.956559	−	0.291539i	$-0.905833\pi$
$953$	− 18.0000i	− 0.583077i	0.956559	−	0.291539i	$-0.0941672\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 18.0000i	− 0.581857i
$958$	0	0
$959$	0	0
$960$	0	0
$961$	18.0000	0.580645
$962$	0	0
$963$	− 30.0000i	− 0.966736i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	16.0000i	0.514525i	0.966342	+	0.257263i	$0.0828206\pi$
$967$	16.0000i	0.514525i	−0.966342	+	0.257263i	$0.917179\pi$
$968$	0	0
$969$	3.00000	0.0963739
$970$	0	0
$971$	51.0000	1.63667	0.818334	−	0.574743i	$-0.194898\pi$
$971$	51.0000	1.63667	0.818334	+	0.574743i	$0.194898\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 27.0000i	− 0.863807i	−0.901920	−	0.431903i	$-0.857842\pi$
$977$	− 27.0000i	− 0.863807i	0.901920	−	0.431903i	$-0.142158\pi$
$978$	0	0
$979$	45.0000	1.43821
$980$	0	0
$981$	2.00000	0.0638551
$982$	0	0
$983$	33.0000i	1.05254i	0.850319	+	0.526268i	$0.176409\pi$
$983$	33.0000i	1.05254i	−0.850319	+	0.526268i	$0.823591\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	12.0000	0.381578
$990$	0	0
$991$	−19.0000	−0.603555	−0.301777	−	0.953378i	$-0.597580\pi$
$991$	−19.0000	−0.603555	−0.301777	+	0.953378i	$0.597580\pi$
$992$	0	0
$993$	− 13.0000i	− 0.412543i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 59.0000i	− 1.86855i	−0.356555	−	0.934274i	$-0.616049\pi$
$997$	− 59.0000i	− 1.86855i	0.356555	−	0.934274i	$-0.383951\pi$
$998$	0	0
$999$	−5.00000	−0.158193

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4900.2.e.i.2549.2		2
5.2	odd	4		196.2.a.b.1.1		1
5.3	odd	4		4900.2.a.g.1.1		1
5.4	even	2	inner	4900.2.e.i.2549.1		2
7.2	even	3		700.2.r.b.249.1		4
7.4	even	3		700.2.r.b.149.2		4
7.6	odd	2		4900.2.e.h.2549.1		2
15.2	even	4		1764.2.a.a.1.1		1
20.7	even	4		784.2.a.d.1.1		1
35.2	odd	12		28.2.e.a.25.1	yes	2
35.4	even	6		700.2.r.b.149.1		4
35.9	even	6		700.2.r.b.249.2		4
35.12	even	12		196.2.e.a.165.1		2
35.13	even	4		4900.2.a.n.1.1		1
35.17	even	12		196.2.e.a.177.1		2
35.18	odd	12		700.2.i.c.401.1		2
35.23	odd	12		700.2.i.c.501.1		2
35.27	even	4		196.2.a.a.1.1		1
35.32	odd	12		28.2.e.a.9.1	✓	2
35.34	odd	2		4900.2.e.h.2549.2		2
40.27	even	4		3136.2.a.s.1.1		1
40.37	odd	4		3136.2.a.h.1.1		1
60.47	odd	4		7056.2.a.f.1.1		1
105.2	even	12		252.2.k.c.109.1		2
105.17	odd	12		1764.2.k.b.1549.1		2
105.32	even	12		252.2.k.c.37.1		2
105.47	odd	12		1764.2.k.b.361.1		2
105.62	odd	4		1764.2.a.j.1.1		1
140.27	odd	4		784.2.a.g.1.1		1
140.47	odd	12		784.2.i.d.753.1		2
140.67	even	12		112.2.i.b.65.1		2
140.87	odd	12		784.2.i.d.177.1		2
140.107	even	12		112.2.i.b.81.1		2
280.27	odd	4		3136.2.a.k.1.1		1
280.37	odd	12		448.2.i.e.193.1		2
280.67	even	12		448.2.i.c.65.1		2
280.107	even	12		448.2.i.c.193.1		2
280.237	even	4		3136.2.a.v.1.1		1
280.277	odd	12		448.2.i.e.65.1		2
315.2	even	12		2268.2.l.a.109.1		2
315.32	even	12		2268.2.l.a.541.1		2
315.67	odd	12		2268.2.l.h.541.1		2
315.137	even	12		2268.2.i.h.2053.1		2
315.142	odd	12		2268.2.l.h.109.1		2
315.212	even	12		2268.2.i.h.865.1		2
315.247	odd	12		2268.2.i.a.865.1		2
315.277	odd	12		2268.2.i.a.2053.1		2
420.107	odd	12		1008.2.s.p.865.1		2
420.167	even	4		7056.2.a.bw.1.1		1
420.347	odd	12		1008.2.s.p.289.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
28.2.e.a.9.1	✓	2	35.32	odd	12
28.2.e.a.25.1	yes	2	35.2	odd	12
112.2.i.b.65.1		2	140.67	even	12
112.2.i.b.81.1		2	140.107	even	12
196.2.a.a.1.1		1	35.27	even	4
196.2.a.b.1.1		1	5.2	odd	4
196.2.e.a.165.1		2	35.12	even	12
196.2.e.a.177.1		2	35.17	even	12
252.2.k.c.37.1		2	105.32	even	12
252.2.k.c.109.1		2	105.2	even	12
448.2.i.c.65.1		2	280.67	even	12
448.2.i.c.193.1		2	280.107	even	12
448.2.i.e.65.1		2	280.277	odd	12
448.2.i.e.193.1		2	280.37	odd	12
700.2.i.c.401.1		2	35.18	odd	12
700.2.i.c.501.1		2	35.23	odd	12
700.2.r.b.149.1		4	35.4	even	6
700.2.r.b.149.2		4	7.4	even	3
700.2.r.b.249.1		4	7.2	even	3
700.2.r.b.249.2		4	35.9	even	6
784.2.a.d.1.1		1	20.7	even	4
784.2.a.g.1.1		1	140.27	odd	4
784.2.i.d.177.1		2	140.87	odd	12
784.2.i.d.753.1		2	140.47	odd	12
1008.2.s.p.289.1		2	420.347	odd	12
1008.2.s.p.865.1		2	420.107	odd	12
1764.2.a.a.1.1		1	15.2	even	4
1764.2.a.j.1.1		1	105.62	odd	4
1764.2.k.b.361.1		2	105.47	odd	12
1764.2.k.b.1549.1		2	105.17	odd	12
2268.2.i.a.865.1		2	315.247	odd	12
2268.2.i.a.2053.1		2	315.277	odd	12
2268.2.i.h.865.1		2	315.212	even	12
2268.2.i.h.2053.1		2	315.137	even	12
2268.2.l.a.109.1		2	315.2	even	12
2268.2.l.a.541.1		2	315.32	even	12
2268.2.l.h.109.1		2	315.142	odd	12
2268.2.l.h.541.1		2	315.67	odd	12
3136.2.a.h.1.1		1	40.37	odd	4
3136.2.a.k.1.1		1	280.27	odd	4
3136.2.a.s.1.1		1	40.27	even	4
3136.2.a.v.1.1		1	280.237	even	4
4900.2.a.g.1.1		1	5.3	odd	4
4900.2.a.n.1.1		1	35.13	even	4
4900.2.e.h.2549.1		2	7.6	odd	2
4900.2.e.h.2549.2		2	35.34	odd	2
4900.2.e.i.2549.1		2	5.4	even	2	inner
4900.2.e.i.2549.2		2	1.1	even	1	trivial
7056.2.a.f.1.1		1	60.47	odd	4
7056.2.a.bw.1.1		1	420.167	even	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:	\( N \)	\(=\)	\( 4900 = 2^{2} \cdot 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4900.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} +2.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{3} +2.00000 q^{9} -3.00000 q^{11} +2.00000i q^{13} -3.00000i q^{17} +1.00000 q^{19} +3.00000i q^{23} +5.00000i q^{27} +6.00000 q^{29} -7.00000 q^{31} -3.00000i q^{33} +1.00000i q^{37} -2.00000 q^{39} +6.00000 q^{41} -4.00000i q^{43} +9.00000i q^{47} +3.00000 q^{51} +3.00000i q^{53} +1.00000i q^{57} -9.00000 q^{59} -1.00000 q^{61} +7.00000i q^{67} -3.00000 q^{69} -1.00000i q^{73} +13.0000 q^{79} +1.00000 q^{81} +12.0000i q^{83} +6.00000i q^{87} -15.0000 q^{89} -7.00000i q^{93} +10.0000i q^{97} -6.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{9}+O(q^{10}) \) 2 * q + 4 * q^9	\( 2 q + 4 q^{9} - 6 q^{11} + 2 q^{19} + 12 q^{29} - 14 q^{31} - 4 q^{39} + 12 q^{41} + 6 q^{51} - 18 q^{59} - 2 q^{61} - 6 q^{69} + 26 q^{79} + 2 q^{81} - 30 q^{89} - 12 q^{99}+O(q^{100}) \) 2 * q + 4 * q^9 - 6 * q^11 + 2 * q^19 + 12 * q^29 - 14 * q^31 - 4 * q^39 + 12 * q^41 + 6 * q^51 - 18 * q^59 - 2 * q^61 - 6 * q^69 + 26 * q^79 + 2 * q^81 - 30 * q^89 - 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more