LMFDB - Embedded newform 2900.2.c.b.349.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2900,2,Mod(349,2900)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2900.349"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2900, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2900 = 2^{2} \cdot 5^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2900.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,0,0,0,0,-2,0,-12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$23.1566165862$
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:	$ 2 $
Twist minimal:	no (minimal twist has level 116)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			349.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	2900.349
Dual form			2900.2.c.b.349.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000i q^{3} +4.00000i q^{7} -1.00000 q^{9} -6.00000 q^{11} -2.00000i q^{13} +2.00000i q^{17} +6.00000 q^{19} +8.00000 q^{21} -4.00000i q^{23} -4.00000i q^{27} +1.00000 q^{29} -6.00000 q^{31} +12.0000i q^{33} +2.00000i q^{37} -4.00000 q^{39} +2.00000 q^{41} -10.0000i q^{43} -2.00000i q^{47} -9.00000 q^{49} +4.00000 q^{51} -10.0000i q^{53} -12.0000i q^{57} +10.0000 q^{61} -4.00000i q^{63} -12.0000i q^{67} -8.00000 q^{69} +8.00000 q^{71} -10.0000i q^{73} -24.0000i q^{77} +6.00000 q^{79} -11.0000 q^{81} -16.0000i q^{83} -2.00000i q^{87} -2.00000 q^{89} +8.00000 q^{91} +12.0000i q^{93} +10.0000i q^{97} +6.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{9} - 12 q^{11} + 12 q^{19} + 16 q^{21} + 2 q^{29} - 12 q^{31} - 8 q^{39} + 4 q^{41} - 18 q^{49} + 8 q^{51} + 20 q^{61} - 16 q^{69} + 16 q^{71} + 12 q^{79} - 22 q^{81} - 4 q^{89} + 16 q^{91} + 12 q^{99}+O(q^{100}) $ 2 * q - 2 * q^9 - 12 * q^11 + 12 * q^19 + 16 * q^21 + 2 * q^29 - 12 * q^31 - 8 * q^39 + 4 * q^41 - 18 * q^49 + 8 * q^51 + 20 * q^61 - 16 * q^69 + 16 * q^71 + 12 * q^79 - 22 * q^81 - 4 * q^89 + 16 * q^91 + 12 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2900\mathbb{Z}\right)^\times$.

$n$	$901$	$1277$	$1451$
$\chi(n)$	$1$	$-1$	$1$

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
$3$	− 2.00000i	− 1.15470i	−0.816497	−	0.577350i	$-0.804087\pi$
$3$	− 2.00000i	− 1.15470i	0.816497	−	0.577350i	$-0.195913\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	4.00000i	1.51186i	0.654654	+	0.755929i	$0.272814\pi$
$7$	4.00000i	1.51186i	−0.654654	+	0.755929i	$0.727186\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	−6.00000	−1.80907	−0.904534	−	0.426401i	$-0.859781\pi$
$11$	−6.00000	−1.80907	−0.904534	+	0.426401i	$0.859781\pi$
$12$	0	0
$13$	− 2.00000i	− 0.554700i	−0.960769	−	0.277350i	$-0.910544\pi$
$13$	− 2.00000i	− 0.554700i	0.960769	−	0.277350i	$-0.0894562\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	2.00000i	0.485071i	0.970143	+	0.242536i	$0.0779791\pi$
$17$	2.00000i	0.485071i	−0.970143	+	0.242536i	$0.922021\pi$
$18$	0	0
$19$	6.00000	1.37649	0.688247	−	0.725476i	$-0.258380\pi$
$19$	6.00000	1.37649	0.688247	+	0.725476i	$0.258380\pi$
$20$	0	0
$21$	8.00000	1.74574
$22$	0	0
$23$	− 4.00000i	− 0.834058i	−0.908893	−	0.417029i	$-0.863071\pi$
$23$	− 4.00000i	− 0.834058i	0.908893	−	0.417029i	$-0.136929\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 4.00000i	− 0.769800i
$28$	0	0
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	−6.00000	−1.07763	−0.538816	−	0.842424i	$-0.681128\pi$
$31$	−6.00000	−1.07763	−0.538816	+	0.842424i	$0.681128\pi$
$32$	0	0
$33$	12.0000i	2.08893i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	2.00000i	0.328798i	0.986394	+	0.164399i	$0.0525685\pi$
$37$	2.00000i	0.328798i	−0.986394	+	0.164399i	$0.947432\pi$
$38$	0	0
$39$	−4.00000	−0.640513
$40$	0	0
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	0	0
$43$	− 10.0000i	− 1.52499i	−0.646997	−	0.762493i	$-0.723975\pi$
$43$	− 10.0000i	− 1.52499i	0.646997	−	0.762493i	$-0.276025\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 2.00000i	− 0.291730i	−0.989305	−	0.145865i	$-0.953403\pi$
$47$	− 2.00000i	− 0.291730i	0.989305	−	0.145865i	$-0.0465965\pi$
$48$	0	0
$49$	−9.00000	−1.28571
$50$	0	0
$51$	4.00000	0.560112
$52$	0	0
$53$	− 10.0000i	− 1.37361i	−0.726844	−	0.686803i	$-0.759014\pi$
$53$	− 10.0000i	− 1.37361i	0.726844	−	0.686803i	$-0.240986\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 12.0000i	− 1.58944i
$58$	0	0
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	10.0000	1.28037	0.640184	−	0.768221i	$-0.278858\pi$
$61$	10.0000	1.28037	0.640184	+	0.768221i	$0.278858\pi$
$62$	0	0
$63$	− 4.00000i	− 0.503953i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 12.0000i	− 1.46603i	−0.680211	−	0.733017i	$-0.738112\pi$
$67$	− 12.0000i	− 1.46603i	0.680211	−	0.733017i	$-0.261888\pi$
$68$	0	0
$69$	−8.00000	−0.963087
$70$	0	0
$71$	8.00000	0.949425	0.474713	−	0.880141i	$-0.342552\pi$
$71$	8.00000	0.949425	0.474713	+	0.880141i	$0.342552\pi$
$72$	0	0
$73$	− 10.0000i	− 1.17041i	−0.810885	−	0.585206i	$-0.801014\pi$
$73$	− 10.0000i	− 1.17041i	0.810885	−	0.585206i	$-0.198986\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 24.0000i	− 2.73505i
$78$	0	0
$79$	6.00000	0.675053	0.337526	−	0.941316i	$-0.390410\pi$
$79$	6.00000	0.675053	0.337526	+	0.941316i	$0.390410\pi$
$80$	0	0
$81$	−11.0000	−1.22222
$82$	0	0
$83$	− 16.0000i	− 1.75623i	−0.478451	−	0.878114i	$-0.658802\pi$
$83$	− 16.0000i	− 1.75623i	0.478451	−	0.878114i	$-0.341198\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 2.00000i	− 0.214423i
$88$	0	0
$89$	−2.00000	−0.212000	−0.106000	−	0.994366i	$-0.533804\pi$
$89$	−2.00000	−0.212000	−0.106000	+	0.994366i	$0.533804\pi$
$90$	0	0
$91$	8.00000	0.838628
$92$	0	0
$93$	12.0000i	1.24434i
$94$	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
$101$	−14.0000	−1.39305	−0.696526	−	0.717532i	$-0.745272\pi$
$101$	−14.0000	−1.39305	−0.696526	+	0.717532i	$0.745272\pi$
$102$	0	0
$103$	4.00000i	0.394132i	0.980390	+	0.197066i	$0.0631413\pi$
$103$	4.00000i	0.394132i	−0.980390	+	0.197066i	$0.936859\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 8.00000i	− 0.773389i	−0.922208	−	0.386695i	$-0.873617\pi$
$107$	− 8.00000i	− 0.773389i	0.922208	−	0.386695i	$-0.126383\pi$
$108$	0	0
$109$	10.0000	0.957826	0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000	0.957826	0.478913	+	0.877862i	$0.341031\pi$
$110$	0	0
$111$	4.00000	0.379663
$112$	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
$117$	2.00000i	0.184900i
$118$	0	0
$119$	−8.00000	−0.733359
$120$	0	0
$121$	25.0000	2.27273
$122$	0	0
$123$	− 4.00000i	− 0.360668i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	2.00000i	0.177471i	0.996055	+	0.0887357i	$0.0282826\pi$
$127$	2.00000i	0.177471i	−0.996055	+	0.0887357i	$0.971717\pi$
$128$	0	0
$129$	−20.0000	−1.76090
$130$	0	0
$131$	−18.0000	−1.57267	−0.786334	−	0.617802i	$-0.788023\pi$
$131$	−18.0000	−1.57267	−0.786334	+	0.617802i	$0.788023\pi$
$132$	0	0
$133$	24.0000i	2.08106i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 6.00000i	− 0.512615i	−0.966595	−	0.256307i	$-0.917494\pi$
$137$	− 6.00000i	− 0.512615i	0.966595	−	0.256307i	$-0.0825059\pi$
$138$	0	0
$139$	8.00000	0.678551	0.339276	−	0.940687i	$-0.389818\pi$
$139$	8.00000	0.678551	0.339276	+	0.940687i	$0.389818\pi$
$140$	0	0
$141$	−4.00000	−0.336861
$142$	0	0
$143$	12.0000i	1.00349i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	18.0000i	1.48461i
$148$	0	0
$149$	6.00000	0.491539	0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000	0.491539	0.245770	+	0.969328i	$0.420959\pi$
$150$	0	0
$151$	−16.0000	−1.30206	−0.651031	−	0.759051i	$-0.725663\pi$
$151$	−16.0000	−1.30206	−0.651031	+	0.759051i	$0.725663\pi$
$152$	0	0
$153$	− 2.00000i	− 0.161690i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 6.00000i	− 0.478852i	−0.970915	−	0.239426i	$-0.923041\pi$
$157$	− 6.00000i	− 0.478852i	0.970915	−	0.239426i	$-0.0769593\pi$
$158$	0	0
$159$	−20.0000	−1.58610
$160$	0	0
$161$	16.0000	1.26098
$162$	0	0
$163$	18.0000i	1.40987i	0.709273	+	0.704934i	$0.249024\pi$
$163$	18.0000i	1.40987i	−0.709273	+	0.704934i	$0.750976\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 8.00000i	− 0.619059i	−0.950890	−	0.309529i	$-0.899829\pi$
$167$	− 8.00000i	− 0.619059i	0.950890	−	0.309529i	$-0.100171\pi$
$168$	0	0
$169$	9.00000	0.692308
$170$	0	0
$171$	−6.00000	−0.458831
$172$	0	0
$173$	− 18.0000i	− 1.36851i	−0.729241	−	0.684257i	$-0.760127\pi$
$173$	− 18.0000i	− 1.36851i	0.729241	−	0.684257i	$-0.239873\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−20.0000	−1.49487	−0.747435	−	0.664335i	$-0.768715\pi$
$179$	−20.0000	−1.49487	−0.747435	+	0.664335i	$0.768715\pi$
$180$	0	0
$181$	22.0000	1.63525	0.817624	−	0.575753i	$-0.195291\pi$
$181$	22.0000	1.63525	0.817624	+	0.575753i	$0.195291\pi$
$182$	0	0
$183$	− 20.0000i	− 1.47844i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 12.0000i	− 0.877527i
$188$	0	0
$189$	16.0000	1.16383
$190$	0	0
$191$	10.0000	0.723575	0.361787	−	0.932261i	$-0.382167\pi$
$191$	10.0000	0.723575	0.361787	+	0.932261i	$0.382167\pi$
$192$	0	0
$193$	− 2.00000i	− 0.143963i	−0.997406	−	0.0719816i	$-0.977068\pi$
$193$	− 2.00000i	− 0.143963i	0.997406	−	0.0719816i	$-0.0229323\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	6.00000i	0.427482i	0.976890	+	0.213741i	$0.0685649\pi$
$197$	6.00000i	0.427482i	−0.976890	+	0.213741i	$0.931435\pi$
$198$	0	0
$199$	−20.0000	−1.41776	−0.708881	−	0.705328i	$-0.750800\pi$
$199$	−20.0000	−1.41776	−0.708881	+	0.705328i	$0.750800\pi$
$200$	0	0
$201$	−24.0000	−1.69283
$202$	0	0
$203$	4.00000i	0.280745i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	4.00000i	0.278019i
$208$	0	0
$209$	−36.0000	−2.49017
$210$	0	0
$211$	−18.0000	−1.23917	−0.619586	−	0.784929i	$-0.712699\pi$
$211$	−18.0000	−1.23917	−0.619586	+	0.784929i	$0.712699\pi$
$212$	0	0
$213$	− 16.0000i	− 1.09630i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 24.0000i	− 1.62923i
$218$	0	0
$219$	−20.0000	−1.35147
$220$	0	0
$221$	4.00000	0.269069
$222$	0	0
$223$	− 4.00000i	− 0.267860i	−0.990991	−	0.133930i	$-0.957240\pi$
$223$	− 4.00000i	− 0.267860i	0.990991	−	0.133930i	$-0.0427597\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	0	0	1.00000			$0$
$227$	0	0	−1.00000			$\pi$
$228$	0	0
$229$	−10.0000	−0.660819	−0.330409	−	0.943838i	$-0.607187\pi$
$229$	−10.0000	−0.660819	−0.330409	+	0.943838i	$0.607187\pi$
$230$	0	0
$231$	−48.0000	−3.15817
$232$	0	0
$233$	22.0000i	1.44127i	0.693316	+	0.720634i	$0.256149\pi$
$233$	22.0000i	1.44127i	−0.693316	+	0.720634i	$0.743851\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 12.0000i	− 0.779484i
$238$	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
$241$	2.00000	0.128831	0.0644157	−	0.997923i	$-0.479482\pi$
$241$	2.00000	0.128831	0.0644157	+	0.997923i	$0.479482\pi$
$242$	0	0
$243$	10.0000i	0.641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 12.0000i	− 0.763542i
$248$	0	0
$249$	−32.0000	−2.02792
$250$	0	0
$251$	−6.00000	−0.378717	−0.189358	−	0.981908i	$-0.560641\pi$
$251$	−6.00000	−0.378717	−0.189358	+	0.981908i	$0.560641\pi$
$252$	0	0
$253$	24.0000i	1.50887i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 2.00000i	− 0.124757i	−0.998053	−	0.0623783i	$-0.980131\pi$
$257$	− 2.00000i	− 0.124757i	0.998053	−	0.0623783i	$-0.0198685\pi$
$258$	0	0
$259$	−8.00000	−0.497096
$260$	0	0
$261$	−1.00000	−0.0618984
$262$	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
$266$	0	0
$267$	4.00000i	0.244796i
$268$	0	0
$269$	6.00000	0.365826	0.182913	−	0.983129i	$-0.441447\pi$
$269$	6.00000	0.365826	0.182913	+	0.983129i	$0.441447\pi$
$270$	0	0
$271$	−10.0000	−0.607457	−0.303728	−	0.952759i	$-0.598232\pi$
$271$	−10.0000	−0.607457	−0.303728	+	0.952759i	$0.598232\pi$
$272$	0	0
$273$	− 16.0000i	− 0.968364i
$274$	0	0
$275$	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	$0.902873\pi$
$278$	0	0
$279$	6.00000	0.359211
$280$	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	$-0.0379328\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	8.00000i	0.472225i
$288$	0	0
$289$	13.0000	0.764706
$290$	0	0
$291$	20.0000	1.17242
$292$	0	0
$293$	− 2.00000i	− 0.116841i	−0.998292	−	0.0584206i	$-0.981394\pi$
$293$	− 2.00000i	− 0.116841i	0.998292	−	0.0584206i	$-0.0186065\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	24.0000i	1.39262i
$298$	0	0
$299$	−8.00000	−0.462652
$300$	0	0
$301$	40.0000	2.30556
$302$	0	0
$303$	28.0000i	1.60856i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	2.00000i	0.114146i	0.998370	+	0.0570730i	$0.0181768\pi$
$307$	2.00000i	0.114146i	−0.998370	+	0.0570730i	$0.981823\pi$
$308$	0	0
$309$	8.00000	0.455104
$310$	0	0
$311$	10.0000	0.567048	0.283524	−	0.958965i	$-0.408496\pi$
$311$	10.0000	0.567048	0.283524	+	0.958965i	$0.408496\pi$
$312$	0	0
$313$	− 6.00000i	− 0.339140i	−0.985518	−	0.169570i	$-0.945762\pi$
$313$	− 6.00000i	− 0.339140i	0.985518	−	0.169570i	$-0.0542379\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	18.0000i	1.01098i	0.862832	+	0.505490i	$0.168688\pi$
$317$	18.0000i	1.01098i	−0.862832	+	0.505490i	$0.831312\pi$
$318$	0	0
$319$	−6.00000	−0.335936
$320$	0	0
$321$	−16.0000	−0.893033
$322$	0	0
$323$	12.0000i	0.667698i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 20.0000i	− 1.10600i
$328$	0	0
$329$	8.00000	0.441054
$330$	0	0
$331$	−2.00000	−0.109930	−0.0549650	−	0.998488i	$-0.517505\pi$
$331$	−2.00000	−0.109930	−0.0549650	+	0.998488i	$0.517505\pi$
$332$	0	0
$333$	− 2.00000i	− 0.109599i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 14.0000i	− 0.762629i	−0.924445	−	0.381314i	$-0.875472\pi$
$337$	− 14.0000i	− 0.762629i	0.924445	−	0.381314i	$-0.124528\pi$
$338$	0	0
$339$	12.0000	0.651751
$340$	0	0
$341$	36.0000	1.94951
$342$	0	0
$343$	− 8.00000i	− 0.431959i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	4.00000i	0.214731i	0.994220	+	0.107366i	$0.0342415\pi$
$347$	4.00000i	0.214731i	−0.994220	+	0.107366i	$0.965758\pi$
$348$	0	0
$349$	30.0000	1.60586	0.802932	−	0.596071i	$-0.203272\pi$
$349$	30.0000	1.60586	0.802932	+	0.596071i	$0.203272\pi$
$350$	0	0
$351$	−8.00000	−0.427008
$352$	0	0
$353$	− 22.0000i	− 1.17094i	−0.810693	−	0.585471i	$-0.800910\pi$
$353$	− 22.0000i	− 1.17094i	0.810693	−	0.585471i	$-0.199090\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	16.0000i	0.846810i
$358$	0	0
$359$	22.0000	1.16112	0.580558	−	0.814219i	$-0.302835\pi$
$359$	22.0000	1.16112	0.580558	+	0.814219i	$0.302835\pi$
$360$	0	0
$361$	17.0000	0.894737
$362$	0	0
$363$	− 50.0000i	− 2.62432i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	10.0000i	0.521996i	0.965339	+	0.260998i	$0.0840516\pi$
$367$	10.0000i	0.521996i	−0.965339	+	0.260998i	$0.915948\pi$
$368$	0	0
$369$	−2.00000	−0.104116
$370$	0	0
$371$	40.0000	2.07670
$372$	0	0
$373$	− 6.00000i	− 0.310668i	−0.987862	−	0.155334i	$-0.950355\pi$
$373$	− 6.00000i	− 0.310668i	0.987862	−	0.155334i	$-0.0496454\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 2.00000i	− 0.103005i
$378$	0	0
$379$	−2.00000	−0.102733	−0.0513665	−	0.998680i	$-0.516358\pi$
$379$	−2.00000	−0.102733	−0.0513665	+	0.998680i	$0.516358\pi$
$380$	0	0
$381$	4.00000	0.204926
$382$	0	0
$383$	16.0000i	0.817562i	0.912633	+	0.408781i	$0.134046\pi$
$383$	16.0000i	0.817562i	−0.912633	+	0.408781i	$0.865954\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	10.0000i	0.508329i
$388$	0	0
$389$	6.00000	0.304212	0.152106	−	0.988364i	$-0.451394\pi$
$389$	6.00000	0.304212	0.152106	+	0.988364i	$0.451394\pi$
$390$	0	0
$391$	8.00000	0.404577
$392$	0	0
$393$	36.0000i	1.81596i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 2.00000i	− 0.100377i	−0.998740	−	0.0501886i	$-0.984018\pi$
$397$	− 2.00000i	− 0.100377i	0.998740	−	0.0501886i	$-0.0159822\pi$
$398$	0	0
$399$	48.0000	2.40301
$400$	0	0
$401$	−30.0000	−1.49813	−0.749064	−	0.662497i	$-0.769497\pi$
$401$	−30.0000	−1.49813	−0.749064	+	0.662497i	$0.769497\pi$
$402$	0	0
$403$	12.0000i	0.597763i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 12.0000i	− 0.594818i
$408$	0	0
$409$	6.00000	0.296681	0.148340	−	0.988936i	$-0.452607\pi$
$409$	6.00000	0.296681	0.148340	+	0.988936i	$0.452607\pi$
$410$	0	0
$411$	−12.0000	−0.591916
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 16.0000i	− 0.783523i
$418$	0	0
$419$	−20.0000	−0.977064	−0.488532	−	0.872546i	$-0.662467\pi$
$419$	−20.0000	−0.977064	−0.488532	+	0.872546i	$0.662467\pi$
$420$	0	0
$421$	2.00000	0.0974740	0.0487370	−	0.998812i	$-0.484480\pi$
$421$	2.00000	0.0974740	0.0487370	+	0.998812i	$0.484480\pi$
$422$	0	0
$423$	2.00000i	0.0972433i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	40.0000i	1.93574i
$428$	0	0
$429$	24.0000	1.15873
$430$	0	0
$431$	−4.00000	−0.192673	−0.0963366	−	0.995349i	$-0.530713\pi$
$431$	−4.00000	−0.192673	−0.0963366	+	0.995349i	$0.530713\pi$
$432$	0	0
$433$	− 2.00000i	− 0.0961139i	−0.998845	−	0.0480569i	$-0.984697\pi$
$433$	− 2.00000i	− 0.0961139i	0.998845	−	0.0480569i	$-0.0153029\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 24.0000i	− 1.14808i
$438$	0	0
$439$	−32.0000	−1.52728	−0.763638	−	0.645644i	$-0.776589\pi$
$439$	−32.0000	−1.52728	−0.763638	+	0.645644i	$0.776589\pi$
$440$	0	0
$441$	9.00000	0.428571
$442$	0	0
$443$	2.00000i	0.0950229i	0.998871	+	0.0475114i	$0.0151291\pi$
$443$	2.00000i	0.0950229i	−0.998871	+	0.0475114i	$0.984871\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 12.0000i	− 0.567581i
$448$	0	0
$449$	−2.00000	−0.0943858	−0.0471929	−	0.998886i	$-0.515028\pi$
$449$	−2.00000	−0.0943858	−0.0471929	+	0.998886i	$0.515028\pi$
$450$	0	0
$451$	−12.0000	−0.565058
$452$	0	0
$453$	32.0000i	1.50349i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 2.00000i	− 0.0935561i	−0.998905	−	0.0467780i	$-0.985105\pi$
$457$	− 2.00000i	− 0.0935561i	0.998905	−	0.0467780i	$-0.0148953\pi$
$458$	0	0
$459$	8.00000	0.373408
$460$	0	0
$461$	−22.0000	−1.02464	−0.512321	−	0.858794i	$-0.671214\pi$
$461$	−22.0000	−1.02464	−0.512321	+	0.858794i	$0.671214\pi$
$462$	0	0
$463$	− 40.0000i	− 1.85896i	−0.368875	−	0.929479i	$-0.620257\pi$
$463$	− 40.0000i	− 1.85896i	0.368875	−	0.929479i	$-0.379743\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	42.0000i	1.94353i	0.235954	+	0.971764i	$0.424178\pi$
$467$	42.0000i	1.94353i	−0.235954	+	0.971764i	$0.575822\pi$
$468$	0	0
$469$	48.0000	2.21643
$470$	0	0
$471$	−12.0000	−0.552931
$472$	0	0
$473$	60.0000i	2.75880i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	10.0000i	0.457869i
$478$	0	0
$479$	2.00000	0.0913823	0.0456912	−	0.998956i	$-0.485451\pi$
$479$	2.00000	0.0913823	0.0456912	+	0.998956i	$0.485451\pi$
$480$	0	0
$481$	4.00000	0.182384
$482$	0	0
$483$	− 32.0000i	− 1.45605i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 28.0000i	− 1.26880i	−0.773004	−	0.634401i	$-0.781247\pi$
$487$	− 28.0000i	− 1.26880i	0.773004	−	0.634401i	$-0.218753\pi$
$488$	0	0
$489$	36.0000	1.62798
$490$	0	0
$491$	−18.0000	−0.812329	−0.406164	−	0.913800i	$-0.633134\pi$
$491$	−18.0000	−0.812329	−0.406164	+	0.913800i	$0.633134\pi$
$492$	0	0
$493$	2.00000i	0.0900755i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	32.0000i	1.43540i
$498$	0	0
$499$	20.0000	0.895323	0.447661	−	0.894203i	$-0.352257\pi$
$499$	20.0000	0.895323	0.447661	+	0.894203i	$0.352257\pi$
$500$	0	0
$501$	−16.0000	−0.714827
$502$	0	0
$503$	2.00000i	0.0891756i	0.999005	+	0.0445878i	$0.0141974\pi$
$503$	2.00000i	0.0891756i	−0.999005	+	0.0445878i	$0.985803\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 18.0000i	− 0.799408i
$508$	0	0
$509$	30.0000	1.32973	0.664863	−	0.746965i	$-0.268490\pi$
$509$	30.0000	1.32973	0.664863	+	0.746965i	$0.268490\pi$
$510$	0	0
$511$	40.0000	1.76950
$512$	0	0
$513$	− 24.0000i	− 1.05963i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	12.0000i	0.527759i
$518$	0	0
$519$	−36.0000	−1.58022
$520$	0	0
$521$	14.0000	0.613351	0.306676	−	0.951814i	$-0.400783\pi$
$521$	14.0000	0.613351	0.306676	+	0.951814i	$0.400783\pi$
$522$	0	0
$523$	0	0	1.00000			$0$
$523$	0	0	−1.00000			$\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 12.0000i	− 0.522728i
$528$	0	0
$529$	7.00000	0.304348
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 4.00000i	− 0.173259i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	40.0000i	1.72613i
$538$	0	0
$539$	54.0000	2.32594
$540$	0	0
$541$	10.0000	0.429934	0.214967	−	0.976621i	$-0.431036\pi$
$541$	10.0000	0.429934	0.214967	+	0.976621i	$0.431036\pi$
$542$	0	0
$543$	− 44.0000i	− 1.88822i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	32.0000i	1.36822i	0.729378	+	0.684111i	$0.239809\pi$
$547$	32.0000i	1.36822i	−0.729378	+	0.684111i	$0.760191\pi$
$548$	0	0
$549$	−10.0000	−0.426790
$550$	0	0
$551$	6.00000	0.255609
$552$	0	0
$553$	24.0000i	1.02058i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	22.0000i	0.932170i	0.884740	+	0.466085i	$0.154336\pi$
$557$	22.0000i	0.932170i	−0.884740	+	0.466085i	$0.845664\pi$
$558$	0	0
$559$	−20.0000	−0.845910
$560$	0	0
$561$	−24.0000	−1.01328
$562$	0	0
$563$	− 22.0000i	− 0.927189i	−0.886047	−	0.463595i	$-0.846559\pi$
$563$	− 22.0000i	− 0.927189i	0.886047	−	0.463595i	$-0.153441\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 44.0000i	− 1.84783i
$568$	0	0
$569$	−42.0000	−1.76073	−0.880366	−	0.474295i	$-0.842703\pi$
$569$	−42.0000	−1.76073	−0.880366	+	0.474295i	$0.842703\pi$
$570$	0	0
$571$	−40.0000	−1.67395	−0.836974	−	0.547243i	$-0.815677\pi$
$571$	−40.0000	−1.67395	−0.836974	+	0.547243i	$0.815677\pi$
$572$	0	0
$573$	− 20.0000i	− 0.835512i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	26.0000i	1.08239i	0.840896	+	0.541197i	$0.182029\pi$
$577$	26.0000i	1.08239i	−0.840896	+	0.541197i	$0.817971\pi$
$578$	0	0
$579$	−4.00000	−0.166234
$580$	0	0
$581$	64.0000	2.65517
$582$	0	0
$583$	60.0000i	2.48495i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 8.00000i	− 0.330195i	−0.986277	−	0.165098i	$-0.947206\pi$
$587$	− 8.00000i	− 0.330195i	0.986277	−	0.165098i	$-0.0527939\pi$
$588$	0	0
$589$	−36.0000	−1.48335
$590$	0	0
$591$	12.0000	0.493614
$592$	0	0
$593$	− 6.00000i	− 0.246390i	−0.992382	−	0.123195i	$-0.960686\pi$
$593$	− 6.00000i	− 0.246390i	0.992382	−	0.123195i	$-0.0393141\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	40.0000i	1.63709i
$598$	0	0
$599$	38.0000	1.55264	0.776319	−	0.630340i	$-0.217085\pi$
$599$	38.0000	1.55264	0.776319	+	0.630340i	$0.217085\pi$
$600$	0	0
$601$	2.00000	0.0815817	0.0407909	−	0.999168i	$-0.487012\pi$
$601$	2.00000	0.0815817	0.0407909	+	0.999168i	$0.487012\pi$
$602$	0	0
$603$	12.0000i	0.488678i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 18.0000i	− 0.730597i	−0.930890	−	0.365299i	$-0.880967\pi$
$607$	− 18.0000i	− 0.730597i	0.930890	−	0.365299i	$-0.119033\pi$
$608$	0	0
$609$	8.00000	0.324176
$610$	0	0
$611$	−4.00000	−0.161823
$612$	0	0
$613$	− 14.0000i	− 0.565455i	−0.959200	−	0.282727i	$-0.908761\pi$
$613$	− 14.0000i	− 0.565455i	0.959200	−	0.282727i	$-0.0912392\pi$
$614$	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
$619$	26.0000	1.04503	0.522514	−	0.852631i	$-0.324994\pi$
$619$	26.0000	1.04503	0.522514	+	0.852631i	$0.324994\pi$
$620$	0	0
$621$	−16.0000	−0.642058
$622$	0	0
$623$	− 8.00000i	− 0.320513i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	72.0000i	2.87540i
$628$	0	0
$629$	−4.00000	−0.159490
$630$	0	0
$631$	16.0000	0.636950	0.318475	−	0.947931i	$-0.396829\pi$
$631$	16.0000	0.636950	0.318475	+	0.947931i	$0.396829\pi$
$632$	0	0
$633$	36.0000i	1.43087i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	18.0000i	0.713186i
$638$	0	0
$639$	−8.00000	−0.316475
$640$	0	0
$641$	−38.0000	−1.50091	−0.750455	−	0.660922i	$-0.770166\pi$
$641$	−38.0000	−1.50091	−0.750455	+	0.660922i	$0.770166\pi$
$642$	0	0
$643$	− 4.00000i	− 0.157745i	−0.996885	−	0.0788723i	$-0.974868\pi$
$643$	− 4.00000i	− 0.157745i	0.996885	−	0.0788723i	$-0.0251319\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 8.00000i	− 0.314512i	−0.987558	−	0.157256i	$-0.949735\pi$
$647$	− 8.00000i	− 0.314512i	0.987558	−	0.157256i	$-0.0502649\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	−48.0000	−1.88127
$652$	0	0
$653$	− 34.0000i	− 1.33052i	−0.746611	−	0.665261i	$-0.768320\pi$
$653$	− 34.0000i	− 1.33052i	0.746611	−	0.665261i	$-0.231680\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	10.0000i	0.390137i
$658$	0	0
$659$	30.0000	1.16863	0.584317	−	0.811525i	$-0.301362\pi$
$659$	30.0000	1.16863	0.584317	+	0.811525i	$0.301362\pi$
$660$	0	0
$661$	42.0000	1.63361	0.816805	−	0.576913i	$-0.195743\pi$
$661$	42.0000	1.63361	0.816805	+	0.576913i	$0.195743\pi$
$662$	0	0
$663$	− 8.00000i	− 0.310694i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 4.00000i	− 0.154881i
$668$	0	0
$669$	−8.00000	−0.309298
$670$	0	0
$671$	−60.0000	−2.31627
$672$	0	0
$673$	30.0000i	1.15642i	0.815890	+	0.578208i	$0.196248\pi$
$673$	30.0000i	1.15642i	−0.815890	+	0.578208i	$0.803752\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 46.0000i	− 1.76792i	−0.467559	−	0.883962i	$-0.654866\pi$
$677$	− 46.0000i	− 1.76792i	0.467559	−	0.883962i	$-0.345134\pi$
$678$	0	0
$679$	−40.0000	−1.53506
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 48.0000i	− 1.83667i	−0.395805	−	0.918334i	$-0.629534\pi$
$683$	− 48.0000i	− 1.83667i	0.395805	−	0.918334i	$-0.370466\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	20.0000i	0.763048i
$688$	0	0
$689$	−20.0000	−0.761939
$690$	0	0
$691$	32.0000	1.21734	0.608669	−	0.793424i	$-0.291704\pi$
$691$	32.0000	1.21734	0.608669	+	0.793424i	$0.291704\pi$
$692$	0	0
$693$	24.0000i	0.911685i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	4.00000i	0.151511i
$698$	0	0
$699$	44.0000	1.66423
$700$	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
$703$	12.0000i	0.452589i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 56.0000i	− 2.10610i
$708$	0	0
$709$	6.00000	0.225335	0.112667	−	0.993633i	$-0.464061\pi$
$709$	6.00000	0.225335	0.112667	+	0.993633i	$0.464061\pi$
$710$	0	0
$711$	−6.00000	−0.225018
$712$	0	0
$713$	24.0000i	0.898807i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 24.0000i	− 0.896296i
$718$	0	0
$719$	44.0000	1.64092	0.820462	−	0.571702i	$-0.193717\pi$
$719$	44.0000	1.64092	0.820462	+	0.571702i	$0.193717\pi$
$720$	0	0
$721$	−16.0000	−0.595871
$722$	0	0
$723$	− 4.00000i	− 0.148762i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	2.00000i	0.0741759i	0.999312	+	0.0370879i	$0.0118082\pi$
$727$	2.00000i	0.0741759i	−0.999312	+	0.0370879i	$0.988192\pi$
$728$	0	0
$729$	−13.0000	−0.481481
$730$	0	0
$731$	20.0000	0.739727
$732$	0	0
$733$	30.0000i	1.10808i	0.832492	+	0.554038i	$0.186914\pi$
$733$	30.0000i	1.10808i	−0.832492	+	0.554038i	$0.813086\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	72.0000i	2.65215i
$738$	0	0
$739$	14.0000	0.514998	0.257499	−	0.966279i	$-0.417102\pi$
$739$	14.0000	0.514998	0.257499	+	0.966279i	$0.417102\pi$
$740$	0	0
$741$	−24.0000	−0.881662
$742$	0	0
$743$	− 38.0000i	− 1.39408i	−0.717030	−	0.697042i	$-0.754499\pi$
$743$	− 38.0000i	− 1.39408i	0.717030	−	0.697042i	$-0.245501\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	16.0000i	0.585409i
$748$	0	0
$749$	32.0000	1.16925
$750$	0	0
$751$	14.0000	0.510867	0.255434	−	0.966827i	$-0.417782\pi$
$751$	14.0000	0.510867	0.255434	+	0.966827i	$0.417782\pi$
$752$	0	0
$753$	12.0000i	0.437304i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	26.0000i	0.944986i	0.881334	+	0.472493i	$0.156646\pi$
$757$	26.0000i	0.944986i	−0.881334	+	0.472493i	$0.843354\pi$
$758$	0	0
$759$	48.0000	1.74229
$760$	0	0
$761$	6.00000	0.217500	0.108750	−	0.994069i	$-0.465315\pi$
$761$	6.00000	0.217500	0.108750	+	0.994069i	$0.465315\pi$
$762$	0	0
$763$	40.0000i	1.44810i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	14.0000	0.504853	0.252426	−	0.967616i	$-0.418771\pi$
$769$	14.0000	0.504853	0.252426	+	0.967616i	$0.418771\pi$
$770$	0	0
$771$	−4.00000	−0.144056
$772$	0	0
$773$	− 26.0000i	− 0.935155i	−0.883952	−	0.467578i	$-0.845127\pi$
$773$	− 26.0000i	− 0.935155i	0.883952	−	0.467578i	$-0.154873\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	16.0000i	0.573997i
$778$	0	0
$779$	12.0000	0.429945
$780$	0	0
$781$	−48.0000	−1.71758
$782$	0	0
$783$	− 4.00000i	− 0.142948i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 4.00000i	− 0.142585i	−0.997455	−	0.0712923i	$-0.977288\pi$
$787$	− 4.00000i	− 0.142585i	0.997455	−	0.0712923i	$-0.0227123\pi$
$788$	0	0
$789$	−36.0000	−1.28163
$790$	0	0
$791$	−24.0000	−0.853342
$792$	0	0
$793$	− 20.0000i	− 0.710221i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	10.0000i	0.354218i	0.984191	+	0.177109i	$0.0566745\pi$
$797$	10.0000i	0.354218i	−0.984191	+	0.177109i	$0.943325\pi$
$798$	0	0
$799$	4.00000	0.141510
$800$	0	0
$801$	2.00000	0.0706665
$802$	0	0
$803$	60.0000i	2.11735i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 12.0000i	− 0.422420i
$808$	0	0
$809$	6.00000	0.210949	0.105474	−	0.994422i	$-0.466364\pi$
$809$	6.00000	0.210949	0.105474	+	0.994422i	$0.466364\pi$
$810$	0	0
$811$	36.0000	1.26413	0.632065	−	0.774915i	$-0.282207\pi$
$811$	36.0000	1.26413	0.632065	+	0.774915i	$0.282207\pi$
$812$	0	0
$813$	20.0000i	0.701431i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 60.0000i	− 2.09913i
$818$	0	0
$819$	−8.00000	−0.279543
$820$	0	0
$821$	42.0000	1.46581	0.732905	−	0.680331i	$-0.238164\pi$
$821$	42.0000	1.46581	0.732905	+	0.680331i	$0.238164\pi$
$822$	0	0
$823$	− 2.00000i	− 0.0697156i	−0.999392	−	0.0348578i	$-0.988902\pi$
$823$	− 2.00000i	− 0.0697156i	0.999392	−	0.0348578i	$-0.0110978\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	22.0000i	0.765015i	0.923952	+	0.382507i	$0.124939\pi$
$827$	22.0000i	0.765015i	−0.923952	+	0.382507i	$0.875061\pi$
$828$	0	0
$829$	−34.0000	−1.18087	−0.590434	−	0.807086i	$-0.701044\pi$
$829$	−34.0000	−1.18087	−0.590434	+	0.807086i	$0.701044\pi$
$830$	0	0
$831$	20.0000	0.693792
$832$	0	0
$833$	− 18.0000i	− 0.623663i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	24.0000i	0.829561i
$838$	0	0
$839$	−54.0000	−1.86429	−0.932144	−	0.362089i	$-0.882064\pi$
$839$	−54.0000	−1.86429	−0.932144	+	0.362089i	$0.882064\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	12.0000i	0.413302i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	100.000i	3.43604i
$848$	0	0
$849$	−8.00000	−0.274559
$850$	0	0
$851$	8.00000	0.274236
$852$	0	0
$853$	22.0000i	0.753266i	0.926363	+	0.376633i	$0.122918\pi$
$853$	22.0000i	0.753266i	−0.926363	+	0.376633i	$0.877082\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	54.0000i	1.84460i	0.386469	+	0.922302i	$0.373695\pi$
$857$	54.0000i	1.84460i	−0.386469	+	0.922302i	$0.626305\pi$
$858$	0	0
$859$	−14.0000	−0.477674	−0.238837	−	0.971060i	$-0.576766\pi$
$859$	−14.0000	−0.477674	−0.238837	+	0.971060i	$0.576766\pi$
$860$	0	0
$861$	16.0000	0.545279
$862$	0	0
$863$	16.0000i	0.544646i	0.962206	+	0.272323i	$0.0877920\pi$
$863$	16.0000i	0.544646i	−0.962206	+	0.272323i	$0.912208\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 26.0000i	− 0.883006i
$868$	0	0
$869$	−36.0000	−1.22122
$870$	0	0
$871$	−24.0000	−0.813209
$872$	0	0
$873$	− 10.0000i	− 0.338449i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 14.0000i	− 0.472746i	−0.971662	−	0.236373i	$-0.924041\pi$
$877$	− 14.0000i	− 0.472746i	0.971662	−	0.236373i	$-0.0759588\pi$
$878$	0	0
$879$	−4.00000	−0.134917
$880$	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
$887$	− 30.0000i	− 1.00730i	−0.863907	−	0.503651i	$-0.831990\pi$
$887$	− 30.0000i	− 1.00730i	0.863907	−	0.503651i	$-0.168010\pi$
$888$	0	0
$889$	−8.00000	−0.268311
$890$	0	0
$891$	66.0000	2.21108
$892$	0	0
$893$	− 12.0000i	− 0.401565i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	16.0000i	0.534224i
$898$	0	0
$899$	−6.00000	−0.200111
$900$	0	0
$901$	20.0000	0.666297
$902$	0	0
$903$	− 80.0000i	− 2.66223i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 26.0000i	− 0.863316i	−0.902037	−	0.431658i	$-0.857929\pi$
$907$	− 26.0000i	− 0.863316i	0.902037	−	0.431658i	$-0.142071\pi$
$908$	0	0
$909$	14.0000	0.464351
$910$	0	0
$911$	−34.0000	−1.12647	−0.563235	−	0.826297i	$-0.690443\pi$
$911$	−34.0000	−1.12647	−0.563235	+	0.826297i	$0.690443\pi$
$912$	0	0
$913$	96.0000i	3.17714i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 72.0000i	− 2.37765i
$918$	0	0
$919$	−12.0000	−0.395843	−0.197922	−	0.980218i	$-0.563419\pi$
$919$	−12.0000	−0.395843	−0.197922	+	0.980218i	$0.563419\pi$
$920$	0	0
$921$	4.00000	0.131804
$922$	0	0
$923$	− 16.0000i	− 0.526646i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 4.00000i	− 0.131377i
$928$	0	0
$929$	18.0000	0.590561	0.295280	−	0.955411i	$-0.404587\pi$
$929$	18.0000	0.590561	0.295280	+	0.955411i	$0.404587\pi$
$930$	0	0
$931$	−54.0000	−1.76978
$932$	0	0
$933$	− 20.0000i	− 0.654771i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 2.00000i	− 0.0653372i	−0.999466	−	0.0326686i	$-0.989599\pi$
$937$	− 2.00000i	− 0.0653372i	0.999466	−	0.0326686i	$-0.0104006\pi$
$938$	0	0
$939$	−12.0000	−0.391605
$940$	0	0
$941$	22.0000	0.717180	0.358590	−	0.933495i	$-0.383258\pi$
$941$	22.0000	0.717180	0.358590	+	0.933495i	$0.383258\pi$
$942$	0	0
$943$	− 8.00000i	− 0.260516i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	42.0000i	1.36482i	0.730971	+	0.682408i	$0.239067\pi$
$947$	42.0000i	1.36482i	−0.730971	+	0.682408i	$0.760933\pi$
$948$	0	0
$949$	−20.0000	−0.649227
$950$	0	0
$951$	36.0000	1.16738
$952$	0	0
$953$	22.0000i	0.712650i	0.934362	+	0.356325i	$0.115970\pi$
$953$	22.0000i	0.712650i	−0.934362	+	0.356325i	$0.884030\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	12.0000i	0.387905i
$958$	0	0
$959$	24.0000	0.775000
$960$	0	0
$961$	5.00000	0.161290
$962$	0	0
$963$	8.00000i	0.257796i
$964$	0	0
$965$	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	$0.884911\pi$
$968$	0	0
$969$	24.0000	0.770991
$970$	0	0
$971$	26.0000	0.834380	0.417190	−	0.908819i	$-0.363015\pi$
$971$	26.0000	0.834380	0.417190	+	0.908819i	$0.363015\pi$
$972$	0	0
$973$	32.0000i	1.02587i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 50.0000i	− 1.59964i	−0.600239	−	0.799821i	$-0.704928\pi$
$977$	− 50.0000i	− 1.59964i	0.600239	−	0.799821i	$-0.295072\pi$
$978$	0	0
$979$	12.0000	0.383522
$980$	0	0
$981$	−10.0000	−0.319275
$982$	0	0
$983$	50.0000i	1.59475i	0.603483	+	0.797376i	$0.293779\pi$
$983$	50.0000i	1.59475i	−0.603483	+	0.797376i	$0.706221\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 16.0000i	− 0.509286i
$988$	0	0
$989$	−40.0000	−1.27193
$990$	0	0
$991$	16.0000	0.508257	0.254128	−	0.967170i	$-0.418211\pi$
$991$	16.0000	0.508257	0.254128	+	0.967170i	$0.418211\pi$
$992$	0	0
$993$	4.00000i	0.126936i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	2.00000i	0.0633406i	0.999498	+	0.0316703i	$0.0100827\pi$
$997$	2.00000i	0.0633406i	−0.999498	+	0.0316703i	$0.989917\pi$
$998$	0	0
$999$	8.00000	0.253109

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2900.2.c.b.349.1		2
5.2	odd	4		2900.2.a.a.1.1		1
5.3	odd	4		116.2.a.c.1.1	✓	1
5.4	even	2	inner	2900.2.c.b.349.2		2
15.8	even	4		1044.2.a.i.1.1		1
20.3	even	4		464.2.a.a.1.1		1
35.13	even	4		5684.2.a.c.1.1		1
40.3	even	4		1856.2.a.n.1.1		1
40.13	odd	4		1856.2.a.c.1.1		1
60.23	odd	4		4176.2.a.x.1.1		1
145.28	odd	4		3364.2.a.a.1.1		1
145.128	even	4		3364.2.c.c.1681.2		2
145.133	even	4		3364.2.c.c.1681.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
116.2.a.c.1.1	✓	1	5.3	odd	4
464.2.a.a.1.1		1	20.3	even	4
1044.2.a.i.1.1		1	15.8	even	4
1856.2.a.c.1.1		1	40.13	odd	4
1856.2.a.n.1.1		1	40.3	even	4
2900.2.a.a.1.1		1	5.2	odd	4
2900.2.c.b.349.1		2	1.1	even	1	trivial
2900.2.c.b.349.2		2	5.4	even	2	inner
3364.2.a.a.1.1		1	145.28	odd	4
3364.2.c.c.1681.1		2	145.133	even	4
3364.2.c.c.1681.2		2	145.128	even	4
4176.2.a.x.1.1		1	60.23	odd	4
5684.2.a.c.1.1		1	35.13	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2900 = 2^{2} \cdot 5^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2900.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.00000i q^{3} +4.00000i q^{7} -1.00000 q^{9} -6.00000 q^{11} -2.00000i q^{13} +2.00000i q^{17} +6.00000 q^{19} +8.00000 q^{21} -4.00000i q^{23} -4.00000i q^{27} +1.00000 q^{29} -6.00000 q^{31} +12.0000i q^{33} +2.00000i q^{37} -4.00000 q^{39} +2.00000 q^{41} -10.0000i q^{43} -2.00000i q^{47} -9.00000 q^{49} +4.00000 q^{51} -10.0000i q^{53} -12.0000i q^{57} +10.0000 q^{61} -4.00000i q^{63} -12.0000i q^{67} -8.00000 q^{69} +8.00000 q^{71} -10.0000i q^{73} -24.0000i q^{77} +6.00000 q^{79} -11.0000 q^{81} -16.0000i q^{83} -2.00000i q^{87} -2.00000 q^{89} +8.00000 q^{91} +12.0000i q^{93} +10.0000i q^{97} +6.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{9} - 12 q^{11} + 12 q^{19} + 16 q^{21} + 2 q^{29} - 12 q^{31} - 8 q^{39} + 4 q^{41} - 18 q^{49} + 8 q^{51} + 20 q^{61} - 16 q^{69} + 16 q^{71} + 12 q^{79} - 22 q^{81} - 4 q^{89} + 16 q^{91} + 12 q^{99}+O(q^{100}) \) 2 * q - 2 * q^9 - 12 * q^11 + 12 * q^19 + 16 * q^21 + 2 * q^29 - 12 * q^31 - 8 * q^39 + 4 * q^41 - 18 * q^49 + 8 * q^51 + 20 * q^61 - 16 * q^69 + 16 * q^71 + 12 * q^79 - 22 * q^81 - 4 * q^89 + 16 * q^91 + 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more