LMFDB - Embedded newform 3200.2.c.x.2049.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3200,2,Mod(2049,3200)] 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("3200.2049"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3200, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 3200 = 2^{7} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3200.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,0,0,-12,0,8,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(29)] == traces)

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

Self dual:	no
Analytic conductor:	$25.5521286468$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{5})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 3x^{2} + 1 $ x^4 + 3*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 640)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2049.2
Root			$-0.618034i$ of defining polynomial
Character	$\chi$	$=$	3200.2049
Dual form			3200.2.c.x.2049.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.23607i q^{3} +3.23607i q^{7} +1.47214 q^{9} +2.00000 q^{11} +4.47214i q^{13} -4.47214i q^{17} -4.47214 q^{19} +4.00000 q^{21} -4.76393i q^{23} -5.52786i q^{27} -2.00000 q^{29} +6.47214 q^{31} -2.47214i q^{33} +6.94427i q^{37} +5.52786 q^{39} +12.4721 q^{41} +7.70820i q^{43} +7.23607i q^{47} -3.47214 q^{49} -5.52786 q^{51} -0.472136i q^{53} +5.52786i q^{57} -8.47214 q^{59} +6.00000 q^{61} +4.76393i q^{63} +7.70820i q^{67} -5.88854 q^{69} -2.47214 q^{71} +4.47214i q^{73} +6.47214i q^{77} -12.9443 q^{79} -2.41641 q^{81} +3.70820i q^{83} +2.47214i q^{87} +14.9443 q^{89} -14.4721 q^{91} -8.00000i q^{93} +16.4721i q^{97} +2.94427 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 12 q^{9} + 8 q^{11} + 16 q^{21} - 8 q^{29} + 8 q^{31} + 40 q^{39} + 32 q^{41} + 4 q^{49} - 40 q^{51} - 16 q^{59} + 24 q^{61} + 48 q^{69} + 8 q^{71} - 16 q^{79} + 44 q^{81} + 24 q^{89} - 40 q^{91}+ \cdots - 24 q^{99}+O(q^{100}) $ 4 * q - 12 * q^9 + 8 * q^11 + 16 * q^21 - 8 * q^29 + 8 * q^31 + 40 * q^39 + 32 * q^41 + 4 * q^49 - 40 * q^51 - 16 * q^59 + 24 * q^61 + 48 * q^69 + 8 * q^71 - 16 * q^79 + 44 * q^81 + 24 * q^89 - 40 * q^91 - 24 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3200\mathbb{Z}\right)^\times$.

$n$	$901$	$1151$	$2177$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	− 1.23607i	− 0.713644i	−0.934172	−	0.356822i	$-0.883860\pi$
$3$	− 1.23607i	− 0.713644i	0.934172	−	0.356822i	$-0.116140\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	3.23607i	1.22312i	0.791199	+	0.611559i	$0.209457\pi$
$7$	3.23607i	1.22312i	−0.791199	+	0.611559i	$0.790543\pi$
$8$	0	0
$9$	1.47214	0.490712
$10$	0	0
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	0	0
$13$	4.47214i	1.24035i	0.784465	+	0.620174i	$0.212938\pi$
$13$	4.47214i	1.24035i	−0.784465	+	0.620174i	$0.787062\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 4.47214i	− 1.08465i	−0.840168	−	0.542326i	$-0.817544\pi$
$17$	− 4.47214i	− 1.08465i	0.840168	−	0.542326i	$-0.182456\pi$
$18$	0	0
$19$	−4.47214	−1.02598	−0.512989	−	0.858395i	$-0.671462\pi$
$19$	−4.47214	−1.02598	−0.512989	+	0.858395i	$0.671462\pi$
$20$	0	0
$21$	4.00000	0.872872
$22$	0	0
$23$	− 4.76393i	− 0.993348i	−0.867937	−	0.496674i	$-0.834554\pi$
$23$	− 4.76393i	− 0.993348i	0.867937	−	0.496674i	$-0.165446\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 5.52786i	− 1.06384i
$28$	0	0
$29$	−2.00000	−0.371391	−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000	−0.371391	−0.185695	+	0.982607i	$0.559454\pi$
$30$	0	0
$31$	6.47214	1.16243	0.581215	−	0.813750i	$-0.302578\pi$
$31$	6.47214	1.16243	0.581215	+	0.813750i	$0.302578\pi$
$32$	0	0
$33$	− 2.47214i	− 0.430344i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	6.94427i	1.14163i	0.821078	+	0.570816i	$0.193373\pi$
$37$	6.94427i	1.14163i	−0.821078	+	0.570816i	$0.806627\pi$
$38$	0	0
$39$	5.52786	0.885167
$40$	0	0
$41$	12.4721	1.94782	0.973910	−	0.226934i	$-0.0728701\pi$
$41$	12.4721	1.94782	0.973910	+	0.226934i	$0.0728701\pi$
$42$	0	0
$43$	7.70820i	1.17549i	0.809046	+	0.587745i	$0.199984\pi$
$43$	7.70820i	1.17549i	−0.809046	+	0.587745i	$0.800016\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	7.23607i	1.05549i	0.849403	+	0.527744i	$0.176962\pi$
$47$	7.23607i	1.05549i	−0.849403	+	0.527744i	$0.823038\pi$
$48$	0	0
$49$	−3.47214	−0.496019
$50$	0	0
$51$	−5.52786	−0.774056
$52$	0	0
$53$	− 0.472136i	− 0.0648529i	−0.999474	−	0.0324264i	$-0.989677\pi$
$53$	− 0.472136i	− 0.0648529i	0.999474	−	0.0324264i	$-0.0103235\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	5.52786i	0.732183i
$58$	0	0
$59$	−8.47214	−1.10298	−0.551489	−	0.834182i	$-0.685940\pi$
$59$	−8.47214	−1.10298	−0.551489	+	0.834182i	$0.685940\pi$
$60$	0	0
$61$	6.00000	0.768221	0.384111	−	0.923287i	$-0.374508\pi$
$61$	6.00000	0.768221	0.384111	+	0.923287i	$0.374508\pi$
$62$	0	0
$63$	4.76393i	0.600199i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	7.70820i	0.941707i	0.882211	+	0.470853i	$0.156054\pi$
$67$	7.70820i	0.941707i	−0.882211	+	0.470853i	$0.843946\pi$
$68$	0	0
$69$	−5.88854	−0.708897
$70$	0	0
$71$	−2.47214	−0.293389	−0.146694	−	0.989182i	$-0.546863\pi$
$71$	−2.47214	−0.293389	−0.146694	+	0.989182i	$0.546863\pi$
$72$	0	0
$73$	4.47214i	0.523424i	0.965146	+	0.261712i	$0.0842870\pi$
$73$	4.47214i	0.523424i	−0.965146	+	0.261712i	$0.915713\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	6.47214i	0.737568i
$78$	0	0
$79$	−12.9443	−1.45634	−0.728172	−	0.685394i	$-0.759630\pi$
$79$	−12.9443	−1.45634	−0.728172	+	0.685394i	$0.759630\pi$
$80$	0	0
$81$	−2.41641	−0.268490
$82$	0	0
$83$	3.70820i	0.407028i	0.979072	+	0.203514i	$0.0652363\pi$
$83$	3.70820i	0.407028i	−0.979072	+	0.203514i	$0.934764\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	2.47214i	0.265041i
$88$	0	0
$89$	14.9443	1.58409	0.792045	−	0.610463i	$-0.209017\pi$
$89$	14.9443	1.58409	0.792045	+	0.610463i	$0.209017\pi$
$90$	0	0
$91$	−14.4721	−1.51709
$92$	0	0
$93$	− 8.00000i	− 0.829561i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	16.4721i	1.67249i	0.548354	+	0.836246i	$0.315254\pi$
$97$	16.4721i	1.67249i	−0.548354	+	0.836246i	$0.684746\pi$
$98$	0	0
$99$	2.94427	0.295910
$100$	0	0
$101$	10.0000	0.995037	0.497519	−	0.867453i	$-0.334245\pi$
$101$	10.0000	0.995037	0.497519	+	0.867453i	$0.334245\pi$
$102$	0	0
$103$	3.23607i	0.318859i	0.987209	+	0.159430i	$0.0509655\pi$
$103$	3.23607i	0.318859i	−0.987209	+	0.159430i	$0.949034\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 17.2361i	− 1.66627i	−0.553067	−	0.833137i	$-0.686543\pi$
$107$	− 17.2361i	− 1.66627i	0.553067	−	0.833137i	$-0.313457\pi$
$108$	0	0
$109$	14.9443	1.43140	0.715701	−	0.698407i	$-0.246107\pi$
$109$	14.9443	1.43140	0.715701	+	0.698407i	$0.246107\pi$
$110$	0	0
$111$	8.58359	0.814719
$112$	0	0
$113$	− 2.94427i	− 0.276974i	−0.990364	−	0.138487i	$-0.955776\pi$
$113$	− 2.94427i	− 0.276974i	0.990364	−	0.138487i	$-0.0442239\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	6.58359i	0.608653i
$118$	0	0
$119$	14.4721	1.32666
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	− 15.4164i	− 1.39005i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	20.1803i	1.79072i	0.445348	+	0.895358i	$0.353080\pi$
$127$	20.1803i	1.79072i	−0.445348	+	0.895358i	$0.646920\pi$
$128$	0	0
$129$	9.52786	0.838882
$130$	0	0
$131$	−14.9443	−1.30569	−0.652844	−	0.757493i	$-0.726424\pi$
$131$	−14.9443	−1.30569	−0.652844	+	0.757493i	$0.726424\pi$
$132$	0	0
$133$	− 14.4721i	− 1.25489i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 2.00000i	− 0.170872i	−0.996344	−	0.0854358i	$-0.972772\pi$
$137$	− 2.00000i	− 0.170872i	0.996344	−	0.0854358i	$-0.0272282\pi$
$138$	0	0
$139$	20.4721	1.73642	0.868212	−	0.496194i	$-0.165269\pi$
$139$	20.4721	1.73642	0.868212	+	0.496194i	$0.165269\pi$
$140$	0	0
$141$	8.94427	0.753244
$142$	0	0
$143$	8.94427i	0.747958i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	4.29180i	0.353981i
$148$	0	0
$149$	6.94427	0.568897	0.284448	−	0.958691i	$-0.408190\pi$
$149$	6.94427	0.568897	0.284448	+	0.958691i	$0.408190\pi$
$150$	0	0
$151$	23.4164	1.90560	0.952800	−	0.303598i	$-0.0981881\pi$
$151$	23.4164	1.90560	0.952800	+	0.303598i	$0.0981881\pi$
$152$	0	0
$153$	− 6.58359i	− 0.532252i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	5.05573i	0.403491i	0.979438	+	0.201746i	$0.0646614\pi$
$157$	5.05573i	0.403491i	−0.979438	+	0.201746i	$0.935339\pi$
$158$	0	0
$159$	−0.583592	−0.0462819
$160$	0	0
$161$	15.4164	1.21498
$162$	0	0
$163$	11.7082i	0.917057i	0.888680	+	0.458529i	$0.151623\pi$
$163$	11.7082i	0.917057i	−0.888680	+	0.458529i	$0.848377\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 14.6525i	− 1.13384i	−0.823772	−	0.566921i	$-0.808134\pi$
$167$	− 14.6525i	− 1.13384i	0.823772	−	0.566921i	$-0.191866\pi$
$168$	0	0
$169$	−7.00000	−0.538462
$170$	0	0
$171$	−6.58359	−0.503460
$172$	0	0
$173$	10.9443i	0.832078i	0.909347	+	0.416039i	$0.136582\pi$
$173$	10.9443i	0.832078i	−0.909347	+	0.416039i	$0.863418\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	10.4721i	0.787134i
$178$	0	0
$179$	−20.4721	−1.53016	−0.765080	−	0.643936i	$-0.777300\pi$
$179$	−20.4721	−1.53016	−0.765080	+	0.643936i	$0.777300\pi$
$180$	0	0
$181$	−10.9443	−0.813481	−0.406741	−	0.913544i	$-0.633335\pi$
$181$	−10.9443	−0.813481	−0.406741	+	0.913544i	$0.633335\pi$
$182$	0	0
$183$	− 7.41641i	− 0.548237i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 8.94427i	− 0.654070i
$188$	0	0
$189$	17.8885	1.30120
$190$	0	0
$191$	11.4164	0.826062	0.413031	−	0.910717i	$-0.364470\pi$
$191$	11.4164	0.826062	0.413031	+	0.910717i	$0.364470\pi$
$192$	0	0
$193$	− 11.5279i	− 0.829794i	−0.909868	−	0.414897i	$-0.863818\pi$
$193$	− 11.5279i	− 0.829794i	0.909868	−	0.414897i	$-0.136182\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	0.472136i	0.0336383i	0.999859	+	0.0168191i	$0.00535395\pi$
$197$	0.472136i	0.0336383i	−0.999859	+	0.0168191i	$0.994646\pi$
$198$	0	0
$199$	0.944272	0.0669377	0.0334688	−	0.999440i	$-0.489345\pi$
$199$	0.944272	0.0669377	0.0334688	+	0.999440i	$0.489345\pi$
$200$	0	0
$201$	9.52786	0.672044
$202$	0	0
$203$	− 6.47214i	− 0.454255i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 7.01316i	− 0.487448i
$208$	0	0
$209$	−8.94427	−0.618688
$210$	0	0
$211$	1.05573	0.0726793	0.0363397	−	0.999339i	$-0.488430\pi$
$211$	1.05573	0.0726793	0.0363397	+	0.999339i	$0.488430\pi$
$212$	0	0
$213$	3.05573i	0.209375i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	20.9443i	1.42179i
$218$	0	0
$219$	5.52786	0.373538
$220$	0	0
$221$	20.0000	1.34535
$222$	0	0
$223$	− 8.76393i	− 0.586876i	−0.955978	−	0.293438i	$-0.905201\pi$
$223$	− 8.76393i	− 0.586876i	0.955978	−	0.293438i	$-0.0947995\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 10.1803i	− 0.675693i	−0.941201	−	0.337846i	$-0.890302\pi$
$227$	− 10.1803i	− 0.675693i	0.941201	−	0.337846i	$-0.109698\pi$
$228$	0	0
$229$	2.94427	0.194563	0.0972815	−	0.995257i	$-0.468985\pi$
$229$	2.94427	0.194563	0.0972815	+	0.995257i	$0.468985\pi$
$230$	0	0
$231$	8.00000	0.526361
$232$	0	0
$233$	15.5279i	1.01726i	0.860984	+	0.508632i	$0.169849\pi$
$233$	15.5279i	1.01726i	−0.860984	+	0.508632i	$0.830151\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	16.0000i	1.03931i
$238$	0	0
$239$	12.9443	0.837295	0.418648	−	0.908149i	$-0.362504\pi$
$239$	12.9443	0.837295	0.418648	+	0.908149i	$0.362504\pi$
$240$	0	0
$241$	26.3607	1.69804	0.849020	−	0.528360i	$-0.177193\pi$
$241$	26.3607	1.69804	0.849020	+	0.528360i	$0.177193\pi$
$242$	0	0
$243$	− 13.5967i	− 0.872232i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 20.0000i	− 1.27257i
$248$	0	0
$249$	4.58359	0.290473
$250$	0	0
$251$	2.00000	0.126239	0.0631194	−	0.998006i	$-0.479895\pi$
$251$	2.00000	0.126239	0.0631194	+	0.998006i	$0.479895\pi$
$252$	0	0
$253$	− 9.52786i	− 0.599012i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	2.94427i	0.183659i	0.995775	+	0.0918293i	$0.0292714\pi$
$257$	2.94427i	0.183659i	−0.995775	+	0.0918293i	$0.970729\pi$
$258$	0	0
$259$	−22.4721	−1.39635
$260$	0	0
$261$	−2.94427	−0.182246
$262$	0	0
$263$	− 17.7082i	− 1.09193i	−0.837806	−	0.545967i	$-0.816162\pi$
$263$	− 17.7082i	− 1.09193i	0.837806	−	0.545967i	$-0.183838\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 18.4721i	− 1.13048i
$268$	0	0
$269$	−23.8885	−1.45651	−0.728255	−	0.685306i	$-0.759668\pi$
$269$	−23.8885	−1.45651	−0.728255	+	0.685306i	$0.759668\pi$
$270$	0	0
$271$	−24.3607	−1.47981	−0.739903	−	0.672714i	$-0.765129\pi$
$271$	−24.3607	−1.47981	−0.739903	+	0.672714i	$0.765129\pi$
$272$	0	0
$273$	17.8885i	1.08266i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 10.9443i	− 0.657578i	−0.944403	−	0.328789i	$-0.893360\pi$
$277$	− 10.9443i	− 0.657578i	0.944403	−	0.328789i	$-0.106640\pi$
$278$	0	0
$279$	9.52786	0.570418
$280$	0	0
$281$	3.52786	0.210455	0.105227	−	0.994448i	$-0.466443\pi$
$281$	3.52786	0.210455	0.105227	+	0.994448i	$0.466443\pi$
$282$	0	0
$283$	− 8.29180i	− 0.492896i	−0.969156	−	0.246448i	$-0.920737\pi$
$283$	− 8.29180i	− 0.492896i	0.969156	−	0.246448i	$-0.0792635\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	40.3607i	2.38242i
$288$	0	0
$289$	−3.00000	−0.176471
$290$	0	0
$291$	20.3607	1.19356
$292$	0	0
$293$	23.8885i	1.39558i	0.716301	+	0.697792i	$0.245834\pi$
$293$	23.8885i	1.39558i	−0.716301	+	0.697792i	$0.754166\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 11.0557i	− 0.641518i
$298$	0	0
$299$	21.3050	1.23210
$300$	0	0
$301$	−24.9443	−1.43776
$302$	0	0
$303$	− 12.3607i	− 0.710102i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 0.291796i	− 0.0166537i	−0.999965	−	0.00832684i	$-0.997349\pi$
$307$	− 0.291796i	− 0.0166537i	0.999965	−	0.00832684i	$-0.00265055\pi$
$308$	0	0
$309$	4.00000	0.227552
$310$	0	0
$311$	−15.4164	−0.874184	−0.437092	−	0.899417i	$-0.643992\pi$
$311$	−15.4164	−0.874184	−0.437092	+	0.899417i	$0.643992\pi$
$312$	0	0
$313$	− 28.8328i	− 1.62973i	−0.579653	−	0.814864i	$-0.696812\pi$
$313$	− 28.8328i	− 1.62973i	0.579653	−	0.814864i	$-0.303188\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 20.4721i	− 1.14983i	−0.818213	−	0.574915i	$-0.805035\pi$
$317$	− 20.4721i	− 1.14983i	0.818213	−	0.574915i	$-0.194965\pi$
$318$	0	0
$319$	−4.00000	−0.223957
$320$	0	0
$321$	−21.3050	−1.18913
$322$	0	0
$323$	20.0000i	1.11283i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 18.4721i	− 1.02151i
$328$	0	0
$329$	−23.4164	−1.29099
$330$	0	0
$331$	−15.8885	−0.873313	−0.436657	−	0.899628i	$-0.643838\pi$
$331$	−15.8885	−0.873313	−0.436657	+	0.899628i	$0.643838\pi$
$332$	0	0
$333$	10.2229i	0.560212i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 5.05573i	− 0.275403i	−0.990474	−	0.137702i	$-0.956029\pi$
$337$	− 5.05573i	− 0.275403i	0.990474	−	0.137702i	$-0.0439715\pi$
$338$	0	0
$339$	−3.63932	−0.197661
$340$	0	0
$341$	12.9443	0.700972
$342$	0	0
$343$	11.4164i	0.616428i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 17.2361i	− 0.925281i	−0.886546	−	0.462640i	$-0.846902\pi$
$347$	− 17.2361i	− 0.925281i	0.886546	−	0.462640i	$-0.153098\pi$
$348$	0	0
$349$	−14.9443	−0.799949	−0.399974	−	0.916526i	$-0.630981\pi$
$349$	−14.9443	−0.799949	−0.399974	+	0.916526i	$0.630981\pi$
$350$	0	0
$351$	24.7214	1.31953
$352$	0	0
$353$	5.05573i	0.269089i	0.990908	+	0.134545i	$0.0429572\pi$
$353$	5.05573i	0.269089i	−0.990908	+	0.134545i	$0.957043\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 17.8885i	− 0.946762i
$358$	0	0
$359$	15.0557	0.794611	0.397305	−	0.917686i	$-0.369945\pi$
$359$	15.0557	0.794611	0.397305	+	0.917686i	$0.369945\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	8.65248i	0.454137i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	18.2918i	0.954824i	0.878680	+	0.477412i	$0.158425\pi$
$367$	18.2918i	0.954824i	−0.878680	+	0.477412i	$0.841575\pi$
$368$	0	0
$369$	18.3607	0.955819
$370$	0	0
$371$	1.52786	0.0793227
$372$	0	0
$373$	− 5.05573i	− 0.261776i	−0.991397	−	0.130888i	$-0.958217\pi$
$373$	− 5.05573i	− 0.261776i	0.991397	−	0.130888i	$-0.0417828\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 8.94427i	− 0.460653i
$378$	0	0
$379$	15.5279	0.797613	0.398806	−	0.917035i	$-0.369425\pi$
$379$	15.5279	0.797613	0.398806	+	0.917035i	$0.369425\pi$
$380$	0	0
$381$	24.9443	1.27793
$382$	0	0
$383$	10.2918i	0.525886i	0.964811	+	0.262943i	$0.0846932\pi$
$383$	10.2918i	0.525886i	−0.964811	+	0.262943i	$0.915307\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	11.3475i	0.576827i
$388$	0	0
$389$	11.8885	0.602773	0.301387	−	0.953502i	$-0.402551\pi$
$389$	11.8885	0.602773	0.301387	+	0.953502i	$0.402551\pi$
$390$	0	0
$391$	−21.3050	−1.07744
$392$	0	0
$393$	18.4721i	0.931796i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 12.4721i	− 0.625959i	−0.949760	−	0.312979i	$-0.898673\pi$
$397$	− 12.4721i	− 0.625959i	0.949760	−	0.312979i	$-0.101327\pi$
$398$	0	0
$399$	−17.8885	−0.895547
$400$	0	0
$401$	−2.00000	−0.0998752	−0.0499376	−	0.998752i	$-0.515902\pi$
$401$	−2.00000	−0.0998752	−0.0499376	+	0.998752i	$0.515902\pi$
$402$	0	0
$403$	28.9443i	1.44182i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	13.8885i	0.688430i
$408$	0	0
$409$	−17.4164	−0.861186	−0.430593	−	0.902546i	$-0.641696\pi$
$409$	−17.4164	−0.861186	−0.430593	+	0.902546i	$0.641696\pi$
$410$	0	0
$411$	−2.47214	−0.121941
$412$	0	0
$413$	− 27.4164i	− 1.34907i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 25.3050i	− 1.23919i
$418$	0	0
$419$	−4.47214	−0.218478	−0.109239	−	0.994016i	$-0.534841\pi$
$419$	−4.47214	−0.218478	−0.109239	+	0.994016i	$0.534841\pi$
$420$	0	0
$421$	28.8328	1.40523	0.702613	−	0.711572i	$-0.252017\pi$
$421$	28.8328	1.40523	0.702613	+	0.711572i	$0.252017\pi$
$422$	0	0
$423$	10.6525i	0.517941i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	19.4164i	0.939626i
$428$	0	0
$429$	11.0557	0.533776
$430$	0	0
$431$	−24.3607	−1.17341	−0.586706	−	0.809800i	$-0.699576\pi$
$431$	−24.3607	−1.17341	−0.586706	+	0.809800i	$0.699576\pi$
$432$	0	0
$433$	− 0.472136i	− 0.0226894i	−0.999936	−	0.0113447i	$-0.996389\pi$
$433$	− 0.472136i	− 0.0226894i	0.999936	−	0.0113447i	$-0.00361121\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	21.3050i	1.01915i
$438$	0	0
$439$	−15.0557	−0.718571	−0.359285	−	0.933228i	$-0.616980\pi$
$439$	−15.0557	−0.718571	−0.359285	+	0.933228i	$0.616980\pi$
$440$	0	0
$441$	−5.11146	−0.243403
$442$	0	0
$443$	4.65248i	0.221046i	0.993874	+	0.110523i	$0.0352526\pi$
$443$	4.65248i	0.221046i	−0.993874	+	0.110523i	$0.964747\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 8.58359i	− 0.405990i
$448$	0	0
$449$	1.41641	0.0668444	0.0334222	−	0.999441i	$-0.489359\pi$
$449$	1.41641	0.0668444	0.0334222	+	0.999441i	$0.489359\pi$
$450$	0	0
$451$	24.9443	1.17458
$452$	0	0
$453$	− 28.9443i	− 1.35992i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 3.88854i	− 0.181898i	−0.995856	−	0.0909492i	$-0.971010\pi$
$457$	− 3.88854i	− 0.181898i	0.995856	−	0.0909492i	$-0.0289901\pi$
$458$	0	0
$459$	−24.7214	−1.15389
$460$	0	0
$461$	−41.7771	−1.94575	−0.972876	−	0.231325i	$-0.925694\pi$
$461$	−41.7771	−1.94575	−0.972876	+	0.231325i	$0.925694\pi$
$462$	0	0
$463$	1.12461i	0.0522651i	0.999658	+	0.0261326i	$0.00831920\pi$
$463$	1.12461i	0.0522651i	−0.999658	+	0.0261326i	$0.991681\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	41.5967i	1.92487i	0.271516	+	0.962434i	$0.412475\pi$
$467$	41.5967i	1.92487i	−0.271516	+	0.962434i	$0.587525\pi$
$468$	0	0
$469$	−24.9443	−1.15182
$470$	0	0
$471$	6.24922	0.287949
$472$	0	0
$473$	15.4164i	0.708847i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 0.695048i	− 0.0318241i
$478$	0	0
$479$	−12.9443	−0.591439	−0.295719	−	0.955275i	$-0.595559\pi$
$479$	−12.9443	−0.591439	−0.295719	+	0.955275i	$0.595559\pi$
$480$	0	0
$481$	−31.0557	−1.41602
$482$	0	0
$483$	− 19.0557i	− 0.867066i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 12.7639i	− 0.578389i	−0.957270	−	0.289194i	$-0.906613\pi$
$487$	− 12.7639i	− 0.578389i	0.957270	−	0.289194i	$-0.0933874\pi$
$488$	0	0
$489$	14.4721	0.654453
$490$	0	0
$491$	21.0557	0.950232	0.475116	−	0.879923i	$-0.342406\pi$
$491$	21.0557	0.950232	0.475116	+	0.879923i	$0.342406\pi$
$492$	0	0
$493$	8.94427i	0.402830i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 8.00000i	− 0.358849i
$498$	0	0
$499$	14.5836	0.652851	0.326426	−	0.945223i	$-0.394156\pi$
$499$	14.5836	0.652851	0.326426	+	0.945223i	$0.394156\pi$
$500$	0	0
$501$	−18.1115	−0.809160
$502$	0	0
$503$	5.12461i	0.228495i	0.993452	+	0.114248i	$0.0364457\pi$
$503$	5.12461i	0.228495i	−0.993452	+	0.114248i	$0.963554\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	8.65248i	0.384270i
$508$	0	0
$509$	−16.8328	−0.746101	−0.373051	−	0.927811i	$-0.621688\pi$
$509$	−16.8328	−0.746101	−0.373051	+	0.927811i	$0.621688\pi$
$510$	0	0
$511$	−14.4721	−0.640210
$512$	0	0
$513$	24.7214i	1.09147i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	14.4721i	0.636484i
$518$	0	0
$519$	13.5279	0.593807
$520$	0	0
$521$	−15.8885	−0.696090	−0.348045	−	0.937478i	$-0.613154\pi$
$521$	−15.8885	−0.696090	−0.348045	+	0.937478i	$0.613154\pi$
$522$	0	0
$523$	− 20.0689i	− 0.877551i	−0.898597	−	0.438776i	$-0.855412\pi$
$523$	− 20.0689i	− 0.877551i	0.898597	−	0.438776i	$-0.144588\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 28.9443i	− 1.26083i
$528$	0	0
$529$	0.304952	0.0132588
$530$	0	0
$531$	−12.4721	−0.541245
$532$	0	0
$533$	55.7771i	2.41597i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	25.3050i	1.09199i
$538$	0	0
$539$	−6.94427	−0.299111
$540$	0	0
$541$	−23.8885	−1.02705	−0.513524	−	0.858075i	$-0.671660\pi$
$541$	−23.8885	−1.02705	−0.513524	+	0.858075i	$0.671660\pi$
$542$	0	0
$543$	13.5279i	0.580536i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	20.6525i	0.883036i	0.897252	+	0.441518i	$0.145560\pi$
$547$	20.6525i	0.883036i	−0.897252	+	0.441518i	$0.854440\pi$
$548$	0	0
$549$	8.83282	0.376975
$550$	0	0
$551$	8.94427	0.381039
$552$	0	0
$553$	− 41.8885i	− 1.78128i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	13.0557i	0.553189i	0.960987	+	0.276594i	$0.0892059\pi$
$557$	13.0557i	0.553189i	−0.960987	+	0.276594i	$0.910794\pi$
$558$	0	0
$559$	−34.4721	−1.45802
$560$	0	0
$561$	−11.0557	−0.466773
$562$	0	0
$563$	− 4.29180i	− 0.180878i	−0.995902	−	0.0904388i	$-0.971173\pi$
$563$	− 4.29180i	− 0.180878i	0.995902	−	0.0904388i	$-0.0288270\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 7.81966i	− 0.328395i
$568$	0	0
$569$	34.3607	1.44047	0.720237	−	0.693728i	$-0.244033\pi$
$569$	34.3607	1.44047	0.720237	+	0.693728i	$0.244033\pi$
$570$	0	0
$571$	16.8328	0.704431	0.352216	−	0.935919i	$-0.385428\pi$
$571$	16.8328	0.704431	0.352216	+	0.935919i	$0.385428\pi$
$572$	0	0
$573$	− 14.1115i	− 0.589515i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 19.8885i	− 0.827971i	−0.910283	−	0.413985i	$-0.864136\pi$
$577$	− 19.8885i	− 0.827971i	0.910283	−	0.413985i	$-0.135864\pi$
$578$	0	0
$579$	−14.2492	−0.592178
$580$	0	0
$581$	−12.0000	−0.497844
$582$	0	0
$583$	− 0.944272i	− 0.0391077i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	8.65248i	0.357126i	0.983928	+	0.178563i	$0.0571448\pi$
$587$	8.65248i	0.357126i	−0.983928	+	0.178563i	$0.942855\pi$
$588$	0	0
$589$	−28.9443	−1.19263
$590$	0	0
$591$	0.583592	0.0240058
$592$	0	0
$593$	− 14.0000i	− 0.574911i	−0.957794	−	0.287456i	$-0.907191\pi$
$593$	− 14.0000i	− 0.574911i	0.957794	−	0.287456i	$-0.0928094\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 1.16718i	− 0.0477697i
$598$	0	0
$599$	21.8885	0.894342	0.447171	−	0.894449i	$-0.352432\pi$
$599$	21.8885	0.894342	0.447171	+	0.894449i	$0.352432\pi$
$600$	0	0
$601$	−22.3607	−0.912111	−0.456056	−	0.889951i	$-0.650738\pi$
$601$	−22.3607	−0.912111	−0.456056	+	0.889951i	$0.650738\pi$
$602$	0	0
$603$	11.3475i	0.462107i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 6.87539i	− 0.279063i	−0.990218	−	0.139532i	$-0.955440\pi$
$607$	− 6.87539i	− 0.279063i	0.990218	−	0.139532i	$-0.0445597\pi$
$608$	0	0
$609$	−8.00000	−0.324176
$610$	0	0
$611$	−32.3607	−1.30917
$612$	0	0
$613$	− 19.5279i	− 0.788723i	−0.918955	−	0.394362i	$-0.870966\pi$
$613$	− 19.5279i	− 0.788723i	0.918955	−	0.394362i	$-0.129034\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 25.4164i	− 1.02323i	−0.859216	−	0.511613i	$-0.829048\pi$
$617$	− 25.4164i	− 1.02323i	0.859216	−	0.511613i	$-0.170952\pi$
$618$	0	0
$619$	20.4721	0.822845	0.411422	−	0.911445i	$-0.365032\pi$
$619$	20.4721	0.822845	0.411422	+	0.911445i	$0.365032\pi$
$620$	0	0
$621$	−26.3344	−1.05676
$622$	0	0
$623$	48.3607i	1.93753i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	11.0557i	0.441523i
$628$	0	0
$629$	31.0557	1.23827
$630$	0	0
$631$	−34.4721	−1.37231	−0.686157	−	0.727453i	$-0.740704\pi$
$631$	−34.4721	−1.37231	−0.686157	+	0.727453i	$0.740704\pi$
$632$	0	0
$633$	− 1.30495i	− 0.0518672i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 15.5279i	− 0.615236i
$638$	0	0
$639$	−3.63932	−0.143969
$640$	0	0
$641$	0.472136	0.0186482	0.00932412	−	0.999957i	$-0.497032\pi$
$641$	0.472136	0.0186482	0.00932412	+	0.999957i	$0.497032\pi$
$642$	0	0
$643$	− 22.1803i	− 0.874707i	−0.899290	−	0.437354i	$-0.855916\pi$
$643$	− 22.1803i	− 0.874707i	0.899290	−	0.437354i	$-0.144084\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 12.7639i	− 0.501802i	−0.968013	−	0.250901i	$-0.919273\pi$
$647$	− 12.7639i	− 0.501802i	0.968013	−	0.250901i	$-0.0807269\pi$
$648$	0	0
$649$	−16.9443	−0.665121
$650$	0	0
$651$	25.8885	1.01465
$652$	0	0
$653$	49.4164i	1.93381i	0.255130	+	0.966907i	$0.417882\pi$
$653$	49.4164i	1.93381i	−0.255130	+	0.966907i	$0.582118\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	6.58359i	0.256850i
$658$	0	0
$659$	21.4164	0.834265	0.417132	−	0.908846i	$-0.363035\pi$
$659$	21.4164	0.834265	0.417132	+	0.908846i	$0.363035\pi$
$660$	0	0
$661$	−35.8885	−1.39590	−0.697951	−	0.716145i	$-0.745905\pi$
$661$	−35.8885	−1.39590	−0.697951	+	0.716145i	$0.745905\pi$
$662$	0	0
$663$	− 24.7214i	− 0.960098i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	9.52786i	0.368920i
$668$	0	0
$669$	−10.8328	−0.418821
$670$	0	0
$671$	12.0000	0.463255
$672$	0	0
$673$	− 23.3050i	− 0.898340i	−0.893446	−	0.449170i	$-0.851720\pi$
$673$	− 23.3050i	− 0.898340i	0.893446	−	0.449170i	$-0.148280\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 30.3607i	− 1.16686i	−0.812165	−	0.583428i	$-0.801711\pi$
$677$	− 30.3607i	− 1.16686i	0.812165	−	0.583428i	$-0.198289\pi$
$678$	0	0
$679$	−53.3050	−2.04566
$680$	0	0
$681$	−12.5836	−0.482204
$682$	0	0
$683$	− 24.2918i	− 0.929500i	−0.885442	−	0.464750i	$-0.846144\pi$
$683$	− 24.2918i	− 0.929500i	0.885442	−	0.464750i	$-0.153856\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 3.63932i	− 0.138849i
$688$	0	0
$689$	2.11146	0.0804401
$690$	0	0
$691$	30.0000	1.14125	0.570627	−	0.821209i	$-0.306700\pi$
$691$	30.0000	1.14125	0.570627	+	0.821209i	$0.306700\pi$
$692$	0	0
$693$	9.52786i	0.361934i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 55.7771i	− 2.11271i
$698$	0	0
$699$	19.1935	0.725965
$700$	0	0
$701$	9.05573	0.342030	0.171015	−	0.985268i	$-0.445295\pi$
$701$	9.05573	0.342030	0.171015	+	0.985268i	$0.445295\pi$
$702$	0	0
$703$	− 31.0557i	− 1.17129i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	32.3607i	1.21705i
$708$	0	0
$709$	−18.0000	−0.676004	−0.338002	−	0.941145i	$-0.609751\pi$
$709$	−18.0000	−0.676004	−0.338002	+	0.941145i	$0.609751\pi$
$710$	0	0
$711$	−19.0557	−0.714646
$712$	0	0
$713$	− 30.8328i	− 1.15470i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 16.0000i	− 0.597531i
$718$	0	0
$719$	−22.8328	−0.851520	−0.425760	−	0.904836i	$-0.639993\pi$
$719$	−22.8328	−0.851520	−0.425760	+	0.904836i	$0.639993\pi$
$720$	0	0
$721$	−10.4721	−0.390003
$722$	0	0
$723$	− 32.5836i	− 1.21180i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 1.70820i	− 0.0633538i	−0.999498	−	0.0316769i	$-0.989915\pi$
$727$	− 1.70820i	− 0.0633538i	0.999498	−	0.0316769i	$-0.0100848\pi$
$728$	0	0
$729$	−24.0557	−0.890953
$730$	0	0
$731$	34.4721	1.27500
$732$	0	0
$733$	− 51.8885i	− 1.91655i	−0.285853	−	0.958274i	$-0.592277\pi$
$733$	− 51.8885i	− 1.91655i	0.285853	−	0.958274i	$-0.407723\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	15.4164i	0.567871i
$738$	0	0
$739$	49.1935	1.80961	0.904806	−	0.425824i	$-0.140016\pi$
$739$	49.1935	1.80961	0.904806	+	0.425824i	$0.140016\pi$
$740$	0	0
$741$	−24.7214	−0.908162
$742$	0	0
$743$	− 30.6525i	− 1.12453i	−0.826957	−	0.562265i	$-0.809930\pi$
$743$	− 30.6525i	− 1.12453i	0.826957	−	0.562265i	$-0.190070\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	5.45898i	0.199734i
$748$	0	0
$749$	55.7771	2.03805
$750$	0	0
$751$	16.3607	0.597010	0.298505	−	0.954408i	$-0.403512\pi$
$751$	16.3607	0.597010	0.298505	+	0.954408i	$0.403512\pi$
$752$	0	0
$753$	− 2.47214i	− 0.0900896i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	19.8885i	0.722861i	0.932399	+	0.361431i	$0.117712\pi$
$757$	19.8885i	0.722861i	−0.932399	+	0.361431i	$0.882288\pi$
$758$	0	0
$759$	−11.7771	−0.427481
$760$	0	0
$761$	3.88854	0.140960	0.0704798	−	0.997513i	$-0.477547\pi$
$761$	3.88854	0.140960	0.0704798	+	0.997513i	$0.477547\pi$
$762$	0	0
$763$	48.3607i	1.75077i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 37.8885i	− 1.36808i
$768$	0	0
$769$	14.9443	0.538904	0.269452	−	0.963014i	$-0.413157\pi$
$769$	14.9443	0.538904	0.269452	+	0.963014i	$0.413157\pi$
$770$	0	0
$771$	3.63932	0.131067
$772$	0	0
$773$	− 18.3607i	− 0.660388i	−0.943913	−	0.330194i	$-0.892886\pi$
$773$	− 18.3607i	− 0.660388i	0.943913	−	0.330194i	$-0.107114\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	27.7771i	0.996497i
$778$	0	0
$779$	−55.7771	−1.99842
$780$	0	0
$781$	−4.94427	−0.176920
$782$	0	0
$783$	11.0557i	0.395099i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 44.0689i	− 1.57089i	−0.618934	−	0.785443i	$-0.712435\pi$
$787$	− 44.0689i	− 1.57089i	0.618934	−	0.785443i	$-0.287565\pi$
$788$	0	0
$789$	−21.8885	−0.779253
$790$	0	0
$791$	9.52786	0.338772
$792$	0	0
$793$	26.8328i	0.952861i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 25.4164i	− 0.900295i	−0.892954	−	0.450148i	$-0.851371\pi$
$797$	− 25.4164i	− 0.900295i	0.892954	−	0.450148i	$-0.148629\pi$
$798$	0	0
$799$	32.3607	1.14484
$800$	0	0
$801$	22.0000	0.777332
$802$	0	0
$803$	8.94427i	0.315637i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	29.5279i	1.03943i
$808$	0	0
$809$	12.1115	0.425816	0.212908	−	0.977072i	$-0.431707\pi$
$809$	12.1115	0.425816	0.212908	+	0.977072i	$0.431707\pi$
$810$	0	0
$811$	2.00000	0.0702295	0.0351147	−	0.999383i	$-0.488820\pi$
$811$	2.00000	0.0702295	0.0351147	+	0.999383i	$0.488820\pi$
$812$	0	0
$813$	30.1115i	1.05605i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 34.4721i	− 1.20603i
$818$	0	0
$819$	−21.3050	−0.744455
$820$	0	0
$821$	−21.0557	−0.734850	−0.367425	−	0.930053i	$-0.619761\pi$
$821$	−21.0557	−0.734850	−0.367425	+	0.930053i	$0.619761\pi$
$822$	0	0
$823$	− 1.70820i	− 0.0595442i	−0.999557	−	0.0297721i	$-0.990522\pi$
$823$	− 1.70820i	− 0.0595442i	0.999557	−	0.0297721i	$-0.00947816\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 6.18034i	− 0.214911i	−0.994210	−	0.107456i	$-0.965730\pi$
$827$	− 6.18034i	− 0.214911i	0.994210	−	0.107456i	$-0.0342704\pi$
$828$	0	0
$829$	−38.0000	−1.31979	−0.659897	−	0.751356i	$-0.729400\pi$
$829$	−38.0000	−1.31979	−0.659897	+	0.751356i	$0.729400\pi$
$830$	0	0
$831$	−13.5279	−0.469276
$832$	0	0
$833$	15.5279i	0.538009i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 35.7771i	− 1.23664i
$838$	0	0
$839$	−4.00000	−0.138095	−0.0690477	−	0.997613i	$-0.521996\pi$
$839$	−4.00000	−0.138095	−0.0690477	+	0.997613i	$0.521996\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	0	0
$843$	− 4.36068i	− 0.150190i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 22.6525i	− 0.778348i
$848$	0	0
$849$	−10.2492	−0.351752
$850$	0	0
$851$	33.0820	1.13404
$852$	0	0
$853$	7.52786i	0.257749i	0.991661	+	0.128875i	$0.0411365\pi$
$853$	7.52786i	0.257749i	−0.991661	+	0.128875i	$0.958864\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 24.8328i	− 0.848273i	−0.905598	−	0.424136i	$-0.860578\pi$
$857$	− 24.8328i	− 0.848273i	0.905598	−	0.424136i	$-0.139422\pi$
$858$	0	0
$859$	49.4164	1.68607	0.843033	−	0.537862i	$-0.180768\pi$
$859$	49.4164	1.68607	0.843033	+	0.537862i	$0.180768\pi$
$860$	0	0
$861$	49.8885	1.70020
$862$	0	0
$863$	18.2918i	0.622660i	0.950302	+	0.311330i	$0.100774\pi$
$863$	18.2918i	0.622660i	−0.950302	+	0.311330i	$0.899226\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	3.70820i	0.125937i
$868$	0	0
$869$	−25.8885	−0.878209
$870$	0	0
$871$	−34.4721	−1.16804
$872$	0	0
$873$	24.2492i	0.820712i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	51.8885i	1.75215i	0.482173	+	0.876076i	$0.339848\pi$
$877$	51.8885i	1.75215i	−0.482173	+	0.876076i	$0.660152\pi$
$878$	0	0
$879$	29.5279	0.995950
$880$	0	0
$881$	−24.4721	−0.824487	−0.412244	−	0.911074i	$-0.635255\pi$
$881$	−24.4721	−0.824487	−0.412244	+	0.911074i	$0.635255\pi$
$882$	0	0
$883$	27.7082i	0.932455i	0.884665	+	0.466228i	$0.154387\pi$
$883$	27.7082i	0.932455i	−0.884665	+	0.466228i	$0.845613\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 11.5967i	− 0.389381i	−0.980865	−	0.194690i	$-0.937630\pi$
$887$	− 11.5967i	− 0.389381i	0.980865	−	0.194690i	$-0.0623702\pi$
$888$	0	0
$889$	−65.3050	−2.19026
$890$	0	0
$891$	−4.83282	−0.161905
$892$	0	0
$893$	− 32.3607i	− 1.08291i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 26.3344i	− 0.879279i
$898$	0	0
$899$	−12.9443	−0.431716
$900$	0	0
$901$	−2.11146	−0.0703428
$902$	0	0
$903$	30.8328i	1.02605i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 9.23607i	− 0.306679i	−0.988174	−	0.153339i	$-0.950997\pi$
$907$	− 9.23607i	− 0.306679i	0.988174	−	0.153339i	$-0.0490027\pi$
$908$	0	0
$909$	14.7214	0.488277
$910$	0	0
$911$	−53.3050	−1.76607	−0.883036	−	0.469305i	$-0.844504\pi$
$911$	−53.3050	−1.76607	−0.883036	+	0.469305i	$0.844504\pi$
$912$	0	0
$913$	7.41641i	0.245447i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 48.3607i	− 1.59701i
$918$	0	0
$919$	5.88854	0.194245	0.0971226	−	0.995272i	$-0.469036\pi$
$919$	5.88854	0.194245	0.0971226	+	0.995272i	$0.469036\pi$
$920$	0	0
$921$	−0.360680	−0.0118848
$922$	0	0
$923$	− 11.0557i	− 0.363904i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	4.76393i	0.156468i
$928$	0	0
$929$	−24.4721	−0.802905	−0.401452	−	0.915880i	$-0.631494\pi$
$929$	−24.4721	−0.802905	−0.401452	+	0.915880i	$0.631494\pi$
$930$	0	0
$931$	15.5279	0.508905
$932$	0	0
$933$	19.0557i	0.623857i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	3.52786i	0.115250i	0.998338	+	0.0576251i	$0.0183528\pi$
$937$	3.52786i	0.115250i	−0.998338	+	0.0576251i	$0.981647\pi$
$938$	0	0
$939$	−35.6393	−1.16305
$940$	0	0
$941$	−44.8328	−1.46151	−0.730754	−	0.682641i	$-0.760831\pi$
$941$	−44.8328	−1.46151	−0.730754	+	0.682641i	$0.760831\pi$
$942$	0	0
$943$	− 59.4164i	− 1.93486i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	21.8197i	0.709044i	0.935048	+	0.354522i	$0.115356\pi$
$947$	21.8197i	0.709044i	−0.935048	+	0.354522i	$0.884644\pi$
$948$	0	0
$949$	−20.0000	−0.649227
$950$	0	0
$951$	−25.3050	−0.820569
$952$	0	0
$953$	45.7771i	1.48287i	0.671027	+	0.741433i	$0.265853\pi$
$953$	45.7771i	1.48287i	−0.671027	+	0.741433i	$0.734147\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	4.94427i	0.159826i
$958$	0	0
$959$	6.47214	0.208996
$960$	0	0
$961$	10.8885	0.351243
$962$	0	0
$963$	− 25.3738i	− 0.817660i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	1.34752i	0.0433335i	0.999765	+	0.0216667i	$0.00689727\pi$
$967$	1.34752i	0.0433335i	−0.999765	+	0.0216667i	$0.993103\pi$
$968$	0	0
$969$	24.7214	0.794164
$970$	0	0
$971$	−23.8885	−0.766620	−0.383310	−	0.923620i	$-0.625216\pi$
$971$	−23.8885	−0.766620	−0.383310	+	0.923620i	$0.625216\pi$
$972$	0	0
$973$	66.2492i	2.12385i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	39.3050i	1.25748i	0.777617	+	0.628738i	$0.216428\pi$
$977$	39.3050i	1.25748i	−0.777617	+	0.628738i	$0.783572\pi$
$978$	0	0
$979$	29.8885	0.955242
$980$	0	0
$981$	22.0000	0.702406
$982$	0	0
$983$	39.0132i	1.24433i	0.782888	+	0.622163i	$0.213746\pi$
$983$	39.0132i	1.24433i	−0.782888	+	0.622163i	$0.786254\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	28.9443i	0.921306i
$988$	0	0
$989$	36.7214	1.16767
$990$	0	0
$991$	−17.5279	−0.556791	−0.278395	−	0.960467i	$-0.589803\pi$
$991$	−17.5279	−0.556791	−0.278395	+	0.960467i	$0.589803\pi$
$992$	0	0
$993$	19.6393i	0.623235i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 34.5836i	− 1.09527i	−0.836716	−	0.547637i	$-0.815528\pi$
$997$	− 34.5836i	− 1.09527i	0.836716	−	0.547637i	$-0.184472\pi$
$998$	0	0
$999$	38.3870	1.21451

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3200.2.c.x.2049.2		4
4.3	odd	2		3200.2.c.v.2049.3		4
5.2	odd	4		640.2.a.l.1.1	yes	2
5.3	odd	4		3200.2.a.bf.1.2		2
5.4	even	2	inner	3200.2.c.x.2049.3		4
8.3	odd	2		3200.2.c.w.2049.2		4
8.5	even	2		3200.2.c.u.2049.3		4
15.2	even	4		5760.2.a.bw.1.1		2
20.3	even	4		3200.2.a.bk.1.1		2
20.7	even	4		640.2.a.j.1.2	yes	2
20.19	odd	2		3200.2.c.v.2049.2		4
40.3	even	4		3200.2.a.be.1.2		2
40.13	odd	4		3200.2.a.bl.1.1		2
40.19	odd	2		3200.2.c.w.2049.3		4
40.27	even	4		640.2.a.k.1.1	yes	2
40.29	even	2		3200.2.c.u.2049.2		4
40.37	odd	4		640.2.a.i.1.2	✓	2
60.47	odd	4		5760.2.a.cd.1.2		2
80.27	even	4		1280.2.d.k.641.2		4
80.37	odd	4		1280.2.d.m.641.3		4
80.67	even	4		1280.2.d.k.641.3		4
80.77	odd	4		1280.2.d.m.641.2		4
120.77	even	4		5760.2.a.ch.1.1		2
120.107	odd	4		5760.2.a.ci.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
640.2.a.i.1.2	✓	2	40.37	odd	4
640.2.a.j.1.2	yes	2	20.7	even	4
640.2.a.k.1.1	yes	2	40.27	even	4
640.2.a.l.1.1	yes	2	5.2	odd	4
1280.2.d.k.641.2		4	80.27	even	4
1280.2.d.k.641.3		4	80.67	even	4
1280.2.d.m.641.2		4	80.77	odd	4
1280.2.d.m.641.3		4	80.37	odd	4
3200.2.a.be.1.2		2	40.3	even	4
3200.2.a.bf.1.2		2	5.3	odd	4
3200.2.a.bk.1.1		2	20.3	even	4
3200.2.a.bl.1.1		2	40.13	odd	4
3200.2.c.u.2049.2		4	40.29	even	2
3200.2.c.u.2049.3		4	8.5	even	2
3200.2.c.v.2049.2		4	20.19	odd	2
3200.2.c.v.2049.3		4	4.3	odd	2
3200.2.c.w.2049.2		4	8.3	odd	2
3200.2.c.w.2049.3		4	40.19	odd	2
3200.2.c.x.2049.2		4	1.1	even	1	trivial
3200.2.c.x.2049.3		4	5.4	even	2	inner
5760.2.a.bw.1.1		2	15.2	even	4
5760.2.a.cd.1.2		2	60.47	odd	4
5760.2.a.ch.1.1		2	120.77	even	4
5760.2.a.ci.1.2		2	120.107	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 3200 = 2^{7} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3200.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.23607i q^{3} +3.23607i q^{7} +1.47214 q^{9} +2.00000 q^{11} +4.47214i q^{13} -4.47214i q^{17} -4.47214 q^{19} +4.00000 q^{21} -4.76393i q^{23} -5.52786i q^{27} -2.00000 q^{29} +6.47214 q^{31} -2.47214i q^{33} +6.94427i q^{37} +5.52786 q^{39} +12.4721 q^{41} +7.70820i q^{43} +7.23607i q^{47} -3.47214 q^{49} -5.52786 q^{51} -0.472136i q^{53} +5.52786i q^{57} -8.47214 q^{59} +6.00000 q^{61} +4.76393i q^{63} +7.70820i q^{67} -5.88854 q^{69} -2.47214 q^{71} +4.47214i q^{73} +6.47214i q^{77} -12.9443 q^{79} -2.41641 q^{81} +3.70820i q^{83} +2.47214i q^{87} +14.9443 q^{89} -14.4721 q^{91} -8.00000i q^{93} +16.4721i q^{97} +2.94427 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 12 q^{9} + 8 q^{11} + 16 q^{21} - 8 q^{29} + 8 q^{31} + 40 q^{39} + 32 q^{41} + 4 q^{49} - 40 q^{51} - 16 q^{59} + 24 q^{61} + 48 q^{69} + 8 q^{71} - 16 q^{79} + 44 q^{81} + 24 q^{89} - 40 q^{91}+ \cdots - 24 q^{99}+O(q^{100}) \) 4 * q - 12 * q^9 + 8 * q^11 + 16 * q^21 - 8 * q^29 + 8 * q^31 + 40 * q^39 + 32 * q^41 + 4 * q^49 - 40 * q^51 - 16 * q^59 + 24 * q^61 + 48 * q^69 + 8 * q^71 - 16 * q^79 + 44 * q^81 + 24 * q^89 - 40 * q^91 - 24 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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