LMFDB - Embedded newform 2600.2.d.m.1249.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2600,2,Mod(1249,2600)] 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("2600.1249"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2600, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2600 = 2^{3} \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2600.d (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,0,0,-4,0,4,0,0,0,0,0,0,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

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

Self dual:	no
Analytic conductor:	$20.7611045255$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 520)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1249.3
Root			$0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	2600.1249
Dual form			2600.2.d.m.1249.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+0.732051i q^{3} -3.46410i q^{7} +2.46410 q^{9} +2.73205 q^{11} +1.00000i q^{13} -7.46410i q^{17} +2.73205 q^{19} +2.53590 q^{21} -0.732051i q^{23} +4.00000i q^{27} -9.46410 q^{29} -0.196152 q^{31} +2.00000i q^{33} -0.732051 q^{39} +0.535898 q^{41} -7.26795i q^{43} -4.53590i q^{47} -5.00000 q^{49} +5.46410 q^{51} -0.535898i q^{53} +2.00000i q^{57} -5.66025 q^{59} -12.3923 q^{61} -8.53590i q^{63} +12.9282i q^{67} +0.535898 q^{69} +9.26795 q^{71} -2.92820i q^{73} -9.46410i q^{77} +6.53590 q^{79} +4.46410 q^{81} -10.3923i q^{83} -6.92820i q^{87} +12.9282 q^{89} +3.46410 q^{91} -0.143594i q^{93} -15.8564i q^{97} +6.73205 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{9} + 4 q^{11} + 4 q^{19} + 24 q^{21} - 24 q^{29} + 20 q^{31} + 4 q^{39} + 16 q^{41} - 20 q^{49} + 8 q^{51} + 12 q^{59} - 8 q^{61} + 16 q^{69} + 44 q^{71} + 40 q^{79} + 4 q^{81} + 24 q^{89} + 20 q^{99}+O(q^{100}) $ 4 * q - 4 * q^9 + 4 * q^11 + 4 * q^19 + 24 * q^21 - 24 * q^29 + 20 * q^31 + 4 * q^39 + 16 * q^41 - 20 * q^49 + 8 * q^51 + 12 * q^59 - 8 * q^61 + 16 * q^69 + 44 * q^71 + 40 * q^79 + 4 * q^81 + 24 * q^89 + 20 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2600\mathbb{Z}\right)^\times$.

$n$	$1301$	$1601$	$1951$	$1977$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$3$	0.732051i	0.422650i	0.977416	+	0.211325i	$0.0677778\pi$
$3$	0.732051i	0.422650i	−0.977416	+	0.211325i	$0.932222\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 3.46410i	− 1.30931i	−0.755929	−	0.654654i	$-0.772814\pi$
$7$	− 3.46410i	− 1.30931i	0.755929	−	0.654654i	$-0.227186\pi$
$8$	0	0
$9$	2.46410	0.821367
$10$	0	0
$11$	2.73205	0.823744	0.411872	−	0.911242i	$-0.364875\pi$
$11$	2.73205	0.823744	0.411872	+	0.911242i	$0.364875\pi$
$12$	0	0
$13$	1.00000i	0.277350i
$13$	1.00000i	0.277350i
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 7.46410i	− 1.81031i	−0.425081	−	0.905155i	$-0.639754\pi$
$17$	− 7.46410i	− 1.81031i	0.425081	−	0.905155i	$-0.360246\pi$
$18$	0	0
$19$	2.73205	0.626775	0.313388	−	0.949625i	$-0.398536\pi$
$19$	2.73205	0.626775	0.313388	+	0.949625i	$0.398536\pi$
$20$	0	0
$21$	2.53590	0.553378
$22$	0	0
$23$	− 0.732051i	− 0.152643i	−0.997083	−	0.0763216i	$-0.975682\pi$
$23$	− 0.732051i	− 0.152643i	0.997083	−	0.0763216i	$-0.0243176\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	4.00000i	0.769800i
$28$	0	0
$29$	−9.46410	−1.75744	−0.878720	−	0.477338i	$-0.841602\pi$
$29$	−9.46410	−1.75744	−0.878720	+	0.477338i	$0.841602\pi$
$30$	0	0
$31$	−0.196152	−0.0352300	−0.0176150	−	0.999845i	$-0.505607\pi$
$31$	−0.196152	−0.0352300	−0.0176150	+	0.999845i	$0.505607\pi$
$32$	0	0
$33$	2.00000i	0.348155i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	0	0
$39$	−0.732051	−0.117222
$40$	0	0
$41$	0.535898	0.0836933	0.0418466	−	0.999124i	$-0.486676\pi$
$41$	0.535898	0.0836933	0.0418466	+	0.999124i	$0.486676\pi$
$42$	0	0
$43$	− 7.26795i	− 1.10835i	−0.832400	−	0.554176i	$-0.813033\pi$
$43$	− 7.26795i	− 1.10835i	0.832400	−	0.554176i	$-0.186967\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 4.53590i	− 0.661629i	−0.943696	−	0.330814i	$-0.892677\pi$
$47$	− 4.53590i	− 0.661629i	0.943696	−	0.330814i	$-0.107323\pi$
$48$	0	0
$49$	−5.00000	−0.714286
$50$	0	0
$51$	5.46410	0.765127
$52$	0	0
$53$	− 0.535898i	− 0.0736113i	−0.999322	−	0.0368057i	$-0.988282\pi$
$53$	− 0.535898i	− 0.0736113i	0.999322	−	0.0368057i	$-0.0117182\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	2.00000i	0.264906i
$58$	0	0
$59$	−5.66025	−0.736902	−0.368451	−	0.929647i	$-0.620112\pi$
$59$	−5.66025	−0.736902	−0.368451	+	0.929647i	$0.620112\pi$
$60$	0	0
$61$	−12.3923	−1.58667	−0.793336	−	0.608784i	$-0.791658\pi$
$61$	−12.3923	−1.58667	−0.793336	+	0.608784i	$0.791658\pi$
$62$	0	0
$63$	− 8.53590i	− 1.07542i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	12.9282i	1.57943i	0.613473	+	0.789716i	$0.289772\pi$
$67$	12.9282i	1.57943i	−0.613473	+	0.789716i	$0.710228\pi$
$68$	0	0
$69$	0.535898	0.0645146
$70$	0	0
$71$	9.26795	1.09990	0.549952	−	0.835197i	$-0.314646\pi$
$71$	9.26795	1.09990	0.549952	+	0.835197i	$0.314646\pi$
$72$	0	0
$73$	− 2.92820i	− 0.342720i	−0.985208	−	0.171360i	$-0.945184\pi$
$73$	− 2.92820i	− 0.342720i	0.985208	−	0.171360i	$-0.0548162\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 9.46410i	− 1.07853i
$78$	0	0
$79$	6.53590	0.735346	0.367673	−	0.929955i	$-0.380155\pi$
$79$	6.53590	0.735346	0.367673	+	0.929955i	$0.380155\pi$
$80$	0	0
$81$	4.46410	0.496011
$82$	0	0
$83$	− 10.3923i	− 1.14070i	−0.821401	−	0.570352i	$-0.806807\pi$
$83$	− 10.3923i	− 1.14070i	0.821401	−	0.570352i	$-0.193193\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 6.92820i	− 0.742781i
$88$	0	0
$89$	12.9282	1.37039	0.685193	−	0.728361i	$-0.259718\pi$
$89$	12.9282	1.37039	0.685193	+	0.728361i	$0.259718\pi$
$90$	0	0
$91$	3.46410	0.363137
$92$	0	0
$93$	− 0.143594i	− 0.0148900i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 15.8564i	− 1.60997i	−0.593292	−	0.804987i	$-0.702172\pi$
$97$	− 15.8564i	− 1.60997i	0.593292	−	0.804987i	$-0.297828\pi$
$98$	0	0
$99$	6.73205	0.676597
$100$	0	0
$101$	4.92820	0.490375	0.245187	−	0.969476i	$-0.421151\pi$
$101$	4.92820	0.490375	0.245187	+	0.969476i	$0.421151\pi$
$102$	0	0
$103$	− 6.19615i	− 0.610525i	−0.952268	−	0.305263i	$-0.901256\pi$
$103$	− 6.19615i	− 0.610525i	0.952268	−	0.305263i	$-0.0987442\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	20.0526i	1.93855i	0.245972	+	0.969277i	$0.420893\pi$
$107$	20.0526i	1.93855i	−0.245972	+	0.969277i	$0.579107\pi$
$108$	0	0
$109$	−15.8564	−1.51877	−0.759384	−	0.650643i	$-0.774500\pi$
$109$	−15.8564	−1.51877	−0.759384	+	0.650643i	$0.774500\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 18.3923i	− 1.73020i	−0.501597	−	0.865101i	$-0.667254\pi$
$113$	− 18.3923i	− 1.73020i	0.501597	−	0.865101i	$-0.332746\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	2.46410i	0.227806i
$118$	0	0
$119$	−25.8564	−2.37025
$120$	0	0
$121$	−3.53590	−0.321445
$122$	0	0
$123$	0.392305i	0.0353729i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	1.80385i	0.160066i	0.996792	+	0.0800328i	$0.0255025\pi$
$127$	1.80385i	0.160066i	−0.996792	+	0.0800328i	$0.974497\pi$
$128$	0	0
$129$	5.32051	0.468445
$130$	0	0
$131$	−2.92820	−0.255838	−0.127919	−	0.991785i	$-0.540830\pi$
$131$	−2.92820	−0.255838	−0.127919	+	0.991785i	$0.540830\pi$
$132$	0	0
$133$	− 9.46410i	− 0.820642i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 8.92820i	− 0.762788i	−0.924413	−	0.381394i	$-0.875444\pi$
$137$	− 8.92820i	− 0.762788i	0.924413	−	0.381394i	$-0.124556\pi$
$138$	0	0
$139$	13.4641	1.14201	0.571005	−	0.820947i	$-0.306554\pi$
$139$	13.4641	1.14201	0.571005	+	0.820947i	$0.306554\pi$
$140$	0	0
$141$	3.32051	0.279637
$142$	0	0
$143$	2.73205i	0.228466i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 3.66025i	− 0.301893i
$148$	0	0
$149$	7.85641	0.643622	0.321811	−	0.946804i	$-0.395708\pi$
$149$	7.85641	0.643622	0.321811	+	0.946804i	$0.395708\pi$
$150$	0	0
$151$	11.8038	0.960583	0.480292	−	0.877109i	$-0.340531\pi$
$151$	11.8038	0.960583	0.480292	+	0.877109i	$0.340531\pi$
$152$	0	0
$153$	− 18.3923i	− 1.48693i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	7.85641i	0.627009i	0.949587	+	0.313505i	$0.101503\pi$
$157$	7.85641i	0.627009i	−0.949587	+	0.313505i	$0.898497\pi$
$158$	0	0
$159$	0.392305	0.0311118
$160$	0	0
$161$	−2.53590	−0.199857
$162$	0	0
$163$	− 8.92820i	− 0.699311i	−0.936878	−	0.349655i	$-0.886299\pi$
$163$	− 8.92820i	− 0.699311i	0.936878	−	0.349655i	$-0.113701\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	21.3205i	1.64983i	0.565256	+	0.824915i	$0.308777\pi$
$167$	21.3205i	1.64983i	−0.565256	+	0.824915i	$0.691223\pi$
$168$	0	0
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	6.73205	0.514813
$172$	0	0
$173$	− 2.39230i	− 0.181884i	−0.995856	−	0.0909418i	$-0.971012\pi$
$173$	− 2.39230i	− 0.181884i	0.995856	−	0.0909418i	$-0.0289877\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 4.14359i	− 0.311452i
$178$	0	0
$179$	21.8564	1.63362	0.816812	−	0.576904i	$-0.195739\pi$
$179$	21.8564	1.63362	0.816812	+	0.576904i	$0.195739\pi$
$180$	0	0
$181$	6.53590	0.485810	0.242905	−	0.970050i	$-0.421900\pi$
$181$	6.53590	0.485810	0.242905	+	0.970050i	$0.421900\pi$
$182$	0	0
$183$	− 9.07180i	− 0.670607i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 20.3923i	− 1.49123i
$188$	0	0
$189$	13.8564	1.00791
$190$	0	0
$191$	18.9282	1.36960	0.684798	−	0.728733i	$-0.259890\pi$
$191$	18.9282	1.36960	0.684798	+	0.728733i	$0.259890\pi$
$192$	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
$197$	− 8.92820i	− 0.636108i	−0.948073	−	0.318054i	$-0.896971\pi$
$197$	− 8.92820i	− 0.636108i	0.948073	−	0.318054i	$-0.103029\pi$
$198$	0	0
$199$	6.92820	0.491127	0.245564	−	0.969380i	$-0.421027\pi$
$199$	6.92820	0.491127	0.245564	+	0.969380i	$0.421027\pi$
$200$	0	0
$201$	−9.46410	−0.667546
$202$	0	0
$203$	32.7846i	2.30103i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 1.80385i	− 0.125376i
$208$	0	0
$209$	7.46410	0.516303
$210$	0	0
$211$	2.92820	0.201586	0.100793	−	0.994907i	$-0.467862\pi$
$211$	2.92820	0.201586	0.100793	+	0.994907i	$0.467862\pi$
$212$	0	0
$213$	6.78461i	0.464874i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0.679492i	0.0461269i
$218$	0	0
$219$	2.14359	0.144851
$220$	0	0
$221$	7.46410	0.502090
$222$	0	0
$223$	25.3205i	1.69559i	0.530327	+	0.847793i	$0.322069\pi$
$223$	25.3205i	1.69559i	−0.530327	+	0.847793i	$0.677931\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 18.0000i	− 1.19470i	−0.801980	−	0.597351i	$-0.796220\pi$
$227$	− 18.0000i	− 1.19470i	0.801980	−	0.597351i	$-0.203780\pi$
$228$	0	0
$229$	−19.4641	−1.28622	−0.643112	−	0.765772i	$-0.722357\pi$
$229$	−19.4641	−1.28622	−0.643112	+	0.765772i	$0.722357\pi$
$230$	0	0
$231$	6.92820	0.455842
$232$	0	0
$233$	18.0000i	1.17922i	0.807688	+	0.589610i	$0.200718\pi$
$233$	18.0000i	1.17922i	−0.807688	+	0.589610i	$0.799282\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	4.78461i	0.310794i
$238$	0	0
$239$	−25.2679	−1.63445	−0.817224	−	0.576320i	$-0.804488\pi$
$239$	−25.2679	−1.63445	−0.817224	+	0.576320i	$0.804488\pi$
$240$	0	0
$241$	20.2487	1.30433	0.652167	−	0.758075i	$-0.273860\pi$
$241$	20.2487	1.30433	0.652167	+	0.758075i	$0.273860\pi$
$242$	0	0
$243$	15.2679i	0.979439i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	2.73205i	0.173836i
$248$	0	0
$249$	7.60770	0.482118
$250$	0	0
$251$	23.3205	1.47198	0.735989	−	0.676994i	$-0.236718\pi$
$251$	23.3205	1.47198	0.735989	+	0.676994i	$0.236718\pi$
$252$	0	0
$253$	− 2.00000i	− 0.125739i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 6.00000i	− 0.374270i	−0.982334	−	0.187135i	$-0.940080\pi$
$257$	− 6.00000i	− 0.374270i	0.982334	−	0.187135i	$-0.0599201\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−23.3205	−1.44350
$262$	0	0
$263$	− 11.6603i	− 0.719002i	−0.933145	−	0.359501i	$-0.882947\pi$
$263$	− 11.6603i	− 0.719002i	0.933145	−	0.359501i	$-0.117053\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	9.46410i	0.579194i
$268$	0	0
$269$	0.143594	0.00875505	0.00437753	−	0.999990i	$-0.498607\pi$
$269$	0.143594	0.00875505	0.00437753	+	0.999990i	$0.498607\pi$
$270$	0	0
$271$	−23.1244	−1.40470	−0.702352	−	0.711830i	$-0.747867\pi$
$271$	−23.1244	−1.40470	−0.702352	+	0.711830i	$0.747867\pi$
$272$	0	0
$273$	2.53590i	0.153480i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	21.3205i	1.28103i	0.767948	+	0.640513i	$0.221278\pi$
$277$	21.3205i	1.28103i	−0.767948	+	0.640513i	$0.778722\pi$
$278$	0	0
$279$	−0.483340	−0.0289368
$280$	0	0
$281$	−4.53590	−0.270589	−0.135295	−	0.990805i	$-0.543198\pi$
$281$	−4.53590	−0.270589	−0.135295	+	0.990805i	$0.543198\pi$
$282$	0	0
$283$	− 24.0526i	− 1.42978i	−0.699239	−	0.714888i	$-0.746478\pi$
$283$	− 24.0526i	− 1.42978i	0.699239	−	0.714888i	$-0.253522\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 1.85641i	− 0.109580i
$288$	0	0
$289$	−38.7128	−2.27722
$290$	0	0
$291$	11.6077	0.680455
$292$	0	0
$293$	6.92820i	0.404750i	0.979308	+	0.202375i	$0.0648660\pi$
$293$	6.92820i	0.404750i	−0.979308	+	0.202375i	$0.935134\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	10.9282i	0.634119i
$298$	0	0
$299$	0.732051	0.0423356
$300$	0	0
$301$	−25.1769	−1.45117
$302$	0	0
$303$	3.60770i	0.207257i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	16.5359i	0.943754i	0.881665	+	0.471877i	$0.156423\pi$
$307$	16.5359i	0.943754i	−0.881665	+	0.471877i	$0.843577\pi$
$308$	0	0
$309$	4.53590	0.258038
$310$	0	0
$311$	31.3205	1.77602	0.888012	−	0.459821i	$-0.152086\pi$
$311$	31.3205	1.77602	0.888012	+	0.459821i	$0.152086\pi$
$312$	0	0
$313$	10.3923i	0.587408i	0.955896	+	0.293704i	$0.0948880\pi$
$313$	10.3923i	0.587408i	−0.955896	+	0.293704i	$0.905112\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	1.85641i	0.104266i	0.998640	+	0.0521331i	$0.0166020\pi$
$317$	1.85641i	0.104266i	−0.998640	+	0.0521331i	$0.983398\pi$
$318$	0	0
$319$	−25.8564	−1.44768
$320$	0	0
$321$	−14.6795	−0.819329
$322$	0	0
$323$	− 20.3923i	− 1.13466i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 11.6077i	− 0.641907i
$328$	0	0
$329$	−15.7128	−0.866275
$330$	0	0
$331$	−8.58846	−0.472064	−0.236032	−	0.971745i	$-0.575847\pi$
$331$	−8.58846	−0.472064	−0.236032	+	0.971745i	$0.575847\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 22.3923i	− 1.21979i	−0.792484	−	0.609893i	$-0.791212\pi$
$337$	− 22.3923i	− 1.21979i	0.792484	−	0.609893i	$-0.208788\pi$
$338$	0	0
$339$	13.4641	0.731270
$340$	0	0
$341$	−0.535898	−0.0290205
$342$	0	0
$343$	− 6.92820i	− 0.374088i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 2.19615i	− 0.117896i	−0.998261	−	0.0589478i	$-0.981225\pi$
$347$	− 2.19615i	− 0.117896i	0.998261	−	0.0589478i	$-0.0187746\pi$
$348$	0	0
$349$	−12.5359	−0.671031	−0.335516	−	0.942035i	$-0.608911\pi$
$349$	−12.5359	−0.671031	−0.335516	+	0.942035i	$0.608911\pi$
$350$	0	0
$351$	−4.00000	−0.213504
$352$	0	0
$353$	− 6.92820i	− 0.368751i	−0.982856	−	0.184376i	$-0.940974\pi$
$353$	− 6.92820i	− 0.368751i	0.982856	−	0.184376i	$-0.0590263\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 18.9282i	− 1.00179i
$358$	0	0
$359$	−11.1244	−0.587121	−0.293561	−	0.955940i	$-0.594840\pi$
$359$	−11.1244	−0.587121	−0.293561	+	0.955940i	$0.594840\pi$
$360$	0	0
$361$	−11.5359	−0.607153
$362$	0	0
$363$	− 2.58846i	− 0.135859i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	1.41154i	0.0736819i	0.999321	+	0.0368410i	$0.0117295\pi$
$367$	1.41154i	0.0736819i	−0.999321	+	0.0368410i	$0.988271\pi$
$368$	0	0
$369$	1.32051	0.0687429
$370$	0	0
$371$	−1.85641	−0.0963798
$372$	0	0
$373$	− 7.85641i	− 0.406789i	−0.979097	−	0.203395i	$-0.934803\pi$
$373$	− 7.85641i	− 0.406789i	0.979097	−	0.203395i	$-0.0651974\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 9.46410i	− 0.487426i
$378$	0	0
$379$	20.1962	1.03741	0.518703	−	0.854954i	$-0.326415\pi$
$379$	20.1962	1.03741	0.518703	+	0.854954i	$0.326415\pi$
$380$	0	0
$381$	−1.32051	−0.0676517
$382$	0	0
$383$	− 1.60770i	− 0.0821494i	−0.999156	−	0.0410747i	$-0.986922\pi$
$383$	− 1.60770i	− 0.0821494i	0.999156	−	0.0410747i	$-0.0130782\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 17.9090i	− 0.910364i
$388$	0	0
$389$	18.0000	0.912636	0.456318	−	0.889817i	$-0.349168\pi$
$389$	18.0000	0.912636	0.456318	+	0.889817i	$0.349168\pi$
$390$	0	0
$391$	−5.46410	−0.276331
$392$	0	0
$393$	− 2.14359i	− 0.108130i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	5.07180i	0.254546i	0.991868	+	0.127273i	$0.0406225\pi$
$397$	5.07180i	0.254546i	−0.991868	+	0.127273i	$0.959378\pi$
$398$	0	0
$399$	6.92820	0.346844
$400$	0	0
$401$	−10.7846	−0.538558	−0.269279	−	0.963062i	$-0.586785\pi$
$401$	−10.7846	−0.538558	−0.269279	+	0.963062i	$0.586785\pi$
$402$	0	0
$403$	− 0.196152i	− 0.00977105i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−33.3205	−1.64759	−0.823797	−	0.566886i	$-0.808148\pi$
$409$	−33.3205	−1.64759	−0.823797	+	0.566886i	$0.808148\pi$
$410$	0	0
$411$	6.53590	0.322392
$412$	0	0
$413$	19.6077i	0.964832i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	9.85641i	0.482670i
$418$	0	0
$419$	25.4641	1.24400	0.622001	−	0.783016i	$-0.286320\pi$
$419$	25.4641	1.24400	0.622001	+	0.783016i	$0.286320\pi$
$420$	0	0
$421$	−19.0718	−0.929503	−0.464751	−	0.885441i	$-0.653856\pi$
$421$	−19.0718	−0.929503	−0.464751	+	0.885441i	$0.653856\pi$
$422$	0	0
$423$	− 11.1769i	− 0.543440i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	42.9282i	2.07744i
$428$	0	0
$429$	−2.00000	−0.0965609
$430$	0	0
$431$	−22.0526	−1.06223	−0.531117	−	0.847298i	$-0.678228\pi$
$431$	−22.0526	−1.06223	−0.531117	+	0.847298i	$0.678228\pi$
$432$	0	0
$433$	15.0718i	0.724304i	0.932119	+	0.362152i	$0.117958\pi$
$433$	15.0718i	0.724304i	−0.932119	+	0.362152i	$0.882042\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 2.00000i	− 0.0956730i
$438$	0	0
$439$	13.8564	0.661330	0.330665	−	0.943748i	$-0.392727\pi$
$439$	13.8564	0.661330	0.330665	+	0.943748i	$0.392727\pi$
$440$	0	0
$441$	−12.3205	−0.586691
$442$	0	0
$443$	28.4449i	1.35146i	0.737151	+	0.675728i	$0.236171\pi$
$443$	28.4449i	1.35146i	−0.737151	+	0.675728i	$0.763829\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	5.75129i	0.272027i
$448$	0	0
$449$	14.3923	0.679215	0.339607	−	0.940567i	$-0.389706\pi$
$449$	14.3923	0.679215	0.339607	+	0.940567i	$0.389706\pi$
$450$	0	0
$451$	1.46410	0.0689419
$452$	0	0
$453$	8.64102i	0.405990i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	30.7846i	1.44004i	0.693952	+	0.720022i	$0.255868\pi$
$457$	30.7846i	1.44004i	−0.693952	+	0.720022i	$0.744132\pi$
$458$	0	0
$459$	29.8564	1.39358
$460$	0	0
$461$	−12.2487	−0.570479	−0.285240	−	0.958456i	$-0.592073\pi$
$461$	−12.2487	−0.570479	−0.285240	+	0.958456i	$0.592073\pi$
$462$	0	0
$463$	27.8564i	1.29460i	0.762237	+	0.647298i	$0.224101\pi$
$463$	27.8564i	1.29460i	−0.762237	+	0.647298i	$0.775899\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 18.5885i	− 0.860171i	−0.902788	−	0.430086i	$-0.858483\pi$
$467$	− 18.5885i	− 0.860171i	0.902788	−	0.430086i	$-0.141517\pi$
$468$	0	0
$469$	44.7846	2.06796
$470$	0	0
$471$	−5.75129	−0.265005
$472$	0	0
$473$	− 19.8564i	− 0.912999i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 1.32051i	− 0.0604619i
$478$	0	0
$479$	−11.1244	−0.508285	−0.254142	−	0.967167i	$-0.581793\pi$
$479$	−11.1244	−0.508285	−0.254142	+	0.967167i	$0.581793\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	− 1.85641i	− 0.0844694i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	39.8564i	1.80607i	0.429571	+	0.903033i	$0.358665\pi$
$487$	39.8564i	1.80607i	−0.429571	+	0.903033i	$0.641335\pi$
$488$	0	0
$489$	6.53590	0.295564
$490$	0	0
$491$	−33.4641	−1.51021	−0.755107	−	0.655602i	$-0.772415\pi$
$491$	−33.4641	−1.51021	−0.755107	+	0.655602i	$0.772415\pi$
$492$	0	0
$493$	70.6410i	3.18151i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 32.1051i	− 1.44011i
$498$	0	0
$499$	−24.1962	−1.08317	−0.541584	−	0.840646i	$-0.682175\pi$
$499$	−24.1962	−1.08317	−0.541584	+	0.840646i	$0.682175\pi$
$500$	0	0
$501$	−15.6077	−0.697300
$502$	0	0
$503$	− 41.1244i	− 1.83364i	−0.399296	−	0.916822i	$-0.630745\pi$
$503$	− 41.1244i	− 1.83364i	0.399296	−	0.916822i	$-0.369255\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 0.732051i	− 0.0325115i
$508$	0	0
$509$	3.46410	0.153544	0.0767718	−	0.997049i	$-0.475539\pi$
$509$	3.46410	0.153544	0.0767718	+	0.997049i	$0.475539\pi$
$510$	0	0
$511$	−10.1436	−0.448726
$512$	0	0
$513$	10.9282i	0.482492i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 12.3923i	− 0.545013i
$518$	0	0
$519$	1.75129	0.0768730
$520$	0	0
$521$	8.39230	0.367674	0.183837	−	0.982957i	$-0.441148\pi$
$521$	8.39230	0.367674	0.183837	+	0.982957i	$0.441148\pi$
$522$	0	0
$523$	17.5167i	0.765950i	0.923759	+	0.382975i	$0.125100\pi$
$523$	17.5167i	0.765950i	−0.923759	+	0.382975i	$0.874900\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1.46410i	0.0637773i
$528$	0	0
$529$	22.4641	0.976700
$530$	0	0
$531$	−13.9474	−0.605267
$532$	0	0
$533$	0.535898i	0.0232123i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	16.0000i	0.690451i
$538$	0	0
$539$	−13.6603	−0.588389
$540$	0	0
$541$	38.3923	1.65061	0.825307	−	0.564684i	$-0.191002\pi$
$541$	38.3923	1.65061	0.825307	+	0.564684i	$0.191002\pi$
$542$	0	0
$543$	4.78461i	0.205327i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	8.33975i	0.356582i	0.983978	+	0.178291i	$0.0570568\pi$
$547$	8.33975i	0.356582i	−0.983978	+	0.178291i	$0.942943\pi$
$548$	0	0
$549$	−30.5359	−1.30324
$550$	0	0
$551$	−25.8564	−1.10152
$552$	0	0
$553$	− 22.6410i	− 0.962794i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 21.8564i	− 0.926086i	−0.886336	−	0.463043i	$-0.846758\pi$
$557$	− 21.8564i	− 0.926086i	0.886336	−	0.463043i	$-0.153242\pi$
$558$	0	0
$559$	7.26795	0.307401
$560$	0	0
$561$	14.9282	0.630269
$562$	0	0
$563$	40.7321i	1.71665i	0.513105	+	0.858326i	$0.328495\pi$
$563$	40.7321i	1.71665i	−0.513105	+	0.858326i	$0.671505\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 15.4641i	− 0.649431i
$568$	0	0
$569$	17.4641	0.732133	0.366067	−	0.930589i	$-0.380704\pi$
$569$	17.4641	0.732133	0.366067	+	0.930589i	$0.380704\pi$
$570$	0	0
$571$	−27.3205	−1.14333	−0.571664	−	0.820488i	$-0.693702\pi$
$571$	−27.3205	−1.14333	−0.571664	+	0.820488i	$0.693702\pi$
$572$	0	0
$573$	13.8564i	0.578860i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	30.6410i	1.27560i	0.770201	+	0.637801i	$0.220156\pi$
$577$	30.6410i	1.27560i	−0.770201	+	0.637801i	$0.779844\pi$
$578$	0	0
$579$	1.46410	0.0608460
$580$	0	0
$581$	−36.0000	−1.49353
$582$	0	0
$583$	− 1.46410i	− 0.0606369i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 1.21539i	− 0.0501645i	−0.999685	−	0.0250823i	$-0.992015\pi$
$587$	− 1.21539i	− 0.0501645i	0.999685	−	0.0250823i	$-0.00798477\pi$
$588$	0	0
$589$	−0.535898	−0.0220813
$590$	0	0
$591$	6.53590	0.268851
$592$	0	0
$593$	40.6410i	1.66893i	0.551064	+	0.834463i	$0.314222\pi$
$593$	40.6410i	1.66893i	−0.551064	+	0.834463i	$0.685778\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	5.07180i	0.207575i
$598$	0	0
$599$	33.4641	1.36731	0.683653	−	0.729807i	$-0.260390\pi$
$599$	33.4641	1.36731	0.683653	+	0.729807i	$0.260390\pi$
$600$	0	0
$601$	−3.85641	−0.157306	−0.0786531	−	0.996902i	$-0.525062\pi$
$601$	−3.85641	−0.157306	−0.0786531	+	0.996902i	$0.525062\pi$
$602$	0	0
$603$	31.8564i	1.29729i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	13.5167i	0.548624i	0.961641	+	0.274312i	$0.0884502\pi$
$607$	13.5167i	0.548624i	−0.961641	+	0.274312i	$0.911550\pi$
$608$	0	0
$609$	−24.0000	−0.972529
$610$	0	0
$611$	4.53590	0.183503
$612$	0	0
$613$	− 0.143594i	− 0.00579969i	−0.999996	−	0.00289984i	$-0.999077\pi$
$613$	− 0.143594i	− 0.00579969i	0.999996	−	0.00289984i	$-0.000923050\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	45.7128i	1.84033i	0.391533	+	0.920164i	$0.371945\pi$
$617$	45.7128i	1.84033i	−0.391533	+	0.920164i	$0.628055\pi$
$618$	0	0
$619$	18.3397	0.737137	0.368568	−	0.929601i	$-0.379848\pi$
$619$	18.3397	0.737137	0.368568	+	0.929601i	$0.379848\pi$
$620$	0	0
$621$	2.92820	0.117505
$622$	0	0
$623$	− 44.7846i	− 1.79426i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	5.46410i	0.218215i
$628$	0	0
$629$	0	0
$630$	0	0
$631$	−7.80385	−0.310666	−0.155333	−	0.987862i	$-0.549645\pi$
$631$	−7.80385	−0.310666	−0.155333	+	0.987862i	$0.549645\pi$
$632$	0	0
$633$	2.14359i	0.0852002i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 5.00000i	− 0.198107i
$638$	0	0
$639$	22.8372	0.903424
$640$	0	0
$641$	−0.928203	−0.0366618	−0.0183309	−	0.999832i	$-0.505835\pi$
$641$	−0.928203	−0.0366618	−0.0183309	+	0.999832i	$0.505835\pi$
$642$	0	0
$643$	− 9.60770i	− 0.378891i	−0.981891	−	0.189445i	$-0.939331\pi$
$643$	− 9.60770i	− 0.378891i	0.981891	−	0.189445i	$-0.0606690\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 0.339746i	− 0.0133568i	−0.999978	−	0.00667840i	$-0.997874\pi$
$647$	− 0.339746i	− 0.0133568i	0.999978	−	0.00667840i	$-0.00212582\pi$
$648$	0	0
$649$	−15.4641	−0.607019
$650$	0	0
$651$	−0.497423	−0.0194955
$652$	0	0
$653$	− 33.7128i	− 1.31928i	−0.751580	−	0.659642i	$-0.770708\pi$
$653$	− 33.7128i	− 1.31928i	0.751580	−	0.659642i	$-0.229292\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 7.21539i	− 0.281499i
$658$	0	0
$659$	−39.3205	−1.53171	−0.765855	−	0.643014i	$-0.777684\pi$
$659$	−39.3205	−1.53171	−0.765855	+	0.643014i	$0.777684\pi$
$660$	0	0
$661$	20.9282	0.814013	0.407006	−	0.913425i	$-0.366573\pi$
$661$	20.9282	0.814013	0.407006	+	0.913425i	$0.366573\pi$
$662$	0	0
$663$	5.46410i	0.212208i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	6.92820i	0.268261i
$668$	0	0
$669$	−18.5359	−0.716639
$670$	0	0
$671$	−33.8564	−1.30701
$672$	0	0
$673$	28.5359i	1.09998i	0.835172	+	0.549989i	$0.185368\pi$
$673$	28.5359i	1.09998i	−0.835172	+	0.549989i	$0.814632\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 9.60770i	− 0.369254i	−0.982809	−	0.184627i	$-0.940892\pi$
$677$	− 9.60770i	− 0.369254i	0.982809	−	0.184627i	$-0.0591077\pi$
$678$	0	0
$679$	−54.9282	−2.10795
$680$	0	0
$681$	13.1769	0.504940
$682$	0	0
$683$	24.9282i	0.953851i	0.878944	+	0.476926i	$0.158249\pi$
$683$	24.9282i	0.953851i	−0.878944	+	0.476926i	$0.841751\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 14.2487i	− 0.543622i
$688$	0	0
$689$	0.535898	0.0204161
$690$	0	0
$691$	43.9090	1.67038	0.835188	−	0.549965i	$-0.185359\pi$
$691$	43.9090	1.67038	0.835188	+	0.549965i	$0.185359\pi$
$692$	0	0
$693$	− 23.3205i	− 0.885873i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 4.00000i	− 0.151511i
$698$	0	0
$699$	−13.1769	−0.498397
$700$	0	0
$701$	29.7128	1.12224	0.561119	−	0.827735i	$-0.310371\pi$
$701$	29.7128	1.12224	0.561119	+	0.827735i	$0.310371\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 17.0718i	− 0.642051i
$708$	0	0
$709$	−5.60770	−0.210601	−0.105301	−	0.994440i	$-0.533581\pi$
$709$	−5.60770	−0.210601	−0.105301	+	0.994440i	$0.533581\pi$
$710$	0	0
$711$	16.1051	0.603989
$712$	0	0
$713$	0.143594i	0.00537762i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 18.4974i	− 0.690799i
$718$	0	0
$719$	34.6410	1.29189	0.645946	−	0.763383i	$-0.276463\pi$
$719$	34.6410	1.29189	0.645946	+	0.763383i	$0.276463\pi$
$720$	0	0
$721$	−21.4641	−0.799365
$722$	0	0
$723$	14.8231i	0.551276i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	34.9808i	1.29736i	0.761059	+	0.648682i	$0.224680\pi$
$727$	34.9808i	1.29736i	−0.761059	+	0.648682i	$0.775320\pi$
$728$	0	0
$729$	2.21539	0.0820515
$730$	0	0
$731$	−54.2487	−2.00646
$732$	0	0
$733$	− 15.8564i	− 0.585670i	−0.956163	−	0.292835i	$-0.905401\pi$
$733$	− 15.8564i	− 0.585670i	0.956163	−	0.292835i	$-0.0945986\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	35.3205i	1.30105i
$738$	0	0
$739$	−1.26795	−0.0466423	−0.0233211	−	0.999728i	$-0.507424\pi$
$739$	−1.26795	−0.0466423	−0.0233211	+	0.999728i	$0.507424\pi$
$740$	0	0
$741$	−2.00000	−0.0734718
$742$	0	0
$743$	25.3205i	0.928919i	0.885594	+	0.464460i	$0.153751\pi$
$743$	25.3205i	0.928919i	−0.885594	+	0.464460i	$0.846249\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 25.6077i	− 0.936937i
$748$	0	0
$749$	69.4641	2.53816
$750$	0	0
$751$	45.4641	1.65901	0.829504	−	0.558500i	$-0.188623\pi$
$751$	45.4641	1.65901	0.829504	+	0.558500i	$0.188623\pi$
$752$	0	0
$753$	17.0718i	0.622131i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	23.4641i	0.852817i	0.904531	+	0.426409i	$0.140221\pi$
$757$	23.4641i	0.852817i	−0.904531	+	0.426409i	$0.859779\pi$
$758$	0	0
$759$	1.46410	0.0531435
$760$	0	0
$761$	−14.0000	−0.507500	−0.253750	−	0.967270i	$-0.581664\pi$
$761$	−14.0000	−0.507500	−0.253750	+	0.967270i	$0.581664\pi$
$762$	0	0
$763$	54.9282i	1.98853i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 5.66025i	− 0.204380i
$768$	0	0
$769$	−15.0718	−0.543503	−0.271751	−	0.962367i	$-0.587603\pi$
$769$	−15.0718	−0.543503	−0.271751	+	0.962367i	$0.587603\pi$
$770$	0	0
$771$	4.39230	0.158185
$772$	0	0
$773$	5.07180i	0.182420i	0.995832	+	0.0912099i	$0.0290734\pi$
$773$	5.07180i	0.182420i	−0.995832	+	0.0912099i	$0.970927\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	1.46410	0.0524569
$780$	0	0
$781$	25.3205	0.906039
$782$	0	0
$783$	− 37.8564i	− 1.35288i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 18.3923i	− 0.655615i	−0.944745	−	0.327807i	$-0.893690\pi$
$787$	− 18.3923i	− 0.655615i	0.944745	−	0.327807i	$-0.106310\pi$
$788$	0	0
$789$	8.53590	0.303886
$790$	0	0
$791$	−63.7128	−2.26537
$792$	0	0
$793$	− 12.3923i	− 0.440064i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 46.7846i	− 1.65720i	−0.559844	−	0.828598i	$-0.689139\pi$
$797$	− 46.7846i	− 1.65720i	0.559844	−	0.828598i	$-0.310861\pi$
$798$	0	0
$799$	−33.8564	−1.19775
$800$	0	0
$801$	31.8564	1.12559
$802$	0	0
$803$	− 8.00000i	− 0.282314i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0.105118i	0.00370032i
$808$	0	0
$809$	−8.67949	−0.305155	−0.152577	−	0.988292i	$-0.548757\pi$
$809$	−8.67949	−0.305155	−0.152577	+	0.988292i	$0.548757\pi$
$810$	0	0
$811$	35.9090	1.26093	0.630467	−	0.776216i	$-0.282863\pi$
$811$	35.9090	1.26093	0.630467	+	0.776216i	$0.282863\pi$
$812$	0	0
$813$	− 16.9282i	− 0.593698i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 19.8564i	− 0.694688i
$818$	0	0
$819$	8.53590	0.298268
$820$	0	0
$821$	−12.9282	−0.451197	−0.225599	−	0.974220i	$-0.572434\pi$
$821$	−12.9282	−0.451197	−0.225599	+	0.974220i	$0.572434\pi$
$822$	0	0
$823$	26.5885i	0.926815i	0.886145	+	0.463408i	$0.153373\pi$
$823$	26.5885i	0.926815i	−0.886145	+	0.463408i	$0.846627\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 3.46410i	− 0.120459i	−0.998185	−	0.0602293i	$-0.980817\pi$
$827$	− 3.46410i	− 0.120459i	0.998185	−	0.0602293i	$-0.0191832\pi$
$828$	0	0
$829$	−44.3923	−1.54181	−0.770904	−	0.636951i	$-0.780195\pi$
$829$	−44.3923	−1.54181	−0.770904	+	0.636951i	$0.780195\pi$
$830$	0	0
$831$	−15.6077	−0.541425
$832$	0	0
$833$	37.3205i	1.29308i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 0.784610i	− 0.0271201i
$838$	0	0
$839$	−34.4449	−1.18917	−0.594584	−	0.804033i	$-0.702683\pi$
$839$	−34.4449	−1.18917	−0.594584	+	0.804033i	$0.702683\pi$
$840$	0	0
$841$	60.5692	2.08859
$842$	0	0
$843$	− 3.32051i	− 0.114364i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	12.2487i	0.420871i
$848$	0	0
$849$	17.6077	0.604295
$850$	0	0
$851$	0	0
$852$	0	0
$853$	− 7.71281i	− 0.264082i	−0.991244	−	0.132041i	$-0.957847\pi$
$853$	− 7.71281i	− 0.264082i	0.991244	−	0.132041i	$-0.0421530\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	4.14359i	0.141542i	0.997493	+	0.0707712i	$0.0225460\pi$
$857$	4.14359i	0.141542i	−0.997493	+	0.0707712i	$0.977454\pi$
$858$	0	0
$859$	12.3923	0.422820	0.211410	−	0.977397i	$-0.432194\pi$
$859$	12.3923	0.422820	0.211410	+	0.977397i	$0.432194\pi$
$860$	0	0
$861$	1.35898	0.0463140
$862$	0	0
$863$	− 33.6077i	− 1.14402i	−0.820247	−	0.572010i	$-0.806164\pi$
$863$	− 33.6077i	− 1.14402i	0.820247	−	0.572010i	$-0.193836\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 28.3397i	− 0.962468i
$868$	0	0
$869$	17.8564	0.605737
$870$	0	0
$871$	−12.9282	−0.438055
$872$	0	0
$873$	− 39.0718i	− 1.32238i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	41.7128i	1.40854i	0.709931	+	0.704271i	$0.248726\pi$
$877$	41.7128i	1.40854i	−0.709931	+	0.704271i	$0.751274\pi$
$878$	0	0
$879$	−5.07180	−0.171067
$880$	0	0
$881$	5.46410	0.184090	0.0920451	−	0.995755i	$-0.470660\pi$
$881$	5.46410	0.184090	0.0920451	+	0.995755i	$0.470660\pi$
$882$	0	0
$883$	22.5885i	0.760162i	0.924953	+	0.380081i	$0.124104\pi$
$883$	22.5885i	0.760162i	−0.924953	+	0.380081i	$0.875896\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 12.4449i	− 0.417858i	−0.977931	−	0.208929i	$-0.933002\pi$
$887$	− 12.4449i	− 0.417858i	0.977931	−	0.208929i	$-0.0669977\pi$
$888$	0	0
$889$	6.24871	0.209575
$890$	0	0
$891$	12.1962	0.408586
$892$	0	0
$893$	− 12.3923i	− 0.414693i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0.535898i	0.0178931i
$898$	0	0
$899$	1.85641	0.0619146
$900$	0	0
$901$	−4.00000	−0.133259
$902$	0	0
$903$	− 18.4308i	− 0.613338i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	36.8372i	1.22316i	0.791183	+	0.611579i	$0.209465\pi$
$907$	36.8372i	1.22316i	−0.791183	+	0.611579i	$0.790535\pi$
$908$	0	0
$909$	12.1436	0.402778
$910$	0	0
$911$	−20.7846	−0.688625	−0.344312	−	0.938855i	$-0.611888\pi$
$911$	−20.7846	−0.688625	−0.344312	+	0.938855i	$0.611888\pi$
$912$	0	0
$913$	− 28.3923i	− 0.939648i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	10.1436i	0.334971i
$918$	0	0
$919$	−39.3205	−1.29706	−0.648532	−	0.761187i	$-0.724617\pi$
$919$	−39.3205	−1.29706	−0.648532	+	0.761187i	$0.724617\pi$
$920$	0	0
$921$	−12.1051	−0.398877
$922$	0	0
$923$	9.26795i	0.305058i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 15.2679i	− 0.501465i
$928$	0	0
$929$	−43.1769	−1.41659	−0.708294	−	0.705917i	$-0.750535\pi$
$929$	−43.1769	−1.41659	−0.708294	+	0.705917i	$0.750535\pi$
$930$	0	0
$931$	−13.6603	−0.447697
$932$	0	0
$933$	22.9282i	0.750636i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	27.0718i	0.884397i	0.896917	+	0.442199i	$0.145801\pi$
$937$	27.0718i	0.884397i	−0.896917	+	0.442199i	$0.854199\pi$
$938$	0	0
$939$	−7.60770	−0.248268
$940$	0	0
$941$	22.3923	0.729968	0.364984	−	0.931014i	$-0.381074\pi$
$941$	22.3923	0.729968	0.364984	+	0.931014i	$0.381074\pi$
$942$	0	0
$943$	− 0.392305i	− 0.0127752i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 22.3923i	− 0.727652i	−0.931467	−	0.363826i	$-0.881470\pi$
$947$	− 22.3923i	− 0.727652i	0.931467	−	0.363826i	$-0.118530\pi$
$948$	0	0
$949$	2.92820	0.0950535
$950$	0	0
$951$	−1.35898	−0.0440681
$952$	0	0
$953$	− 1.21539i	− 0.0393704i	−0.999806	−	0.0196852i	$-0.993734\pi$
$953$	− 1.21539i	− 0.0393704i	0.999806	−	0.0196852i	$-0.00626639\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 18.9282i	− 0.611862i
$958$	0	0
$959$	−30.9282	−0.998724
$960$	0	0
$961$	−30.9615	−0.998759
$962$	0	0
$963$	49.4115i	1.59226i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	18.0000i	0.578841i	0.957202	+	0.289420i	$0.0934626\pi$
$967$	18.0000i	0.578841i	−0.957202	+	0.289420i	$0.906537\pi$
$968$	0	0
$969$	14.9282	0.479563
$970$	0	0
$971$	43.7128	1.40281	0.701405	−	0.712762i	$-0.252556\pi$
$971$	43.7128	1.40281	0.701405	+	0.712762i	$0.252556\pi$
$972$	0	0
$973$	− 46.6410i	− 1.49524i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	2.92820i	0.0936815i	0.998902	+	0.0468408i	$0.0149153\pi$
$977$	2.92820i	0.0936815i	−0.998902	+	0.0468408i	$0.985085\pi$
$978$	0	0
$979$	35.3205	1.12885
$980$	0	0
$981$	−39.0718	−1.24747
$982$	0	0
$983$	− 15.8564i	− 0.505741i	−0.967500	−	0.252870i	$-0.918625\pi$
$983$	− 15.8564i	− 0.505741i	0.967500	−	0.252870i	$-0.0813746\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 11.5026i	− 0.366131i
$988$	0	0
$989$	−5.32051	−0.169182
$990$	0	0
$991$	−53.8564	−1.71081	−0.855403	−	0.517964i	$-0.826690\pi$
$991$	−53.8564	−1.71081	−0.855403	+	0.517964i	$0.826690\pi$
$992$	0	0
$993$	− 6.28719i	− 0.199518i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	9.32051i	0.295183i	0.989048	+	0.147592i	$0.0471522\pi$
$997$	9.32051i	0.295183i	−0.989048	+	0.147592i	$0.952848\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2600.2.d.m.1249.3		4
5.2	odd	4		520.2.a.e.1.2	✓	2
5.3	odd	4		2600.2.a.v.1.1		2
5.4	even	2	inner	2600.2.d.m.1249.2		4
15.2	even	4		4680.2.a.bd.1.2		2
20.3	even	4		5200.2.a.bn.1.2		2
20.7	even	4		1040.2.a.l.1.1		2
40.27	even	4		4160.2.a.w.1.2		2
40.37	odd	4		4160.2.a.bk.1.1		2
60.47	odd	4		9360.2.a.cr.1.1		2
65.12	odd	4		6760.2.a.o.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
520.2.a.e.1.2	✓	2	5.2	odd	4
1040.2.a.l.1.1		2	20.7	even	4
2600.2.a.v.1.1		2	5.3	odd	4
2600.2.d.m.1249.2		4	5.4	even	2	inner
2600.2.d.m.1249.3		4	1.1	even	1	trivial
4160.2.a.w.1.2		2	40.27	even	4
4160.2.a.bk.1.1		2	40.37	odd	4
4680.2.a.bd.1.2		2	15.2	even	4
5200.2.a.bn.1.2		2	20.3	even	4
6760.2.a.o.1.2		2	65.12	odd	4
9360.2.a.cr.1.1		2	60.47	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:	\( N \)	\(=\)	\( 2600 = 2^{3} \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2600.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+0.732051i q^{3} -3.46410i q^{7} +2.46410 q^{9} +2.73205 q^{11} +1.00000i q^{13} -7.46410i q^{17} +2.73205 q^{19} +2.53590 q^{21} -0.732051i q^{23} +4.00000i q^{27} -9.46410 q^{29} -0.196152 q^{31} +2.00000i q^{33} -0.732051 q^{39} +0.535898 q^{41} -7.26795i q^{43} -4.53590i q^{47} -5.00000 q^{49} +5.46410 q^{51} -0.535898i q^{53} +2.00000i q^{57} -5.66025 q^{59} -12.3923 q^{61} -8.53590i q^{63} +12.9282i q^{67} +0.535898 q^{69} +9.26795 q^{71} -2.92820i q^{73} -9.46410i q^{77} +6.53590 q^{79} +4.46410 q^{81} -10.3923i q^{83} -6.92820i q^{87} +12.9282 q^{89} +3.46410 q^{91} -0.143594i q^{93} -15.8564i q^{97} +6.73205 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{9} + 4 q^{11} + 4 q^{19} + 24 q^{21} - 24 q^{29} + 20 q^{31} + 4 q^{39} + 16 q^{41} - 20 q^{49} + 8 q^{51} + 12 q^{59} - 8 q^{61} + 16 q^{69} + 44 q^{71} + 40 q^{79} + 4 q^{81} + 24 q^{89} + 20 q^{99}+O(q^{100}) \) 4 * q - 4 * q^9 + 4 * q^11 + 4 * q^19 + 24 * q^21 - 24 * q^29 + 20 * q^31 + 4 * q^39 + 16 * q^41 - 20 * q^49 + 8 * q^51 + 12 * q^59 - 8 * q^61 + 16 * q^69 + 44 * q^71 + 40 * q^79 + 4 * q^81 + 24 * q^89 + 20 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more