LMFDB - Embedded newform 832.2.b.b.417.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [832,2,Mod(417,832)] 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("832.417"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(832, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 832 = 2^{6} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	832.b (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,12,0,0,0,0,0,0,0,12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(15)] == traces)

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

Self dual:	no
Analytic conductor:	$6.64355344817$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			417.1
Root			$0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	832.417
Dual form			832.2.b.b.417.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.41421i q^{3} +1.82843i q^{5} +4.41421 q^{7} -2.82843 q^{9} -0.828427i q^{11} +1.00000i q^{13} +4.41421 q^{15} +1.00000 q^{17} +5.65685i q^{19} -10.6569i q^{21} +8.82843 q^{23} +1.65685 q^{25} -0.414214i q^{27} +3.65685i q^{29} -3.65685 q^{31} -2.00000 q^{33} +8.07107i q^{35} -7.00000i q^{37} +2.41421 q^{39} -9.65685 q^{41} -8.41421i q^{43} -5.17157i q^{45} -0.757359 q^{47} +12.4853 q^{49} -2.41421i q^{51} -3.65685i q^{53} +1.51472 q^{55} +13.6569 q^{57} +8.00000i q^{59} -12.4853 q^{63} -1.82843 q^{65} -8.82843i q^{67} -21.3137i q^{69} -11.7279 q^{71} -1.65685 q^{73} -4.00000i q^{75} -3.65685i q^{77} +16.1421 q^{79} -9.48528 q^{81} -12.1421i q^{83} +1.82843i q^{85} +8.82843 q^{87} -13.6569 q^{89} +4.41421i q^{91} +8.82843i q^{93} -10.3431 q^{95} -7.65685 q^{97} +2.34315i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 12 q^{7} + 12 q^{15} + 4 q^{17} + 24 q^{23} - 16 q^{25} + 8 q^{31} - 8 q^{33} + 4 q^{39} - 16 q^{41} - 20 q^{47} + 16 q^{49} + 40 q^{55} + 32 q^{57} - 16 q^{63} + 4 q^{65} + 4 q^{71} + 16 q^{73} + 8 q^{79}+ \cdots - 8 q^{97}+O(q^{100}) $ 4 * q + 12 * q^7 + 12 * q^15 + 4 * q^17 + 24 * q^23 - 16 * q^25 + 8 * q^31 - 8 * q^33 + 4 * q^39 - 16 * q^41 - 20 * q^47 + 16 * q^49 + 40 * q^55 + 32 * q^57 - 16 * q^63 + 4 * q^65 + 4 * q^71 + 16 * q^73 + 8 * q^79 - 4 * q^81 + 24 * q^87 - 32 * q^89 - 64 * q^95 - 8 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/832\mathbb{Z}\right)^\times$.

$n$	$261$	$703$	$769$
$\chi(n)$	$-1$	$1$	$1$

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
$3$	− 2.41421i	− 1.39385i	−0.717146	−	0.696923i	$-0.754552\pi$
$3$	− 2.41421i	− 1.39385i	0.717146	−	0.696923i	$-0.245448\pi$
$4$	0	0
$5$	1.82843i	0.817697i	0.912602	+	0.408849i	$0.134070\pi$
$5$	1.82843i	0.817697i	−0.912602	+	0.408849i	$0.865930\pi$
$6$	0	0
$7$	4.41421	1.66842	0.834208	−	0.551450i	$-0.185925\pi$
$7$	4.41421	1.66842	0.834208	+	0.551450i	$0.185925\pi$
$8$	0	0
$9$	−2.82843	−0.942809
$10$	0	0
$11$	− 0.828427i	− 0.249780i	−0.992171	−	0.124890i	$-0.960142\pi$
$11$	− 0.828427i	− 0.249780i	0.992171	−	0.124890i	$-0.0398578\pi$
$12$	0	0
$13$	1.00000i	0.277350i
$13$	1.00000i	0.277350i
$14$	0	0
$15$	4.41421	1.13975
$16$	0	0
$17$	1.00000	0.242536	0.121268	−	0.992620i	$-0.461304\pi$
$17$	1.00000	0.242536	0.121268	+	0.992620i	$0.461304\pi$
$18$	0	0
$19$	5.65685i	1.29777i	0.760886	+	0.648886i	$0.224765\pi$
$19$	5.65685i	1.29777i	−0.760886	+	0.648886i	$0.775235\pi$
$20$	0	0
$21$	− 10.6569i	− 2.32552i
$22$	0	0
$23$	8.82843	1.84085	0.920427	−	0.390914i	$-0.127841\pi$
$23$	8.82843	1.84085	0.920427	+	0.390914i	$0.127841\pi$
$24$	0	0
$25$	1.65685	0.331371
$26$	0	0
$27$	− 0.414214i	− 0.0797154i
$28$	0	0
$29$	3.65685i	0.679061i	0.940595	+	0.339530i	$0.110268\pi$
$29$	3.65685i	0.679061i	−0.940595	+	0.339530i	$0.889732\pi$
$30$	0	0
$31$	−3.65685	−0.656790	−0.328395	−	0.944540i	$-0.606508\pi$
$31$	−3.65685	−0.656790	−0.328395	+	0.944540i	$0.606508\pi$
$32$	0	0
$33$	−2.00000	−0.348155
$34$	0	0
$35$	8.07107i	1.36426i
$36$	0	0
$37$	− 7.00000i	− 1.15079i	−0.817875	−	0.575396i	$-0.804848\pi$
$37$	− 7.00000i	− 1.15079i	0.817875	−	0.575396i	$-0.195152\pi$
$38$	0	0
$39$	2.41421	0.386584
$40$	0	0
$41$	−9.65685	−1.50815	−0.754074	−	0.656790i	$-0.771914\pi$
$41$	−9.65685	−1.50815	−0.754074	+	0.656790i	$0.771914\pi$
$42$	0	0
$43$	− 8.41421i	− 1.28316i	−0.767058	−	0.641578i	$-0.778280\pi$
$43$	− 8.41421i	− 1.28316i	0.767058	−	0.641578i	$-0.221720\pi$
$44$	0	0
$45$	− 5.17157i	− 0.770933i
$46$	0	0
$47$	−0.757359	−0.110472	−0.0552361	−	0.998473i	$-0.517591\pi$
$47$	−0.757359	−0.110472	−0.0552361	+	0.998473i	$0.517591\pi$
$48$	0	0
$49$	12.4853	1.78361
$50$	0	0
$51$	− 2.41421i	− 0.338058i
$52$	0	0
$53$	− 3.65685i	− 0.502308i	−0.967947	−	0.251154i	$-0.919190\pi$
$53$	− 3.65685i	− 0.502308i	0.967947	−	0.251154i	$-0.0808100\pi$
$54$	0	0
$55$	1.51472	0.204245
$56$	0	0
$57$	13.6569	1.80889
$58$	0	0
$59$	8.00000i	1.04151i	0.853706	+	0.520756i	$0.174350\pi$
$59$	8.00000i	1.04151i	−0.853706	+	0.520756i	$0.825650\pi$
$60$	0	0
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	0	0
$63$	−12.4853	−1.57300
$64$	0	0
$65$	−1.82843	−0.226788
$66$	0	0
$67$	− 8.82843i	− 1.07856i	−0.842125	−	0.539282i	$-0.818696\pi$
$67$	− 8.82843i	− 1.07856i	0.842125	−	0.539282i	$-0.181304\pi$
$68$	0	0
$69$	− 21.3137i	− 2.56587i
$70$	0	0
$71$	−11.7279	−1.39185	−0.695924	−	0.718115i	$-0.745005\pi$
$71$	−11.7279	−1.39185	−0.695924	+	0.718115i	$0.745005\pi$
$72$	0	0
$73$	−1.65685	−0.193920	−0.0969601	−	0.995288i	$-0.530912\pi$
$73$	−1.65685	−0.193920	−0.0969601	+	0.995288i	$0.530912\pi$
$74$	0	0
$75$	− 4.00000i	− 0.461880i
$76$	0	0
$77$	− 3.65685i	− 0.416737i
$78$	0	0
$79$	16.1421	1.81613	0.908066	−	0.418827i	$-0.137559\pi$
$79$	16.1421	1.81613	0.908066	+	0.418827i	$0.137559\pi$
$80$	0	0
$81$	−9.48528	−1.05392
$82$	0	0
$83$	− 12.1421i	− 1.33277i	−0.745607	−	0.666386i	$-0.767840\pi$
$83$	− 12.1421i	− 1.33277i	0.745607	−	0.666386i	$-0.232160\pi$
$84$	0	0
$85$	1.82843i	0.198321i
$86$	0	0
$87$	8.82843	0.946507
$88$	0	0
$89$	−13.6569	−1.44762	−0.723812	−	0.689997i	$-0.757612\pi$
$89$	−13.6569	−1.44762	−0.723812	+	0.689997i	$0.757612\pi$
$90$	0	0
$91$	4.41421i	0.462735i
$92$	0	0
$93$	8.82843i	0.915465i
$94$	0	0
$95$	−10.3431	−1.06118
$96$	0	0
$97$	−7.65685	−0.777436	−0.388718	−	0.921357i	$-0.627082\pi$
$97$	−7.65685	−0.777436	−0.388718	+	0.921357i	$0.627082\pi$
$98$	0	0
$99$	2.34315i	0.235495i
$100$	0	0
$101$	17.6569i	1.75692i	0.477813	+	0.878461i	$0.341429\pi$
$101$	17.6569i	1.75692i	−0.477813	+	0.878461i	$0.658571\pi$
$102$	0	0
$103$	16.1421	1.59053	0.795266	−	0.606261i	$-0.207331\pi$
$103$	16.1421	1.59053	0.795266	+	0.606261i	$0.207331\pi$
$104$	0	0
$105$	19.4853	1.90157
$106$	0	0
$107$	− 6.00000i	− 0.580042i	−0.957020	−	0.290021i	$-0.906338\pi$
$107$	− 6.00000i	− 0.580042i	0.957020	−	0.290021i	$-0.0936623\pi$
$108$	0	0
$109$	− 3.34315i	− 0.320215i	−0.987100	−	0.160108i	$-0.948816\pi$
$109$	− 3.34315i	− 0.320215i	0.987100	−	0.160108i	$-0.0511841\pi$
$110$	0	0
$111$	−16.8995	−1.60403
$112$	0	0
$113$	−10.0000	−0.940721	−0.470360	−	0.882474i	$-0.655876\pi$
$113$	−10.0000	−0.940721	−0.470360	+	0.882474i	$0.655876\pi$
$114$	0	0
$115$	16.1421i	1.50526i
$116$	0	0
$117$	− 2.82843i	− 0.261488i
$118$	0	0
$119$	4.41421	0.404650
$120$	0	0
$121$	10.3137	0.937610
$122$	0	0
$123$	23.3137i	2.10213i
$124$	0	0
$125$	12.1716i	1.08866i
$126$	0	0
$127$	−8.82843	−0.783396	−0.391698	−	0.920094i	$-0.628112\pi$
$127$	−8.82843	−0.783396	−0.391698	+	0.920094i	$0.628112\pi$
$128$	0	0
$129$	−20.3137	−1.78852
$130$	0	0
$131$	8.41421i	0.735153i	0.929993	+	0.367577i	$0.119812\pi$
$131$	8.41421i	0.735153i	−0.929993	+	0.367577i	$0.880188\pi$
$132$	0	0
$133$	24.9706i	2.16522i
$134$	0	0
$135$	0.757359	0.0651831
$136$	0	0
$137$	4.34315	0.371060	0.185530	−	0.982639i	$-0.440600\pi$
$137$	4.34315	0.371060	0.185530	+	0.982639i	$0.440600\pi$
$138$	0	0
$139$	1.58579i	0.134505i	0.997736	+	0.0672523i	$0.0214232\pi$
$139$	1.58579i	0.134505i	−0.997736	+	0.0672523i	$0.978577\pi$
$140$	0	0
$141$	1.82843i	0.153981i
$142$	0	0
$143$	0.828427	0.0692766
$144$	0	0
$145$	−6.68629	−0.555266
$146$	0	0
$147$	− 30.1421i	− 2.48608i
$148$	0	0
$149$	21.3137i	1.74609i	0.487642	+	0.873044i	$0.337857\pi$
$149$	21.3137i	1.74609i	−0.487642	+	0.873044i	$0.662143\pi$
$150$	0	0
$151$	−0.757359	−0.0616330	−0.0308165	−	0.999525i	$-0.509811\pi$
$151$	−0.757359	−0.0616330	−0.0308165	+	0.999525i	$0.509811\pi$
$152$	0	0
$153$	−2.82843	−0.228665
$154$	0	0
$155$	− 6.68629i	− 0.537056i
$156$	0	0
$157$	10.3431i	0.825473i	0.910850	+	0.412736i	$0.135427\pi$
$157$	10.3431i	0.825473i	−0.910850	+	0.412736i	$0.864573\pi$
$158$	0	0
$159$	−8.82843	−0.700140
$160$	0	0
$161$	38.9706	3.07131
$162$	0	0
$163$	14.4853i	1.13457i	0.823520	+	0.567287i	$0.192007\pi$
$163$	14.4853i	1.13457i	−0.823520	+	0.567287i	$0.807993\pi$
$164$	0	0
$165$	− 3.65685i	− 0.284686i
$166$	0	0
$167$	3.65685	0.282976	0.141488	−	0.989940i	$-0.454811\pi$
$167$	3.65685	0.282976	0.141488	+	0.989940i	$0.454811\pi$
$168$	0	0
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	− 16.0000i	− 1.22355i
$172$	0	0
$173$	14.0000i	1.06440i	0.846619	+	0.532200i	$0.178635\pi$
$173$	14.0000i	1.06440i	−0.846619	+	0.532200i	$0.821365\pi$
$174$	0	0
$175$	7.31371	0.552864
$176$	0	0
$177$	19.3137	1.45171
$178$	0	0
$179$	− 12.8995i	− 0.964154i	−0.876129	−	0.482077i	$-0.839883\pi$
$179$	− 12.8995i	− 0.964154i	0.876129	−	0.482077i	$-0.160117\pi$
$180$	0	0
$181$	− 3.65685i	− 0.271812i	−0.990722	−	0.135906i	$-0.956606\pi$
$181$	− 3.65685i	− 0.271812i	0.990722	−	0.135906i	$-0.0433945\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	12.7990	0.941000
$186$	0	0
$187$	− 0.828427i	− 0.0605806i
$188$	0	0
$189$	− 1.82843i	− 0.132999i
$190$	0	0
$191$	−19.1716	−1.38721	−0.693603	−	0.720357i	$-0.743978\pi$
$191$	−19.1716	−1.38721	−0.693603	+	0.720357i	$0.743978\pi$
$192$	0	0
$193$	−1.31371	−0.0945628	−0.0472814	−	0.998882i	$-0.515056\pi$
$193$	−1.31371	−0.0945628	−0.0472814	+	0.998882i	$0.515056\pi$
$194$	0	0
$195$	4.41421i	0.316108i
$196$	0	0
$197$	7.00000i	0.498729i	0.968410	+	0.249365i	$0.0802218\pi$
$197$	7.00000i	0.498729i	−0.968410	+	0.249365i	$0.919778\pi$
$198$	0	0
$199$	1.51472	0.107376	0.0536878	−	0.998558i	$-0.482902\pi$
$199$	1.51472	0.107376	0.0536878	+	0.998558i	$0.482902\pi$
$200$	0	0
$201$	−21.3137	−1.50335
$202$	0	0
$203$	16.1421i	1.13296i
$204$	0	0
$205$	− 17.6569i	− 1.23321i
$206$	0	0
$207$	−24.9706	−1.73557
$208$	0	0
$209$	4.68629	0.324158
$210$	0	0
$211$	4.41421i	0.303887i	0.988389	+	0.151943i	$0.0485532\pi$
$211$	4.41421i	0.303887i	−0.988389	+	0.151943i	$0.951447\pi$
$212$	0	0
$213$	28.3137i	1.94002i
$214$	0	0
$215$	15.3848	1.04923
$216$	0	0
$217$	−16.1421	−1.09580
$218$	0	0
$219$	4.00000i	0.270295i
$220$	0	0
$221$	1.00000i	0.0672673i
$222$	0	0
$223$	−9.58579	−0.641912	−0.320956	−	0.947094i	$-0.604004\pi$
$223$	−9.58579	−0.641912	−0.320956	+	0.947094i	$0.604004\pi$
$224$	0	0
$225$	−4.68629	−0.312419
$226$	0	0
$227$	− 12.9706i	− 0.860886i	−0.902618	−	0.430443i	$-0.858357\pi$
$227$	− 12.9706i	− 0.860886i	0.902618	−	0.430443i	$-0.141643\pi$
$228$	0	0
$229$	− 19.4853i	− 1.28762i	−0.765184	−	0.643812i	$-0.777352\pi$
$229$	− 19.4853i	− 1.28762i	0.765184	−	0.643812i	$-0.222648\pi$
$230$	0	0
$231$	−8.82843	−0.580868
$232$	0	0
$233$	−17.8284	−1.16798	−0.583990	−	0.811761i	$-0.698509\pi$
$233$	−17.8284	−1.16798	−0.583990	+	0.811761i	$0.698509\pi$
$234$	0	0
$235$	− 1.38478i	− 0.0903328i
$236$	0	0
$237$	− 38.9706i	− 2.53141i
$238$	0	0
$239$	−22.0711	−1.42766	−0.713829	−	0.700320i	$-0.753041\pi$
$239$	−22.0711	−1.42766	−0.713829	+	0.700320i	$0.753041\pi$
$240$	0	0
$241$	−12.9706	−0.835507	−0.417754	−	0.908560i	$-0.637183\pi$
$241$	−12.9706	−0.835507	−0.417754	+	0.908560i	$0.637183\pi$
$242$	0	0
$243$	21.6569i	1.38929i
$244$	0	0
$245$	22.8284i	1.45845i
$246$	0	0
$247$	−5.65685	−0.359937
$248$	0	0
$249$	−29.3137	−1.85768
$250$	0	0
$251$	− 13.3137i	− 0.840354i	−0.907442	−	0.420177i	$-0.861968\pi$
$251$	− 13.3137i	− 0.840354i	0.907442	−	0.420177i	$-0.138032\pi$
$252$	0	0
$253$	− 7.31371i	− 0.459809i
$254$	0	0
$255$	4.41421	0.276429
$256$	0	0
$257$	22.6569	1.41330	0.706648	−	0.707565i	$-0.250207\pi$
$257$	22.6569	1.41330	0.706648	+	0.707565i	$0.250207\pi$
$258$	0	0
$259$	− 30.8995i	− 1.92000i
$260$	0	0
$261$	− 10.3431i	− 0.640225i
$262$	0	0
$263$	−17.6569	−1.08877	−0.544384	−	0.838836i	$-0.683237\pi$
$263$	−17.6569	−1.08877	−0.544384	+	0.838836i	$0.683237\pi$
$264$	0	0
$265$	6.68629	0.410736
$266$	0	0
$267$	32.9706i	2.01777i
$268$	0	0
$269$	0	0	1.00000			$0$
$269$	0	0	−1.00000			$\pi$
$270$	0	0
$271$	9.58579	0.582295	0.291148	−	0.956678i	$-0.405963\pi$
$271$	9.58579	0.582295	0.291148	+	0.956678i	$0.405963\pi$
$272$	0	0
$273$	10.6569	0.644982
$274$	0	0
$275$	− 1.37258i	− 0.0827699i
$276$	0	0
$277$	24.9706i	1.50034i	0.661247	+	0.750168i	$0.270027\pi$
$277$	24.9706i	1.50034i	−0.661247	+	0.750168i	$0.729973\pi$
$278$	0	0
$279$	10.3431	0.619228
$280$	0	0
$281$	4.00000	0.238620	0.119310	−	0.992857i	$-0.461932\pi$
$281$	4.00000	0.238620	0.119310	+	0.992857i	$0.461932\pi$
$282$	0	0
$283$	− 26.2843i	− 1.56244i	−0.624257	−	0.781219i	$-0.714598\pi$
$283$	− 26.2843i	− 1.56244i	0.624257	−	0.781219i	$-0.285402\pi$
$284$	0	0
$285$	24.9706i	1.47913i
$286$	0	0
$287$	−42.6274	−2.51622
$288$	0	0
$289$	−16.0000	−0.941176
$290$	0	0
$291$	18.4853i	1.08363i
$292$	0	0
$293$	3.34315i	0.195309i	0.995220	+	0.0976543i	$0.0311340\pi$
$293$	3.34315i	0.195309i	−0.995220	+	0.0976543i	$0.968866\pi$
$294$	0	0
$295$	−14.6274	−0.851641
$296$	0	0
$297$	−0.343146	−0.0199113
$298$	0	0
$299$	8.82843i	0.510561i
$300$	0	0
$301$	− 37.1421i	− 2.14084i
$302$	0	0
$303$	42.6274	2.44888
$304$	0	0
$305$	0	0
$306$	0	0
$307$	20.0000i	1.14146i	0.821138	+	0.570730i	$0.193340\pi$
$307$	20.0000i	1.14146i	−0.821138	+	0.570730i	$0.806660\pi$
$308$	0	0
$309$	− 38.9706i	− 2.21696i
$310$	0	0
$311$	−10.3431	−0.586506	−0.293253	−	0.956035i	$-0.594738\pi$
$311$	−10.3431	−0.586506	−0.293253	+	0.956035i	$0.594738\pi$
$312$	0	0
$313$	33.1421	1.87330	0.936652	−	0.350261i	$-0.113907\pi$
$313$	33.1421	1.87330	0.936652	+	0.350261i	$0.113907\pi$
$314$	0	0
$315$	− 22.8284i	− 1.28624i
$316$	0	0
$317$	− 28.6274i	− 1.60788i	−0.594713	−	0.803938i	$-0.702734\pi$
$317$	− 28.6274i	− 1.60788i	0.594713	−	0.803938i	$-0.297266\pi$
$318$	0	0
$319$	3.02944	0.169616
$320$	0	0
$321$	−14.4853	−0.808490
$322$	0	0
$323$	5.65685i	0.314756i
$324$	0	0
$325$	1.65685i	0.0919057i
$326$	0	0
$327$	−8.07107	−0.446331
$328$	0	0
$329$	−3.34315	−0.184314
$330$	0	0
$331$	3.17157i	0.174325i	0.996194	+	0.0871627i	$0.0277800\pi$
$331$	3.17157i	0.174325i	−0.996194	+	0.0871627i	$0.972220\pi$
$332$	0	0
$333$	19.7990i	1.08498i
$334$	0	0
$335$	16.1421	0.881939
$336$	0	0
$337$	−12.1716	−0.663028	−0.331514	−	0.943450i	$-0.607559\pi$
$337$	−12.1716	−0.663028	−0.331514	+	0.943450i	$0.607559\pi$
$338$	0	0
$339$	24.1421i	1.31122i
$340$	0	0
$341$	3.02944i	0.164053i
$342$	0	0
$343$	24.2132	1.30739
$344$	0	0
$345$	38.9706	2.09810
$346$	0	0
$347$	− 20.7574i	− 1.11431i	−0.830407	−	0.557157i	$-0.811892\pi$
$347$	− 20.7574i	− 1.11431i	0.830407	−	0.557157i	$-0.188108\pi$
$348$	0	0
$349$	− 1.82843i	− 0.0978735i	−0.998802	−	0.0489367i	$-0.984417\pi$
$349$	− 1.82843i	− 0.0978735i	0.998802	−	0.0489367i	$-0.0155833\pi$
$350$	0	0
$351$	0.414214	0.0221091
$352$	0	0
$353$	−12.3431	−0.656959	−0.328480	−	0.944511i	$-0.606536\pi$
$353$	−12.3431	−0.656959	−0.328480	+	0.944511i	$0.606536\pi$
$354$	0	0
$355$	− 21.4437i	− 1.13811i
$356$	0	0
$357$	− 10.6569i	− 0.564021i
$358$	0	0
$359$	24.3431	1.28478	0.642391	−	0.766377i	$-0.277943\pi$
$359$	24.3431	1.28478	0.642391	+	0.766377i	$0.277943\pi$
$360$	0	0
$361$	−13.0000	−0.684211
$362$	0	0
$363$	− 24.8995i	− 1.30688i
$364$	0	0
$365$	− 3.02944i	− 0.158568i
$366$	0	0
$367$	23.4558	1.22439	0.612193	−	0.790709i	$-0.290288\pi$
$367$	23.4558	1.22439	0.612193	+	0.790709i	$0.290288\pi$
$368$	0	0
$369$	27.3137	1.42189
$370$	0	0
$371$	− 16.1421i	− 0.838058i
$372$	0	0
$373$	14.6274i	0.757379i	0.925524	+	0.378689i	$0.123625\pi$
$373$	14.6274i	0.757379i	−0.925524	+	0.378689i	$0.876375\pi$
$374$	0	0
$375$	29.3848	1.51742
$376$	0	0
$377$	−3.65685	−0.188338
$378$	0	0
$379$	2.48528i	0.127660i	0.997961	+	0.0638302i	$0.0203316\pi$
$379$	2.48528i	0.127660i	−0.997961	+	0.0638302i	$0.979668\pi$
$380$	0	0
$381$	21.3137i	1.09193i
$382$	0	0
$383$	−2.27208	−0.116098	−0.0580489	−	0.998314i	$-0.518488\pi$
$383$	−2.27208	−0.116098	−0.0580489	+	0.998314i	$0.518488\pi$
$384$	0	0
$385$	6.68629	0.340765
$386$	0	0
$387$	23.7990i	1.20977i
$388$	0	0
$389$	− 28.0000i	− 1.41966i	−0.704375	−	0.709828i	$-0.748773\pi$
$389$	− 28.0000i	− 1.41966i	0.704375	−	0.709828i	$-0.251227\pi$
$390$	0	0
$391$	8.82843	0.446473
$392$	0	0
$393$	20.3137	1.02469
$394$	0	0
$395$	29.5147i	1.48505i
$396$	0	0
$397$	− 6.68629i	− 0.335575i	−0.985823	−	0.167788i	$-0.946338\pi$
$397$	− 6.68629i	− 0.335575i	0.985823	−	0.167788i	$-0.0536623\pi$
$398$	0	0
$399$	60.2843	3.01799
$400$	0	0
$401$	26.2843	1.31257	0.656287	−	0.754511i	$-0.272126\pi$
$401$	26.2843	1.31257	0.656287	+	0.754511i	$0.272126\pi$
$402$	0	0
$403$	− 3.65685i	− 0.182161i
$404$	0	0
$405$	− 17.3431i	− 0.861788i
$406$	0	0
$407$	−5.79899	−0.287445
$408$	0	0
$409$	16.3431	0.808117	0.404058	−	0.914733i	$-0.367599\pi$
$409$	16.3431	0.808117	0.404058	+	0.914733i	$0.367599\pi$
$410$	0	0
$411$	− 10.4853i	− 0.517201i
$412$	0	0
$413$	35.3137i	1.73767i
$414$	0	0
$415$	22.2010	1.08980
$416$	0	0
$417$	3.82843	0.187479
$418$	0	0
$419$	3.24264i	0.158413i	0.996858	+	0.0792067i	$0.0252387\pi$
$419$	3.24264i	0.158413i	−0.996858	+	0.0792067i	$0.974761\pi$
$420$	0	0
$421$	− 12.7990i	− 0.623785i	−0.950117	−	0.311892i	$-0.899037\pi$
$421$	− 12.7990i	− 0.623785i	0.950117	−	0.311892i	$-0.100963\pi$
$422$	0	0
$423$	2.14214	0.104154
$424$	0	0
$425$	1.65685	0.0803692
$426$	0	0
$427$	0	0
$428$	0	0
$429$	− 2.00000i	− 0.0965609i
$430$	0	0
$431$	−29.3848	−1.41541	−0.707707	−	0.706506i	$-0.750271\pi$
$431$	−29.3848	−1.41541	−0.707707	+	0.706506i	$0.750271\pi$
$432$	0	0
$433$	−40.4558	−1.94418	−0.972092	−	0.234600i	$-0.924622\pi$
$433$	−40.4558	−1.94418	−0.972092	+	0.234600i	$0.924622\pi$
$434$	0	0
$435$	16.1421i	0.773956i
$436$	0	0
$437$	49.9411i	2.38901i
$438$	0	0
$439$	−32.2843	−1.54084	−0.770422	−	0.637534i	$-0.779954\pi$
$439$	−32.2843	−1.54084	−0.770422	+	0.637534i	$0.779954\pi$
$440$	0	0
$441$	−35.3137	−1.68161
$442$	0	0
$443$	22.0711i	1.04863i	0.851525	+	0.524314i	$0.175678\pi$
$443$	22.0711i	1.04863i	−0.851525	+	0.524314i	$0.824322\pi$
$444$	0	0
$445$	− 24.9706i	− 1.18372i
$446$	0	0
$447$	51.4558	2.43378
$448$	0	0
$449$	−21.3137	−1.00586	−0.502928	−	0.864328i	$-0.667744\pi$
$449$	−21.3137	−1.00586	−0.502928	+	0.864328i	$0.667744\pi$
$450$	0	0
$451$	8.00000i	0.376705i
$452$	0	0
$453$	1.82843i	0.0859070i
$454$	0	0
$455$	−8.07107	−0.378377
$456$	0	0
$457$	−26.3431	−1.23228	−0.616140	−	0.787637i	$-0.711305\pi$
$457$	−26.3431	−1.23228	−0.616140	+	0.787637i	$0.711305\pi$
$458$	0	0
$459$	− 0.414214i	− 0.0193338i
$460$	0	0
$461$	− 10.6569i	− 0.496339i	−0.968717	−	0.248170i	$-0.920171\pi$
$461$	− 10.6569i	− 0.496339i	0.968717	−	0.248170i	$-0.0798290\pi$
$462$	0	0
$463$	−10.9706	−0.509845	−0.254923	−	0.966961i	$-0.582050\pi$
$463$	−10.9706	−0.509845	−0.254923	+	0.966961i	$0.582050\pi$
$464$	0	0
$465$	−16.1421	−0.748574
$466$	0	0
$467$	− 2.00000i	− 0.0925490i	−0.998929	−	0.0462745i	$-0.985265\pi$
$467$	− 2.00000i	− 0.0925490i	0.998929	−	0.0462745i	$-0.0147349\pi$
$468$	0	0
$469$	− 38.9706i	− 1.79949i
$470$	0	0
$471$	24.9706	1.15058
$472$	0	0
$473$	−6.97056	−0.320507
$474$	0	0
$475$	9.37258i	0.430044i
$476$	0	0
$477$	10.3431i	0.473580i
$478$	0	0
$479$	9.58579	0.437986	0.218993	−	0.975726i	$-0.429723\pi$
$479$	9.58579	0.437986	0.218993	+	0.975726i	$0.429723\pi$
$480$	0	0
$481$	7.00000	0.319173
$482$	0	0
$483$	− 94.0833i	− 4.28094i
$484$	0	0
$485$	− 14.0000i	− 0.635707i
$486$	0	0
$487$	10.9706	0.497124	0.248562	−	0.968616i	$-0.420042\pi$
$487$	10.9706	0.497124	0.248562	+	0.968616i	$0.420042\pi$
$488$	0	0
$489$	34.9706	1.58142
$490$	0	0
$491$	27.8701i	1.25776i	0.777503	+	0.628879i	$0.216486\pi$
$491$	27.8701i	1.25776i	−0.777503	+	0.628879i	$0.783514\pi$
$492$	0	0
$493$	3.65685i	0.164696i
$494$	0	0
$495$	−4.28427	−0.192564
$496$	0	0
$497$	−51.7696	−2.32218
$498$	0	0
$499$	10.3431i	0.463023i	0.972832	+	0.231511i	$0.0743671\pi$
$499$	10.3431i	0.463023i	−0.972832	+	0.231511i	$0.925633\pi$
$500$	0	0
$501$	− 8.82843i	− 0.394425i
$502$	0	0
$503$	−17.6569	−0.787280	−0.393640	−	0.919265i	$-0.628784\pi$
$503$	−17.6569	−0.787280	−0.393640	+	0.919265i	$0.628784\pi$
$504$	0	0
$505$	−32.2843	−1.43663
$506$	0	0
$507$	2.41421i	0.107219i
$508$	0	0
$509$	21.3137i	0.944714i	0.881407	+	0.472357i	$0.156597\pi$
$509$	21.3137i	0.944714i	−0.881407	+	0.472357i	$0.843403\pi$
$510$	0	0
$511$	−7.31371	−0.323539
$512$	0	0
$513$	2.34315	0.103452
$514$	0	0
$515$	29.5147i	1.30057i
$516$	0	0
$517$	0.627417i	0.0275938i
$518$	0	0
$519$	33.7990	1.48361
$520$	0	0
$521$	12.8579	0.563313	0.281657	−	0.959515i	$-0.409116\pi$
$521$	12.8579	0.563313	0.281657	+	0.959515i	$0.409116\pi$
$522$	0	0
$523$	− 2.68629i	− 0.117463i	−0.998274	−	0.0587317i	$-0.981294\pi$
$523$	− 2.68629i	− 0.117463i	0.998274	−	0.0587317i	$-0.0187056\pi$
$524$	0	0
$525$	− 17.6569i	− 0.770608i
$526$	0	0
$527$	−3.65685	−0.159295
$528$	0	0
$529$	54.9411	2.38874
$530$	0	0
$531$	− 22.6274i	− 0.981946i
$532$	0	0
$533$	− 9.65685i	− 0.418285i
$534$	0	0
$535$	10.9706	0.474299
$536$	0	0
$537$	−31.1421	−1.34388
$538$	0	0
$539$	− 10.3431i	− 0.445511i
$540$	0	0
$541$	− 7.00000i	− 0.300954i	−0.988614	−	0.150477i	$-0.951919\pi$
$541$	− 7.00000i	− 0.300954i	0.988614	−	0.150477i	$-0.0480809\pi$
$542$	0	0
$543$	−8.82843	−0.378864
$544$	0	0
$545$	6.11270	0.261839
$546$	0	0
$547$	15.9289i	0.681072i	0.940231	+	0.340536i	$0.110609\pi$
$547$	15.9289i	0.681072i	−0.940231	+	0.340536i	$0.889391\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−20.6863	−0.881266
$552$	0	0
$553$	71.2548	3.03006
$554$	0	0
$555$	− 30.8995i	− 1.31161i
$556$	0	0
$557$	− 31.9706i	− 1.35464i	−0.735690	−	0.677318i	$-0.763142\pi$
$557$	− 31.9706i	− 1.35464i	0.735690	−	0.677318i	$-0.236858\pi$
$558$	0	0
$559$	8.41421	0.355883
$560$	0	0
$561$	−2.00000	−0.0844401
$562$	0	0
$563$	17.0416i	0.718219i	0.933295	+	0.359110i	$0.116920\pi$
$563$	17.0416i	0.718219i	−0.933295	+	0.359110i	$0.883080\pi$
$564$	0	0
$565$	− 18.2843i	− 0.769225i
$566$	0	0
$567$	−41.8701	−1.75838
$568$	0	0
$569$	3.34315	0.140152	0.0700760	−	0.997542i	$-0.477676\pi$
$569$	3.34315	0.140152	0.0700760	+	0.997542i	$0.477676\pi$
$570$	0	0
$571$	− 10.2721i	− 0.429873i	−0.976628	−	0.214937i	$-0.931046\pi$
$571$	− 10.2721i	− 0.429873i	0.976628	−	0.214937i	$-0.0689545\pi$
$572$	0	0
$573$	46.2843i	1.93355i
$574$	0	0
$575$	14.6274	0.610005
$576$	0	0
$577$	9.65685	0.402020	0.201010	−	0.979589i	$-0.435578\pi$
$577$	9.65685	0.402020	0.201010	+	0.979589i	$0.435578\pi$
$578$	0	0
$579$	3.17157i	0.131806i
$580$	0	0
$581$	− 53.5980i	− 2.22362i
$582$	0	0
$583$	−3.02944	−0.125466
$584$	0	0
$585$	5.17157	0.213818
$586$	0	0
$587$	− 18.4853i	− 0.762969i	−0.924375	−	0.381485i	$-0.875413\pi$
$587$	− 18.4853i	− 0.762969i	0.924375	−	0.381485i	$-0.124587\pi$
$588$	0	0
$589$	− 20.6863i	− 0.852364i
$590$	0	0
$591$	16.8995	0.695152
$592$	0	0
$593$	−24.6274	−1.01133	−0.505663	−	0.862731i	$-0.668752\pi$
$593$	−24.6274	−1.01133	−0.505663	+	0.862731i	$0.668752\pi$
$594$	0	0
$595$	8.07107i	0.330882i
$596$	0	0
$597$	− 3.65685i	− 0.149665i
$598$	0	0
$599$	−7.31371	−0.298830	−0.149415	−	0.988775i	$-0.547739\pi$
$599$	−7.31371	−0.298830	−0.149415	+	0.988775i	$0.547739\pi$
$600$	0	0
$601$	−21.9706	−0.896198	−0.448099	−	0.893984i	$-0.647899\pi$
$601$	−21.9706	−0.896198	−0.448099	+	0.893984i	$0.647899\pi$
$602$	0	0
$603$	24.9706i	1.01688i
$604$	0	0
$605$	18.8579i	0.766681i
$606$	0	0
$607$	−7.31371	−0.296854	−0.148427	−	0.988923i	$-0.547421\pi$
$607$	−7.31371	−0.296854	−0.148427	+	0.988923i	$0.547421\pi$
$608$	0	0
$609$	38.9706	1.57917
$610$	0	0
$611$	− 0.757359i	− 0.0306395i
$612$	0	0
$613$	− 0.627417i	− 0.0253411i	−0.999920	−	0.0126706i	$-0.995967\pi$
$613$	− 0.627417i	− 0.0253411i	0.999920	−	0.0126706i	$-0.00403327\pi$
$614$	0	0
$615$	−42.6274	−1.71890
$616$	0	0
$617$	41.2548	1.66086	0.830429	−	0.557125i	$-0.188096\pi$
$617$	41.2548	1.66086	0.830429	+	0.557125i	$0.188096\pi$
$618$	0	0
$619$	− 26.3431i	− 1.05882i	−0.848366	−	0.529410i	$-0.822413\pi$
$619$	− 26.3431i	− 1.05882i	0.848366	−	0.529410i	$-0.177587\pi$
$620$	0	0
$621$	− 3.65685i	− 0.146745i
$622$	0	0
$623$	−60.2843	−2.41524
$624$	0	0
$625$	−13.9706	−0.558823
$626$	0	0
$627$	− 11.3137i	− 0.451826i
$628$	0	0
$629$	− 7.00000i	− 0.279108i
$630$	0	0
$631$	−43.3848	−1.72712	−0.863560	−	0.504246i	$-0.831771\pi$
$631$	−43.3848	−1.72712	−0.863560	+	0.504246i	$0.831771\pi$
$632$	0	0
$633$	10.6569	0.423572
$634$	0	0
$635$	− 16.1421i	− 0.640581i
$636$	0	0
$637$	12.4853i	0.494685i
$638$	0	0
$639$	33.1716	1.31225
$640$	0	0
$641$	−14.0000	−0.552967	−0.276483	−	0.961019i	$-0.589169\pi$
$641$	−14.0000	−0.552967	−0.276483	+	0.961019i	$0.589169\pi$
$642$	0	0
$643$	− 28.8284i	− 1.13688i	−0.822724	−	0.568441i	$-0.807547\pi$
$643$	− 28.8284i	− 1.13688i	0.822724	−	0.568441i	$-0.192453\pi$
$644$	0	0
$645$	− 37.1421i	− 1.46247i
$646$	0	0
$647$	24.9706	0.981694	0.490847	−	0.871246i	$-0.336687\pi$
$647$	24.9706	0.981694	0.490847	+	0.871246i	$0.336687\pi$
$648$	0	0
$649$	6.62742	0.260149
$650$	0	0
$651$	38.9706i	1.52738i
$652$	0	0
$653$	10.3431i	0.404759i	0.979307	+	0.202379i	$0.0648674\pi$
$653$	10.3431i	0.404759i	−0.979307	+	0.202379i	$0.935133\pi$
$654$	0	0
$655$	−15.3848	−0.601133
$656$	0	0
$657$	4.68629	0.182830
$658$	0	0
$659$	− 31.9411i	− 1.24425i	−0.782918	−	0.622125i	$-0.786270\pi$
$659$	− 31.9411i	− 1.24425i	0.782918	−	0.622125i	$-0.213730\pi$
$660$	0	0
$661$	6.68629i	0.260067i	0.991510	+	0.130033i	$0.0415084\pi$
$661$	6.68629i	0.260067i	−0.991510	+	0.130033i	$0.958492\pi$
$662$	0	0
$663$	2.41421	0.0937603
$664$	0	0
$665$	−45.6569	−1.77050
$666$	0	0
$667$	32.2843i	1.25005i
$668$	0	0
$669$	23.1421i	0.894727i
$670$	0	0
$671$	0	0
$672$	0	0
$673$	21.2843	0.820448	0.410224	−	0.911985i	$-0.365450\pi$
$673$	21.2843	0.820448	0.410224	+	0.911985i	$0.365450\pi$
$674$	0	0
$675$	− 0.686292i	− 0.0264154i
$676$	0	0
$677$	46.2843i	1.77885i	0.457082	+	0.889425i	$0.348895\pi$
$677$	46.2843i	1.77885i	−0.457082	+	0.889425i	$0.651105\pi$
$678$	0	0
$679$	−33.7990	−1.29709
$680$	0	0
$681$	−31.3137	−1.19994
$682$	0	0
$683$	8.82843i	0.337810i	0.985632	+	0.168905i	$0.0540232\pi$
$683$	8.82843i	0.337810i	−0.985632	+	0.168905i	$0.945977\pi$
$684$	0	0
$685$	7.94113i	0.303415i
$686$	0	0
$687$	−47.0416	−1.79475
$688$	0	0
$689$	3.65685	0.139315
$690$	0	0
$691$	10.6274i	0.404286i	0.979356	+	0.202143i	$0.0647906\pi$
$691$	10.6274i	0.404286i	−0.979356	+	0.202143i	$0.935209\pi$
$692$	0	0
$693$	10.3431i	0.392904i
$694$	0	0
$695$	−2.89949	−0.109984
$696$	0	0
$697$	−9.65685	−0.365779
$698$	0	0
$699$	43.0416i	1.62798i
$700$	0	0
$701$	14.0000i	0.528773i	0.964417	+	0.264386i	$0.0851694\pi$
$701$	14.0000i	0.528773i	−0.964417	+	0.264386i	$0.914831\pi$
$702$	0	0
$703$	39.5980	1.49347
$704$	0	0
$705$	−3.34315	−0.125910
$706$	0	0
$707$	77.9411i	2.93128i
$708$	0	0
$709$	− 6.68629i	− 0.251109i	−0.992087	−	0.125554i	$-0.959929\pi$
$709$	− 6.68629i	− 0.251109i	0.992087	−	0.125554i	$-0.0400710\pi$
$710$	0	0
$711$	−45.6569	−1.71227
$712$	0	0
$713$	−32.2843	−1.20906
$714$	0	0
$715$	1.51472i	0.0566473i
$716$	0	0
$717$	53.2843i	1.98994i
$718$	0	0
$719$	5.79899	0.216266	0.108133	−	0.994136i	$-0.465513\pi$
$719$	5.79899	0.216266	0.108133	+	0.994136i	$0.465513\pi$
$720$	0	0
$721$	71.2548	2.65367
$722$	0	0
$723$	31.3137i	1.16457i
$724$	0	0
$725$	6.05887i	0.225021i
$726$	0	0
$727$	−36.8284	−1.36589	−0.682945	−	0.730469i	$-0.739301\pi$
$727$	−36.8284	−1.36589	−0.682945	+	0.730469i	$0.739301\pi$
$728$	0	0
$729$	23.8284	0.882534
$730$	0	0
$731$	− 8.41421i	− 0.311211i
$732$	0	0
$733$	47.4853i	1.75391i	0.480574	+	0.876954i	$0.340429\pi$
$733$	47.4853i	1.75391i	−0.480574	+	0.876954i	$0.659571\pi$
$734$	0	0
$735$	55.1127	2.03286
$736$	0	0
$737$	−7.31371	−0.269404
$738$	0	0
$739$	− 15.3137i	− 0.563324i	−0.959514	−	0.281662i	$-0.909114\pi$
$739$	− 15.3137i	− 0.563324i	0.959514	−	0.281662i	$-0.0908857\pi$
$740$	0	0
$741$	13.6569i	0.501697i
$742$	0	0
$743$	25.7279	0.943866	0.471933	−	0.881634i	$-0.343556\pi$
$743$	25.7279	0.943866	0.471933	+	0.881634i	$0.343556\pi$
$744$	0	0
$745$	−38.9706	−1.42777
$746$	0	0
$747$	34.3431i	1.25655i
$748$	0	0
$749$	− 26.4853i	− 0.967751i
$750$	0	0
$751$	28.0000	1.02173	0.510867	−	0.859660i	$-0.329324\pi$
$751$	28.0000	1.02173	0.510867	+	0.859660i	$0.329324\pi$
$752$	0	0
$753$	−32.1421	−1.17132
$754$	0	0
$755$	− 1.38478i	− 0.0503972i
$756$	0	0
$757$	− 28.6274i	− 1.04048i	−0.854020	−	0.520241i	$-0.825842\pi$
$757$	− 28.6274i	− 1.04048i	0.854020	−	0.520241i	$-0.174158\pi$
$758$	0	0
$759$	−17.6569	−0.640903
$760$	0	0
$761$	23.3137	0.845121	0.422561	−	0.906335i	$-0.361131\pi$
$761$	23.3137	0.845121	0.422561	+	0.906335i	$0.361131\pi$
$762$	0	0
$763$	− 14.7574i	− 0.534252i
$764$	0	0
$765$	− 5.17157i	− 0.186979i
$766$	0	0
$767$	−8.00000	−0.288863
$768$	0	0
$769$	−22.6274	−0.815966	−0.407983	−	0.912990i	$-0.633768\pi$
$769$	−22.6274	−0.815966	−0.407983	+	0.912990i	$0.633768\pi$
$770$	0	0
$771$	− 54.6985i	− 1.96992i
$772$	0	0
$773$	26.7990i	0.963893i	0.876201	+	0.481946i	$0.160070\pi$
$773$	26.7990i	0.963893i	−0.876201	+	0.481946i	$0.839930\pi$
$774$	0	0
$775$	−6.05887	−0.217641
$776$	0	0
$777$	−74.5980	−2.67619
$778$	0	0
$779$	− 54.6274i	− 1.95723i
$780$	0	0
$781$	9.71573i	0.347656i
$782$	0	0
$783$	1.51472	0.0541316
$784$	0	0
$785$	−18.9117	−0.674987
$786$	0	0
$787$	− 37.6569i	− 1.34232i	−0.741312	−	0.671161i	$-0.765796\pi$
$787$	− 37.6569i	− 1.34232i	0.741312	−	0.671161i	$-0.234204\pi$
$788$	0	0
$789$	42.6274i	1.51758i
$790$	0	0
$791$	−44.1421	−1.56951
$792$	0	0
$793$	0	0
$794$	0	0
$795$	− 16.1421i	− 0.572503i
$796$	0	0
$797$	− 32.2843i	− 1.14357i	−0.820404	−	0.571784i	$-0.806252\pi$
$797$	− 32.2843i	− 1.14357i	0.820404	−	0.571784i	$-0.193748\pi$
$798$	0	0
$799$	−0.757359	−0.0267934
$800$	0	0
$801$	38.6274	1.36483
$802$	0	0
$803$	1.37258i	0.0484374i
$804$	0	0
$805$	71.2548i	2.51140i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	27.4853	0.966331	0.483166	−	0.875529i	$-0.339487\pi$
$809$	27.4853	0.966331	0.483166	+	0.875529i	$0.339487\pi$
$810$	0	0
$811$	28.9706i	1.01729i	0.860975	+	0.508647i	$0.169854\pi$
$811$	28.9706i	1.01729i	−0.860975	+	0.508647i	$0.830146\pi$
$812$	0	0
$813$	− 23.1421i	− 0.811630i
$814$	0	0
$815$	−26.4853	−0.927739
$816$	0	0
$817$	47.5980	1.66524
$818$	0	0
$819$	− 12.4853i	− 0.436271i
$820$	0	0
$821$	34.1127i	1.19054i	0.803525	+	0.595271i	$0.202955\pi$
$821$	34.1127i	1.19054i	−0.803525	+	0.595271i	$0.797045\pi$
$822$	0	0
$823$	−28.0000	−0.976019	−0.488009	−	0.872838i	$-0.662277\pi$
$823$	−28.0000	−0.976019	−0.488009	+	0.872838i	$0.662277\pi$
$824$	0	0
$825$	−3.31371	−0.115369
$826$	0	0
$827$	− 33.9411i	− 1.18025i	−0.807312	−	0.590124i	$-0.799079\pi$
$827$	− 33.9411i	− 1.18025i	0.807312	−	0.590124i	$-0.200921\pi$
$828$	0	0
$829$	10.9706i	0.381023i	0.981685	+	0.190512i	$0.0610147\pi$
$829$	10.9706i	0.381023i	−0.981685	+	0.190512i	$0.938985\pi$
$830$	0	0
$831$	60.2843	2.09124
$832$	0	0
$833$	12.4853	0.432589
$834$	0	0
$835$	6.68629i	0.231389i
$836$	0	0
$837$	1.51472i	0.0523563i
$838$	0	0
$839$	38.9706	1.34541	0.672707	−	0.739909i	$-0.265132\pi$
$839$	38.9706	1.34541	0.672707	+	0.739909i	$0.265132\pi$
$840$	0	0
$841$	15.6274	0.538876
$842$	0	0
$843$	− 9.65685i	− 0.332600i
$844$	0	0
$845$	− 1.82843i	− 0.0628998i
$846$	0	0
$847$	45.5269	1.56432
$848$	0	0
$849$	−63.4558	−2.17780
$850$	0	0
$851$	− 61.7990i	− 2.11844i
$852$	0	0
$853$	− 16.4558i	− 0.563437i	−0.959497	−	0.281719i	$-0.909096\pi$
$853$	− 16.4558i	− 0.563437i	0.959497	−	0.281719i	$-0.0909045\pi$
$854$	0	0
$855$	29.2548	1.00049
$856$	0	0
$857$	22.6863	0.774949	0.387474	−	0.921880i	$-0.373348\pi$
$857$	22.6863	0.774949	0.387474	+	0.921880i	$0.373348\pi$
$858$	0	0
$859$	16.6274i	0.567320i	0.958925	+	0.283660i	$0.0915487\pi$
$859$	16.6274i	0.567320i	−0.958925	+	0.283660i	$0.908451\pi$
$860$	0	0
$861$	102.912i	3.50722i
$862$	0	0
$863$	−29.3848	−1.00027	−0.500135	−	0.865948i	$-0.666716\pi$
$863$	−29.3848	−1.00027	−0.500135	+	0.865948i	$0.666716\pi$
$864$	0	0
$865$	−25.5980	−0.870357
$866$	0	0
$867$	38.6274i	1.31186i
$868$	0	0
$869$	− 13.3726i	− 0.453634i
$870$	0	0
$871$	8.82843	0.299140
$872$	0	0
$873$	21.6569	0.732973
$874$	0	0
$875$	53.7279i	1.81634i
$876$	0	0
$877$	− 23.1421i	− 0.781454i	−0.920507	−	0.390727i	$-0.872224\pi$
$877$	− 23.1421i	− 0.781454i	0.920507	−	0.390727i	$-0.127776\pi$
$878$	0	0
$879$	8.07107	0.272230
$880$	0	0
$881$	32.3137	1.08868	0.544338	−	0.838866i	$-0.316781\pi$
$881$	32.3137	1.08868	0.544338	+	0.838866i	$0.316781\pi$
$882$	0	0
$883$	26.2132i	0.882145i	0.897472	+	0.441072i	$0.145402\pi$
$883$	26.2132i	0.882145i	−0.897472	+	0.441072i	$0.854598\pi$
$884$	0	0
$885$	35.3137i	1.18706i
$886$	0	0
$887$	10.3431	0.347289	0.173644	−	0.984808i	$-0.444446\pi$
$887$	10.3431	0.347289	0.173644	+	0.984808i	$0.444446\pi$
$888$	0	0
$889$	−38.9706	−1.30703
$890$	0	0
$891$	7.85786i	0.263248i
$892$	0	0
$893$	− 4.28427i	− 0.143368i
$894$	0	0
$895$	23.5858	0.788386
$896$	0	0
$897$	21.3137	0.711644
$898$	0	0
$899$	− 13.3726i	− 0.446001i
$900$	0	0
$901$	− 3.65685i	− 0.121827i
$902$	0	0
$903$	−89.6690	−2.98400
$904$	0	0
$905$	6.68629	0.222260
$906$	0	0
$907$	− 11.4437i	− 0.379980i	−0.981786	−	0.189990i	$-0.939154\pi$
$907$	− 11.4437i	− 0.379980i	0.981786	−	0.189990i	$-0.0608456\pi$
$908$	0	0
$909$	− 49.9411i	− 1.65644i
$910$	0	0
$911$	16.1421	0.534813	0.267406	−	0.963584i	$-0.413833\pi$
$911$	16.1421	0.534813	0.267406	+	0.963584i	$0.413833\pi$
$912$	0	0
$913$	−10.0589	−0.332900
$914$	0	0
$915$	0	0
$916$	0	0
$917$	37.1421i	1.22654i
$918$	0	0
$919$	45.6569	1.50608	0.753040	−	0.657974i	$-0.228586\pi$
$919$	45.6569	1.50608	0.753040	+	0.657974i	$0.228586\pi$
$920$	0	0
$921$	48.2843	1.59102
$922$	0	0
$923$	− 11.7279i	− 0.386029i
$924$	0	0
$925$	− 11.5980i	− 0.381339i
$926$	0	0
$927$	−45.6569	−1.49957
$928$	0	0
$929$	−36.9706	−1.21296	−0.606482	−	0.795097i	$-0.707420\pi$
$929$	−36.9706	−1.21296	−0.606482	+	0.795097i	$0.707420\pi$
$930$	0	0
$931$	70.6274i	2.31472i
$932$	0	0
$933$	24.9706i	0.817500i
$934$	0	0
$935$	1.51472	0.0495366
$936$	0	0
$937$	−14.6863	−0.479780	−0.239890	−	0.970800i	$-0.577111\pi$
$937$	−14.6863	−0.479780	−0.239890	+	0.970800i	$0.577111\pi$
$938$	0	0
$939$	− 80.0122i	− 2.61110i
$940$	0	0
$941$	− 1.20101i	− 0.0391518i	−0.999808	−	0.0195759i	$-0.993768\pi$
$941$	− 1.20101i	− 0.0391518i	0.999808	−	0.0195759i	$-0.00623160\pi$
$942$	0	0
$943$	−85.2548	−2.77628
$944$	0	0
$945$	3.34315	0.108753
$946$	0	0
$947$	− 35.1716i	− 1.14292i	−0.820629	−	0.571461i	$-0.806377\pi$
$947$	− 35.1716i	− 1.14292i	0.820629	−	0.571461i	$-0.193623\pi$
$948$	0	0
$949$	− 1.65685i	− 0.0537838i
$950$	0	0
$951$	−69.1127	−2.24113
$952$	0	0
$953$	42.9411	1.39100	0.695500	−	0.718526i	$-0.255183\pi$
$953$	42.9411	1.39100	0.695500	+	0.718526i	$0.255183\pi$
$954$	0	0
$955$	− 35.0538i	− 1.13432i
$956$	0	0
$957$	− 7.31371i	− 0.236419i
$958$	0	0
$959$	19.1716	0.619082
$960$	0	0
$961$	−17.6274	−0.568626
$962$	0	0
$963$	16.9706i	0.546869i
$964$	0	0
$965$	− 2.40202i	− 0.0773238i
$966$	0	0
$967$	23.5858	0.758468	0.379234	−	0.925301i	$-0.376188\pi$
$967$	23.5858	0.758468	0.379234	+	0.925301i	$0.376188\pi$
$968$	0	0
$969$	13.6569	0.438721
$970$	0	0
$971$	− 42.8995i	− 1.37671i	−0.725374	−	0.688355i	$-0.758333\pi$
$971$	− 42.8995i	− 1.37671i	0.725374	−	0.688355i	$-0.241667\pi$
$972$	0	0
$973$	7.00000i	0.224410i
$974$	0	0
$975$	4.00000	0.128103
$976$	0	0
$977$	1.31371	0.0420293	0.0210146	−	0.999779i	$-0.493310\pi$
$977$	1.31371	0.0420293	0.0210146	+	0.999779i	$0.493310\pi$
$978$	0	0
$979$	11.3137i	0.361588i
$980$	0	0
$981$	9.45584i	0.301902i
$982$	0	0
$983$	4.41421	0.140792	0.0703958	−	0.997519i	$-0.477574\pi$
$983$	4.41421	0.140792	0.0703958	+	0.997519i	$0.477574\pi$
$984$	0	0
$985$	−12.7990	−0.407810
$986$	0	0
$987$	8.07107i	0.256905i
$988$	0	0
$989$	− 74.2843i	− 2.36210i
$990$	0	0
$991$	3.02944	0.0962332	0.0481166	−	0.998842i	$-0.484678\pi$
$991$	3.02944	0.0962332	0.0481166	+	0.998842i	$0.484678\pi$
$992$	0	0
$993$	7.65685	0.242983
$994$	0	0
$995$	2.76955i	0.0878007i
$996$	0	0
$997$	42.0000i	1.33015i	0.746775	+	0.665077i	$0.231601\pi$
$997$	42.0000i	1.33015i	−0.746775	+	0.665077i	$0.768399\pi$
$998$	0	0
$999$	−2.89949	−0.0917360

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	832.2.b.b.417.1	yes	4
4.3	odd	2		832.2.b.a.417.4	yes	4
8.3	odd	2		832.2.b.a.417.1	✓	4
8.5	even	2	inner	832.2.b.b.417.4	yes	4
16.3	odd	4		3328.2.a.p.1.1		2
16.5	even	4		3328.2.a.q.1.1		2
16.11	odd	4		3328.2.a.bb.1.2		2
16.13	even	4		3328.2.a.y.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
832.2.b.a.417.1	✓	4	8.3	odd	2
832.2.b.a.417.4	yes	4	4.3	odd	2
832.2.b.b.417.1	yes	4	1.1	even	1	trivial
832.2.b.b.417.4	yes	4	8.5	even	2	inner
3328.2.a.p.1.1		2	16.3	odd	4
3328.2.a.q.1.1		2	16.5	even	4
3328.2.a.y.1.2		2	16.13	even	4
3328.2.a.bb.1.2		2	16.11	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:	\( N \)	\(=\)	\( 832 = 2^{6} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	832.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.41421i q^{3} +1.82843i q^{5} +4.41421 q^{7} -2.82843 q^{9} -0.828427i q^{11} +1.00000i q^{13} +4.41421 q^{15} +1.00000 q^{17} +5.65685i q^{19} -10.6569i q^{21} +8.82843 q^{23} +1.65685 q^{25} -0.414214i q^{27} +3.65685i q^{29} -3.65685 q^{31} -2.00000 q^{33} +8.07107i q^{35} -7.00000i q^{37} +2.41421 q^{39} -9.65685 q^{41} -8.41421i q^{43} -5.17157i q^{45} -0.757359 q^{47} +12.4853 q^{49} -2.41421i q^{51} -3.65685i q^{53} +1.51472 q^{55} +13.6569 q^{57} +8.00000i q^{59} -12.4853 q^{63} -1.82843 q^{65} -8.82843i q^{67} -21.3137i q^{69} -11.7279 q^{71} -1.65685 q^{73} -4.00000i q^{75} -3.65685i q^{77} +16.1421 q^{79} -9.48528 q^{81} -12.1421i q^{83} +1.82843i q^{85} +8.82843 q^{87} -13.6569 q^{89} +4.41421i q^{91} +8.82843i q^{93} -10.3431 q^{95} -7.65685 q^{97} +2.34315i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 12 q^{7} + 12 q^{15} + 4 q^{17} + 24 q^{23} - 16 q^{25} + 8 q^{31} - 8 q^{33} + 4 q^{39} - 16 q^{41} - 20 q^{47} + 16 q^{49} + 40 q^{55} + 32 q^{57} - 16 q^{63} + 4 q^{65} + 4 q^{71} + 16 q^{73} + 8 q^{79}+ \cdots - 8 q^{97}+O(q^{100}) \) 4 * q + 12 * q^7 + 12 * q^15 + 4 * q^17 + 24 * q^23 - 16 * q^25 + 8 * q^31 - 8 * q^33 + 4 * q^39 - 16 * q^41 - 20 * q^47 + 16 * q^49 + 40 * q^55 + 32 * q^57 - 16 * q^63 + 4 * q^65 + 4 * q^71 + 16 * q^73 + 8 * q^79 - 4 * q^81 + 24 * q^87 - 32 * q^89 - 64 * q^95 - 8 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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