LMFDB - Embedded newform 3328.2.a.q.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3328,2,Mod(1,3328)] 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("3328.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3328, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 3328 = 2^{8} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3328.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,-2,0,2,0,-6,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

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

Self dual:	yes
Analytic conductor:	$26.5742137927$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 832)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	3328.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.41421 q^{3} -1.82843 q^{5} -4.41421 q^{7} +2.82843 q^{9} +0.828427 q^{11} +1.00000 q^{13} +4.41421 q^{15} +1.00000 q^{17} +5.65685 q^{19} +10.6569 q^{21} -8.82843 q^{23} -1.65685 q^{25} +0.414214 q^{27} +3.65685 q^{29} -3.65685 q^{31} -2.00000 q^{33} +8.07107 q^{35} +7.00000 q^{37} -2.41421 q^{39} +9.65685 q^{41} +8.41421 q^{43} -5.17157 q^{45} -0.757359 q^{47} +12.4853 q^{49} -2.41421 q^{51} +3.65685 q^{53} -1.51472 q^{55} -13.6569 q^{57} -8.00000 q^{59} -12.4853 q^{63} -1.82843 q^{65} -8.82843 q^{67} +21.3137 q^{69} +11.7279 q^{71} +1.65685 q^{73} +4.00000 q^{75} -3.65685 q^{77} +16.1421 q^{79} -9.48528 q^{81} -12.1421 q^{83} -1.82843 q^{85} -8.82843 q^{87} +13.6569 q^{89} -4.41421 q^{91} +8.82843 q^{93} -10.3431 q^{95} -7.65685 q^{97} +2.34315 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} + 2 q^{5} - 6 q^{7} - 4 q^{11} + 2 q^{13} + 6 q^{15} + 2 q^{17} + 10 q^{21} - 12 q^{23} + 8 q^{25} - 2 q^{27} - 4 q^{29} + 4 q^{31} - 4 q^{33} + 2 q^{35} + 14 q^{37} - 2 q^{39} + 8 q^{41}+ \cdots + 16 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 + 2 * q^5 - 6 * q^7 - 4 * q^11 + 2 * q^13 + 6 * q^15 + 2 * q^17 + 10 * q^21 - 12 * q^23 + 8 * q^25 - 2 * q^27 - 4 * q^29 + 4 * q^31 - 4 * q^33 + 2 * q^35 + 14 * q^37 - 2 * q^39 + 8 * q^41 + 14 * q^43 - 16 * q^45 - 10 * q^47 + 8 * q^49 - 2 * q^51 - 4 * q^53 - 20 * q^55 - 16 * q^57 - 16 * q^59 - 8 * q^63 + 2 * q^65 - 12 * q^67 + 20 * q^69 - 2 * q^71 - 8 * q^73 + 8 * q^75 + 4 * q^77 + 4 * q^79 - 2 * q^81 + 4 * q^83 + 2 * q^85 - 12 * q^87 + 16 * q^89 - 6 * q^91 + 12 * q^93 - 32 * q^95 - 4 * q^97 + 16 * q^99

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
$3$	−2.41421	−1.39385	−0.696923	−	0.717146i	$-0.745448\pi$
$3$	−2.41421	−1.39385	−0.696923	+	0.717146i	$0.745448\pi$
$4$	0	0
$5$	−1.82843	−0.817697	−0.408849	−	0.912602i	$-0.634070\pi$
$5$	−1.82843	−0.817697	−0.408849	+	0.912602i	$0.634070\pi$
$6$	0	0
$7$	−4.41421	−1.66842	−0.834208	−	0.551450i	$-0.814075\pi$
$7$	−4.41421	−1.66842	−0.834208	+	0.551450i	$0.814075\pi$
$8$	0	0
$9$	2.82843	0.942809
$10$	0	0
$11$	0.828427	0.249780	0.124890	−	0.992171i	$-0.460142\pi$
$11$	0.828427	0.249780	0.124890	+	0.992171i	$0.460142\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	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
$19$	5.65685	1.29777	0.648886	−	0.760886i	$-0.275235\pi$
$19$	5.65685	1.29777	0.648886	+	0.760886i	$0.275235\pi$
$20$	0	0
$21$	10.6569	2.32552
$22$	0	0
$23$	−8.82843	−1.84085	−0.920427	−	0.390914i	$-0.872159\pi$
$23$	−8.82843	−1.84085	−0.920427	+	0.390914i	$0.872159\pi$
$24$	0	0
$25$	−1.65685	−0.331371
$26$	0	0
$27$	0.414214	0.0797154
$28$	0	0
$29$	3.65685	0.679061	0.339530	−	0.940595i	$-0.389732\pi$
$29$	3.65685	0.679061	0.339530	+	0.940595i	$0.389732\pi$
$30$	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
$35$	8.07107	1.36426
$36$	0	0
$37$	7.00000	1.15079	0.575396	−	0.817875i	$-0.304848\pi$
$37$	7.00000	1.15079	0.575396	+	0.817875i	$0.304848\pi$
$38$	0	0
$39$	−2.41421	−0.386584
$40$	0	0
$41$	9.65685	1.50815	0.754074	−	0.656790i	$-0.228086\pi$
$41$	9.65685	1.50815	0.754074	+	0.656790i	$0.228086\pi$
$42$	0	0
$43$	8.41421	1.28316	0.641578	−	0.767058i	$-0.278280\pi$
$43$	8.41421	1.28316	0.641578	+	0.767058i	$0.278280\pi$
$44$	0	0
$45$	−5.17157	−0.770933
$46$	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
$51$	−2.41421	−0.338058
$52$	0	0
$53$	3.65685	0.502308	0.251154	−	0.967947i	$-0.419190\pi$
$53$	3.65685	0.502308	0.251154	+	0.967947i	$0.419190\pi$
$54$	0	0
$55$	−1.51472	−0.204245
$56$	0	0
$57$	−13.6569	−1.80889
$58$	0	0
$59$	−8.00000	−1.04151	−0.520756	−	0.853706i	$-0.674350\pi$
$59$	−8.00000	−1.04151	−0.520756	+	0.853706i	$0.674350\pi$
$60$	0	0
$61$	0	0		−	1.00000i	$-0.5\pi$
$61$	0	0			1.00000i	$0.5\pi$
$62$	0	0
$63$	−12.4853	−1.57300
$64$	0	0
$65$	−1.82843	−0.226788
$66$	0	0
$67$	−8.82843	−1.07856	−0.539282	−	0.842125i	$-0.681304\pi$
$67$	−8.82843	−1.07856	−0.539282	+	0.842125i	$0.681304\pi$
$68$	0	0
$69$	21.3137	2.56587
$70$	0	0
$71$	11.7279	1.39185	0.695924	−	0.718115i	$-0.254995\pi$
$71$	11.7279	1.39185	0.695924	+	0.718115i	$0.254995\pi$
$72$	0	0
$73$	1.65685	0.193920	0.0969601	−	0.995288i	$-0.469088\pi$
$73$	1.65685	0.193920	0.0969601	+	0.995288i	$0.469088\pi$
$74$	0	0
$75$	4.00000	0.461880
$76$	0	0
$77$	−3.65685	−0.416737
$78$	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
$83$	−12.1421	−1.33277	−0.666386	−	0.745607i	$-0.732160\pi$
$83$	−12.1421	−1.33277	−0.666386	+	0.745607i	$0.732160\pi$
$84$	0	0
$85$	−1.82843	−0.198321
$86$	0	0
$87$	−8.82843	−0.946507
$88$	0	0
$89$	13.6569	1.44762	0.723812	−	0.689997i	$-0.242388\pi$
$89$	13.6569	1.44762	0.723812	+	0.689997i	$0.242388\pi$
$90$	0	0
$91$	−4.41421	−0.462735
$92$	0	0
$93$	8.82843	0.915465
$94$	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
$99$	2.34315	0.235495
$100$	0	0
$101$	−17.6569	−1.75692	−0.878461	−	0.477813i	$-0.841429\pi$
$101$	−17.6569	−1.75692	−0.878461	+	0.477813i	$0.841429\pi$
$102$	0	0
$103$	−16.1421	−1.59053	−0.795266	−	0.606261i	$-0.792669\pi$
$103$	−16.1421	−1.59053	−0.795266	+	0.606261i	$0.792669\pi$
$104$	0	0
$105$	−19.4853	−1.90157
$106$	0	0
$107$	6.00000	0.580042	0.290021	−	0.957020i	$-0.406338\pi$
$107$	6.00000	0.580042	0.290021	+	0.957020i	$0.406338\pi$
$108$	0	0
$109$	−3.34315	−0.320215	−0.160108	−	0.987100i	$-0.551184\pi$
$109$	−3.34315	−0.320215	−0.160108	+	0.987100i	$0.551184\pi$
$110$	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
$115$	16.1421	1.50526
$116$	0	0
$117$	2.82843	0.261488
$118$	0	0
$119$	−4.41421	−0.404650
$120$	0	0
$121$	−10.3137	−0.937610
$122$	0	0
$123$	−23.3137	−2.10213
$124$	0	0
$125$	12.1716	1.08866
$126$	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
$131$	8.41421	0.735153	0.367577	−	0.929993i	$-0.380188\pi$
$131$	8.41421	0.735153	0.367577	+	0.929993i	$0.380188\pi$
$132$	0	0
$133$	−24.9706	−2.16522
$134$	0	0
$135$	−0.757359	−0.0651831
$136$	0	0
$137$	−4.34315	−0.371060	−0.185530	−	0.982639i	$-0.559400\pi$
$137$	−4.34315	−0.371060	−0.185530	+	0.982639i	$0.559400\pi$
$138$	0	0
$139$	−1.58579	−0.134505	−0.0672523	−	0.997736i	$-0.521423\pi$
$139$	−1.58579	−0.134505	−0.0672523	+	0.997736i	$0.521423\pi$
$140$	0	0
$141$	1.82843	0.153981
$142$	0	0
$143$	0.828427	0.0692766
$144$	0	0
$145$	−6.68629	−0.555266
$146$	0	0
$147$	−30.1421	−2.48608
$148$	0	0
$149$	−21.3137	−1.74609	−0.873044	−	0.487642i	$-0.837857\pi$
$149$	−21.3137	−1.74609	−0.873044	+	0.487642i	$0.837857\pi$
$150$	0	0
$151$	0.757359	0.0616330	0.0308165	−	0.999525i	$-0.490189\pi$
$151$	0.757359	0.0616330	0.0308165	+	0.999525i	$0.490189\pi$
$152$	0	0
$153$	2.82843	0.228665
$154$	0	0
$155$	6.68629	0.537056
$156$	0	0
$157$	10.3431	0.825473	0.412736	−	0.910850i	$-0.364573\pi$
$157$	10.3431	0.825473	0.412736	+	0.910850i	$0.364573\pi$
$158$	0	0
$159$	−8.82843	−0.700140
$160$	0	0
$161$	38.9706	3.07131
$162$	0	0
$163$	14.4853	1.13457	0.567287	−	0.823520i	$-0.307993\pi$
$163$	14.4853	1.13457	0.567287	+	0.823520i	$0.307993\pi$
$164$	0	0
$165$	3.65685	0.284686
$166$	0	0
$167$	−3.65685	−0.282976	−0.141488	−	0.989940i	$-0.545189\pi$
$167$	−3.65685	−0.282976	−0.141488	+	0.989940i	$0.545189\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	16.0000	1.22355
$172$	0	0
$173$	14.0000	1.06440	0.532200	−	0.846619i	$-0.321365\pi$
$173$	14.0000	1.06440	0.532200	+	0.846619i	$0.321365\pi$
$174$	0	0
$175$	7.31371	0.552864
$176$	0	0
$177$	19.3137	1.45171
$178$	0	0
$179$	−12.8995	−0.964154	−0.482077	−	0.876129i	$-0.660117\pi$
$179$	−12.8995	−0.964154	−0.482077	+	0.876129i	$0.660117\pi$
$180$	0	0
$181$	3.65685	0.271812	0.135906	−	0.990722i	$-0.456606\pi$
$181$	3.65685	0.271812	0.135906	+	0.990722i	$0.456606\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−12.7990	−0.941000
$186$	0	0
$187$	0.828427	0.0605806
$188$	0	0
$189$	−1.82843	−0.132999
$190$	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
$195$	4.41421	0.316108
$196$	0	0
$197$	−7.00000	−0.498729	−0.249365	−	0.968410i	$-0.580222\pi$
$197$	−7.00000	−0.498729	−0.249365	+	0.968410i	$0.580222\pi$
$198$	0	0
$199$	−1.51472	−0.107376	−0.0536878	−	0.998558i	$-0.517098\pi$
$199$	−1.51472	−0.107376	−0.0536878	+	0.998558i	$0.517098\pi$
$200$	0	0
$201$	21.3137	1.50335
$202$	0	0
$203$	−16.1421	−1.13296
$204$	0	0
$205$	−17.6569	−1.23321
$206$	0	0
$207$	−24.9706	−1.73557
$208$	0	0
$209$	4.68629	0.324158
$210$	0	0
$211$	4.41421	0.303887	0.151943	−	0.988389i	$-0.451447\pi$
$211$	4.41421	0.303887	0.151943	+	0.988389i	$0.451447\pi$
$212$	0	0
$213$	−28.3137	−1.94002
$214$	0	0
$215$	−15.3848	−1.04923
$216$	0	0
$217$	16.1421	1.09580
$218$	0	0
$219$	−4.00000	−0.270295
$220$	0	0
$221$	1.00000	0.0672673
$222$	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
$227$	−12.9706	−0.860886	−0.430443	−	0.902618i	$-0.641643\pi$
$227$	−12.9706	−0.860886	−0.430443	+	0.902618i	$0.641643\pi$
$228$	0	0
$229$	19.4853	1.28762	0.643812	−	0.765184i	$-0.277352\pi$
$229$	19.4853	1.28762	0.643812	+	0.765184i	$0.277352\pi$
$230$	0	0
$231$	8.82843	0.580868
$232$	0	0
$233$	17.8284	1.16798	0.583990	−	0.811761i	$-0.301491\pi$
$233$	17.8284	1.16798	0.583990	+	0.811761i	$0.301491\pi$
$234$	0	0
$235$	1.38478	0.0903328
$236$	0	0
$237$	−38.9706	−2.53141
$238$	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
$243$	21.6569	1.38929
$244$	0	0
$245$	−22.8284	−1.45845
$246$	0	0
$247$	5.65685	0.359937
$248$	0	0
$249$	29.3137	1.85768
$250$	0	0
$251$	13.3137	0.840354	0.420177	−	0.907442i	$-0.361968\pi$
$251$	13.3137	0.840354	0.420177	+	0.907442i	$0.361968\pi$
$252$	0	0
$253$	−7.31371	−0.459809
$254$	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
$259$	−30.8995	−1.92000
$260$	0	0
$261$	10.3431	0.640225
$262$	0	0
$263$	17.6569	1.08877	0.544384	−	0.838836i	$-0.316763\pi$
$263$	17.6569	1.08877	0.544384	+	0.838836i	$0.316763\pi$
$264$	0	0
$265$	−6.68629	−0.410736
$266$	0	0
$267$	−32.9706	−2.01777
$268$	0	0
$269$	0	0		−	1.00000i	$-0.5\pi$
$269$	0	0			1.00000i	$0.5\pi$
$270$	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
$275$	−1.37258	−0.0827699
$276$	0	0
$277$	−24.9706	−1.50034	−0.750168	−	0.661247i	$-0.770027\pi$
$277$	−24.9706	−1.50034	−0.750168	+	0.661247i	$0.770027\pi$
$278$	0	0
$279$	−10.3431	−0.619228
$280$	0	0
$281$	−4.00000	−0.238620	−0.119310	−	0.992857i	$-0.538068\pi$
$281$	−4.00000	−0.238620	−0.119310	+	0.992857i	$0.538068\pi$
$282$	0	0
$283$	26.2843	1.56244	0.781219	−	0.624257i	$-0.214598\pi$
$283$	26.2843	1.56244	0.781219	+	0.624257i	$0.214598\pi$
$284$	0	0
$285$	24.9706	1.47913
$286$	0	0
$287$	−42.6274	−2.51622
$288$	0	0
$289$	−16.0000	−0.941176
$290$	0	0
$291$	18.4853	1.08363
$292$	0	0
$293$	−3.34315	−0.195309	−0.0976543	−	0.995220i	$-0.531134\pi$
$293$	−3.34315	−0.195309	−0.0976543	+	0.995220i	$0.531134\pi$
$294$	0	0
$295$	14.6274	0.851641
$296$	0	0
$297$	0.343146	0.0199113
$298$	0	0
$299$	−8.82843	−0.510561
$300$	0	0
$301$	−37.1421	−2.14084
$302$	0	0
$303$	42.6274	2.44888
$304$	0	0
$305$	0	0
$306$	0	0
$307$	20.0000	1.14146	0.570730	−	0.821138i	$-0.306660\pi$
$307$	20.0000	1.14146	0.570730	+	0.821138i	$0.306660\pi$
$308$	0	0
$309$	38.9706	2.21696
$310$	0	0
$311$	10.3431	0.586506	0.293253	−	0.956035i	$-0.405262\pi$
$311$	10.3431	0.586506	0.293253	+	0.956035i	$0.405262\pi$
$312$	0	0
$313$	−33.1421	−1.87330	−0.936652	−	0.350261i	$-0.886093\pi$
$313$	−33.1421	−1.87330	−0.936652	+	0.350261i	$0.886093\pi$
$314$	0	0
$315$	22.8284	1.28624
$316$	0	0
$317$	−28.6274	−1.60788	−0.803938	−	0.594713i	$-0.797266\pi$
$317$	−28.6274	−1.60788	−0.803938	+	0.594713i	$0.797266\pi$
$318$	0	0
$319$	3.02944	0.169616
$320$	0	0
$321$	−14.4853	−0.808490
$322$	0	0
$323$	5.65685	0.314756
$324$	0	0
$325$	−1.65685	−0.0919057
$326$	0	0
$327$	8.07107	0.446331
$328$	0	0
$329$	3.34315	0.184314
$330$	0	0
$331$	−3.17157	−0.174325	−0.0871627	−	0.996194i	$-0.527780\pi$
$331$	−3.17157	−0.174325	−0.0871627	+	0.996194i	$0.527780\pi$
$332$	0	0
$333$	19.7990	1.08498
$334$	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
$339$	24.1421	1.31122
$340$	0	0
$341$	−3.02944	−0.164053
$342$	0	0
$343$	−24.2132	−1.30739
$344$	0	0
$345$	−38.9706	−2.09810
$346$	0	0
$347$	20.7574	1.11431	0.557157	−	0.830407i	$-0.311892\pi$
$347$	20.7574	1.11431	0.557157	+	0.830407i	$0.311892\pi$
$348$	0	0
$349$	−1.82843	−0.0978735	−0.0489367	−	0.998802i	$-0.515583\pi$
$349$	−1.82843	−0.0978735	−0.0489367	+	0.998802i	$0.515583\pi$
$350$	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
$355$	−21.4437	−1.13811
$356$	0	0
$357$	10.6569	0.564021
$358$	0	0
$359$	−24.3431	−1.28478	−0.642391	−	0.766377i	$-0.722057\pi$
$359$	−24.3431	−1.28478	−0.642391	+	0.766377i	$0.722057\pi$
$360$	0	0
$361$	13.0000	0.684211
$362$	0	0
$363$	24.8995	1.30688
$364$	0	0
$365$	−3.02944	−0.158568
$366$	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
$371$	−16.1421	−0.838058
$372$	0	0
$373$	−14.6274	−0.757379	−0.378689	−	0.925524i	$-0.623625\pi$
$373$	−14.6274	−0.757379	−0.378689	+	0.925524i	$0.623625\pi$
$374$	0	0
$375$	−29.3848	−1.51742
$376$	0	0
$377$	3.65685	0.188338
$378$	0	0
$379$	−2.48528	−0.127660	−0.0638302	−	0.997961i	$-0.520332\pi$
$379$	−2.48528	−0.127660	−0.0638302	+	0.997961i	$0.520332\pi$
$380$	0	0
$381$	21.3137	1.09193
$382$	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
$387$	23.7990	1.20977
$388$	0	0
$389$	28.0000	1.41966	0.709828	−	0.704375i	$-0.248773\pi$
$389$	28.0000	1.41966	0.709828	+	0.704375i	$0.248773\pi$
$390$	0	0
$391$	−8.82843	−0.446473
$392$	0	0
$393$	−20.3137	−1.02469
$394$	0	0
$395$	−29.5147	−1.48505
$396$	0	0
$397$	−6.68629	−0.335575	−0.167788	−	0.985823i	$-0.553662\pi$
$397$	−6.68629	−0.335575	−0.167788	+	0.985823i	$0.553662\pi$
$398$	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
$403$	−3.65685	−0.182161
$404$	0	0
$405$	17.3431	0.861788
$406$	0	0
$407$	5.79899	0.287445
$408$	0	0
$409$	−16.3431	−0.808117	−0.404058	−	0.914733i	$-0.632401\pi$
$409$	−16.3431	−0.808117	−0.404058	+	0.914733i	$0.632401\pi$
$410$	0	0
$411$	10.4853	0.517201
$412$	0	0
$413$	35.3137	1.73767
$414$	0	0
$415$	22.2010	1.08980
$416$	0	0
$417$	3.82843	0.187479
$418$	0	0
$419$	3.24264	0.158413	0.0792067	−	0.996858i	$-0.474761\pi$
$419$	3.24264	0.158413	0.0792067	+	0.996858i	$0.474761\pi$
$420$	0	0
$421$	12.7990	0.623785	0.311892	−	0.950117i	$-0.399037\pi$
$421$	12.7990	0.623785	0.311892	+	0.950117i	$0.399037\pi$
$422$	0	0
$423$	−2.14214	−0.104154
$424$	0	0
$425$	−1.65685	−0.0803692
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−2.00000	−0.0965609
$430$	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
$435$	16.1421	0.773956
$436$	0	0
$437$	−49.9411	−2.38901
$438$	0	0
$439$	32.2843	1.54084	0.770422	−	0.637534i	$-0.220046\pi$
$439$	32.2843	1.54084	0.770422	+	0.637534i	$0.220046\pi$
$440$	0	0
$441$	35.3137	1.68161
$442$	0	0
$443$	−22.0711	−1.04863	−0.524314	−	0.851525i	$-0.675678\pi$
$443$	−22.0711	−1.04863	−0.524314	+	0.851525i	$0.675678\pi$
$444$	0	0
$445$	−24.9706	−1.18372
$446$	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
$451$	8.00000	0.376705
$452$	0	0
$453$	−1.82843	−0.0859070
$454$	0	0
$455$	8.07107	0.378377
$456$	0	0
$457$	26.3431	1.23228	0.616140	−	0.787637i	$-0.288695\pi$
$457$	26.3431	1.23228	0.616140	+	0.787637i	$0.288695\pi$
$458$	0	0
$459$	0.414214	0.0193338
$460$	0	0
$461$	−10.6569	−0.496339	−0.248170	−	0.968717i	$-0.579829\pi$
$461$	−10.6569	−0.496339	−0.248170	+	0.968717i	$0.579829\pi$
$462$	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
$467$	−2.00000	−0.0925490	−0.0462745	−	0.998929i	$-0.514735\pi$
$467$	−2.00000	−0.0925490	−0.0462745	+	0.998929i	$0.514735\pi$
$468$	0	0
$469$	38.9706	1.79949
$470$	0	0
$471$	−24.9706	−1.15058
$472$	0	0
$473$	6.97056	0.320507
$474$	0	0
$475$	−9.37258	−0.430044
$476$	0	0
$477$	10.3431	0.473580
$478$	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
$483$	−94.0833	−4.28094
$484$	0	0
$485$	14.0000	0.635707
$486$	0	0
$487$	−10.9706	−0.497124	−0.248562	−	0.968616i	$-0.579958\pi$
$487$	−10.9706	−0.497124	−0.248562	+	0.968616i	$0.579958\pi$
$488$	0	0
$489$	−34.9706	−1.58142
$490$	0	0
$491$	−27.8701	−1.25776	−0.628879	−	0.777503i	$-0.716486\pi$
$491$	−27.8701	−1.25776	−0.628879	+	0.777503i	$0.716486\pi$
$492$	0	0
$493$	3.65685	0.164696
$494$	0	0
$495$	−4.28427	−0.192564
$496$	0	0
$497$	−51.7696	−2.32218
$498$	0	0
$499$	10.3431	0.463023	0.231511	−	0.972832i	$-0.425633\pi$
$499$	10.3431	0.463023	0.231511	+	0.972832i	$0.425633\pi$
$500$	0	0
$501$	8.82843	0.394425
$502$	0	0
$503$	17.6569	0.787280	0.393640	−	0.919265i	$-0.371216\pi$
$503$	17.6569	0.787280	0.393640	+	0.919265i	$0.371216\pi$
$504$	0	0
$505$	32.2843	1.43663
$506$	0	0
$507$	−2.41421	−0.107219
$508$	0	0
$509$	21.3137	0.944714	0.472357	−	0.881407i	$-0.343403\pi$
$509$	21.3137	0.944714	0.472357	+	0.881407i	$0.343403\pi$
$510$	0	0
$511$	−7.31371	−0.323539
$512$	0	0
$513$	2.34315	0.103452
$514$	0	0
$515$	29.5147	1.30057
$516$	0	0
$517$	−0.627417	−0.0275938
$518$	0	0
$519$	−33.7990	−1.48361
$520$	0	0
$521$	−12.8579	−0.563313	−0.281657	−	0.959515i	$-0.590884\pi$
$521$	−12.8579	−0.563313	−0.281657	+	0.959515i	$0.590884\pi$
$522$	0	0
$523$	2.68629	0.117463	0.0587317	−	0.998274i	$-0.481294\pi$
$523$	2.68629	0.117463	0.0587317	+	0.998274i	$0.481294\pi$
$524$	0	0
$525$	−17.6569	−0.770608
$526$	0	0
$527$	−3.65685	−0.159295
$528$	0	0
$529$	54.9411	2.38874
$530$	0	0
$531$	−22.6274	−0.981946
$532$	0	0
$533$	9.65685	0.418285
$534$	0	0
$535$	−10.9706	−0.474299
$536$	0	0
$537$	31.1421	1.34388
$538$	0	0
$539$	10.3431	0.445511
$540$	0	0
$541$	−7.00000	−0.300954	−0.150477	−	0.988614i	$-0.548081\pi$
$541$	−7.00000	−0.300954	−0.150477	+	0.988614i	$0.548081\pi$
$542$	0	0
$543$	−8.82843	−0.378864
$544$	0	0
$545$	6.11270	0.261839
$546$	0	0
$547$	15.9289	0.681072	0.340536	−	0.940231i	$-0.389391\pi$
$547$	15.9289	0.681072	0.340536	+	0.940231i	$0.389391\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	20.6863	0.881266
$552$	0	0
$553$	−71.2548	−3.03006
$554$	0	0
$555$	30.8995	1.31161
$556$	0	0
$557$	−31.9706	−1.35464	−0.677318	−	0.735690i	$-0.736858\pi$
$557$	−31.9706	−1.35464	−0.677318	+	0.735690i	$0.736858\pi$
$558$	0	0
$559$	8.41421	0.355883
$560$	0	0
$561$	−2.00000	−0.0844401
$562$	0	0
$563$	17.0416	0.718219	0.359110	−	0.933295i	$-0.383080\pi$
$563$	17.0416	0.718219	0.359110	+	0.933295i	$0.383080\pi$
$564$	0	0
$565$	18.2843	0.769225
$566$	0	0
$567$	41.8701	1.75838
$568$	0	0
$569$	−3.34315	−0.140152	−0.0700760	−	0.997542i	$-0.522324\pi$
$569$	−3.34315	−0.140152	−0.0700760	+	0.997542i	$0.522324\pi$
$570$	0	0
$571$	10.2721	0.429873	0.214937	−	0.976628i	$-0.431046\pi$
$571$	10.2721	0.429873	0.214937	+	0.976628i	$0.431046\pi$
$572$	0	0
$573$	46.2843	1.93355
$574$	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
$579$	3.17157	0.131806
$580$	0	0
$581$	53.5980	2.22362
$582$	0	0
$583$	3.02944	0.125466
$584$	0	0
$585$	−5.17157	−0.213818
$586$	0	0
$587$	18.4853	0.762969	0.381485	−	0.924375i	$-0.375413\pi$
$587$	18.4853	0.762969	0.381485	+	0.924375i	$0.375413\pi$
$588$	0	0
$589$	−20.6863	−0.852364
$590$	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
$595$	8.07107	0.330882
$596$	0	0
$597$	3.65685	0.149665
$598$	0	0
$599$	7.31371	0.298830	0.149415	−	0.988775i	$-0.452261\pi$
$599$	7.31371	0.298830	0.149415	+	0.988775i	$0.452261\pi$
$600$	0	0
$601$	21.9706	0.896198	0.448099	−	0.893984i	$-0.352101\pi$
$601$	21.9706	0.896198	0.448099	+	0.893984i	$0.352101\pi$
$602$	0	0
$603$	−24.9706	−1.01688
$604$	0	0
$605$	18.8579	0.766681
$606$	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
$611$	−0.757359	−0.0306395
$612$	0	0
$613$	0.627417	0.0253411	0.0126706	−	0.999920i	$-0.495967\pi$
$613$	0.627417	0.0253411	0.0126706	+	0.999920i	$0.495967\pi$
$614$	0	0
$615$	42.6274	1.71890
$616$	0	0
$617$	−41.2548	−1.66086	−0.830429	−	0.557125i	$-0.811904\pi$
$617$	−41.2548	−1.66086	−0.830429	+	0.557125i	$0.811904\pi$
$618$	0	0
$619$	26.3431	1.05882	0.529410	−	0.848366i	$-0.322413\pi$
$619$	26.3431	1.05882	0.529410	+	0.848366i	$0.322413\pi$
$620$	0	0
$621$	−3.65685	−0.146745
$622$	0	0
$623$	−60.2843	−2.41524
$624$	0	0
$625$	−13.9706	−0.558823
$626$	0	0
$627$	−11.3137	−0.451826
$628$	0	0
$629$	7.00000	0.279108
$630$	0	0
$631$	43.3848	1.72712	0.863560	−	0.504246i	$-0.168229\pi$
$631$	43.3848	1.72712	0.863560	+	0.504246i	$0.168229\pi$
$632$	0	0
$633$	−10.6569	−0.423572
$634$	0	0
$635$	16.1421	0.640581
$636$	0	0
$637$	12.4853	0.494685
$638$	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
$643$	−28.8284	−1.13688	−0.568441	−	0.822724i	$-0.692453\pi$
$643$	−28.8284	−1.13688	−0.568441	+	0.822724i	$0.692453\pi$
$644$	0	0
$645$	37.1421	1.46247
$646$	0	0
$647$	−24.9706	−0.981694	−0.490847	−	0.871246i	$-0.663313\pi$
$647$	−24.9706	−0.981694	−0.490847	+	0.871246i	$0.663313\pi$
$648$	0	0
$649$	−6.62742	−0.260149
$650$	0	0
$651$	−38.9706	−1.52738
$652$	0	0
$653$	10.3431	0.404759	0.202379	−	0.979307i	$-0.435133\pi$
$653$	10.3431	0.404759	0.202379	+	0.979307i	$0.435133\pi$
$654$	0	0
$655$	−15.3848	−0.601133
$656$	0	0
$657$	4.68629	0.182830
$658$	0	0
$659$	−31.9411	−1.24425	−0.622125	−	0.782918i	$-0.713730\pi$
$659$	−31.9411	−1.24425	−0.622125	+	0.782918i	$0.713730\pi$
$660$	0	0
$661$	−6.68629	−0.260067	−0.130033	−	0.991510i	$-0.541508\pi$
$661$	−6.68629	−0.260067	−0.130033	+	0.991510i	$0.541508\pi$
$662$	0	0
$663$	−2.41421	−0.0937603
$664$	0	0
$665$	45.6569	1.77050
$666$	0	0
$667$	−32.2843	−1.25005
$668$	0	0
$669$	23.1421	0.894727
$670$	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
$675$	−0.686292	−0.0264154
$676$	0	0
$677$	−46.2843	−1.77885	−0.889425	−	0.457082i	$-0.848895\pi$
$677$	−46.2843	−1.77885	−0.889425	+	0.457082i	$0.848895\pi$
$678$	0	0
$679$	33.7990	1.29709
$680$	0	0
$681$	31.3137	1.19994
$682$	0	0
$683$	−8.82843	−0.337810	−0.168905	−	0.985632i	$-0.554023\pi$
$683$	−8.82843	−0.337810	−0.168905	+	0.985632i	$0.554023\pi$
$684$	0	0
$685$	7.94113	0.303415
$686$	0	0
$687$	−47.0416	−1.79475
$688$	0	0
$689$	3.65685	0.139315
$690$	0	0
$691$	10.6274	0.404286	0.202143	−	0.979356i	$-0.435209\pi$
$691$	10.6274	0.404286	0.202143	+	0.979356i	$0.435209\pi$
$692$	0	0
$693$	−10.3431	−0.392904
$694$	0	0
$695$	2.89949	0.109984
$696$	0	0
$697$	9.65685	0.365779
$698$	0	0
$699$	−43.0416	−1.62798
$700$	0	0
$701$	14.0000	0.528773	0.264386	−	0.964417i	$-0.414831\pi$
$701$	14.0000	0.528773	0.264386	+	0.964417i	$0.414831\pi$
$702$	0	0
$703$	39.5980	1.49347
$704$	0	0
$705$	−3.34315	−0.125910
$706$	0	0
$707$	77.9411	2.93128
$708$	0	0
$709$	6.68629	0.251109	0.125554	−	0.992087i	$-0.459929\pi$
$709$	6.68629	0.251109	0.125554	+	0.992087i	$0.459929\pi$
$710$	0	0
$711$	45.6569	1.71227
$712$	0	0
$713$	32.2843	1.20906
$714$	0	0
$715$	−1.51472	−0.0566473
$716$	0	0
$717$	53.2843	1.98994
$718$	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
$723$	31.3137	1.16457
$724$	0	0
$725$	−6.05887	−0.225021
$726$	0	0
$727$	36.8284	1.36589	0.682945	−	0.730469i	$-0.260699\pi$
$727$	36.8284	1.36589	0.682945	+	0.730469i	$0.260699\pi$
$728$	0	0
$729$	−23.8284	−0.882534
$730$	0	0
$731$	8.41421	0.311211
$732$	0	0
$733$	47.4853	1.75391	0.876954	−	0.480574i	$-0.159571\pi$
$733$	47.4853	1.75391	0.876954	+	0.480574i	$0.159571\pi$
$734$	0	0
$735$	55.1127	2.03286
$736$	0	0
$737$	−7.31371	−0.269404
$738$	0	0
$739$	−15.3137	−0.563324	−0.281662	−	0.959514i	$-0.590886\pi$
$739$	−15.3137	−0.563324	−0.281662	+	0.959514i	$0.590886\pi$
$740$	0	0
$741$	−13.6569	−0.501697
$742$	0	0
$743$	−25.7279	−0.943866	−0.471933	−	0.881634i	$-0.656444\pi$
$743$	−25.7279	−0.943866	−0.471933	+	0.881634i	$0.656444\pi$
$744$	0	0
$745$	38.9706	1.42777
$746$	0	0
$747$	−34.3431	−1.25655
$748$	0	0
$749$	−26.4853	−0.967751
$750$	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
$755$	−1.38478	−0.0503972
$756$	0	0
$757$	28.6274	1.04048	0.520241	−	0.854020i	$-0.325842\pi$
$757$	28.6274	1.04048	0.520241	+	0.854020i	$0.325842\pi$
$758$	0	0
$759$	17.6569	0.640903
$760$	0	0
$761$	−23.3137	−0.845121	−0.422561	−	0.906335i	$-0.638869\pi$
$761$	−23.3137	−0.845121	−0.422561	+	0.906335i	$0.638869\pi$
$762$	0	0
$763$	14.7574	0.534252
$764$	0	0
$765$	−5.17157	−0.186979
$766$	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
$771$	−54.6985	−1.96992
$772$	0	0
$773$	−26.7990	−0.963893	−0.481946	−	0.876201i	$-0.660070\pi$
$773$	−26.7990	−0.963893	−0.481946	+	0.876201i	$0.660070\pi$
$774$	0	0
$775$	6.05887	0.217641
$776$	0	0
$777$	74.5980	2.67619
$778$	0	0
$779$	54.6274	1.95723
$780$	0	0
$781$	9.71573	0.347656
$782$	0	0
$783$	1.51472	0.0541316
$784$	0	0
$785$	−18.9117	−0.674987
$786$	0	0
$787$	−37.6569	−1.34232	−0.671161	−	0.741312i	$-0.734204\pi$
$787$	−37.6569	−1.34232	−0.671161	+	0.741312i	$0.734204\pi$
$788$	0	0
$789$	−42.6274	−1.51758
$790$	0	0
$791$	44.1421	1.56951
$792$	0	0
$793$	0	0
$794$	0	0
$795$	16.1421	0.572503
$796$	0	0
$797$	−32.2843	−1.14357	−0.571784	−	0.820404i	$-0.693748\pi$
$797$	−32.2843	−1.14357	−0.571784	+	0.820404i	$0.693748\pi$
$798$	0	0
$799$	−0.757359	−0.0267934
$800$	0	0
$801$	38.6274	1.36483
$802$	0	0
$803$	1.37258	0.0484374
$804$	0	0
$805$	−71.2548	−2.51140
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−27.4853	−0.966331	−0.483166	−	0.875529i	$-0.660513\pi$
$809$	−27.4853	−0.966331	−0.483166	+	0.875529i	$0.660513\pi$
$810$	0	0
$811$	−28.9706	−1.01729	−0.508647	−	0.860975i	$-0.669854\pi$
$811$	−28.9706	−1.01729	−0.508647	+	0.860975i	$0.669854\pi$
$812$	0	0
$813$	−23.1421	−0.811630
$814$	0	0
$815$	−26.4853	−0.927739
$816$	0	0
$817$	47.5980	1.66524
$818$	0	0
$819$	−12.4853	−0.436271
$820$	0	0
$821$	−34.1127	−1.19054	−0.595271	−	0.803525i	$-0.702955\pi$
$821$	−34.1127	−1.19054	−0.595271	+	0.803525i	$0.702955\pi$
$822$	0	0
$823$	28.0000	0.976019	0.488009	−	0.872838i	$-0.337723\pi$
$823$	28.0000	0.976019	0.488009	+	0.872838i	$0.337723\pi$
$824$	0	0
$825$	3.31371	0.115369
$826$	0	0
$827$	33.9411	1.18025	0.590124	−	0.807312i	$-0.299079\pi$
$827$	33.9411	1.18025	0.590124	+	0.807312i	$0.299079\pi$
$828$	0	0
$829$	10.9706	0.381023	0.190512	−	0.981685i	$-0.438985\pi$
$829$	10.9706	0.381023	0.190512	+	0.981685i	$0.438985\pi$
$830$	0	0
$831$	60.2843	2.09124
$832$	0	0
$833$	12.4853	0.432589
$834$	0	0
$835$	6.68629	0.231389
$836$	0	0
$837$	−1.51472	−0.0523563
$838$	0	0
$839$	−38.9706	−1.34541	−0.672707	−	0.739909i	$-0.734868\pi$
$839$	−38.9706	−1.34541	−0.672707	+	0.739909i	$0.734868\pi$
$840$	0	0
$841$	−15.6274	−0.538876
$842$	0	0
$843$	9.65685	0.332600
$844$	0	0
$845$	−1.82843	−0.0628998
$846$	0	0
$847$	45.5269	1.56432
$848$	0	0
$849$	−63.4558	−2.17780
$850$	0	0
$851$	−61.7990	−2.11844
$852$	0	0
$853$	16.4558	0.563437	0.281719	−	0.959497i	$-0.409096\pi$
$853$	16.4558	0.563437	0.281719	+	0.959497i	$0.409096\pi$
$854$	0	0
$855$	−29.2548	−1.00049
$856$	0	0
$857$	−22.6863	−0.774949	−0.387474	−	0.921880i	$-0.626652\pi$
$857$	−22.6863	−0.774949	−0.387474	+	0.921880i	$0.626652\pi$
$858$	0	0
$859$	−16.6274	−0.567320	−0.283660	−	0.958925i	$-0.591549\pi$
$859$	−16.6274	−0.567320	−0.283660	+	0.958925i	$0.591549\pi$
$860$	0	0
$861$	102.912	3.50722
$862$	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
$867$	38.6274	1.31186
$868$	0	0
$869$	13.3726	0.453634
$870$	0	0
$871$	−8.82843	−0.299140
$872$	0	0
$873$	−21.6569	−0.732973
$874$	0	0
$875$	−53.7279	−1.81634
$876$	0	0
$877$	−23.1421	−0.781454	−0.390727	−	0.920507i	$-0.627776\pi$
$877$	−23.1421	−0.781454	−0.390727	+	0.920507i	$0.627776\pi$
$878$	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
$883$	26.2132	0.882145	0.441072	−	0.897472i	$-0.354598\pi$
$883$	26.2132	0.882145	0.441072	+	0.897472i	$0.354598\pi$
$884$	0	0
$885$	−35.3137	−1.18706
$886$	0	0
$887$	−10.3431	−0.347289	−0.173644	−	0.984808i	$-0.555554\pi$
$887$	−10.3431	−0.347289	−0.173644	+	0.984808i	$0.555554\pi$
$888$	0	0
$889$	38.9706	1.30703
$890$	0	0
$891$	−7.85786	−0.263248
$892$	0	0
$893$	−4.28427	−0.143368
$894$	0	0
$895$	23.5858	0.788386
$896$	0	0
$897$	21.3137	0.711644
$898$	0	0
$899$	−13.3726	−0.446001
$900$	0	0
$901$	3.65685	0.121827
$902$	0	0
$903$	89.6690	2.98400
$904$	0	0
$905$	−6.68629	−0.222260
$906$	0	0
$907$	11.4437	0.379980	0.189990	−	0.981786i	$-0.439154\pi$
$907$	11.4437	0.379980	0.189990	+	0.981786i	$0.439154\pi$
$908$	0	0
$909$	−49.9411	−1.65644
$910$	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
$917$	−37.1421	−1.22654
$918$	0	0
$919$	−45.6569	−1.50608	−0.753040	−	0.657974i	$-0.771414\pi$
$919$	−45.6569	−1.50608	−0.753040	+	0.657974i	$0.771414\pi$
$920$	0	0
$921$	−48.2843	−1.59102
$922$	0	0
$923$	11.7279	0.386029
$924$	0	0
$925$	−11.5980	−0.381339
$926$	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
$931$	70.6274	2.31472
$932$	0	0
$933$	−24.9706	−0.817500
$934$	0	0
$935$	−1.51472	−0.0495366
$936$	0	0
$937$	14.6863	0.479780	0.239890	−	0.970800i	$-0.422889\pi$
$937$	14.6863	0.479780	0.239890	+	0.970800i	$0.422889\pi$
$938$	0	0
$939$	80.0122	2.61110
$940$	0	0
$941$	−1.20101	−0.0391518	−0.0195759	−	0.999808i	$-0.506232\pi$
$941$	−1.20101	−0.0391518	−0.0195759	+	0.999808i	$0.506232\pi$
$942$	0	0
$943$	−85.2548	−2.77628
$944$	0	0
$945$	3.34315	0.108753
$946$	0	0
$947$	−35.1716	−1.14292	−0.571461	−	0.820629i	$-0.693623\pi$
$947$	−35.1716	−1.14292	−0.571461	+	0.820629i	$0.693623\pi$
$948$	0	0
$949$	1.65685	0.0537838
$950$	0	0
$951$	69.1127	2.24113
$952$	0	0
$953$	−42.9411	−1.39100	−0.695500	−	0.718526i	$-0.744817\pi$
$953$	−42.9411	−1.39100	−0.695500	+	0.718526i	$0.744817\pi$
$954$	0	0
$955$	35.0538	1.13432
$956$	0	0
$957$	−7.31371	−0.236419
$958$	0	0
$959$	19.1716	0.619082
$960$	0	0
$961$	−17.6274	−0.568626
$962$	0	0
$963$	16.9706	0.546869
$964$	0	0
$965$	2.40202	0.0773238
$966$	0	0
$967$	−23.5858	−0.758468	−0.379234	−	0.925301i	$-0.623812\pi$
$967$	−23.5858	−0.758468	−0.379234	+	0.925301i	$0.623812\pi$
$968$	0	0
$969$	−13.6569	−0.438721
$970$	0	0
$971$	42.8995	1.37671	0.688355	−	0.725374i	$-0.258333\pi$
$971$	42.8995	1.37671	0.688355	+	0.725374i	$0.258333\pi$
$972$	0	0
$973$	7.00000	0.224410
$974$	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
$979$	11.3137	0.361588
$980$	0	0
$981$	−9.45584	−0.301902
$982$	0	0
$983$	−4.41421	−0.140792	−0.0703958	−	0.997519i	$-0.522426\pi$
$983$	−4.41421	−0.140792	−0.0703958	+	0.997519i	$0.522426\pi$
$984$	0	0
$985$	12.7990	0.407810
$986$	0	0
$987$	−8.07107	−0.256905
$988$	0	0
$989$	−74.2843	−2.36210
$990$	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
$995$	2.76955	0.0878007
$996$	0	0
$997$	−42.0000	−1.33015	−0.665077	−	0.746775i	$-0.731601\pi$
$997$	−42.0000	−1.33015	−0.665077	+	0.746775i	$0.731601\pi$
$998$	0	0
$999$	2.89949	0.0917360

Twists

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

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

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:	\( N \)	\(=\)	\( 3328 = 2^{8} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3328.a (trivial)

\(f(q)\)	\(=\)	\(q-2.41421 q^{3} -1.82843 q^{5} -4.41421 q^{7} +2.82843 q^{9} +0.828427 q^{11} +1.00000 q^{13} +4.41421 q^{15} +1.00000 q^{17} +5.65685 q^{19} +10.6569 q^{21} -8.82843 q^{23} -1.65685 q^{25} +0.414214 q^{27} +3.65685 q^{29} -3.65685 q^{31} -2.00000 q^{33} +8.07107 q^{35} +7.00000 q^{37} -2.41421 q^{39} +9.65685 q^{41} +8.41421 q^{43} -5.17157 q^{45} -0.757359 q^{47} +12.4853 q^{49} -2.41421 q^{51} +3.65685 q^{53} -1.51472 q^{55} -13.6569 q^{57} -8.00000 q^{59} -12.4853 q^{63} -1.82843 q^{65} -8.82843 q^{67} +21.3137 q^{69} +11.7279 q^{71} +1.65685 q^{73} +4.00000 q^{75} -3.65685 q^{77} +16.1421 q^{79} -9.48528 q^{81} -12.1421 q^{83} -1.82843 q^{85} -8.82843 q^{87} +13.6569 q^{89} -4.41421 q^{91} +8.82843 q^{93} -10.3431 q^{95} -7.65685 q^{97} +2.34315 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 2 q^{5} - 6 q^{7} - 4 q^{11} + 2 q^{13} + 6 q^{15} + 2 q^{17} + 10 q^{21} - 12 q^{23} + 8 q^{25} - 2 q^{27} - 4 q^{29} + 4 q^{31} - 4 q^{33} + 2 q^{35} + 14 q^{37} - 2 q^{39} + 8 q^{41}+ \cdots + 16 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 + 2 * q^5 - 6 * q^7 - 4 * q^11 + 2 * q^13 + 6 * q^15 + 2 * q^17 + 10 * q^21 - 12 * q^23 + 8 * q^25 - 2 * q^27 - 4 * q^29 + 4 * q^31 - 4 * q^33 + 2 * q^35 + 14 * q^37 - 2 * q^39 + 8 * q^41 + 14 * q^43 - 16 * q^45 - 10 * q^47 + 8 * q^49 - 2 * q^51 - 4 * q^53 - 20 * q^55 - 16 * q^57 - 16 * q^59 - 8 * q^63 + 2 * q^65 - 12 * q^67 + 20 * q^69 - 2 * q^71 - 8 * q^73 + 8 * q^75 + 4 * q^77 + 4 * q^79 - 2 * q^81 + 4 * q^83 + 2 * q^85 - 12 * q^87 + 16 * q^89 - 6 * q^91 + 12 * q^93 - 32 * q^95 - 4 * q^97 + 16 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more