LMFDB - Embedded newform 8112.2.a.cu.1.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8112,2,Mod(1,8112)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8112, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("8112.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 8112 = 2^{4} \cdot 3 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8112.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,0,-6,0,1,0,5,0,6,0,6,0,0,0,-1,0,9] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	yes
Analytic conductor:	$64.7746461197$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.27700337.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 19x^{4} + 17x^{3} + 103x^{2} - 71x - 127 $ x^6 - x^5 - 19x^4 + 17x^3 + 103x^2 - 71x - 127
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 4056)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-0.920510$ of defining polynomial
Character	$\chi$	$=$	8112.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +2.72245 q^{5} -3.31535 q^{7} +1.00000 q^{9} -0.656645 q^{11} -2.72245 q^{15} -1.16749 q^{17} -6.77137 q^{19} +3.31535 q^{21} -6.90073 q^{23} +2.41172 q^{25} -1.00000 q^{27} -5.82010 q^{29} +0.969574 q^{31} +0.656645 q^{33} -9.02586 q^{35} -9.93482 q^{37} +10.5937 q^{41} +5.07523 q^{43} +2.72245 q^{45} +8.24364 q^{47} +3.99153 q^{49} +1.16749 q^{51} +0.841166 q^{53} -1.78768 q^{55} +6.77137 q^{57} -0.128144 q^{59} +11.7223 q^{61} -3.31535 q^{63} +15.9422 q^{67} +6.90073 q^{69} -5.32056 q^{71} -3.75917 q^{73} -2.41172 q^{75} +2.17701 q^{77} -2.17759 q^{79} +1.00000 q^{81} +10.8237 q^{83} -3.17843 q^{85} +5.82010 q^{87} -4.09193 q^{89} -0.969574 q^{93} -18.4347 q^{95} +16.5008 q^{97} -0.656645 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{3} + q^{5} + 5 q^{7} + 6 q^{9} + 6 q^{11} - q^{15} + 9 q^{17} - 7 q^{19} - 5 q^{21} - 12 q^{23} + 9 q^{25} - 6 q^{27} + 7 q^{29} + 11 q^{31} - 6 q^{33} - 6 q^{35} + 6 q^{37} + 13 q^{41} - 15 q^{43}+ \cdots + 6 q^{99}+O(q^{100}) $ 6 * q - 6 * q^3 + q^5 + 5 * q^7 + 6 * q^9 + 6 * q^11 - q^15 + 9 * q^17 - 7 * q^19 - 5 * q^21 - 12 * q^23 + 9 * q^25 - 6 * q^27 + 7 * q^29 + 11 * q^31 - 6 * q^33 - 6 * q^35 + 6 * q^37 + 13 * q^41 - 15 * q^43 + q^45 + 9 * q^47 + 13 * q^49 - 9 * q^51 + 22 * q^53 - 3 * q^55 + 7 * q^57 + 7 * q^59 + 25 * q^61 + 5 * q^63 - 5 * q^67 + 12 * q^69 - 8 * q^71 + 15 * q^73 - 9 * q^75 + 45 * q^77 - 14 * q^79 + 6 * q^81 + 13 * q^83 - 35 * q^85 - 7 * q^87 + 33 * q^89 - 11 * q^93 - 47 * q^95 + 50 * q^97 + 6 * 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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	2.72245	1.21752	0.608758	−	0.793356i	$-0.291668\pi$
$5$	2.72245	1.21752	0.608758	+	0.793356i	$0.291668\pi$
$6$	0	0
$7$	−3.31535	−1.25308	−0.626542	−	0.779388i	$-0.715530\pi$
$7$	−3.31535	−1.25308	−0.626542	+	0.779388i	$0.715530\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−0.656645	−0.197986	−0.0989930	−	0.995088i	$-0.531562\pi$
$11$	−0.656645	−0.197986	−0.0989930	+	0.995088i	$0.531562\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	−2.72245	−0.702933
$16$	0	0
$17$	−1.16749	−0.283158	−0.141579	−	0.989927i	$-0.545218\pi$
$17$	−1.16749	−0.283158	−0.141579	+	0.989927i	$0.545218\pi$
$18$	0	0
$19$	−6.77137	−1.55346	−0.776729	−	0.629835i	$-0.783122\pi$
$19$	−6.77137	−1.55346	−0.776729	+	0.629835i	$0.783122\pi$
$20$	0	0
$21$	3.31535	0.723468
$22$	0	0
$23$	−6.90073	−1.43890	−0.719451	−	0.694543i	$-0.755607\pi$
$23$	−6.90073	−1.43890	−0.719451	+	0.694543i	$0.755607\pi$
$24$	0	0
$25$	2.41172	0.482344
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−5.82010	−1.08077	−0.540383	−	0.841419i	$-0.681721\pi$
$29$	−5.82010	−1.08077	−0.540383	+	0.841419i	$0.681721\pi$
$30$	0	0
$31$	0.969574	0.174141	0.0870703	−	0.996202i	$-0.472250\pi$
$31$	0.969574	0.174141	0.0870703	+	0.996202i	$0.472250\pi$
$32$	0	0
$33$	0.656645	0.114307
$34$	0	0
$35$	−9.02586	−1.52565
$36$	0	0
$37$	−9.93482	−1.63327	−0.816637	−	0.577151i	$-0.804164\pi$
$37$	−9.93482	−1.63327	−0.816637	+	0.577151i	$0.804164\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	10.5937	1.65445	0.827227	−	0.561869i	$-0.189917\pi$
$41$	10.5937	1.65445	0.827227	+	0.561869i	$0.189917\pi$
$42$	0	0
$43$	5.07523	0.773965	0.386983	−	0.922087i	$-0.373517\pi$
$43$	5.07523	0.773965	0.386983	+	0.922087i	$0.373517\pi$
$44$	0	0
$45$	2.72245	0.405839
$46$	0	0
$47$	8.24364	1.20246	0.601229	−	0.799077i	$-0.294678\pi$
$47$	8.24364	1.20246	0.601229	+	0.799077i	$0.294678\pi$
$48$	0	0
$49$	3.99153	0.570218
$50$	0	0
$51$	1.16749	0.163481
$52$	0	0
$53$	0.841166	0.115543	0.0577715	−	0.998330i	$-0.481601\pi$
$53$	0.841166	0.115543	0.0577715	+	0.998330i	$0.481601\pi$
$54$	0	0
$55$	−1.78768	−0.241051
$56$	0	0
$57$	6.77137	0.896889
$58$	0	0
$59$	−0.128144	−0.0166829	−0.00834145	−	0.999965i	$-0.502655\pi$
$59$	−0.128144	−0.0166829	−0.00834145	+	0.999965i	$0.502655\pi$
$60$	0	0
$61$	11.7223	1.50089	0.750443	−	0.660935i	$-0.229840\pi$
$61$	11.7223	1.50089	0.750443	+	0.660935i	$0.229840\pi$
$62$	0	0
$63$	−3.31535	−0.417694
$64$	0	0
$65$	0	0
$66$	0	0
$67$	15.9422	1.94765	0.973824	−	0.227303i	$-0.0729906\pi$
$67$	15.9422	1.94765	0.973824	+	0.227303i	$0.0729906\pi$
$68$	0	0
$69$	6.90073	0.830750
$70$	0	0
$71$	−5.32056	−0.631434	−0.315717	−	0.948853i	$-0.602245\pi$
$71$	−5.32056	−0.631434	−0.315717	+	0.948853i	$0.602245\pi$
$72$	0	0
$73$	−3.75917	−0.439977	−0.219989	−	0.975502i	$-0.570602\pi$
$73$	−3.75917	−0.439977	−0.219989	+	0.975502i	$0.570602\pi$
$74$	0	0
$75$	−2.41172	−0.278482
$76$	0	0
$77$	2.17701	0.248093
$78$	0	0
$79$	−2.17759	−0.244998	−0.122499	−	0.992469i	$-0.539091\pi$
$79$	−2.17759	−0.244998	−0.122499	+	0.992469i	$0.539091\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	10.8237	1.18806	0.594029	−	0.804444i	$-0.297536\pi$
$83$	10.8237	1.18806	0.594029	+	0.804444i	$0.297536\pi$
$84$	0	0
$85$	−3.17843	−0.344749
$86$	0	0
$87$	5.82010	0.623980
$88$	0	0
$89$	−4.09193	−0.433743	−0.216872	−	0.976200i	$-0.569585\pi$
$89$	−4.09193	−0.433743	−0.216872	+	0.976200i	$0.569585\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−0.969574	−0.100540
$94$	0	0
$95$	−18.4347	−1.89136
$96$	0	0
$97$	16.5008	1.67540	0.837702	−	0.546127i	$-0.183899\pi$
$97$	16.5008	1.67540	0.837702	+	0.546127i	$0.183899\pi$
$98$	0	0
$99$	−0.656645	−0.0659953
$100$	0	0
$101$	14.4134	1.43419	0.717096	−	0.696975i	$-0.245471\pi$
$101$	14.4134	1.43419	0.717096	+	0.696975i	$0.245471\pi$
$102$	0	0
$103$	15.7309	1.55001	0.775007	−	0.631953i	$-0.217747\pi$
$103$	15.7309	1.55001	0.775007	+	0.631953i	$0.217747\pi$
$104$	0	0
$105$	9.02586	0.880834
$106$	0	0
$107$	−9.27509	−0.896656	−0.448328	−	0.893869i	$-0.647980\pi$
$107$	−9.27509	−0.896656	−0.448328	+	0.893869i	$0.647980\pi$
$108$	0	0
$109$	8.82889	0.845655	0.422827	−	0.906210i	$-0.361038\pi$
$109$	8.82889	0.845655	0.422827	+	0.906210i	$0.361038\pi$
$110$	0	0
$111$	9.93482	0.942971
$112$	0	0
$113$	6.45873	0.607586	0.303793	−	0.952738i	$-0.401747\pi$
$113$	6.45873	0.607586	0.303793	+	0.952738i	$0.401747\pi$
$114$	0	0
$115$	−18.7869	−1.75189
$116$	0	0
$117$	0	0
$118$	0	0
$119$	3.87063	0.354820
$120$	0	0
$121$	−10.5688	−0.960802
$122$	0	0
$123$	−10.5937	−0.955199
$124$	0	0
$125$	−7.04645	−0.630254
$126$	0	0
$127$	4.90436	0.435191	0.217596	−	0.976039i	$-0.430179\pi$
$127$	4.90436	0.435191	0.217596	+	0.976039i	$0.430179\pi$
$128$	0	0
$129$	−5.07523	−0.446849
$130$	0	0
$131$	13.9311	1.21717	0.608584	−	0.793489i	$-0.291738\pi$
$131$	13.9311	1.21717	0.608584	+	0.793489i	$0.291738\pi$
$132$	0	0
$133$	22.4494	1.94661
$134$	0	0
$135$	−2.72245	−0.234311
$136$	0	0
$137$	−17.4589	−1.49161	−0.745806	−	0.666163i	$-0.767936\pi$
$137$	−17.4589	−1.49161	−0.745806	+	0.666163i	$0.767936\pi$
$138$	0	0
$139$	−9.93881	−0.842999	−0.421499	−	0.906829i	$-0.638496\pi$
$139$	−9.93881	−0.842999	−0.421499	+	0.906829i	$0.638496\pi$
$140$	0	0
$141$	−8.24364	−0.694239
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−15.8449	−1.31585
$146$	0	0
$147$	−3.99153	−0.329216
$148$	0	0
$149$	−14.7938	−1.21196	−0.605979	−	0.795481i	$-0.707218\pi$
$149$	−14.7938	−1.21196	−0.605979	+	0.795481i	$0.707218\pi$
$150$	0	0
$151$	11.5920	0.943340	0.471670	−	0.881775i	$-0.343651\pi$
$151$	11.5920	0.943340	0.471670	+	0.881775i	$0.343651\pi$
$152$	0	0
$153$	−1.16749	−0.0943859
$154$	0	0
$155$	2.63961	0.212019
$156$	0	0
$157$	−16.3811	−1.30735	−0.653677	−	0.756773i	$-0.726775\pi$
$157$	−16.3811	−1.30735	−0.653677	+	0.756773i	$0.726775\pi$
$158$	0	0
$159$	−0.841166	−0.0667088
$160$	0	0
$161$	22.8783	1.80306
$162$	0	0
$163$	21.2719	1.66615	0.833073	−	0.553163i	$-0.186579\pi$
$163$	21.2719	1.66615	0.833073	+	0.553163i	$0.186579\pi$
$164$	0	0
$165$	1.78768	0.139171
$166$	0	0
$167$	10.5498	0.816367	0.408183	−	0.912900i	$-0.366162\pi$
$167$	10.5498	0.816367	0.408183	+	0.912900i	$0.366162\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	−6.77137	−0.517819
$172$	0	0
$173$	20.4284	1.55314	0.776569	−	0.630032i	$-0.216958\pi$
$173$	20.4284	1.55314	0.776569	+	0.630032i	$0.216958\pi$
$174$	0	0
$175$	−7.99570	−0.604418
$176$	0	0
$177$	0.128144	0.00963188
$178$	0	0
$179$	5.71037	0.426813	0.213406	−	0.976964i	$-0.431544\pi$
$179$	5.71037	0.426813	0.213406	+	0.976964i	$0.431544\pi$
$180$	0	0
$181$	4.70324	0.349589	0.174795	−	0.984605i	$-0.444074\pi$
$181$	4.70324	0.349589	0.174795	+	0.984605i	$0.444074\pi$
$182$	0	0
$183$	−11.7223	−0.866537
$184$	0	0
$185$	−27.0470	−1.98854
$186$	0	0
$187$	0.766626	0.0560613
$188$	0	0
$189$	3.31535	0.241156
$190$	0	0
$191$	−14.7128	−1.06458	−0.532292	−	0.846561i	$-0.678669\pi$
$191$	−14.7128	−1.06458	−0.532292	+	0.846561i	$0.678669\pi$
$192$	0	0
$193$	1.50643	0.108435	0.0542177	−	0.998529i	$-0.482734\pi$
$193$	1.50643	0.108435	0.0542177	+	0.998529i	$0.482734\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−17.4470	−1.24305	−0.621524	−	0.783395i	$-0.713486\pi$
$197$	−17.4470	−1.24305	−0.621524	+	0.783395i	$0.713486\pi$
$198$	0	0
$199$	−17.4032	−1.23368	−0.616841	−	0.787088i	$-0.711588\pi$
$199$	−17.4032	−1.23368	−0.616841	+	0.787088i	$0.711588\pi$
$200$	0	0
$201$	−15.9422	−1.12448
$202$	0	0
$203$	19.2957	1.35429
$204$	0	0
$205$	28.8407	2.01432
$206$	0	0
$207$	−6.90073	−0.479634
$208$	0	0
$209$	4.44638	0.307563
$210$	0	0
$211$	12.4002	0.853662	0.426831	−	0.904331i	$-0.359630\pi$
$211$	12.4002	0.853662	0.426831	+	0.904331i	$0.359630\pi$
$212$	0	0
$213$	5.32056	0.364559
$214$	0	0
$215$	13.8170	0.942315
$216$	0	0
$217$	−3.21447	−0.218213
$218$	0	0
$219$	3.75917	0.254021
$220$	0	0
$221$	0	0
$222$	0	0
$223$	12.5356	0.839448	0.419724	−	0.907652i	$-0.362127\pi$
$223$	12.5356	0.839448	0.419724	+	0.907652i	$0.362127\pi$
$224$	0	0
$225$	2.41172	0.160781
$226$	0	0
$227$	−1.16802	−0.0775243	−0.0387622	−	0.999248i	$-0.512341\pi$
$227$	−1.16802	−0.0775243	−0.0387622	+	0.999248i	$0.512341\pi$
$228$	0	0
$229$	21.9158	1.44824	0.724119	−	0.689675i	$-0.242247\pi$
$229$	21.9158	1.44824	0.724119	+	0.689675i	$0.242247\pi$
$230$	0	0
$231$	−2.17701	−0.143237
$232$	0	0
$233$	−3.00659	−0.196968	−0.0984842	−	0.995139i	$-0.531399\pi$
$233$	−3.00659	−0.196968	−0.0984842	+	0.995139i	$0.531399\pi$
$234$	0	0
$235$	22.4429	1.46401
$236$	0	0
$237$	2.17759	0.141450
$238$	0	0
$239$	−12.7853	−0.827015	−0.413508	−	0.910501i	$-0.635697\pi$
$239$	−12.7853	−0.827015	−0.413508	+	0.910501i	$0.635697\pi$
$240$	0	0
$241$	23.6123	1.52100	0.760499	−	0.649339i	$-0.224954\pi$
$241$	23.6123	1.52100	0.760499	+	0.649339i	$0.224954\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	10.8667	0.694249
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−10.8237	−0.685926
$250$	0	0
$251$	−8.22001	−0.518843	−0.259421	−	0.965764i	$-0.583532\pi$
$251$	−8.22001	−0.518843	−0.259421	+	0.965764i	$0.583532\pi$
$252$	0	0
$253$	4.53133	0.284882
$254$	0	0
$255$	3.17843	0.199041
$256$	0	0
$257$	27.7850	1.73318	0.866592	−	0.499018i	$-0.166306\pi$
$257$	27.7850	1.73318	0.866592	+	0.499018i	$0.166306\pi$
$258$	0	0
$259$	32.9374	2.04663
$260$	0	0
$261$	−5.82010	−0.360255
$262$	0	0
$263$	−10.5270	−0.649120	−0.324560	−	0.945865i	$-0.605216\pi$
$263$	−10.5270	−0.649120	−0.324560	+	0.945865i	$0.605216\pi$
$264$	0	0
$265$	2.29003	0.140676
$266$	0	0
$267$	4.09193	0.250422
$268$	0	0
$269$	−2.51426	−0.153297	−0.0766486	−	0.997058i	$-0.524422\pi$
$269$	−2.51426	−0.153297	−0.0766486	+	0.997058i	$0.524422\pi$
$270$	0	0
$271$	4.83460	0.293681	0.146841	−	0.989160i	$-0.453090\pi$
$271$	4.83460	0.293681	0.146841	+	0.989160i	$0.453090\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−1.58365	−0.0954974
$276$	0	0
$277$	−15.4274	−0.926941	−0.463471	−	0.886112i	$-0.653396\pi$
$277$	−15.4274	−0.926941	−0.463471	+	0.886112i	$0.653396\pi$
$278$	0	0
$279$	0.969574	0.0580469
$280$	0	0
$281$	−6.10796	−0.364370	−0.182185	−	0.983264i	$-0.558317\pi$
$281$	−6.10796	−0.364370	−0.182185	+	0.983264i	$0.558317\pi$
$282$	0	0
$283$	−11.3138	−0.672538	−0.336269	−	0.941766i	$-0.609165\pi$
$283$	−11.3138	−0.672538	−0.336269	+	0.941766i	$0.609165\pi$
$284$	0	0
$285$	18.4347	1.09198
$286$	0	0
$287$	−35.1217	−2.07317
$288$	0	0
$289$	−15.6370	−0.919822
$290$	0	0
$291$	−16.5008	−0.967295
$292$	0	0
$293$	28.0013	1.63585	0.817927	−	0.575322i	$-0.195123\pi$
$293$	28.0013	1.63585	0.817927	+	0.575322i	$0.195123\pi$
$294$	0	0
$295$	−0.348865	−0.0203117
$296$	0	0
$297$	0.656645	0.0381024
$298$	0	0
$299$	0	0
$300$	0	0
$301$	−16.8261	−0.969843
$302$	0	0
$303$	−14.4134	−0.828031
$304$	0	0
$305$	31.9134	1.82735
$306$	0	0
$307$	−18.6077	−1.06199	−0.530997	−	0.847373i	$-0.678183\pi$
$307$	−18.6077	−1.06199	−0.530997	+	0.847373i	$0.678183\pi$
$308$	0	0
$309$	−15.7309	−0.894900
$310$	0	0
$311$	−25.9076	−1.46909	−0.734543	−	0.678563i	$-0.762603\pi$
$311$	−25.9076	−1.46909	−0.734543	+	0.678563i	$0.762603\pi$
$312$	0	0
$313$	−0.799031	−0.0451639	−0.0225819	−	0.999745i	$-0.507189\pi$
$313$	−0.799031	−0.0451639	−0.0225819	+	0.999745i	$0.507189\pi$
$314$	0	0
$315$	−9.02586	−0.508550
$316$	0	0
$317$	32.7072	1.83702	0.918509	−	0.395400i	$-0.129394\pi$
$317$	32.7072	1.83702	0.918509	+	0.395400i	$0.129394\pi$
$318$	0	0
$319$	3.82174	0.213976
$320$	0	0
$321$	9.27509	0.517685
$322$	0	0
$323$	7.90550	0.439874
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−8.82889	−0.488239
$328$	0	0
$329$	−27.3305	−1.50678
$330$	0	0
$331$	−5.69662	−0.313115	−0.156557	−	0.987669i	$-0.550040\pi$
$331$	−5.69662	−0.313115	−0.156557	+	0.987669i	$0.550040\pi$
$332$	0	0
$333$	−9.93482	−0.544425
$334$	0	0
$335$	43.4018	2.37129
$336$	0	0
$337$	−10.4046	−0.566773	−0.283387	−	0.959006i	$-0.591458\pi$
$337$	−10.4046	−0.566773	−0.283387	+	0.959006i	$0.591458\pi$
$338$	0	0
$339$	−6.45873	−0.350790
$340$	0	0
$341$	−0.636666	−0.0344774
$342$	0	0
$343$	9.97413	0.538553
$344$	0	0
$345$	18.7869	1.01145
$346$	0	0
$347$	−9.76863	−0.524408	−0.262204	−	0.965013i	$-0.584449\pi$
$347$	−9.76863	−0.524408	−0.262204	+	0.965013i	$0.584449\pi$
$348$	0	0
$349$	−6.11185	−0.327160	−0.163580	−	0.986530i	$-0.552304\pi$
$349$	−6.11185	−0.327160	−0.163580	+	0.986530i	$0.552304\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	21.1136	1.12376	0.561882	−	0.827217i	$-0.310077\pi$
$353$	21.1136	1.12376	0.561882	+	0.827217i	$0.310077\pi$
$354$	0	0
$355$	−14.4849	−0.768781
$356$	0	0
$357$	−3.87063	−0.204856
$358$	0	0
$359$	−2.37346	−0.125267	−0.0626333	−	0.998037i	$-0.519950\pi$
$359$	−2.37346	−0.125267	−0.0626333	+	0.998037i	$0.519950\pi$
$360$	0	0
$361$	26.8514	1.41323
$362$	0	0
$363$	10.5688	0.554719
$364$	0	0
$365$	−10.2341	−0.535679
$366$	0	0
$367$	34.1977	1.78511	0.892553	−	0.450943i	$-0.148912\pi$
$367$	34.1977	1.78511	0.892553	+	0.450943i	$0.148912\pi$
$368$	0	0
$369$	10.5937	0.551484
$370$	0	0
$371$	−2.78876	−0.144785
$372$	0	0
$373$	4.01148	0.207707	0.103853	−	0.994593i	$-0.466883\pi$
$373$	4.01148	0.207707	0.103853	+	0.994593i	$0.466883\pi$
$374$	0	0
$375$	7.04645	0.363877
$376$	0	0
$377$	0	0
$378$	0	0
$379$	0.282802	0.0145265	0.00726327	−	0.999974i	$-0.497688\pi$
$379$	0.282802	0.0145265	0.00726327	+	0.999974i	$0.497688\pi$
$380$	0	0
$381$	−4.90436	−0.251258
$382$	0	0
$383$	11.7130	0.598508	0.299254	−	0.954173i	$-0.403262\pi$
$383$	11.7130	0.598508	0.299254	+	0.954173i	$0.403262\pi$
$384$	0	0
$385$	5.92679	0.302057
$386$	0	0
$387$	5.07523	0.257988
$388$	0	0
$389$	−11.7485	−0.595672	−0.297836	−	0.954617i	$-0.596265\pi$
$389$	−11.7485	−0.595672	−0.297836	+	0.954617i	$0.596265\pi$
$390$	0	0
$391$	8.05653	0.407436
$392$	0	0
$393$	−13.9311	−0.702733
$394$	0	0
$395$	−5.92838	−0.298289
$396$	0	0
$397$	−1.55897	−0.0782427	−0.0391213	−	0.999234i	$-0.512456\pi$
$397$	−1.55897	−0.0782427	−0.0391213	+	0.999234i	$0.512456\pi$
$398$	0	0
$399$	−22.4494	−1.12388
$400$	0	0
$401$	−5.75468	−0.287375	−0.143687	−	0.989623i	$-0.545896\pi$
$401$	−5.75468	−0.287375	−0.143687	+	0.989623i	$0.545896\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	2.72245	0.135280
$406$	0	0
$407$	6.52365	0.323365
$408$	0	0
$409$	22.5609	1.11557	0.557783	−	0.829987i	$-0.311652\pi$
$409$	22.5609	1.11557	0.557783	+	0.829987i	$0.311652\pi$
$410$	0	0
$411$	17.4589	0.861183
$412$	0	0
$413$	0.424841	0.0209051
$414$	0	0
$415$	29.4670	1.44648
$416$	0	0
$417$	9.93881	0.486706
$418$	0	0
$419$	−14.5676	−0.711675	−0.355837	−	0.934548i	$-0.615804\pi$
$419$	−14.5676	−0.711675	−0.355837	+	0.934548i	$0.615804\pi$
$420$	0	0
$421$	−40.2281	−1.96060	−0.980298	−	0.197525i	$-0.936710\pi$
$421$	−40.2281	−1.96060	−0.980298	+	0.197525i	$0.936710\pi$
$422$	0	0
$423$	8.24364	0.400819
$424$	0	0
$425$	−2.81566	−0.136580
$426$	0	0
$427$	−38.8635	−1.88074
$428$	0	0
$429$	0	0
$430$	0	0
$431$	14.9559	0.720402	0.360201	−	0.932875i	$-0.382708\pi$
$431$	14.9559	0.720402	0.360201	+	0.932875i	$0.382708\pi$
$432$	0	0
$433$	−30.1200	−1.44748	−0.723738	−	0.690074i	$-0.757578\pi$
$433$	−30.1200	−1.44748	−0.723738	+	0.690074i	$0.757578\pi$
$434$	0	0
$435$	15.8449	0.759706
$436$	0	0
$437$	46.7274	2.23527
$438$	0	0
$439$	−10.7914	−0.515043	−0.257522	−	0.966273i	$-0.582906\pi$
$439$	−10.7914	−0.515043	−0.257522	+	0.966273i	$0.582906\pi$
$440$	0	0
$441$	3.99153	0.190073
$442$	0	0
$443$	−30.6498	−1.45622	−0.728108	−	0.685463i	$-0.759600\pi$
$443$	−30.6498	−1.45622	−0.728108	+	0.685463i	$0.759600\pi$
$444$	0	0
$445$	−11.1401	−0.528089
$446$	0	0
$447$	14.7938	0.699724
$448$	0	0
$449$	12.3953	0.584970	0.292485	−	0.956270i	$-0.405518\pi$
$449$	12.3953	0.584970	0.292485	+	0.956270i	$0.405518\pi$
$450$	0	0
$451$	−6.95628	−0.327558
$452$	0	0
$453$	−11.5920	−0.544638
$454$	0	0
$455$	0	0
$456$	0	0
$457$	21.5121	1.00629	0.503147	−	0.864201i	$-0.332175\pi$
$457$	21.5121	1.00629	0.503147	+	0.864201i	$0.332175\pi$
$458$	0	0
$459$	1.16749	0.0544938
$460$	0	0
$461$	−20.2208	−0.941777	−0.470889	−	0.882193i	$-0.656067\pi$
$461$	−20.2208	−0.941777	−0.470889	+	0.882193i	$0.656067\pi$
$462$	0	0
$463$	11.2424	0.522477	0.261239	−	0.965274i	$-0.415869\pi$
$463$	11.2424	0.522477	0.261239	+	0.965274i	$0.415869\pi$
$464$	0	0
$465$	−2.63961	−0.122409
$466$	0	0
$467$	27.1166	1.25481	0.627403	−	0.778695i	$-0.284118\pi$
$467$	27.1166	1.25481	0.627403	+	0.778695i	$0.284118\pi$
$468$	0	0
$469$	−52.8539	−2.44057
$470$	0	0
$471$	16.3811	0.754802
$472$	0	0
$473$	−3.33262	−0.153234
$474$	0	0
$475$	−16.3307	−0.749302
$476$	0	0
$477$	0.841166	0.0385144
$478$	0	0
$479$	13.8701	0.633743	0.316872	−	0.948468i	$-0.397368\pi$
$479$	13.8701	0.633743	0.316872	+	0.948468i	$0.397368\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	−22.8783	−1.04100
$484$	0	0
$485$	44.9226	2.03983
$486$	0	0
$487$	0.0497164	0.00225287	0.00112643	−	0.999999i	$-0.499641\pi$
$487$	0.0497164	0.00225287	0.00112643	+	0.999999i	$0.499641\pi$
$488$	0	0
$489$	−21.2719	−0.961950
$490$	0	0
$491$	21.9646	0.991248	0.495624	−	0.868537i	$-0.334939\pi$
$491$	21.9646	0.991248	0.495624	+	0.868537i	$0.334939\pi$
$492$	0	0
$493$	6.79491	0.306027
$494$	0	0
$495$	−1.78768	−0.0803503
$496$	0	0
$497$	17.6395	0.791240
$498$	0	0
$499$	35.8347	1.60418	0.802091	−	0.597202i	$-0.203721\pi$
$499$	35.8347	1.60418	0.802091	+	0.597202i	$0.203721\pi$
$500$	0	0
$501$	−10.5498	−0.471329
$502$	0	0
$503$	0.104369	0.00465358	0.00232679	−	0.999997i	$-0.499259\pi$
$503$	0.104369	0.00465358	0.00232679	+	0.999997i	$0.499259\pi$
$504$	0	0
$505$	39.2399	1.74615
$506$	0	0
$507$	0	0
$508$	0	0
$509$	1.19452	0.0529464	0.0264732	−	0.999650i	$-0.491572\pi$
$509$	1.19452	0.0529464	0.0264732	+	0.999650i	$0.491572\pi$
$510$	0	0
$511$	12.4629	0.551328
$512$	0	0
$513$	6.77137	0.298963
$514$	0	0
$515$	42.8266	1.88717
$516$	0	0
$517$	−5.41314	−0.238070
$518$	0	0
$519$	−20.4284	−0.896705
$520$	0	0
$521$	22.8446	1.00084	0.500420	−	0.865783i	$-0.333179\pi$
$521$	22.8446	1.00084	0.500420	+	0.865783i	$0.333179\pi$
$522$	0	0
$523$	−38.3582	−1.67728	−0.838642	−	0.544682i	$-0.816650\pi$
$523$	−38.3582	−1.67728	−0.838642	+	0.544682i	$0.816650\pi$
$524$	0	0
$525$	7.99570	0.348961
$526$	0	0
$527$	−1.13197	−0.0493093
$528$	0	0
$529$	24.6201	1.07044
$530$	0	0
$531$	−0.128144	−0.00556097
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−25.2509	−1.09169
$536$	0	0
$537$	−5.71037	−0.246420
$538$	0	0
$539$	−2.62102	−0.112895
$540$	0	0
$541$	−10.5126	−0.451970	−0.225985	−	0.974131i	$-0.572560\pi$
$541$	−10.5126	−0.451970	−0.225985	+	0.974131i	$0.572560\pi$
$542$	0	0
$543$	−4.70324	−0.201835
$544$	0	0
$545$	24.0362	1.02960
$546$	0	0
$547$	−23.9358	−1.02342	−0.511711	−	0.859158i	$-0.670988\pi$
$547$	−23.9358	−1.02342	−0.511711	+	0.859158i	$0.670988\pi$
$548$	0	0
$549$	11.7223	0.500296
$550$	0	0
$551$	39.4100	1.67892
$552$	0	0
$553$	7.21948	0.307003
$554$	0	0
$555$	27.0470	1.14808
$556$	0	0
$557$	0.843060	0.0357216	0.0178608	−	0.999840i	$-0.494314\pi$
$557$	0.843060	0.0357216	0.0178608	+	0.999840i	$0.494314\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	−0.766626	−0.0323670
$562$	0	0
$563$	18.4169	0.776179	0.388090	−	0.921622i	$-0.373135\pi$
$563$	18.4169	0.776179	0.388090	+	0.921622i	$0.373135\pi$
$564$	0	0
$565$	17.5836	0.739746
$566$	0	0
$567$	−3.31535	−0.139231
$568$	0	0
$569$	−27.6396	−1.15871	−0.579356	−	0.815075i	$-0.696696\pi$
$569$	−27.6396	−1.15871	−0.579356	+	0.815075i	$0.696696\pi$
$570$	0	0
$571$	−8.23866	−0.344777	−0.172389	−	0.985029i	$-0.555148\pi$
$571$	−8.23866	−0.344777	−0.172389	+	0.985029i	$0.555148\pi$
$572$	0	0
$573$	14.7128	0.614638
$574$	0	0
$575$	−16.6426	−0.694046
$576$	0	0
$577$	−2.60340	−0.108381	−0.0541906	−	0.998531i	$-0.517258\pi$
$577$	−2.60340	−0.108381	−0.0541906	+	0.998531i	$0.517258\pi$
$578$	0	0
$579$	−1.50643	−0.0626052
$580$	0	0
$581$	−35.8844	−1.48874
$582$	0	0
$583$	−0.552348	−0.0228759
$584$	0	0
$585$	0	0
$586$	0	0
$587$	8.62011	0.355790	0.177895	−	0.984049i	$-0.443071\pi$
$587$	8.62011	0.355790	0.177895	+	0.984049i	$0.443071\pi$
$588$	0	0
$589$	−6.56534	−0.270520
$590$	0	0
$591$	17.4470	0.717674
$592$	0	0
$593$	39.3090	1.61423	0.807114	−	0.590395i	$-0.201028\pi$
$593$	39.3090	1.61423	0.807114	+	0.590395i	$0.201028\pi$
$594$	0	0
$595$	10.5376	0.431999
$596$	0	0
$597$	17.4032	0.712266
$598$	0	0
$599$	−38.7451	−1.58308	−0.791542	−	0.611115i	$-0.790721\pi$
$599$	−38.7451	−1.58308	−0.791542	+	0.611115i	$0.790721\pi$
$600$	0	0
$601$	−31.5673	−1.28766	−0.643828	−	0.765170i	$-0.722655\pi$
$601$	−31.5673	−1.28766	−0.643828	+	0.765170i	$0.722655\pi$
$602$	0	0
$603$	15.9422	0.649216
$604$	0	0
$605$	−28.7731	−1.16979
$606$	0	0
$607$	39.5512	1.60533	0.802667	−	0.596427i	$-0.203414\pi$
$607$	39.5512	1.60533	0.802667	+	0.596427i	$0.203414\pi$
$608$	0	0
$609$	−19.2957	−0.781899
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−42.8092	−1.72905	−0.864524	−	0.502592i	$-0.832380\pi$
$613$	−42.8092	−1.72905	−0.864524	+	0.502592i	$0.832380\pi$
$614$	0	0
$615$	−28.8407	−1.16297
$616$	0	0
$617$	19.9368	0.802626	0.401313	−	0.915941i	$-0.368554\pi$
$617$	19.9368	0.802626	0.401313	+	0.915941i	$0.368554\pi$
$618$	0	0
$619$	−38.1555	−1.53360	−0.766799	−	0.641887i	$-0.778152\pi$
$619$	−38.1555	−1.53360	−0.766799	+	0.641887i	$0.778152\pi$
$620$	0	0
$621$	6.90073	0.276917
$622$	0	0
$623$	13.5662	0.543516
$624$	0	0
$625$	−31.2422	−1.24969
$626$	0	0
$627$	−4.44638	−0.177571
$628$	0	0
$629$	11.5988	0.462474
$630$	0	0
$631$	30.6027	1.21827	0.609137	−	0.793065i	$-0.291516\pi$
$631$	30.6027	1.21827	0.609137	+	0.793065i	$0.291516\pi$
$632$	0	0
$633$	−12.4002	−0.492862
$634$	0	0
$635$	13.3519	0.529852
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−5.32056	−0.210478
$640$	0	0
$641$	25.7211	1.01592	0.507962	−	0.861379i	$-0.330399\pi$
$641$	25.7211	1.01592	0.507962	+	0.861379i	$0.330399\pi$
$642$	0	0
$643$	30.1373	1.18850	0.594249	−	0.804281i	$-0.297449\pi$
$643$	30.1373	1.18850	0.594249	+	0.804281i	$0.297449\pi$
$644$	0	0
$645$	−13.8170	−0.544046
$646$	0	0
$647$	−6.75127	−0.265420	−0.132710	−	0.991155i	$-0.542368\pi$
$647$	−6.75127	−0.265420	−0.132710	+	0.991155i	$0.542368\pi$
$648$	0	0
$649$	0.0841450	0.00330298
$650$	0	0
$651$	3.21447	0.125985
$652$	0	0
$653$	−1.20999	−0.0473507	−0.0236754	−	0.999720i	$-0.507537\pi$
$653$	−1.20999	−0.0473507	−0.0236754	+	0.999720i	$0.507537\pi$
$654$	0	0
$655$	37.9268	1.48192
$656$	0	0
$657$	−3.75917	−0.146659
$658$	0	0
$659$	13.6149	0.530361	0.265181	−	0.964199i	$-0.414568\pi$
$659$	13.6149	0.530361	0.265181	+	0.964199i	$0.414568\pi$
$660$	0	0
$661$	−10.7711	−0.418949	−0.209474	−	0.977814i	$-0.567175\pi$
$661$	−10.7711	−0.418949	−0.209474	+	0.977814i	$0.567175\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	61.1174	2.37003
$666$	0	0
$667$	40.1629	1.55512
$668$	0	0
$669$	−12.5356	−0.484655
$670$	0	0
$671$	−7.69739	−0.297155
$672$	0	0
$673$	−46.0338	−1.77447	−0.887236	−	0.461316i	$-0.847377\pi$
$673$	−46.0338	−1.77447	−0.887236	+	0.461316i	$0.847377\pi$
$674$	0	0
$675$	−2.41172	−0.0928272
$676$	0	0
$677$	1.84261	0.0708172	0.0354086	−	0.999373i	$-0.488727\pi$
$677$	1.84261	0.0708172	0.0354086	+	0.999373i	$0.488727\pi$
$678$	0	0
$679$	−54.7059	−2.09942
$680$	0	0
$681$	1.16802	0.0447587
$682$	0	0
$683$	23.7758	0.909757	0.454878	−	0.890554i	$-0.349683\pi$
$683$	23.7758	0.909757	0.454878	+	0.890554i	$0.349683\pi$
$684$	0	0
$685$	−47.5309	−1.81606
$686$	0	0
$687$	−21.9158	−0.836140
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−1.67973	−0.0639001	−0.0319501	−	0.999489i	$-0.510172\pi$
$691$	−1.67973	−0.0639001	−0.0319501	+	0.999489i	$0.510172\pi$
$692$	0	0
$693$	2.17701	0.0826976
$694$	0	0
$695$	−27.0579	−1.02636
$696$	0	0
$697$	−12.3680	−0.468471
$698$	0	0
$699$	3.00659	0.113720
$700$	0	0
$701$	17.8738	0.675086	0.337543	−	0.941310i	$-0.390404\pi$
$701$	17.8738	0.675086	0.337543	+	0.941310i	$0.390404\pi$
$702$	0	0
$703$	67.2723	2.53722
$704$	0	0
$705$	−22.4429	−0.845247
$706$	0	0
$707$	−47.7856	−1.79716
$708$	0	0
$709$	4.27227	0.160448	0.0802242	−	0.996777i	$-0.474436\pi$
$709$	4.27227	0.160448	0.0802242	+	0.996777i	$0.474436\pi$
$710$	0	0
$711$	−2.17759	−0.0816661
$712$	0	0
$713$	−6.69077	−0.250571
$714$	0	0
$715$	0	0
$716$	0	0
$717$	12.7853	0.477478
$718$	0	0
$719$	27.5341	1.02685	0.513424	−	0.858135i	$-0.328377\pi$
$719$	27.5341	1.02685	0.513424	+	0.858135i	$0.328377\pi$
$720$	0	0
$721$	−52.1534	−1.94230
$722$	0	0
$723$	−23.6123	−0.878149
$724$	0	0
$725$	−14.0365	−0.521301
$726$	0	0
$727$	28.2674	1.04838	0.524189	−	0.851602i	$-0.324369\pi$
$727$	28.2674	1.04838	0.524189	+	0.851602i	$0.324369\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−5.92528	−0.219154
$732$	0	0
$733$	25.2904	0.934123	0.467062	−	0.884225i	$-0.345313\pi$
$733$	25.2904	0.934123	0.467062	+	0.884225i	$0.345313\pi$
$734$	0	0
$735$	−10.8667	−0.400825
$736$	0	0
$737$	−10.4684	−0.385607
$738$	0	0
$739$	−8.06509	−0.296679	−0.148340	−	0.988936i	$-0.547393\pi$
$739$	−8.06509	−0.296679	−0.148340	+	0.988936i	$0.547393\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	14.2716	0.523575	0.261787	−	0.965126i	$-0.415688\pi$
$743$	14.2716	0.523575	0.261787	+	0.965126i	$0.415688\pi$
$744$	0	0
$745$	−40.2754	−1.47558
$746$	0	0
$747$	10.8237	0.396019
$748$	0	0
$749$	30.7501	1.12359
$750$	0	0
$751$	−33.0043	−1.20434	−0.602171	−	0.798367i	$-0.705698\pi$
$751$	−33.0043	−1.20434	−0.602171	+	0.798367i	$0.705698\pi$
$752$	0	0
$753$	8.22001	0.299554
$754$	0	0
$755$	31.5585	1.14853
$756$	0	0
$757$	35.2365	1.28069	0.640347	−	0.768086i	$-0.278791\pi$
$757$	35.2365	1.28069	0.640347	+	0.768086i	$0.278791\pi$
$758$	0	0
$759$	−4.53133	−0.164477
$760$	0	0
$761$	30.7212	1.11364	0.556822	−	0.830632i	$-0.312021\pi$
$761$	30.7212	1.11364	0.556822	+	0.830632i	$0.312021\pi$
$762$	0	0
$763$	−29.2709	−1.05968
$764$	0	0
$765$	−3.17843	−0.114916
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−6.95972	−0.250974	−0.125487	−	0.992095i	$-0.540049\pi$
$769$	−6.95972	−0.250974	−0.125487	+	0.992095i	$0.540049\pi$
$770$	0	0
$771$	−27.7850	−1.00065
$772$	0	0
$773$	−40.5717	−1.45926	−0.729632	−	0.683840i	$-0.760308\pi$
$773$	−40.5717	−1.45926	−0.729632	+	0.683840i	$0.760308\pi$
$774$	0	0
$775$	2.33834	0.0839957
$776$	0	0
$777$	−32.9374	−1.18162
$778$	0	0
$779$	−71.7336	−2.57012
$780$	0	0
$781$	3.49372	0.125015
$782$	0	0
$783$	5.82010	0.207993
$784$	0	0
$785$	−44.5967	−1.59172
$786$	0	0
$787$	4.88731	0.174214	0.0871068	−	0.996199i	$-0.472238\pi$
$787$	4.88731	0.174214	0.0871068	+	0.996199i	$0.472238\pi$
$788$	0	0
$789$	10.5270	0.374769
$790$	0	0
$791$	−21.4129	−0.761356
$792$	0	0
$793$	0	0
$794$	0	0
$795$	−2.29003	−0.0812191
$796$	0	0
$797$	−24.5982	−0.871314	−0.435657	−	0.900113i	$-0.643484\pi$
$797$	−24.5982	−0.871314	−0.435657	+	0.900113i	$0.643484\pi$
$798$	0	0
$799$	−9.62436	−0.340485
$800$	0	0
$801$	−4.09193	−0.144581
$802$	0	0
$803$	2.46844	0.0871093
$804$	0	0
$805$	62.2850	2.19526
$806$	0	0
$807$	2.51426	0.0885061
$808$	0	0
$809$	13.6525	0.479996	0.239998	−	0.970773i	$-0.422853\pi$
$809$	13.6525	0.479996	0.239998	+	0.970773i	$0.422853\pi$
$810$	0	0
$811$	44.2951	1.55541	0.777706	−	0.628628i	$-0.216383\pi$
$811$	44.2951	1.55541	0.777706	+	0.628628i	$0.216383\pi$
$812$	0	0
$813$	−4.83460	−0.169557
$814$	0	0
$815$	57.9117	2.02856
$816$	0	0
$817$	−34.3662	−1.20232
$818$	0	0
$819$	0	0
$820$	0	0
$821$	32.2133	1.12425	0.562127	−	0.827051i	$-0.309983\pi$
$821$	32.2133	1.12425	0.562127	+	0.827051i	$0.309983\pi$
$822$	0	0
$823$	28.6321	0.998052	0.499026	−	0.866587i	$-0.333691\pi$
$823$	28.6321	0.998052	0.499026	+	0.866587i	$0.333691\pi$
$824$	0	0
$825$	1.58365	0.0551355
$826$	0	0
$827$	40.8863	1.42176	0.710879	−	0.703315i	$-0.248298\pi$
$827$	40.8863	1.42176	0.710879	+	0.703315i	$0.248298\pi$
$828$	0	0
$829$	10.7016	0.371681	0.185841	−	0.982580i	$-0.440499\pi$
$829$	10.7016	0.371681	0.185841	+	0.982580i	$0.440499\pi$
$830$	0	0
$831$	15.4274	0.535170
$832$	0	0
$833$	−4.66007	−0.161462
$834$	0	0
$835$	28.7212	0.993939
$836$	0	0
$837$	−0.969574	−0.0335134
$838$	0	0
$839$	16.1362	0.557085	0.278543	−	0.960424i	$-0.410149\pi$
$839$	16.1362	0.557085	0.278543	+	0.960424i	$0.410149\pi$
$840$	0	0
$841$	4.87357	0.168054
$842$	0	0
$843$	6.10796	0.210369
$844$	0	0
$845$	0	0
$846$	0	0
$847$	35.0393	1.20396
$848$	0	0
$849$	11.3138	0.388290
$850$	0	0
$851$	68.5575	2.35012
$852$	0	0
$853$	−15.7776	−0.540215	−0.270108	−	0.962830i	$-0.587059\pi$
$853$	−15.7776	−0.540215	−0.270108	+	0.962830i	$0.587059\pi$
$854$	0	0
$855$	−18.4347	−0.630453
$856$	0	0
$857$	21.5463	0.736006	0.368003	−	0.929825i	$-0.380042\pi$
$857$	21.5463	0.736006	0.368003	+	0.929825i	$0.380042\pi$
$858$	0	0
$859$	4.07749	0.139122	0.0695611	−	0.997578i	$-0.477840\pi$
$859$	4.07749	0.139122	0.0695611	+	0.997578i	$0.477840\pi$
$860$	0	0
$861$	35.1217	1.19694
$862$	0	0
$863$	−17.5883	−0.598712	−0.299356	−	0.954141i	$-0.596772\pi$
$863$	−17.5883	−0.598712	−0.299356	+	0.954141i	$0.596772\pi$
$864$	0	0
$865$	55.6151	1.89097
$866$	0	0
$867$	15.6370	0.531059
$868$	0	0
$869$	1.42991	0.0485062
$870$	0	0
$871$	0	0
$872$	0	0
$873$	16.5008	0.558468
$874$	0	0
$875$	23.3614	0.789761
$876$	0	0
$877$	−5.07451	−0.171354	−0.0856770	−	0.996323i	$-0.527305\pi$
$877$	−5.07451	−0.171354	−0.0856770	+	0.996323i	$0.527305\pi$
$878$	0	0
$879$	−28.0013	−0.944461
$880$	0	0
$881$	49.8471	1.67939	0.839696	−	0.543057i	$-0.182733\pi$
$881$	49.8471	1.67939	0.839696	+	0.543057i	$0.182733\pi$
$882$	0	0
$883$	−46.7712	−1.57398	−0.786989	−	0.616967i	$-0.788361\pi$
$883$	−46.7712	−1.57398	−0.786989	+	0.616967i	$0.788361\pi$
$884$	0	0
$885$	0.348865	0.0117270
$886$	0	0
$887$	19.5889	0.657730	0.328865	−	0.944377i	$-0.393334\pi$
$887$	19.5889	0.657730	0.328865	+	0.944377i	$0.393334\pi$
$888$	0	0
$889$	−16.2596	−0.545331
$890$	0	0
$891$	−0.656645	−0.0219984
$892$	0	0
$893$	−55.8207	−1.86797
$894$	0	0
$895$	15.5462	0.519651
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−5.64302	−0.188205
$900$	0	0
$901$	−0.982053	−0.0327169
$902$	0	0
$903$	16.8261	0.559939
$904$	0	0
$905$	12.8043	0.425630
$906$	0	0
$907$	17.5223	0.581818	0.290909	−	0.956751i	$-0.406042\pi$
$907$	17.5223	0.581818	0.290909	+	0.956751i	$0.406042\pi$
$908$	0	0
$909$	14.4134	0.478064
$910$	0	0
$911$	11.6916	0.387360	0.193680	−	0.981065i	$-0.437958\pi$
$911$	11.6916	0.387360	0.193680	+	0.981065i	$0.437958\pi$
$912$	0	0
$913$	−7.10735	−0.235219
$914$	0	0
$915$	−31.9134	−1.05502
$916$	0	0
$917$	−46.1865	−1.52521
$918$	0	0
$919$	42.7980	1.41177	0.705887	−	0.708324i	$-0.250548\pi$
$919$	42.7980	1.41177	0.705887	+	0.708324i	$0.250548\pi$
$920$	0	0
$921$	18.6077	0.613143
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−23.9600	−0.787801
$926$	0	0
$927$	15.7309	0.516671
$928$	0	0
$929$	−27.8135	−0.912532	−0.456266	−	0.889843i	$-0.650813\pi$
$929$	−27.8135	−0.912532	−0.456266	+	0.889843i	$0.650813\pi$
$930$	0	0
$931$	−27.0281	−0.885810
$932$	0	0
$933$	25.9076	0.848177
$934$	0	0
$935$	2.08710	0.0682555
$936$	0	0
$937$	48.2544	1.57640	0.788201	−	0.615417i	$-0.211013\pi$
$937$	48.2544	1.57640	0.788201	+	0.615417i	$0.211013\pi$
$938$	0	0
$939$	0.799031	0.0260754
$940$	0	0
$941$	13.8449	0.451332	0.225666	−	0.974205i	$-0.427544\pi$
$941$	13.8449	0.451332	0.225666	+	0.974205i	$0.427544\pi$
$942$	0	0
$943$	−73.1041	−2.38060
$944$	0	0
$945$	9.02586	0.293611
$946$	0	0
$947$	−18.6924	−0.607422	−0.303711	−	0.952764i	$-0.598226\pi$
$947$	−18.6924	−0.607422	−0.303711	+	0.952764i	$0.598226\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	−32.7072	−1.06060
$952$	0	0
$953$	44.2242	1.43256	0.716281	−	0.697812i	$-0.245843\pi$
$953$	44.2242	1.43256	0.716281	+	0.697812i	$0.245843\pi$
$954$	0	0
$955$	−40.0550	−1.29615
$956$	0	0
$957$	−3.82174	−0.123539
$958$	0	0
$959$	57.8822	1.86911
$960$	0	0
$961$	−30.0599	−0.969675
$962$	0	0
$963$	−9.27509	−0.298885
$964$	0	0
$965$	4.10118	0.132022
$966$	0	0
$967$	36.5221	1.17447	0.587235	−	0.809416i	$-0.300216\pi$
$967$	36.5221	1.17447	0.587235	+	0.809416i	$0.300216\pi$
$968$	0	0
$969$	−7.90550	−0.253961
$970$	0	0
$971$	−34.0922	−1.09407	−0.547035	−	0.837110i	$-0.684244\pi$
$971$	−34.0922	−1.09407	−0.547035	+	0.837110i	$0.684244\pi$
$972$	0	0
$973$	32.9506	1.05635
$974$	0	0
$975$	0	0
$976$	0	0
$977$	59.7205	1.91063	0.955315	−	0.295591i	$-0.0955166\pi$
$977$	59.7205	1.91063	0.955315	+	0.295591i	$0.0955166\pi$
$978$	0	0
$979$	2.68694	0.0858751
$980$	0	0
$981$	8.82889	0.281885
$982$	0	0
$983$	−49.8000	−1.58837	−0.794186	−	0.607674i	$-0.792103\pi$
$983$	−49.8000	−1.58837	−0.794186	+	0.607674i	$0.792103\pi$
$984$	0	0
$985$	−47.4986	−1.51343
$986$	0	0
$987$	27.3305	0.869940
$988$	0	0
$989$	−35.0228	−1.11366
$990$	0	0
$991$	17.1731	0.545522	0.272761	−	0.962082i	$-0.412063\pi$
$991$	17.1731	0.545522	0.272761	+	0.962082i	$0.412063\pi$
$992$	0	0
$993$	5.69662	0.180777
$994$	0	0
$995$	−47.3794	−1.50203
$996$	0	0
$997$	−34.9494	−1.10686	−0.553430	−	0.832896i	$-0.686681\pi$
$997$	−34.9494	−1.10686	−0.553430	+	0.832896i	$0.686681\pi$
$998$	0	0
$999$	9.93482	0.314324

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8112.2.a.cu.1.5		6
4.3	odd	2		4056.2.a.bi.1.5	yes	6
13.12	even	2		8112.2.a.ct.1.2		6
52.31	even	4		4056.2.c.r.337.3		12
52.47	even	4		4056.2.c.r.337.10		12
52.51	odd	2		4056.2.a.bh.1.2	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4056.2.a.bh.1.2	✓	6	52.51	odd	2
4056.2.a.bi.1.5	yes	6	4.3	odd	2
4056.2.c.r.337.3		12	52.31	even	4
4056.2.c.r.337.10		12	52.47	even	4
8112.2.a.ct.1.2		6	13.12	even	2
8112.2.a.cu.1.5		6	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 8112 = 2^{4} \cdot 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8112.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +2.72245 q^{5} -3.31535 q^{7} +1.00000 q^{9} -0.656645 q^{11} -2.72245 q^{15} -1.16749 q^{17} -6.77137 q^{19} +3.31535 q^{21} -6.90073 q^{23} +2.41172 q^{25} -1.00000 q^{27} -5.82010 q^{29} +0.969574 q^{31} +0.656645 q^{33} -9.02586 q^{35} -9.93482 q^{37} +10.5937 q^{41} +5.07523 q^{43} +2.72245 q^{45} +8.24364 q^{47} +3.99153 q^{49} +1.16749 q^{51} +0.841166 q^{53} -1.78768 q^{55} +6.77137 q^{57} -0.128144 q^{59} +11.7223 q^{61} -3.31535 q^{63} +15.9422 q^{67} +6.90073 q^{69} -5.32056 q^{71} -3.75917 q^{73} -2.41172 q^{75} +2.17701 q^{77} -2.17759 q^{79} +1.00000 q^{81} +10.8237 q^{83} -3.17843 q^{85} +5.82010 q^{87} -4.09193 q^{89} -0.969574 q^{93} -18.4347 q^{95} +16.5008 q^{97} -0.656645 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{3} + q^{5} + 5 q^{7} + 6 q^{9} + 6 q^{11} - q^{15} + 9 q^{17} - 7 q^{19} - 5 q^{21} - 12 q^{23} + 9 q^{25} - 6 q^{27} + 7 q^{29} + 11 q^{31} - 6 q^{33} - 6 q^{35} + 6 q^{37} + 13 q^{41} - 15 q^{43}+ \cdots + 6 q^{99}+O(q^{100}) \) 6 * q - 6 * q^3 + q^5 + 5 * q^7 + 6 * q^9 + 6 * q^11 - q^15 + 9 * q^17 - 7 * q^19 - 5 * q^21 - 12 * q^23 + 9 * q^25 - 6 * q^27 + 7 * q^29 + 11 * q^31 - 6 * q^33 - 6 * q^35 + 6 * q^37 + 13 * q^41 - 15 * q^43 + q^45 + 9 * q^47 + 13 * q^49 - 9 * q^51 + 22 * q^53 - 3 * q^55 + 7 * q^57 + 7 * q^59 + 25 * q^61 + 5 * q^63 - 5 * q^67 + 12 * q^69 - 8 * q^71 + 15 * q^73 - 9 * q^75 + 45 * q^77 - 14 * q^79 + 6 * q^81 + 13 * q^83 - 35 * q^85 - 7 * q^87 + 33 * q^89 - 11 * q^93 - 47 * q^95 + 50 * q^97 + 6 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

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