LMFDB - Embedded newform 2312.2.a.u.1.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2312,2,Mod(1,2312)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2312, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([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("2312.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 2312 = 2^{3} \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2312.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,0,0,0,-6,0,-3,0,0,0,6,0,-6,0,-6] 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:	yes
Analytic conductor:	$18.4614129473$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.3418281.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 9x^{4} - 4x^{3} + 18x^{2} + 12x - 1 $ x^6 - 9x^4 - 4x^3 + 18x^2 + 12x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-2.06104$ of defining polynomial
Character	$\chi$	$=$	2312.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.06104 q^{3} -4.30892 q^{5} +1.72816 q^{7} +1.24788 q^{9} +4.37201 q^{11} -4.54061 q^{13} -8.88085 q^{15} -3.59851 q^{19} +3.56181 q^{21} -3.79125 q^{23} +13.5668 q^{25} -3.61119 q^{27} +0.524140 q^{29} -8.07897 q^{31} +9.01087 q^{33} -7.44650 q^{35} -2.24321 q^{37} -9.35837 q^{39} -7.65013 q^{41} +2.25102 q^{43} -5.37701 q^{45} -1.33083 q^{47} -4.01346 q^{49} +12.5721 q^{53} -18.8386 q^{55} -7.41666 q^{57} +5.17229 q^{59} -4.75321 q^{61} +2.15654 q^{63} +19.5651 q^{65} -5.10542 q^{67} -7.81391 q^{69} -12.7825 q^{71} -1.27999 q^{73} +27.9617 q^{75} +7.55553 q^{77} +0.254999 q^{79} -11.1864 q^{81} +5.55252 q^{83} +1.08027 q^{87} -0.319763 q^{89} -7.84690 q^{91} -16.6511 q^{93} +15.5057 q^{95} -6.35006 q^{97} +5.45574 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{5} - 3 q^{7} + 6 q^{11} - 6 q^{13} - 6 q^{15} + 6 q^{19} + 6 q^{21} - 9 q^{23} + 6 q^{25} - 12 q^{27} - 27 q^{29} - 9 q^{31} + 3 q^{33} + 21 q^{35} - 15 q^{37} + 12 q^{39} - 21 q^{41} - 24 q^{45}+ \cdots + 42 q^{99}+O(q^{100}) $ 6 * q - 6 * q^5 - 3 * q^7 + 6 * q^11 - 6 * q^13 - 6 * q^15 + 6 * q^19 + 6 * q^21 - 9 * q^23 + 6 * q^25 - 12 * q^27 - 27 * q^29 - 9 * q^31 + 3 * q^33 + 21 * q^35 - 15 * q^37 + 12 * q^39 - 21 * q^41 - 24 * q^45 + 3 * q^47 + 9 * q^49 + 33 * q^53 - 51 * q^55 - 30 * q^57 - 18 * q^61 - 24 * q^63 + 9 * q^65 - 9 * q^67 - 3 * q^69 - 6 * q^71 - 27 * q^73 + 24 * q^75 - 45 * q^77 - 33 * q^79 - 18 * q^81 - 12 * q^83 + 15 * q^87 + 3 * q^89 - 30 * q^91 - 30 * q^93 - 3 * q^95 - 51 * q^97 + 42 * 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
$3$	2.06104	1.18994	0.594971	−	0.803747i	$-0.297164\pi$
$3$	2.06104	1.18994	0.594971	+	0.803747i	$0.297164\pi$
$4$	0	0
$5$	−4.30892	−1.92701	−0.963504	−	0.267696i	$-0.913738\pi$
$5$	−4.30892	−1.92701	−0.963504	+	0.267696i	$0.913738\pi$
$6$	0	0
$7$	1.72816	0.653183	0.326592	−	0.945166i	$-0.394100\pi$
$7$	1.72816	0.653183	0.326592	+	0.945166i	$0.394100\pi$
$8$	0	0
$9$	1.24788	0.415960
$10$	0	0
$11$	4.37201	1.31821	0.659105	−	0.752051i	$-0.270935\pi$
$11$	4.37201	1.31821	0.659105	+	0.752051i	$0.270935\pi$
$12$	0	0
$13$	−4.54061	−1.25934	−0.629669	−	0.776864i	$-0.716809\pi$
$13$	−4.54061	−1.25934	−0.629669	+	0.776864i	$0.716809\pi$
$14$	0	0
$15$	−8.88085	−2.29303
$16$	0	0
$17$	0	0
$17$	0	0
$18$	0	0
$19$	−3.59851	−0.825554	−0.412777	−	0.910832i	$-0.635441\pi$
$19$	−3.59851	−0.825554	−0.412777	+	0.910832i	$0.635441\pi$
$20$	0	0
$21$	3.56181	0.777250
$22$	0	0
$23$	−3.79125	−0.790530	−0.395265	−	0.918567i	$-0.629347\pi$
$23$	−3.79125	−0.790530	−0.395265	+	0.918567i	$0.629347\pi$
$24$	0	0
$25$	13.5668	2.71336
$26$	0	0
$27$	−3.61119	−0.694973
$28$	0	0
$29$	0.524140	0.0973304	0.0486652	−	0.998815i	$-0.484503\pi$
$29$	0.524140	0.0973304	0.0486652	+	0.998815i	$0.484503\pi$
$30$	0	0
$31$	−8.07897	−1.45103	−0.725513	−	0.688209i	$-0.758397\pi$
$31$	−8.07897	−1.45103	−0.725513	+	0.688209i	$0.758397\pi$
$32$	0	0
$33$	9.01087	1.56859
$34$	0	0
$35$	−7.44650	−1.25869
$36$	0	0
$37$	−2.24321	−0.368782	−0.184391	−	0.982853i	$-0.559031\pi$
$37$	−2.24321	−0.368782	−0.184391	+	0.982853i	$0.559031\pi$
$38$	0	0
$39$	−9.35837	−1.49854
$40$	0	0
$41$	−7.65013	−1.19475	−0.597375	−	0.801962i	$-0.703790\pi$
$41$	−7.65013	−1.19475	−0.597375	+	0.801962i	$0.703790\pi$
$42$	0	0
$43$	2.25102	0.343277	0.171639	−	0.985160i	$-0.445094\pi$
$43$	2.25102	0.343277	0.171639	+	0.985160i	$0.445094\pi$
$44$	0	0
$45$	−5.37701	−0.801558
$46$	0	0
$47$	−1.33083	−0.194121	−0.0970607	−	0.995278i	$-0.530944\pi$
$47$	−1.33083	−0.194121	−0.0970607	+	0.995278i	$0.530944\pi$
$48$	0	0
$49$	−4.01346	−0.573351
$50$	0	0
$51$	0	0
$52$	0	0
$53$	12.5721	1.72691	0.863457	−	0.504422i	$-0.168294\pi$
$53$	12.5721	1.72691	0.863457	+	0.504422i	$0.168294\pi$
$54$	0	0
$55$	−18.8386	−2.54020
$56$	0	0
$57$	−7.41666	−0.982361
$58$	0	0
$59$	5.17229	0.673375	0.336688	−	0.941616i	$-0.390693\pi$
$59$	5.17229	0.673375	0.336688	+	0.941616i	$0.390693\pi$
$60$	0	0
$61$	−4.75321	−0.608586	−0.304293	−	0.952578i	$-0.598420\pi$
$61$	−4.75321	−0.608586	−0.304293	+	0.952578i	$0.598420\pi$
$62$	0	0
$63$	2.15654	0.271698
$64$	0	0
$65$	19.5651	2.42675
$66$	0	0
$67$	−5.10542	−0.623726	−0.311863	−	0.950127i	$-0.600953\pi$
$67$	−5.10542	−0.623726	−0.311863	+	0.950127i	$0.600953\pi$
$68$	0	0
$69$	−7.81391	−0.940684
$70$	0	0
$71$	−12.7825	−1.51701	−0.758503	−	0.651669i	$-0.774069\pi$
$71$	−12.7825	−1.51701	−0.758503	+	0.651669i	$0.774069\pi$
$72$	0	0
$73$	−1.27999	−0.149811	−0.0749055	−	0.997191i	$-0.523866\pi$
$73$	−1.27999	−0.149811	−0.0749055	+	0.997191i	$0.523866\pi$
$74$	0	0
$75$	27.9617	3.22873
$76$	0	0
$77$	7.55553	0.861033
$78$	0	0
$79$	0.254999	0.0286896	0.0143448	−	0.999897i	$-0.495434\pi$
$79$	0.254999	0.0286896	0.0143448	+	0.999897i	$0.495434\pi$
$80$	0	0
$81$	−11.1864	−1.24294
$82$	0	0
$83$	5.55252	0.609468	0.304734	−	0.952437i	$-0.401432\pi$
$83$	5.55252	0.609468	0.304734	+	0.952437i	$0.401432\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	1.08027	0.115817
$88$	0	0
$89$	−0.319763	−0.0338948	−0.0169474	−	0.999856i	$-0.505395\pi$
$89$	−0.319763	−0.0338948	−0.0169474	+	0.999856i	$0.505395\pi$
$90$	0	0
$91$	−7.84690	−0.822579
$92$	0	0
$93$	−16.6511	−1.72664
$94$	0	0
$95$	15.5057	1.59085
$96$	0	0
$97$	−6.35006	−0.644751	−0.322375	−	0.946612i	$-0.604481\pi$
$97$	−6.35006	−0.644751	−0.322375	+	0.946612i	$0.604481\pi$
$98$	0	0
$99$	5.45574	0.548322
$100$	0	0
$101$	−11.5477	−1.14904	−0.574519	−	0.818492i	$-0.694811\pi$
$101$	−11.5477	−1.14904	−0.574519	+	0.818492i	$0.694811\pi$
$102$	0	0
$103$	−0.797707	−0.0786004	−0.0393002	−	0.999227i	$-0.512513\pi$
$103$	−0.797707	−0.0786004	−0.0393002	+	0.999227i	$0.512513\pi$
$104$	0	0
$105$	−15.3475	−1.49777
$106$	0	0
$107$	3.39489	0.328197	0.164098	−	0.986444i	$-0.447529\pi$
$107$	3.39489	0.328197	0.164098	+	0.986444i	$0.447529\pi$
$108$	0	0
$109$	4.85580	0.465102	0.232551	−	0.972584i	$-0.425293\pi$
$109$	4.85580	0.465102	0.232551	+	0.972584i	$0.425293\pi$
$110$	0	0
$111$	−4.62335	−0.438829
$112$	0	0
$113$	−15.2696	−1.43644	−0.718220	−	0.695816i	$-0.755043\pi$
$113$	−15.2696	−1.43644	−0.718220	+	0.695816i	$0.755043\pi$
$114$	0	0
$115$	16.3362	1.52336
$116$	0	0
$117$	−5.66613	−0.523834
$118$	0	0
$119$	0	0
$120$	0	0
$121$	8.11444	0.737676
$122$	0	0
$123$	−15.7672	−1.42168
$124$	0	0
$125$	−36.9136	−3.30165
$126$	0	0
$127$	−12.3400	−1.09500	−0.547500	−	0.836806i	$-0.684420\pi$
$127$	−12.3400	−1.09500	−0.547500	+	0.836806i	$0.684420\pi$
$128$	0	0
$129$	4.63944	0.408480
$130$	0	0
$131$	−1.03956	−0.0908271	−0.0454135	−	0.998968i	$-0.514461\pi$
$131$	−1.03956	−0.0908271	−0.0454135	+	0.998968i	$0.514461\pi$
$132$	0	0
$133$	−6.21880	−0.539238
$134$	0	0
$135$	15.5603	1.33922
$136$	0	0
$137$	5.01747	0.428672	0.214336	−	0.976760i	$-0.431241\pi$
$137$	5.01747	0.428672	0.214336	+	0.976760i	$0.431241\pi$
$138$	0	0
$139$	−21.4471	−1.81912	−0.909561	−	0.415571i	$-0.863582\pi$
$139$	−21.4471	−1.81912	−0.909561	+	0.415571i	$0.863582\pi$
$140$	0	0
$141$	−2.74289	−0.230993
$142$	0	0
$143$	−19.8516	−1.66007
$144$	0	0
$145$	−2.25848	−0.187556
$146$	0	0
$147$	−8.27190	−0.682254
$148$	0	0
$149$	0.331466	0.0271547	0.0135774	−	0.999908i	$-0.495678\pi$
$149$	0.331466	0.0271547	0.0135774	+	0.999908i	$0.495678\pi$
$150$	0	0
$151$	−18.9835	−1.54486	−0.772429	−	0.635102i	$-0.780958\pi$
$151$	−18.9835	−1.54486	−0.772429	+	0.635102i	$0.780958\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	34.8116	2.79614
$156$	0	0
$157$	7.76102	0.619397	0.309698	−	0.950835i	$-0.399772\pi$
$157$	7.76102	0.619397	0.309698	+	0.950835i	$0.399772\pi$
$158$	0	0
$159$	25.9116	2.05493
$160$	0	0
$161$	−6.55189	−0.516361
$162$	0	0
$163$	−15.5959	−1.22157	−0.610784	−	0.791797i	$-0.709146\pi$
$163$	−15.5959	−1.22157	−0.610784	+	0.791797i	$0.709146\pi$
$164$	0	0
$165$	−38.8271	−3.02269
$166$	0	0
$167$	−20.0905	−1.55465	−0.777324	−	0.629101i	$-0.783423\pi$
$167$	−20.0905	−1.55465	−0.777324	+	0.629101i	$0.783423\pi$
$168$	0	0
$169$	7.61712	0.585932
$170$	0	0
$171$	−4.49051	−0.343398
$172$	0	0
$173$	6.04891	0.459890	0.229945	−	0.973204i	$-0.426145\pi$
$173$	6.04891	0.459890	0.229945	+	0.973204i	$0.426145\pi$
$174$	0	0
$175$	23.4456	1.77232
$176$	0	0
$177$	10.6603	0.801277
$178$	0	0
$179$	18.4058	1.37571	0.687857	−	0.725846i	$-0.258552\pi$
$179$	18.4058	1.37571	0.687857	+	0.725846i	$0.258552\pi$
$180$	0	0
$181$	22.1983	1.64999	0.824995	−	0.565141i	$-0.191178\pi$
$181$	22.1983	1.64999	0.824995	+	0.565141i	$0.191178\pi$
$182$	0	0
$183$	−9.79655	−0.724182
$184$	0	0
$185$	9.66582	0.710645
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−6.24071	−0.453945
$190$	0	0
$191$	12.4592	0.901517	0.450759	−	0.892646i	$-0.351154\pi$
$191$	12.4592	0.901517	0.450759	+	0.892646i	$0.351154\pi$
$192$	0	0
$193$	4.38225	0.315441	0.157721	−	0.987484i	$-0.449585\pi$
$193$	4.38225	0.315441	0.157721	+	0.987484i	$0.449585\pi$
$194$	0	0
$195$	40.3244	2.88769
$196$	0	0
$197$	18.3885	1.31012	0.655062	−	0.755575i	$-0.272643\pi$
$197$	18.3885	1.31012	0.655062	+	0.755575i	$0.272643\pi$
$198$	0	0
$199$	10.8956	0.772371	0.386185	−	0.922421i	$-0.373793\pi$
$199$	10.8956	0.772371	0.386185	+	0.922421i	$0.373793\pi$
$200$	0	0
$201$	−10.5225	−0.742197
$202$	0	0
$203$	0.905799	0.0635746
$204$	0	0
$205$	32.9638	2.30229
$206$	0	0
$207$	−4.73102	−0.328829
$208$	0	0
$209$	−15.7327	−1.08825
$210$	0	0
$211$	22.6468	1.55907	0.779534	−	0.626360i	$-0.215456\pi$
$211$	22.6468	1.55907	0.779534	+	0.626360i	$0.215456\pi$
$212$	0	0
$213$	−26.3453	−1.80515
$214$	0	0
$215$	−9.69946	−0.661498
$216$	0	0
$217$	−13.9618	−0.947786
$218$	0	0
$219$	−2.63810	−0.178266
$220$	0	0
$221$	0	0
$222$	0	0
$223$	7.83363	0.524579	0.262289	−	0.964989i	$-0.415523\pi$
$223$	7.83363	0.524579	0.262289	+	0.964989i	$0.415523\pi$
$224$	0	0
$225$	16.9297	1.12865
$226$	0	0
$227$	24.1208	1.60095	0.800476	−	0.599365i	$-0.204580\pi$
$227$	24.1208	1.60095	0.800476	+	0.599365i	$0.204580\pi$
$228$	0	0
$229$	−6.31906	−0.417575	−0.208788	−	0.977961i	$-0.566952\pi$
$229$	−6.31906	−0.417575	−0.208788	+	0.977961i	$0.566952\pi$
$230$	0	0
$231$	15.5722	1.02458
$232$	0	0
$233$	2.93291	0.192141	0.0960707	−	0.995375i	$-0.469373\pi$
$233$	2.93291	0.192141	0.0960707	+	0.995375i	$0.469373\pi$
$234$	0	0
$235$	5.73443	0.374073
$236$	0	0
$237$	0.525563	0.0341390
$238$	0	0
$239$	21.6824	1.40252	0.701260	−	0.712905i	$-0.252621\pi$
$239$	21.6824	1.40252	0.701260	+	0.712905i	$0.252621\pi$
$240$	0	0
$241$	10.2543	0.660539	0.330269	−	0.943887i	$-0.392860\pi$
$241$	10.2543	0.660539	0.330269	+	0.943887i	$0.392860\pi$
$242$	0	0
$243$	−12.2221	−0.784049
$244$	0	0
$245$	17.2937	1.10485
$246$	0	0
$247$	16.3394	1.03965
$248$	0	0
$249$	11.4440	0.725231
$250$	0	0
$251$	11.1362	0.702912	0.351456	−	0.936204i	$-0.385687\pi$
$251$	11.1362	0.702912	0.351456	+	0.936204i	$0.385687\pi$
$252$	0	0
$253$	−16.5754	−1.04208
$254$	0	0
$255$	0	0
$256$	0	0
$257$	3.49477	0.217998	0.108999	−	0.994042i	$-0.465236\pi$
$257$	3.49477	0.217998	0.108999	+	0.994042i	$0.465236\pi$
$258$	0	0
$259$	−3.87663	−0.240882
$260$	0	0
$261$	0.654064	0.0404856
$262$	0	0
$263$	19.3767	1.19482	0.597408	−	0.801938i	$-0.296197\pi$
$263$	19.3767	1.19482	0.597408	+	0.801938i	$0.296197\pi$
$264$	0	0
$265$	−54.1723	−3.32778
$266$	0	0
$267$	−0.659044	−0.0403329
$268$	0	0
$269$	6.14643	0.374754	0.187377	−	0.982288i	$-0.440001\pi$
$269$	6.14643	0.374754	0.187377	+	0.982288i	$0.440001\pi$
$270$	0	0
$271$	−7.25014	−0.440415	−0.220207	−	0.975453i	$-0.570673\pi$
$271$	−7.25014	−0.440415	−0.220207	+	0.975453i	$0.570673\pi$
$272$	0	0
$273$	−16.1728	−0.978820
$274$	0	0
$275$	59.3140	3.57677
$276$	0	0
$277$	−9.03259	−0.542716	−0.271358	−	0.962479i	$-0.587473\pi$
$277$	−9.03259	−0.542716	−0.271358	+	0.962479i	$0.587473\pi$
$278$	0	0
$279$	−10.0816	−0.603569
$280$	0	0
$281$	16.3386	0.974682	0.487341	−	0.873212i	$-0.337967\pi$
$281$	16.3386	0.974682	0.487341	+	0.873212i	$0.337967\pi$
$282$	0	0
$283$	−18.3573	−1.09123	−0.545614	−	0.838036i	$-0.683704\pi$
$283$	−18.3573	−1.09123	−0.545614	+	0.838036i	$0.683704\pi$
$284$	0	0
$285$	31.9578	1.89302
$286$	0	0
$287$	−13.2207	−0.780391
$288$	0	0
$289$	0	0
$290$	0	0
$291$	−13.0877	−0.767215
$292$	0	0
$293$	6.49150	0.379237	0.189619	−	0.981858i	$-0.439275\pi$
$293$	6.49150	0.379237	0.189619	+	0.981858i	$0.439275\pi$
$294$	0	0
$295$	−22.2870	−1.29760
$296$	0	0
$297$	−15.7881	−0.916120
$298$	0	0
$299$	17.2146	0.995544
$300$	0	0
$301$	3.89012	0.224223
$302$	0	0
$303$	−23.8002	−1.36729
$304$	0	0
$305$	20.4812	1.17275
$306$	0	0
$307$	7.48527	0.427207	0.213603	−	0.976920i	$-0.431480\pi$
$307$	7.48527	0.427207	0.213603	+	0.976920i	$0.431480\pi$
$308$	0	0
$309$	−1.64411	−0.0935299
$310$	0	0
$311$	1.82828	0.103672	0.0518362	−	0.998656i	$-0.483493\pi$
$311$	1.82828	0.103672	0.0518362	+	0.998656i	$0.483493\pi$
$312$	0	0
$313$	−21.3969	−1.20942	−0.604712	−	0.796444i	$-0.706712\pi$
$313$	−21.3969	−1.20942	−0.604712	+	0.796444i	$0.706712\pi$
$314$	0	0
$315$	−9.29234	−0.523564
$316$	0	0
$317$	30.6537	1.72168	0.860841	−	0.508874i	$-0.169938\pi$
$317$	30.6537	1.72168	0.860841	+	0.508874i	$0.169938\pi$
$318$	0	0
$319$	2.29154	0.128302
$320$	0	0
$321$	6.99700	0.390535
$322$	0	0
$323$	0	0
$324$	0	0
$325$	−61.6014	−3.41703
$326$	0	0
$327$	10.0080	0.553444
$328$	0	0
$329$	−2.29989	−0.126797
$330$	0	0
$331$	−15.4692	−0.850263	−0.425132	−	0.905132i	$-0.639772\pi$
$331$	−15.4692	−0.850263	−0.425132	+	0.905132i	$0.639772\pi$
$332$	0	0
$333$	−2.79926	−0.153398
$334$	0	0
$335$	21.9988	1.20192
$336$	0	0
$337$	−7.95343	−0.433251	−0.216625	−	0.976255i	$-0.569505\pi$
$337$	−7.95343	−0.433251	−0.216625	+	0.976255i	$0.569505\pi$
$338$	0	0
$339$	−31.4712	−1.70928
$340$	0	0
$341$	−35.3213	−1.91276
$342$	0	0
$343$	−19.0330	−1.02769
$344$	0	0
$345$	33.6695	1.81270
$346$	0	0
$347$	−16.8002	−0.901882	−0.450941	−	0.892554i	$-0.648912\pi$
$347$	−16.8002	−0.901882	−0.450941	+	0.892554i	$0.648912\pi$
$348$	0	0
$349$	−2.52175	−0.134986	−0.0674932	−	0.997720i	$-0.521500\pi$
$349$	−2.52175	−0.134986	−0.0674932	+	0.997720i	$0.521500\pi$
$350$	0	0
$351$	16.3970	0.875206
$352$	0	0
$353$	−26.5129	−1.41114	−0.705570	−	0.708640i	$-0.749309\pi$
$353$	−26.5129	−1.41114	−0.705570	+	0.708640i	$0.749309\pi$
$354$	0	0
$355$	55.0788	2.92328
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−7.75375	−0.409227	−0.204614	−	0.978843i	$-0.565594\pi$
$359$	−7.75375	−0.409227	−0.204614	+	0.978843i	$0.565594\pi$
$360$	0	0
$361$	−6.05075	−0.318460
$362$	0	0
$363$	16.7242	0.877791
$364$	0	0
$365$	5.51535	0.288687
$366$	0	0
$367$	9.90287	0.516926	0.258463	−	0.966021i	$-0.416784\pi$
$367$	9.90287	0.516926	0.258463	+	0.966021i	$0.416784\pi$
$368$	0	0
$369$	−9.54645	−0.496968
$370$	0	0
$371$	21.7267	1.12799
$372$	0	0
$373$	7.07672	0.366419	0.183209	−	0.983074i	$-0.441351\pi$
$373$	7.07672	0.366419	0.183209	+	0.983074i	$0.441351\pi$
$374$	0	0
$375$	−76.0803	−3.92877
$376$	0	0
$377$	−2.37992	−0.122572
$378$	0	0
$379$	−19.2860	−0.990657	−0.495328	−	0.868706i	$-0.664952\pi$
$379$	−19.2860	−0.990657	−0.495328	+	0.868706i	$0.664952\pi$
$380$	0	0
$381$	−25.4333	−1.30299
$382$	0	0
$383$	−0.0295394	−0.00150939	−0.000754697	−	1.00000i	$-0.500240\pi$
$383$	−0.0295394	−0.00150939	−0.000754697		1.00000i	$0.500240\pi$
$384$	0	0
$385$	−32.5562	−1.65922
$386$	0	0
$387$	2.80900	0.142790
$388$	0	0
$389$	14.5600	0.738221	0.369110	−	0.929386i	$-0.379662\pi$
$389$	14.5600	0.738221	0.369110	+	0.929386i	$0.379662\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−2.14258	−0.108079
$394$	0	0
$395$	−1.09877	−0.0552851
$396$	0	0
$397$	−6.98697	−0.350666	−0.175333	−	0.984509i	$-0.556100\pi$
$397$	−6.98697	−0.350666	−0.175333	+	0.984509i	$0.556100\pi$
$398$	0	0
$399$	−12.8172	−0.641662
$400$	0	0
$401$	−4.67916	−0.233666	−0.116833	−	0.993152i	$-0.537274\pi$
$401$	−4.67916	−0.233666	−0.116833	+	0.993152i	$0.537274\pi$
$402$	0	0
$403$	36.6834	1.82733
$404$	0	0
$405$	48.2014	2.39515
$406$	0	0
$407$	−9.80734	−0.486132
$408$	0	0
$409$	32.4205	1.60309	0.801546	−	0.597933i	$-0.204011\pi$
$409$	32.4205	1.60309	0.801546	+	0.597933i	$0.204011\pi$
$410$	0	0
$411$	10.3412	0.510094
$412$	0	0
$413$	8.93856	0.439838
$414$	0	0
$415$	−23.9254	−1.17445
$416$	0	0
$417$	−44.2034	−2.16465
$418$	0	0
$419$	34.5535	1.68805	0.844024	−	0.536305i	$-0.180180\pi$
$419$	34.5535	1.68805	0.844024	+	0.536305i	$0.180180\pi$
$420$	0	0
$421$	−19.8509	−0.967472	−0.483736	−	0.875214i	$-0.660720\pi$
$421$	−19.8509	−0.967472	−0.483736	+	0.875214i	$0.660720\pi$
$422$	0	0
$423$	−1.66071	−0.0807467
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−8.21431	−0.397518
$428$	0	0
$429$	−40.9148	−1.97539
$430$	0	0
$431$	−3.72269	−0.179315	−0.0896577	−	0.995973i	$-0.528577\pi$
$431$	−3.72269	−0.179315	−0.0896577	+	0.995973i	$0.528577\pi$
$432$	0	0
$433$	26.5298	1.27494	0.637471	−	0.770474i	$-0.279981\pi$
$433$	26.5298	1.27494	0.637471	+	0.770474i	$0.279981\pi$
$434$	0	0
$435$	−4.65481	−0.223181
$436$	0	0
$437$	13.6428	0.652625
$438$	0	0
$439$	28.8706	1.37792	0.688960	−	0.724800i	$-0.258068\pi$
$439$	28.8706	1.37792	0.688960	+	0.724800i	$0.258068\pi$
$440$	0	0
$441$	−5.00832	−0.238491
$442$	0	0
$443$	32.3742	1.53814	0.769072	−	0.639162i	$-0.220719\pi$
$443$	32.3742	1.53814	0.769072	+	0.639162i	$0.220719\pi$
$444$	0	0
$445$	1.37783	0.0653156
$446$	0	0
$447$	0.683163	0.0323125
$448$	0	0
$449$	−17.0269	−0.803548	−0.401774	−	0.915739i	$-0.631606\pi$
$449$	−17.0269	−0.803548	−0.401774	+	0.915739i	$0.631606\pi$
$450$	0	0
$451$	−33.4464	−1.57493
$452$	0	0
$453$	−39.1258	−1.83829
$454$	0	0
$455$	33.8117	1.58511
$456$	0	0
$457$	36.7651	1.71980	0.859900	−	0.510462i	$-0.170526\pi$
$457$	36.7651	1.71980	0.859900	+	0.510462i	$0.170526\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−11.0810	−0.516092	−0.258046	−	0.966133i	$-0.583079\pi$
$461$	−11.0810	−0.516092	−0.258046	+	0.966133i	$0.583079\pi$
$462$	0	0
$463$	−1.20500	−0.0560009	−0.0280005	−	0.999608i	$-0.508914\pi$
$463$	−1.20500	−0.0560009	−0.0280005	+	0.999608i	$0.508914\pi$
$464$	0	0
$465$	71.7481	3.32724
$466$	0	0
$467$	−27.5509	−1.27490	−0.637451	−	0.770491i	$-0.720011\pi$
$467$	−27.5509	−1.27490	−0.637451	+	0.770491i	$0.720011\pi$
$468$	0	0
$469$	−8.82298	−0.407408
$470$	0	0
$471$	15.9958	0.737046
$472$	0	0
$473$	9.84147	0.452511
$474$	0	0
$475$	−48.8202	−2.24002
$476$	0	0
$477$	15.6885	0.718327
$478$	0	0
$479$	20.8405	0.952228	0.476114	−	0.879384i	$-0.342045\pi$
$479$	20.8405	0.952228	0.476114	+	0.879384i	$0.342045\pi$
$480$	0	0
$481$	10.1855	0.464421
$482$	0	0
$483$	−13.5037	−0.614439
$484$	0	0
$485$	27.3619	1.24244
$486$	0	0
$487$	−11.9198	−0.540139	−0.270069	−	0.962841i	$-0.587047\pi$
$487$	−11.9198	−0.540139	−0.270069	+	0.962841i	$0.587047\pi$
$488$	0	0
$489$	−32.1438	−1.45359
$490$	0	0
$491$	−7.03674	−0.317563	−0.158782	−	0.987314i	$-0.550757\pi$
$491$	−7.03674	−0.317563	−0.158782	+	0.987314i	$0.550757\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	−23.5083	−1.05662
$496$	0	0
$497$	−22.0903	−0.990883
$498$	0	0
$499$	−11.0208	−0.493361	−0.246680	−	0.969097i	$-0.579340\pi$
$499$	−11.0208	−0.493361	−0.246680	+	0.969097i	$0.579340\pi$
$500$	0	0
$501$	−41.4072	−1.84994
$502$	0	0
$503$	−10.6130	−0.473211	−0.236606	−	0.971606i	$-0.576035\pi$
$503$	−10.6130	−0.473211	−0.236606	+	0.971606i	$0.576035\pi$
$504$	0	0
$505$	49.7580	2.21420
$506$	0	0
$507$	15.6992	0.697225
$508$	0	0
$509$	4.40973	0.195458	0.0977289	−	0.995213i	$-0.468842\pi$
$509$	4.40973	0.195458	0.0977289	+	0.995213i	$0.468842\pi$
$510$	0	0
$511$	−2.21202	−0.0978541
$512$	0	0
$513$	12.9949	0.573738
$514$	0	0
$515$	3.43726	0.151464
$516$	0	0
$517$	−5.81839	−0.255893
$518$	0	0
$519$	12.4670	0.547242
$520$	0	0
$521$	4.49769	0.197047	0.0985236	−	0.995135i	$-0.468588\pi$
$521$	4.49769	0.197047	0.0985236	+	0.995135i	$0.468588\pi$
$522$	0	0
$523$	−14.6133	−0.638994	−0.319497	−	0.947587i	$-0.603514\pi$
$523$	−14.6133	−0.638994	−0.319497	+	0.947587i	$0.603514\pi$
$524$	0	0
$525$	48.3222	2.10896
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−8.62644	−0.375062
$530$	0	0
$531$	6.45440	0.280097
$532$	0	0
$533$	34.7363	1.50459
$534$	0	0
$535$	−14.6283	−0.632437
$536$	0	0
$537$	37.9351	1.63702
$538$	0	0
$539$	−17.5469	−0.755797
$540$	0	0
$541$	−37.7172	−1.62159	−0.810795	−	0.585331i	$-0.800965\pi$
$541$	−37.7172	−1.62159	−0.810795	+	0.585331i	$0.800965\pi$
$542$	0	0
$543$	45.7516	1.96339
$544$	0	0
$545$	−20.9233	−0.896254
$546$	0	0
$547$	−31.7055	−1.35563	−0.677815	−	0.735233i	$-0.737073\pi$
$547$	−31.7055	−1.35563	−0.677815	+	0.735233i	$0.737073\pi$
$548$	0	0
$549$	−5.93144	−0.253148
$550$	0	0
$551$	−1.88612	−0.0803515
$552$	0	0
$553$	0.440679	0.0187396
$554$	0	0
$555$	19.9216	0.845626
$556$	0	0
$557$	−1.35504	−0.0574148	−0.0287074	−	0.999588i	$-0.509139\pi$
$557$	−1.35504	−0.0574148	−0.0287074	+	0.999588i	$0.509139\pi$
$558$	0	0
$559$	−10.2210	−0.432302
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−28.6570	−1.20775	−0.603875	−	0.797079i	$-0.706377\pi$
$563$	−28.6570	−1.20775	−0.603875	+	0.797079i	$0.706377\pi$
$564$	0	0
$565$	65.7954	2.76803
$566$	0	0
$567$	−19.3320	−0.811866
$568$	0	0
$569$	11.0338	0.462560	0.231280	−	0.972887i	$-0.425709\pi$
$569$	11.0338	0.462560	0.231280	+	0.972887i	$0.425709\pi$
$570$	0	0
$571$	34.9151	1.46115	0.730575	−	0.682833i	$-0.239252\pi$
$571$	34.9151	1.46115	0.730575	+	0.682833i	$0.239252\pi$
$572$	0	0
$573$	25.6789	1.07275
$574$	0	0
$575$	−51.4350	−2.14499
$576$	0	0
$577$	−10.4728	−0.435987	−0.217993	−	0.975950i	$-0.569951\pi$
$577$	−10.4728	−0.435987	−0.217993	+	0.975950i	$0.569951\pi$
$578$	0	0
$579$	9.03199	0.375357
$580$	0	0
$581$	9.59565	0.398095
$582$	0	0
$583$	54.9654	2.27644
$584$	0	0
$585$	24.4149	1.00943
$586$	0	0
$587$	−2.37083	−0.0978547	−0.0489274	−	0.998802i	$-0.515580\pi$
$587$	−2.37083	−0.0978547	−0.0489274	+	0.998802i	$0.515580\pi$
$588$	0	0
$589$	29.0722	1.19790
$590$	0	0
$591$	37.8993	1.55897
$592$	0	0
$593$	−21.9223	−0.900241	−0.450121	−	0.892968i	$-0.648619\pi$
$593$	−21.9223	−0.900241	−0.450121	+	0.892968i	$0.648619\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	22.4563	0.919076
$598$	0	0
$599$	20.9674	0.856705	0.428353	−	0.903612i	$-0.359094\pi$
$599$	20.9674	0.856705	0.428353	+	0.903612i	$0.359094\pi$
$600$	0	0
$601$	−29.2563	−1.19339	−0.596694	−	0.802469i	$-0.703519\pi$
$601$	−29.2563	−1.19339	−0.596694	+	0.802469i	$0.703519\pi$
$602$	0	0
$603$	−6.37095	−0.259445
$604$	0	0
$605$	−34.9645	−1.42151
$606$	0	0
$607$	−41.8212	−1.69747	−0.848734	−	0.528820i	$-0.822635\pi$
$607$	−41.8212	−1.69747	−0.848734	+	0.528820i	$0.822635\pi$
$608$	0	0
$609$	1.86689	0.0756500
$610$	0	0
$611$	6.04277	0.244464
$612$	0	0
$613$	−10.9602	−0.442679	−0.221340	−	0.975197i	$-0.571043\pi$
$613$	−10.9602	−0.442679	−0.221340	+	0.975197i	$0.571043\pi$
$614$	0	0
$615$	67.9397	2.73959
$616$	0	0
$617$	23.9663	0.964846	0.482423	−	0.875938i	$-0.339757\pi$
$617$	23.9663	0.964846	0.482423	+	0.875938i	$0.339757\pi$
$618$	0	0
$619$	2.07982	0.0835950	0.0417975	−	0.999126i	$-0.486692\pi$
$619$	2.07982	0.0835950	0.0417975	+	0.999126i	$0.486692\pi$
$620$	0	0
$621$	13.6909	0.549397
$622$	0	0
$623$	−0.552602	−0.0221395
$624$	0	0
$625$	91.2236	3.64894
$626$	0	0
$627$	−32.4257	−1.29496
$628$	0	0
$629$	0	0
$630$	0	0
$631$	−21.5809	−0.859121	−0.429561	−	0.903038i	$-0.641332\pi$
$631$	−21.5809	−0.859121	−0.429561	+	0.903038i	$0.641332\pi$
$632$	0	0
$633$	46.6759	1.85520
$634$	0	0
$635$	53.1721	2.11007
$636$	0	0
$637$	18.2235	0.722043
$638$	0	0
$639$	−15.9511	−0.631014
$640$	0	0
$641$	−44.1031	−1.74197	−0.870985	−	0.491310i	$-0.836518\pi$
$641$	−44.1031	−1.74197	−0.870985	+	0.491310i	$0.836518\pi$
$642$	0	0
$643$	−19.1552	−0.755406	−0.377703	−	0.925927i	$-0.623286\pi$
$643$	−19.1552	−0.755406	−0.377703	+	0.925927i	$0.623286\pi$
$644$	0	0
$645$	−19.9910	−0.787143
$646$	0	0
$647$	0.248644	0.00977521	0.00488761	−	0.999988i	$-0.498444\pi$
$647$	0.248644	0.00977521	0.00488761	+	0.999988i	$0.498444\pi$
$648$	0	0
$649$	22.6133	0.887650
$650$	0	0
$651$	−28.7757	−1.12781
$652$	0	0
$653$	−23.4225	−0.916592	−0.458296	−	0.888800i	$-0.651540\pi$
$653$	−23.4225	−0.916592	−0.458296	+	0.888800i	$0.651540\pi$
$654$	0	0
$655$	4.47939	0.175024
$656$	0	0
$657$	−1.59727	−0.0623154
$658$	0	0
$659$	−44.0140	−1.71454	−0.857271	−	0.514865i	$-0.827842\pi$
$659$	−44.0140	−1.71454	−0.857271	+	0.514865i	$0.827842\pi$
$660$	0	0
$661$	−51.2354	−1.99283	−0.996414	−	0.0846129i	$-0.973035\pi$
$661$	−51.2354	−1.99283	−0.996414	+	0.0846129i	$0.973035\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	26.7963	1.03912
$666$	0	0
$667$	−1.98715	−0.0769426
$668$	0	0
$669$	16.1454	0.624218
$670$	0	0
$671$	−20.7811	−0.802244
$672$	0	0
$673$	17.6970	0.682170	0.341085	−	0.940032i	$-0.389206\pi$
$673$	17.6970	0.682170	0.341085	+	0.940032i	$0.389206\pi$
$674$	0	0
$675$	−48.9922	−1.88571
$676$	0	0
$677$	−40.6685	−1.56302	−0.781508	−	0.623895i	$-0.785549\pi$
$677$	−40.6685	−1.56302	−0.781508	+	0.623895i	$0.785549\pi$
$678$	0	0
$679$	−10.9739	−0.421140
$680$	0	0
$681$	49.7138	1.90504
$682$	0	0
$683$	−7.35573	−0.281459	−0.140730	−	0.990048i	$-0.544945\pi$
$683$	−7.35573	−0.281459	−0.140730	+	0.990048i	$0.544945\pi$
$684$	0	0
$685$	−21.6199	−0.826053
$686$	0	0
$687$	−13.0238	−0.496890
$688$	0	0
$689$	−57.0851	−2.17477
$690$	0	0
$691$	8.82190	0.335601	0.167800	−	0.985821i	$-0.446334\pi$
$691$	8.82190	0.335601	0.167800	+	0.985821i	$0.446334\pi$
$692$	0	0
$693$	9.42840	0.358155
$694$	0	0
$695$	92.4139	3.50546
$696$	0	0
$697$	0	0
$698$	0	0
$699$	6.04484	0.228637
$700$	0	0
$701$	−35.1943	−1.32927	−0.664636	−	0.747168i	$-0.731413\pi$
$701$	−35.1943	−1.32927	−0.664636	+	0.747168i	$0.731413\pi$
$702$	0	0
$703$	8.07221	0.304449
$704$	0	0
$705$	11.8189	0.445125
$706$	0	0
$707$	−19.9563	−0.750532
$708$	0	0
$709$	23.1526	0.869515	0.434758	−	0.900547i	$-0.356834\pi$
$709$	23.1526	0.869515	0.434758	+	0.900547i	$0.356834\pi$
$710$	0	0
$711$	0.318208	0.0119337
$712$	0	0
$713$	30.6294	1.14708
$714$	0	0
$715$	85.5388	3.19897
$716$	0	0
$717$	44.6883	1.66892
$718$	0	0
$719$	12.9885	0.484391	0.242195	−	0.970228i	$-0.422133\pi$
$719$	12.9885	0.484391	0.242195	+	0.970228i	$0.422133\pi$
$720$	0	0
$721$	−1.37857	−0.0513405
$722$	0	0
$723$	21.1346	0.786003
$724$	0	0
$725$	7.11089	0.264092
$726$	0	0
$727$	47.1886	1.75013	0.875065	−	0.484005i	$-0.160818\pi$
$727$	47.1886	1.75013	0.875065	+	0.484005i	$0.160818\pi$
$728$	0	0
$729$	8.36906	0.309965
$730$	0	0
$731$	0	0
$732$	0	0
$733$	42.3967	1.56596	0.782979	−	0.622048i	$-0.213699\pi$
$733$	42.3967	1.56596	0.782979	+	0.622048i	$0.213699\pi$
$734$	0	0
$735$	35.6429	1.31471
$736$	0	0
$737$	−22.3209	−0.822202
$738$	0	0
$739$	42.3605	1.55826	0.779128	−	0.626865i	$-0.215662\pi$
$739$	42.3605	1.55826	0.779128	+	0.626865i	$0.215662\pi$
$740$	0	0
$741$	33.6762	1.23712
$742$	0	0
$743$	−44.7922	−1.64327	−0.821633	−	0.570017i	$-0.806937\pi$
$743$	−44.7922	−1.64327	−0.821633	+	0.570017i	$0.806937\pi$
$744$	0	0
$745$	−1.42826	−0.0523273
$746$	0	0
$747$	6.92888	0.253514
$748$	0	0
$749$	5.86692	0.214373
$750$	0	0
$751$	−26.5191	−0.967695	−0.483848	−	0.875152i	$-0.660761\pi$
$751$	−26.5191	−0.967695	−0.483848	+	0.875152i	$0.660761\pi$
$752$	0	0
$753$	22.9522	0.836424
$754$	0	0
$755$	81.7984	2.97695
$756$	0	0
$757$	−8.89891	−0.323436	−0.161718	−	0.986837i	$-0.551704\pi$
$757$	−8.89891	−0.323436	−0.161718	+	0.986837i	$0.551704\pi$
$758$	0	0
$759$	−34.1625	−1.24002
$760$	0	0
$761$	21.0320	0.762411	0.381205	−	0.924490i	$-0.375509\pi$
$761$	21.0320	0.762411	0.381205	+	0.924490i	$0.375509\pi$
$762$	0	0
$763$	8.39161	0.303797
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−23.4854	−0.848007
$768$	0	0
$769$	29.9495	1.08001	0.540003	−	0.841663i	$-0.318423\pi$
$769$	29.9495	1.08001	0.540003	+	0.841663i	$0.318423\pi$
$770$	0	0
$771$	7.20285	0.259404
$772$	0	0
$773$	22.8927	0.823391	0.411696	−	0.911321i	$-0.364937\pi$
$773$	22.8927	0.823391	0.411696	+	0.911321i	$0.364937\pi$
$774$	0	0
$775$	−109.606	−3.93715
$776$	0	0
$777$	−7.98989	−0.286636
$778$	0	0
$779$	27.5291	0.986331
$780$	0	0
$781$	−55.8853	−1.99973
$782$	0	0
$783$	−1.89277	−0.0676420
$784$	0	0
$785$	−33.4416	−1.19358
$786$	0	0
$787$	−20.0028	−0.713025	−0.356512	−	0.934291i	$-0.616034\pi$
$787$	−20.0028	−0.713025	−0.356512	+	0.934291i	$0.616034\pi$
$788$	0	0
$789$	39.9360	1.42176
$790$	0	0
$791$	−26.3883	−0.938259
$792$	0	0
$793$	21.5825	0.766416
$794$	0	0
$795$	−111.651	−3.95986
$796$	0	0
$797$	2.69761	0.0955544	0.0477772	−	0.998858i	$-0.484786\pi$
$797$	2.69761	0.0955544	0.0477772	+	0.998858i	$0.484786\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	−0.399026	−0.0140989
$802$	0	0
$803$	−5.59611	−0.197482
$804$	0	0
$805$	28.2315	0.995031
$806$	0	0
$807$	12.6680	0.445936
$808$	0	0
$809$	35.6394	1.25302	0.626508	−	0.779415i	$-0.284484\pi$
$809$	35.6394	1.25302	0.626508	+	0.779415i	$0.284484\pi$
$810$	0	0
$811$	47.4491	1.66616	0.833082	−	0.553150i	$-0.186574\pi$
$811$	47.4491	1.66616	0.833082	+	0.553150i	$0.186574\pi$
$812$	0	0
$813$	−14.9428	−0.524068
$814$	0	0
$815$	67.2017	2.35397
$816$	0	0
$817$	−8.10031	−0.283394
$818$	0	0
$819$	−9.79199	−0.342160
$820$	0	0
$821$	8.29133	0.289370	0.144685	−	0.989478i	$-0.453783\pi$
$821$	8.29133	0.289370	0.144685	+	0.989478i	$0.453783\pi$
$822$	0	0
$823$	−0.376105	−0.0131102	−0.00655510	−	0.999979i	$-0.502087\pi$
$823$	−0.376105	−0.0131102	−0.00655510	+	0.999979i	$0.502087\pi$
$824$	0	0
$825$	122.249	4.25615
$826$	0	0
$827$	37.7588	1.31300	0.656501	−	0.754325i	$-0.272036\pi$
$827$	37.7588	1.31300	0.656501	+	0.754325i	$0.272036\pi$
$828$	0	0
$829$	7.52348	0.261301	0.130651	−	0.991428i	$-0.458293\pi$
$829$	7.52348	0.261301	0.130651	+	0.991428i	$0.458293\pi$
$830$	0	0
$831$	−18.6165	−0.645800
$832$	0	0
$833$	0	0
$834$	0	0
$835$	86.5682	2.99582
$836$	0	0
$837$	29.1747	1.00842
$838$	0	0
$839$	3.51515	0.121356	0.0606782	−	0.998157i	$-0.480674\pi$
$839$	3.51515	0.121356	0.0606782	+	0.998157i	$0.480674\pi$
$840$	0	0
$841$	−28.7253	−0.990527
$842$	0	0
$843$	33.6746	1.15981
$844$	0	0
$845$	−32.8215	−1.12910
$846$	0	0
$847$	14.0231	0.481838
$848$	0	0
$849$	−37.8351	−1.29850
$850$	0	0
$851$	8.50457	0.291533
$852$	0	0
$853$	−30.3585	−1.03945	−0.519727	−	0.854332i	$-0.673966\pi$
$853$	−30.3585	−1.03945	−0.519727	+	0.854332i	$0.673966\pi$
$854$	0	0
$855$	19.3492	0.661729
$856$	0	0
$857$	9.81494	0.335272	0.167636	−	0.985849i	$-0.446387\pi$
$857$	9.81494	0.335272	0.167636	+	0.985849i	$0.446387\pi$
$858$	0	0
$859$	−57.8752	−1.97468	−0.987339	−	0.158627i	$-0.949293\pi$
$859$	−57.8752	−1.97468	−0.987339	+	0.158627i	$0.949293\pi$
$860$	0	0
$861$	−27.2483	−0.928620
$862$	0	0
$863$	−25.3357	−0.862439	−0.431219	−	0.902247i	$-0.641916\pi$
$863$	−25.3357	−0.862439	−0.431219	+	0.902247i	$0.641916\pi$
$864$	0	0
$865$	−26.0643	−0.886212
$866$	0	0
$867$	0	0
$868$	0	0
$869$	1.11486	0.0378189
$870$	0	0
$871$	23.1817	0.785482
$872$	0	0
$873$	−7.92411	−0.268190
$874$	0	0
$875$	−63.7926	−2.15658
$876$	0	0
$877$	22.2380	0.750923	0.375461	−	0.926838i	$-0.377484\pi$
$877$	22.2380	0.750923	0.375461	+	0.926838i	$0.377484\pi$
$878$	0	0
$879$	13.3792	0.451270
$880$	0	0
$881$	−4.30054	−0.144889	−0.0724445	−	0.997372i	$-0.523080\pi$
$881$	−4.30054	−0.144889	−0.0724445	+	0.997372i	$0.523080\pi$
$882$	0	0
$883$	49.5321	1.66689	0.833444	−	0.552603i	$-0.186366\pi$
$883$	49.5321	1.66689	0.833444	+	0.552603i	$0.186366\pi$
$884$	0	0
$885$	−45.9344	−1.54407
$886$	0	0
$887$	2.87025	0.0963735	0.0481868	−	0.998838i	$-0.484656\pi$
$887$	2.87025	0.0963735	0.0481868	+	0.998838i	$0.484656\pi$
$888$	0	0
$889$	−21.3255	−0.715236
$890$	0	0
$891$	−48.9072	−1.63845
$892$	0	0
$893$	4.78900	0.160258
$894$	0	0
$895$	−79.3091	−2.65101
$896$	0	0
$897$	35.4799	1.18464
$898$	0	0
$899$	−4.23451	−0.141229
$900$	0	0
$901$	0	0
$902$	0	0
$903$	8.01769	0.266812
$904$	0	0
$905$	−95.6508	−3.17954
$906$	0	0
$907$	11.1662	0.370767	0.185383	−	0.982666i	$-0.440647\pi$
$907$	11.1662	0.370767	0.185383	+	0.982666i	$0.440647\pi$
$908$	0	0
$909$	−14.4101	−0.477954
$910$	0	0
$911$	6.43009	0.213038	0.106519	−	0.994311i	$-0.466029\pi$
$911$	6.43009	0.213038	0.106519	+	0.994311i	$0.466029\pi$
$912$	0	0
$913$	24.2757	0.803407
$914$	0	0
$915$	42.2125	1.39550
$916$	0	0
$917$	−1.79653	−0.0593267
$918$	0	0
$919$	−15.7732	−0.520308	−0.260154	−	0.965567i	$-0.583773\pi$
$919$	−15.7732	−0.520308	−0.260154	+	0.965567i	$0.583773\pi$
$920$	0	0
$921$	15.4274	0.508351
$922$	0	0
$923$	58.0404	1.91042
$924$	0	0
$925$	−30.4332	−1.00064
$926$	0	0
$927$	−0.995443	−0.0326946
$928$	0	0
$929$	−26.7689	−0.878260	−0.439130	−	0.898423i	$-0.644713\pi$
$929$	−26.7689	−0.878260	−0.439130	+	0.898423i	$0.644713\pi$
$930$	0	0
$931$	14.4425	0.473333
$932$	0	0
$933$	3.76816	0.123364
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−54.2504	−1.77228	−0.886142	−	0.463414i	$-0.846624\pi$
$937$	−54.2504	−1.77228	−0.886142	+	0.463414i	$0.846624\pi$
$938$	0	0
$939$	−44.0998	−1.43914
$940$	0	0
$941$	−42.1735	−1.37482	−0.687408	−	0.726272i	$-0.741251\pi$
$941$	−42.1735	−1.37482	−0.687408	+	0.726272i	$0.741251\pi$
$942$	0	0
$943$	29.0036	0.944486
$944$	0	0
$945$	26.8907	0.874755
$946$	0	0
$947$	−0.685721	−0.0222829	−0.0111415	−	0.999938i	$-0.503547\pi$
$947$	−0.685721	−0.0222829	−0.0111415	+	0.999938i	$0.503547\pi$
$948$	0	0
$949$	5.81191	0.188663
$950$	0	0
$951$	63.1784	2.04870
$952$	0	0
$953$	−49.5946	−1.60653	−0.803263	−	0.595625i	$-0.796904\pi$
$953$	−49.5946	−1.60653	−0.803263	+	0.595625i	$0.796904\pi$
$954$	0	0
$955$	−53.6858	−1.73723
$956$	0	0
$957$	4.72296	0.152672
$958$	0	0
$959$	8.67100	0.280001
$960$	0	0
$961$	34.2697	1.10548
$962$	0	0
$963$	4.23642	0.136517
$964$	0	0
$965$	−18.8828	−0.607858
$966$	0	0
$967$	−34.9795	−1.12487	−0.562433	−	0.826843i	$-0.690135\pi$
$967$	−34.9795	−1.12487	−0.562433	+	0.826843i	$0.690135\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−43.2937	−1.38936	−0.694681	−	0.719318i	$-0.744454\pi$
$971$	−43.2937	−1.38936	−0.694681	+	0.719318i	$0.744454\pi$
$972$	0	0
$973$	−37.0641	−1.18822
$974$	0	0
$975$	−126.963	−4.06607
$976$	0	0
$977$	−53.5695	−1.71384	−0.856920	−	0.515449i	$-0.827625\pi$
$977$	−53.5695	−1.71384	−0.856920	+	0.515449i	$0.827625\pi$
$978$	0	0
$979$	−1.39801	−0.0446805
$980$	0	0
$981$	6.05946	0.193464
$982$	0	0
$983$	−0.928232	−0.0296060	−0.0148030	−	0.999890i	$-0.504712\pi$
$983$	−0.928232	−0.0296060	−0.0148030	+	0.999890i	$0.504712\pi$
$984$	0	0
$985$	−79.2344	−2.52462
$986$	0	0
$987$	−4.74015	−0.150881
$988$	0	0
$989$	−8.53417	−0.271371
$990$	0	0
$991$	−29.2729	−0.929885	−0.464943	−	0.885341i	$-0.653925\pi$
$991$	−29.2729	−0.929885	−0.464943	+	0.885341i	$0.653925\pi$
$992$	0	0
$993$	−31.8826	−1.01176
$994$	0	0
$995$	−46.9484	−1.48836
$996$	0	0
$997$	−42.2269	−1.33734	−0.668669	−	0.743560i	$-0.733136\pi$
$997$	−42.2269	−1.33734	−0.668669	+	0.743560i	$0.733136\pi$
$998$	0	0
$999$	8.10066	0.256293

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2312.2.a.u.1.6	✓	6
4.3	odd	2		4624.2.a.br.1.1		6
17.4	even	4		2312.2.b.o.577.2		12
17.13	even	4		2312.2.b.o.577.11		12
17.16	even	2		2312.2.a.v.1.1	yes	6
68.67	odd	2		4624.2.a.bs.1.6		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2312.2.a.u.1.6	✓	6	1.1	even	1	trivial
2312.2.a.v.1.1	yes	6	17.16	even	2
2312.2.b.o.577.2		12	17.4	even	4
2312.2.b.o.577.11		12	17.13	even	4
4624.2.a.br.1.1		6	4.3	odd	2
4624.2.a.bs.1.6		6	68.67	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:	\( N \)	\(=\)	\( 2312 = 2^{3} \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2312.a (trivial)

\(f(q)\)	\(=\)	\(q+2.06104 q^{3} -4.30892 q^{5} +1.72816 q^{7} +1.24788 q^{9} +4.37201 q^{11} -4.54061 q^{13} -8.88085 q^{15} -3.59851 q^{19} +3.56181 q^{21} -3.79125 q^{23} +13.5668 q^{25} -3.61119 q^{27} +0.524140 q^{29} -8.07897 q^{31} +9.01087 q^{33} -7.44650 q^{35} -2.24321 q^{37} -9.35837 q^{39} -7.65013 q^{41} +2.25102 q^{43} -5.37701 q^{45} -1.33083 q^{47} -4.01346 q^{49} +12.5721 q^{53} -18.8386 q^{55} -7.41666 q^{57} +5.17229 q^{59} -4.75321 q^{61} +2.15654 q^{63} +19.5651 q^{65} -5.10542 q^{67} -7.81391 q^{69} -12.7825 q^{71} -1.27999 q^{73} +27.9617 q^{75} +7.55553 q^{77} +0.254999 q^{79} -11.1864 q^{81} +5.55252 q^{83} +1.08027 q^{87} -0.319763 q^{89} -7.84690 q^{91} -16.6511 q^{93} +15.5057 q^{95} -6.35006 q^{97} +5.45574 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{5} - 3 q^{7} + 6 q^{11} - 6 q^{13} - 6 q^{15} + 6 q^{19} + 6 q^{21} - 9 q^{23} + 6 q^{25} - 12 q^{27} - 27 q^{29} - 9 q^{31} + 3 q^{33} + 21 q^{35} - 15 q^{37} + 12 q^{39} - 21 q^{41} - 24 q^{45}+ \cdots + 42 q^{99}+O(q^{100}) \) 6 * q - 6 * q^5 - 3 * q^7 + 6 * q^11 - 6 * q^13 - 6 * q^15 + 6 * q^19 + 6 * q^21 - 9 * q^23 + 6 * q^25 - 12 * q^27 - 27 * q^29 - 9 * q^31 + 3 * q^33 + 21 * q^35 - 15 * q^37 + 12 * q^39 - 21 * q^41 - 24 * q^45 + 3 * q^47 + 9 * q^49 + 33 * q^53 - 51 * q^55 - 30 * q^57 - 18 * q^61 - 24 * q^63 + 9 * q^65 - 9 * q^67 - 3 * q^69 - 6 * q^71 - 27 * q^73 + 24 * q^75 - 45 * q^77 - 33 * q^79 - 18 * q^81 - 12 * q^83 + 15 * q^87 + 3 * q^89 - 30 * q^91 - 30 * q^93 - 3 * q^95 - 51 * q^97 + 42 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

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