LMFDB - Embedded newform 2312.2.b.o.577.11

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2312,2,Mod(577,2312)] 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, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2312.577"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 2312 = 2^{3} \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2312.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

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

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

Self dual:	no
Analytic conductor:	$18.4614129473$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 18x^{10} + 117x^{8} + 342x^{6} + 438x^{4} + 180x^{2} + 1 $ x^12 + 18x^10 + 117x^8 + 342x^6 + 438x^4 + 180*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			577.11
Root			$2.06104i$ of defining polynomial
Character	$\chi$	$=$	2312.577
Dual form			2312.2.b.o.577.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.06104i q^{3} -4.30892i q^{5} -1.72816i q^{7} -1.24788 q^{9} -4.37201i q^{11} -4.54061 q^{13} +8.88085 q^{15} +3.59851 q^{19} +3.56181 q^{21} +3.79125i q^{23} -13.5668 q^{25} +3.61119i q^{27} +0.524140i q^{29} -8.07897i q^{31} +9.01087 q^{33} -7.44650 q^{35} -2.24321i q^{37} -9.35837i q^{39} +7.65013i q^{41} -2.25102 q^{43} +5.37701i q^{45} -1.33083 q^{47} +4.01346 q^{49} -12.5721 q^{53} -18.8386 q^{55} +7.41666i q^{57} -5.17229 q^{59} +4.75321i q^{61} +2.15654i q^{63} +19.5651i q^{65} -5.10542 q^{67} -7.81391 q^{69} -12.7825i q^{71} -1.27999i q^{73} -27.9617i q^{75} -7.55553 q^{77} -0.254999i q^{79} -11.1864 q^{81} -5.55252 q^{83} -1.08027 q^{87} -0.319763 q^{89} +7.84690i q^{91} +16.6511 q^{93} -15.5057i q^{95} -6.35006i q^{97} +5.45574i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 12 q^{13} + 12 q^{15} - 12 q^{19} + 12 q^{21} - 12 q^{25} + 6 q^{33} + 42 q^{35} + 6 q^{47} - 18 q^{49} - 66 q^{53} - 102 q^{55} - 18 q^{67} - 6 q^{69} + 90 q^{77} - 36 q^{81} + 24 q^{83} - 30 q^{87}+ \cdots + 60 q^{93}+O(q^{100}) $ 12 * q - 12 * q^13 + 12 * q^15 - 12 * q^19 + 12 * q^21 - 12 * q^25 + 6 * q^33 + 42 * q^35 + 6 * q^47 - 18 * q^49 - 66 * q^53 - 102 * q^55 - 18 * q^67 - 6 * q^69 + 90 * q^77 - 36 * q^81 + 24 * q^83 - 30 * q^87 + 6 * q^89 + 60 * q^93

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2312\mathbb{Z}\right)^\times$.

$n$	$1157$	$1735$	$1737$
$\chi(n)$	$1$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	2.06104i	1.18994i	0.803747	+	0.594971i	$0.202836\pi$
$3$	2.06104i	1.18994i	−0.803747	+	0.594971i	$0.797164\pi$
$4$	0	0
$5$	− 4.30892i	− 1.92701i	−0.267696	−	0.963504i	$-0.586262\pi$
$5$	− 4.30892i	− 1.92701i	0.267696	−	0.963504i	$-0.413738\pi$
$6$	0	0
$7$	− 1.72816i	− 0.653183i	−0.945166	−	0.326592i	$-0.894100\pi$
$7$	− 1.72816i	− 0.653183i	0.945166	−	0.326592i	$-0.105900\pi$
$8$	0	0
$9$	−1.24788	−0.415960
$10$	0	0
$11$	− 4.37201i	− 1.31821i	−0.752051	−	0.659105i	$-0.770935\pi$
$11$	− 4.37201i	− 1.31821i	0.752051	−	0.659105i	$-0.229065\pi$
$12$	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	$-0.364559\pi$
$19$	3.59851	0.825554	0.412777	+	0.910832i	$0.364559\pi$
$20$	0	0
$21$	3.56181	0.777250
$22$	0	0
$23$	3.79125i	0.790530i	0.918567	+	0.395265i	$0.129347\pi$
$23$	3.79125i	0.790530i	−0.918567	+	0.395265i	$0.870653\pi$
$24$	0	0
$25$	−13.5668	−2.71336
$26$	0	0
$27$	3.61119i	0.694973i
$28$	0	0
$29$	0.524140i	0.0973304i	0.998815	+	0.0486652i	$0.0154967\pi$
$29$	0.524140i	0.0973304i	−0.998815	+	0.0486652i	$0.984503\pi$
$30$	0	0
$31$	− 8.07897i	− 1.45103i	−0.688209	−	0.725513i	$-0.741603\pi$
$31$	− 8.07897i	− 1.45103i	0.688209	−	0.725513i	$-0.258397\pi$
$32$	0	0
$33$	9.01087	1.56859
$34$	0	0
$35$	−7.44650	−1.25869
$36$	0	0
$37$	− 2.24321i	− 0.368782i	−0.982853	−	0.184391i	$-0.940969\pi$
$37$	− 2.24321i	− 0.368782i	0.982853	−	0.184391i	$-0.0590312\pi$
$38$	0	0
$39$	− 9.35837i	− 1.49854i
$40$	0	0
$41$	7.65013i	1.19475i	0.801962	+	0.597375i	$0.203790\pi$
$41$	7.65013i	1.19475i	−0.801962	+	0.597375i	$0.796210\pi$
$42$	0	0
$43$	−2.25102	−0.343277	−0.171639	−	0.985160i	$-0.554906\pi$
$43$	−2.25102	−0.343277	−0.171639	+	0.985160i	$0.554906\pi$
$44$	0	0
$45$	5.37701i	0.801558i
$46$	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	$-0.831706\pi$
$53$	−12.5721	−1.72691	−0.863457	+	0.504422i	$0.831706\pi$
$54$	0	0
$55$	−18.8386	−2.54020
$56$	0	0
$57$	7.41666i	0.982361i
$58$	0	0
$59$	−5.17229	−0.673375	−0.336688	−	0.941616i	$-0.609307\pi$
$59$	−5.17229	−0.673375	−0.336688	+	0.941616i	$0.609307\pi$
$60$	0	0
$61$	4.75321i	0.608586i	0.952578	+	0.304293i	$0.0984202\pi$
$61$	4.75321i	0.608586i	−0.952578	+	0.304293i	$0.901580\pi$
$62$	0	0
$63$	2.15654i	0.271698i
$64$	0	0
$65$	19.5651i	2.42675i
$66$	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
$71$	− 12.7825i	− 1.51701i	−0.651669	−	0.758503i	$-0.725931\pi$
$71$	− 12.7825i	− 1.51701i	0.651669	−	0.758503i	$-0.274069\pi$
$72$	0	0
$73$	− 1.27999i	− 0.149811i	−0.997191	−	0.0749055i	$-0.976134\pi$
$73$	− 1.27999i	− 0.149811i	0.997191	−	0.0749055i	$-0.0238655\pi$
$74$	0	0
$75$	− 27.9617i	− 3.22873i
$76$	0	0
$77$	−7.55553	−0.861033
$78$	0	0
$79$	− 0.254999i	− 0.0286896i	−0.999897	−	0.0143448i	$-0.995434\pi$
$79$	− 0.254999i	− 0.0286896i	0.999897	−	0.0143448i	$-0.00456625\pi$
$80$	0	0
$81$	−11.1864	−1.24294
$82$	0	0
$83$	−5.55252	−0.609468	−0.304734	−	0.952437i	$-0.598568\pi$
$83$	−5.55252	−0.609468	−0.304734	+	0.952437i	$0.598568\pi$
$84$	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
$91$	7.84690i	0.822579i
$92$	0	0
$93$	16.6511	1.72664
$94$	0	0
$95$	− 15.5057i	− 1.59085i
$96$	0	0
$97$	− 6.35006i	− 0.644751i	−0.946612	−	0.322375i	$-0.895519\pi$
$97$	− 6.35006i	− 0.644751i	0.946612	−	0.322375i	$-0.104481\pi$
$98$	0	0
$99$	5.45574i	0.548322i
$100$	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
$105$	− 15.3475i	− 1.49777i
$106$	0	0
$107$	3.39489i	0.328197i	0.986444	+	0.164098i	$0.0524714\pi$
$107$	3.39489i	0.328197i	−0.986444	+	0.164098i	$0.947529\pi$
$108$	0	0
$109$	− 4.85580i	− 0.465102i	−0.972584	−	0.232551i	$-0.925293\pi$
$109$	− 4.85580i	− 0.465102i	0.972584	−	0.232551i	$-0.0747072\pi$
$110$	0	0
$111$	4.62335	0.438829
$112$	0	0
$113$	15.2696i	1.43644i	0.695816	+	0.718220i	$0.255043\pi$
$113$	15.2696i	1.43644i	−0.695816	+	0.718220i	$0.744957\pi$
$114$	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
$125$	36.9136i	3.30165i
$126$	0	0
$127$	12.3400	1.09500	0.547500	−	0.836806i	$-0.315580\pi$
$127$	12.3400	1.09500	0.547500	+	0.836806i	$0.315580\pi$
$128$	0	0
$129$	− 4.63944i	− 0.408480i
$130$	0	0
$131$	− 1.03956i	− 0.0908271i	−0.998968	−	0.0454135i	$-0.985539\pi$
$131$	− 1.03956i	− 0.0908271i	0.998968	−	0.0454135i	$-0.0144606\pi$
$132$	0	0
$133$	− 6.21880i	− 0.539238i
$134$	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
$139$	− 21.4471i	− 1.81912i	−0.415571	−	0.909561i	$-0.636418\pi$
$139$	− 21.4471i	− 1.81912i	0.415571	−	0.909561i	$-0.363582\pi$
$140$	0	0
$141$	− 2.74289i	− 0.230993i
$142$	0	0
$143$	19.8516i	1.66007i
$144$	0	0
$145$	2.25848	0.187556
$146$	0	0
$147$	8.27190i	0.682254i
$148$	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	$-0.219042\pi$
$151$	18.9835	1.54486	0.772429	+	0.635102i	$0.219042\pi$
$152$	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
$159$	− 25.9116i	− 2.05493i
$160$	0	0
$161$	6.55189	0.516361
$162$	0	0
$163$	15.5959i	1.22157i	0.791797	+	0.610784i	$0.209146\pi$
$163$	15.5959i	1.22157i	−0.791797	+	0.610784i	$0.790854\pi$
$164$	0	0
$165$	− 38.8271i	− 3.02269i
$166$	0	0
$167$	− 20.0905i	− 1.55465i	−0.629101	−	0.777324i	$-0.716577\pi$
$167$	− 20.0905i	− 1.55465i	0.629101	−	0.777324i	$-0.283423\pi$
$168$	0	0
$169$	7.61712	0.585932
$170$	0	0
$171$	−4.49051	−0.343398
$172$	0	0
$173$	6.04891i	0.459890i	0.973204	+	0.229945i	$0.0738547\pi$
$173$	6.04891i	0.459890i	−0.973204	+	0.229945i	$0.926145\pi$
$174$	0	0
$175$	23.4456i	1.77232i
$176$	0	0
$177$	− 10.6603i	− 0.801277i
$178$	0	0
$179$	−18.4058	−1.37571	−0.687857	−	0.725846i	$-0.741448\pi$
$179$	−18.4058	−1.37571	−0.687857	+	0.725846i	$0.741448\pi$
$180$	0	0
$181$	− 22.1983i	− 1.64999i	−0.565141	−	0.824995i	$-0.691178\pi$
$181$	− 22.1983i	− 1.64999i	0.565141	−	0.824995i	$-0.308822\pi$
$182$	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
$193$	− 4.38225i	− 0.315441i	−0.987484	−	0.157721i	$-0.949585\pi$
$193$	− 4.38225i	− 0.315441i	0.987484	−	0.157721i	$-0.0504146\pi$
$194$	0	0
$195$	−40.3244	−2.88769
$196$	0	0
$197$	− 18.3885i	− 1.31012i	−0.755575	−	0.655062i	$-0.772643\pi$
$197$	− 18.3885i	− 1.31012i	0.755575	−	0.655062i	$-0.227357\pi$
$198$	0	0
$199$	10.8956i	0.772371i	0.922421	+	0.386185i	$0.126207\pi$
$199$	10.8956i	0.772371i	−0.922421	+	0.386185i	$0.873793\pi$
$200$	0	0
$201$	− 10.5225i	− 0.742197i
$202$	0	0
$203$	0.905799	0.0635746
$204$	0	0
$205$	32.9638	2.30229
$206$	0	0
$207$	− 4.73102i	− 0.328829i
$208$	0	0
$209$	− 15.7327i	− 1.08825i
$210$	0	0
$211$	− 22.6468i	− 1.55907i	−0.626360	−	0.779534i	$-0.715456\pi$
$211$	− 22.6468i	− 1.55907i	0.626360	−	0.779534i	$-0.284544\pi$
$212$	0	0
$213$	26.3453	1.80515
$214$	0	0
$215$	9.69946i	0.661498i
$216$	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	$-0.584477\pi$
$223$	−7.83363	−0.524579	−0.262289	+	0.964989i	$0.584477\pi$
$224$	0	0
$225$	16.9297	1.12865
$226$	0	0
$227$	− 24.1208i	− 1.60095i	−0.599365	−	0.800476i	$-0.704580\pi$
$227$	− 24.1208i	− 1.60095i	0.599365	−	0.800476i	$-0.295420\pi$
$228$	0	0
$229$	6.31906	0.417575	0.208788	−	0.977961i	$-0.433048\pi$
$229$	6.31906	0.417575	0.208788	+	0.977961i	$0.433048\pi$
$230$	0	0
$231$	− 15.5722i	− 1.02458i
$232$	0	0
$233$	2.93291i	0.192141i	0.995375	+	0.0960707i	$0.0306275\pi$
$233$	2.93291i	0.192141i	−0.995375	+	0.0960707i	$0.969373\pi$
$234$	0	0
$235$	5.73443i	0.374073i
$236$	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
$241$	10.2543i	0.660539i	0.943887	+	0.330269i	$0.107140\pi$
$241$	10.2543i	0.660539i	−0.943887	+	0.330269i	$0.892860\pi$
$242$	0	0
$243$	− 12.2221i	− 0.784049i
$244$	0	0
$245$	− 17.2937i	− 1.10485i
$246$	0	0
$247$	−16.3394	−1.03965
$248$	0	0
$249$	− 11.4440i	− 0.725231i
$250$	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	$-0.534764\pi$
$257$	−3.49477	−0.217998	−0.108999	+	0.994042i	$0.534764\pi$
$258$	0	0
$259$	−3.87663	−0.240882
$260$	0	0
$261$	− 0.654064i	− 0.0404856i
$262$	0	0
$263$	−19.3767	−1.19482	−0.597408	−	0.801938i	$-0.703803\pi$
$263$	−19.3767	−1.19482	−0.597408	+	0.801938i	$0.703803\pi$
$264$	0	0
$265$	54.1723i	3.32778i
$266$	0	0
$267$	− 0.659044i	− 0.0403329i
$268$	0	0
$269$	6.14643i	0.374754i	0.982288	+	0.187377i	$0.0599986\pi$
$269$	6.14643i	0.374754i	−0.982288	+	0.187377i	$0.940001\pi$
$270$	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
$275$	59.3140i	3.57677i
$276$	0	0
$277$	− 9.03259i	− 0.542716i	−0.962479	−	0.271358i	$-0.912527\pi$
$277$	− 9.03259i	− 0.542716i	0.962479	−	0.271358i	$-0.0874727\pi$
$278$	0	0
$279$	10.0816i	0.603569i
$280$	0	0
$281$	−16.3386	−0.974682	−0.487341	−	0.873212i	$-0.662033\pi$
$281$	−16.3386	−0.974682	−0.487341	+	0.873212i	$0.662033\pi$
$282$	0	0
$283$	18.3573i	1.09123i	0.838036	+	0.545614i	$0.183704\pi$
$283$	18.3573i	1.09123i	−0.838036	+	0.545614i	$0.816296\pi$
$284$	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
$295$	22.2870i	1.29760i
$296$	0	0
$297$	15.7881	0.916120
$298$	0	0
$299$	− 17.2146i	− 0.995544i
$300$	0	0
$301$	3.89012i	0.224223i
$302$	0	0
$303$	− 23.8002i	− 1.36729i
$304$	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
$309$	− 1.64411i	− 0.0935299i
$310$	0	0
$311$	1.82828i	0.103672i	0.998656	+	0.0518362i	$0.0165074\pi$
$311$	1.82828i	0.103672i	−0.998656	+	0.0518362i	$0.983493\pi$
$312$	0	0
$313$	21.3969i	1.20942i	0.796444	+	0.604712i	$0.206712\pi$
$313$	21.3969i	1.20942i	−0.796444	+	0.604712i	$0.793288\pi$
$314$	0	0
$315$	9.29234	0.523564
$316$	0	0
$317$	− 30.6537i	− 1.72168i	−0.508874	−	0.860841i	$-0.669938\pi$
$317$	− 30.6537i	− 1.72168i	0.508874	−	0.860841i	$-0.330062\pi$
$318$	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
$329$	2.29989i	0.126797i
$330$	0	0
$331$	15.4692	0.850263	0.425132	−	0.905132i	$-0.360228\pi$
$331$	15.4692	0.850263	0.425132	+	0.905132i	$0.360228\pi$
$332$	0	0
$333$	2.79926i	0.153398i
$334$	0	0
$335$	21.9988i	1.20192i
$336$	0	0
$337$	− 7.95343i	− 0.433251i	−0.976255	−	0.216625i	$-0.930495\pi$
$337$	− 7.95343i	− 0.433251i	0.976255	−	0.216625i	$-0.0695050\pi$
$338$	0	0
$339$	−31.4712	−1.70928
$340$	0	0
$341$	−35.3213	−1.91276
$342$	0	0
$343$	− 19.0330i	− 1.02769i
$344$	0	0
$345$	33.6695i	1.81270i
$346$	0	0
$347$	16.8002i	0.901882i	0.892554	+	0.450941i	$0.148912\pi$
$347$	16.8002i	0.901882i	−0.892554	+	0.450941i	$0.851088\pi$
$348$	0	0
$349$	2.52175	0.134986	0.0674932	−	0.997720i	$-0.478500\pi$
$349$	2.52175	0.134986	0.0674932	+	0.997720i	$0.478500\pi$
$350$	0	0
$351$	− 16.3970i	− 0.875206i
$352$	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	$-0.434406\pi$
$359$	7.75375	0.409227	0.204614	+	0.978843i	$0.434406\pi$
$360$	0	0
$361$	−6.05075	−0.318460
$362$	0	0
$363$	− 16.7242i	− 0.877791i
$364$	0	0
$365$	−5.51535	−0.288687
$366$	0	0
$367$	− 9.90287i	− 0.516926i	−0.966021	−	0.258463i	$-0.916784\pi$
$367$	− 9.90287i	− 0.516926i	0.966021	−	0.258463i	$-0.0832159\pi$
$368$	0	0
$369$	− 9.54645i	− 0.496968i
$370$	0	0
$371$	21.7267i	1.12799i
$372$	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
$377$	− 2.37992i	− 0.122572i
$378$	0	0
$379$	− 19.2860i	− 0.990657i	−0.868706	−	0.495328i	$-0.835048\pi$
$379$	− 19.2860i	− 0.990657i	0.868706	−	0.495328i	$-0.164952\pi$
$380$	0	0
$381$	25.4333i	1.30299i
$382$	0	0
$383$	0.0295394	0.00150939	0.000754697	−	1.00000i	$-0.499760\pi$
$383$	0.0295394	0.00150939	0.000754697		1.00000i	$0.499760\pi$
$384$	0	0
$385$	32.5562i	1.65922i
$386$	0	0
$387$	2.80900	0.142790
$388$	0	0
$389$	−14.5600	−0.738221	−0.369110	−	0.929386i	$-0.620338\pi$
$389$	−14.5600	−0.738221	−0.369110	+	0.929386i	$0.620338\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	2.14258	0.108079
$394$	0	0
$395$	−1.09877	−0.0552851
$396$	0	0
$397$	6.98697i	0.350666i	0.984509	+	0.175333i	$0.0561002\pi$
$397$	6.98697i	0.350666i	−0.984509	+	0.175333i	$0.943900\pi$
$398$	0	0
$399$	12.8172	0.641662
$400$	0	0
$401$	4.67916i	0.233666i	0.993152	+	0.116833i	$0.0372743\pi$
$401$	4.67916i	0.233666i	−0.993152	+	0.116833i	$0.962726\pi$
$402$	0	0
$403$	36.6834i	1.82733i
$404$	0	0
$405$	48.2014i	2.39515i
$406$	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
$411$	10.3412i	0.510094i
$412$	0	0
$413$	8.93856i	0.439838i
$414$	0	0
$415$	23.9254i	1.17445i
$416$	0	0
$417$	44.2034	2.16465
$418$	0	0
$419$	− 34.5535i	− 1.68805i	−0.536305	−	0.844024i	$-0.680180\pi$
$419$	− 34.5535i	− 1.68805i	0.536305	−	0.844024i	$-0.319820\pi$
$420$	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
$431$	3.72269i	0.179315i	0.995973	+	0.0896577i	$0.0285773\pi$
$431$	3.72269i	0.179315i	−0.995973	+	0.0896577i	$0.971423\pi$
$432$	0	0
$433$	−26.5298	−1.27494	−0.637471	−	0.770474i	$-0.720019\pi$
$433$	−26.5298	−1.27494	−0.637471	+	0.770474i	$0.720019\pi$
$434$	0	0
$435$	4.65481i	0.223181i
$436$	0	0
$437$	13.6428i	0.652625i
$438$	0	0
$439$	28.8706i	1.37792i	0.724800	+	0.688960i	$0.241932\pi$
$439$	28.8706i	1.37792i	−0.724800	+	0.688960i	$0.758068\pi$
$440$	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
$445$	1.37783i	0.0653156i
$446$	0	0
$447$	0.683163i	0.0323125i
$448$	0	0
$449$	17.0269i	0.803548i	0.915739	+	0.401774i	$0.131606\pi$
$449$	17.0269i	0.803548i	−0.915739	+	0.401774i	$0.868394\pi$
$450$	0	0
$451$	33.4464	1.57493
$452$	0	0
$453$	39.1258i	1.83829i
$454$	0	0
$455$	33.8117	1.58511
$456$	0	0
$457$	−36.7651	−1.71980	−0.859900	−	0.510462i	$-0.829474\pi$
$457$	−36.7651	−1.71980	−0.859900	+	0.510462i	$0.829474\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	11.0810	0.516092	0.258046	−	0.966133i	$-0.416921\pi$
$461$	11.0810	0.516092	0.258046	+	0.966133i	$0.416921\pi$
$462$	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
$465$	− 71.7481i	− 3.32724i
$466$	0	0
$467$	27.5509	1.27490	0.637451	−	0.770491i	$-0.279989\pi$
$467$	27.5509	1.27490	0.637451	+	0.770491i	$0.279989\pi$
$468$	0	0
$469$	8.82298i	0.407408i
$470$	0	0
$471$	15.9958i	0.737046i
$472$	0	0
$473$	9.84147i	0.452511i
$474$	0	0
$475$	−48.8202	−2.24002
$476$	0	0
$477$	15.6885	0.718327
$478$	0	0
$479$	20.8405i	0.952228i	0.879384	+	0.476114i	$0.157955\pi$
$479$	20.8405i	0.952228i	−0.879384	+	0.476114i	$0.842045\pi$
$480$	0	0
$481$	10.1855i	0.464421i
$482$	0	0
$483$	13.5037i	0.614439i
$484$	0	0
$485$	−27.3619	−1.24244
$486$	0	0
$487$	11.9198i	0.540139i	0.962841	+	0.270069i	$0.0870466\pi$
$487$	11.9198i	0.540139i	−0.962841	+	0.270069i	$0.912953\pi$
$488$	0	0
$489$	−32.1438	−1.45359
$490$	0	0
$491$	7.03674	0.317563	0.158782	−	0.987314i	$-0.449243\pi$
$491$	7.03674	0.317563	0.158782	+	0.987314i	$0.449243\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	23.5083	1.05662
$496$	0	0
$497$	−22.0903	−0.990883
$498$	0	0
$499$	11.0208i	0.493361i	0.969097	+	0.246680i	$0.0793398\pi$
$499$	11.0208i	0.493361i	−0.969097	+	0.246680i	$0.920660\pi$
$500$	0	0
$501$	41.4072	1.84994
$502$	0	0
$503$	10.6130i	0.473211i	0.971606	+	0.236606i	$0.0760349\pi$
$503$	10.6130i	0.473211i	−0.971606	+	0.236606i	$0.923965\pi$
$504$	0	0
$505$	49.7580i	2.21420i
$506$	0	0
$507$	15.6992i	0.697225i
$508$	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
$513$	12.9949i	0.573738i
$514$	0	0
$515$	3.43726i	0.151464i
$516$	0	0
$517$	5.81839i	0.255893i
$518$	0	0
$519$	−12.4670	−0.547242
$520$	0	0
$521$	− 4.49769i	− 0.197047i	−0.995135	−	0.0985236i	$-0.968588\pi$
$521$	− 4.49769i	− 0.197047i	0.995135	−	0.0985236i	$-0.0314120\pi$
$522$	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
$533$	− 34.7363i	− 1.50459i
$534$	0	0
$535$	14.6283	0.632437
$536$	0	0
$537$	− 37.9351i	− 1.63702i
$538$	0	0
$539$	− 17.5469i	− 0.755797i
$540$	0	0
$541$	− 37.7172i	− 1.62159i	−0.585331	−	0.810795i	$-0.699035\pi$
$541$	− 37.7172i	− 1.62159i	0.585331	−	0.810795i	$-0.300965\pi$
$542$	0	0
$543$	45.7516	1.96339
$544$	0	0
$545$	−20.9233	−0.896254
$546$	0	0
$547$	− 31.7055i	− 1.35563i	−0.735233	−	0.677815i	$-0.762927\pi$
$547$	− 31.7055i	− 1.35563i	0.735233	−	0.677815i	$-0.237073\pi$
$548$	0	0
$549$	− 5.93144i	− 0.253148i
$550$	0	0
$551$	1.88612i	0.0803515i
$552$	0	0
$553$	−0.440679	−0.0187396
$554$	0	0
$555$	− 19.9216i	− 0.845626i
$556$	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	$-0.293623\pi$
$563$	28.6570	1.20775	0.603875	+	0.797079i	$0.293623\pi$
$564$	0	0
$565$	65.7954	2.76803
$566$	0	0
$567$	19.3320i	0.811866i
$568$	0	0
$569$	−11.0338	−0.462560	−0.231280	−	0.972887i	$-0.574291\pi$
$569$	−11.0338	−0.462560	−0.231280	+	0.972887i	$0.574291\pi$
$570$	0	0
$571$	− 34.9151i	− 1.46115i	−0.682833	−	0.730575i	$-0.739252\pi$
$571$	− 34.9151i	− 1.46115i	0.682833	−	0.730575i	$-0.260748\pi$
$572$	0	0
$573$	25.6789i	1.07275i
$574$	0	0
$575$	− 51.4350i	− 2.14499i
$576$	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
$581$	9.59565i	0.398095i
$582$	0	0
$583$	54.9654i	2.27644i
$584$	0	0
$585$	− 24.4149i	− 1.00943i
$586$	0	0
$587$	2.37083	0.0978547	0.0489274	−	0.998802i	$-0.484420\pi$
$587$	2.37083	0.0978547	0.0489274	+	0.998802i	$0.484420\pi$
$588$	0	0
$589$	− 29.0722i	− 1.19790i
$590$	0	0
$591$	37.8993	1.55897
$592$	0	0
$593$	21.9223	0.900241	0.450121	−	0.892968i	$-0.351381\pi$
$593$	21.9223	0.900241	0.450121	+	0.892968i	$0.351381\pi$
$594$	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
$601$	29.2563i	1.19339i	0.802469	+	0.596694i	$0.203519\pi$
$601$	29.2563i	1.19339i	−0.802469	+	0.596694i	$0.796481\pi$
$602$	0	0
$603$	6.37095	0.259445
$604$	0	0
$605$	34.9645i	1.42151i
$606$	0	0
$607$	− 41.8212i	− 1.69747i	−0.528820	−	0.848734i	$-0.677365\pi$
$607$	− 41.8212i	− 1.69747i	0.528820	−	0.848734i	$-0.322635\pi$
$608$	0	0
$609$	1.86689i	0.0756500i
$610$	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
$615$	67.9397i	2.73959i
$616$	0	0
$617$	23.9663i	0.964846i	0.875938	+	0.482423i	$0.160243\pi$
$617$	23.9663i	0.964846i	−0.875938	+	0.482423i	$0.839757\pi$
$618$	0	0
$619$	− 2.07982i	− 0.0835950i	−0.999126	−	0.0417975i	$-0.986692\pi$
$619$	− 2.07982i	− 0.0835950i	0.999126	−	0.0417975i	$-0.0133084\pi$
$620$	0	0
$621$	−13.6909	−0.549397
$622$	0	0
$623$	0.552602i	0.0221395i
$624$	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	$-0.358668\pi$
$631$	21.5809	0.859121	0.429561	+	0.903038i	$0.358668\pi$
$632$	0	0
$633$	46.6759	1.85520
$634$	0	0
$635$	− 53.1721i	− 2.11007i
$636$	0	0
$637$	−18.2235	−0.722043
$638$	0	0
$639$	15.9511i	0.631014i
$640$	0	0
$641$	− 44.1031i	− 1.74197i	−0.491310	−	0.870985i	$-0.663482\pi$
$641$	− 44.1031i	− 1.74197i	0.491310	−	0.870985i	$-0.336518\pi$
$642$	0	0
$643$	− 19.1552i	− 0.755406i	−0.925927	−	0.377703i	$-0.876714\pi$
$643$	− 19.1552i	− 0.755406i	0.925927	−	0.377703i	$-0.123286\pi$
$644$	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
$649$	22.6133i	0.887650i
$650$	0	0
$651$	− 28.7757i	− 1.12781i
$652$	0	0
$653$	23.4225i	0.916592i	0.888800	+	0.458296i	$0.151540\pi$
$653$	23.4225i	0.916592i	−0.888800	+	0.458296i	$0.848460\pi$
$654$	0	0
$655$	−4.47939	−0.175024
$656$	0	0
$657$	1.59727i	0.0623154i
$658$	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	$-0.0269654\pi$
$661$	51.2354	1.99283	0.996414	+	0.0846129i	$0.0269654\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−26.7963	−1.03912
$666$	0	0
$667$	−1.98715	−0.0769426
$668$	0	0
$669$	− 16.1454i	− 0.624218i
$670$	0	0
$671$	20.7811	0.802244
$672$	0	0
$673$	− 17.6970i	− 0.682170i	−0.940032	−	0.341085i	$-0.889206\pi$
$673$	− 17.6970i	− 0.682170i	0.940032	−	0.341085i	$-0.110794\pi$
$674$	0	0
$675$	− 48.9922i	− 1.88571i
$676$	0	0
$677$	− 40.6685i	− 1.56302i	−0.623895	−	0.781508i	$-0.714451\pi$
$677$	− 40.6685i	− 1.56302i	0.623895	−	0.781508i	$-0.285549\pi$
$678$	0	0
$679$	−10.9739	−0.421140
$680$	0	0
$681$	49.7138	1.90504
$682$	0	0
$683$	− 7.35573i	− 0.281459i	−0.990048	−	0.140730i	$-0.955055\pi$
$683$	− 7.35573i	− 0.281459i	0.990048	−	0.140730i	$-0.0449448\pi$
$684$	0	0
$685$	− 21.6199i	− 0.826053i
$686$	0	0
$687$	13.0238i	0.496890i
$688$	0	0
$689$	57.0851	2.17477
$690$	0	0
$691$	− 8.82190i	− 0.335601i	−0.985821	−	0.167800i	$-0.946334\pi$
$691$	− 8.82190i	− 0.335601i	0.985821	−	0.167800i	$-0.0536664\pi$
$692$	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
$703$	− 8.07221i	− 0.304449i
$704$	0	0
$705$	−11.8189	−0.445125
$706$	0	0
$707$	19.9563i	0.750532i
$708$	0	0
$709$	23.1526i	0.869515i	0.900547	+	0.434758i	$0.143166\pi$
$709$	23.1526i	0.869515i	−0.900547	+	0.434758i	$0.856834\pi$
$710$	0	0
$711$	0.318208i	0.0119337i
$712$	0	0
$713$	30.6294	1.14708
$714$	0	0
$715$	85.5388	3.19897
$716$	0	0
$717$	44.6883i	1.66892i
$718$	0	0
$719$	12.9885i	0.484391i	0.970228	+	0.242195i	$0.0778675\pi$
$719$	12.9885i	0.484391i	−0.970228	+	0.242195i	$0.922133\pi$
$720$	0	0
$721$	1.37857i	0.0513405i
$722$	0	0
$723$	−21.1346	−0.786003
$724$	0	0
$725$	− 7.11089i	− 0.264092i
$726$	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	$-0.786301\pi$
$733$	−42.3967	−1.56596	−0.782979	+	0.622048i	$0.786301\pi$
$734$	0	0
$735$	35.6429	1.31471
$736$	0	0
$737$	22.3209i	0.822202i
$738$	0	0
$739$	−42.3605	−1.55826	−0.779128	−	0.626865i	$-0.784338\pi$
$739$	−42.3605	−1.55826	−0.779128	+	0.626865i	$0.784338\pi$
$740$	0	0
$741$	− 33.6762i	− 1.23712i
$742$	0	0
$743$	− 44.7922i	− 1.64327i	−0.570017	−	0.821633i	$-0.693063\pi$
$743$	− 44.7922i	− 1.64327i	0.570017	−	0.821633i	$-0.306937\pi$
$744$	0	0
$745$	− 1.42826i	− 0.0523273i
$746$	0	0
$747$	6.92888	0.253514
$748$	0	0
$749$	5.86692	0.214373
$750$	0	0
$751$	− 26.5191i	− 0.967695i	−0.875152	−	0.483848i	$-0.839239\pi$
$751$	− 26.5191i	− 0.967695i	0.875152	−	0.483848i	$-0.160761\pi$
$752$	0	0
$753$	22.9522i	0.836424i
$754$	0	0
$755$	− 81.7984i	− 2.97695i
$756$	0	0
$757$	8.89891	0.323436	0.161718	−	0.986837i	$-0.448296\pi$
$757$	8.89891	0.323436	0.161718	+	0.986837i	$0.448296\pi$
$758$	0	0
$759$	34.1625i	1.24002i
$760$	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
$771$	− 7.20285i	− 0.259404i
$772$	0	0
$773$	−22.8927	−0.823391	−0.411696	−	0.911321i	$-0.635063\pi$
$773$	−22.8927	−0.823391	−0.411696	+	0.911321i	$0.635063\pi$
$774$	0	0
$775$	109.606i	3.93715i
$776$	0	0
$777$	− 7.98989i	− 0.286636i
$778$	0	0
$779$	27.5291i	0.986331i
$780$	0	0
$781$	−55.8853	−1.99973
$782$	0	0
$783$	−1.89277	−0.0676420
$784$	0	0
$785$	− 33.4416i	− 1.19358i
$786$	0	0
$787$	− 20.0028i	− 0.713025i	−0.934291	−	0.356512i	$-0.883966\pi$
$787$	− 20.0028i	− 0.713025i	0.934291	−	0.356512i	$-0.116034\pi$
$788$	0	0
$789$	− 39.9360i	− 1.42176i
$790$	0	0
$791$	26.3883	0.938259
$792$	0	0
$793$	− 21.5825i	− 0.766416i
$794$	0	0
$795$	−111.651	−3.95986
$796$	0	0
$797$	−2.69761	−0.0955544	−0.0477772	−	0.998858i	$-0.515214\pi$
$797$	−2.69761	−0.0955544	−0.0477772	+	0.998858i	$0.515214\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0.399026	0.0140989
$802$	0	0
$803$	−5.59611	−0.197482
$804$	0	0
$805$	− 28.2315i	− 0.995031i
$806$	0	0
$807$	−12.6680	−0.445936
$808$	0	0
$809$	− 35.6394i	− 1.25302i	−0.779415	−	0.626508i	$-0.784484\pi$
$809$	− 35.6394i	− 1.25302i	0.779415	−	0.626508i	$-0.215516\pi$
$810$	0	0
$811$	47.4491i	1.66616i	0.553150	+	0.833082i	$0.313426\pi$
$811$	47.4491i	1.66616i	−0.553150	+	0.833082i	$0.686574\pi$
$812$	0	0
$813$	− 14.9428i	− 0.524068i
$814$	0	0
$815$	67.2017	2.35397
$816$	0	0
$817$	−8.10031	−0.283394
$818$	0	0
$819$	− 9.79199i	− 0.342160i
$820$	0	0
$821$	8.29133i	0.289370i	0.989478	+	0.144685i	$0.0462168\pi$
$821$	8.29133i	0.289370i	−0.989478	+	0.144685i	$0.953783\pi$
$822$	0	0
$823$	0.376105i	0.0131102i	0.999979	+	0.00655510i	$0.00208657\pi$
$823$	0.376105i	0.0131102i	−0.999979	+	0.00655510i	$0.997913\pi$
$824$	0	0
$825$	−122.249	−4.25615
$826$	0	0
$827$	− 37.7588i	− 1.31300i	−0.754325	−	0.656501i	$-0.772036\pi$
$827$	− 37.7588i	− 1.31300i	0.754325	−	0.656501i	$-0.227964\pi$
$828$	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
$839$	− 3.51515i	− 0.121356i	−0.998157	−	0.0606782i	$-0.980674\pi$
$839$	− 3.51515i	− 0.121356i	0.998157	−	0.0606782i	$-0.0193263\pi$
$840$	0	0
$841$	28.7253	0.990527
$842$	0	0
$843$	− 33.6746i	− 1.15981i
$844$	0	0
$845$	− 32.8215i	− 1.12910i
$846$	0	0
$847$	14.0231i	0.481838i
$848$	0	0
$849$	−37.8351	−1.29850
$850$	0	0
$851$	8.50457	0.291533
$852$	0	0
$853$	− 30.3585i	− 1.03945i	−0.854332	−	0.519727i	$-0.826034\pi$
$853$	− 30.3585i	− 1.03945i	0.854332	−	0.519727i	$-0.173966\pi$
$854$	0	0
$855$	19.3492i	0.661729i
$856$	0	0
$857$	− 9.81494i	− 0.335272i	−0.985849	−	0.167636i	$-0.946387\pi$
$857$	− 9.81494i	− 0.335272i	0.985849	−	0.167636i	$-0.0536133\pi$
$858$	0	0
$859$	57.8752	1.97468	0.987339	−	0.158627i	$-0.0507067\pi$
$859$	57.8752	1.97468	0.987339	+	0.158627i	$0.0507067\pi$
$860$	0	0
$861$	27.2483i	0.928620i
$862$	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
$873$	7.92411i	0.268190i
$874$	0	0
$875$	63.7926	2.15658
$876$	0	0
$877$	− 22.2380i	− 0.750923i	−0.926838	−	0.375461i	$-0.877484\pi$
$877$	− 22.2380i	− 0.750923i	0.926838	−	0.375461i	$-0.122516\pi$
$878$	0	0
$879$	13.3792i	0.451270i
$880$	0	0
$881$	− 4.30054i	− 0.144889i	−0.997372	−	0.0724445i	$-0.976920\pi$
$881$	− 4.30054i	− 0.144889i	0.997372	−	0.0724445i	$-0.0230800\pi$
$882$	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
$887$	2.87025i	0.0963735i	0.998838	+	0.0481868i	$0.0153443\pi$
$887$	2.87025i	0.0963735i	−0.998838	+	0.0481868i	$0.984656\pi$
$888$	0	0
$889$	− 21.3255i	− 0.715236i
$890$	0	0
$891$	48.9072i	1.63845i
$892$	0	0
$893$	−4.78900	−0.160258
$894$	0	0
$895$	79.3091i	2.65101i
$896$	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
$907$	− 11.1662i	− 0.370767i	−0.982666	−	0.185383i	$-0.940647\pi$
$907$	− 11.1662i	− 0.370767i	0.982666	−	0.185383i	$-0.0593527\pi$
$908$	0	0
$909$	14.4101	0.477954
$910$	0	0
$911$	− 6.43009i	− 0.213038i	−0.994311	−	0.106519i	$-0.966029\pi$
$911$	− 6.43009i	− 0.213038i	0.994311	−	0.106519i	$-0.0339706\pi$
$912$	0	0
$913$	24.2757i	0.803407i
$914$	0	0
$915$	42.2125i	1.39550i
$916$	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
$921$	15.4274i	0.508351i
$922$	0	0
$923$	58.0404i	1.91042i
$924$	0	0
$925$	30.4332i	1.00064i
$926$	0	0
$927$	0.995443	0.0326946
$928$	0	0
$929$	26.7689i	0.878260i	0.898423	+	0.439130i	$0.144713\pi$
$929$	26.7689i	0.878260i	−0.898423	+	0.439130i	$0.855287\pi$
$930$	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	$-0.153376\pi$
$937$	54.2504	1.77228	0.886142	+	0.463414i	$0.153376\pi$
$938$	0	0
$939$	−44.0998	−1.43914
$940$	0	0
$941$	42.1735i	1.37482i	0.726272	+	0.687408i	$0.241251\pi$
$941$	42.1735i	1.37482i	−0.726272	+	0.687408i	$0.758749\pi$
$942$	0	0
$943$	−29.0036	−0.944486
$944$	0	0
$945$	− 26.8907i	− 0.874755i
$946$	0	0
$947$	− 0.685721i	− 0.0222829i	−0.999938	−	0.0111415i	$-0.996453\pi$
$947$	− 0.685721i	− 0.0222829i	0.999938	−	0.0111415i	$-0.00354651\pi$
$948$	0	0
$949$	5.81191i	0.188663i
$950$	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
$955$	− 53.6858i	− 1.73723i
$956$	0	0
$957$	4.72296i	0.152672i
$958$	0	0
$959$	− 8.67100i	− 0.280001i
$960$	0	0
$961$	−34.2697	−1.10548
$962$	0	0
$963$	− 4.23642i	− 0.136517i
$964$	0	0
$965$	−18.8828	−0.607858
$966$	0	0
$967$	34.9795	1.12487	0.562433	−	0.826843i	$-0.309865\pi$
$967$	34.9795	1.12487	0.562433	+	0.826843i	$0.309865\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	43.2937	1.38936	0.694681	−	0.719318i	$-0.255546\pi$
$971$	43.2937	1.38936	0.694681	+	0.719318i	$0.255546\pi$
$972$	0	0
$973$	−37.0641	−1.18822
$974$	0	0
$975$	126.963i	4.06607i
$976$	0	0
$977$	53.5695	1.71384	0.856920	−	0.515449i	$-0.172375\pi$
$977$	53.5695	1.71384	0.856920	+	0.515449i	$0.172375\pi$
$978$	0	0
$979$	1.39801i	0.0446805i
$980$	0	0
$981$	6.05946i	0.193464i
$982$	0	0
$983$	− 0.928232i	− 0.0296060i	−0.999890	−	0.0148030i	$-0.995288\pi$
$983$	− 0.928232i	− 0.0296060i	0.999890	−	0.0148030i	$-0.00471211\pi$
$984$	0	0
$985$	−79.2344	−2.52462
$986$	0	0
$987$	−4.74015	−0.150881
$988$	0	0
$989$	− 8.53417i	− 0.271371i
$990$	0	0
$991$	− 29.2729i	− 0.929885i	−0.885341	−	0.464943i	$-0.846075\pi$
$991$	− 29.2729i	− 0.929885i	0.885341	−	0.464943i	$-0.153925\pi$
$992$	0	0
$993$	31.8826i	1.01176i
$994$	0	0
$995$	46.9484	1.48836
$996$	0	0
$997$	42.2269i	1.33734i	0.743560	+	0.668669i	$0.233136\pi$
$997$	42.2269i	1.33734i	−0.743560	+	0.668669i	$0.766864\pi$
$998$	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.b.o.577.11		12
17.4	even	4		2312.2.a.u.1.6	✓	6
17.13	even	4		2312.2.a.v.1.1	yes	6
17.16	even	2	inner	2312.2.b.o.577.2		12
68.47	odd	4		4624.2.a.bs.1.6		6
68.55	odd	4		4624.2.a.br.1.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2312.2.a.u.1.6	✓	6	17.4	even	4
2312.2.a.v.1.1	yes	6	17.13	even	4
2312.2.b.o.577.2		12	17.16	even	2	inner
2312.2.b.o.577.11		12	1.1	even	1	trivial
4624.2.a.br.1.1		6	68.55	odd	4
4624.2.a.bs.1.6		6	68.47	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2312 = 2^{3} \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2312.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.06104i q^{3} -4.30892i q^{5} -1.72816i q^{7} -1.24788 q^{9} -4.37201i q^{11} -4.54061 q^{13} +8.88085 q^{15} +3.59851 q^{19} +3.56181 q^{21} +3.79125i q^{23} -13.5668 q^{25} +3.61119i q^{27} +0.524140i q^{29} -8.07897i q^{31} +9.01087 q^{33} -7.44650 q^{35} -2.24321i q^{37} -9.35837i q^{39} +7.65013i q^{41} -2.25102 q^{43} +5.37701i q^{45} -1.33083 q^{47} +4.01346 q^{49} -12.5721 q^{53} -18.8386 q^{55} +7.41666i q^{57} -5.17229 q^{59} +4.75321i q^{61} +2.15654i q^{63} +19.5651i q^{65} -5.10542 q^{67} -7.81391 q^{69} -12.7825i q^{71} -1.27999i q^{73} -27.9617i q^{75} -7.55553 q^{77} -0.254999i q^{79} -11.1864 q^{81} -5.55252 q^{83} -1.08027 q^{87} -0.319763 q^{89} +7.84690i q^{91} +16.6511 q^{93} -15.5057i q^{95} -6.35006i q^{97} +5.45574i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 12 q^{13} + 12 q^{15} - 12 q^{19} + 12 q^{21} - 12 q^{25} + 6 q^{33} + 42 q^{35} + 6 q^{47} - 18 q^{49} - 66 q^{53} - 102 q^{55} - 18 q^{67} - 6 q^{69} + 90 q^{77} - 36 q^{81} + 24 q^{83} - 30 q^{87}+ \cdots + 60 q^{93}+O(q^{100}) \) 12 * q - 12 * q^13 + 12 * q^15 - 12 * q^19 + 12 * q^21 - 12 * q^25 + 6 * q^33 + 42 * q^35 + 6 * q^47 - 18 * q^49 - 66 * q^53 - 102 * q^55 - 18 * q^67 - 6 * q^69 + 90 * q^77 - 36 * q^81 + 24 * q^83 - 30 * q^87 + 6 * q^89 + 60 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more