LMFDB - Embedded newform 2432.2.c.j.1217.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2432,2,Mod(1217,2432)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2432, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 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("2432.1217"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 2432 = 2^{7} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2432.c (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [20,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$19.4196177716$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} + 170x^{16} + 6593x^{12} + 64168x^{8} + 95760x^{4} + 4096 $ x^20 + 170x^16 + 6593x^12 + 64168x^8 + 95760x^4 + 4096
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{19} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1217.6
Root			$1.31430 - 1.31430i$ of defining polynomial
Character	$\chi$	$=$	2432.1217
Dual form			2432.2.c.j.1217.15

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.85871i q^{3} +1.69983i q^{5} -4.99365 q^{7} -0.454788 q^{9} -1.11059i q^{11} -3.29382i q^{13} +3.15948 q^{15} +1.15203 q^{17} -1.00000i q^{19} +9.28173i q^{21} -1.96339 q^{23} +2.11059 q^{25} -4.73080i q^{27} +0.533986i q^{29} -5.01704 q^{31} -2.06426 q^{33} -8.48834i q^{35} -0.331051i q^{37} -6.12225 q^{39} -4.12133 q^{41} +9.29618i q^{43} -0.773061i q^{45} -5.71748 q^{47} +17.9366 q^{49} -2.14129i q^{51} +10.5213i q^{53} +1.88781 q^{55} -1.85871 q^{57} +14.1441i q^{59} +13.2760i q^{61} +2.27105 q^{63} +5.59893 q^{65} +11.9258i q^{67} +3.64936i q^{69} +4.73009 q^{71} -6.28279 q^{73} -3.92296i q^{75} +5.54590i q^{77} -2.23894 q^{79} -10.1575 q^{81} +3.09516i q^{83} +1.95826i q^{85} +0.992524 q^{87} -6.62510 q^{89} +16.4482i q^{91} +9.32520i q^{93} +1.69983 q^{95} -6.46818 q^{97} +0.505083i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 28 q^{9} + 8 q^{17} - 20 q^{25} - 16 q^{33} + 24 q^{41} + 52 q^{49} - 8 q^{57} - 48 q^{65} - 24 q^{73} + 68 q^{81} + 40 q^{89} - 56 q^{97}+O(q^{100}) $ 20 * q - 28 * q^9 + 8 * q^17 - 20 * q^25 - 16 * q^33 + 24 * q^41 + 52 * q^49 - 8 * q^57 - 48 * q^65 - 24 * q^73 + 68 * q^81 + 40 * q^89 - 56 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2432\mathbb{Z}\right)^\times$.

$n$	$1407$	$1921$	$2053$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	− 1.85871i	− 1.07312i	−0.843861	−	0.536562i	$-0.819723\pi$
$3$	− 1.85871i	− 1.07312i	0.843861	−	0.536562i	$-0.180277\pi$
$4$	0	0
$5$	1.69983i	0.760186i	0.924948	+	0.380093i	$0.124108\pi$
$5$	1.69983i	0.760186i	−0.924948	+	0.380093i	$0.875892\pi$
$6$	0	0
$7$	−4.99365	−1.88742	−0.943711	−	0.330770i	$-0.892692\pi$
$7$	−4.99365	−1.88742	−0.943711	+	0.330770i	$0.892692\pi$
$8$	0	0
$9$	−0.454788	−0.151596
$10$	0	0
$11$	− 1.11059i	− 0.334855i	−0.985884	−	0.167428i	$-0.946454\pi$
$11$	− 1.11059i	− 0.334855i	0.985884	−	0.167428i	$-0.0535461\pi$
$12$	0	0
$13$	− 3.29382i	− 0.913543i	−0.889584	−	0.456771i	$-0.849006\pi$
$13$	− 3.29382i	− 0.913543i	0.889584	−	0.456771i	$-0.150994\pi$
$14$	0	0
$15$	3.15948	0.815774
$16$	0	0
$17$	1.15203	0.279409	0.139705	−	0.990193i	$-0.455385\pi$
$17$	1.15203	0.279409	0.139705	+	0.990193i	$0.455385\pi$
$18$	0	0
$19$	− 1.00000i	− 0.229416i
$19$	− 1.00000i	− 0.229416i
$20$	0	0
$21$	9.28173i	2.02544i
$22$	0	0
$23$	−1.96339	−0.409395	−0.204698	−	0.978825i	$-0.565621\pi$
$23$	−1.96339	−0.409395	−0.204698	+	0.978825i	$0.565621\pi$
$24$	0	0
$25$	2.11059	0.422118
$26$	0	0
$27$	− 4.73080i	− 0.910443i
$28$	0	0
$29$	0.533986i	0.0991588i	0.998770	+	0.0495794i	$0.0157881\pi$
$29$	0.533986i	0.0991588i	−0.998770	+	0.0495794i	$0.984212\pi$
$30$	0	0
$31$	−5.01704	−0.901087	−0.450543	−	0.892755i	$-0.648770\pi$
$31$	−5.01704	−0.901087	−0.450543	+	0.892755i	$0.648770\pi$
$32$	0	0
$33$	−2.06426	−0.359341
$34$	0	0
$35$	− 8.48834i	− 1.43479i
$36$	0	0
$37$	− 0.331051i	− 0.0544244i	−0.999630	−	0.0272122i	$-0.991337\pi$
$37$	− 0.331051i	− 0.0544244i	0.999630	−	0.0272122i	$-0.00866299\pi$
$38$	0	0
$39$	−6.12225	−0.980345
$40$	0	0
$41$	−4.12133	−0.643644	−0.321822	−	0.946800i	$-0.604295\pi$
$41$	−4.12133	−0.643644	−0.321822	+	0.946800i	$0.604295\pi$
$42$	0	0
$43$	9.29618i	1.41765i	0.705382	+	0.708827i	$0.250775\pi$
$43$	9.29618i	1.41765i	−0.705382	+	0.708827i	$0.749225\pi$
$44$	0	0
$45$	− 0.773061i	− 0.115241i
$46$	0	0
$47$	−5.71748	−0.833980	−0.416990	−	0.908911i	$-0.636915\pi$
$47$	−5.71748	−0.833980	−0.416990	+	0.908911i	$0.636915\pi$
$48$	0	0
$49$	17.9366	2.56236
$50$	0	0
$51$	− 2.14129i	− 0.299841i
$52$	0	0
$53$	10.5213i	1.44521i	0.691261	+	0.722605i	$0.257055\pi$
$53$	10.5213i	1.44521i	−0.691261	+	0.722605i	$0.742945\pi$
$54$	0	0
$55$	1.88781	0.254552
$56$	0	0
$57$	−1.85871	−0.246192
$58$	0	0
$59$	14.1441i	1.84141i	0.390259	+	0.920705i	$0.372386\pi$
$59$	14.1441i	1.84141i	−0.390259	+	0.920705i	$0.627614\pi$
$60$	0	0
$61$	13.2760i	1.69982i	0.526930	+	0.849909i	$0.323343\pi$
$61$	13.2760i	1.69982i	−0.526930	+	0.849909i	$0.676657\pi$
$62$	0	0
$63$	2.27105	0.286126
$64$	0	0
$65$	5.59893	0.694462
$66$	0	0
$67$	11.9258i	1.45697i	0.685062	+	0.728485i	$0.259775\pi$
$67$	11.9258i	1.45697i	−0.685062	+	0.728485i	$0.740225\pi$
$68$	0	0
$69$	3.64936i	0.439332i
$70$	0	0
$71$	4.73009	0.561358	0.280679	−	0.959802i	$-0.409440\pi$
$71$	4.73009	0.561358	0.280679	+	0.959802i	$0.409440\pi$
$72$	0	0
$73$	−6.28279	−0.735345	−0.367672	−	0.929955i	$-0.619845\pi$
$73$	−6.28279	−0.735345	−0.367672	+	0.929955i	$0.619845\pi$
$74$	0	0
$75$	− 3.92296i	− 0.452985i
$76$	0	0
$77$	5.54590i	0.632013i
$78$	0	0
$79$	−2.23894	−0.251901	−0.125950	−	0.992037i	$-0.540198\pi$
$79$	−2.23894	−0.251901	−0.125950	+	0.992037i	$0.540198\pi$
$80$	0	0
$81$	−10.1575	−1.12861
$82$	0	0
$83$	3.09516i	0.339738i	0.985467	+	0.169869i	$0.0543345\pi$
$83$	3.09516i	0.339738i	−0.985467	+	0.169869i	$0.945665\pi$
$84$	0	0
$85$	1.95826i	0.212403i
$86$	0	0
$87$	0.992524	0.106410
$88$	0	0
$89$	−6.62510	−0.702259	−0.351129	−	0.936327i	$-0.614202\pi$
$89$	−6.62510	−0.702259	−0.351129	+	0.936327i	$0.614202\pi$
$90$	0	0
$91$	16.4482i	1.72424i
$92$	0	0
$93$	9.32520i	0.966978i
$94$	0	0
$95$	1.69983	0.174399
$96$	0	0
$97$	−6.46818	−0.656744	−0.328372	−	0.944549i	$-0.606500\pi$
$97$	−6.46818	−0.656744	−0.328372	+	0.944549i	$0.606500\pi$
$98$	0	0
$99$	0.505083i	0.0507627i
$100$	0	0
$101$	− 7.77688i	− 0.773828i	−0.922116	−	0.386914i	$-0.873541\pi$
$101$	− 7.77688i	− 0.773828i	0.922116	−	0.386914i	$-0.126459\pi$
$102$	0	0
$103$	−1.85756	−0.183031	−0.0915155	−	0.995804i	$-0.529171\pi$
$103$	−1.85756	−0.183031	−0.0915155	+	0.995804i	$0.529171\pi$
$104$	0	0
$105$	−15.7773	−1.53971
$106$	0	0
$107$	− 8.77113i	− 0.847937i	−0.905677	−	0.423969i	$-0.860637\pi$
$107$	− 8.77113i	− 0.847937i	0.905677	−	0.423969i	$-0.139363\pi$
$108$	0	0
$109$	0.00686030i	0 0.000657098i	1.00000		0.000328549i	$0.000104580\pi$
$109$	0.00686030i	0 0.000657098i	−1.00000		0.000328549i	$0.999895\pi$
$110$	0	0
$111$	−0.615326	−0.0584042
$112$	0	0
$113$	−16.7751	−1.57807	−0.789034	−	0.614349i	$-0.789419\pi$
$113$	−16.7751	−1.57807	−0.789034	+	0.614349i	$0.789419\pi$
$114$	0	0
$115$	− 3.33742i	− 0.311216i
$116$	0	0
$117$	1.49799i	0.138489i
$118$	0	0
$119$	−5.75286	−0.527364
$120$	0	0
$121$	9.76659	0.887872
$122$	0	0
$123$	7.66034i	0.690710i
$124$	0	0
$125$	12.0868i	1.08107i
$126$	0	0
$127$	14.0735	1.24883	0.624413	−	0.781095i	$-0.285338\pi$
$127$	14.0735	1.24883	0.624413	+	0.781095i	$0.285338\pi$
$128$	0	0
$129$	17.2789	1.52132
$130$	0	0
$131$	− 11.8828i	− 1.03821i	−0.854711	−	0.519104i	$-0.826266\pi$
$131$	− 11.8828i	− 1.03821i	0.854711	−	0.519104i	$-0.173734\pi$
$132$	0	0
$133$	4.99365i	0.433004i
$134$	0	0
$135$	8.04154	0.692106
$136$	0	0
$137$	−5.95987	−0.509186	−0.254593	−	0.967048i	$-0.581942\pi$
$137$	−5.95987	−0.509186	−0.254593	+	0.967048i	$0.581942\pi$
$138$	0	0
$139$	9.63584i	0.817301i	0.912691	+	0.408651i	$0.134001\pi$
$139$	9.63584i	0.817301i	−0.912691	+	0.408651i	$0.865999\pi$
$140$	0	0
$141$	10.6271i	0.894965i
$142$	0	0
$143$	−3.65809	−0.305905
$144$	0	0
$145$	−0.907684	−0.0753791
$146$	0	0
$147$	− 33.3388i	− 2.74974i
$148$	0	0
$149$	9.08676i	0.744416i	0.928149	+	0.372208i	$0.121399\pi$
$149$	9.08676i	0.744416i	−0.928149	+	0.372208i	$0.878601\pi$
$150$	0	0
$151$	1.45792	0.118644	0.0593220	−	0.998239i	$-0.481106\pi$
$151$	1.45792	0.118644	0.0593220	+	0.998239i	$0.481106\pi$
$152$	0	0
$153$	−0.523932	−0.0423574
$154$	0	0
$155$	− 8.52810i	− 0.684993i
$156$	0	0
$157$	15.2445i	1.21664i	0.793690	+	0.608322i	$0.208157\pi$
$157$	15.2445i	1.21664i	−0.793690	+	0.608322i	$0.791843\pi$
$158$	0	0
$159$	19.5560	1.55089
$160$	0	0
$161$	9.80448	0.772702
$162$	0	0
$163$	− 5.49623i	− 0.430498i	−0.976559	−	0.215249i	$-0.930944\pi$
$163$	− 5.49623i	− 0.430498i	0.976559	−	0.215249i	$-0.0690564\pi$
$164$	0	0
$165$	− 3.50888i	− 0.273166i
$166$	0	0
$167$	−2.94782	−0.228109	−0.114055	−	0.993474i	$-0.536384\pi$
$167$	−2.94782	−0.228109	−0.114055	+	0.993474i	$0.536384\pi$
$168$	0	0
$169$	2.15072	0.165440
$170$	0	0
$171$	0.454788i	0.0347785i
$172$	0	0
$173$	− 20.7338i	− 1.57636i	−0.615443	−	0.788181i	$-0.711023\pi$
$173$	− 20.7338i	− 1.57636i	0.615443	−	0.788181i	$-0.288977\pi$
$174$	0	0
$175$	−10.5395	−0.796715
$176$	0	0
$177$	26.2898	1.97606
$178$	0	0
$179$	− 9.93574i	− 0.742632i	−0.928507	−	0.371316i	$-0.878907\pi$
$179$	− 9.93574i	− 0.742632i	0.928507	−	0.371316i	$-0.121093\pi$
$180$	0	0
$181$	− 14.9317i	− 1.10987i	−0.831895	−	0.554933i	$-0.812744\pi$
$181$	− 14.9317i	− 1.10987i	0.831895	−	0.554933i	$-0.187256\pi$
$182$	0	0
$183$	24.6762	1.82412
$184$	0	0
$185$	0.562729	0.0413727
$186$	0	0
$187$	− 1.27944i	− 0.0935617i
$188$	0	0
$189$	23.6240i	1.71839i
$190$	0	0
$191$	13.9130	1.00671	0.503354	−	0.864080i	$-0.332099\pi$
$191$	13.9130	1.00671	0.503354	+	0.864080i	$0.332099\pi$
$192$	0	0
$193$	−18.0432	−1.29878	−0.649388	−	0.760457i	$-0.724975\pi$
$193$	−18.0432	−1.29878	−0.649388	+	0.760457i	$0.724975\pi$
$194$	0	0
$195$	− 10.4068i	− 0.745244i
$196$	0	0
$197$	18.9494i	1.35009i	0.737778	+	0.675044i	$0.235875\pi$
$197$	18.9494i	1.35009i	−0.737778	+	0.675044i	$0.764125\pi$
$198$	0	0
$199$	23.5571	1.66992	0.834960	−	0.550311i	$-0.185491\pi$
$199$	23.5571	1.66992	0.834960	+	0.550311i	$0.185491\pi$
$200$	0	0
$201$	22.1666	1.56351
$202$	0	0
$203$	− 2.66654i	− 0.187155i
$204$	0	0
$205$	− 7.00555i	− 0.489289i
$206$	0	0
$207$	0.892926	0.0620627
$208$	0	0
$209$	−1.11059	−0.0768211
$210$	0	0
$211$	7.03487i	0.484300i	0.970239	+	0.242150i	$0.0778527\pi$
$211$	7.03487i	0.484300i	−0.970239	+	0.242150i	$0.922147\pi$
$212$	0	0
$213$	− 8.79184i	− 0.602407i
$214$	0	0
$215$	−15.8019	−1.07768
$216$	0	0
$217$	25.0533	1.70073
$218$	0	0
$219$	11.6779i	0.789117i
$220$	0	0
$221$	− 3.79460i	− 0.255252i
$222$	0	0
$223$	−26.5805	−1.77996	−0.889981	−	0.455997i	$-0.849283\pi$
$223$	−26.5805	−1.77996	−0.889981	+	0.455997i	$0.849283\pi$
$224$	0	0
$225$	−0.959871	−0.0639914
$226$	0	0
$227$	− 7.42897i	− 0.493078i	−0.969133	−	0.246539i	$-0.920707\pi$
$227$	− 7.42897i	− 0.493078i	0.969133	−	0.246539i	$-0.0792933\pi$
$228$	0	0
$229$	− 15.2748i	− 1.00938i	−0.863299	−	0.504692i	$-0.831606\pi$
$229$	− 15.2748i	− 1.00938i	0.863299	−	0.504692i	$-0.168394\pi$
$230$	0	0
$231$	10.3082	0.678229
$232$	0	0
$233$	−23.5221	−1.54098	−0.770492	−	0.637450i	$-0.779989\pi$
$233$	−23.5221	−1.54098	−0.770492	+	0.637450i	$0.779989\pi$
$234$	0	0
$235$	− 9.71873i	− 0.633980i
$236$	0	0
$237$	4.16154i	0.270321i
$238$	0	0
$239$	−14.0034	−0.905804	−0.452902	−	0.891560i	$-0.649611\pi$
$239$	−14.0034	−0.905804	−0.452902	+	0.891560i	$0.649611\pi$
$240$	0	0
$241$	−14.3117	−0.921895	−0.460948	−	0.887427i	$-0.652490\pi$
$241$	−14.3117	−0.921895	−0.460948	+	0.887427i	$0.652490\pi$
$242$	0	0
$243$	4.68746i	0.300701i
$244$	0	0
$245$	30.4890i	1.94787i
$246$	0	0
$247$	−3.29382	−0.209581
$248$	0	0
$249$	5.75300	0.364582
$250$	0	0
$251$	18.5407i	1.17028i	0.810933	+	0.585138i	$0.198960\pi$
$251$	18.5407i	1.17028i	−0.810933	+	0.585138i	$0.801040\pi$
$252$	0	0
$253$	2.18052i	0.137088i
$254$	0	0
$255$	3.63983	0.227935
$256$	0	0
$257$	−30.7165	−1.91604	−0.958022	−	0.286693i	$-0.907444\pi$
$257$	−30.7165	−1.91604	−0.958022	+	0.286693i	$0.907444\pi$
$258$	0	0
$259$	1.65315i	0.102722i
$260$	0	0
$261$	− 0.242851i	− 0.0150321i
$262$	0	0
$263$	21.6908	1.33751	0.668757	−	0.743481i	$-0.266827\pi$
$263$	21.6908	1.33751	0.668757	+	0.743481i	$0.266827\pi$
$264$	0	0
$265$	−17.8844	−1.09863
$266$	0	0
$267$	12.3141i	0.753611i
$268$	0	0
$269$	21.4204i	1.30603i	0.757347	+	0.653013i	$0.226495\pi$
$269$	21.4204i	1.30603i	−0.757347	+	0.653013i	$0.773505\pi$
$270$	0	0
$271$	14.3429	0.871267	0.435634	−	0.900124i	$-0.356524\pi$
$271$	14.3429	0.871267	0.435634	+	0.900124i	$0.356524\pi$
$272$	0	0
$273$	30.5724	1.85033
$274$	0	0
$275$	− 2.34400i	− 0.141348i
$276$	0	0
$277$	− 22.3325i	− 1.34183i	−0.741534	−	0.670915i	$-0.765901\pi$
$277$	− 22.3325i	− 1.34183i	0.741534	−	0.670915i	$-0.234099\pi$
$278$	0	0
$279$	2.28169	0.136601
$280$	0	0
$281$	28.1504	1.67931	0.839656	−	0.543118i	$-0.182757\pi$
$281$	28.1504	1.67931	0.839656	+	0.543118i	$0.182757\pi$
$282$	0	0
$283$	18.2272i	1.08350i	0.840541	+	0.541748i	$0.182237\pi$
$283$	18.2272i	1.08350i	−0.840541	+	0.541748i	$0.817763\pi$
$284$	0	0
$285$	− 3.15948i	− 0.187151i
$286$	0	0
$287$	20.5805	1.21483
$288$	0	0
$289$	−15.6728	−0.921930
$290$	0	0
$291$	12.0224i	0.704768i
$292$	0	0
$293$	3.08217i	0.180062i	0.995939	+	0.0900311i	$0.0286966\pi$
$293$	3.08217i	0.180062i	−0.995939	+	0.0900311i	$0.971303\pi$
$294$	0	0
$295$	−24.0426	−1.39981
$296$	0	0
$297$	−5.25398	−0.304867
$298$	0	0
$299$	6.46706i	0.374000i
$300$	0	0
$301$	− 46.4219i	− 2.67571i
$302$	0	0
$303$	−14.4549	−0.830414
$304$	0	0
$305$	−22.5669	−1.29218
$306$	0	0
$307$	− 21.0933i	− 1.20386i	−0.798550	−	0.601928i	$-0.794399\pi$
$307$	− 21.0933i	− 1.20386i	0.798550	−	0.601928i	$-0.205601\pi$
$308$	0	0
$309$	3.45266i	0.196415i
$310$	0	0
$311$	−2.25610	−0.127932	−0.0639658	−	0.997952i	$-0.520375\pi$
$311$	−2.25610	−0.127932	−0.0639658	+	0.997952i	$0.520375\pi$
$312$	0	0
$313$	26.0686	1.47349	0.736743	−	0.676173i	$-0.236363\pi$
$313$	26.0686	1.47349	0.736743	+	0.676173i	$0.236363\pi$
$314$	0	0
$315$	3.86040i	0.217509i
$316$	0	0
$317$	− 20.6543i	− 1.16006i	−0.814594	−	0.580032i	$-0.803040\pi$
$317$	− 20.6543i	− 1.16006i	0.814594	−	0.580032i	$-0.196960\pi$
$318$	0	0
$319$	0.593040	0.0332038
$320$	0	0
$321$	−16.3030	−0.909942
$322$	0	0
$323$	− 1.15203i	− 0.0641009i
$324$	0	0
$325$	− 6.95191i	− 0.385623i
$326$	0	0
$327$	0.0127513	0.000705148 0
$328$	0	0
$329$	28.5511	1.57407
$330$	0	0
$331$	9.74497i	0.535632i	0.963470	+	0.267816i	$0.0863019\pi$
$331$	9.74497i	0.535632i	−0.963470	+	0.267816i	$0.913698\pi$
$332$	0	0
$333$	0.150558i	0.00825053i
$334$	0	0
$335$	−20.2718	−1.10757
$336$	0	0
$337$	22.4739	1.22423	0.612115	−	0.790769i	$-0.290319\pi$
$337$	22.4739	1.22423	0.612115	+	0.790769i	$0.290319\pi$
$338$	0	0
$339$	31.1800i	1.69346i
$340$	0	0
$341$	5.57187i	0.301734i
$342$	0	0
$343$	−54.6133	−2.94884
$344$	0	0
$345$	−6.20329	−0.333974
$346$	0	0
$347$	19.8570i	1.06598i	0.846121	+	0.532990i	$0.178932\pi$
$347$	19.8570i	1.06598i	−0.846121	+	0.532990i	$0.821068\pi$
$348$	0	0
$349$	− 1.09822i	− 0.0587864i	−0.999568	−	0.0293932i	$-0.990643\pi$
$349$	− 1.09822i	− 0.0587864i	0.999568	−	0.0293932i	$-0.00935750\pi$
$350$	0	0
$351$	−15.5824	−0.831729
$352$	0	0
$353$	13.6146	0.724630	0.362315	−	0.932056i	$-0.381987\pi$
$353$	13.6146	0.724630	0.362315	+	0.932056i	$0.381987\pi$
$354$	0	0
$355$	8.04033i	0.426736i
$356$	0	0
$357$	10.6929i	0.565927i
$358$	0	0
$359$	2.11086	0.111407	0.0557036	−	0.998447i	$-0.482260\pi$
$359$	2.11086	0.111407	0.0557036	+	0.998447i	$0.482260\pi$
$360$	0	0
$361$	−1.00000	−0.0526316
$362$	0	0
$363$	− 18.1532i	− 0.952797i
$364$	0	0
$365$	− 10.6797i	− 0.558999i
$366$	0	0
$367$	18.2079	0.950446	0.475223	−	0.879865i	$-0.342367\pi$
$367$	18.2079	0.950446	0.475223	+	0.879865i	$0.342367\pi$
$368$	0	0
$369$	1.87433	0.0975738
$370$	0	0
$371$	− 52.5397i	− 2.72772i
$372$	0	0
$373$	10.4064i	0.538823i	0.963025	+	0.269411i	$0.0868292\pi$
$373$	10.4064i	0.538823i	−0.963025	+	0.269411i	$0.913171\pi$
$374$	0	0
$375$	22.4658	1.16013
$376$	0	0
$377$	1.75886	0.0905858
$378$	0	0
$379$	26.5865i	1.36566i	0.730579	+	0.682828i	$0.239250\pi$
$379$	26.5865i	1.36566i	−0.730579	+	0.682828i	$0.760750\pi$
$380$	0	0
$381$	− 26.1586i	− 1.34014i
$382$	0	0
$383$	−32.1769	−1.64416	−0.822081	−	0.569371i	$-0.807187\pi$
$383$	−32.1769	−1.64416	−0.822081	+	0.569371i	$0.807187\pi$
$384$	0	0
$385$	−9.42706	−0.480448
$386$	0	0
$387$	− 4.22779i	− 0.214911i
$388$	0	0
$389$	12.5671i	0.637178i	0.947893	+	0.318589i	$0.103209\pi$
$389$	12.5671i	0.637178i	−0.947893	+	0.318589i	$0.896791\pi$
$390$	0	0
$391$	−2.26189	−0.114389
$392$	0	0
$393$	−22.0867	−1.11413
$394$	0	0
$395$	− 3.80582i	− 0.191491i
$396$	0	0
$397$	13.9346i	0.699359i	0.936869	+	0.349680i	$0.113710\pi$
$397$	13.9346i	0.699359i	−0.936869	+	0.349680i	$0.886290\pi$
$398$	0	0
$399$	9.28173	0.464668
$400$	0	0
$401$	12.1528	0.606884	0.303442	−	0.952850i	$-0.401864\pi$
$401$	12.1528	0.606884	0.303442	+	0.952850i	$0.401864\pi$
$402$	0	0
$403$	16.5252i	0.823181i
$404$	0	0
$405$	− 17.2660i	− 0.857957i
$406$	0	0
$407$	−0.367662	−0.0182243
$408$	0	0
$409$	−19.2621	−0.952451	−0.476225	−	0.879323i	$-0.657995\pi$
$409$	−19.2621	−0.952451	−0.476225	+	0.879323i	$0.657995\pi$
$410$	0	0
$411$	11.0776i	0.546420i
$412$	0	0
$413$	− 70.6309i	− 3.47552i
$414$	0	0
$415$	−5.26124	−0.258264
$416$	0	0
$417$	17.9102	0.877066
$418$	0	0
$419$	24.7994i	1.21153i	0.795643	+	0.605766i	$0.207133\pi$
$419$	24.7994i	1.21153i	−0.795643	+	0.605766i	$0.792867\pi$
$420$	0	0
$421$	− 8.44804i	− 0.411732i	−0.978580	−	0.205866i	$-0.933999\pi$
$421$	− 8.44804i	− 0.411732i	0.978580	−	0.205866i	$-0.0660012\pi$
$422$	0	0
$423$	2.60024	0.126428
$424$	0	0
$425$	2.43147	0.117944
$426$	0	0
$427$	− 66.2957i	− 3.20827i
$428$	0	0
$429$	6.79931i	0.328274i
$430$	0	0
$431$	13.5241	0.651435	0.325717	−	0.945467i	$-0.394394\pi$
$431$	13.5241	0.651435	0.325717	+	0.945467i	$0.394394\pi$
$432$	0	0
$433$	−22.1120	−1.06263	−0.531317	−	0.847173i	$-0.678303\pi$
$433$	−22.1120	−1.06263	−0.531317	+	0.847173i	$0.678303\pi$
$434$	0	0
$435$	1.68712i	0.0808911i
$436$	0	0
$437$	1.96339i	0.0939217i
$438$	0	0
$439$	−27.3233	−1.30407	−0.652036	−	0.758188i	$-0.726085\pi$
$439$	−27.3233	−1.30407	−0.652036	+	0.758188i	$0.726085\pi$
$440$	0	0
$441$	−8.15733	−0.388444
$442$	0	0
$443$	24.5051i	1.16427i	0.813092	+	0.582136i	$0.197783\pi$
$443$	24.5051i	1.16427i	−0.813092	+	0.582136i	$0.802217\pi$
$444$	0	0
$445$	− 11.2615i	− 0.533847i
$446$	0	0
$447$	16.8896	0.798851
$448$	0	0
$449$	−24.6752	−1.16450	−0.582248	−	0.813011i	$-0.697827\pi$
$449$	−24.6752	−1.16450	−0.582248	+	0.813011i	$0.697827\pi$
$450$	0	0
$451$	4.57710i	0.215527i
$452$	0	0
$453$	− 2.70985i	− 0.127320i
$454$	0	0
$455$	−27.9591	−1.31074
$456$	0	0
$457$	0.949705	0.0444253	0.0222127	−	0.999753i	$-0.492929\pi$
$457$	0.949705	0.0444253	0.0222127	+	0.999753i	$0.492929\pi$
$458$	0	0
$459$	− 5.45005i	− 0.254386i
$460$	0	0
$461$	20.4010i	0.950171i	0.879940	+	0.475085i	$0.157583\pi$
$461$	20.4010i	0.950171i	−0.879940	+	0.475085i	$0.842417\pi$
$462$	0	0
$463$	−6.95349	−0.323156	−0.161578	−	0.986860i	$-0.551658\pi$
$463$	−6.95349	−0.323156	−0.161578	+	0.986860i	$0.551658\pi$
$464$	0	0
$465$	−15.8512	−0.735083
$466$	0	0
$467$	− 33.0618i	− 1.52992i	−0.644078	−	0.764960i	$-0.722759\pi$
$467$	− 33.0618i	− 1.52992i	0.644078	−	0.764960i	$-0.277241\pi$
$468$	0	0
$469$	− 59.5534i	− 2.74992i
$470$	0	0
$471$	28.3351	1.30561
$472$	0	0
$473$	10.3242	0.474709
$474$	0	0
$475$	− 2.11059i	− 0.0968405i
$476$	0	0
$477$	− 4.78496i	− 0.219088i
$478$	0	0
$479$	−40.8353	−1.86581	−0.932907	−	0.360117i	$-0.882737\pi$
$479$	−40.8353	−1.86581	−0.932907	+	0.360117i	$0.882737\pi$
$480$	0	0
$481$	−1.09042	−0.0497191
$482$	0	0
$483$	− 18.2237i	− 0.829205i
$484$	0	0
$485$	− 10.9948i	− 0.499247i
$486$	0	0
$487$	−2.51966	−0.114177	−0.0570884	−	0.998369i	$-0.518182\pi$
$487$	−2.51966	−0.114177	−0.0570884	+	0.998369i	$0.518182\pi$
$488$	0	0
$489$	−10.2159	−0.461978
$490$	0	0
$491$	34.1557i	1.54143i	0.637183	+	0.770713i	$0.280100\pi$
$491$	34.1557i	1.54143i	−0.637183	+	0.770713i	$0.719900\pi$
$492$	0	0
$493$	0.615171i	0.0277059i
$494$	0	0
$495$	−0.858553	−0.0385891
$496$	0	0
$497$	−23.6204	−1.05952
$498$	0	0
$499$	− 3.55199i	− 0.159009i	−0.996835	−	0.0795044i	$-0.974666\pi$
$499$	− 3.55199i	− 0.159009i	0.996835	−	0.0795044i	$-0.0253338\pi$
$500$	0	0
$501$	5.47913i	0.244790i
$502$	0	0
$503$	−17.1375	−0.764121	−0.382061	−	0.924137i	$-0.624785\pi$
$503$	−17.1375	−0.764121	−0.382061	+	0.924137i	$0.624785\pi$
$504$	0	0
$505$	13.2193	0.588253
$506$	0	0
$507$	− 3.99755i	− 0.177538i
$508$	0	0
$509$	17.8510i	0.791233i	0.918416	+	0.395616i	$0.129469\pi$
$509$	17.8510i	0.791233i	−0.918416	+	0.395616i	$0.870531\pi$
$510$	0	0
$511$	31.3741	1.38791
$512$	0	0
$513$	−4.73080	−0.208870
$514$	0	0
$515$	− 3.15753i	− 0.139137i
$516$	0	0
$517$	6.34977i	0.279263i
$518$	0	0
$519$	−38.5381	−1.69163
$520$	0	0
$521$	18.7080	0.819612	0.409806	−	0.912173i	$-0.365596\pi$
$521$	18.7080	0.819612	0.409806	+	0.912173i	$0.365596\pi$
$522$	0	0
$523$	− 25.3363i	− 1.10788i	−0.832557	−	0.553940i	$-0.813124\pi$
$523$	− 25.3363i	− 1.10788i	0.832557	−	0.553940i	$-0.186876\pi$
$524$	0	0
$525$	19.5899i	0.854974i
$526$	0	0
$527$	−5.77980	−0.251772
$528$	0	0
$529$	−19.1451	−0.832396
$530$	0	0
$531$	− 6.43259i	− 0.279151i
$532$	0	0
$533$	13.5749i	0.587996i
$534$	0	0
$535$	14.9094	0.644590
$536$	0	0
$537$	−18.4676	−0.796937
$538$	0	0
$539$	− 19.9201i	− 0.858021i
$540$	0	0
$541$	− 25.2010i	− 1.08348i	−0.840547	−	0.541738i	$-0.817767\pi$
$541$	− 25.2010i	− 1.08348i	0.840547	−	0.541738i	$-0.182233\pi$
$542$	0	0
$543$	−27.7537	−1.19102
$544$	0	0
$545$	−0.0116613	−0.000499516 0
$546$	0	0
$547$	− 38.4387i	− 1.64352i	−0.569834	−	0.821760i	$-0.692993\pi$
$547$	− 38.4387i	− 1.64352i	0.569834	−	0.821760i	$-0.307007\pi$
$548$	0	0
$549$	− 6.03776i	− 0.257686i
$550$	0	0
$551$	0.533986	0.0227486
$552$	0	0
$553$	11.1805	0.475443
$554$	0	0
$555$	− 1.04595i	− 0.0443980i
$556$	0	0
$557$	− 34.4826i	− 1.46107i	−0.682873	−	0.730537i	$-0.739270\pi$
$557$	− 34.4826i	− 1.46107i	0.682873	−	0.730537i	$-0.260730\pi$
$558$	0	0
$559$	30.6200	1.29509
$560$	0	0
$561$	−2.37810	−0.100403
$562$	0	0
$563$	19.3574i	0.815818i	0.913023	+	0.407909i	$0.133742\pi$
$563$	19.3574i	0.815818i	−0.913023	+	0.407909i	$0.866258\pi$
$564$	0	0
$565$	− 28.5148i	− 1.19962i
$566$	0	0
$567$	50.7232	2.13017
$568$	0	0
$569$	10.1856	0.427002	0.213501	−	0.976943i	$-0.431513\pi$
$569$	10.1856	0.427002	0.213501	+	0.976943i	$0.431513\pi$
$570$	0	0
$571$	− 47.1495i	− 1.97315i	−0.163320	−	0.986573i	$-0.552220\pi$
$571$	− 47.1495i	− 1.97315i	0.163320	−	0.986573i	$-0.447780\pi$
$572$	0	0
$573$	− 25.8601i	− 1.08032i
$574$	0	0
$575$	−4.14391	−0.172813
$576$	0	0
$577$	32.4237	1.34982	0.674908	−	0.737902i	$-0.264183\pi$
$577$	32.4237	1.34982	0.674908	+	0.737902i	$0.264183\pi$
$578$	0	0
$579$	33.5370i	1.39375i
$580$	0	0
$581$	− 15.4562i	− 0.641230i
$582$	0	0
$583$	11.6848	0.483936
$584$	0	0
$585$	−2.54633	−0.105278
$586$	0	0
$587$	− 5.31292i	− 0.219288i	−0.993971	−	0.109644i	$-0.965029\pi$
$587$	− 5.31292i	− 0.219288i	0.993971	−	0.109644i	$-0.0349710\pi$
$588$	0	0
$589$	5.01704i	0.206723i
$590$	0	0
$591$	35.2213	1.44881
$592$	0	0
$593$	45.1632	1.85463	0.927316	−	0.374280i	$-0.122110\pi$
$593$	45.1632	1.85463	0.927316	+	0.374280i	$0.122110\pi$
$594$	0	0
$595$	− 9.77887i	− 0.400894i
$596$	0	0
$597$	− 43.7857i	− 1.79203i
$598$	0	0
$599$	1.43343	0.0585685	0.0292842	−	0.999571i	$-0.490677\pi$
$599$	1.43343	0.0585685	0.0292842	+	0.999571i	$0.490677\pi$
$600$	0	0
$601$	−12.5520	−0.512008	−0.256004	−	0.966676i	$-0.582406\pi$
$601$	−12.5520	−0.512008	−0.256004	+	0.966676i	$0.582406\pi$
$602$	0	0
$603$	− 5.42372i	− 0.220871i
$604$	0	0
$605$	16.6015i	0.674947i
$606$	0	0
$607$	−16.1205	−0.654310	−0.327155	−	0.944971i	$-0.606090\pi$
$607$	−16.1205	−0.654310	−0.327155	+	0.944971i	$0.606090\pi$
$608$	0	0
$609$	−4.95632	−0.200840
$610$	0	0
$611$	18.8324i	0.761877i
$612$	0	0
$613$	− 24.3980i	− 0.985426i	−0.870192	−	0.492713i	$-0.836005\pi$
$613$	− 24.3980i	− 0.985426i	0.870192	−	0.492713i	$-0.163995\pi$
$614$	0	0
$615$	−13.0213	−0.525068
$616$	0	0
$617$	−23.3224	−0.938924	−0.469462	−	0.882953i	$-0.655552\pi$
$617$	−23.3224	−0.938924	−0.469462	+	0.882953i	$0.655552\pi$
$618$	0	0
$619$	− 3.63412i	− 0.146068i	−0.997329	−	0.0730338i	$-0.976732\pi$
$619$	− 3.63412i	− 0.146068i	0.997329	−	0.0730338i	$-0.0232681\pi$
$620$	0	0
$621$	9.28841i	0.372731i
$622$	0	0
$623$	33.0834	1.32546
$624$	0	0
$625$	−9.99247	−0.399699
$626$	0	0
$627$	2.06426i	0.0824386i
$628$	0	0
$629$	− 0.381382i	− 0.0152067i
$630$	0	0
$631$	−23.8545	−0.949632	−0.474816	−	0.880085i	$-0.657485\pi$
$631$	−23.8545	−0.949632	−0.474816	+	0.880085i	$0.657485\pi$
$632$	0	0
$633$	13.0758	0.519715
$634$	0	0
$635$	23.9226i	0.949339i
$636$	0	0
$637$	− 59.0799i	− 2.34083i
$638$	0	0
$639$	−2.15119	−0.0850997
$640$	0	0
$641$	4.48926	0.177315	0.0886575	−	0.996062i	$-0.471742\pi$
$641$	4.48926	0.177315	0.0886575	+	0.996062i	$0.471742\pi$
$642$	0	0
$643$	21.1677i	0.834771i	0.908730	+	0.417386i	$0.137054\pi$
$643$	21.1677i	0.834771i	−0.908730	+	0.417386i	$0.862946\pi$
$644$	0	0
$645$	29.3711i	1.15648i
$646$	0	0
$647$	−35.3649	−1.39034	−0.695169	−	0.718847i	$-0.744670\pi$
$647$	−35.3649	−1.39034	−0.695169	+	0.718847i	$0.744670\pi$
$648$	0	0
$649$	15.7083	0.616606
$650$	0	0
$651$	− 46.5668i	− 1.82510i
$652$	0	0
$653$	32.1652i	1.25872i	0.777113	+	0.629361i	$0.216683\pi$
$653$	32.1652i	1.25872i	−0.777113	+	0.629361i	$0.783317\pi$
$654$	0	0
$655$	20.1988	0.789231
$656$	0	0
$657$	2.85734	0.111475
$658$	0	0
$659$	2.75228i	0.107214i	0.998562	+	0.0536068i	$0.0170718\pi$
$659$	2.75228i	0.107214i	−0.998562	+	0.0536068i	$0.982928\pi$
$660$	0	0
$661$	4.60431i	0.179087i	0.995983	+	0.0895434i	$0.0285408\pi$
$661$	4.60431i	0.179087i	−0.995983	+	0.0895434i	$0.971459\pi$
$662$	0	0
$663$	−7.05305	−0.273918
$664$	0	0
$665$	−8.48834	−0.329164
$666$	0	0
$667$	− 1.04842i	− 0.0405951i
$668$	0	0
$669$	49.4054i	1.91012i
$670$	0	0
$671$	14.7442	0.569193
$672$	0	0
$673$	6.59710	0.254299	0.127150	−	0.991884i	$-0.459417\pi$
$673$	6.59710	0.254299	0.127150	+	0.991884i	$0.459417\pi$
$674$	0	0
$675$	− 9.98478i	− 0.384314i
$676$	0	0
$677$	− 19.3836i	− 0.744972i	−0.928038	−	0.372486i	$-0.878506\pi$
$677$	− 19.3836i	− 0.744972i	0.928038	−	0.372486i	$-0.121494\pi$
$678$	0	0
$679$	32.2998	1.23955
$680$	0	0
$681$	−13.8083	−0.529134
$682$	0	0
$683$	22.8042i	0.872577i	0.899807	+	0.436289i	$0.143707\pi$
$683$	22.8042i	0.872577i	−0.899807	+	0.436289i	$0.856293\pi$
$684$	0	0
$685$	− 10.1307i	− 0.387076i
$686$	0	0
$687$	−28.3913	−1.08320
$688$	0	0
$689$	34.6553	1.32026
$690$	0	0
$691$	4.42861i	0.168472i	0.996446	+	0.0842361i	$0.0268450\pi$
$691$	4.42861i	0.168472i	−0.996446	+	0.0842361i	$0.973155\pi$
$692$	0	0
$693$	− 2.52221i	− 0.0958107i
$694$	0	0
$695$	−16.3793	−0.621300
$696$	0	0
$697$	−4.74792	−0.179840
$698$	0	0
$699$	43.7207i	1.65367i
$700$	0	0
$701$	29.2291i	1.10397i	0.833855	+	0.551983i	$0.186129\pi$
$701$	29.2291i	1.10397i	−0.833855	+	0.551983i	$0.813871\pi$
$702$	0	0
$703$	−0.331051	−0.0124858
$704$	0	0
$705$	−18.0643	−0.680339
$706$	0	0
$707$	38.8350i	1.46054i
$708$	0	0
$709$	− 27.6770i	− 1.03943i	−0.854339	−	0.519716i	$-0.826038\pi$
$709$	− 27.6770i	− 1.03943i	0.854339	−	0.519716i	$-0.173962\pi$
$710$	0	0
$711$	1.01824	0.0381872
$712$	0	0
$713$	9.85040	0.368901
$714$	0	0
$715$	− 6.21811i	− 0.232544i
$716$	0	0
$717$	26.0282i	0.972040i
$718$	0	0
$719$	−23.7354	−0.885181	−0.442591	−	0.896724i	$-0.645940\pi$
$719$	−23.7354	−0.885181	−0.442591	+	0.896724i	$0.645940\pi$
$720$	0	0
$721$	9.27601	0.345457
$722$	0	0
$723$	26.6012i	0.989308i
$724$	0	0
$725$	1.12703i	0.0418567i
$726$	0	0
$727$	−15.5963	−0.578434	−0.289217	−	0.957264i	$-0.593395\pi$
$727$	−15.5963	−0.578434	−0.289217	+	0.957264i	$0.593395\pi$
$728$	0	0
$729$	−21.7600	−0.805925
$730$	0	0
$731$	10.7095i	0.396106i
$732$	0	0
$733$	− 33.0714i	− 1.22152i	−0.791816	−	0.610760i	$-0.790864\pi$
$733$	− 33.0714i	− 1.22152i	0.791816	−	0.610760i	$-0.209136\pi$
$734$	0	0
$735$	56.6702	2.09031
$736$	0	0
$737$	13.2447	0.487874
$738$	0	0
$739$	− 12.5146i	− 0.460356i	−0.973149	−	0.230178i	$-0.926069\pi$
$739$	− 12.5146i	− 0.460356i	0.973149	−	0.230178i	$-0.0739308\pi$
$740$	0	0
$741$	6.12225i	0.224907i
$742$	0	0
$743$	34.5320	1.26685	0.633427	−	0.773802i	$-0.281648\pi$
$743$	34.5320	1.26685	0.633427	+	0.773802i	$0.281648\pi$
$744$	0	0
$745$	−15.4459	−0.565894
$746$	0	0
$747$	− 1.40764i	− 0.0515030i
$748$	0	0
$749$	43.8000i	1.60042i
$750$	0	0
$751$	−7.00101	−0.255470	−0.127735	−	0.991808i	$-0.540771\pi$
$751$	−7.00101	−0.255470	−0.127735	+	0.991808i	$0.540771\pi$
$752$	0	0
$753$	34.4617	1.25585
$754$	0	0
$755$	2.47821i	0.0901914i
$756$	0	0
$757$	− 30.4249i	− 1.10581i	−0.833244	−	0.552905i	$-0.813519\pi$
$757$	− 30.4249i	− 1.10581i	0.833244	−	0.552905i	$-0.186481\pi$
$758$	0	0
$759$	4.05294	0.147113
$760$	0	0
$761$	−0.313350	−0.0113589	−0.00567947	−	0.999984i	$-0.501808\pi$
$761$	−0.313350	−0.0113589	−0.00567947	+	0.999984i	$0.501808\pi$
$762$	0	0
$763$	− 0.0342580i	− 0.00124022i
$764$	0	0
$765$	− 0.890593i	− 0.0321995i
$766$	0	0
$767$	46.5883	1.68221
$768$	0	0
$769$	8.26131	0.297910	0.148955	−	0.988844i	$-0.452409\pi$
$769$	8.26131	0.297910	0.148955	+	0.988844i	$0.452409\pi$
$770$	0	0
$771$	57.0930i	2.05615i
$772$	0	0
$773$	− 50.9112i	− 1.83115i	−0.402149	−	0.915574i	$-0.631737\pi$
$773$	− 50.9112i	− 1.83115i	0.402149	−	0.915574i	$-0.368263\pi$
$774$	0	0
$775$	−10.5889	−0.380365
$776$	0	0
$777$	3.07273	0.110233
$778$	0	0
$779$	4.12133i	0.147662i
$780$	0	0
$781$	− 5.25319i	− 0.187974i
$782$	0	0
$783$	2.52618	0.0902784
$784$	0	0
$785$	−25.9130	−0.924876
$786$	0	0
$787$	2.29558i	0.0818286i	0.999163	+	0.0409143i	$0.0130271\pi$
$787$	2.29558i	0.0818286i	−0.999163	+	0.0409143i	$0.986973\pi$
$788$	0	0
$789$	− 40.3169i	− 1.43532i
$790$	0	0
$791$	83.7690	2.97848
$792$	0	0
$793$	43.7288	1.55286
$794$	0	0
$795$	33.2418i	1.17896i
$796$	0	0
$797$	− 11.3651i	− 0.402574i	−0.979532	−	0.201287i	$-0.935488\pi$
$797$	− 11.3651i	− 0.402574i	0.979532	−	0.201287i	$-0.0645124\pi$
$798$	0	0
$799$	−6.58674	−0.233022
$800$	0	0
$801$	3.01301	0.106460
$802$	0	0
$803$	6.97760i	0.246234i
$804$	0	0
$805$	16.6659i	0.587397i
$806$	0	0
$807$	39.8142	1.40153
$808$	0	0
$809$	−43.6307	−1.53397	−0.766986	−	0.641664i	$-0.778244\pi$
$809$	−43.6307	−1.53397	−0.766986	+	0.641664i	$0.778244\pi$
$810$	0	0
$811$	− 24.7308i	− 0.868416i	−0.900813	−	0.434208i	$-0.857028\pi$
$811$	− 24.7308i	− 0.868416i	0.900813	−	0.434208i	$-0.142972\pi$
$812$	0	0
$813$	− 26.6592i	− 0.934978i
$814$	0	0
$815$	9.34264	0.327259
$816$	0	0
$817$	9.29618	0.325232
$818$	0	0
$819$	− 7.48045i	− 0.261388i
$820$	0	0
$821$	33.9492i	1.18484i	0.805631	+	0.592418i	$0.201827\pi$
$821$	33.9492i	1.18484i	−0.805631	+	0.592418i	$0.798173\pi$
$822$	0	0
$823$	11.3696	0.396321	0.198160	−	0.980170i	$-0.436503\pi$
$823$	11.3696	0.396321	0.198160	+	0.980170i	$0.436503\pi$
$824$	0	0
$825$	−4.35680	−0.151684
$826$	0	0
$827$	42.0705i	1.46294i	0.681876	+	0.731468i	$0.261164\pi$
$827$	42.0705i	1.46294i	−0.681876	+	0.731468i	$0.738836\pi$
$828$	0	0
$829$	35.3449i	1.22758i	0.789470	+	0.613790i	$0.210356\pi$
$829$	35.3449i	1.22758i	−0.789470	+	0.613790i	$0.789644\pi$
$830$	0	0
$831$	−41.5096	−1.43995
$832$	0	0
$833$	20.6635	0.715949
$834$	0	0
$835$	− 5.01078i	− 0.173405i
$836$	0	0
$837$	23.7346i	0.820388i
$838$	0	0
$839$	33.1160	1.14329	0.571645	−	0.820501i	$-0.306305\pi$
$839$	33.1160	1.14329	0.571645	+	0.820501i	$0.306305\pi$
$840$	0	0
$841$	28.7149	0.990168
$842$	0	0
$843$	− 52.3233i	− 1.80211i
$844$	0	0
$845$	3.65585i	0.125765i
$846$	0	0
$847$	−48.7710	−1.67579
$848$	0	0
$849$	33.8791	1.16273
$850$	0	0
$851$	0.649982i	0.0222811i
$852$	0	0
$853$	8.43149i	0.288689i	0.989528	+	0.144344i	$0.0461073\pi$
$853$	8.43149i	0.288689i	−0.989528	+	0.144344i	$0.953893\pi$
$854$	0	0
$855$	−0.773061	−0.0264381
$856$	0	0
$857$	29.8421	1.01939	0.509694	−	0.860356i	$-0.329759\pi$
$857$	29.8421	1.01939	0.509694	+	0.860356i	$0.329759\pi$
$858$	0	0
$859$	− 2.21512i	− 0.0755790i	−0.999286	−	0.0377895i	$-0.987968\pi$
$859$	− 2.21512i	− 0.0755790i	0.999286	−	0.0377895i	$-0.0120316\pi$
$860$	0	0
$861$	− 38.2531i	− 1.30366i
$862$	0	0
$863$	−25.5362	−0.869263	−0.434631	−	0.900608i	$-0.643121\pi$
$863$	−25.5362	−0.869263	−0.434631	+	0.900608i	$0.643121\pi$
$864$	0	0
$865$	35.2439	1.19833
$866$	0	0
$867$	29.1312i	0.989346i
$868$	0	0
$869$	2.48655i	0.0843503i
$870$	0	0
$871$	39.2815	1.33100
$872$	0	0
$873$	2.94165	0.0995597
$874$	0	0
$875$	− 60.3571i	− 2.04044i
$876$	0	0
$877$	− 11.9569i	− 0.403757i	−0.979411	−	0.201878i	$-0.935295\pi$
$877$	− 11.9569i	− 0.403757i	0.979411	−	0.201878i	$-0.0647046\pi$
$878$	0	0
$879$	5.72884	0.193229
$880$	0	0
$881$	−27.8077	−0.936866	−0.468433	−	0.883499i	$-0.655181\pi$
$881$	−27.8077	−0.936866	−0.468433	+	0.883499i	$0.655181\pi$
$882$	0	0
$883$	19.8570i	0.668242i	0.942530	+	0.334121i	$0.108439\pi$
$883$	19.8570i	0.668242i	−0.942530	+	0.334121i	$0.891561\pi$
$884$	0	0
$885$	44.6881i	1.50217i
$886$	0	0
$887$	40.6937	1.36636	0.683180	−	0.730250i	$-0.260597\pi$
$887$	40.6937	1.36636	0.683180	+	0.730250i	$0.260597\pi$
$888$	0	0
$889$	−70.2784	−2.35706
$890$	0	0
$891$	11.2808i	0.377923i
$892$	0	0
$893$	5.71748i	0.191328i
$894$	0	0
$895$	16.8890	0.564538
$896$	0	0
$897$	12.0204	0.401348
$898$	0	0
$899$	− 2.67903i	− 0.0893507i
$900$	0	0
$901$	12.1209i	0.403805i
$902$	0	0
$903$	−86.2846	−2.87137
$904$	0	0
$905$	25.3813	0.843704
$906$	0	0
$907$	− 15.8372i	− 0.525866i	−0.964814	−	0.262933i	$-0.915310\pi$
$907$	− 15.8372i	− 0.525866i	0.964814	−	0.262933i	$-0.0846899\pi$
$908$	0	0
$909$	3.53683i	0.117309i
$910$	0	0
$911$	−39.7130	−1.31575	−0.657875	−	0.753127i	$-0.728545\pi$
$911$	−39.7130	−1.31575	−0.657875	+	0.753127i	$0.728545\pi$
$912$	0	0
$913$	3.43746	0.113763
$914$	0	0
$915$	41.9452i	1.38667i
$916$	0	0
$917$	59.3387i	1.95954i
$918$	0	0
$919$	41.7674	1.37778	0.688889	−	0.724867i	$-0.258099\pi$
$919$	41.7674	1.37778	0.688889	+	0.724867i	$0.258099\pi$
$920$	0	0
$921$	−39.2062	−1.29189
$922$	0	0
$923$	− 15.5801i	− 0.512825i
$924$	0	0
$925$	− 0.698713i	− 0.0229735i
$926$	0	0
$927$	0.844797	0.0277468
$928$	0	0
$929$	−4.32933	−0.142041	−0.0710203	−	0.997475i	$-0.522626\pi$
$929$	−4.32933	−0.142041	−0.0710203	+	0.997475i	$0.522626\pi$
$930$	0	0
$931$	− 17.9366i	− 0.587847i
$932$	0	0
$933$	4.19343i	0.137287i
$934$	0	0
$935$	2.17482	0.0711243
$936$	0	0
$937$	−42.6962	−1.39482	−0.697412	−	0.716670i	$-0.745665\pi$
$937$	−42.6962	−1.39482	−0.697412	+	0.716670i	$0.745665\pi$
$938$	0	0
$939$	− 48.4539i	− 1.58123i
$940$	0	0
$941$	− 3.56032i	− 0.116063i	−0.998315	−	0.0580315i	$-0.981518\pi$
$941$	− 3.56032i	− 0.116063i	0.998315	−	0.0580315i	$-0.0184824\pi$
$942$	0	0
$943$	8.09178	0.263505
$944$	0	0
$945$	−40.1567	−1.30630
$946$	0	0
$947$	− 48.9588i	− 1.59095i	−0.605988	−	0.795473i	$-0.707222\pi$
$947$	− 48.9588i	− 1.59095i	0.605988	−	0.795473i	$-0.292778\pi$
$948$	0	0
$949$	20.6944i	0.671769i
$950$	0	0
$951$	−38.3903	−1.24489
$952$	0	0
$953$	7.27695	0.235723	0.117862	−	0.993030i	$-0.462396\pi$
$953$	7.27695	0.235723	0.117862	+	0.993030i	$0.462396\pi$
$954$	0	0
$955$	23.6497i	0.765285i
$956$	0	0
$957$	− 1.10229i	− 0.0356319i
$958$	0	0
$959$	29.7615	0.961050
$960$	0	0
$961$	−5.82932	−0.188043
$962$	0	0
$963$	3.98901i	0.128544i
$964$	0	0
$965$	− 30.6703i	− 0.987311i
$966$	0	0
$967$	−8.44568	−0.271595	−0.135797	−	0.990737i	$-0.543360\pi$
$967$	−8.44568	−0.271595	−0.135797	+	0.990737i	$0.543360\pi$
$968$	0	0
$969$	−2.14129	−0.0687883
$970$	0	0
$971$	6.14737i	0.197278i	0.995123	+	0.0986392i	$0.0314490\pi$
$971$	6.14737i	0.197278i	−0.995123	+	0.0986392i	$0.968551\pi$
$972$	0	0
$973$	− 48.1180i	− 1.54259i
$974$	0	0
$975$	−12.9216	−0.413821
$976$	0	0
$977$	−36.3838	−1.16402	−0.582010	−	0.813181i	$-0.697734\pi$
$977$	−36.3838	−1.16402	−0.582010	+	0.813181i	$0.697734\pi$
$978$	0	0
$979$	7.35776i	0.235155i
$980$	0	0
$981$	− 0.00311998i	0 9.96134e-5i
$982$	0	0
$983$	−34.8229	−1.11068	−0.555340	−	0.831624i	$-0.687412\pi$
$983$	−34.8229	−1.11068	−0.555340	+	0.831624i	$0.687412\pi$
$984$	0	0
$985$	−32.2107	−1.02632
$986$	0	0
$987$	− 53.0681i	− 1.68918i
$988$	0	0
$989$	− 18.2520i	− 0.580380i
$990$	0	0
$991$	−44.9897	−1.42914	−0.714572	−	0.699562i	$-0.753378\pi$
$991$	−44.9897	−1.42914	−0.714572	+	0.699562i	$0.753378\pi$
$992$	0	0
$993$	18.1130	0.574800
$994$	0	0
$995$	40.0430i	1.26945i
$996$	0	0
$997$	48.3676i	1.53182i	0.642949	+	0.765909i	$0.277711\pi$
$997$	48.3676i	1.53182i	−0.642949	+	0.765909i	$0.722289\pi$
$998$	0	0
$999$	−1.56614	−0.0495504

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2432.2.c.j.1217.6	yes	20
4.3	odd	2	inner	2432.2.c.j.1217.16	yes	20
8.3	odd	2	inner	2432.2.c.j.1217.5	✓	20
8.5	even	2	inner	2432.2.c.j.1217.15	yes	20
16.3	odd	4		4864.2.a.bs.1.4		10
16.5	even	4		4864.2.a.bs.1.3		10
16.11	odd	4		4864.2.a.bt.1.7		10
16.13	even	4		4864.2.a.bt.1.8		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2432.2.c.j.1217.5	✓	20	8.3	odd	2	inner
2432.2.c.j.1217.6	yes	20	1.1	even	1	trivial
2432.2.c.j.1217.15	yes	20	8.5	even	2	inner
2432.2.c.j.1217.16	yes	20	4.3	odd	2	inner
4864.2.a.bs.1.3		10	16.5	even	4
4864.2.a.bs.1.4		10	16.3	odd	4
4864.2.a.bt.1.7		10	16.11	odd	4
4864.2.a.bt.1.8		10	16.13	even	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:	\( N \)	\(=\)	\( 2432 = 2^{7} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2432.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.85871i q^{3} +1.69983i q^{5} -4.99365 q^{7} -0.454788 q^{9} -1.11059i q^{11} -3.29382i q^{13} +3.15948 q^{15} +1.15203 q^{17} -1.00000i q^{19} +9.28173i q^{21} -1.96339 q^{23} +2.11059 q^{25} -4.73080i q^{27} +0.533986i q^{29} -5.01704 q^{31} -2.06426 q^{33} -8.48834i q^{35} -0.331051i q^{37} -6.12225 q^{39} -4.12133 q^{41} +9.29618i q^{43} -0.773061i q^{45} -5.71748 q^{47} +17.9366 q^{49} -2.14129i q^{51} +10.5213i q^{53} +1.88781 q^{55} -1.85871 q^{57} +14.1441i q^{59} +13.2760i q^{61} +2.27105 q^{63} +5.59893 q^{65} +11.9258i q^{67} +3.64936i q^{69} +4.73009 q^{71} -6.28279 q^{73} -3.92296i q^{75} +5.54590i q^{77} -2.23894 q^{79} -10.1575 q^{81} +3.09516i q^{83} +1.95826i q^{85} +0.992524 q^{87} -6.62510 q^{89} +16.4482i q^{91} +9.32520i q^{93} +1.69983 q^{95} -6.46818 q^{97} +0.505083i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 28 q^{9} + 8 q^{17} - 20 q^{25} - 16 q^{33} + 24 q^{41} + 52 q^{49} - 8 q^{57} - 48 q^{65} - 24 q^{73} + 68 q^{81} + 40 q^{89} - 56 q^{97}+O(q^{100}) \) 20 * q - 28 * q^9 + 8 * q^17 - 20 * q^25 - 16 * q^33 + 24 * q^41 + 52 * q^49 - 8 * q^57 - 48 * q^65 - 24 * q^73 + 68 * q^81 + 40 * q^89 - 56 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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