LMFDB - Embedded newform 384.4.d.e.193.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [384,4,Mod(193,384)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(384, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("384.193");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 384 = 2^{7} \cdot 3 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	384.d (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$22.6567334422$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{13})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 7x^{2} + 9 $ x^4 + 7*x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			193.4
Root			$1.30278i$ of defining polynomial
Character	$\chi$	$=$	384.193
Dual form			384.4.d.e.193.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.00000i q^{3} +10.4222i q^{5} -6.42221 q^{7} -9.00000 q^{9} +O(q^{10})$	$q+3.00000i q^{3} +10.4222i q^{5} -6.42221 q^{7} -9.00000 q^{9} -61.6888i q^{11} -64.8444i q^{13} -31.2666 q^{15} -75.6888 q^{17} -10.3112i q^{19} -19.2666i q^{21} +156.844 q^{23} +16.3776 q^{25} -27.0000i q^{27} +53.7998i q^{29} +227.489 q^{31} +185.066 q^{33} -66.9335i q^{35} +10.3112i q^{37} +194.533 q^{39} -70.4441 q^{41} -298.311i q^{43} -93.7998i q^{45} -89.9109 q^{47} -301.755 q^{49} -227.066i q^{51} +388.333i q^{53} +642.934 q^{55} +30.9335 q^{57} -324.000i q^{59} -324.000i q^{61} +57.7998 q^{63} +675.822 q^{65} -920.266i q^{67} +470.533i q^{69} +995.156 q^{71} +362.266 q^{73} +49.1329i q^{75} +396.178i q^{77} -1098.91 q^{79} +81.0000 q^{81} -791.822i q^{83} -788.844i q^{85} -161.400 q^{87} -150.622 q^{89} +416.444i q^{91} +682.466i q^{93} +107.465 q^{95} -1879.15 q^{97} +555.199i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 32 q^{7} - 36 q^{9}+O(q^{10}) $ 4 * q + 32 * q^7 - 36 * q^9	$ 4 q + 32 q^{7} - 36 q^{9} + 48 q^{15} - 72 q^{17} + 512 q^{23} - 396 q^{25} + 160 q^{31} + 48 q^{33} + 432 q^{39} + 872 q^{41} + 448 q^{47} - 284 q^{49} + 3264 q^{55} + 816 q^{57} - 288 q^{63} + 1088 q^{65} + 4096 q^{71} - 1320 q^{73} - 992 q^{79} + 324 q^{81} + 912 q^{87} - 1064 q^{89} - 4416 q^{95} - 2440 q^{97}+O(q^{100}) $ 4 * q + 32 * q^7 - 36 * q^9 + 48 * q^15 - 72 * q^17 + 512 * q^23 - 396 * q^25 + 160 * q^31 + 48 * q^33 + 432 * q^39 + 872 * q^41 + 448 * q^47 - 284 * q^49 + 3264 * q^55 + 816 * q^57 - 288 * q^63 + 1088 * q^65 + 4096 * q^71 - 1320 * q^73 - 992 * q^79 + 324 * q^81 + 912 * q^87 - 1064 * q^89 - 4416 * q^95 - 2440 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/384\mathbb{Z}\right)^\times$.

$n$	$127$	$133$	$257$
$\chi(n)$	$1$	$-1$	$1$

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
$3$	3.00000i	0.577350i
$3$	3.00000i	0.577350i
$4$	0	0
$5$	10.4222i	0.932190i	0.884735	+	0.466095i	$0.154340\pi$
$5$	10.4222i	0.932190i	−0.884735	+	0.466095i	$0.845660\pi$
$6$	0	0
$7$	−6.42221	−0.346766	−0.173383	−	0.984854i	$-0.555470\pi$
$7$	−6.42221	−0.346766	−0.173383	+	0.984854i	$0.555470\pi$
$8$	0	0
$9$	−9.00000	−0.333333
$10$	0	0
$11$	− 61.6888i	− 1.69090i	−0.534056	−	0.845449i	$-0.679333\pi$
$11$	− 61.6888i	− 1.69090i	0.534056	−	0.845449i	$-0.320667\pi$
$12$	0	0
$13$	− 64.8444i	− 1.38343i	−0.722170	−	0.691716i	$-0.756855\pi$
$13$	− 64.8444i	− 1.38343i	0.722170	−	0.691716i	$-0.243145\pi$
$14$	0	0
$15$	−31.2666	−0.538200
$16$	0	0
$17$	−75.6888	−1.07984	−0.539919	−	0.841717i	$-0.681545\pi$
$17$	−75.6888	−1.07984	−0.539919	+	0.841717i	$0.681545\pi$
$18$	0	0
$19$	− 10.3112i	− 0.124502i	−0.998061	−	0.0622512i	$-0.980172\pi$
$19$	− 10.3112i	− 0.124502i	0.998061	−	0.0622512i	$-0.0198280\pi$
$20$	0	0
$21$	− 19.2666i	− 0.200206i
$22$	0	0
$23$	156.844	1.42193	0.710963	−	0.703229i	$-0.248259\pi$
$23$	156.844	1.42193	0.710963	+	0.703229i	$0.248259\pi$
$24$	0	0
$25$	16.3776	0.131021
$26$	0	0
$27$	− 27.0000i	− 0.192450i
$28$	0	0
$29$	53.7998i	0.344496i	0.985054	+	0.172248i	$0.0551030\pi$
$29$	53.7998i	0.344496i	−0.985054	+	0.172248i	$0.944897\pi$
$30$	0	0
$31$	227.489	1.31801	0.659003	−	0.752141i	$-0.270979\pi$
$31$	227.489	1.31801	0.659003	+	0.752141i	$0.270979\pi$
$32$	0	0
$33$	185.066	0.976240
$34$	0	0
$35$	− 66.9335i	− 0.323252i
$36$	0	0
$37$	10.3112i	0.0458148i	0.999738	+	0.0229074i	$0.00729229\pi$
$37$	10.3112i	0.0458148i	−0.999738	+	0.0229074i	$0.992708\pi$
$38$	0	0
$39$	194.533	0.798724
$40$	0	0
$41$	−70.4441	−0.268330	−0.134165	−	0.990959i	$-0.542835\pi$
$41$	−70.4441	−0.268330	−0.134165	+	0.990959i	$0.542835\pi$
$42$	0	0
$43$	− 298.311i	− 1.05795i	−0.848636	−	0.528977i	$-0.822576\pi$
$43$	− 298.311i	− 1.05795i	0.848636	−	0.528977i	$-0.177424\pi$
$44$	0	0
$45$	− 93.7998i	− 0.310730i
$46$	0	0
$47$	−89.9109	−0.279039	−0.139520	−	0.990219i	$-0.544556\pi$
$47$	−89.9109	−0.279039	−0.139520	+	0.990219i	$0.544556\pi$
$48$	0	0
$49$	−301.755	−0.879753
$50$	0	0
$51$	− 227.066i	− 0.623444i
$52$	0	0
$53$	388.333i	1.00645i	0.864157	+	0.503223i	$0.167853\pi$
$53$	388.333i	1.00645i	−0.864157	+	0.503223i	$0.832147\pi$
$54$	0	0
$55$	642.934	1.57624
$56$	0	0
$57$	30.9335	0.0718815
$58$	0	0
$59$	− 324.000i	− 0.714936i	−0.933925	−	0.357468i	$-0.883640\pi$
$59$	− 324.000i	− 0.714936i	0.933925	−	0.357468i	$-0.116360\pi$
$60$	0	0
$61$	− 324.000i	− 0.680065i	−0.940414	−	0.340032i	$-0.889562\pi$
$61$	− 324.000i	− 0.680065i	0.940414	−	0.340032i	$-0.110438\pi$
$62$	0	0
$63$	57.7998	0.115589
$64$	0	0
$65$	675.822	1.28962
$66$	0	0
$67$	− 920.266i	− 1.67804i	−0.544104	−	0.839018i	$-0.683130\pi$
$67$	− 920.266i	− 1.67804i	0.544104	−	0.839018i	$-0.316870\pi$
$68$	0	0
$69$	470.533i	0.820950i
$70$	0	0
$71$	995.156	1.66343	0.831713	−	0.555206i	$-0.187361\pi$
$71$	995.156	1.66343	0.831713	+	0.555206i	$0.187361\pi$
$72$	0	0
$73$	362.266	0.580822	0.290411	−	0.956902i	$-0.406208\pi$
$73$	362.266	0.580822	0.290411	+	0.956902i	$0.406208\pi$
$74$	0	0
$75$	49.1329i	0.0756451i
$76$	0	0
$77$	396.178i	0.586347i
$78$	0	0
$79$	−1098.91	−1.56503	−0.782513	−	0.622634i	$-0.786062\pi$
$79$	−1098.91	−1.56503	−0.782513	+	0.622634i	$0.786062\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	− 791.822i	− 1.04715i	−0.851979	−	0.523577i	$-0.824597\pi$
$83$	− 791.822i	− 1.04715i	0.851979	−	0.523577i	$-0.175403\pi$
$84$	0	0
$85$	− 788.844i	− 1.00661i
$86$	0	0
$87$	−161.400	−0.198895
$88$	0	0
$89$	−150.622	−0.179393	−0.0896963	−	0.995969i	$-0.528590\pi$
$89$	−150.622	−0.179393	−0.0896963	+	0.995969i	$0.528590\pi$
$90$	0	0
$91$	416.444i	0.479728i
$92$	0	0
$93$	682.466i	0.760951i
$94$	0	0
$95$	107.465	0.116060
$96$	0	0
$97$	−1879.15	−1.96700	−0.983501	−	0.180903i	$-0.942098\pi$
$97$	−1879.15	−1.96700	−0.983501	+	0.180903i	$0.942098\pi$
$98$	0	0
$99$	555.199i	0.563633i
$100$	0	0
$101$	− 1722.51i	− 1.69699i	−0.529202	−	0.848496i	$-0.677509\pi$
$101$	− 1722.51i	− 1.69699i	0.529202	−	0.848496i	$-0.322491\pi$
$102$	0	0
$103$	1908.82	1.82604	0.913018	−	0.407919i	$-0.133745\pi$
$103$	1908.82	1.82604	0.913018	+	0.407919i	$0.133745\pi$
$104$	0	0
$105$	200.801	0.186630
$106$	0	0
$107$	− 128.622i	− 0.116209i	−0.998310	−	0.0581046i	$-0.981494\pi$
$107$	− 128.622i	− 0.116209i	0.998310	−	0.0581046i	$-0.0185057\pi$
$108$	0	0
$109$	− 758.267i	− 0.666320i	−0.942870	−	0.333160i	$-0.891885\pi$
$109$	− 758.267i	− 0.666320i	0.942870	−	0.333160i	$-0.108115\pi$
$110$	0	0
$111$	−30.9335	−0.0264512
$112$	0	0
$113$	921.643	0.767265	0.383632	−	0.923486i	$-0.374673\pi$
$113$	921.643	0.767265	0.383632	+	0.923486i	$0.374673\pi$
$114$	0	0
$115$	1634.66i	1.32551i
$116$	0	0
$117$	583.600i	0.461144i
$118$	0	0
$119$	486.089	0.374451
$120$	0	0
$121$	−2474.51	−1.85914
$122$	0	0
$123$	− 211.332i	− 0.154920i
$124$	0	0
$125$	1473.47i	1.05433i
$126$	0	0
$127$	−1647.22	−1.15092	−0.575462	−	0.817828i	$-0.695178\pi$
$127$	−1647.22	−1.15092	−0.575462	+	0.817828i	$0.695178\pi$
$128$	0	0
$129$	894.934	0.610810
$130$	0	0
$131$	2000.27i	1.33408i	0.745023	+	0.667038i	$0.232438\pi$
$131$	2000.27i	1.33408i	−0.745023	+	0.667038i	$0.767562\pi$
$132$	0	0
$133$	66.2205i	0.0431733i
$134$	0	0
$135$	281.400	0.179400
$136$	0	0
$137$	2574.71	1.60564	0.802819	−	0.596223i	$-0.203333\pi$
$137$	2574.71	1.60564	0.802819	+	0.596223i	$0.203333\pi$
$138$	0	0
$139$	− 1331.29i	− 0.812362i	−0.913793	−	0.406181i	$-0.866860\pi$
$139$	− 1331.29i	− 0.812362i	0.913793	−	0.406181i	$-0.133140\pi$
$140$	0	0
$141$	− 269.733i	− 0.161103i
$142$	0	0
$143$	−4000.18	−2.33924
$144$	0	0
$145$	−560.713	−0.321136
$146$	0	0
$147$	− 905.266i	− 0.507926i
$148$	0	0
$149$	552.377i	0.303708i	0.988403	+	0.151854i	$0.0485243\pi$
$149$	552.377i	0.303708i	−0.988403	+	0.151854i	$0.951476\pi$
$150$	0	0
$151$	−1307.13	−0.704456	−0.352228	−	0.935914i	$-0.614576\pi$
$151$	−1307.13	−0.704456	−0.352228	+	0.935914i	$0.614576\pi$
$152$	0	0
$153$	681.199	0.359946
$154$	0	0
$155$	2370.93i	1.22863i
$156$	0	0
$157$	− 3527.11i	− 1.79295i	−0.443089	−	0.896477i	$-0.646118\pi$
$157$	− 3527.11i	− 1.79295i	0.443089	−	0.896477i	$-0.353882\pi$
$158$	0	0
$159$	−1165.00	−0.581072
$160$	0	0
$161$	−1007.29	−0.493077
$162$	0	0
$163$	926.045i	0.444991i	0.974934	+	0.222495i	$0.0714202\pi$
$163$	926.045i	0.444991i	−0.974934	+	0.222495i	$0.928580\pi$
$164$	0	0
$165$	1928.80i	0.910042i
$166$	0	0
$167$	−1393.33	−0.645624	−0.322812	−	0.946463i	$-0.604628\pi$
$167$	−1393.33	−0.645624	−0.322812	+	0.946463i	$0.604628\pi$
$168$	0	0
$169$	−2007.80	−0.913881
$170$	0	0
$171$	92.8006i	0.0415008i
$172$	0	0
$173$	3311.71i	1.45540i	0.685894	+	0.727701i	$0.259411\pi$
$173$	3311.71i	1.45540i	−0.685894	+	0.727701i	$0.740589\pi$
$174$	0	0
$175$	−105.181	−0.0454337
$176$	0	0
$177$	972.000	0.412768
$178$	0	0
$179$	− 323.287i	− 0.134992i	−0.997720	−	0.0674961i	$-0.978499\pi$
$179$	− 323.287i	− 0.134992i	0.997720	−	0.0674961i	$-0.0215010\pi$
$180$	0	0
$181$	− 3066.80i	− 1.25941i	−0.776834	−	0.629705i	$-0.783176\pi$
$181$	− 3066.80i	− 1.25941i	0.776834	−	0.629705i	$-0.216824\pi$
$182$	0	0
$183$	972.000	0.392636
$184$	0	0
$185$	−107.465	−0.0427081
$186$	0	0
$187$	4669.15i	1.82589i
$188$	0	0
$189$	173.400i	0.0667352i
$190$	0	0
$191$	−1856.44	−0.703286	−0.351643	−	0.936134i	$-0.614377\pi$
$191$	−1856.44	−0.703286	−0.351643	+	0.936134i	$0.614377\pi$
$192$	0	0
$193$	3008.84	1.12218	0.561091	−	0.827754i	$-0.310382\pi$
$193$	3008.84	1.12218	0.561091	+	0.827754i	$0.310382\pi$
$194$	0	0
$195$	2027.47i	0.744563i
$196$	0	0
$197$	909.309i	0.328861i	0.986389	+	0.164430i	$0.0525786\pi$
$197$	909.309i	0.328861i	−0.986389	+	0.164430i	$0.947421\pi$
$198$	0	0
$199$	−2780.24	−0.990383	−0.495191	−	0.868784i	$-0.664902\pi$
$199$	−2780.24	−0.990383	−0.495191	+	0.868784i	$0.664902\pi$
$200$	0	0
$201$	2760.80	0.968814
$202$	0	0
$203$	− 345.514i	− 0.119460i
$204$	0	0
$205$	− 734.183i	− 0.250134i
$206$	0	0
$207$	−1411.60	−0.473976
$208$	0	0
$209$	−636.085	−0.210521
$210$	0	0
$211$	− 385.511i	− 0.125780i	−0.998020	−	0.0628901i	$-0.979968\pi$
$211$	− 385.511i	− 0.125780i	0.998020	−	0.0628901i	$-0.0200318\pi$
$212$	0	0
$213$	2985.47i	0.960379i
$214$	0	0
$215$	3109.06	0.986215
$216$	0	0
$217$	−1460.98	−0.457040
$218$	0	0
$219$	1086.80i	0.335338i
$220$	0	0
$221$	4908.00i	1.49388i
$222$	0	0
$223$	−601.975	−0.180768	−0.0903839	−	0.995907i	$-0.528809\pi$
$223$	−601.975	−0.180768	−0.0903839	+	0.995907i	$0.528809\pi$
$224$	0	0
$225$	−147.399	−0.0436737
$226$	0	0
$227$	185.779i	0.0543199i	0.999631	+	0.0271600i	$0.00864634\pi$
$227$	185.779i	0.0543199i	−0.999631	+	0.0271600i	$0.991354\pi$
$228$	0	0
$229$	1237.11i	0.356989i	0.983941	+	0.178495i	$0.0571227\pi$
$229$	1237.11i	0.356989i	−0.983941	+	0.178495i	$0.942877\pi$
$230$	0	0
$231$	−1188.53	−0.338527
$232$	0	0
$233$	217.378	0.0611197	0.0305598	−	0.999533i	$-0.490271\pi$
$233$	217.378	0.0611197	0.0305598	+	0.999533i	$0.490271\pi$
$234$	0	0
$235$	− 937.070i	− 0.260118i
$236$	0	0
$237$	− 3296.73i	− 0.903568i
$238$	0	0
$239$	6055.99	1.63904	0.819518	−	0.573053i	$-0.194241\pi$
$239$	6055.99	1.63904	0.819518	+	0.573053i	$0.194241\pi$
$240$	0	0
$241$	−6815.69	−1.82173	−0.910865	−	0.412705i	$-0.864584\pi$
$241$	−6815.69	−1.82173	−0.910865	+	0.412705i	$0.864584\pi$
$242$	0	0
$243$	243.000i	0.0641500i
$244$	0	0
$245$	− 3144.96i	− 0.820097i
$246$	0	0
$247$	−668.622	−0.172241
$248$	0	0
$249$	2375.47	0.604574
$250$	0	0
$251$	2448.53i	0.615738i	0.951429	+	0.307869i	$0.0996158\pi$
$251$	2448.53i	0.615738i	−0.951429	+	0.307869i	$0.900384\pi$
$252$	0	0
$253$	− 9675.55i	− 2.40433i
$254$	0	0
$255$	2366.53	0.581169
$256$	0	0
$257$	2949.55	0.715907	0.357953	−	0.933739i	$-0.383475\pi$
$257$	2949.55	0.715907	0.357953	+	0.933739i	$0.383475\pi$
$258$	0	0
$259$	− 66.2205i	− 0.0158870i
$260$	0	0
$261$	− 484.199i	− 0.114832i
$262$	0	0
$263$	−4831.64	−1.13282	−0.566410	−	0.824124i	$-0.691668\pi$
$263$	−4831.64	−1.13282	−0.566410	+	0.824124i	$0.691668\pi$
$264$	0	0
$265$	−4047.29	−0.938199
$266$	0	0
$267$	− 451.867i	− 0.103572i
$268$	0	0
$269$	− 1093.84i	− 0.247928i	−0.992287	−	0.123964i	$-0.960439\pi$
$269$	− 1093.84i	− 0.247928i	0.992287	−	0.123964i	$-0.0395607\pi$
$270$	0	0
$271$	3079.80	0.690349	0.345175	−	0.938539i	$-0.387820\pi$
$271$	3079.80	0.690349	0.345175	+	0.938539i	$0.387820\pi$
$272$	0	0
$273$	−1249.33	−0.276971
$274$	0	0
$275$	− 1010.32i	− 0.221543i
$276$	0	0
$277$	5600.75i	1.21486i	0.794373	+	0.607430i	$0.207800\pi$
$277$	5600.75i	1.21486i	−0.794373	+	0.607430i	$0.792200\pi$
$278$	0	0
$279$	−2047.40	−0.439335
$280$	0	0
$281$	−203.426	−0.0431864	−0.0215932	−	0.999767i	$-0.506874\pi$
$281$	−203.426	−0.0431864	−0.0215932	+	0.999767i	$0.506874\pi$
$282$	0	0
$283$	− 2214.84i	− 0.465225i	−0.972570	−	0.232612i	$-0.925273\pi$
$283$	− 2214.84i	− 0.465225i	0.972570	−	0.232612i	$-0.0747273\pi$
$284$	0	0
$285$	322.396i	0.0670073i
$286$	0	0
$287$	452.406	0.0930478
$288$	0	0
$289$	815.798	0.166049
$290$	0	0
$291$	− 5637.46i	− 1.13565i
$292$	0	0
$293$	− 5812.37i	− 1.15892i	−0.815002	−	0.579458i	$-0.803264\pi$
$293$	− 5812.37i	− 1.15892i	0.815002	−	0.579458i	$-0.196736\pi$
$294$	0	0
$295$	3376.79	0.666456
$296$	0	0
$297$	−1665.60	−0.325413
$298$	0	0
$299$	− 10170.5i	− 1.96714i
$300$	0	0
$301$	1915.82i	0.366863i
$302$	0	0
$303$	5167.53	0.979758
$304$	0	0
$305$	3376.79	0.633950
$306$	0	0
$307$	− 7337.33i	− 1.36405i	−0.731329	−	0.682025i	$-0.761099\pi$
$307$	− 7337.33i	− 1.36405i	0.731329	−	0.682025i	$-0.238901\pi$
$308$	0	0
$309$	5726.46i	1.05426i
$310$	0	0
$311$	−6575.91	−1.19899	−0.599494	−	0.800379i	$-0.704632\pi$
$311$	−6575.91	−1.19899	−0.599494	+	0.800379i	$0.704632\pi$
$312$	0	0
$313$	1556.67	0.281113	0.140556	−	0.990073i	$-0.455111\pi$
$313$	1556.67	0.281113	0.140556	+	0.990073i	$0.455111\pi$
$314$	0	0
$315$	602.402i	0.107751i
$316$	0	0
$317$	9457.27i	1.67562i	0.545959	+	0.837812i	$0.316166\pi$
$317$	9457.27i	1.67562i	−0.545959	+	0.837812i	$0.683834\pi$
$318$	0	0
$319$	3318.85	0.582507
$320$	0	0
$321$	385.867	0.0670935
$322$	0	0
$323$	780.441i	0.134442i
$324$	0	0
$325$	− 1062.00i	− 0.181259i
$326$	0	0
$327$	2274.80	0.384700
$328$	0	0
$329$	577.426	0.0967615
$330$	0	0
$331$	− 10907.6i	− 1.81129i	−0.424032	−	0.905647i	$-0.639386\pi$
$331$	− 10907.6i	− 1.81129i	0.424032	−	0.905647i	$-0.360614\pi$
$332$	0	0
$333$	− 92.8006i	− 0.0152716i
$334$	0	0
$335$	9591.20	1.56425
$336$	0	0
$337$	4256.22	0.687986	0.343993	−	0.938972i	$-0.388220\pi$
$337$	4256.22	0.687986	0.343993	+	0.938972i	$0.388220\pi$
$338$	0	0
$339$	2764.93i	0.442981i
$340$	0	0
$341$	− 14033.5i	− 2.22861i
$342$	0	0
$343$	4140.75	0.651835
$344$	0	0
$345$	−4903.99	−0.765282
$346$	0	0
$347$	4217.07i	0.652403i	0.945300	+	0.326202i	$0.105769\pi$
$347$	4217.07i	0.652403i	−0.945300	+	0.326202i	$0.894231\pi$
$348$	0	0
$349$	2986.57i	0.458073i	0.973418	+	0.229037i	$0.0735576\pi$
$349$	2986.57i	0.458073i	−0.973418	+	0.229037i	$0.926442\pi$
$350$	0	0
$351$	−1750.80	−0.266241
$352$	0	0
$353$	−8231.42	−1.24112	−0.620558	−	0.784160i	$-0.713094\pi$
$353$	−8231.42	−1.24112	−0.620558	+	0.784160i	$0.713094\pi$
$354$	0	0
$355$	10371.7i	1.55063i
$356$	0	0
$357$	1458.27i	0.216190i
$358$	0	0
$359$	5857.19	0.861088	0.430544	−	0.902569i	$-0.358322\pi$
$359$	5857.19	0.861088	0.430544	+	0.902569i	$0.358322\pi$
$360$	0	0
$361$	6752.68	0.984499
$362$	0	0
$363$	− 7423.53i	− 1.07337i
$364$	0	0
$365$	3775.61i	0.541437i
$366$	0	0
$367$	10715.7	1.52412	0.762061	−	0.647505i	$-0.224188\pi$
$367$	10715.7	1.52412	0.762061	+	0.647505i	$0.224188\pi$
$368$	0	0
$369$	633.997	0.0894433
$370$	0	0
$371$	− 2493.95i	− 0.349002i
$372$	0	0
$373$	1817.52i	0.252299i	0.992011	+	0.126149i	$0.0402619\pi$
$373$	1817.52i	0.252299i	−0.992011	+	0.126149i	$0.959738\pi$
$374$	0	0
$375$	−4420.40	−0.608716
$376$	0	0
$377$	3488.62	0.476586
$378$	0	0
$379$	9789.68i	1.32681i	0.748259	+	0.663407i	$0.230890\pi$
$379$	9789.68i	1.32681i	−0.748259	+	0.663407i	$0.769110\pi$
$380$	0	0
$381$	− 4941.67i	− 0.664486i
$382$	0	0
$383$	11502.3	1.53457	0.767285	−	0.641306i	$-0.221607\pi$
$383$	11502.3	1.53457	0.767285	+	0.641306i	$0.221607\pi$
$384$	0	0
$385$	−4129.05	−0.546587
$386$	0	0
$387$	2684.80i	0.352651i
$388$	0	0
$389$	6.06261i	0 0.000790197i	1.00000		0.000395098i	$0.000125764\pi$
$389$	6.06261i	0 0.000790197i	−1.00000		0.000395098i	$0.999874\pi$
$390$	0	0
$391$	−11871.4	−1.53545
$392$	0	0
$393$	−6000.80	−0.770230
$394$	0	0
$395$	− 11453.1i	− 1.45890i
$396$	0	0
$397$	5982.31i	0.756281i	0.925748	+	0.378141i	$0.123436\pi$
$397$	5982.31i	0.756281i	−0.925748	+	0.378141i	$0.876564\pi$
$398$	0	0
$399$	−198.662	−0.0249261
$400$	0	0
$401$	10443.6	1.30057	0.650284	−	0.759691i	$-0.274650\pi$
$401$	10443.6	1.30057	0.650284	+	0.759691i	$0.274650\pi$
$402$	0	0
$403$	− 14751.4i	− 1.82337i
$404$	0	0
$405$	844.199i	0.103577i
$406$	0	0
$407$	636.085	0.0774682
$408$	0	0
$409$	8141.01	0.984223	0.492112	−	0.870532i	$-0.336225\pi$
$409$	8141.01	0.984223	0.492112	+	0.870532i	$0.336225\pi$
$410$	0	0
$411$	7724.13i	0.927015i
$412$	0	0
$413$	2080.79i	0.247916i
$414$	0	0
$415$	8252.53	0.976146
$416$	0	0
$417$	3993.86	0.469017
$418$	0	0
$419$	− 8716.96i	− 1.01635i	−0.861253	−	0.508176i	$-0.830320\pi$
$419$	− 8716.96i	− 1.01635i	0.861253	−	0.508176i	$-0.169680\pi$
$420$	0	0
$421$	13437.5i	1.55560i	0.628514	+	0.777798i	$0.283663\pi$
$421$	13437.5i	1.55560i	−0.628514	+	0.777798i	$0.716337\pi$
$422$	0	0
$423$	809.198	0.0930131
$424$	0	0
$425$	−1239.60	−0.141482
$426$	0	0
$427$	2080.79i	0.235824i
$428$	0	0
$429$	− 12000.5i	− 1.35056i
$430$	0	0
$431$	−7343.86	−0.820745	−0.410373	−	0.911918i	$-0.634601\pi$
$431$	−7343.86	−0.820745	−0.410373	+	0.911918i	$0.634601\pi$
$432$	0	0
$433$	4490.80	0.498416	0.249208	−	0.968450i	$-0.419830\pi$
$433$	4490.80	0.498416	0.249208	+	0.968450i	$0.419830\pi$
$434$	0	0
$435$	− 1682.14i	− 0.185408i
$436$	0	0
$437$	− 1617.25i	− 0.177033i
$438$	0	0
$439$	−9437.93	−1.02608	−0.513038	−	0.858366i	$-0.671480\pi$
$439$	−9437.93	−1.02608	−0.513038	+	0.858366i	$0.671480\pi$
$440$	0	0
$441$	2715.80	0.293251
$442$	0	0
$443$	12668.3i	1.35866i	0.733832	+	0.679331i	$0.237730\pi$
$443$	12668.3i	1.35866i	−0.733832	+	0.679331i	$0.762270\pi$
$444$	0	0
$445$	− 1569.82i	− 0.167228i
$446$	0	0
$447$	−1657.13	−0.175346
$448$	0	0
$449$	−13052.0	−1.37186	−0.685929	−	0.727669i	$-0.740604\pi$
$449$	−13052.0	−1.37186	−0.685929	+	0.727669i	$0.740604\pi$
$450$	0	0
$451$	4345.61i	0.453718i
$452$	0	0
$453$	− 3921.40i	− 0.406718i
$454$	0	0
$455$	−4340.27	−0.447197
$456$	0	0
$457$	1313.64	0.134463	0.0672316	−	0.997737i	$-0.478583\pi$
$457$	1313.64	0.134463	0.0672316	+	0.997737i	$0.478583\pi$
$458$	0	0
$459$	2043.60i	0.207815i
$460$	0	0
$461$	627.883i	0.0634347i	0.999497	+	0.0317174i	$0.0100976\pi$
$461$	627.883i	0.0634347i	−0.999497	+	0.0317174i	$0.989902\pi$
$462$	0	0
$463$	−7315.03	−0.734251	−0.367126	−	0.930171i	$-0.619658\pi$
$463$	−7315.03	−0.734251	−0.367126	+	0.930171i	$0.619658\pi$
$464$	0	0
$465$	−7112.80	−0.709351
$466$	0	0
$467$	− 759.997i	− 0.0753072i	−0.999291	−	0.0376536i	$-0.988012\pi$
$467$	− 759.997i	− 0.0753072i	0.999291	−	0.0376536i	$-0.0119883\pi$
$468$	0	0
$469$	5910.14i	0.581886i
$470$	0	0
$471$	10581.3	1.03516
$472$	0	0
$473$	−18402.5	−1.78889
$474$	0	0
$475$	− 168.873i	− 0.0163125i
$476$	0	0
$477$	− 3495.00i	− 0.335482i
$478$	0	0
$479$	−7403.51	−0.706211	−0.353106	−	0.935583i	$-0.614874\pi$
$479$	−7403.51	−0.706211	−0.353106	+	0.935583i	$0.614874\pi$
$480$	0	0
$481$	668.622	0.0633816
$482$	0	0
$483$	− 3021.86i	− 0.284678i
$484$	0	0
$485$	− 19584.9i	− 1.83362i
$486$	0	0
$487$	−3488.11	−0.324561	−0.162281	−	0.986745i	$-0.551885\pi$
$487$	−3488.11	−0.324561	−0.162281	+	0.986745i	$0.551885\pi$
$488$	0	0
$489$	−2778.14	−0.256915
$490$	0	0
$491$	6575.73i	0.604396i	0.953245	+	0.302198i	$0.0977204\pi$
$491$	6575.73i	0.604396i	−0.953245	+	0.302198i	$0.902280\pi$
$492$	0	0
$493$	− 4072.05i	− 0.372000i
$494$	0	0
$495$	−5786.40	−0.525413
$496$	0	0
$497$	−6391.09	−0.576820
$498$	0	0
$499$	5187.82i	0.465408i	0.972548	+	0.232704i	$0.0747574\pi$
$499$	5187.82i	0.465408i	−0.972548	+	0.232704i	$0.925243\pi$
$500$	0	0
$501$	− 4180.00i	− 0.372751i
$502$	0	0
$503$	−7248.38	−0.642524	−0.321262	−	0.946990i	$-0.604107\pi$
$503$	−7248.38	−0.642524	−0.321262	+	0.946990i	$0.604107\pi$
$504$	0	0
$505$	17952.4	1.58192
$506$	0	0
$507$	− 6023.39i	− 0.527630i
$508$	0	0
$509$	6613.44i	0.575905i	0.957645	+	0.287952i	$0.0929745\pi$
$509$	6613.44i	0.575905i	−0.957645	+	0.287952i	$0.907026\pi$
$510$	0	0
$511$	−2326.55	−0.201410
$512$	0	0
$513$	−278.402	−0.0239605
$514$	0	0
$515$	19894.1i	1.70221i
$516$	0	0
$517$	5546.50i	0.471827i
$518$	0	0
$519$	−9935.13	−0.840277
$520$	0	0
$521$	−20761.3	−1.74581	−0.872906	−	0.487888i	$-0.837767\pi$
$521$	−20761.3	−1.74581	−0.872906	+	0.487888i	$0.837767\pi$
$522$	0	0
$523$	− 13494.2i	− 1.12822i	−0.825698	−	0.564112i	$-0.809219\pi$
$523$	− 13494.2i	− 1.12822i	0.825698	−	0.564112i	$-0.190781\pi$
$524$	0	0
$525$	− 315.542i	− 0.0262312i
$526$	0	0
$527$	−17218.3	−1.42323
$528$	0	0
$529$	12433.2	1.02188
$530$	0	0
$531$	2916.00i	0.238312i
$532$	0	0
$533$	4567.91i	0.371216i
$534$	0	0
$535$	1340.53	0.108329
$536$	0	0
$537$	969.861	0.0779378
$538$	0	0
$539$	18614.9i	1.48757i
$540$	0	0
$541$	2708.58i	0.215251i	0.994191	+	0.107626i	$0.0343248\pi$
$541$	2708.58i	0.215251i	−0.994191	+	0.107626i	$0.965675\pi$
$542$	0	0
$543$	9200.39	0.727121
$544$	0	0
$545$	7902.82	0.621137
$546$	0	0
$547$	15783.5i	1.23373i	0.787068	+	0.616866i	$0.211598\pi$
$547$	15783.5i	1.23373i	−0.787068	+	0.616866i	$0.788402\pi$
$548$	0	0
$549$	2916.00i	0.226688i
$550$	0	0
$551$	554.740	0.0428906
$552$	0	0
$553$	7057.43	0.542699
$554$	0	0
$555$	− 322.396i	− 0.0246575i
$556$	0	0
$557$	1892.77i	0.143984i	0.997405	+	0.0719922i	$0.0229357\pi$
$557$	1892.77i	0.143984i	−0.997405	+	0.0719922i	$0.977064\pi$
$558$	0	0
$559$	−19343.8	−1.46361
$560$	0	0
$561$	−14007.5	−1.05418
$562$	0	0
$563$	3876.26i	0.290169i	0.989419	+	0.145084i	$0.0463454\pi$
$563$	3876.26i	0.290169i	−0.989419	+	0.145084i	$0.953655\pi$
$564$	0	0
$565$	9605.56i	0.715237i
$566$	0	0
$567$	−520.199	−0.0385296
$568$	0	0
$569$	14900.9	1.09785	0.548927	−	0.835870i	$-0.315037\pi$
$569$	14900.9	1.09785	0.548927	+	0.835870i	$0.315037\pi$
$570$	0	0
$571$	24926.0i	1.82683i	0.407032	+	0.913414i	$0.366564\pi$
$571$	24926.0i	1.82683i	−0.407032	+	0.913414i	$0.633436\pi$
$572$	0	0
$573$	− 5569.33i	− 0.406042i
$574$	0	0
$575$	2568.74	0.186302
$576$	0	0
$577$	21022.4	1.51677	0.758384	−	0.651808i	$-0.225989\pi$
$577$	21022.4	1.51677	0.758384	+	0.651808i	$0.225989\pi$
$578$	0	0
$579$	9026.52i	0.647892i
$580$	0	0
$581$	5085.24i	0.363118i
$582$	0	0
$583$	23955.8	1.70180
$584$	0	0
$585$	−6082.40	−0.429874
$586$	0	0
$587$	− 13847.0i	− 0.973641i	−0.873502	−	0.486821i	$-0.838157\pi$
$587$	− 13847.0i	− 0.973641i	0.873502	−	0.486821i	$-0.161843\pi$
$588$	0	0
$589$	− 2345.68i	− 0.164095i
$590$	0	0
$591$	−2727.93	−0.189868
$592$	0	0
$593$	−4113.01	−0.284825	−0.142413	−	0.989807i	$-0.545486\pi$
$593$	−4113.01	−0.284825	−0.142413	+	0.989807i	$0.545486\pi$
$594$	0	0
$595$	5066.12i	0.349060i
$596$	0	0
$597$	− 8340.73i	− 0.571798i
$598$	0	0
$599$	4863.60	0.331755	0.165878	−	0.986146i	$-0.446954\pi$
$599$	4863.60	0.331755	0.165878	+	0.986146i	$0.446954\pi$
$600$	0	0
$601$	−6827.76	−0.463411	−0.231706	−	0.972786i	$-0.574431\pi$
$601$	−6827.76	−0.463411	−0.231706	+	0.972786i	$0.574431\pi$
$602$	0	0
$603$	8282.39i	0.559345i
$604$	0	0
$605$	− 25789.9i	− 1.73307i
$606$	0	0
$607$	−18178.3	−1.21554	−0.607770	−	0.794113i	$-0.707936\pi$
$607$	−18178.3	−1.21554	−0.607770	+	0.794113i	$0.707936\pi$
$608$	0	0
$609$	1036.54	0.0689700
$610$	0	0
$611$	5830.22i	0.386032i
$612$	0	0
$613$	15687.2i	1.03360i	0.856105	+	0.516802i	$0.172878\pi$
$613$	15687.2i	1.03360i	−0.856105	+	0.516802i	$0.827122\pi$
$614$	0	0
$615$	2202.55	0.144415
$616$	0	0
$617$	16420.2	1.07140	0.535700	−	0.844409i	$-0.320048\pi$
$617$	16420.2	1.07140	0.535700	+	0.844409i	$0.320048\pi$
$618$	0	0
$619$	5517.78i	0.358285i	0.983823	+	0.179142i	$0.0573322\pi$
$619$	5517.78i	0.358285i	−0.983823	+	0.179142i	$0.942668\pi$
$620$	0	0
$621$	− 4234.80i	− 0.273650i
$622$	0	0
$623$	967.328	0.0622073
$624$	0	0
$625$	−13309.6	−0.851812
$626$	0	0
$627$	− 1908.25i	− 0.121544i
$628$	0	0
$629$	− 780.441i	− 0.0494725i
$630$	0	0
$631$	1559.17	0.0983670	0.0491835	−	0.998790i	$-0.484338\pi$
$631$	1559.17	0.0983670	0.0491835	+	0.998790i	$0.484338\pi$
$632$	0	0
$633$	1156.53	0.0726193
$634$	0	0
$635$	− 17167.7i	− 1.07288i
$636$	0	0
$637$	19567.1i	1.21708i
$638$	0	0
$639$	−8956.40	−0.554475
$640$	0	0
$641$	−15188.6	−0.935900	−0.467950	−	0.883755i	$-0.655007\pi$
$641$	−15188.6	−0.935900	−0.467950	+	0.883755i	$0.655007\pi$
$642$	0	0
$643$	− 16666.6i	− 1.02219i	−0.859524	−	0.511095i	$-0.829240\pi$
$643$	− 16666.6i	− 1.02219i	0.859524	−	0.511095i	$-0.170760\pi$
$644$	0	0
$645$	9327.18i	0.569391i
$646$	0	0
$647$	3038.27	0.184616	0.0923081	−	0.995730i	$-0.470576\pi$
$647$	3038.27	0.184616	0.0923081	+	0.995730i	$0.470576\pi$
$648$	0	0
$649$	−19987.2	−1.20888
$650$	0	0
$651$	− 4382.94i	− 0.263872i
$652$	0	0
$653$	9078.99i	0.544086i	0.962285	+	0.272043i	$0.0876994\pi$
$653$	9078.99i	0.544086i	−0.962285	+	0.272043i	$0.912301\pi$
$654$	0	0
$655$	−20847.2	−1.24361
$656$	0	0
$657$	−3260.39	−0.193607
$658$	0	0
$659$	− 16892.0i	− 0.998511i	−0.866455	−	0.499255i	$-0.833607\pi$
$659$	− 16892.0i	− 0.998511i	0.866455	−	0.499255i	$-0.166393\pi$
$660$	0	0
$661$	5064.16i	0.297992i	0.988838	+	0.148996i	$0.0476042\pi$
$661$	5064.16i	0.297992i	−0.988838	+	0.148996i	$0.952396\pi$
$662$	0	0
$663$	−14724.0	−0.862492
$664$	0	0
$665$	−690.164	−0.0402457
$666$	0	0
$667$	8438.21i	0.489848i
$668$	0	0
$669$	− 1805.93i	− 0.104366i
$670$	0	0
$671$	−19987.2	−1.14992
$672$	0	0
$673$	28625.9	1.63959	0.819797	−	0.572654i	$-0.194086\pi$
$673$	28625.9	1.63959	0.819797	+	0.572654i	$0.194086\pi$
$674$	0	0
$675$	− 442.196i	− 0.0252150i
$676$	0	0
$677$	− 16061.2i	− 0.911791i	−0.890033	−	0.455895i	$-0.849319\pi$
$677$	− 16061.2i	− 0.911791i	0.890033	−	0.455895i	$-0.150681\pi$
$678$	0	0
$679$	12068.3	0.682090
$680$	0	0
$681$	−557.338	−0.0313616
$682$	0	0
$683$	− 7868.09i	− 0.440796i	−0.975410	−	0.220398i	$-0.929264\pi$
$683$	− 7868.09i	− 0.440796i	0.975410	−	0.220398i	$-0.0707357\pi$
$684$	0	0
$685$	26834.2i	1.49676i
$686$	0	0
$687$	−3711.33	−0.206108
$688$	0	0
$689$	25181.2	1.39235
$690$	0	0
$691$	16886.2i	0.929641i	0.885405	+	0.464820i	$0.153881\pi$
$691$	16886.2i	0.929641i	−0.885405	+	0.464820i	$0.846119\pi$
$692$	0	0
$693$	− 3565.60i	− 0.195449i
$694$	0	0
$695$	13874.9	0.757276
$696$	0	0
$697$	5331.83	0.289753
$698$	0	0
$699$	652.133i	0.0352875i
$700$	0	0
$701$	− 18293.6i	− 0.985647i	−0.870129	−	0.492824i	$-0.835965\pi$
$701$	− 18293.6i	− 0.985647i	0.870129	−	0.492824i	$-0.164035\pi$
$702$	0	0
$703$	106.320	0.00570406
$704$	0	0
$705$	2811.21	0.150179
$706$	0	0
$707$	11062.3i	0.588460i
$708$	0	0
$709$	− 21555.6i	− 1.14180i	−0.821019	−	0.570900i	$-0.806594\pi$
$709$	− 21555.6i	− 1.14180i	0.821019	−	0.570900i	$-0.193406\pi$
$710$	0	0
$711$	9890.19	0.521675
$712$	0	0
$713$	35680.3	1.87411
$714$	0	0
$715$	− 41690.6i	− 2.18062i
$716$	0	0
$717$	18168.0i	0.946298i
$718$	0	0
$719$	9059.31	0.469896	0.234948	−	0.972008i	$-0.424508\pi$
$719$	9059.31	0.469896	0.234948	+	0.972008i	$0.424508\pi$
$720$	0	0
$721$	−12258.8	−0.633208
$722$	0	0
$723$	− 20447.1i	− 1.05178i
$724$	0	0
$725$	881.115i	0.0451362i
$726$	0	0
$727$	7074.83	0.360923	0.180462	−	0.983582i	$-0.442241\pi$
$727$	7074.83	0.360923	0.180462	+	0.983582i	$0.442241\pi$
$728$	0	0
$729$	−729.000	−0.0370370
$730$	0	0
$731$	22578.8i	1.14242i
$732$	0	0
$733$	17095.9i	0.861459i	0.902481	+	0.430730i	$0.141744\pi$
$733$	17095.9i	0.861459i	−0.902481	+	0.430730i	$0.858256\pi$
$734$	0	0
$735$	9434.87	0.473483
$736$	0	0
$737$	−56770.1	−2.83739
$738$	0	0
$739$	− 28157.4i	− 1.40161i	−0.713355	−	0.700803i	$-0.752825\pi$
$739$	− 28157.4i	− 1.40161i	0.713355	−	0.700803i	$-0.247175\pi$
$740$	0	0
$741$	− 2005.87i	− 0.0994431i
$742$	0	0
$743$	34033.2	1.68043	0.840214	−	0.542255i	$-0.182429\pi$
$743$	34033.2	1.68043	0.840214	+	0.542255i	$0.182429\pi$
$744$	0	0
$745$	−5756.99	−0.283114
$746$	0	0
$747$	7126.40i	0.349051i
$748$	0	0
$749$	826.039i	0.0402975i
$750$	0	0
$751$	34356.8	1.66937	0.834685	−	0.550728i	$-0.185650\pi$
$751$	34356.8	1.66937	0.834685	+	0.550728i	$0.185650\pi$
$752$	0	0
$753$	−7345.60	−0.355496
$754$	0	0
$755$	− 13623.2i	− 0.656687i
$756$	0	0
$757$	− 29464.6i	− 1.41467i	−0.706876	−	0.707337i	$-0.749896\pi$
$757$	− 29464.6i	− 1.41467i	0.706876	−	0.707337i	$-0.250104\pi$
$758$	0	0
$759$	29026.6	1.38814
$760$	0	0
$761$	−27284.5	−1.29969	−0.649843	−	0.760069i	$-0.725165\pi$
$761$	−27284.5	−1.29969	−0.649843	+	0.760069i	$0.725165\pi$
$762$	0	0
$763$	4869.75i	0.231057i
$764$	0	0
$765$	7099.60i	0.335538i
$766$	0	0
$767$	−21009.6	−0.989064
$768$	0	0
$769$	2320.13	0.108798	0.0543992	−	0.998519i	$-0.482676\pi$
$769$	2320.13	0.108798	0.0543992	+	0.998519i	$0.482676\pi$
$770$	0	0
$771$	8848.66i	0.413329i
$772$	0	0
$773$	4217.83i	0.196254i	0.995174	+	0.0981272i	$0.0312852\pi$
$773$	4217.83i	0.196254i	−0.995174	+	0.0981272i	$0.968715\pi$
$774$	0	0
$775$	3725.73	0.172687
$776$	0	0
$777$	198.662	0.00917238
$778$	0	0
$779$	726.362i	0.0334077i
$780$	0	0
$781$	− 61390.0i	− 2.81268i
$782$	0	0
$783$	1452.60	0.0662983
$784$	0	0
$785$	36760.3	1.67138
$786$	0	0
$787$	− 11436.3i	− 0.517991i	−0.965879	−	0.258995i	$-0.916609\pi$
$787$	− 11436.3i	− 0.517991i	0.965879	−	0.258995i	$-0.0833914\pi$
$788$	0	0
$789$	− 14494.9i	− 0.654034i
$790$	0	0
$791$	−5918.98	−0.266062
$792$	0	0
$793$	−21009.6	−0.940823
$794$	0	0
$795$	− 12141.9i	− 0.541670i
$796$	0	0
$797$	14102.3i	0.626761i	0.949628	+	0.313381i	$0.101462\pi$
$797$	14102.3i	0.626761i	−0.949628	+	0.313381i	$0.898538\pi$
$798$	0	0
$799$	6805.25	0.301317
$800$	0	0
$801$	1355.60	0.0597975
$802$	0	0
$803$	− 22347.8i	− 0.982111i
$804$	0	0
$805$	− 10498.2i	− 0.459641i
$806$	0	0
$807$	3281.52	0.143141
$808$	0	0
$809$	−18791.3	−0.816647	−0.408323	−	0.912837i	$-0.633886\pi$
$809$	−18791.3	−0.816647	−0.408323	+	0.912837i	$0.633886\pi$
$810$	0	0
$811$	14452.6i	0.625771i	0.949791	+	0.312886i	$0.101296\pi$
$811$	14452.6i	0.625771i	−0.949791	+	0.312886i	$0.898704\pi$
$812$	0	0
$813$	9239.40i	0.398573i
$814$	0	0
$815$	−9651.43	−0.414816
$816$	0	0
$817$	−3075.94	−0.131718
$818$	0	0
$819$	− 3748.00i	− 0.159909i
$820$	0	0
$821$	27786.1i	1.18117i	0.806975	+	0.590585i	$0.201103\pi$
$821$	27786.1i	1.18117i	−0.806975	+	0.590585i	$0.798897\pi$
$822$	0	0
$823$	39205.8	1.66055	0.830273	−	0.557357i	$-0.188185\pi$
$823$	39205.8	1.66055	0.830273	+	0.557357i	$0.188185\pi$
$824$	0	0
$825$	3030.95	0.127908
$826$	0	0
$827$	− 5929.15i	− 0.249307i	−0.992200	−	0.124653i	$-0.960218\pi$
$827$	− 5929.15i	− 0.249307i	0.992200	−	0.124653i	$-0.0397819\pi$
$828$	0	0
$829$	1269.44i	0.0531840i	0.999646	+	0.0265920i	$0.00846550\pi$
$829$	1269.44i	0.0531840i	−0.999646	+	0.0265920i	$0.991535\pi$
$830$	0	0
$831$	−16802.3	−0.701400
$832$	0	0
$833$	22839.5	0.949990
$834$	0	0
$835$	− 14521.6i	− 0.601845i
$836$	0	0
$837$	− 6142.19i	− 0.253650i
$838$	0	0
$839$	17884.1	0.735910	0.367955	−	0.929844i	$-0.380058\pi$
$839$	17884.1	0.735910	0.367955	+	0.929844i	$0.380058\pi$
$840$	0	0
$841$	21494.6	0.881323
$842$	0	0
$843$	− 610.278i	− 0.0249337i
$844$	0	0
$845$	− 20925.7i	− 0.851911i
$846$	0	0
$847$	15891.8	0.644686
$848$	0	0
$849$	6644.52	0.268598
$850$	0	0
$851$	1617.25i	0.0651453i
$852$	0	0
$853$	16083.4i	0.645585i	0.946470	+	0.322793i	$0.104622\pi$
$853$	16083.4i	0.645585i	−0.946470	+	0.322793i	$0.895378\pi$
$854$	0	0
$855$	−967.187	−0.0386867
$856$	0	0
$857$	−17203.6	−0.685722	−0.342861	−	0.939386i	$-0.611396\pi$
$857$	−17203.6	−0.685722	−0.342861	+	0.939386i	$0.611396\pi$
$858$	0	0
$859$	− 24427.5i	− 0.970264i	−0.874441	−	0.485132i	$-0.838772\pi$
$859$	− 24427.5i	− 0.970264i	0.874441	−	0.485132i	$-0.161228\pi$
$860$	0	0
$861$	1357.22i	0.0537212i
$862$	0	0
$863$	46584.1	1.83747	0.918737	−	0.394870i	$-0.129210\pi$
$863$	46584.1	1.83747	0.918737	+	0.394870i	$0.129210\pi$
$864$	0	0
$865$	−34515.3	−1.35671
$866$	0	0
$867$	2447.39i	0.0958683i
$868$	0	0
$869$	67790.5i	2.64630i
$870$	0	0
$871$	−59674.1	−2.32145
$872$	0	0
$873$	16912.4	0.655667
$874$	0	0
$875$	− 9462.91i	− 0.365605i
$876$	0	0
$877$	7402.05i	0.285005i	0.989794	+	0.142503i	$0.0455149\pi$
$877$	7402.05i	0.285005i	−0.989794	+	0.142503i	$0.954485\pi$
$878$	0	0
$879$	17437.1	0.669101
$880$	0	0
$881$	12044.7	0.460608	0.230304	−	0.973119i	$-0.426028\pi$
$881$	12044.7	0.460608	0.230304	+	0.973119i	$0.426028\pi$
$882$	0	0
$883$	7150.02i	0.272500i	0.990674	+	0.136250i	$0.0435050\pi$
$883$	7150.02i	0.272500i	−0.990674	+	0.136250i	$0.956495\pi$
$884$	0	0
$885$	10130.4i	0.384779i
$886$	0	0
$887$	21897.6	0.828916	0.414458	−	0.910068i	$-0.363971\pi$
$887$	21897.6	0.828916	0.414458	+	0.910068i	$0.363971\pi$
$888$	0	0
$889$	10578.8	0.399102
$890$	0	0
$891$	− 4996.79i	− 0.187878i
$892$	0	0
$893$	927.087i	0.0347411i
$894$	0	0
$895$	3369.36	0.125838
$896$	0	0
$897$	30511.4	1.13573
$898$	0	0
$899$	12238.9i	0.454047i
$900$	0	0
$901$	− 29392.5i	− 1.08680i
$902$	0	0
$903$	−5747.45	−0.211808
$904$	0	0
$905$	31962.8	1.17401
$906$	0	0
$907$	800.885i	0.0293197i	0.999893	+	0.0146598i	$0.00466654\pi$
$907$	800.885i	0.0293197i	−0.999893	+	0.0146598i	$0.995333\pi$
$908$	0	0
$909$	15502.6i	0.565664i
$910$	0	0
$911$	26742.9	0.972593	0.486296	−	0.873794i	$-0.338348\pi$
$911$	26742.9	0.972593	0.486296	+	0.873794i	$0.338348\pi$
$912$	0	0
$913$	−48846.5	−1.77063
$914$	0	0
$915$	10130.4i	0.366011i
$916$	0	0
$917$	− 12846.1i	− 0.462613i
$918$	0	0
$919$	−3744.72	−0.134414	−0.0672072	−	0.997739i	$-0.521409\pi$
$919$	−3744.72	−0.134414	−0.0672072	+	0.997739i	$0.521409\pi$
$920$	0	0
$921$	22012.0	0.787535
$922$	0	0
$923$	− 64530.3i	− 2.30124i
$924$	0	0
$925$	168.873i	0.00600271i
$926$	0	0
$927$	−17179.4	−0.608679
$928$	0	0
$929$	−25667.3	−0.906477	−0.453238	−	0.891389i	$-0.649731\pi$
$929$	−25667.3	−0.906477	−0.453238	+	0.891389i	$0.649731\pi$
$930$	0	0
$931$	3111.45i	0.109531i
$932$	0	0
$933$	− 19727.7i	− 0.692236i
$934$	0	0
$935$	−48662.9	−1.70208
$936$	0	0
$937$	−17978.7	−0.626829	−0.313414	−	0.949616i	$-0.601473\pi$
$937$	−17978.7	−0.626829	−0.313414	+	0.949616i	$0.601473\pi$
$938$	0	0
$939$	4670.01i	0.162300i
$940$	0	0
$941$	− 8120.01i	− 0.281302i	−0.990059	−	0.140651i	$-0.955081\pi$
$941$	− 8120.01i	− 0.281302i	0.990059	−	0.140651i	$-0.0449195\pi$
$942$	0	0
$943$	−11048.8	−0.381545
$944$	0	0
$945$	−1807.21	−0.0622099
$946$	0	0
$947$	7930.95i	0.272145i	0.990699	+	0.136072i	$0.0434480\pi$
$947$	7930.95i	0.272145i	−0.990699	+	0.136072i	$0.956552\pi$
$948$	0	0
$949$	− 23490.9i	− 0.803527i
$950$	0	0
$951$	−28371.8	−0.967422
$952$	0	0
$953$	−29833.3	−1.01406	−0.507028	−	0.861930i	$-0.669256\pi$
$953$	−29833.3	−1.01406	−0.507028	+	0.861930i	$0.669256\pi$
$954$	0	0
$955$	− 19348.2i	− 0.655596i
$956$	0	0
$957$	9956.55i	0.336311i
$958$	0	0
$959$	−16535.3	−0.556781
$960$	0	0
$961$	21960.1	0.737139
$962$	0	0
$963$	1157.60i	0.0387364i
$964$	0	0
$965$	31358.7i	1.04609i
$966$	0	0
$967$	7650.56	0.254421	0.127211	−	0.991876i	$-0.459398\pi$
$967$	7650.56	0.254421	0.127211	+	0.991876i	$0.459398\pi$
$968$	0	0
$969$	−2341.32	−0.0776204
$970$	0	0
$971$	5634.51i	0.186220i	0.995656	+	0.0931102i	$0.0296809\pi$
$971$	5634.51i	0.186220i	−0.995656	+	0.0931102i	$0.970319\pi$
$972$	0	0
$973$	8549.80i	0.281700i
$974$	0	0
$975$	3186.00	0.104650
$976$	0	0
$977$	43983.5	1.44028	0.720142	−	0.693827i	$-0.244077\pi$
$977$	43983.5	1.44028	0.720142	+	0.693827i	$0.244077\pi$
$978$	0	0
$979$	9291.72i	0.303335i
$980$	0	0
$981$	6824.41i	0.222107i
$982$	0	0
$983$	−7759.30	−0.251763	−0.125882	−	0.992045i	$-0.540176\pi$
$983$	−7759.30	−0.251763	−0.125882	+	0.992045i	$0.540176\pi$
$984$	0	0
$985$	−9477.00	−0.306561
$986$	0	0
$987$	1732.28i	0.0558653i
$988$	0	0
$989$	− 46788.4i	− 1.50433i
$990$	0	0
$991$	−10114.8	−0.324226	−0.162113	−	0.986772i	$-0.551831\pi$
$991$	−10114.8	−0.324226	−0.162113	+	0.986772i	$0.551831\pi$
$992$	0	0
$993$	32722.9	1.04575
$994$	0	0
$995$	− 28976.3i	− 0.923225i
$996$	0	0
$997$	− 43842.2i	− 1.39267i	−0.717715	−	0.696337i	$-0.754812\pi$
$997$	− 43842.2i	− 1.39267i	0.717715	−	0.696337i	$-0.245188\pi$
$998$	0	0
$999$	278.402	0.00881706

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.4.d.e.193.4	yes	4
3.2	odd	2		1152.4.d.o.577.2		4
4.3	odd	2		384.4.d.c.193.2	✓	4
8.3	odd	2		384.4.d.c.193.3	yes	4
8.5	even	2	inner	384.4.d.e.193.1	yes	4
12.11	even	2		1152.4.d.i.577.2		4
16.3	odd	4		768.4.a.k.1.2		2
16.5	even	4		768.4.a.p.1.1		2
16.11	odd	4		768.4.a.j.1.1		2
16.13	even	4		768.4.a.e.1.2		2
24.5	odd	2		1152.4.d.o.577.3		4
24.11	even	2		1152.4.d.i.577.3		4
48.5	odd	4		2304.4.a.s.1.2		2
48.11	even	4		2304.4.a.t.1.2		2
48.29	odd	4		2304.4.a.bp.1.1		2
48.35	even	4		2304.4.a.bq.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.4.d.c.193.2	✓	4	4.3	odd	2
384.4.d.c.193.3	yes	4	8.3	odd	2
384.4.d.e.193.1	yes	4	8.5	even	2	inner
384.4.d.e.193.4	yes	4	1.1	even	1	trivial
768.4.a.e.1.2		2	16.13	even	4
768.4.a.j.1.1		2	16.11	odd	4
768.4.a.k.1.2		2	16.3	odd	4
768.4.a.p.1.1		2	16.5	even	4
1152.4.d.i.577.2		4	12.11	even	2
1152.4.d.i.577.3		4	24.11	even	2
1152.4.d.o.577.2		4	3.2	odd	2
1152.4.d.o.577.3		4	24.5	odd	2
2304.4.a.s.1.2		2	48.5	odd	4
2304.4.a.t.1.2		2	48.11	even	4
2304.4.a.bp.1.1		2	48.29	odd	4
2304.4.a.bq.1.1		2	48.35	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 384 = 2^{7} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	384.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+3.00000i q^{3} +10.4222i q^{5} -6.42221 q^{7} -9.00000 q^{9} +O(q^{10})\)	\(q+3.00000i q^{3} +10.4222i q^{5} -6.42221 q^{7} -9.00000 q^{9} -61.6888i q^{11} -64.8444i q^{13} -31.2666 q^{15} -75.6888 q^{17} -10.3112i q^{19} -19.2666i q^{21} +156.844 q^{23} +16.3776 q^{25} -27.0000i q^{27} +53.7998i q^{29} +227.489 q^{31} +185.066 q^{33} -66.9335i q^{35} +10.3112i q^{37} +194.533 q^{39} -70.4441 q^{41} -298.311i q^{43} -93.7998i q^{45} -89.9109 q^{47} -301.755 q^{49} -227.066i q^{51} +388.333i q^{53} +642.934 q^{55} +30.9335 q^{57} -324.000i q^{59} -324.000i q^{61} +57.7998 q^{63} +675.822 q^{65} -920.266i q^{67} +470.533i q^{69} +995.156 q^{71} +362.266 q^{73} +49.1329i q^{75} +396.178i q^{77} -1098.91 q^{79} +81.0000 q^{81} -791.822i q^{83} -788.844i q^{85} -161.400 q^{87} -150.622 q^{89} +416.444i q^{91} +682.466i q^{93} +107.465 q^{95} -1879.15 q^{97} +555.199i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 32 q^{7} - 36 q^{9}+O(q^{10}) \) 4 * q + 32 * q^7 - 36 * q^9	\( 4 q + 32 q^{7} - 36 q^{9} + 48 q^{15} - 72 q^{17} + 512 q^{23} - 396 q^{25} + 160 q^{31} + 48 q^{33} + 432 q^{39} + 872 q^{41} + 448 q^{47} - 284 q^{49} + 3264 q^{55} + 816 q^{57} - 288 q^{63} + 1088 q^{65} + 4096 q^{71} - 1320 q^{73} - 992 q^{79} + 324 q^{81} + 912 q^{87} - 1064 q^{89} - 4416 q^{95} - 2440 q^{97}+O(q^{100}) \) 4 * q + 32 * q^7 - 36 * q^9 + 48 * q^15 - 72 * q^17 + 512 * q^23 - 396 * q^25 + 160 * q^31 + 48 * q^33 + 432 * q^39 + 872 * q^41 + 448 * q^47 - 284 * q^49 + 3264 * q^55 + 816 * q^57 - 288 * q^63 + 1088 * q^65 + 4096 * q^71 - 1320 * q^73 - 992 * q^79 + 324 * q^81 + 912 * q^87 - 1064 * q^89 - 4416 * q^95 - 2440 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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