LMFDB - Embedded newform 384.4.d.c.193.3

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			193.3
Root			$-1.30278i$ of defining polynomial
Character	$\chi$	$=$	384.193
Dual form			384.4.d.c.193.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

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

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	$-0.845660\pi$
$5$	− 10.4222i	− 0.932190i	0.884735	−	0.466095i	$-0.154340\pi$
$6$	0	0
$7$	6.42221	0.346766	0.173383	−	0.984854i	$-0.444530\pi$
$7$	6.42221	0.346766	0.173383	+	0.984854i	$0.444530\pi$
$8$	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	$0.243145\pi$
$13$	64.8444i	1.38343i	−0.722170	+	0.691716i	$0.756855\pi$
$14$	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	$-0.751741\pi$
$23$	−156.844	−1.42193	−0.710963	+	0.703229i	$0.751741\pi$
$24$	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	$-0.944897\pi$
$29$	− 53.7998i	− 0.344496i	0.985054	−	0.172248i	$-0.0551030\pi$
$30$	0	0
$31$	−227.489	−1.31801	−0.659003	−	0.752141i	$-0.729021\pi$
$31$	−227.489	−1.31801	−0.659003	+	0.752141i	$0.729021\pi$
$32$	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	$-0.992708\pi$
$37$	− 10.3112i	− 0.0458148i	0.999738	−	0.0229074i	$-0.00729229\pi$
$38$	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	$-0.455444\pi$
$47$	89.9109	0.279039	0.139520	+	0.990219i	$0.455444\pi$
$48$	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	$-0.832147\pi$
$53$	− 388.333i	− 1.00645i	0.864157	−	0.503223i	$-0.167853\pi$
$54$	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	$0.110438\pi$
$61$	324.000i	0.680065i	−0.940414	+	0.340032i	$0.889562\pi$
$62$	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	$-0.812639\pi$
$71$	−995.156	−1.66343	−0.831713	+	0.555206i	$0.812639\pi$
$72$	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	$-0.213938\pi$
$79$	1098.91	1.56503	0.782513	+	0.622634i	$0.213938\pi$
$80$	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	$0.322491\pi$
$101$	1722.51i	1.69699i	−0.529202	+	0.848496i	$0.677509\pi$
$102$	0	0
$103$	−1908.82	−1.82604	−0.913018	−	0.407919i	$-0.866255\pi$
$103$	−1908.82	−1.82604	−0.913018	+	0.407919i	$0.866255\pi$
$104$	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	$0.108115\pi$
$109$	758.267i	0.666320i	−0.942870	+	0.333160i	$0.891885\pi$
$110$	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	$-0.304822\pi$
$127$	1647.22	1.15092	0.575462	+	0.817828i	$0.304822\pi$
$128$	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	$-0.951476\pi$
$149$	− 552.377i	− 0.303708i	0.988403	−	0.151854i	$-0.0485243\pi$
$150$	0	0
$151$	1307.13	0.704456	0.352228	−	0.935914i	$-0.385424\pi$
$151$	1307.13	0.704456	0.352228	+	0.935914i	$0.385424\pi$
$152$	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	$0.353882\pi$
$157$	3527.11i	1.79295i	−0.443089	+	0.896477i	$0.646118\pi$
$158$	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	$-0.395372\pi$
$167$	1393.33	0.645624	0.322812	+	0.946463i	$0.395372\pi$
$168$	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	$-0.740589\pi$
$173$	− 3311.71i	− 1.45540i	0.685894	−	0.727701i	$-0.259411\pi$
$174$	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	$0.216824\pi$
$181$	3066.80i	1.25941i	−0.776834	+	0.629705i	$0.783176\pi$
$182$	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	$-0.385623\pi$
$191$	1856.44	0.703286	0.351643	+	0.936134i	$0.385623\pi$
$192$	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	$-0.947421\pi$
$197$	− 909.309i	− 0.328861i	0.986389	−	0.164430i	$-0.0525786\pi$
$198$	0	0
$199$	2780.24	0.990383	0.495191	−	0.868784i	$-0.335098\pi$
$199$	2780.24	0.990383	0.495191	+	0.868784i	$0.335098\pi$
$200$	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	$-0.471191\pi$
$223$	601.975	0.180768	0.0903839	+	0.995907i	$0.471191\pi$
$224$	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	$-0.942877\pi$
$229$	− 1237.11i	− 0.356989i	0.983941	−	0.178495i	$-0.0571227\pi$
$230$	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	$-0.805759\pi$
$239$	−6055.99	−1.63904	−0.819518	+	0.573053i	$0.805759\pi$
$240$	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	$-0.308332\pi$
$263$	4831.64	1.13282	0.566410	+	0.824124i	$0.308332\pi$
$264$	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	$0.0395607\pi$
$269$	1093.84i	0.247928i	−0.992287	+	0.123964i	$0.960439\pi$
$270$	0	0
$271$	−3079.80	−0.690349	−0.345175	−	0.938539i	$-0.612180\pi$
$271$	−3079.80	−0.690349	−0.345175	+	0.938539i	$0.612180\pi$
$272$	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	$-0.792200\pi$
$277$	− 5600.75i	− 1.21486i	0.794373	−	0.607430i	$-0.207800\pi$
$278$	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	$0.196736\pi$
$293$	5812.37i	1.15892i	−0.815002	+	0.579458i	$0.803264\pi$
$294$	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	$-0.295368\pi$
$311$	6575.91	1.19899	0.599494	+	0.800379i	$0.295368\pi$
$312$	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	$-0.683834\pi$
$317$	− 9457.27i	− 1.67562i	0.545959	−	0.837812i	$-0.316166\pi$
$318$	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	$-0.926442\pi$
$349$	− 2986.57i	− 0.458073i	0.973418	−	0.229037i	$-0.0735576\pi$
$350$	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	$-0.641678\pi$
$359$	−5857.19	−0.861088	−0.430544	+	0.902569i	$0.641678\pi$
$360$	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	$-0.775812\pi$
$367$	−10715.7	−1.52412	−0.762061	+	0.647505i	$0.775812\pi$
$368$	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	$-0.959738\pi$
$373$	− 1817.52i	− 0.252299i	0.992011	−	0.126149i	$-0.0402619\pi$
$374$	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	$-0.778393\pi$
$383$	−11502.3	−1.53457	−0.767285	+	0.641306i	$0.778393\pi$
$384$	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	$-0.999874\pi$
$389$	− 6.06261i	0 0.000790197i	1.00000		0.000395098i	$-0.000125764\pi$
$390$	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	$-0.876564\pi$
$397$	− 5982.31i	− 0.756281i	0.925748	−	0.378141i	$-0.123436\pi$
$398$	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	$-0.716337\pi$
$421$	− 13437.5i	− 1.55560i	0.628514	−	0.777798i	$-0.283663\pi$
$422$	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	$-0.365399\pi$
$431$	7343.86	0.820745	0.410373	+	0.911918i	$0.365399\pi$
$432$	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	$-0.328520\pi$
$439$	9437.93	1.02608	0.513038	+	0.858366i	$0.328520\pi$
$440$	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	$-0.989902\pi$
$461$	− 627.883i	− 0.0634347i	0.999497	−	0.0317174i	$-0.0100976\pi$
$462$	0	0
$463$	7315.03	0.734251	0.367126	−	0.930171i	$-0.380342\pi$
$463$	7315.03	0.734251	0.367126	+	0.930171i	$0.380342\pi$
$464$	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	$-0.385126\pi$
$479$	7403.51	0.706211	0.353106	+	0.935583i	$0.385126\pi$
$480$	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	$-0.448115\pi$
$487$	3488.11	0.324561	0.162281	+	0.986745i	$0.448115\pi$
$488$	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	$-0.395893\pi$
$503$	7248.38	0.642524	0.321262	+	0.946990i	$0.395893\pi$
$504$	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	$-0.907026\pi$
$509$	− 6613.44i	− 0.575905i	0.957645	−	0.287952i	$-0.0929745\pi$
$510$	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	$-0.965675\pi$
$541$	− 2708.58i	− 0.215251i	0.994191	−	0.107626i	$-0.0343248\pi$
$542$	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	$-0.977064\pi$
$557$	− 1892.77i	− 0.143984i	0.997405	−	0.0719922i	$-0.0229357\pi$
$558$	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	$-0.553046\pi$
$599$	−4863.60	−0.331755	−0.165878	+	0.986146i	$0.553046\pi$
$600$	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	$-0.292064\pi$
$607$	18178.3	1.21554	0.607770	+	0.794113i	$0.292064\pi$
$608$	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	$-0.827122\pi$
$613$	− 15687.2i	− 1.03360i	0.856105	−	0.516802i	$-0.172878\pi$
$614$	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	$-0.515662\pi$
$631$	−1559.17	−0.0983670	−0.0491835	+	0.998790i	$0.515662\pi$
$632$	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	$-0.529424\pi$
$647$	−3038.27	−0.184616	−0.0923081	+	0.995730i	$0.529424\pi$
$648$	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	$-0.912301\pi$
$653$	− 9078.99i	− 0.544086i	0.962285	−	0.272043i	$-0.0876994\pi$
$654$	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	$-0.952396\pi$
$661$	− 5064.16i	− 0.297992i	0.988838	−	0.148996i	$-0.0476042\pi$
$662$	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	$0.150681\pi$
$677$	16061.2i	0.911791i	−0.890033	+	0.455895i	$0.849319\pi$
$678$	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	$0.164035\pi$
$701$	18293.6i	0.985647i	−0.870129	+	0.492824i	$0.835965\pi$
$702$	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	$0.193406\pi$
$709$	21555.6i	1.14180i	−0.821019	+	0.570900i	$0.806594\pi$
$710$	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	$-0.575492\pi$
$719$	−9059.31	−0.469896	−0.234948	+	0.972008i	$0.575492\pi$
$720$	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	$-0.557759\pi$
$727$	−7074.83	−0.360923	−0.180462	+	0.983582i	$0.557759\pi$
$728$	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	$-0.858256\pi$
$733$	− 17095.9i	− 0.861459i	0.902481	−	0.430730i	$-0.141744\pi$
$734$	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	$-0.817571\pi$
$743$	−34033.2	−1.68043	−0.840214	+	0.542255i	$0.817571\pi$
$744$	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	$-0.814350\pi$
$751$	−34356.8	−1.66937	−0.834685	+	0.550728i	$0.814350\pi$
$752$	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	$0.250104\pi$
$757$	29464.6i	1.41467i	−0.706876	+	0.707337i	$0.749896\pi$
$758$	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	$-0.968715\pi$
$773$	− 4217.83i	− 0.196254i	0.995174	−	0.0981272i	$-0.0312852\pi$
$774$	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	$-0.898538\pi$
$797$	− 14102.3i	− 0.626761i	0.949628	−	0.313381i	$-0.101462\pi$
$798$	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	$-0.798897\pi$
$821$	− 27786.1i	− 1.18117i	0.806975	−	0.590585i	$-0.201103\pi$
$822$	0	0
$823$	−39205.8	−1.66055	−0.830273	−	0.557357i	$-0.811815\pi$
$823$	−39205.8	−1.66055	−0.830273	+	0.557357i	$0.811815\pi$
$824$	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	$-0.991535\pi$
$829$	− 1269.44i	− 0.0531840i	0.999646	−	0.0265920i	$-0.00846550\pi$
$830$	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	$-0.619942\pi$
$839$	−17884.1	−0.735910	−0.367955	+	0.929844i	$0.619942\pi$
$840$	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	$-0.895378\pi$
$853$	− 16083.4i	− 0.645585i	0.946470	−	0.322793i	$-0.104622\pi$
$854$	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	$-0.870790\pi$
$863$	−46584.1	−1.83747	−0.918737	+	0.394870i	$0.870790\pi$
$864$	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	$-0.954485\pi$
$877$	− 7402.05i	− 0.285005i	0.989794	−	0.142503i	$-0.0455149\pi$
$878$	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	$-0.636029\pi$
$887$	−21897.6	−0.828916	−0.414458	+	0.910068i	$0.636029\pi$
$888$	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	$-0.661652\pi$
$911$	−26742.9	−0.972593	−0.486296	+	0.873794i	$0.661652\pi$
$912$	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	$-0.478591\pi$
$919$	3744.72	0.134414	0.0672072	+	0.997739i	$0.478591\pi$
$920$	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	$0.0449195\pi$
$941$	8120.01i	0.281302i	−0.990059	+	0.140651i	$0.955081\pi$
$942$	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	$-0.540602\pi$
$967$	−7650.56	−0.254421	−0.127211	+	0.991876i	$0.540602\pi$
$968$	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	$-0.459824\pi$
$983$	7759.30	0.251763	0.125882	+	0.992045i	$0.459824\pi$
$984$	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	$-0.448169\pi$
$991$	10114.8	0.324226	0.162113	+	0.986772i	$0.448169\pi$
$992$	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	$0.245188\pi$
$997$	43842.2i	1.39267i	−0.717715	+	0.696337i	$0.754812\pi$
$998$	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.c.193.3	yes	4
3.2	odd	2		1152.4.d.i.577.3		4
4.3	odd	2		384.4.d.e.193.1	yes	4
8.3	odd	2		384.4.d.e.193.4	yes	4
8.5	even	2	inner	384.4.d.c.193.2	✓	4
12.11	even	2		1152.4.d.o.577.3		4
16.3	odd	4		768.4.a.p.1.1		2
16.5	even	4		768.4.a.k.1.2		2
16.11	odd	4		768.4.a.e.1.2		2
16.13	even	4		768.4.a.j.1.1		2
24.5	odd	2		1152.4.d.i.577.2		4
24.11	even	2		1152.4.d.o.577.2		4
48.5	odd	4		2304.4.a.bq.1.1		2
48.11	even	4		2304.4.a.bp.1.1		2
48.29	odd	4		2304.4.a.t.1.2		2
48.35	even	4		2304.4.a.s.1.2		2

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

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

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