LMFDB - Embedded newform 192.4.d.c.97.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [192,4,Mod(97,192)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(192, 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("192.97");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 192 = 2^{6} \cdot 3 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	192.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:	$11.3283667211$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			97.1
Root			$0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	192.97
Dual form			192.4.d.c.97.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000i q^{3} -3.46410i q^{5} -24.2487 q^{7} -9.00000 q^{9} +O(q^{10})$	$q-3.00000i q^{3} -3.46410i q^{5} -24.2487 q^{7} -9.00000 q^{9} +48.0000i q^{11} +41.5692i q^{13} -10.3923 q^{15} +54.0000 q^{17} +4.00000i q^{19} +72.7461i q^{21} -173.205 q^{23} +113.000 q^{25} +27.0000i q^{27} +162.813i q^{29} -58.8897 q^{31} +144.000 q^{33} +84.0000i q^{35} +325.626i q^{37} +124.708 q^{39} -294.000 q^{41} +188.000i q^{43} +31.1769i q^{45} -505.759 q^{47} +245.000 q^{49} -162.000i q^{51} -744.782i q^{53} +166.277 q^{55} +12.0000 q^{57} +252.000i q^{59} -90.0666i q^{61} +218.238 q^{63} +144.000 q^{65} -628.000i q^{67} +519.615i q^{69} +6.92820 q^{71} -1006.00 q^{73} -339.000i q^{75} -1163.94i q^{77} +1340.61 q^{79} +81.0000 q^{81} +720.000i q^{83} -187.061i q^{85} +488.438 q^{87} -1482.00 q^{89} -1008.00i q^{91} +176.669i q^{93} +13.8564 q^{95} +1822.00 q^{97} -432.000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 36 q^{9}+O(q^{10}) $ 4 * q - 36 * q^9	$ 4 q - 36 q^{9} + 216 q^{17} + 452 q^{25} + 576 q^{33} - 1176 q^{41} + 980 q^{49} + 48 q^{57} + 576 q^{65} - 4024 q^{73} + 324 q^{81} - 5928 q^{89} + 7288 q^{97}+O(q^{100}) $ 4 * q - 36 * q^9 + 216 * q^17 + 452 * q^25 + 576 * q^33 - 1176 * q^41 + 980 * q^49 + 48 * q^57 + 576 * q^65 - 4024 * q^73 + 324 * q^81 - 5928 * q^89 + 7288 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/192\mathbb{Z}\right)^\times$.

$n$	$65$	$127$	$133$
$\chi(n)$	$1$	$1$	$-1$

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
$5$	− 3.46410i	− 0.309839i	−0.987927	−	0.154919i	$-0.950488\pi$
$5$	− 3.46410i	− 0.309839i	0.987927	−	0.154919i	$-0.0495118\pi$
$6$	0	0
$7$	−24.2487	−1.30931	−0.654654	−	0.755929i	$-0.727186\pi$
$7$	−24.2487	−1.30931	−0.654654	+	0.755929i	$0.727186\pi$
$8$	0	0
$9$	−9.00000	−0.333333
$10$	0	0
$11$	48.0000i	1.31569i	0.753155	+	0.657843i	$0.228531\pi$
$11$	48.0000i	1.31569i	−0.753155	+	0.657843i	$0.771469\pi$
$12$	0	0
$13$	41.5692i	0.886864i	0.896308	+	0.443432i	$0.146239\pi$
$13$	41.5692i	0.886864i	−0.896308	+	0.443432i	$0.853761\pi$
$14$	0	0
$15$	−10.3923	−0.178885
$16$	0	0
$17$	54.0000	0.770407	0.385204	−	0.922832i	$-0.374131\pi$
$17$	54.0000	0.770407	0.385204	+	0.922832i	$0.374131\pi$
$18$	0	0
$19$	4.00000i	0.0482980i	0.999708	+	0.0241490i	$0.00768762\pi$
$19$	4.00000i	0.0482980i	−0.999708	+	0.0241490i	$0.992312\pi$
$20$	0	0
$21$	72.7461i	0.755929i
$22$	0	0
$23$	−173.205	−1.57025	−0.785125	−	0.619337i	$-0.787401\pi$
$23$	−173.205	−1.57025	−0.785125	+	0.619337i	$0.787401\pi$
$24$	0	0
$25$	113.000	0.904000
$26$	0	0
$27$	27.0000i	0.192450i
$28$	0	0
$29$	162.813i	1.04254i	0.853393	+	0.521269i	$0.174541\pi$
$29$	162.813i	1.04254i	−0.853393	+	0.521269i	$0.825459\pi$
$30$	0	0
$31$	−58.8897	−0.341191	−0.170595	−	0.985341i	$-0.554569\pi$
$31$	−58.8897	−0.341191	−0.170595	+	0.985341i	$0.554569\pi$
$32$	0	0
$33$	144.000	0.759612
$34$	0	0
$35$	84.0000i	0.405674i
$36$	0	0
$37$	325.626i	1.44682i	0.690416	+	0.723412i	$0.257427\pi$
$37$	325.626i	1.44682i	−0.690416	+	0.723412i	$0.742573\pi$
$38$	0	0
$39$	124.708	0.512031
$40$	0	0
$41$	−294.000	−1.11988	−0.559940	−	0.828533i	$-0.689176\pi$
$41$	−294.000	−1.11988	−0.559940	+	0.828533i	$0.689176\pi$
$42$	0	0
$43$	188.000i	0.666738i	0.942796	+	0.333369i	$0.108185\pi$
$43$	188.000i	0.666738i	−0.942796	+	0.333369i	$0.891815\pi$
$44$	0	0
$45$	31.1769i	0.103280i
$46$	0	0
$47$	−505.759	−1.56963	−0.784814	−	0.619731i	$-0.787242\pi$
$47$	−505.759	−1.56963	−0.784814	+	0.619731i	$0.787242\pi$
$48$	0	0
$49$	245.000	0.714286
$50$	0	0
$51$	− 162.000i	− 0.444795i
$52$	0	0
$53$	− 744.782i	− 1.93026i	−0.261775	−	0.965129i	$-0.584308\pi$
$53$	− 744.782i	− 1.93026i	0.261775	−	0.965129i	$-0.415692\pi$
$54$	0	0
$55$	166.277	0.407650
$56$	0	0
$57$	12.0000	0.0278849
$58$	0	0
$59$	252.000i	0.556061i	0.960572	+	0.278031i	$0.0896817\pi$
$59$	252.000i	0.556061i	−0.960572	+	0.278031i	$0.910318\pi$
$60$	0	0
$61$	− 90.0666i	− 0.189047i	−0.995523	−	0.0945234i	$-0.969867\pi$
$61$	− 90.0666i	− 0.189047i	0.995523	−	0.0945234i	$-0.0301327\pi$
$62$	0	0
$63$	218.238	0.436436
$64$	0	0
$65$	144.000	0.274785
$66$	0	0
$67$	− 628.000i	− 1.14511i	−0.819866	−	0.572555i	$-0.805952\pi$
$67$	− 628.000i	− 1.14511i	0.819866	−	0.572555i	$-0.194048\pi$
$68$	0	0
$69$	519.615i	0.906584i
$70$	0	0
$71$	6.92820	0.0115807	0.00579033	−	0.999983i	$-0.498157\pi$
$71$	6.92820	0.0115807	0.00579033	+	0.999983i	$0.498157\pi$
$72$	0	0
$73$	−1006.00	−1.61292	−0.806462	−	0.591286i	$-0.798620\pi$
$73$	−1006.00	−1.61292	−0.806462	+	0.591286i	$0.798620\pi$
$74$	0	0
$75$	− 339.000i	− 0.521925i
$76$	0	0
$77$	− 1163.94i	− 1.72264i
$78$	0	0
$79$	1340.61	1.90924	0.954621	−	0.297824i	$-0.0962607\pi$
$79$	1340.61	1.90924	0.954621	+	0.297824i	$0.0962607\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	720.000i	0.952172i	0.879399	+	0.476086i	$0.157945\pi$
$83$	720.000i	0.952172i	−0.879399	+	0.476086i	$0.842055\pi$
$84$	0	0
$85$	− 187.061i	− 0.238702i
$86$	0	0
$87$	488.438	0.601909
$88$	0	0
$89$	−1482.00	−1.76508	−0.882538	−	0.470242i	$-0.844167\pi$
$89$	−1482.00	−1.76508	−0.882538	+	0.470242i	$0.844167\pi$
$90$	0	0
$91$	− 1008.00i	− 1.16118i
$92$	0	0
$93$	176.669i	0.196986i
$94$	0	0
$95$	13.8564	0.0149646
$96$	0	0
$97$	1822.00	1.90718	0.953588	−	0.301114i	$-0.0973586\pi$
$97$	1822.00	1.90718	0.953588	+	0.301114i	$0.0973586\pi$
$98$	0	0
$99$	− 432.000i	− 0.438562i
$100$	0	0
$101$	− 911.059i	− 0.897562i	−0.893642	−	0.448781i	$-0.851858\pi$
$101$	− 911.059i	− 0.897562i	0.893642	−	0.448781i	$-0.148142\pi$
$102$	0	0
$103$	453.797	0.434116	0.217058	−	0.976159i	$-0.430354\pi$
$103$	453.797	0.434116	0.217058	+	0.976159i	$0.430354\pi$
$104$	0	0
$105$	252.000	0.234216
$106$	0	0
$107$	− 1188.00i	− 1.07335i	−0.843790	−	0.536674i	$-0.819680\pi$
$107$	− 1188.00i	− 1.07335i	0.843790	−	0.536674i	$-0.180320\pi$
$108$	0	0
$109$	471.118i	0.413990i	0.978342	+	0.206995i	$0.0663684\pi$
$109$	471.118i	0.413990i	−0.978342	+	0.206995i	$0.933632\pi$
$110$	0	0
$111$	976.877	0.835325
$112$	0	0
$113$	−390.000	−0.324674	−0.162337	−	0.986735i	$-0.551903\pi$
$113$	−390.000	−0.324674	−0.162337	+	0.986735i	$0.551903\pi$
$114$	0	0
$115$	600.000i	0.486524i
$116$	0	0
$117$	− 374.123i	− 0.295621i
$118$	0	0
$119$	−1309.43	−1.00870
$120$	0	0
$121$	−973.000	−0.731029
$122$	0	0
$123$	882.000i	0.646563i
$124$	0	0
$125$	− 824.456i	− 0.589933i
$126$	0	0
$127$	−606.218	−0.423568	−0.211784	−	0.977317i	$-0.567927\pi$
$127$	−606.218	−0.423568	−0.211784	+	0.977317i	$0.567927\pi$
$128$	0	0
$129$	564.000	0.384941
$130$	0	0
$131$	1380.00i	0.920391i	0.887818	+	0.460195i	$0.152221\pi$
$131$	1380.00i	0.920391i	−0.887818	+	0.460195i	$0.847779\pi$
$132$	0	0
$133$	− 96.9948i	− 0.0632370i
$134$	0	0
$135$	93.5307	0.0596285
$136$	0	0
$137$	−1158.00	−0.722150	−0.361075	−	0.932537i	$-0.617590\pi$
$137$	−1158.00	−0.722150	−0.361075	+	0.932537i	$0.617590\pi$
$138$	0	0
$139$	1180.00i	0.720045i	0.932944	+	0.360023i	$0.117231\pi$
$139$	1180.00i	0.720045i	−0.932944	+	0.360023i	$0.882769\pi$
$140$	0	0
$141$	1517.28i	0.906225i
$142$	0	0
$143$	−1995.32	−1.16683
$144$	0	0
$145$	564.000	0.323018
$146$	0	0
$147$	− 735.000i	− 0.412393i
$148$	0	0
$149$	2171.99i	1.19420i	0.802165	+	0.597102i	$0.203681\pi$
$149$	2171.99i	1.19420i	−0.802165	+	0.597102i	$0.796319\pi$
$150$	0	0
$151$	142.028	0.0765436	0.0382718	−	0.999267i	$-0.487815\pi$
$151$	142.028	0.0765436	0.0382718	+	0.999267i	$0.487815\pi$
$152$	0	0
$153$	−486.000	−0.256802
$154$	0	0
$155$	204.000i	0.105714i
$156$	0	0
$157$	1337.14i	0.679717i	0.940476	+	0.339859i	$0.110379\pi$
$157$	1337.14i	0.679717i	−0.940476	+	0.339859i	$0.889621\pi$
$158$	0	0
$159$	−2234.35	−1.11443
$160$	0	0
$161$	4200.00	2.05594
$162$	0	0
$163$	− 1748.00i	− 0.839963i	−0.907533	−	0.419981i	$-0.862037\pi$
$163$	− 1748.00i	− 0.839963i	0.907533	−	0.419981i	$-0.137963\pi$
$164$	0	0
$165$	− 498.831i	− 0.235357i
$166$	0	0
$167$	−13.8564	−0.00642060	−0.00321030	−	0.999995i	$-0.501022\pi$
$167$	−13.8564	−0.00642060	−0.00321030	+	0.999995i	$0.501022\pi$
$168$	0	0
$169$	469.000	0.213473
$170$	0	0
$171$	− 36.0000i	− 0.0160993i
$172$	0	0
$173$	599.290i	0.263371i	0.991292	+	0.131685i	$0.0420389\pi$
$173$	599.290i	0.263371i	−0.991292	+	0.131685i	$0.957961\pi$
$174$	0	0
$175$	−2740.10	−1.18361
$176$	0	0
$177$	756.000	0.321042
$178$	0	0
$179$	3228.00i	1.34789i	0.738782	+	0.673944i	$0.235401\pi$
$179$	3228.00i	1.34789i	−0.738782	+	0.673944i	$0.764599\pi$
$180$	0	0
$181$	2023.04i	0.830779i	0.909644	+	0.415390i	$0.136355\pi$
$181$	2023.04i	0.830779i	−0.909644	+	0.415390i	$0.863645\pi$
$182$	0	0
$183$	−270.200	−0.109146
$184$	0	0
$185$	1128.00	0.448282
$186$	0	0
$187$	2592.00i	1.01361i
$188$	0	0
$189$	− 654.715i	− 0.251976i
$190$	0	0
$191$	3477.96	1.31757	0.658786	−	0.752330i	$-0.271070\pi$
$191$	3477.96	1.31757	0.658786	+	0.752330i	$0.271070\pi$
$192$	0	0
$193$	−766.000	−0.285689	−0.142844	−	0.989745i	$-0.545625\pi$
$193$	−766.000	−0.285689	−0.142844	+	0.989745i	$0.545625\pi$
$194$	0	0
$195$	− 432.000i	− 0.158647i
$196$	0	0
$197$	2899.45i	1.04862i	0.851529	+	0.524308i	$0.175676\pi$
$197$	2899.45i	1.04862i	−0.851529	+	0.524308i	$0.824324\pi$
$198$	0	0
$199$	−1735.51	−0.618228	−0.309114	−	0.951025i	$-0.600032\pi$
$199$	−1735.51	−0.618228	−0.309114	+	0.951025i	$0.600032\pi$
$200$	0	0
$201$	−1884.00	−0.661130
$202$	0	0
$203$	− 3948.00i	− 1.36500i
$204$	0	0
$205$	1018.45i	0.346982i
$206$	0	0
$207$	1558.85	0.523417
$208$	0	0
$209$	−192.000	−0.0635451
$210$	0	0
$211$	1100.00i	0.358896i	0.983767	+	0.179448i	$0.0574312\pi$
$211$	1100.00i	0.358896i	−0.983767	+	0.179448i	$0.942569\pi$
$212$	0	0
$213$	− 20.7846i	− 0.00668609i
$214$	0	0
$215$	651.251	0.206581
$216$	0	0
$217$	1428.00	0.446723
$218$	0	0
$219$	3018.00i	0.931222i
$220$	0	0
$221$	2244.74i	0.683246i
$222$	0	0
$223$	391.443	0.117547	0.0587735	−	0.998271i	$-0.481281\pi$
$223$	391.443	0.117547	0.0587735	+	0.998271i	$0.481281\pi$
$224$	0	0
$225$	−1017.00	−0.301333
$226$	0	0
$227$	− 3336.00i	− 0.975410i	−0.873008	−	0.487705i	$-0.837834\pi$
$227$	− 3336.00i	− 0.975410i	0.873008	−	0.487705i	$-0.162166\pi$
$228$	0	0
$229$	− 5999.82i	− 1.73135i	−0.500605	−	0.865676i	$-0.666889\pi$
$229$	− 5999.82i	− 1.73135i	0.500605	−	0.865676i	$-0.333111\pi$
$230$	0	0
$231$	−3491.81	−0.994565
$232$	0	0
$233$	318.000	0.0894115	0.0447057	−	0.999000i	$-0.485765\pi$
$233$	318.000	0.0894115	0.0447057	+	0.999000i	$0.485765\pi$
$234$	0	0
$235$	1752.00i	0.486331i
$236$	0	0
$237$	− 4021.82i	− 1.10230i
$238$	0	0
$239$	859.097	0.232512	0.116256	−	0.993219i	$-0.462911\pi$
$239$	859.097	0.232512	0.116256	+	0.993219i	$0.462911\pi$
$240$	0	0
$241$	−2710.00	−0.724342	−0.362171	−	0.932112i	$-0.617964\pi$
$241$	−2710.00	−0.724342	−0.362171	+	0.932112i	$0.617964\pi$
$242$	0	0
$243$	− 243.000i	− 0.0641500i
$244$	0	0
$245$	− 848.705i	− 0.221313i
$246$	0	0
$247$	−166.277	−0.0428338
$248$	0	0
$249$	2160.00	0.549737
$250$	0	0
$251$	− 5136.00i	− 1.29156i	−0.763524	−	0.645780i	$-0.776532\pi$
$251$	− 5136.00i	− 1.29156i	0.763524	−	0.645780i	$-0.223468\pi$
$252$	0	0
$253$	− 8313.84i	− 2.06596i
$254$	0	0
$255$	−561.184	−0.137815
$256$	0	0
$257$	−4398.00	−1.06747	−0.533735	−	0.845652i	$-0.679212\pi$
$257$	−4398.00	−1.06747	−0.533735	+	0.845652i	$0.679212\pi$
$258$	0	0
$259$	− 7896.00i	− 1.89434i
$260$	0	0
$261$	− 1465.31i	− 0.347512i
$262$	0	0
$263$	6817.35	1.59839	0.799194	−	0.601073i	$-0.205260\pi$
$263$	6817.35	1.59839	0.799194	+	0.601073i	$0.205260\pi$
$264$	0	0
$265$	−2580.00	−0.598068
$266$	0	0
$267$	4446.00i	1.01907i
$268$	0	0
$269$	4624.58i	1.04820i	0.851657	+	0.524099i	$0.175598\pi$
$269$	4624.58i	1.04820i	−0.851657	+	0.524099i	$0.824402\pi$
$270$	0	0
$271$	3883.26	0.870447	0.435223	−	0.900322i	$-0.356669\pi$
$271$	3883.26	0.870447	0.435223	+	0.900322i	$0.356669\pi$
$272$	0	0
$273$	−3024.00	−0.670406
$274$	0	0
$275$	5424.00i	1.18938i
$276$	0	0
$277$	1524.20i	0.330616i	0.986242	+	0.165308i	$0.0528618\pi$
$277$	1524.20i	0.330616i	−0.986242	+	0.165308i	$0.947138\pi$
$278$	0	0
$279$	530.008	0.113730
$280$	0	0
$281$	4398.00	0.933675	0.466838	−	0.884343i	$-0.345393\pi$
$281$	4398.00	0.933675	0.466838	+	0.884343i	$0.345393\pi$
$282$	0	0
$283$	4372.00i	0.918334i	0.888350	+	0.459167i	$0.151852\pi$
$283$	4372.00i	0.918334i	−0.888350	+	0.459167i	$0.848148\pi$
$284$	0	0
$285$	− 41.5692i	− 0.00863982i
$286$	0	0
$287$	7129.12	1.46627
$288$	0	0
$289$	−1997.00	−0.406473
$290$	0	0
$291$	− 5466.00i	− 1.10111i
$292$	0	0
$293$	− 3571.49i	− 0.712111i	−0.934465	−	0.356056i	$-0.884121\pi$
$293$	− 3571.49i	− 0.712111i	0.934465	−	0.356056i	$-0.115879\pi$
$294$	0	0
$295$	872.954	0.172289
$296$	0	0
$297$	−1296.00	−0.253204
$298$	0	0
$299$	− 7200.00i	− 1.39260i
$300$	0	0
$301$	− 4558.76i	− 0.872965i
$302$	0	0
$303$	−2733.18	−0.518207
$304$	0	0
$305$	−312.000	−0.0585740
$306$	0	0
$307$	4172.00i	0.775598i	0.921744	+	0.387799i	$0.126765\pi$
$307$	4172.00i	0.775598i	−0.921744	+	0.387799i	$0.873235\pi$
$308$	0	0
$309$	− 1361.39i	− 0.250637i
$310$	0	0
$311$	6470.94	1.17985	0.589925	−	0.807458i	$-0.299157\pi$
$311$	6470.94	1.17985	0.589925	+	0.807458i	$0.299157\pi$
$312$	0	0
$313$	−74.0000	−0.0133633	−0.00668167	−	0.999978i	$-0.502127\pi$
$313$	−74.0000	−0.0133633	−0.00668167	+	0.999978i	$0.502127\pi$
$314$	0	0
$315$	− 756.000i	− 0.135225i
$316$	0	0
$317$	1964.15i	0.348004i	0.984745	+	0.174002i	$0.0556700\pi$
$317$	1964.15i	0.348004i	−0.984745	+	0.174002i	$0.944330\pi$
$318$	0	0
$319$	−7815.01	−1.37165
$320$	0	0
$321$	−3564.00	−0.619698
$322$	0	0
$323$	216.000i	0.0372092i
$324$	0	0
$325$	4697.32i	0.801725i
$326$	0	0
$327$	1413.35	0.239017
$328$	0	0
$329$	12264.0	2.05513
$330$	0	0
$331$	− 7556.00i	− 1.25473i	−0.778726	−	0.627365i	$-0.784134\pi$
$331$	− 7556.00i	− 1.25473i	0.778726	−	0.627365i	$-0.215866\pi$
$332$	0	0
$333$	− 2930.63i	− 0.482275i
$334$	0	0
$335$	−2175.46	−0.354800
$336$	0	0
$337$	−4106.00	−0.663703	−0.331852	−	0.943332i	$-0.607673\pi$
$337$	−4106.00	−0.663703	−0.331852	+	0.943332i	$0.607673\pi$
$338$	0	0
$339$	1170.00i	0.187450i
$340$	0	0
$341$	− 2826.71i	− 0.448900i
$342$	0	0
$343$	2376.37	0.374088
$344$	0	0
$345$	1800.00	0.280895
$346$	0	0
$347$	− 5256.00i	− 0.813132i	−0.913621	−	0.406566i	$-0.866726\pi$
$347$	− 5256.00i	− 0.813132i	0.913621	−	0.406566i	$-0.133274\pi$
$348$	0	0
$349$	10385.4i	1.59288i	0.604715	+	0.796442i	$0.293287\pi$
$349$	10385.4i	1.59288i	−0.604715	+	0.796442i	$0.706713\pi$
$350$	0	0
$351$	−1122.37	−0.170677
$352$	0	0
$353$	−3942.00	−0.594367	−0.297183	−	0.954820i	$-0.596047\pi$
$353$	−3942.00	−0.594367	−0.297183	+	0.954820i	$0.596047\pi$
$354$	0	0
$355$	− 24.0000i	− 0.00358813i
$356$	0	0
$357$	3928.29i	0.582373i
$358$	0	0
$359$	−6644.15	−0.976782	−0.488391	−	0.872625i	$-0.662416\pi$
$359$	−6644.15	−0.976782	−0.488391	+	0.872625i	$0.662416\pi$
$360$	0	0
$361$	6843.00	0.997667
$362$	0	0
$363$	2919.00i	0.422060i
$364$	0	0
$365$	3484.89i	0.499746i
$366$	0	0
$367$	2906.38	0.413384	0.206692	−	0.978406i	$-0.433730\pi$
$367$	2906.38	0.413384	0.206692	+	0.978406i	$0.433730\pi$
$368$	0	0
$369$	2646.00	0.373293
$370$	0	0
$371$	18060.0i	2.52730i
$372$	0	0
$373$	− 10246.8i	− 1.42241i	−0.702983	−	0.711206i	$-0.748149\pi$
$373$	− 10246.8i	− 1.42241i	0.702983	−	0.711206i	$-0.251851\pi$
$374$	0	0
$375$	−2473.37	−0.340598
$376$	0	0
$377$	−6768.00	−0.924588
$378$	0	0
$379$	13844.0i	1.87630i	0.346226	+	0.938151i	$0.387463\pi$
$379$	13844.0i	1.87630i	−0.346226	+	0.938151i	$0.612537\pi$
$380$	0	0
$381$	1818.65i	0.244547i
$382$	0	0
$383$	−7163.76	−0.955747	−0.477874	−	0.878429i	$-0.658592\pi$
$383$	−7163.76	−0.955747	−0.477874	+	0.878429i	$0.658592\pi$
$384$	0	0
$385$	−4032.00	−0.533740
$386$	0	0
$387$	− 1692.00i	− 0.222246i
$388$	0	0
$389$	12993.8i	1.69361i	0.531904	+	0.846805i	$0.321477\pi$
$389$	12993.8i	1.69361i	−0.531904	+	0.846805i	$0.678523\pi$
$390$	0	0
$391$	−9353.07	−1.20973
$392$	0	0
$393$	4140.00	0.531388
$394$	0	0
$395$	− 4644.00i	− 0.591557i
$396$	0	0
$397$	− 117.779i	− 0.0148896i	−0.999972	−	0.00744481i	$-0.997630\pi$
$397$	− 117.779i	− 0.0148896i	0.999972	−	0.00744481i	$-0.00236978\pi$
$398$	0	0
$399$	−290.985	−0.0365099
$400$	0	0
$401$	−5418.00	−0.674718	−0.337359	−	0.941376i	$-0.609534\pi$
$401$	−5418.00	−0.674718	−0.337359	+	0.941376i	$0.609534\pi$
$402$	0	0
$403$	− 2448.00i	− 0.302589i
$404$	0	0
$405$	− 280.592i	− 0.0344265i
$406$	0	0
$407$	−15630.0	−1.90357
$408$	0	0
$409$	11450.0	1.38427	0.692135	−	0.721768i	$-0.256670\pi$
$409$	11450.0	1.38427	0.692135	+	0.721768i	$0.256670\pi$
$410$	0	0
$411$	3474.00i	0.416934i
$412$	0	0
$413$	− 6110.68i	− 0.728055i
$414$	0	0
$415$	2494.15	0.295020
$416$	0	0
$417$	3540.00	0.415718
$418$	0	0
$419$	− 1176.00i	− 0.137115i	−0.997647	−	0.0685577i	$-0.978160\pi$
$419$	− 1176.00i	− 0.137115i	0.997647	−	0.0685577i	$-0.0218397\pi$
$420$	0	0
$421$	10032.0i	1.16136i	0.814133	+	0.580679i	$0.197213\pi$
$421$	10032.0i	1.16136i	−0.814133	+	0.580679i	$0.802787\pi$
$422$	0	0
$423$	4551.83	0.523209
$424$	0	0
$425$	6102.00	0.696448
$426$	0	0
$427$	2184.00i	0.247520i
$428$	0	0
$429$	5985.97i	0.673672i
$430$	0	0
$431$	838.313	0.0936893	0.0468447	−	0.998902i	$-0.485083\pi$
$431$	838.313	0.0936893	0.0468447	+	0.998902i	$0.485083\pi$
$432$	0	0
$433$	4318.00	0.479237	0.239619	−	0.970867i	$-0.422978\pi$
$433$	4318.00	0.479237	0.239619	+	0.970867i	$0.422978\pi$
$434$	0	0
$435$	− 1692.00i	− 0.186495i
$436$	0	0
$437$	− 692.820i	− 0.0758400i
$438$	0	0
$439$	−1610.81	−0.175124	−0.0875622	−	0.996159i	$-0.527908\pi$
$439$	−1610.81	−0.175124	−0.0875622	+	0.996159i	$0.527908\pi$
$440$	0	0
$441$	−2205.00	−0.238095
$442$	0	0
$443$	− 1032.00i	− 0.110681i	−0.998468	−	0.0553406i	$-0.982376\pi$
$443$	− 1032.00i	− 0.110681i	0.998468	−	0.0553406i	$-0.0176245\pi$
$444$	0	0
$445$	5133.80i	0.546889i
$446$	0	0
$447$	6515.98	0.689474
$448$	0	0
$449$	726.000	0.0763075	0.0381537	−	0.999272i	$-0.487852\pi$
$449$	726.000	0.0763075	0.0381537	+	0.999272i	$0.487852\pi$
$450$	0	0
$451$	− 14112.0i	− 1.47341i
$452$	0	0
$453$	− 426.084i	− 0.0441925i
$454$	0	0
$455$	−3491.81	−0.359778
$456$	0	0
$457$	8666.00	0.887042	0.443521	−	0.896264i	$-0.353729\pi$
$457$	8666.00	0.887042	0.443521	+	0.896264i	$0.353729\pi$
$458$	0	0
$459$	1458.00i	0.148265i
$460$	0	0
$461$	− 14684.3i	− 1.48355i	−0.670648	−	0.741776i	$-0.733984\pi$
$461$	− 14684.3i	− 1.48355i	0.670648	−	0.741776i	$-0.266016\pi$
$462$	0	0
$463$	−4998.70	−0.501748	−0.250874	−	0.968020i	$-0.580718\pi$
$463$	−4998.70	−0.501748	−0.250874	+	0.968020i	$0.580718\pi$
$464$	0	0
$465$	612.000	0.0610340
$466$	0	0
$467$	16824.0i	1.66707i	0.552466	+	0.833535i	$0.313687\pi$
$467$	16824.0i	1.66707i	−0.552466	+	0.833535i	$0.686313\pi$
$468$	0	0
$469$	15228.2i	1.49930i
$470$	0	0
$471$	4011.43	0.392435
$472$	0	0
$473$	−9024.00	−0.877218
$474$	0	0
$475$	452.000i	0.0436614i
$476$	0	0
$477$	6703.04i	0.643419i
$478$	0	0
$479$	−10953.5	−1.04484	−0.522419	−	0.852689i	$-0.674970\pi$
$479$	−10953.5	−1.04484	−0.522419	+	0.852689i	$0.674970\pi$
$480$	0	0
$481$	−13536.0	−1.28314
$482$	0	0
$483$	− 12600.0i	− 1.18700i
$484$	0	0
$485$	− 6311.59i	− 0.590917i
$486$	0	0
$487$	10714.5	0.996959	0.498479	−	0.866902i	$-0.333892\pi$
$487$	10714.5	0.996959	0.498479	+	0.866902i	$0.333892\pi$
$488$	0	0
$489$	−5244.00	−0.484953
$490$	0	0
$491$	852.000i	0.0783100i	0.999233	+	0.0391550i	$0.0124666\pi$
$491$	852.000i	0.0783100i	−0.999233	+	0.0391550i	$0.987533\pi$
$492$	0	0
$493$	8791.89i	0.803178i
$494$	0	0
$495$	−1496.49	−0.135883
$496$	0	0
$497$	−168.000	−0.0151626
$498$	0	0
$499$	11156.0i	1.00082i	0.865787	+	0.500412i	$0.166818\pi$
$499$	11156.0i	1.00082i	−0.865787	+	0.500412i	$0.833182\pi$
$500$	0	0
$501$	41.5692i	0.00370694i
$502$	0	0
$503$	14999.6	1.32962	0.664808	−	0.747014i	$-0.268513\pi$
$503$	14999.6	1.32962	0.664808	+	0.747014i	$0.268513\pi$
$504$	0	0
$505$	−3156.00	−0.278099
$506$	0	0
$507$	− 1407.00i	− 0.123249i
$508$	0	0
$509$	− 9287.26i	− 0.808743i	−0.914595	−	0.404372i	$-0.867490\pi$
$509$	− 9287.26i	− 0.808743i	0.914595	−	0.404372i	$-0.132510\pi$
$510$	0	0
$511$	24394.2	2.11181
$512$	0	0
$513$	−108.000	−0.00929496
$514$	0	0
$515$	− 1572.00i	− 0.134506i
$516$	0	0
$517$	− 24276.4i	− 2.06514i
$518$	0	0
$519$	1797.87	0.152057
$520$	0	0
$521$	−2766.00	−0.232592	−0.116296	−	0.993215i	$-0.537102\pi$
$521$	−2766.00	−0.232592	−0.116296	+	0.993215i	$0.537102\pi$
$522$	0	0
$523$	− 18988.0i	− 1.58755i	−0.608213	−	0.793774i	$-0.708113\pi$
$523$	− 18988.0i	− 1.58755i	0.608213	−	0.793774i	$-0.291887\pi$
$524$	0	0
$525$	8220.31i	0.683360i
$526$	0	0
$527$	−3180.05	−0.262856
$528$	0	0
$529$	17833.0	1.46569
$530$	0	0
$531$	− 2268.00i	− 0.185354i
$532$	0	0
$533$	− 12221.4i	− 0.993181i
$534$	0	0
$535$	−4115.35	−0.332565
$536$	0	0
$537$	9684.00	0.778204
$538$	0	0
$539$	11760.0i	0.939776i
$540$	0	0
$541$	− 12997.3i	− 1.03290i	−0.856318	−	0.516449i	$-0.827254\pi$
$541$	− 12997.3i	− 1.03290i	0.856318	−	0.516449i	$-0.172746\pi$
$542$	0	0
$543$	6069.11	0.479651
$544$	0	0
$545$	1632.00	0.128270
$546$	0	0
$547$	− 21188.0i	− 1.65619i	−0.560591	−	0.828093i	$-0.689426\pi$
$547$	− 21188.0i	− 1.65619i	0.560591	−	0.828093i	$-0.310574\pi$
$548$	0	0
$549$	810.600i	0.0630156i
$550$	0	0
$551$	−651.251	−0.0503525
$552$	0	0
$553$	−32508.0	−2.49978
$554$	0	0
$555$	− 3384.00i	− 0.258816i
$556$	0	0
$557$	− 12231.7i	− 0.930477i	−0.885185	−	0.465238i	$-0.845969\pi$
$557$	− 12231.7i	− 0.930477i	0.885185	−	0.465238i	$-0.154031\pi$
$558$	0	0
$559$	−7815.01	−0.591306
$560$	0	0
$561$	7776.00	0.585210
$562$	0	0
$563$	− 504.000i	− 0.0377284i	−0.999822	−	0.0188642i	$-0.993995\pi$
$563$	− 504.000i	− 0.0377284i	0.999822	−	0.0188642i	$-0.00600501\pi$
$564$	0	0
$565$	1351.00i	0.100596i
$566$	0	0
$567$	−1964.15	−0.145479
$568$	0	0
$569$	−20358.0	−1.49992	−0.749958	−	0.661486i	$-0.769926\pi$
$569$	−20358.0	−1.49992	−0.749958	+	0.661486i	$0.769926\pi$
$570$	0	0
$571$	13300.0i	0.974760i	0.873190	+	0.487380i	$0.162047\pi$
$571$	13300.0i	0.974760i	−0.873190	+	0.487380i	$0.837953\pi$
$572$	0	0
$573$	− 10433.9i	− 0.760700i
$574$	0	0
$575$	−19572.2	−1.41951
$576$	0	0
$577$	4606.00	0.332323	0.166161	−	0.986099i	$-0.446863\pi$
$577$	4606.00	0.332323	0.166161	+	0.986099i	$0.446863\pi$
$578$	0	0
$579$	2298.00i	0.164942i
$580$	0	0
$581$	− 17459.1i	− 1.24669i
$582$	0	0
$583$	35749.5	2.53961
$584$	0	0
$585$	−1296.00	−0.0915949
$586$	0	0
$587$	− 13980.0i	− 0.982992i	−0.870880	−	0.491496i	$-0.836450\pi$
$587$	− 13980.0i	− 0.982992i	0.870880	−	0.491496i	$-0.163550\pi$
$588$	0	0
$589$	− 235.559i	− 0.0164788i
$590$	0	0
$591$	8698.36	0.605419
$592$	0	0
$593$	−12486.0	−0.864652	−0.432326	−	0.901717i	$-0.642307\pi$
$593$	−12486.0	−0.864652	−0.432326	+	0.901717i	$0.642307\pi$
$594$	0	0
$595$	4536.00i	0.312534i
$596$	0	0
$597$	5206.54i	0.356934i
$598$	0	0
$599$	−8778.03	−0.598766	−0.299383	−	0.954133i	$-0.596781\pi$
$599$	−8778.03	−0.598766	−0.299383	+	0.954133i	$0.596781\pi$
$600$	0	0
$601$	6986.00	0.474151	0.237076	−	0.971491i	$-0.423811\pi$
$601$	6986.00	0.474151	0.237076	+	0.971491i	$0.423811\pi$
$602$	0	0
$603$	5652.00i	0.381704i
$604$	0	0
$605$	3370.57i	0.226501i
$606$	0	0
$607$	−4596.86	−0.307382	−0.153691	−	0.988119i	$-0.549116\pi$
$607$	−4596.86	−0.307382	−0.153691	+	0.988119i	$0.549116\pi$
$608$	0	0
$609$	−11844.0	−0.788084
$610$	0	0
$611$	− 21024.0i	− 1.39205i
$612$	0	0
$613$	11092.1i	0.730838i	0.930843	+	0.365419i	$0.119074\pi$
$613$	11092.1i	0.730838i	−0.930843	+	0.365419i	$0.880926\pi$
$614$	0	0
$615$	3055.34	0.200330
$616$	0	0
$617$	−2850.00	−0.185959	−0.0929795	−	0.995668i	$-0.529639\pi$
$617$	−2850.00	−0.185959	−0.0929795	+	0.995668i	$0.529639\pi$
$618$	0	0
$619$	20116.0i	1.30619i	0.757277	+	0.653094i	$0.226529\pi$
$619$	20116.0i	1.30619i	−0.757277	+	0.653094i	$0.773471\pi$
$620$	0	0
$621$	− 4676.54i	− 0.302195i
$622$	0	0
$623$	35936.6	2.31103
$624$	0	0
$625$	11269.0	0.721216
$626$	0	0
$627$	576.000i	0.0366878i
$628$	0	0
$629$	17583.8i	1.11464i
$630$	0	0
$631$	−7271.15	−0.458732	−0.229366	−	0.973340i	$-0.573665\pi$
$631$	−7271.15	−0.458732	−0.229366	+	0.973340i	$0.573665\pi$
$632$	0	0
$633$	3300.00	0.207209
$634$	0	0
$635$	2100.00i	0.131238i
$636$	0	0
$637$	10184.5i	0.633474i
$638$	0	0
$639$	−62.3538	−0.00386022
$640$	0	0
$641$	7230.00	0.445504	0.222752	−	0.974875i	$-0.428496\pi$
$641$	7230.00	0.445504	0.222752	+	0.974875i	$0.428496\pi$
$642$	0	0
$643$	− 2948.00i	− 0.180805i	−0.995905	−	0.0904026i	$-0.971185\pi$
$643$	− 2948.00i	− 0.180805i	0.995905	−	0.0904026i	$-0.0288154\pi$
$644$	0	0
$645$	− 1953.75i	− 0.119270i
$646$	0	0
$647$	−17161.2	−1.04277	−0.521387	−	0.853320i	$-0.674585\pi$
$647$	−17161.2	−1.04277	−0.521387	+	0.853320i	$0.674585\pi$
$648$	0	0
$649$	−12096.0	−0.731602
$650$	0	0
$651$	− 4284.00i	− 0.257916i
$652$	0	0
$653$	10381.9i	0.622168i	0.950382	+	0.311084i	$0.100692\pi$
$653$	10381.9i	0.622168i	−0.950382	+	0.311084i	$0.899308\pi$
$654$	0	0
$655$	4780.46	0.285173
$656$	0	0
$657$	9054.00	0.537641
$658$	0	0
$659$	10308.0i	0.609321i	0.952461	+	0.304661i	$0.0985430\pi$
$659$	10308.0i	0.609321i	−0.952461	+	0.304661i	$0.901457\pi$
$660$	0	0
$661$	− 15803.2i	− 0.929916i	−0.885333	−	0.464958i	$-0.846069\pi$
$661$	− 15803.2i	− 0.929916i	0.885333	−	0.464958i	$-0.153931\pi$
$662$	0	0
$663$	6734.21	0.394472
$664$	0	0
$665$	−336.000	−0.0195933
$666$	0	0
$667$	− 28200.0i	− 1.63704i
$668$	0	0
$669$	− 1174.33i	− 0.0678658i
$670$	0	0
$671$	4323.20	0.248726
$672$	0	0
$673$	30910.0	1.77042	0.885210	−	0.465191i	$-0.154014\pi$
$673$	30910.0	1.77042	0.885210	+	0.465191i	$0.154014\pi$
$674$	0	0
$675$	3051.00i	0.173975i
$676$	0	0
$677$	− 14802.1i	− 0.840312i	−0.907452	−	0.420156i	$-0.861975\pi$
$677$	− 14802.1i	− 0.840312i	0.907452	−	0.420156i	$-0.138025\pi$
$678$	0	0
$679$	−44181.2	−2.49708
$680$	0	0
$681$	−10008.0	−0.563153
$682$	0	0
$683$	528.000i	0.0295803i	0.999891	+	0.0147902i	$0.00470803\pi$
$683$	528.000i	0.0295803i	−0.999891	+	0.0147902i	$0.995292\pi$
$684$	0	0
$685$	4011.43i	0.223750i
$686$	0	0
$687$	−17999.5	−0.999596
$688$	0	0
$689$	30960.0	1.71188
$690$	0	0
$691$	9052.00i	0.498342i	0.968460	+	0.249171i	$0.0801581\pi$
$691$	9052.00i	0.498342i	−0.968460	+	0.249171i	$0.919842\pi$
$692$	0	0
$693$	10475.4i	0.574212i
$694$	0	0
$695$	4087.64	0.223098
$696$	0	0
$697$	−15876.0	−0.862764
$698$	0	0
$699$	− 954.000i	− 0.0516217i
$700$	0	0
$701$	32600.7i	1.75650i	0.478197	+	0.878252i	$0.341290\pi$
$701$	32600.7i	1.75650i	−0.478197	+	0.878252i	$0.658710\pi$
$702$	0	0
$703$	−1302.50	−0.0698788
$704$	0	0
$705$	5256.00	0.280784
$706$	0	0
$707$	22092.0i	1.17518i
$708$	0	0
$709$	27227.8i	1.44226i	0.692799	+	0.721130i	$0.256377\pi$
$709$	27227.8i	1.44226i	−0.692799	+	0.721130i	$0.743623\pi$
$710$	0	0
$711$	−12065.5	−0.636414
$712$	0	0
$713$	10200.0	0.535755
$714$	0	0
$715$	6912.00i	0.361530i
$716$	0	0
$717$	− 2577.29i	− 0.134241i
$718$	0	0
$719$	−685.892	−0.0355764	−0.0177882	−	0.999842i	$-0.505662\pi$
$719$	−685.892	−0.0355764	−0.0177882	+	0.999842i	$0.505662\pi$
$720$	0	0
$721$	−11004.0	−0.568392
$722$	0	0
$723$	8130.00i	0.418199i
$724$	0	0
$725$	18397.8i	0.942453i
$726$	0	0
$727$	20192.2	1.03011	0.515054	−	0.857158i	$-0.327772\pi$
$727$	20192.2	1.03011	0.515054	+	0.857158i	$0.327772\pi$
$728$	0	0
$729$	−729.000	−0.0370370
$730$	0	0
$731$	10152.0i	0.513660i
$732$	0	0
$733$	− 35236.8i	− 1.77558i	−0.460246	−	0.887792i	$-0.652239\pi$
$733$	− 35236.8i	− 1.77558i	0.460246	−	0.887792i	$-0.347761\pi$
$734$	0	0
$735$	−2546.11	−0.127775
$736$	0	0
$737$	30144.0	1.50661
$738$	0	0
$739$	13940.0i	0.693899i	0.937884	+	0.346949i	$0.112782\pi$
$739$	13940.0i	0.693899i	−0.937884	+	0.346949i	$0.887218\pi$
$740$	0	0
$741$	498.831i	0.0247301i
$742$	0	0
$743$	11002.0	0.543235	0.271618	−	0.962405i	$-0.412441\pi$
$743$	11002.0	0.543235	0.271618	+	0.962405i	$0.412441\pi$
$744$	0	0
$745$	7524.00	0.370011
$746$	0	0
$747$	− 6480.00i	− 0.317391i
$748$	0	0
$749$	28807.5i	1.40534i
$750$	0	0
$751$	−33342.0	−1.62006	−0.810031	−	0.586388i	$-0.800550\pi$
$751$	−33342.0	−1.62006	−0.810031	+	0.586388i	$0.800550\pi$
$752$	0	0
$753$	−15408.0	−0.745682
$754$	0	0
$755$	− 492.000i	− 0.0237162i
$756$	0	0
$757$	17445.2i	0.837592i	0.908080	+	0.418796i	$0.137548\pi$
$757$	17445.2i	0.837592i	−0.908080	+	0.418796i	$0.862452\pi$
$758$	0	0
$759$	−24941.5	−1.19278
$760$	0	0
$761$	−30222.0	−1.43961	−0.719807	−	0.694174i	$-0.755770\pi$
$761$	−30222.0	−1.43961	−0.719807	+	0.694174i	$0.755770\pi$
$762$	0	0
$763$	− 11424.0i	− 0.542040i
$764$	0	0
$765$	1683.55i	0.0795673i
$766$	0	0
$767$	−10475.4	−0.493150
$768$	0	0
$769$	−11758.0	−0.551371	−0.275686	−	0.961248i	$-0.588905\pi$
$769$	−11758.0	−0.551371	−0.275686	+	0.961248i	$0.588905\pi$
$770$	0	0
$771$	13194.0i	0.616304i
$772$	0	0
$773$	1874.08i	0.0872004i	0.999049	+	0.0436002i	$0.0138828\pi$
$773$	1874.08i	0.0872004i	−0.999049	+	0.0436002i	$0.986117\pi$
$774$	0	0
$775$	−6654.54	−0.308436
$776$	0	0
$777$	−23688.0	−1.09370
$778$	0	0
$779$	− 1176.00i	− 0.0540880i
$780$	0	0
$781$	332.554i	0.0152365i
$782$	0	0
$783$	−4395.94	−0.200636
$784$	0	0
$785$	4632.00	0.210603
$786$	0	0
$787$	31012.0i	1.40465i	0.711857	+	0.702324i	$0.247854\pi$
$787$	31012.0i	1.40465i	−0.711857	+	0.702324i	$0.752146\pi$
$788$	0	0
$789$	− 20452.1i	− 0.922829i
$790$	0	0
$791$	9457.00	0.425097
$792$	0	0
$793$	3744.00	0.167659
$794$	0	0
$795$	7740.00i	0.345295i
$796$	0	0
$797$	− 7091.02i	− 0.315153i	−0.987507	−	0.157576i	$-0.949632\pi$
$797$	− 7091.02i	− 0.315153i	0.987507	−	0.157576i	$-0.0503680\pi$
$798$	0	0
$799$	−27311.0	−1.20925
$800$	0	0
$801$	13338.0	0.588358
$802$	0	0
$803$	− 48288.0i	− 2.12210i
$804$	0	0
$805$	− 14549.2i	− 0.637010i
$806$	0	0
$807$	13873.7	0.605178
$808$	0	0
$809$	40650.0	1.76660	0.883299	−	0.468810i	$-0.155317\pi$
$809$	40650.0	1.76660	0.883299	+	0.468810i	$0.155317\pi$
$810$	0	0
$811$	8372.00i	0.362492i	0.983438	+	0.181246i	$0.0580130\pi$
$811$	8372.00i	0.362492i	−0.983438	+	0.181246i	$0.941987\pi$
$812$	0	0
$813$	− 11649.8i	− 0.502553i
$814$	0	0
$815$	−6055.25	−0.260253
$816$	0	0
$817$	−752.000	−0.0322021
$818$	0	0
$819$	9072.00i	0.387059i
$820$	0	0
$821$	9370.39i	0.398330i	0.979966	+	0.199165i	$0.0638230\pi$
$821$	9370.39i	0.398330i	−0.979966	+	0.199165i	$0.936177\pi$
$822$	0	0
$823$	−21668.0	−0.917737	−0.458868	−	0.888504i	$-0.651745\pi$
$823$	−21668.0	−0.917737	−0.458868	+	0.888504i	$0.651745\pi$
$824$	0	0
$825$	16272.0	0.686689
$826$	0	0
$827$	6684.00i	0.281046i	0.990077	+	0.140523i	$0.0448785\pi$
$827$	6684.00i	0.281046i	−0.990077	+	0.140523i	$0.955122\pi$
$828$	0	0
$829$	24359.6i	1.02056i	0.860009	+	0.510279i	$0.170458\pi$
$829$	24359.6i	1.02056i	−0.860009	+	0.510279i	$0.829542\pi$
$830$	0	0
$831$	4572.61	0.190881
$832$	0	0
$833$	13230.0	0.550291
$834$	0	0
$835$	48.0000i	0.00198935i
$836$	0	0
$837$	− 1590.02i	− 0.0656622i
$838$	0	0
$839$	19613.7	0.807082	0.403541	−	0.914962i	$-0.367779\pi$
$839$	19613.7	0.807082	0.403541	+	0.914962i	$0.367779\pi$
$840$	0	0
$841$	−2119.00	−0.0868834
$842$	0	0
$843$	− 13194.0i	− 0.539058i
$844$	0	0
$845$	− 1624.66i	− 0.0661422i
$846$	0	0
$847$	23594.0	0.957142
$848$	0	0
$849$	13116.0	0.530200
$850$	0	0
$851$	− 56400.0i	− 2.27188i
$852$	0	0
$853$	− 20569.8i	− 0.825671i	−0.910806	−	0.412836i	$-0.864538\pi$
$853$	− 20569.8i	− 0.825671i	0.910806	−	0.412836i	$-0.135462\pi$
$854$	0	0
$855$	−124.708	−0.00498820
$856$	0	0
$857$	−27222.0	−1.08505	−0.542524	−	0.840040i	$-0.682531\pi$
$857$	−27222.0	−1.08505	−0.542524	+	0.840040i	$0.682531\pi$
$858$	0	0
$859$	3548.00i	0.140927i	0.997514	+	0.0704634i	$0.0224478\pi$
$859$	3548.00i	0.140927i	−0.997514	+	0.0704634i	$0.977552\pi$
$860$	0	0
$861$	− 21387.4i	− 0.846550i
$862$	0	0
$863$	45047.2	1.77685	0.888426	−	0.459019i	$-0.151799\pi$
$863$	45047.2	1.77685	0.888426	+	0.459019i	$0.151799\pi$
$864$	0	0
$865$	2076.00	0.0816024
$866$	0	0
$867$	5991.00i	0.234677i
$868$	0	0
$869$	64349.2i	2.51196i
$870$	0	0
$871$	26105.5	1.01556
$872$	0	0
$873$	−16398.0	−0.635725
$874$	0	0
$875$	19992.0i	0.772403i
$876$	0	0
$877$	− 29022.2i	− 1.11746i	−0.829350	−	0.558729i	$-0.811289\pi$
$877$	− 29022.2i	− 1.11746i	0.829350	−	0.558729i	$-0.188711\pi$
$878$	0	0
$879$	−10714.5	−0.411138
$880$	0	0
$881$	−48318.0	−1.84776	−0.923879	−	0.382685i	$-0.875000\pi$
$881$	−48318.0	−1.84776	−0.923879	+	0.382685i	$0.875000\pi$
$882$	0	0
$883$	14380.0i	0.548047i	0.961723	+	0.274024i	$0.0883546\pi$
$883$	14380.0i	0.548047i	−0.961723	+	0.274024i	$0.911645\pi$
$884$	0	0
$885$	− 2618.86i	− 0.0994712i
$886$	0	0
$887$	34086.8	1.29033	0.645164	−	0.764044i	$-0.276789\pi$
$887$	34086.8	1.29033	0.645164	+	0.764044i	$0.276789\pi$
$888$	0	0
$889$	14700.0	0.554581
$890$	0	0
$891$	3888.00i	0.146187i
$892$	0	0
$893$	− 2023.04i	− 0.0758100i
$894$	0	0
$895$	11182.1	0.417628
$896$	0	0
$897$	−21600.0	−0.804017
$898$	0	0
$899$	− 9588.00i	− 0.355704i
$900$	0	0
$901$	− 40218.2i	− 1.48708i
$902$	0	0
$903$	−13676.3	−0.504007
$904$	0	0
$905$	7008.00	0.257408
$906$	0	0
$907$	− 31252.0i	− 1.14411i	−0.820216	−	0.572054i	$-0.806147\pi$
$907$	− 31252.0i	− 1.14411i	0.820216	−	0.572054i	$-0.193853\pi$
$908$	0	0
$909$	8199.53i	0.299187i
$910$	0	0
$911$	−13080.4	−0.475713	−0.237857	−	0.971300i	$-0.576445\pi$
$911$	−13080.4	−0.475713	−0.237857	+	0.971300i	$0.576445\pi$
$912$	0	0
$913$	−34560.0	−1.25276
$914$	0	0
$915$	936.000i	0.0338177i
$916$	0	0
$917$	− 33463.2i	− 1.20507i
$918$	0	0
$919$	8843.85	0.317445	0.158722	−	0.987323i	$-0.449263\pi$
$919$	8843.85	0.317445	0.158722	+	0.987323i	$0.449263\pi$
$920$	0	0
$921$	12516.0	0.447792
$922$	0	0
$923$	288.000i	0.0102705i
$924$	0	0
$925$	36795.7i	1.30793i
$926$	0	0
$927$	−4084.18	−0.144705
$928$	0	0
$929$	29622.0	1.04614	0.523071	−	0.852289i	$-0.324786\pi$
$929$	29622.0	1.04614	0.523071	+	0.852289i	$0.324786\pi$
$930$	0	0
$931$	980.000i	0.0344986i
$932$	0	0
$933$	− 19412.8i	− 0.681187i
$934$	0	0
$935$	8978.95	0.314057
$936$	0	0
$937$	−23210.0	−0.809218	−0.404609	−	0.914490i	$-0.632592\pi$
$937$	−23210.0	−0.809218	−0.404609	+	0.914490i	$0.632592\pi$
$938$	0	0
$939$	222.000i	0.00771533i
$940$	0	0
$941$	− 19728.1i	− 0.683439i	−0.939802	−	0.341720i	$-0.888991\pi$
$941$	− 19728.1i	− 0.683439i	0.939802	−	0.341720i	$-0.111009\pi$
$942$	0	0
$943$	50922.3	1.75849
$944$	0	0
$945$	−2268.00	−0.0780720
$946$	0	0
$947$	− 1236.00i	− 0.0424125i	−0.999775	−	0.0212062i	$-0.993249\pi$
$947$	− 1236.00i	− 0.0424125i	0.999775	−	0.0212062i	$-0.00675066\pi$
$948$	0	0
$949$	− 41818.6i	− 1.43044i
$950$	0	0
$951$	5892.44	0.200920
$952$	0	0
$953$	18402.0	0.625498	0.312749	−	0.949836i	$-0.398750\pi$
$953$	18402.0	0.625498	0.312749	+	0.949836i	$0.398750\pi$
$954$	0	0
$955$	− 12048.0i	− 0.408235i
$956$	0	0
$957$	23445.0i	0.791923i
$958$	0	0
$959$	28080.0	0.945517
$960$	0	0
$961$	−26323.0	−0.883589
$962$	0	0
$963$	10692.0i	0.357783i
$964$	0	0
$965$	2653.50i	0.0885174i
$966$	0	0
$967$	−41836.0	−1.39127	−0.695633	−	0.718398i	$-0.744876\pi$
$967$	−41836.0	−1.39127	−0.695633	+	0.718398i	$0.744876\pi$
$968$	0	0
$969$	648.000	0.0214827
$970$	0	0
$971$	− 8832.00i	− 0.291897i	−0.989292	−	0.145949i	$-0.953377\pi$
$971$	− 8832.00i	− 0.291897i	0.989292	−	0.145949i	$-0.0466234\pi$
$972$	0	0
$973$	− 28613.5i	− 0.942761i
$974$	0	0
$975$	14092.0	0.462876
$976$	0	0
$977$	−23034.0	−0.754271	−0.377136	−	0.926158i	$-0.623091\pi$
$977$	−23034.0	−0.754271	−0.377136	+	0.926158i	$0.623091\pi$
$978$	0	0
$979$	− 71136.0i	− 2.32228i
$980$	0	0
$981$	− 4240.06i	− 0.137997i
$982$	0	0
$983$	−17791.6	−0.577278	−0.288639	−	0.957438i	$-0.593203\pi$
$983$	−17791.6	−0.577278	−0.288639	+	0.957438i	$0.593203\pi$
$984$	0	0
$985$	10044.0	0.324902
$986$	0	0
$987$	− 36792.0i	− 1.18653i
$988$	0	0
$989$	− 32562.6i	− 1.04695i
$990$	0	0
$991$	3072.66	0.0984926	0.0492463	−	0.998787i	$-0.484318\pi$
$991$	3072.66	0.0984926	0.0492463	+	0.998787i	$0.484318\pi$
$992$	0	0
$993$	−22668.0	−0.724418
$994$	0	0
$995$	6012.00i	0.191551i
$996$	0	0
$997$	52273.3i	1.66049i	0.557396	+	0.830247i	$0.311800\pi$
$997$	52273.3i	1.66049i	−0.557396	+	0.830247i	$0.688200\pi$
$998$	0	0
$999$	−8791.89	−0.278442

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	192.4.d.c.97.1	✓	4
3.2	odd	2		576.4.d.c.289.3		4
4.3	odd	2	inner	192.4.d.c.97.3	yes	4
8.3	odd	2	inner	192.4.d.c.97.2	yes	4
8.5	even	2	inner	192.4.d.c.97.4	yes	4
12.11	even	2		576.4.d.c.289.4		4
16.3	odd	4		768.4.a.f.1.1		2
16.5	even	4		768.4.a.f.1.2		2
16.11	odd	4		768.4.a.o.1.2		2
16.13	even	4		768.4.a.o.1.1		2
24.5	odd	2		576.4.d.c.289.1		4
24.11	even	2		576.4.d.c.289.2		4
48.5	odd	4		2304.4.a.bk.1.1		2
48.11	even	4		2304.4.a.x.1.1		2
48.29	odd	4		2304.4.a.x.1.2		2
48.35	even	4		2304.4.a.bk.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
192.4.d.c.97.1	✓	4	1.1	even	1	trivial
192.4.d.c.97.2	yes	4	8.3	odd	2	inner
192.4.d.c.97.3	yes	4	4.3	odd	2	inner
192.4.d.c.97.4	yes	4	8.5	even	2	inner
576.4.d.c.289.1		4	24.5	odd	2
576.4.d.c.289.2		4	24.11	even	2
576.4.d.c.289.3		4	3.2	odd	2
576.4.d.c.289.4		4	12.11	even	2
768.4.a.f.1.1		2	16.3	odd	4
768.4.a.f.1.2		2	16.5	even	4
768.4.a.o.1.1		2	16.13	even	4
768.4.a.o.1.2		2	16.11	odd	4
2304.4.a.x.1.1		2	48.11	even	4
2304.4.a.x.1.2		2	48.29	odd	4
2304.4.a.bk.1.1		2	48.5	odd	4
2304.4.a.bk.1.2		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:	\( N \)	\(=\)	\( 192 = 2^{6} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	192.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-3.00000i q^{3} -3.46410i q^{5} -24.2487 q^{7} -9.00000 q^{9} +O(q^{10})\)	\(q-3.00000i q^{3} -3.46410i q^{5} -24.2487 q^{7} -9.00000 q^{9} +48.0000i q^{11} +41.5692i q^{13} -10.3923 q^{15} +54.0000 q^{17} +4.00000i q^{19} +72.7461i q^{21} -173.205 q^{23} +113.000 q^{25} +27.0000i q^{27} +162.813i q^{29} -58.8897 q^{31} +144.000 q^{33} +84.0000i q^{35} +325.626i q^{37} +124.708 q^{39} -294.000 q^{41} +188.000i q^{43} +31.1769i q^{45} -505.759 q^{47} +245.000 q^{49} -162.000i q^{51} -744.782i q^{53} +166.277 q^{55} +12.0000 q^{57} +252.000i q^{59} -90.0666i q^{61} +218.238 q^{63} +144.000 q^{65} -628.000i q^{67} +519.615i q^{69} +6.92820 q^{71} -1006.00 q^{73} -339.000i q^{75} -1163.94i q^{77} +1340.61 q^{79} +81.0000 q^{81} +720.000i q^{83} -187.061i q^{85} +488.438 q^{87} -1482.00 q^{89} -1008.00i q^{91} +176.669i q^{93} +13.8564 q^{95} +1822.00 q^{97} -432.000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 36 q^{9}+O(q^{10}) \) 4 * q - 36 * q^9	\( 4 q - 36 q^{9} + 216 q^{17} + 452 q^{25} + 576 q^{33} - 1176 q^{41} + 980 q^{49} + 48 q^{57} + 576 q^{65} - 4024 q^{73} + 324 q^{81} - 5928 q^{89} + 7288 q^{97}+O(q^{100}) \) 4 * q - 36 * q^9 + 216 * q^17 + 452 * q^25 + 576 * q^33 - 1176 * q^41 + 980 * q^49 + 48 * q^57 + 576 * q^65 - 4024 * q^73 + 324 * q^81 - 5928 * q^89 + 7288 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more