LMFDB - Embedded newform 192.4.d.a.97.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [192,4,Mod(97,192)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

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:traces = [4,0,0,0,0,0,0,0,-36,0,0,0,0,0,0,0,-360] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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			97.2
Root			$-0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	192.97
Dual form			192.4.d.a.97.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000i q^{3} +10.3923i q^{5} +3.46410 q^{7} -9.00000 q^{9} +55.4256i q^{13} +31.1769 q^{15} -90.0000 q^{17} +116.000i q^{19} -10.3923i q^{21} +103.923 q^{23} +17.0000 q^{25} +27.0000i q^{27} +259.808i q^{29} +301.377 q^{31} +36.0000i q^{35} +34.6410i q^{37} +166.277 q^{39} -54.0000 q^{41} -20.0000i q^{43} -93.5307i q^{45} -394.908 q^{47} -331.000 q^{49} +270.000i q^{51} +488.438i q^{53} +348.000 q^{57} -324.000i q^{59} -575.041i q^{61} -31.1769 q^{63} -576.000 q^{65} -116.000i q^{67} -311.769i q^{69} -1101.58 q^{71} +1106.00 q^{73} -51.0000i q^{75} +148.956 q^{79} +81.0000 q^{81} -1152.00i q^{83} -935.307i q^{85} +779.423 q^{87} +918.000 q^{89} +192.000i q^{91} -904.131i q^{93} -1205.51 q^{95} +190.000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 36 q^{9} - 360 q^{17} + 68 q^{25} - 216 q^{41} - 1324 q^{49} + 1392 q^{57} - 2304 q^{65} + 4424 q^{73} + 324 q^{81} + 3672 q^{89} + 760 q^{97}+O(q^{100}) $ 4 * q - 36 * q^9 - 360 * q^17 + 68 * q^25 - 216 * q^41 - 1324 * q^49 + 1392 * q^57 - 2304 * q^65 + 4424 * q^73 + 324 * q^81 + 3672 * q^89 + 760 * q^97

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
$5$	10.3923i	0.929516i	0.885438	+	0.464758i	$0.153859\pi$
$5$	10.3923i	0.929516i	−0.885438	+	0.464758i	$0.846141\pi$
$6$	0	0
$7$	3.46410	0.187044	0.0935220	−	0.995617i	$-0.470187\pi$
$7$	3.46410	0.187044	0.0935220	+	0.995617i	$0.470187\pi$
$8$	0	0
$9$	−9.00000	−0.333333
$10$	0	0
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	0	0
$13$	55.4256i	1.18248i	0.806494	+	0.591242i	$0.201362\pi$
$13$	55.4256i	1.18248i	−0.806494	+	0.591242i	$0.798638\pi$
$14$	0	0
$15$	31.1769	0.536656
$16$	0	0
$17$	−90.0000	−1.28401	−0.642006	−	0.766700i	$-0.721898\pi$
$17$	−90.0000	−1.28401	−0.642006	+	0.766700i	$0.721898\pi$
$18$	0	0
$19$	116.000i	1.40064i	0.713827	+	0.700322i	$0.246960\pi$
$19$	116.000i	1.40064i	−0.713827	+	0.700322i	$0.753040\pi$
$20$	0	0
$21$	− 10.3923i	− 0.107990i
$22$	0	0
$23$	103.923	0.942150	0.471075	−	0.882093i	$-0.343866\pi$
$23$	103.923	0.942150	0.471075	+	0.882093i	$0.343866\pi$
$24$	0	0
$25$	17.0000	0.136000
$26$	0	0
$27$	27.0000i	0.192450i
$28$	0	0
$29$	259.808i	1.66362i	0.555058	+	0.831811i	$0.312696\pi$
$29$	259.808i	1.66362i	−0.555058	+	0.831811i	$0.687304\pi$
$30$	0	0
$31$	301.377	1.74609	0.873046	−	0.487637i	$-0.162141\pi$
$31$	301.377	1.74609	0.873046	+	0.487637i	$0.162141\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	36.0000i	0.173860i
$36$	0	0
$37$	34.6410i	0.153918i	0.997034	+	0.0769588i	$0.0245210\pi$
$37$	34.6410i	0.153918i	−0.997034	+	0.0769588i	$0.975479\pi$
$38$	0	0
$39$	166.277	0.682708
$40$	0	0
$41$	−54.0000	−0.205692	−0.102846	−	0.994697i	$-0.532795\pi$
$41$	−54.0000	−0.205692	−0.102846	+	0.994697i	$0.532795\pi$
$42$	0	0
$43$	− 20.0000i	− 0.0709296i	−0.999371	−	0.0354648i	$-0.988709\pi$
$43$	− 20.0000i	− 0.0709296i	0.999371	−	0.0354648i	$-0.0112912\pi$
$44$	0	0
$45$	− 93.5307i	− 0.309839i
$46$	0	0
$47$	−394.908	−1.22560	−0.612800	−	0.790238i	$-0.709957\pi$
$47$	−394.908	−1.22560	−0.612800	+	0.790238i	$0.709957\pi$
$48$	0	0
$49$	−331.000	−0.965015
$50$	0	0
$51$	270.000i	0.741325i
$52$	0	0
$53$	488.438i	1.26589i	0.774197	+	0.632945i	$0.218154\pi$
$53$	488.438i	1.26589i	−0.774197	+	0.632945i	$0.781846\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	348.000	0.808662
$58$	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
$61$	− 575.041i	− 1.20699i	−0.797366	−	0.603495i	$-0.793774\pi$
$61$	− 575.041i	− 1.20699i	0.797366	−	0.603495i	$-0.206226\pi$
$62$	0	0
$63$	−31.1769	−0.0623480
$64$	0	0
$65$	−576.000	−1.09914
$66$	0	0
$67$	− 116.000i	− 0.211517i	−0.994392	−	0.105759i	$-0.966273\pi$
$67$	− 116.000i	− 0.211517i	0.994392	−	0.105759i	$-0.0337271\pi$
$68$	0	0
$69$	− 311.769i	− 0.543951i
$70$	0	0
$71$	−1101.58	−1.84132	−0.920662	−	0.390361i	$-0.872350\pi$
$71$	−1101.58	−1.84132	−0.920662	+	0.390361i	$0.872350\pi$
$72$	0	0
$73$	1106.00	1.77325	0.886627	−	0.462486i	$-0.153042\pi$
$73$	1106.00	1.77325	0.886627	+	0.462486i	$0.153042\pi$
$74$	0	0
$75$	− 51.0000i	− 0.0785196i
$76$	0	0
$77$	0	0
$78$	0	0
$79$	148.956	0.212138	0.106069	−	0.994359i	$-0.466174\pi$
$79$	148.956	0.212138	0.106069	+	0.994359i	$0.466174\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	− 1152.00i	− 1.52348i	−0.647886	−	0.761738i	$-0.724347\pi$
$83$	− 1152.00i	− 1.52348i	0.647886	−	0.761738i	$-0.275653\pi$
$84$	0	0
$85$	− 935.307i	− 1.19351i
$86$	0	0
$87$	779.423	0.960493
$88$	0	0
$89$	918.000	1.09335	0.546673	−	0.837346i	$-0.315894\pi$
$89$	918.000	1.09335	0.546673	+	0.837346i	$0.315894\pi$
$90$	0	0
$91$	192.000i	0.221177i
$92$	0	0
$93$	− 904.131i	− 1.00811i
$94$	0	0
$95$	−1205.51	−1.30192
$96$	0	0
$97$	190.000	0.198882	0.0994411	−	0.995043i	$-0.468295\pi$
$97$	190.000	0.198882	0.0994411	+	0.995043i	$0.468295\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 10.3923i	− 0.0102383i	−0.999987	−	0.00511917i	$-0.998371\pi$
$101$	− 10.3923i	− 0.0102383i	0.999987	−	0.00511917i	$-0.00162949\pi$
$102$	0	0
$103$	−793.279	−0.758875	−0.379438	−	0.925217i	$-0.623882\pi$
$103$	−793.279	−0.758875	−0.379438	+	0.925217i	$0.623882\pi$
$104$	0	0
$105$	108.000	0.100378
$106$	0	0
$107$	252.000i	0.227680i	0.993499	+	0.113840i	$0.0363151\pi$
$107$	252.000i	0.227680i	−0.993499	+	0.113840i	$0.963685\pi$
$108$	0	0
$109$	− 457.261i	− 0.401814i	−0.979610	−	0.200907i	$-0.935611\pi$
$109$	− 457.261i	− 0.401814i	0.979610	−	0.200907i	$-0.0643889\pi$
$110$	0	0
$111$	103.923	0.0888643
$112$	0	0
$113$	−2214.00	−1.84315	−0.921573	−	0.388204i	$-0.873096\pi$
$113$	−2214.00	−1.84315	−0.921573	+	0.388204i	$0.873096\pi$
$114$	0	0
$115$	1080.00i	0.875744i
$116$	0	0
$117$	− 498.831i	− 0.394162i
$118$	0	0
$119$	−311.769	−0.240167
$120$	0	0
$121$	1331.00	1.00000
$122$	0	0
$123$	162.000i	0.118756i
$124$	0	0
$125$	1475.71i	1.05593i
$126$	0	0
$127$	696.284	0.486498	0.243249	−	0.969964i	$-0.421787\pi$
$127$	696.284	0.486498	0.243249	+	0.969964i	$0.421787\pi$
$128$	0	0
$129$	−60.0000	−0.0409512
$130$	0	0
$131$	− 2268.00i	− 1.51264i	−0.654201	−	0.756321i	$-0.726995\pi$
$131$	− 2268.00i	− 1.51264i	0.654201	−	0.756321i	$-0.273005\pi$
$132$	0	0
$133$	401.836i	0.261982i
$134$	0	0
$135$	−280.592	−0.178885
$136$	0	0
$137$	522.000	0.325529	0.162764	−	0.986665i	$-0.447959\pi$
$137$	522.000	0.325529	0.162764	+	0.986665i	$0.447959\pi$
$138$	0	0
$139$	− 676.000i	− 0.412501i	−0.978499	−	0.206250i	$-0.933874\pi$
$139$	− 676.000i	− 0.412501i	0.978499	−	0.206250i	$-0.0661261\pi$
$140$	0	0
$141$	1184.72i	0.707600i
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−2700.00	−1.54636
$146$	0	0
$147$	993.000i	0.557151i
$148$	0	0
$149$	1465.31i	0.805660i	0.915275	+	0.402830i	$0.131973\pi$
$149$	1465.31i	0.805660i	−0.915275	+	0.402830i	$0.868027\pi$
$150$	0	0
$151$	2386.77	1.28631	0.643153	−	0.765738i	$-0.277626\pi$
$151$	2386.77	1.28631	0.643153	+	0.765738i	$0.277626\pi$
$152$	0	0
$153$	810.000	0.428004
$154$	0	0
$155$	3132.00i	1.62302i
$156$	0	0
$157$	2016.11i	1.02486i	0.858729	+	0.512430i	$0.171254\pi$
$157$	2016.11i	1.02486i	−0.858729	+	0.512430i	$0.828746\pi$
$158$	0	0
$159$	1465.31	0.730862
$160$	0	0
$161$	360.000	0.176223
$162$	0	0
$163$	− 388.000i	− 0.186445i	−0.995645	−	0.0932224i	$-0.970283\pi$
$163$	− 388.000i	− 0.186445i	0.995645	−	0.0932224i	$-0.0297168\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	2203.17	1.02088	0.510438	−	0.859915i	$-0.329483\pi$
$167$	2203.17	1.02088	0.510438	+	0.859915i	$0.329483\pi$
$168$	0	0
$169$	−875.000	−0.398270
$170$	0	0
$171$	− 1044.00i	− 0.466881i
$172$	0	0
$173$	197.454i	0.0867753i	0.999058	+	0.0433877i	$0.0138151\pi$
$173$	197.454i	0.0867753i	−0.999058	+	0.0433877i	$0.986185\pi$
$174$	0	0
$175$	58.8897	0.0254380
$176$	0	0
$177$	−972.000	−0.412768
$178$	0	0
$179$	2844.00i	1.18754i	0.804633	+	0.593772i	$0.202362\pi$
$179$	2844.00i	1.18754i	−0.804633	+	0.593772i	$0.797638\pi$
$180$	0	0
$181$	− 96.9948i	− 0.0398319i	−0.999802	−	0.0199159i	$-0.993660\pi$
$181$	− 96.9948i	− 0.0398319i	0.999802	−	0.0199159i	$-0.00633986\pi$
$182$	0	0
$183$	−1725.12	−0.696856
$184$	0	0
$185$	−360.000	−0.143069
$186$	0	0
$187$	0	0
$188$	0	0
$189$	93.5307i	0.0359966i
$190$	0	0
$191$	3200.83	1.21259	0.606293	−	0.795241i	$-0.292656\pi$
$191$	3200.83	1.21259	0.606293	+	0.795241i	$0.292656\pi$
$192$	0	0
$193$	−1342.00	−0.500514	−0.250257	−	0.968179i	$-0.580515\pi$
$193$	−1342.00	−0.500514	−0.250257	+	0.968179i	$0.580515\pi$
$194$	0	0
$195$	1728.00i	0.634588i
$196$	0	0
$197$	− 966.484i	− 0.349539i	−0.984609	−	0.174769i	$-0.944082\pi$
$197$	− 966.484i	− 0.349539i	0.984609	−	0.174769i	$-0.0559180\pi$
$198$	0	0
$199$	1784.01	0.635504	0.317752	−	0.948174i	$-0.397072\pi$
$199$	1784.01	0.635504	0.317752	+	0.948174i	$0.397072\pi$
$200$	0	0
$201$	−348.000	−0.122120
$202$	0	0
$203$	900.000i	0.311171i
$204$	0	0
$205$	− 561.184i	− 0.191194i
$206$	0	0
$207$	−935.307	−0.314050
$208$	0	0
$209$	0	0
$210$	0	0
$211$	2764.00i	0.901809i	0.892572	+	0.450904i	$0.148898\pi$
$211$	2764.00i	0.901809i	−0.892572	+	0.450904i	$0.851102\pi$
$212$	0	0
$213$	3304.75i	1.06309i
$214$	0	0
$215$	207.846	0.0659302
$216$	0	0
$217$	1044.00	0.326596
$218$	0	0
$219$	− 3318.00i	− 1.02379i
$220$	0	0
$221$	− 4988.31i	− 1.51832i
$222$	0	0
$223$	−4292.02	−1.28886	−0.644428	−	0.764665i	$-0.722905\pi$
$223$	−4292.02	−1.28886	−0.644428	+	0.764665i	$0.722905\pi$
$224$	0	0
$225$	−153.000	−0.0453333
$226$	0	0
$227$	− 5688.00i	− 1.66311i	−0.555443	−	0.831555i	$-0.687451\pi$
$227$	− 5688.00i	− 1.66311i	0.555443	−	0.831555i	$-0.312549\pi$
$228$	0	0
$229$	5570.28i	1.60740i	0.595036	+	0.803699i	$0.297138\pi$
$229$	5570.28i	1.60740i	−0.595036	+	0.803699i	$0.702862\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	2718.00	0.764215	0.382108	−	0.924118i	$-0.375198\pi$
$233$	2718.00	0.764215	0.382108	+	0.924118i	$0.375198\pi$
$234$	0	0
$235$	− 4104.00i	− 1.13921i
$236$	0	0
$237$	− 446.869i	− 0.122478i
$238$	0	0
$239$	3574.95	0.967550	0.483775	−	0.875192i	$-0.339265\pi$
$239$	3574.95	0.967550	0.483775	+	0.875192i	$0.339265\pi$
$240$	0	0
$241$	4490.00	1.20011	0.600055	−	0.799959i	$-0.295146\pi$
$241$	4490.00	1.20011	0.600055	+	0.799959i	$0.295146\pi$
$242$	0	0
$243$	− 243.000i	− 0.0641500i
$244$	0	0
$245$	− 3439.85i	− 0.896996i
$246$	0	0
$247$	−6429.37	−1.65624
$248$	0	0
$249$	−3456.00	−0.879579
$250$	0	0
$251$	4608.00i	1.15878i	0.815050	+	0.579391i	$0.196710\pi$
$251$	4608.00i	1.15878i	−0.815050	+	0.579391i	$0.803290\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−2805.92	−0.689073
$256$	0	0
$257$	4626.00	1.12281	0.561405	−	0.827541i	$-0.310261\pi$
$257$	4626.00	1.12281	0.561405	+	0.827541i	$0.310261\pi$
$258$	0	0
$259$	120.000i	0.0287893i
$260$	0	0
$261$	− 2338.27i	− 0.554541i
$262$	0	0
$263$	−1995.32	−0.467821	−0.233910	−	0.972258i	$-0.575152\pi$
$263$	−1995.32	−0.467821	−0.233910	+	0.972258i	$0.575152\pi$
$264$	0	0
$265$	−5076.00	−1.17666
$266$	0	0
$267$	− 2754.00i	− 0.631244i
$268$	0	0
$269$	− 3148.87i	− 0.713717i	−0.934158	−	0.356859i	$-0.883848\pi$
$269$	− 3148.87i	− 0.713717i	0.934158	−	0.356859i	$-0.116152\pi$
$270$	0	0
$271$	−5345.11	−1.19813	−0.599063	−	0.800702i	$-0.704460\pi$
$271$	−5345.11	−1.19813	−0.599063	+	0.800702i	$0.704460\pi$
$272$	0	0
$273$	576.000	0.127696
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 6526.37i	− 1.41564i	−0.706394	−	0.707818i	$-0.749679\pi$
$277$	− 6526.37i	− 1.41564i	0.706394	−	0.707818i	$-0.250321\pi$
$278$	0	0
$279$	−2712.39	−0.582031
$280$	0	0
$281$	−1170.00	−0.248386	−0.124193	−	0.992258i	$-0.539634\pi$
$281$	−1170.00	−0.248386	−0.124193	+	0.992258i	$0.539634\pi$
$282$	0	0
$283$	− 5740.00i	− 1.20568i	−0.797862	−	0.602840i	$-0.794036\pi$
$283$	− 5740.00i	− 1.20568i	0.797862	−	0.602840i	$-0.205964\pi$
$284$	0	0
$285$	3616.52i	0.751664i
$286$	0	0
$287$	−187.061	−0.0384735
$288$	0	0
$289$	3187.00	0.648687
$290$	0	0
$291$	− 570.000i	− 0.114825i
$292$	0	0
$293$	− 7991.68i	− 1.59344i	−0.604346	−	0.796722i	$-0.706566\pi$
$293$	− 7991.68i	− 1.59344i	0.604346	−	0.796722i	$-0.293434\pi$
$294$	0	0
$295$	3367.11	0.664544
$296$	0	0
$297$	0	0
$298$	0	0
$299$	5760.00i	1.11408i
$300$	0	0
$301$	− 69.2820i	− 0.0132669i
$302$	0	0
$303$	−31.1769	−0.00591111
$304$	0	0
$305$	5976.00	1.12192
$306$	0	0
$307$	5452.00i	1.01356i	0.862076	+	0.506779i	$0.169164\pi$
$307$	5452.00i	1.01356i	−0.862076	+	0.506779i	$0.830836\pi$
$308$	0	0
$309$	2379.84i	0.438137i
$310$	0	0
$311$	2203.17	0.401705	0.200852	−	0.979622i	$-0.435629\pi$
$311$	2203.17	0.401705	0.200852	+	0.979622i	$0.435629\pi$
$312$	0	0
$313$	−1034.00	−0.186726	−0.0933628	−	0.995632i	$-0.529762\pi$
$313$	−1034.00	−0.186726	−0.0933628	+	0.995632i	$0.529762\pi$
$314$	0	0
$315$	− 324.000i	− 0.0579534i
$316$	0	0
$317$	− 2650.04i	− 0.469530i	−0.972052	−	0.234765i	$-0.924568\pi$
$317$	− 2650.04i	− 0.469530i	0.972052	−	0.234765i	$-0.0754320\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	756.000	0.131451
$322$	0	0
$323$	− 10440.0i	− 1.79844i
$324$	0	0
$325$	942.236i	0.160818i
$326$	0	0
$327$	−1371.78	−0.231987
$328$	0	0
$329$	−1368.00	−0.229241
$330$	0	0
$331$	− 4132.00i	− 0.686149i	−0.939308	−	0.343074i	$-0.888532\pi$
$331$	− 4132.00i	− 0.686149i	0.939308	−	0.343074i	$-0.111468\pi$
$332$	0	0
$333$	− 311.769i	− 0.0513058i
$334$	0	0
$335$	1205.51	0.196609
$336$	0	0
$337$	−458.000	−0.0740322	−0.0370161	−	0.999315i	$-0.511785\pi$
$337$	−458.000	−0.0740322	−0.0370161	+	0.999315i	$0.511785\pi$
$338$	0	0
$339$	6642.00i	1.06414i
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−2334.80	−0.367544
$344$	0	0
$345$	3240.00	0.505611
$346$	0	0
$347$	11016.0i	1.70424i	0.523350	+	0.852118i	$0.324682\pi$
$347$	11016.0i	1.70424i	−0.523350	+	0.852118i	$0.675318\pi$
$348$	0	0
$349$	2528.79i	0.387860i	0.981015	+	0.193930i	$0.0621235\pi$
$349$	2528.79i	0.387860i	−0.981015	+	0.193930i	$0.937876\pi$
$350$	0	0
$351$	−1496.49	−0.227569
$352$	0	0
$353$	5562.00	0.838627	0.419314	−	0.907841i	$-0.362271\pi$
$353$	5562.00	0.838627	0.419314	+	0.907841i	$0.362271\pi$
$354$	0	0
$355$	− 11448.0i	− 1.71154i
$356$	0	0
$357$	935.307i	0.138660i
$358$	0	0
$359$	8875.03	1.30475	0.652376	−	0.757895i	$-0.273772\pi$
$359$	8875.03	1.30475	0.652376	+	0.757895i	$0.273772\pi$
$360$	0	0
$361$	−6597.00	−0.961802
$362$	0	0
$363$	− 3993.00i	− 0.577350i
$364$	0	0
$365$	11493.9i	1.64827i
$366$	0	0
$367$	12799.9	1.82056	0.910282	−	0.413989i	$-0.135865\pi$
$367$	12799.9	1.82056	0.910282	+	0.413989i	$0.135865\pi$
$368$	0	0
$369$	486.000	0.0685641
$370$	0	0
$371$	1692.00i	0.236777i
$372$	0	0
$373$	4981.38i	0.691491i	0.938328	+	0.345745i	$0.112374\pi$
$373$	4981.38i	0.691491i	−0.938328	+	0.345745i	$0.887626\pi$
$374$	0	0
$375$	4427.12	0.609642
$376$	0	0
$377$	−14400.0	−1.96721
$378$	0	0
$379$	9892.00i	1.34068i	0.742054	+	0.670340i	$0.233852\pi$
$379$	9892.00i	1.34068i	−0.742054	+	0.670340i	$0.766148\pi$
$380$	0	0
$381$	− 2088.85i	− 0.280880i
$382$	0	0
$383$	−8771.11	−1.17019	−0.585095	−	0.810965i	$-0.698943\pi$
$383$	−8771.11	−1.17019	−0.585095	+	0.810965i	$0.698943\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	180.000i	0.0236432i
$388$	0	0
$389$	9903.87i	1.29086i	0.763818	+	0.645432i	$0.223323\pi$
$389$	9903.87i	1.29086i	−0.763818	+	0.645432i	$0.776677\pi$
$390$	0	0
$391$	−9353.07	−1.20973
$392$	0	0
$393$	−6804.00	−0.873324
$394$	0	0
$395$	1548.00i	0.197186i
$396$	0	0
$397$	− 103.923i	− 0.0131379i	−0.999978	−	0.00656895i	$-0.997909\pi$
$397$	− 103.923i	− 0.0131379i	0.999978	−	0.00656895i	$-0.00209098\pi$
$398$	0	0
$399$	1205.51	0.151255
$400$	0	0
$401$	1062.00	0.132254	0.0661269	−	0.997811i	$-0.478936\pi$
$401$	1062.00	0.132254	0.0661269	+	0.997811i	$0.478936\pi$
$402$	0	0
$403$	16704.0i	2.06473i
$404$	0	0
$405$	841.777i	0.103280i
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−8614.00	−1.04141	−0.520703	−	0.853738i	$-0.674330\pi$
$409$	−8614.00	−1.04141	−0.520703	+	0.853738i	$0.674330\pi$
$410$	0	0
$411$	− 1566.00i	− 0.187944i
$412$	0	0
$413$	− 1122.37i	− 0.133724i
$414$	0	0
$415$	11971.9	1.41609
$416$	0	0
$417$	−2028.00	−0.238157
$418$	0	0
$419$	− 10440.0i	− 1.21725i	−0.793458	−	0.608625i	$-0.791722\pi$
$419$	− 10440.0i	− 1.21725i	0.793458	−	0.608625i	$-0.208278\pi$
$420$	0	0
$421$	− 900.666i	− 0.104266i	−0.998640	−	0.0521328i	$-0.983398\pi$
$421$	− 900.666i	− 0.104266i	0.998640	−	0.0521328i	$-0.0166019\pi$
$422$	0	0
$423$	3554.17	0.408533
$424$	0	0
$425$	−1530.00	−0.174626
$426$	0	0
$427$	− 1992.00i	− 0.225760i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	394.908	0.0441346	0.0220673	−	0.999756i	$-0.492975\pi$
$431$	394.908	0.0441346	0.0220673	+	0.999756i	$0.492975\pi$
$432$	0	0
$433$	12958.0	1.43816	0.719078	−	0.694929i	$-0.244564\pi$
$433$	12958.0	1.43816	0.719078	+	0.694929i	$0.244564\pi$
$434$	0	0
$435$	8100.00i	0.892794i
$436$	0	0
$437$	12055.1i	1.31962i
$438$	0	0
$439$	11441.9	1.24395	0.621974	−	0.783038i	$-0.286331\pi$
$439$	11441.9	1.24395	0.621974	+	0.783038i	$0.286331\pi$
$440$	0	0
$441$	2979.00	0.321672
$442$	0	0
$443$	1800.00i	0.193049i	0.995331	+	0.0965244i	$0.0307726\pi$
$443$	1800.00i	0.193049i	−0.995331	+	0.0965244i	$0.969227\pi$
$444$	0	0
$445$	9540.14i	1.01628i
$446$	0	0
$447$	4395.94	0.465148
$448$	0	0
$449$	−13626.0	−1.43218	−0.716092	−	0.698006i	$-0.754071\pi$
$449$	−13626.0	−1.43218	−0.716092	+	0.698006i	$0.754071\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	− 7160.30i	− 0.742649i
$454$	0	0
$455$	−1995.32	−0.205587
$456$	0	0
$457$	12602.0	1.28993	0.644964	−	0.764213i	$-0.276873\pi$
$457$	12602.0	1.28993	0.644964	+	0.764213i	$0.276873\pi$
$458$	0	0
$459$	− 2430.00i	− 0.247108i
$460$	0	0
$461$	− 1839.44i	− 0.185838i	−0.995674	−	0.0929188i	$-0.970380\pi$
$461$	− 1839.44i	− 0.185838i	0.995674	−	0.0929188i	$-0.0296197\pi$
$462$	0	0
$463$	−11012.4	−1.10538	−0.552688	−	0.833389i	$-0.686398\pi$
$463$	−11012.4	−1.10538	−0.552688	+	0.833389i	$0.686398\pi$
$464$	0	0
$465$	9396.00	0.937052
$466$	0	0
$467$	− 9144.00i	− 0.906068i	−0.891493	−	0.453034i	$-0.850342\pi$
$467$	− 9144.00i	− 0.906068i	0.891493	−	0.453034i	$-0.149658\pi$
$468$	0	0
$469$	− 401.836i	− 0.0395630i
$470$	0	0
$471$	6048.32	0.591703
$472$	0	0
$473$	0	0
$474$	0	0
$475$	1972.00i	0.190488i
$476$	0	0
$477$	− 4395.94i	− 0.421963i
$478$	0	0
$479$	6173.03	0.588837	0.294418	−	0.955677i	$-0.404874\pi$
$479$	6173.03	0.588837	0.294418	+	0.955677i	$0.404874\pi$
$480$	0	0
$481$	−1920.00	−0.182005
$482$	0	0
$483$	− 1080.00i	− 0.101743i
$484$	0	0
$485$	1974.54i	0.184864i
$486$	0	0
$487$	3204.29	0.298153	0.149076	−	0.988826i	$-0.452370\pi$
$487$	3204.29	0.298153	0.149076	+	0.988826i	$0.452370\pi$
$488$	0	0
$489$	−1164.00	−0.107644
$490$	0	0
$491$	− 396.000i	− 0.0363976i	−0.999834	−	0.0181988i	$-0.994207\pi$
$491$	− 396.000i	− 0.0363976i	0.999834	−	0.0181988i	$-0.00579318\pi$
$492$	0	0
$493$	− 23382.7i	− 2.13611i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−3816.00	−0.344408
$498$	0	0
$499$	12436.0i	1.11565i	0.829957	+	0.557827i	$0.188365\pi$
$499$	12436.0i	1.11565i	−0.829957	+	0.557827i	$0.811635\pi$
$500$	0	0
$501$	− 6609.51i	− 0.589403i
$502$	0	0
$503$	−16482.2	−1.46104	−0.730522	−	0.682890i	$-0.760723\pi$
$503$	−16482.2	−1.46104	−0.730522	+	0.682890i	$0.760723\pi$
$504$	0	0
$505$	108.000	0.00951671
$506$	0	0
$507$	2625.00i	0.229942i
$508$	0	0
$509$	9155.62i	0.797280i	0.917107	+	0.398640i	$0.130518\pi$
$509$	9155.62i	0.797280i	−0.917107	+	0.398640i	$0.869482\pi$
$510$	0	0
$511$	3831.30	0.331676
$512$	0	0
$513$	−3132.00	−0.269554
$514$	0	0
$515$	− 8244.00i	− 0.705386i
$516$	0	0
$517$	0	0
$518$	0	0
$519$	592.361	0.0500998
$520$	0	0
$521$	7650.00	0.643287	0.321644	−	0.946861i	$-0.395765\pi$
$521$	7650.00	0.643287	0.321644	+	0.946861i	$0.395765\pi$
$522$	0	0
$523$	− 18332.0i	− 1.53270i	−0.642423	−	0.766350i	$-0.722071\pi$
$523$	− 18332.0i	− 1.53270i	0.642423	−	0.766350i	$-0.277929\pi$
$524$	0	0
$525$	− 176.669i	− 0.0146866i
$526$	0	0
$527$	−27123.9	−2.24200
$528$	0	0
$529$	−1367.00	−0.112353
$530$	0	0
$531$	2916.00i	0.238312i
$532$	0	0
$533$	− 2992.98i	− 0.243228i
$534$	0	0
$535$	−2618.86	−0.211632
$536$	0	0
$537$	8532.00	0.685629
$538$	0	0
$539$	0	0
$540$	0	0
$541$	− 16863.2i	− 1.34012i	−0.742305	−	0.670062i	$-0.766267\pi$
$541$	− 16863.2i	− 1.34012i	0.742305	−	0.670062i	$-0.233733\pi$
$542$	0	0
$543$	−290.985	−0.0229969
$544$	0	0
$545$	4752.00	0.373492
$546$	0	0
$547$	− 1684.00i	− 0.131632i	−0.997832	−	0.0658159i	$-0.979035\pi$
$547$	− 1684.00i	− 0.131632i	0.997832	−	0.0658159i	$-0.0209650\pi$
$548$	0	0
$549$	5175.37i	0.402330i
$550$	0	0
$551$	−30137.7	−2.33014
$552$	0	0
$553$	516.000	0.0396791
$554$	0	0
$555$	1080.00i	0.0826008i
$556$	0	0
$557$	2275.91i	0.173130i	0.996246	+	0.0865652i	$0.0275891\pi$
$557$	2275.91i	0.173130i	−0.996246	+	0.0865652i	$0.972411\pi$
$558$	0	0
$559$	1108.51	0.0838731
$560$	0	0
$561$	0	0
$562$	0	0
$563$	7992.00i	0.598264i	0.954212	+	0.299132i	$0.0966971\pi$
$563$	7992.00i	0.598264i	−0.954212	+	0.299132i	$0.903303\pi$
$564$	0	0
$565$	− 23008.6i	− 1.71323i
$566$	0	0
$567$	280.592	0.0207827
$568$	0	0
$569$	−5526.00	−0.407139	−0.203569	−	0.979061i	$-0.565254\pi$
$569$	−5526.00	−0.407139	−0.203569	+	0.979061i	$0.565254\pi$
$570$	0	0
$571$	− 13420.0i	− 0.983554i	−0.870721	−	0.491777i	$-0.836347\pi$
$571$	− 13420.0i	− 0.983554i	0.870721	−	0.491777i	$-0.163653\pi$
$572$	0	0
$573$	− 9602.49i	− 0.700087i
$574$	0	0
$575$	1766.69	0.128132
$576$	0	0
$577$	−10178.0	−0.734343	−0.367171	−	0.930153i	$-0.619674\pi$
$577$	−10178.0	−0.734343	−0.367171	+	0.930153i	$0.619674\pi$
$578$	0	0
$579$	4026.00i	0.288972i
$580$	0	0
$581$	− 3990.65i	− 0.284957i
$582$	0	0
$583$	0	0
$584$	0	0
$585$	5184.00	0.366380
$586$	0	0
$587$	− 18684.0i	− 1.31375i	−0.754000	−	0.656875i	$-0.771878\pi$
$587$	− 18684.0i	− 1.31375i	0.754000	−	0.656875i	$-0.228122\pi$
$588$	0	0
$589$	34959.7i	2.44565i
$590$	0	0
$591$	−2899.45	−0.201806
$592$	0	0
$593$	−5094.00	−0.352758	−0.176379	−	0.984322i	$-0.556438\pi$
$593$	−5094.00	−0.352758	−0.176379	+	0.984322i	$0.556438\pi$
$594$	0	0
$595$	− 3240.00i	− 0.223239i
$596$	0	0
$597$	− 5352.04i	− 0.366908i
$598$	0	0
$599$	19433.6	1.32560	0.662801	−	0.748795i	$-0.269367\pi$
$599$	19433.6	1.32560	0.662801	+	0.748795i	$0.269367\pi$
$600$	0	0
$601$	27722.0	1.88154	0.940769	−	0.339049i	$-0.110105\pi$
$601$	27722.0	1.88154	0.940769	+	0.339049i	$0.110105\pi$
$602$	0	0
$603$	1044.00i	0.0705057i
$604$	0	0
$605$	13832.2i	0.929516i
$606$	0	0
$607$	−26684.0	−1.78430	−0.892149	−	0.451741i	$-0.850803\pi$
$607$	−26684.0	−1.78430	−0.892149	+	0.451741i	$0.850803\pi$
$608$	0	0
$609$	2700.00	0.179654
$610$	0	0
$611$	− 21888.0i	− 1.44925i
$612$	0	0
$613$	− 16911.7i	− 1.11429i	−0.830416	−	0.557144i	$-0.811897\pi$
$613$	− 16911.7i	− 1.11429i	0.830416	−	0.557144i	$-0.188103\pi$
$614$	0	0
$615$	−1683.55	−0.110386
$616$	0	0
$617$	17694.0	1.15451	0.577256	−	0.816563i	$-0.304124\pi$
$617$	17694.0	1.15451	0.577256	+	0.816563i	$0.304124\pi$
$618$	0	0
$619$	13652.0i	0.886462i	0.896407	+	0.443231i	$0.146168\pi$
$619$	13652.0i	0.886462i	−0.896407	+	0.443231i	$0.853832\pi$
$620$	0	0
$621$	2805.92i	0.181317i
$622$	0	0
$623$	3180.05	0.204504
$624$	0	0
$625$	−13211.0	−0.845504
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 3117.69i	− 0.197632i
$630$	0	0
$631$	9162.55	0.578059	0.289030	−	0.957320i	$-0.406667\pi$
$631$	9162.55	0.578059	0.289030	+	0.957320i	$0.406667\pi$
$632$	0	0
$633$	8292.00	0.520659
$634$	0	0
$635$	7236.00i	0.452208i
$636$	0	0
$637$	− 18345.9i	− 1.14112i
$638$	0	0
$639$	9914.26	0.613775
$640$	0	0
$641$	−5202.00	−0.320541	−0.160270	−	0.987073i	$-0.551237\pi$
$641$	−5202.00	−0.320541	−0.160270	+	0.987073i	$0.551237\pi$
$642$	0	0
$643$	− 15892.0i	− 0.974680i	−0.873212	−	0.487340i	$-0.837967\pi$
$643$	− 15892.0i	− 0.974680i	0.873212	−	0.487340i	$-0.162033\pi$
$644$	0	0
$645$	− 623.538i	− 0.0380648i
$646$	0	0
$647$	−478.046	−0.0290478	−0.0145239	−	0.999895i	$-0.504623\pi$
$647$	−478.046	−0.0290478	−0.0145239	+	0.999895i	$0.504623\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	− 3132.00i	− 0.188560i
$652$	0	0
$653$	− 24660.9i	− 1.47788i	−0.673770	−	0.738941i	$-0.735326\pi$
$653$	− 24660.9i	− 1.47788i	0.673770	−	0.738941i	$-0.264674\pi$
$654$	0	0
$655$	23569.7	1.40602
$656$	0	0
$657$	−9954.00	−0.591085
$658$	0	0
$659$	28260.0i	1.67049i	0.549878	+	0.835245i	$0.314674\pi$
$659$	28260.0i	1.67049i	−0.549878	+	0.835245i	$0.685326\pi$
$660$	0	0
$661$	25863.0i	1.52187i	0.648830	+	0.760933i	$0.275258\pi$
$661$	25863.0i	1.52187i	−0.648830	+	0.760933i	$0.724742\pi$
$662$	0	0
$663$	−14964.9	−0.876605
$664$	0	0
$665$	−4176.00	−0.243516
$666$	0	0
$667$	27000.0i	1.56738i
$668$	0	0
$669$	12876.1i	0.744122i
$670$	0	0
$671$	0	0
$672$	0	0
$673$	190.000	0.0108826	0.00544128	−	0.999985i	$-0.498268\pi$
$673$	190.000	0.0108826	0.00544128	+	0.999985i	$0.498268\pi$
$674$	0	0
$675$	459.000i	0.0261732i
$676$	0	0
$677$	4998.70i	0.283775i	0.989883	+	0.141887i	$0.0453171\pi$
$677$	4998.70i	0.283775i	−0.989883	+	0.141887i	$0.954683\pi$
$678$	0	0
$679$	658.179	0.0371997
$680$	0	0
$681$	−17064.0	−0.960197
$682$	0	0
$683$	8064.00i	0.451772i	0.974154	+	0.225886i	$0.0725277\pi$
$683$	8064.00i	0.451772i	−0.974154	+	0.225886i	$0.927472\pi$
$684$	0	0
$685$	5424.78i	0.302584i
$686$	0	0
$687$	16710.8	0.928032
$688$	0	0
$689$	−27072.0	−1.49690
$690$	0	0
$691$	19244.0i	1.05944i	0.848171	+	0.529722i	$0.177704\pi$
$691$	19244.0i	1.05944i	−0.848171	+	0.529722i	$0.822296\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	7025.20	0.383426
$696$	0	0
$697$	4860.00	0.264111
$698$	0	0
$699$	− 8154.00i	− 0.441220i
$700$	0	0
$701$	6204.21i	0.334279i	0.985933	+	0.167140i	$0.0534530\pi$
$701$	6204.21i	0.334279i	−0.985933	+	0.167140i	$0.946547\pi$
$702$	0	0
$703$	−4018.36	−0.215584
$704$	0	0
$705$	−12312.0	−0.657726
$706$	0	0
$707$	− 36.0000i	− 0.00191502i
$708$	0	0
$709$	15020.3i	0.795629i	0.917466	+	0.397814i	$0.130231\pi$
$709$	15020.3i	0.795629i	−0.917466	+	0.397814i	$0.869769\pi$
$710$	0	0
$711$	−1340.61	−0.0707127
$712$	0	0
$713$	31320.0	1.64508
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 10724.9i	− 0.558615i
$718$	0	0
$719$	30740.4	1.59447	0.797236	−	0.603668i	$-0.206295\pi$
$719$	30740.4	1.59447	0.797236	+	0.603668i	$0.206295\pi$
$720$	0	0
$721$	−2748.00	−0.141943
$722$	0	0
$723$	− 13470.0i	− 0.692883i
$724$	0	0
$725$	4416.73i	0.226253i
$726$	0	0
$727$	12127.8	0.618701	0.309351	−	0.950948i	$-0.399888\pi$
$727$	12127.8	0.618701	0.309351	+	0.950948i	$0.399888\pi$
$728$	0	0
$729$	−729.000	−0.0370370
$730$	0	0
$731$	1800.00i	0.0910744i
$732$	0	0
$733$	12387.6i	0.624212i	0.950047	+	0.312106i	$0.101034\pi$
$733$	12387.6i	0.624212i	−0.950047	+	0.312106i	$0.898966\pi$
$734$	0	0
$735$	−10319.6	−0.517881
$736$	0	0
$737$	0	0
$738$	0	0
$739$	− 11180.0i	− 0.556513i	−0.960507	−	0.278256i	$-0.910244\pi$
$739$	− 11180.0i	− 0.556513i	0.960507	−	0.278256i	$-0.0897565\pi$
$740$	0	0
$741$	19288.1i	0.956230i
$742$	0	0
$743$	35500.1	1.75286	0.876429	−	0.481532i	$-0.159919\pi$
$743$	35500.1	1.75286	0.876429	+	0.481532i	$0.159919\pi$
$744$	0	0
$745$	−15228.0	−0.748873
$746$	0	0
$747$	10368.0i	0.507825i
$748$	0	0
$749$	872.954i	0.0425862i
$750$	0	0
$751$	−37970.0	−1.84493	−0.922467	−	0.386076i	$-0.873830\pi$
$751$	−37970.0	−1.84493	−0.922467	+	0.386076i	$0.873830\pi$
$752$	0	0
$753$	13824.0	0.669023
$754$	0	0
$755$	24804.0i	1.19564i
$756$	0	0
$757$	39047.4i	1.87477i	0.348296	+	0.937385i	$0.386760\pi$
$757$	39047.4i	1.87477i	−0.348296	+	0.937385i	$0.613240\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−12222.0	−0.582191	−0.291095	−	0.956694i	$-0.594020\pi$
$761$	−12222.0	−0.582191	−0.291095	+	0.956694i	$0.594020\pi$
$762$	0	0
$763$	− 1584.00i	− 0.0751568i
$764$	0	0
$765$	8417.77i	0.397837i
$766$	0	0
$767$	17957.9	0.845401
$768$	0	0
$769$	−34030.0	−1.59578	−0.797889	−	0.602804i	$-0.794050\pi$
$769$	−34030.0	−1.59578	−0.797889	+	0.602804i	$0.794050\pi$
$770$	0	0
$771$	− 13878.0i	− 0.648254i
$772$	0	0
$773$	− 4873.99i	− 0.226786i	−0.993550	−	0.113393i	$-0.963828\pi$
$773$	− 4873.99i	− 0.226786i	0.993550	−	0.113393i	$-0.0361718\pi$
$774$	0	0
$775$	5123.41	0.237469
$776$	0	0
$777$	360.000	0.0166215
$778$	0	0
$779$	− 6264.00i	− 0.288102i
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−7014.81	−0.320164
$784$	0	0
$785$	−20952.0	−0.952623
$786$	0	0
$787$	− 30988.0i	− 1.40356i	−0.712393	−	0.701781i	$-0.752389\pi$
$787$	− 30988.0i	− 1.40356i	0.712393	−	0.701781i	$-0.247611\pi$
$788$	0	0
$789$	5985.97i	0.270096i
$790$	0	0
$791$	−7669.52	−0.344749
$792$	0	0
$793$	31872.0	1.42725
$794$	0	0
$795$	15228.0i	0.679348i
$796$	0	0
$797$	− 7160.30i	− 0.318232i	−0.987260	−	0.159116i	$-0.949136\pi$
$797$	− 7160.30i	− 0.318232i	0.987260	−	0.159116i	$-0.0508644\pi$
$798$	0	0
$799$	35541.7	1.57369
$800$	0	0
$801$	−8262.00	−0.364449
$802$	0	0
$803$	0	0
$804$	0	0
$805$	3741.23i	0.163803i
$806$	0	0
$807$	−9446.61	−0.412065
$808$	0	0
$809$	37530.0	1.63101	0.815503	−	0.578752i	$-0.196460\pi$
$809$	37530.0	1.63101	0.815503	+	0.578752i	$0.196460\pi$
$810$	0	0
$811$	10852.0i	0.469871i	0.972011	+	0.234935i	$0.0754879\pi$
$811$	10852.0i	0.469871i	−0.972011	+	0.234935i	$0.924512\pi$
$812$	0	0
$813$	16035.3i	0.691739i
$814$	0	0
$815$	4032.21	0.173303
$816$	0	0
$817$	2320.00	0.0993470
$818$	0	0
$819$	− 1728.00i	− 0.0737255i
$820$	0	0
$821$	− 31353.6i	− 1.33282i	−0.745584	−	0.666411i	$-0.767829\pi$
$821$	− 31353.6i	− 1.33282i	0.745584	−	0.666411i	$-0.232171\pi$
$822$	0	0
$823$	−32947.1	−1.39546	−0.697729	−	0.716361i	$-0.745806\pi$
$823$	−32947.1	−1.39546	−0.697729	+	0.716361i	$0.745806\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	10044.0i	0.422327i	0.977451	+	0.211163i	$0.0677252\pi$
$827$	10044.0i	0.422327i	−0.977451	+	0.211163i	$0.932275\pi$
$828$	0	0
$829$	9796.48i	0.410429i	0.978717	+	0.205215i	$0.0657892\pi$
$829$	9796.48i	0.410429i	−0.978717	+	0.205215i	$0.934211\pi$
$830$	0	0
$831$	−19579.1	−0.817318
$832$	0	0
$833$	29790.0	1.23909
$834$	0	0
$835$	22896.0i	0.948921i
$836$	0	0
$837$	8137.17i	0.336036i
$838$	0	0
$839$	21054.8	0.866380	0.433190	−	0.901303i	$-0.357388\pi$
$839$	21054.8	0.866380	0.433190	+	0.901303i	$0.357388\pi$
$840$	0	0
$841$	−43111.0	−1.76764
$842$	0	0
$843$	3510.00i	0.143405i
$844$	0	0
$845$	− 9093.27i	− 0.370199i
$846$	0	0
$847$	4610.72	0.187044
$848$	0	0
$849$	−17220.0	−0.696100
$850$	0	0
$851$	3600.00i	0.145013i
$852$	0	0
$853$	− 40703.2i	− 1.63382i	−0.576763	−	0.816911i	$-0.695684\pi$
$853$	− 40703.2i	− 1.63382i	0.576763	−	0.816911i	$-0.304316\pi$
$854$	0	0
$855$	10849.6	0.433973
$856$	0	0
$857$	−18342.0	−0.731098	−0.365549	−	0.930792i	$-0.619119\pi$
$857$	−18342.0	−0.731098	−0.365549	+	0.930792i	$0.619119\pi$
$858$	0	0
$859$	− 26324.0i	− 1.04559i	−0.852458	−	0.522796i	$-0.824889\pi$
$859$	− 26324.0i	− 1.04559i	0.852458	−	0.522796i	$-0.175111\pi$
$860$	0	0
$861$	561.184i	0.0222127i
$862$	0	0
$863$	−12761.8	−0.503378	−0.251689	−	0.967808i	$-0.580986\pi$
$863$	−12761.8	−0.503378	−0.251689	+	0.967808i	$0.580986\pi$
$864$	0	0
$865$	−2052.00	−0.0806591
$866$	0	0
$867$	− 9561.00i	− 0.374520i
$868$	0	0
$869$	0	0
$870$	0	0
$871$	6429.37	0.250116
$872$	0	0
$873$	−1710.00	−0.0662941
$874$	0	0
$875$	5112.00i	0.197505i
$876$	0	0
$877$	− 2459.51i	− 0.0946999i	−0.998878	−	0.0473500i	$-0.984922\pi$
$877$	− 2459.51i	− 0.0946999i	0.998878	−	0.0473500i	$-0.0150776\pi$
$878$	0	0
$879$	−23975.0	−0.919975
$880$	0	0
$881$	37314.0	1.42695	0.713474	−	0.700682i	$-0.247121\pi$
$881$	37314.0	1.42695	0.713474	+	0.700682i	$0.247121\pi$
$882$	0	0
$883$	− 18244.0i	− 0.695311i	−0.937622	−	0.347655i	$-0.886978\pi$
$883$	− 18244.0i	− 0.695311i	0.937622	−	0.347655i	$-0.113022\pi$
$884$	0	0
$885$	− 10101.3i	− 0.383675i
$886$	0	0
$887$	−17957.9	−0.679783	−0.339891	−	0.940465i	$-0.610390\pi$
$887$	−17957.9	−0.679783	−0.339891	+	0.940465i	$0.610390\pi$
$888$	0	0
$889$	2412.00	0.0909965
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 45809.3i	− 1.71663i
$894$	0	0
$895$	−29555.7	−1.10384
$896$	0	0
$897$	17280.0	0.643213
$898$	0	0
$899$	78300.0i	2.90484i
$900$	0	0
$901$	− 43959.4i	− 1.62542i
$902$	0	0
$903$	−207.846	−0.00765967
$904$	0	0
$905$	1008.00	0.0370244
$906$	0	0
$907$	− 16388.0i	− 0.599950i	−0.953947	−	0.299975i	$-0.903022\pi$
$907$	− 16388.0i	− 0.599950i	0.953947	−	0.299975i	$-0.0969783\pi$
$908$	0	0
$909$	93.5307i	0.00341278i
$910$	0	0
$911$	25107.8	0.913127	0.456564	−	0.889691i	$-0.349080\pi$
$911$	25107.8	0.913127	0.456564	+	0.889691i	$0.349080\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	− 17928.0i	− 0.647739i
$916$	0	0
$917$	− 7856.58i	− 0.282930i
$918$	0	0
$919$	−27155.1	−0.974716	−0.487358	−	0.873202i	$-0.662039\pi$
$919$	−27155.1	−0.974716	−0.487358	+	0.873202i	$0.662039\pi$
$920$	0	0
$921$	16356.0	0.585178
$922$	0	0
$923$	− 61056.0i	− 2.17734i
$924$	0	0
$925$	588.897i	0.0209328i
$926$	0	0
$927$	7139.51	0.252958
$928$	0	0
$929$	48006.0	1.69540	0.847700	−	0.530477i	$-0.177987\pi$
$929$	48006.0	1.69540	0.847700	+	0.530477i	$0.177987\pi$
$930$	0	0
$931$	− 38396.0i	− 1.35164i
$932$	0	0
$933$	− 6609.51i	− 0.231924i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	7894.00	0.275225	0.137612	−	0.990486i	$-0.456057\pi$
$937$	7894.00	0.275225	0.137612	+	0.990486i	$0.456057\pi$
$938$	0	0
$939$	3102.00i	0.107806i
$940$	0	0
$941$	− 2670.82i	− 0.0925253i	−0.998929	−	0.0462627i	$-0.985269\pi$
$941$	− 2670.82i	− 0.0925253i	0.998929	−	0.0462627i	$-0.0147311\pi$
$942$	0	0
$943$	−5611.84	−0.193793
$944$	0	0
$945$	−972.000	−0.0334594
$946$	0	0
$947$	− 22356.0i	− 0.767130i	−0.923514	−	0.383565i	$-0.874696\pi$
$947$	− 22356.0i	− 0.767130i	0.923514	−	0.383565i	$-0.125304\pi$
$948$	0	0
$949$	61300.7i	2.09685i
$950$	0	0
$951$	−7950.11	−0.271083
$952$	0	0
$953$	−14958.0	−0.508434	−0.254217	−	0.967147i	$-0.581818\pi$
$953$	−14958.0	−0.508434	−0.254217	+	0.967147i	$0.581818\pi$
$954$	0	0
$955$	33264.0i	1.12712i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	1808.26	0.0608882
$960$	0	0
$961$	61037.0	2.04884
$962$	0	0
$963$	− 2268.00i	− 0.0758933i
$964$	0	0
$965$	− 13946.5i	− 0.465236i
$966$	0	0
$967$	−17476.4	−0.581182	−0.290591	−	0.956847i	$-0.593852\pi$
$967$	−17476.4	−0.581182	−0.290591	+	0.956847i	$0.593852\pi$
$968$	0	0
$969$	−31320.0	−1.03833
$970$	0	0
$971$	− 48528.0i	− 1.60385i	−0.597425	−	0.801925i	$-0.703810\pi$
$971$	− 48528.0i	− 1.60385i	0.597425	−	0.801925i	$-0.296190\pi$
$972$	0	0
$973$	− 2341.73i	− 0.0771557i
$974$	0	0
$975$	2826.71	0.0928483
$976$	0	0
$977$	−39978.0	−1.30912	−0.654560	−	0.756010i	$-0.727146\pi$
$977$	−39978.0	−1.30912	−0.654560	+	0.756010i	$0.727146\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	4115.35i	0.133938i
$982$	0	0
$983$	−17957.9	−0.582674	−0.291337	−	0.956621i	$-0.594100\pi$
$983$	−17957.9	−0.582674	−0.291337	+	0.956621i	$0.594100\pi$
$984$	0	0
$985$	10044.0	0.324902
$986$	0	0
$987$	4104.00i	0.132352i
$988$	0	0
$989$	− 2078.46i	− 0.0668263i
$990$	0	0
$991$	−1666.23	−0.0534103	−0.0267052	−	0.999643i	$-0.508502\pi$
$991$	−1666.23	−0.0534103	−0.0267052	+	0.999643i	$0.508502\pi$
$992$	0	0
$993$	−12396.0	−0.396148
$994$	0	0
$995$	18540.0i	0.590711i
$996$	0	0
$997$	− 41465.3i	− 1.31717i	−0.752506	−	0.658585i	$-0.771155\pi$
$997$	− 41465.3i	− 1.31717i	0.752506	−	0.658585i	$-0.228845\pi$
$998$	0	0
$999$	−935.307	−0.0296214

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
192.4.d.a.97.1	✓	4	8.3	odd	2	inner
192.4.d.a.97.2	yes	4	1.1	even	1	trivial
192.4.d.a.97.3	yes	4	8.5	even	2	inner
192.4.d.a.97.4	yes	4	4.3	odd	2	inner
576.4.d.g.289.1		4	12.11	even	2
576.4.d.g.289.2		4	3.2	odd	2
576.4.d.g.289.3		4	24.11	even	2
576.4.d.g.289.4		4	24.5	odd	2
768.4.a.g.1.1		2	16.5	even	4
768.4.a.g.1.2		2	16.3	odd	4
768.4.a.n.1.1		2	16.11	odd	4
768.4.a.n.1.2		2	16.13	even	4
2304.4.a.be.1.1		2	48.29	odd	4
2304.4.a.be.1.2		2	48.11	even	4
2304.4.a.bg.1.1		2	48.35	even	4
2304.4.a.bg.1.2		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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 192 = 2^{6} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	192.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-3.00000i q^{3} +10.3923i q^{5} +3.46410 q^{7} -9.00000 q^{9} +55.4256i q^{13} +31.1769 q^{15} -90.0000 q^{17} +116.000i q^{19} -10.3923i q^{21} +103.923 q^{23} +17.0000 q^{25} +27.0000i q^{27} +259.808i q^{29} +301.377 q^{31} +36.0000i q^{35} +34.6410i q^{37} +166.277 q^{39} -54.0000 q^{41} -20.0000i q^{43} -93.5307i q^{45} -394.908 q^{47} -331.000 q^{49} +270.000i q^{51} +488.438i q^{53} +348.000 q^{57} -324.000i q^{59} -575.041i q^{61} -31.1769 q^{63} -576.000 q^{65} -116.000i q^{67} -311.769i q^{69} -1101.58 q^{71} +1106.00 q^{73} -51.0000i q^{75} +148.956 q^{79} +81.0000 q^{81} -1152.00i q^{83} -935.307i q^{85} +779.423 q^{87} +918.000 q^{89} +192.000i q^{91} -904.131i q^{93} -1205.51 q^{95} +190.000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 36 q^{9} - 360 q^{17} + 68 q^{25} - 216 q^{41} - 1324 q^{49} + 1392 q^{57} - 2304 q^{65} + 4424 q^{73} + 324 q^{81} + 3672 q^{89} + 760 q^{97}+O(q^{100}) \) 4 * q - 36 * q^9 - 360 * q^17 + 68 * q^25 - 216 * q^41 - 1324 * q^49 + 1392 * q^57 - 2304 * q^65 + 4424 * q^73 + 324 * q^81 + 3672 * q^89 + 760 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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