LMFDB - Embedded newform 3168.2.o.f.703.18

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3168,2,Mod(703,3168)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3168, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("3168.703");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3168 = 2^{5} \cdot 3^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3168.o (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:	$25.2966073603$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			703.18
Character	$\chi$	$=$	3168.703
Dual form			3168.2.o.f.703.19

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.86960 q^{5} -3.91787 q^{7} +(3.30563 - 0.269894i) q^{11} -3.53618i q^{13} +1.77102i q^{17} -3.02570 q^{19} +7.41031i q^{23} -1.50459 q^{25} +5.55384i q^{29} +9.85431i q^{31} -7.32485 q^{35} +9.85431 q^{37} -8.38227i q^{41} +2.26233 q^{43} +3.13132i q^{47} +8.34972 q^{49} +8.13224 q^{53} +(6.18020 - 0.504595i) q^{55} -10.5416i q^{59} -8.82421i q^{61} -6.61125i q^{65} +4.84513i q^{67} +0.607882i q^{71} +13.8871i q^{73} +(-12.9510 + 1.05741i) q^{77} +3.15450 q^{79} +6.37046 q^{83} +3.31109i q^{85} +15.9002 q^{89} +13.8543i q^{91} -5.65685 q^{95} +9.34972 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 40 q^{25} + 40 q^{49} + 64 q^{97}+O(q^{100}) $ 24 * q + 40 * q^25 + 40 * q^49 + 64 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3168\mathbb{Z}\right)^\times$.

$n$	$353$	$991$	$1189$	$1729$
$\chi(n)$	$1$	$-1$	$1$	$-1$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$	1.86960			0.836111			0.418055	−	0.908422i	$-0.362712\pi$
$5$	1.86960			0.836111			0.418055	+	0.908422i	$0.362712\pi$
$6$		0			0
$7$	−3.91787			−1.48082			−0.740408	−	0.672158i	$-0.765368\pi$
$7$	−3.91787			−1.48082			−0.740408	+	0.672158i	$0.765368\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	3.30563	−	0.269894i	0.996683	−	0.0813762i
$11$	3.30563	−	0.269894i	0.996683	−	0.0813762i
$12$		0			0
$13$		−	3.53618i		−	0.980761i	−0.871508	−	0.490380i	$-0.836858\pi$
$13$		−	3.53618i		−	0.980761i	0.871508	−	0.490380i	$-0.163142\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			1.77102i			0.429534i	0.976665	+	0.214767i	$0.0688993\pi$
$17$			1.77102i			0.429534i	−0.976665	+	0.214767i	$0.931101\pi$
$18$		0			0
$19$	−3.02570			−0.694144			−0.347072	−	0.937839i	$-0.612824\pi$
$19$	−3.02570			−0.694144			−0.347072	+	0.937839i	$0.612824\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			7.41031i			1.54516i	0.634920	+	0.772578i	$0.281033\pi$
$23$			7.41031i			1.54516i	−0.634920	+	0.772578i	$0.718967\pi$
$24$		0			0
$25$	−1.50459			−0.300919
$26$		0			0
$27$		0			0
$28$		0			0
$29$			5.55384i			1.03132i	0.856793	+	0.515661i	$0.172454\pi$
$29$			5.55384i			1.03132i	−0.856793	+	0.515661i	$0.827546\pi$
$30$		0			0
$31$			9.85431i			1.76989i	0.465698	+	0.884944i	$0.345803\pi$
$31$			9.85431i			1.76989i	−0.465698	+	0.884944i	$0.654197\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−7.32485			−1.23813
$36$		0			0
$37$	9.85431			1.62004			0.810020	−	0.586403i	$-0.199456\pi$
$37$	9.85431			1.62004			0.810020	+	0.586403i	$0.199456\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	8.38227i		−	1.30909i	−0.756023	−	0.654545i	$-0.772860\pi$
$41$		−	8.38227i		−	1.30909i	0.756023	−	0.654545i	$-0.227140\pi$
$42$		0			0
$43$	2.26233			0.345001			0.172501	−	0.985009i	$-0.444815\pi$
$43$	2.26233			0.345001			0.172501	+	0.985009i	$0.444815\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			3.13132i			0.456750i	0.973573	+	0.228375i	$0.0733412\pi$
$47$			3.13132i			0.456750i	−0.973573	+	0.228375i	$0.926659\pi$
$48$		0			0
$49$	8.34972			1.19282
$50$		0			0
$51$		0			0
$52$		0			0
$53$	8.13224			1.11705			0.558524	−	0.829488i	$-0.311368\pi$
$53$	8.13224			1.11705			0.558524	+	0.829488i	$0.311368\pi$
$54$		0			0
$55$	6.18020	−	0.504595i	0.833338	−	0.0680395i
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	10.5416i		−	1.37240i	−0.727412	−	0.686201i	$-0.759277\pi$
$59$		−	10.5416i		−	1.37240i	0.727412	−	0.686201i	$-0.240723\pi$
$60$		0			0
$61$		−	8.82421i		−	1.12982i	−0.825151	−	0.564912i	$-0.808910\pi$
$61$		−	8.82421i		−	1.12982i	0.825151	−	0.564912i	$-0.191090\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	6.61125i		−	0.820025i
$66$		0			0
$67$			4.84513i			0.591926i	0.955199	+	0.295963i	$0.0956406\pi$
$67$			4.84513i			0.591926i	−0.955199	+	0.295963i	$0.904359\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			0.607882i			0.0721423i	0.999349	+	0.0360712i	$0.0114843\pi$
$71$			0.607882i			0.0721423i	−0.999349	+	0.0360712i	$0.988516\pi$
$72$		0			0
$73$			13.8871i			1.62537i	0.582705	+	0.812684i	$0.301994\pi$
$73$			13.8871i			1.62537i	−0.582705	+	0.812684i	$0.698006\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−12.9510	+	1.05741i	−1.47591	+	0.120503i
$78$		0			0
$79$	3.15450			0.354908			0.177454	−	0.984129i	$-0.443214\pi$
$79$	3.15450			0.354908			0.177454	+	0.984129i	$0.443214\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	6.37046			0.699249			0.349624	−	0.936890i	$-0.386309\pi$
$83$	6.37046			0.699249			0.349624	+	0.936890i	$0.386309\pi$
$84$		0			0
$85$			3.31109i			0.359138i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	15.9002			1.68542			0.842709	−	0.538370i	$-0.180960\pi$
$89$	15.9002			1.68542			0.842709	+	0.538370i	$0.180960\pi$
$90$		0			0
$91$			13.8543i			1.45233i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−5.65685			−0.580381
$96$		0			0
$97$	9.34972			0.949320			0.474660	−	0.880169i	$-0.342571\pi$
$97$	9.34972			0.949320			0.474660	+	0.880169i	$0.342571\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$			7.66866i			0.763060i	0.924356	+	0.381530i	$0.124603\pi$
$101$			7.66866i			0.763060i	−0.924356	+	0.381530i	$0.875397\pi$
$102$		0			0
$103$			8.84513i			0.871536i	0.900059	+	0.435768i	$0.143523\pi$
$103$			8.84513i			0.871536i	−0.900059	+	0.435768i	$0.856477\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	0.954396			0.0922649			0.0461325	−	0.998935i	$-0.485310\pi$
$107$	0.954396			0.0922649			0.0461325	+	0.998935i	$0.485310\pi$
$108$		0			0
$109$			15.6390i			1.49794i	0.662602	+	0.748972i	$0.269452\pi$
$109$			15.6390i			1.49794i	−0.662602	+	0.748972i	$0.730548\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−13.7410			−1.29265			−0.646324	−	0.763063i	$-0.723695\pi$
$113$	−13.7410			−1.29265			−0.646324	+	0.763063i	$0.723695\pi$
$114$		0			0
$115$			13.8543i			1.29192i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	6.93861i		−	0.636062i
$120$		0			0
$121$	10.8543	−	1.78434i	0.986756	−	0.162213i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−12.1610			−1.08771
$126$		0			0
$127$	4.93883			0.438251			0.219125	−	0.975697i	$-0.429680\pi$
$127$	4.93883			0.438251			0.219125	+	0.975697i	$0.429680\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	0.713604			0.0623479			0.0311740	−	0.999514i	$-0.490075\pi$
$131$	0.713604			0.0623479			0.0311740	+	0.999514i	$0.490075\pi$
$132$		0			0
$133$	11.8543			1.02790
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−10.0018			−0.854515			−0.427257	−	0.904130i	$-0.640520\pi$
$137$	−10.0018			−0.854515			−0.427257	+	0.904130i	$0.640520\pi$
$138$		0			0
$139$	8.57132			0.727010			0.363505	−	0.931592i	$-0.381580\pi$
$139$	8.57132			0.727010			0.363505	+	0.931592i	$0.381580\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−0.954396	−	11.6893i	−0.0798106	−	0.977508i
$144$		0			0
$145$			10.3835i			0.862299i
$146$		0			0
$147$		0			0
$148$		0			0
$149$			13.3255i			1.09167i	0.837893	+	0.545834i	$0.183787\pi$
$149$			13.3255i			1.09167i	−0.837893	+	0.545834i	$0.816213\pi$
$150$		0			0
$151$	−22.3297			−1.81716			−0.908581	−	0.417708i	$-0.862834\pi$
$151$	−22.3297			−1.81716			−0.908581	+	0.417708i	$0.862834\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$			18.4236i			1.47982i
$156$		0			0
$157$	16.6994			1.33276			0.666380	−	0.745612i	$-0.267843\pi$
$157$	16.6994			1.33276			0.666380	+	0.745612i	$0.267843\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	29.0326i		−	2.28809i
$162$		0			0
$163$		−	1.00919i		−	0.0790458i	−0.999219	−	0.0395229i	$-0.987416\pi$
$163$		−	1.00919i		−	0.0790458i	0.999219	−	0.0395229i	$-0.0125838\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	20.3066			1.57137			0.785685	−	0.618627i	$-0.212311\pi$
$167$	20.3066			1.57137			0.785685	+	0.618627i	$0.212311\pi$
$168$		0			0
$169$	0.495405			0.0381081
$170$		0			0
$171$		0			0
$172$		0			0
$173$			13.1195i			0.997456i	0.866759	+	0.498728i	$0.166199\pi$
$173$			13.1195i			0.997456i	−0.866759	+	0.498728i	$0.833801\pi$
$174$		0			0
$175$	5.89481			0.445606
$176$		0			0
$177$		0			0
$178$		0			0
$179$			13.7410i			1.02705i	0.858074	+	0.513527i	$0.171661\pi$
$179$			13.7410i			1.02705i	−0.858074	+	0.513527i	$0.828339\pi$
$180$		0			0
$181$	−3.00919			−0.223671			−0.111836	−	0.993727i	$-0.535673\pi$
$181$	−3.00919			−0.223671			−0.111836	+	0.993727i	$0.535673\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	18.4236			1.35453
$186$		0			0
$187$	0.477987	+	5.85431i	0.0349539	+	0.428110i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	25.2941i		−	1.83022i	−0.403204	−	0.915110i	$-0.632103\pi$
$191$		−	25.2941i		−	1.83022i	0.403204	−	0.915110i	$-0.367897\pi$
$192$		0			0
$193$		−	3.76128i		−	0.270743i	−0.990795	−	0.135371i	$-0.956777\pi$
$193$		−	3.76128i		−	0.270743i	0.990795	−	0.135371i	$-0.0432227\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			18.7763i			1.33776i	0.743371	+	0.668879i	$0.233226\pi$
$197$			18.7763i			1.33776i	−0.743371	+	0.668879i	$0.766774\pi$
$198$		0			0
$199$			17.8543i			1.26566i	0.774291	+	0.632829i	$0.218107\pi$
$199$			17.8543i			1.26566i	−0.774291	+	0.632829i	$0.781893\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		−	21.7592i		−	1.52720i
$204$		0			0
$205$		−	15.6715i		−	1.09454i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−10.0018	+	0.816620i	−0.691842	+	0.0564868i
$210$		0			0
$211$	−10.8614			−0.747733			−0.373866	−	0.927483i	$-0.621968\pi$
$211$	−10.8614			−0.747733			−0.373866	+	0.927483i	$0.621968\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	4.22965			0.288459
$216$		0			0
$217$		−	38.6079i		−	2.62088i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	6.26264			0.421271
$222$		0			0
$223$			1.85431i			0.124174i	0.998071	+	0.0620870i	$0.0197756\pi$
$223$			1.85431i			0.124174i	−0.998071	+	0.0620870i	$0.980224\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−17.9250			−1.18972			−0.594861	−	0.803829i	$-0.702793\pi$
$227$	−17.9250			−1.18972			−0.594861	+	0.803829i	$0.702793\pi$
$228$		0			0
$229$	8.84513			0.584502			0.292251	−	0.956342i	$-0.405596\pi$
$229$	8.84513			0.584502			0.292251	+	0.956342i	$0.405596\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	12.4059i		−	0.812736i	−0.913709	−	0.406368i	$-0.866795\pi$
$233$		−	12.4059i		−	0.812736i	0.913709	−	0.406368i	$-0.133205\pi$
$234$		0			0
$235$			5.85431i			0.381893i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−18.8794			−1.22120			−0.610602	−	0.791938i	$-0.709072\pi$
$239$	−18.8794			−1.22120			−0.610602	+	0.791938i	$0.709072\pi$
$240$		0			0
$241$		−	10.3835i		−	0.668857i	−0.942421	−	0.334429i	$-0.891457\pi$
$241$		−	10.3835i		−	0.668857i	0.942421	−	0.334429i	$-0.108543\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	15.6106			0.997327
$246$		0			0
$247$			10.6994i			0.680789i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			27.1178i			1.71166i	0.517257	+	0.855830i	$0.326953\pi$
$251$			27.1178i			1.71166i	−0.517257	+	0.855830i	$0.673047\pi$
$252$		0			0
$253$	2.00000	+	24.4957i	0.125739	+	1.54003i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−6.26264			−0.390653			−0.195326	−	0.980738i	$-0.562577\pi$
$257$	−6.26264			−0.390653			−0.195326	+	0.980738i	$0.562577\pi$
$258$		0			0
$259$	−38.6079			−2.39898
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−7.08406			−0.436822			−0.218411	−	0.975857i	$-0.570087\pi$
$263$	−7.08406			−0.436822			−0.218411	+	0.975857i	$0.570087\pi$
$264$		0			0
$265$	15.2040			0.933976
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−4.02875			−0.245638			−0.122819	−	0.992429i	$-0.539193\pi$
$269$	−4.02875			−0.245638			−0.122819	+	0.992429i	$0.539193\pi$
$270$		0			0
$271$	0.349194			0.0212120			0.0106060	−	0.999944i	$-0.496624\pi$
$271$	0.349194			0.0212120			0.0106060	+	0.999944i	$0.496624\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−4.97363	+	0.406082i	−0.299921	+	0.0244876i
$276$		0			0
$277$		−	20.9920i		−	1.26129i	−0.776072	−	0.630644i	$-0.782791\pi$
$277$		−	20.9920i		−	1.26129i	0.776072	−	0.630644i	$-0.217209\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	14.7875i		−	0.882147i	−0.897471	−	0.441074i	$-0.854598\pi$
$281$		−	14.7875i		−	0.882147i	0.897471	−	0.441074i	$-0.145402\pi$
$282$		0			0
$283$	8.37872			0.498063			0.249032	−	0.968495i	$-0.419888\pi$
$283$	8.37872			0.498063			0.249032	+	0.968495i	$0.419888\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			32.8406i			1.93852i
$288$		0			0
$289$	13.8635			0.815500
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	12.8439i		−	0.750351i	−0.926954	−	0.375175i	$-0.877583\pi$
$293$		−	12.8439i		−	0.750351i	0.926954	−	0.375175i	$-0.122417\pi$
$294$		0			0
$295$		−	19.7086i		−	1.14748i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	26.2042			1.51543
$300$		0			0
$301$	−8.86350			−0.510884
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	16.4977i		−	0.944658i
$306$		0			0
$307$	20.6741			1.17993			0.589967	−	0.807427i	$-0.299141\pi$
$307$	20.6741			1.17993			0.589967	+	0.807427i	$0.299141\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	10.0699i		−	0.571013i	−0.958377	−	0.285507i	$-0.907838\pi$
$311$		−	10.0699i		−	0.571013i	0.958377	−	0.285507i	$-0.0921619\pi$
$312$		0			0
$313$	−4.84513			−0.273863			−0.136931	−	0.990581i	$-0.543724\pi$
$313$	−4.84513			−0.273863			−0.136931	+	0.990581i	$0.543724\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	9.34800			0.525036			0.262518	−	0.964927i	$-0.415447\pi$
$317$	9.34800			0.525036			0.262518	+	0.964927i	$0.415447\pi$
$318$		0			0
$319$	1.49895	+	18.3589i	0.0839251	+	1.02790i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	5.35857i		−	0.298159i
$324$		0			0
$325$			5.32052i			0.295129i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		−	12.2681i		−	0.676362i
$330$		0			0
$331$		−	26.6994i		−	1.46753i	−0.679401	−	0.733767i	$-0.737760\pi$
$331$		−	26.6994i		−	1.46753i	0.679401	−	0.733767i	$-0.262240\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$			9.05845i			0.494916i
$336$		0			0
$337$			13.8871i			0.756481i	0.925707	+	0.378241i	$0.123471\pi$
$337$			13.8871i			0.756481i	−0.925707	+	0.378241i	$0.876529\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	2.65962	+	32.5747i	0.144027	+	1.76402i
$342$		0			0
$343$	−5.28803			−0.285527
$344$		0			0
$345$		0			0
$346$		0			0
$347$	16.2570			0.872719			0.436360	−	0.899772i	$-0.356268\pi$
$347$	16.2570			0.872719			0.436360	+	0.899772i	$0.356268\pi$
$348$		0			0
$349$			15.7040i			0.840615i	0.907382	+	0.420307i	$0.138078\pi$
$349$			15.7040i			0.840615i	−0.907382	+	0.420307i	$0.861922\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	14.6844			0.781573			0.390787	−	0.920481i	$-0.372203\pi$
$353$	14.6844			0.781573			0.390787	+	0.920481i	$0.372203\pi$
$354$		0			0
$355$			1.13650i			0.0603190i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−3.74806			−0.197815			−0.0989076	−	0.995097i	$-0.531535\pi$
$359$	−3.74806			−0.197815			−0.0989076	+	0.995097i	$0.531535\pi$
$360$		0			0
$361$	−9.84513			−0.518164
$362$		0			0
$363$		0			0
$364$		0			0
$365$			25.9634i			1.35899i
$366$		0			0
$367$		−	0.845125i		−	0.0441152i	−0.999757	−	0.0220576i	$-0.992978\pi$
$367$		−	0.845125i		−	0.0441152i	0.999757	−	0.0220576i	$-0.00702172\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−31.8611			−1.65414
$372$		0			0
$373$			29.7837i			1.54214i	0.636749	+	0.771071i	$0.280279\pi$
$373$			29.7837i			1.54214i	−0.636749	+	0.771071i	$0.719721\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	19.6394			1.01148
$378$		0			0
$379$		−	11.7086i		−	0.601432i	−0.953714	−	0.300716i	$-0.902774\pi$
$379$		−	11.7086i		−	0.601432i	0.953714	−	0.300716i	$-0.0972256\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			17.2760i			0.882760i	0.897320	+	0.441380i	$0.145511\pi$
$383$			17.2760i			0.882760i	−0.897320	+	0.441380i	$0.854489\pi$
$384$		0			0
$385$	−24.2132	+	1.97694i	−1.23402	+	0.100754i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−17.7698			−0.900964			−0.450482	−	0.892785i	$-0.648748\pi$
$389$	−17.7698			−0.900964			−0.450482	+	0.892785i	$0.648748\pi$
$390$		0			0
$391$	−13.1238			−0.663698
$392$		0			0
$393$		0			0
$394$		0			0
$395$	5.89765			0.296743
$396$		0			0
$397$	−18.8635			−0.946732			−0.473366	−	0.880866i	$-0.656961\pi$
$397$	−18.8635			−0.946732			−0.473366	+	0.880866i	$0.656961\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−3.73920			−0.186727			−0.0933634	−	0.995632i	$-0.529762\pi$
$401$	−3.73920			−0.186727			−0.0933634	+	0.995632i	$0.529762\pi$
$402$		0			0
$403$	34.8467			1.73584
$404$		0			0
$405$		0			0
$406$		0			0
$407$	32.5747	−	2.65962i	1.61467	−	0.131833i
$408$		0			0
$409$			24.2706i			1.20010i	0.799961	+	0.600052i	$0.204854\pi$
$409$			24.2706i			1.20010i	−0.799961	+	0.600052i	$0.795146\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			41.3007i			2.03228i
$414$		0			0
$415$	11.9102			0.584649
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	34.5962i		−	1.69013i	−0.534660	−	0.845067i	$-0.679560\pi$
$419$		−	34.5962i		−	1.69013i	0.534660	−	0.845067i	$-0.320440\pi$
$420$		0			0
$421$	−31.3989			−1.53029			−0.765144	−	0.643859i	$-0.777332\pi$
$421$	−31.3989			−1.53029			−0.765144	+	0.643859i	$0.777332\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		−	2.66466i		−	0.129255i
$426$		0			0
$427$			34.5721i			1.67306i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−29.7810			−1.43450			−0.717250	−	0.696816i	$-0.754600\pi$
$431$	−29.7810			−1.43450			−0.717250	+	0.696816i	$0.754600\pi$
$432$		0			0
$433$	6.86350			0.329839			0.164919	−	0.986307i	$-0.447264\pi$
$433$	6.86350			0.329839			0.164919	+	0.986307i	$0.447264\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	22.4214i		−	1.07256i
$438$		0			0
$439$	18.5684			0.886221			0.443111	−	0.896467i	$-0.353875\pi$
$439$	18.5684			0.886221			0.443111	+	0.896467i	$0.353875\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			2.83513i			0.134701i	0.997729	+	0.0673506i	$0.0214546\pi$
$443$			2.83513i			0.134701i	−0.997729	+	0.0673506i	$0.978545\pi$
$444$		0			0
$445$	29.7270			1.40920
$446$		0			0
$447$		0			0
$448$		0			0
$449$	7.11412			0.335736			0.167868	−	0.985809i	$-0.446312\pi$
$449$	7.11412			0.335736			0.167868	+	0.985809i	$0.446312\pi$
$450$		0			0
$451$	−2.26233	−	27.7086i	−0.106529	−	1.30475i
$452$		0			0
$453$		0			0
$454$		0			0
$455$			25.9020i			1.21431i
$456$		0			0
$457$			10.6410i			0.497767i	0.968533	+	0.248884i	$0.0800636\pi$
$457$			10.6410i			0.497767i	−0.968533	+	0.248884i	$0.919936\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	12.6379i		−	0.588606i	−0.955712	−	0.294303i	$-0.904913\pi$
$461$		−	12.6379i		−	0.588606i	0.955712	−	0.294303i	$-0.0950874\pi$
$462$		0			0
$463$			6.14569i			0.285614i	0.989751	+	0.142807i	$0.0456129\pi$
$463$			6.14569i			0.285614i	−0.989751	+	0.142807i	$0.954387\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	24.0546i		−	1.11311i	−0.830810	−	0.556556i	$-0.812122\pi$
$467$		−	24.0546i		−	1.11311i	0.830810	−	0.556556i	$-0.187878\pi$
$468$		0			0
$469$		−	18.9826i		−	0.876534i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	7.47840	−	0.610589i	0.343857	−	0.0280749i
$474$		0			0
$475$	4.55246			0.208881
$476$		0			0
$477$		0			0
$478$		0			0
$479$	24.0546			1.09908			0.549542	−	0.835466i	$-0.314802\pi$
$479$	24.0546			1.09908			0.549542	+	0.835466i	$0.314802\pi$
$480$		0			0
$481$		−	34.8467i		−	1.58887i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	17.4802			0.793737
$486$		0			0
$487$		−	34.8635i		−	1.57982i	−0.613225	−	0.789908i	$-0.710128\pi$
$487$		−	34.8635i		−	1.57982i	0.613225	−	0.789908i	$-0.289872\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−8.45052			−0.381367			−0.190683	−	0.981652i	$-0.561070\pi$
$491$	−8.45052			−0.381367			−0.190683	+	0.981652i	$0.561070\pi$
$492$		0			0
$493$	−9.83594			−0.442988
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	2.38160i		−	0.106830i
$498$		0			0
$499$		−	8.00000i		−	0.358129i	−0.983837	−	0.179065i	$-0.942693\pi$
$499$		−	8.00000i		−	0.358129i	0.983837	−	0.179065i	$-0.0573071\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	18.4673			0.823416			0.411708	−	0.911316i	$-0.364932\pi$
$503$	18.4673			0.823416			0.411708	+	0.911316i	$0.364932\pi$
$504$		0			0
$505$			14.3373i			0.638003i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−25.2482			−1.11911			−0.559553	−	0.828794i	$-0.689027\pi$
$509$	−25.2482			−1.11911			−0.559553	+	0.828794i	$0.689027\pi$
$510$		0			0
$511$		−	54.4081i		−	2.40687i
$512$		0			0
$513$		0			0
$514$		0			0
$515$			16.5368i			0.728701i
$516$		0			0
$517$	0.845125	+	10.3510i	0.0371686	+	0.455235i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−31.2213			−1.36783			−0.683914	−	0.729562i	$-0.739724\pi$
$521$	−31.2213			−1.36783			−0.683914	+	0.729562i	$0.739724\pi$
$522$		0			0
$523$	25.5120			1.11556			0.557780	−	0.829989i	$-0.311653\pi$
$523$	25.5120			1.11556			0.557780	+	0.829989i	$0.311653\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−17.4521			−0.760227
$528$		0			0
$529$	−31.9127			−1.38751
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−29.6412			−1.28390
$534$		0			0
$535$	1.78434			0.0771437
$536$		0			0
$537$		0			0
$538$		0			0
$539$	27.6010	−	2.25354i	1.18886	−	0.0970669i
$540$		0			0
$541$		−	25.7649i		−	1.10772i	−0.832610	−	0.553859i	$-0.813155\pi$
$541$		−	25.7649i		−	1.10772i	0.832610	−	0.553859i	$-0.186845\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$			29.2387i			1.25245i
$546$		0			0
$547$	−31.3152			−1.33894			−0.669470	−	0.742839i	$-0.733479\pi$
$547$	−31.3152			−1.33894			−0.669470	+	0.742839i	$0.733479\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	16.8043i		−	0.715886i
$552$		0			0
$553$	−12.3589			−0.525554
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	11.4167i		−	0.483742i	−0.970308	−	0.241871i	$-0.922239\pi$
$557$		−	11.4167i		−	0.483742i	0.970308	−	0.241871i	$-0.0777611\pi$
$558$		0			0
$559$		−	8.00000i		−	0.338364i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−8.52004			−0.359077			−0.179538	−	0.983751i	$-0.557460\pi$
$563$	−8.52004			−0.359077			−0.179538	+	0.983751i	$0.557460\pi$
$564$		0			0
$565$	−25.6903			−1.08080
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	33.3913i		−	1.39984i	−0.714223	−	0.699918i	$-0.753220\pi$
$569$		−	33.3913i		−	1.39984i	0.714223	−	0.699918i	$-0.246780\pi$
$570$		0			0
$571$	−19.7182			−0.825179			−0.412589	−	0.910917i	$-0.635376\pi$
$571$	−19.7182			−0.825179			−0.412589	+	0.910917i	$0.635376\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	11.1495i		−	0.464967i
$576$		0			0
$577$	−26.2132			−1.09127			−0.545635	−	0.838023i	$-0.683712\pi$
$577$	−26.2132			−1.09127			−0.545635	+	0.838023i	$0.683712\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−24.9586			−1.03546
$582$		0			0
$583$	26.8821	−	2.19484i	1.11334	−	0.0909012i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	16.9405i		−	0.699207i	−0.936898	−	0.349604i	$-0.886316\pi$
$587$		−	16.9405i		−	0.699207i	0.936898	−	0.349604i	$-0.113684\pi$
$588$		0			0
$589$		−	29.8162i		−	1.22856i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	22.5592i		−	0.926394i	−0.886255	−	0.463197i	$-0.846702\pi$
$593$		−	22.5592i		−	0.926394i	0.886255	−	0.463197i	$-0.153298\pi$
$594$		0			0
$595$		−	12.9724i		−	0.531818i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			28.2655i			1.15490i	0.816428	+	0.577448i	$0.195951\pi$
$599$			28.2655i			1.15490i	−0.816428	+	0.577448i	$0.804049\pi$
$600$		0			0
$601$			14.4023i			0.587483i	0.955885	+	0.293741i	$0.0949005\pi$
$601$			14.4023i			0.587483i	−0.955885	+	0.293741i	$0.905100\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	20.2932	−	3.33600i	0.825037	−	0.135628i
$606$		0			0
$607$	3.91787			0.159022			0.0795108	−	0.996834i	$-0.474664\pi$
$607$	3.91787			0.159022			0.0795108	+	0.996834i	$0.474664\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	11.0729			0.447962
$612$		0			0
$613$		−	1.75184i		−	0.0707563i	−0.999374	−	0.0353782i	$-0.988736\pi$
$613$		−	1.75184i		−	0.0707563i	0.999374	−	0.0353782i	$-0.0112636\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−38.0630			−1.53236			−0.766180	−	0.642626i	$-0.777845\pi$
$617$	−38.0630			−1.53236			−0.766180	+	0.642626i	$0.777845\pi$
$618$		0			0
$619$		−	8.55375i		−	0.343804i	−0.985114	−	0.171902i	$-0.945009\pi$
$619$		−	8.55375i		−	0.343804i	0.985114	−	0.171902i	$-0.0549913\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−62.2949			−2.49579
$624$		0			0
$625$	−15.2132			−0.608529
$626$		0			0
$627$		0			0
$628$		0			0
$629$			17.4521i			0.695863i
$630$		0			0
$631$		−	10.5354i		−	0.419407i	−0.977765	−	0.209703i	$-0.932750\pi$
$631$		−	10.5354i		−	0.419407i	0.977765	−	0.209703i	$-0.0672498\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	9.23365			0.366426
$636$		0			0
$637$		−	29.5261i		−	1.16987i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	38.4273			1.51779			0.758894	−	0.651215i	$-0.225740\pi$
$641$	38.4273			1.51779			0.758894	+	0.651215i	$0.225740\pi$
$642$		0			0
$643$			20.8451i			0.822051i	0.911624	+	0.411026i	$0.134829\pi$
$643$			20.8451i			0.822051i	−0.911624	+	0.411026i	$0.865171\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	32.3689i		−	1.27255i	−0.771460	−	0.636277i	$-0.780473\pi$
$647$		−	32.3689i		−	1.27255i	0.771460	−	0.636277i	$-0.219527\pi$
$648$		0			0
$649$	−2.84513	−	34.8467i	−0.111681	−	1.36785i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	41.1484			1.61026			0.805130	−	0.593098i	$-0.202095\pi$
$653$	41.1484			1.61026			0.805130	+	0.593098i	$0.202095\pi$
$654$		0			0
$655$	1.33416			0.0521298
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−25.4906			−0.992973			−0.496486	−	0.868044i	$-0.665377\pi$
$659$	−25.4906			−0.992973			−0.496486	+	0.868044i	$0.665377\pi$
$660$		0			0
$661$	−8.69944			−0.338369			−0.169184	−	0.985584i	$-0.554113\pi$
$661$	−8.69944			−0.338369			−0.169184	+	0.985584i	$0.554113\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	22.1628			0.859438
$666$		0			0
$667$	−41.1557			−1.59355
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−2.38160	−	29.1695i	−0.0919408	−	1.12608i
$672$		0			0
$673$			45.9379i			1.77078i	0.464853	+	0.885388i	$0.346107\pi$
$673$			45.9379i			1.77078i	−0.464853	+	0.885388i	$0.653893\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			24.1576i			0.928454i	0.885716	+	0.464227i	$0.153668\pi$
$677$			24.1576i			0.928454i	−0.885716	+	0.464227i	$0.846332\pi$
$678$		0			0
$679$	−36.6310			−1.40577
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	7.84268i		−	0.300092i	−0.988679	−	0.150046i	$-0.952058\pi$
$683$		−	7.84268i		−	0.300092i	0.988679	−	0.150046i	$-0.0479422\pi$
$684$		0			0
$685$	−18.6994			−0.714469
$686$		0			0
$687$		0			0
$688$		0			0
$689$		−	28.7571i		−	1.09556i
$690$		0			0
$691$		−	42.2440i		−	1.60704i	−0.595279	−	0.803519i	$-0.702959\pi$
$691$		−	42.2440i		−	1.60704i	0.595279	−	0.803519i	$-0.297041\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	16.0249			0.607861
$696$		0			0
$697$	14.8451			0.562299
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	27.6997i		−	1.04620i	−0.852271	−	0.523101i	$-0.824775\pi$
$701$		−	27.6997i		−	1.04620i	0.852271	−	0.523101i	$-0.175225\pi$
$702$		0			0
$703$	−29.8162			−1.12454
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	30.0448i		−	1.12995i
$708$		0			0
$709$	−36.5538			−1.37281			−0.686403	−	0.727222i	$-0.740811\pi$
$709$	−36.5538			−1.37281			−0.686403	+	0.727222i	$0.740811\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−73.0235			−2.73475
$714$		0			0
$715$	−1.78434	−	21.8543i	−0.0667305	−	0.817305i
$716$		0			0
$717$		0			0
$718$		0			0
$719$			17.2366i			0.642818i	0.946940	+	0.321409i	$0.104156\pi$
$719$			17.2366i			0.642818i	−0.946940	+	0.321409i	$0.895844\pi$
$720$		0			0
$721$		−	34.6541i		−	1.29058i
$722$		0			0
$723$		0			0
$724$		0			0
$725$		−	8.35628i		−	0.310344i
$726$		0			0
$727$			6.14569i			0.227931i	0.993485	+	0.113965i	$0.0363553\pi$
$727$			6.14569i			0.227931i	−0.993485	+	0.113965i	$0.963645\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$			4.00661i			0.148190i
$732$		0			0
$733$			15.6390i			0.577639i	0.957384	+	0.288820i	$0.0932628\pi$
$733$			15.6390i			0.577639i	−0.957384	+	0.288820i	$0.906737\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	1.30767	+	16.0162i	0.0481687	+	0.589963i
$738$		0			0
$739$	0.285389			0.0104982			0.00524911	−	0.999986i	$-0.498329\pi$
$739$	0.285389			0.0104982			0.00524911	+	0.999986i	$0.498329\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	48.5908			1.78262			0.891312	−	0.453390i	$-0.149785\pi$
$743$	48.5908			1.78262			0.891312	+	0.453390i	$0.149785\pi$
$744$		0			0
$745$			24.9134i			0.912756i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−3.73920			−0.136627
$750$		0			0
$751$		−	32.8451i		−	1.19854i	−0.800549	−	0.599268i	$-0.795458\pi$
$751$		−	32.8451i		−	1.19854i	0.800549	−	0.599268i	$-0.204542\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−41.7476			−1.51935
$756$		0			0
$757$	−8.69944			−0.316186			−0.158093	−	0.987424i	$-0.550535\pi$
$757$	−8.69944			−0.316186			−0.158093	+	0.987424i	$0.550535\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			46.6138i			1.68975i	0.534965	+	0.844874i	$0.320325\pi$
$761$			46.6138i			1.68975i	−0.534965	+	0.844874i	$0.679675\pi$
$762$		0			0
$763$		−	61.2716i		−	2.21818i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−37.2771			−1.34600
$768$		0			0
$769$		−	7.32995i		−	0.264325i	−0.991228	−	0.132162i	$-0.957808\pi$
$769$		−	7.32995i		−	0.264325i	0.991228	−	0.132162i	$-0.0421921\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−12.8148			−0.460917			−0.230459	−	0.973082i	$-0.574023\pi$
$773$	−12.8148			−0.460917			−0.230459	+	0.973082i	$0.574023\pi$
$774$		0			0
$775$		−	14.8267i		−	0.532592i
$776$		0			0
$777$		0			0
$778$		0			0
$779$			25.3622i			0.908697i
$780$		0			0
$781$	0.164064	+	2.00943i	0.00587067	+	0.0719031i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	31.2213			1.11434
$786$		0			0
$787$	14.8803			0.530426			0.265213	−	0.964190i	$-0.414558\pi$
$787$	14.8803			0.530426			0.265213	+	0.964190i	$0.414558\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	53.8356			1.91417
$792$		0			0
$793$	−31.2040			−1.10809
$794$		0			0
$795$		0			0
$796$		0			0
$797$	23.6681			0.838369			0.419184	−	0.907901i	$-0.362316\pi$
$797$	23.6681			0.838369			0.419184	+	0.907901i	$0.362316\pi$
$798$		0			0
$799$	−5.54561			−0.196190
$800$		0			0
$801$		0			0
$802$		0			0
$803$	3.74806	+	45.9057i	0.132266	+	1.61998i
$804$		0			0
$805$		−	54.2794i		−	1.91310i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			32.4369i			1.14042i	0.821499	+	0.570210i	$0.193138\pi$
$809$			32.4369i			1.14042i	−0.821499	+	0.570210i	$0.806862\pi$
$810$		0			0
$811$	14.9359			0.524471			0.262235	−	0.965004i	$-0.415540\pi$
$811$	14.9359			0.524471			0.262235	+	0.965004i	$0.415540\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		−	1.88678i		−	0.0660910i
$816$		0			0
$817$	−6.84513			−0.239481
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	41.6098i		−	1.45219i	−0.687594	−	0.726096i	$-0.741333\pi$
$821$		−	41.6098i		−	1.45219i	0.687594	−	0.726096i	$-0.258667\pi$
$822$		0			0
$823$			0.845125i			0.0294592i	0.999892	+	0.0147296i	$0.00468875\pi$
$823$			0.845125i			0.0294592i	−0.999892	+	0.0147296i	$0.995311\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	38.4723			1.33781			0.668907	−	0.743346i	$-0.266763\pi$
$827$	38.4723			1.33781			0.668907	+	0.743346i	$0.266763\pi$
$828$		0			0
$829$	−1.85431			−0.0644030			−0.0322015	−	0.999481i	$-0.510252\pi$
$829$	−1.85431			−0.0644030			−0.0322015	+	0.999481i	$0.510252\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$			14.7875i			0.512356i
$834$		0			0
$835$	37.9652			1.31384
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	6.46692i		−	0.223263i	−0.993750	−	0.111631i	$-0.964392\pi$
$839$		−	6.46692i		−	0.223263i	0.993750	−	0.111631i	$-0.0356076\pi$
$840$		0			0
$841$	−1.84513			−0.0636250
$842$		0			0
$843$		0			0
$844$		0			0
$845$	0.926210			0.0318626
$846$		0			0
$847$	−42.5258	+	6.99081i	−1.46120	+	0.240207i
$848$		0			0
$849$		0			0
$850$		0			0
$851$			73.0235i			2.50321i
$852$		0			0
$853$			3.79377i			0.129896i	0.997889	+	0.0649481i	$0.0206882\pi$
$853$			3.79377i			0.129896i	−0.997889	+	0.0649481i	$0.979312\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			34.6212i			1.18264i	0.806437	+	0.591319i	$0.201393\pi$
$857$			34.6212i			1.18264i	−0.806437	+	0.591319i	$0.798607\pi$
$858$		0			0
$859$		−	14.3097i		−	0.488242i	−0.969745	−	0.244121i	$-0.921501\pi$
$859$		−	14.3097i		−	0.488242i	0.969745	−	0.244121i	$-0.0784995\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			1.28386i			0.0437030i	0.999761	+	0.0218515i	$0.00695611\pi$
$863$			1.28386i			0.0437030i	−0.999761	+	0.0218515i	$0.993044\pi$
$864$		0			0
$865$			24.5282i			0.833984i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	10.4276	−	0.851380i	0.353731	−	0.0288811i
$870$		0			0
$871$	17.1333			0.580538
$872$		0			0
$873$		0			0
$874$		0			0
$875$	47.6452			1.61070
$876$		0			0
$877$			33.0948i			1.11753i	0.829325	+	0.558766i	$0.188725\pi$
$877$			33.0948i			1.11753i	−0.829325	+	0.558766i	$0.811275\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−34.6881			−1.16867			−0.584336	−	0.811512i	$-0.698645\pi$
$881$	−34.6881			−1.16867			−0.584336	+	0.811512i	$0.698645\pi$
$882$		0			0
$883$			48.8819i			1.64501i	0.568761	+	0.822503i	$0.307423\pi$
$883$			48.8819i			1.64501i	−0.568761	+	0.822503i	$0.692577\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−22.1458			−0.743584			−0.371792	−	0.928316i	$-0.621257\pi$
$887$	−22.1458			−0.743584			−0.371792	+	0.928316i	$0.621257\pi$
$888$		0			0
$889$	−19.3497			−0.648969
$890$		0			0
$891$		0			0
$892$		0			0
$893$		−	9.47444i		−	0.317050i
$894$		0			0
$895$			25.6903i			0.858730i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−54.7293			−1.82532
$900$		0			0
$901$			14.4023i			0.479811i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−5.62598			−0.187014
$906$		0			0
$907$			19.1549i			0.636027i	0.948086	+	0.318014i	$0.103016\pi$
$907$			19.1549i			0.636027i	−0.948086	+	0.318014i	$0.896984\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	52.0477i		−	1.72442i	−0.506554	−	0.862208i	$-0.669081\pi$
$911$		−	52.0477i		−	1.72442i	0.506554	−	0.862208i	$-0.330919\pi$
$912$		0			0
$913$	21.0583	−	1.71935i	0.696930	−	0.0569022i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−2.79581			−0.0923258
$918$		0			0
$919$	13.4080			0.442288			0.221144	−	0.975241i	$-0.429021\pi$
$919$	13.4080			0.442288			0.221144	+	0.975241i	$0.429021\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	2.14958			0.0707544
$924$		0			0
$925$	−14.8267			−0.487500
$926$		0			0
$927$		0			0
$928$		0			0
$929$	34.5962			1.13506			0.567532	−	0.823351i	$-0.307898\pi$
$929$	34.5962			1.13506			0.567532	+	0.823351i	$0.307898\pi$
$930$		0			0
$931$	−25.2638			−0.827987
$932$		0			0
$933$		0			0
$934$		0			0
$935$	0.893645	+	10.9452i	0.0292253	+	0.357947i
$936$		0			0
$937$		−	42.4342i		−	1.38626i	−0.720810	−	0.693132i	$-0.756230\pi$
$937$		−	42.4342i		−	1.38626i	0.720810	−	0.693132i	$-0.243770\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	14.8222i		−	0.483191i	−0.970377	−	0.241596i	$-0.922329\pi$
$941$		−	14.8222i		−	0.483191i	0.970377	−	0.241596i	$-0.0776708\pi$
$942$		0			0
$943$	62.1152			2.02275
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	24.3220i		−	0.790358i	−0.918604	−	0.395179i	$-0.870683\pi$
$947$		−	24.3220i		−	0.790358i	0.918604	−	0.395179i	$-0.129317\pi$
$948$		0			0
$949$	49.1075			1.59410
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	58.8819i		−	1.90737i	−0.300805	−	0.953686i	$-0.597255\pi$
$953$		−	58.8819i		−	1.90737i	0.300805	−	0.953686i	$-0.402745\pi$
$954$		0			0
$955$		−	47.2899i		−	1.53027i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	39.1859			1.26538
$960$		0			0
$961$	−66.1075			−2.13250
$962$		0			0
$963$		0			0
$964$		0			0
$965$		−	7.03208i		−	0.226371i
$966$		0			0
$967$	36.5394			1.17503			0.587514	−	0.809214i	$-0.300107\pi$
$967$	36.5394			1.17503			0.587514	+	0.809214i	$0.300107\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			36.4437i			1.16953i	0.811202	+	0.584766i	$0.198814\pi$
$971$			36.4437i			1.16953i	−0.811202	+	0.584766i	$0.801186\pi$
$972$		0			0
$973$	−33.5813			−1.07657
$974$		0			0
$975$		0			0
$976$		0			0
$977$	49.0658			1.56975			0.784877	−	0.619652i	$-0.212726\pi$
$977$	49.0658			1.56975			0.784877	+	0.619652i	$0.212726\pi$
$978$		0			0
$979$	52.5601	−	4.29137i	1.67983	−	0.137153i
$980$		0			0
$981$		0			0
$982$		0			0
$983$			16.6836i			0.532124i	0.963956	+	0.266062i	$0.0857225\pi$
$983$			16.6836i			0.532124i	−0.963956	+	0.266062i	$0.914277\pi$
$984$		0			0
$985$			35.1043i			1.11851i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			16.7645i			0.533081i
$990$		0			0
$991$			2.43706i			0.0774157i	0.999251	+	0.0387078i	$0.0123242\pi$
$991$			2.43706i			0.0774157i	−0.999251	+	0.0387078i	$0.987676\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$			33.3804i			1.05823i
$996$		0			0
$997$			17.4233i			0.551802i	0.961186	+	0.275901i	$0.0889763\pi$
$997$			17.4233i			0.551802i	−0.961186	+	0.275901i	$0.911024\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3168.2.o.f.703.18	yes	24
3.2	odd	2	inner	3168.2.o.f.703.5	✓	24
4.3	odd	2	inner	3168.2.o.f.703.17	yes	24
11.10	odd	2	inner	3168.2.o.f.703.20	yes	24
12.11	even	2	inner	3168.2.o.f.703.6	yes	24
33.32	even	2	inner	3168.2.o.f.703.7	yes	24
44.43	even	2	inner	3168.2.o.f.703.19	yes	24
132.131	odd	2	inner	3168.2.o.f.703.8	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3168.2.o.f.703.5	✓	24	3.2	odd	2	inner
3168.2.o.f.703.6	yes	24	12.11	even	2	inner
3168.2.o.f.703.7	yes	24	33.32	even	2	inner
3168.2.o.f.703.8	yes	24	132.131	odd	2	inner
3168.2.o.f.703.17	yes	24	4.3	odd	2	inner
3168.2.o.f.703.18	yes	24	1.1	even	1	trivial
3168.2.o.f.703.19	yes	24	44.43	even	2	inner
3168.2.o.f.703.20	yes	24	11.10	odd	2	inner

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:	\( N \)	\(=\)	\( 3168 = 2^{5} \cdot 3^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3168.o (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.86960 q^{5} -3.91787 q^{7} +(3.30563 - 0.269894i) q^{11} -3.53618i q^{13} +1.77102i q^{17} -3.02570 q^{19} +7.41031i q^{23} -1.50459 q^{25} +5.55384i q^{29} +9.85431i q^{31} -7.32485 q^{35} +9.85431 q^{37} -8.38227i q^{41} +2.26233 q^{43} +3.13132i q^{47} +8.34972 q^{49} +8.13224 q^{53} +(6.18020 - 0.504595i) q^{55} -10.5416i q^{59} -8.82421i q^{61} -6.61125i q^{65} +4.84513i q^{67} +0.607882i q^{71} +13.8871i q^{73} +(-12.9510 + 1.05741i) q^{77} +3.15450 q^{79} +6.37046 q^{83} +3.31109i q^{85} +15.9002 q^{89} +13.8543i q^{91} -5.65685 q^{95} +9.34972 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 40 q^{25} + 40 q^{49} + 64 q^{97}+O(q^{100}) \) 24 * q + 40 * q^25 + 40 * q^49 + 64 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more