LMFDB - Embedded newform 2288.2.b.c.2287.17

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2288,2,Mod(2287,2288)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2288, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2288.2287");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2288 = 2^{4} \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2288.b (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:	$18.2697719825$
Analytic rank:	$0$
Dimension:	$56$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2287.17
Character	$\chi$	$=$	2288.2287
Dual form			2288.2.b.c.2287.40

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.16103i q^{3} -1.21516i q^{5} -0.788902i q^{7} +1.65201 q^{9} +O(q^{10})$	$q-1.16103i q^{3} -1.21516i q^{5} -0.788902i q^{7} +1.65201 q^{9} +(2.36186 - 2.32844i) q^{11} +(3.52325 + 0.765959i) q^{13} -1.41084 q^{15} -7.63589i q^{17} +5.51338i q^{19} -0.915941 q^{21} +7.33120i q^{23} +3.52339 q^{25} -5.40113i q^{27} -5.28741i q^{29} +7.43481 q^{31} +(-2.70339 - 2.74219i) q^{33} -0.958640 q^{35} +8.27242i q^{37} +(0.889302 - 4.09061i) q^{39} -7.07032 q^{41} +6.40708 q^{43} -2.00745i q^{45} -6.28955 q^{47} +6.37763 q^{49} -8.86552 q^{51} +7.01647 q^{53} +(-2.82942 - 2.87003i) q^{55} +6.40121 q^{57} -12.2100 q^{59} -0.392498i q^{61} -1.30327i q^{63} +(0.930760 - 4.28130i) q^{65} -4.27445 q^{67} +8.51175 q^{69} +2.70898 q^{71} -2.28236 q^{73} -4.09077i q^{75} +(-1.83691 - 1.86327i) q^{77} +3.31946 q^{79} -1.31486 q^{81} +7.20296i q^{83} -9.27881 q^{85} -6.13885 q^{87} +2.88842i q^{89} +(0.604267 - 2.77950i) q^{91} -8.63205i q^{93} +6.69962 q^{95} -18.3618i q^{97} +(3.90180 - 3.84660i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 56 q - 72 q^{9}+O(q^{10}) $ 56 * q - 72 * q^9	$ 56 q - 72 q^{9} - 72 q^{25} - 8 q^{49} + 72 q^{53} + 8 q^{69} - 48 q^{77} + 88 q^{81}+O(q^{100}) $ 56 * q - 72 * q^9 - 72 * q^25 - 8 * q^49 + 72 * q^53 + 8 * q^69 - 48 * q^77 + 88 * q^81

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2288\mathbb{Z}\right)^\times$.

$n$	$287$	$353$	$1717$	$2081$
$\chi(n)$	$-1$	$-1$	$1$	$-1$

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
$3$		−	1.16103i		−	0.670322i	−0.942161	−	0.335161i	$-0.891209\pi$
$3$		−	1.16103i		−	0.670322i	0.942161	−	0.335161i	$-0.108791\pi$
$4$		0			0
$5$		−	1.21516i		−	0.543435i	−0.962377	−	0.271717i	$-0.912408\pi$
$5$		−	1.21516i		−	0.543435i	0.962377	−	0.271717i	$-0.0875916\pi$
$6$		0			0
$7$		−	0.788902i		−	0.298177i	−0.988824	−	0.149089i	$-0.952366\pi$
$7$		−	0.788902i		−	0.298177i	0.988824	−	0.149089i	$-0.0476339\pi$
$8$		0			0
$9$	1.65201			0.550668
$10$		0			0
$11$	2.36186	−	2.32844i	0.712126	−	0.702051i
$11$	2.36186	−	2.32844i	0.712126	−	0.702051i
$12$		0			0
$13$	3.52325	+	0.765959i	0.977174	+	0.212439i
$13$	3.52325	+	0.765959i	0.977174	+	0.212439i
$14$		0			0
$15$	−1.41084			−0.364276
$16$		0			0
$17$		−	7.63589i		−	1.85198i	−0.377553	−	0.925988i	$-0.623234\pi$
$17$		−	7.63589i		−	1.85198i	0.377553	−	0.925988i	$-0.376766\pi$
$18$		0			0
$19$			5.51338i			1.26486i	0.774619	+	0.632428i	$0.217942\pi$
$19$			5.51338i			1.26486i	−0.774619	+	0.632428i	$0.782058\pi$
$20$		0			0
$21$	−0.915941			−0.199875
$22$		0			0
$23$			7.33120i			1.52866i	0.644825	+	0.764330i	$0.276930\pi$
$23$			7.33120i			1.52866i	−0.644825	+	0.764330i	$0.723070\pi$
$24$		0			0
$25$	3.52339			0.704679
$26$		0			0
$27$		−	5.40113i		−	1.03945i
$28$		0			0
$29$		−	5.28741i		−	0.981847i	−0.871203	−	0.490923i	$-0.836660\pi$
$29$		−	5.28741i		−	0.981847i	0.871203	−	0.490923i	$-0.163340\pi$
$30$		0			0
$31$	7.43481			1.33533			0.667666	−	0.744461i	$-0.267294\pi$
$31$	7.43481			1.33533			0.667666	+	0.744461i	$0.267294\pi$
$32$		0			0
$33$	−2.70339	−	2.74219i	−0.470600	−	0.477354i
$34$		0			0
$35$	−0.958640			−0.162040
$36$		0			0
$37$			8.27242i			1.35998i	0.733222	+	0.679989i	$0.238016\pi$
$37$			8.27242i			1.35998i	−0.733222	+	0.679989i	$0.761984\pi$
$38$		0			0
$39$	0.889302	−	4.09061i	0.142402	−	0.655022i
$40$		0			0
$41$	−7.07032			−1.10420			−0.552099	−	0.833778i	$-0.686173\pi$
$41$	−7.07032			−1.10420			−0.552099	+	0.833778i	$0.686173\pi$
$42$		0			0
$43$	6.40708			0.977071			0.488536	−	0.872544i	$-0.337531\pi$
$43$	6.40708			0.977071			0.488536	+	0.872544i	$0.337531\pi$
$44$		0			0
$45$		−	2.00745i		−	0.299252i
$46$		0			0
$47$	−6.28955			−0.917426			−0.458713	−	0.888585i	$-0.651689\pi$
$47$	−6.28955			−0.917426			−0.458713	+	0.888585i	$0.651689\pi$
$48$		0			0
$49$	6.37763			0.911090
$50$		0			0
$51$	−8.86552			−1.24142
$52$		0			0
$53$	7.01647			0.963787			0.481893	−	0.876230i	$-0.339949\pi$
$53$	7.01647			0.963787			0.481893	+	0.876230i	$0.339949\pi$
$54$		0			0
$55$	−2.82942	−	2.87003i	−0.381519	−	0.386994i
$56$		0			0
$57$	6.40121			0.847861
$58$		0			0
$59$	−12.2100			−1.58960			−0.794800	−	0.606871i	$-0.792424\pi$
$59$	−12.2100			−1.58960			−0.794800	+	0.606871i	$0.792424\pi$
$60$		0			0
$61$		−	0.392498i		−	0.0502543i	−0.999684	−	0.0251271i	$-0.992001\pi$
$61$		−	0.392498i		−	0.0502543i	0.999684	−	0.0251271i	$-0.00799906\pi$
$62$		0			0
$63$		−	1.30327i		−	0.164197i
$64$		0			0
$65$	0.930760	−	4.28130i	0.115447	−	0.531030i
$66$		0			0
$67$	−4.27445			−0.522207			−0.261104	−	0.965311i	$-0.584086\pi$
$67$	−4.27445			−0.522207			−0.261104	+	0.965311i	$0.584086\pi$
$68$		0			0
$69$	8.51175			1.02469
$70$		0			0
$71$	2.70898			0.321497			0.160748	−	0.986995i	$-0.448609\pi$
$71$	2.70898			0.321497			0.160748	+	0.986995i	$0.448609\pi$
$72$		0			0
$73$	−2.28236			−0.267130			−0.133565	−	0.991040i	$-0.542642\pi$
$73$	−2.28236			−0.267130			−0.133565	+	0.991040i	$0.542642\pi$
$74$		0			0
$75$		−	4.09077i		−	0.472362i
$76$		0			0
$77$	−1.83691	−	1.86327i	−0.209336	−	0.212340i
$78$		0			0
$79$	3.31946			0.373468			0.186734	−	0.982410i	$-0.440210\pi$
$79$	3.31946			0.373468			0.186734	+	0.982410i	$0.440210\pi$
$80$		0			0
$81$	−1.31486			−0.146096
$82$		0			0
$83$			7.20296i			0.790628i	0.918546	+	0.395314i	$0.129364\pi$
$83$			7.20296i			0.790628i	−0.918546	+	0.395314i	$0.870636\pi$
$84$		0			0
$85$	−9.27881			−1.00643
$86$		0			0
$87$	−6.13885			−0.658153
$88$		0			0
$89$			2.88842i			0.306172i	0.988213	+	0.153086i	$0.0489211\pi$
$89$			2.88842i			0.306172i	−0.988213	+	0.153086i	$0.951079\pi$
$90$		0			0
$91$	0.604267	−	2.77950i	0.0633443	−	0.291371i
$92$		0			0
$93$		−	8.63205i		−	0.895102i
$94$		0			0
$95$	6.69962			0.687367
$96$		0			0
$97$		−	18.3618i		−	1.86436i	−0.361991	−	0.932182i	$-0.617903\pi$
$97$		−	18.3618i		−	1.86436i	0.361991	−	0.932182i	$-0.382097\pi$
$98$		0			0
$99$	3.90180	−	3.84660i	0.392145	−	0.386597i
$100$		0			0
$101$		−	14.7924i		−	1.47190i	−0.677038	−	0.735948i	$-0.736737\pi$
$101$		−	14.7924i		−	1.47190i	0.677038	−	0.735948i	$-0.263263\pi$
$102$		0			0
$103$		−	5.68100i		−	0.559766i	−0.960034	−	0.279883i	$-0.909704\pi$
$103$		−	5.68100i		−	0.559766i	0.960034	−	0.279883i	$-0.0902956\pi$
$104$		0			0
$105$			1.11301i			0.108619i
$106$		0			0
$107$	−10.1381			−0.980090			−0.490045	−	0.871697i	$-0.663020\pi$
$107$	−10.1381			−0.980090			−0.490045	+	0.871697i	$0.663020\pi$
$108$		0			0
$109$	0.748805			0.0717225			0.0358613	−	0.999357i	$-0.488583\pi$
$109$	0.748805			0.0717225			0.0358613	+	0.999357i	$0.488583\pi$
$110$		0			0
$111$	9.60455			0.911623
$112$		0			0
$113$	10.3831			0.976764			0.488382	−	0.872630i	$-0.337587\pi$
$113$	10.3831			0.976764			0.488382	+	0.872630i	$0.337587\pi$
$114$		0			0
$115$	8.90855			0.830727
$116$		0			0
$117$	5.82043	+	1.26537i	0.538099	+	0.116983i
$118$		0			0
$119$	−6.02397			−0.552217
$120$		0			0
$121$	0.156727	−	10.9989i	0.0142479	−	0.999898i
$122$		0			0
$123$			8.20887i			0.740169i
$124$		0			0
$125$		−	10.3573i		−	0.926382i
$126$		0			0
$127$	−0.560777			−0.0497609			−0.0248804	−	0.999690i	$-0.507921\pi$
$127$	−0.560777			−0.0497609			−0.0248804	+	0.999690i	$0.507921\pi$
$128$		0			0
$129$		−	7.43883i		−	0.654952i
$130$		0			0
$131$	−22.3838			−1.95568			−0.977841	−	0.209347i	$-0.932866\pi$
$131$	−22.3838			−1.95568			−0.977841	+	0.209347i	$0.932866\pi$
$132$		0			0
$133$	4.34952			0.377151
$134$		0			0
$135$	−6.56322			−0.564872
$136$		0			0
$137$			7.67123i			0.655397i	0.944782	+	0.327699i	$0.106273\pi$
$137$			7.67123i			0.655397i	−0.944782	+	0.327699i	$0.893727\pi$
$138$		0			0
$139$	−15.6705			−1.32915			−0.664576	−	0.747221i	$-0.731388\pi$
$139$	−15.6705			−1.32915			−0.664576	+	0.747221i	$0.731388\pi$
$140$		0			0
$141$			7.30237i			0.614971i
$142$		0			0
$143$	10.1049	−	6.39460i	0.845015	−	0.534743i
$144$		0			0
$145$	−6.42503			−0.533569
$146$		0			0
$147$		−	7.40464i		−	0.610724i
$148$		0			0
$149$	−14.3374			−1.17457			−0.587284	−	0.809381i	$-0.699803\pi$
$149$	−14.3374			−1.17457			−0.587284	+	0.809381i	$0.699803\pi$
$150$		0			0
$151$		−	4.36582i		−	0.355285i	−0.984095	−	0.177643i	$-0.943153\pi$
$151$		−	4.36582i		−	0.355285i	0.984095	−	0.177643i	$-0.0568471\pi$
$152$		0			0
$153$		−	12.6145i		−	1.01982i
$154$		0			0
$155$		−	9.03446i		−	0.725665i
$156$		0			0
$157$	−5.23806			−0.418042			−0.209021	−	0.977911i	$-0.567028\pi$
$157$	−5.23806			−0.418042			−0.209021	+	0.977911i	$0.567028\pi$
$158$		0			0
$159$		−	8.14635i		−	0.646047i
$160$		0			0
$161$	5.78360			0.455811
$162$		0			0
$163$	−4.72408			−0.370019			−0.185009	−	0.982737i	$-0.559232\pi$
$163$	−4.72408			−0.370019			−0.185009	+	0.982737i	$0.559232\pi$
$164$		0			0
$165$	−3.33219	+	3.28505i	−0.259411	+	0.255741i
$166$		0			0
$167$			18.9098i			1.46328i	0.681690	+	0.731641i	$0.261245\pi$
$167$			18.9098i			1.46328i	−0.681690	+	0.731641i	$0.738755\pi$
$168$		0			0
$169$	11.8266	+	5.39733i	0.909740	+	0.415179i
$170$		0			0
$171$			9.10813i			0.696516i
$172$		0			0
$173$			19.9829i			1.51927i	0.650351	+	0.759634i	$0.274622\pi$
$173$			19.9829i			1.51927i	−0.650351	+	0.759634i	$0.725378\pi$
$174$		0			0
$175$		−	2.77961i		−	0.210119i
$176$		0			0
$177$			14.1761i			1.06554i
$178$		0			0
$179$		−	3.09345i		−	0.231215i	−0.993295	−	0.115608i	$-0.963118\pi$
$179$		−	3.09345i		−	0.231215i	0.993295	−	0.115608i	$-0.0368815\pi$
$180$		0			0
$181$	−13.8212			−1.02732			−0.513659	−	0.857994i	$-0.671710\pi$
$181$	−13.8212			−1.02732			−0.513659	+	0.857994i	$0.671710\pi$
$182$		0			0
$183$	−0.455703			−0.0336865
$184$		0			0
$185$	10.0523			0.739059
$186$		0			0
$187$	−17.7797	−	18.0349i	−1.30018	−	1.31884i
$188$		0			0
$189$	−4.26096			−0.309939
$190$		0			0
$191$		−	7.36292i		−	0.532762i	−0.963868	−	0.266381i	$-0.914172\pi$
$191$		−	7.36292i		−	0.532762i	0.963868	−	0.266381i	$-0.0858280\pi$
$192$		0			0
$193$	−3.52627			−0.253826			−0.126913	−	0.991914i	$-0.540507\pi$
$193$	−3.52627			−0.253826			−0.126913	+	0.991914i	$0.540507\pi$
$194$		0			0
$195$	−4.97073	−	1.08064i	−0.355961	−	0.0773864i
$196$		0			0
$197$	13.4215			0.956241			0.478120	−	0.878294i	$-0.341318\pi$
$197$	13.4215			0.956241			0.478120	+	0.878294i	$0.341318\pi$
$198$		0			0
$199$			7.41159i			0.525394i	0.964878	+	0.262697i	$0.0846119\pi$
$199$			7.41159i			0.525394i	−0.964878	+	0.262697i	$0.915388\pi$
$200$		0			0
$201$			4.96277i			0.350047i
$202$		0			0
$203$	−4.17125			−0.292764
$204$		0			0
$205$			8.59155i			0.600060i
$206$		0			0
$207$			12.1112i			0.841785i
$208$		0			0
$209$	12.8376	+	13.0218i	0.887994	+	0.900737i
$210$		0			0
$211$	11.5189			0.792992			0.396496	−	0.918036i	$-0.370226\pi$
$211$	11.5189			0.792992			0.396496	+	0.918036i	$0.370226\pi$
$212$		0			0
$213$		−	3.14521i		−	0.215506i
$214$		0			0
$215$		−	7.78561i		−	0.530974i
$216$		0			0
$217$		−	5.86534i		−	0.398165i
$218$		0			0
$219$			2.64989i			0.179063i
$220$		0			0
$221$	5.84878	−	26.9032i	0.393432	−	1.80970i
$222$		0			0
$223$	17.9338			1.20094			0.600468	−	0.799649i	$-0.294981\pi$
$223$	17.9338			1.20094			0.600468	+	0.799649i	$0.294981\pi$
$224$		0			0
$225$	5.82066			0.388044
$226$		0			0
$227$			0.695842i			0.0461846i	0.999733	+	0.0230923i	$0.00735116\pi$
$227$			0.695842i			0.0461846i	−0.999733	+	0.0230923i	$0.992649\pi$
$228$		0			0
$229$			24.0233i			1.58751i	0.608239	+	0.793754i	$0.291876\pi$
$229$			24.0233i			1.58751i	−0.608239	+	0.793754i	$0.708124\pi$
$230$		0			0
$231$	−2.16332	+	2.13271i	−0.142336	+	0.140322i
$232$		0			0
$233$		−	5.15891i		−	0.337971i	−0.985619	−	0.168986i	$-0.945951\pi$
$233$		−	5.15891i		−	0.337971i	0.985619	−	0.168986i	$-0.0540492\pi$
$234$		0			0
$235$			7.64280i			0.498561i
$236$		0			0
$237$		−	3.85400i		−	0.250344i
$238$		0			0
$239$		−	22.1858i		−	1.43508i	−0.696517	−	0.717541i	$-0.745268\pi$
$239$		−	22.1858i		−	1.43508i	0.696517	−	0.717541i	$-0.254732\pi$
$240$		0			0
$241$	18.0897			1.16526			0.582630	−	0.812737i	$-0.302024\pi$
$241$	18.0897			1.16526			0.582630	+	0.812737i	$0.302024\pi$
$242$		0			0
$243$		−	14.6768i		−	0.941516i
$244$		0			0
$245$		−	7.74982i		−	0.495118i
$246$		0			0
$247$	−4.22302	+	19.4250i	−0.268704	+	1.23599i
$248$		0			0
$249$	8.36287			0.529975
$250$		0			0
$251$		−	18.5331i		−	1.16980i	−0.811106	−	0.584899i	$-0.801134\pi$
$251$		−	18.5331i		−	1.16980i	0.811106	−	0.584899i	$-0.198866\pi$
$252$		0			0
$253$	17.0703	+	17.3152i	1.07320	+	1.08860i
$254$		0			0
$255$			10.7730i			0.674631i
$256$		0			0
$257$	−12.6567			−0.789505			−0.394752	−	0.918788i	$-0.629170\pi$
$257$	−12.6567			−0.789505			−0.394752	+	0.918788i	$0.629170\pi$
$258$		0			0
$259$	6.52613			0.405514
$260$		0			0
$261$		−	8.73482i		−	0.540672i
$262$		0			0
$263$	16.9932			1.04785			0.523923	−	0.851766i	$-0.324468\pi$
$263$	16.9932			1.04785			0.523923	+	0.851766i	$0.324468\pi$
$264$		0			0
$265$		−	8.52611i		−	0.523755i
$266$		0			0
$267$	3.35355			0.205234
$268$		0			0
$269$	−14.3781			−0.876650			−0.438325	−	0.898817i	$-0.644428\pi$
$269$	−14.3781			−0.876650			−0.438325	+	0.898817i	$0.644428\pi$
$270$		0			0
$271$			20.6942i			1.25709i	0.777775	+	0.628543i	$0.216348\pi$
$271$			20.6942i			1.25709i	−0.777775	+	0.628543i	$0.783652\pi$
$272$		0			0
$273$	−3.22709	−	0.701573i	−0.195312	−	0.0424611i
$274$		0			0
$275$	8.32175	−	8.20401i	0.501820	−	0.494721i
$276$		0			0
$277$			3.06741i			0.184303i	0.995745	+	0.0921515i	$0.0293744\pi$
$277$			3.06741i			0.184303i	−0.995745	+	0.0921515i	$0.970626\pi$
$278$		0			0
$279$	12.2823			0.735325
$280$		0			0
$281$	−22.8789			−1.36484			−0.682420	−	0.730961i	$-0.739072\pi$
$281$	−22.8789			−1.36484			−0.682420	+	0.730961i	$0.739072\pi$
$282$		0			0
$283$	−8.78163			−0.522014			−0.261007	−	0.965337i	$-0.584055\pi$
$283$	−8.78163			−0.522014			−0.261007	+	0.965337i	$0.584055\pi$
$284$		0			0
$285$		−	7.77847i		−	0.460757i
$286$		0			0
$287$			5.57779i			0.329247i
$288$		0			0
$289$	−41.3069			−2.42982
$290$		0			0
$291$	−21.3187			−1.24972
$292$		0			0
$293$	11.0370			0.644790			0.322395	−	0.946605i	$-0.395512\pi$
$293$	11.0370			0.644790			0.322395	+	0.946605i	$0.395512\pi$
$294$		0			0
$295$			14.8370i			0.863844i
$296$		0			0
$297$	−12.5762	−	12.7567i	−0.729745	−	0.740218i
$298$		0			0
$299$	−5.61539	+	25.8297i	−0.324747	+	1.49377i
$300$		0			0
$301$		−	5.05456i		−	0.291340i
$302$		0			0
$303$	−17.1744			−0.986644
$304$		0			0
$305$	−0.476947			−0.0273099
$306$		0			0
$307$		−	18.9787i		−	1.08317i	−0.840646	−	0.541585i	$-0.817824\pi$
$307$		−	18.9787i		−	1.08317i	0.840646	−	0.541585i	$-0.182176\pi$
$308$		0			0
$309$	−6.59582			−0.375223
$310$		0			0
$311$			12.4803i			0.707693i	0.935303	+	0.353847i	$0.115127\pi$
$311$			12.4803i			0.707693i	−0.935303	+	0.353847i	$0.884873\pi$
$312$		0			0
$313$	7.48469			0.423060			0.211530	−	0.977372i	$-0.432155\pi$
$313$	7.48469			0.423060			0.211530	+	0.977372i	$0.432155\pi$
$314$		0			0
$315$	−1.58368			−0.0892302
$316$		0			0
$317$			0.874419i			0.0491123i	0.999698	+	0.0245561i	$0.00781725\pi$
$317$			0.874419i			0.0491123i	−0.999698	+	0.0245561i	$0.992183\pi$
$318$		0			0
$319$	−12.3114	−	12.4881i	−0.689307	−	0.699199i
$320$		0			0
$321$			11.7707i			0.656976i
$322$		0			0
$323$	42.0996			2.34248
$324$		0			0
$325$	12.4138	+	2.69877i	0.688594	+	0.149701i
$326$		0			0
$327$		−	0.869386i		−	0.0480772i
$328$		0			0
$329$			4.96184i			0.273555i
$330$		0			0
$331$	1.27356			0.0700011			0.0350005	−	0.999387i	$-0.488857\pi$
$331$	1.27356			0.0700011			0.0350005	+	0.999387i	$0.488857\pi$
$332$		0			0
$333$			13.6661i			0.748897i
$334$		0			0
$335$			5.19413i			0.283786i
$336$		0			0
$337$			11.9388i			0.650347i	0.945654	+	0.325173i	$0.105423\pi$
$337$			11.9388i			0.650347i	−0.945654	+	0.325173i	$0.894577\pi$
$338$		0			0
$339$		−	12.0552i		−	0.654746i
$340$		0			0
$341$	17.5599	−	17.3115i	0.950925	−	0.937471i
$342$		0			0
$343$		−	10.5536i		−	0.569843i
$344$		0			0
$345$		−	10.3431i		−	0.556854i
$346$		0			0
$347$	24.8130			1.33203			0.666017	−	0.745937i	$-0.267998\pi$
$347$	24.8130			1.33203			0.666017	+	0.745937i	$0.267998\pi$
$348$		0			0
$349$	−31.6198			−1.69257			−0.846284	−	0.532733i	$-0.821165\pi$
$349$	−31.6198			−1.69257			−0.846284	+	0.532733i	$0.821165\pi$
$350$		0			0
$351$	4.13704	−	19.0295i	0.220819	−	1.01572i
$352$		0			0
$353$			15.4123i			0.820311i	0.912015	+	0.410156i	$0.134526\pi$
$353$			15.4123i			0.820311i	−0.912015	+	0.410156i	$0.865474\pi$
$354$		0			0
$355$		−	3.29184i		−	0.174712i
$356$		0			0
$357$			6.99403i			0.370163i
$358$		0			0
$359$			6.58762i			0.347681i	0.984774	+	0.173841i	$0.0556178\pi$
$359$			6.58762i			0.347681i	−0.984774	+	0.173841i	$0.944382\pi$
$360$		0			0
$361$	−11.3974			−0.599861
$362$		0			0
$363$	−12.7701	−	0.181964i	−0.670254	−	0.00955066i
$364$		0			0
$365$			2.77342i			0.145168i
$366$		0			0
$367$			32.0933i			1.67526i	0.546241	+	0.837628i	$0.316058\pi$
$367$			32.0933i			1.67526i	−0.546241	+	0.837628i	$0.683942\pi$
$368$		0			0
$369$	−11.6802			−0.608047
$370$		0			0
$371$		−	5.53531i		−	0.287379i
$372$		0			0
$373$			28.9247i			1.49767i	0.662759	+	0.748833i	$0.269386\pi$
$373$			28.9247i			1.49767i	−0.662759	+	0.748833i	$0.730614\pi$
$374$		0			0
$375$	−12.0251			−0.620974
$376$		0			0
$377$	4.04993	−	18.6289i	0.208582	−	0.959435i
$378$		0			0
$379$	5.31401			0.272962			0.136481	−	0.990643i	$-0.456421\pi$
$379$	5.31401			0.272962			0.136481	+	0.990643i	$0.456421\pi$
$380$		0			0
$381$			0.651080i			0.0333558i
$382$		0			0
$383$	−10.8625			−0.555048			−0.277524	−	0.960719i	$-0.589514\pi$
$383$	−10.8625			−0.555048			−0.277524	+	0.960719i	$0.589514\pi$
$384$		0			0
$385$	−2.26417	+	2.23214i	−0.115393	+	0.113760i
$386$		0			0
$387$	10.5845			0.538042
$388$		0			0
$389$	26.9462			1.36623			0.683114	−	0.730312i	$-0.260625\pi$
$389$	26.9462			1.36623			0.683114	+	0.730312i	$0.260625\pi$
$390$		0			0
$391$	55.9802			2.83104
$392$		0			0
$393$			25.9883i			1.31094i
$394$		0			0
$395$		−	4.03366i		−	0.202956i
$396$		0			0
$397$			15.5880i			0.782337i	0.920319	+	0.391168i	$0.127929\pi$
$397$			15.5880i			0.782337i	−0.920319	+	0.391168i	$0.872071\pi$
$398$		0			0
$399$		−	5.04993i		−	0.252813i
$400$		0			0
$401$			23.5765i			1.17736i	0.808368	+	0.588678i	$0.200351\pi$
$401$			23.5765i			1.17736i	−0.808368	+	0.588678i	$0.799649\pi$
$402$		0			0
$403$	26.1947	+	5.69476i	1.30485	+	0.283676i
$404$		0			0
$405$			1.59777i			0.0793936i
$406$		0			0
$407$	19.2619	+	19.5383i	0.954775	+	0.968476i
$408$		0			0
$409$	36.5350			1.80654			0.903269	−	0.429074i	$-0.141160\pi$
$409$	36.5350			1.80654			0.903269	+	0.429074i	$0.141160\pi$
$410$		0			0
$411$	8.90654			0.439327
$412$		0			0
$413$			9.63246i			0.473982i
$414$		0			0
$415$	8.75273			0.429655
$416$		0			0
$417$			18.1939i			0.890959i
$418$		0			0
$419$		−	2.77950i		−	0.135788i	−0.997693	−	0.0678938i	$-0.978372\pi$
$419$		−	2.77950i		−	0.135788i	0.997693	−	0.0678938i	$-0.0216279\pi$
$420$		0			0
$421$		−	25.5954i		−	1.24744i	−0.781646	−	0.623722i	$-0.785620\pi$
$421$		−	25.5954i		−	1.24744i	0.781646	−	0.623722i	$-0.214380\pi$
$422$		0			0
$423$	−10.3904			−0.505197
$424$		0			0
$425$		−	26.9043i		−	1.30505i
$426$		0			0
$427$	−0.309643			−0.0149847
$428$		0			0
$429$	−7.42433	−	11.7321i	−0.358450	−	0.566432i
$430$		0			0
$431$			7.78996i			0.375229i	0.982243	+	0.187614i	$0.0600756\pi$
$431$			7.78996i			0.375229i	−0.982243	+	0.187614i	$0.939924\pi$
$432$		0			0
$433$	−20.8785			−1.00336			−0.501679	−	0.865054i	$-0.667284\pi$
$433$	−20.8785			−1.00336			−0.501679	+	0.865054i	$0.667284\pi$
$434$		0			0
$435$			7.45966i			0.357663i
$436$		0			0
$437$	−40.4197			−1.93354
$438$		0			0
$439$	12.7959			0.610717			0.305359	−	0.952237i	$-0.401224\pi$
$439$	12.7959			0.610717			0.305359	+	0.952237i	$0.401224\pi$
$440$		0			0
$441$	10.5359			0.501709
$442$		0			0
$443$			2.22523i			0.105724i	0.998602	+	0.0528619i	$0.0168343\pi$
$443$			2.22523i			0.105724i	−0.998602	+	0.0528619i	$0.983166\pi$
$444$		0			0
$445$	3.50988			0.166384
$446$		0			0
$447$			16.6462i			0.787338i
$448$		0			0
$449$		−	21.0846i		−	0.995045i	−0.867451	−	0.497522i	$-0.834243\pi$
$449$		−	21.0846i		−	0.995045i	0.867451	−	0.497522i	$-0.165757\pi$
$450$		0			0
$451$	−16.6991	+	16.4628i	−0.786329	+	0.775204i
$452$		0			0
$453$	−5.06885			−0.238156
$454$		0			0
$455$	−3.37753	−	0.734279i	−0.158341	−	0.0344235i
$456$		0			0
$457$	22.1641			1.03679			0.518396	−	0.855141i	$-0.326530\pi$
$457$	22.1641			1.03679			0.518396	+	0.855141i	$0.326530\pi$
$458$		0			0
$459$	−41.2424			−1.92503
$460$		0			0
$461$	−4.71687			−0.219687			−0.109843	−	0.993949i	$-0.535035\pi$
$461$	−4.71687			−0.219687			−0.109843	+	0.993949i	$0.535035\pi$
$462$		0			0
$463$	−22.7460			−1.05710			−0.528548	−	0.848904i	$-0.677263\pi$
$463$	−22.7460			−1.05710			−0.528548	+	0.848904i	$0.677263\pi$
$464$		0			0
$465$	−10.4893			−0.486429
$466$		0			0
$467$			30.2510i			1.39985i	0.714216	+	0.699925i	$0.246783\pi$
$467$			30.2510i			1.39985i	−0.714216	+	0.699925i	$0.753217\pi$
$468$		0			0
$469$			3.37212i			0.155710i
$470$		0			0
$471$			6.08155i			0.280223i
$472$		0			0
$473$	15.1326	−	14.9185i	0.695798	−	0.685954i
$474$		0			0
$475$			19.4258i			0.891317i
$476$		0			0
$477$	11.5912			0.530727
$478$		0			0
$479$			23.7311i			1.08430i	0.840281	+	0.542150i	$0.182390\pi$
$479$			23.7311i			1.08430i	−0.840281	+	0.542150i	$0.817610\pi$
$480$		0			0
$481$	−6.33634	+	29.1458i	−0.288912	+	1.32894i
$482$		0			0
$483$		−	6.71494i		−	0.305540i
$484$		0			0
$485$	−22.3125			−1.01316
$486$		0			0
$487$	32.0799			1.45368			0.726838	−	0.686808i	$-0.240989\pi$
$487$	32.0799			1.45368			0.726838	+	0.686808i	$0.240989\pi$
$488$		0			0
$489$			5.48481i			0.248032i
$490$		0			0
$491$	−18.4000			−0.830380			−0.415190	−	0.909735i	$-0.636285\pi$
$491$	−18.4000			−0.830380			−0.415190	+	0.909735i	$0.636285\pi$
$492$		0			0
$493$	−40.3741			−1.81836
$494$		0			0
$495$	−4.67422	−	4.74130i	−0.210090	−	0.213105i
$496$		0			0
$497$		−	2.13712i		−	0.0958629i
$498$		0			0
$499$	−26.3436			−1.17930			−0.589651	−	0.807658i	$-0.700735\pi$
$499$	−26.3436			−1.17930			−0.589651	+	0.807658i	$0.700735\pi$
$500$		0			0
$501$	21.9549			0.980870
$502$		0			0
$503$	31.5403			1.40631			0.703156	−	0.711035i	$-0.251773\pi$
$503$	31.5403			1.40631			0.703156	+	0.711035i	$0.251773\pi$
$504$		0			0
$505$	−17.9750			−0.799879
$506$		0			0
$507$	6.26647	−	13.7311i	0.278304	−	0.609818i
$508$		0			0
$509$			43.1924i			1.91447i	0.289309	+	0.957236i	$0.406575\pi$
$509$			43.1924i			1.91447i	−0.289309	+	0.957236i	$0.593425\pi$
$510$		0			0
$511$			1.80056i			0.0796520i
$512$		0			0
$513$	29.7785			1.31475
$514$		0			0
$515$	−6.90331			−0.304196
$516$		0			0
$517$	−14.8550	+	14.6449i	−0.653323	+	0.644080i
$518$		0			0
$519$	23.2007			1.01840
$520$		0			0
$521$	−23.5659			−1.03244			−0.516220	−	0.856456i	$-0.672661\pi$
$521$	−23.5659			−1.03244			−0.516220	+	0.856456i	$0.672661\pi$
$522$		0			0
$523$	−34.3780			−1.50325			−0.751623	−	0.659593i	$-0.770729\pi$
$523$	−34.3780			−1.50325			−0.751623	+	0.659593i	$0.770729\pi$
$524$		0			0
$525$	−3.22722			−0.140847
$526$		0			0
$527$		−	56.7714i		−	2.47300i
$528$		0			0
$529$	−30.7464			−1.33680
$530$		0			0
$531$	−20.1709			−0.875343
$532$		0			0
$533$	−24.9105	−	5.41557i	−1.07899	−	0.234575i
$534$		0			0
$535$			12.3194i			0.532615i
$536$		0			0
$537$	−3.59159			−0.154989
$538$		0			0
$539$	15.0631	−	14.8499i	0.648812	−	0.639632i
$540$		0			0
$541$	22.6260			0.972769			0.486384	−	0.873745i	$-0.338315\pi$
$541$	22.6260			0.972769			0.486384	+	0.873745i	$0.338315\pi$
$542$		0			0
$543$			16.0468i			0.688634i
$544$		0			0
$545$		−	0.909915i		−	0.0389765i
$546$		0			0
$547$	42.9678			1.83717			0.918586	−	0.395221i	$-0.129332\pi$
$547$	42.9678			1.83717			0.918586	+	0.395221i	$0.129332\pi$
$548$		0			0
$549$		−	0.648409i		−	0.0276734i
$550$		0			0
$551$	29.1515			1.24189
$552$		0			0
$553$		−	2.61873i		−	0.111360i
$554$		0			0
$555$		−	11.6710i		−	0.495408i
$556$		0			0
$557$	16.5661			0.701929			0.350964	−	0.936389i	$-0.385854\pi$
$557$	16.5661			0.701929			0.350964	+	0.936389i	$0.385854\pi$
$558$		0			0
$559$	22.5738	+	4.90756i	0.954769	+	0.207568i
$560$		0			0
$561$	−20.9391	+	20.6428i	−0.884048	+	0.871541i
$562$		0			0
$563$	−10.0480			−0.423474			−0.211737	−	0.977327i	$-0.567912\pi$
$563$	−10.0480			−0.423474			−0.211737	+	0.977327i	$0.567912\pi$
$564$		0			0
$565$		−	12.6171i		−	0.530807i
$566$		0			0
$567$			1.03730i			0.0435624i
$568$		0			0
$569$			7.50844i			0.314770i	0.987537	+	0.157385i	$0.0503064\pi$
$569$			7.50844i			0.314770i	−0.987537	+	0.157385i	$0.949694\pi$
$570$		0			0
$571$	−10.7066			−0.448058			−0.224029	−	0.974582i	$-0.571921\pi$
$571$	−10.7066			−0.448058			−0.224029	+	0.974582i	$0.571921\pi$
$572$		0			0
$573$	−8.54859			−0.357122
$574$		0			0
$575$			25.8307i			1.07721i
$576$		0			0
$577$		−	16.0426i		−	0.667860i	−0.942598	−	0.333930i	$-0.891625\pi$
$577$		−	16.0426i		−	0.667860i	0.942598	−	0.333930i	$-0.108375\pi$
$578$		0			0
$579$			4.09411i			0.170145i
$580$		0			0
$581$	5.68243			0.235747
$582$		0			0
$583$	16.5719	−	16.3374i	0.686338	−	0.676628i
$584$		0			0
$585$	1.53762	−	7.07274i	0.0635728	−	0.292422i
$586$		0			0
$587$	−4.91491			−0.202860			−0.101430	−	0.994843i	$-0.532342\pi$
$587$	−4.91491			−0.202860			−0.101430	+	0.994843i	$0.532342\pi$
$588$		0			0
$589$			40.9909i			1.68900i
$590$		0			0
$591$		−	15.5828i		−	0.640989i
$592$		0			0
$593$	−32.1947			−1.32208			−0.661039	−	0.750352i	$-0.729884\pi$
$593$	−32.1947			−1.32208			−0.661039	+	0.750352i	$0.729884\pi$
$594$		0			0
$595$			7.32007i			0.300094i
$596$		0			0
$597$	8.60509			0.352183
$598$		0			0
$599$			0.575615i			0.0235190i	0.999931	+	0.0117595i	$0.00374325\pi$
$599$			0.575615i			0.0235190i	−0.999931	+	0.0117595i	$0.996257\pi$
$600$		0			0
$601$			21.8560i			0.891527i	0.895151	+	0.445763i	$0.147068\pi$
$601$			21.8560i			0.891527i	−0.895151	+	0.445763i	$0.852932\pi$
$602$		0			0
$603$	−7.06142			−0.287563
$604$		0			0
$605$	−13.3654	−	0.190447i	−0.543380	−	0.00774278i
$606$		0			0
$607$	37.4171			1.51871			0.759356	−	0.650675i	$-0.225514\pi$
$607$	37.4171			1.51871			0.759356	+	0.650675i	$0.225514\pi$
$608$		0			0
$609$			4.84295i			0.196246i
$610$		0			0
$611$	−22.1597	−	4.81754i	−0.896485	−	0.194897i
$612$		0			0
$613$	−40.6164			−1.64048			−0.820241	−	0.572018i	$-0.806161\pi$
$613$	−40.6164			−1.64048			−0.820241	+	0.572018i	$0.806161\pi$
$614$		0			0
$615$	9.97506			0.402233
$616$		0			0
$617$		−	5.25178i		−	0.211429i	−0.994397	−	0.105714i	$-0.966287\pi$
$617$		−	5.25178i		−	0.211429i	0.994397	−	0.105714i	$-0.0337129\pi$
$618$		0			0
$619$	12.8579			0.516803			0.258402	−	0.966038i	$-0.416804\pi$
$619$	12.8579			0.516803			0.258402	+	0.966038i	$0.416804\pi$
$620$		0			0
$621$	39.5967			1.58896
$622$		0			0
$623$	2.27868			0.0912934
$624$		0			0
$625$	5.03127			0.201251
$626$		0			0
$627$	15.1187	−	14.9048i	0.603784	−	0.595242i
$628$		0			0
$629$	63.1674			2.51865
$630$		0			0
$631$	−20.5713			−0.818929			−0.409465	−	0.912326i	$-0.634284\pi$
$631$	−20.5713			−0.818929			−0.409465	+	0.912326i	$0.634284\pi$
$632$		0			0
$633$		−	13.3738i		−	0.531560i
$634$		0			0
$635$			0.681432i			0.0270418i
$636$		0			0
$637$	22.4700	+	4.88500i	0.890294	+	0.193551i
$638$		0			0
$639$	4.47525			0.177038
$640$		0			0
$641$	38.0380			1.50241			0.751205	−	0.660068i	$-0.229473\pi$
$641$	38.0380			1.50241			0.751205	+	0.660068i	$0.229473\pi$
$642$		0			0
$643$	19.6862			0.776349			0.388174	−	0.921586i	$-0.373106\pi$
$643$	19.6862			0.776349			0.388174	+	0.921586i	$0.373106\pi$
$644$		0			0
$645$	−9.03934			−0.355924
$646$		0			0
$647$			0.748903i			0.0294424i	0.999892	+	0.0147212i	$0.00468607\pi$
$647$			0.748903i			0.0294424i	−0.999892	+	0.0147212i	$0.995314\pi$
$648$		0			0
$649$	−28.8381	+	28.4301i	−1.13200	+	1.11598i
$650$		0			0
$651$	−6.80984			−0.266899
$652$		0			0
$653$	−27.3010			−1.06837			−0.534186	−	0.845367i	$-0.679382\pi$
$653$	−27.3010			−1.06837			−0.534186	+	0.845367i	$0.679382\pi$
$654$		0			0
$655$			27.1998i			1.06279i
$656$		0			0
$657$	−3.77047			−0.147100
$658$		0			0
$659$	−24.9290			−0.971095			−0.485548	−	0.874210i	$-0.661380\pi$
$659$	−24.9290			−0.971095			−0.485548	+	0.874210i	$0.661380\pi$
$660$		0			0
$661$			16.9779i			0.660365i	0.943917	+	0.330183i	$0.107110\pi$
$661$			16.9779i			0.660365i	−0.943917	+	0.330183i	$0.892890\pi$
$662$		0			0
$663$	−31.2354	−	6.79062i	−1.21308	−	0.263726i
$664$		0			0
$665$		−	5.28535i		−	0.204957i
$666$		0			0
$667$	38.7630			1.50091
$668$		0			0
$669$		−	20.8217i		−	0.805013i
$670$		0			0
$671$	−0.913909	−	0.927025i	−0.0352811	−	0.0357874i
$672$		0			0
$673$			13.9892i			0.539245i	0.962966	+	0.269623i	$0.0868990\pi$
$673$			13.9892i			0.539245i	−0.962966	+	0.269623i	$0.913101\pi$
$674$		0			0
$675$		−	19.0303i		−	0.732476i
$676$		0			0
$677$		−	12.0255i		−	0.462179i	−0.972933	−	0.231089i	$-0.925771\pi$
$677$		−	12.0255i		−	0.462179i	0.972933	−	0.231089i	$-0.0742289\pi$
$678$		0			0
$679$	−14.4857			−0.555910
$680$		0			0
$681$	0.807894			0.0309586
$682$		0			0
$683$	−25.4179			−0.972589			−0.486295	−	0.873795i	$-0.661652\pi$
$683$	−25.4179			−0.972589			−0.486295	+	0.873795i	$0.661652\pi$
$684$		0			0
$685$	9.32174			0.356165
$686$		0			0
$687$	27.8919			1.06414
$688$		0			0
$689$	24.7208	+	5.37433i	0.941788	+	0.204746i
$690$		0			0
$691$	24.8346			0.944753			0.472377	−	0.881397i	$-0.343396\pi$
$691$	24.8346			0.944753			0.472377	+	0.881397i	$0.343396\pi$
$692$		0			0
$693$	−3.03459	−	3.07814i	−0.115274	−	0.116929i
$694$		0			0
$695$			19.0421i			0.722307i
$696$		0			0
$697$			53.9882i			2.04495i
$698$		0			0
$699$	−5.98966			−0.226550
$700$		0			0
$701$			5.88584i			0.222305i	0.993803	+	0.111153i	$0.0354542\pi$
$701$			5.88584i			0.222305i	−0.993803	+	0.111153i	$0.964546\pi$
$702$		0			0
$703$	−45.6090			−1.72018
$704$		0			0
$705$	8.87353			0.334196
$706$		0			0
$707$	−11.6697			−0.438885
$708$		0			0
$709$		−	33.0877i		−	1.24263i	−0.783560	−	0.621317i	$-0.786598\pi$
$709$		−	33.0877i		−	1.24263i	0.783560	−	0.621317i	$-0.213402\pi$
$710$		0			0
$711$	5.48377			0.205657
$712$		0			0
$713$			54.5060i			2.04127i
$714$		0			0
$715$	−7.77044	−	12.2790i	−0.290598	−	0.459210i
$716$		0			0
$717$	−25.7584			−0.961966
$718$		0			0
$719$		−	14.7257i		−	0.549176i	−0.961562	−	0.274588i	$-0.911459\pi$
$719$		−	14.7257i		−	0.549176i	0.961562	−	0.274588i	$-0.0885415\pi$
$720$		0			0
$721$	−4.48175			−0.166909
$722$		0			0
$723$		−	21.0027i		−	0.781100i
$724$		0			0
$725$		−	18.6296i		−	0.691886i
$726$		0			0
$727$			2.97380i			0.110292i	0.998478	+	0.0551461i	$0.0175625\pi$
$727$			2.97380i			0.110292i	−0.998478	+	0.0551461i	$0.982438\pi$
$728$		0			0
$729$	−20.9848			−0.777215
$730$		0			0
$731$		−	48.9238i		−	1.80951i
$732$		0			0
$733$	36.5741			1.35090			0.675448	−	0.737408i	$-0.263950\pi$
$733$	36.5741			1.35090			0.675448	+	0.737408i	$0.263950\pi$
$734$		0			0
$735$	−8.99779			−0.331889
$736$		0			0
$737$	−10.0956	+	9.95281i	−0.371878	+	0.366616i
$738$		0			0
$739$			25.2529i			0.928942i	0.885588	+	0.464471i	$0.153755\pi$
$739$			25.2529i			0.928942i	−0.885588	+	0.464471i	$0.846245\pi$
$740$		0			0
$741$	22.5531	+	4.90306i	0.828508	+	0.180119i
$742$		0			0
$743$			13.9894i			0.513220i	0.966515	+	0.256610i	$0.0826056\pi$
$743$			13.9894i			0.513220i	−0.966515	+	0.256610i	$0.917394\pi$
$744$		0			0
$745$			17.4222i			0.638300i
$746$		0			0
$747$			11.8993i			0.435374i
$748$		0			0
$749$			7.99800i			0.292240i
$750$		0			0
$751$		−	21.0924i		−	0.769674i	−0.922985	−	0.384837i	$-0.874258\pi$
$751$		−	21.0924i		−	0.769674i	0.922985	−	0.384837i	$-0.125742\pi$
$752$		0			0
$753$	−21.5175			−0.784141
$754$		0			0
$755$	−5.30515			−0.193074
$756$		0			0
$757$	−5.04972			−0.183535			−0.0917675	−	0.995780i	$-0.529252\pi$
$757$	−5.04972			−0.183535			−0.0917675	+	0.995780i	$0.529252\pi$
$758$		0			0
$759$	20.1035	−	19.8191i	0.729712	−	0.719388i
$760$		0			0
$761$	5.79871			0.210203			0.105102	−	0.994461i	$-0.466483\pi$
$761$	5.79871			0.210203			0.105102	+	0.994461i	$0.466483\pi$
$762$		0			0
$763$		−	0.590734i		−	0.0213860i
$764$		0			0
$765$	−15.3286			−0.554208
$766$		0			0
$767$	−43.0187	−	9.35232i	−1.55332	−	0.337693i
$768$		0			0
$769$	−14.8965			−0.537181			−0.268590	−	0.963254i	$-0.586558\pi$
$769$	−14.8965			−0.537181			−0.268590	+	0.963254i	$0.586558\pi$
$770$		0			0
$771$			14.6949i			0.529222i
$772$		0			0
$773$		−	19.4126i		−	0.698223i	−0.937081	−	0.349112i	$-0.886483\pi$
$773$		−	19.4126i		−	0.698223i	0.937081	−	0.349112i	$-0.113517\pi$
$774$		0			0
$775$	26.1958			0.940980
$776$		0			0
$777$		−	7.57705i		−	0.271825i
$778$		0			0
$779$		−	38.9814i		−	1.39665i
$780$		0			0
$781$	6.39822	−	6.30770i	0.228946	−	0.225707i
$782$		0			0
$783$	−28.5579			−1.02058
$784$		0			0
$785$			6.36506i			0.227179i
$786$		0			0
$787$		−	35.7931i		−	1.27589i	−0.770084	−	0.637943i	$-0.779786\pi$
$787$		−	35.7931i		−	1.27589i	0.770084	−	0.637943i	$-0.220214\pi$
$788$		0			0
$789$		−	19.7297i		−	0.702394i
$790$		0			0
$791$		−	8.19128i		−	0.291248i
$792$		0			0
$793$	0.300638	−	1.38287i	0.0106760	−	0.0491072i
$794$		0			0
$795$	−9.89909			−0.351085
$796$		0			0
$797$	3.80492			0.134777			0.0673885	−	0.997727i	$-0.478533\pi$
$797$	3.80492			0.134777			0.0673885	+	0.997727i	$0.478533\pi$
$798$		0			0
$799$			48.0264i			1.69905i
$800$		0			0
$801$			4.77168i			0.168599i
$802$		0			0
$803$	−5.39060	+	5.31434i	−0.190230	+	0.187539i
$804$		0			0
$805$		−	7.02798i		−	0.247704i
$806$		0			0
$807$			16.6935i			0.587638i
$808$		0			0
$809$			13.3210i			0.468340i	0.972196	+	0.234170i	$0.0752373\pi$
$809$			13.3210i			0.468340i	−0.972196	+	0.234170i	$0.924763\pi$
$810$		0			0
$811$		−	42.6768i		−	1.49858i	−0.662239	−	0.749292i	$-0.730394\pi$
$811$		−	42.6768i		−	1.49858i	0.662239	−	0.749292i	$-0.269606\pi$
$812$		0			0
$813$	24.0267			0.842652
$814$		0			0
$815$			5.74050i			0.201081i
$816$		0			0
$817$			35.3247i			1.23585i
$818$		0			0
$819$	0.998251	−	4.59175i	0.0348817	−	0.160449i
$820$		0			0
$821$	−7.75759			−0.270742			−0.135371	−	0.990795i	$-0.543223\pi$
$821$	−7.75759			−0.270742			−0.135371	+	0.990795i	$0.543223\pi$
$822$		0			0
$823$			28.7754i			1.00305i	0.865144	+	0.501524i	$0.167227\pi$
$823$			28.7754i			1.00305i	−0.865144	+	0.501524i	$0.832773\pi$
$824$		0			0
$825$	−9.52512	−	9.66181i	−0.331622	−	0.336381i
$826$		0			0
$827$			47.6262i			1.65613i	0.560635	+	0.828063i	$0.310557\pi$
$827$			47.6262i			1.65613i	−0.560635	+	0.828063i	$0.689443\pi$
$828$		0			0
$829$	−19.9714			−0.693635			−0.346818	−	0.937933i	$-0.612738\pi$
$829$	−19.9714			−0.693635			−0.346818	+	0.937933i	$0.612738\pi$
$830$		0			0
$831$	3.56136			0.123542
$832$		0			0
$833$		−	48.6989i		−	1.68732i
$834$		0			0
$835$	22.9783			0.795198
$836$		0			0
$837$		−	40.1563i		−	1.38801i
$838$		0			0
$839$	−38.4640			−1.32792			−0.663962	−	0.747767i	$-0.731126\pi$
$839$	−38.4640			−1.32792			−0.663962	+	0.747767i	$0.731126\pi$
$840$		0			0
$841$	1.04335			0.0359774
$842$		0			0
$843$			26.5631i			0.914882i
$844$		0			0
$845$	6.55860	−	14.3712i	0.225623	−	0.494384i
$846$		0			0
$847$	−8.67704	−	0.123642i	−0.298147	−	0.00424839i
$848$		0			0
$849$			10.1958i			0.349917i
$850$		0			0
$851$	−60.6468			−2.07894
$852$		0			0
$853$	−43.9507			−1.50484			−0.752422	−	0.658682i	$-0.771114\pi$
$853$	−43.9507			−1.50484			−0.752422	+	0.658682i	$0.771114\pi$
$854$		0			0
$855$	11.0678			0.378511
$856$		0			0
$857$			7.18533i			0.245446i	0.992441	+	0.122723i	$0.0391627\pi$
$857$			7.18533i			0.245446i	−0.992441	+	0.122723i	$0.960837\pi$
$858$		0			0
$859$			33.3448i			1.13771i	0.822438	+	0.568855i	$0.192613\pi$
$859$			33.3448i			1.13771i	−0.822438	+	0.568855i	$0.807387\pi$
$860$		0			0
$861$	6.47599			0.220701
$862$		0			0
$863$	−16.9093			−0.575600			−0.287800	−	0.957691i	$-0.592924\pi$
$863$	−16.9093			−0.575600			−0.287800	+	0.957691i	$0.592924\pi$
$864$		0			0
$865$	24.2823			0.825623
$866$		0			0
$867$			47.9586i			1.62876i
$868$		0			0
$869$	7.84009	−	7.72917i	0.265957	−	0.262194i
$870$		0			0
$871$	−15.0600	−	3.27405i	−0.510288	−	0.110937i
$872$		0			0
$873$		−	30.3339i		−	1.02665i
$874$		0			0
$875$	−8.17087			−0.276226
$876$		0			0
$877$	−5.21826			−0.176208			−0.0881040	−	0.996111i	$-0.528081\pi$
$877$	−5.21826			−0.176208			−0.0881040	+	0.996111i	$0.528081\pi$
$878$		0			0
$879$		−	12.8143i		−	0.432217i
$880$		0			0
$881$	8.48058			0.285718			0.142859	−	0.989743i	$-0.454370\pi$
$881$	8.48058			0.285718			0.142859	+	0.989743i	$0.454370\pi$
$882$		0			0
$883$			8.04028i			0.270577i	0.990806	+	0.135289i	$0.0431961\pi$
$883$			8.04028i			0.270577i	−0.990806	+	0.135289i	$0.956804\pi$
$884$		0			0
$885$	17.2262			0.579054
$886$		0			0
$887$	21.3918			0.718267			0.359134	−	0.933286i	$-0.383072\pi$
$887$	21.3918			0.718267			0.359134	+	0.933286i	$0.383072\pi$
$888$		0			0
$889$			0.442398i			0.0148376i
$890$		0			0
$891$	−3.10552	+	3.06158i	−0.104039	+	0.102567i
$892$		0			0
$893$		−	34.6767i		−	1.16041i
$894$		0			0
$895$	−3.75903			−0.125650
$896$		0			0
$897$	29.9890	+	6.51965i	1.00131	+	0.217685i
$898$		0			0
$899$		−	39.3109i		−	1.31109i
$900$		0			0
$901$		−	53.5770i		−	1.78491i
$902$		0			0
$903$	−5.86851			−0.195292
$904$		0			0
$905$			16.7949i			0.558280i
$906$		0			0
$907$		−	20.5572i		−	0.682589i	−0.939956	−	0.341295i	$-0.889135\pi$
$907$		−	20.5572i		−	0.682589i	0.939956	−	0.341295i	$-0.110865\pi$
$908$		0			0
$909$		−	24.4371i		−	0.810526i
$910$		0			0
$911$		−	40.4836i		−	1.34128i	−0.741782	−	0.670641i	$-0.766019\pi$
$911$		−	40.4836i		−	1.34128i	0.741782	−	0.670641i	$-0.233981\pi$
$912$		0			0
$913$	16.7717	+	17.0124i	0.555061	+	0.563027i
$914$		0			0
$915$			0.553751i			0.0183064i
$916$		0			0
$917$			17.6586i			0.583140i
$918$		0			0
$919$	10.7144			0.353435			0.176717	−	0.984262i	$-0.443452\pi$
$919$	10.7144			0.353435			0.176717	+	0.984262i	$0.443452\pi$
$920$		0			0
$921$	−22.0349			−0.726073
$922$		0			0
$923$	9.54442	+	2.07497i	0.314158	+	0.0682984i
$924$		0			0
$925$			29.1470i			0.958348i
$926$		0			0
$927$		−	9.38504i		−	0.308245i
$928$		0			0
$929$		−	45.4526i		−	1.49125i	−0.666364	−	0.745626i	$-0.732150\pi$
$929$		−	45.4526i		−	1.49125i	0.666364	−	0.745626i	$-0.267850\pi$
$930$		0			0
$931$			35.1623i			1.15240i
$932$		0			0
$933$	14.4900			0.474382
$934$		0			0
$935$	−21.9152	+	21.6052i	−0.716704	+	0.706564i
$936$		0			0
$937$			5.42920i			0.177364i	0.996060	+	0.0886822i	$0.0282655\pi$
$937$			5.42920i			0.177364i	−0.996060	+	0.0886822i	$0.971734\pi$
$938$		0			0
$939$		−	8.68996i		−	0.283586i
$940$		0			0
$941$	−8.12114			−0.264741			−0.132371	−	0.991200i	$-0.542259\pi$
$941$	−8.12114			−0.264741			−0.132371	+	0.991200i	$0.542259\pi$
$942$		0			0
$943$		−	51.8339i		−	1.68794i
$944$		0			0
$945$			5.17774i			0.168432i
$946$		0			0
$947$	51.9766			1.68901			0.844506	−	0.535546i	$-0.179894\pi$
$947$	51.9766			1.68901			0.844506	+	0.535546i	$0.179894\pi$
$948$		0			0
$949$	−8.04133	−	1.74819i	−0.261033	−	0.0567487i
$950$		0			0
$951$	1.01523			0.0329210
$952$		0			0
$953$		−	35.3823i		−	1.14614i	−0.819505	−	0.573072i	$-0.805751\pi$
$953$		−	35.3823i		−	1.14614i	0.819505	−	0.573072i	$-0.194249\pi$
$954$		0			0
$955$	−8.94710			−0.289521
$956$		0			0
$957$	−14.4991	+	14.2939i	−0.468688	+	0.462057i
$958$		0			0
$959$	6.05185			0.195424
$960$		0			0
$961$	24.2764			0.783109
$962$		0			0
$963$	−16.7482			−0.539705
$964$		0			0
$965$			4.28497i			0.137938i
$966$		0			0
$967$		−	32.4785i		−	1.04444i	−0.852812	−	0.522218i	$-0.825105\pi$
$967$		−	32.4785i		−	1.04444i	0.852812	−	0.522218i	$-0.174895\pi$
$968$		0			0
$969$		−	48.8790i		−	1.57022i
$970$		0			0
$971$		−	25.1998i		−	0.808700i	−0.914604	−	0.404350i	$-0.867498\pi$
$971$		−	25.1998i		−	0.808700i	0.914604	−	0.404350i	$-0.132502\pi$
$972$		0			0
$973$			12.3625i			0.396322i
$974$		0			0
$975$	3.13336	−	14.4128i	0.100348	−	0.461580i
$976$		0			0
$977$		−	5.72742i		−	0.183236i	−0.995794	−	0.0916182i	$-0.970796\pi$
$977$		−	5.72742i		−	0.183236i	0.995794	−	0.0916182i	$-0.0292039\pi$
$978$		0			0
$979$	6.72551	+	6.82203i	0.214948	+	0.218033i
$980$		0			0
$981$	1.23703			0.0394953
$982$		0			0
$983$	27.5070			0.877338			0.438669	−	0.898649i	$-0.355450\pi$
$983$	27.5070			0.877338			0.438669	+	0.898649i	$0.355450\pi$
$984$		0			0
$985$		−	16.3092i		−	0.519654i
$986$		0			0
$987$	5.76086			0.183370
$988$		0			0
$989$			46.9716i			1.49361i
$990$		0			0
$991$		−	34.3394i		−	1.09083i	−0.838167	−	0.545413i	$-0.816373\pi$
$991$		−	34.3394i		−	1.09083i	0.838167	−	0.545413i	$-0.183627\pi$
$992$		0			0
$993$		−	1.47864i		−	0.0469233i
$994$		0			0
$995$	9.00624			0.285517
$996$		0			0
$997$			48.4507i			1.53445i	0.641379	+	0.767224i	$0.278363\pi$
$997$			48.4507i			1.53445i	−0.641379	+	0.767224i	$0.721637\pi$
$998$		0			0
$999$	44.6804			1.41363

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2288.2.b.c.2287.17	✓	56
4.3	odd	2	inner	2288.2.b.c.2287.38	yes	56
11.10	odd	2	inner	2288.2.b.c.2287.18	yes	56
13.12	even	2	inner	2288.2.b.c.2287.20	yes	56
44.43	even	2	inner	2288.2.b.c.2287.37	yes	56
52.51	odd	2	inner	2288.2.b.c.2287.39	yes	56
143.142	odd	2	inner	2288.2.b.c.2287.19	yes	56
572.571	even	2	inner	2288.2.b.c.2287.40	yes	56

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2288.2.b.c.2287.17	✓	56	1.1	even	1	trivial
2288.2.b.c.2287.18	yes	56	11.10	odd	2	inner
2288.2.b.c.2287.19	yes	56	143.142	odd	2	inner
2288.2.b.c.2287.20	yes	56	13.12	even	2	inner
2288.2.b.c.2287.37	yes	56	44.43	even	2	inner
2288.2.b.c.2287.38	yes	56	4.3	odd	2	inner
2288.2.b.c.2287.39	yes	56	52.51	odd	2	inner
2288.2.b.c.2287.40	yes	56	572.571	even	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:	\( N \)	\(=\)	\( 2288 = 2^{4} \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2288.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.16103i q^{3} -1.21516i q^{5} -0.788902i q^{7} +1.65201 q^{9} +O(q^{10})\)	\(q-1.16103i q^{3} -1.21516i q^{5} -0.788902i q^{7} +1.65201 q^{9} +(2.36186 - 2.32844i) q^{11} +(3.52325 + 0.765959i) q^{13} -1.41084 q^{15} -7.63589i q^{17} +5.51338i q^{19} -0.915941 q^{21} +7.33120i q^{23} +3.52339 q^{25} -5.40113i q^{27} -5.28741i q^{29} +7.43481 q^{31} +(-2.70339 - 2.74219i) q^{33} -0.958640 q^{35} +8.27242i q^{37} +(0.889302 - 4.09061i) q^{39} -7.07032 q^{41} +6.40708 q^{43} -2.00745i q^{45} -6.28955 q^{47} +6.37763 q^{49} -8.86552 q^{51} +7.01647 q^{53} +(-2.82942 - 2.87003i) q^{55} +6.40121 q^{57} -12.2100 q^{59} -0.392498i q^{61} -1.30327i q^{63} +(0.930760 - 4.28130i) q^{65} -4.27445 q^{67} +8.51175 q^{69} +2.70898 q^{71} -2.28236 q^{73} -4.09077i q^{75} +(-1.83691 - 1.86327i) q^{77} +3.31946 q^{79} -1.31486 q^{81} +7.20296i q^{83} -9.27881 q^{85} -6.13885 q^{87} +2.88842i q^{89} +(0.604267 - 2.77950i) q^{91} -8.63205i q^{93} +6.69962 q^{95} -18.3618i q^{97} +(3.90180 - 3.84660i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q - 72 q^{9}+O(q^{10}) \) 56 * q - 72 * q^9	\( 56 q - 72 q^{9} - 72 q^{25} - 8 q^{49} + 72 q^{53} + 8 q^{69} - 48 q^{77} + 88 q^{81}+O(q^{100}) \) 56 * q - 72 * q^9 - 72 * q^25 - 8 * q^49 + 72 * q^53 + 8 * q^69 - 48 * q^77 + 88 * q^81

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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