LMFDB - Embedded newform 288.10.d.d.145.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [288,10,Mod(145,288)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(288, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 10, names="a")

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

chi := DirichletCharacter("288.145");

S:= CuspForms(chi, 10);

N := Newforms(S);

Level:	$ N $	$=$	$ 288 = 2^{5} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	288.d (of order $2$, degree $1$, not 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:	$148.330320815$
Analytic rank:	$0$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} + 5058217 x^{16} + 10504880354852 x^{14} + \cdots + 26\!\cdots\!25 $ x^18 + 5058217x^16 + 10504880354852x^14 + 11547743283519467732x^12 + 7226508878828480992045182x^10 + 2579729575666602678677474677566x^8 + 498427594358587281769332022194607700x^6 + 45246234294859640050885317802771131342500x^4 + 1272616059551332143482031903024533629596015625x^2 + 2697474089409566173619796125333482360928175390625
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{149}\cdot 3^{28} $
Twist minimal:	no (minimal twist has level 24)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			145.11
Root			$208.075i$ of defining polynomial
Character	$\chi$	$=$	288.145
Dual form			288.10.d.d.145.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+416.150i q^{5} -10561.7 q^{7} +O(q^{10})$	$q+416.150i q^{5} -10561.7 q^{7} -54750.4i q^{11} +103511. i q^{13} -293266. q^{17} -759080. i q^{19} -875669. q^{23} +1.77994e6 q^{25} -2.12690e6i q^{29} -8.70915e6 q^{31} -4.39526e6i q^{35} -9.75881e6i q^{37} -1.02465e7 q^{41} -2.38973e7i q^{43} +1.08649e7 q^{47} +7.11967e7 q^{49} +2.54371e7i q^{53} +2.27844e7 q^{55} +1.17130e8i q^{59} +8.80779e6i q^{61} -4.30761e7 q^{65} +1.03646e8i q^{67} -4.10162e8 q^{71} +2.25467e8 q^{73} +5.78259e8i q^{77} +5.26393e8 q^{79} -6.56694e8i q^{83} -1.22043e8i q^{85} -3.92379e8 q^{89} -1.09326e9i q^{91} +3.15891e8 q^{95} -4.27535e8 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q + 9604 q^{7}+O(q^{10}) $ 18 * q + 9604 * q^7	$ 18 q + 9604 q^{7} - 203996 q^{17} - 3465976 q^{23} - 5309486 q^{25} - 4737540 q^{31} + 3780956 q^{41} - 112458264 q^{47} + 90859338 q^{49} + 76355584 q^{55} - 300816016 q^{65} - 408942312 q^{71} + 826124308 q^{73} - 382409492 q^{79} + 458464820 q^{89} + 331883504 q^{95} + 264098780 q^{97}+O(q^{100}) $ 18 * q + 9604 * q^7 - 203996 * q^17 - 3465976 * q^23 - 5309486 * q^25 - 4737540 * q^31 + 3780956 * q^41 - 112458264 * q^47 + 90859338 * q^49 + 76355584 * q^55 - 300816016 * q^65 - 408942312 * q^71 + 826124308 * q^73 - 382409492 * q^79 + 458464820 * q^89 + 331883504 * q^95 + 264098780 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/288\mathbb{Z}\right)^\times$.

$n$	$37$	$65$	$127$
$\chi(n)$	$-1$	$1$	$1$

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	416.150i	0.297773i	0.988854	+	0.148886i	$0.0475688\pi$
$5$	416.150i	0.297773i	−0.988854	+	0.148886i	$0.952431\pi$
$6$	0	0
$7$	−10561.7	−1.66262	−0.831312	−	0.555806i	$-0.812410\pi$
$7$	−10561.7	−1.66262	−0.831312	+	0.555806i	$0.812410\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 54750.4i	− 1.12751i	−0.825942	−	0.563755i	$-0.809356\pi$
$11$	− 54750.4i	− 1.12751i	0.825942	−	0.563755i	$-0.190644\pi$
$12$	0	0
$13$	103511.i	1.00517i	0.864527	+	0.502587i	$0.167618\pi$
$13$	103511.i	1.00517i	−0.864527	+	0.502587i	$0.832382\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−293266.	−0.851613	−0.425806	−	0.904814i	$-0.640009\pi$
$17$	−293266.	−0.851613	−0.425806	+	0.904814i	$0.640009\pi$
$18$	0	0
$19$	− 759080.i	− 1.33628i	−0.744037	−	0.668139i	$-0.767091\pi$
$19$	− 759080.i	− 1.33628i	0.744037	−	0.668139i	$-0.232909\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−875669.	−0.652476	−0.326238	−	0.945288i	$-0.605781\pi$
$23$	−875669.	−0.652476	−0.326238	+	0.945288i	$0.605781\pi$
$24$	0	0
$25$	1.77994e6	0.911332
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 2.12690e6i	− 0.558414i	−0.960231	−	0.279207i	$-0.909928\pi$
$29$	− 2.12690e6i	− 0.558414i	0.960231	−	0.279207i	$-0.0900716\pi$
$30$	0	0
$31$	−8.70915e6	−1.69374	−0.846872	−	0.531796i	$-0.821517\pi$
$31$	−8.70915e6	−1.69374	−0.846872	+	0.531796i	$0.821517\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 4.39526e6i	− 0.495084i
$36$	0	0
$37$	− 9.75881e6i	− 0.856030i	−0.903772	−	0.428015i	$-0.859213\pi$
$37$	− 9.75881e6i	− 0.856030i	0.903772	−	0.428015i	$-0.140787\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−1.02465e7	−0.566304	−0.283152	−	0.959075i	$-0.591380\pi$
$41$	−1.02465e7	−0.566304	−0.283152	+	0.959075i	$0.591380\pi$
$42$	0	0
$43$	− 2.38973e7i	− 1.06596i	−0.846128	−	0.532980i	$-0.821072\pi$
$43$	− 2.38973e7i	− 1.06596i	0.846128	−	0.532980i	$-0.178928\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	1.08649e7	0.324778	0.162389	−	0.986727i	$-0.448080\pi$
$47$	1.08649e7	0.324778	0.162389	+	0.986727i	$0.448080\pi$
$48$	0	0
$49$	7.11967e7	1.76432
$50$	0	0
$51$	0	0
$52$	0	0
$53$	2.54371e7i	0.442819i	0.975181	+	0.221409i	$0.0710657\pi$
$53$	2.54371e7i	0.442819i	−0.975181	+	0.221409i	$0.928934\pi$
$54$	0	0
$55$	2.27844e7	0.335741
$56$	0	0
$57$	0	0
$58$	0	0
$59$	1.17130e8i	1.25844i	0.777226	+	0.629221i	$0.216626\pi$
$59$	1.17130e8i	1.25844i	−0.777226	+	0.629221i	$0.783374\pi$
$60$	0	0
$61$	8.80779e6i	0.0814484i	0.999170	+	0.0407242i	$0.0129665\pi$
$61$	8.80779e6i	0.0814484i	−0.999170	+	0.0407242i	$0.987033\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−4.30761e7	−0.299313
$66$	0	0
$67$	1.03646e8i	0.628372i	0.949361	+	0.314186i	$0.101732\pi$
$67$	1.03646e8i	0.628372i	−0.949361	+	0.314186i	$0.898268\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−4.10162e8	−1.91555	−0.957773	−	0.287525i	$-0.907168\pi$
$71$	−4.10162e8	−1.91555	−0.957773	+	0.287525i	$0.907168\pi$
$72$	0	0
$73$	2.25467e8	0.929246	0.464623	−	0.885509i	$-0.346190\pi$
$73$	2.25467e8	0.929246	0.464623	+	0.885509i	$0.346190\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	5.78259e8i	1.87462i
$78$	0	0
$79$	5.26393e8	1.52051	0.760253	−	0.649627i	$-0.225075\pi$
$79$	5.26393e8	1.52051	0.760253	+	0.649627i	$0.225075\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 6.56694e8i	− 1.51884i	−0.650601	−	0.759420i	$-0.725483\pi$
$83$	− 6.56694e8i	− 1.51884i	0.650601	−	0.759420i	$-0.274517\pi$
$84$	0	0
$85$	− 1.22043e8i	− 0.253587i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−3.92379e8	−0.662904	−0.331452	−	0.943472i	$-0.607538\pi$
$89$	−3.92379e8	−0.662904	−0.331452	+	0.943472i	$0.607538\pi$
$90$	0	0
$91$	− 1.09326e9i	− 1.67123i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	3.15891e8	0.397907
$96$	0	0
$97$	−4.27535e8	−0.490342	−0.245171	−	0.969480i	$-0.578844\pi$
$97$	−4.27535e8	−0.490342	−0.245171	+	0.969480i	$0.578844\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	6.57171e8i	0.628394i	0.949358	+	0.314197i	$0.101735\pi$
$101$	6.57171e8i	0.628394i	−0.949358	+	0.314197i	$0.898265\pi$
$102$	0	0
$103$	7.28209e7	0.0637512	0.0318756	−	0.999492i	$-0.489852\pi$
$103$	7.28209e7	0.0637512	0.0318756	+	0.999492i	$0.489852\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	7.80852e8i	0.575893i	0.957647	+	0.287946i	$0.0929725\pi$
$107$	7.80852e8i	0.575893i	−0.957647	+	0.287946i	$0.907027\pi$
$108$	0	0
$109$	6.17128e8i	0.418751i	0.977835	+	0.209376i	$0.0671432\pi$
$109$	6.17128e8i	0.418751i	−0.977835	+	0.209376i	$0.932857\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−3.01950e9	−1.74213	−0.871067	−	0.491164i	$-0.836572\pi$
$113$	−3.01950e9	−1.74213	−0.871067	+	0.491164i	$0.836572\pi$
$114$	0	0
$115$	− 3.64409e8i	− 0.194290i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	3.09740e9	1.41591
$120$	0	0
$121$	−6.39658e8	−0.271277
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.55352e9i	0.569142i
$126$	0	0
$127$	1.26724e9	0.432256	0.216128	−	0.976365i	$-0.430657\pi$
$127$	1.26724e9	0.432256	0.216128	+	0.976365i	$0.430657\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 4.74712e9i	− 1.40835i	−0.710028	−	0.704173i	$-0.751318\pi$
$131$	− 4.74712e9i	− 1.40835i	0.710028	−	0.704173i	$-0.248682\pi$
$132$	0	0
$133$	8.01721e9i	2.22173i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	6.92089e9	1.67849	0.839247	−	0.543751i	$-0.182996\pi$
$137$	6.92089e9	1.67849	0.839247	+	0.543751i	$0.182996\pi$
$138$	0	0
$139$	8.00426e9i	1.81867i	0.416061	+	0.909337i	$0.363410\pi$
$139$	8.00426e9i	1.81867i	−0.416061	+	0.909337i	$0.636590\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	5.66727e9	1.13334
$144$	0	0
$145$	8.85110e8	0.166280
$146$	0	0
$147$	0	0
$148$	0	0
$149$	4.29056e9i	0.713142i	0.934268	+	0.356571i	$0.116054\pi$
$149$	4.29056e9i	0.713142i	−0.934268	+	0.356571i	$0.883946\pi$
$150$	0	0
$151$	3.45621e9	0.541008	0.270504	−	0.962719i	$-0.412810\pi$
$151$	3.45621e9	0.541008	0.270504	+	0.962719i	$0.412810\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 3.62431e9i	− 0.504351i
$156$	0	0
$157$	2.17477e8i	0.0285670i	0.999898	+	0.0142835i	$0.00454673\pi$
$157$	2.17477e8i	0.0285670i	−0.999898	+	0.0142835i	$0.995453\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	9.24858e9	1.08482
$162$	0	0
$163$	− 4.03995e8i	− 0.0448262i	−0.999749	−	0.0224131i	$-0.992865\pi$
$163$	− 4.03995e8i	− 0.0448262i	0.999749	−	0.0224131i	$-0.00713490\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−1.00000e10	−0.994893	−0.497446	−	0.867495i	$-0.665729\pi$
$167$	−1.00000e10	−0.994893	−0.497446	+	0.867495i	$0.665729\pi$
$168$	0	0
$169$	−1.10019e8	−0.0103747
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 8.68720e9i	− 0.737348i	−0.929559	−	0.368674i	$-0.879812\pi$
$173$	− 8.68720e9i	− 0.737348i	0.929559	−	0.368674i	$-0.120188\pi$
$174$	0	0
$175$	−1.87993e10	−1.51520
$176$	0	0
$177$	0	0
$178$	0	0
$179$	1.63457e10i	1.19005i	0.803707	+	0.595025i	$0.202858\pi$
$179$	1.63457e10i	1.19005i	−0.803707	+	0.595025i	$0.797142\pi$
$180$	0	0
$181$	8.18574e9i	0.566897i	0.958987	+	0.283449i	$0.0914785\pi$
$181$	8.18574e9i	0.566897i	−0.958987	+	0.283449i	$0.908521\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	4.06113e9	0.254902
$186$	0	0
$187$	1.60564e10i	0.960201i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	5.80329e9	0.315518	0.157759	−	0.987478i	$-0.449573\pi$
$191$	5.80329e9	0.315518	0.157759	+	0.987478i	$0.449573\pi$
$192$	0	0
$193$	7.78576e9	0.403918	0.201959	−	0.979394i	$-0.435269\pi$
$193$	7.78576e9	0.403918	0.201959	+	0.979394i	$0.435269\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	1.83650e10i	0.868748i	0.900733	+	0.434374i	$0.143030\pi$
$197$	1.83650e10i	0.868748i	−0.900733	+	0.434374i	$0.856970\pi$
$198$	0	0
$199$	2.35677e10	1.06532	0.532658	−	0.846331i	$-0.321193\pi$
$199$	2.35677e10	1.06532	0.532658	+	0.846331i	$0.321193\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	2.24638e10i	0.928433i
$204$	0	0
$205$	− 4.26409e9i	− 0.168630i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−4.15600e10	−1.50667
$210$	0	0
$211$	3.43201e10i	1.19200i	0.802983	+	0.596001i	$0.203245\pi$
$211$	3.43201e10i	1.19200i	−0.802983	+	0.596001i	$0.796755\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	9.94485e9	0.317413
$216$	0	0
$217$	9.19837e10	2.81606
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 3.03563e10i	− 0.856019i
$222$	0	0
$223$	5.62113e10	1.52213	0.761065	−	0.648676i	$-0.224677\pi$
$223$	5.62113e10	1.52213	0.761065	+	0.648676i	$0.224677\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	6.25184e10i	1.56276i	0.624057	+	0.781379i	$0.285483\pi$
$227$	6.25184e10i	1.56276i	−0.624057	+	0.781379i	$0.714517\pi$
$228$	0	0
$229$	5.27164e10i	1.26674i	0.773851	+	0.633368i	$0.218328\pi$
$229$	5.27164e10i	1.26674i	−0.773851	+	0.633368i	$0.781672\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	7.37673e10	1.63969	0.819846	−	0.572583i	$-0.194059\pi$
$233$	7.37673e10	1.63969	0.819846	+	0.572583i	$0.194059\pi$
$234$	0	0
$235$	4.52144e9i	0.0967100i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−1.34146e10	−0.265942	−0.132971	−	0.991120i	$-0.542452\pi$
$239$	−1.34146e10	−0.265942	−0.132971	+	0.991120i	$0.542452\pi$
$240$	0	0
$241$	−2.65736e10	−0.507428	−0.253714	−	0.967279i	$-0.581652\pi$
$241$	−2.65736e10	−0.507428	−0.253714	+	0.967279i	$0.581652\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	2.96285e10i	0.525366i
$246$	0	0
$247$	7.85731e10	1.34319
$248$	0	0
$249$	0	0
$250$	0	0
$251$	6.65264e10i	1.05794i	0.848639	+	0.528972i	$0.177422\pi$
$251$	6.65264e10i	1.05794i	−0.848639	+	0.528972i	$0.822578\pi$
$252$	0	0
$253$	4.79432e10i	0.735673i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−7.31724e10	−1.04628	−0.523141	−	0.852246i	$-0.675240\pi$
$257$	−7.31724e10	−1.04628	−0.523141	+	0.852246i	$0.675240\pi$
$258$	0	0
$259$	1.03070e11i	1.42326i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−7.24021e10	−0.933147	−0.466574	−	0.884482i	$-0.654512\pi$
$263$	−7.24021e10	−0.933147	−0.466574	+	0.884482i	$0.654512\pi$
$264$	0	0
$265$	−1.05856e10	−0.131859
$266$	0	0
$267$	0	0
$268$	0	0
$269$	1.35446e11i	1.57718i	0.614917	+	0.788592i	$0.289190\pi$
$269$	1.35446e11i	1.57718i	−0.614917	+	0.788592i	$0.710810\pi$
$270$	0	0
$271$	−1.79847e10	−0.202555	−0.101277	−	0.994858i	$-0.532293\pi$
$271$	−1.79847e10	−0.202555	−0.101277	+	0.994858i	$0.532293\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 9.74527e10i	− 1.02753i
$276$	0	0
$277$	− 1.57486e11i	− 1.60725i	−0.595137	−	0.803624i	$-0.702902\pi$
$277$	− 1.57486e11i	− 1.60725i	0.595137	−	0.803624i	$-0.297098\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	1.64916e11	1.57792	0.788958	−	0.614448i	$-0.210621\pi$
$281$	1.64916e11	1.57792	0.788958	+	0.614448i	$0.210621\pi$
$282$	0	0
$283$	− 5.98665e10i	− 0.554811i	−0.960753	−	0.277406i	$-0.910525\pi$
$283$	− 5.98665e10i	− 0.554811i	0.960753	−	0.277406i	$-0.0894746\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	1.08221e11	0.941551
$288$	0	0
$289$	−3.25827e10	−0.274756
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 1.84626e11i	− 1.46349i	−0.681580	−	0.731743i	$-0.738707\pi$
$293$	− 1.84626e11i	− 1.46349i	0.681580	−	0.731743i	$-0.261293\pi$
$294$	0	0
$295$	−4.87435e10	−0.374730
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 9.06413e10i	− 0.655852i
$300$	0	0
$301$	2.52397e11i	1.77229i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−3.66536e9	−0.0242531
$306$	0	0
$307$	8.95668e10i	0.575472i	0.957710	+	0.287736i	$0.0929026\pi$
$307$	8.95668e10i	0.575472i	−0.957710	+	0.287736i	$0.907097\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	1.93571e11	1.17333	0.586663	−	0.809831i	$-0.300441\pi$
$311$	1.93571e11	1.17333	0.586663	+	0.809831i	$0.300441\pi$
$312$	0	0
$313$	1.28784e11	0.758423	0.379211	−	0.925310i	$-0.376195\pi$
$313$	1.28784e11	0.758423	0.379211	+	0.925310i	$0.376195\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 8.44213e10i	− 0.469554i	−0.972049	−	0.234777i	$-0.924564\pi$
$317$	− 8.44213e10i	− 0.469554i	0.972049	−	0.234777i	$-0.0754360\pi$
$318$	0	0
$319$	−1.16449e11	−0.629617
$320$	0	0
$321$	0	0
$322$	0	0
$323$	2.22613e11i	1.13799i
$324$	0	0
$325$	1.84244e11i	0.916047i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−1.14753e11	−0.539984
$330$	0	0
$331$	1.47944e11i	0.677440i	0.940887	+	0.338720i	$0.109994\pi$
$331$	1.47944e11i	0.677440i	−0.940887	+	0.338720i	$0.890006\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−4.31324e10	−0.187112
$336$	0	0
$337$	3.22312e10	0.136126	0.0680631	−	0.997681i	$-0.478318\pi$
$337$	3.22312e10	0.136126	0.0680631	+	0.997681i	$0.478318\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	4.76829e11i	1.90971i
$342$	0	0
$343$	−3.25756e11	−1.27078
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 4.93727e11i	− 1.82812i	−0.405581	−	0.914059i	$-0.632931\pi$
$347$	− 4.93727e11i	− 1.82812i	0.405581	−	0.914059i	$-0.367069\pi$
$348$	0	0
$349$	3.00694e11i	1.08495i	0.840071	+	0.542476i	$0.182513\pi$
$349$	3.00694e11i	1.08495i	−0.840071	+	0.542476i	$0.817487\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	1.48977e9	0.00510661	0.00255330	−	0.999997i	$-0.499187\pi$
$353$	1.48977e9	0.00510661	0.00255330	+	0.999997i	$0.499187\pi$
$354$	0	0
$355$	− 1.70689e11i	− 0.570397i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	1.45640e11	0.462759	0.231380	−	0.972864i	$-0.425676\pi$
$359$	1.45640e11	0.462759	0.231380	+	0.972864i	$0.425676\pi$
$360$	0	0
$361$	−2.53515e11	−0.785637
$362$	0	0
$363$	0	0
$364$	0	0
$365$	9.38282e10i	0.276704i
$366$	0	0
$367$	3.98300e11	1.14608	0.573038	−	0.819529i	$-0.305765\pi$
$367$	3.98300e11	1.14608	0.573038	+	0.819529i	$0.305765\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 2.68660e11i	− 0.736241i
$372$	0	0
$373$	− 6.46674e11i	− 1.72980i	−0.501945	−	0.864900i	$-0.667382\pi$
$373$	− 6.46674e11i	− 1.72980i	0.501945	−	0.864900i	$-0.332618\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	2.20158e11	0.561303
$378$	0	0
$379$	− 3.77422e11i	− 0.939616i	−0.882768	−	0.469808i	$-0.844323\pi$
$379$	− 3.77422e11i	− 0.939616i	0.882768	−	0.469808i	$-0.155677\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−1.17245e11	−0.278419	−0.139210	−	0.990263i	$-0.544456\pi$
$383$	−1.17245e11	−0.278419	−0.139210	+	0.990263i	$0.544456\pi$
$384$	0	0
$385$	−2.40642e11	−0.558212
$386$	0	0
$387$	0	0
$388$	0	0
$389$	5.70541e10i	0.126332i	0.998003	+	0.0631661i	$0.0201198\pi$
$389$	5.70541e10i	0.126332i	−0.998003	+	0.0631661i	$0.979880\pi$
$390$	0	0
$391$	2.56804e11	0.555657
$392$	0	0
$393$	0	0
$394$	0	0
$395$	2.19058e11i	0.452765i
$396$	0	0
$397$	− 5.50745e11i	− 1.11274i	−0.830935	−	0.556370i	$-0.812194\pi$
$397$	− 5.50745e11i	− 1.11274i	0.830935	−	0.556370i	$-0.187806\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−6.57952e11	−1.27070	−0.635352	−	0.772223i	$-0.719145\pi$
$401$	−6.57952e11	−1.27070	−0.635352	+	0.772223i	$0.719145\pi$
$402$	0	0
$403$	− 9.01492e11i	− 1.70251i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−5.34299e11	−0.965182
$408$	0	0
$409$	−4.15936e11	−0.734973	−0.367486	−	0.930029i	$-0.619782\pi$
$409$	−4.15936e11	−0.734973	−0.367486	+	0.930029i	$0.619782\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 1.23709e12i	− 2.09232i
$414$	0	0
$415$	2.73283e11	0.452269
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 1.16764e11i	− 0.185074i	−0.995709	−	0.0925369i	$-0.970502\pi$
$419$	− 1.16764e11i	− 0.185074i	0.995709	−	0.0925369i	$-0.0294976\pi$
$420$	0	0
$421$	− 5.40488e11i	− 0.838527i	−0.907865	−	0.419263i	$-0.862288\pi$
$421$	− 5.40488e11i	− 0.838527i	0.907865	−	0.419263i	$-0.137712\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−5.21998e11	−0.776101
$426$	0	0
$427$	− 9.30255e10i	− 0.135418i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−6.12492e11	−0.854974	−0.427487	−	0.904022i	$-0.640601\pi$
$431$	−6.12492e11	−0.854974	−0.427487	+	0.904022i	$0.640601\pi$
$432$	0	0
$433$	−8.42258e10	−0.115146	−0.0575731	−	0.998341i	$-0.518336\pi$
$433$	−8.42258e10	−0.115146	−0.0575731	+	0.998341i	$0.518336\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	6.64703e11i	0.871889i
$438$	0	0
$439$	−2.78791e11	−0.358252	−0.179126	−	0.983826i	$-0.557327\pi$
$439$	−2.78791e11	−0.358252	−0.179126	+	0.983826i	$0.557327\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	6.63629e11i	0.818669i	0.912384	+	0.409335i	$0.134239\pi$
$443$	6.63629e11i	0.818669i	−0.912384	+	0.409335i	$0.865761\pi$
$444$	0	0
$445$	− 1.63288e11i	− 0.197395i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	5.42621e10	0.0630068	0.0315034	−	0.999504i	$-0.489970\pi$
$449$	5.42621e10	0.0630068	0.0315034	+	0.999504i	$0.489970\pi$
$450$	0	0
$451$	5.61002e11i	0.638513i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	4.54958e11	0.497645
$456$	0	0
$457$	5.47436e11	0.587097	0.293549	−	0.955944i	$-0.405164\pi$
$457$	5.47436e11	0.587097	0.293549	+	0.955944i	$0.405164\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 5.32000e11i	− 0.548602i	−0.961644	−	0.274301i	$-0.911554\pi$
$461$	− 5.32000e11i	− 0.548602i	0.961644	−	0.274301i	$-0.0884465\pi$
$462$	0	0
$463$	−1.22274e12	−1.23658	−0.618288	−	0.785951i	$-0.712174\pi$
$463$	−1.22274e12	−1.23658	−0.618288	+	0.785951i	$0.712174\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 4.68729e11i	− 0.456033i	−0.973657	−	0.228016i	$-0.926776\pi$
$467$	− 4.68729e11i	− 0.456033i	0.973657	−	0.228016i	$-0.0732239\pi$
$468$	0	0
$469$	− 1.09468e12i	− 1.04475i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−1.30839e12	−1.20188
$474$	0	0
$475$	− 1.35112e12i	− 1.21779i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−1.17454e12	−1.01943	−0.509715	−	0.860343i	$-0.670249\pi$
$479$	−1.17454e12	−1.01943	−0.509715	+	0.860343i	$0.670249\pi$
$480$	0	0
$481$	1.01014e12	0.860460
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 1.77919e11i	− 0.146010i
$486$	0	0
$487$	−6.70816e11	−0.540410	−0.270205	−	0.962803i	$-0.587091\pi$
$487$	−6.70816e11	−0.540410	−0.270205	+	0.962803i	$0.587091\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	1.62509e12i	1.26186i	0.775841	+	0.630929i	$0.217326\pi$
$491$	1.62509e12i	1.26186i	−0.775841	+	0.630929i	$0.782674\pi$
$492$	0	0
$493$	6.23749e11i	0.475553i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	4.33202e12	3.18483
$498$	0	0
$499$	8.66354e11i	0.625523i	0.949832	+	0.312761i	$0.101254\pi$
$499$	8.66354e11i	0.625523i	−0.949832	+	0.312761i	$0.898746\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−6.42584e10	−0.0447583	−0.0223792	−	0.999750i	$-0.507124\pi$
$503$	−6.42584e10	−0.0447583	−0.0223792	+	0.999750i	$0.507124\pi$
$504$	0	0
$505$	−2.73481e11	−0.187118
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 2.27663e12i	− 1.50336i	−0.659531	−	0.751678i	$-0.729245\pi$
$509$	− 2.27663e12i	− 1.50336i	0.659531	−	0.751678i	$-0.270755\pi$
$510$	0	0
$511$	−2.38133e12	−1.54499
$512$	0	0
$513$	0	0
$514$	0	0
$515$	3.03044e10i	0.0189834i
$516$	0	0
$517$	− 5.94860e11i	− 0.366191i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−1.15540e12	−0.687011	−0.343505	−	0.939151i	$-0.611614\pi$
$521$	−1.15540e12	−0.687011	−0.343505	+	0.939151i	$0.611614\pi$
$522$	0	0
$523$	6.71579e11i	0.392500i	0.980554	+	0.196250i	$0.0628764\pi$
$523$	6.71579e11i	0.392500i	−0.980554	+	0.196250i	$0.937124\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	2.55410e12	1.44241
$528$	0	0
$529$	−1.03436e12	−0.574275
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 1.06063e12i	− 0.569234i
$534$	0	0
$535$	−3.24951e11	−0.171485
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 3.89805e12i	− 1.98929i
$540$	0	0
$541$	− 7.08583e11i	− 0.355634i	−0.984064	−	0.177817i	$-0.943097\pi$
$541$	− 7.08583e11i	− 0.355634i	0.984064	−	0.177817i	$-0.0569035\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−2.56818e11	−0.124693
$546$	0	0
$547$	− 5.39817e11i	− 0.257812i	−0.991657	−	0.128906i	$-0.958853\pi$
$547$	− 5.39817e11i	− 0.257812i	0.991657	−	0.128906i	$-0.0411466\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−1.61449e12	−0.746196
$552$	0	0
$553$	−5.55962e12	−2.52803
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 1.59724e12i	− 0.703106i	−0.936168	−	0.351553i	$-0.885654\pi$
$557$	− 1.59724e12i	− 0.703106i	0.936168	−	0.351553i	$-0.114346\pi$
$558$	0	0
$559$	2.47363e12	1.07147
$560$	0	0
$561$	0	0
$562$	0	0
$563$	9.72484e11i	0.407939i	0.978977	+	0.203969i	$0.0653843\pi$
$563$	9.72484e11i	0.407939i	−0.978977	+	0.203969i	$0.934616\pi$
$564$	0	0
$565$	− 1.25656e12i	− 0.518760i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−2.51375e11	−0.100535	−0.0502675	−	0.998736i	$-0.516007\pi$
$569$	−2.51375e11	−0.100535	−0.0502675	+	0.998736i	$0.516007\pi$
$570$	0	0
$571$	− 5.79852e11i	− 0.228273i	−0.993465	−	0.114136i	$-0.963590\pi$
$571$	− 5.79852e11i	− 0.228273i	0.993465	−	0.114136i	$-0.0364101\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−1.55864e12	−0.594622
$576$	0	0
$577$	6.58678e11	0.247390	0.123695	−	0.992320i	$-0.460526\pi$
$577$	6.58678e11	0.247390	0.123695	+	0.992320i	$0.460526\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	6.93583e12i	2.52526i
$582$	0	0
$583$	1.39269e12	0.499282
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 1.53421e12i	− 0.533353i	−0.963786	−	0.266676i	$-0.914075\pi$
$587$	− 1.53421e12i	− 0.533353i	0.963786	−	0.266676i	$-0.0859255\pi$
$588$	0	0
$589$	6.61094e12i	2.26331i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	2.64973e11	0.0879946	0.0439973	−	0.999032i	$-0.485991\pi$
$593$	2.64973e11	0.0879946	0.0439973	+	0.999032i	$0.485991\pi$
$594$	0	0
$595$	1.28898e12i	0.421620i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	4.65955e12	1.47885	0.739423	−	0.673241i	$-0.235098\pi$
$599$	4.65955e12	1.47885	0.739423	+	0.673241i	$0.235098\pi$
$600$	0	0
$601$	3.23954e12	1.01286	0.506428	−	0.862282i	$-0.330965\pi$
$601$	3.23954e12	1.01286	0.506428	+	0.862282i	$0.330965\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 2.66194e11i	− 0.0807790i
$606$	0	0
$607$	2.37269e12	0.709401	0.354700	−	0.934980i	$-0.384583\pi$
$607$	2.37269e12	0.709401	0.354700	+	0.934980i	$0.384583\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	1.12464e12i	0.326459i
$612$	0	0
$613$	− 3.48233e12i	− 0.996088i	−0.867152	−	0.498044i	$-0.834052\pi$
$613$	− 3.48233e12i	− 0.996088i	0.867152	−	0.498044i	$-0.165948\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	1.26692e12	0.351939	0.175969	−	0.984396i	$-0.443694\pi$
$617$	1.26692e12	0.351939	0.175969	+	0.984396i	$0.443694\pi$
$618$	0	0
$619$	3.81076e12i	1.04329i	0.853164	+	0.521643i	$0.174681\pi$
$619$	3.81076e12i	1.04329i	−0.853164	+	0.521643i	$0.825319\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	4.14420e12	1.10216
$624$	0	0
$625$	2.82996e12	0.741857
$626$	0	0
$627$	0	0
$628$	0	0
$629$	2.86193e12i	0.729006i
$630$	0	0
$631$	1.66134e12	0.417184	0.208592	−	0.978003i	$-0.433112\pi$
$631$	1.66134e12	0.417184	0.208592	+	0.978003i	$0.433112\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	5.27361e11i	0.128714i
$636$	0	0
$637$	7.36963e12i	1.77345i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−5.23434e12	−1.22462	−0.612309	−	0.790618i	$-0.709759\pi$
$641$	−5.23434e12	−1.22462	−0.612309	+	0.790618i	$0.709759\pi$
$642$	0	0
$643$	− 7.76746e12i	− 1.79197i	−0.444089	−	0.895983i	$-0.646473\pi$
$643$	− 7.76746e12i	− 1.79197i	0.444089	−	0.895983i	$-0.353527\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−4.78238e12	−1.07294	−0.536470	−	0.843920i	$-0.680242\pi$
$647$	−4.78238e12	−1.07294	−0.536470	+	0.843920i	$0.680242\pi$
$648$	0	0
$649$	6.41290e12	1.41891
$650$	0	0
$651$	0	0
$652$	0	0
$653$	2.68837e12i	0.578601i	0.957238	+	0.289301i	$0.0934227\pi$
$653$	2.68837e12i	0.578601i	−0.957238	+	0.289301i	$0.906577\pi$
$654$	0	0
$655$	1.97551e12	0.419367
$656$	0	0
$657$	0	0
$658$	0	0
$659$	3.65807e12i	0.755557i	0.925896	+	0.377778i	$0.123312\pi$
$659$	3.65807e12i	0.755557i	−0.925896	+	0.377778i	$0.876688\pi$
$660$	0	0
$661$	3.85004e12i	0.784439i	0.919872	+	0.392219i	$0.128293\pi$
$661$	3.85004e12i	0.784439i	−0.919872	+	0.392219i	$0.871707\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−3.33636e12	−0.661569
$666$	0	0
$667$	1.86246e12i	0.364352i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	4.82230e11	0.0918339
$672$	0	0
$673$	3.09913e10	0.00582333	0.00291167	−	0.999996i	$-0.499073\pi$
$673$	3.09913e10	0.00582333	0.00291167	+	0.999996i	$0.499073\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	9.89140e12i	1.80971i	0.425720	+	0.904855i	$0.360021\pi$
$677$	9.89140e12i	1.80971i	−0.425720	+	0.904855i	$0.639979\pi$
$678$	0	0
$679$	4.51551e12	0.815254
$680$	0	0
$681$	0	0
$682$	0	0
$683$	2.51437e12i	0.442115i	0.975261	+	0.221058i	$0.0709509\pi$
$683$	2.51437e12i	0.442115i	−0.975261	+	0.221058i	$0.929049\pi$
$684$	0	0
$685$	2.88013e12i	0.499809i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−2.63302e12	−0.445110
$690$	0	0
$691$	7.61485e12i	1.27060i	0.772264	+	0.635302i	$0.219124\pi$
$691$	7.61485e12i	1.27060i	−0.772264	+	0.635302i	$0.780876\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−3.33097e12	−0.541551
$696$	0	0
$697$	3.00496e12	0.482272
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 5.46754e12i	− 0.855187i	−0.903971	−	0.427594i	$-0.859361\pi$
$701$	− 5.46754e12i	− 0.855187i	0.903971	−	0.427594i	$-0.140639\pi$
$702$	0	0
$703$	−7.40772e12	−1.14389
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 6.94086e12i	− 1.04478i
$708$	0	0
$709$	− 1.18804e13i	− 1.76572i	−0.469638	−	0.882859i	$-0.655616\pi$
$709$	− 1.18804e13i	− 1.76572i	0.469638	−	0.882859i	$-0.344384\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	7.62633e12	1.10513
$714$	0	0
$715$	2.35843e12i	0.337478i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−1.13520e12	−0.158414	−0.0792071	−	0.996858i	$-0.525239\pi$
$719$	−1.13520e12	−0.158414	−0.0792071	+	0.996858i	$0.525239\pi$
$720$	0	0
$721$	−7.69115e11	−0.105994
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 3.78577e12i	− 0.508900i
$726$	0	0
$727$	8.53103e12	1.13265	0.566326	−	0.824181i	$-0.308364\pi$
$727$	8.53103e12	1.13265	0.566326	+	0.824181i	$0.308364\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	7.00827e12i	0.907784i
$732$	0	0
$733$	− 2.95638e11i	− 0.0378262i	−0.999821	−	0.0189131i	$-0.993979\pi$
$733$	− 2.95638e11i	− 0.0378262i	0.999821	−	0.0189131i	$-0.00602059\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	5.67467e12	0.708496
$738$	0	0
$739$	8.95692e12i	1.10474i	0.833600	+	0.552369i	$0.186276\pi$
$739$	8.95692e12i	1.10474i	−0.833600	+	0.552369i	$0.813724\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	1.50988e12	0.181758	0.0908789	−	0.995862i	$-0.471032\pi$
$743$	1.50988e12	0.181758	0.0908789	+	0.995862i	$0.471032\pi$
$744$	0	0
$745$	−1.78552e12	−0.212354
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 8.24715e12i	− 0.957494i
$750$	0	0
$751$	3.27872e12	0.376118	0.188059	−	0.982158i	$-0.439780\pi$
$751$	3.27872e12	0.376118	0.188059	+	0.982158i	$0.439780\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	1.43830e12i	0.161097i
$756$	0	0
$757$	1.03608e12i	0.114673i	0.998355	+	0.0573364i	$0.0182608\pi$
$757$	1.03608e12i	0.114673i	−0.998355	+	0.0573364i	$0.981739\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	1.14256e13	1.23495	0.617475	−	0.786591i	$-0.288156\pi$
$761$	1.14256e13	1.23495	0.617475	+	0.786591i	$0.288156\pi$
$762$	0	0
$763$	− 6.51795e12i	− 0.696226i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−1.21242e13	−1.26495
$768$	0	0
$769$	3.90710e12	0.402889	0.201445	−	0.979500i	$-0.435436\pi$
$769$	3.90710e12	0.402889	0.201445	+	0.979500i	$0.435436\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	5.78500e12i	0.582768i	0.956606	+	0.291384i	$0.0941157\pi$
$773$	5.78500e12i	0.582768i	−0.956606	+	0.291384i	$0.905884\pi$
$774$	0	0
$775$	−1.55018e13	−1.54356
$776$	0	0
$777$	0	0
$778$	0	0
$779$	7.77795e12i	0.756739i
$780$	0	0
$781$	2.24565e13i	2.15980i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−9.05029e10	−0.00850647
$786$	0	0
$787$	9.01542e11i	0.0837722i	0.999122	+	0.0418861i	$0.0133367\pi$
$787$	9.01542e11i	0.0837722i	−0.999122	+	0.0418861i	$0.986663\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	3.18911e13	2.89652
$792$	0	0
$793$	−9.11703e11	−0.0818698
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 1.98771e12i	− 0.174498i	−0.996187	−	0.0872489i	$-0.972192\pi$
$797$	− 1.98771e12i	− 0.174498i	0.996187	−	0.0872489i	$-0.0278076\pi$
$798$	0	0
$799$	−3.18632e12	−0.276585
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 1.23444e13i	− 1.04773i
$804$	0	0
$805$	3.84880e12i	0.323030i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	1.59599e13	1.30997	0.654986	−	0.755641i	$-0.272674\pi$
$809$	1.59599e13	1.30997	0.654986	+	0.755641i	$0.272674\pi$
$810$	0	0
$811$	2.09687e13i	1.70207i	0.525106	+	0.851037i	$0.324026\pi$
$811$	2.09687e13i	1.70207i	−0.525106	+	0.851037i	$0.675974\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	1.68122e11	0.0133480
$816$	0	0
$817$	−1.81400e13	−1.42442
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 1.06480e13i	− 0.817945i	−0.912547	−	0.408973i	$-0.865887\pi$
$821$	− 1.06480e13i	− 0.817945i	0.912547	−	0.408973i	$-0.134113\pi$
$822$	0	0
$823$	5.17550e12	0.393236	0.196618	−	0.980480i	$-0.437004\pi$
$823$	5.17550e12	0.393236	0.196618	+	0.980480i	$0.437004\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	1.21139e13i	0.900552i	0.892889	+	0.450276i	$0.148674\pi$
$827$	1.21139e13i	0.900552i	−0.892889	+	0.450276i	$0.851326\pi$
$828$	0	0
$829$	− 9.02964e12i	− 0.664010i	−0.943278	−	0.332005i	$-0.892275\pi$
$829$	− 9.02964e12i	− 0.664010i	0.943278	−	0.332005i	$-0.107725\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−2.08796e13	−1.50252
$834$	0	0
$835$	− 4.16150e12i	− 0.296252i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	3.30696e12	0.230409	0.115205	−	0.993342i	$-0.463248\pi$
$839$	3.30696e12	0.230409	0.115205	+	0.993342i	$0.463248\pi$
$840$	0	0
$841$	9.98343e12	0.688174
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 4.57844e10i	− 0.00308931i
$846$	0	0
$847$	6.75590e12	0.451033
$848$	0	0
$849$	0	0
$850$	0	0
$851$	8.54549e12i	0.558540i
$852$	0	0
$853$	− 1.60549e13i	− 1.03834i	−0.854672	−	0.519168i	$-0.826242\pi$
$853$	− 1.60549e13i	− 1.03834i	0.854672	−	0.519168i	$-0.173758\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−1.33261e13	−0.843897	−0.421949	−	0.906620i	$-0.638654\pi$
$857$	−1.33261e13	−0.843897	−0.421949	+	0.906620i	$0.638654\pi$
$858$	0	0
$859$	− 5.29072e12i	− 0.331547i	−0.986164	−	0.165774i	$-0.946988\pi$
$859$	− 5.29072e12i	− 0.331547i	0.986164	−	0.165774i	$-0.0530121\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−1.30813e13	−0.802793	−0.401397	−	0.915904i	$-0.631475\pi$
$863$	−1.30813e13	−0.802793	−0.401397	+	0.915904i	$0.631475\pi$
$864$	0	0
$865$	3.61518e12	0.219562
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 2.88202e13i	− 1.71438i
$870$	0	0
$871$	−1.07285e13	−0.631623
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 1.64078e13i	− 0.946269i
$876$	0	0
$877$	− 1.27920e13i	− 0.730199i	−0.930969	−	0.365099i	$-0.881035\pi$
$877$	− 1.27920e13i	− 0.730199i	0.930969	−	0.365099i	$-0.118965\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−7.26763e12	−0.406444	−0.203222	−	0.979133i	$-0.565141\pi$
$881$	−7.26763e12	−0.406444	−0.203222	+	0.979133i	$0.565141\pi$
$882$	0	0
$883$	− 1.11873e13i	− 0.619304i	−0.950850	−	0.309652i	$-0.899787\pi$
$883$	− 1.11873e13i	− 0.619304i	0.950850	−	0.309652i	$-0.100213\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−3.38540e13	−1.83634	−0.918171	−	0.396185i	$-0.870334\pi$
$887$	−3.38540e13	−1.83634	−0.918171	+	0.396185i	$0.870334\pi$
$888$	0	0
$889$	−1.33842e13	−0.718680
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 8.24736e12i	− 0.433994i
$894$	0	0
$895$	−6.80226e12	−0.354364
$896$	0	0
$897$	0	0
$898$	0	0
$899$	1.85235e13i	0.945811i
$900$	0	0
$901$	− 7.45984e12i	− 0.377110i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−3.40649e12	−0.168806
$906$	0	0
$907$	− 2.04441e13i	− 1.00308i	−0.865134	−	0.501540i	$-0.832767\pi$
$907$	− 2.04441e13i	− 1.00308i	0.865134	−	0.501540i	$-0.167233\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	3.21964e13	1.54873	0.774363	−	0.632742i	$-0.218071\pi$
$911$	3.21964e13	1.54873	0.774363	+	0.632742i	$0.218071\pi$
$912$	0	0
$913$	−3.59543e13	−1.71251
$914$	0	0
$915$	0	0
$916$	0	0
$917$	5.01379e13i	2.34155i
$918$	0	0
$919$	−1.16757e13	−0.539960	−0.269980	−	0.962866i	$-0.587017\pi$
$919$	−1.16757e13	−0.539960	−0.269980	+	0.962866i	$0.587017\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 4.24562e13i	− 1.92546i
$924$	0	0
$925$	− 1.73701e13i	− 0.780128i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	3.09973e12	0.136538	0.0682690	−	0.997667i	$-0.478252\pi$
$929$	3.09973e12	0.136538	0.0682690	+	0.997667i	$0.478252\pi$
$930$	0	0
$931$	− 5.40440e13i	− 2.35762i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−6.68189e12	−0.285922
$936$	0	0
$937$	1.33365e13	0.565216	0.282608	−	0.959236i	$-0.408801\pi$
$937$	1.33365e13	0.565216	0.282608	+	0.959236i	$0.408801\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	3.02157e12i	0.125626i	0.998025	+	0.0628130i	$0.0200072\pi$
$941$	3.02157e12i	0.125626i	−0.998025	+	0.0628130i	$0.979993\pi$
$942$	0	0
$943$	8.97257e12	0.369500
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 9.61342e12i	− 0.388421i	−0.980960	−	0.194211i	$-0.937785\pi$
$947$	− 9.61342e12i	− 0.388421i	0.980960	−	0.194211i	$-0.0622145\pi$
$948$	0	0
$949$	2.33383e13i	0.934054i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−3.02125e12	−0.118650	−0.0593251	−	0.998239i	$-0.518895\pi$
$953$	−3.02125e12	−0.118650	−0.0593251	+	0.998239i	$0.518895\pi$
$954$	0	0
$955$	2.41504e12i	0.0939526i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−7.30967e13	−2.79070
$960$	0	0
$961$	4.94096e13	1.86877
$962$	0	0
$963$	0	0
$964$	0	0
$965$	3.24004e12i	0.120276i
$966$	0	0
$967$	2.09650e13	0.771038	0.385519	−	0.922700i	$-0.374022\pi$
$967$	2.09650e13	0.771038	0.385519	+	0.922700i	$0.374022\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	3.32447e12i	0.120015i	0.998198	+	0.0600076i	$0.0191125\pi$
$971$	3.32447e12i	0.120015i	−0.998198	+	0.0600076i	$0.980888\pi$
$972$	0	0
$973$	− 8.45389e13i	− 3.02377i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	3.35185e13	1.17695	0.588477	−	0.808514i	$-0.299728\pi$
$977$	3.35185e13	1.17695	0.588477	+	0.808514i	$0.299728\pi$
$978$	0	0
$979$	2.14829e13i	0.747430i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	9.60348e12	0.328048	0.164024	−	0.986456i	$-0.447553\pi$
$983$	9.60348e12	0.328048	0.164024	+	0.986456i	$0.447553\pi$
$984$	0	0
$985$	−7.64261e12	−0.258689
$986$	0	0
$987$	0	0
$988$	0	0
$989$	2.09261e13i	0.695513i
$990$	0	0
$991$	−1.42518e13	−0.469394	−0.234697	−	0.972069i	$-0.575410\pi$
$991$	−1.42518e13	−0.469394	−0.234697	+	0.972069i	$0.575410\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	9.80769e12i	0.317222i
$996$	0	0
$997$	2.92635e13i	0.937988i	0.883201	+	0.468994i	$0.155383\pi$
$997$	2.92635e13i	0.937988i	−0.883201	+	0.468994i	$0.844617\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	288.10.d.d.145.11		18
3.2	odd	2		96.10.d.a.49.5		18
4.3	odd	2		72.10.d.d.37.13		18
8.3	odd	2		72.10.d.d.37.14		18
8.5	even	2	inner	288.10.d.d.145.8		18
12.11	even	2		24.10.d.a.13.6	yes	18
24.5	odd	2		96.10.d.a.49.14		18
24.11	even	2		24.10.d.a.13.5	✓	18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
24.10.d.a.13.5	✓	18	24.11	even	2
24.10.d.a.13.6	yes	18	12.11	even	2
72.10.d.d.37.13		18	4.3	odd	2
72.10.d.d.37.14		18	8.3	odd	2
96.10.d.a.49.5		18	3.2	odd	2
96.10.d.a.49.14		18	24.5	odd	2
288.10.d.d.145.8		18	8.5	even	2	inner
288.10.d.d.145.11		18	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 288 = 2^{5} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	288.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+416.150i q^{5} -10561.7 q^{7} +O(q^{10})\)	\(q+416.150i q^{5} -10561.7 q^{7} -54750.4i q^{11} +103511. i q^{13} -293266. q^{17} -759080. i q^{19} -875669. q^{23} +1.77994e6 q^{25} -2.12690e6i q^{29} -8.70915e6 q^{31} -4.39526e6i q^{35} -9.75881e6i q^{37} -1.02465e7 q^{41} -2.38973e7i q^{43} +1.08649e7 q^{47} +7.11967e7 q^{49} +2.54371e7i q^{53} +2.27844e7 q^{55} +1.17130e8i q^{59} +8.80779e6i q^{61} -4.30761e7 q^{65} +1.03646e8i q^{67} -4.10162e8 q^{71} +2.25467e8 q^{73} +5.78259e8i q^{77} +5.26393e8 q^{79} -6.56694e8i q^{83} -1.22043e8i q^{85} -3.92379e8 q^{89} -1.09326e9i q^{91} +3.15891e8 q^{95} -4.27535e8 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 9604 q^{7}+O(q^{10}) \) 18 * q + 9604 * q^7	\( 18 q + 9604 q^{7} - 203996 q^{17} - 3465976 q^{23} - 5309486 q^{25} - 4737540 q^{31} + 3780956 q^{41} - 112458264 q^{47} + 90859338 q^{49} + 76355584 q^{55} - 300816016 q^{65} - 408942312 q^{71} + 826124308 q^{73} - 382409492 q^{79} + 458464820 q^{89} + 331883504 q^{95} + 264098780 q^{97}+O(q^{100}) \) 18 * q + 9604 * q^7 - 203996 * q^17 - 3465976 * q^23 - 5309486 * q^25 - 4737540 * q^31 + 3780956 * q^41 - 112458264 * q^47 + 90859338 * q^49 + 76355584 * q^55 - 300816016 * q^65 - 408942312 * q^71 + 826124308 * q^73 - 382409492 * q^79 + 458464820 * q^89 + 331883504 * q^95 + 264098780 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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