LMFDB - Embedded newform 144.12.a.t.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [144,12,Mod(1,144)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(144, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 12, names="a")

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

chi := DirichletCharacter("144.1");

S:= CuspForms(chi, 12);

N := Newforms(S);

Level:	$ N $	$=$	$ 144 = 2^{4} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 12 $
Character orbit:	$[\chi]$	$=$	144.a (trivial)

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:	yes
Analytic conductor:	$110.641418001$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	$\mathbb{Q}[x]/(x^{3} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 829x - 6375 $ x^3 - 829*x - 6375
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{12}\cdot 3^{4} $
Twist minimal:	no (minimal twist has level 72)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$32.0600$ of defining polynomial
Character	$\chi$	$=$	144.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+10562.2 q^{5} +36458.0 q^{7} +O(q^{10})$	$q+10562.2 q^{5} +36458.0 q^{7} -286883. q^{11} -1.97472e6 q^{13} +1.34298e6 q^{17} +1.68726e7 q^{19} +3.95976e7 q^{23} +6.27316e7 q^{25} +1.70464e8 q^{29} -1.59742e8 q^{31} +3.85076e8 q^{35} -2.92345e8 q^{37} -2.22597e8 q^{41} +8.28318e8 q^{43} +6.71760e8 q^{47} -6.48141e8 q^{49} -3.76569e9 q^{53} -3.03011e9 q^{55} -1.13907e9 q^{59} +9.13871e9 q^{61} -2.08573e10 q^{65} +8.39308e9 q^{67} +2.73512e10 q^{71} -1.62220e10 q^{73} -1.04592e10 q^{77} +1.95906e10 q^{79} +4.79771e10 q^{83} +1.41848e10 q^{85} -2.86128e9 q^{89} -7.19942e10 q^{91} +1.78212e11 q^{95} +6.47159e10 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 1584 q^{5} + 17796 q^{7}+O(q^{10}) $ 3 * q + 1584 * q^5 + 17796 * q^7	$ 3 q + 1584 q^{5} + 17796 q^{7} - 238272 q^{11} - 49458 q^{13} - 766368 q^{17} + 4590936 q^{19} + 13188480 q^{23} + 18034281 q^{25} + 8005200 q^{29} - 10185564 q^{31} + 235065408 q^{35} - 73215510 q^{37} - 1032516960 q^{41} - 28458216 q^{43} + 2840606592 q^{47} - 109741845 q^{49} - 6755828784 q^{53} + 846590976 q^{55} + 15928599936 q^{59} + 5136563970 q^{61} - 37415401248 q^{65} + 6119899728 q^{67} + 58699811328 q^{71} + 2561705778 q^{73} - 86561454336 q^{77} + 17842143972 q^{79} + 134316444096 q^{83} + 43096069632 q^{85} - 152402442048 q^{89} + 56268770664 q^{91} + 207495767424 q^{95} + 106677733482 q^{97}+O(q^{100}) $ 3 * q + 1584 * q^5 + 17796 * q^7 - 238272 * q^11 - 49458 * q^13 - 766368 * q^17 + 4590936 * q^19 + 13188480 * q^23 + 18034281 * q^25 + 8005200 * q^29 - 10185564 * q^31 + 235065408 * q^35 - 73215510 * q^37 - 1032516960 * q^41 - 28458216 * q^43 + 2840606592 * q^47 - 109741845 * q^49 - 6755828784 * q^53 + 846590976 * q^55 + 15928599936 * q^59 + 5136563970 * q^61 - 37415401248 * q^65 + 6119899728 * q^67 + 58699811328 * q^71 + 2561705778 * q^73 - 86561454336 * q^77 + 17842143972 * q^79 + 134316444096 * q^83 + 43096069632 * q^85 - 152402442048 * q^89 + 56268770664 * q^91 + 207495767424 * q^95 + 106677733482 * q^97

Display 100 coefficients 10 coefficients

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
$5$	10562.2	1.51154	0.755768	−	0.654839i	$-0.227264\pi$
$5$	10562.2	1.51154	0.755768	+	0.654839i	$0.227264\pi$
$6$	0	0
$7$	36458.0	0.819886	0.409943	−	0.912111i	$-0.365549\pi$
$7$	36458.0	0.819886	0.409943	+	0.912111i	$0.365549\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−286883.	−0.537087	−0.268544	−	0.963268i	$-0.586542\pi$
$11$	−286883.	−0.537087	−0.268544	+	0.963268i	$0.586542\pi$
$12$	0	0
$13$	−1.97472e6	−1.47508	−0.737542	−	0.675302i	$-0.764013\pi$
$13$	−1.97472e6	−1.47508	−0.737542	+	0.675302i	$0.764013\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1.34298e6	0.229404	0.114702	−	0.993400i	$-0.463409\pi$
$17$	1.34298e6	0.229404	0.114702	+	0.993400i	$0.463409\pi$
$18$	0	0
$19$	1.68726e7	1.56328	0.781642	−	0.623727i	$-0.214382\pi$
$19$	1.68726e7	1.56328	0.781642	+	0.623727i	$0.214382\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	3.95976e7	1.28282	0.641410	−	0.767198i	$-0.278350\pi$
$23$	3.95976e7	1.28282	0.641410	+	0.767198i	$0.278350\pi$
$24$	0	0
$25$	6.27316e7	1.28474
$26$	0	0
$27$	0	0
$28$	0	0
$29$	1.70464e8	1.54328	0.771639	−	0.636061i	$-0.219437\pi$
$29$	1.70464e8	1.54328	0.771639	+	0.636061i	$0.219437\pi$
$30$	0	0
$31$	−1.59742e8	−1.00215	−0.501073	−	0.865405i	$-0.667061\pi$
$31$	−1.59742e8	−1.00215	−0.501073	+	0.865405i	$0.667061\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	3.85076e8	1.23929
$36$	0	0
$37$	−2.92345e8	−0.693084	−0.346542	−	0.938034i	$-0.612644\pi$
$37$	−2.92345e8	−0.693084	−0.346542	+	0.938034i	$0.612644\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−2.22597e8	−0.300060	−0.150030	−	0.988681i	$-0.547937\pi$
$41$	−2.22597e8	−0.300060	−0.150030	+	0.988681i	$0.547937\pi$
$42$	0	0
$43$	8.28318e8	0.859252	0.429626	−	0.903007i	$-0.358645\pi$
$43$	8.28318e8	0.859252	0.429626	+	0.903007i	$0.358645\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	6.71760e8	0.427244	0.213622	−	0.976916i	$-0.431474\pi$
$47$	6.71760e8	0.427244	0.213622	+	0.976916i	$0.431474\pi$
$48$	0	0
$49$	−6.48141e8	−0.327787
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−3.76569e9	−1.23688	−0.618439	−	0.785833i	$-0.712235\pi$
$53$	−3.76569e9	−1.23688	−0.618439	+	0.785833i	$0.712235\pi$
$54$	0	0
$55$	−3.03011e9	−0.811827
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−1.13907e9	−0.207426	−0.103713	−	0.994607i	$-0.533072\pi$
$59$	−1.13907e9	−0.207426	−0.103713	+	0.994607i	$0.533072\pi$
$60$	0	0
$61$	9.13871e9	1.38539	0.692693	−	0.721233i	$-0.256424\pi$
$61$	9.13871e9	1.38539	0.692693	+	0.721233i	$0.256424\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−2.08573e10	−2.22964
$66$	0	0
$67$	8.39308e9	0.759468	0.379734	−	0.925096i	$-0.376015\pi$
$67$	8.39308e9	0.759468	0.379734	+	0.925096i	$0.376015\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	2.73512e10	1.79910	0.899550	−	0.436817i	$-0.143894\pi$
$71$	2.73512e10	1.79910	0.899550	+	0.436817i	$0.143894\pi$
$72$	0	0
$73$	−1.62220e10	−0.915861	−0.457930	−	0.888988i	$-0.651409\pi$
$73$	−1.62220e10	−0.915861	−0.457930	+	0.888988i	$0.651409\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−1.04592e10	−0.440350
$78$	0	0
$79$	1.95906e10	0.716305	0.358153	−	0.933663i	$-0.383407\pi$
$79$	1.95906e10	0.716305	0.358153	+	0.933663i	$0.383407\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	4.79771e10	1.33692	0.668459	−	0.743749i	$-0.266954\pi$
$83$	4.79771e10	1.33692	0.668459	+	0.743749i	$0.266954\pi$
$84$	0	0
$85$	1.41848e10	0.346753
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−2.86128e9	−0.0543144	−0.0271572	−	0.999631i	$-0.508645\pi$
$89$	−2.86128e9	−0.0543144	−0.0271572	+	0.999631i	$0.508645\pi$
$90$	0	0
$91$	−7.19942e10	−1.20940
$92$	0	0
$93$	0	0
$94$	0	0
$95$	1.78212e11	2.36296
$96$	0	0
$97$	6.47159e10	0.765185	0.382593	−	0.923917i	$-0.375031\pi$
$97$	6.47159e10	0.765185	0.382593	+	0.923917i	$0.375031\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−1.39683e11	−1.32244	−0.661218	−	0.750194i	$-0.729960\pi$
$101$	−1.39683e11	−1.32244	−0.661218	+	0.750194i	$0.729960\pi$
$102$	0	0
$103$	1.67404e11	1.42286	0.711429	−	0.702758i	$-0.248048\pi$
$103$	1.67404e11	1.42286	0.711429	+	0.702758i	$0.248048\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−2.71875e11	−1.87395	−0.936976	−	0.349393i	$-0.886388\pi$
$107$	−2.71875e11	−1.87395	−0.936976	+	0.349393i	$0.886388\pi$
$108$	0	0
$109$	2.15539e11	1.34178	0.670889	−	0.741558i	$-0.265913\pi$
$109$	2.15539e11	1.34178	0.670889	+	0.741558i	$0.265913\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−4.79467e10	−0.244809	−0.122404	−	0.992480i	$-0.539060\pi$
$113$	−4.79467e10	−0.244809	−0.122404	+	0.992480i	$0.539060\pi$
$114$	0	0
$115$	4.18237e11	1.93903
$116$	0	0
$117$	0	0
$118$	0	0
$119$	4.89624e10	0.188085
$120$	0	0
$121$	−2.03010e11	−0.711538
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.46851e11	0.430400
$126$	0	0
$127$	−3.12237e11	−0.838619	−0.419309	−	0.907843i	$-0.637728\pi$
$127$	−3.12237e11	−0.838619	−0.419309	+	0.907843i	$0.637728\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−7.95055e11	−1.80055	−0.900275	−	0.435322i	$-0.856635\pi$
$131$	−7.95055e11	−1.80055	−0.900275	+	0.435322i	$0.856635\pi$
$132$	0	0
$133$	6.15142e11	1.28172
$134$	0	0
$135$	0	0
$136$	0	0
$137$	4.30008e11	0.761226	0.380613	−	0.924734i	$-0.375713\pi$
$137$	4.30008e11	0.761226	0.380613	+	0.924734i	$0.375713\pi$
$138$	0	0
$139$	4.51927e11	0.738732	0.369366	−	0.929284i	$-0.379575\pi$
$139$	4.51927e11	0.738732	0.369366	+	0.929284i	$0.379575\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	5.66512e11	0.792248
$144$	0	0
$145$	1.80047e12	2.33272
$146$	0	0
$147$	0	0
$148$	0	0
$149$	1.76669e12	1.97078	0.985388	−	0.170327i	$-0.0544823\pi$
$149$	1.76669e12	1.97078	0.985388	+	0.170327i	$0.0544823\pi$
$150$	0	0
$151$	1.66143e12	1.72230	0.861149	−	0.508353i	$-0.169745\pi$
$151$	1.66143e12	1.72230	0.861149	+	0.508353i	$0.169745\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−1.68723e12	−1.51478
$156$	0	0
$157$	−1.14752e12	−0.960087	−0.480043	−	0.877245i	$-0.659379\pi$
$157$	−1.14752e12	−0.960087	−0.480043	+	0.877245i	$0.659379\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	1.44365e12	1.05177
$162$	0	0
$163$	1.42914e12	0.972840	0.486420	−	0.873725i	$-0.338302\pi$
$163$	1.42914e12	0.972840	0.486420	+	0.873725i	$0.338302\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	6.86980e11	0.409264	0.204632	−	0.978839i	$-0.434400\pi$
$167$	6.86980e11	0.409264	0.204632	+	0.978839i	$0.434400\pi$
$168$	0	0
$169$	2.10735e12	1.17587
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−1.25585e12	−0.616147	−0.308074	−	0.951363i	$-0.599684\pi$
$173$	−1.25585e12	−0.616147	−0.308074	+	0.951363i	$0.599684\pi$
$174$	0	0
$175$	2.28707e12	1.05334
$176$	0	0
$177$	0	0
$178$	0	0
$179$	4.37178e11	0.177814	0.0889071	−	0.996040i	$-0.471663\pi$
$179$	4.37178e11	0.177814	0.0889071	+	0.996040i	$0.471663\pi$
$180$	0	0
$181$	4.43992e12	1.69880	0.849401	−	0.527748i	$-0.176964\pi$
$181$	4.43992e12	1.69880	0.849401	+	0.527748i	$0.176964\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−3.08780e12	−1.04762
$186$	0	0
$187$	−3.85278e11	−0.123210
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−5.01464e12	−1.42743	−0.713717	−	0.700435i	$-0.752990\pi$
$191$	−5.01464e12	−1.42743	−0.713717	+	0.700435i	$0.752990\pi$
$192$	0	0
$193$	1.96862e12	0.529171	0.264585	−	0.964362i	$-0.414765\pi$
$193$	1.96862e12	0.529171	0.264585	+	0.964362i	$0.414765\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−3.32598e12	−0.798648	−0.399324	−	0.916810i	$-0.630755\pi$
$197$	−3.32598e12	−0.798648	−0.399324	+	0.916810i	$0.630755\pi$
$198$	0	0
$199$	3.79596e11	0.0862243	0.0431122	−	0.999070i	$-0.486273\pi$
$199$	3.79596e11	0.0862243	0.0431122	+	0.999070i	$0.486273\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	6.21478e12	1.26531
$204$	0	0
$205$	−2.35111e12	−0.453551
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−4.84047e12	−0.839620
$210$	0	0
$211$	3.25865e12	0.536395	0.268198	−	0.963364i	$-0.413572\pi$
$211$	3.25865e12	0.536395	0.268198	+	0.963364i	$0.413572\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	8.74885e12	1.29879
$216$	0	0
$217$	−5.82389e12	−0.821646
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−2.65201e12	−0.338390
$222$	0	0
$223$	4.13883e12	0.502575	0.251287	−	0.967913i	$-0.419146\pi$
$223$	4.13883e12	0.502575	0.251287	+	0.967913i	$0.419146\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	1.76315e13	1.94154	0.970771	−	0.240007i	$-0.0771499\pi$
$227$	1.76315e13	1.94154	0.970771	+	0.240007i	$0.0771499\pi$
$228$	0	0
$229$	8.91402e12	0.935359	0.467679	−	0.883898i	$-0.345090\pi$
$229$	8.91402e12	0.935359	0.467679	+	0.883898i	$0.345090\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	7.70724e12	0.735260	0.367630	−	0.929972i	$-0.380169\pi$
$233$	7.70724e12	0.735260	0.367630	+	0.929972i	$0.380169\pi$
$234$	0	0
$235$	7.09525e12	0.645795
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−8.44101e12	−0.700174	−0.350087	−	0.936717i	$-0.613848\pi$
$239$	−8.44101e12	−0.700174	−0.350087	+	0.936717i	$0.613848\pi$
$240$	0	0
$241$	8.51074e12	0.674332	0.337166	−	0.941445i	$-0.390532\pi$
$241$	8.51074e12	0.674332	0.337166	+	0.941445i	$0.390532\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−6.84579e12	−0.495462
$246$	0	0
$247$	−3.33187e13	−2.30597
$248$	0	0
$249$	0	0
$250$	0	0
$251$	1.19272e13	0.755670	0.377835	−	0.925873i	$-0.376669\pi$
$251$	1.19272e13	0.755670	0.377835	+	0.925873i	$0.376669\pi$
$252$	0	0
$253$	−1.13599e13	−0.688986
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−1.12710e11	−0.00627091	−0.00313546	−	0.999995i	$-0.500998\pi$
$257$	−1.12710e11	−0.00627091	−0.00313546	+	0.999995i	$0.500998\pi$
$258$	0	0
$259$	−1.06583e13	−0.568250
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−3.90634e13	−1.91432	−0.957158	−	0.289567i	$-0.906489\pi$
$263$	−3.90634e13	−1.91432	−0.957158	+	0.289567i	$0.906489\pi$
$264$	0	0
$265$	−3.97739e13	−1.86959
$266$	0	0
$267$	0	0
$268$	0	0
$269$	1.46989e13	0.636279	0.318140	−	0.948044i	$-0.396942\pi$
$269$	1.46989e13	0.636279	0.318140	+	0.948044i	$0.396942\pi$
$270$	0	0
$271$	1.17345e13	0.487680	0.243840	−	0.969815i	$-0.421593\pi$
$271$	1.17345e13	0.487680	0.243840	+	0.969815i	$0.421593\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−1.79966e13	−0.690019
$276$	0	0
$277$	−2.37233e13	−0.874049	−0.437024	−	0.899450i	$-0.643968\pi$
$277$	−2.37233e13	−0.874049	−0.437024	+	0.899450i	$0.643968\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−3.44266e13	−1.17222	−0.586111	−	0.810231i	$-0.699342\pi$
$281$	−3.44266e13	−1.17222	−0.586111	+	0.810231i	$0.699342\pi$
$282$	0	0
$283$	−3.62114e13	−1.18582	−0.592911	−	0.805268i	$-0.702021\pi$
$283$	−3.62114e13	−1.18582	−0.592911	+	0.805268i	$0.702021\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−8.11543e12	−0.246015
$288$	0	0
$289$	−3.24683e13	−0.947374
$290$	0	0
$291$	0	0
$292$	0	0
$293$	6.11226e13	1.65360	0.826800	−	0.562497i	$-0.190159\pi$
$293$	6.11226e13	1.65360	0.826800	+	0.562497i	$0.190159\pi$
$294$	0	0
$295$	−1.20311e13	−0.313533
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−7.81941e13	−1.89227
$300$	0	0
$301$	3.01988e13	0.704489
$302$	0	0
$303$	0	0
$304$	0	0
$305$	9.65247e13	2.09406
$306$	0	0
$307$	−2.91668e13	−0.610420	−0.305210	−	0.952285i	$-0.598727\pi$
$307$	−2.91668e13	−0.610420	−0.305210	+	0.952285i	$0.598727\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	3.57264e13	0.696317	0.348158	−	0.937436i	$-0.386807\pi$
$311$	3.57264e13	0.696317	0.348158	+	0.937436i	$0.386807\pi$
$312$	0	0
$313$	4.12304e13	0.775754	0.387877	−	0.921711i	$-0.373209\pi$
$313$	4.12304e13	0.775754	0.387877	+	0.921711i	$0.373209\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−5.66206e13	−0.993456	−0.496728	−	0.867906i	$-0.665465\pi$
$317$	−5.66206e13	−0.993456	−0.496728	+	0.867906i	$0.665465\pi$
$318$	0	0
$319$	−4.89032e13	−0.828875
$320$	0	0
$321$	0	0
$322$	0	0
$323$	2.26596e13	0.358624
$324$	0	0
$325$	−1.23877e14	−1.89510
$326$	0	0
$327$	0	0
$328$	0	0
$329$	2.44910e13	0.350291
$330$	0	0
$331$	−1.05843e13	−0.146422	−0.0732110	−	0.997316i	$-0.523325\pi$
$331$	−1.05843e13	−0.146422	−0.0732110	+	0.997316i	$0.523325\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	8.86492e13	1.14796
$336$	0	0
$337$	−4.49814e13	−0.563726	−0.281863	−	0.959455i	$-0.590952\pi$
$337$	−4.49814e13	−0.563726	−0.281863	+	0.959455i	$0.590952\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	4.58273e13	0.538240
$342$	0	0
$343$	−9.57193e13	−1.08863
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−3.99816e13	−0.426627	−0.213314	−	0.976984i	$-0.568426\pi$
$347$	−3.99816e13	−0.426627	−0.213314	+	0.976984i	$0.568426\pi$
$348$	0	0
$349$	−9.96383e13	−1.03012	−0.515059	−	0.857155i	$-0.672230\pi$
$349$	−9.96383e13	−1.03012	−0.515059	+	0.857155i	$0.672230\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−4.80930e13	−0.467005	−0.233502	−	0.972356i	$-0.575019\pi$
$353$	−4.80930e13	−0.467005	−0.233502	+	0.972356i	$0.575019\pi$
$354$	0	0
$355$	2.88888e14	2.71941
$356$	0	0
$357$	0	0
$358$	0	0
$359$	1.14554e14	1.01389	0.506944	−	0.861979i	$-0.330775\pi$
$359$	1.14554e14	1.01389	0.506944	+	0.861979i	$0.330775\pi$
$360$	0	0
$361$	1.68195e14	1.44386
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−1.71340e14	−1.38436
$366$	0	0
$367$	−1.08181e14	−0.848175	−0.424088	−	0.905621i	$-0.639405\pi$
$367$	−1.08181e14	−0.848175	−0.424088	+	0.905621i	$0.639405\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−1.37289e14	−1.01410
$372$	0	0
$373$	1.14333e14	0.819920	0.409960	−	0.912103i	$-0.365543\pi$
$373$	1.14333e14	0.819920	0.409960	+	0.912103i	$0.365543\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−3.36619e14	−2.27646
$378$	0	0
$379$	−1.85226e14	−1.21671	−0.608355	−	0.793665i	$-0.708170\pi$
$379$	−1.85226e14	−1.21671	−0.608355	+	0.793665i	$0.708170\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	8.25520e13	0.511840	0.255920	−	0.966698i	$-0.417622\pi$
$383$	8.25520e13	0.511840	0.255920	+	0.966698i	$0.417622\pi$
$384$	0	0
$385$	−1.10472e14	−0.665606
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−7.96002e13	−0.453097	−0.226549	−	0.974000i	$-0.572744\pi$
$389$	−7.96002e13	−0.453097	−0.226549	+	0.974000i	$0.572744\pi$
$390$	0	0
$391$	5.31789e13	0.294284
$392$	0	0
$393$	0	0
$394$	0	0
$395$	2.06919e14	1.08272
$396$	0	0
$397$	3.80424e14	1.93606	0.968032	−	0.250827i	$-0.0807026\pi$
$397$	3.80424e14	1.93606	0.968032	+	0.250827i	$0.0807026\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	1.48481e14	0.715118	0.357559	−	0.933891i	$-0.383609\pi$
$401$	1.48481e14	0.715118	0.357559	+	0.933891i	$0.383609\pi$
$402$	0	0
$403$	3.15446e14	1.47825
$404$	0	0
$405$	0	0
$406$	0	0
$407$	8.38687e13	0.372247
$408$	0	0
$409$	−1.46026e13	−0.0630890	−0.0315445	−	0.999502i	$-0.510043\pi$
$409$	−1.46026e13	−0.0630890	−0.0315445	+	0.999502i	$0.510043\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−4.15282e13	−0.170066
$414$	0	0
$415$	5.06743e14	2.02080
$416$	0	0
$417$	0	0
$418$	0	0
$419$	5.44477e13	0.205969	0.102985	−	0.994683i	$-0.467161\pi$
$419$	5.44477e13	0.205969	0.102985	+	0.994683i	$0.467161\pi$
$420$	0	0
$421$	8.29696e13	0.305751	0.152875	−	0.988245i	$-0.451147\pi$
$421$	8.29696e13	0.305751	0.152875	+	0.988245i	$0.451147\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	8.42474e13	0.294725
$426$	0	0
$427$	3.33179e14	1.13586
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−2.22381e14	−0.720233	−0.360117	−	0.932907i	$-0.617263\pi$
$431$	−2.22381e14	−0.720233	−0.360117	+	0.932907i	$0.617263\pi$
$432$	0	0
$433$	1.39447e14	0.440276	0.220138	−	0.975469i	$-0.429349\pi$
$433$	1.39447e14	0.440276	0.220138	+	0.975469i	$0.429349\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	6.68116e14	2.00541
$438$	0	0
$439$	−5.06899e14	−1.48377	−0.741885	−	0.670527i	$-0.766068\pi$
$439$	−5.06899e14	−1.48377	−0.741885	+	0.670527i	$0.766068\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	5.35170e14	1.49029	0.745146	−	0.666902i	$-0.232380\pi$
$443$	5.35170e14	1.49029	0.745146	+	0.666902i	$0.232380\pi$
$444$	0	0
$445$	−3.02213e13	−0.0820982
$446$	0	0
$447$	0	0
$448$	0	0
$449$	3.18755e14	0.824331	0.412166	−	0.911109i	$-0.364772\pi$
$449$	3.18755e14	0.824331	0.412166	+	0.911109i	$0.364772\pi$
$450$	0	0
$451$	6.38592e13	0.161158
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−7.60416e14	−1.82805
$456$	0	0
$457$	3.04115e14	0.713673	0.356836	−	0.934167i	$-0.383855\pi$
$457$	3.04115e14	0.713673	0.356836	+	0.934167i	$0.383855\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−4.21640e14	−0.943163	−0.471581	−	0.881823i	$-0.656317\pi$
$461$	−4.21640e14	−0.943163	−0.471581	+	0.881823i	$0.656317\pi$
$462$	0	0
$463$	−4.08415e14	−0.892086	−0.446043	−	0.895012i	$-0.647167\pi$
$463$	−4.08415e14	−0.892086	−0.446043	+	0.895012i	$0.647167\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−4.16892e14	−0.868522	−0.434261	−	0.900787i	$-0.642990\pi$
$467$	−4.16892e14	−0.868522	−0.434261	+	0.900787i	$0.642990\pi$
$468$	0	0
$469$	3.05995e14	0.622678
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−2.37630e14	−0.461493
$474$	0	0
$475$	1.05845e15	2.00842
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−2.99432e14	−0.542567	−0.271283	−	0.962500i	$-0.587448\pi$
$479$	−2.99432e14	−0.542567	−0.271283	+	0.962500i	$0.587448\pi$
$480$	0	0
$481$	5.77298e14	1.02236
$482$	0	0
$483$	0	0
$484$	0	0
$485$	6.83542e14	1.15661
$486$	0	0
$487$	2.87895e13	0.0476239	0.0238120	−	0.999716i	$-0.492420\pi$
$487$	2.87895e13	0.0476239	0.0238120	+	0.999716i	$0.492420\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−8.28170e14	−1.30970	−0.654849	−	0.755760i	$-0.727268\pi$
$491$	−8.28170e14	−1.30970	−0.654849	+	0.755760i	$0.727268\pi$
$492$	0	0
$493$	2.28930e14	0.354034
$494$	0	0
$495$	0	0
$496$	0	0
$497$	9.97170e14	1.47506
$498$	0	0
$499$	7.80229e14	1.12894	0.564468	−	0.825455i	$-0.309081\pi$
$499$	7.80229e14	1.12894	0.564468	+	0.825455i	$0.309081\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−1.05944e15	−1.46707	−0.733537	−	0.679649i	$-0.762132\pi$
$503$	−1.05944e15	−1.46707	−0.733537	+	0.679649i	$0.762132\pi$
$504$	0	0
$505$	−1.47535e15	−1.99891
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−7.39773e14	−0.959734	−0.479867	−	0.877341i	$-0.659315\pi$
$509$	−7.39773e14	−0.959734	−0.479867	+	0.877341i	$0.659315\pi$
$510$	0	0
$511$	−5.91423e14	−0.750902
$512$	0	0
$513$	0	0
$514$	0	0
$515$	1.76815e15	2.15070
$516$	0	0
$517$	−1.92716e14	−0.229467
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−1.62637e15	−1.85614	−0.928071	−	0.372403i	$-0.878534\pi$
$521$	−1.62637e15	−1.85614	−0.928071	+	0.372403i	$0.878534\pi$
$522$	0	0
$523$	5.12775e14	0.573018	0.286509	−	0.958078i	$-0.407505\pi$
$523$	5.12775e14	0.573018	0.286509	+	0.958078i	$0.407505\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−2.14531e14	−0.229896
$528$	0	0
$529$	6.15162e14	0.645629
$530$	0	0
$531$	0	0
$532$	0	0
$533$	4.39566e14	0.442613
$534$	0	0
$535$	−2.87159e15	−2.83255
$536$	0	0
$537$	0	0
$538$	0	0
$539$	1.85941e14	0.176050
$540$	0	0
$541$	−1.53592e15	−1.42490	−0.712450	−	0.701723i	$-0.752414\pi$
$541$	−1.53592e15	−1.42490	−0.712450	+	0.701723i	$0.752414\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	2.27657e15	2.02815
$546$	0	0
$547$	−1.62787e14	−0.142131	−0.0710656	−	0.997472i	$-0.522640\pi$
$547$	−1.62787e14	−0.142131	−0.0710656	+	0.997472i	$0.522640\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	2.87618e15	2.41258
$552$	0	0
$553$	7.14233e14	0.587289
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−1.63255e15	−1.29021	−0.645107	−	0.764092i	$-0.723187\pi$
$557$	−1.63255e15	−1.29021	−0.645107	+	0.764092i	$0.723187\pi$
$558$	0	0
$559$	−1.63569e15	−1.26747
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−9.06207e14	−0.675198	−0.337599	−	0.941290i	$-0.609615\pi$
$563$	−9.06207e14	−0.675198	−0.337599	+	0.941290i	$0.609615\pi$
$564$	0	0
$565$	−5.06422e14	−0.370038
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−1.96813e15	−1.38336	−0.691681	−	0.722203i	$-0.743129\pi$
$569$	−1.96813e15	−1.38336	−0.691681	+	0.722203i	$0.743129\pi$
$570$	0	0
$571$	4.10069e14	0.282721	0.141361	−	0.989958i	$-0.454852\pi$
$571$	4.10069e14	0.282721	0.141361	+	0.989958i	$0.454852\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	2.48402e15	1.64810
$576$	0	0
$577$	−8.81776e14	−0.573973	−0.286986	−	0.957935i	$-0.592653\pi$
$577$	−8.81776e14	−0.573973	−0.286986	+	0.957935i	$0.592653\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	1.74915e15	1.09612
$582$	0	0
$583$	1.08031e15	0.664311
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−6.65206e13	−0.0393955	−0.0196978	−	0.999806i	$-0.506270\pi$
$587$	−6.65206e13	−0.0393955	−0.0196978	+	0.999806i	$0.506270\pi$
$588$	0	0
$589$	−2.69528e15	−1.56664
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−2.22598e15	−1.24658	−0.623290	−	0.781991i	$-0.714204\pi$
$593$	−2.22598e15	−1.24658	−0.623290	+	0.781991i	$0.714204\pi$
$594$	0	0
$595$	5.17150e14	0.284298
$596$	0	0
$597$	0	0
$598$	0	0
$599$	5.96380e14	0.315991	0.157996	−	0.987440i	$-0.449497\pi$
$599$	5.96380e14	0.315991	0.157996	+	0.987440i	$0.449497\pi$
$600$	0	0
$601$	−1.48139e15	−0.770656	−0.385328	−	0.922780i	$-0.625912\pi$
$601$	−1.48139e15	−0.770656	−0.385328	+	0.922780i	$0.625912\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−2.14423e15	−1.07552
$606$	0	0
$607$	−2.84316e15	−1.40044	−0.700219	−	0.713928i	$-0.746914\pi$
$607$	−2.84316e15	−1.40044	−0.700219	+	0.713928i	$0.746914\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−1.32654e15	−0.630220
$612$	0	0
$613$	1.61248e15	0.752423	0.376212	−	0.926534i	$-0.377227\pi$
$613$	1.61248e15	0.752423	0.376212	+	0.926534i	$0.377227\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−1.46250e15	−0.658456	−0.329228	−	0.944251i	$-0.606788\pi$
$617$	−1.46250e15	−0.658456	−0.329228	+	0.944251i	$0.606788\pi$
$618$	0	0
$619$	2.02458e14	0.0895439	0.0447720	−	0.998997i	$-0.485744\pi$
$619$	2.02458e14	0.0895439	0.0447720	+	0.998997i	$0.485744\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−1.04316e14	−0.0445316
$624$	0	0
$625$	−1.51200e15	−0.634177
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−3.92614e14	−0.158996
$630$	0	0
$631$	−4.30930e15	−1.71493	−0.857463	−	0.514545i	$-0.827961\pi$
$631$	−4.30930e15	−1.71493	−0.857463	+	0.514545i	$0.827961\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−3.29791e15	−1.26760
$636$	0	0
$637$	1.27990e15	0.483513
$638$	0	0
$639$	0	0
$640$	0	0
$641$	6.28934e14	0.229555	0.114777	−	0.993391i	$-0.463385\pi$
$641$	6.28934e14	0.229555	0.114777	+	0.993391i	$0.463385\pi$
$642$	0	0
$643$	3.69823e15	1.32689	0.663443	−	0.748226i	$-0.269094\pi$
$643$	3.69823e15	1.32689	0.663443	+	0.748226i	$0.269094\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−8.93928e14	−0.309977	−0.154988	−	0.987916i	$-0.549534\pi$
$647$	−8.93928e14	−0.309977	−0.154988	+	0.987916i	$0.549534\pi$
$648$	0	0
$649$	3.26779e14	0.111406
$650$	0	0
$651$	0	0
$652$	0	0
$653$	2.62786e15	0.866123	0.433062	−	0.901364i	$-0.357433\pi$
$653$	2.62786e15	0.866123	0.433062	+	0.901364i	$0.357433\pi$
$654$	0	0
$655$	−8.39752e15	−2.72160
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−4.73523e14	−0.148413	−0.0742065	−	0.997243i	$-0.523642\pi$
$659$	−4.73523e14	−0.148413	−0.0742065	+	0.997243i	$0.523642\pi$
$660$	0	0
$661$	1.50188e14	0.0462944	0.0231472	−	0.999732i	$-0.492631\pi$
$661$	1.50188e14	0.0462944	0.0231472	+	0.999732i	$0.492631\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	6.49725e15	1.93736
$666$	0	0
$667$	6.74998e15	1.97975
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−2.62174e15	−0.744073
$672$	0	0
$673$	−3.92035e15	−1.09457	−0.547283	−	0.836948i	$-0.684338\pi$
$673$	−3.92035e15	−1.09457	−0.547283	+	0.836948i	$0.684338\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−1.03347e15	−0.279292	−0.139646	−	0.990201i	$-0.544596\pi$
$677$	−1.03347e15	−0.279292	−0.139646	+	0.990201i	$0.544596\pi$
$678$	0	0
$679$	2.35941e15	0.627365
$680$	0	0
$681$	0	0
$682$	0	0
$683$	3.38320e15	0.870990	0.435495	−	0.900191i	$-0.356573\pi$
$683$	3.38320e15	0.870990	0.435495	+	0.900191i	$0.356573\pi$
$684$	0	0
$685$	4.54183e15	1.15062
$686$	0	0
$687$	0	0
$688$	0	0
$689$	7.43617e15	1.82450
$690$	0	0
$691$	4.94668e15	1.19450	0.597248	−	0.802056i	$-0.296261\pi$
$691$	4.94668e15	1.19450	0.597248	+	0.802056i	$0.296261\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	4.77334e15	1.11662
$696$	0	0
$697$	−2.98943e14	−0.0688349
$698$	0	0
$699$	0	0
$700$	0	0
$701$	2.16125e15	0.482232	0.241116	−	0.970496i	$-0.422487\pi$
$701$	2.16125e15	0.482232	0.241116	+	0.970496i	$0.422487\pi$
$702$	0	0
$703$	−4.93263e15	−1.08349
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−5.09255e15	−1.08425
$708$	0	0
$709$	−5.70891e15	−1.19674	−0.598369	−	0.801221i	$-0.704184\pi$
$709$	−5.70891e15	−1.19674	−0.598369	+	0.801221i	$0.704184\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−6.32542e15	−1.28557
$714$	0	0
$715$	5.98361e15	1.19751
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−8.22653e15	−1.59664	−0.798321	−	0.602232i	$-0.794278\pi$
$719$	−8.22653e15	−1.59664	−0.798321	+	0.602232i	$0.794278\pi$
$720$	0	0
$721$	6.10322e15	1.16658
$722$	0	0
$723$	0	0
$724$	0	0
$725$	1.06935e16	1.98272
$726$	0	0
$727$	2.98100e15	0.544405	0.272203	−	0.962240i	$-0.412248\pi$
$727$	2.98100e15	0.544405	0.272203	+	0.962240i	$0.412248\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	1.11242e15	0.197116
$732$	0	0
$733$	1.08275e16	1.88997	0.944985	−	0.327115i	$-0.106076\pi$
$733$	1.08275e16	1.88997	0.944985	+	0.327115i	$0.106076\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−2.40783e15	−0.407901
$738$	0	0
$739$	−2.45500e15	−0.409739	−0.204869	−	0.978789i	$-0.565677\pi$
$739$	−2.45500e15	−0.409739	−0.204869	+	0.978789i	$0.565677\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−1.17874e16	−1.90976	−0.954880	−	0.296993i	$-0.904016\pi$
$743$	−1.17874e16	−1.90976	−0.954880	+	0.296993i	$0.904016\pi$
$744$	0	0
$745$	1.86602e16	2.97890
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−9.91202e15	−1.53643
$750$	0	0
$751$	7.70607e15	1.17710	0.588550	−	0.808461i	$-0.299699\pi$
$751$	7.70607e15	1.17710	0.588550	+	0.808461i	$0.299699\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	1.75483e16	2.60332
$756$	0	0
$757$	−4.81274e15	−0.703663	−0.351832	−	0.936063i	$-0.614441\pi$
$757$	−4.81274e15	−0.703663	−0.351832	+	0.936063i	$0.614441\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−1.54814e15	−0.219884	−0.109942	−	0.993938i	$-0.535067\pi$
$761$	−1.54814e15	−0.219884	−0.109942	+	0.993938i	$0.535067\pi$
$762$	0	0
$763$	7.85813e15	1.10011
$764$	0	0
$765$	0	0
$766$	0	0
$767$	2.24934e15	0.305971
$768$	0	0
$769$	−3.08434e15	−0.413587	−0.206794	−	0.978385i	$-0.566303\pi$
$769$	−3.08434e15	−0.413587	−0.206794	+	0.978385i	$0.566303\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−8.21442e15	−1.07051	−0.535253	−	0.844692i	$-0.679784\pi$
$773$	−8.21442e15	−1.07051	−0.535253	+	0.844692i	$0.679784\pi$
$774$	0	0
$775$	−1.00209e16	−1.28750
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−3.75579e15	−0.469079
$780$	0	0
$781$	−7.84659e15	−0.966274
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−1.21203e16	−1.45121
$786$	0	0
$787$	3.31392e15	0.391274	0.195637	−	0.980676i	$-0.437323\pi$
$787$	3.31392e15	0.391274	0.195637	+	0.980676i	$0.437323\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−1.74804e15	−0.200715
$792$	0	0
$793$	−1.80464e16	−2.04356
$794$	0	0
$795$	0	0
$796$	0	0
$797$	2.67164e15	0.294278	0.147139	−	0.989116i	$-0.452994\pi$
$797$	2.67164e15	0.294278	0.147139	+	0.989116i	$0.452994\pi$
$798$	0	0
$799$	9.02161e14	0.0980114
$800$	0	0
$801$	0	0
$802$	0	0
$803$	4.65382e15	0.491897
$804$	0	0
$805$	1.52481e16	1.58978
$806$	0	0
$807$	0	0
$808$	0	0
$809$	7.98563e15	0.810200	0.405100	−	0.914272i	$-0.367237\pi$
$809$	7.98563e15	0.810200	0.405100	+	0.914272i	$0.367237\pi$
$810$	0	0
$811$	1.90733e16	1.90902	0.954510	−	0.298179i	$-0.0963792\pi$
$811$	1.90733e16	1.90902	0.954510	+	0.298179i	$0.0963792\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	1.50948e16	1.47048
$816$	0	0
$817$	1.39759e16	1.34326
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−1.59813e16	−1.49529	−0.747643	−	0.664101i	$-0.768814\pi$
$821$	−1.59813e16	−1.49529	−0.747643	+	0.664101i	$0.768814\pi$
$822$	0	0
$823$	−6.21210e15	−0.573508	−0.286754	−	0.958004i	$-0.592576\pi$
$823$	−6.21210e15	−0.573508	−0.286754	+	0.958004i	$0.592576\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	1.65883e16	1.49115	0.745577	−	0.666420i	$-0.232174\pi$
$827$	1.65883e16	1.49115	0.745577	+	0.666420i	$0.232174\pi$
$828$	0	0
$829$	2.78832e15	0.247339	0.123669	−	0.992323i	$-0.460534\pi$
$829$	2.78832e15	0.247339	0.123669	+	0.992323i	$0.460534\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−8.70442e14	−0.0751956
$834$	0	0
$835$	7.25601e15	0.618617
$836$	0	0
$837$	0	0
$838$	0	0
$839$	3.20857e15	0.266453	0.133227	−	0.991086i	$-0.457466\pi$
$839$	3.20857e15	0.266453	0.133227	+	0.991086i	$0.457466\pi$
$840$	0	0
$841$	1.68575e16	1.38171
$842$	0	0
$843$	0	0
$844$	0	0
$845$	2.22582e16	1.77737
$846$	0	0
$847$	−7.40134e15	−0.583380
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−1.15762e16	−0.889103
$852$	0	0
$853$	−8.05451e14	−0.0610688	−0.0305344	−	0.999534i	$-0.509721\pi$
$853$	−8.05451e14	−0.0610688	−0.0305344	+	0.999534i	$0.509721\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−2.33561e15	−0.172586	−0.0862932	−	0.996270i	$-0.527502\pi$
$857$	−2.33561e15	−0.172586	−0.0862932	+	0.996270i	$0.527502\pi$
$858$	0	0
$859$	−1.27674e16	−0.931411	−0.465705	−	0.884940i	$-0.654199\pi$
$859$	−1.27674e16	−0.931411	−0.465705	+	0.884940i	$0.654199\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	9.84316e15	0.699963	0.349982	−	0.936757i	$-0.386188\pi$
$863$	9.84316e15	0.699963	0.349982	+	0.936757i	$0.386188\pi$
$864$	0	0
$865$	−1.32645e16	−0.931329
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−5.62020e15	−0.384718
$870$	0	0
$871$	−1.65740e16	−1.12028
$872$	0	0
$873$	0	0
$874$	0	0
$875$	5.35390e15	0.352879
$876$	0	0
$877$	−1.92649e16	−1.25392	−0.626959	−	0.779052i	$-0.715701\pi$
$877$	−1.92649e16	−1.25392	−0.626959	+	0.779052i	$0.715701\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−1.81889e16	−1.15462	−0.577309	−	0.816526i	$-0.695897\pi$
$881$	−1.81889e16	−1.15462	−0.577309	+	0.816526i	$0.695897\pi$
$882$	0	0
$883$	7.25519e15	0.454847	0.227423	−	0.973796i	$-0.426970\pi$
$883$	7.25519e15	0.454847	0.227423	+	0.973796i	$0.426970\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	1.50496e15	0.0920332	0.0460166	−	0.998941i	$-0.485347\pi$
$887$	1.50496e15	0.0920332	0.0460166	+	0.998941i	$0.485347\pi$
$888$	0	0
$889$	−1.13835e16	−0.687572
$890$	0	0
$891$	0	0
$892$	0	0
$893$	1.13344e16	0.667904
$894$	0	0
$895$	4.61755e15	0.268773
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−2.72304e16	−1.54659
$900$	0	0
$901$	−5.05725e15	−0.283745
$902$	0	0
$903$	0	0
$904$	0	0
$905$	4.68952e16	2.56780
$906$	0	0
$907$	−1.88696e16	−1.02076	−0.510379	−	0.859949i	$-0.670495\pi$
$907$	−1.88696e16	−1.02076	−0.510379	+	0.859949i	$0.670495\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−1.29750e16	−0.685102	−0.342551	−	0.939499i	$-0.611291\pi$
$911$	−1.29750e16	−0.685102	−0.342551	+	0.939499i	$0.611291\pi$
$912$	0	0
$913$	−1.37638e16	−0.718041
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−2.89861e16	−1.47625
$918$	0	0
$919$	9.68239e15	0.487245	0.243622	−	0.969870i	$-0.421664\pi$
$919$	9.68239e15	0.487245	0.243622	+	0.969870i	$0.421664\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−5.40109e16	−2.65382
$924$	0	0
$925$	−1.83393e16	−0.890435
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−9.03404e15	−0.428347	−0.214173	−	0.976796i	$-0.568706\pi$
$929$	−9.03404e15	−0.428347	−0.214173	+	0.976796i	$0.568706\pi$
$930$	0	0
$931$	−1.09359e16	−0.512424
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−4.06938e15	−0.186236
$936$	0	0
$937$	−2.55497e16	−1.15563	−0.577814	−	0.816168i	$-0.696094\pi$
$937$	−2.55497e16	−1.15563	−0.577814	+	0.816168i	$0.696094\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	1.38962e16	0.613976	0.306988	−	0.951713i	$-0.400679\pi$
$941$	1.38962e16	0.613976	0.306988	+	0.951713i	$0.400679\pi$
$942$	0	0
$943$	−8.81430e15	−0.384923
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−1.96888e16	−0.840028	−0.420014	−	0.907518i	$-0.637975\pi$
$947$	−1.96888e16	−0.840028	−0.420014	+	0.907518i	$0.637975\pi$
$948$	0	0
$949$	3.20339e16	1.35097
$950$	0	0
$951$	0	0
$952$	0	0
$953$	4.96679e15	0.204675	0.102337	−	0.994750i	$-0.467368\pi$
$953$	4.96679e15	0.204675	0.102337	+	0.994750i	$0.467368\pi$
$954$	0	0
$955$	−5.29655e16	−2.15762
$956$	0	0
$957$	0	0
$958$	0	0
$959$	1.56772e16	0.624119
$960$	0	0
$961$	1.09165e14	0.00429641
$962$	0	0
$963$	0	0
$964$	0	0
$965$	2.07929e16	0.799861
$966$	0	0
$967$	−2.12822e15	−0.0809416	−0.0404708	−	0.999181i	$-0.512886\pi$
$967$	−2.12822e15	−0.0809416	−0.0404708	+	0.999181i	$0.512886\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	1.41899e16	0.527563	0.263781	−	0.964582i	$-0.415030\pi$
$971$	1.41899e16	0.527563	0.263781	+	0.964582i	$0.415030\pi$
$972$	0	0
$973$	1.64764e16	0.605676
$974$	0	0
$975$	0	0
$976$	0	0
$977$	4.28897e14	0.0154146	0.00770731	−	0.999970i	$-0.497547\pi$
$977$	4.28897e14	0.0154146	0.00770731	+	0.999970i	$0.497547\pi$
$978$	0	0
$979$	8.20851e14	0.0291716
$980$	0	0
$981$	0	0
$982$	0	0
$983$	3.77409e16	1.31150	0.655750	−	0.754978i	$-0.272353\pi$
$983$	3.77409e16	1.31150	0.655750	+	0.754978i	$0.272353\pi$
$984$	0	0
$985$	−3.51296e16	−1.20719
$986$	0	0
$987$	0	0
$988$	0	0
$989$	3.27994e16	1.10227
$990$	0	0
$991$	−1.52204e16	−0.505851	−0.252925	−	0.967486i	$-0.581393\pi$
$991$	−1.52204e16	−0.505851	−0.252925	+	0.967486i	$0.581393\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	4.00936e15	0.130331
$996$	0	0
$997$	−1.15391e16	−0.370980	−0.185490	−	0.982646i	$-0.559387\pi$
$997$	−1.15391e16	−0.370980	−0.185490	+	0.982646i	$0.559387\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	144.12.a.t.1.3		3
3.2	odd	2		144.12.a.s.1.1		3
4.3	odd	2		72.12.a.h.1.3	yes	3
12.11	even	2		72.12.a.g.1.1	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
72.12.a.g.1.1	✓	3	12.11	even	2
72.12.a.h.1.3	yes	3	4.3	odd	2
144.12.a.s.1.1		3	3.2	odd	2
144.12.a.t.1.3		3	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:	\( N \)	\(=\)	\( 144 = 2^{4} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	144.a (trivial)

\(f(q)\)	\(=\)	\(q+10562.2 q^{5} +36458.0 q^{7} +O(q^{10})\)	\(q+10562.2 q^{5} +36458.0 q^{7} -286883. q^{11} -1.97472e6 q^{13} +1.34298e6 q^{17} +1.68726e7 q^{19} +3.95976e7 q^{23} +6.27316e7 q^{25} +1.70464e8 q^{29} -1.59742e8 q^{31} +3.85076e8 q^{35} -2.92345e8 q^{37} -2.22597e8 q^{41} +8.28318e8 q^{43} +6.71760e8 q^{47} -6.48141e8 q^{49} -3.76569e9 q^{53} -3.03011e9 q^{55} -1.13907e9 q^{59} +9.13871e9 q^{61} -2.08573e10 q^{65} +8.39308e9 q^{67} +2.73512e10 q^{71} -1.62220e10 q^{73} -1.04592e10 q^{77} +1.95906e10 q^{79} +4.79771e10 q^{83} +1.41848e10 q^{85} -2.86128e9 q^{89} -7.19942e10 q^{91} +1.78212e11 q^{95} +6.47159e10 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 1584 q^{5} + 17796 q^{7}+O(q^{10}) \) 3 * q + 1584 * q^5 + 17796 * q^7	\( 3 q + 1584 q^{5} + 17796 q^{7} - 238272 q^{11} - 49458 q^{13} - 766368 q^{17} + 4590936 q^{19} + 13188480 q^{23} + 18034281 q^{25} + 8005200 q^{29} - 10185564 q^{31} + 235065408 q^{35} - 73215510 q^{37} - 1032516960 q^{41} - 28458216 q^{43} + 2840606592 q^{47} - 109741845 q^{49} - 6755828784 q^{53} + 846590976 q^{55} + 15928599936 q^{59} + 5136563970 q^{61} - 37415401248 q^{65} + 6119899728 q^{67} + 58699811328 q^{71} + 2561705778 q^{73} - 86561454336 q^{77} + 17842143972 q^{79} + 134316444096 q^{83} + 43096069632 q^{85} - 152402442048 q^{89} + 56268770664 q^{91} + 207495767424 q^{95} + 106677733482 q^{97}+O(q^{100}) \) 3 * q + 1584 * q^5 + 17796 * q^7 - 238272 * q^11 - 49458 * q^13 - 766368 * q^17 + 4590936 * q^19 + 13188480 * q^23 + 18034281 * q^25 + 8005200 * q^29 - 10185564 * q^31 + 235065408 * q^35 - 73215510 * q^37 - 1032516960 * q^41 - 28458216 * q^43 + 2840606592 * q^47 - 109741845 * q^49 - 6755828784 * q^53 + 846590976 * q^55 + 15928599936 * q^59 + 5136563970 * q^61 - 37415401248 * q^65 + 6119899728 * q^67 + 58699811328 * q^71 + 2561705778 * q^73 - 86561454336 * q^77 + 17842143972 * q^79 + 134316444096 * q^83 + 43096069632 * q^85 - 152402442048 * q^89 + 56268770664 * q^91 + 207495767424 * q^95 + 106677733482 * q^97

Introduction

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