LMFDB - Embedded newform 144.11.g.e.127.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [144,11,Mod(127,144)]

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([1, 0, 0]))

N = Newforms(chi, 11, names="a")

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

chi := DirichletCharacter("144.127");

S:= CuspForms(chi, 11);

N := Newforms(S);

Level:	$ N $	$=$	$ 144 = 2^{4} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 11 $
Character orbit:	$[\chi]$	$=$	144.g (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:	$91.4914443850$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{2545})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} + 637x^{2} + 636x + 404496 $ x^4 - x^3 + 637x^2 + 636x + 404496
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{12}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 48)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			127.2
Root			$-12.3620 - 21.4116i$ of defining polynomial
Character	$\chi$	$=$	144.127
Dual form			144.11.g.e.127.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1960.75 q^{5} +28003.4i q^{7} +O(q^{10})$	$q-1960.75 q^{5} +28003.4i q^{7} +43532.8i q^{11} -139927. q^{13} -2.12441e6 q^{17} +2.98220e6i q^{19} +4.60241e6i q^{23} -5.92108e6 q^{25} +2.90324e7 q^{29} +1.52638e7i q^{31} -5.49077e7i q^{35} -3.18138e7 q^{37} +2.12718e8 q^{41} +1.02316e8i q^{43} -3.05477e8i q^{47} -5.01715e8 q^{49} -2.21185e8 q^{53} -8.53569e7i q^{55} +6.12527e8i q^{59} -6.10376e8 q^{61} +2.74362e8 q^{65} +5.40101e7i q^{67} -2.86449e9i q^{71} +2.92732e9 q^{73} -1.21907e9 q^{77} -2.00200e9i q^{79} +3.29668e9i q^{83} +4.16543e9 q^{85} -8.15974e9 q^{89} -3.91843e9i q^{91} -5.84735e9i q^{95} -5.41238e9 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 3000 q^{5}+O(q^{10}) $ 4 * q - 3000 * q^5	$ 4 q - 3000 q^{5} + 864136 q^{13} + 55128 q^{17} - 30948820 q^{25} + 80790312 q^{29} - 212511640 q^{37} + 544453272 q^{41} - 1845182972 q^{49} + 888804072 q^{53} - 208335256 q^{61} + 1075819920 q^{65} + 4881669448 q^{73} - 18371224512 q^{77} + 10313912880 q^{85} - 21540382536 q^{89} + 14965768456 q^{97}+O(q^{100}) $ 4 * q - 3000 * q^5 + 864136 * q^13 + 55128 * q^17 - 30948820 * q^25 + 80790312 * q^29 - 212511640 * q^37 + 544453272 * q^41 - 1845182972 * q^49 + 888804072 * q^53 - 208335256 * q^61 + 1075819920 * q^65 + 4881669448 * q^73 - 18371224512 * q^77 + 10313912880 * q^85 - 21540382536 * q^89 + 14965768456 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/144\mathbb{Z}\right)^\times$.

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

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
$5$	−1960.75	−0.627441	−0.313720	−	0.949515i	$-0.601575\pi$
$5$	−1960.75	−0.627441	−0.313720	+	0.949515i	$0.601575\pi$
$6$	0	0
$7$	28003.4i	1.66618i	0.553141	+	0.833088i	$0.313429\pi$
$7$	28003.4i	1.66618i	−0.553141	+	0.833088i	$0.686571\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	43532.8i	0.270304i	0.990825	+	0.135152i	$0.0431523\pi$
$11$	43532.8i	0.270304i	−0.990825	+	0.135152i	$0.956848\pi$
$12$	0	0
$13$	−139927.	−0.376864	−0.188432	−	0.982086i	$-0.560341\pi$
$13$	−139927.	−0.376864	−0.188432	+	0.982086i	$0.560341\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−2.12441e6	−1.49621	−0.748106	−	0.663580i	$-0.769036\pi$
$17$	−2.12441e6	−1.49621	−0.748106	+	0.663580i	$0.769036\pi$
$18$	0	0
$19$	2.98220e6i	1.20439i	0.798347	+	0.602197i	$0.205708\pi$
$19$	2.98220e6i	1.20439i	−0.798347	+	0.602197i	$0.794292\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	4.60241e6i	0.715066i	0.933900	+	0.357533i	$0.116382\pi$
$23$	4.60241e6i	0.715066i	−0.933900	+	0.357533i	$0.883618\pi$
$24$	0	0
$25$	−5.92108e6	−0.606318
$26$	0	0
$27$	0	0
$28$	0	0
$29$	2.90324e7	1.41545	0.707723	−	0.706490i	$-0.249722\pi$
$29$	2.90324e7	1.41545	0.707723	+	0.706490i	$0.249722\pi$
$30$	0	0
$31$	1.52638e7i	0.533156i	0.963813	+	0.266578i	$0.0858929\pi$
$31$	1.52638e7i	0.533156i	−0.963813	+	0.266578i	$0.914107\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 5.49077e7i	− 1.04543i
$36$	0	0
$37$	−3.18138e7	−0.458783	−0.229392	−	0.973334i	$-0.573674\pi$
$37$	−3.18138e7	−0.458783	−0.229392	+	0.973334i	$0.573674\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	2.12718e8	1.83605	0.918024	−	0.396524i	$-0.129784\pi$
$41$	2.12718e8	1.83605	0.918024	+	0.396524i	$0.129784\pi$
$42$	0	0
$43$	1.02316e8i	0.695989i	0.937497	+	0.347995i	$0.113137\pi$
$43$	1.02316e8i	0.695989i	−0.937497	+	0.347995i	$0.886863\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 3.05477e8i	− 1.33195i	−0.745972	−	0.665977i	$-0.768015\pi$
$47$	− 3.05477e8i	− 1.33195i	0.745972	−	0.665977i	$-0.231985\pi$
$48$	0	0
$49$	−5.01715e8	−1.77614
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−2.21185e8	−0.528903	−0.264451	−	0.964399i	$-0.585191\pi$
$53$	−2.21185e8	−0.528903	−0.264451	+	0.964399i	$0.585191\pi$
$54$	0	0
$55$	− 8.53569e7i	− 0.169600i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	6.12527e8i	0.856772i	0.903596	+	0.428386i	$0.140918\pi$
$59$	6.12527e8i	0.856772i	−0.903596	+	0.428386i	$0.859082\pi$
$60$	0	0
$61$	−6.10376e8	−0.722684	−0.361342	−	0.932433i	$-0.617681\pi$
$61$	−6.10376e8	−0.722684	−0.361342	+	0.932433i	$0.617681\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	2.74362e8	0.236460
$66$	0	0
$67$	5.40101e7i	0.0400038i	0.999800	+	0.0200019i	$0.00636722\pi$
$67$	5.40101e7i	0.0400038i	−0.999800	+	0.0200019i	$0.993633\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 2.86449e9i	− 1.58765i	−0.608144	−	0.793827i	$-0.708086\pi$
$71$	− 2.86449e9i	− 1.58765i	0.608144	−	0.793827i	$-0.291914\pi$
$72$	0	0
$73$	2.92732e9	1.41207	0.706033	−	0.708178i	$-0.250483\pi$
$73$	2.92732e9	1.41207	0.706033	+	0.708178i	$0.250483\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−1.21907e9	−0.450374
$78$	0	0
$79$	− 2.00200e9i	− 0.650620i	−0.945607	−	0.325310i	$-0.894531\pi$
$79$	− 2.00200e9i	− 0.650620i	0.945607	−	0.325310i	$-0.105469\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	3.29668e9i	0.836924i	0.908234	+	0.418462i	$0.137431\pi$
$83$	3.29668e9i	0.836924i	−0.908234	+	0.418462i	$0.862569\pi$
$84$	0	0
$85$	4.16543e9	0.938784
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−8.15974e9	−1.46126	−0.730628	−	0.682776i	$-0.760772\pi$
$89$	−8.15974e9	−1.46126	−0.730628	+	0.682776i	$0.760772\pi$
$90$	0	0
$91$	− 3.91843e9i	− 0.627922i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	− 5.84735e9i	− 0.755686i
$96$	0	0
$97$	−5.41238e9	−0.630274	−0.315137	−	0.949046i	$-0.602051\pi$
$97$	−5.41238e9	−0.630274	−0.315137	+	0.949046i	$0.602051\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−1.69718e10	−1.61481	−0.807405	−	0.589997i	$-0.799129\pi$
$101$	−1.69718e10	−1.61481	−0.807405	+	0.589997i	$0.799129\pi$
$102$	0	0
$103$	− 1.12732e10i	− 0.972432i	−0.873839	−	0.486216i	$-0.838377\pi$
$103$	− 1.12732e10i	− 0.972432i	0.873839	−	0.486216i	$-0.161623\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 8.02650e9i	− 0.572278i	−0.958188	−	0.286139i	$-0.907628\pi$
$107$	− 8.02650e9i	− 0.572278i	0.958188	−	0.286139i	$-0.0923719\pi$
$108$	0	0
$109$	−2.15037e10	−1.39759	−0.698797	−	0.715320i	$-0.746281\pi$
$109$	−2.15037e10	−1.39759	−0.698797	+	0.715320i	$0.746281\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	5.08540e9	0.276015	0.138008	−	0.990431i	$-0.455930\pi$
$113$	5.08540e9	0.276015	0.138008	+	0.990431i	$0.455930\pi$
$114$	0	0
$115$	− 9.02419e9i	− 0.448662i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 5.94906e10i	− 2.49295i
$120$	0	0
$121$	2.40423e10	0.926936
$122$	0	0
$123$	0	0
$124$	0	0
$125$	3.07577e10	1.00787
$126$	0	0
$127$	3.17676e10i	0.961537i	0.876847	+	0.480769i	$0.159642\pi$
$127$	3.17676e10i	0.961537i	−0.876847	+	0.480769i	$0.840358\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 6.33920e10i	− 1.64315i	−0.570098	−	0.821577i	$-0.693095\pi$
$131$	− 6.33920e10i	− 1.64315i	0.570098	−	0.821577i	$-0.306905\pi$
$132$	0	0
$133$	−8.35117e10	−2.00673
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−4.99756e9	−0.103551	−0.0517756	−	0.998659i	$-0.516488\pi$
$137$	−4.99756e9	−0.103551	−0.0517756	+	0.998659i	$0.516488\pi$
$138$	0	0
$139$	− 8.93247e10i	− 1.72146i	−0.509059	−	0.860732i	$-0.670007\pi$
$139$	− 8.93247e10i	− 1.72146i	0.509059	−	0.860732i	$-0.329993\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 6.09141e9i	− 0.101868i
$144$	0	0
$145$	−5.69254e10	−0.888109
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−1.01995e11	−1.38882	−0.694410	−	0.719579i	$-0.744335\pi$
$149$	−1.01995e11	−1.38882	−0.694410	+	0.719579i	$0.744335\pi$
$150$	0	0
$151$	1.05116e11i	1.33901i	0.742809	+	0.669504i	$0.233493\pi$
$151$	1.05116e11i	1.33901i	−0.742809	+	0.669504i	$0.766507\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 2.99285e10i	− 0.334523i
$156$	0	0
$157$	1.37587e11	1.44237	0.721187	−	0.692740i	$-0.243597\pi$
$157$	1.37587e11	1.44237	0.721187	+	0.692740i	$0.243597\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−1.28883e11	−1.19143
$162$	0	0
$163$	− 1.87748e11i	− 1.63169i	−0.578271	−	0.815845i	$-0.696272\pi$
$163$	− 1.87748e11i	− 1.63169i	0.578271	−	0.815845i	$-0.303728\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 1.51202e11i	− 1.16406i	−0.813167	−	0.582030i	$-0.802259\pi$
$167$	− 1.51202e11i	− 1.16406i	0.813167	−	0.582030i	$-0.197741\pi$
$168$	0	0
$169$	−1.18279e11	−0.857973
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−1.32108e11	−0.852510	−0.426255	−	0.904603i	$-0.640167\pi$
$173$	−1.32108e11	−0.852510	−0.426255	+	0.904603i	$0.640167\pi$
$174$	0	0
$175$	− 1.65810e11i	− 1.01023i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	4.21702e10i	0.229478i	0.993396	+	0.114739i	$0.0366031\pi$
$179$	4.21702e10i	0.229478i	−0.993396	+	0.114739i	$0.963397\pi$
$180$	0	0
$181$	1.69890e11	0.874530	0.437265	−	0.899333i	$-0.355947\pi$
$181$	1.69890e11	0.874530	0.437265	+	0.899333i	$0.355947\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	6.23790e10	0.287859
$186$	0	0
$187$	− 9.24812e10i	− 0.404432i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 2.64701e11i	− 1.04133i	−0.853760	−	0.520666i	$-0.825684\pi$
$191$	− 2.64701e11i	− 1.04133i	0.853760	−	0.520666i	$-0.174316\pi$
$192$	0	0
$193$	3.57937e11	1.33666	0.668329	−	0.743866i	$-0.267010\pi$
$193$	3.57937e11	1.33666	0.668329	+	0.743866i	$0.267010\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	9.62491e10	0.324389	0.162194	−	0.986759i	$-0.448143\pi$
$197$	9.62491e10	0.324389	0.162194	+	0.986759i	$0.448143\pi$
$198$	0	0
$199$	1.65299e11i	0.529670i	0.964294	+	0.264835i	$0.0853175\pi$
$199$	1.65299e11i	0.529670i	−0.964294	+	0.264835i	$0.914683\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	8.13007e11i	2.35838i
$204$	0	0
$205$	−4.17086e11	−1.15201
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−1.29823e11	−0.325553
$210$	0	0
$211$	3.71076e10i	0.0887259i	0.999015	+	0.0443630i	$0.0141258\pi$
$211$	3.71076e10i	0.0887259i	−0.999015	+	0.0443630i	$0.985874\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	− 2.00617e11i	− 0.436692i
$216$	0	0
$217$	−4.27438e11	−0.888331
$218$	0	0
$219$	0	0
$220$	0	0
$221$	2.97262e11	0.563868
$222$	0	0
$223$	1.05438e12i	1.91193i	0.293487	+	0.955963i	$0.405184\pi$
$223$	1.05438e12i	1.91193i	−0.293487	+	0.955963i	$0.594816\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	9.28471e10i	0.154042i	0.997029	+	0.0770210i	$0.0245408\pi$
$227$	9.28471e10i	0.154042i	−0.997029	+	0.0770210i	$0.975459\pi$
$228$	0	0
$229$	2.71477e11	0.431077	0.215538	−	0.976495i	$-0.430849\pi$
$229$	2.71477e11	0.431077	0.215538	+	0.976495i	$0.430849\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	8.90552e11	1.29682	0.648410	−	0.761291i	$-0.275434\pi$
$233$	8.90552e11	1.29682	0.648410	+	0.761291i	$0.275434\pi$
$234$	0	0
$235$	5.98965e11i	0.835722i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	1.25803e10i	0.0161326i	0.999967	+	0.00806628i	$0.00256760\pi$
$239$	1.25803e10i	0.0161326i	−0.999967	+	0.00806628i	$0.997432\pi$
$240$	0	0
$241$	−9.96269e10	−0.122544	−0.0612719	−	0.998121i	$-0.519516\pi$
$241$	−9.96269e10	−0.122544	−0.0612719	+	0.998121i	$0.519516\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	9.83740e11	1.11442
$246$	0	0
$247$	− 4.17290e11i	− 0.453893i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	1.60779e12i	1.61384i	0.590659	+	0.806921i	$0.298868\pi$
$251$	1.60779e12i	1.61384i	−0.590659	+	0.806921i	$0.701132\pi$
$252$	0	0
$253$	−2.00356e11	−0.193285
$254$	0	0
$255$	0	0
$256$	0	0
$257$	1.00716e12	0.898320	0.449160	−	0.893451i	$-0.351723\pi$
$257$	1.00716e12	0.898320	0.449160	+	0.893451i	$0.351723\pi$
$258$	0	0
$259$	− 8.90896e11i	− 0.764413i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	1.16466e11i	0.0925594i	0.998929	+	0.0462797i	$0.0147365\pi$
$263$	1.16466e11i	0.0925594i	−0.998929	+	0.0462797i	$0.985263\pi$
$264$	0	0
$265$	4.33688e11	0.331855
$266$	0	0
$267$	0	0
$268$	0	0
$269$	2.19817e11	0.156063	0.0780313	−	0.996951i	$-0.475137\pi$
$269$	2.19817e11	0.156063	0.0780313	+	0.996951i	$0.475137\pi$
$270$	0	0
$271$	4.67966e11i	0.320161i	0.987104	+	0.160080i	$0.0511753\pi$
$271$	4.67966e11i	0.320161i	−0.987104	+	0.160080i	$0.948825\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 2.57761e11i	− 0.163890i
$276$	0	0
$277$	1.67183e12	1.02516	0.512580	−	0.858639i	$-0.328690\pi$
$277$	1.67183e12	1.02516	0.512580	+	0.858639i	$0.328690\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−1.59078e12	−0.907982	−0.453991	−	0.891006i	$-0.650000\pi$
$281$	−1.59078e12	−0.907982	−0.453991	+	0.891006i	$0.650000\pi$
$282$	0	0
$283$	2.30728e12i	1.27107i	0.772074	+	0.635533i	$0.219219\pi$
$283$	2.30728e12i	1.27107i	−0.772074	+	0.635533i	$0.780781\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	5.95682e12i	3.05918i
$288$	0	0
$289$	2.49711e12	1.23865
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−3.93730e12	−1.82331	−0.911655	−	0.410957i	$-0.865195\pi$
$293$	−3.93730e12	−1.82331	−0.911655	+	0.410957i	$0.865195\pi$
$294$	0	0
$295$	− 1.20101e12i	− 0.537573i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 6.44002e11i	− 0.269483i
$300$	0	0
$301$	−2.86521e12	−1.15964
$302$	0	0
$303$	0	0
$304$	0	0
$305$	1.19680e12	0.453441
$306$	0	0
$307$	− 1.46935e10i	− 0.00538808i	−0.999996	−	0.00269404i	$-0.999142\pi$
$307$	− 1.46935e10i	− 0.00538808i	0.999996	−	0.00269404i	$-0.000857541\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	1.88632e12i	0.648356i	0.945996	+	0.324178i	$0.105088\pi$
$311$	1.88632e12i	0.648356i	−0.945996	+	0.324178i	$0.894912\pi$
$312$	0	0
$313$	4.10199e12	1.36544	0.682721	−	0.730679i	$-0.260797\pi$
$313$	4.10199e12	1.36544	0.682721	+	0.730679i	$0.260797\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−3.75215e11	−0.117215	−0.0586077	−	0.998281i	$-0.518666\pi$
$317$	−3.75215e11	−0.117215	−0.0586077	+	0.998281i	$0.518666\pi$
$318$	0	0
$319$	1.26386e12i	0.382601i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 6.33540e12i	− 1.80203i
$324$	0	0
$325$	8.28519e11	0.228500
$326$	0	0
$327$	0	0
$328$	0	0
$329$	8.55440e12	2.21927
$330$	0	0
$331$	− 1.70388e12i	− 0.428843i	−0.976741	−	0.214422i	$-0.931213\pi$
$331$	− 1.70388e12i	− 0.428843i	0.976741	−	0.214422i	$-0.0687867\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	− 1.05900e11i	− 0.0251000i
$336$	0	0
$337$	−7.60571e12	−1.74981	−0.874903	−	0.484298i	$-0.839075\pi$
$337$	−7.60571e12	−1.74981	−0.874903	+	0.484298i	$0.839075\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−6.64475e11	−0.144114
$342$	0	0
$343$	− 6.13947e12i	− 1.29318i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 3.05813e12i	− 0.607867i	−0.952693	−	0.303933i	$-0.901700\pi$
$347$	− 3.05813e12i	− 0.607867i	0.952693	−	0.303933i	$-0.0983000\pi$
$348$	0	0
$349$	−1.03496e12	−0.199892	−0.0999458	−	0.994993i	$-0.531867\pi$
$349$	−1.03496e12	−0.199892	−0.0999458	+	0.994993i	$0.531867\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−3.06228e12	−0.558690	−0.279345	−	0.960191i	$-0.590117\pi$
$353$	−3.06228e12	−0.558690	−0.279345	+	0.960191i	$0.590117\pi$
$354$	0	0
$355$	5.61655e12i	0.996158i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 1.08477e13i	− 1.81913i	−0.415562	−	0.909565i	$-0.636415\pi$
$359$	− 1.08477e13i	− 1.81913i	0.415562	−	0.909565i	$-0.363585\pi$
$360$	0	0
$361$	−2.76245e12	−0.450566
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−5.73974e12	−0.885988
$366$	0	0
$367$	− 2.16676e12i	− 0.325447i	−0.986672	−	0.162723i	$-0.947972\pi$
$367$	− 2.16676e12i	− 0.325447i	0.986672	−	0.162723i	$-0.0520278\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 6.19393e12i	− 0.881245i
$372$	0	0
$373$	−1.23356e13	−1.70850	−0.854252	−	0.519859i	$-0.825984\pi$
$373$	−1.23356e13	−1.70850	−0.854252	+	0.519859i	$0.825984\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−4.06242e12	−0.533431
$378$	0	0
$379$	5.12686e12i	0.655624i	0.944743	+	0.327812i	$0.106311\pi$
$379$	5.12686e12i	0.655624i	−0.944743	+	0.327812i	$0.893689\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 8.94417e11i	− 0.108529i	−0.998527	−	0.0542645i	$-0.982719\pi$
$383$	− 8.94417e11i	− 0.108529i	0.998527	−	0.0542645i	$-0.0172814\pi$
$384$	0	0
$385$	2.39028e12	0.282583
$386$	0	0
$387$	0	0
$388$	0	0
$389$	2.57641e12	0.289246	0.144623	−	0.989487i	$-0.453803\pi$
$389$	2.57641e12	0.289246	0.144623	+	0.989487i	$0.453803\pi$
$390$	0	0
$391$	− 9.77739e12i	− 1.06989i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	3.92542e12i	0.408226i
$396$	0	0
$397$	−2.08854e12	−0.211783	−0.105891	−	0.994378i	$-0.533770\pi$
$397$	−2.08854e12	−0.211783	−0.105891	+	0.994378i	$0.533770\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−4.73346e12	−0.456517	−0.228259	−	0.973601i	$-0.573303\pi$
$401$	−4.73346e12	−0.456517	−0.228259	+	0.973601i	$0.573303\pi$
$402$	0	0
$403$	− 2.13582e12i	− 0.200927i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 1.38494e12i	− 0.124011i
$408$	0	0
$409$	2.02257e13	1.76721	0.883603	−	0.468237i	$-0.155111\pi$
$409$	2.02257e13	1.76721	0.883603	+	0.468237i	$0.155111\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−1.71528e13	−1.42753
$414$	0	0
$415$	− 6.46397e12i	− 0.525120i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 1.51587e13i	− 1.17380i	−0.809661	−	0.586898i	$-0.800349\pi$
$419$	− 1.51587e13i	− 1.17380i	0.809661	−	0.586898i	$-0.199651\pi$
$420$	0	0
$421$	5.82326e12	0.440307	0.220154	−	0.975465i	$-0.429344\pi$
$421$	5.82326e12	0.440307	0.220154	+	0.975465i	$0.429344\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	1.25788e13	0.907180
$426$	0	0
$427$	− 1.70926e13i	− 1.20412i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 1.05176e13i	− 0.707183i	−0.935400	−	0.353591i	$-0.884960\pi$
$431$	− 1.05176e13i	− 0.707183i	0.935400	−	0.353591i	$-0.115040\pi$
$432$	0	0
$433$	−2.83926e12	−0.186537	−0.0932687	−	0.995641i	$-0.529732\pi$
$433$	−2.83926e12	−0.186537	−0.0932687	+	0.995641i	$0.529732\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−1.37253e13	−0.861222
$438$	0	0
$439$	3.91291e12i	0.239981i	0.992775	+	0.119990i	$0.0382864\pi$
$439$	3.91291e12i	0.239981i	−0.992775	+	0.119990i	$0.961714\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	2.18258e13i	1.27924i	0.768691	+	0.639621i	$0.220909\pi$
$443$	2.18258e13i	1.27924i	−0.768691	+	0.639621i	$0.779091\pi$
$444$	0	0
$445$	1.59992e13	0.916851
$446$	0	0
$447$	0	0
$448$	0	0
$449$	1.19109e13	0.652699	0.326350	−	0.945249i	$-0.394181\pi$
$449$	1.19109e13	0.652699	0.326350	+	0.945249i	$0.394181\pi$
$450$	0	0
$451$	9.26018e12i	0.496292i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	7.68308e12i	0.393984i
$456$	0	0
$457$	−1.98444e13	−0.995534	−0.497767	−	0.867311i	$-0.665846\pi$
$457$	−1.98444e13	−0.995534	−0.497767	+	0.867311i	$0.665846\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−3.59865e13	−1.72836	−0.864182	−	0.503179i	$-0.832164\pi$
$461$	−3.59865e13	−1.72836	−0.864182	+	0.503179i	$0.832164\pi$
$462$	0	0
$463$	1.02191e13i	0.480295i	0.970736	+	0.240148i	$0.0771958\pi$
$463$	1.02191e13i	0.480295i	−0.970736	+	0.240148i	$0.922804\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	8.51827e11i	0.0383501i	0.999816	+	0.0191751i	$0.00610399\pi$
$467$	8.51827e11i	0.0383501i	−0.999816	+	0.0191751i	$0.993896\pi$
$468$	0	0
$469$	−1.51247e12	−0.0666533
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−4.45411e12	−0.188129
$474$	0	0
$475$	− 1.76578e13i	− 0.730246i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 9.36597e12i	− 0.371428i	−0.982604	−	0.185714i	$-0.940540\pi$
$479$	− 9.36597e12i	− 0.371428i	0.982604	−	0.185714i	$-0.0594599\pi$
$480$	0	0
$481$	4.45162e12	0.172899
$482$	0	0
$483$	0	0
$484$	0	0
$485$	1.06123e13	0.395460
$486$	0	0
$487$	1.74139e13i	0.635700i	0.948141	+	0.317850i	$0.102961\pi$
$487$	1.74139e13i	0.635700i	−0.948141	+	0.317850i	$0.897039\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	1.83308e13i	0.642354i	0.947019	+	0.321177i	$0.104078\pi$
$491$	1.83308e13i	0.642354i	−0.947019	+	0.321177i	$0.895922\pi$
$492$	0	0
$493$	−6.16767e13	−2.11781
$494$	0	0
$495$	0	0
$496$	0	0
$497$	8.02155e13	2.64531
$498$	0	0
$499$	− 1.56274e13i	− 0.505107i	−0.967583	−	0.252553i	$-0.918730\pi$
$499$	− 1.56274e13i	− 0.505107i	0.967583	−	0.252553i	$-0.0812704\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	3.43952e13i	1.06821i	0.845417	+	0.534107i	$0.179352\pi$
$503$	3.43952e13i	1.06821i	−0.845417	+	0.534107i	$0.820648\pi$
$504$	0	0
$505$	3.32775e13	1.01320
$506$	0	0
$507$	0	0
$508$	0	0
$509$	4.78947e13	1.40184	0.700921	−	0.713239i	$-0.252773\pi$
$509$	4.78947e13	1.40184	0.700921	+	0.713239i	$0.252773\pi$
$510$	0	0
$511$	8.19748e13i	2.35275i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	2.21039e13i	0.610143i
$516$	0	0
$517$	1.32983e13	0.360033
$518$	0	0
$519$	0	0
$520$	0	0
$521$	4.18118e13	1.08921	0.544603	−	0.838694i	$-0.316680\pi$
$521$	4.18118e13	1.08921	0.544603	+	0.838694i	$0.316680\pi$
$522$	0	0
$523$	2.21493e13i	0.566047i	0.959113	+	0.283023i	$0.0913374\pi$
$523$	2.21493e13i	0.566047i	−0.959113	+	0.283023i	$0.908663\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 3.24265e13i	− 0.797713i
$528$	0	0
$529$	2.02443e13	0.488680
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−2.97649e13	−0.691941
$534$	0	0
$535$	1.57380e13i	0.359071i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 2.18411e13i	− 0.480098i
$540$	0	0
$541$	−1.49671e13	−0.322962	−0.161481	−	0.986876i	$-0.551627\pi$
$541$	−1.49671e13	−0.322962	−0.161481	+	0.986876i	$0.551627\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	4.21634e13	0.876907
$546$	0	0
$547$	9.38678e12i	0.191681i	0.995397	+	0.0958407i	$0.0305539\pi$
$547$	9.38678e12i	0.191681i	−0.995397	+	0.0958407i	$0.969446\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	8.65805e13i	1.70476i
$552$	0	0
$553$	5.60627e13	1.08405
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−6.82168e13	−1.27238	−0.636188	−	0.771534i	$-0.719490\pi$
$557$	−6.82168e13	−1.27238	−0.636188	+	0.771534i	$0.719490\pi$
$558$	0	0
$559$	− 1.43168e13i	− 0.262293i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	1.03609e14i	1.83170i	0.401516	+	0.915852i	$0.368483\pi$
$563$	1.03609e14i	1.83170i	−0.401516	+	0.915852i	$0.631517\pi$
$564$	0	0
$565$	−9.97122e12	−0.173183
$566$	0	0
$567$	0	0
$568$	0	0
$569$	2.65090e13	0.444460	0.222230	−	0.974994i	$-0.428666\pi$
$569$	2.65090e13	0.444460	0.222230	+	0.974994i	$0.428666\pi$
$570$	0	0
$571$	− 7.96650e13i	− 1.31246i	−0.754560	−	0.656231i	$-0.772150\pi$
$571$	− 7.96650e13i	− 1.31246i	0.754560	−	0.656231i	$-0.227850\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 2.72512e13i	− 0.433558i
$576$	0	0
$577$	1.76404e13	0.275822	0.137911	−	0.990445i	$-0.455961\pi$
$577$	1.76404e13	0.275822	0.137911	+	0.990445i	$0.455961\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−9.23182e13	−1.39446
$582$	0	0
$583$	− 9.62878e12i	− 0.142965i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	6.21258e13i	0.891418i	0.895178	+	0.445709i	$0.147048\pi$
$587$	6.21258e13i	0.891418i	−0.895178	+	0.445709i	$0.852952\pi$
$588$	0	0
$589$	−4.55197e13	−0.642130
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−5.41591e13	−0.738581	−0.369291	−	0.929314i	$-0.620399\pi$
$593$	−5.41591e13	−0.738581	−0.369291	+	0.929314i	$0.620399\pi$
$594$	0	0
$595$	1.16646e14i	1.56418i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	1.17345e13i	0.152170i	0.997101	+	0.0760850i	$0.0242420\pi$
$599$	1.17345e13i	0.152170i	−0.997101	+	0.0760850i	$0.975758\pi$
$600$	0	0
$601$	8.93118e13	1.13903	0.569517	−	0.821980i	$-0.307130\pi$
$601$	8.93118e13	1.13903	0.569517	+	0.821980i	$0.307130\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−4.71410e13	−0.581597
$606$	0	0
$607$	− 4.48549e13i	− 0.544335i	−0.962250	−	0.272167i	$-0.912260\pi$
$607$	− 4.48549e13i	− 0.544335i	0.962250	−	0.272167i	$-0.0877405\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	4.27445e13i	0.501966i
$612$	0	0
$613$	4.54738e12	0.0525362	0.0262681	−	0.999655i	$-0.491638\pi$
$613$	4.54738e12	0.0525362	0.0262681	+	0.999655i	$0.491638\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	8.06819e13	0.902299	0.451149	−	0.892448i	$-0.351014\pi$
$617$	8.06819e13	0.902299	0.451149	+	0.892448i	$0.351014\pi$
$618$	0	0
$619$	− 1.71647e14i	− 1.88879i	−0.328815	−	0.944394i	$-0.606649\pi$
$619$	− 1.71647e14i	− 1.88879i	0.328815	−	0.944394i	$-0.393351\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 2.28500e14i	− 2.43471i
$624$	0	0
$625$	−2.48526e12	−0.0260598
$626$	0	0
$627$	0	0
$628$	0	0
$629$	6.75855e13	0.686436
$630$	0	0
$631$	− 1.26821e14i	− 1.26778i	−0.773422	−	0.633892i	$-0.781456\pi$
$631$	− 1.26821e14i	− 1.26778i	0.773422	−	0.633892i	$-0.218544\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 6.22884e13i	− 0.603308i
$636$	0	0
$637$	7.02036e13	0.669363
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−1.07570e13	−0.0994032	−0.0497016	−	0.998764i	$-0.515827\pi$
$641$	−1.07570e13	−0.0994032	−0.0497016	+	0.998764i	$0.515827\pi$
$642$	0	0
$643$	1.97002e14i	1.79232i	0.443730	+	0.896161i	$0.353655\pi$
$643$	1.97002e14i	1.79232i	−0.443730	+	0.896161i	$0.646345\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	5.12762e13i	0.452267i	0.974096	+	0.226133i	$0.0726085\pi$
$647$	5.12762e13i	0.452267i	−0.974096	+	0.226133i	$0.927391\pi$
$648$	0	0
$649$	−2.66650e13	−0.231589
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−7.25652e13	−0.611171	−0.305585	−	0.952165i	$-0.598852\pi$
$653$	−7.25652e13	−0.611171	−0.305585	+	0.952165i	$0.598852\pi$
$654$	0	0
$655$	1.24296e14i	1.03098i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 2.28320e14i	− 1.83703i	−0.395384	−	0.918516i	$-0.629388\pi$
$659$	− 2.28320e14i	− 1.83703i	0.395384	−	0.918516i	$-0.370612\pi$
$660$	0	0
$661$	−1.92435e14	−1.52503	−0.762514	−	0.646972i	$-0.776035\pi$
$661$	−1.92435e14	−1.52503	−0.762514	+	0.646972i	$0.776035\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	1.63746e14	1.25910
$666$	0	0
$667$	1.33619e14i	1.01214i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	− 2.65714e13i	− 0.195344i
$672$	0	0
$673$	−6.05450e13	−0.438534	−0.219267	−	0.975665i	$-0.570367\pi$
$673$	−6.05450e13	−0.438534	−0.219267	+	0.975665i	$0.570367\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−1.35241e14	−0.950965	−0.475483	−	0.879725i	$-0.657727\pi$
$677$	−1.35241e14	−0.950965	−0.475483	+	0.879725i	$0.657727\pi$
$678$	0	0
$679$	− 1.51565e14i	− 1.05015i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 1.81334e13i	− 0.122005i	−0.998138	−	0.0610023i	$-0.980570\pi$
$683$	− 1.81334e13i	− 0.122005i	0.998138	−	0.0610023i	$-0.0194297\pi$
$684$	0	0
$685$	9.79897e12	0.0649722
$686$	0	0
$687$	0	0
$688$	0	0
$689$	3.09497e13	0.199325
$690$	0	0
$691$	− 1.86141e14i	− 1.18155i	−0.806836	−	0.590775i	$-0.798822\pi$
$691$	− 1.86141e14i	− 1.18155i	0.806836	−	0.590775i	$-0.201178\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	1.75144e14i	1.08012i
$696$	0	0
$697$	−4.51898e14	−2.74712
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−1.30460e14	−0.770705	−0.385353	−	0.922769i	$-0.625920\pi$
$701$	−1.30460e14	−0.770705	−0.385353	+	0.922769i	$0.625920\pi$
$702$	0	0
$703$	− 9.48752e13i	− 0.552556i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 4.75269e14i	− 2.69056i
$708$	0	0
$709$	−1.34678e13	−0.0751734	−0.0375867	−	0.999293i	$-0.511967\pi$
$709$	−1.34678e13	−0.0751734	−0.0375867	+	0.999293i	$0.511967\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−7.02503e13	−0.381242
$714$	0	0
$715$	1.19437e13i	0.0639161i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	2.37711e13i	0.123710i	0.998085	+	0.0618551i	$0.0197017\pi$
$719$	2.37711e13i	0.123710i	−0.998085	+	0.0618551i	$0.980298\pi$
$720$	0	0
$721$	3.15687e14	1.62024
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−1.71903e14	−0.858211
$726$	0	0
$727$	− 2.82784e13i	− 0.139246i	−0.997573	−	0.0696229i	$-0.977820\pi$
$727$	− 2.82784e13i	− 0.139246i	0.997573	−	0.0696229i	$-0.0221796\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 2.17361e14i	− 1.04135i
$732$	0	0
$733$	−9.57811e13	−0.452647	−0.226324	−	0.974052i	$-0.572671\pi$
$733$	−9.57811e13	−0.452647	−0.226324	+	0.974052i	$0.572671\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−2.35121e12	−0.0108132
$738$	0	0
$739$	2.94507e14i	1.33621i	0.744069	+	0.668103i	$0.232893\pi$
$739$	2.94507e14i	1.33621i	−0.744069	+	0.668103i	$0.767107\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	1.05112e14i	0.464201i	0.972692	+	0.232101i	$0.0745599\pi$
$743$	1.05112e14i	0.464201i	−0.972692	+	0.232101i	$0.925440\pi$
$744$	0	0
$745$	1.99986e14	0.871402
$746$	0	0
$747$	0	0
$748$	0	0
$749$	2.24769e14	0.953516
$750$	0	0
$751$	− 2.11066e14i	− 0.883524i	−0.897132	−	0.441762i	$-0.854354\pi$
$751$	− 2.11066e14i	− 0.883524i	0.897132	−	0.441762i	$-0.145646\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 2.06106e14i	− 0.840148i
$756$	0	0
$757$	1.46472e14	0.589217	0.294609	−	0.955618i	$-0.404811\pi$
$757$	1.46472e14	0.589217	0.294609	+	0.955618i	$0.404811\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−1.59301e14	−0.624160	−0.312080	−	0.950056i	$-0.601026\pi$
$761$	−1.59301e14	−0.624160	−0.312080	+	0.950056i	$0.601026\pi$
$762$	0	0
$763$	− 6.02177e14i	− 2.32864i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 8.57091e13i	− 0.322887i
$768$	0	0
$769$	7.73317e13	0.287558	0.143779	−	0.989610i	$-0.454075\pi$
$769$	7.73317e13	0.287558	0.143779	+	0.989610i	$0.454075\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	1.30206e14	0.471775	0.235888	−	0.971780i	$-0.424200\pi$
$773$	1.30206e14	0.471775	0.235888	+	0.971780i	$0.424200\pi$
$774$	0	0
$775$	− 9.03781e13i	− 0.323262i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	6.34366e14i	2.21133i
$780$	0	0
$781$	1.24699e14	0.429149
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−2.69773e14	−0.905005
$786$	0	0
$787$	3.11665e13i	0.103232i	0.998667	+	0.0516161i	$0.0164372\pi$
$787$	3.11665e13i	0.103232i	−0.998667	+	0.0516161i	$0.983563\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	1.42409e14i	0.459890i
$792$	0	0
$793$	8.54081e13	0.272354
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−4.30484e13	−0.133864	−0.0669322	−	0.997758i	$-0.521321\pi$
$797$	−4.30484e13	−0.133864	−0.0669322	+	0.997758i	$0.521321\pi$
$798$	0	0
$799$	6.48957e14i	1.99288i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	1.27434e14i	0.381688i
$804$	0	0
$805$	2.52708e14	0.747549
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−5.13586e14	−1.48208	−0.741038	−	0.671464i	$-0.765666\pi$
$809$	−5.13586e14	−1.48208	−0.741038	+	0.671464i	$0.765666\pi$
$810$	0	0
$811$	4.73516e14i	1.34968i	0.737964	+	0.674840i	$0.235787\pi$
$811$	4.73516e14i	1.34968i	−0.737964	+	0.674840i	$0.764213\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	3.68127e14i	1.02379i
$816$	0	0
$817$	−3.05128e14	−0.838246
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−1.80616e14	−0.484218	−0.242109	−	0.970249i	$-0.577839\pi$
$821$	−1.80616e14	−0.484218	−0.242109	+	0.970249i	$0.577839\pi$
$822$	0	0
$823$	− 5.69478e14i	− 1.50826i	−0.656723	−	0.754132i	$-0.728058\pi$
$823$	− 5.69478e14i	− 1.50826i	0.656723	−	0.754132i	$-0.271942\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	3.66425e14i	0.947235i	0.880731	+	0.473617i	$0.157052\pi$
$827$	3.66425e14i	0.947235i	−0.880731	+	0.473617i	$0.842948\pi$
$828$	0	0
$829$	−3.56256e14	−0.909891	−0.454945	−	0.890519i	$-0.650341\pi$
$829$	−3.56256e14	−0.909891	−0.454945	+	0.890519i	$0.650341\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	1.06585e15	2.65748
$834$	0	0
$835$	2.96470e14i	0.730378i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	1.22645e13i	0.0295012i	0.999891	+	0.0147506i	$0.00469543\pi$
$839$	1.22645e13i	0.0295012i	−0.999891	+	0.0147506i	$0.995305\pi$
$840$	0	0
$841$	4.22175e14	1.00349
$842$	0	0
$843$	0	0
$844$	0	0
$845$	2.31916e14	0.538327
$846$	0	0
$847$	6.73267e14i	1.54444i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 1.46420e14i	− 0.328060i
$852$	0	0
$853$	−4.11937e14	−0.912190	−0.456095	−	0.889931i	$-0.650752\pi$
$853$	−4.11937e14	−0.912190	−0.456095	+	0.889931i	$0.650752\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	6.28656e14	1.35991	0.679953	−	0.733255i	$-0.262000\pi$
$857$	6.28656e14	1.35991	0.679953	+	0.733255i	$0.262000\pi$
$858$	0	0
$859$	− 3.82662e14i	− 0.818180i	−0.912494	−	0.409090i	$-0.865846\pi$
$859$	− 3.82662e14i	− 0.818180i	0.912494	−	0.409090i	$-0.134154\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 5.82551e14i	− 1.21697i	−0.793565	−	0.608485i	$-0.791778\pi$
$863$	− 5.82551e14i	− 1.21697i	0.793565	−	0.608485i	$-0.208222\pi$
$864$	0	0
$865$	2.59032e14	0.534900
$866$	0	0
$867$	0	0
$868$	0	0
$869$	8.71524e13	0.175865
$870$	0	0
$871$	− 7.55747e12i	− 0.0150760i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	8.61321e14i	1.67929i
$876$	0	0
$877$	1.76512e14	0.340233	0.170116	−	0.985424i	$-0.445586\pi$
$877$	1.76512e14	0.340233	0.170116	+	0.985424i	$0.445586\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−3.11184e12	−0.00586324	−0.00293162	−	0.999996i	$-0.500933\pi$
$881$	−3.11184e12	−0.00586324	−0.00293162	+	0.999996i	$0.500933\pi$
$882$	0	0
$883$	6.42282e14i	1.19653i	0.801300	+	0.598263i	$0.204142\pi$
$883$	6.42282e14i	1.19653i	−0.801300	+	0.598263i	$0.795858\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 7.46478e14i	− 1.35956i	−0.733416	−	0.679781i	$-0.762075\pi$
$887$	− 7.46478e14i	− 1.35956i	0.733416	−	0.679781i	$-0.237925\pi$
$888$	0	0
$889$	−8.89602e14	−1.60209
$890$	0	0
$891$	0	0
$892$	0	0
$893$	9.10993e14	1.60420
$894$	0	0
$895$	− 8.26852e13i	− 0.143984i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	4.43145e14i	0.754653i
$900$	0	0
$901$	4.69886e14	0.791350
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−3.33112e14	−0.548716
$906$	0	0
$907$	9.63392e12i	0.0156952i	0.999969	+	0.00784760i	$0.00249799\pi$
$907$	9.63392e12i	0.0156952i	−0.999969	+	0.00784760i	$0.997502\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	5.46133e14i	0.870375i	0.900340	+	0.435188i	$0.143318\pi$
$911$	5.46133e14i	0.870375i	−0.900340	+	0.435188i	$0.856682\pi$
$912$	0	0
$913$	−1.43513e14	−0.226224
$914$	0	0
$915$	0	0
$916$	0	0
$917$	1.77519e15	2.73778
$918$	0	0
$919$	4.59623e14i	0.701171i	0.936531	+	0.350586i	$0.114017\pi$
$919$	4.59623e14i	0.701171i	−0.936531	+	0.350586i	$0.885983\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	4.00820e14i	0.598330i
$924$	0	0
$925$	1.88372e14	0.278169
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−7.06601e14	−1.02116	−0.510582	−	0.859829i	$-0.670570\pi$
$929$	−7.06601e14	−1.02116	−0.510582	+	0.859829i	$0.670570\pi$
$930$	0	0
$931$	− 1.49622e15i	− 2.13917i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	1.81333e14i	0.253757i
$936$	0	0
$937$	−3.51470e14	−0.486621	−0.243311	−	0.969948i	$-0.578233\pi$
$937$	−3.51470e14	−0.486621	−0.243311	+	0.969948i	$0.578233\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−1.00424e15	−1.36110	−0.680548	−	0.732704i	$-0.738258\pi$
$941$	−1.00424e15	−1.36110	−0.680548	+	0.732704i	$0.738258\pi$
$942$	0	0
$943$	9.79014e14i	1.31290i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1.10610e14i	0.145226i	0.997360	+	0.0726128i	$0.0231337\pi$
$947$	1.10610e14i	0.145226i	−0.997360	+	0.0726128i	$0.976866\pi$
$948$	0	0
$949$	−4.09611e14	−0.532157
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−1.36100e15	−1.73138	−0.865690	−	0.500580i	$-0.833120\pi$
$953$	−1.36100e15	−1.73138	−0.865690	+	0.500580i	$0.833120\pi$
$954$	0	0
$955$	5.19014e14i	0.653374i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	− 1.39949e14i	− 0.172534i
$960$	0	0
$961$	5.86645e14	0.715745
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−7.01826e14	−0.838674
$966$	0	0
$967$	4.16847e13i	0.0492998i	0.999696	+	0.0246499i	$0.00784709\pi$
$967$	4.16847e13i	0.0492998i	−0.999696	+	0.0246499i	$0.992153\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	7.32216e14i	0.848288i	0.905595	+	0.424144i	$0.139425\pi$
$971$	7.32216e14i	0.848288i	−0.905595	+	0.424144i	$0.860575\pi$
$972$	0	0
$973$	2.50140e15	2.86826
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−2.03714e14	−0.228849	−0.114424	−	0.993432i	$-0.536502\pi$
$977$	−2.03714e14	−0.228849	−0.114424	+	0.993432i	$0.536502\pi$
$978$	0	0
$979$	− 3.55216e14i	− 0.394983i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	2.48032e14i	0.270234i	0.990830	+	0.135117i	$0.0431410\pi$
$983$	2.48032e14i	0.270234i	−0.990830	+	0.135117i	$0.956859\pi$
$984$	0	0
$985$	−1.88721e14	−0.203535
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−4.70902e14	−0.497679
$990$	0	0
$991$	5.10805e14i	0.534425i	0.963638	+	0.267213i	$0.0861026\pi$
$991$	5.10805e14i	0.534425i	−0.963638	+	0.267213i	$0.913897\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 3.24111e14i	− 0.332336i
$996$	0	0
$997$	−4.44756e14	−0.451488	−0.225744	−	0.974187i	$-0.572481\pi$
$997$	−4.44756e14	−0.451488	−0.225744	+	0.974187i	$0.572481\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	144.11.g.e.127.2		4
3.2	odd	2		48.11.g.c.31.4	yes	4
4.3	odd	2	inner	144.11.g.e.127.1		4
12.11	even	2		48.11.g.c.31.2	✓	4
24.5	odd	2		192.11.g.b.127.1		4
24.11	even	2		192.11.g.b.127.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
48.11.g.c.31.2	✓	4	12.11	even	2
48.11.g.c.31.4	yes	4	3.2	odd	2
144.11.g.e.127.1		4	4.3	odd	2	inner
144.11.g.e.127.2		4	1.1	even	1	trivial
192.11.g.b.127.1		4	24.5	odd	2
192.11.g.b.127.3		4	24.11	even	2

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:	\( k \)	\(=\)	\( 11 \)
Character orbit:	\([\chi]\)	\(=\)	144.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1960.75 q^{5} +28003.4i q^{7} +O(q^{10})\)	\(q-1960.75 q^{5} +28003.4i q^{7} +43532.8i q^{11} -139927. q^{13} -2.12441e6 q^{17} +2.98220e6i q^{19} +4.60241e6i q^{23} -5.92108e6 q^{25} +2.90324e7 q^{29} +1.52638e7i q^{31} -5.49077e7i q^{35} -3.18138e7 q^{37} +2.12718e8 q^{41} +1.02316e8i q^{43} -3.05477e8i q^{47} -5.01715e8 q^{49} -2.21185e8 q^{53} -8.53569e7i q^{55} +6.12527e8i q^{59} -6.10376e8 q^{61} +2.74362e8 q^{65} +5.40101e7i q^{67} -2.86449e9i q^{71} +2.92732e9 q^{73} -1.21907e9 q^{77} -2.00200e9i q^{79} +3.29668e9i q^{83} +4.16543e9 q^{85} -8.15974e9 q^{89} -3.91843e9i q^{91} -5.84735e9i q^{95} -5.41238e9 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 3000 q^{5}+O(q^{10}) \) 4 * q - 3000 * q^5	\( 4 q - 3000 q^{5} + 864136 q^{13} + 55128 q^{17} - 30948820 q^{25} + 80790312 q^{29} - 212511640 q^{37} + 544453272 q^{41} - 1845182972 q^{49} + 888804072 q^{53} - 208335256 q^{61} + 1075819920 q^{65} + 4881669448 q^{73} - 18371224512 q^{77} + 10313912880 q^{85} - 21540382536 q^{89} + 14965768456 q^{97}+O(q^{100}) \) 4 * q - 3000 * q^5 + 864136 * q^13 + 55128 * q^17 - 30948820 * q^25 + 80790312 * q^29 - 212511640 * q^37 + 544453272 * q^41 - 1845182972 * q^49 + 888804072 * q^53 - 208335256 * q^61 + 1075819920 * q^65 + 4881669448 * q^73 - 18371224512 * q^77 + 10313912880 * q^85 - 21540382536 * q^89 + 14965768456 * q^97

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