LMFDB - Embedded newform 132.12.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [132,12,Mod(1,132)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(132, 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("132.1");

S:= CuspForms(chi, 12);

N := Newforms(S);

Level:	$ N $	$=$	$ 132 = 2^{2} \cdot 3 \cdot 11 $
Weight:	$ k $	$=$	$ 12 $
Character orbit:	$[\chi]$	$=$	132.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:	$101.421299834$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 243453x^{2} + 8521201x + 11492037452 $ x^4 - x^3 - 243453x^2 + 8521201x + 11492037452
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{5}\cdot 3\cdot 5 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-453.854$ of defining polynomial
Character	$\chi$	$=$	132.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+243.000 q^{3} -10438.7 q^{5} +22390.8 q^{7} +59049.0 q^{9} +161051. q^{11} +378099. q^{13} -2.53661e6 q^{15} -8.44623e6 q^{17} +1.03187e7 q^{19} +5.44097e6 q^{21} +6.71020e6 q^{23} +6.01392e7 q^{25} +1.43489e7 q^{27} +5.25055e7 q^{29} +5.56053e7 q^{31} +3.91354e7 q^{33} -2.33732e8 q^{35} +1.06153e8 q^{37} +9.18781e7 q^{39} +4.48381e8 q^{41} -4.06725e8 q^{43} -6.16397e8 q^{45} +5.05987e8 q^{47} -1.47598e9 q^{49} -2.05243e9 q^{51} -5.92170e9 q^{53} -1.68117e9 q^{55} +2.50744e9 q^{57} -7.40580e9 q^{59} -4.22853e8 q^{61} +1.32216e9 q^{63} -3.94688e9 q^{65} +1.42882e10 q^{67} +1.63058e9 q^{69} -9.01648e9 q^{71} +1.28539e10 q^{73} +1.46138e10 q^{75} +3.60606e9 q^{77} -1.54192e10 q^{79} +3.48678e9 q^{81} -5.96607e10 q^{83} +8.81680e10 q^{85} +1.27588e10 q^{87} +6.43244e10 q^{89} +8.46595e9 q^{91} +1.35121e10 q^{93} -1.07714e11 q^{95} -1.23779e11 q^{97} +9.50990e9 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 972 q^{3} - 7020 q^{5} - 10934 q^{7} + 236196 q^{9} + 644204 q^{11} - 2559568 q^{13} - 1705860 q^{15} - 1606814 q^{17} - 5621902 q^{19} - 2656962 q^{21} + 19030288 q^{23} + 14585700 q^{25} + 57395628 q^{27}+ \cdots + 38039601996 q^{99}+O(q^{100}) $ 4 * q + 972 * q^3 - 7020 * q^5 - 10934 * q^7 + 236196 * q^9 + 644204 * q^11 - 2559568 * q^13 - 1705860 * q^15 - 1606814 * q^17 - 5621902 * q^19 - 2656962 * q^21 + 19030288 * q^23 + 14585700 * q^25 + 57395628 * q^27 + 146818034 * q^29 - 158103664 * q^31 + 156541572 * q^33 - 108199300 * q^35 - 770947012 * q^37 - 621975024 * q^39 - 582529706 * q^41 - 2203144746 * q^43 - 414523980 * q^45 - 1693221664 * q^47 - 1348303344 * q^49 - 390455802 * q^51 - 2342022320 * q^53 - 1130578020 * q^55 - 1366122186 * q^57 - 8715508320 * q^59 - 2069677236 * q^61 - 645641766 * q^63 - 15698175680 * q^65 - 5402316304 * q^67 + 4624359984 * q^69 - 13967424104 * q^71 + 3355935748 * q^73 + 3544325100 * q^75 - 1760931634 * q^77 + 32830867762 * q^79 + 13947137604 * q^81 + 10761308844 * q^83 - 7372813140 * q^85 + 35676782262 * q^87 - 51453668352 * q^89 - 15456476028 * q^91 - 38419190352 * q^93 - 56741221900 * q^95 - 118256509380 * q^97 + 38039601996 * q^99

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
$3$	243.000	0.577350
$3$	243.000	0.577350
$4$	0	0
$5$	−10438.7	−1.49387	−0.746935	−	0.664897i	$-0.768476\pi$
$5$	−10438.7	−1.49387	−0.746935	+	0.664897i	$0.768476\pi$
$6$	0	0
$7$	22390.8	0.503536	0.251768	−	0.967788i	$-0.418988\pi$
$7$	22390.8	0.503536	0.251768	+	0.967788i	$0.418988\pi$
$8$	0	0
$9$	59049.0	0.333333
$10$	0	0
$11$	161051.	0.301511
$11$	161051.	0.301511
$12$	0	0
$13$	378099.	0.282434	0.141217	−	0.989979i	$-0.454898\pi$
$13$	378099.	0.282434	0.141217	+	0.989979i	$0.454898\pi$
$14$	0	0
$15$	−2.53661e6	−0.862487
$16$	0	0
$17$	−8.44623e6	−1.44276	−0.721380	−	0.692540i	$-0.756492\pi$
$17$	−8.44623e6	−1.44276	−0.721380	+	0.692540i	$0.756492\pi$
$18$	0	0
$19$	1.03187e7	0.956048	0.478024	−	0.878347i	$-0.341353\pi$
$19$	1.03187e7	0.956048	0.478024	+	0.878347i	$0.341353\pi$
$20$	0	0
$21$	5.44097e6	0.290717
$22$	0	0
$23$	6.71020e6	0.217386	0.108693	−	0.994075i	$-0.465333\pi$
$23$	6.71020e6	0.217386	0.108693	+	0.994075i	$0.465333\pi$
$24$	0	0
$25$	6.01392e7	1.23165
$26$	0	0
$27$	1.43489e7	0.192450
$28$	0	0
$29$	5.25055e7	0.475352	0.237676	−	0.971344i	$-0.423614\pi$
$29$	5.25055e7	0.475352	0.237676	+	0.971344i	$0.423614\pi$
$30$	0	0
$31$	5.56053e7	0.348840	0.174420	−	0.984671i	$-0.444195\pi$
$31$	5.56053e7	0.348840	0.174420	+	0.984671i	$0.444195\pi$
$32$	0	0
$33$	3.91354e7	0.174078
$34$	0	0
$35$	−2.33732e8	−0.752218
$36$	0	0
$37$	1.06153e8	0.251666	0.125833	−	0.992051i	$-0.459840\pi$
$37$	1.06153e8	0.251666	0.125833	+	0.992051i	$0.459840\pi$
$38$	0	0
$39$	9.18781e7	0.163063
$40$	0	0
$41$	4.48381e8	0.604416	0.302208	−	0.953242i	$-0.402276\pi$
$41$	4.48381e8	0.604416	0.302208	+	0.953242i	$0.402276\pi$
$42$	0	0
$43$	−4.06725e8	−0.421915	−0.210957	−	0.977495i	$-0.567658\pi$
$43$	−4.06725e8	−0.421915	−0.210957	+	0.977495i	$0.567658\pi$
$44$	0	0
$45$	−6.16397e8	−0.497957
$46$	0	0
$47$	5.05987e8	0.321812	0.160906	−	0.986970i	$-0.448558\pi$
$47$	5.05987e8	0.321812	0.160906	+	0.986970i	$0.448558\pi$
$48$	0	0
$49$	−1.47598e9	−0.746451
$50$	0	0
$51$	−2.05243e9	−0.832978
$52$	0	0
$53$	−5.92170e9	−1.94504	−0.972521	−	0.232817i	$-0.925206\pi$
$53$	−5.92170e9	−1.94504	−0.972521	+	0.232817i	$0.925206\pi$
$54$	0	0
$55$	−1.68117e9	−0.450419
$56$	0	0
$57$	2.50744e9	0.551975
$58$	0	0
$59$	−7.40580e9	−1.34861	−0.674304	−	0.738454i	$-0.735556\pi$
$59$	−7.40580e9	−1.34861	−0.674304	+	0.738454i	$0.735556\pi$
$60$	0	0
$61$	−4.22853e8	−0.0641026	−0.0320513	−	0.999486i	$-0.510204\pi$
$61$	−4.22853e8	−0.0641026	−0.0320513	+	0.999486i	$0.510204\pi$
$62$	0	0
$63$	1.32216e9	0.167845
$64$	0	0
$65$	−3.94688e9	−0.421920
$66$	0	0
$67$	1.42882e10	1.29291	0.646453	−	0.762954i	$-0.276252\pi$
$67$	1.42882e10	1.29291	0.646453	+	0.762954i	$0.276252\pi$
$68$	0	0
$69$	1.63058e9	0.125508
$70$	0	0
$71$	−9.01648e9	−0.593084	−0.296542	−	0.955020i	$-0.595833\pi$
$71$	−9.01648e9	−0.593084	−0.296542	+	0.955020i	$0.595833\pi$
$72$	0	0
$73$	1.28539e10	0.725703	0.362851	−	0.931847i	$-0.381803\pi$
$73$	1.28539e10	0.725703	0.362851	+	0.931847i	$0.381803\pi$
$74$	0	0
$75$	1.46138e10	0.711094
$76$	0	0
$77$	3.60606e9	0.151822
$78$	0	0
$79$	−1.54192e10	−0.563785	−0.281892	−	0.959446i	$-0.590962\pi$
$79$	−1.54192e10	−0.563785	−0.281892	+	0.959446i	$0.590962\pi$
$80$	0	0
$81$	3.48678e9	0.111111
$82$	0	0
$83$	−5.96607e10	−1.66249	−0.831244	−	0.555908i	$-0.812371\pi$
$83$	−5.96607e10	−1.66249	−0.831244	+	0.555908i	$0.812371\pi$
$84$	0	0
$85$	8.81680e10	2.15530
$86$	0	0
$87$	1.27588e10	0.274445
$88$	0	0
$89$	6.43244e10	1.22104	0.610521	−	0.792000i	$-0.290960\pi$
$89$	6.43244e10	1.22104	0.610521	+	0.792000i	$0.290960\pi$
$90$	0	0
$91$	8.46595e9	0.142216
$92$	0	0
$93$	1.35121e10	0.201403
$94$	0	0
$95$	−1.07714e11	−1.42821
$96$	0	0
$97$	−1.23779e11	−1.46353	−0.731767	−	0.681555i	$-0.761304\pi$
$97$	−1.23779e11	−1.46353	−0.731767	+	0.681555i	$0.761304\pi$
$98$	0	0
$99$	9.50990e9	0.100504
$100$	0	0
$101$	1.52226e9	0.0144119	0.00720594	−	0.999974i	$-0.497706\pi$
$101$	1.52226e9	0.0144119	0.00720594	+	0.999974i	$0.497706\pi$
$102$	0	0
$103$	−9.60602e10	−0.816467	−0.408234	−	0.912878i	$-0.633855\pi$
$103$	−9.60602e10	−0.816467	−0.408234	+	0.912878i	$0.633855\pi$
$104$	0	0
$105$	−5.67968e10	−0.434293
$106$	0	0
$107$	−2.10323e11	−1.44969	−0.724847	−	0.688910i	$-0.758089\pi$
$107$	−2.10323e11	−1.44969	−0.724847	+	0.688910i	$0.758089\pi$
$108$	0	0
$109$	−4.12468e10	−0.256770	−0.128385	−	0.991724i	$-0.540979\pi$
$109$	−4.12468e10	−0.256770	−0.128385	+	0.991724i	$0.540979\pi$
$110$	0	0
$111$	2.57952e10	0.145299
$112$	0	0
$113$	−1.50432e11	−0.768084	−0.384042	−	0.923316i	$-0.625468\pi$
$113$	−1.50432e11	−0.768084	−0.384042	+	0.923316i	$0.625468\pi$
$114$	0	0
$115$	−7.00460e10	−0.324747
$116$	0	0
$117$	2.23264e10	0.0941447
$118$	0	0
$119$	−1.89118e11	−0.726482
$120$	0	0
$121$	2.59374e10	0.0909091
$122$	0	0
$123$	1.08957e11	0.348960
$124$	0	0
$125$	−1.18073e11	−0.346056
$126$	0	0
$127$	−5.52674e11	−1.48439	−0.742196	−	0.670183i	$-0.766215\pi$
$127$	−5.52674e11	−1.48439	−0.742196	+	0.670183i	$0.766215\pi$
$128$	0	0
$129$	−9.88342e10	−0.243593
$130$	0	0
$131$	−2.36207e11	−0.534935	−0.267467	−	0.963567i	$-0.586187\pi$
$131$	−2.36207e11	−0.534935	−0.267467	+	0.963567i	$0.586187\pi$
$132$	0	0
$133$	2.31044e11	0.481405
$134$	0	0
$135$	−1.49785e11	−0.287496
$136$	0	0
$137$	3.36516e10	0.0595720	0.0297860	−	0.999556i	$-0.490517\pi$
$137$	3.36516e10	0.0595720	0.0297860	+	0.999556i	$0.490517\pi$
$138$	0	0
$139$	−2.84016e11	−0.464261	−0.232130	−	0.972685i	$-0.574570\pi$
$139$	−2.84016e11	−0.464261	−0.232130	+	0.972685i	$0.574570\pi$
$140$	0	0
$141$	1.22955e11	0.185798
$142$	0	0
$143$	6.08932e10	0.0851571
$144$	0	0
$145$	−5.48091e11	−0.710115
$146$	0	0
$147$	−3.58663e11	−0.430964
$148$	0	0
$149$	1.23946e12	1.38264	0.691319	−	0.722550i	$-0.257030\pi$
$149$	1.23946e12	1.38264	0.691319	+	0.722550i	$0.257030\pi$
$150$	0	0
$151$	−1.50322e12	−1.55830	−0.779148	−	0.626840i	$-0.784348\pi$
$151$	−1.50322e12	−1.55830	−0.779148	+	0.626840i	$0.784348\pi$
$152$	0	0
$153$	−4.98742e11	−0.480920
$154$	0	0
$155$	−5.80449e11	−0.521123
$156$	0	0
$157$	−3.36324e11	−0.281391	−0.140695	−	0.990053i	$-0.544934\pi$
$157$	−3.36324e11	−0.281391	−0.140695	+	0.990053i	$0.544934\pi$
$158$	0	0
$159$	−1.43897e12	−1.12297
$160$	0	0
$161$	1.50247e11	0.109462
$162$	0	0
$163$	3.12389e11	0.212649	0.106325	−	0.994331i	$-0.466092\pi$
$163$	3.12389e11	0.212649	0.106325	+	0.994331i	$0.466092\pi$
$164$	0	0
$165$	−4.08524e11	−0.260050
$166$	0	0
$167$	−2.22401e12	−1.32494	−0.662471	−	0.749087i	$-0.730492\pi$
$167$	−2.22401e12	−1.32494	−0.662471	+	0.749087i	$0.730492\pi$
$168$	0	0
$169$	−1.64920e12	−0.920231
$170$	0	0
$171$	6.09308e11	0.318683
$172$	0	0
$173$	−1.58618e12	−0.778212	−0.389106	−	0.921193i	$-0.627216\pi$
$173$	−1.58618e12	−0.778212	−0.389106	+	0.921193i	$0.627216\pi$
$174$	0	0
$175$	1.34657e12	0.620180
$176$	0	0
$177$	−1.79961e12	−0.778619
$178$	0	0
$179$	1.75536e12	0.713963	0.356981	−	0.934112i	$-0.383806\pi$
$179$	1.75536e12	0.713963	0.356981	+	0.934112i	$0.383806\pi$
$180$	0	0
$181$	4.17813e11	0.159864	0.0799318	−	0.996800i	$-0.474530\pi$
$181$	4.17813e11	0.159864	0.0799318	+	0.996800i	$0.474530\pi$
$182$	0	0
$183$	−1.02753e11	−0.0370097
$184$	0	0
$185$	−1.10811e12	−0.375956
$186$	0	0
$187$	−1.36027e12	−0.435008
$188$	0	0
$189$	3.21284e11	0.0969056
$190$	0	0
$191$	1.26664e12	0.360555	0.180277	−	0.983616i	$-0.442300\pi$
$191$	1.26664e12	0.360555	0.180277	+	0.983616i	$0.442300\pi$
$192$	0	0
$193$	8.96872e10	0.0241082	0.0120541	−	0.999927i	$-0.496163\pi$
$193$	8.96872e10	0.0241082	0.0120541	+	0.999927i	$0.496163\pi$
$194$	0	0
$195$	−9.59092e11	−0.243596
$196$	0	0
$197$	−6.61977e11	−0.158957	−0.0794784	−	0.996837i	$-0.525325\pi$
$197$	−6.61977e11	−0.158957	−0.0794784	+	0.996837i	$0.525325\pi$
$198$	0	0
$199$	1.43435e12	0.325809	0.162904	−	0.986642i	$-0.447914\pi$
$199$	1.43435e12	0.325809	0.162904	+	0.986642i	$0.447914\pi$
$200$	0	0
$201$	3.47204e12	0.746459
$202$	0	0
$203$	1.17564e12	0.239357
$204$	0	0
$205$	−4.68053e12	−0.902919
$206$	0	0
$207$	3.96230e11	0.0724621
$208$	0	0
$209$	1.66184e12	0.288259
$210$	0	0
$211$	2.40720e12	0.396241	0.198120	−	0.980178i	$-0.436516\pi$
$211$	2.40720e12	0.396241	0.198120	+	0.980178i	$0.436516\pi$
$212$	0	0
$213$	−2.19100e12	−0.342417
$214$	0	0
$215$	4.24570e12	0.630286
$216$	0	0
$217$	1.24505e12	0.175654
$218$	0	0
$219$	3.12349e12	0.418985
$220$	0	0
$221$	−3.19351e12	−0.407485
$222$	0	0
$223$	6.09097e12	0.739621	0.369811	−	0.929107i	$-0.379423\pi$
$223$	6.09097e12	0.739621	0.369811	+	0.929107i	$0.379423\pi$
$224$	0	0
$225$	3.55116e12	0.410550
$226$	0	0
$227$	−1.11126e13	−1.22369	−0.611847	−	0.790976i	$-0.709573\pi$
$227$	−1.11126e13	−1.22369	−0.611847	+	0.790976i	$0.709573\pi$
$228$	0	0
$229$	1.15305e13	1.20991	0.604954	−	0.796260i	$-0.293191\pi$
$229$	1.15305e13	1.20991	0.604954	+	0.796260i	$0.293191\pi$
$230$	0	0
$231$	8.76273e11	0.0876544
$232$	0	0
$233$	−3.26716e12	−0.311683	−0.155841	−	0.987782i	$-0.549809\pi$
$233$	−3.26716e12	−0.311683	−0.155841	+	0.987782i	$0.549809\pi$
$234$	0	0
$235$	−5.28187e12	−0.480745
$236$	0	0
$237$	−3.74687e12	−0.325501
$238$	0	0
$239$	−5.36323e12	−0.444875	−0.222438	−	0.974947i	$-0.571401\pi$
$239$	−5.36323e12	−0.444875	−0.222438	+	0.974947i	$0.571401\pi$
$240$	0	0
$241$	9.10400e12	0.721337	0.360669	−	0.932694i	$-0.382549\pi$
$241$	9.10400e12	0.721337	0.360669	+	0.932694i	$0.382549\pi$
$242$	0	0
$243$	8.47289e11	0.0641500
$244$	0	0
$245$	1.54074e13	1.11510
$246$	0	0
$247$	3.90149e12	0.270021
$248$	0	0
$249$	−1.44975e13	−0.959838
$250$	0	0
$251$	−2.38054e13	−1.50824	−0.754120	−	0.656736i	$-0.771936\pi$
$251$	−2.38054e13	−1.50824	−0.754120	+	0.656736i	$0.771936\pi$
$252$	0	0
$253$	1.08068e12	0.0655444
$254$	0	0
$255$	2.14248e13	1.24436
$256$	0	0
$257$	−2.51861e13	−1.40129	−0.700647	−	0.713508i	$-0.747105\pi$
$257$	−2.51861e13	−1.40129	−0.700647	+	0.713508i	$0.747105\pi$
$258$	0	0
$259$	2.37686e12	0.126723
$260$	0	0
$261$	3.10039e12	0.158451
$262$	0	0
$263$	1.62806e13	0.797836	0.398918	−	0.916987i	$-0.369386\pi$
$263$	1.62806e13	0.797836	0.398918	+	0.916987i	$0.369386\pi$
$264$	0	0
$265$	6.18150e13	2.90564
$266$	0	0
$267$	1.56308e13	0.704969
$268$	0	0
$269$	1.96610e13	0.851075	0.425537	−	0.904941i	$-0.360085\pi$
$269$	1.96610e13	0.851075	0.425537	+	0.904941i	$0.360085\pi$
$270$	0	0
$271$	3.65342e12	0.151834	0.0759169	−	0.997114i	$-0.475812\pi$
$271$	3.65342e12	0.151834	0.0759169	+	0.997114i	$0.475812\pi$
$272$	0	0
$273$	2.05723e12	0.0821083
$274$	0	0
$275$	9.68547e12	0.371357
$276$	0	0
$277$	−4.51525e13	−1.66358	−0.831788	−	0.555093i	$-0.812683\pi$
$277$	−4.51525e13	−1.66358	−0.831788	+	0.555093i	$0.812683\pi$
$278$	0	0
$279$	3.28344e12	0.116280
$280$	0	0
$281$	2.36882e13	0.806579	0.403289	−	0.915073i	$-0.367867\pi$
$281$	2.36882e13	0.806579	0.403289	+	0.915073i	$0.367867\pi$
$282$	0	0
$283$	−2.64951e12	−0.0867641	−0.0433820	−	0.999059i	$-0.513813\pi$
$283$	−2.64951e12	−0.0867641	−0.0433820	+	0.999059i	$0.513813\pi$
$284$	0	0
$285$	−2.61745e13	−0.824579
$286$	0	0
$287$	1.00396e13	0.304345
$288$	0	0
$289$	3.70669e13	1.08155
$290$	0	0
$291$	−3.00783e13	−0.844971
$292$	0	0
$293$	2.05989e13	0.557280	0.278640	−	0.960396i	$-0.410116\pi$
$293$	2.05989e13	0.557280	0.278640	+	0.960396i	$0.410116\pi$
$294$	0	0
$295$	7.73072e13	2.01465
$296$	0	0
$297$	2.31091e12	0.0580259
$298$	0	0
$299$	2.53712e12	0.0613973
$300$	0	0
$301$	−9.10691e12	−0.212449
$302$	0	0
$303$	3.69908e11	0.00832070
$304$	0	0
$305$	4.41406e12	0.0957610
$306$	0	0
$307$	8.90777e13	1.86427	0.932133	−	0.362117i	$-0.117946\pi$
$307$	8.90777e13	1.86427	0.932133	+	0.362117i	$0.117946\pi$
$308$	0	0
$309$	−2.33426e13	−0.471387
$310$	0	0
$311$	9.51670e13	1.85483	0.927415	−	0.374034i	$-0.122026\pi$
$311$	9.51670e13	1.85483	0.927415	+	0.374034i	$0.122026\pi$
$312$	0	0
$313$	2.25209e13	0.423733	0.211867	−	0.977299i	$-0.432046\pi$
$313$	2.25209e13	0.423733	0.211867	+	0.977299i	$0.432046\pi$
$314$	0	0
$315$	−1.38016e13	−0.250739
$316$	0	0
$317$	8.31665e13	1.45923	0.729613	−	0.683860i	$-0.239700\pi$
$317$	8.31665e13	1.45923	0.729613	+	0.683860i	$0.239700\pi$
$318$	0	0
$319$	8.45606e12	0.143324
$320$	0	0
$321$	−5.11085e13	−0.836981
$322$	0	0
$323$	−8.71540e13	−1.37935
$324$	0	0
$325$	2.27386e13	0.347860
$326$	0	0
$327$	−1.00230e13	−0.148246
$328$	0	0
$329$	1.13295e13	0.162044
$330$	0	0
$331$	−3.05096e13	−0.422068	−0.211034	−	0.977479i	$-0.567683\pi$
$331$	−3.05096e13	−0.422068	−0.211034	+	0.977479i	$0.567683\pi$
$332$	0	0
$333$	6.26825e12	0.0838885
$334$	0	0
$335$	−1.49151e14	−1.93143
$336$	0	0
$337$	−7.81104e13	−0.978914	−0.489457	−	0.872027i	$-0.662805\pi$
$337$	−7.81104e13	−0.978914	−0.489457	+	0.872027i	$0.662805\pi$
$338$	0	0
$339$	−3.65550e13	−0.443453
$340$	0	0
$341$	8.95529e12	0.105179
$342$	0	0
$343$	−7.73223e13	−0.879401
$344$	0	0
$345$	−1.70212e13	−0.187493
$346$	0	0
$347$	−3.82809e13	−0.408479	−0.204240	−	0.978921i	$-0.565472\pi$
$347$	−3.82809e13	−0.408479	−0.204240	+	0.978921i	$0.565472\pi$
$348$	0	0
$349$	−1.49220e14	−1.54272	−0.771360	−	0.636399i	$-0.780423\pi$
$349$	−1.49220e14	−1.54272	−0.771360	+	0.636399i	$0.780423\pi$
$350$	0	0
$351$	5.42531e12	0.0543545
$352$	0	0
$353$	−5.79312e13	−0.562537	−0.281269	−	0.959629i	$-0.590755\pi$
$353$	−5.79312e13	−0.562537	−0.281269	+	0.959629i	$0.590755\pi$
$354$	0	0
$355$	9.41206e13	0.885990
$356$	0	0
$357$	−4.59557e13	−0.419434
$358$	0	0
$359$	−8.27326e13	−0.732246	−0.366123	−	0.930566i	$-0.619315\pi$
$359$	−8.27326e13	−0.732246	−0.366123	+	0.930566i	$0.619315\pi$
$360$	0	0
$361$	−1.00149e13	−0.0859719
$362$	0	0
$363$	6.30279e12	0.0524864
$364$	0	0
$365$	−1.34178e14	−1.08411
$366$	0	0
$367$	1.12200e14	0.879688	0.439844	−	0.898074i	$-0.355034\pi$
$367$	1.12200e14	0.879688	0.439844	+	0.898074i	$0.355034\pi$
$368$	0	0
$369$	2.64764e13	0.201472
$370$	0	0
$371$	−1.32592e14	−0.979399
$372$	0	0
$373$	8.99382e13	0.644979	0.322489	−	0.946573i	$-0.395480\pi$
$373$	8.99382e13	0.644979	0.322489	+	0.946573i	$0.395480\pi$
$374$	0	0
$375$	−2.86918e13	−0.199795
$376$	0	0
$377$	1.98523e13	0.134256
$378$	0	0
$379$	1.16348e14	0.764264	0.382132	−	0.924108i	$-0.375190\pi$
$379$	1.16348e14	0.764264	0.382132	+	0.924108i	$0.375190\pi$
$380$	0	0
$381$	−1.34300e14	−0.857014
$382$	0	0
$383$	7.22004e13	0.447658	0.223829	−	0.974628i	$-0.428144\pi$
$383$	7.22004e13	0.447658	0.223829	+	0.974628i	$0.428144\pi$
$384$	0	0
$385$	−3.76428e13	−0.226802
$386$	0	0
$387$	−2.40167e13	−0.140638
$388$	0	0
$389$	−1.92782e14	−1.09735	−0.548673	−	0.836037i	$-0.684867\pi$
$389$	−1.92782e14	−1.09735	−0.548673	+	0.836037i	$0.684867\pi$
$390$	0	0
$391$	−5.66759e13	−0.313636
$392$	0	0
$393$	−5.73983e13	−0.308845
$394$	0	0
$395$	1.60957e14	0.842222
$396$	0	0
$397$	1.21840e14	0.620073	0.310036	−	0.950725i	$-0.399659\pi$
$397$	1.21840e14	0.620073	0.310036	+	0.950725i	$0.399659\pi$
$398$	0	0
$399$	5.61437e13	0.277939
$400$	0	0
$401$	−7.13186e12	−0.0343486	−0.0171743	−	0.999853i	$-0.505467\pi$
$401$	−7.13186e12	−0.0343486	−0.0171743	+	0.999853i	$0.505467\pi$
$402$	0	0
$403$	2.10243e13	0.0985245
$404$	0	0
$405$	−3.63976e13	−0.165986
$406$	0	0
$407$	1.70961e13	0.0758800
$408$	0	0
$409$	−1.10009e14	−0.475280	−0.237640	−	0.971353i	$-0.576374\pi$
$409$	−1.10009e14	−0.475280	−0.237640	+	0.971353i	$0.576374\pi$
$410$	0	0
$411$	8.17734e12	0.0343939
$412$	0	0
$413$	−1.65822e14	−0.679073
$414$	0	0
$415$	6.22782e14	2.48354
$416$	0	0
$417$	−6.90160e13	−0.268041
$418$	0	0
$419$	−3.50326e14	−1.32524	−0.662622	−	0.748954i	$-0.730556\pi$
$419$	−3.50326e14	−1.32524	−0.662622	+	0.748954i	$0.730556\pi$
$420$	0	0
$421$	2.48364e14	0.915245	0.457623	−	0.889146i	$-0.348701\pi$
$421$	2.48364e14	0.915245	0.457623	+	0.889146i	$0.348701\pi$
$422$	0	0
$423$	2.98781e13	0.107271
$424$	0	0
$425$	−5.07949e14	−1.77698
$426$	0	0
$427$	−9.46803e12	−0.0322780
$428$	0	0
$429$	1.47971e13	0.0491655
$430$	0	0
$431$	−3.46773e13	−0.112310	−0.0561552	−	0.998422i	$-0.517884\pi$
$431$	−3.46773e13	−0.112310	−0.0561552	+	0.998422i	$0.517884\pi$
$432$	0	0
$433$	−2.19214e14	−0.692125	−0.346062	−	0.938211i	$-0.612481\pi$
$433$	−2.19214e14	−0.692125	−0.346062	+	0.938211i	$0.612481\pi$
$434$	0	0
$435$	−1.33186e14	−0.409985
$436$	0	0
$437$	6.92404e13	0.207832
$438$	0	0
$439$	3.99626e14	1.16977	0.584883	−	0.811118i	$-0.301140\pi$
$439$	3.99626e14	1.16977	0.584883	+	0.811118i	$0.301140\pi$
$440$	0	0
$441$	−8.71550e13	−0.248817
$442$	0	0
$443$	−3.21705e14	−0.895853	−0.447926	−	0.894070i	$-0.647837\pi$
$443$	−3.21705e14	−0.895853	−0.447926	+	0.894070i	$0.647837\pi$
$444$	0	0
$445$	−6.71465e14	−1.82408
$446$	0	0
$447$	3.01189e14	0.798266
$448$	0	0
$449$	−4.53471e14	−1.17272	−0.586361	−	0.810050i	$-0.699440\pi$
$449$	−4.53471e14	−1.17272	−0.586361	+	0.810050i	$0.699440\pi$
$450$	0	0
$451$	7.22122e13	0.182238
$452$	0	0
$453$	−3.65283e14	−0.899682
$454$	0	0
$455$	−8.83738e13	−0.212452
$456$	0	0
$457$	7.64224e14	1.79342	0.896709	−	0.442620i	$-0.145951\pi$
$457$	7.64224e14	1.79342	0.896709	+	0.442620i	$0.145951\pi$
$458$	0	0
$459$	−1.21194e14	−0.277659
$460$	0	0
$461$	−3.22239e14	−0.720814	−0.360407	−	0.932795i	$-0.617362\pi$
$461$	−3.22239e14	−0.720814	−0.360407	+	0.932795i	$0.617362\pi$
$462$	0	0
$463$	1.60826e14	0.351285	0.175643	−	0.984454i	$-0.443800\pi$
$463$	1.60826e14	0.351285	0.175643	+	0.984454i	$0.443800\pi$
$464$	0	0
$465$	−1.41049e14	−0.300870
$466$	0	0
$467$	−8.56388e14	−1.78414	−0.892068	−	0.451902i	$-0.850746\pi$
$467$	−8.56388e14	−1.78414	−0.892068	+	0.451902i	$0.850746\pi$
$468$	0	0
$469$	3.19925e14	0.651025
$470$	0	0
$471$	−8.17268e13	−0.162461
$472$	0	0
$473$	−6.55035e13	−0.127212
$474$	0	0
$475$	6.20558e14	1.17752
$476$	0	0
$477$	−3.49670e14	−0.648347
$478$	0	0
$479$	1.98704e14	0.360048	0.180024	−	0.983662i	$-0.442382\pi$
$479$	1.98704e14	0.360048	0.180024	+	0.983662i	$0.442382\pi$
$480$	0	0
$481$	4.01365e13	0.0710790
$482$	0	0
$483$	3.65100e13	0.0631978
$484$	0	0
$485$	1.29210e15	2.18633
$486$	0	0
$487$	6.13525e14	1.01490	0.507449	−	0.861682i	$-0.330588\pi$
$487$	6.13525e14	1.01490	0.507449	+	0.861682i	$0.330588\pi$
$488$	0	0
$489$	7.59105e13	0.122773
$490$	0	0
$491$	7.59527e14	1.20114	0.600572	−	0.799571i	$-0.294940\pi$
$491$	7.59527e14	1.20114	0.600572	+	0.799571i	$0.294940\pi$
$492$	0	0
$493$	−4.43473e14	−0.685819
$494$	0	0
$495$	−9.92714e13	−0.150140
$496$	0	0
$497$	−2.01886e14	−0.298639
$498$	0	0
$499$	2.19010e14	0.316892	0.158446	−	0.987368i	$-0.449352\pi$
$499$	2.19010e14	0.316892	0.158446	+	0.987368i	$0.449352\pi$
$500$	0	0
$501$	−5.40436e14	−0.764956
$502$	0	0
$503$	1.14520e15	1.58584	0.792920	−	0.609326i	$-0.208560\pi$
$503$	1.14520e15	1.58584	0.792920	+	0.609326i	$0.208560\pi$
$504$	0	0
$505$	−1.58904e13	−0.0215295
$506$	0	0
$507$	−4.00756e14	−0.531296
$508$	0	0
$509$	−4.32106e14	−0.560586	−0.280293	−	0.959914i	$-0.590432\pi$
$509$	−4.32106e14	−0.560586	−0.280293	+	0.959914i	$0.590432\pi$
$510$	0	0
$511$	2.87809e14	0.365417
$512$	0	0
$513$	1.48062e14	0.183992
$514$	0	0
$515$	1.00275e15	1.21970
$516$	0	0
$517$	8.14898e13	0.0970298
$518$	0	0
$519$	−3.85441e14	−0.449301
$520$	0	0
$521$	8.60763e14	0.982372	0.491186	−	0.871055i	$-0.336564\pi$
$521$	8.60763e14	0.982372	0.491186	+	0.871055i	$0.336564\pi$
$522$	0	0
$523$	4.03592e14	0.451007	0.225503	−	0.974242i	$-0.427597\pi$
$523$	4.03592e14	0.451007	0.225503	+	0.974242i	$0.427597\pi$
$524$	0	0
$525$	3.27215e14	0.358061
$526$	0	0
$527$	−4.69655e14	−0.503293
$528$	0	0
$529$	−9.07783e14	−0.952743
$530$	0	0
$531$	−4.37305e14	−0.449536
$532$	0	0
$533$	1.69532e14	0.170708
$534$	0	0
$535$	2.19551e15	2.16565
$536$	0	0
$537$	4.26553e14	0.412207
$538$	0	0
$539$	−2.37708e14	−0.225064
$540$	0	0
$541$	1.76236e15	1.63497	0.817487	−	0.575948i	$-0.195367\pi$
$541$	1.76236e15	1.63497	0.817487	+	0.575948i	$0.195367\pi$
$542$	0	0
$543$	1.01529e14	0.0922973
$544$	0	0
$545$	4.30564e14	0.383581
$546$	0	0
$547$	−1.08828e15	−0.950192	−0.475096	−	0.879934i	$-0.657587\pi$
$547$	−1.08828e15	−0.950192	−0.475096	+	0.879934i	$0.657587\pi$
$548$	0	0
$549$	−2.49691e13	−0.0213675
$550$	0	0
$551$	5.41788e14	0.454460
$552$	0	0
$553$	−3.45249e14	−0.283886
$554$	0	0
$555$	−2.69270e14	−0.217058
$556$	0	0
$557$	3.51566e14	0.277846	0.138923	−	0.990303i	$-0.455636\pi$
$557$	3.51566e14	0.277846	0.138923	+	0.990303i	$0.455636\pi$
$558$	0	0
$559$	−1.53782e14	−0.119163
$560$	0	0
$561$	−3.30547e14	−0.251152
$562$	0	0
$563$	−8.68064e14	−0.646779	−0.323389	−	0.946266i	$-0.604822\pi$
$563$	−8.68064e14	−0.646779	−0.323389	+	0.946266i	$0.604822\pi$
$564$	0	0
$565$	1.57032e15	1.14742
$566$	0	0
$567$	7.80719e13	0.0559485
$568$	0	0
$569$	3.16339e14	0.222349	0.111174	−	0.993801i	$-0.464539\pi$
$569$	3.16339e14	0.222349	0.111174	+	0.993801i	$0.464539\pi$
$570$	0	0
$571$	−8.87299e14	−0.611747	−0.305873	−	0.952072i	$-0.598948\pi$
$571$	−8.87299e14	−0.611747	−0.305873	+	0.952072i	$0.598948\pi$
$572$	0	0
$573$	3.07795e14	0.208166
$574$	0	0
$575$	4.03546e14	0.267744
$576$	0	0
$577$	−1.51333e14	−0.0985072	−0.0492536	−	0.998786i	$-0.515684\pi$
$577$	−1.51333e14	−0.0985072	−0.0492536	+	0.998786i	$0.515684\pi$
$578$	0	0
$579$	2.17940e13	0.0139189
$580$	0	0
$581$	−1.33585e15	−0.837123
$582$	0	0
$583$	−9.53695e14	−0.586452
$584$	0	0
$585$	−2.33059e14	−0.140640
$586$	0	0
$587$	3.31226e13	0.0196162	0.00980809	−	0.999952i	$-0.496878\pi$
$587$	3.31226e13	0.0196162	0.00980809	+	0.999952i	$0.496878\pi$
$588$	0	0
$589$	5.73774e14	0.333508
$590$	0	0
$591$	−1.60861e14	−0.0917737
$592$	0	0
$593$	−1.69874e15	−0.951319	−0.475660	−	0.879629i	$-0.657791\pi$
$593$	−1.69874e15	−0.951319	−0.475660	+	0.879629i	$0.657791\pi$
$594$	0	0
$595$	1.97415e15	1.08527
$596$	0	0
$597$	3.48546e14	0.188106
$598$	0	0
$599$	1.92371e15	1.01927	0.509637	−	0.860389i	$-0.329780\pi$
$599$	1.92371e15	1.01927	0.509637	+	0.860389i	$0.329780\pi$
$600$	0	0
$601$	1.23456e15	0.642246	0.321123	−	0.947037i	$-0.395940\pi$
$601$	1.23456e15	0.642246	0.321123	+	0.947037i	$0.395940\pi$
$602$	0	0
$603$	8.43705e14	0.430969
$604$	0	0
$605$	−2.70754e14	−0.135806
$606$	0	0
$607$	8.68452e14	0.427768	0.213884	−	0.976859i	$-0.431389\pi$
$607$	8.68452e14	0.427768	0.213884	+	0.976859i	$0.431389\pi$
$608$	0	0
$609$	2.85681e14	0.138193
$610$	0	0
$611$	1.91313e14	0.0908906
$612$	0	0
$613$	3.38779e15	1.58083	0.790414	−	0.612574i	$-0.209866\pi$
$613$	3.38779e15	1.58083	0.790414	+	0.612574i	$0.209866\pi$
$614$	0	0
$615$	−1.13737e15	−0.521301
$616$	0	0
$617$	−3.21326e15	−1.44670	−0.723348	−	0.690483i	$-0.757398\pi$
$617$	−3.21326e15	−1.44670	−0.723348	+	0.690483i	$0.757398\pi$
$618$	0	0
$619$	−2.96780e14	−0.131261	−0.0656306	−	0.997844i	$-0.520906\pi$
$619$	−2.96780e14	−0.131261	−0.0656306	+	0.997844i	$0.520906\pi$
$620$	0	0
$621$	9.62840e13	0.0418360
$622$	0	0
$623$	1.44027e15	0.614839
$624$	0	0
$625$	−1.70395e15	−0.714688
$626$	0	0
$627$	4.03826e14	0.166427
$628$	0	0
$629$	−8.96595e14	−0.363093
$630$	0	0
$631$	5.79336e14	0.230552	0.115276	−	0.993333i	$-0.463225\pi$
$631$	5.79336e14	0.230552	0.115276	+	0.993333i	$0.463225\pi$
$632$	0	0
$633$	5.84950e14	0.228770
$634$	0	0
$635$	5.76922e15	2.21749
$636$	0	0
$637$	−5.58066e14	−0.210823
$638$	0	0
$639$	−5.32414e14	−0.197695
$640$	0	0
$641$	−4.02969e15	−1.47080	−0.735398	−	0.677636i	$-0.763005\pi$
$641$	−4.02969e15	−1.47080	−0.735398	+	0.677636i	$0.763005\pi$
$642$	0	0
$643$	−1.48626e15	−0.533255	−0.266627	−	0.963800i	$-0.585909\pi$
$643$	−1.48626e15	−0.533255	−0.266627	+	0.963800i	$0.585909\pi$
$644$	0	0
$645$	1.03170e15	0.363896
$646$	0	0
$647$	−1.19732e15	−0.415179	−0.207589	−	0.978216i	$-0.566562\pi$
$647$	−1.19732e15	−0.415179	−0.207589	+	0.978216i	$0.566562\pi$
$648$	0	0
$649$	−1.19271e15	−0.406621
$650$	0	0
$651$	3.02547e14	0.101414
$652$	0	0
$653$	−5.78485e15	−1.90664	−0.953322	−	0.301956i	$-0.902361\pi$
$653$	−5.78485e15	−1.90664	−0.953322	+	0.301956i	$0.902361\pi$
$654$	0	0
$655$	2.46570e15	0.799123
$656$	0	0
$657$	7.59009e14	0.241901
$658$	0	0
$659$	−1.94159e15	−0.608538	−0.304269	−	0.952586i	$-0.598412\pi$
$659$	−1.94159e15	−0.608538	−0.304269	+	0.952586i	$0.598412\pi$
$660$	0	0
$661$	−3.84086e15	−1.18392	−0.591958	−	0.805969i	$-0.701645\pi$
$661$	−3.84086e15	−1.18392	−0.591958	+	0.805969i	$0.701645\pi$
$662$	0	0
$663$	−7.76024e14	−0.235261
$664$	0	0
$665$	−2.41181e15	−0.719157
$666$	0	0
$667$	3.52322e14	0.103335
$668$	0	0
$669$	1.48011e15	0.427021
$670$	0	0
$671$	−6.81010e13	−0.0193277
$672$	0	0
$673$	−3.75645e15	−1.04881	−0.524403	−	0.851470i	$-0.675711\pi$
$673$	−3.75645e15	−1.04881	−0.524403	+	0.851470i	$0.675711\pi$
$674$	0	0
$675$	8.62931e14	0.237031
$676$	0	0
$677$	−5.28022e15	−1.42697	−0.713485	−	0.700671i	$-0.752884\pi$
$677$	−5.28022e15	−1.42697	−0.713485	+	0.700671i	$0.752884\pi$
$678$	0	0
$679$	−2.77151e15	−0.736942
$680$	0	0
$681$	−2.70036e15	−0.706500
$682$	0	0
$683$	2.81860e15	0.725637	0.362818	−	0.931860i	$-0.381815\pi$
$683$	2.81860e15	0.725637	0.362818	+	0.931860i	$0.381815\pi$
$684$	0	0
$685$	−3.51280e14	−0.0889929
$686$	0	0
$687$	2.80191e15	0.698541
$688$	0	0
$689$	−2.23899e15	−0.549346
$690$	0	0
$691$	−6.58716e15	−1.59063	−0.795315	−	0.606196i	$-0.792695\pi$
$691$	−6.58716e15	−1.59063	−0.795315	+	0.606196i	$0.792695\pi$
$692$	0	0
$693$	2.12934e14	0.0506073
$694$	0	0
$695$	2.96477e15	0.693545
$696$	0	0
$697$	−3.78713e15	−0.872026
$698$	0	0
$699$	−7.93920e14	−0.179950
$700$	0	0
$701$	1.77328e15	0.395666	0.197833	−	0.980236i	$-0.436610\pi$
$701$	1.77328e15	0.395666	0.197833	+	0.980236i	$0.436610\pi$
$702$	0	0
$703$	1.09536e15	0.240604
$704$	0	0
$705$	−1.28349e15	−0.277558
$706$	0	0
$707$	3.40846e13	0.00725690
$708$	0	0
$709$	−7.55761e15	−1.58427	−0.792137	−	0.610344i	$-0.791031\pi$
$709$	−7.55761e15	−1.58427	−0.792137	+	0.610344i	$0.791031\pi$
$710$	0	0
$711$	−9.10489e14	−0.187928
$712$	0	0
$713$	3.73122e14	0.0758331
$714$	0	0
$715$	−6.35649e14	−0.127214
$716$	0	0
$717$	−1.30327e15	−0.256849
$718$	0	0
$719$	−4.35299e15	−0.844849	−0.422424	−	0.906398i	$-0.638821\pi$
$719$	−4.35299e15	−0.844849	−0.422424	+	0.906398i	$0.638821\pi$
$720$	0	0
$721$	−2.15087e15	−0.411121
$722$	0	0
$723$	2.21227e15	0.416464
$724$	0	0
$725$	3.15764e15	0.585468
$726$	0	0
$727$	1.01683e16	1.85698	0.928492	−	0.371354i	$-0.121106\pi$
$727$	1.01683e16	1.85698	0.928492	+	0.371354i	$0.121106\pi$
$728$	0	0
$729$	2.05891e14	0.0370370
$730$	0	0
$731$	3.43530e15	0.608721
$732$	0	0
$733$	−8.02622e15	−1.40100	−0.700502	−	0.713651i	$-0.747040\pi$
$733$	−8.02622e15	−1.40100	−0.700502	+	0.713651i	$0.747040\pi$
$734$	0	0
$735$	3.74399e15	0.643804
$736$	0	0
$737$	2.30113e15	0.389826
$738$	0	0
$739$	3.26270e14	0.0544544	0.0272272	−	0.999629i	$-0.491332\pi$
$739$	3.26270e14	0.0544544	0.0272272	+	0.999629i	$0.491332\pi$
$740$	0	0
$741$	9.48062e14	0.155897
$742$	0	0
$743$	3.72057e15	0.602797	0.301398	−	0.953498i	$-0.402547\pi$
$743$	3.72057e15	0.602797	0.301398	+	0.953498i	$0.402547\pi$
$744$	0	0
$745$	−1.29384e16	−2.06548
$746$	0	0
$747$	−3.52290e15	−0.554163
$748$	0	0
$749$	−4.70930e15	−0.729973
$750$	0	0
$751$	−9.93996e15	−1.51833	−0.759163	−	0.650900i	$-0.774392\pi$
$751$	−9.93996e15	−1.51833	−0.759163	+	0.650900i	$0.774392\pi$
$752$	0	0
$753$	−5.78472e15	−0.870783
$754$	0	0
$755$	1.56917e16	2.32789
$756$	0	0
$757$	6.83176e15	0.998861	0.499431	−	0.866354i	$-0.333543\pi$
$757$	6.83176e15	0.998861	0.499431	+	0.866354i	$0.333543\pi$
$758$	0	0
$759$	2.62606e14	0.0378421
$760$	0	0
$761$	9.13669e15	1.29770	0.648848	−	0.760918i	$-0.275251\pi$
$761$	9.13669e15	1.29770	0.648848	+	0.760918i	$0.275251\pi$
$762$	0	0
$763$	−9.23548e14	−0.129293
$764$	0	0
$765$	5.20623e15	0.718432
$766$	0	0
$767$	−2.80013e15	−0.380893
$768$	0	0
$769$	5.66650e15	0.759836	0.379918	−	0.925020i	$-0.375952\pi$
$769$	5.66650e15	0.759836	0.379918	+	0.925020i	$0.375952\pi$
$770$	0	0
$771$	−6.12023e15	−0.809038
$772$	0	0
$773$	1.23541e16	1.61000	0.804998	−	0.593278i	$-0.202166\pi$
$773$	1.23541e16	1.61000	0.804998	+	0.593278i	$0.202166\pi$
$774$	0	0
$775$	3.34406e15	0.429649
$776$	0	0
$777$	5.77577e14	0.0731634
$778$	0	0
$779$	4.62670e15	0.577850
$780$	0	0
$781$	−1.45211e15	−0.178821
$782$	0	0
$783$	7.53396e14	0.0914816
$784$	0	0
$785$	3.51080e15	0.420362
$786$	0	0
$787$	−4.26312e15	−0.503346	−0.251673	−	0.967812i	$-0.580981\pi$
$787$	−4.26312e15	−0.503346	−0.251673	+	0.967812i	$0.580981\pi$
$788$	0	0
$789$	3.95618e15	0.460631
$790$	0	0
$791$	−3.36829e15	−0.386758
$792$	0	0
$793$	−1.59880e14	−0.0181048
$794$	0	0
$795$	1.50211e16	1.67757
$796$	0	0
$797$	6.37811e15	0.702541	0.351270	−	0.936274i	$-0.385750\pi$
$797$	6.37811e15	0.702541	0.351270	+	0.936274i	$0.385750\pi$
$798$	0	0
$799$	−4.27369e15	−0.464297
$800$	0	0
$801$	3.79829e15	0.407014
$802$	0	0
$803$	2.07013e15	0.218808
$804$	0	0
$805$	−1.56839e15	−0.163522
$806$	0	0
$807$	4.77762e15	0.491368
$808$	0	0
$809$	−9.67817e15	−0.981921	−0.490960	−	0.871182i	$-0.663354\pi$
$809$	−9.67817e15	−0.981921	−0.490960	+	0.871182i	$0.663354\pi$
$810$	0	0
$811$	−1.32716e16	−1.32833	−0.664167	−	0.747584i	$-0.731214\pi$
$811$	−1.32716e16	−1.32833	−0.664167	+	0.747584i	$0.731214\pi$
$812$	0	0
$813$	8.87781e14	0.0876613
$814$	0	0
$815$	−3.26095e15	−0.317671
$816$	0	0
$817$	−4.19687e15	−0.403371
$818$	0	0
$819$	4.99906e14	0.0474053
$820$	0	0
$821$	−1.83531e16	−1.71720	−0.858602	−	0.512642i	$-0.828667\pi$
$821$	−1.83531e16	−1.71720	−0.858602	+	0.512642i	$0.828667\pi$
$822$	0	0
$823$	−1.53641e16	−1.41843	−0.709214	−	0.704993i	$-0.750950\pi$
$823$	−1.53641e16	−1.41843	−0.709214	+	0.704993i	$0.750950\pi$
$824$	0	0
$825$	2.35357e15	0.214403
$826$	0	0
$827$	−1.33219e16	−1.19752	−0.598762	−	0.800927i	$-0.704341\pi$
$827$	−1.33219e16	−1.19752	−0.598762	+	0.800927i	$0.704341\pi$
$828$	0	0
$829$	3.99889e15	0.354723	0.177362	−	0.984146i	$-0.443244\pi$
$829$	3.99889e15	0.354723	0.177362	+	0.984146i	$0.443244\pi$
$830$	0	0
$831$	−1.09721e16	−0.960466
$832$	0	0
$833$	1.24665e16	1.07695
$834$	0	0
$835$	2.32159e16	1.97929
$836$	0	0
$837$	7.97875e14	0.0671344
$838$	0	0
$839$	1.50683e16	1.25134	0.625669	−	0.780089i	$-0.284826\pi$
$839$	1.50683e16	1.25134	0.625669	+	0.780089i	$0.284826\pi$
$840$	0	0
$841$	−9.44369e15	−0.774040
$842$	0	0
$843$	5.75623e15	0.465678
$844$	0	0
$845$	1.72156e16	1.37471
$846$	0	0
$847$	5.80760e14	0.0457760
$848$	0	0
$849$	−6.43830e14	−0.0500933
$850$	0	0
$851$	7.12309e14	0.0547087
$852$	0	0
$853$	4.90526e15	0.371914	0.185957	−	0.982558i	$-0.440461\pi$
$853$	4.90526e15	0.371914	0.185957	+	0.982558i	$0.440461\pi$
$854$	0	0
$855$	−6.36041e15	−0.476071
$856$	0	0
$857$	1.51583e16	1.12010	0.560048	−	0.828460i	$-0.310783\pi$
$857$	1.51583e16	1.12010	0.560048	+	0.828460i	$0.310783\pi$
$858$	0	0
$859$	−1.34328e15	−0.0979948	−0.0489974	−	0.998799i	$-0.515603\pi$
$859$	−1.34328e15	−0.0979948	−0.0489974	+	0.998799i	$0.515603\pi$
$860$	0	0
$861$	2.43962e15	0.175714
$862$	0	0
$863$	−1.82817e16	−1.30004	−0.650022	−	0.759915i	$-0.725240\pi$
$863$	−1.82817e16	−1.30004	−0.650022	+	0.759915i	$0.725240\pi$
$864$	0	0
$865$	1.65577e16	1.16255
$866$	0	0
$867$	9.00726e15	0.624436
$868$	0	0
$869$	−2.48328e15	−0.169988
$870$	0	0
$871$	5.40237e15	0.365161
$872$	0	0
$873$	−7.30903e15	−0.487844
$874$	0	0
$875$	−2.64375e15	−0.174252
$876$	0	0
$877$	−2.65127e16	−1.72566	−0.862831	−	0.505492i	$-0.831311\pi$
$877$	−2.65127e16	−1.72566	−0.862831	+	0.505492i	$0.831311\pi$
$878$	0	0
$879$	5.00554e15	0.321746
$880$	0	0
$881$	1.16283e16	0.738154	0.369077	−	0.929399i	$-0.379674\pi$
$881$	1.16283e16	0.738154	0.369077	+	0.929399i	$0.379674\pi$
$882$	0	0
$883$	2.27267e16	1.42480	0.712399	−	0.701775i	$-0.247609\pi$
$883$	2.27267e16	1.42480	0.712399	+	0.701775i	$0.247609\pi$
$884$	0	0
$885$	1.87857e16	1.16316
$886$	0	0
$887$	−2.30074e16	−1.40698	−0.703491	−	0.710704i	$-0.748376\pi$
$887$	−2.30074e16	−1.40698	−0.703491	+	0.710704i	$0.748376\pi$
$888$	0	0
$889$	−1.23748e16	−0.747445
$890$	0	0
$891$	5.61550e14	0.0335013
$892$	0	0
$893$	5.22113e15	0.307667
$894$	0	0
$895$	−1.83238e16	−1.06657
$896$	0	0
$897$	6.16520e14	0.0354478
$898$	0	0
$899$	2.91958e15	0.165822
$900$	0	0
$901$	5.00160e16	2.80623
$902$	0	0
$903$	−2.21298e15	−0.122658
$904$	0	0
$905$	−4.36144e15	−0.238816
$906$	0	0
$907$	−4.05537e15	−0.219377	−0.109688	−	0.993966i	$-0.534985\pi$
$907$	−4.05537e15	−0.219377	−0.109688	+	0.993966i	$0.534985\pi$
$908$	0	0
$909$	8.98877e13	0.00480396
$910$	0	0
$911$	1.65687e16	0.874860	0.437430	−	0.899253i	$-0.355889\pi$
$911$	1.65687e16	0.874860	0.437430	+	0.899253i	$0.355889\pi$
$912$	0	0
$913$	−9.60841e15	−0.501259
$914$	0	0
$915$	1.07262e15	0.0552877
$916$	0	0
$917$	−5.28887e15	−0.269359
$918$	0	0
$919$	−1.06338e15	−0.0535120	−0.0267560	−	0.999642i	$-0.508518\pi$
$919$	−1.06338e15	−0.0535120	−0.0267560	+	0.999642i	$0.508518\pi$
$920$	0	0
$921$	2.16459e16	1.07633
$922$	0	0
$923$	−3.40912e15	−0.167507
$924$	0	0
$925$	6.38397e15	0.309964
$926$	0	0
$927$	−5.67226e15	−0.272156
$928$	0	0
$929$	4.05410e16	1.92224	0.961120	−	0.276130i	$-0.0890520\pi$
$929$	4.05410e16	1.92224	0.961120	+	0.276130i	$0.0890520\pi$
$930$	0	0
$931$	−1.52302e16	−0.713643
$932$	0	0
$933$	2.31256e16	1.07089
$934$	0	0
$935$	1.41995e16	0.649846
$936$	0	0
$937$	2.21854e16	1.00346	0.501729	−	0.865025i	$-0.332697\pi$
$937$	2.21854e16	1.00346	0.501729	+	0.865025i	$0.332697\pi$
$938$	0	0
$939$	5.47259e15	0.244643
$940$	0	0
$941$	1.38588e16	0.612325	0.306163	−	0.951979i	$-0.400955\pi$
$941$	1.38588e16	0.612325	0.306163	+	0.951979i	$0.400955\pi$
$942$	0	0
$943$	3.00872e15	0.131392
$944$	0	0
$945$	−3.35380e15	−0.144764
$946$	0	0
$947$	−6.17388e15	−0.263410	−0.131705	−	0.991289i	$-0.542045\pi$
$947$	−6.17388e15	−0.263410	−0.131705	+	0.991289i	$0.542045\pi$
$948$	0	0
$949$	4.86004e15	0.204963
$950$	0	0
$951$	2.02095e16	0.842484
$952$	0	0
$953$	−1.43645e16	−0.591941	−0.295970	−	0.955197i	$-0.595643\pi$
$953$	−1.43645e16	−0.591941	−0.295970	+	0.955197i	$0.595643\pi$
$954$	0	0
$955$	−1.32222e16	−0.538622
$956$	0	0
$957$	2.05482e15	0.0827482
$958$	0	0
$959$	7.53486e14	0.0299967
$960$	0	0
$961$	−2.23165e16	−0.878310
$962$	0	0
$963$	−1.24194e16	−0.483231
$964$	0	0
$965$	−9.36221e14	−0.0360146
$966$	0	0
$967$	−4.28323e15	−0.162902	−0.0814509	−	0.996677i	$-0.525955\pi$
$967$	−4.28323e15	−0.162902	−0.0814509	+	0.996677i	$0.525955\pi$
$968$	0	0
$969$	−2.11784e16	−0.796367
$970$	0	0
$971$	−2.86642e16	−1.06570	−0.532850	−	0.846210i	$-0.678879\pi$
$971$	−2.86642e16	−1.06570	−0.532850	+	0.846210i	$0.678879\pi$
$972$	0	0
$973$	−6.35936e15	−0.233772
$974$	0	0
$975$	5.52547e15	0.200837
$976$	0	0
$977$	2.87970e16	1.03497	0.517484	−	0.855693i	$-0.326869\pi$
$977$	2.87970e16	1.03497	0.517484	+	0.855693i	$0.326869\pi$
$978$	0	0
$979$	1.03595e16	0.368158
$980$	0	0
$981$	−2.43558e15	−0.0855900
$982$	0	0
$983$	−2.16613e16	−0.752732	−0.376366	−	0.926471i	$-0.622827\pi$
$983$	−2.16613e16	−0.752732	−0.376366	+	0.926471i	$0.622827\pi$
$984$	0	0
$985$	6.91021e15	0.237461
$986$	0	0
$987$	2.75306e15	0.0935560
$988$	0	0
$989$	−2.72921e15	−0.0917184
$990$	0	0
$991$	6.94182e15	0.230711	0.115356	−	0.993324i	$-0.463199\pi$
$991$	6.94182e15	0.230711	0.115356	+	0.993324i	$0.463199\pi$
$992$	0	0
$993$	−7.41383e15	−0.243681
$994$	0	0
$995$	−1.49728e16	−0.486716
$996$	0	0
$997$	−5.11186e16	−1.64345	−0.821723	−	0.569887i	$-0.806987\pi$
$997$	−5.11186e16	−1.64345	−0.821723	+	0.569887i	$0.806987\pi$
$998$	0	0
$999$	1.52318e15	0.0484331

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	132.12.a.b.1.1	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
132.12.a.b.1.1	✓	4	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:	\( N \)	\(=\)	\( 132 = 2^{2} \cdot 3 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	132.a (trivial)

\(f(q)\)	\(=\)	\(q+243.000 q^{3} -10438.7 q^{5} +22390.8 q^{7} +59049.0 q^{9} +161051. q^{11} +378099. q^{13} -2.53661e6 q^{15} -8.44623e6 q^{17} +1.03187e7 q^{19} +5.44097e6 q^{21} +6.71020e6 q^{23} +6.01392e7 q^{25} +1.43489e7 q^{27} +5.25055e7 q^{29} +5.56053e7 q^{31} +3.91354e7 q^{33} -2.33732e8 q^{35} +1.06153e8 q^{37} +9.18781e7 q^{39} +4.48381e8 q^{41} -4.06725e8 q^{43} -6.16397e8 q^{45} +5.05987e8 q^{47} -1.47598e9 q^{49} -2.05243e9 q^{51} -5.92170e9 q^{53} -1.68117e9 q^{55} +2.50744e9 q^{57} -7.40580e9 q^{59} -4.22853e8 q^{61} +1.32216e9 q^{63} -3.94688e9 q^{65} +1.42882e10 q^{67} +1.63058e9 q^{69} -9.01648e9 q^{71} +1.28539e10 q^{73} +1.46138e10 q^{75} +3.60606e9 q^{77} -1.54192e10 q^{79} +3.48678e9 q^{81} -5.96607e10 q^{83} +8.81680e10 q^{85} +1.27588e10 q^{87} +6.43244e10 q^{89} +8.46595e9 q^{91} +1.35121e10 q^{93} -1.07714e11 q^{95} -1.23779e11 q^{97} +9.50990e9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 972 q^{3} - 7020 q^{5} - 10934 q^{7} + 236196 q^{9} + 644204 q^{11} - 2559568 q^{13} - 1705860 q^{15} - 1606814 q^{17} - 5621902 q^{19} - 2656962 q^{21} + 19030288 q^{23} + 14585700 q^{25} + 57395628 q^{27}+ \cdots + 38039601996 q^{99}+O(q^{100}) \) 4 * q + 972 * q^3 - 7020 * q^5 - 10934 * q^7 + 236196 * q^9 + 644204 * q^11 - 2559568 * q^13 - 1705860 * q^15 - 1606814 * q^17 - 5621902 * q^19 - 2656962 * q^21 + 19030288 * q^23 + 14585700 * q^25 + 57395628 * q^27 + 146818034 * q^29 - 158103664 * q^31 + 156541572 * q^33 - 108199300 * q^35 - 770947012 * q^37 - 621975024 * q^39 - 582529706 * q^41 - 2203144746 * q^43 - 414523980 * q^45 - 1693221664 * q^47 - 1348303344 * q^49 - 390455802 * q^51 - 2342022320 * q^53 - 1130578020 * q^55 - 1366122186 * q^57 - 8715508320 * q^59 - 2069677236 * q^61 - 645641766 * q^63 - 15698175680 * q^65 - 5402316304 * q^67 + 4624359984 * q^69 - 13967424104 * q^71 + 3355935748 * q^73 + 3544325100 * q^75 - 1760931634 * q^77 + 32830867762 * q^79 + 13947137604 * q^81 + 10761308844 * q^83 - 7372813140 * q^85 + 35676782262 * q^87 - 51453668352 * q^89 - 15456476028 * q^91 - 38419190352 * q^93 - 56741221900 * q^95 - 118256509380 * q^97 + 38039601996 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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