LMFDB - Embedded newform 2070.2.a.z.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2070,2,Mod(1,2070)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2070, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2070.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2070 = 2 \cdot 3^{2} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2070.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:	$16.5290332184$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.1101.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 9x + 12 $ x^3 - x^2 - 9*x + 12
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 230)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.68740$ of defining polynomial
Character	$\chi$	$=$	2070.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -4.59692 q^{7} -1.00000 q^{8} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -4.59692 q^{7} -1.00000 q^{8} -1.00000 q^{10} -5.13163 q^{11} -1.22212 q^{13} +4.59692 q^{14} +1.00000 q^{16} +4.68740 q^{17} -4.59692 q^{19} +1.00000 q^{20} +5.13163 q^{22} +1.00000 q^{23} +1.00000 q^{25} +1.22212 q^{26} -4.59692 q^{28} -3.37480 q^{29} -0.777884 q^{31} -1.00000 q^{32} -4.68740 q^{34} -4.59692 q^{35} +5.81903 q^{37} +4.59692 q^{38} -1.00000 q^{40} +8.50643 q^{41} +8.00000 q^{43} -5.13163 q^{44} -1.00000 q^{46} +6.44423 q^{47} +14.1316 q^{49} -1.00000 q^{50} -1.22212 q^{52} +6.00000 q^{53} -5.13163 q^{55} +4.59692 q^{56} +3.37480 q^{58} -9.37480 q^{59} +10.9507 q^{61} +0.777884 q^{62} +1.00000 q^{64} -1.22212 q^{65} +15.6381 q^{67} +4.68740 q^{68} +4.59692 q^{70} -1.31260 q^{71} -4.44423 q^{73} -5.81903 q^{74} -4.59692 q^{76} +23.5897 q^{77} -4.88847 q^{79} +1.00000 q^{80} -8.50643 q^{82} +3.81903 q^{83} +4.68740 q^{85} -8.00000 q^{86} +5.13163 q^{88} -8.93057 q^{89} +5.61797 q^{91} +1.00000 q^{92} -6.44423 q^{94} -4.59692 q^{95} -18.0622 q^{97} -14.1316 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{2} + 3 q^{4} + 3 q^{5} + 3 q^{7} - 3 q^{8}+O(q^{10}) $ 3 * q - 3 * q^2 + 3 * q^4 + 3 * q^5 + 3 * q^7 - 3 * q^8	$ 3 q - 3 q^{2} + 3 q^{4} + 3 q^{5} + 3 q^{7} - 3 q^{8} - 3 q^{10} - 3 q^{11} - q^{13} - 3 q^{14} + 3 q^{16} + 7 q^{17} + 3 q^{19} + 3 q^{20} + 3 q^{22} + 3 q^{23} + 3 q^{25} + q^{26} + 3 q^{28} + 4 q^{29} - 5 q^{31} - 3 q^{32} - 7 q^{34} + 3 q^{35} - 2 q^{37} - 3 q^{38} - 3 q^{40} - q^{41} + 24 q^{43} - 3 q^{44} - 3 q^{46} + 14 q^{47} + 30 q^{49} - 3 q^{50} - q^{52} + 18 q^{53} - 3 q^{55} - 3 q^{56} - 4 q^{58} - 14 q^{59} + q^{61} + 5 q^{62} + 3 q^{64} - q^{65} + 8 q^{67} + 7 q^{68} - 3 q^{70} - 11 q^{71} - 8 q^{73} + 2 q^{74} + 3 q^{76} + 24 q^{77} - 4 q^{79} + 3 q^{80} + q^{82} - 8 q^{83} + 7 q^{85} - 24 q^{86} + 3 q^{88} - 18 q^{89} + q^{91} + 3 q^{92} - 14 q^{94} + 3 q^{95} - 33 q^{97} - 30 q^{98}+O(q^{100}) $ 3 * q - 3 * q^2 + 3 * q^4 + 3 * q^5 + 3 * q^7 - 3 * q^8 - 3 * q^10 - 3 * q^11 - q^13 - 3 * q^14 + 3 * q^16 + 7 * q^17 + 3 * q^19 + 3 * q^20 + 3 * q^22 + 3 * q^23 + 3 * q^25 + q^26 + 3 * q^28 + 4 * q^29 - 5 * q^31 - 3 * q^32 - 7 * q^34 + 3 * q^35 - 2 * q^37 - 3 * q^38 - 3 * q^40 - q^41 + 24 * q^43 - 3 * q^44 - 3 * q^46 + 14 * q^47 + 30 * q^49 - 3 * q^50 - q^52 + 18 * q^53 - 3 * q^55 - 3 * q^56 - 4 * q^58 - 14 * q^59 + q^61 + 5 * q^62 + 3 * q^64 - q^65 + 8 * q^67 + 7 * q^68 - 3 * q^70 - 11 * q^71 - 8 * q^73 + 2 * q^74 + 3 * q^76 + 24 * q^77 - 4 * q^79 + 3 * q^80 + q^82 - 8 * q^83 + 7 * q^85 - 24 * q^86 + 3 * q^88 - 18 * q^89 + q^91 + 3 * q^92 - 14 * q^94 + 3 * q^95 - 33 * q^97 - 30 * q^98

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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−4.59692	−1.73747	−0.868735	−	0.495277i	$-0.835067\pi$
$7$	−4.59692	−1.73747	−0.868735	+	0.495277i	$0.835067\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	−1.00000	−0.316228
$11$	−5.13163	−1.54725	−0.773623	−	0.633647i	$-0.781557\pi$
$11$	−5.13163	−1.54725	−0.773623	+	0.633647i	$0.781557\pi$
$12$	0	0
$13$	−1.22212	−0.338954	−0.169477	−	0.985534i	$-0.554208\pi$
$13$	−1.22212	−0.338954	−0.169477	+	0.985534i	$0.554208\pi$
$14$	4.59692	1.22858
$15$	0	0
$16$	1.00000	0.250000
$17$	4.68740	1.13686	0.568431	−	0.822731i	$-0.307551\pi$
$17$	4.68740	1.13686	0.568431	+	0.822731i	$0.307551\pi$
$18$	0	0
$19$	−4.59692	−1.05460	−0.527302	−	0.849678i	$-0.676796\pi$
$19$	−4.59692	−1.05460	−0.527302	+	0.849678i	$0.676796\pi$
$20$	1.00000	0.223607
$21$	0	0
$22$	5.13163	1.09407
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	1.00000	0.200000
$26$	1.22212	0.239677
$27$	0	0
$28$	−4.59692	−0.868735
$29$	−3.37480	−0.626684	−0.313342	−	0.949640i	$-0.601449\pi$
$29$	−3.37480	−0.626684	−0.313342	+	0.949640i	$0.601449\pi$
$30$	0	0
$31$	−0.777884	−0.139712	−0.0698560	−	0.997557i	$-0.522254\pi$
$31$	−0.777884	−0.139712	−0.0698560	+	0.997557i	$0.522254\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−4.68740	−0.803882
$35$	−4.59692	−0.777021
$36$	0	0
$37$	5.81903	0.956643	0.478321	−	0.878185i	$-0.341245\pi$
$37$	5.81903	0.956643	0.478321	+	0.878185i	$0.341245\pi$
$38$	4.59692	0.745718
$39$	0	0
$40$	−1.00000	−0.158114
$41$	8.50643	1.32848	0.664241	−	0.747519i	$-0.268755\pi$
$41$	8.50643	1.32848	0.664241	+	0.747519i	$0.268755\pi$
$42$	0	0
$43$	8.00000	1.21999	0.609994	−	0.792406i	$-0.291172\pi$
$43$	8.00000	1.21999	0.609994	+	0.792406i	$0.291172\pi$
$44$	−5.13163	−0.773623
$45$	0	0
$46$	−1.00000	−0.147442
$47$	6.44423	0.939988	0.469994	−	0.882670i	$-0.344256\pi$
$47$	6.44423	0.939988	0.469994	+	0.882670i	$0.344256\pi$
$48$	0	0
$49$	14.1316	2.01880
$50$	−1.00000	−0.141421
$51$	0	0
$52$	−1.22212	−0.169477
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	0	0
$55$	−5.13163	−0.691949
$56$	4.59692	0.614289
$57$	0	0
$58$	3.37480	0.443133
$59$	−9.37480	−1.22049	−0.610247	−	0.792211i	$-0.708930\pi$
$59$	−9.37480	−1.22049	−0.610247	+	0.792211i	$0.708930\pi$
$60$	0	0
$61$	10.9507	1.40209	0.701044	−	0.713118i	$-0.252717\pi$
$61$	10.9507	1.40209	0.701044	+	0.713118i	$0.252717\pi$
$62$	0.777884	0.0987913
$63$	0	0
$64$	1.00000	0.125000
$65$	−1.22212	−0.151585
$66$	0	0
$67$	15.6381	1.91049	0.955247	−	0.295810i	$-0.0955895\pi$
$67$	15.6381	1.91049	0.955247	+	0.295810i	$0.0955895\pi$
$68$	4.68740	0.568431
$69$	0	0
$70$	4.59692	0.549436
$71$	−1.31260	−0.155777	−0.0778885	−	0.996962i	$-0.524818\pi$
$71$	−1.31260	−0.155777	−0.0778885	+	0.996962i	$0.524818\pi$
$72$	0	0
$73$	−4.44423	−0.520158	−0.260079	−	0.965587i	$-0.583749\pi$
$73$	−4.44423	−0.520158	−0.260079	+	0.965587i	$0.583749\pi$
$74$	−5.81903	−0.676449
$75$	0	0
$76$	−4.59692	−0.527302
$77$	23.5897	2.68829
$78$	0	0
$79$	−4.88847	−0.549995	−0.274998	−	0.961445i	$-0.588677\pi$
$79$	−4.88847	−0.549995	−0.274998	+	0.961445i	$0.588677\pi$
$80$	1.00000	0.111803
$81$	0	0
$82$	−8.50643	−0.939378
$83$	3.81903	0.419193	0.209597	−	0.977788i	$-0.432785\pi$
$83$	3.81903	0.419193	0.209597	+	0.977788i	$0.432785\pi$
$84$	0	0
$85$	4.68740	0.508420
$86$	−8.00000	−0.862662
$87$	0	0
$88$	5.13163	0.547034
$89$	−8.93057	−0.946638	−0.473319	−	0.880891i	$-0.656944\pi$
$89$	−8.93057	−0.946638	−0.473319	+	0.880891i	$0.656944\pi$
$90$	0	0
$91$	5.61797	0.588923
$92$	1.00000	0.104257
$93$	0	0
$94$	−6.44423	−0.664672
$95$	−4.59692	−0.471634
$96$	0	0
$97$	−18.0622	−1.83394	−0.916969	−	0.398958i	$-0.869372\pi$
$97$	−18.0622	−1.83394	−0.916969	+	0.398958i	$0.869372\pi$
$98$	−14.1316	−1.42751
$99$	0	0
$100$	1.00000	0.100000
$101$	−3.37480	−0.335805	−0.167903	−	0.985804i	$-0.553699\pi$
$101$	−3.37480	−0.335805	−0.167903	+	0.985804i	$0.553699\pi$
$102$	0	0
$103$	13.1316	1.29390	0.646949	−	0.762533i	$-0.276045\pi$
$103$	13.1316	1.29390	0.646949	+	0.762533i	$0.276045\pi$
$104$	1.22212	0.119838
$105$	0	0
$106$	−6.00000	−0.582772
$107$	−12.3054	−1.18960	−0.594802	−	0.803872i	$-0.702770\pi$
$107$	−12.3054	−1.18960	−0.594802	+	0.803872i	$0.702770\pi$
$108$	0	0
$109$	−10.5969	−1.01500	−0.507500	−	0.861652i	$-0.669430\pi$
$109$	−10.5969	−1.01500	−0.507500	+	0.861652i	$0.669430\pi$
$110$	5.13163	0.489282
$111$	0	0
$112$	−4.59692	−0.434368
$113$	6.00000	0.564433	0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000	0.564433	0.282216	+	0.959351i	$0.408930\pi$
$114$	0	0
$115$	1.00000	0.0932505
$116$	−3.37480	−0.313342
$117$	0	0
$118$	9.37480	0.863020
$119$	−21.5476	−1.97526
$120$	0	0
$121$	15.3337	1.39397
$122$	−10.9507	−0.991427
$123$	0	0
$124$	−0.777884	−0.0698560
$125$	1.00000	0.0894427
$126$	0	0
$127$	−2.26326	−0.200832	−0.100416	−	0.994946i	$-0.532017\pi$
$127$	−2.26326	−0.200832	−0.100416	+	0.994946i	$0.532017\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	1.22212	0.107187
$131$	−2.93057	−0.256045	−0.128022	−	0.991771i	$-0.540863\pi$
$131$	−2.93057	−0.256045	−0.128022	+	0.991771i	$0.540863\pi$
$132$	0	0
$133$	21.1316	1.83234
$134$	−15.6381	−1.35092
$135$	0	0
$136$	−4.68740	−0.401941
$137$	19.7907	1.69084	0.845419	−	0.534104i	$-0.179351\pi$
$137$	19.7907	1.69084	0.845419	+	0.534104i	$0.179351\pi$
$138$	0	0
$139$	6.26326	0.531243	0.265622	−	0.964077i	$-0.414423\pi$
$139$	6.26326	0.531243	0.265622	+	0.964077i	$0.414423\pi$
$140$	−4.59692	−0.388510
$141$	0	0
$142$	1.31260	0.110151
$143$	6.27145	0.524445
$144$	0	0
$145$	−3.37480	−0.280262
$146$	4.44423	0.367807
$147$	0	0
$148$	5.81903	0.478321
$149$	19.7907	1.62132	0.810661	−	0.585516i	$-0.199108\pi$
$149$	19.7907	1.62132	0.810661	+	0.585516i	$0.199108\pi$
$150$	0	0
$151$	4.06220	0.330577	0.165289	−	0.986245i	$-0.447144\pi$
$151$	4.06220	0.330577	0.165289	+	0.986245i	$0.447144\pi$
$152$	4.59692	0.372859
$153$	0	0
$154$	−23.5897	−1.90091
$155$	−0.777884	−0.0624811
$156$	0	0
$157$	4.62520	0.369131	0.184566	−	0.982820i	$-0.440912\pi$
$157$	4.62520	0.369131	0.184566	+	0.982820i	$0.440912\pi$
$158$	4.88847	0.388905
$159$	0	0
$160$	−1.00000	−0.0790569
$161$	−4.59692	−0.362288
$162$	0	0
$163$	12.4159	0.972492	0.486246	−	0.873822i	$-0.338366\pi$
$163$	12.4159	0.972492	0.486246	+	0.873822i	$0.338366\pi$
$164$	8.50643	0.664241
$165$	0	0
$166$	−3.81903	−0.296414
$167$	−12.8885	−0.997339	−0.498670	−	0.866792i	$-0.666178\pi$
$167$	−12.8885	−0.997339	−0.498670	+	0.866792i	$0.666178\pi$
$168$	0	0
$169$	−11.5064	−0.885110
$170$	−4.68740	−0.359507
$171$	0	0
$172$	8.00000	0.609994
$173$	10.2432	0.778774	0.389387	−	0.921074i	$-0.372687\pi$
$173$	10.2432	0.778774	0.389387	+	0.921074i	$0.372687\pi$
$174$	0	0
$175$	−4.59692	−0.347494
$176$	−5.13163	−0.386811
$177$	0	0
$178$	8.93057	0.669374
$179$	13.1938	0.986153	0.493077	−	0.869986i	$-0.335872\pi$
$179$	13.1938	0.986153	0.493077	+	0.869986i	$0.335872\pi$
$180$	0	0
$181$	15.7907	1.17372	0.586858	−	0.809690i	$-0.300364\pi$
$181$	15.7907	1.17372	0.586858	+	0.809690i	$0.300364\pi$
$182$	−5.61797	−0.416431
$183$	0	0
$184$	−1.00000	−0.0737210
$185$	5.81903	0.427824
$186$	0	0
$187$	−24.0540	−1.75900
$188$	6.44423	0.469994
$189$	0	0
$190$	4.59692	0.333495
$191$	−16.1244	−1.16672	−0.583360	−	0.812214i	$-0.698262\pi$
$191$	−16.1244	−1.16672	−0.583360	+	0.812214i	$0.698262\pi$
$192$	0	0
$193$	−17.9434	−1.29160	−0.645798	−	0.763508i	$-0.723475\pi$
$193$	−17.9434	−1.29160	−0.645798	+	0.763508i	$0.723475\pi$
$194$	18.0622	1.29679
$195$	0	0
$196$	14.1316	1.00940
$197$	−5.88123	−0.419020	−0.209510	−	0.977806i	$-0.567187\pi$
$197$	−5.88123	−0.419020	−0.209510	+	0.977806i	$0.567187\pi$
$198$	0	0
$199$	6.56863	0.465638	0.232819	−	0.972520i	$-0.425205\pi$
$199$	6.56863	0.465638	0.232819	+	0.972520i	$0.425205\pi$
$200$	−1.00000	−0.0707107
$201$	0	0
$202$	3.37480	0.237450
$203$	15.5137	1.08885
$204$	0	0
$205$	8.50643	0.594115
$206$	−13.1316	−0.914924
$207$	0	0
$208$	−1.22212	−0.0847385
$209$	23.5897	1.63173
$210$	0	0
$211$	−15.4571	−1.06411	−0.532055	−	0.846710i	$-0.678580\pi$
$211$	−15.4571	−1.06411	−0.532055	+	0.846710i	$0.678580\pi$
$212$	6.00000	0.412082
$213$	0	0
$214$	12.3054	0.841177
$215$	8.00000	0.545595
$216$	0	0
$217$	3.57587	0.242746
$218$	10.5969	0.717714
$219$	0	0
$220$	−5.13163	−0.345975
$221$	−5.72855	−0.385344
$222$	0	0
$223$	−10.7496	−0.719846	−0.359923	−	0.932982i	$-0.617197\pi$
$223$	−10.7496	−0.719846	−0.359923	+	0.932982i	$0.617197\pi$
$224$	4.59692	0.307144
$225$	0	0
$226$	−6.00000	−0.399114
$227$	12.3054	0.816736	0.408368	−	0.912817i	$-0.366098\pi$
$227$	12.3054	0.816736	0.408368	+	0.912817i	$0.366098\pi$
$228$	0	0
$229$	9.63806	0.636901	0.318451	−	0.947939i	$-0.396838\pi$
$229$	9.63806	0.636901	0.318451	+	0.947939i	$0.396838\pi$
$230$	−1.00000	−0.0659380
$231$	0	0
$232$	3.37480	0.221566
$233$	−13.9434	−0.913464	−0.456732	−	0.889604i	$-0.650980\pi$
$233$	−13.9434	−0.913464	−0.456732	+	0.889604i	$0.650980\pi$
$234$	0	0
$235$	6.44423	0.420375
$236$	−9.37480	−0.610247
$237$	0	0
$238$	21.5476	1.39672
$239$	26.3877	1.70688	0.853438	−	0.521194i	$-0.174513\pi$
$239$	26.3877	1.70688	0.853438	+	0.521194i	$0.174513\pi$
$240$	0	0
$241$	−7.06943	−0.455382	−0.227691	−	0.973733i	$-0.573118\pi$
$241$	−7.06943	−0.455382	−0.227691	+	0.973733i	$0.573118\pi$
$242$	−15.3337	−0.985684
$243$	0	0
$244$	10.9507	0.701044
$245$	14.1316	0.902837
$246$	0	0
$247$	5.61797	0.357463
$248$	0.777884	0.0493957
$249$	0	0
$250$	−1.00000	−0.0632456
$251$	24.9023	1.57182	0.785909	−	0.618342i	$-0.212195\pi$
$251$	24.9023	1.57182	0.785909	+	0.618342i	$0.212195\pi$
$252$	0	0
$253$	−5.13163	−0.322623
$254$	2.26326	0.142010
$255$	0	0
$256$	1.00000	0.0625000
$257$	−0.444233	−0.0277105	−0.0138552	−	0.999904i	$-0.504410\pi$
$257$	−0.444233	−0.0277105	−0.0138552	+	0.999904i	$0.504410\pi$
$258$	0	0
$259$	−26.7496	−1.66214
$260$	−1.22212	−0.0757924
$261$	0	0
$262$	2.93057	0.181051
$263$	23.8812	1.47258	0.736290	−	0.676666i	$-0.236576\pi$
$263$	23.8812	1.47258	0.736290	+	0.676666i	$0.236576\pi$
$264$	0	0
$265$	6.00000	0.368577
$266$	−21.1316	−1.29566
$267$	0	0
$268$	15.6381	0.955247
$269$	−16.2633	−0.991589	−0.495794	−	0.868440i	$-0.665123\pi$
$269$	−16.2633	−0.991589	−0.495794	+	0.868440i	$0.665123\pi$
$270$	0	0
$271$	25.6098	1.55568	0.777842	−	0.628460i	$-0.216315\pi$
$271$	25.6098	1.55568	0.777842	+	0.628460i	$0.216315\pi$
$272$	4.68740	0.284215
$273$	0	0
$274$	−19.7907	−1.19560
$275$	−5.13163	−0.309449
$276$	0	0
$277$	2.88847	0.173551	0.0867755	−	0.996228i	$-0.472344\pi$
$277$	2.88847	0.173551	0.0867755	+	0.996228i	$0.472344\pi$
$278$	−6.26326	−0.375646
$279$	0	0
$280$	4.59692	0.274718
$281$	−4.26326	−0.254325	−0.127163	−	0.991882i	$-0.540587\pi$
$281$	−4.26326	−0.254325	−0.127163	+	0.991882i	$0.540587\pi$
$282$	0	0
$283$	30.5686	1.81712	0.908558	−	0.417758i	$-0.137184\pi$
$283$	30.5686	1.81712	0.908558	+	0.417758i	$0.137184\pi$
$284$	−1.31260	−0.0778885
$285$	0	0
$286$	−6.27145	−0.370839
$287$	−39.1033	−2.30820
$288$	0	0
$289$	4.97171	0.292454
$290$	3.37480	0.198175
$291$	0	0
$292$	−4.44423	−0.260079
$293$	6.00000	0.350524	0.175262	−	0.984522i	$-0.443923\pi$
$293$	6.00000	0.350524	0.175262	+	0.984522i	$0.443923\pi$
$294$	0	0
$295$	−9.37480	−0.545822
$296$	−5.81903	−0.338224
$297$	0	0
$298$	−19.7907	−1.14645
$299$	−1.22212	−0.0706768
$300$	0	0
$301$	−36.7753	−2.11969
$302$	−4.06220	−0.233753
$303$	0	0
$304$	−4.59692	−0.263651
$305$	10.9507	0.627033
$306$	0	0
$307$	−8.54853	−0.487891	−0.243945	−	0.969789i	$-0.578442\pi$
$307$	−8.54853	−0.487891	−0.243945	+	0.969789i	$0.578442\pi$
$308$	23.5897	1.34415
$309$	0	0
$310$	0.777884	0.0441808
$311$	7.63806	0.433115	0.216557	−	0.976270i	$-0.430517\pi$
$311$	7.63806	0.433115	0.216557	+	0.976270i	$0.430517\pi$
$312$	0	0
$313$	−18.2350	−1.03070	−0.515351	−	0.856979i	$-0.672338\pi$
$313$	−18.2350	−1.03070	−0.515351	+	0.856979i	$0.672338\pi$
$314$	−4.62520	−0.261015
$315$	0	0
$316$	−4.88847	−0.274998
$317$	16.8602	0.946962	0.473481	−	0.880804i	$-0.342997\pi$
$317$	16.8602	0.946962	0.473481	+	0.880804i	$0.342997\pi$
$318$	0	0
$319$	17.3182	0.969635
$320$	1.00000	0.0559017
$321$	0	0
$322$	4.59692	0.256176
$323$	−21.5476	−1.19894
$324$	0	0
$325$	−1.22212	−0.0677908
$326$	−12.4159	−0.687656
$327$	0	0
$328$	−8.50643	−0.469689
$329$	−29.6236	−1.63320
$330$	0	0
$331$	19.1517	1.05267	0.526337	−	0.850276i	$-0.323565\pi$
$331$	19.1517	1.05267	0.526337	+	0.850276i	$0.323565\pi$
$332$	3.81903	0.209597
$333$	0	0
$334$	12.8885	0.705225
$335$	15.6381	0.854399
$336$	0	0
$337$	1.70845	0.0930652	0.0465326	−	0.998917i	$-0.485183\pi$
$337$	1.70845	0.0930652	0.0465326	+	0.998917i	$0.485183\pi$
$338$	11.5064	0.625867
$339$	0	0
$340$	4.68740	0.254210
$341$	3.99181	0.216169
$342$	0	0
$343$	−32.7835	−1.77014
$344$	−8.00000	−0.431331
$345$	0	0
$346$	−10.2432	−0.550676
$347$	−10.6874	−0.573730	−0.286865	−	0.957971i	$-0.592613\pi$
$347$	−10.6874	−0.573730	−0.286865	+	0.957971i	$0.592613\pi$
$348$	0	0
$349$	−12.3877	−0.663096	−0.331548	−	0.943438i	$-0.607571\pi$
$349$	−12.3877	−0.663096	−0.331548	+	0.943438i	$0.607571\pi$
$350$	4.59692	0.245715
$351$	0	0
$352$	5.13163	0.273517
$353$	−5.45710	−0.290452	−0.145226	−	0.989399i	$-0.546391\pi$
$353$	−5.45710	−0.290452	−0.145226	+	0.989399i	$0.546391\pi$
$354$	0	0
$355$	−1.31260	−0.0696656
$356$	−8.93057	−0.473319
$357$	0	0
$358$	−13.1938	−0.697316
$359$	−20.1810	−1.06511	−0.532555	−	0.846395i	$-0.678768\pi$
$359$	−20.1810	−1.06511	−0.532555	+	0.846395i	$0.678768\pi$
$360$	0	0
$361$	2.13163	0.112191
$362$	−15.7907	−0.829943
$363$	0	0
$364$	5.61797	0.294461
$365$	−4.44423	−0.232622
$366$	0	0
$367$	2.74960	0.143528	0.0717639	−	0.997422i	$-0.477137\pi$
$367$	2.74960	0.143528	0.0717639	+	0.997422i	$0.477137\pi$
$368$	1.00000	0.0521286
$369$	0	0
$370$	−5.81903	−0.302517
$371$	−27.5815	−1.43196
$372$	0	0
$373$	−12.0823	−0.625598	−0.312799	−	0.949819i	$-0.601267\pi$
$373$	−12.0823	−0.625598	−0.312799	+	0.949819i	$0.601267\pi$
$374$	24.0540	1.24380
$375$	0	0
$376$	−6.44423	−0.332336
$377$	4.12440	0.212417
$378$	0	0
$379$	5.25603	0.269984	0.134992	−	0.990847i	$-0.456899\pi$
$379$	5.25603	0.269984	0.134992	+	0.990847i	$0.456899\pi$
$380$	−4.59692	−0.235817
$381$	0	0
$382$	16.1244	0.824996
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	0	0
$385$	23.5897	1.20224
$386$	17.9434	0.913297
$387$	0	0
$388$	−18.0622	−0.916969
$389$	−0.325463	−0.0165017	−0.00825083	−	0.999966i	$-0.502626\pi$
$389$	−0.325463	−0.0165017	−0.00825083	+	0.999966i	$0.502626\pi$
$390$	0	0
$391$	4.68740	0.237052
$392$	−14.1316	−0.713755
$393$	0	0
$394$	5.88123	0.296292
$395$	−4.88847	−0.245965
$396$	0	0
$397$	−17.7568	−0.891190	−0.445595	−	0.895235i	$-0.647008\pi$
$397$	−17.7568	−0.891190	−0.445595	+	0.895235i	$0.647008\pi$
$398$	−6.56863	−0.329256
$399$	0	0
$400$	1.00000	0.0500000
$401$	−3.91770	−0.195641	−0.0978204	−	0.995204i	$-0.531187\pi$
$401$	−3.91770	−0.195641	−0.0978204	+	0.995204i	$0.531187\pi$
$402$	0	0
$403$	0.950664	0.0473560
$404$	−3.37480	−0.167903
$405$	0	0
$406$	−15.5137	−0.769930
$407$	−29.8611	−1.48016
$408$	0	0
$409$	9.58405	0.473901	0.236950	−	0.971522i	$-0.423852\pi$
$409$	9.58405	0.473901	0.236950	+	0.971522i	$0.423852\pi$
$410$	−8.50643	−0.420103
$411$	0	0
$412$	13.1316	0.646949
$413$	43.0952	2.12057
$414$	0	0
$415$	3.81903	0.187469
$416$	1.22212	0.0599192
$417$	0	0
$418$	−23.5897	−1.15381
$419$	20.5265	1.00279	0.501393	−	0.865219i	$-0.332821\pi$
$419$	20.5265	1.00279	0.501393	+	0.865219i	$0.332821\pi$
$420$	0	0
$421$	2.29155	0.111683	0.0558417	−	0.998440i	$-0.482216\pi$
$421$	2.29155	0.111683	0.0558417	+	0.998440i	$0.482216\pi$
$422$	15.4571	0.752440
$423$	0	0
$424$	−6.00000	−0.291386
$425$	4.68740	0.227372
$426$	0	0
$427$	−50.3393	−2.43609
$428$	−12.3054	−0.594802
$429$	0	0
$430$	−8.00000	−0.385794
$431$	8.83189	0.425417	0.212709	−	0.977116i	$-0.431771\pi$
$431$	8.83189	0.425417	0.212709	+	0.977116i	$0.431771\pi$
$432$	0	0
$433$	33.3465	1.60253	0.801266	−	0.598309i	$-0.204160\pi$
$433$	33.3465	1.60253	0.801266	+	0.598309i	$0.204160\pi$
$434$	−3.57587	−0.171647
$435$	0	0
$436$	−10.5969	−0.507500
$437$	−4.59692	−0.219900
$438$	0	0
$439$	−6.02829	−0.287714	−0.143857	−	0.989598i	$-0.545951\pi$
$439$	−6.02829	−0.287714	−0.143857	+	0.989598i	$0.545951\pi$
$440$	5.13163	0.244641
$441$	0	0
$442$	5.72855	0.272479
$443$	15.5275	0.737733	0.368866	−	0.929482i	$-0.379746\pi$
$443$	15.5275	0.737733	0.368866	+	0.929482i	$0.379746\pi$
$444$	0	0
$445$	−8.93057	−0.423349
$446$	10.7496	0.509008
$447$	0	0
$448$	−4.59692	−0.217184
$449$	18.3594	0.866433	0.433216	−	0.901290i	$-0.357379\pi$
$449$	18.3594	0.866433	0.433216	+	0.901290i	$0.357379\pi$
$450$	0	0
$451$	−43.6519	−2.05549
$452$	6.00000	0.282216
$453$	0	0
$454$	−12.3054	−0.577519
$455$	5.61797	0.263374
$456$	0	0
$457$	−11.4992	−0.537910	−0.268955	−	0.963153i	$-0.586678\pi$
$457$	−11.4992	−0.537910	−0.268955	+	0.963153i	$0.586678\pi$
$458$	−9.63806	−0.450357
$459$	0	0
$460$	1.00000	0.0466252
$461$	−1.33270	−0.0620700	−0.0310350	−	0.999518i	$-0.509880\pi$
$461$	−1.33270	−0.0620700	−0.0310350	+	0.999518i	$0.509880\pi$
$462$	0	0
$463$	35.8190	1.66465	0.832326	−	0.554287i	$-0.187009\pi$
$463$	35.8190	1.66465	0.832326	+	0.554287i	$0.187009\pi$
$464$	−3.37480	−0.156671
$465$	0	0
$466$	13.9434	0.645917
$467$	23.7625	1.09960	0.549798	−	0.835298i	$-0.314705\pi$
$467$	23.7625	1.09960	0.549798	+	0.835298i	$0.314705\pi$
$468$	0	0
$469$	−71.8869	−3.31943
$470$	−6.44423	−0.297250
$471$	0	0
$472$	9.37480	0.431510
$473$	−41.0531	−1.88762
$474$	0	0
$475$	−4.59692	−0.210921
$476$	−21.5476	−0.987632
$477$	0	0
$478$	−26.3877	−1.20694
$479$	2.04210	0.0933060	0.0466530	−	0.998911i	$-0.485145\pi$
$479$	2.04210	0.0933060	0.0466530	+	0.998911i	$0.485145\pi$
$480$	0	0
$481$	−7.11153	−0.324258
$482$	7.06943	0.322004
$483$	0	0
$484$	15.3337	0.696984
$485$	−18.0622	−0.820162
$486$	0	0
$487$	−18.6252	−0.843988	−0.421994	−	0.906599i	$-0.638670\pi$
$487$	−18.6252	−0.843988	−0.421994	+	0.906599i	$0.638670\pi$
$488$	−10.9507	−0.495713
$489$	0	0
$490$	−14.1316	−0.638402
$491$	−33.7204	−1.52178	−0.760889	−	0.648882i	$-0.775237\pi$
$491$	−33.7204	−1.52178	−0.760889	+	0.648882i	$0.775237\pi$
$492$	0	0
$493$	−15.8190	−0.712453
$494$	−5.61797	−0.252764
$495$	0	0
$496$	−0.777884	−0.0349280
$497$	6.03391	0.270658
$498$	0	0
$499$	−16.8885	−0.756032	−0.378016	−	0.925799i	$-0.623393\pi$
$499$	−16.8885	−0.756032	−0.378016	+	0.925799i	$0.623393\pi$
$500$	1.00000	0.0447214
$501$	0	0
$502$	−24.9023	−1.11144
$503$	−11.4031	−0.508438	−0.254219	−	0.967147i	$-0.581818\pi$
$503$	−11.4031	−0.508438	−0.254219	+	0.967147i	$0.581818\pi$
$504$	0	0
$505$	−3.37480	−0.150177
$506$	5.13163	0.228129
$507$	0	0
$508$	−2.26326	−0.100416
$509$	1.87560	0.0831346	0.0415673	−	0.999136i	$-0.486765\pi$
$509$	1.87560	0.0831346	0.0415673	+	0.999136i	$0.486765\pi$
$510$	0	0
$511$	20.4298	0.903759
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	0.444233	0.0195943
$515$	13.1316	0.578649
$516$	0	0
$517$	−33.0694	−1.45439
$518$	26.7496	1.17531
$519$	0	0
$520$	1.22212	0.0535933
$521$	−5.11153	−0.223940	−0.111970	−	0.993712i	$-0.535716\pi$
$521$	−5.11153	−0.223940	−0.111970	+	0.993712i	$0.535716\pi$
$522$	0	0
$523$	19.4571	0.850799	0.425400	−	0.905006i	$-0.360134\pi$
$523$	19.4571	0.850799	0.425400	+	0.905006i	$0.360134\pi$
$524$	−2.93057	−0.128022
$525$	0	0
$526$	−23.8812	−1.04127
$527$	−3.64625	−0.158833
$528$	0	0
$529$	1.00000	0.0434783
$530$	−6.00000	−0.260623
$531$	0	0
$532$	21.1316	0.916172
$533$	−10.3958	−0.450294
$534$	0	0
$535$	−12.3054	−0.532007
$536$	−15.6381	−0.675461
$537$	0	0
$538$	16.2633	0.701159
$539$	−72.5183	−3.12359
$540$	0	0
$541$	−2.70750	−0.116404	−0.0582022	−	0.998305i	$-0.518537\pi$
$541$	−2.70750	−0.116404	−0.0582022	+	0.998305i	$0.518537\pi$
$542$	−25.6098	−1.10003
$543$	0	0
$544$	−4.68740	−0.200971
$545$	−10.5969	−0.453922
$546$	0	0
$547$	−1.66635	−0.0712479	−0.0356240	−	0.999365i	$-0.511342\pi$
$547$	−1.66635	−0.0712479	−0.0356240	+	0.999365i	$0.511342\pi$
$548$	19.7907	0.845419
$549$	0	0
$550$	5.13163	0.218814
$551$	15.5137	0.660904
$552$	0	0
$553$	22.4719	0.955601
$554$	−2.88847	−0.122719
$555$	0	0
$556$	6.26326	0.265622
$557$	20.9306	0.886857	0.443428	−	0.896310i	$-0.353762\pi$
$557$	20.9306	0.886857	0.443428	+	0.896310i	$0.353762\pi$
$558$	0	0
$559$	−9.77693	−0.413520
$560$	−4.59692	−0.194255
$561$	0	0
$562$	4.26326	0.179835
$563$	15.2761	0.643812	0.321906	−	0.946772i	$-0.395676\pi$
$563$	15.2761	0.643812	0.321906	+	0.946772i	$0.395676\pi$
$564$	0	0
$565$	6.00000	0.252422
$566$	−30.5686	−1.28490
$567$	0	0
$568$	1.31260	0.0550755
$569$	−20.3877	−0.854695	−0.427348	−	0.904087i	$-0.640552\pi$
$569$	−20.3877	−0.854695	−0.427348	+	0.904087i	$0.640552\pi$
$570$	0	0
$571$	5.49357	0.229899	0.114949	−	0.993371i	$-0.463329\pi$
$571$	5.49357	0.229899	0.114949	+	0.993371i	$0.463329\pi$
$572$	6.27145	0.262223
$573$	0	0
$574$	39.1033	1.63214
$575$	1.00000	0.0417029
$576$	0	0
$577$	17.2761	0.719215	0.359607	−	0.933104i	$-0.382911\pi$
$577$	17.2761	0.719215	0.359607	+	0.933104i	$0.382911\pi$
$578$	−4.97171	−0.206796
$579$	0	0
$580$	−3.37480	−0.140131
$581$	−17.5558	−0.728336
$582$	0	0
$583$	−30.7898	−1.27518
$584$	4.44423	0.183904
$585$	0	0
$586$	−6.00000	−0.247858
$587$	−31.8247	−1.31354	−0.656772	−	0.754089i	$-0.728079\pi$
$587$	−31.8247	−1.31354	−0.656772	+	0.754089i	$0.728079\pi$
$588$	0	0
$589$	3.57587	0.147341
$590$	9.37480	0.385954
$591$	0	0
$592$	5.81903	0.239161
$593$	12.4442	0.511023	0.255512	−	0.966806i	$-0.417756\pi$
$593$	12.4442	0.511023	0.255512	+	0.966806i	$0.417756\pi$
$594$	0	0
$595$	−21.5476	−0.883365
$596$	19.7907	0.810661
$597$	0	0
$598$	1.22212	0.0499761
$599$	−17.4370	−0.712456	−0.356228	−	0.934399i	$-0.615937\pi$
$599$	−17.4370	−0.712456	−0.356228	+	0.934399i	$0.615937\pi$
$600$	0	0
$601$	−0.916751	−0.0373951	−0.0186975	−	0.999825i	$-0.505952\pi$
$601$	−0.916751	−0.0373951	−0.0186975	+	0.999825i	$0.505952\pi$
$602$	36.7753	1.49885
$603$	0	0
$604$	4.06220	0.165289
$605$	15.3337	0.623402
$606$	0	0
$607$	36.7753	1.49266	0.746332	−	0.665574i	$-0.231813\pi$
$607$	36.7753	1.49266	0.746332	+	0.665574i	$0.231813\pi$
$608$	4.59692	0.186430
$609$	0	0
$610$	−10.9507	−0.443379
$611$	−7.87560	−0.318613
$612$	0	0
$613$	4.38766	0.177216	0.0886080	−	0.996067i	$-0.471758\pi$
$613$	4.38766	0.177216	0.0886080	+	0.996067i	$0.471758\pi$
$614$	8.54853	0.344991
$615$	0	0
$616$	−23.5897	−0.950455
$617$	−40.4499	−1.62845	−0.814225	−	0.580549i	$-0.802838\pi$
$617$	−40.4499	−1.62845	−0.814225	+	0.580549i	$0.802838\pi$
$618$	0	0
$619$	−39.8165	−1.60036	−0.800180	−	0.599760i	$-0.795263\pi$
$619$	−39.8165	−1.60036	−0.800180	+	0.599760i	$0.795263\pi$
$620$	−0.777884	−0.0312406
$621$	0	0
$622$	−7.63806	−0.306258
$623$	41.0531	1.64476
$624$	0	0
$625$	1.00000	0.0400000
$626$	18.2350	0.728816
$627$	0	0
$628$	4.62520	0.184566
$629$	27.2761	1.08757
$630$	0	0
$631$	25.9013	1.03112	0.515558	−	0.856855i	$-0.327585\pi$
$631$	25.9013	1.03112	0.515558	+	0.856855i	$0.327585\pi$
$632$	4.88847	0.194453
$633$	0	0
$634$	−16.8602	−0.669603
$635$	−2.26326	−0.0898149
$636$	0	0
$637$	−17.2705	−0.684282
$638$	−17.3182	−0.685635
$639$	0	0
$640$	−1.00000	−0.0395285
$641$	43.8448	1.73176	0.865882	−	0.500248i	$-0.166758\pi$
$641$	43.8448	1.73176	0.865882	+	0.500248i	$0.166758\pi$
$642$	0	0
$643$	3.94343	0.155514	0.0777568	−	0.996972i	$-0.475224\pi$
$643$	3.94343	0.155514	0.0777568	+	0.996972i	$0.475224\pi$
$644$	−4.59692	−0.181144
$645$	0	0
$646$	21.5476	0.847778
$647$	−24.3456	−0.957123	−0.478561	−	0.878054i	$-0.658842\pi$
$647$	−24.3456	−0.957123	−0.478561	+	0.878054i	$0.658842\pi$
$648$	0	0
$649$	48.1080	1.88841
$650$	1.22212	0.0479353
$651$	0	0
$652$	12.4159	0.486246
$653$	−37.6921	−1.47500	−0.737502	−	0.675344i	$-0.763995\pi$
$653$	−37.6921	−1.47500	−0.737502	+	0.675344i	$0.763995\pi$
$654$	0	0
$655$	−2.93057	−0.114507
$656$	8.50643	0.332120
$657$	0	0
$658$	29.6236	1.15485
$659$	−24.6107	−0.958698	−0.479349	−	0.877624i	$-0.659127\pi$
$659$	−24.6107	−0.958698	−0.479349	+	0.877624i	$0.659127\pi$
$660$	0	0
$661$	27.3126	1.06234	0.531169	−	0.847266i	$-0.321753\pi$
$661$	27.3126	1.06234	0.531169	+	0.847266i	$0.321753\pi$
$662$	−19.1517	−0.744353
$663$	0	0
$664$	−3.81903	−0.148207
$665$	21.1316	0.819450
$666$	0	0
$667$	−3.37480	−0.130673
$668$	−12.8885	−0.498670
$669$	0	0
$670$	−15.6381	−0.604151
$671$	−56.1948	−2.16938
$672$	0	0
$673$	22.5265	0.868334	0.434167	−	0.900832i	$-0.357043\pi$
$673$	22.5265	0.868334	0.434167	+	0.900832i	$0.357043\pi$
$674$	−1.70845	−0.0658070
$675$	0	0
$676$	−11.5064	−0.442555
$677$	6.00000	0.230599	0.115299	−	0.993331i	$-0.463217\pi$
$677$	6.00000	0.230599	0.115299	+	0.993331i	$0.463217\pi$
$678$	0	0
$679$	83.0304	3.18641
$680$	−4.68740	−0.179754
$681$	0	0
$682$	−3.99181	−0.152854
$683$	31.9974	1.22435	0.612174	−	0.790723i	$-0.290295\pi$
$683$	31.9974	1.22435	0.612174	+	0.790723i	$0.290295\pi$
$684$	0	0
$685$	19.7907	0.756166
$686$	32.7835	1.25168
$687$	0	0
$688$	8.00000	0.304997
$689$	−7.33270	−0.279354
$690$	0	0
$691$	−2.80617	−0.106752	−0.0533758	−	0.998574i	$-0.516998\pi$
$691$	−2.80617	−0.106752	−0.0533758	+	0.998574i	$0.516998\pi$
$692$	10.2432	0.389387
$693$	0	0
$694$	10.6874	0.405688
$695$	6.26326	0.237579
$696$	0	0
$697$	39.8730	1.51030
$698$	12.3877	0.468880
$699$	0	0
$700$	−4.59692	−0.173747
$701$	−8.37385	−0.316276	−0.158138	−	0.987417i	$-0.550549\pi$
$701$	−8.37385	−0.316276	−0.158138	+	0.987417i	$0.550549\pi$
$702$	0	0
$703$	−26.7496	−1.00888
$704$	−5.13163	−0.193406
$705$	0	0
$706$	5.45710	0.205381
$707$	15.5137	0.583451
$708$	0	0
$709$	21.2139	0.796706	0.398353	−	0.917232i	$-0.369582\pi$
$709$	21.2139	0.796706	0.398353	+	0.917232i	$0.369582\pi$
$710$	1.31260	0.0492610
$711$	0	0
$712$	8.93057	0.334687
$713$	−0.777884	−0.0291320
$714$	0	0
$715$	6.27145	0.234539
$716$	13.1938	0.493077
$717$	0	0
$718$	20.1810	0.753147
$719$	28.1106	1.04835	0.524174	−	0.851611i	$-0.324374\pi$
$719$	28.1106	1.04835	0.524174	+	0.851611i	$0.324374\pi$
$720$	0	0
$721$	−60.3650	−2.24811
$722$	−2.13163	−0.0793311
$723$	0	0
$724$	15.7907	0.586858
$725$	−3.37480	−0.125337
$726$	0	0
$727$	−39.0330	−1.44765	−0.723826	−	0.689982i	$-0.757618\pi$
$727$	−39.0330	−1.44765	−0.723826	+	0.689982i	$0.757618\pi$
$728$	−5.61797	−0.208216
$729$	0	0
$730$	4.44423	0.164488
$731$	37.4992	1.38696
$732$	0	0
$733$	47.1373	1.74105	0.870527	−	0.492120i	$-0.163778\pi$
$733$	47.1373	1.74105	0.870527	+	0.492120i	$0.163778\pi$
$734$	−2.74960	−0.101490
$735$	0	0
$736$	−1.00000	−0.0368605
$737$	−80.2488	−2.95600
$738$	0	0
$739$	−24.5265	−0.902223	−0.451111	−	0.892468i	$-0.648972\pi$
$739$	−24.5265	−0.902223	−0.451111	+	0.892468i	$0.648972\pi$
$740$	5.81903	0.213912
$741$	0	0
$742$	27.5815	1.01255
$743$	7.34651	0.269517	0.134759	−	0.990878i	$-0.456974\pi$
$743$	7.34651	0.269517	0.134759	+	0.990878i	$0.456974\pi$
$744$	0	0
$745$	19.7907	0.725077
$746$	12.0823	0.442364
$747$	0	0
$748$	−24.0540	−0.879502
$749$	56.5667	2.06690
$750$	0	0
$751$	11.2359	0.410005	0.205002	−	0.978761i	$-0.434280\pi$
$751$	11.2359	0.410005	0.205002	+	0.978761i	$0.434280\pi$
$752$	6.44423	0.234997
$753$	0	0
$754$	−4.12440	−0.150202
$755$	4.06220	0.147839
$756$	0	0
$757$	49.1794	1.78745	0.893727	−	0.448611i	$-0.148081\pi$
$757$	49.1794	1.78745	0.893727	+	0.448611i	$0.148081\pi$
$758$	−5.25603	−0.190908
$759$	0	0
$760$	4.59692	0.166748
$761$	−43.2478	−1.56773	−0.783867	−	0.620929i	$-0.786755\pi$
$761$	−43.2478	−1.56773	−0.783867	+	0.620929i	$0.786755\pi$
$762$	0	0
$763$	48.7131	1.76353
$764$	−16.1244	−0.583360
$765$	0	0
$766$	0	0
$767$	11.4571	0.413692
$768$	0	0
$769$	−39.6638	−1.43031	−0.715156	−	0.698964i	$-0.753645\pi$
$769$	−39.6638	−1.43031	−0.715156	+	0.698964i	$0.753645\pi$
$770$	−23.5897	−0.850113
$771$	0	0
$772$	−17.9434	−0.645798
$773$	2.18097	0.0784440	0.0392220	−	0.999231i	$-0.487512\pi$
$773$	2.18097	0.0784440	0.0392220	+	0.999231i	$0.487512\pi$
$774$	0	0
$775$	−0.777884	−0.0279424
$776$	18.0622	0.648395
$777$	0	0
$778$	0.325463	0.0116684
$779$	−39.1033	−1.40102
$780$	0	0
$781$	6.73578	0.241025
$782$	−4.68740	−0.167621
$783$	0	0
$784$	14.1316	0.504701
$785$	4.62520	0.165080
$786$	0	0
$787$	4.76407	0.169821	0.0849103	−	0.996389i	$-0.472940\pi$
$787$	4.76407	0.169821	0.0849103	+	0.996389i	$0.472940\pi$
$788$	−5.88123	−0.209510
$789$	0	0
$790$	4.88847	0.173924
$791$	−27.5815	−0.980685
$792$	0	0
$793$	−13.3830	−0.475244
$794$	17.7568	0.630166
$795$	0	0
$796$	6.56863	0.232819
$797$	−39.7204	−1.40697	−0.703484	−	0.710711i	$-0.748373\pi$
$797$	−39.7204	−1.40697	−0.703484	+	0.710711i	$0.748373\pi$
$798$	0	0
$799$	30.2067	1.06864
$800$	−1.00000	−0.0353553
$801$	0	0
$802$	3.91770	0.138339
$803$	22.8062	0.804812
$804$	0	0
$805$	−4.59692	−0.162020
$806$	−0.950664	−0.0334857
$807$	0	0
$808$	3.37480	0.118725
$809$	−46.8941	−1.64871	−0.824354	−	0.566074i	$-0.808462\pi$
$809$	−46.8941	−1.64871	−0.824354	+	0.566074i	$0.808462\pi$
$810$	0	0
$811$	−36.8319	−1.29334	−0.646671	−	0.762769i	$-0.723839\pi$
$811$	−36.8319	−1.29334	−0.646671	+	0.762769i	$0.723839\pi$
$812$	15.5137	0.544423
$813$	0	0
$814$	29.8611	1.04663
$815$	12.4159	0.434912
$816$	0	0
$817$	−36.7753	−1.28661
$818$	−9.58405	−0.335099
$819$	0	0
$820$	8.50643	0.297057
$821$	18.6107	0.649519	0.324759	−	0.945797i	$-0.394717\pi$
$821$	18.6107	0.649519	0.324759	+	0.945797i	$0.394717\pi$
$822$	0	0
$823$	−9.31823	−0.324813	−0.162407	−	0.986724i	$-0.551926\pi$
$823$	−9.31823	−0.324813	−0.162407	+	0.986724i	$0.551926\pi$
$824$	−13.1316	−0.457462
$825$	0	0
$826$	−43.0952	−1.49947
$827$	−53.0129	−1.84344	−0.921719	−	0.387858i	$-0.873215\pi$
$827$	−53.0129	−1.84344	−0.921719	+	0.387858i	$0.873215\pi$
$828$	0	0
$829$	−25.2761	−0.877876	−0.438938	−	0.898517i	$-0.644645\pi$
$829$	−25.2761	−0.877876	−0.438938	+	0.898517i	$0.644645\pi$
$830$	−3.81903	−0.132561
$831$	0	0
$832$	−1.22212	−0.0423693
$833$	66.2406	2.29510
$834$	0	0
$835$	−12.8885	−0.446024
$836$	23.5897	0.815866
$837$	0	0
$838$	−20.5265	−0.709077
$839$	−11.6946	−0.403744	−0.201872	−	0.979412i	$-0.564702\pi$
$839$	−11.6946	−0.403744	−0.201872	+	0.979412i	$0.564702\pi$
$840$	0	0
$841$	−17.6107	−0.607267
$842$	−2.29155	−0.0789720
$843$	0	0
$844$	−15.4571	−0.532055
$845$	−11.5064	−0.395833
$846$	0	0
$847$	−70.4875	−2.42198
$848$	6.00000	0.206041
$849$	0	0
$850$	−4.68740	−0.160776
$851$	5.81903	0.199474
$852$	0	0
$853$	38.6371	1.32291	0.661455	−	0.749985i	$-0.269939\pi$
$853$	38.6371	1.32291	0.661455	+	0.749985i	$0.269939\pi$
$854$	50.3393	1.72257
$855$	0	0
$856$	12.3054	0.420589
$857$	−26.2488	−0.896642	−0.448321	−	0.893873i	$-0.647978\pi$
$857$	−26.2488	−0.896642	−0.448321	+	0.893873i	$0.647978\pi$
$858$	0	0
$859$	37.5558	1.28139	0.640693	−	0.767797i	$-0.278647\pi$
$859$	37.5558	1.28139	0.640693	+	0.767797i	$0.278647\pi$
$860$	8.00000	0.272798
$861$	0	0
$862$	−8.83189	−0.300816
$863$	21.9855	0.748396	0.374198	−	0.927349i	$-0.377918\pi$
$863$	21.9855	0.748396	0.374198	+	0.927349i	$0.377918\pi$
$864$	0	0
$865$	10.2432	0.348278
$866$	−33.3465	−1.13316
$867$	0	0
$868$	3.57587	0.121373
$869$	25.0858	0.850978
$870$	0	0
$871$	−19.1115	−0.647570
$872$	10.5969	0.358857
$873$	0	0
$874$	4.59692	0.155493
$875$	−4.59692	−0.155404
$876$	0	0
$877$	−36.5064	−1.23273	−0.616367	−	0.787459i	$-0.711396\pi$
$877$	−36.5064	−1.23273	−0.616367	+	0.787459i	$0.711396\pi$
$878$	6.02829	0.203445
$879$	0	0
$880$	−5.13163	−0.172987
$881$	17.4571	0.588145	0.294072	−	0.955783i	$-0.404989\pi$
$881$	17.4571	0.588145	0.294072	+	0.955783i	$0.404989\pi$
$882$	0	0
$883$	22.9444	0.772140	0.386070	−	0.922470i	$-0.373832\pi$
$883$	22.9444	0.772140	0.386070	+	0.922470i	$0.373832\pi$
$884$	−5.72855	−0.192672
$885$	0	0
$886$	−15.5275	−0.521656
$887$	11.7223	0.393595	0.196798	−	0.980444i	$-0.436946\pi$
$887$	11.7223	0.393595	0.196798	+	0.980444i	$0.436946\pi$
$888$	0	0
$889$	10.4040	0.348940
$890$	8.93057	0.299353
$891$	0	0
$892$	−10.7496	−0.359923
$893$	−29.6236	−0.991316
$894$	0	0
$895$	13.1938	0.441021
$896$	4.59692	0.153572
$897$	0	0
$898$	−18.3594	−0.612660
$899$	2.62520	0.0875554
$900$	0	0
$901$	28.1244	0.936960
$902$	43.6519	1.45345
$903$	0	0
$904$	−6.00000	−0.199557
$905$	15.7907	0.524902
$906$	0	0
$907$	−1.91770	−0.0636763	−0.0318381	−	0.999493i	$-0.510136\pi$
$907$	−1.91770	−0.0636763	−0.0318381	+	0.999493i	$0.510136\pi$
$908$	12.3054	0.408368
$909$	0	0
$910$	−5.61797	−0.186234
$911$	56.8319	1.88292	0.941462	−	0.337118i	$-0.109452\pi$
$911$	56.8319	1.88292	0.941462	+	0.337118i	$0.109452\pi$
$912$	0	0
$913$	−19.5979	−0.648595
$914$	11.4992	0.380360
$915$	0	0
$916$	9.63806	0.318451
$917$	13.4716	0.444870
$918$	0	0
$919$	−26.0257	−0.858509	−0.429255	−	0.903183i	$-0.641224\pi$
$919$	−26.0257	−0.858509	−0.429255	+	0.903183i	$0.641224\pi$
$920$	−1.00000	−0.0329690
$921$	0	0
$922$	1.33270	0.0438901
$923$	1.60415	0.0528013
$924$	0	0
$925$	5.81903	0.191329
$926$	−35.8190	−1.17709
$927$	0	0
$928$	3.37480	0.110783
$929$	17.1115	0.561411	0.280706	−	0.959794i	$-0.409432\pi$
$929$	17.1115	0.561411	0.280706	+	0.959794i	$0.409432\pi$
$930$	0	0
$931$	−64.9619	−2.12904
$932$	−13.9434	−0.456732
$933$	0	0
$934$	−23.7625	−0.777531
$935$	−24.0540	−0.786650
$936$	0	0
$937$	16.3738	0.534910	0.267455	−	0.963570i	$-0.413817\pi$
$937$	16.3738	0.534910	0.267455	+	0.963570i	$0.413817\pi$
$938$	71.8869	2.34719
$939$	0	0
$940$	6.44423	0.210188
$941$	16.4097	0.534940	0.267470	−	0.963566i	$-0.413812\pi$
$941$	16.4097	0.534940	0.267470	+	0.963566i	$0.413812\pi$
$942$	0	0
$943$	8.50643	0.277008
$944$	−9.37480	−0.305124
$945$	0	0
$946$	41.0531	1.33475
$947$	21.3886	0.695037	0.347518	−	0.937673i	$-0.387024\pi$
$947$	21.3886	0.695037	0.347518	+	0.937673i	$0.387024\pi$
$948$	0	0
$949$	5.43137	0.176310
$950$	4.59692	0.149144
$951$	0	0
$952$	21.5476	0.698361
$953$	16.7276	0.541860	0.270930	−	0.962599i	$-0.412669\pi$
$953$	16.7276	0.541860	0.270930	+	0.962599i	$0.412669\pi$
$954$	0	0
$955$	−16.1244	−0.521773
$956$	26.3877	0.853438
$957$	0	0
$958$	−2.04210	−0.0659773
$959$	−90.9764	−2.93778
$960$	0	0
$961$	−30.3949	−0.980481
$962$	7.11153	0.229285
$963$	0	0
$964$	−7.06943	−0.227691
$965$	−17.9434	−0.577619
$966$	0	0
$967$	45.2617	1.45552	0.727758	−	0.685834i	$-0.240562\pi$
$967$	45.2617	1.45552	0.727758	+	0.685834i	$0.240562\pi$
$968$	−15.3337	−0.492842
$969$	0	0
$970$	18.0622	0.579942
$971$	−45.2560	−1.45234	−0.726168	−	0.687518i	$-0.758700\pi$
$971$	−45.2560	−1.45234	−0.726168	+	0.687518i	$0.758700\pi$
$972$	0	0
$973$	−28.7917	−0.923019
$974$	18.6252	0.596790
$975$	0	0
$976$	10.9507	0.350522
$977$	13.3465	0.426993	0.213496	−	0.976944i	$-0.431515\pi$
$977$	13.3465	0.426993	0.213496	+	0.976944i	$0.431515\pi$
$978$	0	0
$979$	45.8284	1.46468
$980$	14.1316	0.451418
$981$	0	0
$982$	33.7204	1.07606
$983$	−5.13163	−0.163674	−0.0818368	−	0.996646i	$-0.526079\pi$
$983$	−5.13163	−0.163674	−0.0818368	+	0.996646i	$0.526079\pi$
$984$	0	0
$985$	−5.88123	−0.187392
$986$	15.8190	0.503781
$987$	0	0
$988$	5.61797	0.178731
$989$	8.00000	0.254385
$990$	0	0
$991$	17.7989	0.565402	0.282701	−	0.959208i	$-0.408770\pi$
$991$	17.7989	0.565402	0.282701	+	0.959208i	$0.408770\pi$
$992$	0.777884	0.0246978
$993$	0	0
$994$	−6.03391	−0.191384
$995$	6.56863	0.208240
$996$	0	0
$997$	40.3877	1.27909	0.639545	−	0.768754i	$-0.279123\pi$
$997$	40.3877	1.27909	0.639545	+	0.768754i	$0.279123\pi$
$998$	16.8885	0.534595
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2070.2.a.z.1.1		3
3.2	odd	2		230.2.a.d.1.3	✓	3
12.11	even	2		1840.2.a.r.1.1		3
15.2	even	4		1150.2.b.j.599.4		6
15.8	even	4		1150.2.b.j.599.3		6
15.14	odd	2		1150.2.a.q.1.1		3
24.5	odd	2		7360.2.a.bz.1.1		3
24.11	even	2		7360.2.a.ce.1.3		3
60.59	even	2		9200.2.a.cf.1.3		3
69.68	even	2		5290.2.a.r.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
230.2.a.d.1.3	✓	3	3.2	odd	2
1150.2.a.q.1.1		3	15.14	odd	2
1150.2.b.j.599.3		6	15.8	even	4
1150.2.b.j.599.4		6	15.2	even	4
1840.2.a.r.1.1		3	12.11	even	2
2070.2.a.z.1.1		3	1.1	even	1	trivial
5290.2.a.r.1.3		3	69.68	even	2
7360.2.a.bz.1.1		3	24.5	odd	2
7360.2.a.ce.1.3		3	24.11	even	2
9200.2.a.cf.1.3		3	60.59	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:	\( N \)	\(=\)	\( 2070 = 2 \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2070.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -4.59692 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -4.59692 q^{7} -1.00000 q^{8} -1.00000 q^{10} -5.13163 q^{11} -1.22212 q^{13} +4.59692 q^{14} +1.00000 q^{16} +4.68740 q^{17} -4.59692 q^{19} +1.00000 q^{20} +5.13163 q^{22} +1.00000 q^{23} +1.00000 q^{25} +1.22212 q^{26} -4.59692 q^{28} -3.37480 q^{29} -0.777884 q^{31} -1.00000 q^{32} -4.68740 q^{34} -4.59692 q^{35} +5.81903 q^{37} +4.59692 q^{38} -1.00000 q^{40} +8.50643 q^{41} +8.00000 q^{43} -5.13163 q^{44} -1.00000 q^{46} +6.44423 q^{47} +14.1316 q^{49} -1.00000 q^{50} -1.22212 q^{52} +6.00000 q^{53} -5.13163 q^{55} +4.59692 q^{56} +3.37480 q^{58} -9.37480 q^{59} +10.9507 q^{61} +0.777884 q^{62} +1.00000 q^{64} -1.22212 q^{65} +15.6381 q^{67} +4.68740 q^{68} +4.59692 q^{70} -1.31260 q^{71} -4.44423 q^{73} -5.81903 q^{74} -4.59692 q^{76} +23.5897 q^{77} -4.88847 q^{79} +1.00000 q^{80} -8.50643 q^{82} +3.81903 q^{83} +4.68740 q^{85} -8.00000 q^{86} +5.13163 q^{88} -8.93057 q^{89} +5.61797 q^{91} +1.00000 q^{92} -6.44423 q^{94} -4.59692 q^{95} -18.0622 q^{97} -14.1316 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{2} + 3 q^{4} + 3 q^{5} + 3 q^{7} - 3 q^{8}+O(q^{10}) \) 3 * q - 3 * q^2 + 3 * q^4 + 3 * q^5 + 3 * q^7 - 3 * q^8	\( 3 q - 3 q^{2} + 3 q^{4} + 3 q^{5} + 3 q^{7} - 3 q^{8} - 3 q^{10} - 3 q^{11} - q^{13} - 3 q^{14} + 3 q^{16} + 7 q^{17} + 3 q^{19} + 3 q^{20} + 3 q^{22} + 3 q^{23} + 3 q^{25} + q^{26} + 3 q^{28} + 4 q^{29} - 5 q^{31} - 3 q^{32} - 7 q^{34} + 3 q^{35} - 2 q^{37} - 3 q^{38} - 3 q^{40} - q^{41} + 24 q^{43} - 3 q^{44} - 3 q^{46} + 14 q^{47} + 30 q^{49} - 3 q^{50} - q^{52} + 18 q^{53} - 3 q^{55} - 3 q^{56} - 4 q^{58} - 14 q^{59} + q^{61} + 5 q^{62} + 3 q^{64} - q^{65} + 8 q^{67} + 7 q^{68} - 3 q^{70} - 11 q^{71} - 8 q^{73} + 2 q^{74} + 3 q^{76} + 24 q^{77} - 4 q^{79} + 3 q^{80} + q^{82} - 8 q^{83} + 7 q^{85} - 24 q^{86} + 3 q^{88} - 18 q^{89} + q^{91} + 3 q^{92} - 14 q^{94} + 3 q^{95} - 33 q^{97} - 30 q^{98}+O(q^{100}) \) 3 * q - 3 * q^2 + 3 * q^4 + 3 * q^5 + 3 * q^7 - 3 * q^8 - 3 * q^10 - 3 * q^11 - q^13 - 3 * q^14 + 3 * q^16 + 7 * q^17 + 3 * q^19 + 3 * q^20 + 3 * q^22 + 3 * q^23 + 3 * q^25 + q^26 + 3 * q^28 + 4 * q^29 - 5 * q^31 - 3 * q^32 - 7 * q^34 + 3 * q^35 - 2 * q^37 - 3 * q^38 - 3 * q^40 - q^41 + 24 * q^43 - 3 * q^44 - 3 * q^46 + 14 * q^47 + 30 * q^49 - 3 * q^50 - q^52 + 18 * q^53 - 3 * q^55 - 3 * q^56 - 4 * q^58 - 14 * q^59 + q^61 + 5 * q^62 + 3 * q^64 - q^65 + 8 * q^67 + 7 * q^68 - 3 * q^70 - 11 * q^71 - 8 * q^73 + 2 * q^74 + 3 * q^76 + 24 * q^77 - 4 * q^79 + 3 * q^80 + q^82 - 8 * q^83 + 7 * q^85 - 24 * q^86 + 3 * q^88 - 18 * q^89 + q^91 + 3 * q^92 - 14 * q^94 + 3 * q^95 - 33 * q^97 - 30 * q^98

Introduction

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