LMFDB - Embedded newform 4050.2.a.bm.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4050,2,Mod(1,4050)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4050, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("4050.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4050 = 2 \cdot 3^{4} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4050.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:	$32.3394128186$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{6}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 6 $ x^2 - 6
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 90)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.44949$ of defining polynomial
Character	$\chi$	$=$	4050.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -4.44949 q^{7} -1.00000 q^{8} -1.44949 q^{11} +2.44949 q^{13} +4.44949 q^{14} +1.00000 q^{16} +3.89898 q^{17} -0.550510 q^{19} +1.44949 q^{22} -2.89898 q^{23} -2.44949 q^{26} -4.44949 q^{28} +6.00000 q^{29} +6.44949 q^{31} -1.00000 q^{32} -3.89898 q^{34} -8.00000 q^{37} +0.550510 q^{38} +1.00000 q^{41} -7.44949 q^{43} -1.44949 q^{44} +2.89898 q^{46} +0.449490 q^{47} +12.7980 q^{49} +2.44949 q^{52} +8.44949 q^{53} +4.44949 q^{56} -6.00000 q^{58} +11.2474 q^{59} -0.449490 q^{61} -6.44949 q^{62} +1.00000 q^{64} -9.44949 q^{67} +3.89898 q^{68} -2.44949 q^{71} +4.79796 q^{73} +8.00000 q^{74} -0.550510 q^{76} +6.44949 q^{77} -7.34847 q^{79} -1.00000 q^{82} -4.00000 q^{83} +7.44949 q^{86} +1.44949 q^{88} -12.8990 q^{89} -10.8990 q^{91} -2.89898 q^{92} -0.449490 q^{94} -13.0000 q^{97} -12.7980 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} - 4 q^{7} - 2 q^{8} + 2 q^{11} + 4 q^{14} + 2 q^{16} - 2 q^{17} - 6 q^{19} - 2 q^{22} + 4 q^{23} - 4 q^{28} + 12 q^{29} + 8 q^{31} - 2 q^{32} + 2 q^{34} - 16 q^{37} + 6 q^{38} + 2 q^{41}+ \cdots - 6 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 - 4 * q^7 - 2 * q^8 + 2 * q^11 + 4 * q^14 + 2 * q^16 - 2 * q^17 - 6 * q^19 - 2 * q^22 + 4 * q^23 - 4 * q^28 + 12 * q^29 + 8 * q^31 - 2 * q^32 + 2 * q^34 - 16 * q^37 + 6 * q^38 + 2 * q^41 - 10 * q^43 + 2 * q^44 - 4 * q^46 - 4 * q^47 + 6 * q^49 + 12 * q^53 + 4 * q^56 - 12 * q^58 - 2 * q^59 + 4 * q^61 - 8 * q^62 + 2 * q^64 - 14 * q^67 - 2 * q^68 - 10 * q^73 + 16 * q^74 - 6 * q^76 + 8 * q^77 - 2 * q^82 - 8 * q^83 + 10 * q^86 - 2 * q^88 - 16 * q^89 - 12 * q^91 + 4 * q^92 + 4 * q^94 - 26 * q^97 - 6 * q^98

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
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−4.44949	−1.68175	−0.840875	−	0.541230i	$-0.817959\pi$
$7$	−4.44949	−1.68175	−0.840875	+	0.541230i	$0.817959\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	0	0
$11$	−1.44949	−0.437038	−0.218519	−	0.975833i	$-0.570122\pi$
$11$	−1.44949	−0.437038	−0.218519	+	0.975833i	$0.570122\pi$
$12$	0	0
$13$	2.44949	0.679366	0.339683	−	0.940540i	$-0.389680\pi$
$13$	2.44949	0.679366	0.339683	+	0.940540i	$0.389680\pi$
$14$	4.44949	1.18918
$15$	0	0
$16$	1.00000	0.250000
$17$	3.89898	0.945641	0.472821	−	0.881159i	$-0.343236\pi$
$17$	3.89898	0.945641	0.472821	+	0.881159i	$0.343236\pi$
$18$	0	0
$19$	−0.550510	−0.126296	−0.0631479	−	0.998004i	$-0.520114\pi$
$19$	−0.550510	−0.126296	−0.0631479	+	0.998004i	$0.520114\pi$
$20$	0	0
$21$	0	0
$22$	1.44949	0.309032
$23$	−2.89898	−0.604479	−0.302240	−	0.953232i	$-0.597734\pi$
$23$	−2.89898	−0.604479	−0.302240	+	0.953232i	$0.597734\pi$
$24$	0	0
$25$	0	0
$26$	−2.44949	−0.480384
$27$	0	0
$28$	−4.44949	−0.840875
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	0	0
$31$	6.44949	1.15836	0.579181	−	0.815199i	$-0.303372\pi$
$31$	6.44949	1.15836	0.579181	+	0.815199i	$0.303372\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−3.89898	−0.668669
$35$	0	0
$36$	0	0
$37$	−8.00000	−1.31519	−0.657596	−	0.753371i	$-0.728427\pi$
$37$	−8.00000	−1.31519	−0.657596	+	0.753371i	$0.728427\pi$
$38$	0.550510	0.0893046
$39$	0	0
$40$	0	0
$41$	1.00000	0.156174	0.0780869	−	0.996947i	$-0.475119\pi$
$41$	1.00000	0.156174	0.0780869	+	0.996947i	$0.475119\pi$
$42$	0	0
$43$	−7.44949	−1.13604	−0.568018	−	0.823016i	$-0.692290\pi$
$43$	−7.44949	−1.13604	−0.568018	+	0.823016i	$0.692290\pi$
$44$	−1.44949	−0.218519
$45$	0	0
$46$	2.89898	0.427431
$47$	0.449490	0.0655648	0.0327824	−	0.999463i	$-0.489563\pi$
$47$	0.449490	0.0655648	0.0327824	+	0.999463i	$0.489563\pi$
$48$	0	0
$49$	12.7980	1.82828
$50$	0	0
$51$	0	0
$52$	2.44949	0.339683
$53$	8.44949	1.16063	0.580313	−	0.814393i	$-0.302930\pi$
$53$	8.44949	1.16063	0.580313	+	0.814393i	$0.302930\pi$
$54$	0	0
$55$	0	0
$56$	4.44949	0.594588
$57$	0	0
$58$	−6.00000	−0.787839
$59$	11.2474	1.46429	0.732147	−	0.681147i	$-0.238519\pi$
$59$	11.2474	1.46429	0.732147	+	0.681147i	$0.238519\pi$
$60$	0	0
$61$	−0.449490	−0.0575513	−0.0287756	−	0.999586i	$-0.509161\pi$
$61$	−0.449490	−0.0575513	−0.0287756	+	0.999586i	$0.509161\pi$
$62$	−6.44949	−0.819086
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	−9.44949	−1.15444	−0.577219	−	0.816589i	$-0.695862\pi$
$67$	−9.44949	−1.15444	−0.577219	+	0.816589i	$0.695862\pi$
$68$	3.89898	0.472821
$69$	0	0
$70$	0	0
$71$	−2.44949	−0.290701	−0.145350	−	0.989380i	$-0.546431\pi$
$71$	−2.44949	−0.290701	−0.145350	+	0.989380i	$0.546431\pi$
$72$	0	0
$73$	4.79796	0.561559	0.280779	−	0.959772i	$-0.409407\pi$
$73$	4.79796	0.561559	0.280779	+	0.959772i	$0.409407\pi$
$74$	8.00000	0.929981
$75$	0	0
$76$	−0.550510	−0.0631479
$77$	6.44949	0.734988
$78$	0	0
$79$	−7.34847	−0.826767	−0.413384	−	0.910557i	$-0.635653\pi$
$79$	−7.34847	−0.826767	−0.413384	+	0.910557i	$0.635653\pi$
$80$	0	0
$81$	0	0
$82$	−1.00000	−0.110432
$83$	−4.00000	−0.439057	−0.219529	−	0.975606i	$-0.570452\pi$
$83$	−4.00000	−0.439057	−0.219529	+	0.975606i	$0.570452\pi$
$84$	0	0
$85$	0	0
$86$	7.44949	0.803299
$87$	0	0
$88$	1.44949	0.154516
$89$	−12.8990	−1.36729	−0.683645	−	0.729815i	$-0.739606\pi$
$89$	−12.8990	−1.36729	−0.683645	+	0.729815i	$0.739606\pi$
$90$	0	0
$91$	−10.8990	−1.14252
$92$	−2.89898	−0.302240
$93$	0	0
$94$	−0.449490	−0.0463613
$95$	0	0
$96$	0	0
$97$	−13.0000	−1.31995	−0.659975	−	0.751288i	$-0.729433\pi$
$97$	−13.0000	−1.31995	−0.659975	+	0.751288i	$0.729433\pi$
$98$	−12.7980	−1.29279
$99$	0	0
$100$	0	0
$101$	8.00000	0.796030	0.398015	−	0.917379i	$-0.369699\pi$
$101$	8.00000	0.796030	0.398015	+	0.917379i	$0.369699\pi$
$102$	0	0
$103$	−10.2474	−1.00971	−0.504856	−	0.863204i	$-0.668454\pi$
$103$	−10.2474	−1.00971	−0.504856	+	0.863204i	$0.668454\pi$
$104$	−2.44949	−0.240192
$105$	0	0
$106$	−8.44949	−0.820687
$107$	1.65153	0.159660	0.0798298	−	0.996809i	$-0.474562\pi$
$107$	1.65153	0.159660	0.0798298	+	0.996809i	$0.474562\pi$
$108$	0	0
$109$	−8.00000	−0.766261	−0.383131	−	0.923694i	$-0.625154\pi$
$109$	−8.00000	−0.766261	−0.383131	+	0.923694i	$0.625154\pi$
$110$	0	0
$111$	0	0
$112$	−4.44949	−0.420437
$113$	4.89898	0.460857	0.230429	−	0.973089i	$-0.425987\pi$
$113$	4.89898	0.460857	0.230429	+	0.973089i	$0.425987\pi$
$114$	0	0
$115$	0	0
$116$	6.00000	0.557086
$117$	0	0
$118$	−11.2474	−1.03541
$119$	−17.3485	−1.59033
$120$	0	0
$121$	−8.89898	−0.808998
$122$	0.449490	0.0406949
$123$	0	0
$124$	6.44949	0.579181
$125$	0	0
$126$	0	0
$127$	2.89898	0.257243	0.128621	−	0.991694i	$-0.458945\pi$
$127$	2.89898	0.257243	0.128621	+	0.991694i	$0.458945\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	0	0
$131$	4.89898	0.428026	0.214013	−	0.976831i	$-0.431347\pi$
$131$	4.89898	0.428026	0.214013	+	0.976831i	$0.431347\pi$
$132$	0	0
$133$	2.44949	0.212398
$134$	9.44949	0.816312
$135$	0	0
$136$	−3.89898	−0.334335
$137$	3.00000	0.256307	0.128154	−	0.991754i	$-0.459095\pi$
$137$	3.00000	0.256307	0.128154	+	0.991754i	$0.459095\pi$
$138$	0	0
$139$	−11.2474	−0.953996	−0.476998	−	0.878904i	$-0.658275\pi$
$139$	−11.2474	−0.953996	−0.476998	+	0.878904i	$0.658275\pi$
$140$	0	0
$141$	0	0
$142$	2.44949	0.205557
$143$	−3.55051	−0.296909
$144$	0	0
$145$	0	0
$146$	−4.79796	−0.397082
$147$	0	0
$148$	−8.00000	−0.657596
$149$	−16.2474	−1.33104	−0.665521	−	0.746379i	$-0.731791\pi$
$149$	−16.2474	−1.33104	−0.665521	+	0.746379i	$0.731791\pi$
$150$	0	0
$151$	−6.89898	−0.561431	−0.280715	−	0.959791i	$-0.590572\pi$
$151$	−6.89898	−0.561431	−0.280715	+	0.959791i	$0.590572\pi$
$152$	0.550510	0.0446523
$153$	0	0
$154$	−6.44949	−0.519715
$155$	0	0
$156$	0	0
$157$	−16.0000	−1.27694	−0.638470	−	0.769647i	$-0.720432\pi$
$157$	−16.0000	−1.27694	−0.638470	+	0.769647i	$0.720432\pi$
$158$	7.34847	0.584613
$159$	0	0
$160$	0	0
$161$	12.8990	1.01658
$162$	0	0
$163$	0.898979	0.0704135	0.0352068	−	0.999380i	$-0.488791\pi$
$163$	0.898979	0.0704135	0.0352068	+	0.999380i	$0.488791\pi$
$164$	1.00000	0.0780869
$165$	0	0
$166$	4.00000	0.310460
$167$	−24.2474	−1.87632	−0.938162	−	0.346197i	$-0.887473\pi$
$167$	−24.2474	−1.87632	−0.938162	+	0.346197i	$0.887473\pi$
$168$	0	0
$169$	−7.00000	−0.538462
$170$	0	0
$171$	0	0
$172$	−7.44949	−0.568018
$173$	7.79796	0.592868	0.296434	−	0.955053i	$-0.404203\pi$
$173$	7.79796	0.592868	0.296434	+	0.955053i	$0.404203\pi$
$174$	0	0
$175$	0	0
$176$	−1.44949	−0.109259
$177$	0	0
$178$	12.8990	0.966819
$179$	−8.89898	−0.665141	−0.332570	−	0.943078i	$-0.607916\pi$
$179$	−8.89898	−0.665141	−0.332570	+	0.943078i	$0.607916\pi$
$180$	0	0
$181$	10.4495	0.776704	0.388352	−	0.921511i	$-0.373044\pi$
$181$	10.4495	0.776704	0.388352	+	0.921511i	$0.373044\pi$
$182$	10.8990	0.807886
$183$	0	0
$184$	2.89898	0.213716
$185$	0	0
$186$	0	0
$187$	−5.65153	−0.413281
$188$	0.449490	0.0327824
$189$	0	0
$190$	0	0
$191$	6.24745	0.452050	0.226025	−	0.974122i	$-0.427427\pi$
$191$	6.24745	0.452050	0.226025	+	0.974122i	$0.427427\pi$
$192$	0	0
$193$	15.6969	1.12989	0.564945	−	0.825128i	$-0.308897\pi$
$193$	15.6969	1.12989	0.564945	+	0.825128i	$0.308897\pi$
$194$	13.0000	0.933346
$195$	0	0
$196$	12.7980	0.914140
$197$	−8.00000	−0.569976	−0.284988	−	0.958531i	$-0.591990\pi$
$197$	−8.00000	−0.569976	−0.284988	+	0.958531i	$0.591990\pi$
$198$	0	0
$199$	20.4495	1.44963	0.724813	−	0.688946i	$-0.241926\pi$
$199$	20.4495	1.44963	0.724813	+	0.688946i	$0.241926\pi$
$200$	0	0
$201$	0	0
$202$	−8.00000	−0.562878
$203$	−26.6969	−1.87376
$204$	0	0
$205$	0	0
$206$	10.2474	0.713974
$207$	0	0
$208$	2.44949	0.169842
$209$	0.797959	0.0551960
$210$	0	0
$211$	−15.7980	−1.08758	−0.543788	−	0.839223i	$-0.683011\pi$
$211$	−15.7980	−1.08758	−0.543788	+	0.839223i	$0.683011\pi$
$212$	8.44949	0.580313
$213$	0	0
$214$	−1.65153	−0.112896
$215$	0	0
$216$	0	0
$217$	−28.6969	−1.94808
$218$	8.00000	0.541828
$219$	0	0
$220$	0	0
$221$	9.55051	0.642437
$222$	0	0
$223$	−18.8990	−1.26557	−0.632785	−	0.774328i	$-0.718088\pi$
$223$	−18.8990	−1.26557	−0.632785	+	0.774328i	$0.718088\pi$
$224$	4.44949	0.297294
$225$	0	0
$226$	−4.89898	−0.325875
$227$	1.44949	0.0962060	0.0481030	−	0.998842i	$-0.484682\pi$
$227$	1.44949	0.0962060	0.0481030	+	0.998842i	$0.484682\pi$
$228$	0	0
$229$	−13.5505	−0.895443	−0.447721	−	0.894173i	$-0.647764\pi$
$229$	−13.5505	−0.895443	−0.447721	+	0.894173i	$0.647764\pi$
$230$	0	0
$231$	0	0
$232$	−6.00000	−0.393919
$233$	15.6969	1.02834	0.514170	−	0.857688i	$-0.328100\pi$
$233$	15.6969	1.02834	0.514170	+	0.857688i	$0.328100\pi$
$234$	0	0
$235$	0	0
$236$	11.2474	0.732147
$237$	0	0
$238$	17.3485	1.12453
$239$	−28.6969	−1.85625	−0.928125	−	0.372268i	$-0.878580\pi$
$239$	−28.6969	−1.85625	−0.928125	+	0.372268i	$0.878580\pi$
$240$	0	0
$241$	1.00000	0.0644157	0.0322078	−	0.999481i	$-0.489746\pi$
$241$	1.00000	0.0644157	0.0322078	+	0.999481i	$0.489746\pi$
$242$	8.89898	0.572048
$243$	0	0
$244$	−0.449490	−0.0287756
$245$	0	0
$246$	0	0
$247$	−1.34847	−0.0858010
$248$	−6.44949	−0.409543
$249$	0	0
$250$	0	0
$251$	−11.4495	−0.722685	−0.361343	−	0.932433i	$-0.617682\pi$
$251$	−11.4495	−0.722685	−0.361343	+	0.932433i	$0.617682\pi$
$252$	0	0
$253$	4.20204	0.264180
$254$	−2.89898	−0.181898
$255$	0	0
$256$	1.00000	0.0625000
$257$	19.8990	1.24126	0.620632	−	0.784102i	$-0.286876\pi$
$257$	19.8990	1.24126	0.620632	+	0.784102i	$0.286876\pi$
$258$	0	0
$259$	35.5959	2.21182
$260$	0	0
$261$	0	0
$262$	−4.89898	−0.302660
$263$	−7.55051	−0.465584	−0.232792	−	0.972526i	$-0.574786\pi$
$263$	−7.55051	−0.465584	−0.232792	+	0.972526i	$0.574786\pi$
$264$	0	0
$265$	0	0
$266$	−2.44949	−0.150188
$267$	0	0
$268$	−9.44949	−0.577219
$269$	28.0454	1.70996	0.854979	−	0.518662i	$-0.173570\pi$
$269$	28.0454	1.70996	0.854979	+	0.518662i	$0.173570\pi$
$270$	0	0
$271$	23.5959	1.43335	0.716675	−	0.697407i	$-0.245663\pi$
$271$	23.5959	1.43335	0.716675	+	0.697407i	$0.245663\pi$
$272$	3.89898	0.236410
$273$	0	0
$274$	−3.00000	−0.181237
$275$	0	0
$276$	0	0
$277$	−9.59592	−0.576563	−0.288281	−	0.957546i	$-0.593084\pi$
$277$	−9.59592	−0.576563	−0.288281	+	0.957546i	$0.593084\pi$
$278$	11.2474	0.674577
$279$	0	0
$280$	0	0
$281$	−12.0000	−0.715860	−0.357930	−	0.933748i	$-0.616517\pi$
$281$	−12.0000	−0.715860	−0.357930	+	0.933748i	$0.616517\pi$
$282$	0	0
$283$	−4.00000	−0.237775	−0.118888	−	0.992908i	$-0.537933\pi$
$283$	−4.00000	−0.237775	−0.118888	+	0.992908i	$0.537933\pi$
$284$	−2.44949	−0.145350
$285$	0	0
$286$	3.55051	0.209946
$287$	−4.44949	−0.262645
$288$	0	0
$289$	−1.79796	−0.105762
$290$	0	0
$291$	0	0
$292$	4.79796	0.280779
$293$	18.0000	1.05157	0.525786	−	0.850617i	$-0.323771\pi$
$293$	18.0000	1.05157	0.525786	+	0.850617i	$0.323771\pi$
$294$	0	0
$295$	0	0
$296$	8.00000	0.464991
$297$	0	0
$298$	16.2474	0.941189
$299$	−7.10102	−0.410663
$300$	0	0
$301$	33.1464	1.91053
$302$	6.89898	0.396992
$303$	0	0
$304$	−0.550510	−0.0315739
$305$	0	0
$306$	0	0
$307$	23.9444	1.36658	0.683289	−	0.730148i	$-0.260549\pi$
$307$	23.9444	1.36658	0.683289	+	0.730148i	$0.260549\pi$
$308$	6.44949	0.367494
$309$	0	0
$310$	0	0
$311$	−22.8990	−1.29848	−0.649241	−	0.760583i	$-0.724913\pi$
$311$	−22.8990	−1.29848	−0.649241	+	0.760583i	$0.724913\pi$
$312$	0	0
$313$	−7.69694	−0.435057	−0.217528	−	0.976054i	$-0.569799\pi$
$313$	−7.69694	−0.435057	−0.217528	+	0.976054i	$0.569799\pi$
$314$	16.0000	0.902932
$315$	0	0
$316$	−7.34847	−0.413384
$317$	−30.9444	−1.73801	−0.869005	−	0.494803i	$-0.835240\pi$
$317$	−30.9444	−1.73801	−0.869005	+	0.494803i	$0.835240\pi$
$318$	0	0
$319$	−8.69694	−0.486935
$320$	0	0
$321$	0	0
$322$	−12.8990	−0.718832
$323$	−2.14643	−0.119430
$324$	0	0
$325$	0	0
$326$	−0.898979	−0.0497899
$327$	0	0
$328$	−1.00000	−0.0552158
$329$	−2.00000	−0.110264
$330$	0	0
$331$	33.3939	1.83549	0.917747	−	0.397166i	$-0.130006\pi$
$331$	33.3939	1.83549	0.917747	+	0.397166i	$0.130006\pi$
$332$	−4.00000	−0.219529
$333$	0	0
$334$	24.2474	1.32676
$335$	0	0
$336$	0	0
$337$	20.5959	1.12193	0.560966	−	0.827839i	$-0.310430\pi$
$337$	20.5959	1.12193	0.560966	+	0.827839i	$0.310430\pi$
$338$	7.00000	0.380750
$339$	0	0
$340$	0	0
$341$	−9.34847	−0.506248
$342$	0	0
$343$	−25.7980	−1.39296
$344$	7.44949	0.401650
$345$	0	0
$346$	−7.79796	−0.419221
$347$	−15.2474	−0.818526	−0.409263	−	0.912416i	$-0.634214\pi$
$347$	−15.2474	−0.818526	−0.409263	+	0.912416i	$0.634214\pi$
$348$	0	0
$349$	11.5959	0.620715	0.310358	−	0.950620i	$-0.399551\pi$
$349$	11.5959	0.620715	0.310358	+	0.950620i	$0.399551\pi$
$350$	0	0
$351$	0	0
$352$	1.44949	0.0772581
$353$	−6.59592	−0.351065	−0.175533	−	0.984474i	$-0.556165\pi$
$353$	−6.59592	−0.351065	−0.175533	+	0.984474i	$0.556165\pi$
$354$	0	0
$355$	0	0
$356$	−12.8990	−0.683645
$357$	0	0
$358$	8.89898	0.470326
$359$	−8.44949	−0.445947	−0.222974	−	0.974825i	$-0.571576\pi$
$359$	−8.44949	−0.445947	−0.222974	+	0.974825i	$0.571576\pi$
$360$	0	0
$361$	−18.6969	−0.984049
$362$	−10.4495	−0.549213
$363$	0	0
$364$	−10.8990	−0.571262
$365$	0	0
$366$	0	0
$367$	11.5959	0.605302	0.302651	−	0.953101i	$-0.402128\pi$
$367$	11.5959	0.605302	0.302651	+	0.953101i	$0.402128\pi$
$368$	−2.89898	−0.151120
$369$	0	0
$370$	0	0
$371$	−37.5959	−1.95188
$372$	0	0
$373$	−25.5959	−1.32531	−0.662653	−	0.748926i	$-0.730570\pi$
$373$	−25.5959	−1.32531	−0.662653	+	0.748926i	$0.730570\pi$
$374$	5.65153	0.292234
$375$	0	0
$376$	−0.449490	−0.0231807
$377$	14.6969	0.756931
$378$	0	0
$379$	−30.1464	−1.54852	−0.774259	−	0.632869i	$-0.781877\pi$
$379$	−30.1464	−1.54852	−0.774259	+	0.632869i	$0.781877\pi$
$380$	0	0
$381$	0	0
$382$	−6.24745	−0.319647
$383$	1.79796	0.0918714	0.0459357	−	0.998944i	$-0.485373\pi$
$383$	1.79796	0.0918714	0.0459357	+	0.998944i	$0.485373\pi$
$384$	0	0
$385$	0	0
$386$	−15.6969	−0.798953
$387$	0	0
$388$	−13.0000	−0.659975
$389$	−22.4495	−1.13823	−0.569117	−	0.822256i	$-0.692715\pi$
$389$	−22.4495	−1.13823	−0.569117	+	0.822256i	$0.692715\pi$
$390$	0	0
$391$	−11.3031	−0.571620
$392$	−12.7980	−0.646395
$393$	0	0
$394$	8.00000	0.403034
$395$	0	0
$396$	0	0
$397$	−17.7980	−0.893254	−0.446627	−	0.894720i	$-0.647375\pi$
$397$	−17.7980	−0.893254	−0.446627	+	0.894720i	$0.647375\pi$
$398$	−20.4495	−1.02504
$399$	0	0
$400$	0	0
$401$	−28.7980	−1.43810	−0.719051	−	0.694958i	$-0.755423\pi$
$401$	−28.7980	−1.43810	−0.719051	+	0.694958i	$0.755423\pi$
$402$	0	0
$403$	15.7980	0.786952
$404$	8.00000	0.398015
$405$	0	0
$406$	26.6969	1.32495
$407$	11.5959	0.574788
$408$	0	0
$409$	6.10102	0.301676	0.150838	−	0.988558i	$-0.451803\pi$
$409$	6.10102	0.301676	0.150838	+	0.988558i	$0.451803\pi$
$410$	0	0
$411$	0	0
$412$	−10.2474	−0.504856
$413$	−50.0454	−2.46257
$414$	0	0
$415$	0	0
$416$	−2.44949	−0.120096
$417$	0	0
$418$	−0.797959	−0.0390294
$419$	−0.898979	−0.0439180	−0.0219590	−	0.999759i	$-0.506990\pi$
$419$	−0.898979	−0.0439180	−0.0219590	+	0.999759i	$0.506990\pi$
$420$	0	0
$421$	−16.4495	−0.801699	−0.400850	−	0.916144i	$-0.631285\pi$
$421$	−16.4495	−0.801699	−0.400850	+	0.916144i	$0.631285\pi$
$422$	15.7980	0.769033
$423$	0	0
$424$	−8.44949	−0.410343
$425$	0	0
$426$	0	0
$427$	2.00000	0.0967868
$428$	1.65153	0.0798298
$429$	0	0
$430$	0	0
$431$	−13.7526	−0.662437	−0.331219	−	0.943554i	$-0.607460\pi$
$431$	−13.7526	−0.662437	−0.331219	+	0.943554i	$0.607460\pi$
$432$	0	0
$433$	−23.0000	−1.10531	−0.552655	−	0.833410i	$-0.686385\pi$
$433$	−23.0000	−1.10531	−0.552655	+	0.833410i	$0.686385\pi$
$434$	28.6969	1.37750
$435$	0	0
$436$	−8.00000	−0.383131
$437$	1.59592	0.0763431
$438$	0	0
$439$	−22.0454	−1.05217	−0.526085	−	0.850432i	$-0.676341\pi$
$439$	−22.0454	−1.05217	−0.526085	+	0.850432i	$0.676341\pi$
$440$	0	0
$441$	0	0
$442$	−9.55051	−0.454271
$443$	−3.24745	−0.154291	−0.0771455	−	0.997020i	$-0.524581\pi$
$443$	−3.24745	−0.154291	−0.0771455	+	0.997020i	$0.524581\pi$
$444$	0	0
$445$	0	0
$446$	18.8990	0.894893
$447$	0	0
$448$	−4.44949	−0.210219
$449$	0.797959	0.0376580	0.0188290	−	0.999823i	$-0.494006\pi$
$449$	0.797959	0.0376580	0.0188290	+	0.999823i	$0.494006\pi$
$450$	0	0
$451$	−1.44949	−0.0682538
$452$	4.89898	0.230429
$453$	0	0
$454$	−1.44949	−0.0680279
$455$	0	0
$456$	0	0
$457$	8.10102	0.378950	0.189475	−	0.981886i	$-0.439321\pi$
$457$	8.10102	0.378950	0.189475	+	0.981886i	$0.439321\pi$
$458$	13.5505	0.633174
$459$	0	0
$460$	0	0
$461$	−2.44949	−0.114084	−0.0570421	−	0.998372i	$-0.518167\pi$
$461$	−2.44949	−0.114084	−0.0570421	+	0.998372i	$0.518167\pi$
$462$	0	0
$463$	−24.0000	−1.11537	−0.557687	−	0.830051i	$-0.688311\pi$
$463$	−24.0000	−1.11537	−0.557687	+	0.830051i	$0.688311\pi$
$464$	6.00000	0.278543
$465$	0	0
$466$	−15.6969	−0.727147
$467$	4.34847	0.201223	0.100612	−	0.994926i	$-0.467920\pi$
$467$	4.34847	0.201223	0.100612	+	0.994926i	$0.467920\pi$
$468$	0	0
$469$	42.0454	1.94148
$470$	0	0
$471$	0	0
$472$	−11.2474	−0.517706
$473$	10.7980	0.496491
$474$	0	0
$475$	0	0
$476$	−17.3485	−0.795166
$477$	0	0
$478$	28.6969	1.31257
$479$	12.6969	0.580138	0.290069	−	0.957006i	$-0.406322\pi$
$479$	12.6969	0.580138	0.290069	+	0.957006i	$0.406322\pi$
$480$	0	0
$481$	−19.5959	−0.893497
$482$	−1.00000	−0.0455488
$483$	0	0
$484$	−8.89898	−0.404499
$485$	0	0
$486$	0	0
$487$	34.8990	1.58142	0.790712	−	0.612188i	$-0.209711\pi$
$487$	34.8990	1.58142	0.790712	+	0.612188i	$0.209711\pi$
$488$	0.449490	0.0203474
$489$	0	0
$490$	0	0
$491$	−23.4495	−1.05826	−0.529130	−	0.848541i	$-0.677482\pi$
$491$	−23.4495	−1.05826	−0.529130	+	0.848541i	$0.677482\pi$
$492$	0	0
$493$	23.3939	1.05361
$494$	1.34847	0.0606705
$495$	0	0
$496$	6.44949	0.289591
$497$	10.8990	0.488886
$498$	0	0
$499$	3.24745	0.145376	0.0726879	−	0.997355i	$-0.476842\pi$
$499$	3.24745	0.145376	0.0726879	+	0.997355i	$0.476842\pi$
$500$	0	0
$501$	0	0
$502$	11.4495	0.511016
$503$	9.55051	0.425836	0.212918	−	0.977070i	$-0.431703\pi$
$503$	9.55051	0.425836	0.212918	+	0.977070i	$0.431703\pi$
$504$	0	0
$505$	0	0
$506$	−4.20204	−0.186804
$507$	0	0
$508$	2.89898	0.128621
$509$	7.59592	0.336683	0.168342	−	0.985729i	$-0.446159\pi$
$509$	7.59592	0.336683	0.168342	+	0.985729i	$0.446159\pi$
$510$	0	0
$511$	−21.3485	−0.944401
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−19.8990	−0.877706
$515$	0	0
$516$	0	0
$517$	−0.651531	−0.0286543
$518$	−35.5959	−1.56399
$519$	0	0
$520$	0	0
$521$	7.69694	0.337209	0.168604	−	0.985684i	$-0.446074\pi$
$521$	7.69694	0.337209	0.168604	+	0.985684i	$0.446074\pi$
$522$	0	0
$523$	29.7980	1.30297	0.651487	−	0.758660i	$-0.274146\pi$
$523$	29.7980	1.30297	0.651487	+	0.758660i	$0.274146\pi$
$524$	4.89898	0.214013
$525$	0	0
$526$	7.55051	0.329218
$527$	25.1464	1.09540
$528$	0	0
$529$	−14.5959	−0.634605
$530$	0	0
$531$	0	0
$532$	2.44949	0.106199
$533$	2.44949	0.106099
$534$	0	0
$535$	0	0
$536$	9.44949	0.408156
$537$	0	0
$538$	−28.0454	−1.20912
$539$	−18.5505	−0.799027
$540$	0	0
$541$	−39.5959	−1.70236	−0.851181	−	0.524873i	$-0.824113\pi$
$541$	−39.5959	−1.70236	−0.851181	+	0.524873i	$0.824113\pi$
$542$	−23.5959	−1.01353
$543$	0	0
$544$	−3.89898	−0.167167
$545$	0	0
$546$	0	0
$547$	7.24745	0.309879	0.154939	−	0.987924i	$-0.450482\pi$
$547$	7.24745	0.309879	0.154939	+	0.987924i	$0.450482\pi$
$548$	3.00000	0.128154
$549$	0	0
$550$	0	0
$551$	−3.30306	−0.140715
$552$	0	0
$553$	32.6969	1.39042
$554$	9.59592	0.407691
$555$	0	0
$556$	−11.2474	−0.476998
$557$	−38.9444	−1.65013	−0.825063	−	0.565040i	$-0.808861\pi$
$557$	−38.9444	−1.65013	−0.825063	+	0.565040i	$0.808861\pi$
$558$	0	0
$559$	−18.2474	−0.771785
$560$	0	0
$561$	0	0
$562$	12.0000	0.506189
$563$	36.1464	1.52339	0.761695	−	0.647936i	$-0.224367\pi$
$563$	36.1464	1.52339	0.761695	+	0.647936i	$0.224367\pi$
$564$	0	0
$565$	0	0
$566$	4.00000	0.168133
$567$	0	0
$568$	2.44949	0.102778
$569$	−17.4949	−0.733424	−0.366712	−	0.930334i	$-0.619517\pi$
$569$	−17.4949	−0.733424	−0.366712	+	0.930334i	$0.619517\pi$
$570$	0	0
$571$	12.7526	0.533678	0.266839	−	0.963741i	$-0.414021\pi$
$571$	12.7526	0.533678	0.266839	+	0.963741i	$0.414021\pi$
$572$	−3.55051	−0.148454
$573$	0	0
$574$	4.44949	0.185718
$575$	0	0
$576$	0	0
$577$	−13.6969	−0.570211	−0.285106	−	0.958496i	$-0.592029\pi$
$577$	−13.6969	−0.570211	−0.285106	+	0.958496i	$0.592029\pi$
$578$	1.79796	0.0747852
$579$	0	0
$580$	0	0
$581$	17.7980	0.738384
$582$	0	0
$583$	−12.2474	−0.507237
$584$	−4.79796	−0.198541
$585$	0	0
$586$	−18.0000	−0.743573
$587$	29.9444	1.23594	0.617969	−	0.786203i	$-0.287956\pi$
$587$	29.9444	1.23594	0.617969	+	0.786203i	$0.287956\pi$
$588$	0	0
$589$	−3.55051	−0.146296
$590$	0	0
$591$	0	0
$592$	−8.00000	−0.328798
$593$	41.3939	1.69984	0.849921	−	0.526910i	$-0.176649\pi$
$593$	41.3939	1.69984	0.849921	+	0.526910i	$0.176649\pi$
$594$	0	0
$595$	0	0
$596$	−16.2474	−0.665521
$597$	0	0
$598$	7.10102	0.290382
$599$	−14.2020	−0.580280	−0.290140	−	0.956984i	$-0.593702\pi$
$599$	−14.2020	−0.580280	−0.290140	+	0.956984i	$0.593702\pi$
$600$	0	0
$601$	19.2020	0.783268	0.391634	−	0.920121i	$-0.371910\pi$
$601$	19.2020	0.783268	0.391634	+	0.920121i	$0.371910\pi$
$602$	−33.1464	−1.35095
$603$	0	0
$604$	−6.89898	−0.280715
$605$	0	0
$606$	0	0
$607$	−11.5959	−0.470664	−0.235332	−	0.971915i	$-0.575618\pi$
$607$	−11.5959	−0.470664	−0.235332	+	0.971915i	$0.575618\pi$
$608$	0.550510	0.0223261
$609$	0	0
$610$	0	0
$611$	1.10102	0.0445425
$612$	0	0
$613$	−16.9444	−0.684377	−0.342189	−	0.939631i	$-0.611168\pi$
$613$	−16.9444	−0.684377	−0.342189	+	0.939631i	$0.611168\pi$
$614$	−23.9444	−0.966317
$615$	0	0
$616$	−6.44949	−0.259857
$617$	37.6969	1.51762	0.758811	−	0.651311i	$-0.225781\pi$
$617$	37.6969	1.51762	0.758811	+	0.651311i	$0.225781\pi$
$618$	0	0
$619$	−33.4495	−1.34445	−0.672224	−	0.740348i	$-0.734661\pi$
$619$	−33.4495	−1.34445	−0.672224	+	0.740348i	$0.734661\pi$
$620$	0	0
$621$	0	0
$622$	22.8990	0.918165
$623$	57.3939	2.29944
$624$	0	0
$625$	0	0
$626$	7.69694	0.307632
$627$	0	0
$628$	−16.0000	−0.638470
$629$	−31.1918	−1.24370
$630$	0	0
$631$	3.34847	0.133300	0.0666502	−	0.997776i	$-0.478769\pi$
$631$	3.34847	0.133300	0.0666502	+	0.997776i	$0.478769\pi$
$632$	7.34847	0.292306
$633$	0	0
$634$	30.9444	1.22896
$635$	0	0
$636$	0	0
$637$	31.3485	1.24207
$638$	8.69694	0.344315
$639$	0	0
$640$	0	0
$641$	37.0000	1.46141	0.730706	−	0.682692i	$-0.239191\pi$
$641$	37.0000	1.46141	0.730706	+	0.682692i	$0.239191\pi$
$642$	0	0
$643$	9.24745	0.364684	0.182342	−	0.983235i	$-0.441632\pi$
$643$	9.24745	0.364684	0.182342	+	0.983235i	$0.441632\pi$
$644$	12.8990	0.508291
$645$	0	0
$646$	2.14643	0.0844501
$647$	25.1010	0.986823	0.493411	−	0.869796i	$-0.335750\pi$
$647$	25.1010	0.986823	0.493411	+	0.869796i	$0.335750\pi$
$648$	0	0
$649$	−16.3031	−0.639951
$650$	0	0
$651$	0	0
$652$	0.898979	0.0352068
$653$	−34.0000	−1.33052	−0.665261	−	0.746611i	$-0.731680\pi$
$653$	−34.0000	−1.33052	−0.665261	+	0.746611i	$0.731680\pi$
$654$	0	0
$655$	0	0
$656$	1.00000	0.0390434
$657$	0	0
$658$	2.00000	0.0779681
$659$	16.2020	0.631142	0.315571	−	0.948902i	$-0.397804\pi$
$659$	16.2020	0.631142	0.315571	+	0.948902i	$0.397804\pi$
$660$	0	0
$661$	−13.7980	−0.536678	−0.268339	−	0.963324i	$-0.586475\pi$
$661$	−13.7980	−0.536678	−0.268339	+	0.963324i	$0.586475\pi$
$662$	−33.3939	−1.29789
$663$	0	0
$664$	4.00000	0.155230
$665$	0	0
$666$	0	0
$667$	−17.3939	−0.673494
$668$	−24.2474	−0.938162
$669$	0	0
$670$	0	0
$671$	0.651531	0.0251521
$672$	0	0
$673$	19.1010	0.736290	0.368145	−	0.929768i	$-0.379993\pi$
$673$	19.1010	0.736290	0.368145	+	0.929768i	$0.379993\pi$
$674$	−20.5959	−0.793325
$675$	0	0
$676$	−7.00000	−0.269231
$677$	0.404082	0.0155301	0.00776507	−	0.999970i	$-0.497528\pi$
$677$	0.404082	0.0155301	0.00776507	+	0.999970i	$0.497528\pi$
$678$	0	0
$679$	57.8434	2.21982
$680$	0	0
$681$	0	0
$682$	9.34847	0.357971
$683$	−40.5505	−1.55162	−0.775811	−	0.630965i	$-0.782659\pi$
$683$	−40.5505	−1.55162	−0.775811	+	0.630965i	$0.782659\pi$
$684$	0	0
$685$	0	0
$686$	25.7980	0.984971
$687$	0	0
$688$	−7.44949	−0.284009
$689$	20.6969	0.788491
$690$	0	0
$691$	−21.5959	−0.821547	−0.410774	−	0.911737i	$-0.634741\pi$
$691$	−21.5959	−0.821547	−0.410774	+	0.911737i	$0.634741\pi$
$692$	7.79796	0.296434
$693$	0	0
$694$	15.2474	0.578785
$695$	0	0
$696$	0	0
$697$	3.89898	0.147684
$698$	−11.5959	−0.438912
$699$	0	0
$700$	0	0
$701$	−19.3939	−0.732497	−0.366248	−	0.930517i	$-0.619358\pi$
$701$	−19.3939	−0.732497	−0.366248	+	0.930517i	$0.619358\pi$
$702$	0	0
$703$	4.40408	0.166103
$704$	−1.44949	−0.0546297
$705$	0	0
$706$	6.59592	0.248241
$707$	−35.5959	−1.33872
$708$	0	0
$709$	22.6515	0.850696	0.425348	−	0.905030i	$-0.360152\pi$
$709$	22.6515	0.850696	0.425348	+	0.905030i	$0.360152\pi$
$710$	0	0
$711$	0	0
$712$	12.8990	0.483410
$713$	−18.6969	−0.700206
$714$	0	0
$715$	0	0
$716$	−8.89898	−0.332570
$717$	0	0
$718$	8.44949	0.315332
$719$	−22.2020	−0.827996	−0.413998	−	0.910278i	$-0.635868\pi$
$719$	−22.2020	−0.827996	−0.413998	+	0.910278i	$0.635868\pi$
$720$	0	0
$721$	45.5959	1.69808
$722$	18.6969	0.695828
$723$	0	0
$724$	10.4495	0.388352
$725$	0	0
$726$	0	0
$727$	10.6515	0.395043	0.197522	−	0.980299i	$-0.436711\pi$
$727$	10.6515	0.395043	0.197522	+	0.980299i	$0.436711\pi$
$728$	10.8990	0.403943
$729$	0	0
$730$	0	0
$731$	−29.0454	−1.07428
$732$	0	0
$733$	13.1464	0.485574	0.242787	−	0.970080i	$-0.421938\pi$
$733$	13.1464	0.485574	0.242787	+	0.970080i	$0.421938\pi$
$734$	−11.5959	−0.428013
$735$	0	0
$736$	2.89898	0.106858
$737$	13.6969	0.504533
$738$	0	0
$739$	7.24745	0.266602	0.133301	−	0.991076i	$-0.457442\pi$
$739$	7.24745	0.266602	0.133301	+	0.991076i	$0.457442\pi$
$740$	0	0
$741$	0	0
$742$	37.5959	1.38019
$743$	−13.1010	−0.480630	−0.240315	−	0.970695i	$-0.577251\pi$
$743$	−13.1010	−0.480630	−0.240315	+	0.970695i	$0.577251\pi$
$744$	0	0
$745$	0	0
$746$	25.5959	0.937133
$747$	0	0
$748$	−5.65153	−0.206640
$749$	−7.34847	−0.268507
$750$	0	0
$751$	44.9898	1.64170	0.820850	−	0.571143i	$-0.193500\pi$
$751$	44.9898	1.64170	0.820850	+	0.571143i	$0.193500\pi$
$752$	0.449490	0.0163912
$753$	0	0
$754$	−14.6969	−0.535231
$755$	0	0
$756$	0	0
$757$	10.0000	0.363456	0.181728	−	0.983349i	$-0.441831\pi$
$757$	10.0000	0.363456	0.181728	+	0.983349i	$0.441831\pi$
$758$	30.1464	1.09497
$759$	0	0
$760$	0	0
$761$	48.4949	1.75794	0.878969	−	0.476878i	$-0.158232\pi$
$761$	48.4949	1.75794	0.878969	+	0.476878i	$0.158232\pi$
$762$	0	0
$763$	35.5959	1.28866
$764$	6.24745	0.226025
$765$	0	0
$766$	−1.79796	−0.0649629
$767$	27.5505	0.994791
$768$	0	0
$769$	−24.4949	−0.883309	−0.441654	−	0.897185i	$-0.645608\pi$
$769$	−24.4949	−0.883309	−0.441654	+	0.897185i	$0.645608\pi$
$770$	0	0
$771$	0	0
$772$	15.6969	0.564945
$773$	−23.3939	−0.841419	−0.420710	−	0.907195i	$-0.638219\pi$
$773$	−23.3939	−0.841419	−0.420710	+	0.907195i	$0.638219\pi$
$774$	0	0
$775$	0	0
$776$	13.0000	0.466673
$777$	0	0
$778$	22.4495	0.804853
$779$	−0.550510	−0.0197241
$780$	0	0
$781$	3.55051	0.127047
$782$	11.3031	0.404197
$783$	0	0
$784$	12.7980	0.457070
$785$	0	0
$786$	0	0
$787$	−7.39388	−0.263563	−0.131782	−	0.991279i	$-0.542070\pi$
$787$	−7.39388	−0.263563	−0.131782	+	0.991279i	$0.542070\pi$
$788$	−8.00000	−0.284988
$789$	0	0
$790$	0	0
$791$	−21.7980	−0.775046
$792$	0	0
$793$	−1.10102	−0.0390984
$794$	17.7980	0.631626
$795$	0	0
$796$	20.4495	0.724813
$797$	−35.5959	−1.26087	−0.630436	−	0.776241i	$-0.717124\pi$
$797$	−35.5959	−1.26087	−0.630436	+	0.776241i	$0.717124\pi$
$798$	0	0
$799$	1.75255	0.0620008
$800$	0	0
$801$	0	0
$802$	28.7980	1.01689
$803$	−6.95459	−0.245422
$804$	0	0
$805$	0	0
$806$	−15.7980	−0.556459
$807$	0	0
$808$	−8.00000	−0.281439
$809$	−47.0908	−1.65562	−0.827812	−	0.561005i	$-0.810415\pi$
$809$	−47.0908	−1.65562	−0.827812	+	0.561005i	$0.810415\pi$
$810$	0	0
$811$	−17.2474	−0.605640	−0.302820	−	0.953048i	$-0.597928\pi$
$811$	−17.2474	−0.605640	−0.302820	+	0.953048i	$0.597928\pi$
$812$	−26.6969	−0.936879
$813$	0	0
$814$	−11.5959	−0.406437
$815$	0	0
$816$	0	0
$817$	4.10102	0.143477
$818$	−6.10102	−0.213317
$819$	0	0
$820$	0	0
$821$	42.0454	1.46739	0.733697	−	0.679476i	$-0.237793\pi$
$821$	42.0454	1.46739	0.733697	+	0.679476i	$0.237793\pi$
$822$	0	0
$823$	−7.59592	−0.264777	−0.132389	−	0.991198i	$-0.542265\pi$
$823$	−7.59592	−0.264777	−0.132389	+	0.991198i	$0.542265\pi$
$824$	10.2474	0.356987
$825$	0	0
$826$	50.0454	1.74130
$827$	−13.7980	−0.479802	−0.239901	−	0.970797i	$-0.577115\pi$
$827$	−13.7980	−0.479802	−0.239901	+	0.970797i	$0.577115\pi$
$828$	0	0
$829$	−21.5505	−0.748480	−0.374240	−	0.927332i	$-0.622096\pi$
$829$	−21.5505	−0.748480	−0.374240	+	0.927332i	$0.622096\pi$
$830$	0	0
$831$	0	0
$832$	2.44949	0.0849208
$833$	49.8990	1.72890
$834$	0	0
$835$	0	0
$836$	0.797959	0.0275980
$837$	0	0
$838$	0.898979	0.0310547
$839$	−15.7526	−0.543838	−0.271919	−	0.962320i	$-0.587658\pi$
$839$	−15.7526	−0.543838	−0.271919	+	0.962320i	$0.587658\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	16.4495	0.566887
$843$	0	0
$844$	−15.7980	−0.543788
$845$	0	0
$846$	0	0
$847$	39.5959	1.36053
$848$	8.44949	0.290157
$849$	0	0
$850$	0	0
$851$	23.1918	0.795006
$852$	0	0
$853$	25.1464	0.860997	0.430499	−	0.902591i	$-0.358338\pi$
$853$	25.1464	0.860997	0.430499	+	0.902591i	$0.358338\pi$
$854$	−2.00000	−0.0684386
$855$	0	0
$856$	−1.65153	−0.0564482
$857$	−53.3939	−1.82390	−0.911950	−	0.410301i	$-0.865424\pi$
$857$	−53.3939	−1.82390	−0.911950	+	0.410301i	$0.865424\pi$
$858$	0	0
$859$	35.7423	1.21951	0.609757	−	0.792589i	$-0.291267\pi$
$859$	35.7423	1.21951	0.609757	+	0.792589i	$0.291267\pi$
$860$	0	0
$861$	0	0
$862$	13.7526	0.468414
$863$	21.5505	0.733588	0.366794	−	0.930302i	$-0.380455\pi$
$863$	21.5505	0.733588	0.366794	+	0.930302i	$0.380455\pi$
$864$	0	0
$865$	0	0
$866$	23.0000	0.781572
$867$	0	0
$868$	−28.6969	−0.974038
$869$	10.6515	0.361328
$870$	0	0
$871$	−23.1464	−0.784287
$872$	8.00000	0.270914
$873$	0	0
$874$	−1.59592	−0.0539827
$875$	0	0
$876$	0	0
$877$	55.1464	1.86216	0.931081	−	0.364812i	$-0.118867\pi$
$877$	55.1464	1.86216	0.931081	+	0.364812i	$0.118867\pi$
$878$	22.0454	0.743996
$879$	0	0
$880$	0	0
$881$	30.0000	1.01073	0.505363	−	0.862907i	$-0.331359\pi$
$881$	30.0000	1.01073	0.505363	+	0.862907i	$0.331359\pi$
$882$	0	0
$883$	11.4495	0.385306	0.192653	−	0.981267i	$-0.438291\pi$
$883$	11.4495	0.385306	0.192653	+	0.981267i	$0.438291\pi$
$884$	9.55051	0.321218
$885$	0	0
$886$	3.24745	0.109100
$887$	4.65153	0.156183	0.0780916	−	0.996946i	$-0.475117\pi$
$887$	4.65153	0.156183	0.0780916	+	0.996946i	$0.475117\pi$
$888$	0	0
$889$	−12.8990	−0.432618
$890$	0	0
$891$	0	0
$892$	−18.8990	−0.632785
$893$	−0.247449	−0.00828056
$894$	0	0
$895$	0	0
$896$	4.44949	0.148647
$897$	0	0
$898$	−0.797959	−0.0266282
$899$	38.6969	1.29062
$900$	0	0
$901$	32.9444	1.09754
$902$	1.44949	0.0482627
$903$	0	0
$904$	−4.89898	−0.162938
$905$	0	0
$906$	0	0
$907$	−33.7423	−1.12040	−0.560198	−	0.828359i	$-0.689275\pi$
$907$	−33.7423	−1.12040	−0.560198	+	0.828359i	$0.689275\pi$
$908$	1.44949	0.0481030
$909$	0	0
$910$	0	0
$911$	0.247449	0.00819834	0.00409917	−	0.999992i	$-0.498695\pi$
$911$	0.247449	0.00819834	0.00409917	+	0.999992i	$0.498695\pi$
$912$	0	0
$913$	5.79796	0.191884
$914$	−8.10102	−0.267958
$915$	0	0
$916$	−13.5505	−0.447721
$917$	−21.7980	−0.719832
$918$	0	0
$919$	10.8990	0.359524	0.179762	−	0.983710i	$-0.442467\pi$
$919$	10.8990	0.359524	0.179762	+	0.983710i	$0.442467\pi$
$920$	0	0
$921$	0	0
$922$	2.44949	0.0806696
$923$	−6.00000	−0.197492
$924$	0	0
$925$	0	0
$926$	24.0000	0.788689
$927$	0	0
$928$	−6.00000	−0.196960
$929$	55.5959	1.82404	0.912021	−	0.410143i	$-0.134521\pi$
$929$	55.5959	1.82404	0.912021	+	0.410143i	$0.134521\pi$
$930$	0	0
$931$	−7.04541	−0.230904
$932$	15.6969	0.514170
$933$	0	0
$934$	−4.34847	−0.142286
$935$	0	0
$936$	0	0
$937$	39.5959	1.29354	0.646771	−	0.762684i	$-0.276119\pi$
$937$	39.5959	1.29354	0.646771	+	0.762684i	$0.276119\pi$
$938$	−42.0454	−1.37283
$939$	0	0
$940$	0	0
$941$	49.7980	1.62337	0.811684	−	0.584097i	$-0.198551\pi$
$941$	49.7980	1.62337	0.811684	+	0.584097i	$0.198551\pi$
$942$	0	0
$943$	−2.89898	−0.0944038
$944$	11.2474	0.366073
$945$	0	0
$946$	−10.7980	−0.351072
$947$	−21.2474	−0.690449	−0.345225	−	0.938520i	$-0.612197\pi$
$947$	−21.2474	−0.690449	−0.345225	+	0.938520i	$0.612197\pi$
$948$	0	0
$949$	11.7526	0.381504
$950$	0	0
$951$	0	0
$952$	17.3485	0.562267
$953$	−50.7980	−1.64551	−0.822754	−	0.568398i	$-0.807563\pi$
$953$	−50.7980	−1.64551	−0.822754	+	0.568398i	$0.807563\pi$
$954$	0	0
$955$	0	0
$956$	−28.6969	−0.928125
$957$	0	0
$958$	−12.6969	−0.410219
$959$	−13.3485	−0.431045
$960$	0	0
$961$	10.5959	0.341804
$962$	19.5959	0.631798
$963$	0	0
$964$	1.00000	0.0322078
$965$	0	0
$966$	0	0
$967$	28.6969	0.922831	0.461416	−	0.887184i	$-0.347342\pi$
$967$	28.6969	0.922831	0.461416	+	0.887184i	$0.347342\pi$
$968$	8.89898	0.286024
$969$	0	0
$970$	0	0
$971$	23.3939	0.750745	0.375373	−	0.926874i	$-0.377515\pi$
$971$	23.3939	0.750745	0.375373	+	0.926874i	$0.377515\pi$
$972$	0	0
$973$	50.0454	1.60438
$974$	−34.8990	−1.11824
$975$	0	0
$976$	−0.449490	−0.0143878
$977$	37.8990	1.21250	0.606248	−	0.795276i	$-0.292674\pi$
$977$	37.8990	1.21250	0.606248	+	0.795276i	$0.292674\pi$
$978$	0	0
$979$	18.6969	0.597557
$980$	0	0
$981$	0	0
$982$	23.4495	0.748303
$983$	−37.3939	−1.19268	−0.596340	−	0.802732i	$-0.703379\pi$
$983$	−37.3939	−1.19268	−0.596340	+	0.802732i	$0.703379\pi$
$984$	0	0
$985$	0	0
$986$	−23.3939	−0.745013
$987$	0	0
$988$	−1.34847	−0.0429005
$989$	21.5959	0.686710
$990$	0	0
$991$	4.00000	0.127064	0.0635321	−	0.997980i	$-0.479763\pi$
$991$	4.00000	0.127064	0.0635321	+	0.997980i	$0.479763\pi$
$992$	−6.44949	−0.204772
$993$	0	0
$994$	−10.8990	−0.345695
$995$	0	0
$996$	0	0
$997$	32.9444	1.04336	0.521680	−	0.853141i	$-0.325306\pi$
$997$	32.9444	1.04336	0.521680	+	0.853141i	$0.325306\pi$
$998$	−3.24745	−0.102796
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4050.2.a.bm.1.1		2
3.2	odd	2		4050.2.a.bs.1.1		2
5.2	odd	4		810.2.c.e.649.2		4
5.3	odd	4		810.2.c.e.649.4		4
5.4	even	2		4050.2.a.bz.1.2		2
9.2	odd	6		450.2.e.k.301.1		4
9.4	even	3		1350.2.e.m.451.2		4
9.5	odd	6		450.2.e.k.151.1		4
9.7	even	3		1350.2.e.m.901.2		4
15.2	even	4		810.2.c.f.649.3		4
15.8	even	4		810.2.c.f.649.1		4
15.14	odd	2		4050.2.a.bq.1.2		2
45.2	even	12		90.2.i.b.49.4	yes	8
45.4	even	6		1350.2.e.j.451.1		4
45.7	odd	12		270.2.i.b.199.2		8
45.13	odd	12		270.2.i.b.19.2		8
45.14	odd	6		450.2.e.n.151.2		4
45.22	odd	12		270.2.i.b.19.3		8
45.23	even	12		90.2.i.b.79.4	yes	8
45.29	odd	6		450.2.e.n.301.2		4
45.32	even	12		90.2.i.b.79.1	yes	8
45.34	even	6		1350.2.e.j.901.1		4
45.38	even	12		90.2.i.b.49.1	✓	8
45.43	odd	12		270.2.i.b.199.3		8
180.7	even	12		2160.2.by.d.1009.4		8
180.23	odd	12		720.2.by.c.529.1		8
180.43	even	12		2160.2.by.d.1009.1		8
180.47	odd	12		720.2.by.c.49.1		8
180.67	even	12		2160.2.by.d.289.1		8
180.83	odd	12		720.2.by.c.49.4		8
180.103	even	12		2160.2.by.d.289.4		8
180.167	odd	12		720.2.by.c.529.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
90.2.i.b.49.1	✓	8	45.38	even	12
90.2.i.b.49.4	yes	8	45.2	even	12
90.2.i.b.79.1	yes	8	45.32	even	12
90.2.i.b.79.4	yes	8	45.23	even	12
270.2.i.b.19.2		8	45.13	odd	12
270.2.i.b.19.3		8	45.22	odd	12
270.2.i.b.199.2		8	45.7	odd	12
270.2.i.b.199.3		8	45.43	odd	12
450.2.e.k.151.1		4	9.5	odd	6
450.2.e.k.301.1		4	9.2	odd	6
450.2.e.n.151.2		4	45.14	odd	6
450.2.e.n.301.2		4	45.29	odd	6
720.2.by.c.49.1		8	180.47	odd	12
720.2.by.c.49.4		8	180.83	odd	12
720.2.by.c.529.1		8	180.23	odd	12
720.2.by.c.529.4		8	180.167	odd	12
810.2.c.e.649.2		4	5.2	odd	4
810.2.c.e.649.4		4	5.3	odd	4
810.2.c.f.649.1		4	15.8	even	4
810.2.c.f.649.3		4	15.2	even	4
1350.2.e.j.451.1		4	45.4	even	6
1350.2.e.j.901.1		4	45.34	even	6
1350.2.e.m.451.2		4	9.4	even	3
1350.2.e.m.901.2		4	9.7	even	3
2160.2.by.d.289.1		8	180.67	even	12
2160.2.by.d.289.4		8	180.103	even	12
2160.2.by.d.1009.1		8	180.43	even	12
2160.2.by.d.1009.4		8	180.7	even	12
4050.2.a.bm.1.1		2	1.1	even	1	trivial
4050.2.a.bq.1.2		2	15.14	odd	2
4050.2.a.bs.1.1		2	3.2	odd	2
4050.2.a.bz.1.2		2	5.4	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:	\( N \)	\(=\)	\( 4050 = 2 \cdot 3^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4050.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -4.44949 q^{7} -1.00000 q^{8} -1.44949 q^{11} +2.44949 q^{13} +4.44949 q^{14} +1.00000 q^{16} +3.89898 q^{17} -0.550510 q^{19} +1.44949 q^{22} -2.89898 q^{23} -2.44949 q^{26} -4.44949 q^{28} +6.00000 q^{29} +6.44949 q^{31} -1.00000 q^{32} -3.89898 q^{34} -8.00000 q^{37} +0.550510 q^{38} +1.00000 q^{41} -7.44949 q^{43} -1.44949 q^{44} +2.89898 q^{46} +0.449490 q^{47} +12.7980 q^{49} +2.44949 q^{52} +8.44949 q^{53} +4.44949 q^{56} -6.00000 q^{58} +11.2474 q^{59} -0.449490 q^{61} -6.44949 q^{62} +1.00000 q^{64} -9.44949 q^{67} +3.89898 q^{68} -2.44949 q^{71} +4.79796 q^{73} +8.00000 q^{74} -0.550510 q^{76} +6.44949 q^{77} -7.34847 q^{79} -1.00000 q^{82} -4.00000 q^{83} +7.44949 q^{86} +1.44949 q^{88} -12.8990 q^{89} -10.8990 q^{91} -2.89898 q^{92} -0.449490 q^{94} -13.0000 q^{97} -12.7980 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} - 4 q^{7} - 2 q^{8} + 2 q^{11} + 4 q^{14} + 2 q^{16} - 2 q^{17} - 6 q^{19} - 2 q^{22} + 4 q^{23} - 4 q^{28} + 12 q^{29} + 8 q^{31} - 2 q^{32} + 2 q^{34} - 16 q^{37} + 6 q^{38} + 2 q^{41}+ \cdots - 6 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 - 4 * q^7 - 2 * q^8 + 2 * q^11 + 4 * q^14 + 2 * q^16 - 2 * q^17 - 6 * q^19 - 2 * q^22 + 4 * q^23 - 4 * q^28 + 12 * q^29 + 8 * q^31 - 2 * q^32 + 2 * q^34 - 16 * q^37 + 6 * q^38 + 2 * q^41 - 10 * q^43 + 2 * q^44 - 4 * q^46 - 4 * q^47 + 6 * q^49 + 12 * q^53 + 4 * q^56 - 12 * q^58 - 2 * q^59 + 4 * q^61 - 8 * q^62 + 2 * q^64 - 14 * q^67 - 2 * q^68 - 10 * q^73 + 16 * q^74 - 6 * q^76 + 8 * q^77 - 2 * q^82 - 8 * q^83 + 10 * q^86 - 2 * q^88 - 16 * q^89 - 12 * q^91 + 4 * q^92 + 4 * q^94 - 26 * q^97 - 6 * q^98

Introduction

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