LMFDB - Embedded newform 9450.2.a.ek.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9450,2,Mod(1,9450)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9450, 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("9450.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9450 = 2 \cdot 3^{3} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9450.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:	$75.4586299101$
Analytic rank:	$0$
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:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1890)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.44949$ of defining polynomial
Character	$\chi$	$=$	9450.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{7} -1.00000 q^{8} +O(q^{10})$	$q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{7} -1.00000 q^{8} -2.44949 q^{11} -1.44949 q^{13} -1.00000 q^{14} +1.00000 q^{16} -1.00000 q^{17} -4.44949 q^{19} +2.44949 q^{22} -1.44949 q^{23} +1.44949 q^{26} +1.00000 q^{28} -2.55051 q^{29} -1.44949 q^{31} -1.00000 q^{32} +1.00000 q^{34} -3.34847 q^{37} +4.44949 q^{38} -4.89898 q^{41} +3.89898 q^{43} -2.44949 q^{44} +1.44949 q^{46} -11.7980 q^{47} +1.00000 q^{49} -1.44949 q^{52} +5.44949 q^{53} -1.00000 q^{56} +2.55051 q^{58} -0.101021 q^{59} +10.8990 q^{61} +1.44949 q^{62} +1.00000 q^{64} +5.00000 q^{67} -1.00000 q^{68} +5.44949 q^{71} +11.7980 q^{73} +3.34847 q^{74} -4.44949 q^{76} -2.44949 q^{77} -6.44949 q^{79} +4.89898 q^{82} +10.8990 q^{83} -3.89898 q^{86} +2.44949 q^{88} -11.0000 q^{89} -1.44949 q^{91} -1.44949 q^{92} +11.7980 q^{94} +9.55051 q^{97} -1.00000 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} + 2 q^{7} - 2 q^{8}+O(q^{10}) $ 2 * q - 2 * q^2 + 2 * q^4 + 2 * q^7 - 2 * q^8	$ 2 q - 2 q^{2} + 2 q^{4} + 2 q^{7} - 2 q^{8} + 2 q^{13} - 2 q^{14} + 2 q^{16} - 2 q^{17} - 4 q^{19} + 2 q^{23} - 2 q^{26} + 2 q^{28} - 10 q^{29} + 2 q^{31} - 2 q^{32} + 2 q^{34} + 8 q^{37} + 4 q^{38} - 2 q^{43} - 2 q^{46} - 4 q^{47} + 2 q^{49} + 2 q^{52} + 6 q^{53} - 2 q^{56} + 10 q^{58} - 10 q^{59} + 12 q^{61} - 2 q^{62} + 2 q^{64} + 10 q^{67} - 2 q^{68} + 6 q^{71} + 4 q^{73} - 8 q^{74} - 4 q^{76} - 8 q^{79} + 12 q^{83} + 2 q^{86} - 22 q^{89} + 2 q^{91} + 2 q^{92} + 4 q^{94} + 24 q^{97} - 2 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 + 2 * q^7 - 2 * q^8 + 2 * q^13 - 2 * q^14 + 2 * q^16 - 2 * q^17 - 4 * q^19 + 2 * q^23 - 2 * q^26 + 2 * q^28 - 10 * q^29 + 2 * q^31 - 2 * q^32 + 2 * q^34 + 8 * q^37 + 4 * q^38 - 2 * q^43 - 2 * q^46 - 4 * q^47 + 2 * q^49 + 2 * q^52 + 6 * q^53 - 2 * q^56 + 10 * q^58 - 10 * q^59 + 12 * q^61 - 2 * q^62 + 2 * q^64 + 10 * q^67 - 2 * q^68 + 6 * q^71 + 4 * q^73 - 8 * q^74 - 4 * q^76 - 8 * q^79 + 12 * q^83 + 2 * q^86 - 22 * q^89 + 2 * q^91 + 2 * q^92 + 4 * q^94 + 24 * q^97 - 2 * 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
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−1.00000	−0.353553
$9$	0	0
$10$	0	0
$11$	−2.44949	−0.738549	−0.369274	−	0.929320i	$-0.620394\pi$
$11$	−2.44949	−0.738549	−0.369274	+	0.929320i	$0.620394\pi$
$12$	0	0
$13$	−1.44949	−0.402016	−0.201008	−	0.979590i	$-0.564422\pi$
$13$	−1.44949	−0.402016	−0.201008	+	0.979590i	$0.564422\pi$
$14$	−1.00000	−0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	−1.00000	−0.242536	−0.121268	−	0.992620i	$-0.538696\pi$
$17$	−1.00000	−0.242536	−0.121268	+	0.992620i	$0.538696\pi$
$18$	0	0
$19$	−4.44949	−1.02078	−0.510391	−	0.859942i	$-0.670499\pi$
$19$	−4.44949	−1.02078	−0.510391	+	0.859942i	$0.670499\pi$
$20$	0	0
$21$	0	0
$22$	2.44949	0.522233
$23$	−1.44949	−0.302240	−0.151120	−	0.988515i	$-0.548288\pi$
$23$	−1.44949	−0.302240	−0.151120	+	0.988515i	$0.548288\pi$
$24$	0	0
$25$	0	0
$26$	1.44949	0.284268
$27$	0	0
$28$	1.00000	0.188982
$29$	−2.55051	−0.473618	−0.236809	−	0.971556i	$-0.576102\pi$
$29$	−2.55051	−0.473618	−0.236809	+	0.971556i	$0.576102\pi$
$30$	0	0
$31$	−1.44949	−0.260336	−0.130168	−	0.991492i	$-0.541552\pi$
$31$	−1.44949	−0.260336	−0.130168	+	0.991492i	$0.541552\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	1.00000	0.171499
$35$	0	0
$36$	0	0
$37$	−3.34847	−0.550485	−0.275242	−	0.961375i	$-0.588758\pi$
$37$	−3.34847	−0.550485	−0.275242	+	0.961375i	$0.588758\pi$
$38$	4.44949	0.721803
$39$	0	0
$40$	0	0
$41$	−4.89898	−0.765092	−0.382546	−	0.923936i	$-0.624953\pi$
$41$	−4.89898	−0.765092	−0.382546	+	0.923936i	$0.624953\pi$
$42$	0	0
$43$	3.89898	0.594589	0.297294	−	0.954786i	$-0.403916\pi$
$43$	3.89898	0.594589	0.297294	+	0.954786i	$0.403916\pi$
$44$	−2.44949	−0.369274
$45$	0	0
$46$	1.44949	0.213716
$47$	−11.7980	−1.72091	−0.860455	−	0.509527i	$-0.829820\pi$
$47$	−11.7980	−1.72091	−0.860455	+	0.509527i	$0.829820\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	−1.44949	−0.201008
$53$	5.44949	0.748545	0.374272	−	0.927319i	$-0.377892\pi$
$53$	5.44949	0.748545	0.374272	+	0.927319i	$0.377892\pi$
$54$	0	0
$55$	0	0
$56$	−1.00000	−0.133631
$57$	0	0
$58$	2.55051	0.334898
$59$	−0.101021	−0.0131518	−0.00657588	−	0.999978i	$-0.502093\pi$
$59$	−0.101021	−0.0131518	−0.00657588	+	0.999978i	$0.502093\pi$
$60$	0	0
$61$	10.8990	1.39547	0.697736	−	0.716355i	$-0.254191\pi$
$61$	10.8990	1.39547	0.697736	+	0.716355i	$0.254191\pi$
$62$	1.44949	0.184085
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	5.00000	0.610847	0.305424	−	0.952217i	$-0.401202\pi$
$67$	5.00000	0.610847	0.305424	+	0.952217i	$0.401202\pi$
$68$	−1.00000	−0.121268
$69$	0	0
$70$	0	0
$71$	5.44949	0.646735	0.323368	−	0.946273i	$-0.395185\pi$
$71$	5.44949	0.646735	0.323368	+	0.946273i	$0.395185\pi$
$72$	0	0
$73$	11.7980	1.38085	0.690423	−	0.723406i	$-0.257424\pi$
$73$	11.7980	1.38085	0.690423	+	0.723406i	$0.257424\pi$
$74$	3.34847	0.389252
$75$	0	0
$76$	−4.44949	−0.510391
$77$	−2.44949	−0.279145
$78$	0	0
$79$	−6.44949	−0.725624	−0.362812	−	0.931862i	$-0.618183\pi$
$79$	−6.44949	−0.725624	−0.362812	+	0.931862i	$0.618183\pi$
$80$	0	0
$81$	0	0
$82$	4.89898	0.541002
$83$	10.8990	1.19632	0.598159	−	0.801377i	$-0.295899\pi$
$83$	10.8990	1.19632	0.598159	+	0.801377i	$0.295899\pi$
$84$	0	0
$85$	0	0
$86$	−3.89898	−0.420438
$87$	0	0
$88$	2.44949	0.261116
$89$	−11.0000	−1.16600	−0.582999	−	0.812473i	$-0.698121\pi$
$89$	−11.0000	−1.16600	−0.582999	+	0.812473i	$0.698121\pi$
$90$	0	0
$91$	−1.44949	−0.151948
$92$	−1.44949	−0.151120
$93$	0	0
$94$	11.7980	1.21687
$95$	0	0
$96$	0	0
$97$	9.55051	0.969707	0.484854	−	0.874595i	$-0.338873\pi$
$97$	9.55051	0.969707	0.484854	+	0.874595i	$0.338873\pi$
$98$	−1.00000	−0.101015
$99$	0	0
$100$	0	0
$101$	−12.2474	−1.21867	−0.609333	−	0.792914i	$-0.708563\pi$
$101$	−12.2474	−1.21867	−0.609333	+	0.792914i	$0.708563\pi$
$102$	0	0
$103$	15.4495	1.52228	0.761142	−	0.648586i	$-0.224639\pi$
$103$	15.4495	1.52228	0.761142	+	0.648586i	$0.224639\pi$
$104$	1.44949	0.142134
$105$	0	0
$106$	−5.44949	−0.529301
$107$	−3.34847	−0.323709	−0.161854	−	0.986815i	$-0.551747\pi$
$107$	−3.34847	−0.323709	−0.161854	+	0.986815i	$0.551747\pi$
$108$	0	0
$109$	−15.1464	−1.45076	−0.725382	−	0.688346i	$-0.758337\pi$
$109$	−15.1464	−1.45076	−0.725382	+	0.688346i	$0.758337\pi$
$110$	0	0
$111$	0	0
$112$	1.00000	0.0944911
$113$	6.44949	0.606717	0.303358	−	0.952877i	$-0.401892\pi$
$113$	6.44949	0.606717	0.303358	+	0.952877i	$0.401892\pi$
$114$	0	0
$115$	0	0
$116$	−2.55051	−0.236809
$117$	0	0
$118$	0.101021	0.00929969
$119$	−1.00000	−0.0916698
$120$	0	0
$121$	−5.00000	−0.454545
$122$	−10.8990	−0.986747
$123$	0	0
$124$	−1.44949	−0.130168
$125$	0	0
$126$	0	0
$127$	8.89898	0.789657	0.394828	−	0.918755i	$-0.370804\pi$
$127$	8.89898	0.789657	0.394828	+	0.918755i	$0.370804\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	0	0
$131$	16.7980	1.46764	0.733822	−	0.679342i	$-0.237734\pi$
$131$	16.7980	1.46764	0.733822	+	0.679342i	$0.237734\pi$
$132$	0	0
$133$	−4.44949	−0.385820
$134$	−5.00000	−0.431934
$135$	0	0
$136$	1.00000	0.0857493
$137$	−5.10102	−0.435810	−0.217905	−	0.975970i	$-0.569922\pi$
$137$	−5.10102	−0.435810	−0.217905	+	0.975970i	$0.569922\pi$
$138$	0	0
$139$	22.2474	1.88700	0.943502	−	0.331367i	$-0.107510\pi$
$139$	22.2474	1.88700	0.943502	+	0.331367i	$0.107510\pi$
$140$	0	0
$141$	0	0
$142$	−5.44949	−0.457311
$143$	3.55051	0.296909
$144$	0	0
$145$	0	0
$146$	−11.7980	−0.976406
$147$	0	0
$148$	−3.34847	−0.275242
$149$	−0.550510	−0.0450996	−0.0225498	−	0.999746i	$-0.507178\pi$
$149$	−0.550510	−0.0450996	−0.0225498	+	0.999746i	$0.507178\pi$
$150$	0	0
$151$	12.2474	0.996683	0.498342	−	0.866981i	$-0.333943\pi$
$151$	12.2474	0.996683	0.498342	+	0.866981i	$0.333943\pi$
$152$	4.44949	0.360901
$153$	0	0
$154$	2.44949	0.197386
$155$	0	0
$156$	0	0
$157$	5.44949	0.434917	0.217458	−	0.976070i	$-0.430223\pi$
$157$	5.44949	0.434917	0.217458	+	0.976070i	$0.430223\pi$
$158$	6.44949	0.513094
$159$	0	0
$160$	0	0
$161$	−1.44949	−0.114236
$162$	0	0
$163$	4.10102	0.321217	0.160608	−	0.987018i	$-0.448654\pi$
$163$	4.10102	0.321217	0.160608	+	0.987018i	$0.448654\pi$
$164$	−4.89898	−0.382546
$165$	0	0
$166$	−10.8990	−0.845925
$167$	−3.55051	−0.274747	−0.137373	−	0.990519i	$-0.543866\pi$
$167$	−3.55051	−0.274747	−0.137373	+	0.990519i	$0.543866\pi$
$168$	0	0
$169$	−10.8990	−0.838383
$170$	0	0
$171$	0	0
$172$	3.89898	0.297294
$173$	11.1464	0.847447	0.423724	−	0.905792i	$-0.360723\pi$
$173$	11.1464	0.847447	0.423724	+	0.905792i	$0.360723\pi$
$174$	0	0
$175$	0	0
$176$	−2.44949	−0.184637
$177$	0	0
$178$	11.0000	0.824485
$179$	0.898979	0.0671929	0.0335964	−	0.999435i	$-0.489304\pi$
$179$	0.898979	0.0671929	0.0335964	+	0.999435i	$0.489304\pi$
$180$	0	0
$181$	7.44949	0.553716	0.276858	−	0.960911i	$-0.410707\pi$
$181$	7.44949	0.553716	0.276858	+	0.960911i	$0.410707\pi$
$182$	1.44949	0.107443
$183$	0	0
$184$	1.44949	0.106858
$185$	0	0
$186$	0	0
$187$	2.44949	0.179124
$188$	−11.7980	−0.860455
$189$	0	0
$190$	0	0
$191$	22.6969	1.64229	0.821146	−	0.570718i	$-0.193335\pi$
$191$	22.6969	1.64229	0.821146	+	0.570718i	$0.193335\pi$
$192$	0	0
$193$	7.20204	0.518414	0.259207	−	0.965822i	$-0.416539\pi$
$193$	7.20204	0.518414	0.259207	+	0.965822i	$0.416539\pi$
$194$	−9.55051	−0.685687
$195$	0	0
$196$	1.00000	0.0714286
$197$	12.8990	0.919014	0.459507	−	0.888174i	$-0.348026\pi$
$197$	12.8990	0.919014	0.459507	+	0.888174i	$0.348026\pi$
$198$	0	0
$199$	9.24745	0.655534	0.327767	−	0.944759i	$-0.393704\pi$
$199$	9.24745	0.655534	0.327767	+	0.944759i	$0.393704\pi$
$200$	0	0
$201$	0	0
$202$	12.2474	0.861727
$203$	−2.55051	−0.179011
$204$	0	0
$205$	0	0
$206$	−15.4495	−1.07642
$207$	0	0
$208$	−1.44949	−0.100504
$209$	10.8990	0.753898
$210$	0	0
$211$	−15.0000	−1.03264	−0.516321	−	0.856395i	$-0.672699\pi$
$211$	−15.0000	−1.03264	−0.516321	+	0.856395i	$0.672699\pi$
$212$	5.44949	0.374272
$213$	0	0
$214$	3.34847	0.228897
$215$	0	0
$216$	0	0
$217$	−1.44949	−0.0983978
$218$	15.1464	1.02585
$219$	0	0
$220$	0	0
$221$	1.44949	0.0975032
$222$	0	0
$223$	4.00000	0.267860	0.133930	−	0.990991i	$-0.457240\pi$
$223$	4.00000	0.267860	0.133930	+	0.990991i	$0.457240\pi$
$224$	−1.00000	−0.0668153
$225$	0	0
$226$	−6.44949	−0.429014
$227$	11.6969	0.776353	0.388177	−	0.921585i	$-0.373105\pi$
$227$	11.6969	0.776353	0.388177	+	0.921585i	$0.373105\pi$
$228$	0	0
$229$	−3.10102	−0.204921	−0.102461	−	0.994737i	$-0.532672\pi$
$229$	−3.10102	−0.204921	−0.102461	+	0.994737i	$0.532672\pi$
$230$	0	0
$231$	0	0
$232$	2.55051	0.167449
$233$	13.1464	0.861251	0.430626	−	0.902531i	$-0.358293\pi$
$233$	13.1464	0.861251	0.430626	+	0.902531i	$0.358293\pi$
$234$	0	0
$235$	0	0
$236$	−0.101021	−0.00657588
$237$	0	0
$238$	1.00000	0.0648204
$239$	−20.4949	−1.32570	−0.662852	−	0.748750i	$-0.730654\pi$
$239$	−20.4949	−1.32570	−0.662852	+	0.748750i	$0.730654\pi$
$240$	0	0
$241$	10.6515	0.686125	0.343063	−	0.939313i	$-0.388536\pi$
$241$	10.6515	0.686125	0.343063	+	0.939313i	$0.388536\pi$
$242$	5.00000	0.321412
$243$	0	0
$244$	10.8990	0.697736
$245$	0	0
$246$	0	0
$247$	6.44949	0.410371
$248$	1.44949	0.0920427
$249$	0	0
$250$	0	0
$251$	−27.5959	−1.74184	−0.870919	−	0.491426i	$-0.836476\pi$
$251$	−27.5959	−1.74184	−0.870919	+	0.491426i	$0.836476\pi$
$252$	0	0
$253$	3.55051	0.223219
$254$	−8.89898	−0.558372
$255$	0	0
$256$	1.00000	0.0625000
$257$	12.0000	0.748539	0.374270	−	0.927320i	$-0.377893\pi$
$257$	12.0000	0.748539	0.374270	+	0.927320i	$0.377893\pi$
$258$	0	0
$259$	−3.34847	−0.208064
$260$	0	0
$261$	0	0
$262$	−16.7980	−1.03778
$263$	−23.9444	−1.47647	−0.738237	−	0.674541i	$-0.764341\pi$
$263$	−23.9444	−1.47647	−0.738237	+	0.674541i	$0.764341\pi$
$264$	0	0
$265$	0	0
$266$	4.44949	0.272816
$267$	0	0
$268$	5.00000	0.305424
$269$	−3.10102	−0.189073	−0.0945363	−	0.995521i	$-0.530137\pi$
$269$	−3.10102	−0.189073	−0.0945363	+	0.995521i	$0.530137\pi$
$270$	0	0
$271$	−4.34847	−0.264151	−0.132075	−	0.991240i	$-0.542164\pi$
$271$	−4.34847	−0.264151	−0.132075	+	0.991240i	$0.542164\pi$
$272$	−1.00000	−0.0606339
$273$	0	0
$274$	5.10102	0.308164
$275$	0	0
$276$	0	0
$277$	6.20204	0.372645	0.186322	−	0.982489i	$-0.440343\pi$
$277$	6.20204	0.372645	0.186322	+	0.982489i	$0.440343\pi$
$278$	−22.2474	−1.33431
$279$	0	0
$280$	0	0
$281$	13.3485	0.796303	0.398151	−	0.917320i	$-0.369652\pi$
$281$	13.3485	0.796303	0.398151	+	0.917320i	$0.369652\pi$
$282$	0	0
$283$	17.5505	1.04327	0.521635	−	0.853169i	$-0.325322\pi$
$283$	17.5505	1.04327	0.521635	+	0.853169i	$0.325322\pi$
$284$	5.44949	0.323368
$285$	0	0
$286$	−3.55051	−0.209946
$287$	−4.89898	−0.289178
$288$	0	0
$289$	−16.0000	−0.941176
$290$	0	0
$291$	0	0
$292$	11.7980	0.690423
$293$	−30.4949	−1.78153	−0.890765	−	0.454463i	$-0.849831\pi$
$293$	−30.4949	−1.78153	−0.890765	+	0.454463i	$0.849831\pi$
$294$	0	0
$295$	0	0
$296$	3.34847	0.194626
$297$	0	0
$298$	0.550510	0.0318902
$299$	2.10102	0.121505
$300$	0	0
$301$	3.89898	0.224733
$302$	−12.2474	−0.704761
$303$	0	0
$304$	−4.44949	−0.255196
$305$	0	0
$306$	0	0
$307$	−24.4495	−1.39541	−0.697703	−	0.716387i	$-0.745794\pi$
$307$	−24.4495	−1.39541	−0.697703	+	0.716387i	$0.745794\pi$
$308$	−2.44949	−0.139573
$309$	0	0
$310$	0	0
$311$	13.3485	0.756922	0.378461	−	0.925617i	$-0.376453\pi$
$311$	13.3485	0.756922	0.378461	+	0.925617i	$0.376453\pi$
$312$	0	0
$313$	28.9444	1.63603	0.818017	−	0.575194i	$-0.195074\pi$
$313$	28.9444	1.63603	0.818017	+	0.575194i	$0.195074\pi$
$314$	−5.44949	−0.307532
$315$	0	0
$316$	−6.44949	−0.362812
$317$	4.20204	0.236010	0.118005	−	0.993013i	$-0.462350\pi$
$317$	4.20204	0.236010	0.118005	+	0.993013i	$0.462350\pi$
$318$	0	0
$319$	6.24745	0.349790
$320$	0	0
$321$	0	0
$322$	1.44949	0.0807769
$323$	4.44949	0.247576
$324$	0	0
$325$	0	0
$326$	−4.10102	−0.227135
$327$	0	0
$328$	4.89898	0.270501
$329$	−11.7980	−0.650443
$330$	0	0
$331$	−13.0000	−0.714545	−0.357272	−	0.934000i	$-0.616293\pi$
$331$	−13.0000	−0.714545	−0.357272	+	0.934000i	$0.616293\pi$
$332$	10.8990	0.598159
$333$	0	0
$334$	3.55051	0.194275
$335$	0	0
$336$	0	0
$337$	22.7980	1.24188	0.620942	−	0.783857i	$-0.286750\pi$
$337$	22.7980	1.24188	0.620942	+	0.783857i	$0.286750\pi$
$338$	10.8990	0.592826
$339$	0	0
$340$	0	0
$341$	3.55051	0.192271
$342$	0	0
$343$	1.00000	0.0539949
$344$	−3.89898	−0.210219
$345$	0	0
$346$	−11.1464	−0.599236
$347$	13.5959	0.729867	0.364934	−	0.931034i	$-0.381092\pi$
$347$	13.5959	0.729867	0.364934	+	0.931034i	$0.381092\pi$
$348$	0	0
$349$	−20.1464	−1.07841	−0.539207	−	0.842173i	$-0.681276\pi$
$349$	−20.1464	−1.07841	−0.539207	+	0.842173i	$0.681276\pi$
$350$	0	0
$351$	0	0
$352$	2.44949	0.130558
$353$	−15.4949	−0.824710	−0.412355	−	0.911023i	$-0.635294\pi$
$353$	−15.4949	−0.824710	−0.412355	+	0.911023i	$0.635294\pi$
$354$	0	0
$355$	0	0
$356$	−11.0000	−0.582999
$357$	0	0
$358$	−0.898979	−0.0475125
$359$	−21.4495	−1.13206	−0.566030	−	0.824384i	$-0.691522\pi$
$359$	−21.4495	−1.13206	−0.566030	+	0.824384i	$0.691522\pi$
$360$	0	0
$361$	0.797959	0.0419978
$362$	−7.44949	−0.391536
$363$	0	0
$364$	−1.44949	−0.0759739
$365$	0	0
$366$	0	0
$367$	−15.0454	−0.785364	−0.392682	−	0.919674i	$-0.628453\pi$
$367$	−15.0454	−0.785364	−0.392682	+	0.919674i	$0.628453\pi$
$368$	−1.44949	−0.0755599
$369$	0	0
$370$	0	0
$371$	5.44949	0.282923
$372$	0	0
$373$	27.3485	1.41605	0.708025	−	0.706187i	$-0.249586\pi$
$373$	27.3485	1.41605	0.708025	+	0.706187i	$0.249586\pi$
$374$	−2.44949	−0.126660
$375$	0	0
$376$	11.7980	0.608433
$377$	3.69694	0.190402
$378$	0	0
$379$	10.0000	0.513665	0.256833	−	0.966456i	$-0.417321\pi$
$379$	10.0000	0.513665	0.256833	+	0.966456i	$0.417321\pi$
$380$	0	0
$381$	0	0
$382$	−22.6969	−1.16128
$383$	−7.10102	−0.362845	−0.181423	−	0.983405i	$-0.558070\pi$
$383$	−7.10102	−0.362845	−0.181423	+	0.983405i	$0.558070\pi$
$384$	0	0
$385$	0	0
$386$	−7.20204	−0.366574
$387$	0	0
$388$	9.55051	0.484854
$389$	28.8990	1.46524	0.732618	−	0.680640i	$-0.238298\pi$
$389$	28.8990	1.46524	0.732618	+	0.680640i	$0.238298\pi$
$390$	0	0
$391$	1.44949	0.0733038
$392$	−1.00000	−0.0505076
$393$	0	0
$394$	−12.8990	−0.649841
$395$	0	0
$396$	0	0
$397$	−22.6969	−1.13913	−0.569563	−	0.821947i	$-0.692888\pi$
$397$	−22.6969	−1.13913	−0.569563	+	0.821947i	$0.692888\pi$
$398$	−9.24745	−0.463533
$399$	0	0
$400$	0	0
$401$	8.69694	0.434304	0.217152	−	0.976138i	$-0.430323\pi$
$401$	8.69694	0.434304	0.217152	+	0.976138i	$0.430323\pi$
$402$	0	0
$403$	2.10102	0.104659
$404$	−12.2474	−0.609333
$405$	0	0
$406$	2.55051	0.126580
$407$	8.20204	0.406560
$408$	0	0
$409$	−9.59592	−0.474488	−0.237244	−	0.971450i	$-0.576244\pi$
$409$	−9.59592	−0.474488	−0.237244	+	0.971450i	$0.576244\pi$
$410$	0	0
$411$	0	0
$412$	15.4495	0.761142
$413$	−0.101021	−0.00497089
$414$	0	0
$415$	0	0
$416$	1.44949	0.0710671
$417$	0	0
$418$	−10.8990	−0.533087
$419$	38.5959	1.88553	0.942767	−	0.333452i	$-0.108214\pi$
$419$	38.5959	1.88553	0.942767	+	0.333452i	$0.108214\pi$
$420$	0	0
$421$	19.7980	0.964893	0.482447	−	0.875925i	$-0.339748\pi$
$421$	19.7980	0.964893	0.482447	+	0.875925i	$0.339748\pi$
$422$	15.0000	0.730189
$423$	0	0
$424$	−5.44949	−0.264651
$425$	0	0
$426$	0	0
$427$	10.8990	0.527439
$428$	−3.34847	−0.161854
$429$	0	0
$430$	0	0
$431$	7.79796	0.375614	0.187807	−	0.982206i	$-0.439862\pi$
$431$	7.79796	0.375614	0.187807	+	0.982206i	$0.439862\pi$
$432$	0	0
$433$	23.5959	1.13395	0.566974	−	0.823736i	$-0.308114\pi$
$433$	23.5959	1.13395	0.566974	+	0.823736i	$0.308114\pi$
$434$	1.44949	0.0695777
$435$	0	0
$436$	−15.1464	−0.725382
$437$	6.44949	0.308521
$438$	0	0
$439$	6.34847	0.302996	0.151498	−	0.988458i	$-0.451590\pi$
$439$	6.34847	0.302996	0.151498	+	0.988458i	$0.451590\pi$
$440$	0	0
$441$	0	0
$442$	−1.44949	−0.0689452
$443$	−27.5505	−1.30896	−0.654482	−	0.756077i	$-0.727113\pi$
$443$	−27.5505	−1.30896	−0.654482	+	0.756077i	$0.727113\pi$
$444$	0	0
$445$	0	0
$446$	−4.00000	−0.189405
$447$	0	0
$448$	1.00000	0.0472456
$449$	−32.0000	−1.51017	−0.755087	−	0.655625i	$-0.772405\pi$
$449$	−32.0000	−1.51017	−0.755087	+	0.655625i	$0.772405\pi$
$450$	0	0
$451$	12.0000	0.565058
$452$	6.44949	0.303358
$453$	0	0
$454$	−11.6969	−0.548965
$455$	0	0
$456$	0	0
$457$	0.101021	0.00472554	0.00236277	−	0.999997i	$-0.499248\pi$
$457$	0.101021	0.00472554	0.00236277	+	0.999997i	$0.499248\pi$
$458$	3.10102	0.144901
$459$	0	0
$460$	0	0
$461$	−10.0000	−0.465746	−0.232873	−	0.972507i	$-0.574813\pi$
$461$	−10.0000	−0.465746	−0.232873	+	0.972507i	$0.574813\pi$
$462$	0	0
$463$	3.75255	0.174396	0.0871979	−	0.996191i	$-0.472209\pi$
$463$	3.75255	0.174396	0.0871979	+	0.996191i	$0.472209\pi$
$464$	−2.55051	−0.118404
$465$	0	0
$466$	−13.1464	−0.608997
$467$	−12.2020	−0.564643	−0.282322	−	0.959320i	$-0.591105\pi$
$467$	−12.2020	−0.564643	−0.282322	+	0.959320i	$0.591105\pi$
$468$	0	0
$469$	5.00000	0.230879
$470$	0	0
$471$	0	0
$472$	0.101021	0.00464985
$473$	−9.55051	−0.439133
$474$	0	0
$475$	0	0
$476$	−1.00000	−0.0458349
$477$	0	0
$478$	20.4949	0.937415
$479$	36.0000	1.64488	0.822441	−	0.568850i	$-0.192612\pi$
$479$	36.0000	1.64488	0.822441	+	0.568850i	$0.192612\pi$
$480$	0	0
$481$	4.85357	0.221304
$482$	−10.6515	−0.485164
$483$	0	0
$484$	−5.00000	−0.227273
$485$	0	0
$486$	0	0
$487$	39.3939	1.78511	0.892553	−	0.450942i	$-0.148912\pi$
$487$	39.3939	1.78511	0.892553	+	0.450942i	$0.148912\pi$
$488$	−10.8990	−0.493374
$489$	0	0
$490$	0	0
$491$	4.40408	0.198753	0.0993767	−	0.995050i	$-0.468315\pi$
$491$	4.40408	0.198753	0.0993767	+	0.995050i	$0.468315\pi$
$492$	0	0
$493$	2.55051	0.114869
$494$	−6.44949	−0.290176
$495$	0	0
$496$	−1.44949	−0.0650840
$497$	5.44949	0.244443
$498$	0	0
$499$	2.89898	0.129776	0.0648881	−	0.997893i	$-0.479331\pi$
$499$	2.89898	0.129776	0.0648881	+	0.997893i	$0.479331\pi$
$500$	0	0
$501$	0	0
$502$	27.5959	1.23167
$503$	6.65153	0.296577	0.148289	−	0.988944i	$-0.452624\pi$
$503$	6.65153	0.296577	0.148289	+	0.988944i	$0.452624\pi$
$504$	0	0
$505$	0	0
$506$	−3.55051	−0.157839
$507$	0	0
$508$	8.89898	0.394828
$509$	−13.1464	−0.582705	−0.291353	−	0.956616i	$-0.594105\pi$
$509$	−13.1464	−0.582705	−0.291353	+	0.956616i	$0.594105\pi$
$510$	0	0
$511$	11.7980	0.521911
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−12.0000	−0.529297
$515$	0	0
$516$	0	0
$517$	28.8990	1.27098
$518$	3.34847	0.147123
$519$	0	0
$520$	0	0
$521$	−3.00000	−0.131432	−0.0657162	−	0.997838i	$-0.520933\pi$
$521$	−3.00000	−0.131432	−0.0657162	+	0.997838i	$0.520933\pi$
$522$	0	0
$523$	−3.59592	−0.157239	−0.0786193	−	0.996905i	$-0.525051\pi$
$523$	−3.59592	−0.157239	−0.0786193	+	0.996905i	$0.525051\pi$
$524$	16.7980	0.733822
$525$	0	0
$526$	23.9444	1.04402
$527$	1.44949	0.0631408
$528$	0	0
$529$	−20.8990	−0.908651
$530$	0	0
$531$	0	0
$532$	−4.44949	−0.192910
$533$	7.10102	0.307579
$534$	0	0
$535$	0	0
$536$	−5.00000	−0.215967
$537$	0	0
$538$	3.10102	0.133694
$539$	−2.44949	−0.105507
$540$	0	0
$541$	−9.75255	−0.419295	−0.209647	−	0.977777i	$-0.567232\pi$
$541$	−9.75255	−0.419295	−0.209647	+	0.977777i	$0.567232\pi$
$542$	4.34847	0.186783
$543$	0	0
$544$	1.00000	0.0428746
$545$	0	0
$546$	0	0
$547$	0.696938	0.0297989	0.0148995	−	0.999889i	$-0.495257\pi$
$547$	0.696938	0.0297989	0.0148995	+	0.999889i	$0.495257\pi$
$548$	−5.10102	−0.217905
$549$	0	0
$550$	0	0
$551$	11.3485	0.483461
$552$	0	0
$553$	−6.44949	−0.274260
$554$	−6.20204	−0.263499
$555$	0	0
$556$	22.2474	0.943502
$557$	−33.0454	−1.40018	−0.700089	−	0.714055i	$-0.746857\pi$
$557$	−33.0454	−1.40018	−0.700089	+	0.714055i	$0.746857\pi$
$558$	0	0
$559$	−5.65153	−0.239034
$560$	0	0
$561$	0	0
$562$	−13.3485	−0.563071
$563$	9.69694	0.408677	0.204339	−	0.978900i	$-0.434496\pi$
$563$	9.69694	0.408677	0.204339	+	0.978900i	$0.434496\pi$
$564$	0	0
$565$	0	0
$566$	−17.5505	−0.737703
$567$	0	0
$568$	−5.44949	−0.228656
$569$	−31.8434	−1.33494	−0.667472	−	0.744635i	$-0.732623\pi$
$569$	−31.8434	−1.33494	−0.667472	+	0.744635i	$0.732623\pi$
$570$	0	0
$571$	−19.4949	−0.815836	−0.407918	−	0.913019i	$-0.633745\pi$
$571$	−19.4949	−0.815836	−0.407918	+	0.913019i	$0.633745\pi$
$572$	3.55051	0.148454
$573$	0	0
$574$	4.89898	0.204479
$575$	0	0
$576$	0	0
$577$	−19.5505	−0.813898	−0.406949	−	0.913451i	$-0.633407\pi$
$577$	−19.5505	−0.813898	−0.406949	+	0.913451i	$0.633407\pi$
$578$	16.0000	0.665512
$579$	0	0
$580$	0	0
$581$	10.8990	0.452166
$582$	0	0
$583$	−13.3485	−0.552837
$584$	−11.7980	−0.488203
$585$	0	0
$586$	30.4949	1.25973
$587$	35.4949	1.46503	0.732516	−	0.680750i	$-0.238346\pi$
$587$	35.4949	1.46503	0.732516	+	0.680750i	$0.238346\pi$
$588$	0	0
$589$	6.44949	0.265747
$590$	0	0
$591$	0	0
$592$	−3.34847	−0.137621
$593$	−16.8990	−0.693958	−0.346979	−	0.937873i	$-0.612792\pi$
$593$	−16.8990	−0.693958	−0.346979	+	0.937873i	$0.612792\pi$
$594$	0	0
$595$	0	0
$596$	−0.550510	−0.0225498
$597$	0	0
$598$	−2.10102	−0.0859171
$599$	5.44949	0.222660	0.111330	−	0.993783i	$-0.464489\pi$
$599$	5.44949	0.222660	0.111330	+	0.993783i	$0.464489\pi$
$600$	0	0
$601$	−13.3031	−0.542643	−0.271322	−	0.962489i	$-0.587461\pi$
$601$	−13.3031	−0.542643	−0.271322	+	0.962489i	$0.587461\pi$
$602$	−3.89898	−0.158911
$603$	0	0
$604$	12.2474	0.498342
$605$	0	0
$606$	0	0
$607$	−23.0454	−0.935384	−0.467692	−	0.883891i	$-0.654914\pi$
$607$	−23.0454	−0.935384	−0.467692	+	0.883891i	$0.654914\pi$
$608$	4.44949	0.180451
$609$	0	0
$610$	0	0
$611$	17.1010	0.691833
$612$	0	0
$613$	−48.2474	−1.94870	−0.974348	−	0.225046i	$-0.927747\pi$
$613$	−48.2474	−1.94870	−0.974348	+	0.225046i	$0.927747\pi$
$614$	24.4495	0.986701
$615$	0	0
$616$	2.44949	0.0986928
$617$	38.8990	1.56601	0.783007	−	0.622013i	$-0.213685\pi$
$617$	38.8990	1.56601	0.783007	+	0.622013i	$0.213685\pi$
$618$	0	0
$619$	43.7980	1.76039	0.880194	−	0.474614i	$-0.157412\pi$
$619$	43.7980	1.76039	0.880194	+	0.474614i	$0.157412\pi$
$620$	0	0
$621$	0	0
$622$	−13.3485	−0.535225
$623$	−11.0000	−0.440706
$624$	0	0
$625$	0	0
$626$	−28.9444	−1.15685
$627$	0	0
$628$	5.44949	0.217458
$629$	3.34847	0.133512
$630$	0	0
$631$	4.89898	0.195025	0.0975126	−	0.995234i	$-0.468911\pi$
$631$	4.89898	0.195025	0.0975126	+	0.995234i	$0.468911\pi$
$632$	6.44949	0.256547
$633$	0	0
$634$	−4.20204	−0.166884
$635$	0	0
$636$	0	0
$637$	−1.44949	−0.0574309
$638$	−6.24745	−0.247339
$639$	0	0
$640$	0	0
$641$	5.55051	0.219232	0.109616	−	0.993974i	$-0.465038\pi$
$641$	5.55051	0.219232	0.109616	+	0.993974i	$0.465038\pi$
$642$	0	0
$643$	10.8536	0.428023	0.214012	−	0.976831i	$-0.431347\pi$
$643$	10.8536	0.428023	0.214012	+	0.976831i	$0.431347\pi$
$644$	−1.44949	−0.0571179
$645$	0	0
$646$	−4.44949	−0.175063
$647$	−13.5959	−0.534511	−0.267255	−	0.963626i	$-0.586117\pi$
$647$	−13.5959	−0.534511	−0.267255	+	0.963626i	$0.586117\pi$
$648$	0	0
$649$	0.247449	0.00971321
$650$	0	0
$651$	0	0
$652$	4.10102	0.160608
$653$	−15.2474	−0.596679	−0.298339	−	0.954460i	$-0.596433\pi$
$653$	−15.2474	−0.596679	−0.298339	+	0.954460i	$0.596433\pi$
$654$	0	0
$655$	0	0
$656$	−4.89898	−0.191273
$657$	0	0
$658$	11.7980	0.459932
$659$	23.7526	0.925268	0.462634	−	0.886549i	$-0.346904\pi$
$659$	23.7526	0.925268	0.462634	+	0.886549i	$0.346904\pi$
$660$	0	0
$661$	34.4949	1.34170	0.670848	−	0.741595i	$-0.265930\pi$
$661$	34.4949	1.34170	0.670848	+	0.741595i	$0.265930\pi$
$662$	13.0000	0.505259
$663$	0	0
$664$	−10.8990	−0.422962
$665$	0	0
$666$	0	0
$667$	3.69694	0.143146
$668$	−3.55051	−0.137373
$669$	0	0
$670$	0	0
$671$	−26.6969	−1.03062
$672$	0	0
$673$	47.4949	1.83079	0.915397	−	0.402553i	$-0.131877\pi$
$673$	47.4949	1.83079	0.915397	+	0.402553i	$0.131877\pi$
$674$	−22.7980	−0.878145
$675$	0	0
$676$	−10.8990	−0.419192
$677$	−13.7980	−0.530299	−0.265149	−	0.964207i	$-0.585421\pi$
$677$	−13.7980	−0.530299	−0.265149	+	0.964207i	$0.585421\pi$
$678$	0	0
$679$	9.55051	0.366515
$680$	0	0
$681$	0	0
$682$	−3.55051	−0.135956
$683$	5.55051	0.212384	0.106192	−	0.994346i	$-0.466134\pi$
$683$	5.55051	0.212384	0.106192	+	0.994346i	$0.466134\pi$
$684$	0	0
$685$	0	0
$686$	−1.00000	−0.0381802
$687$	0	0
$688$	3.89898	0.148647
$689$	−7.89898	−0.300927
$690$	0	0
$691$	−18.2929	−0.695893	−0.347947	−	0.937514i	$-0.613121\pi$
$691$	−18.2929	−0.695893	−0.347947	+	0.937514i	$0.613121\pi$
$692$	11.1464	0.423724
$693$	0	0
$694$	−13.5959	−0.516094
$695$	0	0
$696$	0	0
$697$	4.89898	0.185562
$698$	20.1464	0.762554
$699$	0	0
$700$	0	0
$701$	39.3939	1.48789	0.743943	−	0.668243i	$-0.232953\pi$
$701$	39.3939	1.48789	0.743943	+	0.668243i	$0.232953\pi$
$702$	0	0
$703$	14.8990	0.561926
$704$	−2.44949	−0.0923186
$705$	0	0
$706$	15.4949	0.583158
$707$	−12.2474	−0.460613
$708$	0	0
$709$	23.1464	0.869282	0.434641	−	0.900604i	$-0.356875\pi$
$709$	23.1464	0.869282	0.434641	+	0.900604i	$0.356875\pi$
$710$	0	0
$711$	0	0
$712$	11.0000	0.412242
$713$	2.10102	0.0786838
$714$	0	0
$715$	0	0
$716$	0.898979	0.0335964
$717$	0	0
$718$	21.4495	0.800488
$719$	15.1464	0.564866	0.282433	−	0.959287i	$-0.408858\pi$
$719$	15.1464	0.564866	0.282433	+	0.959287i	$0.408858\pi$
$720$	0	0
$721$	15.4495	0.575369
$722$	−0.797959	−0.0296970
$723$	0	0
$724$	7.44949	0.276858
$725$	0	0
$726$	0	0
$727$	−3.65153	−0.135428	−0.0677139	−	0.997705i	$-0.521571\pi$
$727$	−3.65153	−0.135428	−0.0677139	+	0.997705i	$0.521571\pi$
$728$	1.44949	0.0537217
$729$	0	0
$730$	0	0
$731$	−3.89898	−0.144209
$732$	0	0
$733$	37.4495	1.38323	0.691614	−	0.722267i	$-0.256900\pi$
$733$	37.4495	1.38323	0.691614	+	0.722267i	$0.256900\pi$
$734$	15.0454	0.555336
$735$	0	0
$736$	1.44949	0.0534289
$737$	−12.2474	−0.451141
$738$	0	0
$739$	43.7980	1.61113	0.805567	−	0.592505i	$-0.201861\pi$
$739$	43.7980	1.61113	0.805567	+	0.592505i	$0.201861\pi$
$740$	0	0
$741$	0	0
$742$	−5.44949	−0.200057
$743$	40.3485	1.48024	0.740121	−	0.672474i	$-0.234768\pi$
$743$	40.3485	1.48024	0.740121	+	0.672474i	$0.234768\pi$
$744$	0	0
$745$	0	0
$746$	−27.3485	−1.00130
$747$	0	0
$748$	2.44949	0.0895622
$749$	−3.34847	−0.122350
$750$	0	0
$751$	−46.4495	−1.69497	−0.847483	−	0.530823i	$-0.821883\pi$
$751$	−46.4495	−1.69497	−0.847483	+	0.530823i	$0.821883\pi$
$752$	−11.7980	−0.430227
$753$	0	0
$754$	−3.69694	−0.134635
$755$	0	0
$756$	0	0
$757$	−43.5959	−1.58452	−0.792260	−	0.610183i	$-0.791096\pi$
$757$	−43.5959	−1.58452	−0.792260	+	0.610183i	$0.791096\pi$
$758$	−10.0000	−0.363216
$759$	0	0
$760$	0	0
$761$	29.4949	1.06919	0.534595	−	0.845109i	$-0.320464\pi$
$761$	29.4949	1.06919	0.534595	+	0.845109i	$0.320464\pi$
$762$	0	0
$763$	−15.1464	−0.548338
$764$	22.6969	0.821146
$765$	0	0
$766$	7.10102	0.256570
$767$	0.146428	0.00528722
$768$	0	0
$769$	−20.7423	−0.747988	−0.373994	−	0.927431i	$-0.622012\pi$
$769$	−20.7423	−0.747988	−0.373994	+	0.927431i	$0.622012\pi$
$770$	0	0
$771$	0	0
$772$	7.20204	0.259207
$773$	38.6969	1.39183	0.695916	−	0.718123i	$-0.254999\pi$
$773$	38.6969	1.39183	0.695916	+	0.718123i	$0.254999\pi$
$774$	0	0
$775$	0	0
$776$	−9.55051	−0.342843
$777$	0	0
$778$	−28.8990	−1.03608
$779$	21.7980	0.780993
$780$	0	0
$781$	−13.3485	−0.477646
$782$	−1.44949	−0.0518336
$783$	0	0
$784$	1.00000	0.0357143
$785$	0	0
$786$	0	0
$787$	52.2929	1.86404	0.932020	−	0.362408i	$-0.118045\pi$
$787$	52.2929	1.86404	0.932020	+	0.362408i	$0.118045\pi$
$788$	12.8990	0.459507
$789$	0	0
$790$	0	0
$791$	6.44949	0.229317
$792$	0	0
$793$	−15.7980	−0.561002
$794$	22.6969	0.805484
$795$	0	0
$796$	9.24745	0.327767
$797$	2.49490	0.0883738	0.0441869	−	0.999023i	$-0.485930\pi$
$797$	2.49490	0.0883738	0.0441869	+	0.999023i	$0.485930\pi$
$798$	0	0
$799$	11.7980	0.417382
$800$	0	0
$801$	0	0
$802$	−8.69694	−0.307100
$803$	−28.8990	−1.01982
$804$	0	0
$805$	0	0
$806$	−2.10102	−0.0740053
$807$	0	0
$808$	12.2474	0.430864
$809$	−6.20204	−0.218052	−0.109026	−	0.994039i	$-0.534773\pi$
$809$	−6.20204	−0.218052	−0.109026	+	0.994039i	$0.534773\pi$
$810$	0	0
$811$	−45.1918	−1.58690	−0.793450	−	0.608635i	$-0.791717\pi$
$811$	−45.1918	−1.58690	−0.793450	+	0.608635i	$0.791717\pi$
$812$	−2.55051	−0.0895054
$813$	0	0
$814$	−8.20204	−0.287481
$815$	0	0
$816$	0	0
$817$	−17.3485	−0.606946
$818$	9.59592	0.335513
$819$	0	0
$820$	0	0
$821$	53.4495	1.86540	0.932700	−	0.360653i	$-0.117446\pi$
$821$	53.4495	1.86540	0.932700	+	0.360653i	$0.117446\pi$
$822$	0	0
$823$	55.8434	1.94658	0.973289	−	0.229585i	$-0.0737368\pi$
$823$	55.8434	1.94658	0.973289	+	0.229585i	$0.0737368\pi$
$824$	−15.4495	−0.538208
$825$	0	0
$826$	0.101021	0.00351495
$827$	−36.2929	−1.26203	−0.631013	−	0.775772i	$-0.717361\pi$
$827$	−36.2929	−1.26203	−0.631013	+	0.775772i	$0.717361\pi$
$828$	0	0
$829$	−6.89898	−0.239611	−0.119806	−	0.992797i	$-0.538227\pi$
$829$	−6.89898	−0.239611	−0.119806	+	0.992797i	$0.538227\pi$
$830$	0	0
$831$	0	0
$832$	−1.44949	−0.0502520
$833$	−1.00000	−0.0346479
$834$	0	0
$835$	0	0
$836$	10.8990	0.376949
$837$	0	0
$838$	−38.5959	−1.33327
$839$	42.4495	1.46552	0.732760	−	0.680488i	$-0.238232\pi$
$839$	42.4495	1.46552	0.732760	+	0.680488i	$0.238232\pi$
$840$	0	0
$841$	−22.4949	−0.775686
$842$	−19.7980	−0.682283
$843$	0	0
$844$	−15.0000	−0.516321
$845$	0	0
$846$	0	0
$847$	−5.00000	−0.171802
$848$	5.44949	0.187136
$849$	0	0
$850$	0	0
$851$	4.85357	0.166378
$852$	0	0
$853$	−13.2474	−0.453584	−0.226792	−	0.973943i	$-0.572824\pi$
$853$	−13.2474	−0.453584	−0.226792	+	0.973943i	$0.572824\pi$
$854$	−10.8990	−0.372955
$855$	0	0
$856$	3.34847	0.114448
$857$	−0.101021	−0.00345080	−0.00172540	−	0.999999i	$-0.500549\pi$
$857$	−0.101021	−0.00345080	−0.00172540	+	0.999999i	$0.500549\pi$
$858$	0	0
$859$	20.9444	0.714613	0.357307	−	0.933987i	$-0.383695\pi$
$859$	20.9444	0.714613	0.357307	+	0.933987i	$0.383695\pi$
$860$	0	0
$861$	0	0
$862$	−7.79796	−0.265600
$863$	0.348469	0.0118620	0.00593102	−	0.999982i	$-0.498112\pi$
$863$	0.348469	0.0118620	0.00593102	+	0.999982i	$0.498112\pi$
$864$	0	0
$865$	0	0
$866$	−23.5959	−0.801822
$867$	0	0
$868$	−1.44949	−0.0491989
$869$	15.7980	0.535909
$870$	0	0
$871$	−7.24745	−0.245570
$872$	15.1464	0.512923
$873$	0	0
$874$	−6.44949	−0.218157
$875$	0	0
$876$	0	0
$877$	13.3031	0.449212	0.224606	−	0.974450i	$-0.427890\pi$
$877$	13.3031	0.449212	0.224606	+	0.974450i	$0.427890\pi$
$878$	−6.34847	−0.214250
$879$	0	0
$880$	0	0
$881$	−13.2929	−0.447848	−0.223924	−	0.974607i	$-0.571887\pi$
$881$	−13.2929	−0.447848	−0.223924	+	0.974607i	$0.571887\pi$
$882$	0	0
$883$	2.10102	0.0707050	0.0353525	−	0.999375i	$-0.488745\pi$
$883$	2.10102	0.0707050	0.0353525	+	0.999375i	$0.488745\pi$
$884$	1.44949	0.0487516
$885$	0	0
$886$	27.5505	0.925577
$887$	0	0		−	1.00000i	$-0.5\pi$
$887$	0	0			1.00000i	$0.5\pi$
$888$	0	0
$889$	8.89898	0.298462
$890$	0	0
$891$	0	0
$892$	4.00000	0.133930
$893$	52.4949	1.75667
$894$	0	0
$895$	0	0
$896$	−1.00000	−0.0334077
$897$	0	0
$898$	32.0000	1.06785
$899$	3.69694	0.123300
$900$	0	0
$901$	−5.44949	−0.181549
$902$	−12.0000	−0.399556
$903$	0	0
$904$	−6.44949	−0.214507
$905$	0	0
$906$	0	0
$907$	−18.2020	−0.604389	−0.302194	−	0.953246i	$-0.597719\pi$
$907$	−18.2020	−0.604389	−0.302194	+	0.953246i	$0.597719\pi$
$908$	11.6969	0.388177
$909$	0	0
$910$	0	0
$911$	6.20204	0.205483	0.102741	−	0.994708i	$-0.467239\pi$
$911$	6.20204	0.205483	0.102741	+	0.994708i	$0.467239\pi$
$912$	0	0
$913$	−26.6969	−0.883540
$914$	−0.101021	−0.00334146
$915$	0	0
$916$	−3.10102	−0.102461
$917$	16.7980	0.554717
$918$	0	0
$919$	−17.5959	−0.580436	−0.290218	−	0.956961i	$-0.593728\pi$
$919$	−17.5959	−0.580436	−0.290218	+	0.956961i	$0.593728\pi$
$920$	0	0
$921$	0	0
$922$	10.0000	0.329332
$923$	−7.89898	−0.259998
$924$	0	0
$925$	0	0
$926$	−3.75255	−0.123316
$927$	0	0
$928$	2.55051	0.0837246
$929$	31.5959	1.03663	0.518314	−	0.855190i	$-0.326560\pi$
$929$	31.5959	1.03663	0.518314	+	0.855190i	$0.326560\pi$
$930$	0	0
$931$	−4.44949	−0.145826
$932$	13.1464	0.430626
$933$	0	0
$934$	12.2020	0.399263
$935$	0	0
$936$	0	0
$937$	−34.9444	−1.14158	−0.570792	−	0.821095i	$-0.693364\pi$
$937$	−34.9444	−1.14158	−0.570792	+	0.821095i	$0.693364\pi$
$938$	−5.00000	−0.163256
$939$	0	0
$940$	0	0
$941$	29.5959	0.964799	0.482400	−	0.875951i	$-0.339765\pi$
$941$	29.5959	0.964799	0.482400	+	0.875951i	$0.339765\pi$
$942$	0	0
$943$	7.10102	0.231241
$944$	−0.101021	−0.00328794
$945$	0	0
$946$	9.55051	0.310514
$947$	−28.0454	−0.911353	−0.455677	−	0.890145i	$-0.650603\pi$
$947$	−28.0454	−0.911353	−0.455677	+	0.890145i	$0.650603\pi$
$948$	0	0
$949$	−17.1010	−0.555123
$950$	0	0
$951$	0	0
$952$	1.00000	0.0324102
$953$	46.2929	1.49957	0.749786	−	0.661680i	$-0.230156\pi$
$953$	46.2929	1.49957	0.749786	+	0.661680i	$0.230156\pi$
$954$	0	0
$955$	0	0
$956$	−20.4949	−0.662852
$957$	0	0
$958$	−36.0000	−1.16311
$959$	−5.10102	−0.164721
$960$	0	0
$961$	−28.8990	−0.932225
$962$	−4.85357	−0.156485
$963$	0	0
$964$	10.6515	0.343063
$965$	0	0
$966$	0	0
$967$	24.8990	0.800697	0.400349	−	0.916363i	$-0.368889\pi$
$967$	24.8990	0.800697	0.400349	+	0.916363i	$0.368889\pi$
$968$	5.00000	0.160706
$969$	0	0
$970$	0	0
$971$	−9.49490	−0.304706	−0.152353	−	0.988326i	$-0.548685\pi$
$971$	−9.49490	−0.304706	−0.152353	+	0.988326i	$0.548685\pi$
$972$	0	0
$973$	22.2474	0.713220
$974$	−39.3939	−1.26226
$975$	0	0
$976$	10.8990	0.348868
$977$	−35.3939	−1.13235	−0.566175	−	0.824285i	$-0.691577\pi$
$977$	−35.3939	−1.13235	−0.566175	+	0.824285i	$0.691577\pi$
$978$	0	0
$979$	26.9444	0.861146
$980$	0	0
$981$	0	0
$982$	−4.40408	−0.140540
$983$	34.6969	1.10666	0.553330	−	0.832962i	$-0.313357\pi$
$983$	34.6969	1.10666	0.553330	+	0.832962i	$0.313357\pi$
$984$	0	0
$985$	0	0
$986$	−2.55051	−0.0812248
$987$	0	0
$988$	6.44949	0.205186
$989$	−5.65153	−0.179708
$990$	0	0
$991$	27.3939	0.870195	0.435098	−	0.900383i	$-0.356714\pi$
$991$	27.3939	0.870195	0.435098	+	0.900383i	$0.356714\pi$
$992$	1.44949	0.0460213
$993$	0	0
$994$	−5.44949	−0.172847
$995$	0	0
$996$	0	0
$997$	49.7423	1.57536	0.787678	−	0.616087i	$-0.211283\pi$
$997$	49.7423	1.57536	0.787678	+	0.616087i	$0.211283\pi$
$998$	−2.89898	−0.0917656
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9450.2.a.ek.1.1		2
3.2	odd	2		9450.2.a.ev.1.2		2
5.2	odd	4		1890.2.g.m.379.1	✓	4
5.3	odd	4		1890.2.g.m.379.3	yes	4
5.4	even	2		9450.2.a.ep.1.1		2
15.2	even	4		1890.2.g.r.379.4	yes	4
15.8	even	4		1890.2.g.r.379.2	yes	4
15.14	odd	2		9450.2.a.ea.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1890.2.g.m.379.1	✓	4	5.2	odd	4
1890.2.g.m.379.3	yes	4	5.3	odd	4
1890.2.g.r.379.2	yes	4	15.8	even	4
1890.2.g.r.379.4	yes	4	15.2	even	4
9450.2.a.ea.1.2		2	15.14	odd	2
9450.2.a.ek.1.1		2	1.1	even	1	trivial
9450.2.a.ep.1.1		2	5.4	even	2
9450.2.a.ev.1.2		2	3.2	odd	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:	\( N \)	\(=\)	\( 9450 = 2 \cdot 3^{3} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9450.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{7} -1.00000 q^{8} -2.44949 q^{11} -1.44949 q^{13} -1.00000 q^{14} +1.00000 q^{16} -1.00000 q^{17} -4.44949 q^{19} +2.44949 q^{22} -1.44949 q^{23} +1.44949 q^{26} +1.00000 q^{28} -2.55051 q^{29} -1.44949 q^{31} -1.00000 q^{32} +1.00000 q^{34} -3.34847 q^{37} +4.44949 q^{38} -4.89898 q^{41} +3.89898 q^{43} -2.44949 q^{44} +1.44949 q^{46} -11.7980 q^{47} +1.00000 q^{49} -1.44949 q^{52} +5.44949 q^{53} -1.00000 q^{56} +2.55051 q^{58} -0.101021 q^{59} +10.8990 q^{61} +1.44949 q^{62} +1.00000 q^{64} +5.00000 q^{67} -1.00000 q^{68} +5.44949 q^{71} +11.7980 q^{73} +3.34847 q^{74} -4.44949 q^{76} -2.44949 q^{77} -6.44949 q^{79} +4.89898 q^{82} +10.8990 q^{83} -3.89898 q^{86} +2.44949 q^{88} -11.0000 q^{89} -1.44949 q^{91} -1.44949 q^{92} +11.7980 q^{94} +9.55051 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} + 2 q^{7} - 2 q^{8}+O(q^{10}) \) 2 * q - 2 * q^2 + 2 * q^4 + 2 * q^7 - 2 * q^8	\( 2 q - 2 q^{2} + 2 q^{4} + 2 q^{7} - 2 q^{8} + 2 q^{13} - 2 q^{14} + 2 q^{16} - 2 q^{17} - 4 q^{19} + 2 q^{23} - 2 q^{26} + 2 q^{28} - 10 q^{29} + 2 q^{31} - 2 q^{32} + 2 q^{34} + 8 q^{37} + 4 q^{38} - 2 q^{43} - 2 q^{46} - 4 q^{47} + 2 q^{49} + 2 q^{52} + 6 q^{53} - 2 q^{56} + 10 q^{58} - 10 q^{59} + 12 q^{61} - 2 q^{62} + 2 q^{64} + 10 q^{67} - 2 q^{68} + 6 q^{71} + 4 q^{73} - 8 q^{74} - 4 q^{76} - 8 q^{79} + 12 q^{83} + 2 q^{86} - 22 q^{89} + 2 q^{91} + 2 q^{92} + 4 q^{94} + 24 q^{97} - 2 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 + 2 * q^7 - 2 * q^8 + 2 * q^13 - 2 * q^14 + 2 * q^16 - 2 * q^17 - 4 * q^19 + 2 * q^23 - 2 * q^26 + 2 * q^28 - 10 * q^29 + 2 * q^31 - 2 * q^32 + 2 * q^34 + 8 * q^37 + 4 * q^38 - 2 * q^43 - 2 * q^46 - 4 * q^47 + 2 * q^49 + 2 * q^52 + 6 * q^53 - 2 * q^56 + 10 * q^58 - 10 * q^59 + 12 * q^61 - 2 * q^62 + 2 * q^64 + 10 * q^67 - 2 * q^68 + 6 * q^71 + 4 * q^73 - 8 * q^74 - 4 * q^76 - 8 * q^79 + 12 * q^83 + 2 * q^86 - 22 * q^89 + 2 * q^91 + 2 * q^92 + 4 * q^94 + 24 * q^97 - 2 * q^98

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