LMFDB - Embedded newform 2175.2.c.p.349.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2175,2,Mod(349,2175)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2175, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

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

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

chi := DirichletCharacter("2175.349");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2175 = 3 \cdot 5^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2175.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$17.3674624396$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 28x^{14} + 308x^{12} + 1671x^{10} + 4568x^{8} + 5616x^{6} + 2105x^{4} + 256x^{2} + 4 $ x^16 + 28x^14 + 308x^12 + 1671x^10 + 4568x^8 + 5616x^6 + 2105x^4 + 256*x^2 + 4
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			349.11
Root			$0.510732i$ of defining polynomial
Character	$\chi$	$=$	2175.349
Dual form			2175.2.c.p.349.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.510732i q^{2} -1.00000i q^{3} +1.73915 q^{4} +0.510732 q^{6} -4.82343i q^{7} +1.90970i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q+0.510732i q^{2} -1.00000i q^{3} +1.73915 q^{4} +0.510732 q^{6} -4.82343i q^{7} +1.90970i q^{8} -1.00000 q^{9} +4.88439 q^{11} -1.73915i q^{12} +4.59669i q^{13} +2.46348 q^{14} +2.50296 q^{16} -6.50987i q^{17} -0.510732i q^{18} -3.09205 q^{19} -4.82343 q^{21} +2.49461i q^{22} -5.62549i q^{23} +1.90970 q^{24} -2.34767 q^{26} +1.00000i q^{27} -8.38869i q^{28} -1.00000 q^{29} +9.24375 q^{31} +5.09775i q^{32} -4.88439i q^{33} +3.32480 q^{34} -1.73915 q^{36} +11.1261i q^{37} -1.57921i q^{38} +4.59669 q^{39} -2.84537 q^{41} -2.46348i q^{42} -4.58611i q^{43} +8.49470 q^{44} +2.87311 q^{46} -3.62713i q^{47} -2.50296i q^{48} -16.2655 q^{49} -6.50987 q^{51} +7.99434i q^{52} -0.967845i q^{53} -0.510732 q^{54} +9.21132 q^{56} +3.09205i q^{57} -0.510732i q^{58} +0.298882 q^{59} -0.786908 q^{61} +4.72108i q^{62} +4.82343i q^{63} +2.40234 q^{64} +2.49461 q^{66} -4.86742i q^{67} -11.3217i q^{68} -5.62549 q^{69} +0.741689 q^{71} -1.90970i q^{72} +5.52981i q^{73} -5.68246 q^{74} -5.37755 q^{76} -23.5595i q^{77} +2.34767i q^{78} +2.96278 q^{79} +1.00000 q^{81} -1.45322i q^{82} -13.6633i q^{83} -8.38869 q^{84} +2.34227 q^{86} +1.00000i q^{87} +9.32773i q^{88} -3.67835 q^{89} +22.1718 q^{91} -9.78358i q^{92} -9.24375i q^{93} +1.85249 q^{94} +5.09775 q^{96} +2.87658i q^{97} -8.30730i q^{98} -4.88439 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 24 q^{4} + 4 q^{6} - 16 q^{9}+O(q^{10}) $ 16 * q - 24 * q^4 + 4 * q^6 - 16 * q^9	$ 16 q - 24 q^{4} + 4 q^{6} - 16 q^{9} + 12 q^{11} - 18 q^{14} + 64 q^{16} - 4 q^{21} - 6 q^{24} + 36 q^{26} - 16 q^{29} + 16 q^{31} + 26 q^{34} + 24 q^{36} + 12 q^{39} + 4 q^{41} + 30 q^{44} + 48 q^{46} - 76 q^{49} + 24 q^{51} - 4 q^{54} + 116 q^{56} - 36 q^{59} + 24 q^{61} - 42 q^{64} - 6 q^{66} - 28 q^{69} + 48 q^{71} + 44 q^{74} - 20 q^{79} + 16 q^{81} + 28 q^{84} + 16 q^{86} - 68 q^{89} + 52 q^{91} + 86 q^{94} - 4 q^{96} - 12 q^{99}+O(q^{100}) $ 16 * q - 24 * q^4 + 4 * q^6 - 16 * q^9 + 12 * q^11 - 18 * q^14 + 64 * q^16 - 4 * q^21 - 6 * q^24 + 36 * q^26 - 16 * q^29 + 16 * q^31 + 26 * q^34 + 24 * q^36 + 12 * q^39 + 4 * q^41 + 30 * q^44 + 48 * q^46 - 76 * q^49 + 24 * q^51 - 4 * q^54 + 116 * q^56 - 36 * q^59 + 24 * q^61 - 42 * q^64 - 6 * q^66 - 28 * q^69 + 48 * q^71 + 44 * q^74 - 20 * q^79 + 16 * q^81 + 28 * q^84 + 16 * q^86 - 68 * q^89 + 52 * q^91 + 86 * q^94 - 4 * q^96 - 12 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2175\mathbb{Z}\right)^\times$.

$n$	$901$	$1451$	$2002$
$\chi(n)$	$1$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0.510732i	0.361142i	0.983562	+	0.180571i	$0.0577945\pi$
$2$	0.510732i	0.361142i	−0.983562	+	0.180571i	$0.942205\pi$
$3$	− 1.00000i	− 0.577350i
$3$	− 1.00000i	− 0.577350i
$4$	1.73915	0.869577
$5$	0	0
$5$	0	0
$6$	0.510732	0.208505
$7$	− 4.82343i	− 1.82309i	−0.411205	−	0.911543i	$-0.634892\pi$
$7$	− 4.82343i	− 1.82309i	0.411205	−	0.911543i	$-0.365108\pi$
$8$	1.90970i	0.675182i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	4.88439	1.47270	0.736349	−	0.676602i	$-0.236548\pi$
$11$	4.88439	1.47270	0.736349	+	0.676602i	$0.236548\pi$
$12$	− 1.73915i	− 0.502050i
$13$	4.59669i	1.27489i	0.770495	+	0.637446i	$0.220009\pi$
$13$	4.59669i	1.27489i	−0.770495	+	0.637446i	$0.779991\pi$
$14$	2.46348	0.658392
$15$	0	0
$16$	2.50296	0.625740
$17$	− 6.50987i	− 1.57888i	−0.613830	−	0.789438i	$-0.710372\pi$
$17$	− 6.50987i	− 1.57888i	0.613830	−	0.789438i	$-0.289628\pi$
$18$	− 0.510732i	− 0.120381i
$19$	−3.09205	−0.709364	−0.354682	−	0.934987i	$-0.615411\pi$
$19$	−3.09205	−0.709364	−0.354682	+	0.934987i	$0.615411\pi$
$20$	0	0
$21$	−4.82343	−1.05256
$22$	2.49461i	0.531853i
$23$	− 5.62549i	− 1.17299i	−0.809951	−	0.586497i	$-0.800506\pi$
$23$	− 5.62549i	− 1.17299i	0.809951	−	0.586497i	$-0.199494\pi$
$24$	1.90970	0.389817
$25$	0	0
$26$	−2.34767	−0.460416
$27$	1.00000i	0.192450i
$28$	− 8.38869i	− 1.58531i
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	9.24375	1.66023	0.830113	−	0.557595i	$-0.188276\pi$
$31$	9.24375	1.66023	0.830113	+	0.557595i	$0.188276\pi$
$32$	5.09775i	0.901163i
$33$	− 4.88439i	− 0.850262i
$34$	3.32480	0.570198
$35$	0	0
$36$	−1.73915	−0.289859
$37$	11.1261i	1.82912i	0.404448	+	0.914561i	$0.367464\pi$
$37$	11.1261i	1.82912i	−0.404448	+	0.914561i	$0.632536\pi$
$38$	− 1.57921i	− 0.256181i
$39$	4.59669	0.736059
$40$	0	0
$41$	−2.84537	−0.444372	−0.222186	−	0.975004i	$-0.571319\pi$
$41$	−2.84537	−0.444372	−0.222186	+	0.975004i	$0.571319\pi$
$42$	− 2.46348i	− 0.380123i
$43$	− 4.58611i	− 0.699375i	−0.936866	−	0.349688i	$-0.886288\pi$
$43$	− 4.58611i	− 0.699375i	0.936866	−	0.349688i	$-0.113712\pi$
$44$	8.49470	1.28062
$45$	0	0
$46$	2.87311	0.423617
$47$	− 3.62713i	− 0.529071i	−0.964376	−	0.264536i	$-0.914781\pi$
$47$	− 3.62713i	− 0.529071i	0.964376	−	0.264536i	$-0.0852187\pi$
$48$	− 2.50296i	− 0.361271i
$49$	−16.2655	−2.32364
$50$	0	0
$51$	−6.50987	−0.911564
$52$	7.99434i	1.10862i
$53$	− 0.967845i	− 0.132944i	−0.997788	−	0.0664719i	$-0.978826\pi$
$53$	− 0.967845i	− 0.132944i	0.997788	−	0.0664719i	$-0.0211743\pi$
$54$	−0.510732	−0.0695018
$55$	0	0
$56$	9.21132	1.23091
$57$	3.09205i	0.409552i
$58$	− 0.510732i	− 0.0670623i
$59$	0.298882	0.0389112	0.0194556	−	0.999811i	$-0.493807\pi$
$59$	0.298882	0.0389112	0.0194556	+	0.999811i	$0.493807\pi$
$60$	0	0
$61$	−0.786908	−0.100753	−0.0503766	−	0.998730i	$-0.516042\pi$
$61$	−0.786908	−0.100753	−0.0503766	+	0.998730i	$0.516042\pi$
$62$	4.72108i	0.599577i
$63$	4.82343i	0.607695i
$64$	2.40234	0.300293
$65$	0	0
$66$	2.49461	0.307065
$67$	− 4.86742i	− 0.594650i	−0.954776	−	0.297325i	$-0.903906\pi$
$67$	− 4.86742i	− 0.594650i	0.954776	−	0.297325i	$-0.0960945\pi$
$68$	− 11.3217i	− 1.37295i
$69$	−5.62549	−0.677229
$70$	0	0
$71$	0.741689	0.0880223	0.0440112	−	0.999031i	$-0.485986\pi$
$71$	0.741689	0.0880223	0.0440112	+	0.999031i	$0.485986\pi$
$72$	− 1.90970i	− 0.225061i
$73$	5.52981i	0.647215i	0.946191	+	0.323608i	$0.104896\pi$
$73$	5.52981i	0.647215i	−0.946191	+	0.323608i	$0.895104\pi$
$74$	−5.68246	−0.660572
$75$	0	0
$76$	−5.37755	−0.616847
$77$	− 23.5595i	− 2.68485i
$78$	2.34767i	0.265822i
$79$	2.96278	0.333339	0.166669	−	0.986013i	$-0.446699\pi$
$79$	2.96278	0.333339	0.166669	+	0.986013i	$0.446699\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	− 1.45322i	− 0.160481i
$83$	− 13.6633i	− 1.49975i	−0.661581	−	0.749874i	$-0.730114\pi$
$83$	− 13.6633i	− 1.49975i	0.661581	−	0.749874i	$-0.269886\pi$
$84$	−8.38869	−0.915281
$85$	0	0
$86$	2.34227	0.252574
$87$	1.00000i	0.107211i
$88$	9.32773i	0.994339i
$89$	−3.67835	−0.389905	−0.194952	−	0.980813i	$-0.562455\pi$
$89$	−3.67835	−0.389905	−0.194952	+	0.980813i	$0.562455\pi$
$90$	0	0
$91$	22.1718	2.32424
$92$	− 9.78358i	− 1.02001i
$93$	− 9.24375i	− 0.958532i
$94$	1.85249	0.191070
$95$	0	0
$96$	5.09775	0.520287
$97$	2.87658i	0.292072i	0.989279	+	0.146036i	$0.0466515\pi$
$97$	2.87658i	0.292072i	−0.989279	+	0.146036i	$0.953348\pi$
$98$	− 8.30730i	− 0.839164i
$99$	−4.88439	−0.490899
$100$	0	0
$101$	−14.6780	−1.46051	−0.730255	−	0.683174i	$-0.760599\pi$
$101$	−14.6780	−1.46051	−0.730255	+	0.683174i	$0.760599\pi$
$102$	− 3.32480i	− 0.329204i
$103$	− 2.48154i	− 0.244514i	−0.992498	−	0.122257i	$-0.960987\pi$
$103$	− 2.48154i	− 0.244514i	0.992498	−	0.122257i	$-0.0390132\pi$
$104$	−8.77831	−0.860784
$105$	0	0
$106$	0.494309	0.0480115
$107$	4.48423i	0.433507i	0.976226	+	0.216753i	$0.0695467\pi$
$107$	4.48423i	0.433507i	−0.976226	+	0.216753i	$0.930453\pi$
$108$	1.73915i	0.167350i
$109$	−3.28140	−0.314301	−0.157151	−	0.987575i	$-0.550231\pi$
$109$	−3.28140	−0.314301	−0.157151	+	0.987575i	$0.550231\pi$
$110$	0	0
$111$	11.1261	1.05604
$112$	− 12.0729i	− 1.14078i
$113$	− 17.9820i	− 1.69160i	−0.533500	−	0.845800i	$-0.679123\pi$
$113$	− 17.9820i	− 1.69160i	0.533500	−	0.845800i	$-0.320877\pi$
$114$	−1.57921	−0.147906
$115$	0	0
$116$	−1.73915	−0.161476
$117$	− 4.59669i	− 0.424964i
$118$	0.152649i	0.0140524i
$119$	−31.3999	−2.87843
$120$	0	0
$121$	12.8572	1.16884
$122$	− 0.401899i	− 0.0363862i
$123$	2.84537i	0.256558i
$124$	16.0763	1.44369
$125$	0	0
$126$	−2.46348	−0.219464
$127$	13.5099i	1.19881i	0.800446	+	0.599405i	$0.204596\pi$
$127$	13.5099i	1.19881i	−0.800446	+	0.599405i	$0.795404\pi$
$128$	11.4224i	1.00961i
$129$	−4.58611	−0.403784
$130$	0	0
$131$	12.1450	1.06111	0.530556	−	0.847650i	$-0.321983\pi$
$131$	12.1450	1.06111	0.530556	+	0.847650i	$0.321983\pi$
$132$	− 8.49470i	− 0.739368i
$133$	14.9143i	1.29323i
$134$	2.48594	0.214753
$135$	0	0
$136$	12.4319	1.06603
$137$	9.97466i	0.852193i	0.904678	+	0.426096i	$0.140112\pi$
$137$	9.97466i	0.852193i	−0.904678	+	0.426096i	$0.859888\pi$
$138$	− 2.87311i	− 0.244576i
$139$	−2.82971	−0.240013	−0.120007	−	0.992773i	$-0.538292\pi$
$139$	−2.82971	−0.240013	−0.120007	+	0.992773i	$0.538292\pi$
$140$	0	0
$141$	−3.62713	−0.305459
$142$	0.378804i	0.0317885i
$143$	22.4520i	1.87753i
$144$	−2.50296	−0.208580
$145$	0	0
$146$	−2.82425	−0.233736
$147$	16.2655i	1.34155i
$148$	19.3500i	1.59056i
$149$	22.5396	1.84652	0.923259	−	0.384178i	$-0.125515\pi$
$149$	22.5396	1.84652	0.923259	+	0.384178i	$0.125515\pi$
$150$	0	0
$151$	−14.2205	−1.15725	−0.578626	−	0.815593i	$-0.696411\pi$
$151$	−14.2205	−1.15725	−0.578626	+	0.815593i	$0.696411\pi$
$152$	− 5.90490i	− 0.478950i
$153$	6.50987i	0.526292i
$154$	12.0326	0.969613
$155$	0	0
$156$	7.99434	0.640059
$157$	− 10.7712i	− 0.859632i	−0.902917	−	0.429816i	$-0.858579\pi$
$157$	− 10.7712i	− 0.859632i	0.902917	−	0.429816i	$-0.141421\pi$
$158$	1.51319i	0.120383i
$159$	−0.967845	−0.0767551
$160$	0	0
$161$	−27.1341	−2.13847
$162$	0.510732i	0.0401269i
$163$	− 10.6908i	− 0.837365i	−0.908133	−	0.418682i	$-0.862492\pi$
$163$	− 10.6908i	− 0.837365i	0.908133	−	0.418682i	$-0.137508\pi$
$164$	−4.94853	−0.386415
$165$	0	0
$166$	6.97830	0.541621
$167$	− 7.90643i	− 0.611818i	−0.952061	−	0.305909i	$-0.901040\pi$
$167$	− 7.90643i	− 0.611818i	0.952061	−	0.305909i	$-0.0989603\pi$
$168$	− 9.21132i	− 0.710669i
$169$	−8.12952	−0.625347
$170$	0	0
$171$	3.09205	0.236455
$172$	− 7.97595i	− 0.608160i
$173$	3.76760i	0.286445i	0.989690	+	0.143223i	$0.0457465\pi$
$173$	3.76760i	0.286445i	−0.989690	+	0.143223i	$0.954253\pi$
$174$	−0.510732	−0.0387185
$175$	0	0
$176$	12.2254	0.921526
$177$	− 0.298882i	− 0.0224654i
$178$	− 1.87865i	− 0.140811i
$179$	−10.3282	−0.771964	−0.385982	−	0.922506i	$-0.626137\pi$
$179$	−10.3282	−0.771964	−0.385982	+	0.922506i	$0.626137\pi$
$180$	0	0
$181$	14.4533	1.07431	0.537154	−	0.843484i	$-0.319500\pi$
$181$	14.4533	1.07431	0.537154	+	0.843484i	$0.319500\pi$
$182$	11.3238i	0.839378i
$183$	0.786908i	0.0581699i
$184$	10.7430	0.791985
$185$	0	0
$186$	4.72108	0.346166
$187$	− 31.7967i	− 2.32521i
$188$	− 6.30813i	− 0.460068i
$189$	4.82343	0.350853
$190$	0	0
$191$	8.81550	0.637867	0.318934	−	0.947777i	$-0.396675\pi$
$191$	8.81550	0.637867	0.318934	+	0.947777i	$0.396675\pi$
$192$	− 2.40234i	− 0.173374i
$193$	17.2543i	1.24199i	0.783814	+	0.620996i	$0.213272\pi$
$193$	17.2543i	1.24199i	−0.783814	+	0.620996i	$0.786728\pi$
$194$	−1.46916	−0.105479
$195$	0	0
$196$	−28.2882	−2.02058
$197$	22.5719i	1.60818i	0.594508	+	0.804090i	$0.297347\pi$
$197$	22.5719i	1.60818i	−0.594508	+	0.804090i	$0.702653\pi$
$198$	− 2.49461i	− 0.177284i
$199$	13.7975	0.978078	0.489039	−	0.872262i	$-0.337348\pi$
$199$	13.7975	0.978078	0.489039	+	0.872262i	$0.337348\pi$
$200$	0	0
$201$	−4.86742	−0.343321
$202$	− 7.49649i	− 0.527451i
$203$	4.82343i	0.338538i
$204$	−11.3217	−0.792675
$205$	0	0
$206$	1.26740	0.0883041
$207$	5.62549i	0.390998i
$208$	11.5053i	0.797751i
$209$	−15.1028	−1.04468
$210$	0	0
$211$	16.4705	1.13387	0.566937	−	0.823761i	$-0.308128\pi$
$211$	16.4705	1.13387	0.566937	+	0.823761i	$0.308128\pi$
$212$	− 1.68323i	− 0.115605i
$213$	− 0.741689i	− 0.0508197i
$214$	−2.29024	−0.156557
$215$	0	0
$216$	−1.90970	−0.129939
$217$	− 44.5866i	− 3.02674i
$218$	− 1.67592i	− 0.113507i
$219$	5.52981	0.373670
$220$	0	0
$221$	29.9238	2.01289
$222$	5.68246i	0.381382i
$223$	− 4.58903i	− 0.307304i	−0.988125	−	0.153652i	$-0.950896\pi$
$223$	− 4.58903i	− 0.307304i	0.988125	−	0.153652i	$-0.0491035\pi$
$224$	24.5886	1.64290
$225$	0	0
$226$	9.18396	0.610908
$227$	25.5254i	1.69418i	0.531452	+	0.847089i	$0.321647\pi$
$227$	25.5254i	1.69418i	−0.531452	+	0.847089i	$0.678353\pi$
$228$	5.37755i	0.356137i
$229$	8.04982	0.531947	0.265974	−	0.963980i	$-0.414307\pi$
$229$	8.04982	0.531947	0.265974	+	0.963980i	$0.414307\pi$
$230$	0	0
$231$	−23.5595	−1.55010
$232$	− 1.90970i	− 0.125378i
$233$	3.75903i	0.246262i	0.992390	+	0.123131i	$0.0392935\pi$
$233$	3.75903i	0.246262i	−0.992390	+	0.123131i	$0.960706\pi$
$234$	2.34767	0.153472
$235$	0	0
$236$	0.519802	0.0338362
$237$	− 2.96278i	− 0.192453i
$238$	− 16.0369i	− 1.03952i
$239$	3.53192	0.228461	0.114231	−	0.993454i	$-0.463560\pi$
$239$	3.53192	0.228461	0.114231	+	0.993454i	$0.463560\pi$
$240$	0	0
$241$	3.05553	0.196824	0.0984118	−	0.995146i	$-0.468624\pi$
$241$	3.05553	0.196824	0.0984118	+	0.995146i	$0.468624\pi$
$242$	6.56659i	0.422117i
$243$	− 1.00000i	− 0.0641500i
$244$	−1.36855	−0.0876126
$245$	0	0
$246$	−1.45322	−0.0926538
$247$	− 14.2132i	− 0.904363i
$248$	17.6528i	1.12096i
$249$	−13.6633	−0.865880
$250$	0	0
$251$	−23.3283	−1.47247	−0.736233	−	0.676728i	$-0.763397\pi$
$251$	−23.3283	−1.47247	−0.736233	+	0.676728i	$0.763397\pi$
$252$	8.38869i	0.528437i
$253$	− 27.4770i	− 1.72747i
$254$	−6.89994	−0.432941
$255$	0	0
$256$	−1.02912	−0.0643202
$257$	− 28.2268i	− 1.76074i	−0.474289	−	0.880369i	$-0.657295\pi$
$257$	− 28.2268i	− 1.76074i	0.474289	−	0.880369i	$-0.342705\pi$
$258$	− 2.34227i	− 0.145823i
$259$	53.6660	3.33465
$260$	0	0
$261$	1.00000	0.0618984
$262$	6.20283i	0.383212i
$263$	− 27.6688i	− 1.70613i	−0.521802	−	0.853067i	$-0.674740\pi$
$263$	− 27.6688i	− 1.70613i	0.521802	−	0.853067i	$-0.325260\pi$
$264$	9.32773	0.574082
$265$	0	0
$266$	−7.61719	−0.467040
$267$	3.67835i	0.225112i
$268$	− 8.46519i	− 0.517094i
$269$	−0.839052	−0.0511579	−0.0255790	−	0.999673i	$-0.508143\pi$
$269$	−0.839052	−0.0511579	−0.0255790	+	0.999673i	$0.508143\pi$
$270$	0	0
$271$	1.27423	0.0774037	0.0387018	−	0.999251i	$-0.487678\pi$
$271$	1.27423	0.0774037	0.0387018	+	0.999251i	$0.487678\pi$
$272$	− 16.2940i	− 0.987966i
$273$	− 22.1718i	− 1.34190i
$274$	−5.09437	−0.307762
$275$	0	0
$276$	−9.78358	−0.588902
$277$	26.7387i	1.60657i	0.595592	+	0.803287i	$0.296917\pi$
$277$	26.7387i	1.60657i	−0.595592	+	0.803287i	$0.703083\pi$
$278$	− 1.44522i	− 0.0866788i
$279$	−9.24375	−0.553409
$280$	0	0
$281$	−16.3446	−0.975038	−0.487519	−	0.873112i	$-0.662098\pi$
$281$	−16.3446	−0.975038	−0.487519	+	0.873112i	$0.662098\pi$
$282$	− 1.85249i	− 0.110314i
$283$	23.7207i	1.41005i	0.709183	+	0.705025i	$0.249064\pi$
$283$	23.7207i	1.41005i	−0.709183	+	0.705025i	$0.750936\pi$
$284$	1.28991	0.0765422
$285$	0	0
$286$	−11.4669	−0.678054
$287$	13.7244i	0.810127i
$288$	− 5.09775i	− 0.300388i
$289$	−25.3784	−1.49285
$290$	0	0
$291$	2.87658	0.168628
$292$	9.61718i	0.562803i
$293$	− 8.28535i	− 0.484035i	−0.970272	−	0.242018i	$-0.922191\pi$
$293$	− 8.28535i	− 0.484035i	0.970272	−	0.242018i	$-0.0778092\pi$
$294$	−8.30730	−0.484491
$295$	0	0
$296$	−21.2476	−1.23499
$297$	4.88439i	0.283421i
$298$	11.5117i	0.666855i
$299$	25.8586	1.49544
$300$	0	0
$301$	−22.1208	−1.27502
$302$	− 7.26288i	− 0.417932i
$303$	14.6780i	0.843226i
$304$	−7.73927	−0.443878
$305$	0	0
$306$	−3.32480	−0.190066
$307$	16.3024i	0.930425i	0.885199	+	0.465213i	$0.154022\pi$
$307$	16.3024i	0.930425i	−0.885199	+	0.465213i	$0.845978\pi$
$308$	− 40.9736i	− 2.33469i
$309$	−2.48154	−0.141170
$310$	0	0
$311$	−7.72085	−0.437809	−0.218905	−	0.975746i	$-0.570248\pi$
$311$	−7.72085	−0.437809	−0.218905	+	0.975746i	$0.570248\pi$
$312$	8.77831i	0.496974i
$313$	− 6.22393i	− 0.351797i	−0.984408	−	0.175899i	$-0.943717\pi$
$313$	− 6.22393i	− 0.351797i	0.984408	−	0.175899i	$-0.0562831\pi$
$314$	5.50117	0.310449
$315$	0	0
$316$	5.15273	0.289864
$317$	31.0672i	1.74491i	0.488696	+	0.872454i	$0.337473\pi$
$317$	31.0672i	1.74491i	−0.488696	+	0.872454i	$0.662527\pi$
$318$	− 0.494309i	− 0.0277195i
$319$	−4.88439	−0.273473
$320$	0	0
$321$	4.48423	0.250285
$322$	− 13.8583i	− 0.772291i
$323$	20.1288i	1.12000i
$324$	1.73915	0.0966196
$325$	0	0
$326$	5.46011	0.302407
$327$	3.28140i	0.181462i
$328$	− 5.43381i	− 0.300032i
$329$	−17.4952	−0.964542
$330$	0	0
$331$	34.8731	1.91680	0.958399	−	0.285431i	$-0.0921366\pi$
$331$	34.8731	1.91680	0.958399	+	0.285431i	$0.0921366\pi$
$332$	− 23.7627i	− 1.30415i
$333$	− 11.1261i	− 0.609707i
$334$	4.03806	0.220953
$335$	0	0
$336$	−12.0729	−0.658628
$337$	16.5875i	0.903579i	0.892124	+	0.451790i	$0.149214\pi$
$337$	16.5875i	0.903579i	−0.892124	+	0.451790i	$0.850786\pi$
$338$	− 4.15200i	− 0.225839i
$339$	−17.9820	−0.976646
$340$	0	0
$341$	45.1500	2.44501
$342$	1.57921i	0.0853937i
$343$	44.6914i	2.41311i
$344$	8.75811	0.472206
$345$	0	0
$346$	−1.92423	−0.103447
$347$	− 0.413605i	− 0.0222035i	−0.999938	−	0.0111017i	$-0.996466\pi$
$347$	− 0.413605i	− 0.0222035i	0.999938	−	0.0111017i	$-0.00353387\pi$
$348$	1.73915i	0.0932284i
$349$	−35.5590	−1.90343	−0.951715	−	0.306982i	$-0.900681\pi$
$349$	−35.5590	−1.90343	−0.951715	+	0.306982i	$0.900681\pi$
$350$	0	0
$351$	−4.59669	−0.245353
$352$	24.8994i	1.32714i
$353$	3.08030i	0.163948i	0.996634	+	0.0819740i	$0.0261225\pi$
$353$	3.08030i	0.163948i	−0.996634	+	0.0819740i	$0.973878\pi$
$354$	0.152649	0.00811318
$355$	0	0
$356$	−6.39722	−0.339052
$357$	31.3999i	1.66186i
$358$	− 5.27492i	− 0.278788i
$359$	14.3219	0.755879	0.377940	−	0.925830i	$-0.376633\pi$
$359$	14.3219	0.755879	0.377940	+	0.925830i	$0.376633\pi$
$360$	0	0
$361$	−9.43924	−0.496802
$362$	7.38177i	0.387977i
$363$	− 12.8572i	− 0.674829i
$364$	38.5601	2.02110
$365$	0	0
$366$	−0.401899	−0.0210076
$367$	− 17.3009i	− 0.903102i	−0.892245	−	0.451551i	$-0.850871\pi$
$367$	− 17.3009i	− 0.903102i	0.892245	−	0.451551i	$-0.149129\pi$
$368$	− 14.0804i	− 0.733990i
$369$	2.84537	0.148124
$370$	0	0
$371$	−4.66833	−0.242368
$372$	− 16.0763i	− 0.833517i
$373$	31.8314i	1.64817i	0.566469	+	0.824083i	$0.308309\pi$
$373$	31.8314i	1.64817i	−0.566469	+	0.824083i	$0.691691\pi$
$374$	16.2396	0.839729
$375$	0	0
$376$	6.92674	0.357219
$377$	− 4.59669i	− 0.236741i
$378$	2.46348i	0.126708i
$379$	18.1855	0.934127	0.467064	−	0.884224i	$-0.345312\pi$
$379$	18.1855	0.934127	0.467064	+	0.884224i	$0.345312\pi$
$380$	0	0
$381$	13.5099	0.692134
$382$	4.50236i	0.230361i
$383$	32.3026i	1.65059i	0.564704	+	0.825293i	$0.308990\pi$
$383$	32.3026i	1.65059i	−0.564704	+	0.825293i	$0.691010\pi$
$384$	11.4224	0.582899
$385$	0	0
$386$	−8.81232	−0.448535
$387$	4.58611i	0.233125i
$388$	5.00281i	0.253979i
$389$	26.5696	1.34713	0.673567	−	0.739126i	$-0.264761\pi$
$389$	26.5696	1.34713	0.673567	+	0.739126i	$0.264761\pi$
$390$	0	0
$391$	−36.6212	−1.85201
$392$	− 31.0622i	− 1.56888i
$393$	− 12.1450i	− 0.612634i
$394$	−11.5282	−0.580781
$395$	0	0
$396$	−8.49470	−0.426875
$397$	− 8.87832i	− 0.445590i	−0.974865	−	0.222795i	$-0.928482\pi$
$397$	− 8.87832i	− 0.445590i	0.974865	−	0.222795i	$-0.0715181\pi$
$398$	7.04681i	0.353225i
$399$	14.9143	0.746648
$400$	0	0
$401$	25.6502	1.28091	0.640455	−	0.767995i	$-0.278746\pi$
$401$	25.6502	1.28091	0.640455	+	0.767995i	$0.278746\pi$
$402$	− 2.48594i	− 0.123988i
$403$	42.4906i	2.11661i
$404$	−25.5272	−1.27003
$405$	0	0
$406$	−2.46348	−0.122260
$407$	54.3442i	2.69374i
$408$	− 12.4319i	− 0.615472i
$409$	−9.92575	−0.490797	−0.245398	−	0.969422i	$-0.578919\pi$
$409$	−9.92575	−0.490797	−0.245398	+	0.969422i	$0.578919\pi$
$410$	0	0
$411$	9.97466	0.492014
$412$	− 4.31579i	− 0.212624i
$413$	− 1.44164i	− 0.0709384i
$414$	−2.87311	−0.141206
$415$	0	0
$416$	−23.4327	−1.14888
$417$	2.82971i	0.138572i
$418$	− 7.71345i	− 0.377277i
$419$	−8.12620	−0.396991	−0.198495	−	0.980102i	$-0.563606\pi$
$419$	−8.12620	−0.396991	−0.198495	+	0.980102i	$0.563606\pi$
$420$	0	0
$421$	−8.37826	−0.408332	−0.204166	−	0.978936i	$-0.565448\pi$
$421$	−8.37826	−0.408332	−0.204166	+	0.978936i	$0.565448\pi$
$422$	8.41198i	0.409489i
$423$	3.62713i	0.176357i
$424$	1.84830	0.0897612
$425$	0	0
$426$	0.378804	0.0183531
$427$	3.79559i	0.183682i
$428$	7.79876i	0.376967i
$429$	22.4520	1.08399
$430$	0	0
$431$	−32.3877	−1.56006	−0.780030	−	0.625742i	$-0.784796\pi$
$431$	−32.3877	−1.56006	−0.780030	+	0.625742i	$0.784796\pi$
$432$	2.50296i	0.120424i
$433$	9.89328i	0.475441i	0.971334	+	0.237720i	$0.0764002\pi$
$433$	9.89328i	0.475441i	−0.971334	+	0.237720i	$0.923600\pi$
$434$	22.7718	1.09308
$435$	0	0
$436$	−5.70686	−0.273309
$437$	17.3943i	0.832081i
$438$	2.82425i	0.134948i
$439$	−5.84726	−0.279075	−0.139537	−	0.990217i	$-0.544562\pi$
$439$	−5.84726	−0.279075	−0.139537	+	0.990217i	$0.544562\pi$
$440$	0	0
$441$	16.2655	0.774547
$442$	15.2830i	0.726940i
$443$	3.14176i	0.149270i	0.997211	+	0.0746348i	$0.0237791\pi$
$443$	3.14176i	0.149270i	−0.997211	+	0.0746348i	$0.976221\pi$
$444$	19.3500	0.918311
$445$	0	0
$446$	2.34376	0.110980
$447$	− 22.5396i	− 1.06609i
$448$	− 11.5875i	− 0.547459i
$449$	−27.8643	−1.31500	−0.657500	−	0.753455i	$-0.728386\pi$
$449$	−27.8643	−1.31500	−0.657500	+	0.753455i	$0.728386\pi$
$450$	0	0
$451$	−13.8979	−0.654425
$452$	− 31.2734i	− 1.47098i
$453$	14.2205i	0.668139i
$454$	−13.0366	−0.611838
$455$	0	0
$456$	−5.90490	−0.276522
$457$	31.8928i	1.49188i	0.666013	+	0.745940i	$0.268000\pi$
$457$	31.8928i	1.49188i	−0.666013	+	0.745940i	$0.732000\pi$
$458$	4.11130i	0.192108i
$459$	6.50987	0.303855
$460$	0	0
$461$	5.64660	0.262989	0.131494	−	0.991317i	$-0.458022\pi$
$461$	5.64660	0.262989	0.131494	+	0.991317i	$0.458022\pi$
$462$	− 12.0326i	− 0.559806i
$463$	32.7602i	1.52250i	0.648460	+	0.761248i	$0.275413\pi$
$463$	32.7602i	1.52250i	−0.648460	+	0.761248i	$0.724587\pi$
$464$	−2.50296	−0.116197
$465$	0	0
$466$	−1.91985	−0.0889355
$467$	37.4129i	1.73126i	0.500680	+	0.865632i	$0.333083\pi$
$467$	37.4129i	1.73126i	−0.500680	+	0.865632i	$0.666917\pi$
$468$	− 7.99434i	− 0.369539i
$469$	−23.4777	−1.08410
$470$	0	0
$471$	−10.7712	−0.496308
$472$	0.570777i	0.0262721i
$473$	− 22.4003i	− 1.02997i
$474$	1.51319	0.0695029
$475$	0	0
$476$	−54.6093	−2.50301
$477$	0.967845i	0.0443146i
$478$	1.80387i	0.0825069i
$479$	13.0770	0.597503	0.298751	−	0.954331i	$-0.403430\pi$
$479$	13.0770	0.597503	0.298751	+	0.954331i	$0.403430\pi$
$480$	0	0
$481$	−51.1432	−2.33193
$482$	1.56055i	0.0710813i
$483$	27.1341i	1.23465i
$484$	22.3607	1.01639
$485$	0	0
$486$	0.510732	0.0231673
$487$	− 19.4492i	− 0.881328i	−0.897672	−	0.440664i	$-0.854743\pi$
$487$	− 19.4492i	− 0.881328i	0.897672	−	0.440664i	$-0.145257\pi$
$488$	− 1.50276i	− 0.0680268i
$489$	−10.6908	−0.483453
$490$	0	0
$491$	15.8221	0.714041	0.357020	−	0.934097i	$-0.383793\pi$
$491$	15.8221	0.714041	0.357020	+	0.934097i	$0.383793\pi$
$492$	4.94853i	0.223097i
$493$	6.50987i	0.293190i
$494$	7.25912	0.326603
$495$	0	0
$496$	23.1367	1.03887
$497$	− 3.57749i	− 0.160472i
$498$	− 6.97830i	− 0.312705i
$499$	−4.88592	−0.218724	−0.109362	−	0.994002i	$-0.534881\pi$
$499$	−4.88592	−0.218724	−0.109362	+	0.994002i	$0.534881\pi$
$500$	0	0
$501$	−7.90643	−0.353233
$502$	− 11.9145i	− 0.531769i
$503$	− 16.9719i	− 0.756742i	−0.925654	−	0.378371i	$-0.876484\pi$
$503$	− 16.9719i	− 0.756742i	0.925654	−	0.378371i	$-0.123516\pi$
$504$	−9.21132	−0.410305
$505$	0	0
$506$	14.0334	0.623860
$507$	8.12952i	0.361045i
$508$	23.4958i	1.04246i
$509$	7.53223	0.333860	0.166930	−	0.985969i	$-0.446615\pi$
$509$	7.53223	0.333860	0.166930	+	0.985969i	$0.446615\pi$
$510$	0	0
$511$	26.6726	1.17993
$512$	22.3193i	0.986383i
$513$	− 3.09205i	− 0.136517i
$514$	14.4163	0.635876
$515$	0	0
$516$	−7.97595	−0.351122
$517$	− 17.7163i	− 0.779162i
$518$	27.4089i	1.20428i
$519$	3.76760	0.165379
$520$	0	0
$521$	−0.748655	−0.0327992	−0.0163996	−	0.999866i	$-0.505220\pi$
$521$	−0.748655	−0.0327992	−0.0163996	+	0.999866i	$0.505220\pi$
$522$	0.510732i	0.0223541i
$523$	18.3909i	0.804178i	0.915601	+	0.402089i	$0.131716\pi$
$523$	18.3909i	0.804178i	−0.915601	+	0.402089i	$0.868284\pi$
$524$	21.1220	0.922719
$525$	0	0
$526$	14.1313	0.616156
$527$	− 60.1756i	− 2.62129i
$528$	− 12.2254i	− 0.532043i
$529$	−8.64609	−0.375917
$530$	0	0
$531$	−0.298882	−0.0129704
$532$	25.9382i	1.12456i
$533$	− 13.0793i	− 0.566525i
$534$	−1.87865	−0.0812972
$535$	0	0
$536$	9.29533	0.401497
$537$	10.3282i	0.445693i
$538$	− 0.428531i	− 0.0184753i
$539$	−79.4469	−3.42202
$540$	0	0
$541$	33.4160	1.43667	0.718334	−	0.695698i	$-0.244905\pi$
$541$	33.4160	1.43667	0.718334	+	0.695698i	$0.244905\pi$
$542$	0.650787i	0.0279537i
$543$	− 14.4533i	− 0.620251i
$544$	33.1857	1.42282
$545$	0	0
$546$	11.3238	0.484615
$547$	− 3.00020i	− 0.128279i	−0.997941	−	0.0641396i	$-0.979570\pi$
$547$	− 3.00020i	− 0.128279i	0.997941	−	0.0641396i	$-0.0204303\pi$
$548$	17.3475i	0.741047i
$549$	0.786908	0.0335844
$550$	0	0
$551$	3.09205	0.131726
$552$	− 10.7430i	− 0.457253i
$553$	− 14.2908i	− 0.607705i
$554$	−13.6563	−0.580201
$555$	0	0
$556$	−4.92131	−0.208710
$557$	− 4.11112i	− 0.174194i	−0.996200	−	0.0870969i	$-0.972241\pi$
$557$	− 4.11112i	− 0.174194i	0.996200	−	0.0870969i	$-0.0277590\pi$
$558$	− 4.72108i	− 0.199859i
$559$	21.0809	0.891627
$560$	0	0
$561$	−31.7967	−1.34246
$562$	− 8.34771i	− 0.352127i
$563$	− 11.5127i	− 0.485204i	−0.970126	−	0.242602i	$-0.921999\pi$
$563$	− 11.5127i	− 0.485204i	0.970126	−	0.242602i	$-0.0780009\pi$
$564$	−6.30813	−0.265620
$565$	0	0
$566$	−12.1149	−0.509228
$567$	− 4.82343i	− 0.202565i
$568$	1.41641i	0.0594311i
$569$	−32.9470	−1.38121	−0.690605	−	0.723232i	$-0.742656\pi$
$569$	−32.9470	−1.38121	−0.690605	+	0.723232i	$0.742656\pi$
$570$	0	0
$571$	−22.3213	−0.934117	−0.467059	−	0.884226i	$-0.654686\pi$
$571$	−22.3213	−0.934117	−0.467059	+	0.884226i	$0.654686\pi$
$572$	39.0474i	1.63266i
$573$	− 8.81550i	− 0.368273i
$574$	−7.00950	−0.292571
$575$	0	0
$576$	−2.40234	−0.100098
$577$	39.8197i	1.65772i	0.559458	+	0.828859i	$0.311009\pi$
$577$	39.8197i	1.65772i	−0.559458	+	0.828859i	$0.688991\pi$
$578$	− 12.9616i	− 0.539130i
$579$	17.2543	0.717064
$580$	0	0
$581$	−65.9042	−2.73417
$582$	1.46916i	0.0608986i
$583$	− 4.72733i	− 0.195786i
$584$	−10.5603	−0.436988
$585$	0	0
$586$	4.23159	0.174805
$587$	28.7487i	1.18659i	0.804987	+	0.593293i	$0.202172\pi$
$587$	28.7487i	1.18659i	−0.804987	+	0.593293i	$0.797828\pi$
$588$	28.2882i	1.16658i
$589$	−28.5821	−1.17771
$590$	0	0
$591$	22.5719	0.928483
$592$	27.8482i	1.14455i
$593$	27.5400i	1.13093i	0.824771	+	0.565467i	$0.191304\pi$
$593$	27.5400i	1.13093i	−0.824771	+	0.565467i	$0.808696\pi$
$594$	−2.49461	−0.102355
$595$	0	0
$596$	39.1999	1.60569
$597$	− 13.7975i	− 0.564693i
$598$	13.2068i	0.540066i
$599$	4.11339	0.168069	0.0840343	−	0.996463i	$-0.473219\pi$
$599$	4.11339	0.168069	0.0840343	+	0.996463i	$0.473219\pi$
$600$	0	0
$601$	−0.656799	−0.0267914	−0.0133957	−	0.999910i	$-0.504264\pi$
$601$	−0.656799	−0.0267914	−0.0133957	+	0.999910i	$0.504264\pi$
$602$	− 11.2978i	− 0.460463i
$603$	4.86742i	0.198217i
$604$	−24.7317	−1.00632
$605$	0	0
$606$	−7.49649	−0.304524
$607$	18.4210i	0.747684i	0.927492	+	0.373842i	$0.121960\pi$
$607$	18.4210i	0.747684i	−0.927492	+	0.373842i	$0.878040\pi$
$608$	− 15.7625i	− 0.639253i
$609$	4.82343	0.195455
$610$	0	0
$611$	16.6728	0.674508
$612$	11.3217i	0.457651i
$613$	21.3356i	0.861737i	0.902415	+	0.430869i	$0.141793\pi$
$613$	21.3356i	0.861737i	−0.902415	+	0.430869i	$0.858207\pi$
$614$	−8.32613	−0.336015
$615$	0	0
$616$	44.9917	1.81277
$617$	23.2463i	0.935861i	0.883765	+	0.467931i	$0.155000\pi$
$617$	23.2463i	0.935861i	−0.883765	+	0.467931i	$0.845000\pi$
$618$	− 1.26740i	− 0.0509824i
$619$	4.11413	0.165361	0.0826804	−	0.996576i	$-0.473652\pi$
$619$	4.11413	0.165361	0.0826804	+	0.996576i	$0.473652\pi$
$620$	0	0
$621$	5.62549	0.225743
$622$	− 3.94328i	− 0.158111i
$623$	17.7423i	0.710830i
$624$	11.5053	0.460582
$625$	0	0
$626$	3.17876	0.127049
$627$	15.1028i	0.603146i
$628$	− 18.7327i	− 0.747515i
$629$	72.4296	2.88796
$630$	0	0
$631$	−45.7221	−1.82017	−0.910083	−	0.414426i	$-0.863982\pi$
$631$	−45.7221	−1.82017	−0.910083	+	0.414426i	$0.863982\pi$
$632$	5.65803i	0.225065i
$633$	− 16.4705i	− 0.654642i
$634$	−15.8670	−0.630159
$635$	0	0
$636$	−1.68323	−0.0667444
$637$	− 74.7673i	− 2.96239i
$638$	− 2.49461i	− 0.0987626i
$639$	−0.741689	−0.0293408
$640$	0	0
$641$	28.3443	1.11953	0.559767	−	0.828650i	$-0.310891\pi$
$641$	28.3443	1.11953	0.559767	+	0.828650i	$0.310891\pi$
$642$	2.29024i	0.0903884i
$643$	− 17.4545i	− 0.688337i	−0.938908	−	0.344168i	$-0.888161\pi$
$643$	− 17.4545i	− 0.688337i	0.938908	−	0.344168i	$-0.111839\pi$
$644$	−47.1904	−1.85956
$645$	0	0
$646$	−10.2804	−0.404478
$647$	− 17.3874i	− 0.683569i	−0.939778	−	0.341784i	$-0.888969\pi$
$647$	− 17.3874i	− 0.683569i	0.939778	−	0.341784i	$-0.111031\pi$
$648$	1.90970i	0.0750202i
$649$	1.45986	0.0573044
$650$	0	0
$651$	−44.5866	−1.74749
$652$	− 18.5929i	− 0.728153i
$653$	− 30.2052i	− 1.18202i	−0.806664	−	0.591010i	$-0.798729\pi$
$653$	− 30.2052i	− 1.18202i	0.806664	−	0.591010i	$-0.201271\pi$
$654$	−1.67592	−0.0655335
$655$	0	0
$656$	−7.12184	−0.278061
$657$	− 5.52981i	− 0.215738i
$658$	− 8.93535i	− 0.348336i
$659$	−26.3659	−1.02707	−0.513535	−	0.858069i	$-0.671664\pi$
$659$	−26.3659	−1.02707	−0.513535	+	0.858069i	$0.671664\pi$
$660$	0	0
$661$	15.8187	0.615274	0.307637	−	0.951504i	$-0.400462\pi$
$661$	15.8187	0.615274	0.307637	+	0.951504i	$0.400462\pi$
$662$	17.8108i	0.692236i
$663$	− 29.9238i	− 1.16215i
$664$	26.0929	1.01260
$665$	0	0
$666$	5.68246	0.220191
$667$	5.62549i	0.217820i
$668$	− 13.7505i	− 0.532023i
$669$	−4.58903	−0.177422
$670$	0	0
$671$	−3.84356	−0.148379
$672$	− 24.5886i	− 0.948527i
$673$	43.1070i	1.66165i	0.556530	+	0.830827i	$0.312132\pi$
$673$	43.1070i	1.66165i	−0.556530	+	0.830827i	$0.687868\pi$
$674$	−8.47177	−0.326320
$675$	0	0
$676$	−14.1385	−0.543788
$677$	− 40.6978i	− 1.56415i	−0.623187	−	0.782073i	$-0.714163\pi$
$677$	− 40.6978i	− 1.56415i	0.623187	−	0.782073i	$-0.285837\pi$
$678$	− 9.18396i	− 0.352708i
$679$	13.8750	0.532472
$680$	0	0
$681$	25.5254	0.978134
$682$	23.0596i	0.882996i
$683$	− 21.0217i	− 0.804372i	−0.915558	−	0.402186i	$-0.868250\pi$
$683$	− 21.0217i	− 0.804372i	0.915558	−	0.402186i	$-0.131750\pi$
$684$	5.37755	0.205616
$685$	0	0
$686$	−22.8253	−0.871475
$687$	− 8.04982i	− 0.307120i
$688$	− 11.4789i	− 0.437627i
$689$	4.44888	0.169489
$690$	0	0
$691$	6.98613	0.265765	0.132882	−	0.991132i	$-0.457577\pi$
$691$	6.98613	0.265765	0.132882	+	0.991132i	$0.457577\pi$
$692$	6.55244i	0.249086i
$693$	23.5595i	0.894951i
$694$	0.211241	0.00801860
$695$	0	0
$696$	−1.90970	−0.0723871
$697$	18.5230i	0.701608i
$698$	− 18.1611i	− 0.687408i
$699$	3.75903	0.142179
$700$	0	0
$701$	1.40572	0.0530934	0.0265467	−	0.999648i	$-0.491549\pi$
$701$	1.40572	0.0530934	0.0265467	+	0.999648i	$0.491549\pi$
$702$	− 2.34767i	− 0.0886072i
$703$	− 34.4025i	− 1.29751i
$704$	11.7340	0.442240
$705$	0	0
$706$	−1.57321	−0.0592085
$707$	70.7981i	2.66264i
$708$	− 0.519802i	− 0.0195354i
$709$	−1.27110	−0.0477372	−0.0238686	−	0.999715i	$-0.507598\pi$
$709$	−1.27110	−0.0477372	−0.0238686	+	0.999715i	$0.507598\pi$
$710$	0	0
$711$	−2.96278	−0.111113
$712$	− 7.02457i	− 0.263257i
$713$	− 52.0006i	− 1.94744i
$714$	−16.0369	−0.600167
$715$	0	0
$716$	−17.9623	−0.671282
$717$	− 3.53192i	− 0.131902i
$718$	7.31463i	0.272979i
$719$	−27.8439	−1.03840	−0.519201	−	0.854652i	$-0.673770\pi$
$719$	−27.8439	−1.03840	−0.519201	+	0.854652i	$0.673770\pi$
$720$	0	0
$721$	−11.9696	−0.445770
$722$	− 4.82092i	− 0.179416i
$723$	− 3.05553i	− 0.113636i
$724$	25.1365	0.934192
$725$	0	0
$726$	6.56659	0.243709
$727$	8.97787i	0.332971i	0.986044	+	0.166485i	$0.0532419\pi$
$727$	8.97787i	0.332971i	−0.986044	+	0.166485i	$0.946758\pi$
$728$	42.3416i	1.56928i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	−29.8550	−1.10423
$732$	1.36855i	0.0505832i
$733$	− 34.7466i	− 1.28339i	−0.766958	−	0.641697i	$-0.778231\pi$
$733$	− 34.7466i	− 1.28339i	0.766958	−	0.641697i	$-0.221769\pi$
$734$	8.83614	0.326148
$735$	0	0
$736$	28.6773	1.05706
$737$	− 23.7744i	− 0.875740i
$738$	1.45322i	0.0534937i
$739$	3.32518	0.122319	0.0611594	−	0.998128i	$-0.480520\pi$
$739$	3.32518	0.122319	0.0611594	+	0.998128i	$0.480520\pi$
$740$	0	0
$741$	−14.2132	−0.522134
$742$	− 2.38426i	− 0.0875291i
$743$	− 1.06314i	− 0.0390029i	−0.999810	−	0.0195015i	$-0.993792\pi$
$743$	− 1.06314i	− 0.0390029i	0.999810	−	0.0195015i	$-0.00620790\pi$
$744$	17.6528	0.647184
$745$	0	0
$746$	−16.2573	−0.595222
$747$	13.6633i	0.499916i
$748$	− 55.2994i	− 2.02195i
$749$	21.6294	0.790320
$750$	0	0
$751$	−38.7756	−1.41494	−0.707472	−	0.706742i	$-0.750164\pi$
$751$	−38.7756	−1.41494	−0.707472	+	0.706742i	$0.750164\pi$
$752$	− 9.07856i	− 0.331061i
$753$	23.3283i	0.850129i
$754$	2.34767	0.0854972
$755$	0	0
$756$	8.38869	0.305094
$757$	− 40.9695i	− 1.48906i	−0.667588	−	0.744531i	$-0.732673\pi$
$757$	− 40.9695i	− 1.48906i	0.667588	−	0.744531i	$-0.267327\pi$
$758$	9.28792i	0.337352i
$759$	−27.4770	−0.997354
$760$	0	0
$761$	13.8561	0.502283	0.251142	−	0.967950i	$-0.419194\pi$
$761$	13.8561	0.502283	0.251142	+	0.967950i	$0.419194\pi$
$762$	6.89994i	0.249958i
$763$	15.8276i	0.572998i
$764$	15.3315	0.554675
$765$	0	0
$766$	−16.4980	−0.596096
$767$	1.37387i	0.0496075i
$768$	1.02912i	0.0371353i
$769$	6.73919	0.243021	0.121511	−	0.992590i	$-0.461226\pi$
$769$	6.73919	0.243021	0.121511	+	0.992590i	$0.461226\pi$
$770$	0	0
$771$	−28.2268	−1.01656
$772$	30.0079i	1.08001i
$773$	7.45730i	0.268220i	0.990966	+	0.134110i	$0.0428176\pi$
$773$	7.45730i	0.268220i	−0.990966	+	0.134110i	$0.957182\pi$
$774$	−2.34227	−0.0841912
$775$	0	0
$776$	−5.49341	−0.197202
$777$	− 53.6660i	− 1.92526i
$778$	13.5700i	0.486506i
$779$	8.79801	0.315221
$780$	0	0
$781$	3.62270	0.129630
$782$	− 18.7036i	− 0.668839i
$783$	− 1.00000i	− 0.0357371i
$784$	−40.7119	−1.45400
$785$	0	0
$786$	6.20283	0.221248
$787$	7.81608i	0.278613i	0.990249	+	0.139307i	$0.0444873\pi$
$787$	7.81608i	0.278613i	−0.990249	+	0.139307i	$0.955513\pi$
$788$	39.2559i	1.39844i
$789$	−27.6688	−0.985037
$790$	0	0
$791$	−86.7347	−3.08393
$792$	− 9.32773i	− 0.331446i
$793$	− 3.61717i	− 0.128449i
$794$	4.53444	0.160921
$795$	0	0
$796$	23.9959	0.850514
$797$	− 21.8313i	− 0.773305i	−0.922225	−	0.386653i	$-0.873631\pi$
$797$	− 21.8313i	− 0.773305i	0.922225	−	0.386653i	$-0.126369\pi$
$798$	7.61719i	0.269646i
$799$	−23.6121	−0.835338
$800$	0	0
$801$	3.67835	0.129968
$802$	13.1004i	0.462590i
$803$	27.0097i	0.953152i
$804$	−8.46519	−0.298544
$805$	0	0
$806$	−21.7013	−0.764396
$807$	0.839052i	0.0295360i
$808$	− 28.0305i	− 0.986111i
$809$	−5.71715	−0.201004	−0.100502	−	0.994937i	$-0.532045\pi$
$809$	−5.71715	−0.201004	−0.100502	+	0.994937i	$0.532045\pi$
$810$	0	0
$811$	−44.8661	−1.57546	−0.787732	−	0.616019i	$-0.788745\pi$
$811$	−44.8661	−1.57546	−0.787732	+	0.616019i	$0.788745\pi$
$812$	8.38869i	0.294385i
$813$	− 1.27423i	− 0.0446890i
$814$	−27.7553	−0.972823
$815$	0	0
$816$	−16.2940	−0.570402
$817$	14.1805i	0.496112i
$818$	− 5.06939i	− 0.177247i
$819$	−22.1718	−0.774745
$820$	0	0
$821$	37.3699	1.30422	0.652110	−	0.758125i	$-0.273884\pi$
$821$	37.3699	1.30422	0.652110	+	0.758125i	$0.273884\pi$
$822$	5.09437i	0.177687i
$823$	− 31.6982i	− 1.10493i	−0.833537	−	0.552464i	$-0.813688\pi$
$823$	− 31.6982i	− 1.10493i	0.833537	−	0.552464i	$-0.186312\pi$
$824$	4.73901	0.165091
$825$	0	0
$826$	0.736290	0.0256188
$827$	14.4089i	0.501046i	0.968111	+	0.250523i	$0.0806025\pi$
$827$	14.4089i	0.501046i	−0.968111	+	0.250523i	$0.919397\pi$
$828$	9.78358i	0.340003i
$829$	20.6622	0.717630	0.358815	−	0.933409i	$-0.383181\pi$
$829$	20.6622	0.717630	0.358815	+	0.933409i	$0.383181\pi$
$830$	0	0
$831$	26.7387	0.927556
$832$	11.0428i	0.382840i
$833$	105.886i	3.66874i
$834$	−1.44522	−0.0500440
$835$	0	0
$836$	−26.2660	−0.908429
$837$	9.24375i	0.319511i
$838$	− 4.15031i	− 0.143370i
$839$	−1.64970	−0.0569541	−0.0284771	−	0.999594i	$-0.509066\pi$
$839$	−1.64970	−0.0569541	−0.0284771	+	0.999594i	$0.509066\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	− 4.27904i	− 0.147466i
$843$	16.3446i	0.562938i
$844$	28.6447	0.985990
$845$	0	0
$846$	−1.85249	−0.0636899
$847$	− 62.0159i	− 2.13089i
$848$	− 2.42248i	− 0.0831882i
$849$	23.7207	0.814092
$850$	0	0
$851$	62.5898	2.14555
$852$	− 1.28991i	− 0.0441916i
$853$	− 23.4066i	− 0.801428i	−0.916203	−	0.400714i	$-0.868762\pi$
$853$	− 23.4066i	− 0.801428i	0.916203	−	0.400714i	$-0.131238\pi$
$854$	−1.93853	−0.0663351
$855$	0	0
$856$	−8.56355	−0.292696
$857$	− 18.7479i	− 0.640415i	−0.947347	−	0.320207i	$-0.896247\pi$
$857$	− 18.7479i	− 0.640415i	0.947347	−	0.320207i	$-0.103753\pi$
$858$	11.4669i	0.391475i
$859$	29.1326	0.993990	0.496995	−	0.867753i	$-0.334437\pi$
$859$	29.1326	0.993990	0.496995	+	0.867753i	$0.334437\pi$
$860$	0	0
$861$	13.7244	0.467727
$862$	− 16.5414i	− 0.563403i
$863$	25.5033i	0.868142i	0.900879	+	0.434071i	$0.142923\pi$
$863$	25.5033i	0.868142i	−0.900879	+	0.434071i	$0.857077\pi$
$864$	−5.09775	−0.173429
$865$	0	0
$866$	−5.05281	−0.171701
$867$	25.3784i	0.861897i
$868$	− 77.5429i	− 2.63198i
$869$	14.4714	0.490908
$870$	0	0
$871$	22.3740	0.758114
$872$	− 6.26651i	− 0.212211i
$873$	− 2.87658i	− 0.0973573i
$874$	−8.88380	−0.300499
$875$	0	0
$876$	9.61718	0.324935
$877$	− 24.3134i	− 0.821005i	−0.911860	−	0.410502i	$-0.865353\pi$
$877$	− 24.3134i	− 0.821005i	0.911860	−	0.410502i	$-0.134647\pi$
$878$	− 2.98638i	− 0.100786i
$879$	−8.28535	−0.279458
$880$	0	0
$881$	51.3387	1.72965	0.864823	−	0.502077i	$-0.167431\pi$
$881$	51.3387	1.72965	0.864823	+	0.502077i	$0.167431\pi$
$882$	8.30730i	0.279721i
$883$	17.1764i	0.578033i	0.957324	+	0.289016i	$0.0933282\pi$
$883$	17.1764i	0.578033i	−0.957324	+	0.289016i	$0.906672\pi$
$884$	52.0421	1.75037
$885$	0	0
$886$	−1.60460	−0.0539075
$887$	− 38.2392i	− 1.28395i	−0.766727	−	0.641974i	$-0.778116\pi$
$887$	− 38.2392i	− 1.28395i	0.766727	−	0.641974i	$-0.221884\pi$
$888$	21.2476i	0.713022i
$889$	65.1641	2.18553
$890$	0	0
$891$	4.88439	0.163633
$892$	− 7.98103i	− 0.267225i
$893$	11.2153i	0.375304i
$894$	11.5117	0.385009
$895$	0	0
$896$	55.0954	1.84061
$897$	− 25.8586i	− 0.863393i
$898$	− 14.2312i	− 0.474901i
$899$	−9.24375	−0.308296
$900$	0	0
$901$	−6.30055	−0.209902
$902$	− 7.09808i	− 0.236340i
$903$	22.1208i	0.736134i
$904$	34.3402	1.14214
$905$	0	0
$906$	−7.26288	−0.241293
$907$	− 9.37007i	− 0.311128i	−0.987826	−	0.155564i	$-0.950281\pi$
$907$	− 9.37007i	− 0.311128i	0.987826	−	0.155564i	$-0.0497195\pi$
$908$	44.3925i	1.47322i
$909$	14.6780	0.486837
$910$	0	0
$911$	−10.5220	−0.348611	−0.174305	−	0.984692i	$-0.555768\pi$
$911$	−10.5220	−0.348611	−0.174305	+	0.984692i	$0.555768\pi$
$912$	7.73927i	0.256273i
$913$	− 66.7371i	− 2.20867i
$914$	−16.2886	−0.538780
$915$	0	0
$916$	13.9999	0.462569
$917$	− 58.5805i	− 1.93450i
$918$	3.32480i	0.109735i
$919$	47.9043	1.58022	0.790109	−	0.612967i	$-0.210024\pi$
$919$	47.9043	1.58022	0.790109	+	0.612967i	$0.210024\pi$
$920$	0	0
$921$	16.3024	0.537181
$922$	2.88390i	0.0949762i
$923$	3.40931i	0.112219i
$924$	−40.9736	−1.34793
$925$	0	0
$926$	−16.7317	−0.549837
$927$	2.48154i	0.0815046i
$928$	− 5.09775i	− 0.167342i
$929$	−32.0568	−1.05175	−0.525875	−	0.850562i	$-0.676262\pi$
$929$	−32.0568	−1.05175	−0.525875	+	0.850562i	$0.676262\pi$
$930$	0	0
$931$	50.2937	1.64831
$932$	6.53752i	0.214144i
$933$	7.72085i	0.252769i
$934$	−19.1080	−0.625232
$935$	0	0
$936$	8.77831	0.286928
$937$	− 23.8183i	− 0.778109i	−0.921215	−	0.389054i	$-0.872802\pi$
$937$	− 23.8183i	− 0.778109i	0.921215	−	0.389054i	$-0.127198\pi$
$938$	− 11.9908i	− 0.391513i
$939$	−6.22393	−0.203110
$940$	0	0
$941$	45.7836	1.49250	0.746251	−	0.665664i	$-0.231852\pi$
$941$	45.7836	1.49250	0.746251	+	0.665664i	$0.231852\pi$
$942$	− 5.50117i	− 0.179238i
$943$	16.0066i	0.521246i
$944$	0.748091	0.0243483
$945$	0	0
$946$	11.4406	0.371965
$947$	4.69130i	0.152447i	0.997091	+	0.0762233i	$0.0242862\pi$
$947$	4.69130i	0.152447i	−0.997091	+	0.0762233i	$0.975714\pi$
$948$	− 5.15273i	− 0.167353i
$949$	−25.4188	−0.825129
$950$	0	0
$951$	31.0672	1.00742
$952$	− 59.9645i	− 1.94346i
$953$	56.4381i	1.82821i	0.405479	+	0.914104i	$0.367105\pi$
$953$	56.4381i	1.82821i	−0.405479	+	0.914104i	$0.632895\pi$
$954$	−0.494309	−0.0160038
$955$	0	0
$956$	6.14256	0.198665
$957$	4.88439i	0.157890i
$958$	6.67883i	0.215783i
$959$	48.1121	1.55362
$960$	0	0
$961$	54.4469	1.75635
$962$	− 26.1205i	− 0.842158i
$963$	− 4.48423i	− 0.144502i
$964$	5.31403	0.171153
$965$	0	0
$966$	−13.8583	−0.445882
$967$	42.2611i	1.35902i	0.733664	+	0.679512i	$0.237808\pi$
$967$	42.2611i	1.35902i	−0.733664	+	0.679512i	$0.762192\pi$
$968$	24.5535i	0.789179i
$969$	20.1288	0.646631
$970$	0	0
$971$	25.6613	0.823510	0.411755	−	0.911295i	$-0.364916\pi$
$971$	25.6613	0.823510	0.411755	+	0.911295i	$0.364916\pi$
$972$	− 1.73915i	− 0.0557834i
$973$	13.6489i	0.437565i
$974$	9.93333	0.318284
$975$	0	0
$976$	−1.96960	−0.0630453
$977$	− 7.35242i	− 0.235225i	−0.993060	−	0.117612i	$-0.962476\pi$
$977$	− 7.35242i	− 0.235225i	0.993060	−	0.117612i	$-0.0375240\pi$
$978$	− 5.46011i	− 0.174595i
$979$	−17.9665	−0.574212
$980$	0	0
$981$	3.28140	0.104767
$982$	8.08084i	0.257870i
$983$	3.23911i	0.103312i	0.998665	+	0.0516558i	$0.0164499\pi$
$983$	3.23911i	0.103312i	−0.998665	+	0.0516558i	$0.983550\pi$
$984$	−5.43381	−0.173223
$985$	0	0
$986$	−3.32480	−0.105883
$987$	17.4952i	0.556879i
$988$	− 24.7189i	− 0.786413i
$989$	−25.7991	−0.820364
$990$	0	0
$991$	3.19230	0.101407	0.0507034	−	0.998714i	$-0.483854\pi$
$991$	3.19230	0.101407	0.0507034	+	0.998714i	$0.483854\pi$
$992$	47.1223i	1.49614i
$993$	− 34.8731i	− 1.10666i
$994$	1.82714	0.0579532
$995$	0	0
$996$	−23.7627	−0.752949
$997$	49.0253i	1.55265i	0.630335	+	0.776323i	$0.282917\pi$
$997$	49.0253i	1.55265i	−0.630335	+	0.776323i	$0.717083\pi$
$998$	− 2.49539i	− 0.0789903i
$999$	−11.1261	−0.352015

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2175.2.c.p.349.11		16
5.2	odd	4		2175.2.a.bc.1.4	✓	8
5.3	odd	4		2175.2.a.bd.1.5	yes	8
5.4	even	2	inner	2175.2.c.p.349.6		16
15.2	even	4		6525.2.a.bz.1.5		8
15.8	even	4		6525.2.a.by.1.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2175.2.a.bc.1.4	✓	8	5.2	odd	4
2175.2.a.bd.1.5	yes	8	5.3	odd	4
2175.2.c.p.349.6		16	5.4	even	2	inner
2175.2.c.p.349.11		16	1.1	even	1	trivial
6525.2.a.by.1.4		8	15.8	even	4
6525.2.a.bz.1.5		8	15.2	even	4

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:	\( N \)	\(=\)	\( 2175 = 3 \cdot 5^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2175.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+0.510732i q^{2} -1.00000i q^{3} +1.73915 q^{4} +0.510732 q^{6} -4.82343i q^{7} +1.90970i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q+0.510732i q^{2} -1.00000i q^{3} +1.73915 q^{4} +0.510732 q^{6} -4.82343i q^{7} +1.90970i q^{8} -1.00000 q^{9} +4.88439 q^{11} -1.73915i q^{12} +4.59669i q^{13} +2.46348 q^{14} +2.50296 q^{16} -6.50987i q^{17} -0.510732i q^{18} -3.09205 q^{19} -4.82343 q^{21} +2.49461i q^{22} -5.62549i q^{23} +1.90970 q^{24} -2.34767 q^{26} +1.00000i q^{27} -8.38869i q^{28} -1.00000 q^{29} +9.24375 q^{31} +5.09775i q^{32} -4.88439i q^{33} +3.32480 q^{34} -1.73915 q^{36} +11.1261i q^{37} -1.57921i q^{38} +4.59669 q^{39} -2.84537 q^{41} -2.46348i q^{42} -4.58611i q^{43} +8.49470 q^{44} +2.87311 q^{46} -3.62713i q^{47} -2.50296i q^{48} -16.2655 q^{49} -6.50987 q^{51} +7.99434i q^{52} -0.967845i q^{53} -0.510732 q^{54} +9.21132 q^{56} +3.09205i q^{57} -0.510732i q^{58} +0.298882 q^{59} -0.786908 q^{61} +4.72108i q^{62} +4.82343i q^{63} +2.40234 q^{64} +2.49461 q^{66} -4.86742i q^{67} -11.3217i q^{68} -5.62549 q^{69} +0.741689 q^{71} -1.90970i q^{72} +5.52981i q^{73} -5.68246 q^{74} -5.37755 q^{76} -23.5595i q^{77} +2.34767i q^{78} +2.96278 q^{79} +1.00000 q^{81} -1.45322i q^{82} -13.6633i q^{83} -8.38869 q^{84} +2.34227 q^{86} +1.00000i q^{87} +9.32773i q^{88} -3.67835 q^{89} +22.1718 q^{91} -9.78358i q^{92} -9.24375i q^{93} +1.85249 q^{94} +5.09775 q^{96} +2.87658i q^{97} -8.30730i q^{98} -4.88439 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 24 q^{4} + 4 q^{6} - 16 q^{9}+O(q^{10}) \) 16 * q - 24 * q^4 + 4 * q^6 - 16 * q^9	\( 16 q - 24 q^{4} + 4 q^{6} - 16 q^{9} + 12 q^{11} - 18 q^{14} + 64 q^{16} - 4 q^{21} - 6 q^{24} + 36 q^{26} - 16 q^{29} + 16 q^{31} + 26 q^{34} + 24 q^{36} + 12 q^{39} + 4 q^{41} + 30 q^{44} + 48 q^{46} - 76 q^{49} + 24 q^{51} - 4 q^{54} + 116 q^{56} - 36 q^{59} + 24 q^{61} - 42 q^{64} - 6 q^{66} - 28 q^{69} + 48 q^{71} + 44 q^{74} - 20 q^{79} + 16 q^{81} + 28 q^{84} + 16 q^{86} - 68 q^{89} + 52 q^{91} + 86 q^{94} - 4 q^{96} - 12 q^{99}+O(q^{100}) \) 16 * q - 24 * q^4 + 4 * q^6 - 16 * q^9 + 12 * q^11 - 18 * q^14 + 64 * q^16 - 4 * q^21 - 6 * q^24 + 36 * q^26 - 16 * q^29 + 16 * q^31 + 26 * q^34 + 24 * q^36 + 12 * q^39 + 4 * q^41 + 30 * q^44 + 48 * q^46 - 76 * q^49 + 24 * q^51 - 4 * q^54 + 116 * q^56 - 36 * q^59 + 24 * q^61 - 42 * q^64 - 6 * q^66 - 28 * q^69 + 48 * q^71 + 44 * q^74 - 20 * q^79 + 16 * q^81 + 28 * q^84 + 16 * q^86 - 68 * q^89 + 52 * q^91 + 86 * q^94 - 4 * q^96 - 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more