LMFDB - Embedded newform 1413.2.b.e.784.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1413,2,Mod(784,1413)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1413, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1413.784");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1413 = 3^{2} \cdot 157 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1413.b (of order $2$, degree $1$, 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:	$11.2828618056$
Analytic rank:	$0$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} + 24x^{12} + 224x^{10} + 1027x^{8} + 2399x^{6} + 2652x^{4} + 1094x^{2} + 147 $ x^14 + 24x^12 + 224x^10 + 1027x^8 + 2399x^6 + 2652x^4 + 1094x^2 + 147
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 471)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			784.12
Root			$2.14537i$ of defining polynomial
Character	$\chi$	$=$	1413.784
Dual form			1413.2.b.e.784.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.14537i q^{2} -2.60262 q^{4} -4.18337i q^{5} -4.37069i q^{7} -1.29283i q^{8} +O(q^{10})$	\(q+2.14537i q^{2} -2.60262 q^{4} -4.18337i q^{5} -4.37069i q^{7} -1.29283i q^{8} +8.97487 q^{10} -2.50855 q^{11} +2.82658 q^{13} +9.37675 q^{14} -2.43162 q^{16} -6.39890 q^{17} -0.233469 q^{19} +10.8877i q^{20} -5.38177i q^{22} +3.94841i q^{23} -12.5006 q^{25} +6.06406i q^{26} +11.3752i q^{28} +4.95950i q^{29} +2.27508 q^{31} -7.80240i q^{32} -13.7280i q^{34} -18.2842 q^{35} -7.10166 q^{37} -0.500876i q^{38} -5.40840 q^{40} +5.04389i q^{41} +11.5053i q^{43} +6.52880 q^{44} -8.47081 q^{46} +4.02024 q^{47} -12.1029 q^{49} -26.8183i q^{50} -7.35650 q^{52} -11.3180i q^{53} +10.4942i q^{55} -5.65058 q^{56} -10.6400 q^{58} -11.9276i q^{59} -5.58233i q^{61} +4.88090i q^{62} +11.8758 q^{64} -11.8246i q^{65} +2.69668 q^{67} +16.6539 q^{68} -39.2264i q^{70} -14.2640 q^{71} +3.28911i q^{73} -15.2357i q^{74} +0.607629 q^{76} +10.9641i q^{77} -1.98333i q^{79} +10.1724i q^{80} -10.8210 q^{82} -3.82072i q^{83} +26.7689i q^{85} -24.6831 q^{86} +3.24314i q^{88} +11.0637 q^{89} -12.3541i q^{91} -10.2762i q^{92} +8.62492i q^{94} +0.976684i q^{95} -2.62451i q^{97} -25.9653i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q - 20 q^{4}+O(q^{10}) $ 14 * q - 20 * q^4	$ 14 q - 20 q^{4} + 6 q^{10} + 2 q^{11} + 24 q^{16} - 18 q^{17} - 12 q^{19} - 18 q^{25} - 14 q^{31} - 16 q^{35} - 14 q^{37} - 36 q^{40} - 24 q^{44} - 8 q^{46} - 22 q^{47} - 48 q^{49} - 50 q^{52} + 62 q^{56} + 20 q^{58} - 34 q^{64} + 42 q^{67} + 56 q^{68} - 38 q^{71} + 52 q^{76} + 10 q^{82} - 34 q^{86} + 48 q^{89}+O(q^{100}) $ 14 * q - 20 * q^4 + 6 * q^10 + 2 * q^11 + 24 * q^16 - 18 * q^17 - 12 * q^19 - 18 * q^25 - 14 * q^31 - 16 * q^35 - 14 * q^37 - 36 * q^40 - 24 * q^44 - 8 * q^46 - 22 * q^47 - 48 * q^49 - 50 * q^52 + 62 * q^56 + 20 * q^58 - 34 * q^64 + 42 * q^67 + 56 * q^68 - 38 * q^71 + 52 * q^76 + 10 * q^82 - 34 * q^86 + 48 * q^89

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1413\mathbb{Z}\right)^\times$.

$n$	$1100$	$1261$
$\chi(n)$	$1$	$-1$

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 $

$2$			2.14537i			1.51701i	0.651669	+	0.758503i	$0.274069\pi$
$2$			2.14537i			1.51701i	−0.651669	+	0.758503i	$0.725931\pi$
$3$		0			0
$3$		0			0
$4$	−2.60262			−1.30131
$5$		−	4.18337i		−	1.87086i	−0.353514	−	0.935429i	$-0.615013\pi$
$5$		−	4.18337i		−	1.87086i	0.353514	−	0.935429i	$-0.384987\pi$
$6$		0			0
$7$		−	4.37069i		−	1.65197i	−0.563696	−	0.825983i	$-0.690621\pi$
$7$		−	4.37069i		−	1.65197i	0.563696	−	0.825983i	$-0.309379\pi$
$8$		−	1.29283i		−	0.457086i
$9$		0			0
$10$	8.97487			2.83810
$11$	−2.50855			−0.756357			−0.378178	−	0.925733i	$-0.623449\pi$
$11$	−2.50855			−0.756357			−0.378178	+	0.925733i	$0.623449\pi$
$12$		0			0
$13$	2.82658			0.783952			0.391976	−	0.919975i	$-0.371792\pi$
$13$	2.82658			0.783952			0.391976	+	0.919975i	$0.371792\pi$
$14$	9.37675			2.50604
$15$		0			0
$16$	−2.43162			−0.607906
$17$	−6.39890			−1.55196			−0.775980	−	0.630757i	$-0.782744\pi$
$17$	−6.39890			−1.55196			−0.775980	+	0.630757i	$0.782744\pi$
$18$		0			0
$19$	−0.233469			−0.0535613			−0.0267807	−	0.999641i	$-0.508526\pi$
$19$	−0.233469			−0.0535613			−0.0267807	+	0.999641i	$0.508526\pi$
$20$			10.8877i			2.43456i
$21$		0			0
$22$		−	5.38177i		−	1.14740i
$23$			3.94841i			0.823301i	0.911342	+	0.411651i	$0.135048\pi$
$23$			3.94841i			0.823301i	−0.911342	+	0.411651i	$0.864952\pi$
$24$		0			0
$25$	−12.5006			−2.50011
$26$			6.06406i			1.18926i
$27$		0			0
$28$			11.3752i			2.14972i
$29$			4.95950i			0.920956i	0.887671	+	0.460478i	$0.152322\pi$
$29$			4.95950i			0.920956i	−0.887671	+	0.460478i	$0.847678\pi$
$30$		0			0
$31$	2.27508			0.408617			0.204309	−	0.978907i	$-0.434505\pi$
$31$	2.27508			0.408617			0.204309	+	0.978907i	$0.434505\pi$
$32$		−	7.80240i		−	1.37928i
$33$		0			0
$34$		−	13.7280i		−	2.35433i
$35$	−18.2842			−3.09059
$36$		0			0
$37$	−7.10166			−1.16751			−0.583753	−	0.811931i	$-0.698416\pi$
$37$	−7.10166			−1.16751			−0.583753	+	0.811931i	$0.698416\pi$
$38$		−	0.500876i		−	0.0812529i
$39$		0			0
$40$	−5.40840			−0.855143
$41$			5.04389i			0.787723i	0.919170	+	0.393862i	$0.128861\pi$
$41$			5.04389i			0.787723i	−0.919170	+	0.393862i	$0.871139\pi$
$42$		0			0
$43$			11.5053i			1.75454i	0.479998	+	0.877269i	$0.340637\pi$
$43$			11.5053i			1.75454i	−0.479998	+	0.877269i	$0.659363\pi$
$44$	6.52880			0.984253
$45$		0			0
$46$	−8.47081			−1.24895
$47$	4.02024			0.586413			0.293206	−	0.956049i	$-0.405278\pi$
$47$	4.02024			0.586413			0.293206	+	0.956049i	$0.405278\pi$
$48$		0			0
$49$	−12.1029			−1.72899
$50$		−	26.8183i		−	3.79269i
$51$		0			0
$52$	−7.35650			−1.02016
$53$		−	11.3180i		−	1.55464i	−0.629105	−	0.777321i	$-0.716578\pi$
$53$		−	11.3180i		−	1.55464i	0.629105	−	0.777321i	$-0.283422\pi$
$54$		0			0
$55$			10.4942i			1.41504i
$56$	−5.65058			−0.755090
$57$		0			0
$58$	−10.6400			−1.39710
$59$		−	11.9276i		−	1.55284i	−0.630219	−	0.776418i	$-0.717035\pi$
$59$		−	11.9276i		−	1.55284i	0.630219	−	0.776418i	$-0.282965\pi$
$60$		0			0
$61$		−	5.58233i		−	0.714744i	−0.933962	−	0.357372i	$-0.883673\pi$
$61$		−	5.58233i		−	0.714744i	0.933962	−	0.357372i	$-0.116327\pi$
$62$			4.88090i			0.619875i
$63$		0			0
$64$	11.8758			1.48447
$65$		−	11.8246i		−	1.46666i
$66$		0			0
$67$	2.69668			0.329452			0.164726	−	0.986339i	$-0.447326\pi$
$67$	2.69668			0.329452			0.164726	+	0.986339i	$0.447326\pi$
$68$	16.6539			2.01958
$69$		0			0
$70$		−	39.2264i		−	4.68845i
$71$	−14.2640			−1.69282			−0.846410	−	0.532532i	$-0.821241\pi$
$71$	−14.2640			−1.69282			−0.846410	+	0.532532i	$0.821241\pi$
$72$		0			0
$73$			3.28911i			0.384961i	0.981301	+	0.192480i	$0.0616532\pi$
$73$			3.28911i			0.384961i	−0.981301	+	0.192480i	$0.938347\pi$
$74$		−	15.2357i		−	1.77111i
$75$		0			0
$76$	0.607629			0.0696998
$77$			10.9641i			1.24948i
$78$		0			0
$79$		−	1.98333i		−	0.223142i	−0.993756	−	0.111571i	$-0.964412\pi$
$79$		−	1.98333i		−	0.223142i	0.993756	−	0.111571i	$-0.0355882\pi$
$80$			10.1724i			1.13731i
$81$		0			0
$82$	−10.8210			−1.19498
$83$		−	3.82072i		−	0.419379i	−0.977768	−	0.209689i	$-0.932755\pi$
$83$		−	3.82072i		−	0.419379i	0.977768	−	0.209689i	$-0.0672453\pi$
$84$		0			0
$85$			26.7689i			2.90350i
$86$	−24.6831			−2.66165
$87$		0			0
$88$			3.24314i			0.345720i
$89$	11.0637			1.17275			0.586373	−	0.810041i	$-0.300555\pi$
$89$	11.0637			1.17275			0.586373	+	0.810041i	$0.300555\pi$
$90$		0			0
$91$		−	12.3541i		−	1.29506i
$92$		−	10.2762i		−	1.07137i
$93$		0			0
$94$			8.62492i			0.889592i
$95$			0.976684i			0.100206i
$96$		0			0
$97$		−	2.62451i		−	0.266479i	−0.991084	−	0.133239i	$-0.957462\pi$
$97$		−	2.62451i		−	0.266479i	0.991084	−	0.133239i	$-0.0425379\pi$
$98$		−	25.9653i		−	2.62289i
$99$		0			0
$100$	32.5342			3.25342
$101$	4.15015			0.412955			0.206478	−	0.978451i	$-0.433800\pi$
$101$	4.15015			0.412955			0.206478	+	0.978451i	$0.433800\pi$
$102$		0			0
$103$		−	1.85327i		−	0.182608i	−0.995823	−	0.0913040i	$-0.970896\pi$
$103$		−	1.85327i		−	0.182608i	0.995823	−	0.0913040i	$-0.0291035\pi$
$104$		−	3.65430i		−	0.358334i
$105$		0			0
$106$	24.2812			2.35840
$107$			6.17144i			0.596615i	0.954470	+	0.298308i	$0.0964222\pi$
$107$			6.17144i			0.596615i	−0.954470	+	0.298308i	$0.903578\pi$
$108$		0			0
$109$	−6.10480			−0.584734			−0.292367	−	0.956306i	$-0.594443\pi$
$109$	−6.10480			−0.584734			−0.292367	+	0.956306i	$0.594443\pi$
$110$	−22.5139			−2.14662
$111$		0			0
$112$			10.6279i			1.00424i
$113$	20.1010			1.89094			0.945472	−	0.325702i	$-0.105601\pi$
$113$	20.1010			1.89094			0.945472	+	0.325702i	$0.105601\pi$
$114$		0			0
$115$	16.5177			1.54028
$116$		−	12.9077i		−	1.19845i
$117$		0			0
$118$	25.5890			2.35566
$119$			27.9676i			2.56378i
$120$		0			0
$121$	−4.70717			−0.427924
$122$	11.9762			1.08427
$123$		0			0
$124$	−5.92117			−0.531737
$125$			31.3776i			2.80650i
$126$		0			0
$127$	−10.9577			−0.972342			−0.486171	−	0.873864i	$-0.661607\pi$
$127$	−10.9577			−0.972342			−0.486171	+	0.873864i	$0.661607\pi$
$128$			9.87318i			0.872674i
$129$		0			0
$130$	25.3682			2.22494
$131$		−	13.5960i		−	1.18789i	−0.804507	−	0.593943i	$-0.797570\pi$
$131$		−	13.5960i		−	1.18789i	0.804507	−	0.593943i	$-0.202430\pi$
$132$		0			0
$133$			1.02042i			0.0884815i
$134$			5.78538i			0.499780i
$135$		0			0
$136$			8.27271i			0.709379i
$137$		−	20.0726i		−	1.71492i	−0.514553	−	0.857458i	$-0.672042\pi$
$137$		−	20.0726i		−	1.71492i	0.514553	−	0.857458i	$-0.327958\pi$
$138$		0			0
$139$		−	7.56565i		−	0.641710i	−0.947128	−	0.320855i	$-0.896030\pi$
$139$		−	7.56565i		−	0.641710i	0.947128	−	0.320855i	$-0.103970\pi$
$140$	47.5867			4.02181
$141$		0			0
$142$		−	30.6015i		−	2.56802i
$143$	−7.09063			−0.592948
$144$		0			0
$145$	20.7474			1.72298
$146$	−7.05636			−0.583988
$147$		0			0
$148$	18.4829			1.51929
$149$			10.4097i			0.852795i	0.904536	+	0.426398i	$0.140218\pi$
$149$			10.4097i			0.852795i	−0.904536	+	0.426398i	$0.859782\pi$
$150$		0			0
$151$		−	1.48402i		−	0.120768i	−0.998175	−	0.0603838i	$-0.980768\pi$
$151$		−	1.48402i		−	0.120768i	0.998175	−	0.0603838i	$-0.0192324\pi$
$152$			0.301836i			0.0244821i
$153$		0			0
$154$	−23.5221			−1.89546
$155$		−	9.51751i		−	0.764465i
$156$		0			0
$157$	−2.53968	+	12.2699i	−0.202689	+	0.979243i
$157$	−2.53968	+	12.2699i	−0.202689	+	0.979243i
$158$	4.25497			0.338507
$159$		0			0
$160$	−32.6403			−2.58044
$161$	17.2573			1.36006
$162$		0			0
$163$		−	5.83590i		−	0.457103i	−0.973532	−	0.228551i	$-0.926601\pi$
$163$		−	5.83590i		−	0.457103i	0.973532	−	0.228551i	$-0.0733989\pi$
$164$		−	13.1273i		−	1.02507i
$165$		0			0
$166$	8.19687			0.636200
$167$	1.18984			0.0920726			0.0460363	−	0.998940i	$-0.485341\pi$
$167$	1.18984			0.0920726			0.0460363	+	0.998940i	$0.485341\pi$
$168$		0			0
$169$	−5.01044			−0.385419
$170$	−57.4293			−4.40462
$171$		0			0
$172$		−	29.9438i		−	2.28320i
$173$	−18.0773			−1.37439			−0.687194	−	0.726474i	$-0.741158\pi$
$173$	−18.0773			−1.37439			−0.687194	+	0.726474i	$0.741158\pi$
$174$		0			0
$175$			54.6361i			4.13010i
$176$	6.09985			0.459794
$177$		0			0
$178$			23.7357i			1.77906i
$179$		−	6.47464i		−	0.483937i	−0.970284	−	0.241969i	$-0.922207\pi$
$179$		−	6.47464i		−	0.483937i	0.970284	−	0.241969i	$-0.0777931\pi$
$180$		0			0
$181$		−	2.34627i		−	0.174397i	−0.996191	−	0.0871985i	$-0.972209\pi$
$181$		−	2.34627i		−	0.174397i	0.996191	−	0.0871985i	$-0.0277914\pi$
$182$	26.5041			1.96462
$183$		0			0
$184$	5.10464			0.376319
$185$			29.7089i			2.18424i
$186$		0			0
$187$	16.0520			1.17384
$188$	−10.4632			−0.763104
$189$		0			0
$190$	−2.09535			−0.152013
$191$		−	15.9374i		−	1.15319i	−0.817031	−	0.576594i	$-0.804381\pi$
$191$		−	15.9374i		−	1.15319i	0.817031	−	0.576594i	$-0.195619\pi$
$192$		0			0
$193$	−5.18386			−0.373142			−0.186571	−	0.982441i	$-0.559738\pi$
$193$	−5.18386			−0.373142			−0.186571	+	0.982441i	$0.559738\pi$
$194$	5.63055			0.404250
$195$		0			0
$196$	31.4993			2.24995
$197$	1.72286			0.122749			0.0613743	−	0.998115i	$-0.480452\pi$
$197$	1.72286			0.122749			0.0613743	+	0.998115i	$0.480452\pi$
$198$		0			0
$199$	−3.01049			−0.213408			−0.106704	−	0.994291i	$-0.534030\pi$
$199$	−3.01049			−0.213408			−0.106704	+	0.994291i	$0.534030\pi$
$200$			16.1611i			1.14277i
$201$		0			0
$202$			8.90360i			0.626455i
$203$	21.6764			1.52139
$204$		0			0
$205$	21.1004			1.47372
$206$	3.97595			0.277018
$207$		0			0
$208$	−6.87318			−0.476569
$209$	0.585668			0.0405115
$210$		0			0
$211$		−	24.8157i		−	1.70838i	−0.519961	−	0.854190i	$-0.674054\pi$
$211$		−	24.8157i		−	1.70838i	0.519961	−	0.854190i	$-0.325946\pi$
$212$			29.4563i			2.02307i
$213$		0			0
$214$	−13.2400			−0.905069
$215$	48.1308			3.28249
$216$		0			0
$217$		−	9.94368i		−	0.675021i
$218$		−	13.0971i		−	0.887046i
$219$		0			0
$220$		−	27.3124i		−	1.84140i
$221$	−18.0870			−1.21666
$222$		0			0
$223$		−	9.65832i		−	0.646769i	−0.946268	−	0.323384i	$-0.895179\pi$
$223$		−	9.65832i		−	0.646769i	0.946268	−	0.323384i	$-0.104821\pi$
$224$	−34.1019			−2.27853
$225$		0			0
$226$			43.1241i			2.86857i
$227$		−	16.9716i		−	1.12644i	−0.826306	−	0.563221i	$-0.809562\pi$
$227$		−	16.9716i		−	1.12644i	0.826306	−	0.563221i	$-0.190438\pi$
$228$		0			0
$229$		−	16.6857i		−	1.10262i	−0.834299	−	0.551312i	$-0.814127\pi$
$229$		−	16.6857i		−	1.10262i	0.834299	−	0.551312i	$-0.185873\pi$
$230$			35.4365i			2.33661i
$231$		0			0
$232$	6.41181			0.420956
$233$	−8.21075			−0.537904			−0.268952	−	0.963154i	$-0.586677\pi$
$233$	−8.21075			−0.537904			−0.268952	+	0.963154i	$0.586677\pi$
$234$		0			0
$235$		−	16.8182i		−	1.09710i
$236$			31.0428i			2.02072i
$237$		0			0
$238$	−60.0008			−3.88928
$239$	−19.1336			−1.23765			−0.618825	−	0.785529i	$-0.712391\pi$
$239$	−19.1336			−1.23765			−0.618825	+	0.785529i	$0.712391\pi$
$240$		0			0
$241$		−	6.45705i		−	0.415935i	−0.978136	−	0.207967i	$-0.933315\pi$
$241$		−	6.45705i		−	0.415935i	0.978136	−	0.207967i	$-0.0666848\pi$
$242$		−	10.0986i		−	0.649164i
$243$		0			0
$244$			14.5287i			0.930102i
$245$			50.6310i			3.23469i
$246$		0			0
$247$	−0.659918			−0.0419895
$248$		−	2.94131i		−	0.186773i
$249$		0			0
$250$	−67.3166			−4.25747
$251$		−	14.4833i		−	0.914179i	−0.889421	−	0.457089i	$-0.848892\pi$
$251$		−	14.4833i		−	0.914179i	0.889421	−	0.457089i	$-0.151108\pi$
$252$		0			0
$253$		−	9.90480i		−	0.622710i
$254$		−	23.5084i		−	1.47505i
$255$		0			0
$256$	2.56995			0.160622
$257$	1.04509			0.0651910			0.0325955	−	0.999469i	$-0.489623\pi$
$257$	1.04509			0.0651910			0.0325955	+	0.999469i	$0.489623\pi$
$258$		0			0
$259$			31.0392i			1.92868i
$260$			30.7750i			1.90858i
$261$		0			0
$262$	29.1684			1.80203
$263$	−20.1065			−1.23982			−0.619909	−	0.784674i	$-0.712831\pi$
$263$	−20.1065			−1.23982			−0.619909	+	0.784674i	$0.712831\pi$
$264$		0			0
$265$	−47.3472			−2.90851
$266$	−2.18918			−0.134227
$267$		0			0
$268$	−7.01842			−0.428718
$269$			9.39200i			0.572640i	0.958134	+	0.286320i	$0.0924321\pi$
$269$			9.39200i			0.572640i	−0.958134	+	0.286320i	$0.907568\pi$
$270$		0			0
$271$		−	5.18044i		−	0.314690i	−0.987544	−	0.157345i	$-0.949707\pi$
$271$		−	5.18044i		−	0.314690i	0.987544	−	0.157345i	$-0.0502934\pi$
$272$	15.5597			0.943446
$273$		0			0
$274$	43.0631			2.60154
$275$	31.3583			1.89098
$276$		0			0
$277$	−8.66035			−0.520350			−0.260175	−	0.965561i	$-0.583780\pi$
$277$	−8.66035			−0.520350			−0.260175	+	0.965561i	$0.583780\pi$
$278$	16.2311			0.973478
$279$		0			0
$280$			23.6384i			1.41267i
$281$	17.0715			1.01840			0.509201	−	0.860647i	$-0.329941\pi$
$281$	17.0715			1.01840			0.509201	+	0.860647i	$0.329941\pi$
$282$		0			0
$283$	−6.45547			−0.383738			−0.191869	−	0.981421i	$-0.561455\pi$
$283$	−6.45547			−0.383738			−0.191869	+	0.981421i	$0.561455\pi$
$284$	37.1236			2.20288
$285$		0			0
$286$		−	15.2120i		−	0.899506i
$287$	22.0453			1.30129
$288$		0			0
$289$	23.9459			1.40858
$290$			44.5109i			2.61377i
$291$		0			0
$292$		−	8.56028i		−	0.500953i
$293$		−	1.58808i		−	0.0927765i	−0.998923	−	0.0463882i	$-0.985229\pi$
$293$		−	1.58808i		−	0.0927765i	0.998923	−	0.0463882i	$-0.0147711\pi$
$294$		0			0
$295$	−49.8973			−2.90514
$296$			9.18127i			0.533651i
$297$		0			0
$298$	−22.3326			−1.29370
$299$			11.1605i			0.645429i
$300$		0			0
$301$	50.2860			2.89844
$302$	3.18377			0.183205
$303$		0			0
$304$	0.567707			0.0325603
$305$	−23.3529			−1.33718
$306$		0			0
$307$		−	20.1431i		−	1.14963i	−0.818283	−	0.574815i	$-0.805074\pi$
$307$		−	20.1431i		−	1.14963i	0.818283	−	0.574815i	$-0.194926\pi$
$308$		−	28.5353i		−	1.62595i
$309$		0			0
$310$	20.4186			1.15970
$311$	10.1624			0.576259			0.288130	−	0.957591i	$-0.406967\pi$
$311$	10.1624			0.576259			0.288130	+	0.957591i	$0.406967\pi$
$312$		0			0
$313$	−0.161676			−0.00913845			−0.00456923	−	0.999990i	$-0.501454\pi$
$313$	−0.161676			−0.00913845			−0.00456923	+	0.999990i	$0.501454\pi$
$314$	−26.3234	−	5.44856i	−1.48552	−	0.307480i
$315$		0			0
$316$			5.16184i			0.290376i
$317$	−26.4817			−1.48736			−0.743681	−	0.668535i	$-0.766922\pi$
$317$	−26.4817			−1.48736			−0.743681	+	0.668535i	$0.766922\pi$
$318$		0			0
$319$		−	12.4412i		−	0.696571i
$320$		−	49.6808i		−	2.77724i
$321$		0			0
$322$			37.0233i			2.06323i
$323$	1.49394			0.0831251
$324$		0			0
$325$	−35.3338			−1.95997
$326$	12.5202			0.693427
$327$		0			0
$328$	6.52091			0.360057
$329$		−	17.5712i		−	0.968734i
$330$		0			0
$331$	26.8355			1.47501			0.737506	−	0.675341i	$-0.236003\pi$
$331$	26.8355			1.47501			0.737506	+	0.675341i	$0.236003\pi$
$332$			9.94387i			0.545741i
$333$		0			0
$334$			2.55265i			0.139675i
$335$		−	11.2812i		−	0.616358i
$336$		0			0
$337$			9.70019i			0.528403i	0.964468	+	0.264201i	$0.0851084\pi$
$337$			9.70019i			0.528403i	−0.964468	+	0.264201i	$0.914892\pi$
$338$		−	10.7493i		−	0.584682i
$339$		0			0
$340$		−	69.6692i		−	3.77834i
$341$	−5.70717			−0.309060
$342$		0			0
$343$			22.3033i			1.20426i
$344$	14.8744			0.801975
$345$		0			0
$346$		−	38.7824i		−	2.08496i
$347$	−6.17647			−0.331570			−0.165785	−	0.986162i	$-0.553016\pi$
$347$	−6.17647			−0.331570			−0.165785	+	0.986162i	$0.553016\pi$
$348$		0			0
$349$	12.1263			0.649106			0.324553	−	0.945867i	$-0.394786\pi$
$349$	12.1263			0.649106			0.324553	+	0.945867i	$0.394786\pi$
$350$	−117.215			−6.26538
$351$		0			0
$352$			19.5727i			1.04323i
$353$	5.10423			0.271671			0.135835	−	0.990731i	$-0.456628\pi$
$353$	5.10423			0.271671			0.135835	+	0.990731i	$0.456628\pi$
$354$		0			0
$355$			59.6713i			3.16703i
$356$	−28.7945			−1.52610
$357$		0			0
$358$	13.8905			0.734136
$359$			19.7492i			1.04232i	0.853458	+	0.521161i	$0.174501\pi$
$359$			19.7492i			1.04232i	−0.853458	+	0.521161i	$0.825499\pi$
$360$		0			0
$361$	−18.9455			−0.997131
$362$	5.03362			0.264561
$363$		0			0
$364$			32.1530i			1.68527i
$365$	13.7595			0.720208
$366$		0			0
$367$		−	21.3765i		−	1.11584i	−0.829894	−	0.557921i	$-0.811599\pi$
$367$		−	21.3765i		−	1.11584i	0.829894	−	0.557921i	$-0.188401\pi$
$368$		−	9.60105i		−	0.500490i
$369$		0			0
$370$	−63.7365			−3.31350
$371$	−49.4673			−2.56821
$372$		0			0
$373$			0.449226i			0.0232600i	0.999932	+	0.0116300i	$0.00370203\pi$
$373$			0.449226i			0.0232600i	−0.999932	+	0.0116300i	$0.996298\pi$
$374$			34.4374i			1.78072i
$375$		0			0
$376$		−	5.19751i		−	0.268041i
$377$			14.0184i			0.721986i
$378$		0			0
$379$			5.10286i			0.262117i	0.991375	+	0.131058i	$0.0418375\pi$
$379$			5.10286i			0.262117i	−0.991375	+	0.131058i	$0.958163\pi$
$380$		−	2.54193i		−	0.130398i
$381$		0			0
$382$	34.1916			1.74939
$383$		−	28.4601i		−	1.45424i	−0.686509	−	0.727121i	$-0.740858\pi$
$383$		−	28.4601i		−	1.45424i	0.686509	−	0.727121i	$-0.259142\pi$
$384$		0			0
$385$	45.8669			2.33759
$386$		−	11.1213i		−	0.566059i
$387$		0			0
$388$			6.83060i			0.346771i
$389$	2.88222			0.146134			0.0730672	−	0.997327i	$-0.476721\pi$
$389$	2.88222			0.146134			0.0730672	+	0.997327i	$0.476721\pi$
$390$		0			0
$391$		−	25.2655i		−	1.27773i
$392$			15.6471i			0.790296i
$393$		0			0
$394$			3.69617i			0.186210i
$395$	−8.29698			−0.417466
$396$		0			0
$397$			13.1892i			0.661947i	0.943640	+	0.330974i	$0.107377\pi$
$397$			13.1892i			0.661947i	−0.943640	+	0.330974i	$0.892623\pi$
$398$		−	6.45861i		−	0.323741i
$399$		0			0
$400$	30.3967			1.51983
$401$		−	3.31822i		−	0.165704i	−0.996562	−	0.0828520i	$-0.973597\pi$
$401$		−	3.31822i		−	0.165704i	0.996562	−	0.0828520i	$-0.0264029\pi$
$402$		0			0
$403$	6.43071			0.320336
$404$	−10.8012			−0.537382
$405$		0			0
$406$			46.5040i			2.30795i
$407$	17.8149			0.883052
$408$		0			0
$409$			21.5264i			1.06441i	0.846615	+	0.532206i	$0.178637\pi$
$409$			21.5264i			1.06441i	−0.846615	+	0.532206i	$0.821363\pi$
$410$			45.2683i			2.23564i
$411$		0			0
$412$			4.82335i			0.237629i
$413$	−52.1316			−2.56523
$414$		0			0
$415$	−15.9835			−0.784599
$416$		−	22.0541i		−	1.08129i
$417$		0			0
$418$			1.25647i			0.0614562i
$419$	−27.8656			−1.36132			−0.680662	−	0.732597i	$-0.738308\pi$
$419$	−27.8656			−1.36132			−0.680662	+	0.732597i	$0.738308\pi$
$420$		0			0
$421$			11.5865i			0.564690i	0.959313	+	0.282345i	$0.0911123\pi$
$421$			11.5865i			0.564690i	−0.959313	+	0.282345i	$0.908888\pi$
$422$	53.2388			2.59162
$423$		0			0
$424$	−14.6322			−0.710604
$425$	79.9898			3.88007
$426$		0			0
$427$	−24.3986			−1.18073
$428$		−	16.0619i		−	0.776380i
$429$		0			0
$430$			103.258i			4.97956i
$431$	27.8186			1.33998			0.669988	−	0.742372i	$-0.266299\pi$
$431$	27.8186			1.33998			0.669988	+	0.742372i	$0.266299\pi$
$432$		0			0
$433$			36.2528i			1.74220i	0.491106	+	0.871100i	$0.336593\pi$
$433$			36.2528i			1.74220i	−0.491106	+	0.871100i	$0.663407\pi$
$434$	21.3329			1.02401
$435$		0			0
$436$	15.8885			0.760919
$437$		−	0.921830i		−	0.0440971i
$438$		0			0
$439$			12.1305i			0.578955i	0.957185	+	0.289478i	$0.0934816\pi$
$439$			12.1305i			0.578955i	−0.957185	+	0.289478i	$0.906518\pi$
$440$	13.5673			0.646793
$441$		0			0
$442$		−	38.8033i		−	1.84569i
$443$		−	14.9540i		−	0.710488i	−0.934774	−	0.355244i	$-0.884398\pi$
$443$		−	14.9540i		−	0.710488i	0.934774	−	0.355244i	$-0.115602\pi$
$444$		0			0
$445$		−	46.2834i		−	2.19404i
$446$	20.7207			0.981152
$447$		0			0
$448$		−	51.9054i		−	2.45230i
$449$			25.0896i			1.18405i	0.805920	+	0.592025i	$0.201671\pi$
$449$			25.0896i			1.18405i	−0.805920	+	0.592025i	$0.798329\pi$
$450$		0			0
$451$		−	12.6529i		−	0.595800i
$452$	−52.3152			−2.46070
$453$		0			0
$454$	36.4103			1.70882
$455$	−51.6818			−2.42288
$456$		0			0
$457$	−20.9128			−0.978261			−0.489131	−	0.872211i	$-0.662686\pi$
$457$	−20.9128			−0.978261			−0.489131	+	0.872211i	$0.662686\pi$
$458$	35.7971			1.67269
$459$		0			0
$460$	−42.9891			−2.00438
$461$	3.06541			0.142770			0.0713852	−	0.997449i	$-0.477258\pi$
$461$	3.06541			0.142770			0.0713852	+	0.997449i	$0.477258\pi$
$462$		0			0
$463$		−	11.8540i		−	0.550903i	−0.961315	−	0.275451i	$-0.911173\pi$
$463$		−	11.8540i		−	0.550903i	0.961315	−	0.275451i	$-0.0888273\pi$
$464$		−	12.0596i		−	0.559854i
$465$		0			0
$466$		−	17.6151i		−	0.816004i
$467$	18.1090			0.837983			0.418992	−	0.907990i	$-0.362384\pi$
$467$	18.1090			0.837983			0.418992	+	0.907990i	$0.362384\pi$
$468$		0			0
$469$		−	11.7863i		−	0.544243i
$470$	36.0812			1.66430
$471$		0			0
$472$	−15.4203			−0.709779
$473$		−	28.8616i		−	1.32706i
$474$		0			0
$475$	2.91849			0.133909
$476$		−	72.7889i		−	3.33627i
$477$		0			0
$478$		−	41.0487i		−	1.87752i
$479$			32.7776i			1.49765i	0.662769	+	0.748824i	$0.269381\pi$
$479$			32.7776i			1.49765i	−0.662769	+	0.748824i	$0.730619\pi$
$480$		0			0
$481$	−20.0734			−0.915269
$482$	13.8528			0.630976
$483$		0			0
$484$	12.2509			0.556861
$485$	−10.9793			−0.498544
$486$		0			0
$487$	35.2211			1.59602			0.798010	−	0.602644i	$-0.205886\pi$
$487$	35.2211			1.59602			0.798010	+	0.602644i	$0.205886\pi$
$488$	−7.21702			−0.326699
$489$		0			0
$490$	−108.622			−4.90705
$491$			14.3095i			0.645780i	0.946437	+	0.322890i	$0.104654\pi$
$491$			14.3095i			0.645780i	−0.946437	+	0.322890i	$0.895346\pi$
$492$		0			0
$493$		−	31.7353i		−	1.42929i
$494$		−	1.41577i		−	0.0636984i
$495$		0			0
$496$	−5.53215			−0.248401
$497$			62.3433i			2.79648i
$498$		0			0
$499$			7.44211i			0.333154i	0.986028	+	0.166577i	$0.0532715\pi$
$499$			7.44211i			0.333154i	−0.986028	+	0.166577i	$0.946729\pi$
$500$		−	81.6638i		−	3.65212i
$501$		0			0
$502$	31.0721			1.38682
$503$			38.2057i			1.70351i	0.523943	+	0.851754i	$0.324461\pi$
$503$			38.2057i			1.70351i	−0.523943	+	0.851754i	$0.675539\pi$
$504$		0			0
$505$		−	17.3616i		−	0.772580i
$506$	21.2495			0.944654
$507$		0			0
$508$	28.5188			1.26532
$509$			30.3752i			1.34636i	0.739479	+	0.673180i	$0.235072\pi$
$509$			30.3752i			1.34636i	−0.739479	+	0.673180i	$0.764928\pi$
$510$		0			0
$511$	14.3757			0.635942
$512$			25.2599i			1.11634i
$513$		0			0
$514$			2.24211i			0.0988952i
$515$	−7.75291			−0.341634
$516$		0			0
$517$	−10.0850			−0.443538
$518$	−66.5905			−2.92582
$519$		0			0
$520$	−15.2873			−0.670391
$521$		−	4.19942i		−	0.183980i	−0.995760	−	0.0919899i	$-0.970677\pi$
$521$		−	4.19942i		−	0.183980i	0.995760	−	0.0919899i	$-0.0293228\pi$
$522$		0			0
$523$	7.09312			0.310160			0.155080	−	0.987902i	$-0.450436\pi$
$523$	7.09312			0.310160			0.155080	+	0.987902i	$0.450436\pi$
$524$			35.3851i			1.54581i
$525$		0			0
$526$		−	43.1358i		−	1.88081i
$527$	−14.5580			−0.634157
$528$		0			0
$529$	7.41003			0.322175
$530$		−	101.577i		−	4.41223i
$531$		0			0
$532$		−	2.65576i		−	0.115142i
$533$			14.2570i			0.617538i
$534$		0			0
$535$	25.8174			1.11618
$536$		−	3.48636i		−	0.150588i
$537$		0			0
$538$	−20.1493			−0.868699
$539$	30.3608			1.30773
$540$		0			0
$541$			23.7651i			1.02174i	0.859657	+	0.510872i	$0.170677\pi$
$541$			23.7651i			1.02174i	−0.859657	+	0.510872i	$0.829323\pi$
$542$	11.1140			0.477386
$543$		0			0
$544$			49.9267i			2.14059i
$545$			25.5386i			1.09396i
$546$		0			0
$547$	−21.4055			−0.915233			−0.457616	−	0.889150i	$-0.651297\pi$
$547$	−21.4055			−0.915233			−0.457616	+	0.889150i	$0.651297\pi$
$548$			52.2412i			2.23163i
$549$		0			0
$550$			67.2752i			2.86862i
$551$		−	1.15789i		−	0.0493276i
$552$		0			0
$553$	−8.66850			−0.368622
$554$		−	18.5797i		−	0.789374i
$555$		0			0
$556$			19.6905i			0.835063i
$557$	−0.447534			−0.0189626			−0.00948131	−	0.999955i	$-0.503018\pi$
$557$	−0.447534			−0.0189626			−0.00948131	+	0.999955i	$0.503018\pi$
$558$		0			0
$559$			32.5206i			1.37547i
$560$	44.4603			1.87879
$561$		0			0
$562$			36.6248i			1.54492i
$563$			14.6706i			0.618291i	0.951015	+	0.309146i	$0.100043\pi$
$563$			14.6706i			0.618291i	−0.951015	+	0.309146i	$0.899957\pi$
$564$		0			0
$565$		−	84.0899i		−	3.53769i
$566$		−	13.8494i		−	0.582133i
$567$		0			0
$568$			18.4409i			0.773764i
$569$		−	20.3893i		−	0.854764i	−0.904071	−	0.427382i	$-0.859436\pi$
$569$		−	20.3893i		−	0.854764i	0.904071	−	0.427382i	$-0.140564\pi$
$570$		0			0
$571$	16.6685			0.697555			0.348778	−	0.937206i	$-0.386597\pi$
$571$	16.6685			0.697555			0.348778	+	0.937206i	$0.386597\pi$
$572$	18.4542			0.771608
$573$		0			0
$574$			47.2953i			1.97407i
$575$		−	49.3574i		−	2.05835i
$576$		0			0
$577$	27.7508			1.15528			0.577640	−	0.816291i	$-0.303974\pi$
$577$	27.7508			1.15528			0.577640	+	0.816291i	$0.303974\pi$
$578$			51.3728i			2.13682i
$579$		0			0
$580$	−53.9975			−2.24213
$581$	−16.6992			−0.692799
$582$		0			0
$583$			28.3917i			1.17586i
$584$	4.25227			0.175960
$585$		0			0
$586$	3.40702			0.140743
$587$			19.7450i			0.814963i	0.913213	+	0.407482i	$0.133593\pi$
$587$			19.7450i			0.814963i	−0.913213	+	0.407482i	$0.866407\pi$
$588$		0			0
$589$	−0.531160			−0.0218861
$590$		−	107.048i		−	4.40711i
$591$		0			0
$592$	17.2686			0.709734
$593$	−31.8141			−1.30645			−0.653224	−	0.757165i	$-0.726584\pi$
$593$	−31.8141			−1.30645			−0.653224	+	0.757165i	$0.726584\pi$
$594$		0			0
$595$	116.999			4.79648
$596$		−	27.0924i		−	1.10975i
$597$		0			0
$598$	−23.9434			−0.979120
$599$		−	6.89452i		−	0.281702i	−0.990031	−	0.140851i	$-0.955016\pi$
$599$		−	6.89452i		−	0.281702i	0.990031	−	0.140851i	$-0.0449839\pi$
$600$		0			0
$601$	−33.8318			−1.38003			−0.690014	−	0.723796i	$-0.742396\pi$
$601$	−33.8318			−1.38003			−0.690014	+	0.723796i	$0.742396\pi$
$602$			107.882i			4.39695i
$603$		0			0
$604$			3.86232i			0.157156i
$605$			19.6918i			0.800586i
$606$		0			0
$607$			0.893978i			0.0362854i	0.999835	+	0.0181427i	$0.00577532\pi$
$607$			0.893978i			0.0362854i	−0.999835	+	0.0181427i	$0.994225\pi$
$608$			1.82161i			0.0738762i
$609$		0			0
$610$		−	50.1007i		−	2.02852i
$611$	11.3635			0.459720
$612$		0			0
$613$			18.7214i			0.756150i	0.925775	+	0.378075i	$0.123414\pi$
$613$			18.7214i			0.756150i	−0.925775	+	0.378075i	$0.876586\pi$
$614$	43.2145			1.74400
$615$		0			0
$616$	14.1748			0.571117
$617$	−37.0201			−1.49037			−0.745187	−	0.666855i	$-0.767640\pi$
$617$	−37.0201			−1.49037			−0.745187	+	0.666855i	$0.767640\pi$
$618$		0			0
$619$	28.2634			1.13600			0.568000	−	0.823029i	$-0.307717\pi$
$619$	28.2634			1.13600			0.568000	+	0.823029i	$0.307717\pi$
$620$			24.7704i			0.994804i
$621$		0			0
$622$			21.8022i			0.874189i
$623$		−	48.3559i		−	1.93734i
$624$		0			0
$625$	68.7612			2.75045
$626$		−	0.346854i		−	0.0138631i
$627$		0			0
$628$	6.60981	−	31.9338i	0.263760	−	1.27430i
$629$	45.4428			1.81192
$630$		0			0
$631$	34.4834			1.37276			0.686381	−	0.727242i	$-0.259198\pi$
$631$	34.4834			1.37276			0.686381	+	0.727242i	$0.259198\pi$
$632$	−2.56411			−0.101995
$633$		0			0
$634$		−	56.8131i		−	2.25634i
$635$			45.8402i			1.81911i
$636$		0			0
$637$	−34.2099			−1.35544
$638$	26.6909			1.05670
$639$		0			0
$640$	41.3032			1.63265
$641$	−14.4323			−0.570043			−0.285022	−	0.958521i	$-0.592001\pi$
$641$	−14.4323			−0.570043			−0.285022	+	0.958521i	$0.592001\pi$
$642$		0			0
$643$		−	27.9415i		−	1.10190i	−0.834537	−	0.550952i	$-0.814265\pi$
$643$		−	27.9415i		−	1.10190i	0.834537	−	0.550952i	$-0.185735\pi$
$644$	−44.9141			−1.76986
$645$		0			0
$646$			3.20506i			0.126101i
$647$	15.1986			0.597519			0.298760	−	0.954328i	$-0.403427\pi$
$647$	15.1986			0.597519			0.298760	+	0.954328i	$0.403427\pi$
$648$		0			0
$649$			29.9209i			1.17450i
$650$		−	75.8042i		−	2.97328i
$651$		0			0
$652$			15.1886i			0.594831i
$653$	48.2888			1.88969			0.944843	−	0.327524i	$-0.106214\pi$
$653$	48.2888			1.88969			0.944843	+	0.327524i	$0.106214\pi$
$654$		0			0
$655$	−56.8770			−2.22237
$656$		−	12.2648i		−	0.478862i
$657$		0			0
$658$	37.6968			1.46958
$659$	40.7990			1.58930			0.794651	−	0.607066i	$-0.207654\pi$
$659$	40.7990			1.58930			0.794651	+	0.607066i	$0.207654\pi$
$660$		0			0
$661$	−39.0880			−1.52035			−0.760173	−	0.649721i	$-0.774886\pi$
$661$	−39.0880			−1.52035			−0.760173	+	0.649721i	$0.774886\pi$
$662$			57.5721i			2.23760i
$663$		0			0
$664$	−4.93956			−0.191692
$665$	4.26878			0.165536
$666$		0			0
$667$	−19.5822			−0.758224
$668$	−3.09670			−0.119815
$669$		0			0
$670$	24.2024			0.935018
$671$			14.0036i			0.540601i
$672$		0			0
$673$		−	49.3944i		−	1.90402i	−0.306074	−	0.952008i	$-0.599016\pi$
$673$		−	49.3944i		−	1.90402i	0.306074	−	0.952008i	$-0.400984\pi$
$674$	−20.8105			−0.801591
$675$		0			0
$676$	13.0403			0.501548
$677$	−48.6523			−1.86986			−0.934930	−	0.354831i	$-0.884538\pi$
$677$	−48.6523			−1.86986			−0.934930	+	0.354831i	$0.884538\pi$
$678$		0			0
$679$	−11.4709			−0.440214
$680$	34.6078			1.32715
$681$		0			0
$682$		−	12.2440i		−	0.468846i
$683$		−	20.5725i		−	0.787185i	−0.919285	−	0.393593i	$-0.871232\pi$
$683$		−	20.5725i		−	0.787185i	0.919285	−	0.393593i	$-0.128768\pi$
$684$		0			0
$685$	−83.9710			−3.20837
$686$	−47.8488			−1.82688
$687$		0			0
$688$		−	27.9765i		−	1.06659i
$689$		−	31.9911i		−	1.21876i
$690$		0			0
$691$			20.7136i			0.787982i	0.919114	+	0.393991i	$0.128906\pi$
$691$			20.7136i			0.787982i	−0.919114	+	0.393991i	$0.871094\pi$
$692$	47.0481			1.78850
$693$		0			0
$694$		−	13.2508i		−	0.502994i
$695$	−31.6499			−1.20055
$696$		0			0
$697$		−	32.2753i		−	1.22252i
$698$			26.0154i			0.984698i
$699$		0			0
$700$		−	142.197i		−	5.37453i
$701$			13.5127i			0.510368i	0.966893	+	0.255184i	$0.0821360\pi$
$701$			13.5127i			0.510368i	−0.966893	+	0.255184i	$0.917864\pi$
$702$		0			0
$703$	1.65801			0.0625332
$704$	−29.7911			−1.12279
$705$		0			0
$706$			10.9505i			0.412126i
$707$		−	18.1390i		−	0.682187i
$708$		0			0
$709$	28.6697			1.07671			0.538356	−	0.842718i	$-0.319046\pi$
$709$	28.6697			1.07671			0.538356	+	0.842718i	$0.319046\pi$
$710$	−128.017			−4.80440
$711$		0			0
$712$		−	14.3035i		−	0.536046i
$713$			8.98297i			0.336415i
$714$		0			0
$715$			29.6627i			1.10932i
$716$			16.8510i			0.629751i
$717$		0			0
$718$	−42.3693			−1.58121
$719$		−	37.8635i		−	1.41207i	−0.708177	−	0.706034i	$-0.750482\pi$
$719$		−	37.8635i		−	1.41207i	0.708177	−	0.706034i	$-0.249518\pi$
$720$		0			0
$721$	−8.10006			−0.301662
$722$		−	40.6451i		−	1.51265i
$723$		0			0
$724$			6.10644i			0.226944i
$725$		−	61.9965i		−	2.30249i
$726$		0			0
$727$	−1.08434			−0.0402161			−0.0201080	−	0.999798i	$-0.506401\pi$
$727$	−1.08434			−0.0402161			−0.0201080	+	0.999798i	$0.506401\pi$
$728$	−15.9718			−0.591955
$729$		0			0
$730$			29.5193i			1.09256i
$731$		−	73.6211i		−	2.72297i
$732$		0			0
$733$	−7.89261			−0.291520			−0.145760	−	0.989320i	$-0.546563\pi$
$733$	−7.89261			−0.291520			−0.145760	+	0.989320i	$0.546563\pi$
$734$	45.8604			1.69274
$735$		0			0
$736$	30.8071			1.13557
$737$	−6.76476			−0.249183
$738$		0			0
$739$	14.8833			0.547493			0.273746	−	0.961802i	$-0.411737\pi$
$739$	14.8833			0.547493			0.273746	+	0.961802i	$0.411737\pi$
$740$		−	77.3208i		−	2.84237i
$741$		0			0
$742$		−	106.126i		−	3.89599i
$743$	−44.5130			−1.63303			−0.816513	−	0.577328i	$-0.804096\pi$
$743$	−44.5130			−1.63303			−0.816513	+	0.577328i	$0.804096\pi$
$744$		0			0
$745$	43.5476			1.59546
$746$	−0.963755			−0.0352856
$747$		0			0
$748$	−41.7771			−1.52752
$749$	26.9734			0.985588
$750$		0			0
$751$		−	44.4622i		−	1.62245i	−0.584734	−	0.811225i	$-0.698801\pi$
$751$		−	44.4622i		−	1.62245i	0.584734	−	0.811225i	$-0.301199\pi$
$752$	−9.77572			−0.356484
$753$		0			0
$754$	−30.0747			−1.09526
$755$	−6.20818			−0.225939
$756$		0			0
$757$			25.1291i			0.913334i	0.889638	+	0.456667i	$0.150957\pi$
$757$			25.1291i			0.913334i	−0.889638	+	0.456667i	$0.849043\pi$
$758$	−10.9475			−0.397632
$759$		0			0
$760$	1.26269			0.0458026
$761$			30.7813i			1.11582i	0.829901	+	0.557911i	$0.188397\pi$
$761$			30.7813i			1.11582i	−0.829901	+	0.557911i	$0.811603\pi$
$762$		0			0
$763$			26.6822i			0.965961i
$764$			41.4789i			1.50065i
$765$		0			0
$766$	61.0574			2.20609
$767$		−	33.7142i		−	1.21735i
$768$		0			0
$769$	25.8839			0.933396			0.466698	−	0.884417i	$-0.345443\pi$
$769$	25.8839			0.933396			0.466698	+	0.884417i	$0.345443\pi$
$770$			98.4014i			3.54614i
$771$		0			0
$772$	13.4916			0.485573
$773$	46.4528			1.67079			0.835396	−	0.549648i	$-0.185238\pi$
$773$	46.4528			1.67079			0.835396	+	0.549648i	$0.185238\pi$
$774$		0			0
$775$	−28.4398			−1.02159
$776$	−3.39306			−0.121804
$777$		0			0
$778$			6.18343i			0.221687i
$779$		−	1.17759i		−	0.0421915i
$780$		0			0
$781$	35.7819			1.28038
$782$	54.2038			1.93833
$783$		0			0
$784$	29.4297			1.05106
$785$	51.3294	+	10.6244i	1.83203	+	0.379202i
$786$		0			0
$787$		−	34.0895i		−	1.21516i	−0.794258	−	0.607580i	$-0.792140\pi$
$787$		−	34.0895i		−	1.21516i	0.794258	−	0.607580i	$-0.207860\pi$
$788$	−4.48394			−0.159734
$789$		0			0
$790$		−	17.8001i		−	0.633299i
$791$		−	87.8553i		−	3.12377i
$792$		0			0
$793$		−	15.7789i		−	0.560325i
$794$	−28.2957			−1.00418
$795$		0			0
$796$	7.83514			0.277709
$797$	20.7326			0.734385			0.367193	−	0.930145i	$-0.380319\pi$
$797$	20.7326			0.734385			0.367193	+	0.930145i	$0.380319\pi$
$798$		0			0
$799$	−25.7251			−0.910090
$800$			97.5344i			3.44836i
$801$		0			0
$802$	7.11881			0.251374
$803$		−	8.25090i		−	0.291168i
$804$		0			0
$805$		−	72.1936i		−	2.54449i
$806$			13.7963i			0.485952i
$807$		0			0
$808$		−	5.36545i		−	0.188756i
$809$		−	20.3924i		−	0.716960i	−0.933537	−	0.358480i	$-0.883295\pi$
$809$		−	20.3924i		−	0.716960i	0.933537	−	0.358480i	$-0.116705\pi$
$810$		0			0
$811$		−	23.3885i		−	0.821282i	−0.911797	−	0.410641i	$-0.865305\pi$
$811$		−	23.3885i		−	0.821282i	0.911797	−	0.410641i	$-0.134695\pi$
$812$	−56.4154			−1.97979
$813$		0			0
$814$			38.2196i			1.33959i
$815$	−24.4137			−0.855174
$816$		0			0
$817$		−	2.68612i		−	0.0939755i
$818$	−46.1821			−1.61472
$819$		0			0
$820$	−54.9163			−1.91776
$821$	19.1379			0.667919			0.333959	−	0.942587i	$-0.391615\pi$
$821$	19.1379			0.667919			0.333959	+	0.942587i	$0.391615\pi$
$822$		0			0
$823$		−	27.1706i		−	0.947107i	−0.880765	−	0.473554i	$-0.842971\pi$
$823$		−	27.1706i		−	0.947107i	0.880765	−	0.473554i	$-0.157029\pi$
$824$	−2.39597			−0.0834676
$825$		0			0
$826$		−	111.842i		−	3.89147i
$827$	21.9330			0.762687			0.381343	−	0.924433i	$-0.375462\pi$
$827$	21.9330			0.762687			0.381343	+	0.924433i	$0.375462\pi$
$828$		0			0
$829$	47.8383			1.66149			0.830746	−	0.556652i	$-0.187914\pi$
$829$	47.8383			1.66149			0.830746	+	0.556652i	$0.187914\pi$
$830$		−	34.2905i		−	1.19024i
$831$		0			0
$832$	33.5679			1.16376
$833$	77.4453			2.68332
$834$		0			0
$835$		−	4.97754i		−	0.172255i
$836$	−1.52427			−0.0527179
$837$		0			0
$838$		−	59.7821i		−	2.06514i
$839$		−	9.67209i		−	0.333918i	−0.985964	−	0.166959i	$-0.946605\pi$
$839$		−	9.67209i		−	0.333918i	0.985964	−	0.166959i	$-0.0533947\pi$
$840$		0			0
$841$	4.40337			0.151840
$842$	−24.8573			−0.856639
$843$		0			0
$844$			64.5856i			2.22313i
$845$			20.9605i			0.721064i
$846$		0			0
$847$			20.5736i			0.706916i
$848$			27.5210i			0.945075i
$849$		0			0
$850$			171.608i			5.88610i
$851$		−	28.0403i		−	0.961209i
$852$		0			0
$853$	−29.5930			−1.01325			−0.506623	−	0.862168i	$-0.669106\pi$
$853$	−29.5930			−1.01325			−0.506623	+	0.862168i	$0.669106\pi$
$854$		−	52.3441i		−	1.79118i
$855$		0			0
$856$	7.97864			0.272704
$857$		−	1.93475i		−	0.0660896i	−0.999454	−	0.0330448i	$-0.989480\pi$
$857$		−	1.93475i		−	0.0660896i	0.999454	−	0.0330448i	$-0.0105204\pi$
$858$		0			0
$859$		−	19.4660i		−	0.664172i	−0.943249	−	0.332086i	$-0.892248\pi$
$859$		−	19.4660i		−	0.664172i	0.943249	−	0.332086i	$-0.107752\pi$
$860$	−125.266			−4.27154
$861$		0			0
$862$			59.6813i			2.03275i
$863$			48.1603i			1.63940i	0.572796	+	0.819698i	$0.305859\pi$
$863$			48.1603i			1.63940i	−0.572796	+	0.819698i	$0.694141\pi$
$864$		0			0
$865$			75.6238i			2.57129i
$866$	−77.7757			−2.64293
$867$		0			0
$868$			25.8796i			0.878410i
$869$			4.97528i			0.168775i
$870$		0			0
$871$	7.62238			0.258275
$872$			7.89250i			0.267274i
$873$		0			0
$874$	1.97767			0.0668956
$875$	137.142			4.63624
$876$		0			0
$877$			21.8441i			0.737623i	0.929504	+	0.368812i	$0.120235\pi$
$877$			21.8441i			0.737623i	−0.929504	+	0.368812i	$0.879765\pi$
$878$	−26.0243			−0.878279
$879$		0			0
$880$		−	25.5179i		−	0.860209i
$881$		−	36.5250i		−	1.23056i	−0.788309	−	0.615279i	$-0.789043\pi$
$881$		−	36.5250i		−	1.23056i	0.788309	−	0.615279i	$-0.210957\pi$
$882$		0			0
$883$			20.6965i			0.696494i	0.937403	+	0.348247i	$0.113223\pi$
$883$			20.6965i			0.696494i	−0.937403	+	0.348247i	$0.886777\pi$
$884$	47.0735			1.58325
$885$		0			0
$886$	32.0820			1.07781
$887$			24.7047i			0.829501i	0.909935	+	0.414751i	$0.136131\pi$
$887$			24.7047i			0.829501i	−0.909935	+	0.414751i	$0.863869\pi$
$888$		0			0
$889$			47.8929i			1.60628i
$890$	99.2950			3.32838
$891$		0			0
$892$			25.1369i			0.841645i
$893$	−0.938601			−0.0314091
$894$		0			0
$895$	−27.0858			−0.905378
$896$	43.1526			1.44163
$897$		0			0
$898$	−53.8264			−1.79621
$899$			11.2833i			0.376318i
$900$		0			0
$901$			72.4224i			2.41274i
$902$	27.1451			0.903832
$903$		0			0
$904$		−	25.9873i		−	0.864324i
$905$	−9.81531			−0.326272
$906$		0			0
$907$	−9.33125			−0.309839			−0.154920	−	0.987927i	$-0.549512\pi$
$907$	−9.33125			−0.309839			−0.154920	+	0.987927i	$0.549512\pi$
$908$			44.1704i			1.46585i
$909$		0			0
$910$		−	110.877i		−	3.67552i
$911$	3.16690			0.104924			0.0524620	−	0.998623i	$-0.483293\pi$
$911$	3.16690			0.104924			0.0524620	+	0.998623i	$0.483293\pi$
$912$		0			0
$913$			9.58448i			0.317200i
$914$		−	44.8658i		−	1.48403i
$915$		0			0
$916$			43.4265i			1.43485i
$917$	−59.4238			−1.96235
$918$		0			0
$919$		−	36.3838i		−	1.20019i	−0.799929	−	0.600095i	$-0.795129\pi$
$919$		−	36.3838i		−	1.20019i	0.799929	−	0.600095i	$-0.204871\pi$
$920$		−	21.3546i		−	0.704040i
$921$		0			0
$922$			6.57644i			0.216584i
$923$	−40.3182			−1.32709
$924$		0			0
$925$	88.7748			2.91890
$926$	25.4313			0.835723
$927$		0			0
$928$	38.6960			1.27026
$929$	−42.4516			−1.39279			−0.696396	−	0.717658i	$-0.745214\pi$
$929$	−42.4516			−1.39279			−0.696396	+	0.717658i	$0.745214\pi$
$930$		0			0
$931$	2.82565			0.0926070
$932$	21.3694			0.699979
$933$		0			0
$934$			38.8504i			1.27123i
$935$		−	67.1513i		−	2.19608i
$936$		0			0
$937$			57.6428i			1.88311i	0.336863	+	0.941554i	$0.390634\pi$
$937$			57.6428i			1.88311i	−0.336863	+	0.941554i	$0.609366\pi$
$938$	25.2861			0.825620
$939$		0			0
$940$			43.7712i			1.42766i
$941$	37.6453			1.22720			0.613600	−	0.789617i	$-0.289721\pi$
$941$	37.6453			1.22720			0.613600	+	0.789617i	$0.289721\pi$
$942$		0			0
$943$	−19.9154			−0.648534
$944$			29.0033i			0.943978i
$945$		0			0
$946$	61.9188			2.01315
$947$			21.4641i			0.697488i	0.937218	+	0.348744i	$0.113392\pi$
$947$			21.4641i			0.697488i	−0.937218	+	0.348744i	$0.886608\pi$
$948$		0			0
$949$			9.29693i			0.301791i
$950$			6.26124i			0.203141i
$951$		0			0
$952$	36.1574			1.17187
$953$	22.2678			0.721323			0.360662	−	0.932697i	$-0.382551\pi$
$953$	22.2678			0.721323			0.360662	+	0.932697i	$0.382551\pi$
$954$		0			0
$955$	−66.6719			−2.15745
$956$	49.7974			1.61056
$957$		0			0
$958$	−70.3202			−2.27194
$959$	−87.7310			−2.83298
$960$		0			0
$961$	−25.8240			−0.833032
$962$		−	43.0649i		−	1.38847i
$963$		0			0
$964$			16.8052i			0.541259i
$965$			21.6860i			0.698097i
$966$		0			0
$967$	9.70998			0.312252			0.156126	−	0.987737i	$-0.450099\pi$
$967$	9.70998			0.312252			0.156126	+	0.987737i	$0.450099\pi$
$968$			6.08558i			0.195598i
$969$		0			0
$970$		−	23.5547i		−	0.756295i
$971$			3.69839i			0.118687i	0.998238	+	0.0593435i	$0.0189007\pi$
$971$			3.69839i			0.118687i	−0.998238	+	0.0593435i	$0.981099\pi$
$972$		0			0
$973$	−33.0671			−1.06008
$974$			75.5623i			2.42117i
$975$		0			0
$976$			13.5741i			0.434497i
$977$	17.4945			0.559698			0.279849	−	0.960044i	$-0.409716\pi$
$977$	17.4945			0.559698			0.279849	+	0.960044i	$0.409716\pi$
$978$		0			0
$979$	−27.7538			−0.887015
$980$		−	131.773i		−	4.20933i
$981$		0			0
$982$	−30.6992			−0.979652
$983$		−	23.1172i		−	0.737324i	−0.929564	−	0.368662i	$-0.879816\pi$
$983$		−	23.1172i		−	0.737324i	0.929564	−	0.368662i	$-0.120184\pi$
$984$		0			0
$985$		−	7.20735i		−	0.229645i
$986$	68.0840			2.16824
$987$		0			0
$988$	1.71751			0.0546413
$989$	−45.4276			−1.44451
$990$		0			0
$991$	−28.6482			−0.910040			−0.455020	−	0.890481i	$-0.650368\pi$
$991$	−28.6482			−0.910040			−0.455020	+	0.890481i	$0.650368\pi$
$992$		−	17.7511i		−	0.563598i
$993$		0			0
$994$	−133.749			−4.24228
$995$			12.5940i			0.399256i
$996$		0			0
$997$			0.774548i			0.0245302i	0.999925	+	0.0122651i	$0.00390420\pi$
$997$			0.774548i			0.0245302i	−0.999925	+	0.0122651i	$0.996096\pi$
$998$	−15.9661			−0.505397
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1413.2.b.e.784.12		14
3.2	odd	2		471.2.b.b.313.3	✓	14
157.156	even	2	inner	1413.2.b.e.784.3		14
471.470	odd	2		471.2.b.b.313.12	yes	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
471.2.b.b.313.3	✓	14	3.2	odd	2
471.2.b.b.313.12	yes	14	471.470	odd	2
1413.2.b.e.784.3		14	157.156	even	2	inner
1413.2.b.e.784.12		14	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1413 = 3^{2} \cdot 157 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1413.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.14537i q^{2} -2.60262 q^{4} -4.18337i q^{5} -4.37069i q^{7} -1.29283i q^{8} +O(q^{10})\)	\(q+2.14537i q^{2} -2.60262 q^{4} -4.18337i q^{5} -4.37069i q^{7} -1.29283i q^{8} +8.97487 q^{10} -2.50855 q^{11} +2.82658 q^{13} +9.37675 q^{14} -2.43162 q^{16} -6.39890 q^{17} -0.233469 q^{19} +10.8877i q^{20} -5.38177i q^{22} +3.94841i q^{23} -12.5006 q^{25} +6.06406i q^{26} +11.3752i q^{28} +4.95950i q^{29} +2.27508 q^{31} -7.80240i q^{32} -13.7280i q^{34} -18.2842 q^{35} -7.10166 q^{37} -0.500876i q^{38} -5.40840 q^{40} +5.04389i q^{41} +11.5053i q^{43} +6.52880 q^{44} -8.47081 q^{46} +4.02024 q^{47} -12.1029 q^{49} -26.8183i q^{50} -7.35650 q^{52} -11.3180i q^{53} +10.4942i q^{55} -5.65058 q^{56} -10.6400 q^{58} -11.9276i q^{59} -5.58233i q^{61} +4.88090i q^{62} +11.8758 q^{64} -11.8246i q^{65} +2.69668 q^{67} +16.6539 q^{68} -39.2264i q^{70} -14.2640 q^{71} +3.28911i q^{73} -15.2357i q^{74} +0.607629 q^{76} +10.9641i q^{77} -1.98333i q^{79} +10.1724i q^{80} -10.8210 q^{82} -3.82072i q^{83} +26.7689i q^{85} -24.6831 q^{86} +3.24314i q^{88} +11.0637 q^{89} -12.3541i q^{91} -10.2762i q^{92} +8.62492i q^{94} +0.976684i q^{95} -2.62451i q^{97} -25.9653i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 20 q^{4}+O(q^{10}) \) 14 * q - 20 * q^4	\( 14 q - 20 q^{4} + 6 q^{10} + 2 q^{11} + 24 q^{16} - 18 q^{17} - 12 q^{19} - 18 q^{25} - 14 q^{31} - 16 q^{35} - 14 q^{37} - 36 q^{40} - 24 q^{44} - 8 q^{46} - 22 q^{47} - 48 q^{49} - 50 q^{52} + 62 q^{56} + 20 q^{58} - 34 q^{64} + 42 q^{67} + 56 q^{68} - 38 q^{71} + 52 q^{76} + 10 q^{82} - 34 q^{86} + 48 q^{89}+O(q^{100}) \) 14 * q - 20 * q^4 + 6 * q^10 + 2 * q^11 + 24 * q^16 - 18 * q^17 - 12 * q^19 - 18 * q^25 - 14 * q^31 - 16 * q^35 - 14 * q^37 - 36 * q^40 - 24 * q^44 - 8 * q^46 - 22 * q^47 - 48 * q^49 - 50 * q^52 + 62 * q^56 + 20 * q^58 - 34 * q^64 + 42 * q^67 + 56 * q^68 - 38 * q^71 + 52 * q^76 + 10 * q^82 - 34 * q^86 + 48 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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