LMFDB - Embedded newform 810.3.b.a.809.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [810,3,Mod(809,810)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(810, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("810.809");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 810 = 2 \cdot 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	810.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:	$22.0709014132$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\Q(\zeta_{24})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 90)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			809.7
Root			$-0.258819 + 0.965926i$ of defining polynomial
Character	$\chi$	$=$	810.809
Dual form			810.3.b.a.809.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421 q^{2} +2.00000 q^{4} +(4.87832 - 1.09638i) q^{5} -2.10102i q^{7} +2.82843 q^{8} +O(q^{10})$	\(q+1.41421 q^{2} +2.00000 q^{4} +(4.87832 - 1.09638i) q^{5} -2.10102i q^{7} +2.82843 q^{8} +(6.89898 - 1.55051i) q^{10} -13.3636i q^{11} -23.1464i q^{13} -2.97129i q^{14} +4.00000 q^{16} -27.5057 q^{17} -15.5505 q^{19} +(9.75663 - 2.19275i) q^{20} -18.8990i q^{22} +14.9528 q^{23} +(22.5959 - 10.6969i) q^{25} -32.7340i q^{26} -4.20204i q^{28} +3.28913i q^{29} -9.84337 q^{31} +5.65685 q^{32} -38.8990 q^{34} +(-2.30351 - 10.2494i) q^{35} +33.3485i q^{37} -21.9917 q^{38} +(13.7980 - 3.10102i) q^{40} -10.8530i q^{41} +43.1918i q^{43} -26.7272i q^{44} +21.1464 q^{46} +36.0873 q^{47} +44.5857 q^{49} +(31.9555 - 15.1278i) q^{50} -46.2929i q^{52} +53.1687 q^{53} +(-14.6515 - 65.1918i) q^{55} -5.94258i q^{56} +4.65153i q^{58} -101.252i q^{59} +0.101021 q^{61} -13.9206 q^{62} +8.00000 q^{64} +(-25.3772 - 112.916i) q^{65} +55.9898i q^{67} -55.0115 q^{68} +(-3.25765 - 14.4949i) q^{70} +79.2457i q^{71} +20.8990i q^{73} +47.1619i q^{74} -31.1010 q^{76} -28.0772 q^{77} -1.48469 q^{79} +(19.5133 - 4.38551i) q^{80} -15.3485i q^{82} +9.93160 q^{83} +(-134.182 + 30.1566i) q^{85} +61.0825i q^{86} -37.7980i q^{88} -152.974i q^{89} -48.6311 q^{91} +29.9056 q^{92} +51.0352 q^{94} +(-75.8603 + 17.0492i) q^{95} +38.7878i q^{97} +63.0537 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 16 q^{4}+O(q^{10}) $ 8 * q + 16 * q^4	$ 8 q + 16 q^{4} + 16 q^{10} + 32 q^{16} - 144 q^{19} + 24 q^{25} + 176 q^{31} - 272 q^{34} + 32 q^{40} + 32 q^{46} - 192 q^{49} - 176 q^{55} + 40 q^{61} + 64 q^{64} - 320 q^{70} - 288 q^{76} + 576 q^{79} - 368 q^{85} + 336 q^{91} - 160 q^{94}+O(q^{100}) $ 8 * q + 16 * q^4 + 16 * q^10 + 32 * q^16 - 144 * q^19 + 24 * q^25 + 176 * q^31 - 272 * q^34 + 32 * q^40 + 32 * q^46 - 192 * q^49 - 176 * q^55 + 40 * q^61 + 64 * q^64 - 320 * q^70 - 288 * q^76 + 576 * q^79 - 368 * q^85 + 336 * q^91 - 160 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/810\mathbb{Z}\right)^\times$.

$n$	$487$	$731$
$\chi(n)$	$-1$	$-1$

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 $

$2$	1.41421			0.707107
$2$	1.41421			0.707107
$3$		0			0
$3$		0			0
$4$	2.00000			0.500000
$5$	4.87832	−	1.09638i	0.975663	−	0.219275i
$5$	4.87832	−	1.09638i	0.975663	−	0.219275i
$6$		0			0
$7$		−	2.10102i		−	0.300146i	−0.988675	−	0.150073i	$-0.952049\pi$
$7$		−	2.10102i		−	0.300146i	0.988675	−	0.150073i	$-0.0479508\pi$
$8$	2.82843			0.353553
$9$		0			0
$10$	6.89898	−	1.55051i	0.689898	−	0.155051i
$11$		−	13.3636i		−	1.21487i	−0.794368	−	0.607436i	$-0.792198\pi$
$11$		−	13.3636i		−	1.21487i	0.794368	−	0.607436i	$-0.207802\pi$
$12$		0			0
$13$		−	23.1464i		−	1.78049i	−0.455478	−	0.890247i	$-0.650531\pi$
$13$		−	23.1464i		−	1.78049i	0.455478	−	0.890247i	$-0.349469\pi$
$14$		−	2.97129i		−	0.212235i
$15$		0			0
$16$	4.00000			0.250000
$17$	−27.5057			−1.61798			−0.808992	−	0.587819i	$-0.799987\pi$
$17$	−27.5057			−1.61798			−0.808992	+	0.587819i	$0.799987\pi$
$18$		0			0
$19$	−15.5505			−0.818448			−0.409224	−	0.912434i	$-0.634201\pi$
$19$	−15.5505			−0.818448			−0.409224	+	0.912434i	$0.634201\pi$
$20$	9.75663	−	2.19275i	0.487832	−	0.109638i
$21$		0			0
$22$		−	18.8990i		−	0.859045i
$23$	14.9528			0.650121			0.325060	−	0.945693i	$-0.394615\pi$
$23$	14.9528			0.650121			0.325060	+	0.945693i	$0.394615\pi$
$24$		0			0
$25$	22.5959	−	10.6969i	0.903837	−	0.427878i
$26$		−	32.7340i		−	1.25900i
$27$		0			0
$28$		−	4.20204i		−	0.150073i
$29$			3.28913i			0.113418i	0.998391	+	0.0567091i	$0.0180608\pi$
$29$			3.28913i			0.113418i	−0.998391	+	0.0567091i	$0.981939\pi$
$30$		0			0
$31$	−9.84337			−0.317528			−0.158764	−	0.987317i	$-0.550751\pi$
$31$	−9.84337			−0.317528			−0.158764	+	0.987317i	$0.550751\pi$
$32$	5.65685			0.176777
$33$		0			0
$34$	−38.8990			−1.14409
$35$	−2.30351	−	10.2494i	−0.0658145	−	0.292841i
$36$		0			0
$37$			33.3485i			0.901310i	0.892698	+	0.450655i	$0.148810\pi$
$37$			33.3485i			0.901310i	−0.892698	+	0.450655i	$0.851190\pi$
$38$	−21.9917			−0.578730
$39$		0			0
$40$	13.7980	−	3.10102i	0.344949	−	0.0775255i
$41$		−	10.8530i		−	0.264707i	−0.991203	−	0.132354i	$-0.957747\pi$
$41$		−	10.8530i		−	0.264707i	0.991203	−	0.132354i	$-0.0422535\pi$
$42$		0			0
$43$			43.1918i			1.00446i	0.864734	+	0.502231i	$0.167487\pi$
$43$			43.1918i			1.00446i	−0.864734	+	0.502231i	$0.832513\pi$
$44$		−	26.7272i		−	0.607436i
$45$		0			0
$46$	21.1464			0.459705
$47$	36.0873			0.767816			0.383908	−	0.923371i	$-0.374578\pi$
$47$	36.0873			0.767816			0.383908	+	0.923371i	$0.374578\pi$
$48$		0			0
$49$	44.5857			0.909913
$50$	31.9555	−	15.1278i	0.639109	−	0.302555i
$51$		0			0
$52$		−	46.2929i		−	0.890247i
$53$	53.1687			1.00318			0.501591	−	0.865105i	$-0.332748\pi$
$53$	53.1687			1.00318			0.501591	+	0.865105i	$0.332748\pi$
$54$		0			0
$55$	−14.6515	−	65.1918i	−0.266391	−	1.18531i
$56$		−	5.94258i		−	0.106118i
$57$		0			0
$58$			4.65153i			0.0801988i
$59$		−	101.252i		−	1.71613i	−0.513538	−	0.858067i	$-0.671665\pi$
$59$		−	101.252i		−	1.71613i	0.513538	−	0.858067i	$-0.328335\pi$
$60$		0			0
$61$	0.101021			0.00165607			0.000828037	−	1.00000i	$-0.499736\pi$
$61$	0.101021			0.00165607			0.000828037		1.00000i	$0.499736\pi$
$62$	−13.9206			−0.224526
$63$		0			0
$64$	8.00000			0.125000
$65$	−25.3772	−	112.916i	−0.390418	−	1.73716i
$66$		0			0
$67$			55.9898i			0.835669i	0.908523	+	0.417834i	$0.137211\pi$
$67$			55.9898i			0.835669i	−0.908523	+	0.417834i	$0.862789\pi$
$68$	−55.0115			−0.808992
$69$		0			0
$70$	−3.25765	−	14.4949i	−0.0465379	−	0.207070i
$71$			79.2457i			1.11614i	0.829795	+	0.558069i	$0.188457\pi$
$71$			79.2457i			1.11614i	−0.829795	+	0.558069i	$0.811543\pi$
$72$		0			0
$73$			20.8990i			0.286287i	0.989702	+	0.143144i	$0.0457211\pi$
$73$			20.8990i			0.286287i	−0.989702	+	0.143144i	$0.954279\pi$
$74$			47.1619i			0.637322i
$75$		0			0
$76$	−31.1010			−0.409224
$77$	−28.0772			−0.364639
$78$		0			0
$79$	−1.48469			−0.0187936			−0.00939679	−	0.999956i	$-0.502991\pi$
$79$	−1.48469			−0.0187936			−0.00939679	+	0.999956i	$0.502991\pi$
$80$	19.5133	−	4.38551i	0.243916	−	0.0548188i
$81$		0			0
$82$		−	15.3485i		−	0.187176i
$83$	9.93160			0.119658			0.0598289	−	0.998209i	$-0.480944\pi$
$83$	9.93160			0.119658			0.0598289	+	0.998209i	$0.480944\pi$
$84$		0			0
$85$	−134.182	+	30.1566i	−1.57861	+	0.354784i
$86$			61.0825i			0.710261i
$87$		0			0
$88$		−	37.7980i		−	0.429522i
$89$		−	152.974i		−	1.71881i	−0.511294	−	0.859406i	$-0.670834\pi$
$89$		−	152.974i		−	1.71881i	0.511294	−	0.859406i	$-0.329166\pi$
$90$		0			0
$91$	−48.6311			−0.534408
$92$	29.9056			0.325060
$93$		0			0
$94$	51.0352			0.542928
$95$	−75.8603	+	17.0492i	−0.798529	+	0.179465i
$96$		0			0
$97$			38.7878i			0.399874i	0.979809	+	0.199937i	$0.0640737\pi$
$97$			38.7878i			0.399874i	−0.979809	+	0.199937i	$0.935926\pi$
$98$	63.0537			0.643405
$99$		0			0
$100$	45.1918	−	21.3939i	0.451918	−	0.213939i
$101$		−	81.9602i		−	0.811487i	−0.913987	−	0.405743i	$-0.867013\pi$
$101$		−	81.9602i		−	0.811487i	0.913987	−	0.405743i	$-0.132987\pi$
$102$		0			0
$103$			115.778i			1.12405i	0.827119	+	0.562027i	$0.189978\pi$
$103$			115.778i			1.12405i	−0.827119	+	0.562027i	$0.810022\pi$
$104$		−	65.4680i		−	0.629500i
$105$		0			0
$106$	75.1918			0.709357
$107$	119.512			1.11693			0.558465	−	0.829528i	$-0.311391\pi$
$107$	119.512			1.11693			0.558465	+	0.829528i	$0.311391\pi$
$108$		0			0
$109$	62.9092			0.577148			0.288574	−	0.957458i	$-0.406819\pi$
$109$	62.9092			0.577148			0.288574	+	0.957458i	$0.406819\pi$
$110$	−20.7204	−	92.1952i	−0.188367	−	0.838138i
$111$		0			0
$112$		−	8.40408i		−	0.0750364i
$113$	80.0243			0.708179			0.354090	−	0.935211i	$-0.384791\pi$
$113$	80.0243			0.708179			0.354090	+	0.935211i	$0.384791\pi$
$114$		0			0
$115$	72.9444	−	16.3939i	0.634299	−	0.142555i
$116$			6.57826i			0.0567091i
$117$		0			0
$118$		−	143.192i		−	1.21349i
$119$			57.7901i			0.485631i
$120$		0			0
$121$	−57.5857			−0.475915
$122$	0.142865			0.00117102
$123$		0			0
$124$	−19.6867			−0.158764
$125$	98.5021	−	76.9567i	0.788017	−	0.615653i
$126$		0			0
$127$			122.687i			0.966037i	0.875610	+	0.483019i	$0.160460\pi$
$127$			122.687i			0.966037i	−0.875610	+	0.483019i	$0.839540\pi$
$128$	11.3137			0.0883883
$129$		0			0
$130$	−35.8888	−	159.687i	−0.276067	−	1.22836i
$131$		−	22.3417i		−	0.170547i	−0.996358	−	0.0852736i	$-0.972824\pi$
$131$		−	22.3417i		−	0.170547i	0.996358	−	0.0852736i	$-0.0271764\pi$
$132$		0			0
$133$			32.6719i			0.245654i
$134$			79.1815i			0.590907i
$135$		0			0
$136$	−77.7980			−0.572044
$137$	−0.857187			−0.00625684			−0.00312842	−	0.999995i	$-0.500996\pi$
$137$	−0.857187			−0.00625684			−0.00312842	+	0.999995i	$0.500996\pi$
$138$		0			0
$139$	235.283			1.69268			0.846340	−	0.532642i	$-0.178801\pi$
$139$	235.283			1.69268			0.846340	+	0.532642i	$0.178801\pi$
$140$	−4.60702	−	20.4989i	−0.0329073	−	0.146421i
$141$		0			0
$142$			112.070i			0.789228i
$143$	−309.320			−2.16307
$144$		0			0
$145$	3.60612	+	16.0454i	0.0248698	+	0.110658i
$146$			29.5556i			0.202436i
$147$		0			0
$148$			66.6969i			0.450655i
$149$		−	131.204i		−	0.880564i	−0.897860	−	0.440282i	$-0.854878\pi$
$149$		−	131.204i		−	0.880564i	0.897860	−	0.440282i	$-0.145122\pi$
$150$		0			0
$151$	−88.4495			−0.585758			−0.292879	−	0.956149i	$-0.594613\pi$
$151$	−88.4495			−0.585758			−0.292879	+	0.956149i	$0.594613\pi$
$152$	−43.9835			−0.289365
$153$		0			0
$154$	−39.7071			−0.257839
$155$	−48.0190	+	10.7920i	−0.309800	+	0.0696260i
$156$		0			0
$157$			162.697i			1.03629i	0.855294	+	0.518143i	$0.173377\pi$
$157$			162.697i			1.03629i	−0.855294	+	0.518143i	$0.826623\pi$
$158$	−2.09967			−0.0132891
$159$		0			0
$160$	27.5959	−	6.20204i	0.172474	−	0.0387628i
$161$		−	31.4161i		−	0.195131i
$162$		0			0
$163$			275.192i			1.68829i	0.536112	+	0.844147i	$0.319892\pi$
$163$			275.192i			1.68829i	−0.536112	+	0.844147i	$0.680108\pi$
$164$		−	21.7060i		−	0.132354i
$165$		0			0
$166$	14.0454			0.0846109
$167$	136.989			0.820295			0.410148	−	0.912019i	$-0.365477\pi$
$167$	136.989			0.820295			0.410148	+	0.912019i	$0.365477\pi$
$168$		0			0
$169$	−366.757			−2.17016
$170$	−189.761	+	42.6479i	−1.11624	+	0.250870i
$171$		0			0
$172$			86.3837i			0.502231i
$173$	92.7666			0.536223			0.268112	−	0.963388i	$-0.413600\pi$
$173$	92.7666			0.536223			0.268112	+	0.963388i	$0.413600\pi$
$174$		0			0
$175$	−22.4745	−	47.4745i	−0.128426	−	0.271283i
$176$		−	53.4544i		−	0.303718i
$177$		0			0
$178$		−	216.338i		−	1.21538i
$179$		−	103.273i		−	0.576944i	−0.957488	−	0.288472i	$-0.906853\pi$
$179$		−	103.273i		−	0.576944i	0.957488	−	0.288472i	$-0.0931472\pi$
$180$		0			0
$181$	215.050			1.18812			0.594061	−	0.804420i	$-0.297524\pi$
$181$	215.050			1.18812			0.594061	+	0.804420i	$0.297524\pi$
$182$	−68.7748			−0.377883
$183$		0			0
$184$	42.2929			0.229852
$185$	36.5625	+	162.684i	0.197635	+	0.879375i
$186$		0			0
$187$			367.576i			1.96564i
$188$	72.1747			0.383908
$189$		0			0
$190$	−107.283	+	24.1112i	−0.564646	+	0.126901i
$191$			272.693i			1.42771i	0.700293	+	0.713856i	$0.253053\pi$
$191$			272.693i			1.42771i	−0.700293	+	0.713856i	$0.746947\pi$
$192$		0			0
$193$		−	321.621i		−	1.66643i	−0.552949	−	0.833215i	$-0.686498\pi$
$193$		−	321.621i		−	1.66643i	0.552949	−	0.833215i	$-0.313502\pi$
$194$			54.8542i			0.282753i
$195$		0			0
$196$	89.1714			0.454956
$197$	37.6267			0.190999			0.0954993	−	0.995429i	$-0.469555\pi$
$197$	37.6267			0.190999			0.0954993	+	0.995429i	$0.469555\pi$
$198$		0			0
$199$	−318.520			−1.60060			−0.800301	−	0.599598i	$-0.795327\pi$
$199$	−318.520			−1.60060			−0.800301	+	0.599598i	$0.795327\pi$
$200$	63.9109	−	30.2555i	0.319555	−	0.151278i
$201$		0			0
$202$		−	115.909i		−	0.573808i
$203$	6.91053			0.0340420
$204$		0			0
$205$	−11.8990	−	52.9444i	−0.0580438	−	0.258265i
$206$			163.734i			0.794826i
$207$		0			0
$208$		−	92.5857i		−	0.445124i
$209$			207.811i			0.994310i
$210$		0			0
$211$	5.75255			0.0272633			0.0136316	−	0.999907i	$-0.495661\pi$
$211$	5.75255			0.0272633			0.0136316	+	0.999907i	$0.495661\pi$
$212$	106.337			0.501591
$213$		0			0
$214$	169.015			0.789789
$215$	47.3545	+	210.703i	0.220254	+	0.980016i
$216$		0			0
$217$			20.6811i			0.0953047i
$218$	88.9670			0.408106
$219$		0			0
$220$	−29.3031	−	130.384i	−0.133196	−	0.592653i
$221$			636.659i			2.88081i
$222$		0			0
$223$			23.8990i			0.107170i	0.998563	+	0.0535852i	$0.0170649\pi$
$223$			23.8990i			0.107170i	−0.998563	+	0.0535852i	$0.982935\pi$
$224$		−	11.8852i		−	0.0530588i
$225$		0			0
$226$	113.171			0.500759
$227$	−212.931			−0.938024			−0.469012	−	0.883192i	$-0.655390\pi$
$227$	−212.931			−0.938024			−0.469012	+	0.883192i	$0.655390\pi$
$228$		0			0
$229$	−337.474			−1.47369			−0.736844	−	0.676063i	$-0.763685\pi$
$229$	−337.474			−1.47369			−0.736844	+	0.676063i	$0.763685\pi$
$230$	103.159	−	23.1844i	0.448517	−	0.100802i
$231$		0			0
$232$			9.30306i			0.0400994i
$233$	−198.996			−0.854062			−0.427031	−	0.904237i	$-0.640440\pi$
$233$	−198.996			−0.854062			−0.427031	+	0.904237i	$0.640440\pi$
$234$		0			0
$235$	176.045	−	39.5653i	0.749129	−	0.168363i
$236$		−	202.504i		−	0.858067i
$237$		0			0
$238$			81.7276i			0.343393i
$239$			76.1107i			0.318455i	0.987242	+	0.159227i	$0.0509003\pi$
$239$			76.1107i			0.318455i	−0.987242	+	0.159227i	$0.949100\pi$
$240$		0			0
$241$	326.757			1.35584			0.677919	−	0.735136i	$-0.262882\pi$
$241$	326.757			1.35584			0.677919	+	0.735136i	$0.262882\pi$
$242$	−81.4385			−0.336523
$243$		0			0
$244$	0.202041			0.000828037
$245$	217.503	−	48.8827i	0.887768	−	0.199521i
$246$		0			0
$247$			359.939i			1.45724i
$248$	−27.8412			−0.112263
$249$		0			0
$250$	139.303	−	108.833i	0.557212	−	0.435333i
$251$			289.663i			1.15404i	0.816731	+	0.577019i	$0.195784\pi$
$251$			289.663i			1.15404i	−0.816731	+	0.577019i	$0.804216\pi$
$252$		0			0
$253$		−	199.823i		−	0.789814i
$254$			173.505i			0.683092i
$255$		0			0
$256$	16.0000			0.0625000
$257$	−460.533			−1.79196			−0.895978	−	0.444098i	$-0.853524\pi$
$257$	−460.533			−1.79196			−0.895978	+	0.444098i	$0.853524\pi$
$258$		0			0
$259$	70.0658			0.270524
$260$	−50.7544	−	225.831i	−0.195209	−	0.868581i
$261$		0			0
$262$		−	31.5959i		−	0.120595i
$263$	−211.053			−0.802484			−0.401242	−	0.915972i	$-0.631421\pi$
$263$	−211.053			−0.802484			−0.401242	+	0.915972i	$0.631421\pi$
$264$		0			0
$265$	259.373	−	58.2929i	0.978768	−	0.219973i
$266$			46.2051i			0.173703i
$267$		0			0
$268$			111.980i			0.417834i
$269$			187.201i			0.695915i	0.937510	+	0.347957i	$0.113125\pi$
$269$			187.201i			0.695915i	−0.937510	+	0.347957i	$0.886875\pi$
$270$		0			0
$271$	−199.914			−0.737689			−0.368845	−	0.929491i	$-0.620247\pi$
$271$	−199.914			−0.737689			−0.368845	+	0.929491i	$0.620247\pi$
$272$	−110.023			−0.404496
$273$		0			0
$274$	−1.21225			−0.00442426
$275$	−142.950	−	301.963i	−0.519817	−	1.09805i
$276$		0			0
$277$		−	107.187i		−	0.386958i	−0.981104	−	0.193479i	$-0.938023\pi$
$277$		−	107.187i		−	0.386958i	0.981104	−	0.193479i	$-0.0619771\pi$
$278$	332.740			1.19691
$279$		0			0
$280$	−6.51531	−	28.9898i	−0.0232690	−	0.103535i
$281$			330.786i			1.17718i	0.808433	+	0.588588i	$0.200316\pi$
$281$			330.786i			1.17718i	−0.808433	+	0.588588i	$0.799684\pi$
$282$		0			0
$283$		−	316.576i		−	1.11864i	−0.828951	−	0.559321i	$-0.811062\pi$
$283$		−	316.576i		−	1.11864i	0.828951	−	0.559321i	$-0.188938\pi$
$284$			158.491i			0.558069i
$285$		0			0
$286$	−437.444			−1.52952
$287$	−22.8024			−0.0794508
$288$		0			0
$289$	467.565			1.61787
$290$	5.09983	+	22.6916i	0.0175856	+	0.0782470i
$291$		0			0
$292$			41.7980i			0.143144i
$293$	85.0022			0.290110			0.145055	−	0.989424i	$-0.453664\pi$
$293$	85.0022			0.290110			0.145055	+	0.989424i	$0.453664\pi$
$294$		0			0
$295$	−111.010	−	493.939i	−0.376306	−	1.67437i
$296$			94.3237i			0.318661i
$297$		0			0
$298$		−	185.551i		−	0.622653i
$299$		−	346.104i		−	1.15754i
$300$		0			0
$301$	90.7469			0.301485
$302$	−125.086			−0.414194
$303$		0			0
$304$	−62.2020			−0.204612
$305$	0.492810	−	0.110756i	0.00161577	−	0.000363136i
$306$		0			0
$307$			34.5959i			0.112690i	0.998411	+	0.0563451i	$0.0179447\pi$
$307$			34.5959i			0.112690i	−0.998411	+	0.0563451i	$0.982055\pi$
$308$	−56.1544			−0.182319
$309$		0			0
$310$	−67.9092	+	15.2622i	−0.219062	+	0.0492330i
$311$		−	42.3478i		−	0.136166i	−0.997680	−	0.0680832i	$-0.978312\pi$
$311$		−	42.3478i		−	0.136166i	0.997680	−	0.0680832i	$-0.0216883\pi$
$312$		0			0
$313$		−	366.924i		−	1.17228i	−0.810209	−	0.586141i	$-0.800647\pi$
$313$		−	366.924i		−	1.17228i	0.810209	−	0.586141i	$-0.199353\pi$
$314$			230.088i			0.732765i
$315$		0			0
$316$	−2.96938			−0.00939679
$317$	−87.1307			−0.274860			−0.137430	−	0.990511i	$-0.543884\pi$
$317$	−87.1307			−0.274860			−0.137430	+	0.990511i	$0.543884\pi$
$318$		0			0
$319$	43.9546			0.137789
$320$	39.0265	−	8.77101i	0.121958	−	0.0274094i
$321$		0			0
$322$		−	44.4291i		−	0.137979i
$323$	427.728			1.32424
$324$		0			0
$325$	−247.596	−	523.015i	−0.761834	−	1.60928i
$326$			389.180i			1.19380i
$327$		0			0
$328$		−	30.6969i		−	0.0935882i
$329$		−	75.8202i		−	0.230457i
$330$		0			0
$331$	−127.333			−0.384691			−0.192345	−	0.981327i	$-0.561609\pi$
$331$	−127.333			−0.384691			−0.192345	+	0.981327i	$0.561609\pi$
$332$	19.8632			0.0598289
$333$		0			0
$334$	193.732			0.580036
$335$	61.3859	+	273.136i	0.183241	+	0.815331i
$336$		0			0
$337$			428.595i			1.27179i	0.771774	+	0.635897i	$0.219370\pi$
$337$			428.595i			1.27179i	−0.771774	+	0.635897i	$0.780630\pi$
$338$	−518.673			−1.53454
$339$		0			0
$340$	−268.363	+	60.3133i	−0.789304	+	0.177392i
$341$			131.543i			0.385756i
$342$		0			0
$343$		−	196.626i		−	0.573252i
$344$			122.165i			0.355131i
$345$		0			0
$346$	131.192			0.379167
$347$	350.503			1.01010			0.505048	−	0.863091i	$-0.331475\pi$
$347$	350.503			1.01010			0.505048	+	0.863091i	$0.331475\pi$
$348$		0			0
$349$	603.170			1.72828			0.864141	−	0.503250i	$-0.167862\pi$
$349$	603.170			1.72828			0.864141	+	0.503250i	$0.167862\pi$
$350$	−31.7837	−	67.1391i	−0.0908106	−	0.191826i
$351$		0			0
$352$		−	75.5959i		−	0.214761i
$353$	−7.81426			−0.0221367			−0.0110684	−	0.999939i	$-0.503523\pi$
$353$	−7.81426			−0.0221367			−0.0110684	+	0.999939i	$0.503523\pi$
$354$		0			0
$355$	86.8832	+	386.586i	0.244741	+	1.08897i
$356$		−	305.949i		−	0.859406i
$357$		0			0
$358$		−	146.050i		−	0.407961i
$359$			159.113i			0.443211i	0.975136	+	0.221605i	$0.0711297\pi$
$359$			159.113i			0.443211i	−0.975136	+	0.221605i	$0.928870\pi$
$360$		0			0
$361$	−119.182			−0.330143
$362$	304.127			0.840129
$363$		0			0
$364$	−97.2622			−0.267204
$365$	22.9131	+	101.952i	0.0627757	+	0.279320i
$366$		0			0
$367$			563.292i			1.53486i	0.641136	+	0.767428i	$0.278464\pi$
$367$			563.292i			1.53486i	−0.641136	+	0.767428i	$0.721536\pi$
$368$	59.8111			0.162530
$369$		0			0
$370$	51.7071	+	230.070i	0.139749	+	0.621812i
$371$		−	111.708i		−	0.301101i
$372$		0			0
$373$		−	469.312i		−	1.25821i	−0.777320	−	0.629105i	$-0.783422\pi$
$373$		−	469.312i		−	1.25821i	0.777320	−	0.629105i	$-0.216578\pi$
$374$			519.830i			1.38992i
$375$		0			0
$376$	102.070			0.271464
$377$	76.1316			0.201941
$378$		0			0
$379$	−339.101			−0.894726			−0.447363	−	0.894353i	$-0.647637\pi$
$379$	−339.101			−0.894726			−0.447363	+	0.894353i	$0.647637\pi$
$380$	−151.721	+	34.0984i	−0.399265	+	0.0897327i
$381$		0			0
$382$			385.646i			1.00954i
$383$	5.62150			0.0146776			0.00733878	−	0.999973i	$-0.497664\pi$
$383$	5.62150			0.0146776			0.00733878	+	0.999973i	$0.497664\pi$
$384$		0			0
$385$	−136.969	+	30.7832i	−0.355765	+	0.0799563i
$386$		−	454.841i		−	1.17834i
$387$		0			0
$388$			77.5755i			0.199937i
$389$		−	635.621i		−	1.63399i	−0.576647	−	0.816993i	$-0.695639\pi$
$389$		−	635.621i		−	1.63399i	0.576647	−	0.816993i	$-0.304361\pi$
$390$		0			0
$391$	−411.287			−1.05189
$392$	126.107			0.321703
$393$		0			0
$394$	53.2122			0.135056
$395$	−7.24280	+	1.62778i	−0.0183362	+	0.00412097i
$396$		0			0
$397$			626.656i			1.57848i	0.614086	+	0.789239i	$0.289525\pi$
$397$			626.656i			1.57848i	−0.614086	+	0.789239i	$0.710475\pi$
$398$	−450.455			−1.13180
$399$		0			0
$400$	90.3837	−	42.7878i	0.225959	−	0.106969i
$401$		−	459.826i		−	1.14670i	−0.819311	−	0.573350i	$-0.805644\pi$
$401$		−	459.826i		−	1.14670i	0.819311	−	0.573350i	$-0.194356\pi$
$402$		0			0
$403$			227.839i			0.565357i
$404$		−	163.920i		−	0.405743i
$405$		0			0
$406$	9.77296			0.0240713
$407$	445.655			1.09498
$408$		0			0
$409$	−59.9592			−0.146599			−0.0732997	−	0.997310i	$-0.523353\pi$
$409$	−59.9592			−0.146599			−0.0732997	+	0.997310i	$0.523353\pi$
$410$	−16.8277	−	74.8747i	−0.0410432	−	0.182621i
$411$		0			0
$412$			231.555i			0.562027i
$413$	−212.732			−0.515090
$414$		0			0
$415$	48.4495	−	10.8888i	0.116746	−	0.0262380i
$416$		−	130.936i		−	0.314750i
$417$		0			0
$418$			293.889i			0.703083i
$419$			221.438i			0.528491i	0.964455	+	0.264245i	$0.0851229\pi$
$419$			221.438i			0.528491i	−0.964455	+	0.264245i	$0.914877\pi$
$420$		0			0
$421$	−133.171			−0.316322			−0.158161	−	0.987413i	$-0.550556\pi$
$421$	−133.171			−0.316322			−0.158161	+	0.987413i	$0.550556\pi$
$422$	8.13534			0.0192780
$423$		0			0
$424$	150.384			0.354678
$425$	−621.517	+	294.227i	−1.46239	+	0.692299i
$426$		0			0
$427$		−	0.212246i		−	0.000497064i
$428$	239.023			0.558465
$429$		0			0
$430$	66.9694	+	297.980i	0.155743	+	0.692976i
$431$		−	458.090i		−	1.06285i	−0.847104	−	0.531427i	$-0.821656\pi$
$431$		−	458.090i		−	1.06285i	0.847104	−	0.531427i	$-0.178344\pi$
$432$		0			0
$433$		−	240.318i		−	0.555007i	−0.960725	−	0.277503i	$-0.910493\pi$
$433$		−	240.318i		−	0.555007i	0.960725	−	0.277503i	$-0.0895069\pi$
$434$			29.2475i			0.0673906i
$435$		0			0
$436$	125.818			0.288574
$437$	−232.523			−0.532090
$438$		0			0
$439$	440.908			1.00435			0.502173	−	0.864767i	$-0.332534\pi$
$439$	440.908			1.00435			0.502173	+	0.864767i	$0.332534\pi$
$440$	−41.4408	−	184.390i	−0.0941836	−	0.419069i
$441$		0			0
$442$			900.372i			2.03704i
$443$	607.808			1.37203			0.686014	−	0.727588i	$-0.259359\pi$
$443$	607.808			1.37203			0.686014	+	0.727588i	$0.259359\pi$
$444$		0			0
$445$	−167.717	−	746.257i	−0.376893	−	1.67698i
$446$			33.7983i			0.0757809i
$447$		0			0
$448$		−	16.8082i		−	0.0375182i
$449$		−	56.1962i		−	0.125159i	−0.998040	−	0.0625793i	$-0.980067\pi$
$449$		−	56.1962i		−	0.125159i	0.998040	−	0.0625793i	$-0.0199326\pi$
$450$		0			0
$451$	−145.035			−0.321586
$452$	160.049			0.354090
$453$		0			0
$454$	−301.131			−0.663283
$455$	−237.238	+	53.3180i	−0.521402	+	0.117182i
$456$		0			0
$457$		−	213.975i		−	0.468217i	−0.972211	−	0.234108i	$-0.924783\pi$
$457$		−	213.975i		−	0.468217i	0.972211	−	0.234108i	$-0.0752170\pi$
$458$	−477.261			−1.04205
$459$		0			0
$460$	145.889	−	32.7878i	0.317150	−	0.0712777i
$461$		−	217.749i		−	0.472340i	−0.971712	−	0.236170i	$-0.924108\pi$
$461$		−	217.749i		−	0.472340i	0.971712	−	0.236170i	$-0.0758923\pi$
$462$		0			0
$463$			141.637i			0.305911i	0.988233	+	0.152955i	$0.0488791\pi$
$463$			141.637i			0.305911i	−0.988233	+	0.152955i	$0.951121\pi$
$464$			13.1565i			0.0283546i
$465$		0			0
$466$	−281.423			−0.603913
$467$	−254.279			−0.544495			−0.272248	−	0.962227i	$-0.587767\pi$
$467$	−254.279			−0.544495			−0.272248	+	0.962227i	$0.587767\pi$
$468$		0			0
$469$	117.636			0.250822
$470$	248.966	−	55.9538i	0.529714	−	0.119051i
$471$		0			0
$472$		−	286.384i		−	0.606745i
$473$	577.198			1.22029
$474$		0			0
$475$	−351.378	+	166.343i	−0.739743	+	0.350195i
$476$			115.580i			0.242816i
$477$		0			0
$478$			107.637i			0.225181i
$479$		−	853.370i		−	1.78157i	−0.454430	−	0.890783i	$-0.650157\pi$
$479$		−	853.370i		−	1.78157i	0.454430	−	0.890783i	$-0.349843\pi$
$480$		0			0
$481$	771.898			1.60478
$482$	462.104			0.958723
$483$		0			0
$484$	−115.171			−0.237957
$485$	42.5260	+	189.219i	0.0876824	+	0.390142i
$486$		0			0
$487$			504.261i			1.03544i	0.855549	+	0.517722i	$0.173220\pi$
$487$			504.261i			1.03544i	−0.855549	+	0.517722i	$0.826780\pi$
$488$	0.285729			0.000585511
$489$		0			0
$490$	307.596	−	69.1306i	0.627747	−	0.141083i
$491$		−	439.748i		−	0.895618i	−0.894129	−	0.447809i	$-0.852205\pi$
$491$		−	439.748i		−	0.895618i	0.894129	−	0.447809i	$-0.147795\pi$
$492$		0			0
$493$		−	90.4699i		−	0.183509i
$494$			509.030i			1.03043i
$495$		0			0
$496$	−39.3735			−0.0793820
$497$	166.497			0.335004
$498$		0			0
$499$	−807.614			−1.61847			−0.809233	−	0.587488i	$-0.800117\pi$
$499$	−807.614			−1.61847			−0.809233	+	0.587488i	$0.800117\pi$
$500$	197.004	−	153.913i	0.394009	−	0.307827i
$501$		0			0
$502$			409.646i			0.816028i
$503$	−526.962			−1.04764			−0.523819	−	0.851829i	$-0.675493\pi$
$503$	−526.962			−1.04764			−0.523819	+	0.851829i	$0.675493\pi$
$504$		0			0
$505$	−89.8592	−	399.828i	−0.177939	−	0.791738i
$506$		−	282.592i		−	0.558483i
$507$		0			0
$508$			245.373i			0.483019i
$509$		−	120.439i		−	0.236620i	−0.992977	−	0.118310i	$-0.962252\pi$
$509$		−	120.439i		−	0.236620i	0.992977	−	0.118310i	$-0.0377476\pi$
$510$		0			0
$511$	43.9092			0.0859280
$512$	22.6274			0.0441942
$513$		0			0
$514$	−651.292			−1.26710
$515$	126.936	+	564.799i	0.246477	+	1.09670i
$516$		0			0
$517$		−	482.257i		−	0.932798i
$518$	99.0880			0.191290
$519$		0			0
$520$	−71.7775	−	319.373i	−0.138034	−	0.614180i
$521$			34.7871i			0.0667699i	0.999443	+	0.0333850i	$0.0106287\pi$
$521$			34.7871i			0.0667699i	−0.999443	+	0.0333850i	$0.989371\pi$
$522$		0			0
$523$			210.535i			0.402552i	0.979535	+	0.201276i	$0.0645088\pi$
$523$			210.535i			0.402552i	−0.979535	+	0.201276i	$0.935491\pi$
$524$		−	44.6834i		−	0.0852736i
$525$		0			0
$526$	−298.474			−0.567442
$527$	270.749			0.513755
$528$		0			0
$529$	−305.414			−0.577343
$530$	366.809	−	82.4385i	0.692093	−	0.155544i
$531$		0			0
$532$			65.3439i			0.122827i
$533$	−251.208			−0.471310
$534$		0			0
$535$	583.015	−	131.030i	1.08975	−	0.244915i
$536$			158.363i			0.295453i
$537$		0			0
$538$			264.742i			0.492086i
$539$		−	595.825i		−	1.10543i
$540$		0			0
$541$	−123.070			−0.227487			−0.113743	−	0.993510i	$-0.536284\pi$
$541$	−123.070			−0.227487			−0.113743	+	0.993510i	$0.536284\pi$
$542$	−282.721			−0.521625
$543$		0			0
$544$	−155.596			−0.286022
$545$	306.891	−	68.9721i	0.563102	−	0.126554i
$546$		0			0
$547$			219.627i			0.401511i	0.979641	+	0.200756i	$0.0643397\pi$
$547$			219.627i			0.401511i	−0.979641	+	0.200756i	$0.935660\pi$
$548$	−1.71437			−0.00312842
$549$		0			0
$550$	−202.161	−	427.040i	−0.367566	−	0.776436i
$551$		−	51.1476i		−	0.0928269i
$552$		0			0
$553$			3.11937i			0.00564081i
$554$		−	151.586i		−	0.273620i
$555$		0			0
$556$	470.565			0.846340
$557$	−351.396			−0.630873			−0.315436	−	0.948947i	$-0.602151\pi$
$557$	−351.396			−0.630873			−0.315436	+	0.948947i	$0.602151\pi$
$558$		0			0
$559$	999.737			1.78844
$560$	−9.21404	−	40.9978i	−0.0164536	−	0.0732103i
$561$		0			0
$562$			467.803i			0.832389i
$563$	−338.321			−0.600926			−0.300463	−	0.953793i	$-0.597141\pi$
$563$	−338.321			−0.600926			−0.300463	+	0.953793i	$0.597141\pi$
$564$		0			0
$565$	390.384	−	87.7367i	0.690945	−	0.155286i
$566$		−	447.705i		−	0.790999i
$567$		0			0
$568$			224.141i			0.394614i
$569$			75.6741i			0.132995i	0.997787	+	0.0664975i	$0.0211824\pi$
$569$			75.6741i			0.132995i	−0.997787	+	0.0664975i	$0.978818\pi$
$570$		0			0
$571$	503.596			0.881954			0.440977	−	0.897518i	$-0.354632\pi$
$571$	503.596			0.881954			0.440977	+	0.897518i	$0.354632\pi$
$572$	−618.639			−1.08154
$573$		0			0
$574$	−32.2474			−0.0561802
$575$	337.872	−	159.949i	0.587603	−	0.278172i
$576$		0			0
$577$		−	665.090i		−	1.15267i	−0.817214	−	0.576334i	$-0.804483\pi$
$577$		−	665.090i		−	1.15267i	0.817214	−	0.576334i	$-0.195517\pi$
$578$	661.237			1.14401
$579$		0			0
$580$	7.21225	+	32.0908i	0.0124349	+	0.0553290i
$581$		−	20.8665i		−	0.0359148i
$582$		0			0
$583$		−	710.524i		−	1.21874i
$584$			59.1112i			0.101218i
$585$		0			0
$586$	120.211			0.205139
$587$	−460.672			−0.784791			−0.392396	−	0.919797i	$-0.628354\pi$
$587$	−460.672			−0.784791			−0.392396	+	0.919797i	$0.628354\pi$
$588$		0			0
$589$	153.069			0.259880
$590$	−156.992	−	698.535i	−0.266088	−	1.18396i
$591$		0			0
$592$			133.394i			0.225327i
$593$	585.162			0.986782			0.493391	−	0.869808i	$-0.335757\pi$
$593$	585.162			0.986782			0.493391	+	0.869808i	$0.335757\pi$
$594$		0			0
$595$	63.3597	+	281.918i	0.106487	+	0.473812i
$596$		−	262.408i		−	0.440282i
$597$		0			0
$598$		−	489.464i		−	0.818502i
$599$		−	2.79956i		−	0.00467373i	−0.999997	−	0.00233686i	$-0.999256\pi$
$599$		−	2.79956i		−	0.00467373i	0.999997	−	0.00233686i	$-0.000743847\pi$
$600$		0			0
$601$	−498.869			−0.830066			−0.415033	−	0.909806i	$-0.636230\pi$
$601$	−498.869			−0.830066			−0.415033	+	0.909806i	$0.636230\pi$
$602$	128.336			0.213182
$603$		0			0
$604$	−176.899			−0.292879
$605$	−280.921	+	63.1356i	−0.464333	+	0.104356i
$606$		0			0
$607$		−	30.3847i		−	0.0500572i	−0.999687	−	0.0250286i	$-0.992032\pi$
$607$		−	30.3847i		−	0.0500572i	0.999687	−	0.0250286i	$-0.00796768\pi$
$608$	−87.9670			−0.144683
$609$		0			0
$610$	0.696938	−	0.156633i	0.00114252	−	0.000256776i
$611$		−	835.293i		−	1.36709i
$612$		0			0
$613$		−	448.449i		−	0.731565i	−0.930700	−	0.365783i	$-0.880801\pi$
$613$		−	448.449i		−	0.731565i	0.930700	−	0.365783i	$-0.119199\pi$
$614$			48.9260i			0.0796841i
$615$		0			0
$616$	−79.4143			−0.128919
$617$	−170.733			−0.276715			−0.138357	−	0.990382i	$-0.544182\pi$
$617$	−170.733			−0.276715			−0.138357	+	0.990382i	$0.544182\pi$
$618$		0			0
$619$	238.247			0.384891			0.192445	−	0.981308i	$-0.438358\pi$
$619$	238.247			0.384891			0.192445	+	0.981308i	$0.438358\pi$
$620$	−96.0381	+	21.5841i	−0.154900	+	0.0348130i
$621$		0			0
$622$		−	59.8888i		−	0.0962842i
$623$	−321.402			−0.515894
$624$		0			0
$625$	396.151	−	483.414i	0.633842	−	0.773463i
$626$		−	518.909i		−	0.828928i
$627$		0			0
$628$			325.394i			0.518143i
$629$		−	917.274i		−	1.45831i
$630$		0			0
$631$	1148.15			1.81957			0.909786	−	0.415078i	$-0.136246\pi$
$631$	1148.15			1.81957			0.909786	+	0.415078i	$0.136246\pi$
$632$	−4.19934			−0.00664453
$633$		0			0
$634$	−123.221			−0.194356
$635$	134.511	+	598.505i	0.211828	+	0.942527i
$636$		0			0
$637$		−	1032.00i		−	1.62009i
$638$	62.1612			0.0974313
$639$		0			0
$640$	55.1918	−	12.4041i	0.0862372	−	0.0193814i
$641$			388.533i			0.606136i	0.952969	+	0.303068i	$0.0980109\pi$
$641$			388.533i			0.606136i	−0.952969	+	0.303068i	$0.901989\pi$
$642$		0			0
$643$			928.980i			1.44476i	0.691497	+	0.722379i	$0.256951\pi$
$643$			928.980i			1.44476i	−0.691497	+	0.722379i	$0.743049\pi$
$644$		−	62.8322i		−	0.0975655i
$645$		0			0
$646$	604.899			0.936376
$647$	485.464			0.750330			0.375165	−	0.926958i	$-0.377586\pi$
$647$	485.464			0.750330			0.375165	+	0.926958i	$0.377586\pi$
$648$		0			0
$649$	−1353.09			−2.08488
$650$	−350.154	−	739.655i	−0.538698	−	1.13793i
$651$		0			0
$652$			550.384i			0.844147i
$653$	33.9620			0.0520093			0.0260046	−	0.999662i	$-0.491722\pi$
$653$	33.9620			0.0520093			0.0260046	+	0.999662i	$0.491722\pi$
$654$		0			0
$655$	−24.4949	−	108.990i	−0.0373968	−	0.166397i
$656$		−	43.4120i		−	0.0661769i
$657$		0			0
$658$		−	107.226i		−	0.162957i
$659$			67.9818i			0.103159i	0.998669	+	0.0515795i	$0.0164256\pi$
$659$			67.9818i			0.103159i	−0.998669	+	0.0515795i	$0.983574\pi$
$660$		0			0
$661$	−883.959			−1.33731			−0.668653	−	0.743575i	$-0.733129\pi$
$661$	−883.959			−1.33731			−0.668653	+	0.743575i	$0.733129\pi$
$662$	−180.076			−0.272017
$663$		0			0
$664$	28.0908			0.0423054
$665$	35.8207	+	159.384i	0.0538658	+	0.239675i
$666$		0			0
$667$			49.1816i			0.0737356i
$668$	273.979			0.410148
$669$		0			0
$670$	86.8128	+	386.272i	0.129571	+	0.576526i
$671$		−	1.35000i		−	0.00201192i
$672$		0			0
$673$		−	217.448i		−	0.323103i	−0.986864	−	0.161552i	$-0.948350\pi$
$673$		−	217.448i		−	0.323103i	0.986864	−	0.161552i	$-0.0516498\pi$
$674$			606.125i			0.899295i
$675$		0			0
$676$	−733.514			−1.08508
$677$	926.345			1.36831			0.684155	−	0.729337i	$-0.260171\pi$
$677$	926.345			1.36831			0.684155	+	0.729337i	$0.260171\pi$
$678$		0			0
$679$	81.4939			0.120020
$680$	−379.523	+	85.2958i	−0.558122	+	0.125435i
$681$		0			0
$682$			186.030i			0.272771i
$683$	586.563			0.858804			0.429402	−	0.903113i	$-0.358724\pi$
$683$	586.563			0.858804			0.429402	+	0.903113i	$0.358724\pi$
$684$		0			0
$685$	−4.18163	+	0.939800i	−0.00610457	+	0.00137197i
$686$		−	278.070i		−	0.405351i
$687$		0			0
$688$			172.767i			0.251115i
$689$		−	1230.66i		−	1.78616i
$690$		0			0
$691$	−830.161			−1.20139			−0.600696	−	0.799478i	$-0.705110\pi$
$691$	−830.161			−1.20139			−0.600696	+	0.799478i	$0.705110\pi$
$692$	185.533			0.268112
$693$		0			0
$694$	495.687			0.714246
$695$	1147.78	−	257.958i	1.65149	−	0.371163i
$696$		0			0
$697$			298.520i			0.428293i
$698$	853.012			1.22208
$699$		0			0
$700$	−44.9490	−	94.9490i	−0.0642128	−	0.135641i
$701$			915.199i			1.30556i	0.757547	+	0.652781i	$0.226398\pi$
$701$			915.199i			1.30556i	−0.757547	+	0.652781i	$0.773602\pi$
$702$		0			0
$703$		−	518.586i		−	0.737675i
$704$		−	106.909i		−	0.151859i
$705$		0			0
$706$	−11.0510			−0.0156530
$707$	−172.200			−0.243564
$708$		0			0
$709$	−18.8480			−0.0265839			−0.0132919	−	0.999912i	$-0.504231\pi$
$709$	−18.8480			−0.0265839			−0.0132919	+	0.999912i	$0.504231\pi$
$710$	122.871	+	546.715i	0.173058	+	0.770021i
$711$		0			0
$712$		−	432.677i		−	0.607692i
$713$	−147.186			−0.206432
$714$		0			0
$715$	−1508.96	+	339.131i	−2.11043	+	0.474309i
$716$		−	206.546i		−	0.288472i
$717$		0			0
$718$			225.019i			0.313397i
$719$			65.5964i			0.0912328i	0.998959	+	0.0456164i	$0.0145252\pi$
$719$			65.5964i			0.0912328i	−0.998959	+	0.0456164i	$0.985475\pi$
$720$		0			0
$721$	243.251			0.337380
$722$	−168.548			−0.233446
$723$		0			0
$724$	430.100			0.594061
$725$	35.1836	+	74.3209i	0.0485291	+	0.102512i
$726$		0			0
$727$			1249.52i			1.71873i	0.511364	+	0.859364i	$0.329140\pi$
$727$			1249.52i			1.71873i	−0.511364	+	0.859364i	$0.670860\pi$
$728$	−137.550			−0.188942
$729$		0			0
$730$	32.4041	+	144.182i	0.0443892	+	0.197509i
$731$		−	1188.02i		−	1.62520i
$732$		0			0
$733$			195.991i			0.267382i	0.991023	+	0.133691i	$0.0426829\pi$
$733$			195.991i			0.267382i	−0.991023	+	0.133691i	$0.957317\pi$
$734$			796.615i			1.08531i
$735$		0			0
$736$	84.5857			0.114926
$737$	748.225			1.01523
$738$		0			0
$739$	376.536			0.509521			0.254760	−	0.967004i	$-0.418003\pi$
$739$	376.536			0.509521			0.254760	+	0.967004i	$0.418003\pi$
$740$	73.1249	+	325.369i	0.0988175	+	0.439687i
$741$		0			0
$742$		−	157.980i		−	0.212910i
$743$	−1293.12			−1.74041			−0.870204	−	0.492692i	$-0.836013\pi$
$743$	−1293.12			−1.74041			−0.870204	+	0.492692i	$0.836013\pi$
$744$		0			0
$745$	−143.849	−	640.055i	−0.193086	−	0.859134i
$746$		−	663.708i		−	0.889689i
$747$		0			0
$748$			735.151i			0.982822i
$749$		−	251.096i		−	0.335242i
$750$		0			0
$751$	8.50866			0.0113298			0.00566489	−	0.999984i	$-0.498197\pi$
$751$	8.50866			0.0113298			0.00566489	+	0.999984i	$0.498197\pi$
$752$	144.349			0.191954
$753$		0			0
$754$	107.666			0.142794
$755$	−431.484	+	96.9739i	−0.571503	+	0.128442i
$756$		0			0
$757$		−	759.469i		−	1.00326i	−0.865082	−	0.501631i	$-0.832734\pi$
$757$		−	759.469i		−	1.00326i	0.865082	−	0.501631i	$-0.167266\pi$
$758$	−479.561			−0.632667
$759$		0			0
$760$	−214.565	+	48.2225i	−0.282323	+	0.0634506i
$761$		−	452.801i		−	0.595007i	−0.954721	−	0.297504i	$-0.903846\pi$
$761$		−	452.801i		−	0.595007i	0.954721	−	0.297504i	$-0.0961540\pi$
$762$		0			0
$763$		−	132.173i		−	0.173229i
$764$			545.386i			0.713856i
$765$		0			0
$766$	7.95001			0.0103786
$767$	−2343.62			−3.05557
$768$		0			0
$769$	1196.72			1.55620			0.778099	−	0.628142i	$-0.216184\pi$
$769$	1196.72			1.55620			0.778099	+	0.628142i	$0.216184\pi$
$770$	−193.704	+	43.5340i	−0.251564	+	0.0565376i
$771$		0			0
$772$		−	643.242i		−	0.833215i
$773$	287.556			0.372000			0.186000	−	0.982550i	$-0.440448\pi$
$773$	287.556			0.372000			0.186000	+	0.982550i	$0.440448\pi$
$774$		0			0
$775$	−222.420	+	105.294i	−0.286993	+	0.135863i
$776$			109.708i			0.141377i
$777$		0			0
$778$		−	898.904i		−	1.15540i
$779$			168.770i			0.216649i
$780$		0			0
$781$	1059.01			1.35596
$782$	−581.648			−0.743795
$783$		0			0
$784$	178.343			0.227478
$785$	178.377	+	793.687i	0.227232	+	1.01107i
$786$		0			0
$787$		−	341.757i		−	0.434253i	−0.976143	−	0.217127i	$-0.930332\pi$
$787$		−	341.757i		−	0.434253i	0.976143	−	0.217127i	$-0.0696684\pi$
$788$	75.2535			0.0954993
$789$		0			0
$790$	−10.2429	+	2.30203i	−0.0129656	+	0.00291396i
$791$		−	168.133i		−	0.212557i
$792$		0			0
$793$		−	2.33826i		−	0.00294863i
$794$			886.226i			1.11615i
$795$		0			0
$796$	−637.040			−0.800301
$797$	423.701			0.531619			0.265810	−	0.964026i	$-0.414361\pi$
$797$	423.701			0.531619			0.265810	+	0.964026i	$0.414361\pi$
$798$		0			0
$799$	−992.609			−1.24231
$800$	127.822	−	60.5110i	0.159777	−	0.0756388i
$801$		0			0
$802$		−	650.293i		−	0.810839i
$803$	279.286			0.347803
$804$		0			0
$805$	−34.4439	−	153.258i	−0.0427874	−	0.190382i
$806$			322.213i			0.399768i
$807$		0			0
$808$		−	231.818i		−	0.286904i
$809$			46.4331i			0.0573957i	0.999588	+	0.0286978i	$0.00913606\pi$
$809$			46.4331i			0.0573957i	−0.999588	+	0.0286978i	$0.990864\pi$
$810$		0			0
$811$	−511.160			−0.630284			−0.315142	−	0.949045i	$-0.602052\pi$
$811$	−511.160			−0.630284			−0.315142	+	0.949045i	$0.602052\pi$
$812$	13.8211			0.0170210
$813$		0			0
$814$	630.252			0.774265
$815$	301.714	+	1342.47i	0.370201	+	1.64721i
$816$		0			0
$817$		−	671.655i		−	0.822099i
$818$	−84.7951			−0.103661
$819$		0			0
$820$	−23.7980	−	105.889i	−0.0290219	−	0.129133i
$821$		−	154.461i		−	0.188137i	−0.995566	−	0.0940686i	$-0.970013\pi$
$821$		−	154.461i		−	0.188137i	0.995566	−	0.0940686i	$-0.0299873\pi$
$822$		0			0
$823$		−	1038.95i		−	1.26239i	−0.775624	−	0.631196i	$-0.782565\pi$
$823$		−	1038.95i		−	1.26239i	0.775624	−	0.631196i	$-0.217435\pi$
$824$			327.468i			0.397413i
$825$		0			0
$826$	−300.849			−0.364224
$827$	−792.713			−0.958540			−0.479270	−	0.877668i	$-0.659099\pi$
$827$	−792.713			−0.958540			−0.479270	+	0.877668i	$0.659099\pi$
$828$		0			0
$829$	1143.31			1.37914			0.689572	−	0.724217i	$-0.257798\pi$
$829$	1143.31			1.37914			0.689572	+	0.724217i	$0.257798\pi$
$830$	68.5179	−	15.3991i	0.0825517	−	0.0185531i
$831$		0			0
$832$		−	185.171i		−	0.222562i
$833$	−1226.36			−1.47222
$834$		0			0
$835$	668.277	−	150.192i	0.800332	−	0.179870i
$836$			415.621i			0.497155i
$837$		0			0
$838$			313.160i			0.373700i
$839$			1121.25i			1.33641i	0.743976	+	0.668206i	$0.232938\pi$
$839$			1121.25i			1.33641i	−0.743976	+	0.668206i	$0.767062\pi$
$840$		0			0
$841$	830.182			0.987136
$842$	−188.333			−0.223673
$843$		0			0
$844$	11.5051			0.0136316
$845$	−1789.16	+	402.104i	−2.11735	+	0.475863i
$846$		0			0
$847$			120.989i			0.142844i
$848$	212.675			0.250796
$849$		0			0
$850$	−878.958	+	416.100i	−1.03407	+	0.489529i
$851$			498.652i			0.585961i
$852$		0			0
$853$			1571.03i			1.84177i	0.389830	+	0.920887i	$0.372534\pi$
$853$			1571.03i			1.84177i	−0.389830	+	0.920887i	$0.627466\pi$
$854$		−	0.300161i		−	0.000351477i
$855$		0			0
$856$	338.030			0.394894
$857$	1566.41			1.82778			0.913892	−	0.405957i	$-0.133062\pi$
$857$	1566.41			1.82778			0.913892	+	0.405957i	$0.133062\pi$
$858$		0			0
$859$	−54.9240			−0.0639394			−0.0319697	−	0.999489i	$-0.510178\pi$
$859$	−54.9240			−0.0639394			−0.0319697	+	0.999489i	$0.510178\pi$
$860$	94.7090	+	421.407i	0.110127	+	0.490008i
$861$		0			0
$862$		−	647.837i		−	0.751551i
$863$	397.076			0.460111			0.230056	−	0.973177i	$-0.426109\pi$
$863$	397.076			0.460111			0.230056	+	0.973177i	$0.426109\pi$
$864$		0			0
$865$	452.545	−	101.707i	0.523173	−	0.117581i
$866$		−	339.861i		−	0.392449i
$867$		0			0
$868$			41.3622i			0.0476523i
$869$			19.8408i			0.0228318i
$870$		0			0
$871$	1295.96			1.48790
$872$	177.934			0.204053
$873$		0			0
$874$	−328.838			−0.376245
$875$	−161.688	−	206.955i	−0.184786	−	0.236520i
$876$		0			0
$877$			94.0250i			0.107212i	0.998562	+	0.0536060i	$0.0170715\pi$
$877$			94.0250i			0.107212i	−0.998562	+	0.0536060i	$0.982928\pi$
$878$	623.538			0.710180
$879$		0			0
$880$	−58.6061	−	260.767i	−0.0665979	−	0.296327i
$881$		−	283.024i		−	0.321253i	−0.987015	−	0.160627i	$-0.948648\pi$
$881$		−	283.024i		−	0.321253i	0.987015	−	0.160627i	$-0.0513515\pi$
$882$		0			0
$883$			1194.90i			1.35323i	0.736339	+	0.676613i	$0.236553\pi$
$883$			1194.90i			1.35323i	−0.736339	+	0.676613i	$0.763447\pi$
$884$			1273.32i			1.44041i
$885$		0			0
$886$	859.571			0.970170
$887$	714.584			0.805619			0.402810	−	0.915284i	$-0.368034\pi$
$887$	714.584			0.805619			0.402810	+	0.915284i	$0.368034\pi$
$888$		0			0
$889$	257.767			0.289952
$890$	−237.188	−	1055.37i	−0.266504	−	1.18580i
$891$		0			0
$892$			47.7980i			0.0535852i
$893$	−561.177			−0.628417
$894$		0			0
$895$	−113.226	−	503.798i	−0.126510	−	0.562903i
$896$		−	23.7703i		−	0.0265294i
$897$		0			0
$898$		−	79.4735i		−	0.0885005i
$899$		−	32.3761i		−	0.0360135i
$900$		0			0
$901$	−1462.44			−1.62313
$902$	−205.111			−0.227396
$903$		0			0
$904$	226.343			0.250379
$905$	1049.08	−	235.776i	1.15921	−	0.260526i
$906$		0			0
$907$			645.000i			0.711136i	0.934650	+	0.355568i	$0.115712\pi$
$907$			645.000i			0.711136i	−0.934650	+	0.355568i	$0.884288\pi$
$908$	−425.863			−0.469012
$909$		0			0
$910$	−335.505	+	75.4031i	−0.368687	+	0.0828605i
$911$			1002.58i			1.10052i	0.834992	+	0.550261i	$0.185472\pi$
$911$			1002.58i			1.10052i	−0.834992	+	0.550261i	$0.814528\pi$
$912$		0			0
$913$		−	132.722i		−	0.145369i
$914$		−	302.606i		−	0.331079i
$915$		0			0
$916$	−674.949			−0.736844
$917$	−46.9403			−0.0511890
$918$		0			0
$919$	−1583.98			−1.72359			−0.861794	−	0.507258i	$-0.830659\pi$
$919$	−1583.98			−1.72359			−0.861794	+	0.507258i	$0.830659\pi$
$920$	206.318	−	46.3689i	0.224259	−	0.0504010i
$921$		0			0
$922$		−	307.943i		−	0.333995i
$923$	1834.26			1.98728
$924$		0			0
$925$	356.727	+	753.539i	0.385650	+	0.814637i
$926$			200.305i			0.216312i
$927$		0			0
$928$			18.6061i			0.0200497i
$929$			636.903i			0.685579i	0.939412	+	0.342790i	$0.111372\pi$
$929$			636.903i			0.685579i	−0.939412	+	0.342790i	$0.888628\pi$
$930$		0			0
$931$	−693.331			−0.744716
$932$	−397.993			−0.427031
$933$		0			0
$934$	−359.605			−0.385016
$935$	403.001	+	1793.15i	0.431017	+	1.91781i
$936$		0			0
$937$			398.688i			0.425494i	0.977107	+	0.212747i	$0.0682410\pi$
$937$			398.688i			0.425494i	−0.977107	+	0.212747i	$0.931759\pi$
$938$	166.362			0.177358
$939$		0			0
$940$	352.091	−	79.1306i	0.374565	−	0.0841815i
$941$			523.582i			0.556410i	0.960522	+	0.278205i	$0.0897395\pi$
$941$			523.582i			0.556410i	−0.960522	+	0.278205i	$0.910261\pi$
$942$		0			0
$943$		−	162.283i		−	0.172092i
$944$		−	405.008i		−	0.429034i
$945$		0			0
$946$	816.282			0.862877
$947$	317.944			0.335739			0.167869	−	0.985809i	$-0.446311\pi$
$947$	317.944			0.335739			0.167869	+	0.985809i	$0.446311\pi$
$948$		0			0
$949$	483.737			0.509733
$950$	−496.924	+	235.244i	−0.523077	+	0.247626i
$951$		0			0
$952$			163.455i			0.171697i
$953$	−848.148			−0.889977			−0.444988	−	0.895536i	$-0.646792\pi$
$953$	−848.148			−0.889977			−0.444988	+	0.895536i	$0.646792\pi$
$954$		0			0
$955$	298.974	+	1330.28i	0.313062	+	1.39297i
$956$			152.221i			0.159227i
$957$		0			0
$958$		−	1206.85i		−	1.25976i
$959$			1.80097i			0.00187797i
$960$		0			0
$961$	−864.108			−0.899176
$962$	1091.63			1.13475
$963$		0			0
$964$	653.514			0.677919
$965$	−352.618	−	1568.97i	−0.365407	−	1.62587i
$966$		0			0
$967$		−	998.725i		−	1.03281i	−0.856345	−	0.516404i	$-0.827270\pi$
$967$		−	998.725i		−	1.03281i	0.856345	−	0.516404i	$-0.172730\pi$
$968$	−162.877			−0.168261
$969$		0			0
$970$	60.1408	+	267.596i	0.0620008	+	0.275872i
$971$			1392.89i			1.43449i	0.696820	+	0.717246i	$0.254598\pi$
$971$			1392.89i			1.43449i	−0.696820	+	0.717246i	$0.745402\pi$
$972$		0			0
$973$		−	494.334i		−	0.508051i
$974$			713.133i			0.732169i
$975$		0			0
$976$	0.404082			0.000414019
$977$	618.875			0.633444			0.316722	−	0.948518i	$-0.397418\pi$
$977$	618.875			0.633444			0.316722	+	0.948518i	$0.397418\pi$
$978$		0			0
$979$	−2044.29			−2.08814
$980$	435.006	−	97.7654i	0.443884	−	0.0997607i
$981$		0			0
$982$		−	621.898i		−	0.633297i
$983$	−192.813			−0.196148			−0.0980739	−	0.995179i	$-0.531268\pi$
$983$	−192.813			−0.196148			−0.0980739	+	0.995179i	$0.531268\pi$
$984$		0			0
$985$	183.555	−	41.2531i	0.186350	−	0.0418813i
$986$		−	127.944i		−	0.129760i
$987$		0			0
$988$			719.878i			0.728621i
$989$			645.838i			0.653021i
$990$		0			0
$991$	410.243			0.413969			0.206984	−	0.978344i	$-0.433635\pi$
$991$	410.243			0.413969			0.206984	+	0.978344i	$0.433635\pi$
$992$	−55.6825			−0.0561315
$993$		0			0
$994$	235.462			0.236884
$995$	−1553.84	+	349.218i	−1.56165	+	0.350973i
$996$		0			0
$997$		−	453.648i		−	0.455013i	−0.973777	−	0.227507i	$-0.926943\pi$
$997$		−	453.648i		−	0.455013i	0.973777	−	0.227507i	$-0.0730573\pi$
$998$	−1142.14			−1.14443
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	810.3.b.a.809.7		8
3.2	odd	2	inner	810.3.b.a.809.2		8
5.4	even	2	inner	810.3.b.a.809.1		8
9.2	odd	6		270.3.j.a.179.3		8
9.4	even	3		270.3.j.a.89.1		8
9.5	odd	6		90.3.j.a.29.4	yes	8
9.7	even	3		90.3.j.a.59.1	yes	8
15.14	odd	2	inner	810.3.b.a.809.8		8
45.2	even	12		1350.3.i.a.1151.1		4
45.4	even	6		270.3.j.a.89.3		8
45.7	odd	12		450.3.i.a.401.2		4
45.13	odd	12		1350.3.i.c.251.2		4
45.14	odd	6		90.3.j.a.29.1	✓	8
45.22	odd	12		1350.3.i.a.251.1		4
45.23	even	12		450.3.i.c.101.1		4
45.29	odd	6		270.3.j.a.179.1		8
45.32	even	12		450.3.i.a.101.2		4
45.34	even	6		90.3.j.a.59.4	yes	8
45.38	even	12		1350.3.i.c.1151.2		4
45.43	odd	12		450.3.i.c.401.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
90.3.j.a.29.1	✓	8	45.14	odd	6
90.3.j.a.29.4	yes	8	9.5	odd	6
90.3.j.a.59.1	yes	8	9.7	even	3
90.3.j.a.59.4	yes	8	45.34	even	6
270.3.j.a.89.1		8	9.4	even	3
270.3.j.a.89.3		8	45.4	even	6
270.3.j.a.179.1		8	45.29	odd	6
270.3.j.a.179.3		8	9.2	odd	6
450.3.i.a.101.2		4	45.32	even	12
450.3.i.a.401.2		4	45.7	odd	12
450.3.i.c.101.1		4	45.23	even	12
450.3.i.c.401.1		4	45.43	odd	12
810.3.b.a.809.1		8	5.4	even	2	inner
810.3.b.a.809.2		8	3.2	odd	2	inner
810.3.b.a.809.7		8	1.1	even	1	trivial
810.3.b.a.809.8		8	15.14	odd	2	inner
1350.3.i.a.251.1		4	45.22	odd	12
1350.3.i.a.1151.1		4	45.2	even	12
1350.3.i.c.251.2		4	45.13	odd	12
1350.3.i.c.1151.2		4	45.38	even	12

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:	\( N \)	\(=\)	\( 810 = 2 \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	810.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.41421 q^{2} +2.00000 q^{4} +(4.87832 - 1.09638i) q^{5} -2.10102i q^{7} +2.82843 q^{8} +O(q^{10})\)	\(q+1.41421 q^{2} +2.00000 q^{4} +(4.87832 - 1.09638i) q^{5} -2.10102i q^{7} +2.82843 q^{8} +(6.89898 - 1.55051i) q^{10} -13.3636i q^{11} -23.1464i q^{13} -2.97129i q^{14} +4.00000 q^{16} -27.5057 q^{17} -15.5505 q^{19} +(9.75663 - 2.19275i) q^{20} -18.8990i q^{22} +14.9528 q^{23} +(22.5959 - 10.6969i) q^{25} -32.7340i q^{26} -4.20204i q^{28} +3.28913i q^{29} -9.84337 q^{31} +5.65685 q^{32} -38.8990 q^{34} +(-2.30351 - 10.2494i) q^{35} +33.3485i q^{37} -21.9917 q^{38} +(13.7980 - 3.10102i) q^{40} -10.8530i q^{41} +43.1918i q^{43} -26.7272i q^{44} +21.1464 q^{46} +36.0873 q^{47} +44.5857 q^{49} +(31.9555 - 15.1278i) q^{50} -46.2929i q^{52} +53.1687 q^{53} +(-14.6515 - 65.1918i) q^{55} -5.94258i q^{56} +4.65153i q^{58} -101.252i q^{59} +0.101021 q^{61} -13.9206 q^{62} +8.00000 q^{64} +(-25.3772 - 112.916i) q^{65} +55.9898i q^{67} -55.0115 q^{68} +(-3.25765 - 14.4949i) q^{70} +79.2457i q^{71} +20.8990i q^{73} +47.1619i q^{74} -31.1010 q^{76} -28.0772 q^{77} -1.48469 q^{79} +(19.5133 - 4.38551i) q^{80} -15.3485i q^{82} +9.93160 q^{83} +(-134.182 + 30.1566i) q^{85} +61.0825i q^{86} -37.7980i q^{88} -152.974i q^{89} -48.6311 q^{91} +29.9056 q^{92} +51.0352 q^{94} +(-75.8603 + 17.0492i) q^{95} +38.7878i q^{97} +63.0537 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 16 q^{4}+O(q^{10}) \) 8 * q + 16 * q^4	\( 8 q + 16 q^{4} + 16 q^{10} + 32 q^{16} - 144 q^{19} + 24 q^{25} + 176 q^{31} - 272 q^{34} + 32 q^{40} + 32 q^{46} - 192 q^{49} - 176 q^{55} + 40 q^{61} + 64 q^{64} - 320 q^{70} - 288 q^{76} + 576 q^{79} - 368 q^{85} + 336 q^{91} - 160 q^{94}+O(q^{100}) \) 8 * q + 16 * q^4 + 16 * q^10 + 32 * q^16 - 144 * q^19 + 24 * q^25 + 176 * q^31 - 272 * q^34 + 32 * q^40 + 32 * q^46 - 192 * q^49 - 176 * q^55 + 40 * q^61 + 64 * q^64 - 320 * q^70 - 288 * q^76 + 576 * q^79 - 368 * q^85 + 336 * q^91 - 160 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more