LMFDB - Embedded newform 1232.2.e.e.769.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1232,2,Mod(769,1232)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1232, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1232.769");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1232 = 2^{4} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1232.e (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$9.83756952902$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.12745506816.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 23x^{4} + 1 $ x^8 + 23*x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 154)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			769.3
Root			$0.323042 - 0.323042i$ of defining polynomial
Character	$\chi$	$=$	1232.769
Dual form			1232.2.e.e.769.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.646084i q^{3} +3.09557i q^{5} +(-2.44949 - 1.00000i) q^{7} +2.58258 q^{9} +O(q^{10})$	$q-0.646084i q^{3} +3.09557i q^{5} +(-2.44949 - 1.00000i) q^{7} +2.58258 q^{9} +(1.79129 + 2.79129i) q^{11} +0.646084 q^{13} +2.00000 q^{15} -3.74166 q^{17} +1.80341 q^{19} +(-0.646084 + 1.58258i) q^{21} -4.00000 q^{23} -4.58258 q^{25} -3.60681i q^{27} +1.58258i q^{29} +8.64064i q^{31} +(1.80341 - 1.15732i) q^{33} +(3.09557 - 7.58258i) q^{35} +3.58258 q^{37} -0.417424i q^{39} -9.93280 q^{41} +7.16515i q^{43} +7.99455i q^{45} +9.93280i q^{47} +(5.00000 + 4.89898i) q^{49} +2.41742i q^{51} -11.5826 q^{53} +(-8.64064 + 5.54506i) q^{55} -1.16515i q^{57} +0.646084i q^{59} +1.93825 q^{61} +(-6.32599 - 2.58258i) q^{63} +2.00000i q^{65} +7.58258 q^{67} +2.58434i q^{69} -2.00000 q^{71} +16.1240 q^{73} +2.96073i q^{75} +(-1.59645 - 8.62852i) q^{77} +4.00000i q^{79} +5.41742 q^{81} -12.8935 q^{83} -11.5826i q^{85} +1.02248 q^{87} +9.79796i q^{89} +(-1.58258 - 0.646084i) q^{91} +5.58258 q^{93} +5.58258i q^{95} -8.50579i q^{97} +(4.62614 + 7.20871i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 16 q^{9}+O(q^{10}) $ 8 * q - 16 * q^9	$ 8 q - 16 q^{9} - 4 q^{11} + 16 q^{15} - 32 q^{23} - 8 q^{37} + 40 q^{49} - 56 q^{53} + 24 q^{67} - 16 q^{71} + 4 q^{77} + 80 q^{81} + 24 q^{91} + 8 q^{93} + 92 q^{99}+O(q^{100}) $ 8 * q - 16 * q^9 - 4 * q^11 + 16 * q^15 - 32 * q^23 - 8 * q^37 + 40 * q^49 - 56 * q^53 + 24 * q^67 - 16 * q^71 + 4 * q^77 + 80 * q^81 + 24 * q^91 + 8 * q^93 + 92 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1232\mathbb{Z}\right)^\times$.

$n$	$309$	$353$	$463$	$673$
$\chi(n)$	$1$	$-1$	$1$	$-1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		−	0.646084i		−	0.373017i	−0.982453	−	0.186508i	$-0.940283\pi$
$3$		−	0.646084i		−	0.373017i	0.982453	−	0.186508i	$-0.0597171\pi$
$4$		0			0
$5$			3.09557i			1.38438i	0.721714	+	0.692191i	$0.243355\pi$
$5$			3.09557i			1.38438i	−0.721714	+	0.692191i	$0.756645\pi$
$6$		0			0
$7$	−2.44949	−	1.00000i	−0.925820	−	0.377964i
$7$	−2.44949	−	1.00000i	−0.925820	−	0.377964i
$8$		0			0
$9$	2.58258			0.860859
$10$		0			0
$11$	1.79129	+	2.79129i	0.540094	+	0.841605i
$11$	1.79129	+	2.79129i	0.540094	+	0.841605i
$12$		0			0
$13$	0.646084			0.179191			0.0895957	−	0.995978i	$-0.471443\pi$
$13$	0.646084			0.179191			0.0895957	+	0.995978i	$0.471443\pi$
$14$		0			0
$15$	2.00000			0.516398
$16$		0			0
$17$	−3.74166			−0.907485			−0.453743	−	0.891133i	$-0.649911\pi$
$17$	−3.74166			−0.907485			−0.453743	+	0.891133i	$0.649911\pi$
$18$		0			0
$19$	1.80341			0.413730			0.206865	−	0.978370i	$-0.433674\pi$
$19$	1.80341			0.413730			0.206865	+	0.978370i	$0.433674\pi$
$20$		0			0
$21$	−0.646084	+	1.58258i	−0.140987	+	0.345346i
$22$		0			0
$23$	−4.00000			−0.834058			−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000			−0.834058			−0.417029	+	0.908893i	$0.636929\pi$
$24$		0			0
$25$	−4.58258			−0.916515
$26$		0			0
$27$		−	3.60681i		−	0.694131i
$28$		0			0
$29$			1.58258i			0.293877i	0.989146	+	0.146938i	$0.0469419\pi$
$29$			1.58258i			0.293877i	−0.989146	+	0.146938i	$0.953058\pi$
$30$		0			0
$31$			8.64064i			1.55190i	0.630792	+	0.775952i	$0.282730\pi$
$31$			8.64064i			1.55190i	−0.630792	+	0.775952i	$0.717270\pi$
$32$		0			0
$33$	1.80341	−	1.15732i	0.313933	−	0.201464i
$34$		0			0
$35$	3.09557	−	7.58258i	0.523247	−	1.28169i
$36$		0			0
$37$	3.58258			0.588972			0.294486	−	0.955656i	$-0.404852\pi$
$37$	3.58258			0.588972			0.294486	+	0.955656i	$0.404852\pi$
$38$		0			0
$39$		−	0.417424i		−	0.0668414i
$40$		0			0
$41$	−9.93280			−1.55124			−0.775622	−	0.631198i	$-0.782564\pi$
$41$	−9.93280			−1.55124			−0.775622	+	0.631198i	$0.782564\pi$
$42$		0			0
$43$			7.16515i			1.09268i	0.837565	+	0.546338i	$0.183978\pi$
$43$			7.16515i			1.09268i	−0.837565	+	0.546338i	$0.816022\pi$
$44$		0			0
$45$			7.99455i			1.19176i
$46$		0			0
$47$			9.93280i			1.44885i	0.689354	+	0.724424i	$0.257894\pi$
$47$			9.93280i			1.44885i	−0.689354	+	0.724424i	$0.742106\pi$
$48$		0			0
$49$	5.00000	+	4.89898i	0.714286	+	0.699854i
$50$		0			0
$51$			2.41742i			0.338507i
$52$		0			0
$53$	−11.5826			−1.59099			−0.795495	−	0.605961i	$-0.792789\pi$
$53$	−11.5826			−1.59099			−0.795495	+	0.605961i	$0.792789\pi$
$54$		0			0
$55$	−8.64064	+	5.54506i	−1.16510	+	0.747696i
$56$		0			0
$57$		−	1.16515i		−	0.154328i
$58$		0			0
$59$			0.646084i			0.0841129i	0.999115	+	0.0420565i	$0.0133909\pi$
$59$			0.646084i			0.0841129i	−0.999115	+	0.0420565i	$0.986609\pi$
$60$		0			0
$61$	1.93825			0.248168			0.124084	−	0.992272i	$-0.460401\pi$
$61$	1.93825			0.248168			0.124084	+	0.992272i	$0.460401\pi$
$62$		0			0
$63$	−6.32599	−	2.58258i	−0.797000	−	0.325374i
$64$		0			0
$65$			2.00000i			0.248069i
$66$		0			0
$67$	7.58258			0.926359			0.463180	−	0.886264i	$-0.346709\pi$
$67$	7.58258			0.926359			0.463180	+	0.886264i	$0.346709\pi$
$68$		0			0
$69$			2.58434i			0.311117i
$70$		0			0
$71$	−2.00000			−0.237356			−0.118678	−	0.992933i	$-0.537866\pi$
$71$	−2.00000			−0.237356			−0.118678	+	0.992933i	$0.537866\pi$
$72$		0			0
$73$	16.1240			1.88717			0.943583	−	0.331136i	$-0.107432\pi$
$73$	16.1240			1.88717			0.943583	+	0.331136i	$0.107432\pi$
$74$		0			0
$75$			2.96073i			0.341875i
$76$		0			0
$77$	−1.59645	−	8.62852i	−0.181933	−	0.983311i
$78$		0			0
$79$			4.00000i			0.450035i	0.974355	+	0.225018i	$0.0722440\pi$
$79$			4.00000i			0.450035i	−0.974355	+	0.225018i	$0.927756\pi$
$80$		0			0
$81$	5.41742			0.601936
$82$		0			0
$83$	−12.8935			−1.41525			−0.707625	−	0.706589i	$-0.750233\pi$
$83$	−12.8935			−1.41525			−0.707625	+	0.706589i	$0.750233\pi$
$84$		0			0
$85$		−	11.5826i		−	1.25631i
$86$		0			0
$87$	1.02248			0.109621
$88$		0			0
$89$			9.79796i			1.03858i	0.854598	+	0.519291i	$0.173804\pi$
$89$			9.79796i			1.03858i	−0.854598	+	0.519291i	$0.826196\pi$
$90$		0			0
$91$	−1.58258	−	0.646084i	−0.165899	−	0.0677280i
$92$		0			0
$93$	5.58258			0.578886
$94$		0			0
$95$			5.58258i			0.572760i
$96$		0			0
$97$		−	8.50579i		−	0.863632i	−0.901962	−	0.431816i	$-0.857873\pi$
$97$		−	8.50579i		−	0.863632i	0.901962	−	0.431816i	$-0.142127\pi$
$98$		0			0
$99$	4.62614	+	7.20871i	0.464944	+	0.724503i
$100$		0			0
$101$	−11.7362			−1.16780			−0.583898	−	0.811827i	$-0.698473\pi$
$101$	−11.7362			−1.16780			−0.583898	+	0.811827i	$0.698473\pi$
$102$		0			0
$103$		−	6.05630i		−	0.596745i	−0.954449	−	0.298373i	$-0.903556\pi$
$103$		−	6.05630i		−	0.596745i	0.954449	−	0.298373i	$-0.0964438\pi$
$104$		0			0
$105$	−4.89898	−	2.00000i	−0.478091	−	0.195180i
$106$		0			0
$107$		−	17.5826i		−	1.69977i	−0.526967	−	0.849886i	$-0.676671\pi$
$107$		−	17.5826i		−	1.69977i	0.526967	−	0.849886i	$-0.323329\pi$
$108$		0			0
$109$			17.1652i			1.64412i	0.569398	+	0.822062i	$0.307176\pi$
$109$			17.1652i			1.64412i	−0.569398	+	0.822062i	$0.692824\pi$
$110$		0			0
$111$		−	2.31464i		−	0.219696i
$112$		0			0
$113$	10.7477			1.01106			0.505531	−	0.862809i	$-0.331297\pi$
$113$	10.7477			1.01106			0.505531	+	0.862809i	$0.331297\pi$
$114$		0			0
$115$		−	12.3823i		−	1.15465i
$116$		0			0
$117$	1.66856			0.154258
$118$		0			0
$119$	9.16515	+	3.74166i	0.840168	+	0.342997i
$120$		0			0
$121$	−4.58258	+	10.0000i	−0.416598	+	0.909091i
$122$		0			0
$123$			6.41742i			0.578640i
$124$		0			0
$125$			1.29217i			0.115575i
$126$		0			0
$127$			3.58258i			0.317902i	0.987286	+	0.158951i	$0.0508112\pi$
$127$			3.58258i			0.317902i	−0.987286	+	0.158951i	$0.949189\pi$
$128$		0			0
$129$	4.62929			0.407586
$130$		0			0
$131$	16.7700			1.46520			0.732602	−	0.680657i	$-0.238306\pi$
$131$	16.7700			1.46520			0.732602	+	0.680657i	$0.238306\pi$
$132$		0			0
$133$	−4.41742	−	1.80341i	−0.383039	−	0.156375i
$134$		0			0
$135$	11.1652			0.960943
$136$		0			0
$137$	−3.16515			−0.270417			−0.135209	−	0.990817i	$-0.543170\pi$
$137$	−3.16515			−0.270417			−0.135209	+	0.990817i	$0.543170\pi$
$138$		0			0
$139$	15.4779			1.31282			0.656408	−	0.754406i	$-0.272075\pi$
$139$	15.4779			1.31282			0.656408	+	0.754406i	$0.272075\pi$
$140$		0			0
$141$	6.41742			0.540445
$142$		0			0
$143$	1.15732	+	1.80341i	0.0967801	+	0.150808i
$144$		0			0
$145$	−4.89898			−0.406838
$146$		0			0
$147$	3.16515	−	3.23042i	0.261057	−	0.266440i
$148$		0			0
$149$		−	14.0000i		−	1.14692i	−0.819232	−	0.573462i	$-0.805600\pi$
$149$		−	14.0000i		−	1.14692i	0.819232	−	0.573462i	$-0.194400\pi$
$150$		0			0
$151$		−	5.58258i		−	0.454304i	−0.973859	−	0.227152i	$-0.927059\pi$
$151$		−	5.58258i		−	0.454304i	0.973859	−	0.227152i	$-0.0729414\pi$
$152$		0			0
$153$	−9.66311			−0.781216
$154$		0			0
$155$	−26.7477			−2.14843
$156$		0			0
$157$		−	13.1632i		−	1.05054i	−0.850936	−	0.525270i	$-0.823964\pi$
$157$		−	13.1632i		−	1.05054i	0.850936	−	0.525270i	$-0.176036\pi$
$158$		0			0
$159$			7.48331i			0.593465i
$160$		0			0
$161$	9.79796	+	4.00000i	0.772187	+	0.315244i
$162$		0			0
$163$	−4.00000			−0.313304			−0.156652	−	0.987654i	$-0.550070\pi$
$163$	−4.00000			−0.313304			−0.156652	+	0.987654i	$0.550070\pi$
$164$		0			0
$165$	3.58258	+	5.58258i	0.278903	+	0.434603i
$166$		0			0
$167$	−1.29217			−0.0999909			−0.0499955	−	0.998749i	$-0.515921\pi$
$167$	−1.29217			−0.0999909			−0.0499955	+	0.998749i	$0.515921\pi$
$168$		0			0
$169$	−12.5826			−0.967890
$170$		0			0
$171$	4.65743			0.356163
$172$		0			0
$173$	10.7137			0.814550			0.407275	−	0.913306i	$-0.366479\pi$
$173$	10.7137			0.814550			0.407275	+	0.913306i	$0.366479\pi$
$174$		0			0
$175$	11.2250	+	4.58258i	0.848528	+	0.346410i
$176$		0			0
$177$	0.417424			0.0313755
$178$		0			0
$179$	−11.1652			−0.834523			−0.417261	−	0.908787i	$-0.637010\pi$
$179$	−11.1652			−0.834523			−0.417261	+	0.908787i	$0.637010\pi$
$180$		0			0
$181$			5.41022i			0.402138i	0.979577	+	0.201069i	$0.0644416\pi$
$181$			5.41022i			0.402138i	−0.979577	+	0.201069i	$0.935558\pi$
$182$		0			0
$183$		−	1.25227i		−	0.0925707i
$184$		0			0
$185$			11.0901i			0.815362i
$186$		0			0
$187$	−6.70239	−	10.4440i	−0.490127	−	0.763744i
$188$		0			0
$189$	−3.60681	+	8.83485i	−0.262357	+	0.642641i
$190$		0			0
$191$	18.0000			1.30243			0.651217	−	0.758891i	$-0.274259\pi$
$191$	18.0000			1.30243			0.651217	+	0.758891i	$0.274259\pi$
$192$		0			0
$193$			16.3303i			1.17548i	0.809050	+	0.587740i	$0.199982\pi$
$193$			16.3303i			1.17548i	−0.809050	+	0.587740i	$0.800018\pi$
$194$		0			0
$195$	1.29217			0.0925340
$196$		0			0
$197$			8.74773i			0.623250i	0.950205	+	0.311625i	$0.100873\pi$
$197$			8.74773i			0.623250i	−0.950205	+	0.311625i	$0.899127\pi$
$198$		0			0
$199$			9.66311i			0.685000i	0.939518	+	0.342500i	$0.111274\pi$
$199$			9.66311i			0.685000i	−0.939518	+	0.342500i	$0.888726\pi$
$200$		0			0
$201$		−	4.89898i		−	0.345547i
$202$		0			0
$203$	1.58258	−	3.87650i	0.111075	−	0.272077i
$204$		0			0
$205$		−	30.7477i		−	2.14751i
$206$		0			0
$207$	−10.3303			−0.718006
$208$		0			0
$209$	3.23042	+	5.03383i	0.223453	+	0.348197i
$210$		0			0
$211$		−	16.7477i		−	1.15296i	−0.817111	−	0.576481i	$-0.804426\pi$
$211$		−	16.7477i		−	1.15296i	0.817111	−	0.576481i	$-0.195574\pi$
$212$		0			0
$213$			1.29217i			0.0885379i
$214$		0			0
$215$	−22.1803			−1.51268
$216$		0			0
$217$	8.64064	−	21.1652i	0.586565	−	1.43678i
$218$		0			0
$219$		−	10.4174i		−	0.703944i
$220$		0			0
$221$	−2.41742			−0.162614
$222$		0			0
$223$		−	16.1240i		−	1.07974i	−0.841749	−	0.539870i	$-0.818473\pi$
$223$		−	16.1240i		−	1.07974i	0.841749	−	0.539870i	$-0.181527\pi$
$224$		0			0
$225$	−11.8348			−0.788990
$226$		0			0
$227$	0.511238			0.0339321			0.0169660	−	0.999856i	$-0.494599\pi$
$227$	0.511238			0.0339321			0.0169660	+	0.999856i	$0.494599\pi$
$228$		0			0
$229$		−	5.67991i		−	0.375339i	−0.982232	−	0.187669i	$-0.939907\pi$
$229$		−	5.67991i		−	0.375339i	0.982232	−	0.187669i	$-0.0600934\pi$
$230$		0			0
$231$	−5.57475	+	1.03144i	−0.366791	+	0.0678639i
$232$		0			0
$233$		−	27.1652i		−	1.77965i	−0.456304	−	0.889824i	$-0.650827\pi$
$233$		−	27.1652i		−	1.77965i	0.456304	−	0.889824i	$-0.349173\pi$
$234$		0			0
$235$	−30.7477			−2.00576
$236$		0			0
$237$	2.58434			0.167871
$238$		0			0
$239$			8.41742i			0.544478i	0.962230	+	0.272239i	$0.0877641\pi$
$239$			8.41742i			0.544478i	−0.962230	+	0.272239i	$0.912236\pi$
$240$		0			0
$241$	−3.47197			−0.223649			−0.111825	−	0.993728i	$-0.535669\pi$
$241$	−3.47197			−0.223649			−0.111825	+	0.993728i	$0.535669\pi$
$242$		0			0
$243$		−	14.3205i		−	0.918663i
$244$		0			0
$245$	−15.1652	+	15.4779i	−0.968866	+	0.988845i
$246$		0			0
$247$	1.16515			0.0741368
$248$		0			0
$249$			8.33030i			0.527911i
$250$		0			0
$251$		−	24.1185i		−	1.52235i	−0.648549	−	0.761173i	$-0.724624\pi$
$251$		−	24.1185i		−	1.52235i	0.648549	−	0.761173i	$-0.275376\pi$
$252$		0			0
$253$	−7.16515	−	11.1652i	−0.450469	−	0.701947i
$254$		0			0
$255$	−7.48331			−0.468623
$256$		0			0
$257$		−	4.89898i		−	0.305590i	−0.988258	−	0.152795i	$-0.951173\pi$
$257$		−	4.89898i		−	0.305590i	0.988258	−	0.152795i	$-0.0488274\pi$
$258$		0			0
$259$	−8.77548	−	3.58258i	−0.545282	−	0.222610i
$260$		0			0
$261$			4.08712i			0.252986i
$262$		0			0
$263$			8.33030i			0.513668i	0.966455	+	0.256834i	$0.0826794\pi$
$263$			8.33030i			0.513668i	−0.966455	+	0.256834i	$0.917321\pi$
$264$		0			0
$265$		−	35.8547i		−	2.20254i
$266$		0			0
$267$	6.33030			0.387408
$268$		0			0
$269$		−	16.5003i		−	1.00604i	−0.864274	−	0.503022i	$-0.832222\pi$
$269$		−	16.5003i		−	1.00604i	0.864274	−	0.503022i	$-0.167778\pi$
$270$		0			0
$271$	4.89898			0.297592			0.148796	−	0.988868i	$-0.452460\pi$
$271$	4.89898			0.297592			0.148796	+	0.988868i	$0.452460\pi$
$272$		0			0
$273$	−0.417424	+	1.02248i	−0.0252637	+	0.0618831i
$274$		0			0
$275$	−8.20871	−	12.7913i	−0.495004	−	0.771344i
$276$		0			0
$277$			2.00000i			0.120168i	0.998193	+	0.0600842i	$0.0191369\pi$
$277$			2.00000i			0.120168i	−0.998193	+	0.0600842i	$0.980863\pi$
$278$		0			0
$279$			22.3151i			1.33597i
$280$		0			0
$281$			11.1652i			0.666057i	0.942917	+	0.333029i	$0.108071\pi$
$281$			11.1652i			0.666057i	−0.942917	+	0.333029i	$0.891929\pi$
$282$		0			0
$283$	18.0622			1.07369			0.536843	−	0.843682i	$-0.319617\pi$
$283$	18.0622			1.07369			0.536843	+	0.843682i	$0.319617\pi$
$284$		0			0
$285$	3.60681			0.213649
$286$		0			0
$287$	24.3303	+	9.93280i	1.43617	+	0.586315i
$288$		0			0
$289$	−3.00000			−0.176471
$290$		0			0
$291$	−5.49545			−0.322149
$292$		0			0
$293$	25.1410			1.46875			0.734376	−	0.678743i	$-0.237475\pi$
$293$	25.1410			1.46875			0.734376	+	0.678743i	$0.237475\pi$
$294$		0			0
$295$	−2.00000			−0.116445
$296$		0			0
$297$	10.0677	−	6.46084i	0.584184	−	0.374896i
$298$		0			0
$299$	−2.58434			−0.149456
$300$		0			0
$301$	7.16515	−	17.5510i	0.412992	−	1.01162i
$302$		0			0
$303$			7.58258i			0.435608i
$304$		0			0
$305$			6.00000i			0.343559i
$306$		0			0
$307$	10.5789			0.603769			0.301885	−	0.953344i	$-0.402384\pi$
$307$	10.5789			0.603769			0.301885	+	0.953344i	$0.402384\pi$
$308$		0			0
$309$	−3.91288			−0.222596
$310$		0			0
$311$		−	5.03383i		−	0.285442i	−0.989763	−	0.142721i	$-0.954415\pi$
$311$		−	5.03383i		−	0.285442i	0.989763	−	0.142721i	$-0.0455852\pi$
$312$		0			0
$313$		−	23.4724i		−	1.32674i	−0.748292	−	0.663370i	$-0.769126\pi$
$313$		−	23.4724i		−	1.32674i	0.748292	−	0.663370i	$-0.230874\pi$
$314$		0			0
$315$	7.99455	−	19.5826i	0.450442	−	1.10335i
$316$		0			0
$317$	9.16515			0.514766			0.257383	−	0.966309i	$-0.417140\pi$
$317$	9.16515			0.514766			0.257383	+	0.966309i	$0.417140\pi$
$318$		0			0
$319$	−4.41742	+	2.83485i	−0.247328	+	0.158721i
$320$		0			0
$321$	−11.3598			−0.634043
$322$		0			0
$323$	−6.74773			−0.375454
$324$		0			0
$325$	−2.96073			−0.164232
$326$		0			0
$327$	11.0901			0.613285
$328$		0			0
$329$	9.93280	−	24.3303i	0.547613	−	1.34137i
$330$		0			0
$331$	23.5826			1.29622			0.648108	−	0.761549i	$-0.275561\pi$
$331$	23.5826			1.29622			0.648108	+	0.761549i	$0.275561\pi$
$332$		0			0
$333$	9.25227			0.507021
$334$		0			0
$335$			23.4724i			1.28244i
$336$		0			0
$337$			10.8348i			0.590212i	0.955465	+	0.295106i	$0.0953549\pi$
$337$			10.8348i			0.590212i	−0.955465	+	0.295106i	$0.904645\pi$
$338$		0			0
$339$		−	6.94393i		−	0.377143i
$340$		0			0
$341$	−24.1185	+	15.4779i	−1.30609	+	0.838174i
$342$		0			0
$343$	−7.34847	−	17.0000i	−0.396780	−	0.917914i
$344$		0			0
$345$	−8.00000			−0.430706
$346$		0			0
$347$			19.1652i			1.02884i	0.857539	+	0.514420i	$0.171993\pi$
$347$			19.1652i			1.02884i	−0.857539	+	0.514420i	$0.828007\pi$
$348$		0			0
$349$	24.1185			1.29103			0.645517	−	0.763746i	$-0.276642\pi$
$349$	24.1185			1.29103			0.645517	+	0.763746i	$0.276642\pi$
$350$		0			0
$351$		−	2.33030i		−	0.124382i
$352$		0			0
$353$			7.48331i			0.398297i	0.979969	+	0.199148i	$0.0638176\pi$
$353$			7.48331i			0.398297i	−0.979969	+	0.199148i	$0.936182\pi$
$354$		0			0
$355$		−	6.19115i		−	0.328592i
$356$		0			0
$357$	2.41742	−	5.92146i	0.127944	−	0.313397i
$358$		0			0
$359$			9.58258i			0.505749i	0.967499	+	0.252875i	$0.0813760\pi$
$359$			9.58258i			0.505749i	−0.967499	+	0.252875i	$0.918624\pi$
$360$		0			0
$361$	−15.7477			−0.828828
$362$		0			0
$363$	6.46084	+	2.96073i	0.339106	+	0.155398i
$364$		0			0
$365$			49.9129i			2.61256i
$366$		0			0
$367$		−	21.0229i		−	1.09739i	−0.836023	−	0.548694i	$-0.815125\pi$
$367$		−	21.0229i		−	1.09739i	0.836023	−	0.548694i	$-0.184875\pi$
$368$		0			0
$369$	−25.6522			−1.33540
$370$		0			0
$371$	28.3714	+	11.5826i	1.47297	+	0.601337i
$372$		0			0
$373$			9.58258i			0.496167i	0.968739	+	0.248083i	$0.0798007\pi$
$373$			9.58258i			0.496167i	−0.968739	+	0.248083i	$0.920199\pi$
$374$		0			0
$375$	0.834849			0.0431114
$376$		0			0
$377$			1.02248i			0.0526602i
$378$		0			0
$379$	22.3303			1.14703			0.573515	−	0.819195i	$-0.305579\pi$
$379$	22.3303			1.14703			0.573515	+	0.819195i	$0.305579\pi$
$380$		0			0
$381$	2.31464			0.118583
$382$		0			0
$383$		−	19.7308i		−	1.00819i	−0.863647	−	0.504097i	$-0.831825\pi$
$383$		−	19.7308i		−	1.00819i	0.863647	−	0.504097i	$-0.168175\pi$
$384$		0			0
$385$	26.7102	−	4.94194i	1.36128	−	0.251865i
$386$		0			0
$387$			18.5045i			0.940639i
$388$		0			0
$389$	10.0000			0.507020			0.253510	−	0.967333i	$-0.418415\pi$
$389$	10.0000			0.507020			0.253510	+	0.967333i	$0.418415\pi$
$390$		0			0
$391$	14.9666			0.756895
$392$		0			0
$393$		−	10.8348i		−	0.546546i
$394$		0			0
$395$	−12.3823			−0.623021
$396$		0			0
$397$		−	27.5905i		−	1.38473i	−0.721549	−	0.692363i	$-0.756570\pi$
$397$		−	27.5905i		−	1.38473i	0.721549	−	0.692363i	$-0.243430\pi$
$398$		0			0
$399$	−1.16515	+	2.85403i	−0.0583305	+	0.142880i
$400$		0			0
$401$	7.58258			0.378656			0.189328	−	0.981914i	$-0.439369\pi$
$401$	7.58258			0.378656			0.189328	+	0.981914i	$0.439369\pi$
$402$		0			0
$403$			5.58258i			0.278088i
$404$		0			0
$405$			16.7700i			0.833310i
$406$		0			0
$407$	6.41742	+	10.0000i	0.318100	+	0.495682i
$408$		0			0
$409$	−5.03383			−0.248907			−0.124453	−	0.992225i	$-0.539718\pi$
$409$	−5.03383			−0.248907			−0.124453	+	0.992225i	$0.539718\pi$
$410$		0			0
$411$			2.04495i			0.100870i
$412$		0			0
$413$	0.646084	−	1.58258i	0.0317917	−	0.0778735i
$414$		0			0
$415$		−	39.9129i		−	1.95925i
$416$		0			0
$417$		−	10.0000i		−	0.489702i
$418$		0			0
$419$		−	13.0284i		−	0.636478i	−0.948011	−	0.318239i	$-0.896909\pi$
$419$		−	13.0284i		−	0.636478i	0.948011	−	0.318239i	$-0.103091\pi$
$420$		0			0
$421$	7.58258			0.369552			0.184776	−	0.982781i	$-0.440844\pi$
$421$	7.58258			0.369552			0.184776	+	0.982781i	$0.440844\pi$
$422$		0			0
$423$			25.6522i			1.24725i
$424$		0			0
$425$	17.1464			0.831724
$426$		0			0
$427$	−4.74773	−	1.93825i	−0.229759	−	0.0937986i
$428$		0			0
$429$	1.16515	−	0.747727i	0.0562540	−	0.0361006i
$430$		0			0
$431$			27.9129i			1.34452i	0.740317	+	0.672258i	$0.234675\pi$
$431$			27.9129i			1.34452i	−0.740317	+	0.672258i	$0.765325\pi$
$432$		0			0
$433$			35.5850i			1.71011i	0.518540	+	0.855054i	$0.326476\pi$
$433$			35.5850i			1.71011i	−0.518540	+	0.855054i	$0.673524\pi$
$434$		0			0
$435$			3.16515i			0.151757i
$436$		0			0
$437$	−7.21362			−0.345074
$438$		0			0
$439$	−17.2813			−0.824790			−0.412395	−	0.911005i	$-0.635308\pi$
$439$	−17.2813			−0.824790			−0.412395	+	0.911005i	$0.635308\pi$
$440$		0			0
$441$	12.9129	+	12.6520i	0.614899	+	0.602475i
$442$		0			0
$443$	−15.1652			−0.720518			−0.360259	−	0.932852i	$-0.617312\pi$
$443$	−15.1652			−0.720518			−0.360259	+	0.932852i	$0.617312\pi$
$444$		0			0
$445$	−30.3303			−1.43779
$446$		0			0
$447$	−9.04517			−0.427822
$448$		0			0
$449$	31.1652			1.47077			0.735387	−	0.677647i	$-0.237000\pi$
$449$	31.1652			1.47077			0.735387	+	0.677647i	$0.237000\pi$
$450$		0			0
$451$	−17.7925	−	27.7253i	−0.837817	−	1.30553i
$452$		0			0
$453$	−3.60681			−0.169463
$454$		0			0
$455$	2.00000	−	4.89898i	0.0937614	−	0.229668i
$456$		0			0
$457$			33.1652i			1.55140i	0.631102	+	0.775700i	$0.282603\pi$
$457$			33.1652i			1.55140i	−0.631102	+	0.775700i	$0.717397\pi$
$458$		0			0
$459$			13.4955i			0.629914i
$460$		0			0
$461$	30.0400			1.39910			0.699550	−	0.714583i	$-0.253384\pi$
$461$	30.0400			1.39910			0.699550	+	0.714583i	$0.253384\pi$
$462$		0			0
$463$	−5.16515			−0.240045			−0.120022	−	0.992771i	$-0.538297\pi$
$463$	−5.16515			−0.240045			−0.120022	+	0.992771i	$0.538297\pi$
$464$		0			0
$465$			17.2813i			0.801400i
$466$		0			0
$467$			11.7362i			0.543087i	0.962426	+	0.271544i	$0.0875341\pi$
$467$			11.7362i			0.543087i	−0.962426	+	0.271544i	$0.912466\pi$
$468$		0			0
$469$	−18.5734	−	7.58258i	−0.857642	−	0.350131i
$470$		0			0
$471$	−8.50455			−0.391869
$472$		0			0
$473$	−20.0000	+	12.8348i	−0.919601	+	0.590147i
$474$		0			0
$475$	−8.26424			−0.379190
$476$		0			0
$477$	−29.9129			−1.36962
$478$		0			0
$479$	−30.9557			−1.41440			−0.707202	−	0.707012i	$-0.750043\pi$
$479$	−30.9557			−1.41440			−0.707202	+	0.707012i	$0.750043\pi$
$480$		0			0
$481$	2.31464			0.105539
$482$		0			0
$483$	2.58434	−	6.33030i	0.117591	−	0.288039i
$484$		0			0
$485$	26.3303			1.19560
$486$		0			0
$487$	−21.4955			−0.974052			−0.487026	−	0.873387i	$-0.661918\pi$
$487$	−21.4955			−0.974052			−0.487026	+	0.873387i	$0.661918\pi$
$488$		0			0
$489$			2.58434i			0.116868i
$490$		0			0
$491$		−	33.4955i		−	1.51163i	−0.654786	−	0.755814i	$-0.727241\pi$
$491$		−	33.4955i		−	1.51163i	0.654786	−	0.755814i	$-0.272759\pi$
$492$		0			0
$493$		−	5.92146i		−	0.266689i
$494$		0			0
$495$	−22.3151	+	14.3205i	−1.00299	+	0.643661i
$496$		0			0
$497$	4.89898	+	2.00000i	0.219749	+	0.0897123i
$498$		0			0
$499$	17.9129			0.801891			0.400945	−	0.916102i	$-0.368682\pi$
$499$	17.9129			0.801891			0.400945	+	0.916102i	$0.368682\pi$
$500$		0			0
$501$			0.834849i			0.0372983i
$502$		0			0
$503$	17.0116			0.758509			0.379254	−	0.925292i	$-0.376180\pi$
$503$	17.0116			0.758509			0.379254	+	0.925292i	$0.376180\pi$
$504$		0			0
$505$		−	36.3303i		−	1.61668i
$506$		0			0
$507$			8.12940i			0.361039i
$508$		0			0
$509$			0.780929i			0.0346141i	0.999850	+	0.0173070i	$0.00550928\pi$
$509$			0.780929i			0.0346141i	−0.999850	+	0.0173070i	$0.994491\pi$
$510$		0			0
$511$	−39.4955	−	16.1240i	−1.74718	−	0.713282i
$512$		0			0
$513$		−	6.50455i		−	0.287183i
$514$		0			0
$515$	18.7477			0.826124
$516$		0			0
$517$	−27.7253	+	17.7925i	−1.21936	+	0.782514i
$518$		0			0
$519$		−	6.92197i		−	0.303841i
$520$		0			0
$521$		−	28.1017i		−	1.23116i	−0.788075	−	0.615579i	$-0.788922\pi$
$521$		−	28.1017i		−	1.23116i	0.788075	−	0.615579i	$-0.211078\pi$
$522$		0			0
$523$	14.4554			0.632090			0.316045	−	0.948744i	$-0.397645\pi$
$523$	14.4554			0.632090			0.316045	+	0.948744i	$0.397645\pi$
$524$		0			0
$525$	2.96073	−	7.25227i	0.129217	−	0.316515i
$526$		0			0
$527$		−	32.3303i		−	1.40833i
$528$		0			0
$529$	−7.00000			−0.304348
$530$		0			0
$531$			1.66856i			0.0724094i
$532$		0			0
$533$	−6.41742			−0.277970
$534$		0			0
$535$	54.4282			2.35313
$536$		0			0
$537$			7.21362i			0.311291i
$538$		0			0
$539$	−4.71802	+	22.7319i	−0.203220	+	0.979133i
$540$		0			0
$541$		−	36.7477i		−	1.57991i	−0.613166	−	0.789954i	$-0.710104\pi$
$541$		−	36.7477i		−	1.57991i	0.613166	−	0.789954i	$-0.289896\pi$
$542$		0			0
$543$	3.49545			0.150004
$544$		0			0
$545$	−53.1360			−2.27610
$546$		0			0
$547$			24.7477i			1.05814i	0.848579	+	0.529068i	$0.177458\pi$
$547$			24.7477i			1.05814i	−0.848579	+	0.529068i	$0.822542\pi$
$548$		0			0
$549$	5.00568			0.213637
$550$		0			0
$551$			2.85403i			0.121586i
$552$		0			0
$553$	4.00000	−	9.79796i	0.170097	−	0.416652i
$554$		0			0
$555$	7.16515			0.304144
$556$		0			0
$557$		−	35.9129i		−	1.52168i	−0.648941	−	0.760839i	$-0.724788\pi$
$557$		−	35.9129i		−	1.52168i	0.648941	−	0.760839i	$-0.275212\pi$
$558$		0			0
$559$			4.62929i			0.195798i
$560$		0			0
$561$	−6.74773	+	4.33030i	−0.284889	+	0.182826i
$562$		0			0
$563$	−23.7140			−0.999425			−0.499712	−	0.866191i	$-0.666561\pi$
$563$	−23.7140			−0.999425			−0.499712	+	0.866191i	$0.666561\pi$
$564$		0			0
$565$			33.2704i			1.39970i
$566$		0			0
$567$	−13.2699	−	5.41742i	−0.557284	−	0.227510i
$568$		0			0
$569$			39.4955i			1.65574i	0.560923	+	0.827868i	$0.310446\pi$
$569$			39.4955i			1.65574i	−0.560923	+	0.827868i	$0.689554\pi$
$570$		0			0
$571$			5.58258i			0.233624i	0.993154	+	0.116812i	$0.0372674\pi$
$571$			5.58258i			0.233624i	−0.993154	+	0.116812i	$0.962733\pi$
$572$		0			0
$573$		−	11.6295i		−	0.485830i
$574$		0			0
$575$	18.3303			0.764426
$576$		0			0
$577$			29.6636i			1.23491i	0.786606	+	0.617455i	$0.211836\pi$
$577$			29.6636i			1.23491i	−0.786606	+	0.617455i	$0.788164\pi$
$578$		0			0
$579$	10.5507			0.438474
$580$		0			0
$581$	31.5826	+	12.8935i	1.31027	+	0.534914i
$582$		0			0
$583$	−20.7477	−	32.3303i	−0.859283	−	1.33898i
$584$		0			0
$585$			5.16515i			0.213553i
$586$		0			0
$587$			39.0851i			1.61322i	0.591087	+	0.806608i	$0.298699\pi$
$587$			39.0851i			1.61322i	−0.591087	+	0.806608i	$0.701301\pi$
$588$		0			0
$589$			15.5826i			0.642069i
$590$		0			0
$591$	5.65176			0.232483
$592$		0			0
$593$	−14.8318			−0.609068			−0.304534	−	0.952501i	$-0.598501\pi$
$593$	−14.8318			−0.609068			−0.304534	+	0.952501i	$0.598501\pi$
$594$		0			0
$595$	−11.5826	+	28.3714i	−0.474839	+	1.16311i
$596$		0			0
$597$	6.24318			0.255516
$598$		0			0
$599$	21.1652			0.864785			0.432392	−	0.901686i	$-0.357670\pi$
$599$	21.1652			0.864785			0.432392	+	0.901686i	$0.357670\pi$
$600$		0			0
$601$	21.0229			0.857543			0.428772	−	0.903413i	$-0.358947\pi$
$601$	21.0229			0.857543			0.428772	+	0.903413i	$0.358947\pi$
$602$		0			0
$603$	19.5826			0.797464
$604$		0			0
$605$	−30.9557	−	14.1857i	−1.25853	−	0.576731i
$606$		0			0
$607$	−2.04495			−0.0830021			−0.0415010	−	0.999138i	$-0.513214\pi$
$607$	−2.04495			−0.0830021			−0.0415010	+	0.999138i	$0.513214\pi$
$608$		0			0
$609$	−2.50455	−	1.02248i	−0.101489	−	0.0414328i
$610$		0			0
$611$			6.41742i			0.259621i
$612$		0			0
$613$			31.9129i			1.28895i	0.764626	+	0.644475i	$0.222924\pi$
$613$			31.9129i			1.28895i	−0.764626	+	0.644475i	$0.777076\pi$
$614$		0			0
$615$	−19.8656			−0.801059
$616$		0			0
$617$	−19.9129			−0.801662			−0.400831	−	0.916152i	$-0.631279\pi$
$617$	−19.9129			−0.801662			−0.400831	+	0.916152i	$0.631279\pi$
$618$		0			0
$619$			17.9274i			0.720561i	0.932844	+	0.360281i	$0.117319\pi$
$619$			17.9274i			0.720561i	−0.932844	+	0.360281i	$0.882681\pi$
$620$		0			0
$621$			14.4272i			0.578945i
$622$		0			0
$623$	9.79796	−	24.0000i	0.392547	−	0.961540i
$624$		0			0
$625$	−26.9129			−1.07652
$626$		0			0
$627$	3.25227	−	2.08712i	0.129883	−	0.0833516i
$628$		0			0
$629$	−13.4048			−0.534483
$630$		0			0
$631$	−13.1652			−0.524096			−0.262048	−	0.965055i	$-0.584398\pi$
$631$	−13.1652			−0.524096			−0.262048	+	0.965055i	$0.584398\pi$
$632$		0			0
$633$	−10.8204			−0.430074
$634$		0			0
$635$	−11.0901			−0.440098
$636$		0			0
$637$	3.23042	+	3.16515i	0.127994	+	0.125408i
$638$		0			0
$639$	−5.16515			−0.204330
$640$		0			0
$641$	−15.9129			−0.628521			−0.314260	−	0.949337i	$-0.601757\pi$
$641$	−15.9129			−0.628521			−0.314260	+	0.949337i	$0.601757\pi$
$642$		0			0
$643$			9.42157i			0.371550i	0.982592	+	0.185775i	$0.0594796\pi$
$643$			9.42157i			0.371550i	−0.982592	+	0.185775i	$0.940520\pi$
$644$		0			0
$645$			14.3303i			0.564255i
$646$		0			0
$647$			30.0681i			1.18210i	0.806635	+	0.591050i	$0.201286\pi$
$647$			30.0681i			1.18210i	−0.806635	+	0.591050i	$0.798714\pi$
$648$		0			0
$649$	−1.80341	+	1.15732i	−0.0707899	+	0.0454289i
$650$		0			0
$651$	−13.6745	−	5.58258i	−0.535944	−	0.218798i
$652$		0			0
$653$	−6.00000			−0.234798			−0.117399	−	0.993085i	$-0.537456\pi$
$653$	−6.00000			−0.234798			−0.117399	+	0.993085i	$0.537456\pi$
$654$		0			0
$655$			51.9129i			2.02840i
$656$		0			0
$657$	41.6413			1.62458
$658$		0			0
$659$			6.33030i			0.246594i	0.992370	+	0.123297i	$0.0393467\pi$
$659$			6.33030i			0.246594i	−0.992370	+	0.123297i	$0.960653\pi$
$660$		0			0
$661$			8.26424i			0.321442i	0.987000	+	0.160721i	$0.0513819\pi$
$661$			8.26424i			0.321442i	−0.987000	+	0.160721i	$0.948618\pi$
$662$		0			0
$663$			1.56186i			0.0606576i
$664$		0			0
$665$	5.58258	−	13.6745i	0.216483	−	0.530273i
$666$		0			0
$667$		−	6.33030i		−	0.245110i
$668$		0			0
$669$	−10.4174			−0.402761
$670$		0			0
$671$	3.47197	+	5.41022i	0.134034	+	0.208859i
$672$		0			0
$673$		−	18.3303i		−	0.706581i	−0.935514	−	0.353291i	$-0.885063\pi$
$673$		−	18.3303i		−	0.706581i	0.935514	−	0.353291i	$-0.114937\pi$
$674$		0			0
$675$			16.5285i			0.636182i
$676$		0			0
$677$	5.27537			0.202749			0.101375	−	0.994848i	$-0.467676\pi$
$677$	5.27537			0.202749			0.101375	+	0.994848i	$0.467676\pi$
$678$		0			0
$679$	−8.50579	+	20.8348i	−0.326422	+	0.799568i
$680$		0			0
$681$		−	0.330303i		−	0.0126572i
$682$		0			0
$683$	9.49545			0.363333			0.181667	−	0.983360i	$-0.441851\pi$
$683$	9.49545			0.363333			0.181667	+	0.983360i	$0.441851\pi$
$684$		0			0
$685$		−	9.79796i		−	0.374361i
$686$		0			0
$687$	−3.66970			−0.140008
$688$		0			0
$689$	−7.48331			−0.285092
$690$		0			0
$691$			3.98320i			0.151528i	0.997126	+	0.0757641i	$0.0241396\pi$
$691$			3.98320i			0.151528i	−0.997126	+	0.0757641i	$0.975860\pi$
$692$		0			0
$693$	−4.12296	−	22.2838i	−0.156618	−	0.846492i
$694$		0			0
$695$			47.9129i			1.81744i
$696$		0			0
$697$	37.1652			1.40773
$698$		0			0
$699$	−17.5510			−0.663838
$700$		0			0
$701$			12.3303i			0.465709i	0.972512	+	0.232855i	$0.0748066\pi$
$701$			12.3303i			0.465709i	−0.972512	+	0.232855i	$0.925193\pi$
$702$		0			0
$703$	6.46084			0.243675
$704$		0			0
$705$			19.8656i			0.748182i
$706$		0			0
$707$	28.7477	+	11.7362i	1.08117	+	0.441386i
$708$		0			0
$709$	52.3303			1.96531			0.982653	−	0.185454i	$-0.0593757\pi$
$709$	52.3303			1.96531			0.982653	+	0.185454i	$0.0593757\pi$
$710$		0			0
$711$			10.3303i			0.387417i
$712$		0			0
$713$		−	34.5625i		−	1.29438i
$714$		0			0
$715$	−5.58258	+	3.58258i	−0.208776	+	0.133981i
$716$		0			0
$717$	5.43836			0.203099
$718$		0			0
$719$			23.3376i			0.870345i	0.900347	+	0.435172i	$0.143313\pi$
$719$			23.3376i			0.870345i	−0.900347	+	0.435172i	$0.856687\pi$
$720$		0			0
$721$	−6.05630	+	14.8348i	−0.225548	+	0.552479i
$722$		0			0
$723$			2.24318i			0.0834248i
$724$		0			0
$725$		−	7.25227i		−	0.269343i
$726$		0			0
$727$		−	37.5514i		−	1.39271i	−0.717700	−	0.696353i	$-0.754805\pi$
$727$		−	37.5514i		−	1.39271i	0.717700	−	0.696353i	$-0.245195\pi$
$728$		0			0
$729$	7.00000			0.259259
$730$		0			0
$731$		−	26.8095i		−	0.991587i
$732$		0			0
$733$	10.7137			0.395721			0.197860	−	0.980230i	$-0.436601\pi$
$733$	10.7137			0.395721			0.197860	+	0.980230i	$0.436601\pi$
$734$		0			0
$735$	10.0000	+	9.79796i	0.368856	+	0.361403i
$736$		0			0
$737$	13.5826	+	21.1652i	0.500321	+	0.779628i
$738$		0			0
$739$		−	1.58258i		−	0.0582160i	−0.999576	−	0.0291080i	$-0.990733\pi$
$739$		−	1.58258i		−	0.0582160i	0.999576	−	0.0291080i	$-0.00926667\pi$
$740$		0			0
$741$		−	0.752785i		−	0.0276543i
$742$		0			0
$743$		−	1.91288i		−	0.0701767i	−0.999384	−	0.0350884i	$-0.988829\pi$
$743$		−	1.91288i		−	0.0701767i	0.999384	−	0.0350884i	$-0.0111713\pi$
$744$		0			0
$745$	43.3380			1.58778
$746$		0			0
$747$	−33.2985			−1.21833
$748$		0			0
$749$	−17.5826	+	43.0683i	−0.642453	+	1.57368i
$750$		0			0
$751$	41.4955			1.51419			0.757095	−	0.653304i	$-0.226618\pi$
$751$	41.4955			1.51419			0.757095	+	0.653304i	$0.226618\pi$
$752$		0			0
$753$	−15.5826			−0.567861
$754$		0			0
$755$	17.2813			0.628930
$756$		0			0
$757$	−30.8348			−1.12071			−0.560356	−	0.828252i	$-0.689336\pi$
$757$	−30.8348			−1.12071			−0.560356	+	0.828252i	$0.689336\pi$
$758$		0			0
$759$	−7.21362	+	4.62929i	−0.261838	+	0.168033i
$760$		0			0
$761$	28.2366			1.02357			0.511787	−	0.859112i	$-0.328984\pi$
$761$	28.2366			1.02357			0.511787	+	0.859112i	$0.328984\pi$
$762$		0			0
$763$	17.1652	−	42.0459i	0.621420	−	1.52216i
$764$		0			0
$765$		−	29.9129i		−	1.08150i
$766$		0			0
$767$			0.417424i			0.0150723i
$768$		0			0
$769$	−15.1015			−0.544573			−0.272287	−	0.962216i	$-0.587780\pi$
$769$	−15.1015			−0.544573			−0.272287	+	0.962216i	$0.587780\pi$
$770$		0			0
$771$	−3.16515			−0.113990
$772$		0			0
$773$			10.0395i			0.361096i	0.983566	+	0.180548i	$0.0577871\pi$
$773$			10.0395i			0.361096i	−0.983566	+	0.180548i	$0.942213\pi$
$774$		0			0
$775$		−	39.5964i		−	1.42234i
$776$		0			0
$777$	−2.31464	+	5.66970i	−0.0830374	+	0.203399i
$778$		0			0
$779$	−17.9129			−0.641795
$780$		0			0
$781$	−3.58258	−	5.58258i	−0.128195	−	0.199760i
$782$		0			0
$783$	5.70805			0.203989
$784$		0			0
$785$	40.7477			1.45435
$786$		0			0
$787$	−23.2309			−0.828091			−0.414046	−	0.910256i	$-0.635885\pi$
$787$	−23.2309			−0.828091			−0.414046	+	0.910256i	$0.635885\pi$
$788$		0			0
$789$	5.38207			0.191607
$790$		0			0
$791$	−26.3264	−	10.7477i	−0.936061	−	0.382145i
$792$		0			0
$793$	1.25227			0.0444695
$794$		0			0
$795$	−23.1652			−0.821583
$796$		0			0
$797$			25.0061i			0.885763i	0.896580	+	0.442881i	$0.146044\pi$
$797$			25.0061i			0.885763i	−0.896580	+	0.442881i	$0.853956\pi$
$798$		0			0
$799$		−	37.1652i		−	1.31481i
$800$		0			0
$801$			25.3040i			0.894072i
$802$		0			0
$803$	28.8826	+	45.0066i	1.01925	+	1.58825i
$804$		0			0
$805$	−12.3823	+	30.3303i	−0.436419	+	1.06900i
$806$		0			0
$807$	−10.6606			−0.375271
$808$		0			0
$809$			14.8348i			0.521566i	0.965398	+	0.260783i	$0.0839806\pi$
$809$			14.8348i			0.521566i	−0.965398	+	0.260783i	$0.916019\pi$
$810$		0			0
$811$	−3.36526			−0.118170			−0.0590852	−	0.998253i	$-0.518818\pi$
$811$	−3.36526			−0.118170			−0.0590852	+	0.998253i	$0.518818\pi$
$812$		0			0
$813$		−	3.16515i		−	0.111007i
$814$		0			0
$815$		−	12.3823i		−	0.433733i
$816$		0			0
$817$			12.9217i			0.452072i
$818$		0			0
$819$	−4.08712	−	1.66856i	−0.142816	−	0.0583042i
$820$		0			0
$821$		−	43.4955i		−	1.51800i	−0.651090	−	0.759001i	$-0.725688\pi$
$821$		−	43.4955i		−	1.51800i	0.651090	−	0.759001i	$-0.274312\pi$
$822$		0			0
$823$	3.66970			0.127918			0.0639588	−	0.997953i	$-0.479627\pi$
$823$	3.66970			0.127918			0.0639588	+	0.997953i	$0.479627\pi$
$824$		0			0
$825$	−8.26424	+	5.30352i	−0.287724	+	0.184645i
$826$		0			0
$827$		−	3.16515i		−	0.110063i	−0.998485	−	0.0550315i	$-0.982474\pi$
$827$		−	3.16515i		−	0.110063i	0.998485	−	0.0550315i	$-0.0175259\pi$
$828$		0			0
$829$			7.99455i			0.277662i	0.990316	+	0.138831i	$0.0443345\pi$
$829$			7.99455i			0.277662i	−0.990316	+	0.138831i	$0.955665\pi$
$830$		0			0
$831$	1.29217			0.0448248
$832$		0			0
$833$	−18.7083	−	18.3303i	−0.648204	−	0.635107i
$834$		0			0
$835$		−	4.00000i		−	0.138426i
$836$		0			0
$837$	31.1652			1.07723
$838$		0			0
$839$			9.66311i			0.333608i	0.985990	+	0.166804i	$0.0533447\pi$
$839$			9.66311i			0.333608i	−0.985990	+	0.166804i	$0.946655\pi$
$840$		0			0
$841$	26.4955			0.913636
$842$		0			0
$843$	7.21362			0.248450
$844$		0			0
$845$		−	38.9503i		−	1.33993i
$846$		0			0
$847$	21.2250	−	19.9123i	0.729299	−	0.684195i
$848$		0			0
$849$		−	11.6697i		−	0.400503i
$850$		0			0
$851$	−14.3303			−0.491236
$852$		0			0
$853$	11.4665			0.392606			0.196303	−	0.980543i	$-0.437106\pi$
$853$	11.4665			0.392606			0.196303	+	0.980543i	$0.437106\pi$
$854$		0			0
$855$			14.4174i			0.493066i
$856$		0			0
$857$	−31.0906			−1.06203			−0.531017	−	0.847361i	$-0.678190\pi$
$857$	−31.0906			−1.06203			−0.531017	+	0.847361i	$0.678190\pi$
$858$		0			0
$859$			24.3882i			0.832115i	0.909338	+	0.416057i	$0.136588\pi$
$859$			24.3882i			0.832115i	−0.909338	+	0.416057i	$0.863412\pi$
$860$		0			0
$861$	6.41742	−	15.7194i	0.218705	−	0.535716i
$862$		0			0
$863$	14.8348			0.504984			0.252492	−	0.967599i	$-0.418750\pi$
$863$	14.8348			0.504984			0.252492	+	0.967599i	$0.418750\pi$
$864$		0			0
$865$			33.1652i			1.12765i
$866$		0			0
$867$			1.93825i			0.0658265i
$868$		0			0
$869$	−11.1652	+	7.16515i	−0.378752	+	0.243061i
$870$		0			0
$871$	4.89898			0.165996
$872$		0			0
$873$		−	21.9668i		−	0.743465i
$874$		0			0
$875$	1.29217	−	3.16515i	0.0436832	−	0.107002i
$876$		0			0
$877$		−	6.83485i		−	0.230796i	−0.993319	−	0.115398i	$-0.963186\pi$
$877$		−	6.83485i		−	0.230796i	0.993319	−	0.115398i	$-0.0368144\pi$
$878$		0			0
$879$		−	16.2432i		−	0.547869i
$880$		0			0
$881$			7.21362i			0.243033i	0.992589	+	0.121517i	$0.0387758\pi$
$881$			7.21362i			0.243033i	−0.992589	+	0.121517i	$0.961224\pi$
$882$		0			0
$883$	−12.8348			−0.431927			−0.215964	−	0.976401i	$-0.569289\pi$
$883$	−12.8348			−0.431927			−0.215964	+	0.976401i	$0.569289\pi$
$884$		0			0
$885$			1.29217i			0.0434357i
$886$		0			0
$887$	53.4057			1.79319			0.896594	−	0.442854i	$-0.146034\pi$
$887$	53.4057			1.79319			0.896594	+	0.442854i	$0.146034\pi$
$888$		0			0
$889$	3.58258	−	8.77548i	0.120156	−	0.294320i
$890$		0			0
$891$	9.70417	+	15.1216i	0.325102	+	0.506592i
$892$		0			0
$893$			17.9129i			0.599432i
$894$		0			0
$895$		−	34.5625i		−	1.15530i
$896$		0			0
$897$			1.66970i			0.0557496i
$898$		0			0
$899$	−13.6745			−0.456069
$900$		0			0
$901$	43.3380			1.44380
$902$		0			0
$903$	−11.3394	−	4.62929i	−0.377351	−	0.154053i
$904$		0			0
$905$	−16.7477			−0.556713
$906$		0			0
$907$	−5.91288			−0.196334			−0.0981670	−	0.995170i	$-0.531298\pi$
$907$	−5.91288			−0.196334			−0.0981670	+	0.995170i	$0.531298\pi$
$908$		0			0
$909$	−30.3097			−1.00531
$910$		0			0
$911$	−3.16515			−0.104866			−0.0524331	−	0.998624i	$-0.516698\pi$
$911$	−3.16515			−0.104866			−0.0524331	+	0.998624i	$0.516698\pi$
$912$		0			0
$913$	−23.0960	−	35.9896i	−0.764367	−	1.19108i
$914$		0			0
$915$	3.87650			0.128153
$916$		0			0
$917$	−41.0780	−	16.7700i	−1.35652	−	0.553795i
$918$		0			0
$919$		−	19.4955i		−	0.643096i	−0.946893	−	0.321548i	$-0.895797\pi$
$919$		−	19.4955i		−	0.643096i	0.946893	−	0.321548i	$-0.104203\pi$
$920$		0			0
$921$		−	6.83485i		−	0.225216i
$922$		0			0
$923$	−1.29217			−0.0425322
$924$		0			0
$925$	−16.4174			−0.539802
$926$		0			0
$927$		−	15.6409i		−	0.513713i
$928$		0			0
$929$			43.6077i			1.43072i	0.698755	+	0.715361i	$0.253738\pi$
$929$			43.6077i			1.43072i	−0.698755	+	0.715361i	$0.746262\pi$
$930$		0			0
$931$	9.01703	+	8.83485i	0.295521	+	0.289550i
$932$		0			0
$933$	−3.25227			−0.106475
$934$		0			0
$935$	32.3303	−	20.7477i	1.05731	−	0.678523i
$936$		0			0
$937$	30.0681			0.982282			0.491141	−	0.871080i	$-0.336580\pi$
$937$	30.0681			0.982282			0.491141	+	0.871080i	$0.336580\pi$
$938$		0			0
$939$	−15.1652			−0.494896
$940$		0			0
$941$	−5.27537			−0.171972			−0.0859861	−	0.996296i	$-0.527404\pi$
$941$	−5.27537			−0.171972			−0.0859861	+	0.996296i	$0.527404\pi$
$942$		0			0
$943$	39.7312			1.29383
$944$		0			0
$945$	−27.3489	−	11.1652i	−0.889661	−	0.363202i
$946$		0			0
$947$	−3.58258			−0.116418			−0.0582090	−	0.998304i	$-0.518539\pi$
$947$	−3.58258			−0.116418			−0.0582090	+	0.998304i	$0.518539\pi$
$948$		0			0
$949$	10.4174			0.338164
$950$		0			0
$951$		−	5.92146i		−	0.192016i
$952$		0			0
$953$		−	18.3303i		−	0.593777i	−0.954912	−	0.296888i	$-0.904051\pi$
$953$		−	18.3303i		−	0.593777i	0.954912	−	0.296888i	$-0.0959489\pi$
$954$		0			0
$955$			55.7203i			1.80307i
$956$		0			0
$957$	1.83155	+	2.85403i	0.0592056	+	0.0922576i
$958$		0			0
$959$	7.75301	+	3.16515i	0.250358	+	0.102208i
$960$		0			0
$961$	−43.6606			−1.40841
$962$		0			0
$963$		−	45.4083i		−	1.46326i
$964$		0			0
$965$	−50.5517			−1.62732
$966$		0			0
$967$			22.4174i			0.720896i	0.932779	+	0.360448i	$0.117376\pi$
$967$			22.4174i			0.720896i	−0.932779	+	0.360448i	$0.882624\pi$
$968$		0			0
$969$			4.35960i			0.140050i
$970$		0			0
$971$		−	14.0509i		−	0.450913i	−0.974253	−	0.225457i	$-0.927613\pi$
$971$		−	14.0509i		−	0.450913i	0.974253	−	0.225457i	$-0.0723874\pi$
$972$		0			0
$973$	−37.9129	−	15.4779i	−1.21543	−	0.496198i
$974$		0			0
$975$			1.91288i			0.0612611i
$976$		0			0
$977$	9.16515			0.293219			0.146610	−	0.989194i	$-0.453164\pi$
$977$	9.16515			0.293219			0.146610	+	0.989194i	$0.453164\pi$
$978$		0			0
$979$	−27.3489	+	17.5510i	−0.874075	+	0.560931i
$980$		0			0
$981$			44.3303i			1.41536i
$982$		0			0
$983$		−	50.6865i		−	1.61665i	−0.588738	−	0.808324i	$-0.700375\pi$
$983$		−	50.6865i		−	1.61665i	0.588738	−	0.808324i	$-0.299625\pi$
$984$		0			0
$985$	−27.0792			−0.862816
$986$		0			0
$987$	−15.7194	−	6.41742i	−0.500354	−	0.204269i
$988$		0			0
$989$		−	28.6606i		−	0.911354i
$990$		0			0
$991$	−46.6606			−1.48222			−0.741111	−	0.671382i	$-0.765701\pi$
$991$	−46.6606			−1.48222			−0.741111	+	0.671382i	$0.765701\pi$
$992$		0			0
$993$		−	15.2363i		−	0.483510i
$994$		0			0
$995$	−29.9129			−0.948302
$996$		0			0
$997$	22.5566			0.714376			0.357188	−	0.934032i	$-0.383736\pi$
$997$	22.5566			0.714376			0.357188	+	0.934032i	$0.383736\pi$
$998$		0			0
$999$		−	12.9217i		−	0.408824i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1232.2.e.e.769.3		8
4.3	odd	2		154.2.c.a.153.3	yes	8
7.6	odd	2	inner	1232.2.e.e.769.6		8
11.10	odd	2	inner	1232.2.e.e.769.4		8
12.11	even	2		1386.2.e.b.307.5		8
28.3	even	6		1078.2.i.b.901.6		16
28.11	odd	6		1078.2.i.b.901.7		16
28.19	even	6		1078.2.i.b.1011.3		16
28.23	odd	6		1078.2.i.b.1011.2		16
28.27	even	2		154.2.c.a.153.2	✓	8
44.43	even	2		154.2.c.a.153.7	yes	8
77.76	even	2	inner	1232.2.e.e.769.5		8
84.83	odd	2		1386.2.e.b.307.8		8
132.131	odd	2		1386.2.e.b.307.1		8
308.87	odd	6		1078.2.i.b.901.2		16
308.131	odd	6		1078.2.i.b.1011.7		16
308.219	even	6		1078.2.i.b.1011.6		16
308.263	even	6		1078.2.i.b.901.3		16
308.307	odd	2		154.2.c.a.153.6	yes	8
924.923	even	2		1386.2.e.b.307.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
154.2.c.a.153.2	✓	8	28.27	even	2
154.2.c.a.153.3	yes	8	4.3	odd	2
154.2.c.a.153.6	yes	8	308.307	odd	2
154.2.c.a.153.7	yes	8	44.43	even	2
1078.2.i.b.901.2		16	308.87	odd	6
1078.2.i.b.901.3		16	308.263	even	6
1078.2.i.b.901.6		16	28.3	even	6
1078.2.i.b.901.7		16	28.11	odd	6
1078.2.i.b.1011.2		16	28.23	odd	6
1078.2.i.b.1011.3		16	28.19	even	6
1078.2.i.b.1011.6		16	308.219	even	6
1078.2.i.b.1011.7		16	308.131	odd	6
1232.2.e.e.769.3		8	1.1	even	1	trivial
1232.2.e.e.769.4		8	11.10	odd	2	inner
1232.2.e.e.769.5		8	77.76	even	2	inner
1232.2.e.e.769.6		8	7.6	odd	2	inner
1386.2.e.b.307.1		8	132.131	odd	2
1386.2.e.b.307.4		8	924.923	even	2
1386.2.e.b.307.5		8	12.11	even	2
1386.2.e.b.307.8		8	84.83	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1232 = 2^{4} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1232.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-0.646084i q^{3} +3.09557i q^{5} +(-2.44949 - 1.00000i) q^{7} +2.58258 q^{9} +O(q^{10})\)	\(q-0.646084i q^{3} +3.09557i q^{5} +(-2.44949 - 1.00000i) q^{7} +2.58258 q^{9} +(1.79129 + 2.79129i) q^{11} +0.646084 q^{13} +2.00000 q^{15} -3.74166 q^{17} +1.80341 q^{19} +(-0.646084 + 1.58258i) q^{21} -4.00000 q^{23} -4.58258 q^{25} -3.60681i q^{27} +1.58258i q^{29} +8.64064i q^{31} +(1.80341 - 1.15732i) q^{33} +(3.09557 - 7.58258i) q^{35} +3.58258 q^{37} -0.417424i q^{39} -9.93280 q^{41} +7.16515i q^{43} +7.99455i q^{45} +9.93280i q^{47} +(5.00000 + 4.89898i) q^{49} +2.41742i q^{51} -11.5826 q^{53} +(-8.64064 + 5.54506i) q^{55} -1.16515i q^{57} +0.646084i q^{59} +1.93825 q^{61} +(-6.32599 - 2.58258i) q^{63} +2.00000i q^{65} +7.58258 q^{67} +2.58434i q^{69} -2.00000 q^{71} +16.1240 q^{73} +2.96073i q^{75} +(-1.59645 - 8.62852i) q^{77} +4.00000i q^{79} +5.41742 q^{81} -12.8935 q^{83} -11.5826i q^{85} +1.02248 q^{87} +9.79796i q^{89} +(-1.58258 - 0.646084i) q^{91} +5.58258 q^{93} +5.58258i q^{95} -8.50579i q^{97} +(4.62614 + 7.20871i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 16 q^{9}+O(q^{10}) \) 8 * q - 16 * q^9	\( 8 q - 16 q^{9} - 4 q^{11} + 16 q^{15} - 32 q^{23} - 8 q^{37} + 40 q^{49} - 56 q^{53} + 24 q^{67} - 16 q^{71} + 4 q^{77} + 80 q^{81} + 24 q^{91} + 8 q^{93} + 92 q^{99}+O(q^{100}) \) 8 * q - 16 * q^9 - 4 * q^11 + 16 * q^15 - 32 * q^23 - 8 * q^37 + 40 * q^49 - 56 * q^53 + 24 * q^67 - 16 * q^71 + 4 * q^77 + 80 * q^81 + 24 * q^91 + 8 * q^93 + 92 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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