LMFDB - Embedded newform 208.10.f.d.129.29

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [208,10,Mod(129,208)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("208.129"); S:= CuspForms(chi, 10); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(208, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 10, names="a")

Level:	$ N $	$=$	$ 208 = 2^{4} \cdot 13 $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	208.f (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [32,0,162] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	no
Analytic conductor:	$107.127453922$
Analytic rank:	$0$
Dimension:	$32$
Twist minimal:	no (minimal twist has level 104)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			129.29
Character	$\chi$	$=$	208.129
Dual form			208.10.f.d.129.30

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+241.055 q^{3} -999.605i q^{5} +3393.35i q^{7} +38424.4 q^{9} +54529.9i q^{11} +(-99437.9 - 26769.3i) q^{13} -240960. i q^{15} +643353. q^{17} -1.06231e6i q^{19} +817984. i q^{21} -781294. q^{23} +953914. q^{25} +4.51771e6 q^{27} +1.44821e6 q^{29} -453270. i q^{31} +1.31447e7i q^{33} +3.39202e6 q^{35} +1.33508e7i q^{37} +(-2.39700e7 - 6.45288e6i) q^{39} -2.91030e7i q^{41} +1.95194e7 q^{43} -3.84093e7i q^{45} +5.93105e7i q^{47} +2.88388e7 q^{49} +1.55083e8 q^{51} +9.92934e7 q^{53} +5.45084e7 q^{55} -2.56076e8i q^{57} +3.96876e7i q^{59} +9.22380e7 q^{61} +1.30388e8i q^{63} +(-2.67588e7 + 9.93987e7i) q^{65} -1.69984e8i q^{67} -1.88335e8 q^{69} +7.66552e7i q^{71} -2.76853e8i q^{73} +2.29946e8 q^{75} -1.85039e8 q^{77} -2.12394e8 q^{79} +3.32709e8 q^{81} +1.34193e8i q^{83} -6.43100e8i q^{85} +3.49098e8 q^{87} +5.95176e8i q^{89} +(9.08378e7 - 3.37428e8i) q^{91} -1.09263e8i q^{93} -1.06189e9 q^{95} +3.79670e8i q^{97} +2.09528e9i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q + 162 q^{3} + 223074 q^{9} + 66270 q^{13} - 487902 q^{17} - 3171556 q^{23} - 13526722 q^{25} + 3694974 q^{27} + 8833508 q^{29} + 8281126 q^{35} + 12056860 q^{39} - 89959038 q^{43} - 172344874 q^{49}+ \cdots - 1741143356 q^{95}+O(q^{100}) $ 32 * q + 162 * q^3 + 223074 * q^9 + 66270 * q^13 - 487902 * q^17 - 3171556 * q^23 - 13526722 * q^25 + 3694974 * q^27 + 8833508 * q^29 + 8281126 * q^35 + 12056860 * q^39 - 89959038 * q^43 - 172344874 * q^49 - 51206562 * q^51 + 139521112 * q^53 + 84550176 * q^55 + 372036836 * q^61 + 241471810 * q^65 - 486052516 * q^69 + 604473728 * q^75 - 1666197144 * q^77 - 1165982916 * q^79 + 859636544 * q^81 + 1738691296 * q^87 + 291718146 * q^91 - 1741143356 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/208\mathbb{Z}\right)^\times$.

$n$	$53$	$79$	$145$
$\chi(n)$	$1$	$1$	$-1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	241.055			1.71819			0.859093	−	0.511819i	$-0.171028\pi$
$3$	241.055			1.71819			0.859093	+	0.511819i	$0.171028\pi$
$4$		0			0
$5$		−	999.605i		−	0.715259i	−0.933863	−	0.357630i	$-0.883585\pi$
$5$		−	999.605i		−	0.715259i	0.933863	−	0.357630i	$-0.116415\pi$
$6$		0			0
$7$			3393.35i			0.534180i	0.963672	+	0.267090i	$0.0860622\pi$
$7$			3393.35i			0.534180i	−0.963672	+	0.267090i	$0.913938\pi$
$8$		0			0
$9$	38424.4			1.95216
$10$		0			0
$11$			54529.9i			1.12297i	0.827487	+	0.561485i	$0.189770\pi$
$11$			54529.9i			1.12297i	−0.827487	+	0.561485i	$0.810230\pi$
$12$		0			0
$13$	−99437.9	−	26769.3i	−0.965622	−	0.259952i
$13$	−99437.9	−	26769.3i	−0.965622	−	0.259952i
$14$		0			0
$15$		−	240960.i		−	1.22895i
$16$		0			0
$17$	643353.			1.86823			0.934113	−	0.356977i	$-0.116193\pi$
$17$	643353.			1.86823			0.934113	+	0.356977i	$0.116193\pi$
$18$		0			0
$19$		−	1.06231e6i		−	1.87008i	−0.354537	−	0.935042i	$-0.615362\pi$
$19$		−	1.06231e6i		−	1.87008i	0.354537	−	0.935042i	$-0.384638\pi$
$20$		0			0
$21$			817984.i			0.917822i
$22$		0			0
$23$	−781294.			−0.582156			−0.291078	−	0.956699i	$-0.594014\pi$
$23$	−781294.			−0.582156			−0.291078	+	0.956699i	$0.594014\pi$
$24$		0			0
$25$	953914.			0.488404
$26$		0			0
$27$	4.51771e6			1.63599
$28$		0			0
$29$	1.44821e6			0.380225			0.190112	−	0.981762i	$-0.439115\pi$
$29$	1.44821e6			0.380225			0.190112	+	0.981762i	$0.439115\pi$
$30$		0			0
$31$		−	453270.i		−	0.0881514i	−0.999028	−	0.0440757i	$-0.985966\pi$
$31$		−	453270.i		−	0.0881514i	0.999028	−	0.0440757i	$-0.0140343\pi$
$32$		0			0
$33$			1.31447e7i			1.92947i
$34$		0			0
$35$	3.39202e6			0.382078
$36$		0			0
$37$			1.33508e7i			1.17111i	0.810631	+	0.585557i	$0.199124\pi$
$37$			1.33508e7i			1.17111i	−0.810631	+	0.585557i	$0.800876\pi$
$38$		0			0
$39$	−2.39700e7	−	6.45288e6i	−1.65912	−	0.446645i
$40$		0			0
$41$		−	2.91030e7i		−	1.60846i	−0.594319	−	0.804230i	$-0.702578\pi$
$41$		−	2.91030e7i		−	1.60846i	0.594319	−	0.804230i	$-0.297422\pi$
$42$		0			0
$43$	1.95194e7			0.870680			0.435340	−	0.900266i	$-0.356628\pi$
$43$	1.95194e7			0.870680			0.435340	+	0.900266i	$0.356628\pi$
$44$		0			0
$45$		−	3.84093e7i		−	1.39630i
$46$		0			0
$47$			5.93105e7i			1.77293i	0.462798	+	0.886464i	$0.346845\pi$
$47$			5.93105e7i			1.77293i	−0.462798	+	0.886464i	$0.653155\pi$
$48$		0			0
$49$	2.88388e7			0.714651
$50$		0			0
$51$	1.55083e8			3.20996
$52$		0			0
$53$	9.92934e7			1.72854			0.864269	−	0.503029i	$-0.167781\pi$
$53$	9.92934e7			1.72854			0.864269	+	0.503029i	$0.167781\pi$
$54$		0			0
$55$	5.45084e7			0.803215
$56$		0			0
$57$		−	2.56076e8i		−	3.21315i
$58$		0			0
$59$			3.96876e7i			0.426403i	0.977008	+	0.213202i	$0.0683891\pi$
$59$			3.96876e7i			0.426403i	−0.977008	+	0.213202i	$0.931611\pi$
$60$		0			0
$61$	9.22380e7			0.852954			0.426477	−	0.904498i	$-0.359754\pi$
$61$	9.22380e7			0.852954			0.426477	+	0.904498i	$0.359754\pi$
$62$		0			0
$63$			1.30388e8i			1.04281i
$64$		0			0
$65$	−2.67588e7	+	9.93987e7i	−0.185933	+	0.690670i
$66$		0			0
$67$		−	1.69984e8i		−	1.03055i	−0.857024	−	0.515276i	$-0.827689\pi$
$67$		−	1.69984e8i		−	1.03055i	0.857024	−	0.515276i	$-0.172311\pi$
$68$		0			0
$69$	−1.88335e8			−1.00025
$70$		0			0
$71$			7.66552e7i			0.357996i	0.983849	+	0.178998i	$0.0572856\pi$
$71$			7.66552e7i			0.357996i	−0.983849	+	0.178998i	$0.942714\pi$
$72$		0			0
$73$		−	2.76853e8i		−	1.14103i	−0.821287	−	0.570515i	$-0.806744\pi$
$73$		−	2.76853e8i		−	1.14103i	0.821287	−	0.570515i	$-0.193256\pi$
$74$		0			0
$75$	2.29946e8			0.839169
$76$		0			0
$77$	−1.85039e8			−0.599868
$78$		0			0
$79$	−2.12394e8			−0.613508			−0.306754	−	0.951789i	$-0.599243\pi$
$79$	−2.12394e8			−0.613508			−0.306754	+	0.951789i	$0.599243\pi$
$80$		0			0
$81$	3.32709e8			0.858779
$82$		0			0
$83$			1.34193e8i			0.310368i	0.987886	+	0.155184i	$0.0495971\pi$
$83$			1.34193e8i			0.310368i	−0.987886	+	0.155184i	$0.950403\pi$
$84$		0			0
$85$		−	6.43100e8i		−	1.33627i
$86$		0			0
$87$	3.49098e8			0.653297
$88$		0			0
$89$			5.95176e8i			1.00552i	0.864426	+	0.502759i	$0.167682\pi$
$89$			5.95176e8i			1.00552i	−0.864426	+	0.502759i	$0.832318\pi$
$90$		0			0
$91$	9.08378e7	−	3.37428e8i	0.138861	−	0.515816i
$92$		0			0
$93$		−	1.09263e8i		−	0.151461i
$94$		0			0
$95$	−1.06189e9			−1.33759
$96$		0			0
$97$			3.79670e8i			0.435445i	0.976011	+	0.217723i	$0.0698628\pi$
$97$			3.79670e8i			0.435445i	−0.976011	+	0.217723i	$0.930137\pi$
$98$		0			0
$99$			2.09528e9i			2.19222i
$100$		0			0
$101$	5.16174e7			0.0493571			0.0246786	−	0.999695i	$-0.492144\pi$
$101$	5.16174e7			0.0493571			0.0246786	+	0.999695i	$0.492144\pi$
$102$		0			0
$103$	1.30859e9			1.14561			0.572804	−	0.819692i	$-0.305856\pi$
$103$	1.30859e9			1.14561			0.572804	+	0.819692i	$0.305856\pi$
$104$		0			0
$105$	8.17662e8			0.656481
$106$		0			0
$107$	2.93978e8			0.216814			0.108407	−	0.994107i	$-0.465425\pi$
$107$	2.93978e8			0.216814			0.108407	+	0.994107i	$0.465425\pi$
$108$		0			0
$109$		−	1.71333e9i		−	1.16258i	−0.813698	−	0.581288i	$-0.802549\pi$
$109$		−	1.71333e9i		−	1.16258i	0.813698	−	0.581288i	$-0.197451\pi$
$110$		0			0
$111$			3.21827e9i			2.01219i
$112$		0			0
$113$	−1.39521e9			−0.804983			−0.402492	−	0.915424i	$-0.631856\pi$
$113$	−1.39521e9			−0.804983			−0.402492	+	0.915424i	$0.631856\pi$
$114$		0			0
$115$			7.80986e8i			0.416392i
$116$		0			0
$117$	−3.82085e9	−	1.02860e9i	−1.88505	−	0.507468i
$118$		0			0
$119$			2.18313e9i			0.997970i
$120$		0			0
$121$	−6.15568e8			−0.261061
$122$		0			0
$123$		−	7.01541e9i		−	2.76363i
$124$		0			0
$125$		−	2.90589e9i		−	1.06459i
$126$		0			0
$127$	1.91222e9			0.652260			0.326130	−	0.945325i	$-0.394255\pi$
$127$	1.91222e9			0.652260			0.326130	+	0.945325i	$0.394255\pi$
$128$		0			0
$129$	4.70525e9			1.49599
$130$		0			0
$131$	−2.29794e9			−0.681739			−0.340869	−	0.940111i	$-0.610721\pi$
$131$	−2.29794e9			−0.681739			−0.340869	+	0.940111i	$0.610721\pi$
$132$		0			0
$133$	3.60480e9			0.998962
$134$		0			0
$135$		−	4.51593e9i		−	1.17016i
$136$		0			0
$137$		−	3.65972e9i		−	0.887574i	−0.896132	−	0.443787i	$-0.853635\pi$
$137$		−	3.65972e9i		−	0.887574i	0.896132	−	0.443787i	$-0.146365\pi$
$138$		0			0
$139$	−3.01567e9			−0.685200			−0.342600	−	0.939481i	$-0.611308\pi$
$139$	−3.01567e9			−0.685200			−0.342600	+	0.939481i	$0.611308\pi$
$140$		0			0
$141$			1.42971e10i			3.04622i
$142$		0			0
$143$	1.45973e9	−	5.42235e9i	0.291918	−	1.08436i
$144$		0			0
$145$		−	1.44764e9i		−	0.271959i
$146$		0			0
$147$	6.95172e9			1.22790
$148$		0			0
$149$			4.56748e9i			0.759168i	0.925157	+	0.379584i	$0.123933\pi$
$149$			4.56748e9i			0.759168i	−0.925157	+	0.379584i	$0.876067\pi$
$150$		0			0
$151$		−	6.66876e9i		−	1.04388i	−0.852984	−	0.521938i	$-0.825209\pi$
$151$		−	6.66876e9i		−	1.04388i	0.852984	−	0.521938i	$-0.174791\pi$
$152$		0			0
$153$	2.47205e10			3.64708
$154$		0			0
$155$	−4.53091e8			−0.0630511
$156$		0			0
$157$	−1.44990e9			−0.190454			−0.0952269	−	0.995456i	$-0.530358\pi$
$157$	−1.44990e9			−0.190454			−0.0952269	+	0.995456i	$0.530358\pi$
$158$		0			0
$159$	2.39352e10			2.96995
$160$		0			0
$161$		−	2.65121e9i		−	0.310976i
$162$		0			0
$163$		−	1.29657e9i		−	0.143864i	−0.997410	−	0.0719322i	$-0.977083\pi$
$163$		−	1.29657e9i		−	0.143864i	0.997410	−	0.0719322i	$-0.0229165\pi$
$164$		0			0
$165$	1.31395e10			1.38007
$166$		0			0
$167$			3.52591e9i			0.350790i	0.984498	+	0.175395i	$0.0561203\pi$
$167$			3.52591e9i			0.350790i	−0.984498	+	0.175395i	$0.943880\pi$
$168$		0			0
$169$	9.17131e9	+	5.32377e9i	0.864850	+	0.502030i
$170$		0			0
$171$		−	4.08188e10i		−	3.65071i
$172$		0			0
$173$	9.73757e9			0.826500			0.413250	−	0.910618i	$-0.364394\pi$
$173$	9.73757e9			0.826500			0.413250	+	0.910618i	$0.364394\pi$
$174$		0			0
$175$			3.23697e9i			0.260896i
$176$		0			0
$177$			9.56688e9i			0.732640i
$178$		0			0
$179$	1.71262e10			1.24687			0.623436	−	0.781874i	$-0.285736\pi$
$179$	1.71262e10			1.24687			0.623436	+	0.781874i	$0.285736\pi$
$180$		0			0
$181$	−2.14459e10			−1.48522			−0.742611	−	0.669723i	$-0.766413\pi$
$181$	−2.14459e10			−1.48522			−0.742611	+	0.669723i	$0.766413\pi$
$182$		0			0
$183$	2.22344e10			1.46553
$184$		0			0
$185$	1.33455e10			0.837650
$186$		0			0
$187$			3.50820e10i			2.09796i
$188$		0			0
$189$			1.53302e10i			0.873916i
$190$		0			0
$191$	−1.37072e10			−0.745242			−0.372621	−	0.927984i	$-0.621541\pi$
$191$	−1.37072e10			−0.745242			−0.372621	+	0.927984i	$0.621541\pi$
$192$		0			0
$193$		−	9.25493e9i		−	0.480137i	−0.970756	−	0.240069i	$-0.922830\pi$
$193$		−	9.25493e9i		−	0.480137i	0.970756	−	0.240069i	$-0.0771699\pi$
$194$		0			0
$195$	−6.45033e9	+	2.39605e10i	−0.319467	+	1.18670i
$196$		0			0
$197$		−	3.37885e10i		−	1.59835i	−0.601101	−	0.799173i	$-0.705271\pi$
$197$		−	3.37885e10i		−	1.59835i	0.601101	−	0.799173i	$-0.294729\pi$
$198$		0			0
$199$	−1.41480e10			−0.639524			−0.319762	−	0.947498i	$-0.603603\pi$
$199$	−1.41480e10			−0.639524			−0.319762	+	0.947498i	$0.603603\pi$
$200$		0			0
$201$		−	4.09754e10i		−	1.77068i
$202$		0			0
$203$			4.91429e9i			0.203109i
$204$		0			0
$205$	−2.90915e10			−1.15047
$206$		0			0
$207$	−3.00208e10			−1.13646
$208$		0			0
$209$	5.79278e10			2.10005
$210$		0			0
$211$	4.50949e10			1.56623			0.783116	−	0.621875i	$-0.213629\pi$
$211$	4.50949e10			1.56623			0.783116	+	0.621875i	$0.213629\pi$
$212$		0			0
$213$			1.84781e10i			0.615105i
$214$		0			0
$215$		−	1.95117e10i		−	0.622762i
$216$		0			0
$217$	1.53811e9			0.0470888
$218$		0			0
$219$		−	6.67369e10i		−	1.96050i
$220$		0			0
$221$	−6.39737e10	−	1.72221e10i	−1.80400	−	0.485648i
$222$		0			0
$223$			1.41963e10i			0.384417i	0.981354	+	0.192208i	$0.0615650\pi$
$223$			1.41963e10i			0.384417i	−0.981354	+	0.192208i	$0.938435\pi$
$224$		0			0
$225$	3.66536e10			0.953444
$226$		0			0
$227$			3.49962e10i			0.874791i	0.899269	+	0.437395i	$0.144099\pi$
$227$			3.49962e10i			0.874791i	−0.899269	+	0.437395i	$0.855901\pi$
$228$		0			0
$229$			7.84361e10i			1.88476i	0.334541	+	0.942381i	$0.391419\pi$
$229$			7.84361e10i			1.88476i	−0.334541	+	0.942381i	$0.608581\pi$
$230$		0			0
$231$	−4.46047e10			−1.03069
$232$		0			0
$233$	−2.76296e10			−0.614147			−0.307074	−	0.951686i	$-0.599350\pi$
$233$	−2.76296e10			−0.614147			−0.307074	+	0.951686i	$0.599350\pi$
$234$		0			0
$235$	5.92871e10			1.26810
$236$		0			0
$237$	−5.11986e10			−1.05412
$238$		0			0
$239$			2.97447e10i			0.589683i	0.955546	+	0.294842i	$0.0952669\pi$
$239$			2.97447e10i			0.589683i	−0.955546	+	0.294842i	$0.904733\pi$
$240$		0			0
$241$		−	3.45661e10i		−	0.660046i	−0.943973	−	0.330023i	$-0.892944\pi$
$241$		−	3.45661e10i		−	0.660046i	0.943973	−	0.330023i	$-0.107056\pi$
$242$		0			0
$243$	−8.72113e9			−0.160452
$244$		0			0
$245$		−	2.88274e10i		−	0.511161i
$246$		0			0
$247$	−2.84374e10	+	1.05634e11i	−0.486131	+	1.80579i
$248$		0			0
$249$			3.23478e10i			0.533271i
$250$		0			0
$251$	−4.66764e10			−0.742276			−0.371138	−	0.928578i	$-0.621032\pi$
$251$	−4.66764e10			−0.742276			−0.371138	+	0.928578i	$0.621032\pi$
$252$		0			0
$253$		−	4.26039e10i		−	0.653743i
$254$		0			0
$255$		−	1.55022e11i		−	2.29595i
$256$		0			0
$257$	−3.62948e10			−0.518973			−0.259487	−	0.965747i	$-0.583553\pi$
$257$	−3.62948e10			−0.518973			−0.259487	+	0.965747i	$0.583553\pi$
$258$		0			0
$259$	−4.53039e10			−0.625586
$260$		0			0
$261$	5.56466e10			0.742261
$262$		0			0
$263$	5.40730e9			0.0696914			0.0348457	−	0.999393i	$-0.488906\pi$
$263$	5.40730e9			0.0696914			0.0348457	+	0.999393i	$0.488906\pi$
$264$		0			0
$265$		−	9.92543e10i		−	1.23635i
$266$		0			0
$267$			1.43470e11i			1.72767i
$268$		0			0
$269$	−1.36543e11			−1.58995			−0.794977	−	0.606639i	$-0.792517\pi$
$269$	−1.36543e11			−1.58995			−0.794977	+	0.606639i	$0.792517\pi$
$270$		0			0
$271$			7.07968e10i			0.797355i	0.917091	+	0.398678i	$0.130531\pi$
$271$			7.07968e10i			0.797355i	−0.917091	+	0.398678i	$0.869469\pi$
$272$		0			0
$273$	2.18969e10	−	8.13387e10i	0.238589	−	0.886268i
$274$		0			0
$275$			5.20169e10i			0.548463i
$276$		0			0
$277$	−1.45023e9			−0.0148005			−0.00740027	−	0.999973i	$-0.502356\pi$
$277$	−1.45023e9			−0.0148005			−0.00740027	+	0.999973i	$0.502356\pi$
$278$		0			0
$279$		−	1.74166e10i		−	0.172086i
$280$		0			0
$281$			1.15279e11i			1.10299i	0.834179	+	0.551494i	$0.185942\pi$
$281$			1.15279e11i			1.10299i	−0.834179	+	0.551494i	$0.814058\pi$
$282$		0			0
$283$	−7.85405e10			−0.727872			−0.363936	−	0.931424i	$-0.618567\pi$
$283$	−7.85405e10			−0.727872			−0.363936	+	0.931424i	$0.618567\pi$
$284$		0			0
$285$	−2.55975e11			−2.29824
$286$		0			0
$287$	9.87567e10			0.859208
$288$		0			0
$289$	2.95316e11			2.49027
$290$		0			0
$291$			9.15213e10i			0.748176i
$292$		0			0
$293$		−	1.83406e11i		−	1.45382i	−0.686734	−	0.726909i	$-0.740956\pi$
$293$		−	1.83406e11i		−	1.45382i	0.686734	−	0.726909i	$-0.259044\pi$
$294$		0			0
$295$	3.96719e10			0.304989
$296$		0			0
$297$			2.46351e11i			1.83717i
$298$		0			0
$299$	7.76902e10	+	2.09147e10i	0.562142	+	0.151332i
$300$		0			0
$301$			6.62363e10i			0.465100i
$302$		0			0
$303$	1.24426e10			0.0848047
$304$		0			0
$305$		−	9.22016e10i		−	0.610084i
$306$		0			0
$307$			1.82595e11i			1.17319i	0.809882	+	0.586593i	$0.199531\pi$
$307$			1.82595e11i			1.17319i	−0.809882	+	0.586593i	$0.800469\pi$
$308$		0			0
$309$	3.15442e11			1.96837
$310$		0			0
$311$	−2.19529e10			−0.133067			−0.0665335	−	0.997784i	$-0.521194\pi$
$311$	−2.19529e10			−0.133067			−0.0665335	+	0.997784i	$0.521194\pi$
$312$		0			0
$313$	−1.74741e11			−1.02907			−0.514535	−	0.857469i	$-0.672036\pi$
$313$	−1.74741e11			−1.02907			−0.514535	+	0.857469i	$0.672036\pi$
$314$		0			0
$315$	1.30336e11			0.745878
$316$		0			0
$317$		−	2.20673e11i		−	1.22739i	−0.789544	−	0.613695i	$-0.789683\pi$
$317$		−	2.20673e11i		−	1.22739i	0.789544	−	0.613695i	$-0.210317\pi$
$318$		0			0
$319$			7.89707e10i			0.426981i
$320$		0			0
$321$	7.08649e10			0.372527
$322$		0			0
$323$		−	6.83442e11i		−	3.49374i
$324$		0			0
$325$	−9.48552e10	−	2.55356e10i	−0.471613	−	0.126961i
$326$		0			0
$327$		−	4.13006e11i		−	1.99752i
$328$		0			0
$329$	−2.01261e11			−0.947063
$330$		0			0
$331$		−	2.69594e11i		−	1.23448i	−0.786774	−	0.617241i	$-0.788250\pi$
$331$		−	2.69594e11i		−	1.23448i	0.786774	−	0.617241i	$-0.211750\pi$
$332$		0			0
$333$			5.12996e11i			2.28620i
$334$		0			0
$335$	−1.69916e11			−0.737113
$336$		0			0
$337$	6.80145e10			0.287255			0.143627	−	0.989632i	$-0.454123\pi$
$337$	6.80145e10			0.287255			0.143627	+	0.989632i	$0.454123\pi$
$338$		0			0
$339$	−3.36322e11			−1.38311
$340$		0			0
$341$	2.47168e10			0.0989914
$342$		0			0
$343$			2.34794e11i			0.915933i
$344$		0			0
$345$			1.88260e11i			0.715440i
$346$		0			0
$347$	−3.28948e11			−1.21799			−0.608997	−	0.793173i	$-0.708428\pi$
$347$	−3.28948e11			−1.21799			−0.608997	+	0.793173i	$0.708428\pi$
$348$		0			0
$349$			3.68799e11i			1.33068i	0.746539	+	0.665342i	$0.231714\pi$
$349$			3.68799e11i			1.33068i	−0.746539	+	0.665342i	$0.768286\pi$
$350$		0			0
$351$	−4.49232e11	−	1.20936e11i	−1.57975	−	0.425279i
$352$		0			0
$353$			5.28320e11i			1.81097i	0.424381	+	0.905484i	$0.360492\pi$
$353$			5.28320e11i			1.81097i	−0.424381	+	0.905484i	$0.639508\pi$
$354$		0			0
$355$	7.66249e10			0.256060
$356$		0			0
$357$			5.26253e11i			1.71470i
$358$		0			0
$359$			3.00778e11i			0.955697i	0.878442	+	0.477849i	$0.158583\pi$
$359$			3.00778e11i			0.955697i	−0.878442	+	0.477849i	$0.841417\pi$
$360$		0			0
$361$	−8.05820e11			−2.49721
$362$		0			0
$363$	−1.48386e11			−0.448551
$364$		0			0
$365$	−2.76744e11			−0.816133
$366$		0			0
$367$	−4.73248e11			−1.36173			−0.680866	−	0.732408i	$-0.738397\pi$
$367$	−4.73248e11			−1.36173			−0.680866	+	0.732408i	$0.738397\pi$
$368$		0			0
$369$		−	1.11827e12i		−	3.13998i
$370$		0			0
$371$			3.36938e11i			0.923352i
$372$		0			0
$373$	−4.68473e10			−0.125313			−0.0626563	−	0.998035i	$-0.519957\pi$
$373$	−4.68473e10			−0.125313			−0.0626563	+	0.998035i	$0.519957\pi$
$374$		0			0
$375$		−	7.00479e11i		−	1.82917i
$376$		0			0
$377$	−1.44007e11	−	3.87676e10i	−0.367153	−	0.0988399i
$378$		0			0
$379$			1.03319e11i			0.257220i	0.991695	+	0.128610i	$0.0410515\pi$
$379$			1.03319e11i			0.257220i	−0.991695	+	0.128610i	$0.958948\pi$
$380$		0			0
$381$	4.60949e11			1.12070
$382$		0			0
$383$			5.99429e11i			1.42345i	0.702457	+	0.711726i	$0.252086\pi$
$383$			5.99429e11i			1.42345i	−0.702457	+	0.711726i	$0.747914\pi$
$384$		0			0
$385$			1.84966e11i			0.429062i
$386$		0			0
$387$	7.50022e11			1.69971
$388$		0			0
$389$	4.83508e11			1.07061			0.535304	−	0.844659i	$-0.320197\pi$
$389$	4.83508e11			1.07061			0.535304	+	0.844659i	$0.320197\pi$
$390$		0			0
$391$	−5.02648e11			−1.08760
$392$		0			0
$393$	−5.53930e11			−1.17135
$394$		0			0
$395$			2.12310e11i			0.438817i
$396$		0			0
$397$		−	9.08247e11i		−	1.83504i	−0.397685	−	0.917522i	$-0.630186\pi$
$397$		−	9.08247e11i		−	1.83504i	0.397685	−	0.917522i	$-0.369814\pi$
$398$		0			0
$399$	8.68955e11			1.71640
$400$		0			0
$401$			6.49944e11i			1.25524i	0.778521	+	0.627619i	$0.215970\pi$
$401$			6.49944e11i			1.25524i	−0.778521	+	0.627619i	$0.784030\pi$
$402$		0			0
$403$	−1.21337e10	+	4.50722e10i	−0.0229151	+	0.0851209i
$404$		0			0
$405$		−	3.32577e11i		−	0.614250i
$406$		0			0
$407$	−7.28017e11			−1.31512
$408$		0			0
$409$		−	1.74848e11i		−	0.308962i	−0.987996	−	0.154481i	$-0.950629\pi$
$409$		−	1.74848e11i		−	0.308962i	0.987996	−	0.154481i	$-0.0493705\pi$
$410$		0			0
$411$		−	8.82192e11i		−	1.52502i
$412$		0			0
$413$	−1.34674e11			−0.227776
$414$		0			0
$415$	1.34140e11			0.221994
$416$		0			0
$417$	−7.26942e11			−1.17730
$418$		0			0
$419$	6.33675e11			1.00439			0.502196	−	0.864754i	$-0.332525\pi$
$419$	6.33675e11			1.00439			0.502196	+	0.864754i	$0.332525\pi$
$420$		0			0
$421$			2.62942e11i			0.407934i	0.978978	+	0.203967i	$0.0653835\pi$
$421$			2.62942e11i			0.407934i	−0.978978	+	0.203967i	$0.934616\pi$
$422$		0			0
$423$			2.27897e12i			3.46104i
$424$		0			0
$425$	6.13704e11			0.912449
$426$		0			0
$427$			3.12996e11i			0.455632i
$428$		0			0
$429$	3.51875e11	−	1.30708e12i	0.501569	−	1.86314i
$430$		0			0
$431$		−	5.32577e11i		−	0.743421i	−0.928349	−	0.371710i	$-0.878771\pi$
$431$		−	5.32577e11i		−	0.743421i	0.928349	−	0.371710i	$-0.121229\pi$
$432$		0			0
$433$	−1.15456e12			−1.57842			−0.789210	−	0.614123i	$-0.789510\pi$
$433$	−1.15456e12			−1.57842			−0.789210	+	0.614123i	$0.789510\pi$
$434$		0			0
$435$		−	3.48960e11i		−	0.467277i
$436$		0			0
$437$			8.29978e11i			1.08868i
$438$		0			0
$439$	−6.73385e11			−0.865313			−0.432657	−	0.901559i	$-0.642424\pi$
$439$	−6.73385e11			−0.865313			−0.432657	+	0.901559i	$0.642424\pi$
$440$		0			0
$441$	1.10811e12			1.39512
$442$		0			0
$443$	−1.00519e12			−1.24003			−0.620015	−	0.784590i	$-0.712874\pi$
$443$	−1.00519e12			−1.24003			−0.620015	+	0.784590i	$0.712874\pi$
$444$		0			0
$445$	5.94941e11			0.719207
$446$		0			0
$447$			1.10101e12i			1.30439i
$448$		0			0
$449$			6.78172e11i			0.787465i	0.919225	+	0.393733i	$0.128816\pi$
$449$			6.78172e11i			0.787465i	−0.919225	+	0.393733i	$0.871184\pi$
$450$		0			0
$451$	1.58698e12			1.80625
$452$		0			0
$453$		−	1.60754e12i		−	1.79357i
$454$		0			0
$455$	−3.37295e11	−	9.08020e10i	−0.368942	−	0.0993217i
$456$		0			0
$457$			2.63711e11i			0.282817i	0.989951	+	0.141408i	$0.0451631\pi$
$457$			2.63711e11i			0.282817i	−0.989951	+	0.141408i	$0.954837\pi$
$458$		0			0
$459$	2.90649e12			3.05641
$460$		0			0
$461$		−	2.80546e11i		−	0.289301i	−0.989483	−	0.144650i	$-0.953794\pi$
$461$		−	2.80546e11i		−	0.289301i	0.989483	−	0.144650i	$-0.0462057\pi$
$462$		0			0
$463$		−	9.80047e11i		−	0.991134i	−0.868570	−	0.495567i	$-0.834960\pi$
$463$		−	9.80047e11i		−	0.991134i	0.868570	−	0.495567i	$-0.165040\pi$
$464$		0			0
$465$	−1.09220e11			−0.108334
$466$		0			0
$467$	9.71824e11			0.945500			0.472750	−	0.881197i	$-0.343261\pi$
$467$	9.71824e11			0.945500			0.472750	+	0.881197i	$0.343261\pi$
$468$		0			0
$469$	5.76814e11			0.550501
$470$		0			0
$471$	−3.49506e11			−0.327235
$472$		0			0
$473$			1.06439e12i			0.977747i
$474$		0			0
$475$		−	1.01335e12i		−	0.913356i
$476$		0			0
$477$	3.81529e12			3.37439
$478$		0			0
$479$		−	1.57710e12i		−	1.36883i	−0.729092	−	0.684416i	$-0.760057\pi$
$479$		−	1.57710e12i		−	1.36883i	0.729092	−	0.684416i	$-0.239943\pi$
$480$		0			0
$481$	3.57391e11	−	1.32757e12i	0.304433	−	1.13085i
$482$		0			0
$483$		−	6.39086e11i		−	0.534315i
$484$		0			0
$485$	3.79520e11			0.311456
$486$		0			0
$487$			3.96093e11i			0.319093i	0.987190	+	0.159546i	$0.0510031\pi$
$487$			3.96093e11i			0.319093i	−0.987190	+	0.159546i	$0.948997\pi$
$488$		0			0
$489$		−	3.12546e11i		−	0.247186i
$490$		0			0
$491$	−1.23825e12			−0.961479			−0.480740	−	0.876863i	$-0.659632\pi$
$491$	−1.23825e12			−0.961479			−0.480740	+	0.876863i	$0.659632\pi$
$492$		0			0
$493$	9.31710e11			0.710346
$494$		0			0
$495$	2.09446e12			1.56801
$496$		0			0
$497$	−2.60118e11			−0.191235
$498$		0			0
$499$			1.28906e12i			0.930723i	0.885121	+	0.465361i	$0.154076\pi$
$499$			1.28906e12i			0.930723i	−0.885121	+	0.465361i	$0.845924\pi$
$500$		0			0
$501$			8.49939e11i			0.602723i
$502$		0			0
$503$	3.21065e11			0.223634			0.111817	−	0.993729i	$-0.464333\pi$
$503$	3.21065e11			0.223634			0.111817	+	0.993729i	$0.464333\pi$
$504$		0			0
$505$		−	5.15970e10i		−	0.0353032i
$506$		0			0
$507$	2.21079e12	+	1.28332e12i	1.48597	+	0.862580i
$508$		0			0
$509$		−	2.88071e12i		−	1.90226i	−0.308797	−	0.951128i	$-0.599927\pi$
$509$		−	2.88071e12i		−	1.90226i	0.308797	−	0.951128i	$-0.400073\pi$
$510$		0			0
$511$	9.39462e11			0.609516
$512$		0			0
$513$		−	4.79922e12i		−	3.05945i
$514$		0			0
$515$		−	1.30807e12i		−	0.819407i
$516$		0			0
$517$	−3.23420e12			−1.99094
$518$		0			0
$519$	2.34729e12			1.42008
$520$		0			0
$521$	2.84526e11			0.169181			0.0845907	−	0.996416i	$-0.473042\pi$
$521$	2.84526e11			0.169181			0.0845907	+	0.996416i	$0.473042\pi$
$522$		0			0
$523$	−1.67324e11			−0.0977915			−0.0488957	−	0.998804i	$-0.515570\pi$
$523$	−1.67324e11			−0.0977915			−0.0488957	+	0.998804i	$0.515570\pi$
$524$		0			0
$525$			7.80287e11i			0.448268i
$526$		0			0
$527$		−	2.91613e11i		−	0.164687i
$528$		0			0
$529$	−1.19073e12			−0.661095
$530$		0			0
$531$			1.52497e12i			0.832409i
$532$		0			0
$533$	−7.79067e11	+	2.89394e12i	−0.418121	+	1.55316i
$534$		0			0
$535$		−	2.93862e11i		−	0.155079i
$536$		0			0
$537$	4.12835e12			2.14236
$538$		0			0
$539$			1.57258e12i			0.802532i
$540$		0			0
$541$			2.03454e12i			1.02113i	0.859840	+	0.510563i	$0.170563\pi$
$541$			2.03454e12i			1.02113i	−0.859840	+	0.510563i	$0.829437\pi$
$542$		0			0
$543$	−5.16964e12			−2.55189
$544$		0			0
$545$	−1.71265e12			−0.831543
$546$		0			0
$547$	−2.30343e12			−1.10010			−0.550049	−	0.835132i	$-0.685391\pi$
$547$	−2.30343e12			−1.10010			−0.550049	+	0.835132i	$0.685391\pi$
$548$		0			0
$549$	3.54419e12			1.66511
$550$		0			0
$551$		−	1.53845e12i		−	0.711052i
$552$		0			0
$553$		−	7.20728e11i		−	0.327724i
$554$		0			0
$555$	3.21700e12			1.43924
$556$		0			0
$557$		−	1.06563e12i		−	0.469090i	−0.972105	−	0.234545i	$-0.924640\pi$
$557$		−	1.06563e12i		−	0.469090i	0.972105	−	0.234545i	$-0.0753601\pi$
$558$		0			0
$559$	−1.94097e12	−	5.22521e11i	−0.840748	−	0.226335i
$560$		0			0
$561$			8.45669e12i			3.60469i
$562$		0			0
$563$	9.98781e11			0.418969			0.209485	−	0.977812i	$-0.432821\pi$
$563$	9.98781e11			0.418969			0.209485	+	0.977812i	$0.432821\pi$
$564$		0			0
$565$			1.39466e12i			0.575772i
$566$		0			0
$567$			1.12900e12i			0.458743i
$568$		0			0
$569$	−1.07382e11			−0.0429464			−0.0214732	−	0.999769i	$-0.506836\pi$
$569$	−1.07382e11			−0.0429464			−0.0214732	+	0.999769i	$0.506836\pi$
$570$		0			0
$571$	−4.60685e12			−1.81360			−0.906800	−	0.421561i	$-0.861482\pi$
$571$	−4.60685e12			−1.81360			−0.906800	+	0.421561i	$0.861482\pi$
$572$		0			0
$573$	−3.30418e12			−1.28046
$574$		0			0
$575$	−7.45287e11			−0.284327
$576$		0			0
$577$		−	3.03741e12i		−	1.14081i	−0.821365	−	0.570403i	$-0.806787\pi$
$577$		−	3.03741e12i		−	1.14081i	0.821365	−	0.570403i	$-0.193213\pi$
$578$		0			0
$579$		−	2.23095e12i		−	0.824965i
$580$		0			0
$581$	−4.55363e11			−0.165793
$582$		0			0
$583$			5.41447e12i			1.94110i
$584$		0			0
$585$	−1.02819e12	+	3.81934e12i	−0.362971	+	1.34830i
$586$		0			0
$587$		−	5.09566e11i		−	0.177145i	−0.996070	−	0.0885726i	$-0.971769\pi$
$587$		−	5.09566e11i		−	0.177145i	0.996070	−	0.0885726i	$-0.0282305\pi$
$588$		0			0
$589$	−4.81514e11			−0.164851
$590$		0			0
$591$		−	8.14488e12i		−	2.74626i
$592$		0			0
$593$			2.02750e12i			0.673310i	0.941628	+	0.336655i	$0.109296\pi$
$593$			2.02750e12i			0.673310i	−0.941628	+	0.336655i	$0.890704\pi$
$594$		0			0
$595$	2.18227e12			0.713808
$596$		0			0
$597$	−3.41045e12			−1.09882
$598$		0			0
$599$	−6.18048e12			−1.96156			−0.980779	−	0.195120i	$-0.937490\pi$
$599$	−6.18048e12			−1.96156			−0.980779	+	0.195120i	$0.937490\pi$
$600$		0			0
$601$	9.62398e11			0.300898			0.150449	−	0.988618i	$-0.451928\pi$
$601$	9.62398e11			0.300898			0.150449	+	0.988618i	$0.451928\pi$
$602$		0			0
$603$		−	6.53152e12i		−	2.01181i
$604$		0			0
$605$			6.15325e11i			0.186726i
$606$		0			0
$607$	7.73991e11			0.231413			0.115706	−	0.993283i	$-0.463087\pi$
$607$	7.73991e11			0.231413			0.115706	+	0.993283i	$0.463087\pi$
$608$		0			0
$609$			1.18461e12i			0.348978i
$610$		0			0
$611$	1.58770e12	−	5.89771e12i	0.460875	−	1.71198i
$612$		0			0
$613$		−	2.10077e12i		−	0.600905i	−0.953797	−	0.300453i	$-0.902862\pi$
$613$		−	2.10077e12i		−	0.600905i	0.953797	−	0.300453i	$-0.0971377\pi$
$614$		0			0
$615$	−7.01265e12			−1.97671
$616$		0			0
$617$		−	1.45302e11i		−	0.0403633i	−0.999796	−	0.0201817i	$-0.993576\pi$
$617$		−	1.45302e11i		−	0.0403633i	0.999796	−	0.0201817i	$-0.00642446\pi$
$618$		0			0
$619$			6.77807e12i			1.85566i	0.373005	+	0.927829i	$0.378328\pi$
$619$			6.77807e12i			1.85566i	−0.373005	+	0.927829i	$0.621672\pi$
$620$		0			0
$621$	−3.52966e12			−0.952403
$622$		0			0
$623$	−2.01964e12			−0.537129
$624$		0			0
$625$	−1.04163e12			−0.273058
$626$		0			0
$627$	1.39638e13			3.60827
$628$		0			0
$629$			8.58927e12i			2.18790i
$630$		0			0
$631$			8.59928e11i			0.215939i	0.994154	+	0.107969i	$0.0344348\pi$
$631$			8.59928e11i			0.215939i	−0.994154	+	0.107969i	$0.965565\pi$
$632$		0			0
$633$	1.08703e13			2.69108
$634$		0			0
$635$		−	1.91146e12i		−	0.466535i
$636$		0			0
$637$	−2.86767e12	−	7.71994e11i	−0.690083	−	0.185775i
$638$		0			0
$639$			2.94543e12i			0.698868i
$640$		0			0
$641$	4.66144e12			1.09058			0.545292	−	0.838246i	$-0.316419\pi$
$641$	4.66144e12			1.09058			0.545292	+	0.838246i	$0.316419\pi$
$642$		0			0
$643$		−	1.20355e12i		−	0.277660i	−0.990316	−	0.138830i	$-0.955666\pi$
$643$		−	1.20355e12i		−	0.277660i	0.990316	−	0.138830i	$-0.0443341\pi$
$644$		0			0
$645$		−	4.70339e12i		−	1.07002i
$646$		0			0
$647$	−3.84558e12			−0.862765			−0.431382	−	0.902169i	$-0.641974\pi$
$647$	−3.84558e12			−0.862765			−0.431382	+	0.902169i	$0.641974\pi$
$648$		0			0
$649$	−2.16416e12			−0.478838
$650$		0			0
$651$	3.70768e11			0.0809073
$652$		0			0
$653$	−3.00668e12			−0.647110			−0.323555	−	0.946209i	$-0.604878\pi$
$653$	−3.00668e12			−0.647110			−0.323555	+	0.946209i	$0.604878\pi$
$654$		0			0
$655$			2.29703e12i			0.487620i
$656$		0			0
$657$		−	1.06379e13i		−	2.22748i
$658$		0			0
$659$	8.55776e12			1.76757			0.883783	−	0.467897i	$-0.154988\pi$
$659$	8.55776e12			1.76757			0.883783	+	0.467897i	$0.154988\pi$
$660$		0			0
$661$		−	7.73594e12i		−	1.57618i	−0.615558	−	0.788092i	$-0.711069\pi$
$661$		−	7.73594e12i		−	1.57618i	0.615558	−	0.788092i	$-0.288931\pi$
$662$		0			0
$663$	−1.54212e13	−	4.15148e12i	−3.09961	−	0.834434i
$664$		0			0
$665$		−	3.60338e12i		−	0.714517i
$666$		0			0
$667$	−1.13148e12			−0.221350
$668$		0			0
$669$			3.42208e12i			0.660500i
$670$		0			0
$671$			5.02973e12i			0.957842i
$672$		0			0
$673$	−1.12825e12			−0.212000			−0.106000	−	0.994366i	$-0.533804\pi$
$673$	−1.12825e12			−0.212000			−0.106000	+	0.994366i	$0.533804\pi$
$674$		0			0
$675$	4.30951e12			0.799026
$676$		0			0
$677$	−2.65351e12			−0.485481			−0.242740	−	0.970091i	$-0.578046\pi$
$677$	−2.65351e12			−0.485481			−0.242740	+	0.970091i	$0.578046\pi$
$678$		0			0
$679$	−1.28835e12			−0.232606
$680$		0			0
$681$			8.43600e12i			1.50305i
$682$		0			0
$683$			7.10336e12i			1.24902i	0.781015	+	0.624512i	$0.214702\pi$
$683$			7.10336e12i			1.24902i	−0.781015	+	0.624512i	$0.785298\pi$
$684$		0			0
$685$	−3.65827e12			−0.634846
$686$		0			0
$687$			1.89074e13i			3.23837i
$688$		0			0
$689$	−9.87354e12	−	2.65802e12i	−1.66911	−	0.449336i
$690$		0			0
$691$			1.13125e12i			0.188759i	0.995536	+	0.0943795i	$0.0300867\pi$
$691$			1.13125e12i			0.188759i	−0.995536	+	0.0943795i	$0.969913\pi$
$692$		0			0
$693$	−7.11004e12			−1.17104
$694$		0			0
$695$			3.01448e12i			0.490096i
$696$		0			0
$697$		−	1.87235e13i		−	3.00497i
$698$		0			0
$699$	−6.66024e12			−1.05522
$700$		0			0
$701$	2.76494e12			0.432469			0.216234	−	0.976341i	$-0.430622\pi$
$701$	2.76494e12			0.432469			0.216234	+	0.976341i	$0.430622\pi$
$702$		0			0
$703$	1.41827e13			2.19008
$704$		0			0
$705$	1.42914e13			2.17884
$706$		0			0
$707$			1.75156e11i			0.0263656i
$708$		0			0
$709$		−	2.15735e12i		−	0.320636i	−0.987065	−	0.160318i	$-0.948748\pi$
$709$		−	2.15735e12i		−	0.320636i	0.987065	−	0.160318i	$-0.0512520\pi$
$710$		0			0
$711$	−8.16112e12			−1.19767
$712$		0			0
$713$			3.54137e11i			0.0513178i
$714$		0			0
$715$	−5.42021e12	−	1.45915e12i	−0.775601	−	0.208797i
$716$		0			0
$717$			7.17010e12i			1.01319i
$718$		0			0
$719$	8.17722e12			1.14111			0.570553	−	0.821261i	$-0.306729\pi$
$719$	8.17722e12			1.14111			0.570553	+	0.821261i	$0.306729\pi$
$720$		0			0
$721$			4.44051e12i			0.611961i
$722$		0			0
$723$		−	8.33233e12i		−	1.13408i
$724$		0			0
$725$	1.38147e12			0.185703
$726$		0			0
$727$	2.62778e12			0.348886			0.174443	−	0.984667i	$-0.444187\pi$
$727$	2.62778e12			0.348886			0.174443	+	0.984667i	$0.444187\pi$
$728$		0			0
$729$	−8.65098e12			−1.13447
$730$		0			0
$731$	1.25579e13			1.62663
$732$		0			0
$733$		−	3.57647e11i		−	0.0457601i	−0.999738	−	0.0228800i	$-0.992716\pi$
$733$		−	3.57647e11i		−	0.0457601i	0.999738	−	0.0228800i	$-0.00728358\pi$
$734$		0			0
$735$		−	6.94898e12i		−	0.878270i
$736$		0			0
$737$	9.26919e12			1.15728
$738$		0			0
$739$			8.15940e12i			1.00637i	0.864178	+	0.503186i	$0.167839\pi$
$739$			8.15940e12i			1.00637i	−0.864178	+	0.503186i	$0.832161\pi$
$740$		0			0
$741$	−6.85497e12	+	2.54636e13i	−0.835264	+	3.10269i
$742$		0			0
$743$			9.41025e10i			0.0113280i	0.999984	+	0.00566398i	$0.00180291\pi$
$743$			9.41025e10i			0.0113280i	−0.999984	+	0.00566398i	$0.998197\pi$
$744$		0			0
$745$	4.56567e12			0.543002
$746$		0			0
$747$			5.15628e12i			0.605890i
$748$		0			0
$749$			9.97572e11i			0.115818i
$750$		0			0
$751$	−1.05448e13			−1.20965			−0.604823	−	0.796360i	$-0.706756\pi$
$751$	−1.05448e13			−1.20965			−0.604823	+	0.796360i	$0.706756\pi$
$752$		0			0
$753$	−1.12516e13			−1.27537
$754$		0			0
$755$	−6.66613e12			−0.746642
$756$		0			0
$757$	5.97398e12			0.661199			0.330600	−	0.943771i	$-0.392749\pi$
$757$	5.97398e12			0.661199			0.330600	+	0.943771i	$0.392749\pi$
$758$		0			0
$759$		−	1.02699e13i		−	1.12325i
$760$		0			0
$761$			5.96477e12i			0.644707i	0.946619	+	0.322354i	$0.104474\pi$
$761$			5.96477e12i			0.644707i	−0.946619	+	0.322354i	$0.895526\pi$
$762$		0			0
$763$	5.81393e12			0.621025
$764$		0			0
$765$		−	2.47107e13i		−	2.60861i
$766$		0			0
$767$	1.06241e12	−	3.94645e12i	0.110844	−	0.411744i
$768$		0			0
$769$		−	6.03902e11i		−	0.0622728i	−0.999515	−	0.0311364i	$-0.990087\pi$
$769$		−	6.03902e11i		−	0.0622728i	0.999515	−	0.0311364i	$-0.00991262\pi$
$770$		0			0
$771$	−8.74903e12			−0.891693
$772$		0			0
$773$		−	1.15311e13i		−	1.16162i	−0.814040	−	0.580809i	$-0.802736\pi$
$773$		−	1.15311e13i		−	1.16162i	0.814040	−	0.580809i	$-0.197264\pi$
$774$		0			0
$775$		−	4.32381e11i		−	0.0430535i
$776$		0			0
$777$	−1.09207e13			−1.07487
$778$		0			0
$779$	−3.09165e13			−3.00795
$780$		0			0
$781$	−4.18000e12			−0.402019
$782$		0			0
$783$	6.54259e12			0.622045
$784$		0			0
$785$			1.44933e12i			0.136224i
$786$		0			0
$787$			9.67693e12i			0.899189i	0.893233	+	0.449595i	$0.148432\pi$
$787$			9.67693e12i			0.899189i	−0.893233	+	0.449595i	$0.851568\pi$
$788$		0			0
$789$	1.30346e12			0.119743
$790$		0			0
$791$		−	4.73444e12i		−	0.430006i
$792$		0			0
$793$	−9.17196e12	−	2.46915e12i	−0.823631	−	0.221727i
$794$		0			0
$795$		−	2.39257e13i		−	2.12429i
$796$		0			0
$797$	−3.94698e11			−0.0346500			−0.0173250	−	0.999850i	$-0.505515\pi$
$797$	−3.94698e11			−0.0346500			−0.0173250	+	0.999850i	$0.505515\pi$
$798$		0			0
$799$			3.81576e13i			3.31223i
$800$		0			0
$801$			2.28693e13i			1.96294i
$802$		0			0
$803$	1.50968e13			1.28134
$804$		0			0
$805$	−2.65016e12			−0.222429
$806$		0			0
$807$	−3.29144e13			−2.73184
$808$		0			0
$809$	8.51462e12			0.698871			0.349435	−	0.936960i	$-0.386373\pi$
$809$	8.51462e12			0.698871			0.349435	+	0.936960i	$0.386373\pi$
$810$		0			0
$811$		−	7.06843e12i		−	0.573758i	−0.957967	−	0.286879i	$-0.907382\pi$
$811$		−	7.06843e12i		−	0.573758i	0.957967	−	0.286879i	$-0.0926178\pi$
$812$		0			0
$813$			1.70659e13i			1.37001i
$814$		0			0
$815$	−1.29606e12			−0.102900
$816$		0			0
$817$		−	2.07357e13i		−	1.62824i
$818$		0			0
$819$	3.49039e12	−	1.29655e13i	0.271079	−	1.00696i
$820$		0			0
$821$			3.58345e12i			0.275269i	0.990483	+	0.137634i	$0.0439499\pi$
$821$			3.58345e12i			0.275269i	−0.990483	+	0.137634i	$0.956050\pi$
$822$		0			0
$823$	1.28284e13			0.974702			0.487351	−	0.873206i	$-0.337963\pi$
$823$	1.28284e13			0.974702			0.487351	+	0.873206i	$0.337963\pi$
$824$		0			0
$825$			1.25389e13i			0.942361i
$826$		0			0
$827$		−	9.12355e12i		−	0.678249i	−0.940742	−	0.339124i	$-0.889869\pi$
$827$		−	9.12355e12i		−	0.678249i	0.940742	−	0.339124i	$-0.110131\pi$
$828$		0			0
$829$	−1.95044e13			−1.43429			−0.717147	−	0.696922i	$-0.754552\pi$
$829$	−1.95044e13			−1.43429			−0.717147	+	0.696922i	$0.754552\pi$
$830$		0			0
$831$	−3.49585e11			−0.0254301
$832$		0			0
$833$	1.85535e13			1.33513
$834$		0			0
$835$	3.52452e12			0.250906
$836$		0			0
$837$		−	2.04774e12i		−	0.144215i
$838$		0			0
$839$		−	2.99062e12i		−	0.208368i	−0.994558	−	0.104184i	$-0.966777\pi$
$839$		−	2.99062e12i		−	0.208368i	0.994558	−	0.104184i	$-0.0332231\pi$
$840$		0			0
$841$	−1.24098e13			−0.855429
$842$		0			0
$843$			2.77885e13i			1.89514i
$844$		0			0
$845$	5.32167e12	−	9.16769e12i	0.359081	−	0.618592i
$846$		0			0
$847$		−	2.08884e12i		−	0.139454i
$848$		0			0
$849$	−1.89326e13			−1.25062
$850$		0			0
$851$		−	1.04309e13i		−	0.681770i
$852$		0			0
$853$		−	1.21453e13i		−	0.785484i	−0.919649	−	0.392742i	$-0.871527\pi$
$853$		−	1.21453e13i		−	0.785484i	0.919649	−	0.392742i	$-0.128473\pi$
$854$		0			0
$855$	−4.08027e13			−2.61120
$856$		0			0
$857$	8.03500e12			0.508829			0.254415	−	0.967095i	$-0.418117\pi$
$857$	8.03500e12			0.508829			0.254415	+	0.967095i	$0.418117\pi$
$858$		0			0
$859$	7.86012e12			0.492561			0.246281	−	0.969199i	$-0.420792\pi$
$859$	7.86012e12			0.492561			0.246281	+	0.969199i	$0.420792\pi$
$860$		0			0
$861$	2.38058e13			1.47628
$862$		0			0
$863$		−	1.94302e13i		−	1.19242i	−0.802829	−	0.596209i	$-0.796673\pi$
$863$		−	1.94302e13i		−	1.19242i	0.802829	−	0.596209i	$-0.203327\pi$
$864$		0			0
$865$		−	9.73373e12i		−	0.591162i
$866$		0			0
$867$	7.11873e13			4.27875
$868$		0			0
$869$		−	1.15818e13i		−	0.688951i
$870$		0			0
$871$	−4.55034e12	+	1.69028e13i	−0.267894	+	0.995124i
$872$		0			0
$873$			1.45886e13i			0.850060i
$874$		0			0
$875$	9.86072e12			0.568686
$876$		0			0
$877$			4.08869e12i			0.233392i	0.993168	+	0.116696i	$0.0372303\pi$
$877$			4.08869e12i			0.233392i	−0.993168	+	0.116696i	$0.962770\pi$
$878$		0			0
$879$		−	4.42110e13i		−	2.49793i
$880$		0			0
$881$	−1.23517e13			−0.690772			−0.345386	−	0.938461i	$-0.612252\pi$
$881$	−1.23517e13			−0.690772			−0.345386	+	0.938461i	$0.612252\pi$
$882$		0			0
$883$	2.86217e13			1.58443			0.792214	−	0.610244i	$-0.208929\pi$
$883$	2.86217e13			1.58443			0.792214	+	0.610244i	$0.208929\pi$
$884$		0			0
$885$	9.56311e12			0.524028
$886$		0			0
$887$	5.28822e12			0.286849			0.143425	−	0.989661i	$-0.454189\pi$
$887$	5.28822e12			0.286849			0.143425	+	0.989661i	$0.454189\pi$
$888$		0			0
$889$			6.48883e12i			0.348424i
$890$		0			0
$891$			1.81426e13i			0.964383i
$892$		0			0
$893$	6.30062e13			3.31552
$894$		0			0
$895$		−	1.71194e13i		−	0.891837i
$896$		0			0
$897$	1.87276e13	+	5.04159e12i	0.965865	+	0.260017i
$898$		0			0
$899$		−	6.56429e11i		−	0.0335173i
$900$		0			0
$901$	6.38808e13			3.22930
$902$		0			0
$903$			1.59666e13i			0.799129i
$904$		0			0
$905$			2.14375e13i			1.06232i
$906$		0			0
$907$	4.70499e12			0.230848			0.115424	−	0.993316i	$-0.463177\pi$
$907$	4.70499e12			0.230848			0.115424	+	0.993316i	$0.463177\pi$
$908$		0			0
$909$	1.98337e12			0.0963532
$910$		0			0
$911$	8.42934e12			0.405472			0.202736	−	0.979233i	$-0.435017\pi$
$911$	8.42934e12			0.405472			0.202736	+	0.979233i	$0.435017\pi$
$912$		0			0
$913$	−7.31752e12			−0.348534
$914$		0			0
$915$		−	2.22257e13i		−	1.04824i
$916$		0			0
$917$		−	7.79772e12i		−	0.364171i
$918$		0			0
$919$	−3.51131e13			−1.62386			−0.811932	−	0.583752i	$-0.801584\pi$
$919$	−3.51131e13			−1.62386			−0.811932	+	0.583752i	$0.801584\pi$
$920$		0			0
$921$			4.40155e13i			2.01575i
$922$		0			0
$923$	2.05201e12	−	7.62243e12i	0.0930617	−	0.345689i
$924$		0			0
$925$			1.27355e13i			0.571976i
$926$		0			0
$927$	5.02818e13			2.23641
$928$		0			0
$929$			8.89009e12i			0.391594i	0.980644	+	0.195797i	$0.0627293\pi$
$929$			8.89009e12i			0.391594i	−0.980644	+	0.195797i	$0.937271\pi$
$930$		0			0
$931$		−	3.06358e13i		−	1.33646i
$932$		0			0
$933$	−5.29185e12			−0.228634
$934$		0			0
$935$	3.50682e13			1.50059
$936$		0			0
$937$	4.82026e12			0.204288			0.102144	−	0.994770i	$-0.467430\pi$
$937$	4.82026e12			0.204288			0.102144	+	0.994770i	$0.467430\pi$
$938$		0			0
$939$	−4.21221e13			−1.76813
$940$		0			0
$941$		−	2.71110e12i		−	0.112718i	−0.998411	−	0.0563588i	$-0.982051\pi$
$941$		−	2.71110e12i		−	0.112718i	0.998411	−	0.0563588i	$-0.0179491\pi$
$942$		0			0
$943$			2.27380e13i			0.936374i
$944$		0			0
$945$	1.53242e13			0.625077
$946$		0			0
$947$			1.95635e12i			0.0790445i	0.999219	+	0.0395223i	$0.0125836\pi$
$947$			1.95635e12i			0.0790445i	−0.999219	+	0.0395223i	$0.987416\pi$
$948$		0			0
$949$	−7.41118e12	+	2.75297e13i	−0.296613	+	1.10180i
$950$		0			0
$951$		−	5.31943e13i		−	2.10888i
$952$		0			0
$953$	−3.34572e13			−1.31393			−0.656964	−	0.753922i	$-0.728160\pi$
$953$	−3.34572e13			−1.31393			−0.656964	+	0.753922i	$0.728160\pi$
$954$		0			0
$955$			1.37017e13i			0.533041i
$956$		0			0
$957$			1.90363e13i			0.733632i
$958$		0			0
$959$	1.24187e13			0.474125
$960$		0			0
$961$	2.62342e13			0.992229
$962$		0			0
$963$	1.12959e13			0.423257
$964$		0			0
$965$	−9.25128e12			−0.343423
$966$		0			0
$967$			2.58352e13i			0.950151i	0.879945	+	0.475075i	$0.157579\pi$
$967$			2.58352e13i			0.950151i	−0.879945	+	0.475075i	$0.842421\pi$
$968$		0			0
$969$		−	1.64747e14i		−	6.00290i
$970$		0			0
$971$	4.31980e13			1.55947			0.779736	−	0.626109i	$-0.215353\pi$
$971$	4.31980e13			1.55947			0.779736	+	0.626109i	$0.215353\pi$
$972$		0			0
$973$		−	1.02332e13i		−	0.366020i
$974$		0			0
$975$	−2.28653e13	−	6.15549e12i	−0.810320	−	0.218143i
$976$		0			0
$977$			4.54896e13i			1.59730i	0.601796	+	0.798650i	$0.294452\pi$
$977$			4.54896e13i			1.59730i	−0.601796	+	0.798650i	$0.705548\pi$
$978$		0			0
$979$	−3.24549e13			−1.12917
$980$		0			0
$981$		−	6.58337e13i		−	2.26954i
$982$		0			0
$983$			3.66857e13i			1.25316i	0.779358	+	0.626579i	$0.215545\pi$
$983$			3.66857e13i			1.25316i	−0.779358	+	0.626579i	$0.784455\pi$
$984$		0			0
$985$	−3.37752e13			−1.14323
$986$		0			0
$987$	−4.85150e13			−1.62723
$988$		0			0
$989$	−1.52504e13			−0.506871
$990$		0			0
$991$	1.90942e13			0.628883			0.314442	−	0.949277i	$-0.398183\pi$
$991$	1.90942e13			0.628883			0.314442	+	0.949277i	$0.398183\pi$
$992$		0			0
$993$		−	6.49870e13i		−	2.12107i
$994$		0			0
$995$			1.41424e13i			0.457425i
$996$		0			0
$997$	3.28282e13			1.05225			0.526124	−	0.850408i	$-0.323645\pi$
$997$	3.28282e13			1.05225			0.526124	+	0.850408i	$0.323645\pi$
$998$		0			0
$999$			6.03150e13i			1.91593i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	208.10.f.d.129.29		32
4.3	odd	2		104.10.f.a.25.4	yes	32
13.12	even	2	inner	208.10.f.d.129.30		32
52.51	odd	2		104.10.f.a.25.3	✓	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
104.10.f.a.25.3	✓	32	52.51	odd	2
104.10.f.a.25.4	yes	32	4.3	odd	2
208.10.f.d.129.29		32	1.1	even	1	trivial
208.10.f.d.129.30		32	13.12	even	2	inner

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 208 = 2^{4} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	208.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+241.055 q^{3} -999.605i q^{5} +3393.35i q^{7} +38424.4 q^{9} +54529.9i q^{11} +(-99437.9 - 26769.3i) q^{13} -240960. i q^{15} +643353. q^{17} -1.06231e6i q^{19} +817984. i q^{21} -781294. q^{23} +953914. q^{25} +4.51771e6 q^{27} +1.44821e6 q^{29} -453270. i q^{31} +1.31447e7i q^{33} +3.39202e6 q^{35} +1.33508e7i q^{37} +(-2.39700e7 - 6.45288e6i) q^{39} -2.91030e7i q^{41} +1.95194e7 q^{43} -3.84093e7i q^{45} +5.93105e7i q^{47} +2.88388e7 q^{49} +1.55083e8 q^{51} +9.92934e7 q^{53} +5.45084e7 q^{55} -2.56076e8i q^{57} +3.96876e7i q^{59} +9.22380e7 q^{61} +1.30388e8i q^{63} +(-2.67588e7 + 9.93987e7i) q^{65} -1.69984e8i q^{67} -1.88335e8 q^{69} +7.66552e7i q^{71} -2.76853e8i q^{73} +2.29946e8 q^{75} -1.85039e8 q^{77} -2.12394e8 q^{79} +3.32709e8 q^{81} +1.34193e8i q^{83} -6.43100e8i q^{85} +3.49098e8 q^{87} +5.95176e8i q^{89} +(9.08378e7 - 3.37428e8i) q^{91} -1.09263e8i q^{93} -1.06189e9 q^{95} +3.79670e8i q^{97} +2.09528e9i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 162 q^{3} + 223074 q^{9} + 66270 q^{13} - 487902 q^{17} - 3171556 q^{23} - 13526722 q^{25} + 3694974 q^{27} + 8833508 q^{29} + 8281126 q^{35} + 12056860 q^{39} - 89959038 q^{43} - 172344874 q^{49}+ \cdots - 1741143356 q^{95}+O(q^{100}) \) 32 * q + 162 * q^3 + 223074 * q^9 + 66270 * q^13 - 487902 * q^17 - 3171556 * q^23 - 13526722 * q^25 + 3694974 * q^27 + 8833508 * q^29 + 8281126 * q^35 + 12056860 * q^39 - 89959038 * q^43 - 172344874 * q^49 - 51206562 * q^51 + 139521112 * q^53 + 84550176 * q^55 + 372036836 * q^61 + 241471810 * q^65 - 486052516 * q^69 + 604473728 * q^75 - 1666197144 * q^77 - 1165982916 * q^79 + 859636544 * q^81 + 1738691296 * q^87 + 291718146 * q^91 - 1741143356 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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