LMFDB - Embedded newform 104.10.f.a.25.20

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [104,10,Mod(25,104)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(104, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 10, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("104.25"); S:= CuspForms(chi, 10); N := Newforms(S);

Level:	$ N $	$=$	$ 104 = 2^{3} \cdot 13 $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	104.f (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

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

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

Self dual:	no
Analytic conductor:	$53.5637269610$
Analytic rank:	$0$
Dimension:	$32$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			25.20
Character	$\chi$	$=$	104.25
Dual form			104.10.f.a.25.19

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+35.4013 q^{3} +405.839i q^{5} +7970.77i q^{7} -18429.7 q^{9} -20385.0i q^{11} +(-102968. + 1410.13i) q^{13} +14367.2i q^{15} -92739.6 q^{17} -547568. i q^{19} +282175. i q^{21} +1.27351e6 q^{23} +1.78842e6 q^{25} -1.34924e6 q^{27} -2.11242e6 q^{29} -3.12563e6i q^{31} -721656. i q^{33} -3.23485e6 q^{35} +6.50775e6i q^{37} +(-3.64522e6 + 49920.6i) q^{39} -3.29275e7i q^{41} +1.71254e7 q^{43} -7.47951e6i q^{45} +2.00526e7i q^{47} -2.31795e7 q^{49} -3.28310e6 q^{51} +5.66621e7 q^{53} +8.27303e6 q^{55} -1.93846e7i q^{57} -6.94997e7i q^{59} -1.29165e8 q^{61} -1.46899e8i q^{63} +(-572287. - 4.17886e7i) q^{65} -7.67824e7i q^{67} +4.50838e7 q^{69} -1.90253e8i q^{71} -2.21355e8i q^{73} +6.33124e7 q^{75} +1.62484e8 q^{77} -2.74140e8 q^{79} +3.14988e8 q^{81} -2.37271e8i q^{83} -3.76373e7i q^{85} -7.47826e7 q^{87} -8.62842e8i q^{89} +(-1.12399e7 - 8.20738e8i) q^{91} -1.10651e8i q^{93} +2.22224e8 q^{95} -1.15595e9i q^{97} +3.75691e8i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/104\mathbb{Z}\right)^\times$.

$n$	$41$	$53$	$79$
$\chi(n)$	$-1$	$1$	$1$

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
$3$	35.4013			0.252333			0.126166	−	0.992009i	$-0.459733\pi$
$3$	35.4013			0.252333			0.126166	+	0.992009i	$0.459733\pi$
$4$		0			0
$5$			405.839i			0.290395i	0.989403	+	0.145197i	$0.0463817\pi$
$5$			405.839i			0.290395i	−0.989403	+	0.145197i	$0.953618\pi$
$6$		0			0
$7$			7970.77i			1.25475i	0.778715	+	0.627377i	$0.215872\pi$
$7$			7970.77i			1.25475i	−0.778715	+	0.627377i	$0.784128\pi$
$8$		0			0
$9$	−18429.7			−0.936328
$10$		0			0
$11$		−	20385.0i		−	0.419801i	−0.977723	−	0.209901i	$-0.932686\pi$
$11$		−	20385.0i		−	0.419801i	0.977723	−	0.209901i	$-0.0673141\pi$
$12$		0			0
$13$	−102968.	+	1410.13i	−0.999906	+	0.0136935i
$13$	−102968.	+	1410.13i	−0.999906	+	0.0136935i
$14$		0			0
$15$			14367.2i			0.0732761i
$16$		0			0
$17$	−92739.6			−0.269305			−0.134653	−	0.990893i	$-0.542992\pi$
$17$	−92739.6			−0.269305			−0.134653	+	0.990893i	$0.542992\pi$
$18$		0			0
$19$		−	547568.i		−	0.963932i	−0.876190	−	0.481966i	$-0.839923\pi$
$19$		−	547568.i		−	0.963932i	0.876190	−	0.481966i	$-0.160077\pi$
$20$		0			0
$21$			282175.i			0.316616i
$22$		0			0
$23$	1.27351e6			0.948912			0.474456	−	0.880279i	$-0.342645\pi$
$23$	1.27351e6			0.948912			0.474456	+	0.880279i	$0.342645\pi$
$24$		0			0
$25$	1.78842e6			0.915671
$26$		0			0
$27$	−1.34924e6			−0.488599
$28$		0			0
$29$	−2.11242e6			−0.554613			−0.277307	−	0.960781i	$-0.589442\pi$
$29$	−2.11242e6			−0.554613			−0.277307	+	0.960781i	$0.589442\pi$
$30$		0			0
$31$		−	3.12563e6i		−	0.607869i	−0.952693	−	0.303934i	$-0.901700\pi$
$31$		−	3.12563e6i		−	0.607869i	0.952693	−	0.303934i	$-0.0983004\pi$
$32$		0			0
$33$		−	721656.i		−	0.105930i
$34$		0			0
$35$	−3.23485e6			−0.364374
$36$		0			0
$37$			6.50775e6i			0.570851i	0.958401	+	0.285426i	$0.0921350\pi$
$37$			6.50775e6i			0.570851i	−0.958401	+	0.285426i	$0.907865\pi$
$38$		0			0
$39$	−3.64522e6	+	49920.6i	−0.252309	+	0.00345532i
$40$		0			0
$41$		−	3.29275e7i		−	1.81983i	−0.414793	−	0.909916i	$-0.636146\pi$
$41$		−	3.29275e7i		−	1.81983i	0.414793	−	0.909916i	$-0.363854\pi$
$42$		0			0
$43$	1.71254e7			0.763895			0.381947	−	0.924184i	$-0.375254\pi$
$43$	1.71254e7			0.763895			0.381947	+	0.924184i	$0.375254\pi$
$44$		0			0
$45$		−	7.47951e6i		−	0.271905i
$46$		0			0
$47$			2.00526e7i			0.599418i	0.954031	+	0.299709i	$0.0968896\pi$
$47$			2.00526e7i			0.599418i	−0.954031	+	0.299709i	$0.903110\pi$
$48$		0			0
$49$	−2.31795e7			−0.574410
$50$		0			0
$51$	−3.28310e6			−0.0679546
$52$		0			0
$53$	5.66621e7			0.986395			0.493198	−	0.869917i	$-0.335828\pi$
$53$	5.66621e7			0.986395			0.493198	+	0.869917i	$0.335828\pi$
$54$		0			0
$55$	8.27303e6			0.121908
$56$		0			0
$57$		−	1.93846e7i		−	0.243232i
$58$		0			0
$59$		−	6.94997e7i		−	0.746705i	−0.927690	−	0.373352i	$-0.878208\pi$
$59$		−	6.94997e7i		−	0.746705i	0.927690	−	0.373352i	$-0.121792\pi$
$60$		0			0
$61$	−1.29165e8			−1.19443			−0.597214	−	0.802082i	$-0.703726\pi$
$61$	−1.29165e8			−1.19443			−0.597214	+	0.802082i	$0.703726\pi$
$62$		0			0
$63$		−	1.46899e8i		−	1.17486i
$64$		0			0
$65$	−572287.	−	4.17886e7i	−0.00397653	−	0.290367i
$66$		0			0
$67$		−	7.67824e7i		−	0.465506i	−0.972536	−	0.232753i	$-0.925227\pi$
$67$		−	7.67824e7i		−	0.465506i	0.972536	−	0.232753i	$-0.0747733\pi$
$68$		0			0
$69$	4.50838e7			0.239441
$70$		0			0
$71$		−	1.90253e8i		−	0.888523i	−0.895897	−	0.444262i	$-0.853466\pi$
$71$		−	1.90253e8i		−	0.888523i	0.895897	−	0.444262i	$-0.146534\pi$
$72$		0			0
$73$		−	2.21355e8i		−	0.912299i	−0.889903	−	0.456149i	$-0.849228\pi$
$73$		−	2.21355e8i		−	0.912299i	0.889903	−	0.456149i	$-0.150772\pi$
$74$		0			0
$75$	6.33124e7			0.231054
$76$		0			0
$77$	1.62484e8			0.526748
$78$		0			0
$79$	−2.74140e8			−0.791863			−0.395932	−	0.918280i	$-0.629578\pi$
$79$	−2.74140e8			−0.791863			−0.395932	+	0.918280i	$0.629578\pi$
$80$		0			0
$81$	3.14988e8			0.813039
$82$		0			0
$83$		−	2.37271e8i		−	0.548773i	−0.961620	−	0.274386i	$-0.911525\pi$
$83$		−	2.37271e8i		−	0.548773i	0.961620	−	0.274386i	$-0.0884747\pi$
$84$		0			0
$85$		−	3.76373e7i		−	0.0782049i
$86$		0			0
$87$	−7.47826e7			−0.139947
$88$		0			0
$89$		−	8.62842e8i		−	1.45773i	−0.684659	−	0.728864i	$-0.740049\pi$
$89$		−	8.62842e8i		−	1.45773i	0.684659	−	0.728864i	$-0.259951\pi$
$90$		0			0
$91$	−1.12399e7	−	8.20738e8i	−0.0171820	−	1.25464i
$92$		0			0
$93$		−	1.10651e8i		−	0.153385i
$94$		0			0
$95$	2.22224e8			0.279921
$96$		0			0
$97$		−	1.15595e9i		−	1.32576i	−0.748726	−	0.662879i	$-0.769334\pi$
$97$		−	1.15595e9i		−	1.32576i	0.748726	−	0.662879i	$-0.230666\pi$
$98$		0			0
$99$			3.75691e8i			0.393072i
$100$		0			0
$101$	9.80569e7			0.0937631			0.0468816	−	0.998900i	$-0.485072\pi$
$101$	9.80569e7			0.0937631			0.0468816	+	0.998900i	$0.485072\pi$
$102$		0			0
$103$	−2.15960e9			−1.89063			−0.945314	−	0.326160i	$-0.894245\pi$
$103$	−2.15960e9			−1.89063			−0.945314	+	0.326160i	$0.894245\pi$
$104$		0			0
$105$	−1.14518e8			−0.0919435
$106$		0			0
$107$	6.18567e8			0.456205			0.228102	−	0.973637i	$-0.426748\pi$
$107$	6.18567e8			0.456205			0.228102	+	0.973637i	$0.426748\pi$
$108$		0			0
$109$			4.86390e8i			0.330039i	0.986290	+	0.165020i	$0.0527687\pi$
$109$			4.86390e8i			0.330039i	−0.986290	+	0.165020i	$0.947231\pi$
$110$		0			0
$111$			2.30383e8i			0.144044i
$112$		0			0
$113$	2.23211e7			0.0128784			0.00643921	−	0.999979i	$-0.497950\pi$
$113$	2.23211e7			0.0128784			0.00643921	+	0.999979i	$0.497950\pi$
$114$		0			0
$115$			5.16839e8i			0.275559i
$116$		0			0
$117$	1.89768e9	−	2.59884e7i	0.936240	−	0.0128216i
$118$		0			0
$119$		−	7.39206e8i		−	0.337912i
$120$		0			0
$121$	1.94240e9			0.823767
$122$		0			0
$123$		−	1.16568e9i		−	0.459203i
$124$		0			0
$125$			1.51846e9i			0.556301i
$126$		0			0
$127$	6.98905e8			0.238397			0.119199	−	0.992870i	$-0.461967\pi$
$127$	6.98905e8			0.238397			0.119199	+	0.992870i	$0.461967\pi$
$128$		0			0
$129$	6.06262e8			0.192756
$130$		0			0
$131$	−4.27516e9			−1.26833			−0.634164	−	0.773199i	$-0.718655\pi$
$131$	−4.27516e9			−1.26833			−0.634164	+	0.773199i	$0.718655\pi$
$132$		0			0
$133$	4.36453e9			1.20950
$134$		0			0
$135$		−	5.47574e8i		−	0.141886i
$136$		0			0
$137$			5.21732e9i			1.26533i	0.774425	+	0.632666i	$0.218040\pi$
$137$			5.21732e9i			1.26533i	−0.774425	+	0.632666i	$0.781960\pi$
$138$		0			0
$139$	−8.06337e9			−1.83210			−0.916051	−	0.401061i	$-0.868642\pi$
$139$	−8.06337e9			−1.83210			−0.916051	+	0.401061i	$0.868642\pi$
$140$		0			0
$141$			7.09887e8i			0.151253i
$142$		0			0
$143$	2.87456e7	+	2.09901e9i	0.00574856	+	0.419762i
$144$		0			0
$145$		−	8.57304e8i		−	0.161057i
$146$		0			0
$147$	−8.20585e8			−0.144942
$148$		0			0
$149$		−	2.14203e8i		−	0.0356031i	−0.999842	−	0.0178015i	$-0.994333\pi$
$149$		−	2.14203e8i		−	0.0356031i	0.999842	−	0.0178015i	$-0.00566670\pi$
$150$		0			0
$151$			8.81757e9i			1.38023i	0.723698	+	0.690117i	$0.242441\pi$
$151$			8.81757e9i			1.38023i	−0.723698	+	0.690117i	$0.757559\pi$
$152$		0			0
$153$	1.70917e9			0.252158
$154$		0			0
$155$	1.26850e9			0.176522
$156$		0			0
$157$	−6.64966e9			−0.873476			−0.436738	−	0.899589i	$-0.643866\pi$
$157$	−6.64966e9			−0.873476			−0.436738	+	0.899589i	$0.643866\pi$
$158$		0			0
$159$	2.00591e9			0.248900
$160$		0			0
$161$			1.01508e10i			1.19065i
$162$		0			0
$163$			4.44643e9i			0.493364i	0.969097	+	0.246682i	$0.0793403\pi$
$163$			4.44643e9i			0.493364i	−0.969097	+	0.246682i	$0.920660\pi$
$164$		0			0
$165$	2.92876e8			0.0307614
$166$		0			0
$167$		−	6.59999e9i		−	0.656627i	−0.944569	−	0.328314i	$-0.893520\pi$
$167$		−	6.59999e9i		−	0.656627i	0.944569	−	0.328314i	$-0.106480\pi$
$168$		0			0
$169$	1.06005e10	−	2.90399e8i	0.999625	−	0.0273845i
$170$		0			0
$171$			1.00915e10i			0.902557i
$172$		0			0
$173$	2.06249e10			1.75059			0.875297	−	0.483586i	$-0.160666\pi$
$173$	2.06249e10			1.75059			0.875297	+	0.483586i	$0.160666\pi$
$174$		0			0
$175$			1.42551e10i			1.14894i
$176$		0			0
$177$		−	2.46038e9i		−	0.188418i
$178$		0			0
$179$	−1.67324e10			−1.21820			−0.609100	−	0.793093i	$-0.708469\pi$
$179$	−1.67324e10			−1.21820			−0.609100	+	0.793093i	$0.708469\pi$
$180$		0			0
$181$	6.90129e9			0.477944			0.238972	−	0.971027i	$-0.423190\pi$
$181$	6.90129e9			0.477944			0.238972	+	0.971027i	$0.423190\pi$
$182$		0			0
$183$	−4.57260e9			−0.301393
$184$		0			0
$185$	−2.64110e9			−0.165772
$186$		0			0
$187$			1.89050e9i			0.113055i
$188$		0			0
$189$		−	1.07545e10i		−	0.613072i
$190$		0			0
$191$	−2.28751e10			−1.24369			−0.621846	−	0.783139i	$-0.713617\pi$
$191$	−2.28751e10			−1.24369			−0.621846	+	0.783139i	$0.713617\pi$
$192$		0			0
$193$			2.04898e10i			1.06299i	0.847060	+	0.531497i	$0.178370\pi$
$193$			2.04898e10i			1.06299i	−0.847060	+	0.531497i	$0.821630\pi$
$194$		0			0
$195$	−2.02597e7	−	1.47937e9i	−0.00100341	−	0.0732692i
$196$		0			0
$197$			1.91207e10i			0.904495i	0.891893	+	0.452247i	$0.149378\pi$
$197$			1.91207e10i			0.904495i	−0.891893	+	0.452247i	$0.850622\pi$
$198$		0			0
$199$	−3.62244e9			−0.163743			−0.0818713	−	0.996643i	$-0.526090\pi$
$199$	−3.62244e9			−0.163743			−0.0818713	+	0.996643i	$0.526090\pi$
$200$		0			0
$201$		−	2.71820e9i		−	0.117462i
$202$		0			0
$203$		−	1.68376e10i		−	0.695904i
$204$		0			0
$205$	1.33633e10			0.528469
$206$		0			0
$207$	−2.34704e10			−0.888493
$208$		0			0
$209$	−1.11622e10			−0.404660
$210$		0			0
$211$	2.48853e10			0.864313			0.432157	−	0.901799i	$-0.357753\pi$
$211$	2.48853e10			0.864313			0.432157	+	0.901799i	$0.357753\pi$
$212$		0			0
$213$		−	6.73520e9i		−	0.224203i
$214$		0			0
$215$			6.95017e9i			0.221831i
$216$		0			0
$217$	2.49137e10			0.762726
$218$		0			0
$219$		−	7.83626e9i		−	0.230203i
$220$		0			0
$221$	9.54926e9	−	1.30775e8i	0.269280	−	0.00368774i
$222$		0			0
$223$		−	6.23843e10i		−	1.68929i	−0.535330	−	0.844643i	$-0.679813\pi$
$223$		−	6.23843e10i		−	1.68929i	0.535330	−	0.844643i	$-0.320187\pi$
$224$		0			0
$225$	−3.29601e10			−0.857369
$226$		0			0
$227$			1.05915e9i			0.0264753i	0.999912	+	0.0132377i	$0.00421380\pi$
$227$			1.05915e9i			0.0264753i	−0.999912	+	0.0132377i	$0.995786\pi$
$228$		0			0
$229$		−	5.32362e10i		−	1.27923i	−0.768697	−	0.639613i	$-0.779095\pi$
$229$		−	5.32362e10i		−	1.27923i	0.768697	−	0.639613i	$-0.220905\pi$
$230$		0			0
$231$	5.75215e9			0.132916
$232$		0			0
$233$	5.28232e10			1.17415			0.587074	−	0.809533i	$-0.300280\pi$
$233$	5.28232e10			1.17415			0.587074	+	0.809533i	$0.300280\pi$
$234$		0			0
$235$	−8.13812e9			−0.174068
$236$		0			0
$237$	−9.70490e9			−0.199813
$238$		0			0
$239$		−	4.70943e10i		−	0.933636i	−0.884353	−	0.466818i	$-0.845400\pi$
$239$		−	4.70943e10i		−	0.933636i	0.884353	−	0.466818i	$-0.154600\pi$
$240$		0			0
$241$		−	7.33799e10i		−	1.40120i	−0.713554	−	0.700601i	$-0.752916\pi$
$241$		−	7.33799e10i		−	1.40120i	0.713554	−	0.700601i	$-0.247084\pi$
$242$		0			0
$243$	3.77081e10			0.693755
$244$		0			0
$245$		−	9.40715e9i		−	0.166806i
$246$		0			0
$247$	7.72144e8	+	5.63822e10i	0.0131996	+	0.963842i
$248$		0			0
$249$		−	8.39968e9i		−	0.138473i
$250$		0			0
$251$	−3.73074e10			−0.593285			−0.296643	−	0.954989i	$-0.595867\pi$
$251$	−3.73074e10			−0.593285			−0.296643	+	0.954989i	$0.595867\pi$
$252$		0			0
$253$		−	2.59605e10i		−	0.398355i
$254$		0			0
$255$		−	1.33241e9i		−	0.0197336i
$256$		0			0
$257$	−6.83900e10			−0.977898			−0.488949	−	0.872312i	$-0.662620\pi$
$257$	−6.83900e10			−0.977898			−0.488949	+	0.872312i	$0.662620\pi$
$258$		0			0
$259$	−5.18717e10			−0.716279
$260$		0			0
$261$	3.89315e10			0.519300
$262$		0			0
$263$	1.04625e10			0.134845			0.0674224	−	0.997725i	$-0.478522\pi$
$263$	1.04625e10			0.134845			0.0674224	+	0.997725i	$0.478522\pi$
$264$		0			0
$265$			2.29957e10i			0.286444i
$266$		0			0
$267$		−	3.05457e10i		−	0.367832i
$268$		0			0
$269$	−2.10392e10			−0.244988			−0.122494	−	0.992469i	$-0.539089\pi$
$269$	−2.10392e10			−0.244988			−0.122494	+	0.992469i	$0.539089\pi$
$270$		0			0
$271$		−	1.05185e11i		−	1.18466i	−0.805697	−	0.592328i	$-0.798209\pi$
$271$		−	1.05185e11i		−	1.18466i	0.805697	−	0.592328i	$-0.201791\pi$
$272$		0			0
$273$	−3.97905e8	−	2.90552e10i	−0.00433559	−	0.316586i
$274$		0			0
$275$		−	3.64570e10i		−	0.384400i
$276$		0			0
$277$	−6.52009e10			−0.665418			−0.332709	−	0.943030i	$-0.607963\pi$
$277$	−6.52009e10			−0.665418			−0.332709	+	0.943030i	$0.607963\pi$
$278$		0			0
$279$			5.76046e10i			0.569165i
$280$		0			0
$281$		−	9.04034e10i		−	0.864980i	−0.901639	−	0.432490i	$-0.857635\pi$
$281$		−	9.04034e10i		−	0.864980i	0.901639	−	0.432490i	$-0.142365\pi$
$282$		0			0
$283$	8.56797e10			0.794034			0.397017	−	0.917811i	$-0.370045\pi$
$283$	8.56797e10			0.794034			0.397017	+	0.917811i	$0.370045\pi$
$284$		0			0
$285$	7.86703e9			0.0706332
$286$		0			0
$287$	2.62457e11			2.28344
$288$		0			0
$289$	−1.09987e11			−0.927475
$290$		0			0
$291$		−	4.09220e10i		−	0.334532i
$292$		0			0
$293$			1.37879e11i			1.09293i	0.837481	+	0.546467i	$0.184028\pi$
$293$			1.37879e11i			1.09293i	−0.837481	+	0.546467i	$0.815972\pi$
$294$		0			0
$295$	2.82057e10			0.216839
$296$		0			0
$297$			2.75043e10i			0.205114i
$298$		0			0
$299$	−1.31131e11	+	1.79582e9i	−0.948823	+	0.0129940i
$300$		0			0
$301$			1.36503e11i			0.958501i
$302$		0			0
$303$	3.47134e9			0.0236595
$304$		0			0
$305$		−	5.24201e10i		−	0.346856i
$306$		0			0
$307$			1.84069e11i			1.18265i	0.806432	+	0.591327i	$0.201396\pi$
$307$			1.84069e11i			1.18265i	−0.806432	+	0.591327i	$0.798604\pi$
$308$		0			0
$309$	−7.64527e10			−0.477067
$310$		0			0
$311$	1.74875e11			1.06000			0.530000	−	0.847998i	$-0.322192\pi$
$311$	1.74875e11			1.06000			0.530000	+	0.847998i	$0.322192\pi$
$312$		0			0
$313$	−6.31641e10			−0.371981			−0.185990	−	0.982552i	$-0.559549\pi$
$313$	−6.31641e10			−0.371981			−0.185990	+	0.982552i	$0.559549\pi$
$314$		0			0
$315$	5.96174e10			0.341174
$316$		0			0
$317$			1.59121e11i			0.885038i	0.896759	+	0.442519i	$0.145915\pi$
$317$			1.59121e11i			0.885038i	−0.896759	+	0.442519i	$0.854085\pi$
$318$		0			0
$319$			4.30618e10i			0.232827i
$320$		0			0
$321$	2.18981e10			0.115115
$322$		0			0
$323$			5.07812e10i			0.259592i
$324$		0			0
$325$	−1.84151e11	+	2.52191e9i	−0.915585	+	0.0125388i
$326$		0			0
$327$			1.72188e10i			0.0832797i
$328$		0			0
$329$	−1.59834e11			−0.752123
$330$		0			0
$331$			6.92757e10i			0.317216i	0.987342	+	0.158608i	$0.0507006\pi$
$331$			6.92757e10i			0.317216i	−0.987342	+	0.158608i	$0.949299\pi$
$332$		0			0
$333$		−	1.19936e11i		−	0.534504i
$334$		0			0
$335$	3.11613e10			0.135180
$336$		0			0
$337$	−1.97496e10			−0.0834111			−0.0417055	−	0.999130i	$-0.513279\pi$
$337$	−1.97496e10			−0.0834111			−0.0417055	+	0.999130i	$0.513279\pi$
$338$		0			0
$339$	7.90195e8			0.00324964
$340$		0			0
$341$	−6.37160e10			−0.255184
$342$		0			0
$343$			1.36891e11i			0.534011i
$344$		0			0
$345$			1.82968e10i			0.0695325i
$346$		0			0
$347$	1.39636e11			0.517031			0.258515	−	0.966007i	$-0.416767\pi$
$347$	1.39636e11			0.517031			0.258515	+	0.966007i	$0.416767\pi$
$348$		0			0
$349$		−	9.66133e10i		−	0.348596i	−0.984693	−	0.174298i	$-0.944234\pi$
$349$		−	9.66133e10i		−	0.348596i	0.984693	−	0.174298i	$-0.0557656\pi$
$350$		0			0
$351$	1.38929e11	−	1.90261e9i	0.488553	−	0.00669064i
$352$		0			0
$353$			2.60504e11i			0.892953i	0.894795	+	0.446476i	$0.147321\pi$
$353$			2.60504e11i			0.892953i	−0.894795	+	0.446476i	$0.852679\pi$
$354$		0			0
$355$	7.72121e10			0.258022
$356$		0			0
$357$		−	2.61688e10i		−	0.0852663i
$358$		0			0
$359$			4.22394e11i			1.34212i	0.741401	+	0.671062i	$0.234162\pi$
$359$			4.22394e11i			1.34212i	−0.741401	+	0.671062i	$0.765838\pi$
$360$		0			0
$361$	2.28573e10			0.0708342
$362$		0			0
$363$	6.87634e10			0.207863
$364$		0			0
$365$	8.98346e10			0.264927
$366$		0			0
$367$	−7.04874e10			−0.202821			−0.101411	−	0.994845i	$-0.532336\pi$
$367$	−7.04874e10			−0.202821			−0.101411	+	0.994845i	$0.532336\pi$
$368$		0			0
$369$			6.06845e11i			1.70396i
$370$		0			0
$371$			4.51640e11i			1.23768i
$372$		0			0
$373$	3.08853e10			0.0826155			0.0413078	−	0.999146i	$-0.486848\pi$
$373$	3.08853e10			0.0826155			0.0413078	+	0.999146i	$0.486848\pi$
$374$		0			0
$375$			5.37556e10i			0.140373i
$376$		0			0
$377$	2.17513e11	−	2.97880e9i	0.554561	−	0.00759461i
$378$		0			0
$379$		−	6.24509e11i		−	1.55476i	−0.629033	−	0.777378i	$-0.716549\pi$
$379$		−	6.24509e11i		−	1.55476i	0.629033	−	0.777378i	$-0.283451\pi$
$380$		0			0
$381$	2.47422e10			0.0601555
$382$		0			0
$383$		−	3.70555e11i		−	0.879951i	−0.898010	−	0.439976i	$-0.854987\pi$
$383$		−	3.70555e11i		−	0.879951i	0.898010	−	0.439976i	$-0.145013\pi$
$384$		0			0
$385$			6.59424e10i			0.152965i
$386$		0			0
$387$	−3.15617e11			−0.715256
$388$		0			0
$389$	−6.07471e11			−1.34509			−0.672547	−	0.740055i	$-0.734800\pi$
$389$	−6.07471e11			−1.34509			−0.672547	+	0.740055i	$0.734800\pi$
$390$		0			0
$391$	−1.18105e11			−0.255547
$392$		0			0
$393$	−1.51346e11			−0.320040
$394$		0			0
$395$		−	1.11257e11i		−	0.229953i
$396$		0			0
$397$		−	5.21028e10i		−	0.105270i	−0.998614	−	0.0526349i	$-0.983238\pi$
$397$		−	5.21028e10i		−	0.105270i	0.998614	−	0.0526349i	$-0.0167620\pi$
$398$		0			0
$399$	1.54510e11			0.305196
$400$		0			0
$401$		−	3.09075e11i		−	0.596918i	−0.954422	−	0.298459i	$-0.903527\pi$
$401$		−	3.09075e11i		−	0.596918i	0.954422	−	0.298459i	$-0.0964726\pi$
$402$		0			0
$403$	4.40756e9	+	3.21841e11i	0.00832387	+	0.607812i
$404$		0			0
$405$			1.27834e11i			0.236102i
$406$		0			0
$407$	1.32661e11			0.239644
$408$		0			0
$409$			1.93544e11i			0.341999i	0.985271	+	0.170999i	$0.0546996\pi$
$409$			1.93544e11i			0.341999i	−0.985271	+	0.170999i	$0.945300\pi$
$410$		0			0
$411$			1.84700e11i			0.319285i
$412$		0			0
$413$	5.53966e11			0.936931
$414$		0			0
$415$	9.62936e10			0.159361
$416$		0			0
$417$	−2.85454e11			−0.462299
$418$		0			0
$419$	3.18535e10			0.0504886			0.0252443	−	0.999681i	$-0.491964\pi$
$419$	3.18535e10			0.0504886			0.0252443	+	0.999681i	$0.491964\pi$
$420$		0			0
$421$		−	9.17798e11i		−	1.42389i	−0.702233	−	0.711947i	$-0.747813\pi$
$421$		−	9.17798e11i		−	1.42389i	0.702233	−	0.711947i	$-0.252187\pi$
$422$		0			0
$423$		−	3.69564e11i		−	0.561252i
$424$		0			0
$425$	−1.65857e11			−0.246595
$426$		0			0
$427$		−	1.02954e12i		−	1.49872i
$428$		0			0
$429$	1.01763e9	+	7.43078e10i	0.00145055	+	0.105920i
$430$		0			0
$431$			5.31203e11i			0.741503i	0.928732	+	0.370752i	$0.120900\pi$
$431$			5.31203e11i			0.741503i	−0.928732	+	0.370752i	$0.879100\pi$
$432$		0			0
$433$	9.71070e11			1.32756			0.663781	−	0.747927i	$-0.268951\pi$
$433$	9.71070e11			1.32756			0.663781	+	0.747927i	$0.268951\pi$
$434$		0			0
$435$		−	3.03497e10i		−	0.0406399i
$436$		0			0
$437$		−	6.97331e11i		−	0.914687i
$438$		0			0
$439$	6.50058e11			0.835337			0.417669	−	0.908599i	$-0.362847\pi$
$439$	6.50058e11			0.835337			0.417669	+	0.908599i	$0.362847\pi$
$440$		0			0
$441$	4.27193e11			0.537836
$442$		0			0
$443$	−1.61505e12			−1.99236			−0.996180	−	0.0873188i	$-0.972170\pi$
$443$	−1.61505e12			−1.99236			−0.996180	+	0.0873188i	$0.972170\pi$
$444$		0			0
$445$	3.50175e11			0.423316
$446$		0			0
$447$		−	7.58306e9i		−	0.00898382i
$448$		0			0
$449$		−	6.85301e11i		−	0.795744i	−0.917441	−	0.397872i	$-0.869749\pi$
$449$		−	6.85301e11i		−	0.795744i	0.917441	−	0.397872i	$-0.130251\pi$
$450$		0			0
$451$	−6.71227e11			−0.763968
$452$		0			0
$453$			3.12153e11i			0.348278i
$454$		0			0
$455$	3.33087e11	−	4.56157e9i	0.364340	−	0.00498957i
$456$		0			0
$457$		−	5.61235e11i		−	0.601897i	−0.953640	−	0.300948i	$-0.902697\pi$
$457$		−	5.61235e11i		−	0.601897i	0.953640	−	0.300948i	$-0.0973032\pi$
$458$		0			0
$459$	1.25128e11			0.131582
$460$		0			0
$461$		−	1.84465e11i		−	0.190221i	−0.995467	−	0.0951107i	$-0.969679\pi$
$461$		−	1.84465e11i		−	0.190221i	0.995467	−	0.0951107i	$-0.0303205\pi$
$462$		0			0
$463$		−	1.23213e12i		−	1.24607i	−0.782192	−	0.623037i	$-0.785899\pi$
$463$		−	1.23213e12i		−	1.24607i	0.782192	−	0.623037i	$-0.214101\pi$
$464$		0			0
$465$	4.49066e10			0.0445422
$466$		0			0
$467$	−8.86777e11			−0.862757			−0.431378	−	0.902171i	$-0.641973\pi$
$467$	−8.86777e11			−0.862757			−0.431378	+	0.902171i	$0.641973\pi$
$468$		0			0
$469$	6.12014e11			0.584096
$470$		0			0
$471$	−2.35406e11			−0.220406
$472$		0			0
$473$		−	3.49102e11i		−	0.320684i
$474$		0			0
$475$		−	9.79281e11i		−	0.882645i
$476$		0			0
$477$	−1.04427e12			−0.923590
$478$		0			0
$479$		−	1.89581e11i		−	0.164545i	−0.996610	−	0.0822724i	$-0.973782\pi$
$479$		−	1.89581e11i		−	0.164545i	0.996610	−	0.0822724i	$-0.0262178\pi$
$480$		0			0
$481$	−9.17680e9	−	6.70093e11i	−0.00781697	−	0.570798i
$482$		0			0
$483$			3.59352e11i			0.300440i
$484$		0			0
$485$	4.69128e11			0.384993
$486$		0			0
$487$		−	3.45018e11i		−	0.277947i	−0.990296	−	0.138973i	$-0.955620\pi$
$487$		−	3.45018e11i		−	0.277947i	0.990296	−	0.138973i	$-0.0443802\pi$
$488$		0			0
$489$			1.57409e11i			0.124492i
$490$		0			0
$491$	−1.63167e12			−1.26697			−0.633484	−	0.773756i	$-0.718376\pi$
$491$	−1.63167e12			−1.26697			−0.633484	+	0.773756i	$0.718376\pi$
$492$		0			0
$493$	1.95905e11			0.149360
$494$		0			0
$495$	−1.52470e11			−0.114146
$496$		0			0
$497$	1.51646e12			1.11488
$498$		0			0
$499$			5.03796e11i			0.363750i	0.983322	+	0.181875i	$0.0582166\pi$
$499$			5.03796e11i			0.363750i	−0.983322	+	0.181875i	$0.941783\pi$
$500$		0			0
$501$		−	2.33648e11i		−	0.165689i
$502$		0			0
$503$	7.71114e10			0.0537109			0.0268555	−	0.999639i	$-0.491451\pi$
$503$	7.71114e10			0.0537109			0.0268555	+	0.999639i	$0.491451\pi$
$504$		0			0
$505$			3.97953e10i			0.0272283i
$506$		0			0
$507$	3.75272e11	−	1.02805e10i	0.252238	−	0.00691000i
$508$		0			0
$509$			6.08188e11i			0.401613i	0.979631	+	0.200807i	$0.0643563\pi$
$509$			6.08188e11i			0.401613i	−0.979631	+	0.200807i	$0.935644\pi$
$510$		0			0
$511$	1.76437e12			1.14471
$512$		0			0
$513$			7.38800e11i			0.470976i
$514$		0			0
$515$		−	8.76451e11i		−	0.549029i
$516$		0			0
$517$	4.08772e11			0.251637
$518$		0			0
$519$	7.30150e11			0.441732
$520$		0			0
$521$	8.00279e11			0.475851			0.237926	−	0.971283i	$-0.423533\pi$
$521$	8.00279e11			0.475851			0.237926	+	0.971283i	$0.423533\pi$
$522$		0			0
$523$	−2.02217e12			−1.18184			−0.590921	−	0.806729i	$-0.701236\pi$
$523$	−2.02217e12			−1.18184			−0.590921	+	0.806729i	$0.701236\pi$
$524$		0			0
$525$			5.04648e11i			0.289916i
$526$		0			0
$527$			2.89870e11i			0.163702i
$528$		0			0
$529$	−1.79333e11			−0.0995658
$530$		0			0
$531$			1.28086e12i			0.699161i
$532$		0			0
$533$	4.64322e10	+	3.39049e12i	0.0249199	+	1.81966i
$534$		0			0
$535$			2.51039e11i			0.132479i
$536$		0			0
$537$	−5.92348e11			−0.307392
$538$		0			0
$539$			4.72515e11i			0.241138i
$540$		0			0
$541$			9.90774e11i			0.497264i	0.968598	+	0.248632i	$0.0799810\pi$
$541$			9.90774e11i			0.497264i	−0.968598	+	0.248632i	$0.920019\pi$
$542$		0			0
$543$	2.44314e11			0.120601
$544$		0			0
$545$	−1.97396e11			−0.0958416
$546$		0			0
$547$	−9.54629e11			−0.455923			−0.227962	−	0.973670i	$-0.573206\pi$
$547$	−9.54629e11			−0.455923			−0.227962	+	0.973670i	$0.573206\pi$
$548$		0			0
$549$	2.38048e12			1.11838
$550$		0			0
$551$			1.15670e12i			0.534610i
$552$		0			0
$553$		−	2.18510e12i		−	0.993595i
$554$		0			0
$555$	−9.34983e10			−0.0418297
$556$		0			0
$557$		−	5.53796e11i		−	0.243782i	−0.992543	−	0.121891i	$-0.961104\pi$
$557$		−	5.53796e11i		−	0.243782i	0.992543	−	0.121891i	$-0.0388958\pi$
$558$		0			0
$559$	−1.76338e12	+	2.41492e10i	−0.763823	+	0.0104604i
$560$		0			0
$561$			6.69261e10i			0.0285274i
$562$		0			0
$563$	−3.41221e12			−1.43136			−0.715679	−	0.698429i	$-0.753883\pi$
$563$	−3.41221e12			−1.43136			−0.715679	+	0.698429i	$0.753883\pi$
$564$		0			0
$565$			9.05877e9i			0.00373982i
$566$		0			0
$567$			2.51070e12i			1.02016i
$568$		0			0
$569$	1.63786e12			0.655044			0.327522	−	0.944843i	$-0.393786\pi$
$569$	1.63786e12			0.655044			0.327522	+	0.944843i	$0.393786\pi$
$570$		0			0
$571$	−2.91449e12			−1.14736			−0.573681	−	0.819079i	$-0.694485\pi$
$571$	−2.91449e12			−1.14736			−0.573681	+	0.819079i	$0.694485\pi$
$572$		0			0
$573$	−8.09808e11			−0.313824
$574$		0			0
$575$	2.27756e12			0.868891
$576$		0			0
$577$		−	1.02435e12i		−	0.384732i	−0.981323	−	0.192366i	$-0.938384\pi$
$577$		−	1.02435e12i		−	0.384732i	0.981323	−	0.192366i	$-0.0616161\pi$
$578$		0			0
$579$			7.25366e11i			0.268228i
$580$		0			0
$581$	1.89123e12			0.688575
$582$		0			0
$583$		−	1.15506e12i		−	0.414090i
$584$		0			0
$585$	1.05471e10	+	7.70154e11i	0.00372334	+	0.271879i
$586$		0			0
$587$			5.13805e12i			1.78619i	0.449871	+	0.893093i	$0.351470\pi$
$587$			5.13805e12i			1.78619i	−0.449871	+	0.893093i	$0.648530\pi$
$588$		0			0
$589$	−1.71149e12			−0.585944
$590$		0			0
$591$			6.76898e11i			0.228234i
$592$		0			0
$593$			4.83296e12i			1.60497i	0.596673	+	0.802485i	$0.296489\pi$
$593$			4.83296e12i			1.60497i	−0.596673	+	0.802485i	$0.703511\pi$
$594$		0			0
$595$	2.99999e11			0.0981280
$596$		0			0
$597$	−1.28239e11			−0.0413176
$598$		0			0
$599$	3.86439e12			1.22648			0.613240	−	0.789897i	$-0.289866\pi$
$599$	3.86439e12			1.22648			0.613240	+	0.789897i	$0.289866\pi$
$600$		0			0
$601$	−4.72912e12			−1.47858			−0.739290	−	0.673387i	$-0.764839\pi$
$601$	−4.72912e12			−1.47858			−0.739290	+	0.673387i	$0.764839\pi$
$602$		0			0
$603$			1.41508e12i			0.435866i
$604$		0			0
$605$			7.88301e11i			0.239217i
$606$		0			0
$607$	2.84261e12			0.849900			0.424950	−	0.905217i	$-0.360292\pi$
$607$	2.84261e12			0.849900			0.424950	+	0.905217i	$0.360292\pi$
$608$		0			0
$609$		−	5.96074e11i		−	0.175599i
$610$		0			0
$611$	−2.82768e10	−	2.06478e12i	−0.00820815	−	0.599362i
$612$		0			0
$613$		−	1.34200e12i		−	0.383868i	−0.981408	−	0.191934i	$-0.938524\pi$
$613$		−	1.34200e12i		−	0.383868i	0.981408	−	0.191934i	$-0.0614759\pi$
$614$		0			0
$615$	4.73076e11			0.133350
$616$		0			0
$617$			2.17503e12i			0.604202i	0.953276	+	0.302101i	$0.0976881\pi$
$617$			2.17503e12i			0.604202i	−0.953276	+	0.302101i	$0.902312\pi$
$618$		0			0
$619$			4.41531e11i			0.120880i	0.998172	+	0.0604398i	$0.0192503\pi$
$619$			4.41531e11i			0.120880i	−0.998172	+	0.0604398i	$0.980750\pi$
$620$		0			0
$621$	−1.71827e12			−0.463637
$622$		0			0
$623$	6.87752e12			1.82909
$624$		0			0
$625$	2.87676e12			0.754124
$626$		0			0
$627$	−3.95155e11			−0.102109
$628$		0			0
$629$		−	6.03526e11i		−	0.153733i
$630$		0			0
$631$		−	4.13256e12i		−	1.03774i	−0.854854	−	0.518869i	$-0.826353\pi$
$631$		−	4.13256e12i		−	1.03774i	0.854854	−	0.518869i	$-0.173647\pi$
$632$		0			0
$633$	8.80970e11			0.218094
$634$		0			0
$635$			2.83643e11i			0.0692294i
$636$		0			0
$637$	2.38676e12	−	3.26862e10i	0.574356	−	0.00786570i
$638$		0			0
$639$			3.50631e12i			0.831950i
$640$		0			0
$641$	−3.41769e12			−0.799597			−0.399799	−	0.916603i	$-0.630920\pi$
$641$	−3.41769e12			−0.799597			−0.399799	+	0.916603i	$0.630920\pi$
$642$		0			0
$643$		−	7.30533e12i		−	1.68535i	−0.538422	−	0.842675i	$-0.680979\pi$
$643$		−	7.30533e12i		−	1.68535i	0.538422	−	0.842675i	$-0.319021\pi$
$644$		0			0
$645$			2.46045e11i			0.0559752i
$646$		0			0
$647$	−3.39956e12			−0.762700			−0.381350	−	0.924431i	$-0.624541\pi$
$647$	−3.39956e12			−0.762700			−0.381350	+	0.924431i	$0.624541\pi$
$648$		0			0
$649$	−1.41675e12			−0.313468
$650$		0			0
$651$	8.81976e11			0.192461
$652$		0			0
$653$	−7.51768e12			−1.61799			−0.808993	−	0.587819i	$-0.799987\pi$
$653$	−7.51768e12			−1.61799			−0.808993	+	0.587819i	$0.799987\pi$
$654$		0			0
$655$		−	1.73503e12i		−	0.368316i
$656$		0			0
$657$			4.07952e12i			0.854211i
$658$		0			0
$659$	−7.81712e11			−0.161459			−0.0807295	−	0.996736i	$-0.525725\pi$
$659$	−7.81712e11			−0.161459			−0.0807295	+	0.996736i	$0.525725\pi$
$660$		0			0
$661$		−	5.22505e12i		−	1.06459i	−0.846558	−	0.532297i	$-0.821329\pi$
$661$		−	5.22505e12i		−	1.06459i	0.846558	−	0.532297i	$-0.178671\pi$
$662$		0			0
$663$	3.38056e11	−	4.62962e9i	0.0679482	−	0.000930538i
$664$		0			0
$665$			1.77130e12i			0.351232i
$666$		0			0
$667$	−2.69019e12			−0.526279
$668$		0			0
$669$		−	2.20848e12i		−	0.426262i
$670$		0			0
$671$			2.63303e12i			0.501423i
$672$		0			0
$673$	−1.88696e12			−0.354565			−0.177283	−	0.984160i	$-0.556731\pi$
$673$	−1.88696e12			−0.354565			−0.177283	+	0.984160i	$0.556731\pi$
$674$		0			0
$675$	−2.41301e12			−0.447396
$676$		0			0
$677$	7.28706e11			0.133323			0.0666613	−	0.997776i	$-0.478765\pi$
$677$	7.28706e11			0.133323			0.0666613	+	0.997776i	$0.478765\pi$
$678$		0			0
$679$	9.21377e12			1.66350
$680$		0			0
$681$			3.74953e10i			0.00668059i
$682$		0			0
$683$		−	4.74210e12i		−	0.833831i	−0.908945	−	0.416915i	$-0.863111\pi$
$683$		−	4.74210e12i		−	0.833831i	0.908945	−	0.416915i	$-0.136889\pi$
$684$		0			0
$685$	−2.11739e12			−0.367446
$686$		0			0
$687$		−	1.88463e12i		−	0.322790i
$688$		0			0
$689$	−5.83441e12	+	7.99011e10i	−0.986303	+	0.0135072i
$690$		0			0
$691$			3.71783e12i			0.620352i	0.950679	+	0.310176i	$0.100388\pi$
$691$			3.71783e12i			0.620352i	−0.950679	+	0.310176i	$0.899612\pi$
$692$		0			0
$693$	−2.99454e12			−0.493209
$694$		0			0
$695$		−	3.27243e12i		−	0.532033i
$696$		0			0
$697$			3.05368e12i			0.490091i
$698$		0			0
$699$	1.87001e12			0.296276
$700$		0			0
$701$	8.94976e12			1.39985			0.699923	−	0.714218i	$-0.253218\pi$
$701$	8.94976e12			1.39985			0.699923	+	0.714218i	$0.253218\pi$
$702$		0			0
$703$	3.56343e12			0.550262
$704$		0			0
$705$	−2.88100e11			−0.0439230
$706$		0			0
$707$			7.81589e11i			0.117650i
$708$		0			0
$709$		−	1.16230e13i		−	1.72747i	−0.503947	−	0.863734i	$-0.668119\pi$
$709$		−	1.16230e13i		−	1.72747i	0.503947	−	0.863734i	$-0.331881\pi$
$710$		0			0
$711$	5.05233e12			0.741444
$712$		0			0
$713$		−	3.98051e12i		−	0.576814i
$714$		0			0
$715$	−8.51862e11	+	1.16661e10i	−0.121897	+	0.00166935i
$716$		0			0
$717$		−	1.66720e12i		−	0.235587i
$718$		0			0
$719$	6.62136e12			0.923990			0.461995	−	0.886882i	$-0.347134\pi$
$719$	6.62136e12			0.923990			0.461995	+	0.886882i	$0.347134\pi$
$720$		0			0
$721$		−	1.72137e13i		−	2.37228i
$722$		0			0
$723$		−	2.59774e12i		−	0.353569i
$724$		0			0
$725$	−3.77790e12			−0.507843
$726$		0			0
$727$	5.23622e12			0.695206			0.347603	−	0.937642i	$-0.386996\pi$
$727$	5.23622e12			0.695206			0.347603	+	0.937642i	$0.386996\pi$
$728$		0			0
$729$	−4.86499e12			−0.637982
$730$		0			0
$731$	−1.58821e12			−0.205721
$732$		0			0
$733$		−	1.77038e12i		−	0.226515i	−0.993566	−	0.113258i	$-0.963871\pi$
$733$		−	1.77038e12i		−	0.226515i	0.993566	−	0.113258i	$-0.0361285\pi$
$734$		0			0
$735$		−	3.33025e11i		−	0.0420905i
$736$		0			0
$737$	−1.56521e12			−0.195420
$738$		0			0
$739$		−	7.16924e12i		−	0.884246i	−0.896954	−	0.442123i	$-0.854225\pi$
$739$		−	7.16924e12i		−	0.884246i	0.896954	−	0.442123i	$-0.145775\pi$
$740$		0			0
$741$	2.73349e10	+	1.99600e12i	0.00333070	+	0.243209i
$742$		0			0
$743$		−	1.26107e12i		−	0.151806i	−0.997115	−	0.0759031i	$-0.975816\pi$
$743$		−	1.26107e12i		−	0.151806i	0.997115	−	0.0759031i	$-0.0241840\pi$
$744$		0			0
$745$	8.69319e10			0.0103389
$746$		0			0
$747$			4.37284e12i			0.513831i
$748$		0			0
$749$			4.93046e12i			0.572425i
$750$		0			0
$751$	8.62586e12			0.989515			0.494758	−	0.869031i	$-0.335257\pi$
$751$	8.62586e12			0.989515			0.494758	+	0.869031i	$0.335257\pi$
$752$		0			0
$753$	−1.32073e12			−0.149705
$754$		0			0
$755$	−3.57851e12			−0.400812
$756$		0			0
$757$	−5.83015e12			−0.645280			−0.322640	−	0.946522i	$-0.604570\pi$
$757$	−5.83015e12			−0.645280			−0.322640	+	0.946522i	$0.604570\pi$
$758$		0			0
$759$		−	9.19034e11i		−	0.100518i
$760$		0			0
$761$		−	1.36137e13i		−	1.47144i	−0.677283	−	0.735722i	$-0.736843\pi$
$761$		−	1.36137e13i		−	1.47144i	0.677283	−	0.735722i	$-0.263157\pi$
$762$		0			0
$763$	−3.87690e12			−0.414118
$764$		0			0
$765$			6.93647e11i			0.0732254i
$766$		0			0
$767$	9.80039e10	+	7.15628e12i	0.0102250	+	0.746635i
$768$		0			0
$769$			8.20944e12i			0.846535i	0.906005	+	0.423267i	$0.139117\pi$
$769$			8.20944e12i			0.846535i	−0.906005	+	0.423267i	$0.860883\pi$
$770$		0			0
$771$	−2.42109e12			−0.246756
$772$		0			0
$773$			1.62191e13i			1.63388i	0.576724	+	0.816939i	$0.304331\pi$
$773$			1.62191e13i			1.63388i	−0.576724	+	0.816939i	$0.695669\pi$
$774$		0			0
$775$		−	5.58994e12i		−	0.556608i
$776$		0			0
$777$	−1.83633e12			−0.180740
$778$		0			0
$779$	−1.80300e13			−1.75419
$780$		0			0
$781$	−3.87831e12			−0.373003
$782$		0			0
$783$	2.85017e12			0.270983
$784$		0			0
$785$		−	2.69869e12i		−	0.253653i
$786$		0			0
$787$		−	1.60716e13i		−	1.49338i	−0.665169	−	0.746692i	$-0.731641\pi$
$787$		−	1.60716e13i		−	1.49338i	0.665169	−	0.746692i	$-0.268359\pi$
$788$		0			0
$789$	3.70386e11			0.0340257
$790$		0			0
$791$			1.77916e11i			0.0161593i
$792$		0			0
$793$	1.32999e13	−	1.82140e11i	1.19432	−	0.0163559i
$794$		0			0
$795$			8.14076e11i			0.0722791i
$796$		0			0
$797$	7.80731e12			0.685392			0.342696	−	0.939446i	$-0.388660\pi$
$797$	7.80731e12			0.685392			0.342696	+	0.939446i	$0.388660\pi$
$798$		0			0
$799$		−	1.85967e12i		−	0.161427i
$800$		0			0
$801$			1.59020e13i			1.36491i
$802$		0			0
$803$	−4.51233e12			−0.382984
$804$		0			0
$805$	−4.11960e12			−0.345759
$806$		0			0
$807$	−7.44816e11			−0.0618185
$808$		0			0
$809$	1.15678e13			0.949473			0.474736	−	0.880128i	$-0.342543\pi$
$809$	1.15678e13			0.949473			0.474736	+	0.880128i	$0.342543\pi$
$810$		0			0
$811$		−	7.68937e12i		−	0.624161i	−0.950056	−	0.312081i	$-0.898974\pi$
$811$		−	7.68937e12i		−	0.624161i	0.950056	−	0.312081i	$-0.101026\pi$
$812$		0			0
$813$		−	3.72369e12i		−	0.298927i
$814$		0			0
$815$	−1.80453e12			−0.143270
$816$		0			0
$817$		−	9.37734e12i		−	0.736343i
$818$		0			0
$819$	2.07148e11	+	1.51260e13i	0.0160880	+	1.17475i
$820$		0			0
$821$			2.57253e13i			1.97614i	0.154016	+	0.988068i	$0.450779\pi$
$821$			2.57253e13i			1.97614i	−0.154016	+	0.988068i	$0.549221\pi$
$822$		0			0
$823$	1.37181e13			1.04230			0.521151	−	0.853464i	$-0.325503\pi$
$823$	1.37181e13			1.04230			0.521151	+	0.853464i	$0.325503\pi$
$824$		0			0
$825$		−	1.29062e12i		−	0.0969967i
$826$		0			0
$827$			2.52857e13i			1.87975i	0.341517	+	0.939875i	$0.389059\pi$
$827$			2.52857e13i			1.87975i	−0.341517	+	0.939875i	$0.610941\pi$
$828$		0			0
$829$	7.67414e12			0.564331			0.282166	−	0.959366i	$-0.408947\pi$
$829$	7.67414e12			0.564331			0.282166	+	0.959366i	$0.408947\pi$
$830$		0			0
$831$	−2.30820e12			−0.167907
$832$		0			0
$833$	2.14966e12			0.154692
$834$		0			0
$835$	2.67853e12			0.190681
$836$		0			0
$837$			4.21723e12i			0.297004i
$838$		0			0
$839$		−	7.36016e12i		−	0.512812i	−0.966569	−	0.256406i	$-0.917462\pi$
$839$		−	7.36016e12i		−	0.512812i	0.966569	−	0.256406i	$-0.0825384\pi$
$840$		0			0
$841$	−1.00448e13			−0.692404
$842$		0			0
$843$		−	3.20040e12i		−	0.218263i
$844$		0			0
$845$	1.17855e11	+	4.30210e12i	0.00795231	+	0.290286i
$846$		0			0
$847$			1.54824e13i			1.03363i
$848$		0			0
$849$	3.03317e12			0.200361
$850$		0			0
$851$			8.28766e12i			0.541688i
$852$		0			0
$853$		−	2.20754e13i		−	1.42770i	−0.700296	−	0.713852i	$-0.746949\pi$
$853$		−	2.20754e13i		−	1.42770i	0.700296	−	0.713852i	$-0.253051\pi$
$854$		0			0
$855$	−4.09554e12			−0.262098
$856$		0			0
$857$	−4.08294e12			−0.258559			−0.129279	−	0.991608i	$-0.541266\pi$
$857$	−4.08294e12			−0.258559			−0.129279	+	0.991608i	$0.541266\pi$
$858$		0			0
$859$	−2.39156e13			−1.49869			−0.749345	−	0.662180i	$-0.769631\pi$
$859$	−2.39156e13			−1.49869			−0.749345	+	0.662180i	$0.769631\pi$
$860$		0			0
$861$	9.29133e12			0.576187
$862$		0			0
$863$			1.74827e13i			1.07290i	0.843932	+	0.536451i	$0.180235\pi$
$863$			1.74827e13i			1.07290i	−0.843932	+	0.536451i	$0.819765\pi$
$864$		0			0
$865$			8.37041e12i			0.508363i
$866$		0			0
$867$	−3.89369e12			−0.234032
$868$		0			0
$869$			5.58835e12i			0.332425i
$870$		0			0
$871$	1.08273e11	+	7.90617e12i	0.00637442	+	0.465462i
$872$		0			0
$873$			2.13038e13i			1.24135i
$874$		0			0
$875$	−1.21033e13			−0.698021
$876$		0			0
$877$			2.67493e11i			0.0152691i	0.999971	+	0.00763456i	$0.00243018\pi$
$877$			2.67493e11i			0.0152691i	−0.999971	+	0.00763456i	$0.997570\pi$
$878$		0			0
$879$			4.88110e12i			0.275783i
$880$		0			0
$881$	2.11841e13			1.18473			0.592363	−	0.805671i	$-0.298195\pi$
$881$	2.11841e13			1.18473			0.592363	+	0.805671i	$0.298195\pi$
$882$		0			0
$883$	5.78367e11			0.0320170			0.0160085	−	0.999872i	$-0.494904\pi$
$883$	5.78367e11			0.0320170			0.0160085	+	0.999872i	$0.494904\pi$
$884$		0			0
$885$	9.98518e11			0.0547156
$886$		0			0
$887$	−3.01462e12			−0.163522			−0.0817611	−	0.996652i	$-0.526054\pi$
$887$	−3.01462e12			−0.163522			−0.0817611	+	0.996652i	$0.526054\pi$
$888$		0			0
$889$			5.57081e12i			0.299130i
$890$		0			0
$891$		−	6.42103e12i		−	0.341315i
$892$		0			0
$893$	1.09801e13			0.577798
$894$		0			0
$895$		−	6.79065e12i		−	0.353759i
$896$		0			0
$897$	−4.64221e12	+	6.35742e10i	−0.239419	+	0.00327880i
$898$		0			0
$899$			6.60266e12i			0.337132i
$900$		0			0
$901$	−5.25482e12			−0.265642
$902$		0			0
$903$			4.83238e12i			0.241861i
$904$		0			0
$905$			2.80081e12i			0.138792i
$906$		0			0
$907$	3.38548e12			0.166107			0.0830534	−	0.996545i	$-0.473533\pi$
$907$	3.38548e12			0.166107			0.0830534	+	0.996545i	$0.473533\pi$
$908$		0			0
$909$	−1.80716e12			−0.0877931
$910$		0			0
$911$	−3.25079e12			−0.156371			−0.0781856	−	0.996939i	$-0.524913\pi$
$911$	−3.25079e12			−0.156371			−0.0781856	+	0.996939i	$0.524913\pi$
$912$		0			0
$913$	−4.83677e12			−0.230376
$914$		0			0
$915$		−	1.85574e12i		−	0.0875230i
$916$		0			0
$917$		−	3.40763e13i		−	1.59144i
$918$		0			0
$919$	−3.87645e13			−1.79273			−0.896365	−	0.443317i	$-0.853802\pi$
$919$	−3.87645e13			−1.79273			−0.896365	+	0.443317i	$0.853802\pi$
$920$		0			0
$921$			6.51628e12i			0.298422i
$922$		0			0
$923$	2.68282e11	+	1.95901e13i	0.0121670	+	0.888440i
$924$		0			0
$925$			1.16386e13i			0.522712i
$926$		0			0
$927$	3.98009e13			1.77025
$928$		0			0
$929$		−	3.88346e13i		−	1.71060i	−0.518134	−	0.855299i	$-0.673373\pi$
$929$		−	3.88346e13i		−	1.71060i	0.518134	−	0.855299i	$-0.326627\pi$
$930$		0			0
$931$			1.26924e13i			0.553692i
$932$		0			0
$933$	6.19079e12			0.267472
$934$		0			0
$935$	−7.67238e11			−0.0328305
$936$		0			0
$937$	1.67560e13			0.710137			0.355068	−	0.934840i	$-0.384458\pi$
$937$	1.67560e13			0.710137			0.355068	+	0.934840i	$0.384458\pi$
$938$		0			0
$939$	−2.23609e12			−0.0938629
$940$		0			0
$941$			3.65356e13i			1.51902i	0.650497	+	0.759509i	$0.274561\pi$
$941$			3.65356e13i			1.51902i	−0.650497	+	0.759509i	$0.725439\pi$
$942$		0			0
$943$		−	4.19334e13i		−	1.72686i
$944$		0			0
$945$	4.36459e12			0.178033
$946$		0			0
$947$		−	3.32320e13i		−	1.34271i	−0.741137	−	0.671354i	$-0.765713\pi$
$947$		−	3.32320e13i		−	1.34271i	0.741137	−	0.671354i	$-0.234287\pi$
$948$		0			0
$949$	3.12141e11	+	2.27926e13i	0.0124926	+	0.912213i
$950$		0			0
$951$			5.63310e12i			0.223324i
$952$		0			0
$953$	1.98917e13			0.781186			0.390593	−	0.920564i	$-0.372270\pi$
$953$	1.98917e13			0.781186			0.390593	+	0.920564i	$0.372270\pi$
$954$		0			0
$955$		−	9.28361e12i		−	0.361162i
$956$		0			0
$957$			1.52444e12i			0.0587500i
$958$		0			0
$959$	−4.15860e13			−1.58768
$960$		0			0
$961$	1.66701e13			0.630496
$962$		0			0
$963$	−1.14000e13			−0.427158
$964$		0			0
$965$	−8.31557e12			−0.308688
$966$		0			0
$967$		−	3.34720e13i		−	1.23101i	−0.788132	−	0.615507i	$-0.788951\pi$
$967$		−	3.34720e13i		−	1.23101i	0.788132	−	0.615507i	$-0.211049\pi$
$968$		0			0
$969$			1.79772e12i			0.0655036i
$970$		0			0
$971$	4.61975e13			1.66775			0.833876	−	0.551951i	$-0.186117\pi$
$971$	4.61975e13			1.66775			0.833876	+	0.551951i	$0.186117\pi$
$972$		0			0
$973$		−	6.42712e13i		−	2.29884i
$974$		0			0
$975$	−6.51918e12	+	8.92789e10i	−0.231032	+	0.00316394i
$976$		0			0
$977$			2.33586e13i			0.820203i	0.912040	+	0.410102i	$0.134507\pi$
$977$			2.33586e13i			0.820203i	−0.912040	+	0.410102i	$0.865493\pi$
$978$		0			0
$979$	−1.75891e13			−0.611956
$980$		0			0
$981$		−	8.96405e12i		−	0.309025i
$982$		0			0
$983$			3.55761e13i			1.21526i	0.794222	+	0.607628i	$0.207879\pi$
$983$			3.55761e13i			1.21526i	−0.794222	+	0.607628i	$0.792121\pi$
$984$		0			0
$985$	−7.75993e12			−0.262661
$986$		0			0
$987$	−5.65834e12			−0.189785
$988$		0			0
$989$	2.18094e13			0.724869
$990$		0			0
$991$	−2.68063e13			−0.882889			−0.441444	−	0.897289i	$-0.645534\pi$
$991$	−2.68063e13			−0.882889			−0.441444	+	0.897289i	$0.645534\pi$
$992$		0			0
$993$			2.45245e12i			0.0800440i
$994$		0			0
$995$		−	1.47013e12i		−	0.0475500i
$996$		0			0
$997$	−1.07918e13			−0.345913			−0.172957	−	0.984929i	$-0.555332\pi$
$997$	−1.07918e13			−0.345913			−0.172957	+	0.984929i	$0.555332\pi$
$998$		0			0
$999$		−	8.78052e12i		−	0.278917i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	104.10.f.a.25.20	yes	32
4.3	odd	2		208.10.f.d.129.13		32
13.12	even	2	inner	104.10.f.a.25.19	✓	32
52.51	odd	2		208.10.f.d.129.14		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
104.10.f.a.25.19	✓	32	13.12	even	2	inner
104.10.f.a.25.20	yes	32	1.1	even	1	trivial
208.10.f.d.129.13		32	4.3	odd	2
208.10.f.d.129.14		32	52.51	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 104 = 2^{3} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	104.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+35.4013 q^{3} +405.839i q^{5} +7970.77i q^{7} -18429.7 q^{9} -20385.0i q^{11} +(-102968. + 1410.13i) q^{13} +14367.2i q^{15} -92739.6 q^{17} -547568. i q^{19} +282175. i q^{21} +1.27351e6 q^{23} +1.78842e6 q^{25} -1.34924e6 q^{27} -2.11242e6 q^{29} -3.12563e6i q^{31} -721656. i q^{33} -3.23485e6 q^{35} +6.50775e6i q^{37} +(-3.64522e6 + 49920.6i) q^{39} -3.29275e7i q^{41} +1.71254e7 q^{43} -7.47951e6i q^{45} +2.00526e7i q^{47} -2.31795e7 q^{49} -3.28310e6 q^{51} +5.66621e7 q^{53} +8.27303e6 q^{55} -1.93846e7i q^{57} -6.94997e7i q^{59} -1.29165e8 q^{61} -1.46899e8i q^{63} +(-572287. - 4.17886e7i) q^{65} -7.67824e7i q^{67} +4.50838e7 q^{69} -1.90253e8i q^{71} -2.21355e8i q^{73} +6.33124e7 q^{75} +1.62484e8 q^{77} -2.74140e8 q^{79} +3.14988e8 q^{81} -2.37271e8i q^{83} -3.76373e7i q^{85} -7.47826e7 q^{87} -8.62842e8i q^{89} +(-1.12399e7 - 8.20738e8i) q^{91} -1.10651e8i q^{93} +2.22224e8 q^{95} -1.15595e9i q^{97} +3.75691e8i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 162 q^{3} + 223074 q^{9} + 66270 q^{13} - 487902 q^{17} + 3171556 q^{23} - 13526722 q^{25} - 3694974 q^{27} + 8833508 q^{29} - 8281126 q^{35} - 12056860 q^{39} + 89959038 q^{43} - 172344874 q^{49}+ \cdots + 1741143356 q^{95}+O(q^{100}) \) 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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