LMFDB - Embedded newform 104.10.f.a.25.17

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			25.17
Character	$\chi$	$=$	104.25
Dual form			104.10.f.a.25.18

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-12.4770 q^{3} +1341.60i q^{5} -1173.85i q^{7} -19527.3 q^{9} -70941.0i q^{11} +(89435.4 + 51047.1i) q^{13} -16739.1i q^{15} +499392. q^{17} -65647.1i q^{19} +14646.1i q^{21} -1.08432e6 q^{23} +153237. q^{25} +489225. q^{27} -2.16430e6 q^{29} -1.72498e6i q^{31} +885128. i q^{33} +1.57484e6 q^{35} +1.04421e7i q^{37} +(-1.11588e6 - 636912. i) q^{39} +2.64490e7i q^{41} +180930. q^{43} -2.61978e7i q^{45} -6.80472e6i q^{47} +3.89757e7 q^{49} -6.23089e6 q^{51} +2.42430e7 q^{53} +9.51744e7 q^{55} +819076. i q^{57} +1.74914e8i q^{59} -4.64763e7 q^{61} +2.29222e7i q^{63} +(-6.84847e7 + 1.19986e8i) q^{65} +1.58494e8i q^{67} +1.35291e7 q^{69} -3.28213e7i q^{71} +2.15107e8i q^{73} -1.91193e6 q^{75} -8.32744e7 q^{77} +6.10120e8 q^{79} +3.78252e8 q^{81} +2.98207e7i q^{83} +6.69984e8i q^{85} +2.70038e7 q^{87} +2.33514e8i q^{89} +(5.99218e7 - 1.04984e8i) q^{91} +2.15225e7i q^{93} +8.80721e7 q^{95} +1.03780e8i q^{97} +1.38529e9i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−12.4770			−0.0889330			−0.0444665	−	0.999011i	$-0.514159\pi$
$3$	−12.4770			−0.0889330			−0.0444665	+	0.999011i	$0.514159\pi$
$4$		0			0
$5$			1341.60i			0.959970i	0.877277	+	0.479985i	$0.159358\pi$
$5$			1341.60i			0.959970i	−0.877277	+	0.479985i	$0.840642\pi$
$6$		0			0
$7$		−	1173.85i		−	0.184788i	−0.995723	−	0.0923938i	$-0.970548\pi$
$7$		−	1173.85i		−	0.184788i	0.995723	−	0.0923938i	$-0.0294519\pi$
$8$		0			0
$9$	−19527.3			−0.992091
$10$		0			0
$11$		−	70941.0i		−	1.46093i	−0.682948	−	0.730467i	$-0.739303\pi$
$11$		−	70941.0i		−	1.46093i	0.682948	−	0.730467i	$-0.260697\pi$
$12$		0			0
$13$	89435.4	+	51047.1i	0.868489	+	0.495708i
$13$	89435.4	+	51047.1i	0.868489	+	0.495708i
$14$		0			0
$15$		−	16739.1i		−	0.0853730i
$16$		0			0
$17$	499392.			1.45018			0.725089	−	0.688655i	$-0.241799\pi$
$17$	499392.			1.45018			0.725089	+	0.688655i	$0.241799\pi$
$18$		0			0
$19$		−	65647.1i		−	0.115565i	−0.998329	−	0.0577823i	$-0.981597\pi$
$19$		−	65647.1i		−	0.115565i	0.998329	−	0.0577823i	$-0.0184029\pi$
$20$		0			0
$21$			14646.1i			0.0164337i
$22$		0			0
$23$	−1.08432e6			−0.807949			−0.403975	−	0.914770i	$-0.632372\pi$
$23$	−1.08432e6			−0.807949			−0.403975	+	0.914770i	$0.632372\pi$
$24$		0			0
$25$	153237.			0.0784573
$26$		0			0
$27$	489225.			0.177163
$28$		0			0
$29$	−2.16430e6			−0.568232			−0.284116	−	0.958790i	$-0.591700\pi$
$29$	−2.16430e6			−0.568232			−0.284116	+	0.958790i	$0.591700\pi$
$30$		0			0
$31$		−	1.72498e6i		−	0.335472i	−0.985832	−	0.167736i	$-0.946354\pi$
$31$		−	1.72498e6i		−	0.335472i	0.985832	−	0.167736i	$-0.0536457\pi$
$32$		0			0
$33$			885128.i			0.129925i
$34$		0			0
$35$	1.57484e6			0.177391
$36$		0			0
$37$			1.04421e7i			0.915967i	0.888961	+	0.457983i	$0.151428\pi$
$37$			1.04421e7i			0.915967i	−0.888961	+	0.457983i	$0.848572\pi$
$38$		0			0
$39$	−1.11588e6	−	636912.i	−0.0772374	−	0.0440848i
$40$		0			0
$41$			2.64490e7i			1.46178i	0.682496	+	0.730890i	$0.260895\pi$
$41$			2.64490e7i			1.46178i	−0.682496	+	0.730890i	$0.739105\pi$
$42$		0			0
$43$	180930.			0.00807054			0.00403527	−	0.999992i	$-0.498716\pi$
$43$	180930.			0.00807054			0.00403527	+	0.999992i	$0.498716\pi$
$44$		0			0
$45$		−	2.61978e7i		−	0.952378i
$46$		0			0
$47$		−	6.80472e6i		−	0.203409i	−0.994815	−	0.101705i	$-0.967570\pi$
$47$		−	6.80472e6i		−	0.203409i	0.994815	−	0.101705i	$-0.0324296\pi$
$48$		0			0
$49$	3.89757e7			0.965854
$50$		0			0
$51$	−6.23089e6			−0.128969
$52$		0			0
$53$	2.42430e7			0.422032			0.211016	−	0.977483i	$-0.432323\pi$
$53$	2.42430e7			0.422032			0.211016	+	0.977483i	$0.432323\pi$
$54$		0			0
$55$	9.51744e7			1.40245
$56$		0			0
$57$			819076.i			0.0102775i
$58$		0			0
$59$			1.74914e8i			1.87927i	0.342173	+	0.939637i	$0.388837\pi$
$59$			1.74914e8i			1.87927i	−0.342173	+	0.939637i	$0.611163\pi$
$60$		0			0
$61$	−4.64763e7			−0.429781			−0.214890	−	0.976638i	$-0.568939\pi$
$61$	−4.64763e7			−0.429781			−0.214890	+	0.976638i	$0.568939\pi$
$62$		0			0
$63$			2.29222e7i			0.183326i
$64$		0			0
$65$	−6.84847e7	+	1.19986e8i	−0.475865	+	0.833724i
$66$		0			0
$67$			1.58494e8i			0.960893i	0.877024	+	0.480446i	$0.159525\pi$
$67$			1.58494e8i			0.960893i	−0.877024	+	0.480446i	$0.840475\pi$
$68$		0			0
$69$	1.35291e7			0.0718533
$70$		0			0
$71$		−	3.28213e7i		−	0.153283i	−0.997059	−	0.0766413i	$-0.975580\pi$
$71$		−	3.28213e7i		−	0.153283i	0.997059	−	0.0766413i	$-0.0244196\pi$
$72$		0			0
$73$			2.15107e8i			0.886549i	0.896386	+	0.443274i	$0.146183\pi$
$73$			2.15107e8i			0.886549i	−0.896386	+	0.443274i	$0.853817\pi$
$74$		0			0
$75$	−1.91193e6			−0.00697745
$76$		0			0
$77$	−8.32744e7			−0.269962
$78$		0			0
$79$	6.10120e8			1.76235			0.881177	−	0.472786i	$-0.156752\pi$
$79$	6.10120e8			1.76235			0.881177	+	0.472786i	$0.156752\pi$
$80$		0			0
$81$	3.78252e8			0.976335
$82$		0			0
$83$			2.98207e7i			0.0689710i	0.999405	+	0.0344855i	$0.0109793\pi$
$83$			2.98207e7i			0.0689710i	−0.999405	+	0.0344855i	$0.989021\pi$
$84$		0			0
$85$			6.69984e8i			1.39213i
$86$		0			0
$87$	2.70038e7			0.0505346
$88$		0			0
$89$			2.33514e8i			0.394510i	0.980352	+	0.197255i	$0.0632027\pi$
$89$			2.33514e8i			0.394510i	−0.980352	+	0.197255i	$0.936797\pi$
$90$		0			0
$91$	5.99218e7	−	1.04984e8i	0.0916007	−	0.160486i
$92$		0			0
$93$			2.15225e7i			0.0298346i
$94$		0			0
$95$	8.80721e7			0.110939
$96$		0			0
$97$			1.03780e8i			0.119026i	0.998228	+	0.0595130i	$0.0189548\pi$
$97$			1.03780e8i			0.119026i	−0.998228	+	0.0595130i	$0.981045\pi$
$98$		0			0
$99$			1.38529e9i			1.44938i
$100$		0			0
$101$	1.16333e9			1.11239			0.556196	−	0.831051i	$-0.312260\pi$
$101$	1.16333e9			1.11239			0.556196	+	0.831051i	$0.312260\pi$
$102$		0			0
$103$	−8.60387e8			−0.753228			−0.376614	−	0.926370i	$-0.622912\pi$
$103$	−8.60387e8			−0.753228			−0.376614	+	0.926370i	$0.622912\pi$
$104$		0			0
$105$	−1.96492e7			−0.0157759
$106$		0			0
$107$	2.34920e9			1.73258			0.866290	−	0.499541i	$-0.166498\pi$
$107$	2.34920e9			1.73258			0.866290	+	0.499541i	$0.166498\pi$
$108$		0			0
$109$			1.88936e9i			1.28202i	0.767533	+	0.641010i	$0.221484\pi$
$109$			1.88936e9i			1.28202i	−0.767533	+	0.641010i	$0.778516\pi$
$110$		0			0
$111$		−	1.30286e8i		−	0.0814597i
$112$		0			0
$113$	−1.14370e9			−0.659871			−0.329936	−	0.944003i	$-0.607027\pi$
$113$	−1.14370e9			−0.659871			−0.329936	+	0.944003i	$0.607027\pi$
$114$		0			0
$115$		−	1.45473e9i		−	0.775607i
$116$		0			0
$117$	−1.74643e9	−	9.96812e8i	−0.861621	−	0.491787i
$118$		0			0
$119$		−	5.86213e8i		−	0.267975i
$120$		0			0
$121$	−2.67468e9			−1.13433
$122$		0			0
$123$		−	3.30003e8i		−	0.130000i
$124$		0			0
$125$			2.82589e9i			1.03529i
$126$		0			0
$127$	−1.77859e9			−0.606679			−0.303340	−	0.952883i	$-0.598102\pi$
$127$	−1.77859e9			−0.606679			−0.303340	+	0.952883i	$0.598102\pi$
$128$		0			0
$129$	−2.25746e6			−0.000717738
$130$		0			0
$131$	5.62697e9			1.66937			0.834687	−	0.550724i	$-0.185648\pi$
$131$	5.62697e9			1.66937			0.834687	+	0.550724i	$0.185648\pi$
$132$		0			0
$133$	−7.70602e7			−0.0213549
$134$		0			0
$135$			6.56344e8i			0.170071i
$136$		0			0
$137$		−	1.80239e9i		−	0.437126i	−0.975823	−	0.218563i	$-0.929863\pi$
$137$		−	1.80239e9i		−	0.437126i	0.975823	−	0.218563i	$-0.0701369\pi$
$138$		0			0
$139$	−6.03680e8			−0.137164			−0.0685820	−	0.997645i	$-0.521847\pi$
$139$	−6.03680e8			−0.137164			−0.0685820	+	0.997645i	$0.521847\pi$
$140$		0			0
$141$			8.49022e7i			0.0180898i
$142$		0			0
$143$	3.62133e9	−	6.34464e9i	0.724196	−	1.26881i
$144$		0			0
$145$		−	2.90362e9i		−	0.545486i
$146$		0			0
$147$	−4.86298e8			−0.0858963
$148$		0			0
$149$		−	8.13305e9i		−	1.35181i	−0.736990	−	0.675904i	$-0.763753\pi$
$149$		−	8.13305e9i		−	1.35181i	0.736990	−	0.675904i	$-0.236247\pi$
$150$		0			0
$151$		−	6.52751e9i		−	1.02177i	−0.859651	−	0.510883i	$-0.829319\pi$
$151$		−	6.52751e9i		−	1.02177i	0.859651	−	0.510883i	$-0.170681\pi$
$152$		0			0
$153$	−9.75179e9			−1.43871
$154$		0			0
$155$	2.31423e9			0.322043
$156$		0			0
$157$	−1.29030e10			−1.69490			−0.847449	−	0.530877i	$-0.821863\pi$
$157$	−1.29030e10			−1.69490			−0.847449	+	0.530877i	$0.821863\pi$
$158$		0			0
$159$	−3.02479e8			−0.0375326
$160$		0			0
$161$			1.27284e9i			0.149299i
$162$		0			0
$163$			2.70651e9i			0.300307i	0.988663	+	0.150154i	$0.0479768\pi$
$163$			2.70651e9i			0.300307i	−0.988663	+	0.150154i	$0.952023\pi$
$164$		0			0
$165$	−1.18749e9			−0.124724
$166$		0			0
$167$		−	1.50039e10i		−	1.49272i	−0.665540	−	0.746362i	$-0.731799\pi$
$167$		−	1.50039e10i		−	1.49272i	0.665540	−	0.746362i	$-0.268201\pi$
$168$		0			0
$169$	5.39290e9	+	9.13083e9i	0.508548	+	0.861034i
$170$		0			0
$171$			1.28191e9i			0.114651i
$172$		0			0
$173$	−2.05346e10			−1.74292			−0.871461	−	0.490465i	$-0.836827\pi$
$173$	−2.05346e10			−1.74292			−0.871461	+	0.490465i	$0.836827\pi$
$174$		0			0
$175$		−	1.79878e8i		−	0.0144979i
$176$		0			0
$177$		−	2.18239e9i		−	0.167129i
$178$		0			0
$179$	1.27422e8			0.00927694			0.00463847	−	0.999989i	$-0.498524\pi$
$179$	1.27422e8			0.00927694			0.00463847	+	0.999989i	$0.498524\pi$
$180$		0			0
$181$	2.04762e10			1.41806			0.709031	−	0.705178i	$-0.249133\pi$
$181$	2.04762e10			1.41806			0.709031	+	0.705178i	$0.249133\pi$
$182$		0			0
$183$	5.79882e8			0.0382217
$184$		0			0
$185$	−1.40091e10			−0.879301
$186$		0			0
$187$		−	3.54274e10i		−	2.11861i
$188$		0			0
$189$		−	5.74279e8i		−	0.0327375i
$190$		0			0
$191$	5.61922e9			0.305510			0.152755	−	0.988264i	$-0.451185\pi$
$191$	5.61922e9			0.305510			0.152755	+	0.988264i	$0.451185\pi$
$192$		0			0
$193$			2.62549e10i			1.36208i	0.732246	+	0.681041i	$0.238472\pi$
$193$			2.62549e10i			1.36208i	−0.732246	+	0.681041i	$0.761528\pi$
$194$		0			0
$195$	8.54480e8	−	1.49707e9i	0.0423201	−	0.0741456i
$196$		0			0
$197$		−	8.27421e9i		−	0.391407i	−0.980663	−	0.195704i	$-0.937301\pi$
$197$		−	8.27421e9i		−	0.391407i	0.980663	−	0.195704i	$-0.0626990\pi$
$198$		0			0
$199$	−3.06455e9			−0.138525			−0.0692625	−	0.997598i	$-0.522065\pi$
$199$	−3.06455e9			−0.138525			−0.0692625	+	0.997598i	$0.522065\pi$
$200$		0			0
$201$		−	1.97752e9i		−	0.0854551i
$202$		0			0
$203$			2.54057e9i			0.105002i
$204$		0			0
$205$	−3.54839e10			−1.40326
$206$		0			0
$207$	2.11740e10			0.801559
$208$		0			0
$209$	−4.65708e9			−0.168832
$210$		0			0
$211$	3.94857e10			1.37141			0.685707	−	0.727877i	$-0.259493\pi$
$211$	3.94857e10			1.37141			0.685707	+	0.727877i	$0.259493\pi$
$212$		0			0
$213$			4.09510e8i			0.0136319i
$214$		0			0
$215$			2.42736e8i			0.00774748i
$216$		0			0
$217$	−2.02488e9			−0.0619911
$218$		0			0
$219$		−	2.68389e9i		−	0.0788434i
$220$		0			0
$221$	4.46633e10	+	2.54925e10i	1.25946	+	0.718864i
$222$		0			0
$223$			4.19352e10i			1.13555i	0.823183	+	0.567776i	$0.192196\pi$
$223$			4.19352e10i			1.13555i	−0.823183	+	0.567776i	$0.807804\pi$
$224$		0			0
$225$	−2.99231e9			−0.0778368
$226$		0			0
$227$			3.55412e10i			0.888415i	0.895924	+	0.444207i	$0.146515\pi$
$227$			3.55412e10i			0.888415i	−0.895924	+	0.444207i	$0.853485\pi$
$228$		0			0
$229$		−	1.69032e10i		−	0.406170i	−0.979161	−	0.203085i	$-0.934903\pi$
$229$		−	1.69032e10i		−	0.406170i	0.979161	−	0.203085i	$-0.0650968\pi$
$230$		0			0
$231$	1.03901e9			0.0240086
$232$		0			0
$233$	1.36761e10			0.303991			0.151996	−	0.988381i	$-0.451430\pi$
$233$	1.36761e10			0.303991			0.151996	+	0.988381i	$0.451430\pi$
$234$		0			0
$235$	9.12921e9			0.195267
$236$		0			0
$237$	−7.61244e9			−0.156731
$238$		0			0
$239$			7.07265e10i			1.40214i	0.713092	+	0.701071i	$0.247294\pi$
$239$			7.07265e10i			1.40214i	−0.713092	+	0.701071i	$0.752706\pi$
$240$		0			0
$241$		−	3.43929e10i		−	0.656739i	−0.944549	−	0.328369i	$-0.893501\pi$
$241$		−	3.43929e10i		−	0.656739i	0.944549	−	0.328369i	$-0.106499\pi$
$242$		0			0
$243$	−1.43489e10			−0.263991
$244$		0			0
$245$			5.22897e10i			0.927191i
$246$		0			0
$247$	3.35109e9	−	5.87118e9i	0.0572862	−	0.100367i
$248$		0			0
$249$		−	3.72072e8i		−	0.00613380i
$250$		0			0
$251$	−8.96466e10			−1.42561			−0.712807	−	0.701360i	$-0.752576\pi$
$251$	−8.96466e10			−1.42561			−0.712807	+	0.701360i	$0.752576\pi$
$252$		0			0
$253$			7.69231e10i			1.18036i
$254$		0			0
$255$		−	8.35936e9i		−	0.123806i
$256$		0			0
$257$	−1.12810e10			−0.161305			−0.0806525	−	0.996742i	$-0.525700\pi$
$257$	−1.12810e10			−0.161305			−0.0806525	+	0.996742i	$0.525700\pi$
$258$		0			0
$259$	1.22575e10			0.169259
$260$		0			0
$261$	4.22629e10			0.563738
$262$		0			0
$263$	1.23948e11			1.59749			0.798744	−	0.601671i	$-0.205498\pi$
$263$	1.23948e11			1.59749			0.798744	+	0.601671i	$0.205498\pi$
$264$		0			0
$265$			3.25244e10i			0.405138i
$266$		0			0
$267$		−	2.91355e9i		−	0.0350850i
$268$		0			0
$269$	4.23467e10			0.493099			0.246550	−	0.969130i	$-0.420703\pi$
$269$	4.23467e10			0.493099			0.246550	+	0.969130i	$0.420703\pi$
$270$		0			0
$271$			1.79741e10i			0.202435i	0.994864	+	0.101218i	$0.0322738\pi$
$271$			1.79741e10i			0.202435i	−0.994864	+	0.101218i	$0.967726\pi$
$272$		0			0
$273$	−7.47642e8	+	1.30988e9i	−0.00814632	+	0.0142725i
$274$		0			0
$275$		−	1.08708e10i		−	0.114621i
$276$		0			0
$277$	6.61079e10			0.674675			0.337337	−	0.941384i	$-0.390474\pi$
$277$	6.61079e10			0.674675			0.337337	+	0.941384i	$0.390474\pi$
$278$		0			0
$279$			3.36843e10i			0.332819i
$280$		0			0
$281$			1.99778e10i			0.191148i	0.995422	+	0.0955740i	$0.0304687\pi$
$281$			1.99778e10i			0.191148i	−0.995422	+	0.0955740i	$0.969531\pi$
$282$		0			0
$283$	5.17876e10			0.479940			0.239970	−	0.970780i	$-0.422862\pi$
$283$	5.17876e10			0.479940			0.239970	+	0.970780i	$0.422862\pi$
$284$		0			0
$285$	−1.09887e9			−0.00986609
$286$		0			0
$287$	3.10473e10			0.270119
$288$		0			0
$289$	1.30804e11			1.10302
$290$		0			0
$291$		−	1.29486e9i		−	0.0105853i
$292$		0			0
$293$		−	8.98975e10i		−	0.712596i	−0.934372	−	0.356298i	$-0.884039\pi$
$293$		−	8.98975e10i		−	0.712596i	0.934372	−	0.356298i	$-0.115961\pi$
$294$		0			0
$295$	−2.34664e11			−1.80405
$296$		0			0
$297$		−	3.47062e10i		−	0.258823i
$298$		0			0
$299$	−9.69770e10	−	5.53516e10i	−0.701695	−	0.400507i
$300$		0			0
$301$		−	2.12386e8i		−	0.00149134i
$302$		0			0
$303$	−1.45149e10			−0.0989284
$304$		0			0
$305$		−	6.23525e10i		−	0.412577i
$306$		0			0
$307$			1.34965e11i			0.867161i	0.901115	+	0.433580i	$0.142750\pi$
$307$			1.34965e11i			0.867161i	−0.901115	+	0.433580i	$0.857250\pi$
$308$		0			0
$309$	1.07350e10			0.0669868
$310$		0			0
$311$	−2.82687e11			−1.71350			−0.856751	−	0.515731i	$-0.827520\pi$
$311$	−2.82687e11			−1.71350			−0.856751	+	0.515731i	$0.827520\pi$
$312$		0			0
$313$	5.02489e10			0.295922			0.147961	−	0.988993i	$-0.452729\pi$
$313$	5.02489e10			0.295922			0.147961	+	0.988993i	$0.452729\pi$
$314$		0			0
$315$	−3.07524e10			−0.175988
$316$		0			0
$317$		−	1.65692e11i		−	0.921584i	−0.887508	−	0.460792i	$-0.847565\pi$
$317$		−	1.65692e11i		−	0.921584i	0.887508	−	0.460792i	$-0.152435\pi$
$318$		0			0
$319$			1.53537e11i			0.830149i
$320$		0			0
$321$	−2.93109e10			−0.154084
$322$		0			0
$323$		−	3.27836e10i		−	0.167589i
$324$		0			0
$325$	1.37048e10	+	7.82230e9i	0.0681394	+	0.0388919i
$326$		0			0
$327$		−	2.35734e10i		−	0.114014i
$328$		0			0
$329$	−7.98775e9			−0.0375875
$330$		0			0
$331$		−	3.69684e11i		−	1.69280i	−0.532551	−	0.846398i	$-0.678767\pi$
$331$		−	3.69684e11i		−	1.69280i	0.532551	−	0.846398i	$-0.321233\pi$
$332$		0			0
$333$		−	2.03906e11i		−	0.908722i
$334$		0			0
$335$	−2.12635e11			−0.922429
$336$		0			0
$337$	1.16361e10			0.0491441			0.0245720	−	0.999698i	$-0.492178\pi$
$337$	1.16361e10			0.0491441			0.0245720	+	0.999698i	$0.492178\pi$
$338$		0			0
$339$	1.42699e10			0.0586844
$340$		0			0
$341$	−1.22372e11			−0.490103
$342$		0			0
$343$		−	9.31210e10i		−	0.363265i
$344$		0			0
$345$			1.81506e10i			0.0689771i
$346$		0			0
$347$	1.75332e11			0.649201			0.324600	−	0.945851i	$-0.394770\pi$
$347$	1.75332e11			0.649201			0.324600	+	0.945851i	$0.394770\pi$
$348$		0			0
$349$		−	3.30557e11i		−	1.19270i	−0.802723	−	0.596352i	$-0.796616\pi$
$349$		−	3.30557e11i		−	1.19270i	0.802723	−	0.596352i	$-0.203384\pi$
$350$		0			0
$351$	4.37541e10	+	2.49735e10i	0.153864	+	0.0878209i
$352$		0			0
$353$		−	8.89138e10i		−	0.304778i	−0.988321	−	0.152389i	$-0.951303\pi$
$353$		−	8.89138e10i		−	0.304778i	0.988321	−	0.152389i	$-0.0486966\pi$
$354$		0			0
$355$	4.40330e10			0.147147
$356$		0			0
$357$			7.31416e9i			0.0238318i
$358$		0			0
$359$			3.61315e11i			1.14805i	0.818837	+	0.574025i	$0.194619\pi$
$359$			3.61315e11i			1.14805i	−0.818837	+	0.574025i	$0.805381\pi$
$360$		0			0
$361$	3.18378e11			0.986645
$362$		0			0
$363$	3.33719e10			0.100879
$364$		0			0
$365$	−2.88588e11			−0.851060
$366$		0			0
$367$	−4.62890e11			−1.33193			−0.665963	−	0.745985i	$-0.731979\pi$
$367$	−4.62890e11			−1.33193			−0.665963	+	0.745985i	$0.731979\pi$
$368$		0			0
$369$		−	5.16478e11i		−	1.45022i
$370$		0			0
$371$		−	2.84578e10i		−	0.0779864i
$372$		0			0
$373$	1.75397e11			0.469171			0.234586	−	0.972095i	$-0.424627\pi$
$373$	1.75397e11			0.469171			0.234586	+	0.972095i	$0.424627\pi$
$374$		0			0
$375$		−	3.52585e10i		−	0.0920712i
$376$		0			0
$377$	−1.93565e11	−	1.10481e11i	−0.493504	−	0.281677i
$378$		0			0
$379$			3.55221e10i			0.0884345i	0.999022	+	0.0442172i	$0.0140794\pi$
$379$			3.55221e10i			0.0884345i	−0.999022	+	0.0442172i	$0.985921\pi$
$380$		0			0
$381$	2.21914e10			0.0539538
$382$		0			0
$383$		−	7.85980e11i		−	1.86645i	−0.359289	−	0.933226i	$-0.616981\pi$
$383$		−	7.85980e11i		−	1.86645i	0.359289	−	0.933226i	$-0.383019\pi$
$384$		0			0
$385$		−	1.11721e11i		−	0.259156i
$386$		0			0
$387$	−3.53308e9			−0.00800671
$388$		0			0
$389$	3.74700e11			0.829679			0.414839	−	0.909895i	$-0.363838\pi$
$389$	3.74700e11			0.829679			0.414839	+	0.909895i	$0.363838\pi$
$390$		0			0
$391$	−5.41503e11			−1.17167
$392$		0			0
$393$	−7.02074e10			−0.148462
$394$		0			0
$395$			8.18536e11i			1.69181i
$396$		0			0
$397$			8.78858e10i			0.177567i	0.996051	+	0.0887834i	$0.0282979\pi$
$397$			8.78858e10i			0.177567i	−0.996051	+	0.0887834i	$0.971702\pi$
$398$		0			0
$399$	9.61476e8			0.00189916
$400$		0			0
$401$			2.08722e11i			0.403105i	0.979478	+	0.201553i	$0.0645987\pi$
$401$			2.08722e11i			0.403105i	−0.979478	+	0.201553i	$0.935401\pi$
$402$		0			0
$403$	8.80552e10	−	1.54274e11i	0.166296	−	0.291354i
$404$		0			0
$405$			5.07463e11i			0.937253i
$406$		0			0
$407$	7.40773e11			1.33817
$408$		0			0
$409$			9.73857e11i			1.72084i	0.509586	+	0.860420i	$0.329798\pi$
$409$			9.73857e11i			1.72084i	−0.509586	+	0.860420i	$0.670202\pi$
$410$		0			0
$411$			2.24884e10i			0.0388749i
$412$		0			0
$413$	2.05323e11			0.347267
$414$		0			0
$415$	−4.00075e10			−0.0662101
$416$		0			0
$417$	7.53209e9			0.0121984
$418$		0			0
$419$	−9.21103e11			−1.45997			−0.729987	−	0.683461i	$-0.760474\pi$
$419$	−9.21103e11			−1.45997			−0.729987	+	0.683461i	$0.760474\pi$
$420$		0			0
$421$			5.16727e11i			0.801663i	0.916152	+	0.400832i	$0.131279\pi$
$421$			5.16727e11i			0.801663i	−0.916152	+	0.400832i	$0.868721\pi$
$422$		0			0
$423$			1.32878e11i			0.201800i
$424$		0			0
$425$	7.65253e10			0.113777
$426$		0			0
$427$			5.45564e10i			0.0794182i
$428$		0			0
$429$	−4.51832e10	+	7.91618e10i	−0.0644049	+	0.112839i
$430$		0			0
$431$		−	5.07960e11i		−	0.709057i	−0.935045	−	0.354529i	$-0.884641\pi$
$431$		−	5.07960e11i		−	0.709057i	0.935045	−	0.354529i	$-0.115359\pi$
$432$		0			0
$433$	3.81600e11			0.521691			0.260845	−	0.965381i	$-0.415999\pi$
$433$	3.81600e11			0.521691			0.260845	+	0.965381i	$0.415999\pi$
$434$		0			0
$435$			3.62283e10i			0.0485117i
$436$		0			0
$437$			7.11828e10i			0.0933703i
$438$		0			0
$439$	3.17556e11			0.408065			0.204033	−	0.978964i	$-0.434595\pi$
$439$	3.17556e11			0.408065			0.204033	+	0.978964i	$0.434595\pi$
$440$		0			0
$441$	−7.61091e11			−0.958215
$442$		0			0
$443$	1.01864e12			1.25662			0.628311	−	0.777963i	$-0.283747\pi$
$443$	1.01864e12			1.25662			0.628311	+	0.777963i	$0.283747\pi$
$444$		0			0
$445$	−3.13282e11			−0.378718
$446$		0			0
$447$			1.01476e11i			0.120220i
$448$		0			0
$449$		−	1.43267e12i		−	1.66355i	−0.555110	−	0.831777i	$-0.687324\pi$
$449$		−	1.43267e12i		−	1.66355i	0.555110	−	0.831777i	$-0.312676\pi$
$450$		0			0
$451$	1.87632e12			2.13556
$452$		0			0
$453$			8.14434e10i			0.0908686i
$454$		0			0
$455$	1.40847e11	+	8.03910e10i	0.154062	+	0.0879339i
$456$		0			0
$457$			1.08497e12i			1.16358i	0.813339	+	0.581790i	$0.197647\pi$
$457$			1.08497e12i			1.16358i	−0.813339	+	0.581790i	$0.802353\pi$
$458$		0			0
$459$	2.44315e11			0.256917
$460$		0			0
$461$			7.48131e11i			0.771478i	0.922608	+	0.385739i	$0.126053\pi$
$461$			7.48131e11i			0.771478i	−0.922608	+	0.385739i	$0.873947\pi$
$462$		0			0
$463$			1.29921e12i			1.31390i	0.753933	+	0.656952i	$0.228154\pi$
$463$			1.29921e12i			1.31390i	−0.753933	+	0.656952i	$0.771846\pi$
$464$		0			0
$465$	−2.88746e10			−0.0286403
$466$		0			0
$467$	1.30622e12			1.27084			0.635421	−	0.772166i	$-0.280827\pi$
$467$	1.30622e12			1.27084			0.635421	+	0.772166i	$0.280827\pi$
$468$		0			0
$469$	1.86048e11			0.177561
$470$		0			0
$471$	1.60991e11			0.150732
$472$		0			0
$473$		−	1.28354e10i		−	0.0117905i
$474$		0			0
$475$		−	1.00596e10i		−	0.00906689i
$476$		0			0
$477$	−4.73402e11			−0.418694
$478$		0			0
$479$			3.33053e11i			0.289070i	0.989500	+	0.144535i	$0.0461686\pi$
$479$			3.33053e11i			0.289070i	−0.989500	+	0.144535i	$0.953831\pi$
$480$		0			0
$481$	−5.33038e11	+	9.33893e11i	−0.454052	+	0.795508i
$482$		0			0
$483$		−	1.58812e10i		−	0.0132776i
$484$		0			0
$485$	−1.39231e11			−0.114261
$486$		0			0
$487$			1.22120e12i			0.983798i	0.870652	+	0.491899i	$0.163697\pi$
$487$			1.22120e12i			0.983798i	−0.870652	+	0.491899i	$0.836303\pi$
$488$		0			0
$489$		−	3.37690e10i		−	0.0267072i
$490$		0			0
$491$	1.26215e12			0.980039			0.490020	−	0.871711i	$-0.336990\pi$
$491$	1.26215e12			0.980039			0.490020	+	0.871711i	$0.336990\pi$
$492$		0			0
$493$	−1.08083e12			−0.824038
$494$		0			0
$495$	−1.85850e12			−1.39136
$496$		0			0
$497$	−3.85274e10			−0.0283247
$498$		0			0
$499$		−	1.29029e12i		−	0.931609i	−0.884888	−	0.465804i	$-0.845765\pi$
$499$		−	1.29029e12i		−	0.931609i	0.884888	−	0.465804i	$-0.154235\pi$
$500$		0			0
$501$			1.87203e11i			0.132752i
$502$		0			0
$503$	−1.70917e12			−1.19050			−0.595250	−	0.803540i	$-0.702947\pi$
$503$	−1.70917e12			−1.19050			−0.595250	+	0.803540i	$0.702947\pi$
$504$		0			0
$505$			1.56073e12i			1.06786i
$506$		0			0
$507$	−6.72869e10	−	1.13925e11i	−0.0452267	−	0.0765743i
$508$		0			0
$509$		−	1.03330e12i		−	0.682331i	−0.940003	−	0.341166i	$-0.889178\pi$
$509$		−	1.03330e12i		−	0.682331i	0.940003	−	0.341166i	$-0.110822\pi$
$510$		0			0
$511$	2.52505e11			0.163823
$512$		0			0
$513$		−	3.21163e10i		−	0.0204737i
$514$		0			0
$515$		−	1.15429e12i		−	0.723076i
$516$		0			0
$517$	−4.82734e11			−0.297167
$518$		0			0
$519$	2.56209e11			0.155003
$520$		0			0
$521$	1.72125e12			1.02347			0.511735	−	0.859143i	$-0.329003\pi$
$521$	1.72125e12			1.02347			0.511735	+	0.859143i	$0.329003\pi$
$522$		0			0
$523$	2.12871e12			1.24411			0.622056	−	0.782973i	$-0.286298\pi$
$523$	2.12871e12			1.24411			0.622056	+	0.782973i	$0.286298\pi$
$524$		0			0
$525$			2.24433e9i			0.00128935i
$526$		0			0
$527$		−	8.61441e11i		−	0.486495i
$528$		0			0
$529$	−6.25393e11			−0.347218
$530$		0			0
$531$		−	3.41560e12i		−	1.86441i
$532$		0			0
$533$	−1.35014e12	+	2.36548e12i	−0.724615	+	1.26954i
$534$		0			0
$535$			3.15169e12i			1.66323i
$536$		0			0
$537$	−1.58983e9			−0.000825026
$538$		0			0
$539$		−	2.76497e12i		−	1.41105i
$540$		0			0
$541$		−	2.44843e12i		−	1.22885i	−0.788974	−	0.614426i	$-0.789388\pi$
$541$		−	2.44843e12i		−	1.22885i	0.788974	−	0.614426i	$-0.210612\pi$
$542$		0			0
$543$	−2.55480e11			−0.126112
$544$		0			0
$545$	−2.53476e12			−1.23070
$546$		0			0
$547$	2.97232e12			1.41956			0.709778	−	0.704425i	$-0.248795\pi$
$547$	2.97232e12			1.41956			0.709778	+	0.704425i	$0.248795\pi$
$548$		0			0
$549$	9.07557e11			0.426382
$550$		0			0
$551$			1.42080e11i			0.0656675i
$552$		0			0
$553$		−	7.16192e11i		−	0.325661i
$554$		0			0
$555$	1.74791e11			0.0781989
$556$		0			0
$557$			5.60623e10i			0.0246787i	0.999924	+	0.0123394i	$0.00392784\pi$
$557$			5.60623e10i			0.0246787i	−0.999924	+	0.0123394i	$0.996072\pi$
$558$		0			0
$559$	1.61816e10	+	9.23595e9i	0.00700918	+	0.00400063i
$560$		0			0
$561$			4.42026e11i			0.188415i
$562$		0			0
$563$	−3.22969e12			−1.35479			−0.677396	−	0.735619i	$-0.736892\pi$
$563$	−3.22969e12			−1.35479			−0.677396	+	0.735619i	$0.736892\pi$
$564$		0			0
$565$		−	1.53439e12i		−	0.633457i
$566$		0			0
$567$		−	4.44013e11i		−	0.180415i
$568$		0			0
$569$	−1.93587e12			−0.774233			−0.387117	−	0.922031i	$-0.626529\pi$
$569$	−1.93587e12			−0.774233			−0.387117	+	0.922031i	$0.626529\pi$
$570$		0			0
$571$	−9.23235e11			−0.363454			−0.181727	−	0.983349i	$-0.558169\pi$
$571$	−9.23235e11			−0.363454			−0.181727	+	0.983349i	$0.558169\pi$
$572$		0			0
$573$	−7.01108e10			−0.0271699
$574$		0			0
$575$	−1.66159e11			−0.0633895
$576$		0			0
$577$		−	1.65036e12i		−	0.619852i	−0.950761	−	0.309926i	$-0.899696\pi$
$577$		−	1.65036e12i		−	0.619852i	0.950761	−	0.309926i	$-0.100304\pi$
$578$		0			0
$579$		−	3.27582e11i		−	0.121134i
$580$		0			0
$581$	3.50052e10			0.0127450
$582$		0			0
$583$		−	1.71983e12i		−	0.616561i
$584$		0			0
$585$	1.33732e12	−	2.34302e12i	0.472101	−	0.827130i
$586$		0			0
$587$		−	1.41354e12i		−	0.491400i	−0.969346	−	0.245700i	$-0.920982\pi$
$587$		−	1.41354e12i		−	0.491400i	0.969346	−	0.245700i	$-0.0790179\pi$
$588$		0			0
$589$	−1.13240e11			−0.0387687
$590$		0			0
$591$			1.03237e11i			0.0348090i
$592$		0			0
$593$			1.81879e12i			0.604000i	0.953308	+	0.302000i	$0.0976543\pi$
$593$			1.81879e12i			0.604000i	−0.953308	+	0.302000i	$0.902346\pi$
$594$		0			0
$595$	7.86463e11			0.257248
$596$		0			0
$597$	3.82363e10			0.0123195
$598$		0			0
$599$	−3.27051e12			−1.03799			−0.518996	−	0.854777i	$-0.673694\pi$
$599$	−3.27051e12			−1.03799			−0.518996	+	0.854777i	$0.673694\pi$
$600$		0			0
$601$	1.60924e12			0.503136			0.251568	−	0.967840i	$-0.419054\pi$
$601$	1.60924e12			0.503136			0.251568	+	0.967840i	$0.419054\pi$
$602$		0			0
$603$		−	3.09496e12i		−	0.953293i
$604$		0			0
$605$		−	3.58835e12i		−	1.08892i
$606$		0			0
$607$	−4.27251e12			−1.27742			−0.638711	−	0.769447i	$-0.720532\pi$
$607$	−4.27251e12			−1.27742			−0.638711	+	0.769447i	$0.720532\pi$
$608$		0			0
$609$		−	3.16986e10i		−	0.00933817i
$610$		0			0
$611$	3.47361e11	−	6.08583e11i	0.100831	−	0.176659i
$612$		0			0
$613$			1.28996e12i			0.368981i	0.982834	+	0.184491i	$0.0590635\pi$
$613$			1.28996e12i			0.368981i	−0.982834	+	0.184491i	$0.940936\pi$
$614$		0			0
$615$	4.42732e11			0.124797
$616$		0			0
$617$		−	4.64376e12i		−	1.28999i	−0.764186	−	0.644996i	$-0.776859\pi$
$617$		−	4.64376e12i		−	1.28999i	0.764186	−	0.644996i	$-0.223141\pi$
$618$		0			0
$619$			4.79393e12i			1.31245i	0.754564	+	0.656226i	$0.227848\pi$
$619$			4.79393e12i			1.31245i	−0.754564	+	0.656226i	$0.772152\pi$
$620$		0			0
$621$	−5.30479e11			−0.143138
$622$		0			0
$623$	2.74112e11			0.0729006
$624$		0			0
$625$	−3.49192e12			−0.915387
$626$		0			0
$627$	5.81061e10			0.0150147
$628$		0			0
$629$			5.21470e12i			1.32832i
$630$		0			0
$631$			1.51419e12i			0.380232i	0.981762	+	0.190116i	$0.0608865\pi$
$631$			1.51419e12i			0.380232i	−0.981762	+	0.190116i	$0.939114\pi$
$632$		0			0
$633$	−4.92661e11			−0.121964
$634$		0			0
$635$		−	2.38615e12i		−	0.582394i
$636$		0			0
$637$	3.48581e12	+	1.98959e12i	0.838834	+	0.478781i
$638$		0			0
$639$			6.40912e11i			0.152070i
$640$		0			0
$641$	−6.04474e12			−1.41422			−0.707110	−	0.707104i	$-0.750001\pi$
$641$	−6.04474e12			−1.41422			−0.707110	+	0.707104i	$0.750001\pi$
$642$		0			0
$643$			3.84554e12i			0.887171i	0.896232	+	0.443586i	$0.146294\pi$
$643$			3.84554e12i			0.887171i	−0.896232	+	0.443586i	$0.853706\pi$
$644$		0			0
$645$		−	3.02860e9i		−	0.000689007i
$646$		0			0
$647$	−6.90687e12			−1.54957			−0.774787	−	0.632223i	$-0.782143\pi$
$647$	−6.90687e12			−1.54957			−0.774787	+	0.632223i	$0.782143\pi$
$648$		0			0
$649$	1.24086e13			2.74549
$650$		0			0
$651$	2.52643e10			0.00551306
$652$		0			0
$653$	−2.16289e10			−0.00465506			−0.00232753	−	0.999997i	$-0.500741\pi$
$653$	−2.16289e10			−0.00465506			−0.00232753	+	0.999997i	$0.500741\pi$
$654$		0			0
$655$			7.54914e12i			1.60255i
$656$		0			0
$657$		−	4.20047e12i		−	0.879537i
$658$		0			0
$659$	7.02993e12			1.45200			0.726000	−	0.687694i	$-0.241377\pi$
$659$	7.02993e12			1.45200			0.726000	+	0.687694i	$0.241377\pi$
$660$		0			0
$661$		−	8.46497e12i		−	1.72472i	−0.506294	−	0.862361i	$-0.668985\pi$
$661$		−	8.46497e12i		−	1.72472i	0.506294	−	0.862361i	$-0.331015\pi$
$662$		0			0
$663$	−5.57262e11	−	3.18069e11i	−0.112008	−	0.0639308i
$664$		0			0
$665$		−	1.03384e11i		−	0.0205001i
$666$		0			0
$667$	2.34680e12			0.459103
$668$		0			0
$669$		−	5.23224e11i		−	0.100988i
$670$		0			0
$671$			3.29708e12i			0.627881i
$672$		0			0
$673$	1.79580e12			0.337434			0.168717	−	0.985665i	$-0.446038\pi$
$673$	1.79580e12			0.337434			0.168717	+	0.985665i	$0.446038\pi$
$674$		0			0
$675$	7.49674e10			0.0138997
$676$		0			0
$677$	2.81797e12			0.515570			0.257785	−	0.966202i	$-0.417007\pi$
$677$	2.81797e12			0.515570			0.257785	+	0.966202i	$0.417007\pi$
$678$		0			0
$679$	1.21823e11			0.0219945
$680$		0			0
$681$		−	4.43446e11i		−	0.0790094i
$682$		0			0
$683$		−	5.44241e12i		−	0.956969i	−0.878096	−	0.478485i	$-0.841186\pi$
$683$		−	5.44241e12i		−	0.956969i	0.878096	−	0.478485i	$-0.158814\pi$
$684$		0			0
$685$	2.41809e12			0.419628
$686$		0			0
$687$			2.10900e11i			0.0361220i
$688$		0			0
$689$	2.16819e12	+	1.23754e12i	0.366531	+	0.209205i
$690$		0			0
$691$			7.31934e12i			1.22129i	0.791903	+	0.610647i	$0.209091\pi$
$691$			7.31934e12i			1.22129i	−0.791903	+	0.610647i	$0.790909\pi$
$692$		0			0
$693$	1.62613e12			0.267827
$694$		0			0
$695$		−	8.09897e11i		−	0.131673i
$696$		0			0
$697$			1.32084e13i			2.11984i
$698$		0			0
$699$	−1.70636e11			−0.0270349
$700$		0			0
$701$	3.97007e12			0.620966			0.310483	−	0.950579i	$-0.399509\pi$
$701$	3.97007e12			0.620966			0.310483	+	0.950579i	$0.399509\pi$
$702$		0			0
$703$	6.85493e11			0.105853
$704$		0			0
$705$	−1.13905e11			−0.0173656
$706$		0			0
$707$		−	1.36558e12i		−	0.205556i
$708$		0			0
$709$			7.25576e12i			1.07839i	0.842182	+	0.539194i	$0.181271\pi$
$709$			7.25576e12i			1.07839i	−0.842182	+	0.539194i	$0.818729\pi$
$710$		0			0
$711$	−1.19140e13			−1.74842
$712$		0			0
$713$			1.87044e12i			0.271045i
$714$		0			0
$715$	8.51197e12	+	4.85837e12i	1.21802	+	0.695206i
$716$		0			0
$717$		−	8.82452e11i		−	0.124697i
$718$		0			0
$719$	−5.81879e12			−0.811994			−0.405997	−	0.913874i	$-0.633076\pi$
$719$	−5.81879e12			−0.811994			−0.405997	+	0.913874i	$0.633076\pi$
$720$		0			0
$721$			1.00997e12i			0.139187i
$722$		0			0
$723$			4.29119e11i			0.0584057i
$724$		0			0
$725$	−3.31650e11			−0.0445820
$726$		0			0
$727$	−6.90571e12			−0.916860			−0.458430	−	0.888730i	$-0.651588\pi$
$727$	−6.90571e12			−0.916860			−0.458430	+	0.888730i	$0.651588\pi$
$728$		0			0
$729$	−7.26611e12			−0.952858
$730$		0			0
$731$	9.03550e10			0.0117037
$732$		0			0
$733$			8.14525e12i			1.04217i	0.853506	+	0.521083i	$0.174472\pi$
$733$			8.14525e12i			1.04217i	−0.853506	+	0.521083i	$0.825528\pi$
$734$		0			0
$735$		−	6.52417e11i		−	0.0824578i
$736$		0			0
$737$	1.12437e13			1.40380
$738$		0			0
$739$			7.08797e11i			0.0874223i	0.999044	+	0.0437111i	$0.0139181\pi$
$739$			7.08797e11i			0.0874223i	−0.999044	+	0.0437111i	$0.986082\pi$
$740$		0			0
$741$	−4.18114e10	+	7.32545e10i	−0.00509464	+	0.00892590i
$742$		0			0
$743$			7.10966e12i			0.855852i	0.903814	+	0.427926i	$0.140756\pi$
$743$			7.10966e12i			0.855852i	−0.903814	+	0.427926i	$0.859244\pi$
$744$		0			0
$745$	1.09113e13			1.29770
$746$		0			0
$747$		−	5.82319e11i		−	0.0684255i
$748$		0			0
$749$		−	2.75762e12i		−	0.320160i
$750$		0			0
$751$	−3.11883e12			−0.357777			−0.178889	−	0.983869i	$-0.557250\pi$
$751$	−3.11883e12			−0.357777			−0.178889	+	0.983869i	$0.557250\pi$
$752$		0			0
$753$	1.11852e12			0.126784
$754$		0			0
$755$	8.75730e12			0.980864
$756$		0			0
$757$	4.43974e12			0.491389			0.245695	−	0.969347i	$-0.420984\pi$
$757$	4.43974e12			0.491389			0.245695	+	0.969347i	$0.420984\pi$
$758$		0			0
$759$		−	9.59766e11i		−	0.104973i
$760$		0			0
$761$		−	6.04789e12i		−	0.653692i	−0.945078	−	0.326846i	$-0.894014\pi$
$761$		−	6.04789e12i		−	0.653692i	0.945078	−	0.326846i	$-0.105986\pi$
$762$		0			0
$763$	2.21783e12			0.236901
$764$		0			0
$765$		−	1.30830e13i		−	1.38112i
$766$		0			0
$767$	−8.92883e12	+	1.56435e13i	−0.931570	+	1.63213i
$768$		0			0
$769$			1.90950e13i			1.96903i	0.175302	+	0.984515i	$0.443910\pi$
$769$			1.90950e13i			1.96903i	−0.175302	+	0.984515i	$0.556090\pi$
$770$		0			0
$771$	1.40752e11			0.0143453
$772$		0			0
$773$		−	9.52318e12i		−	0.959344i	−0.877448	−	0.479672i	$-0.840756\pi$
$773$		−	9.52318e12i		−	0.959344i	0.877448	−	0.479672i	$-0.159244\pi$
$774$		0			0
$775$		−	2.64331e11i		−	0.0263203i
$776$		0			0
$777$	−1.52936e11			−0.0150527
$778$		0			0
$779$	1.73630e12			0.168930
$780$		0			0
$781$	−2.32838e12			−0.223936
$782$		0			0
$783$	−1.05883e12			−0.100669
$784$		0			0
$785$		−	1.73107e13i		−	1.62705i
$786$		0			0
$787$		−	1.36839e13i		−	1.27152i	−0.771888	−	0.635758i	$-0.780687\pi$
$787$		−	1.36839e13i		−	1.27152i	0.771888	−	0.635758i	$-0.219313\pi$
$788$		0			0
$789$	−1.54649e12			−0.142069
$790$		0			0
$791$			1.34254e12i			0.121936i
$792$		0			0
$793$	−4.15663e12	−	2.37248e12i	−0.373260	−	0.213046i
$794$		0			0
$795$		−	4.05806e11i		−	0.0360302i
$796$		0			0
$797$	−9.97822e12			−0.875972			−0.437986	−	0.898982i	$-0.644308\pi$
$797$	−9.97822e12			−0.875972			−0.437986	+	0.898982i	$0.644308\pi$
$798$		0			0
$799$		−	3.39822e12i		−	0.294979i
$800$		0			0
$801$		−	4.55991e12i		−	0.391390i
$802$		0			0
$803$	1.52599e13			1.29519
$804$		0			0
$805$	−1.70764e12			−0.143323
$806$		0			0
$807$	−5.28358e11			−0.0438528
$808$		0			0
$809$	−1.63375e13			−1.34096			−0.670480	−	0.741927i	$-0.733912\pi$
$809$	−1.63375e13			−1.34096			−0.670480	+	0.741927i	$0.733912\pi$
$810$		0			0
$811$			6.10157e12i			0.495276i	0.968853	+	0.247638i	$0.0796544\pi$
$811$			6.10157e12i			0.495276i	−0.968853	+	0.247638i	$0.920346\pi$
$812$		0			0
$813$		−	2.24262e11i		−	0.0180032i
$814$		0			0
$815$	−3.63105e12			−0.288286
$816$		0			0
$817$		−	1.18775e10i		−	0.000932669i
$818$		0			0
$819$	−1.17011e12	+	2.05006e12i	−0.0908762	+	0.159217i
$820$		0			0
$821$			1.47439e13i			1.13258i	0.824206	+	0.566290i	$0.191622\pi$
$821$			1.47439e13i			1.13258i	−0.824206	+	0.566290i	$0.808378\pi$
$822$		0			0
$823$	5.38198e12			0.408924			0.204462	−	0.978875i	$-0.434456\pi$
$823$	5.38198e12			0.408924			0.204462	+	0.978875i	$0.434456\pi$
$824$		0			0
$825$			1.35634e11i			0.0101936i
$826$		0			0
$827$		−	1.53477e13i		−	1.14095i	−0.821313	−	0.570477i	$-0.806758\pi$
$827$		−	1.53477e13i		−	1.14095i	0.821313	−	0.570477i	$-0.193242\pi$
$828$		0			0
$829$	4.78439e12			0.351829			0.175914	−	0.984405i	$-0.443712\pi$
$829$	4.78439e12			0.351829			0.175914	+	0.984405i	$0.443712\pi$
$830$		0			0
$831$	−8.24825e11			−0.0600008
$832$		0			0
$833$	1.94641e13			1.40066
$834$		0			0
$835$	2.01292e13			1.43297
$836$		0			0
$837$		−	8.43905e11i		−	0.0594332i
$838$		0			0
$839$		−	2.46071e13i		−	1.71448i	−0.514918	−	0.857239i	$-0.672178\pi$
$839$		−	2.46071e13i		−	1.71448i	0.514918	−	0.857239i	$-0.327822\pi$
$840$		0			0
$841$	−9.82297e12			−0.677112
$842$		0			0
$843$		−	2.49263e11i		−	0.0169994i
$844$		0			0
$845$	−1.22499e13	+	7.23510e12i	−0.826567	+	0.488191i
$846$		0			0
$847$			3.13969e12i			0.209609i
$848$		0			0
$849$	−6.46151e11			−0.0426825
$850$		0			0
$851$		−	1.13226e13i		−	0.740055i
$852$		0			0
$853$			1.86042e13i			1.20321i	0.798794	+	0.601605i	$0.205472\pi$
$853$			1.86042e13i			1.20321i	−0.798794	+	0.601605i	$0.794528\pi$
$854$		0			0
$855$	−1.71981e12			−0.110061
$856$		0			0
$857$	−3.04247e13			−1.92669			−0.963347	−	0.268257i	$-0.913552\pi$
$857$	−3.04247e13			−1.92669			−0.963347	+	0.268257i	$0.913552\pi$
$858$		0			0
$859$	−1.92904e13			−1.20885			−0.604425	−	0.796662i	$-0.706597\pi$
$859$	−1.92904e13			−1.20885			−0.604425	+	0.796662i	$0.706597\pi$
$860$		0			0
$861$	−3.87375e11			−0.0240225
$862$		0			0
$863$		−	1.15531e13i		−	0.709007i	−0.935055	−	0.354504i	$-0.884650\pi$
$863$		−	1.15531e13i		−	0.709007i	0.935055	−	0.354504i	$-0.115350\pi$
$864$		0			0
$865$		−	2.75491e13i		−	1.67315i
$866$		0			0
$867$	−1.63204e12			−0.0980946
$868$		0			0
$869$		−	4.32825e13i		−	2.57468i
$870$		0			0
$871$	−8.09063e12	+	1.41749e13i	−0.476322	+	0.834525i
$872$		0			0
$873$		−	2.02655e12i		−	0.118085i
$874$		0			0
$875$	3.31719e12			0.191308
$876$		0			0
$877$			1.69329e13i			0.966569i	0.875463	+	0.483285i	$0.160556\pi$
$877$			1.69329e13i			0.966569i	−0.875463	+	0.483285i	$0.839444\pi$
$878$		0			0
$879$			1.12165e12i			0.0633733i
$880$		0			0
$881$	−1.39025e13			−0.777503			−0.388752	−	0.921343i	$-0.627094\pi$
$881$	−1.39025e13			−0.777503			−0.388752	+	0.921343i	$0.627094\pi$
$882$		0			0
$883$	−2.22403e13			−1.23117			−0.615584	−	0.788071i	$-0.711080\pi$
$883$	−2.22403e13			−1.23117			−0.615584	+	0.788071i	$0.711080\pi$
$884$		0			0
$885$	2.92789e12			0.160439
$886$		0			0
$887$	−1.95373e13			−1.05976			−0.529881	−	0.848072i	$-0.677764\pi$
$887$	−1.95373e13			−1.05976			−0.529881	+	0.848072i	$0.677764\pi$
$888$		0			0
$889$			2.08781e12i			0.112107i
$890$		0			0
$891$		−	2.68336e13i		−	1.42636i
$892$		0			0
$893$	−4.46711e11			−0.0235069
$894$		0			0
$895$			1.70949e11i			0.00890558i
$896$		0			0
$897$	1.20998e12	+	6.90619e11i	0.0624039	+	0.0356182i
$898$		0			0
$899$			3.73337e12i			0.190626i
$900$		0			0
$901$	1.21068e13			0.612022
$902$		0			0
$903$			2.64992e9i			0.000132629i
$904$		0			0
$905$			2.74708e13i			1.36130i
$906$		0			0
$907$	2.79807e13			1.37286			0.686429	−	0.727197i	$-0.259177\pi$
$907$	2.79807e13			1.37286			0.686429	+	0.727197i	$0.259177\pi$
$908$		0			0
$909$	−2.27168e13			−1.10359
$910$		0			0
$911$	2.56689e13			1.23474			0.617369	−	0.786674i	$-0.288199\pi$
$911$	2.56689e13			1.23474			0.617369	+	0.786674i	$0.288199\pi$
$912$		0			0
$913$	2.11551e12			0.100762
$914$		0			0
$915$			7.77970e11i			0.0366917i
$916$		0			0
$917$		−	6.60524e12i		−	0.308480i
$918$		0			0
$919$	1.28985e13			0.596514			0.298257	−	0.954486i	$-0.403595\pi$
$919$	1.28985e13			0.596514			0.298257	+	0.954486i	$0.403595\pi$
$920$		0			0
$921$		−	1.68396e12i		−	0.0771192i
$922$		0			0
$923$	1.67543e12	−	2.93539e12i	0.0759834	−	0.133124i
$924$		0			0
$925$			1.60011e12i			0.0718643i
$926$		0			0
$927$	1.68011e13			0.747271
$928$		0			0
$929$		−	1.41528e13i		−	0.623408i	−0.950179	−	0.311704i	$-0.899100\pi$
$929$		−	1.41528e13i		−	0.623408i	0.950179	−	0.311704i	$-0.100900\pi$
$930$		0			0
$931$		−	2.55864e12i		−	0.111618i
$932$		0			0
$933$	3.52708e12			0.152387
$934$		0			0
$935$	4.75293e13			2.03381
$936$		0			0
$937$	−4.26783e12			−0.180875			−0.0904375	−	0.995902i	$-0.528827\pi$
$937$	−4.26783e12			−0.180875			−0.0904375	+	0.995902i	$0.528827\pi$
$938$		0			0
$939$	−6.26953e11			−0.0263172
$940$		0			0
$941$			3.48965e13i			1.45087i	0.688291	+	0.725435i	$0.258361\pi$
$941$			3.48965e13i			1.45087i	−0.688291	+	0.725435i	$0.741639\pi$
$942$		0			0
$943$		−	2.86793e13i		−	1.18104i
$944$		0			0
$945$	7.70453e11			0.0314270
$946$		0			0
$947$		−	4.60914e13i		−	1.86228i	−0.364660	−	0.931141i	$-0.618815\pi$
$947$		−	4.60914e13i		−	1.86228i	0.364660	−	0.931141i	$-0.381185\pi$
$948$		0			0
$949$	−1.09806e13	+	1.92382e13i	−0.439469	+	0.769958i
$950$		0			0
$951$			2.06733e12i			0.0819593i
$952$		0			0
$953$	−1.17601e13			−0.461843			−0.230921	−	0.972972i	$-0.574174\pi$
$953$	−1.17601e13			−0.461843			−0.230921	+	0.972972i	$0.574174\pi$
$954$		0			0
$955$			7.53874e12i			0.293281i
$956$		0			0
$957$		−	1.91568e12i		−	0.0738277i
$958$		0			0
$959$	−2.11574e12			−0.0807755
$960$		0			0
$961$	2.34641e13			0.887458
$962$		0			0
$963$	−4.58736e13			−1.71888
$964$		0			0
$965$	−3.52236e13			−1.30756
$966$		0			0
$967$			1.76861e13i			0.650448i	0.945637	+	0.325224i	$0.105440\pi$
$967$			1.76861e13i			0.650448i	−0.945637	+	0.325224i	$0.894560\pi$
$968$		0			0
$969$			4.09040e11i			0.0149042i
$970$		0			0
$971$	1.73464e13			0.626215			0.313107	−	0.949718i	$-0.398630\pi$
$971$	1.73464e13			0.626215			0.313107	+	0.949718i	$0.398630\pi$
$972$		0			0
$973$			7.08633e11i			0.0253462i
$974$		0			0
$975$	−1.70994e11	−	9.75984e10i	−0.00605984	−	0.00345877i
$976$		0			0
$977$			4.87286e13i			1.71103i	0.517776	+	0.855516i	$0.326760\pi$
$977$			4.87286e13i			1.71103i	−0.517776	+	0.855516i	$0.673240\pi$
$978$		0			0
$979$	1.65657e13			0.576353
$980$		0			0
$981$		−	3.68941e13i		−	1.27188i
$982$		0			0
$983$			4.46964e13i			1.52680i	0.645926	+	0.763400i	$0.276471\pi$
$983$			4.46964e13i			1.52680i	−0.645926	+	0.763400i	$0.723529\pi$
$984$		0			0
$985$	1.11007e13			0.375739
$986$		0			0
$987$	9.96628e10			0.00334277
$988$		0			0
$989$	−1.96187e11			−0.00652059
$990$		0			0
$991$	−2.74484e13			−0.904037			−0.452019	−	0.892009i	$-0.649296\pi$
$991$	−2.74484e13			−0.904037			−0.452019	+	0.892009i	$0.649296\pi$
$992$		0			0
$993$			4.61253e12i			0.150545i
$994$		0			0
$995$		−	4.11140e12i		−	0.132980i
$996$		0			0
$997$	5.96082e12			0.191064			0.0955318	−	0.995426i	$-0.469545\pi$
$997$	5.96082e12			0.191064			0.0955318	+	0.995426i	$0.469545\pi$
$998$		0			0
$999$			5.10854e12i			0.162275i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	104.10.f.a.25.17	✓	32
4.3	odd	2		208.10.f.d.129.16		32
13.12	even	2	inner	104.10.f.a.25.18	yes	32
52.51	odd	2		208.10.f.d.129.15		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
104.10.f.a.25.17	✓	32	1.1	even	1	trivial
104.10.f.a.25.18	yes	32	13.12	even	2	inner
208.10.f.d.129.15		32	52.51	odd	2
208.10.f.d.129.16		32	4.3	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)

\(f(q)\)	\(=\)	\(q-12.4770 q^{3} +1341.60i q^{5} -1173.85i q^{7} -19527.3 q^{9} -70941.0i q^{11} +(89435.4 + 51047.1i) q^{13} -16739.1i q^{15} +499392. q^{17} -65647.1i q^{19} +14646.1i q^{21} -1.08432e6 q^{23} +153237. q^{25} +489225. q^{27} -2.16430e6 q^{29} -1.72498e6i q^{31} +885128. i q^{33} +1.57484e6 q^{35} +1.04421e7i q^{37} +(-1.11588e6 - 636912. i) q^{39} +2.64490e7i q^{41} +180930. q^{43} -2.61978e7i q^{45} -6.80472e6i q^{47} +3.89757e7 q^{49} -6.23089e6 q^{51} +2.42430e7 q^{53} +9.51744e7 q^{55} +819076. i q^{57} +1.74914e8i q^{59} -4.64763e7 q^{61} +2.29222e7i q^{63} +(-6.84847e7 + 1.19986e8i) q^{65} +1.58494e8i q^{67} +1.35291e7 q^{69} -3.28213e7i q^{71} +2.15107e8i q^{73} -1.91193e6 q^{75} -8.32744e7 q^{77} +6.10120e8 q^{79} +3.78252e8 q^{81} +2.98207e7i q^{83} +6.69984e8i q^{85} +2.70038e7 q^{87} +2.33514e8i q^{89} +(5.99218e7 - 1.04984e8i) q^{91} +2.15225e7i q^{93} +8.80721e7 q^{95} +1.03780e8i q^{97} +1.38529e9i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 162 q^{3} + 223074 q^{9} + 66270 q^{13} - 487902 q^{17} + 3171556 q^{23} - 13526722 q^{25} - 3694974 q^{27} + 8833508 q^{29} - 8281126 q^{35} - 12056860 q^{39} + 89959038 q^{43} - 172344874 q^{49}+ \cdots + 1741143356 q^{95}+O(q^{100}) \) 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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