LMFDB - Embedded newform 104.10.f.a.25.11

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			25.11
Character	$\chi$	$=$	104.25
Dual form			104.10.f.a.25.12

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-76.3670 q^{3} -1713.06i q^{5} -425.676i q^{7} -13851.1 q^{9} -24500.1i q^{11} +(-102978. - 314.799i) q^{13} +130821. i q^{15} -139497. q^{17} -572688. i q^{19} +32507.6i q^{21} -1.45112e6 q^{23} -981447. q^{25} +2.56090e6 q^{27} +5.71538e6 q^{29} -8.04731e6i q^{31} +1.87100e6i q^{33} -729208. q^{35} +1.92934e7i q^{37} +(7.86410e6 + 24040.3i) q^{39} -2.04134e7i q^{41} -3.39230e7 q^{43} +2.37277e7i q^{45} +4.53942e7i q^{47} +4.01724e7 q^{49} +1.06530e7 q^{51} -8.54616e7 q^{53} -4.19702e7 q^{55} +4.37345e7i q^{57} +7.95866e7i q^{59} +1.38028e8 q^{61} +5.89607e6i q^{63} +(-539270. + 1.76407e8i) q^{65} +2.43776e8i q^{67} +1.10818e8 q^{69} +1.14544e8i q^{71} -1.44800e8i q^{73} +7.49501e7 q^{75} -1.04291e7 q^{77} -7.25337e7 q^{79} +7.70627e7 q^{81} +2.40348e8i q^{83} +2.38966e8i q^{85} -4.36467e8 q^{87} +4.23885e8i q^{89} +(-134002. + 4.38351e7i) q^{91} +6.14549e8i q^{93} -9.81049e8 q^{95} +9.39888e8i q^{97} +3.39353e8i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−76.3670			−0.544327			−0.272164	−	0.962251i	$-0.587739\pi$
$3$	−76.3670			−0.544327			−0.272164	+	0.962251i	$0.587739\pi$
$4$		0			0
$5$		−	1713.06i		−	1.22577i	−0.790174	−	0.612883i	$-0.790010\pi$
$5$		−	1713.06i		−	1.22577i	0.790174	−	0.612883i	$-0.209990\pi$
$6$		0			0
$7$		−	425.676i		−	0.0670098i	−0.999439	−	0.0335049i	$-0.989333\pi$
$7$		−	425.676i		−	0.0670098i	0.999439	−	0.0335049i	$-0.0106669\pi$
$8$		0			0
$9$	−13851.1			−0.703708
$10$		0			0
$11$		−	24500.1i		−	0.504547i	−0.967656	−	0.252273i	$-0.918822\pi$
$11$		−	24500.1i		−	0.504547i	0.967656	−	0.252273i	$-0.0811782\pi$
$12$		0			0
$13$	−102978.	−	314.799i	−0.999995	−	0.00305695i
$13$	−102978.	−	314.799i	−0.999995	−	0.00305695i
$14$		0			0
$15$			130821.i			0.667218i
$16$		0			0
$17$	−139497.			−0.405083			−0.202542	−	0.979274i	$-0.564920\pi$
$17$	−139497.			−0.405083			−0.202542	+	0.979274i	$0.564920\pi$
$18$		0			0
$19$		−	572688.i		−	1.00815i	−0.863659	−	0.504077i	$-0.831833\pi$
$19$		−	572688.i		−	1.00815i	0.863659	−	0.504077i	$-0.168167\pi$
$20$		0			0
$21$			32507.6i			0.0364752i
$22$		0			0
$23$	−1.45112e6			−1.08125			−0.540627	−	0.841262i	$-0.681813\pi$
$23$	−1.45112e6			−1.08125			−0.540627	+	0.841262i	$0.681813\pi$
$24$		0			0
$25$	−981447.			−0.502501
$26$		0			0
$27$	2.56090e6			0.927375
$28$		0			0
$29$	5.71538e6			1.50056			0.750281	−	0.661119i	$-0.229918\pi$
$29$	5.71538e6			1.50056			0.750281	+	0.661119i	$0.229918\pi$
$30$		0			0
$31$		−	8.04731e6i		−	1.56503i	−0.622630	−	0.782516i	$-0.713936\pi$
$31$		−	8.04731e6i		−	1.56503i	0.622630	−	0.782516i	$-0.286064\pi$
$32$		0			0
$33$			1.87100e6i			0.274639i
$34$		0			0
$35$	−729208.			−0.0821382
$36$		0			0
$37$			1.92934e7i			1.69240i	0.532868	+	0.846198i	$0.321114\pi$
$37$			1.92934e7i			1.69240i	−0.532868	+	0.846198i	$0.678886\pi$
$38$		0			0
$39$	7.86410e6	+	24040.3i	0.544325	+	0.00166398i
$40$		0			0
$41$		−	2.04134e7i		−	1.12820i	−0.825705	−	0.564102i	$-0.809223\pi$
$41$		−	2.04134e7i		−	1.12820i	0.825705	−	0.564102i	$-0.190777\pi$
$42$		0			0
$43$	−3.39230e7			−1.51316			−0.756582	−	0.653899i	$-0.773132\pi$
$43$	−3.39230e7			−1.51316			−0.756582	+	0.653899i	$0.773132\pi$
$44$		0			0
$45$			2.37277e7i			0.862580i
$46$		0			0
$47$			4.53942e7i			1.35694i	0.734629	+	0.678469i	$0.237356\pi$
$47$			4.53942e7i			1.35694i	−0.734629	+	0.678469i	$0.762644\pi$
$48$		0			0
$49$	4.01724e7			0.995510
$50$		0			0
$51$	1.06530e7			0.220498
$52$		0			0
$53$	−8.54616e7			−1.48775			−0.743874	−	0.668320i	$-0.767014\pi$
$53$	−8.54616e7			−1.48775			−0.743874	+	0.668320i	$0.767014\pi$
$54$		0			0
$55$	−4.19702e7			−0.618456
$56$		0			0
$57$			4.37345e7i			0.548766i
$58$		0			0
$59$			7.95866e7i			0.855078i	0.903997	+	0.427539i	$0.140619\pi$
$59$			7.95866e7i			0.855078i	−0.903997	+	0.427539i	$0.859381\pi$
$60$		0			0
$61$	1.38028e8			1.27639			0.638195	−	0.769874i	$-0.279681\pi$
$61$	1.38028e8			1.27639			0.638195	+	0.769874i	$0.279681\pi$
$62$		0			0
$63$			5.89607e6i			0.0471553i
$64$		0			0
$65$	−539270.	+	1.76407e8i	−0.00374710	+	1.22576i
$66$		0			0
$67$			2.43776e8i			1.47793i	0.673742	+	0.738967i	$0.264686\pi$
$67$			2.43776e8i			1.47793i	−0.673742	+	0.738967i	$0.735314\pi$
$68$		0			0
$69$	1.10818e8			0.588556
$70$		0			0
$71$			1.14544e8i			0.534944i	0.963566	+	0.267472i	$0.0861883\pi$
$71$			1.14544e8i			0.534944i	−0.963566	+	0.267472i	$0.913812\pi$
$72$		0			0
$73$		−	1.44800e8i		−	0.596782i	−0.954444	−	0.298391i	$-0.903550\pi$
$73$		−	1.44800e8i		−	0.596782i	0.954444	−	0.298391i	$-0.0964500\pi$
$74$		0			0
$75$	7.49501e7			0.273525
$76$		0			0
$77$	−1.04291e7			−0.0338095
$78$		0			0
$79$	−7.25337e7			−0.209516			−0.104758	−	0.994498i	$-0.533407\pi$
$79$	−7.25337e7			−0.209516			−0.104758	+	0.994498i	$0.533407\pi$
$80$		0			0
$81$	7.70627e7			0.198912
$82$		0			0
$83$			2.40348e8i			0.555890i	0.960597	+	0.277945i	$0.0896533\pi$
$83$			2.40348e8i			0.555890i	−0.960597	+	0.277945i	$0.910347\pi$
$84$		0			0
$85$			2.38966e8i			0.496537i
$86$		0			0
$87$	−4.36467e8			−0.816798
$88$		0			0
$89$			4.23885e8i			0.716132i	0.933696	+	0.358066i	$0.116564\pi$
$89$			4.23885e8i			0.716132i	−0.933696	+	0.358066i	$0.883436\pi$
$90$		0			0
$91$	−134002.	+	4.38351e7i	−0.000204846	+	0.0670094i
$92$		0			0
$93$			6.14549e8i			0.851890i
$94$		0			0
$95$	−9.81049e8			−1.23576
$96$		0			0
$97$			9.39888e8i			1.07796i	0.842318	+	0.538981i	$0.181191\pi$
$97$			9.39888e8i			1.07796i	−0.842318	+	0.538981i	$0.818809\pi$
$98$		0			0
$99$			3.39353e8i			0.355053i
$100$		0			0
$101$	−4.57905e8			−0.437854			−0.218927	−	0.975741i	$-0.570256\pi$
$101$	−4.57905e8			−0.437854			−0.218927	+	0.975741i	$0.570256\pi$
$102$		0			0
$103$	1.57302e9			1.37710			0.688550	−	0.725189i	$-0.258248\pi$
$103$	1.57302e9			1.37710			0.688550	+	0.725189i	$0.258248\pi$
$104$		0			0
$105$	5.56875e7			0.0447101
$106$		0			0
$107$	2.30348e8			0.169886			0.0849428	−	0.996386i	$-0.472929\pi$
$107$	2.30348e8			0.169886			0.0849428	+	0.996386i	$0.472929\pi$
$108$		0			0
$109$			7.54070e8i			0.511673i	0.966720	+	0.255836i	$0.0823508\pi$
$109$			7.54070e8i			0.511673i	−0.966720	+	0.255836i	$0.917649\pi$
$110$		0			0
$111$		−	1.47338e9i		−	0.921218i
$112$		0			0
$113$	2.92337e8			0.168667			0.0843336	−	0.996438i	$-0.473124\pi$
$113$	2.92337e8			0.168667			0.0843336	+	0.996438i	$0.473124\pi$
$114$		0			0
$115$			2.48585e9i			1.32536i
$116$		0			0
$117$	1.42635e9	+	4.36031e6i	0.703704	+	0.00215120i
$118$		0			0
$119$			5.93805e7i			0.0271445i
$120$		0			0
$121$	1.75769e9			0.745433
$122$		0			0
$123$			1.55891e9i			0.614112i
$124$		0			0
$125$		−	1.66454e9i		−	0.609817i
$126$		0			0
$127$	−4.40722e9			−1.50331			−0.751654	−	0.659557i	$-0.770744\pi$
$127$	−4.40722e9			−1.50331			−0.751654	+	0.659557i	$0.770744\pi$
$128$		0			0
$129$	2.59060e9			0.823657
$130$		0			0
$131$	−1.52816e9			−0.453366			−0.226683	−	0.973969i	$-0.572788\pi$
$131$	−1.52816e9			−0.453366			−0.226683	+	0.973969i	$0.572788\pi$
$132$		0			0
$133$	−2.43780e8			−0.0675562
$134$		0			0
$135$		−	4.38697e9i		−	1.13674i
$136$		0			0
$137$			8.54102e8i			0.207141i	0.994622	+	0.103571i	$0.0330268\pi$
$137$			8.54102e8i			0.207141i	−0.994622	+	0.103571i	$0.966973\pi$
$138$		0			0
$139$	−2.38628e9			−0.542195			−0.271097	−	0.962552i	$-0.587387\pi$
$139$	−2.38628e9			−0.542195			−0.271097	+	0.962552i	$0.587387\pi$
$140$		0			0
$141$		−	3.46662e9i		−	0.738618i
$142$		0			0
$143$	−7.71262e6	+	2.52297e9i	−0.00154237	+	0.504544i
$144$		0			0
$145$		−	9.79079e9i		−	1.83934i
$146$		0			0
$147$	−3.06785e9			−0.541883
$148$		0			0
$149$		−	3.49957e9i		−	0.581669i	−0.956773	−	0.290835i	$-0.906067\pi$
$149$		−	3.49957e9i		−	0.581669i	0.956773	−	0.290835i	$-0.0939329\pi$
$150$		0			0
$151$		−	5.22200e9i		−	0.817411i	−0.912666	−	0.408706i	$-0.865980\pi$
$151$		−	5.22200e9i		−	0.817411i	0.912666	−	0.408706i	$-0.134020\pi$
$152$		0			0
$153$	1.93218e9			0.285060
$154$		0			0
$155$	−1.37855e10			−1.91836
$156$		0			0
$157$	1.05387e10			1.38433			0.692166	−	0.721738i	$-0.256656\pi$
$157$	1.05387e10			1.38433			0.692166	+	0.721738i	$0.256656\pi$
$158$		0			0
$159$	6.52645e9			0.809822
$160$		0			0
$161$			6.17707e8i			0.0724546i
$162$		0			0
$163$		−	1.01036e10i		−	1.12107i	−0.828132	−	0.560533i	$-0.810596\pi$
$163$		−	1.01036e10i		−	1.12107i	0.828132	−	0.560533i	$-0.189404\pi$
$164$		0			0
$165$	3.20514e9			0.336642
$166$		0			0
$167$		−	1.90716e10i		−	1.89741i	−0.316155	−	0.948707i	$-0.602392\pi$
$167$		−	1.90716e10i		−	1.89741i	0.316155	−	0.948707i	$-0.397608\pi$
$168$		0			0
$169$	1.06043e10	+	6.48346e7i	0.999981	+	0.00611387i
$170$		0			0
$171$			7.93235e9i			0.709446i
$172$		0			0
$173$	−1.74243e10			−1.47893			−0.739467	−	0.673193i	$-0.764922\pi$
$173$	−1.74243e10			−1.47893			−0.739467	+	0.673193i	$0.764922\pi$
$174$		0			0
$175$			4.17778e8i			0.0336724i
$176$		0			0
$177$		−	6.07779e9i		−	0.465442i
$178$		0			0
$179$	−5.31675e9			−0.387086			−0.193543	−	0.981092i	$-0.561998\pi$
$179$	−5.31675e9			−0.387086			−0.193543	+	0.981092i	$0.561998\pi$
$180$		0			0
$181$	2.27345e10			1.57446			0.787230	−	0.616659i	$-0.211514\pi$
$181$	2.27345e10			1.57446			0.787230	+	0.616659i	$0.211514\pi$
$182$		0			0
$183$	−1.05408e10			−0.694775
$184$		0			0
$185$	3.30508e10			2.07448
$186$		0			0
$187$			3.41769e9i			0.204383i
$188$		0			0
$189$		−	1.09011e9i		−	0.0621432i
$190$		0			0
$191$	−1.76412e10			−0.959130			−0.479565	−	0.877506i	$-0.659206\pi$
$191$	−1.76412e10			−0.959130			−0.479565	+	0.877506i	$0.659206\pi$
$192$		0			0
$193$			2.15736e10i			1.11922i	0.828756	+	0.559610i	$0.189049\pi$
$193$			2.15736e10i			1.11922i	−0.828756	+	0.559610i	$0.810951\pi$
$194$		0			0
$195$	4.11824e7	−	1.34717e10i	0.00203965	−	0.667215i
$196$		0			0
$197$			1.12930e10i			0.534210i	0.963667	+	0.267105i	$0.0860671\pi$
$197$			1.12930e10i			0.534210i	−0.963667	+	0.267105i	$0.913933\pi$
$198$		0			0
$199$	−3.73738e10			−1.68938			−0.844692	−	0.535253i	$-0.820216\pi$
$199$	−3.73738e10			−1.68938			−0.844692	+	0.535253i	$0.820216\pi$
$200$		0			0
$201$		−	1.86165e10i		−	0.804480i
$202$		0			0
$203$		−	2.43290e9i		−	0.100552i
$204$		0			0
$205$	−3.49693e10			−1.38291
$206$		0			0
$207$	2.00996e10			0.760887
$208$		0			0
$209$	−1.40309e10			−0.508661
$210$		0			0
$211$	1.19444e10			0.414851			0.207426	−	0.978251i	$-0.433491\pi$
$211$	1.19444e10			0.414851			0.207426	+	0.978251i	$0.433491\pi$
$212$		0			0
$213$		−	8.74736e9i		−	0.291185i
$214$		0			0
$215$			5.81121e10i			1.85478i
$216$		0			0
$217$	−3.42555e9			−0.104872
$218$		0			0
$219$			1.10580e10i			0.324845i
$220$		0			0
$221$	1.43651e10	+	4.39135e7i	0.405081	+	0.00123832i
$222$		0			0
$223$		−	2.12181e10i		−	0.574558i	−0.957847	−	0.287279i	$-0.907249\pi$
$223$		−	2.12181e10i		−	0.574558i	0.957847	−	0.287279i	$-0.0927506\pi$
$224$		0			0
$225$	1.35941e10			0.353614
$226$		0			0
$227$			3.47026e9i			0.0867453i	0.999059	+	0.0433727i	$0.0138103\pi$
$227$			3.47026e9i			0.0867453i	−0.999059	+	0.0433727i	$0.986190\pi$
$228$		0			0
$229$			4.81403e10i			1.15678i	0.815762	+	0.578388i	$0.196318\pi$
$229$			4.81403e10i			1.15678i	−0.815762	+	0.578388i	$0.803682\pi$
$230$		0			0
$231$	7.96441e8			0.0184035
$232$		0			0
$233$	1.69111e10			0.375899			0.187949	−	0.982179i	$-0.439816\pi$
$233$	1.69111e10			0.375899			0.187949	+	0.982179i	$0.439816\pi$
$234$		0			0
$235$	7.77629e10			1.66329
$236$		0			0
$237$	5.53919e9			0.114046
$238$		0			0
$239$			5.85031e10i			1.15981i	0.814683	+	0.579907i	$0.196911\pi$
$239$			5.85031e10i			1.15981i	−0.814683	+	0.579907i	$0.803089\pi$
$240$		0			0
$241$			2.05744e10i			0.392871i	0.980517	+	0.196435i	$0.0629366\pi$
$241$			2.05744e10i			0.392871i	−0.980517	+	0.196435i	$0.937063\pi$
$242$		0			0
$243$	−5.62912e10			−1.03565
$244$		0			0
$245$		−	6.88177e10i		−	1.22026i
$246$		0			0
$247$	−1.80282e8	+	5.89741e10i	−0.00308188	+	1.00815i
$248$		0			0
$249$		−	1.83547e10i		−	0.302586i
$250$		0			0
$251$	−2.53163e10			−0.402595			−0.201297	−	0.979530i	$-0.564516\pi$
$251$	−2.53163e10			−0.402595			−0.201297	+	0.979530i	$0.564516\pi$
$252$		0			0
$253$			3.55526e10i			0.545543i
$254$		0			0
$255$		−	1.82491e10i		−	0.270279i
$256$		0			0
$257$	1.36643e10			0.195383			0.0976916	−	0.995217i	$-0.468854\pi$
$257$	1.36643e10			0.195383			0.0976916	+	0.995217i	$0.468854\pi$
$258$		0			0
$259$	8.21276e9			0.113407
$260$		0			0
$261$	−7.91642e10			−1.05596
$262$		0			0
$263$	1.02747e11			1.32424			0.662120	−	0.749398i	$-0.269657\pi$
$263$	1.02747e11			1.32424			0.662120	+	0.749398i	$0.269657\pi$
$264$		0			0
$265$			1.46401e11i			1.82363i
$266$		0			0
$267$		−	3.23708e10i		−	0.389810i
$268$		0			0
$269$	8.33227e9			0.0970238			0.0485119	−	0.998823i	$-0.484552\pi$
$269$	8.33227e9			0.0970238			0.0485119	+	0.998823i	$0.484552\pi$
$270$		0			0
$271$		−	1.49932e10i		−	0.168862i	−0.996429	−	0.0844310i	$-0.973093\pi$
$271$		−	1.49932e10i		−	0.168862i	0.996429	−	0.0844310i	$-0.0269073\pi$
$272$		0			0
$273$	1.02334e7	−	3.34756e9i	0.000111503	−	0.0364751i
$274$		0			0
$275$			2.40456e10i			0.253535i
$276$		0			0
$277$	−1.20096e11			−1.22566			−0.612829	−	0.790215i	$-0.709969\pi$
$277$	−1.20096e11			−1.22566			−0.612829	+	0.790215i	$0.709969\pi$
$278$		0			0
$279$			1.11464e11i			1.10133i
$280$		0			0
$281$			8.70624e10i			0.833014i	0.909132	+	0.416507i	$0.136746\pi$
$281$			8.70624e10i			0.833014i	−0.909132	+	0.416507i	$0.863254\pi$
$282$		0			0
$283$	−1.64657e11			−1.52596			−0.762979	−	0.646424i	$-0.776264\pi$
$283$	−1.64657e11			−1.52596			−0.762979	+	0.646424i	$0.776264\pi$
$284$		0			0
$285$	7.49198e10			0.672658
$286$		0			0
$287$	−8.68948e9			−0.0756006
$288$		0			0
$289$	−9.91285e10			−0.835908
$290$		0			0
$291$		−	7.17765e10i		−	0.586764i
$292$		0			0
$293$			1.78258e11i			1.41301i	0.707709	+	0.706504i	$0.249729\pi$
$293$			1.78258e11i			1.41301i	−0.707709	+	0.706504i	$0.750271\pi$
$294$		0			0
$295$	1.36337e11			1.04813
$296$		0			0
$297$		−	6.27423e10i		−	0.467904i
$298$		0			0
$299$	1.49433e11	+	4.56811e8i	1.08125	+	0.00330534i
$300$		0			0
$301$			1.44402e10i			0.101397i
$302$		0			0
$303$	3.49688e10			0.238336
$304$		0			0
$305$		−	2.36451e11i		−	1.56456i
$306$		0			0
$307$		−	2.16364e11i		−	1.39015i	−0.718937	−	0.695076i	$-0.755371\pi$
$307$		−	2.16364e11i		−	1.39015i	0.718937	−	0.695076i	$-0.244629\pi$
$308$		0			0
$309$	−1.20126e11			−0.749593
$310$		0			0
$311$	4.31714e10			0.261683			0.130841	−	0.991403i	$-0.458232\pi$
$311$	4.31714e10			0.261683			0.130841	+	0.991403i	$0.458232\pi$
$312$		0			0
$313$	1.02683e11			0.604711			0.302355	−	0.953195i	$-0.402227\pi$
$313$	1.02683e11			0.604711			0.302355	+	0.953195i	$0.402227\pi$
$314$		0			0
$315$	1.01003e10			0.0578013
$316$		0			0
$317$		−	9.71219e10i		−	0.540195i	−0.962833	−	0.270097i	$-0.912944\pi$
$317$		−	9.71219e10i		−	0.540195i	0.962833	−	0.270097i	$-0.0870559\pi$
$318$		0			0
$319$		−	1.40028e11i		−	0.757104i
$320$		0			0
$321$	−1.75910e10			−0.0924734
$322$		0			0
$323$			7.98882e10i			0.408386i
$324$		0			0
$325$	1.01067e11	+	3.08959e8i	0.502498	+	0.00153612i
$326$		0			0
$327$		−	5.75861e10i		−	0.278517i
$328$		0			0
$329$	1.93232e10			0.0909280
$330$		0			0
$331$			1.28537e11i			0.588577i	0.955717	+	0.294288i	$0.0950826\pi$
$331$			1.28537e11i			0.588577i	−0.955717	+	0.294288i	$0.904917\pi$
$332$		0			0
$333$		−	2.67235e11i		−	1.19095i
$334$		0			0
$335$	4.17603e11			1.81160
$336$		0			0
$337$	−3.35448e11			−1.41674			−0.708371	−	0.705841i	$-0.750569\pi$
$337$	−3.35448e11			−1.41674			−0.708371	+	0.705841i	$0.750569\pi$
$338$		0			0
$339$	−2.23249e10			−0.0918102
$340$		0			0
$341$	−1.97160e11			−0.789632
$342$		0			0
$343$		−	3.42780e10i		−	0.133719i
$344$		0			0
$345$		−	1.89837e11i		−	0.721432i
$346$		0			0
$347$	5.38880e9			0.0199531			0.00997654	−	0.999950i	$-0.496824\pi$
$347$	5.38880e9			0.0199531			0.00997654	+	0.999950i	$0.496824\pi$
$348$		0			0
$349$		−	3.49011e11i		−	1.25929i	−0.776884	−	0.629644i	$-0.783201\pi$
$349$		−	3.49011e11i		−	1.25929i	0.776884	−	0.629644i	$-0.216799\pi$
$350$		0			0
$351$	−2.63715e11	−	8.06168e8i	−0.927370	−	0.00283494i
$352$		0			0
$353$		−	2.43649e11i		−	0.835176i	−0.908637	−	0.417588i	$-0.862876\pi$
$353$		−	2.43649e11i		−	0.835176i	0.908637	−	0.417588i	$-0.137124\pi$
$354$		0			0
$355$	1.96220e11			0.655716
$356$		0			0
$357$		−	4.53471e9i		−	0.0147755i
$358$		0			0
$359$		−	4.86575e11i		−	1.54605i	−0.634373	−	0.773027i	$-0.718742\pi$
$359$		−	4.86575e11i		−	1.54605i	0.634373	−	0.773027i	$-0.281258\pi$
$360$		0			0
$361$	−5.28408e9			−0.0163752
$362$		0			0
$363$	−1.34230e11			−0.405759
$364$		0			0
$365$	−2.48051e11			−0.731515
$366$		0			0
$367$	−3.67290e8			−0.00105685			−0.000528423	−	1.00000i	$-0.500168\pi$
$367$	−3.67290e8			−0.00105685			−0.000528423		1.00000i	$0.500168\pi$
$368$		0			0
$369$			2.82747e11i			0.793925i
$370$		0			0
$371$			3.63790e10i			0.0996937i
$372$		0			0
$373$	−4.13188e10			−0.110524			−0.0552622	−	0.998472i	$-0.517599\pi$
$373$	−4.13188e10			−0.110524			−0.0552622	+	0.998472i	$0.517599\pi$
$374$		0			0
$375$			1.27116e11i			0.331940i
$376$		0			0
$377$	−5.88557e11	−	1.79920e9i	−1.50056	−	0.00458715i
$378$		0			0
$379$		−	2.88524e11i		−	0.718299i	−0.933280	−	0.359150i	$-0.883067\pi$
$379$		−	2.88524e11i		−	0.718299i	0.933280	−	0.359150i	$-0.116933\pi$
$380$		0			0
$381$	3.36566e11			0.818292
$382$		0			0
$383$		−	1.88448e11i		−	0.447504i	−0.974646	−	0.223752i	$-0.928169\pi$
$383$		−	1.88448e11i		−	0.447504i	0.974646	−	0.223752i	$-0.0718306\pi$
$384$		0			0
$385$			1.78657e10i			0.0414426i
$386$		0			0
$387$	4.69870e11			1.06483
$388$		0			0
$389$	1.61250e11			0.357048			0.178524	−	0.983936i	$-0.442868\pi$
$389$	1.61250e11			0.357048			0.178524	+	0.983936i	$0.442868\pi$
$390$		0			0
$391$	2.02427e11			0.437998
$392$		0			0
$393$	1.16701e11			0.246779
$394$		0			0
$395$			1.24255e11i			0.256818i
$396$		0			0
$397$			8.85792e11i			1.78968i	0.446390	+	0.894838i	$0.352709\pi$
$397$			8.85792e11i			1.78968i	−0.446390	+	0.894838i	$0.647291\pi$
$398$		0			0
$399$	1.86167e10			0.0367727
$400$		0			0
$401$		−	4.43502e11i		−	0.856536i	−0.903652	−	0.428268i	$-0.859124\pi$
$401$		−	4.43502e11i		−	0.856536i	0.903652	−	0.428268i	$-0.140876\pi$
$402$		0			0
$403$	−2.53329e9	+	8.28694e11i	−0.00478423	+	1.56503i
$404$		0			0
$405$		−	1.32013e11i		−	0.243820i
$406$		0			0
$407$	4.72692e11			0.853893
$408$		0			0
$409$			6.71077e11i			1.18582i	0.805270	+	0.592908i	$0.202020\pi$
$409$			6.71077e11i			1.18582i	−0.805270	+	0.592908i	$0.797980\pi$
$410$		0			0
$411$		−	6.52252e10i		−	0.112753i
$412$		0			0
$413$	3.38781e10			0.0572986
$414$		0			0
$415$	4.11730e11			0.681391
$416$		0			0
$417$	1.82233e11			0.295131
$418$		0			0
$419$	−1.17956e12			−1.86964			−0.934819	−	0.355125i	$-0.884438\pi$
$419$	−1.17956e12			−1.86964			−0.934819	+	0.355125i	$0.884438\pi$
$420$		0			0
$421$			6.27955e10i			0.0974225i	0.998813	+	0.0487112i	$0.0155114\pi$
$421$			6.27955e10i			0.0974225i	−0.998813	+	0.0487112i	$0.984489\pi$
$422$		0			0
$423$		−	6.28758e11i		−	0.954887i
$424$		0			0
$425$	1.36909e11			0.203555
$426$		0			0
$427$		−	5.87553e10i		−	0.0855306i
$428$		0			0
$429$	5.88990e8	−	1.92671e11i	0.000839556	−	0.274637i
$430$		0			0
$431$			3.62937e11i			0.506621i	0.967385	+	0.253310i	$0.0815194\pi$
$431$			3.62937e11i			0.506621i	−0.967385	+	0.253310i	$0.918481\pi$
$432$		0			0
$433$	1.24949e11			0.170819			0.0854096	−	0.996346i	$-0.472780\pi$
$433$	1.24949e11			0.170819			0.0854096	+	0.996346i	$0.472780\pi$
$434$		0			0
$435$			7.47693e11i			1.00120i
$436$		0			0
$437$			8.31039e11i			1.09007i
$438$		0			0
$439$	−6.52823e10			−0.0838891			−0.0419445	−	0.999120i	$-0.513355\pi$
$439$	−6.52823e10			−0.0838891			−0.0419445	+	0.999120i	$0.513355\pi$
$440$		0			0
$441$	−5.56431e11			−0.700548
$442$		0			0
$443$	−6.09884e11			−0.752368			−0.376184	−	0.926545i	$-0.622764\pi$
$443$	−6.09884e11			−0.752368			−0.376184	+	0.926545i	$0.622764\pi$
$444$		0			0
$445$	7.26140e11			0.877810
$446$		0			0
$447$			2.67251e11i			0.316618i
$448$		0			0
$449$			7.71029e11i			0.895287i	0.894212	+	0.447643i	$0.147737\pi$
$449$			7.71029e11i			0.895287i	−0.894212	+	0.447643i	$0.852263\pi$
$450$		0			0
$451$	−5.00130e11			−0.569231
$452$		0			0
$453$			3.98788e11i			0.444939i
$454$		0			0
$455$	7.50922e10	+	2.29554e8i	0.0821379	+	0.000251093i
$456$		0			0
$457$			4.79089e11i			0.513799i	0.966438	+	0.256899i	$0.0827009\pi$
$457$			4.79089e11i			0.513799i	−0.966438	+	0.256899i	$0.917299\pi$
$458$		0			0
$459$	−3.57237e11			−0.375664
$460$		0			0
$461$		−	1.54060e12i		−	1.58868i	−0.607476	−	0.794338i	$-0.707818\pi$
$461$		−	1.54060e12i		−	1.58868i	0.607476	−	0.794338i	$-0.292182\pi$
$462$		0			0
$463$		−	2.83548e11i		−	0.286756i	−0.989668	−	0.143378i	$-0.954204\pi$
$463$		−	2.83548e11i		−	0.286756i	0.989668	−	0.143378i	$-0.0457964\pi$
$464$		0			0
$465$	1.05276e12			1.04422
$466$		0			0
$467$	−1.70101e12			−1.65494			−0.827468	−	0.561513i	$-0.810220\pi$
$467$	−1.70101e12			−1.65494			−0.827468	+	0.561513i	$0.810220\pi$
$468$		0			0
$469$	1.03770e11			0.0990360
$470$		0			0
$471$	−8.04813e11			−0.753530
$472$		0			0
$473$			8.31118e11i			0.763462i
$474$		0			0
$475$			5.62063e11i			0.506598i
$476$		0			0
$477$	1.18374e12			1.04694
$478$		0			0
$479$			7.64990e11i			0.663966i	0.943285	+	0.331983i	$0.107718\pi$
$479$			7.64990e11i			0.663966i	−0.943285	+	0.331983i	$0.892282\pi$
$480$		0			0
$481$	6.07356e9	−	1.98679e12i	0.00517357	−	1.69239i
$482$		0			0
$483$		−	4.71724e10i		−	0.0394390i
$484$		0			0
$485$	1.61008e12			1.32133
$486$		0			0
$487$			1.91214e12i			1.54042i	0.637790	+	0.770211i	$0.279849\pi$
$487$			1.91214e12i			1.54042i	−0.637790	+	0.770211i	$0.720151\pi$
$488$		0			0
$489$			7.71580e11i			0.610227i
$490$		0			0
$491$	−3.14035e11			−0.243843			−0.121922	−	0.992540i	$-0.538906\pi$
$491$	−3.14035e11			−0.243843			−0.121922	+	0.992540i	$0.538906\pi$
$492$		0			0
$493$	−7.97278e11			−0.607853
$494$		0			0
$495$	5.81332e11			0.435212
$496$		0			0
$497$	4.87585e10			0.0358465
$498$		0			0
$499$			1.05884e12i			0.764502i	0.924059	+	0.382251i	$0.124851\pi$
$499$			1.05884e12i			0.764502i	−0.924059	+	0.382251i	$0.875149\pi$
$500$		0			0
$501$			1.45644e12i			1.03281i
$502$		0			0
$503$	8.28369e11			0.576990			0.288495	−	0.957481i	$-0.406845\pi$
$503$	8.28369e11			0.576990			0.288495	+	0.957481i	$0.406845\pi$
$504$		0			0
$505$			7.84418e11i			0.536706i
$506$		0			0
$507$	−8.09819e11	−	4.95122e9i	−0.544317	−	0.00332795i
$508$		0			0
$509$		−	1.72270e12i		−	1.13757i	−0.822485	−	0.568787i	$-0.807413\pi$
$509$		−	1.72270e12i		−	1.13757i	0.822485	−	0.568787i	$-0.192587\pi$
$510$		0			0
$511$	−6.16379e10			−0.0399902
$512$		0			0
$513$		−	1.46660e12i		−	0.934937i
$514$		0			0
$515$		−	2.69467e12i		−	1.68800i
$516$		0			0
$517$	1.11216e12			0.684638
$518$		0			0
$519$	1.33064e12			0.805024
$520$		0			0
$521$	4.85856e11			0.288894			0.144447	−	0.989513i	$-0.453860\pi$
$521$	4.85856e11			0.288894			0.144447	+	0.989513i	$0.453860\pi$
$522$		0			0
$523$	−3.84027e11			−0.224442			−0.112221	−	0.993683i	$-0.535796\pi$
$523$	−3.84027e11			−0.224442			−0.112221	+	0.993683i	$0.535796\pi$
$524$		0			0
$525$		−	3.19045e10i		−	0.0183288i
$526$		0			0
$527$			1.12258e12i			0.633968i
$528$		0			0
$529$	3.04595e11			0.169111
$530$		0			0
$531$		−	1.10236e12i		−	0.601725i
$532$		0			0
$533$	−6.42611e9	+	2.10212e12i	−0.00344886	+	1.12820i
$534$		0			0
$535$		−	3.94599e11i		−	0.208240i
$536$		0			0
$537$	4.06025e11			0.210702
$538$		0			0
$539$		−	9.84229e11i		−	0.502281i
$540$		0			0
$541$			1.38946e12i			0.697361i	0.937242	+	0.348680i	$0.113370\pi$
$541$			1.38946e12i			0.697361i	−0.937242	+	0.348680i	$0.886630\pi$
$542$		0			0
$543$	−1.73617e12			−0.857022
$544$		0			0
$545$	1.29177e12			0.627191
$546$		0			0
$547$	−3.27868e12			−1.56587			−0.782936	−	0.622102i	$-0.786279\pi$
$547$	−3.27868e12			−1.56587			−0.782936	+	0.622102i	$0.786279\pi$
$548$		0			0
$549$	−1.91184e12			−0.898206
$550$		0			0
$551$		−	3.27313e12i		−	1.51280i
$552$		0			0
$553$			3.08759e10i			0.0140396i
$554$		0			0
$555$	−2.52399e12			−1.12920
$556$		0			0
$557$		−	2.21272e12i		−	0.974044i	−0.873390	−	0.487022i	$-0.838083\pi$
$557$		−	2.21272e12i		−	0.974044i	0.873390	−	0.487022i	$-0.161917\pi$
$558$		0			0
$559$	3.49331e12	+	1.06789e10i	1.51316	+	0.00462567i
$560$		0			0
$561$		−	2.60999e11i		−	0.111251i
$562$		0			0
$563$	−2.20696e12			−0.925777			−0.462889	−	0.886416i	$-0.653187\pi$
$563$	−2.20696e12			−0.925777			−0.462889	+	0.886416i	$0.653187\pi$
$564$		0			0
$565$		−	5.00790e11i		−	0.206746i
$566$		0			0
$567$		−	3.28037e10i		−	0.0133291i
$568$		0			0
$569$	−3.20319e12			−1.28108			−0.640541	−	0.767924i	$-0.721290\pi$
$569$	−3.20319e12			−1.28108			−0.640541	+	0.767924i	$0.721290\pi$
$570$		0			0
$571$	1.45480e12			0.572717			0.286359	−	0.958123i	$-0.407555\pi$
$571$	1.45480e12			0.572717			0.286359	+	0.958123i	$0.407555\pi$
$572$		0			0
$573$	1.34720e12			0.522081
$574$		0			0
$575$	1.42420e12			0.543331
$576$		0			0
$577$		−	4.31781e11i		−	0.162171i	−0.996707	−	0.0810853i	$-0.974161\pi$
$577$		−	4.31781e11i		−	0.162171i	0.996707	−	0.0810853i	$-0.0258386\pi$
$578$		0			0
$579$		−	1.64751e12i		−	0.609222i
$580$		0			0
$581$	1.02310e11			0.0372501
$582$		0			0
$583$			2.09382e12i			0.750638i
$584$		0			0
$585$	7.46946e9	−	2.44342e12i	0.00263687	−	0.862576i
$586$		0			0
$587$		−	2.89662e12i		−	1.00698i	−0.864002	−	0.503489i	$-0.832050\pi$
$587$		−	2.89662e12i		−	1.00698i	0.864002	−	0.503489i	$-0.167950\pi$
$588$		0			0
$589$	−4.60860e12			−1.57779
$590$		0			0
$591$		−	8.62414e11i		−	0.290785i
$592$		0			0
$593$		−	2.94073e12i		−	0.976581i	−0.872681	−	0.488291i	$-0.837621\pi$
$593$		−	2.94073e12i		−	0.976581i	0.872681	−	0.488291i	$-0.162379\pi$
$594$		0			0
$595$	1.01722e11			0.0332728
$596$		0			0
$597$	2.85412e12			0.919578
$598$		0			0
$599$	5.01755e12			1.59247			0.796234	−	0.604989i	$-0.206822\pi$
$599$	5.01755e12			1.59247			0.796234	+	0.604989i	$0.206822\pi$
$600$		0			0
$601$	5.93082e12			1.85430			0.927150	−	0.374691i	$-0.122251\pi$
$601$	5.93082e12			1.85430			0.927150	+	0.374691i	$0.122251\pi$
$602$		0			0
$603$		−	3.37657e12i		−	1.04003i
$604$		0			0
$605$		−	3.01103e12i		−	0.913726i
$606$		0			0
$607$	−8.39821e11			−0.251095			−0.125547	−	0.992088i	$-0.540069\pi$
$607$	−8.39821e11			−0.251095			−0.125547	+	0.992088i	$0.540069\pi$
$608$		0			0
$609$			1.85793e11i			0.0547334i
$610$		0			0
$611$	1.42900e10	−	4.67459e12i	0.00414809	−	1.35693i
$612$		0			0
$613$		−	1.33280e12i		−	0.381234i	−0.981664	−	0.190617i	$-0.938951\pi$
$613$		−	1.33280e12i		−	0.381234i	0.981664	−	0.190617i	$-0.0610489\pi$
$614$		0			0
$615$	2.67050e12			0.752757
$616$		0			0
$617$			3.16126e12i			0.878166i	0.898447	+	0.439083i	$0.144697\pi$
$617$			3.16126e12i			0.878166i	−0.898447	+	0.439083i	$0.855303\pi$
$618$		0			0
$619$			3.30760e12i			0.905534i	0.891629	+	0.452767i	$0.149563\pi$
$619$			3.30760e12i			0.905534i	−0.891629	+	0.452767i	$0.850437\pi$
$620$		0			0
$621$	−3.71617e12			−1.00273
$622$		0			0
$623$	1.80438e11			0.0479878
$624$		0			0
$625$	−4.76835e12			−1.24999
$626$		0			0
$627$	1.07150e12			0.276878
$628$		0			0
$629$		−	2.69138e12i		−	0.685562i
$630$		0			0
$631$			1.98059e12i			0.497350i	0.968587	+	0.248675i	$0.0799951\pi$
$631$			1.98059e12i			0.497350i	−0.968587	+	0.248675i	$0.920005\pi$
$632$		0			0
$633$	−9.12156e11			−0.225815
$634$		0			0
$635$			7.54983e12i			1.84270i
$636$		0			0
$637$	−4.13686e12	−	1.26462e10i	−0.995505	−	0.00304322i
$638$		0			0
$639$		−	1.58655e12i		−	0.376444i
$640$		0			0
$641$	−1.34461e12			−0.314582			−0.157291	−	0.987552i	$-0.550276\pi$
$641$	−1.34461e12			−0.314582			−0.157291	+	0.987552i	$0.550276\pi$
$642$		0			0
$643$			3.55018e12i			0.819031i	0.912303	+	0.409516i	$0.134302\pi$
$643$			3.55018e12i			0.819031i	−0.912303	+	0.409516i	$0.865698\pi$
$644$		0			0
$645$		−	4.43785e12i		−	1.00961i
$646$		0			0
$647$	8.29159e12			1.86024			0.930119	−	0.367258i	$-0.119703\pi$
$647$	8.29159e12			1.86024			0.930119	+	0.367258i	$0.119703\pi$
$648$		0			0
$649$	1.94988e12			0.431427
$650$		0			0
$651$	2.61599e11			0.0570849
$652$		0			0
$653$	4.48358e12			0.964973			0.482487	−	0.875903i	$-0.339734\pi$
$653$	4.48358e12			0.964973			0.482487	+	0.875903i	$0.339734\pi$
$654$		0			0
$655$			2.61783e12i			0.555720i
$656$		0			0
$657$			2.00564e12i			0.419960i
$658$		0			0
$659$	−3.15620e12			−0.651899			−0.325950	−	0.945387i	$-0.605684\pi$
$659$	−3.15620e12			−0.651899			−0.325950	+	0.945387i	$0.605684\pi$
$660$		0			0
$661$			8.75602e12i			1.78402i	0.452013	+	0.892011i	$0.350706\pi$
$661$			8.75602e12i			1.78402i	−0.452013	+	0.892011i	$0.649294\pi$
$662$		0			0
$663$	−1.09702e12	−	3.35354e9i	−0.220497	−	0.000674051i
$664$		0			0
$665$			4.17609e11i			0.0828080i
$666$		0			0
$667$	−8.29370e12			−1.62249
$668$		0			0
$669$			1.62036e12i			0.312747i
$670$		0			0
$671$		−	3.38171e12i		−	0.643999i
$672$		0			0
$673$	−5.24324e12			−0.985218			−0.492609	−	0.870251i	$-0.663957\pi$
$673$	−5.24324e12			−0.985218			−0.492609	+	0.870251i	$0.663957\pi$
$674$		0			0
$675$	−2.51338e12			−0.466006
$676$		0			0
$677$	−3.75361e12			−0.686752			−0.343376	−	0.939198i	$-0.611571\pi$
$677$	−3.75361e12			−0.686752			−0.343376	+	0.939198i	$0.611571\pi$
$678$		0			0
$679$	4.00088e11			0.0722340
$680$		0			0
$681$		−	2.65014e11i		−	0.0472179i
$682$		0			0
$683$			1.87161e12i			0.329095i	0.986369	+	0.164548i	$0.0526164\pi$
$683$			1.87161e12i			0.329095i	−0.986369	+	0.164548i	$0.947384\pi$
$684$		0			0
$685$	1.46313e12			0.253907
$686$		0			0
$687$		−	3.67633e12i		−	0.629665i
$688$		0			0
$689$	8.80064e12	+	2.69032e10i	1.48774	+	0.00454797i
$690$		0			0
$691$			5.87987e12i			0.981107i	0.871411	+	0.490553i	$0.163205\pi$
$691$			5.87987e12i			0.981107i	−0.871411	+	0.490553i	$0.836795\pi$
$692$		0			0
$693$	1.44455e11			0.0237920
$694$		0			0
$695$			4.08784e12i			0.664603i
$696$		0			0
$697$			2.84760e12i			0.457016i
$698$		0			0
$699$	−1.29145e12			−0.204612
$700$		0			0
$701$	−8.67254e12			−1.35649			−0.678243	−	0.734838i	$-0.737258\pi$
$701$	−8.67254e12			−1.35649			−0.678243	+	0.734838i	$0.737258\pi$
$702$		0			0
$703$	1.10491e13			1.70620
$704$		0			0
$705$	−5.93852e12			−0.905372
$706$		0			0
$707$			1.94919e11i			0.0293405i
$708$		0			0
$709$			1.16380e13i			1.72969i	0.502037	+	0.864846i	$0.332584\pi$
$709$			1.16380e13i			1.72969i	−0.502037	+	0.864846i	$0.667416\pi$
$710$		0			0
$711$	1.00467e12			0.147438
$712$		0			0
$713$			1.16776e13i			1.69220i
$714$		0			0
$715$	4.32199e12	+	1.32122e10i	0.618453	+	0.00189059i
$716$		0			0
$717$		−	4.46771e12i		−	0.631318i
$718$		0			0
$719$	2.21919e12			0.309681			0.154841	−	0.987939i	$-0.450514\pi$
$719$	2.21919e12			0.309681			0.154841	+	0.987939i	$0.450514\pi$
$720$		0			0
$721$		−	6.69595e11i		−	0.0922791i
$722$		0			0
$723$		−	1.57120e12i		−	0.213850i
$724$		0			0
$725$	−5.60934e12			−0.754034
$726$		0			0
$727$	1.03358e13			1.37227			0.686134	−	0.727475i	$-0.259306\pi$
$727$	1.03358e13			1.37227			0.686134	+	0.727475i	$0.259306\pi$
$728$		0			0
$729$	2.78197e12			0.364819
$730$		0			0
$731$	4.73215e12			0.612958
$732$		0			0
$733$			6.58421e12i			0.842433i	0.906960	+	0.421217i	$0.138397\pi$
$733$			6.58421e12i			0.842433i	−0.906960	+	0.421217i	$0.861603\pi$
$734$		0			0
$735$			5.25540e12i			0.664222i
$736$		0			0
$737$	5.97255e12			0.745686
$738$		0			0
$739$			1.06694e13i			1.31595i	0.753039	+	0.657976i	$0.228587\pi$
$739$			1.06694e13i			1.31595i	−0.753039	+	0.657976i	$0.771413\pi$
$740$		0			0
$741$	1.37676e10	−	4.50368e12i	0.00167755	−	0.548763i
$742$		0			0
$743$		−	9.24168e12i		−	1.11250i	−0.831014	−	0.556251i	$-0.812239\pi$
$743$		−	9.24168e12i		−	1.11250i	0.831014	−	0.556251i	$-0.187761\pi$
$744$		0			0
$745$	−5.99496e12			−0.712990
$746$		0			0
$747$		−	3.32908e12i		−	0.391184i
$748$		0			0
$749$		−	9.80535e10i		−	0.0113840i
$750$		0			0
$751$	1.15521e11			0.0132520			0.00662602	−	0.999978i	$-0.497891\pi$
$751$	1.15521e11			0.0132520			0.00662602	+	0.999978i	$0.497891\pi$
$752$		0			0
$753$	1.93333e12			0.219143
$754$		0			0
$755$	−8.94559e12			−1.00195
$756$		0			0
$757$	2.53437e11			0.0280503			0.0140252	−	0.999902i	$-0.495536\pi$
$757$	2.53437e11			0.0280503			0.0140252	+	0.999902i	$0.495536\pi$
$758$		0			0
$759$		−	2.71505e12i		−	0.296954i
$760$		0			0
$761$		−	1.04156e13i		−	1.12578i	−0.826533	−	0.562889i	$-0.809690\pi$
$761$		−	1.04156e13i		−	1.12578i	0.826533	−	0.562889i	$-0.190310\pi$
$762$		0			0
$763$	3.20989e11			0.0342871
$764$		0			0
$765$		−	3.30994e12i		−	0.349417i
$766$		0			0
$767$	2.50538e10	−	8.19564e12i	0.00261393	−	0.855074i
$768$		0			0
$769$		−	9.65857e10i		−	0.00995966i	−0.999988	−	0.00497983i	$-0.998415\pi$
$769$		−	9.65857e10i		−	0.00995966i	0.999988	−	0.00497983i	$-0.00158514\pi$
$770$		0			0
$771$	−1.04350e12			−0.106352
$772$		0			0
$773$			1.54930e13i			1.56073i	0.625327	+	0.780363i	$0.284966\pi$
$773$			1.54930e13i			1.56073i	−0.625327	+	0.780363i	$0.715034\pi$
$774$		0			0
$775$			7.89801e12i			0.786430i
$776$		0			0
$777$	−6.27184e11			−0.0617306
$778$		0			0
$779$	−1.16905e13			−1.13740
$780$		0			0
$781$	2.80633e12			0.269904
$782$		0			0
$783$	1.46365e13			1.39158
$784$		0			0
$785$		−	1.80535e13i		−	1.69687i
$786$		0			0
$787$			1.63867e13i			1.52267i	0.648359	+	0.761335i	$0.275456\pi$
$787$			1.63867e13i			1.52267i	−0.648359	+	0.761335i	$0.724544\pi$
$788$		0			0
$789$	−7.84645e12			−0.720820
$790$		0			0
$791$		−	1.24441e11i		−	0.0113023i
$792$		0			0
$793$	−1.42138e13	−	4.34512e10i	−1.27638	−	0.00390186i
$794$		0			0
$795$		−	1.11802e13i		−	0.992652i
$796$		0			0
$797$	8.42824e12			0.739903			0.369951	−	0.929051i	$-0.379374\pi$
$797$	8.42824e12			0.739903			0.369951	+	0.929051i	$0.379374\pi$
$798$		0			0
$799$		−	6.33234e12i		−	0.549672i
$800$		0			0
$801$		−	5.87126e12i		−	0.503947i
$802$		0			0
$803$	−3.54762e12			−0.301104
$804$		0			0
$805$	1.05817e12			0.0888123
$806$		0			0
$807$	−6.36311e11			−0.0528127
$808$		0			0
$809$	−1.18753e13			−0.974712			−0.487356	−	0.873203i	$-0.662039\pi$
$809$	−1.18753e13			−0.974712			−0.487356	+	0.873203i	$0.662039\pi$
$810$		0			0
$811$		−	2.84361e12i		−	0.230822i	−0.993318	−	0.115411i	$-0.963182\pi$
$811$		−	2.84361e12i		−	0.230822i	0.993318	−	0.115411i	$-0.0368185\pi$
$812$		0			0
$813$			1.14498e12i			0.0919162i
$814$		0			0
$815$	−1.73080e13			−1.37416
$816$		0			0
$817$			1.94273e13i			1.52550i
$818$		0			0
$819$	1.85608e9	−	6.07164e11i	0.000144151	−	0.0471551i
$820$		0			0
$821$		−	1.05095e13i		−	0.807303i	−0.914913	−	0.403652i	$-0.867741\pi$
$821$		−	1.05095e13i		−	0.807303i	0.914913	−	0.403652i	$-0.132259\pi$
$822$		0			0
$823$	1.98701e13			1.50973			0.754867	−	0.655878i	$-0.227702\pi$
$823$	1.98701e13			1.50973			0.754867	+	0.655878i	$0.227702\pi$
$824$		0			0
$825$		−	1.83629e12i		−	0.138006i
$826$		0			0
$827$			1.31516e13i			0.977699i	0.872368	+	0.488850i	$0.162583\pi$
$827$			1.31516e13i			0.977699i	−0.872368	+	0.488850i	$0.837417\pi$
$828$		0			0
$829$	1.47154e13			1.08213			0.541063	−	0.840982i	$-0.318022\pi$
$829$	1.47154e13			1.08213			0.541063	+	0.840982i	$0.318022\pi$
$830$		0			0
$831$	9.17137e12			0.667160
$832$		0			0
$833$	−5.60392e12			−0.403264
$834$		0			0
$835$	−3.26707e13			−2.32579
$836$		0			0
$837$		−	2.06083e13i		−	1.45137i
$838$		0			0
$839$			1.49293e13i			1.04019i	0.854110	+	0.520093i	$0.174103\pi$
$839$			1.49293e13i			1.04019i	−0.854110	+	0.520093i	$0.825897\pi$
$840$		0			0
$841$	1.81584e13			1.25169
$842$		0			0
$843$		−	6.64870e12i		−	0.453432i
$844$		0			0
$845$	1.11065e11	−	1.81658e13i	0.00749417	−	1.22574i
$846$		0			0
$847$		−	7.48207e11i		−	0.0499513i
$848$		0			0
$849$	1.25744e13			0.830620
$850$		0			0
$851$		−	2.79971e13i		−	1.82991i
$852$		0			0
$853$		−	2.52397e13i		−	1.63235i	−0.577803	−	0.816176i	$-0.696090\pi$
$853$		−	2.52397e13i		−	1.63235i	0.577803	−	0.816176i	$-0.303910\pi$
$854$		0			0
$855$	1.35886e13			0.869614
$856$		0			0
$857$	1.12215e13			0.710617			0.355308	−	0.934749i	$-0.384376\pi$
$857$	1.12215e13			0.710617			0.355308	+	0.934749i	$0.384376\pi$
$858$		0			0
$859$	9.07217e12			0.568515			0.284258	−	0.958748i	$-0.408253\pi$
$859$	9.07217e12			0.568515			0.284258	+	0.958748i	$0.408253\pi$
$860$		0			0
$861$	6.63590e11			0.0411515
$862$		0			0
$863$			1.49491e13i			0.917415i	0.888587	+	0.458707i	$0.151687\pi$
$863$			1.49491e13i			0.917415i	−0.888587	+	0.458707i	$0.848313\pi$
$864$		0			0
$865$			2.98489e13i			1.81283i
$866$		0			0
$867$	7.57015e12			0.455007
$868$		0			0
$869$			1.77709e12i			0.105711i
$870$		0			0
$871$	7.67406e10	−	2.51035e13i	0.00451797	−	1.47793i
$872$		0			0
$873$		−	1.30185e13i		−	0.758570i
$874$		0			0
$875$	−7.08556e11			−0.0408637
$876$		0			0
$877$			2.88188e13i			1.64505i	0.568732	+	0.822523i	$0.307434\pi$
$877$			2.88188e13i			1.64505i	−0.568732	+	0.822523i	$0.692566\pi$
$878$		0			0
$879$		−	1.36130e13i		−	0.769139i
$880$		0			0
$881$	5.34093e12			0.298693			0.149347	−	0.988785i	$-0.452283\pi$
$881$	5.34093e12			0.298693			0.149347	+	0.988785i	$0.452283\pi$
$882$		0			0
$883$	−3.05434e13			−1.69080			−0.845402	−	0.534130i	$-0.820639\pi$
$883$	−3.05434e13			−1.69080			−0.845402	+	0.534130i	$0.820639\pi$
$884$		0			0
$885$	−1.04116e13			−0.570523
$886$		0			0
$887$	−2.14512e13			−1.16358			−0.581788	−	0.813340i	$-0.697647\pi$
$887$	−2.14512e13			−1.16358			−0.581788	+	0.813340i	$0.697647\pi$
$888$		0			0
$889$			1.87605e12i			0.100736i
$890$		0			0
$891$		−	1.88804e12i		−	0.100360i
$892$		0			0
$893$	2.59967e13			1.36800
$894$		0			0
$895$			9.10791e12i			0.474477i
$896$		0			0
$897$	−1.14117e13	−	3.48853e10i	−0.588554	−	0.00179919i
$898$		0			0
$899$		−	4.59935e13i		−	2.34843i
$900$		0			0
$901$	1.19216e13			0.602662
$902$		0			0
$903$		−	1.10276e12i		−	0.0551930i
$904$		0			0
$905$		−	3.89455e13i		−	1.92992i
$906$		0			0
$907$	8.20371e12			0.402511			0.201255	−	0.979539i	$-0.435498\pi$
$907$	8.20371e12			0.402511			0.201255	+	0.979539i	$0.435498\pi$
$908$		0			0
$909$	6.34248e12			0.308121
$910$		0			0
$911$	−3.25403e13			−1.56527			−0.782635	−	0.622481i	$-0.786125\pi$
$911$	−3.25403e13			−1.56527			−0.782635	+	0.622481i	$0.786125\pi$
$912$		0			0
$913$	5.88856e12			0.280473
$914$		0			0
$915$			1.80570e13i			0.851631i
$916$		0			0
$917$			6.50502e11i			0.0303799i
$918$		0			0
$919$	−4.85596e12			−0.224572			−0.112286	−	0.993676i	$-0.535817\pi$
$919$	−4.85596e12			−0.224572			−0.112286	+	0.993676i	$0.535817\pi$
$920$		0			0
$921$			1.65231e13i			0.756697i
$922$		0			0
$923$	3.60583e10	−	1.17954e13i	0.00163530	−	0.534942i
$924$		0			0
$925$		−	1.89355e13i		−	0.850430i
$926$		0			0
$927$	−2.17880e13			−0.969076
$928$		0			0
$929$		−	3.99103e12i		−	0.175798i	−0.996129	−	0.0878991i	$-0.971985\pi$
$929$		−	3.99103e12i		−	0.175798i	0.996129	−	0.0878991i	$-0.0280153\pi$
$930$		0			0
$931$		−	2.30063e13i		−	1.00363i
$932$		0			0
$933$	−3.29687e12			−0.142441
$934$		0			0
$935$	5.85471e12			0.250526
$936$		0			0
$937$	1.06502e13			0.451366			0.225683	−	0.974201i	$-0.427539\pi$
$937$	1.06502e13			0.451366			0.225683	+	0.974201i	$0.427539\pi$
$938$		0			0
$939$	−7.84157e12			−0.329161
$940$		0			0
$941$			3.63674e13i			1.51202i	0.654558	+	0.756012i	$0.272855\pi$
$941$			3.63674e13i			1.51202i	−0.654558	+	0.756012i	$0.727145\pi$
$942$		0			0
$943$			2.96222e13i			1.21988i
$944$		0			0
$945$	−1.86743e12			−0.0761729
$946$		0			0
$947$		−	4.33387e12i		−	0.175106i	−0.996160	−	0.0875530i	$-0.972095\pi$
$947$		−	4.33387e12i		−	0.175106i	0.996160	−	0.0875530i	$-0.0279047\pi$
$948$		0			0
$949$	−4.55829e10	+	1.49112e13i	−0.00182433	+	0.596780i
$950$		0			0
$951$			7.41691e12i			0.294043i
$952$		0			0
$953$	1.22670e13			0.481748			0.240874	−	0.970556i	$-0.422566\pi$
$953$	1.22670e13			0.481748			0.240874	+	0.970556i	$0.422566\pi$
$954$		0			0
$955$			3.02204e13i			1.17567i
$956$		0			0
$957$			1.06935e13i			0.412112i
$958$		0			0
$959$	3.63571e11			0.0138805
$960$		0			0
$961$	−3.83196e13			−1.44933
$962$		0			0
$963$	−3.19056e12			−0.119550
$964$		0			0
$965$	3.69569e13			1.37190
$966$		0			0
$967$		−	3.83239e13i		−	1.40945i	−0.709479	−	0.704727i	$-0.751070\pi$
$967$		−	3.83239e13i		−	1.40945i	0.709479	−	0.704727i	$-0.248930\pi$
$968$		0			0
$969$		−	6.10082e12i		−	0.222296i
$970$		0			0
$971$	−1.67255e13			−0.603799			−0.301900	−	0.953340i	$-0.597621\pi$
$971$	−1.67255e13			−0.603799			−0.301900	+	0.953340i	$0.597621\pi$
$972$		0			0
$973$			1.01578e12i			0.0363323i
$974$		0			0
$975$	−7.71819e12	−	2.35942e10i	−0.273524	−	0.000836152i
$976$		0			0
$977$		−	2.01291e13i		−	0.706805i	−0.935471	−	0.353402i	$-0.885025\pi$
$977$		−	2.01291e13i		−	0.706805i	0.935471	−	0.353402i	$-0.114975\pi$
$978$		0			0
$979$	1.03852e13			0.361322
$980$		0			0
$981$		−	1.04447e13i		−	0.360068i
$982$		0			0
$983$			1.62112e13i			0.553765i	0.960904	+	0.276883i	$0.0893013\pi$
$983$			1.62112e13i			0.553765i	−0.960904	+	0.276883i	$0.910699\pi$
$984$		0			0
$985$	1.93456e13			0.654816
$986$		0			0
$987$	−1.47566e12			−0.0494946
$988$		0			0
$989$	4.92263e13			1.63612
$990$		0			0
$991$	−2.94051e12			−0.0968483			−0.0484241	−	0.998827i	$-0.515420\pi$
$991$	−2.94051e12			−0.0968483			−0.0484241	+	0.998827i	$0.515420\pi$
$992$		0			0
$993$		−	9.81601e12i		−	0.320379i
$994$		0			0
$995$			6.40235e13i			2.07079i
$996$		0			0
$997$	−5.67456e12			−0.181888			−0.0909439	−	0.995856i	$-0.528988\pi$
$997$	−5.67456e12			−0.181888			−0.0909439	+	0.995856i	$0.528988\pi$
$998$		0			0
$999$			4.94085e13i			1.56949i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	104.10.f.a.25.11	✓	32
4.3	odd	2		208.10.f.d.129.22		32
13.12	even	2	inner	104.10.f.a.25.12	yes	32
52.51	odd	2		208.10.f.d.129.21		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
104.10.f.a.25.11	✓	32	1.1	even	1	trivial
104.10.f.a.25.12	yes	32	13.12	even	2	inner
208.10.f.d.129.21		32	52.51	odd	2
208.10.f.d.129.22		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)

\(f(q)\)	\(=\)	\(q-76.3670 q^{3} -1713.06i q^{5} -425.676i q^{7} -13851.1 q^{9} -24500.1i q^{11} +(-102978. - 314.799i) q^{13} +130821. i q^{15} -139497. q^{17} -572688. i q^{19} +32507.6i q^{21} -1.45112e6 q^{23} -981447. q^{25} +2.56090e6 q^{27} +5.71538e6 q^{29} -8.04731e6i q^{31} +1.87100e6i q^{33} -729208. q^{35} +1.92934e7i q^{37} +(7.86410e6 + 24040.3i) q^{39} -2.04134e7i q^{41} -3.39230e7 q^{43} +2.37277e7i q^{45} +4.53942e7i q^{47} +4.01724e7 q^{49} +1.06530e7 q^{51} -8.54616e7 q^{53} -4.19702e7 q^{55} +4.37345e7i q^{57} +7.95866e7i q^{59} +1.38028e8 q^{61} +5.89607e6i q^{63} +(-539270. + 1.76407e8i) q^{65} +2.43776e8i q^{67} +1.10818e8 q^{69} +1.14544e8i q^{71} -1.44800e8i q^{73} +7.49501e7 q^{75} -1.04291e7 q^{77} -7.25337e7 q^{79} +7.70627e7 q^{81} +2.40348e8i q^{83} +2.38966e8i q^{85} -4.36467e8 q^{87} +4.23885e8i q^{89} +(-134002. + 4.38351e7i) q^{91} +6.14549e8i q^{93} -9.81049e8 q^{95} +9.39888e8i q^{97} +3.39353e8i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 162 q^{3} + 223074 q^{9} + 66270 q^{13} - 487902 q^{17} + 3171556 q^{23} - 13526722 q^{25} - 3694974 q^{27} + 8833508 q^{29} - 8281126 q^{35} - 12056860 q^{39} + 89959038 q^{43} - 172344874 q^{49}+ \cdots + 1741143356 q^{95}+O(q^{100}) \) 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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