LMFDB - Embedded newform 32.10.b.a.17.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [32,10,Mod(17,32)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(32, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 10, names="a")

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

chi := DirichletCharacter("32.17");

S:= CuspForms(chi, 10);

N := Newforms(S);

Level:	$ N $	$=$	$ 32 = 2^{5} $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	32.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$16.4811467572$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} + 59x^{6} - 313x^{5} - 315x^{4} - 92091x^{3} + 1261649x^{2} - 16074123x + 251007534 $ x^8 - x^7 + 59x^6 - 313x^5 - 315x^4 - 92091x^3 + 1261649x^2 - 16074123x + 251007534
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{56}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 8)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			17.3
Root			$9.73909 - 3.55976i$ of defining polynomial
Character	$\chi$	$=$	32.17
Dual form			32.10.b.a.17.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-100.481i q^{3} -2583.09i q^{5} -6967.65 q^{7} +9586.48 q^{9} +O(q^{10})$	$q-100.481i q^{3} -2583.09i q^{5} -6967.65 q^{7} +9586.48 q^{9} +25054.5i q^{11} -29701.4i q^{13} -259553. q^{15} -138527. q^{17} +489569. i q^{19} +700119. i q^{21} -847079. q^{23} -4.71924e6 q^{25} -2.94104e6i q^{27} +1.13646e6i q^{29} -4.35747e6 q^{31} +2.51751e6 q^{33} +1.79981e7i q^{35} -535315. i q^{37} -2.98444e6 q^{39} +1.45816e7 q^{41} -3.96441e7i q^{43} -2.47628e7i q^{45} +4.48997e7 q^{47} +8.19448e6 q^{49} +1.39194e7i q^{51} +4.85666e7i q^{53} +6.47180e7 q^{55} +4.91926e7 q^{57} -4.19685e6i q^{59} -6.38659e7i q^{61} -6.67952e7 q^{63} -7.67215e7 q^{65} -5.79621e7i q^{67} +8.51157e7i q^{69} -2.74912e8 q^{71} -9.16969e7 q^{73} +4.74196e8i q^{75} -1.74571e8i q^{77} -2.02396e8 q^{79} -1.06829e8 q^{81} -6.11053e8i q^{83} +3.57829e8i q^{85} +1.14193e8 q^{87} -7.71588e8 q^{89} +2.06949e8i q^{91} +4.37845e8i q^{93} +1.26460e9 q^{95} +1.08292e9 q^{97} +2.40184e8i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 4800 q^{7} - 39368 q^{9}+O(q^{10}) $ 8 * q - 4800 * q^7 - 39368 * q^9	$ 8 q - 4800 q^{7} - 39368 q^{9} + 163136 q^{15} - 102000 q^{17} - 3412032 q^{23} - 2423384 q^{25} - 803584 q^{31} + 58272 q^{33} + 17590208 q^{39} - 2180784 q^{41} - 7432320 q^{47} + 24436680 q^{49} - 7056832 q^{55} + 134003744 q^{57} + 223198400 q^{63} - 146501760 q^{65} - 560234688 q^{71} - 523987120 q^{73} + 248943744 q^{79} + 231960296 q^{81} - 540527424 q^{87} + 744827856 q^{89} + 1465245504 q^{95} - 9932784 q^{97}+O(q^{100}) $ 8 * q - 4800 * q^7 - 39368 * q^9 + 163136 * q^15 - 102000 * q^17 - 3412032 * q^23 - 2423384 * q^25 - 803584 * q^31 + 58272 * q^33 + 17590208 * q^39 - 2180784 * q^41 - 7432320 * q^47 + 24436680 * q^49 - 7056832 * q^55 + 134003744 * q^57 + 223198400 * q^63 - 146501760 * q^65 - 560234688 * q^71 - 523987120 * q^73 + 248943744 * q^79 + 231960296 * q^81 - 540527424 * q^87 + 744827856 * q^89 + 1465245504 * q^95 - 9932784 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/32\mathbb{Z}\right)^\times$.

$n$	$5$	$31$
$\chi(n)$	$-1$	$1$

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
$3$	− 100.481i	− 0.716210i	−0.933681	−	0.358105i	$-0.883423\pi$
$3$	− 100.481i	− 0.716210i	0.933681	−	0.358105i	$-0.116577\pi$
$4$	0	0
$5$	− 2583.09i	− 1.84831i	−0.382017	−	0.924155i	$-0.624770\pi$
$5$	− 2583.09i	− 1.84831i	0.382017	−	0.924155i	$-0.375230\pi$
$6$	0	0
$7$	−6967.65	−1.09684	−0.548422	−	0.836202i	$-0.684771\pi$
$7$	−6967.65	−1.09684	−0.548422	+	0.836202i	$0.684771\pi$
$8$	0	0
$9$	9586.48	0.487044
$10$	0	0
$11$	25054.5i	0.515962i	0.966150	+	0.257981i	$0.0830572\pi$
$11$	25054.5i	0.515962i	−0.966150	+	0.257981i	$0.916943\pi$
$12$	0	0
$13$	− 29701.4i	− 0.288425i	−0.989547	−	0.144212i	$-0.953935\pi$
$13$	− 29701.4i	− 0.288425i	0.989547	−	0.144212i	$-0.0460648\pi$
$14$	0	0
$15$	−259553.	−1.32378
$16$	0	0
$17$	−138527.	−0.402268	−0.201134	−	0.979564i	$-0.564463\pi$
$17$	−138527.	−0.402268	−0.201134	+	0.979564i	$0.564463\pi$
$18$	0	0
$19$	489569.i	0.861832i	0.902392	+	0.430916i	$0.141809\pi$
$19$	489569.i	0.861832i	−0.902392	+	0.430916i	$0.858191\pi$
$20$	0	0
$21$	700119.i	0.785570i
$22$	0	0
$23$	−847079.	−0.631173	−0.315587	−	0.948897i	$-0.602201\pi$
$23$	−847079.	−0.631173	−0.315587	+	0.948897i	$0.602201\pi$
$24$	0	0
$25$	−4.71924e6	−2.41625
$26$	0	0
$27$	− 2.94104e6i	− 1.06504i
$28$	0	0
$29$	1.13646e6i	0.298376i	0.988809	+	0.149188i	$0.0476660\pi$
$29$	1.13646e6i	0.298376i	−0.988809	+	0.149188i	$0.952334\pi$
$30$	0	0
$31$	−4.35747e6	−0.847436	−0.423718	−	0.905794i	$-0.639275\pi$
$31$	−4.35747e6	−0.847436	−0.423718	+	0.905794i	$0.639275\pi$
$32$	0	0
$33$	2.51751e6	0.369537
$34$	0	0
$35$	1.79981e7i	2.02731i
$36$	0	0
$37$	− 535315.i	− 0.0469571i	−0.999724	−	0.0234786i	$-0.992526\pi$
$37$	− 535315.i	− 0.0469571i	0.999724	−	0.0234786i	$-0.00747415\pi$
$38$	0	0
$39$	−2.98444e6	−0.206572
$40$	0	0
$41$	1.45816e7	0.805894	0.402947	−	0.915223i	$-0.367986\pi$
$41$	1.45816e7	0.805894	0.402947	+	0.915223i	$0.367986\pi$
$42$	0	0
$43$	− 3.96441e7i	− 1.76836i	−0.467148	−	0.884179i	$-0.654719\pi$
$43$	− 3.96441e7i	− 1.76836i	0.467148	−	0.884179i	$-0.345281\pi$
$44$	0	0
$45$	− 2.47628e7i	− 0.900208i
$46$	0	0
$47$	4.48997e7	1.34216	0.671078	−	0.741387i	$-0.265832\pi$
$47$	4.48997e7	1.34216	0.671078	+	0.741387i	$0.265832\pi$
$48$	0	0
$49$	8.19448e6	0.203067
$50$	0	0
$51$	1.39194e7i	0.288108i
$52$	0	0
$53$	4.85666e7i	0.845466i	0.906254	+	0.422733i	$0.138929\pi$
$53$	4.85666e7i	0.845466i	−0.906254	+	0.422733i	$0.861071\pi$
$54$	0	0
$55$	6.47180e7	0.953658
$56$	0	0
$57$	4.91926e7	0.617252
$58$	0	0
$59$	− 4.19685e6i	− 0.0450910i	−0.999746	−	0.0225455i	$-0.992823\pi$
$59$	− 4.19685e6i	− 0.0450910i	0.999746	−	0.0225455i	$-0.00717706\pi$
$60$	0	0
$61$	− 6.38659e7i	− 0.590588i	−0.955406	−	0.295294i	$-0.904582\pi$
$61$	− 6.38659e7i	− 0.590588i	0.955406	−	0.295294i	$-0.0954176\pi$
$62$	0	0
$63$	−6.67952e7	−0.534211
$64$	0	0
$65$	−7.67215e7	−0.533098
$66$	0	0
$67$	− 5.79621e7i	− 0.351404i	−0.984443	−	0.175702i	$-0.943780\pi$
$67$	− 5.79621e7i	− 0.351404i	0.984443	−	0.175702i	$-0.0562196\pi$
$68$	0	0
$69$	8.51157e7i	0.452052i
$70$	0	0
$71$	−2.74912e8	−1.28390	−0.641950	−	0.766746i	$-0.721875\pi$
$71$	−2.74912e8	−1.28390	−0.641950	+	0.766746i	$0.721875\pi$
$72$	0	0
$73$	−9.16969e7	−0.377921	−0.188961	−	0.981985i	$-0.560512\pi$
$73$	−9.16969e7	−0.377921	−0.188961	+	0.981985i	$0.560512\pi$
$74$	0	0
$75$	4.74196e8i	1.73054i
$76$	0	0
$77$	− 1.74571e8i	− 0.565930i
$78$	0	0
$79$	−2.02396e8	−0.584627	−0.292314	−	0.956322i	$-0.594425\pi$
$79$	−2.02396e8	−0.584627	−0.292314	+	0.956322i	$0.594425\pi$
$80$	0	0
$81$	−1.06829e8	−0.275744
$82$	0	0
$83$	− 6.11053e8i	− 1.41328i	−0.707574	−	0.706639i	$-0.750210\pi$
$83$	− 6.11053e8i	− 1.41328i	0.707574	−	0.706639i	$-0.249790\pi$
$84$	0	0
$85$	3.57829e8i	0.743516i
$86$	0	0
$87$	1.14193e8	0.213700
$88$	0	0
$89$	−7.71588e8	−1.30356	−0.651779	−	0.758409i	$-0.725977\pi$
$89$	−7.71588e8	−1.30356	−0.651779	+	0.758409i	$0.725977\pi$
$90$	0	0
$91$	2.06949e8i	0.316357i
$92$	0	0
$93$	4.37845e8i	0.606942i
$94$	0	0
$95$	1.26460e9	1.59293
$96$	0	0
$97$	1.08292e9	1.24200	0.621001	−	0.783810i	$-0.286726\pi$
$97$	1.08292e9	1.24200	0.621001	+	0.783810i	$0.286726\pi$
$98$	0	0
$99$	2.40184e8i	0.251296i
$100$	0	0
$101$	− 6.22244e8i	− 0.594996i	−0.954722	−	0.297498i	$-0.903848\pi$
$101$	− 6.22244e8i	− 0.594996i	0.954722	−	0.297498i	$-0.0961522\pi$
$102$	0	0
$103$	−6.34170e8	−0.555186	−0.277593	−	0.960699i	$-0.589537\pi$
$103$	−6.34170e8	−0.555186	−0.277593	+	0.960699i	$0.589537\pi$
$104$	0	0
$105$	1.80847e9	1.45198
$106$	0	0
$107$	− 1.54145e9i	− 1.13685i	−0.822735	−	0.568425i	$-0.807553\pi$
$107$	− 1.54145e9i	− 1.13685i	0.822735	−	0.568425i	$-0.192447\pi$
$108$	0	0
$109$	− 2.19786e9i	− 1.49135i	−0.666309	−	0.745676i	$-0.732127\pi$
$109$	− 2.19786e9i	− 1.49135i	0.666309	−	0.745676i	$-0.267873\pi$
$110$	0	0
$111$	−5.37892e7	−0.0336311
$112$	0	0
$113$	−5.43476e8	−0.313565	−0.156782	−	0.987633i	$-0.550112\pi$
$113$	−5.43476e8	−0.313565	−0.156782	+	0.987633i	$0.550112\pi$
$114$	0	0
$115$	2.18808e9i	1.16660i
$116$	0	0
$117$	− 2.84732e8i	− 0.140475i
$118$	0	0
$119$	9.65210e8	0.441225
$120$	0	0
$121$	1.73022e9	0.733783
$122$	0	0
$123$	− 1.46518e9i	− 0.577189i
$124$	0	0
$125$	7.14513e9i	2.61767i
$126$	0	0
$127$	2.41305e9	0.823093	0.411547	−	0.911389i	$-0.364989\pi$
$127$	2.41305e9	0.823093	0.411547	+	0.911389i	$0.364989\pi$
$128$	0	0
$129$	−3.98349e9	−1.26652
$130$	0	0
$131$	1.57746e9i	0.467990i	0.972238	+	0.233995i	$0.0751799\pi$
$131$	1.57746e9i	0.467990i	−0.972238	+	0.233995i	$0.924820\pi$
$132$	0	0
$133$	− 3.41114e9i	− 0.945295i
$134$	0	0
$135$	−7.59698e9	−1.96852
$136$	0	0
$137$	−6.39076e9	−1.54992	−0.774961	−	0.632009i	$-0.782231\pi$
$137$	−6.39076e9	−1.54992	−0.774961	+	0.632009i	$0.782231\pi$
$138$	0	0
$139$	− 2.45897e9i	− 0.558710i	−0.960188	−	0.279355i	$-0.909879\pi$
$139$	− 2.45897e9i	− 0.558710i	0.960188	−	0.279355i	$-0.0901206\pi$
$140$	0	0
$141$	− 4.51158e9i	− 0.961265i
$142$	0	0
$143$	7.44153e8	0.148816
$144$	0	0
$145$	2.93559e9	0.551492
$146$	0	0
$147$	− 8.23393e8i	− 0.145438i
$148$	0	0
$149$	3.84540e9i	0.639150i	0.947561	+	0.319575i	$0.103540\pi$
$149$	3.84540e9i	0.639150i	−0.947561	+	0.319575i	$0.896460\pi$
$150$	0	0
$151$	−4.07627e9	−0.638067	−0.319034	−	0.947743i	$-0.603358\pi$
$151$	−4.07627e9	−0.638067	−0.319034	+	0.947743i	$0.603358\pi$
$152$	0	0
$153$	−1.32799e9	−0.195922
$154$	0	0
$155$	1.12557e10i	1.56632i
$156$	0	0
$157$	7.59409e9i	0.997533i	0.866736	+	0.498767i	$0.166214\pi$
$157$	7.59409e9i	0.997533i	−0.866736	+	0.498767i	$0.833786\pi$
$158$	0	0
$159$	4.88004e9	0.605531
$160$	0	0
$161$	5.90214e9	0.692298
$162$	0	0
$163$	− 7.71277e9i	− 0.855788i	−0.903829	−	0.427894i	$-0.859256\pi$
$163$	− 7.71277e9i	− 0.855788i	0.903829	−	0.427894i	$-0.140744\pi$
$164$	0	0
$165$	− 6.50295e9i	− 0.683019i
$166$	0	0
$167$	−1.01170e10	−1.00653	−0.503265	−	0.864132i	$-0.667868\pi$
$167$	−1.01170e10	−1.00653	−0.503265	+	0.864132i	$0.667868\pi$
$168$	0	0
$169$	9.72232e9	0.916811
$170$	0	0
$171$	4.69324e9i	0.419750i
$172$	0	0
$173$	− 1.89724e10i	− 1.61033i	−0.593053	−	0.805164i	$-0.702078\pi$
$173$	− 1.89724e10i	− 1.61033i	0.593053	−	0.805164i	$-0.297922\pi$
$174$	0	0
$175$	3.28820e10	2.65025
$176$	0	0
$177$	−4.21706e8	−0.0322946
$178$	0	0
$179$	− 1.72674e10i	− 1.25716i	−0.777746	−	0.628578i	$-0.783637\pi$
$179$	− 1.72674e10i	− 1.25716i	0.777746	−	0.628578i	$-0.216363\pi$
$180$	0	0
$181$	1.03363e10i	0.715830i	0.933754	+	0.357915i	$0.116512\pi$
$181$	1.03363e10i	0.715830i	−0.933754	+	0.357915i	$0.883488\pi$
$182$	0	0
$183$	−6.41733e9	−0.422985
$184$	0	0
$185$	−1.38277e9	−0.0867913
$186$	0	0
$187$	− 3.47073e9i	− 0.207555i
$188$	0	0
$189$	2.04921e10i	1.16818i
$190$	0	0
$191$	9.14299e9	0.497094	0.248547	−	0.968620i	$-0.420047\pi$
$191$	9.14299e9	0.497094	0.248547	+	0.968620i	$0.420047\pi$
$192$	0	0
$193$	8.45502e8	0.0438639	0.0219319	−	0.999759i	$-0.493018\pi$
$193$	8.45502e8	0.0438639	0.0219319	+	0.999759i	$0.493018\pi$
$194$	0	0
$195$	7.70909e9i	0.381810i
$196$	0	0
$197$	− 2.12407e9i	− 0.100478i	−0.998737	−	0.0502390i	$-0.984002\pi$
$197$	− 2.12407e9i	− 0.100478i	0.998737	−	0.0502390i	$-0.0159983\pi$
$198$	0	0
$199$	2.92879e9	0.132388	0.0661941	−	0.997807i	$-0.478914\pi$
$199$	2.92879e9	0.132388	0.0661941	+	0.997807i	$0.478914\pi$
$200$	0	0
$201$	−5.82411e9	−0.251679
$202$	0	0
$203$	− 7.91848e9i	− 0.327272i
$204$	0	0
$205$	− 3.76656e10i	− 1.48954i
$206$	0	0
$207$	−8.12051e9	−0.307409
$208$	0	0
$209$	−1.22659e10	−0.444673
$210$	0	0
$211$	3.16571e10i	1.09951i	0.835325	+	0.549757i	$0.185280\pi$
$211$	3.16571e10i	1.09951i	−0.835325	+	0.549757i	$0.814720\pi$
$212$	0	0
$213$	2.76236e10i	0.919542i
$214$	0	0
$215$	−1.02404e11	−3.26847
$216$	0	0
$217$	3.03613e10	0.929505
$218$	0	0
$219$	9.21383e9i	0.270671i
$220$	0	0
$221$	4.11446e9i	0.116024i
$222$	0	0
$223$	4.92453e10	1.33350	0.666750	−	0.745282i	$-0.267685\pi$
$223$	4.92453e10	1.33350	0.666750	+	0.745282i	$0.267685\pi$
$224$	0	0
$225$	−4.52409e10	−1.17682
$226$	0	0
$227$	5.67209e10i	1.41784i	0.705290	+	0.708919i	$0.250817\pi$
$227$	5.67209e10i	1.41784i	−0.705290	+	0.708919i	$0.749183\pi$
$228$	0	0
$229$	3.33645e9i	0.0801723i	0.999196	+	0.0400862i	$0.0127632\pi$
$229$	3.33645e9i	0.0801723i	−0.999196	+	0.0400862i	$0.987237\pi$
$230$	0	0
$231$	−1.75411e10	−0.405324
$232$	0	0
$233$	−4.60006e10	−1.02250	−0.511249	−	0.859433i	$-0.670817\pi$
$233$	−4.60006e10	−1.02250	−0.511249	+	0.859433i	$0.670817\pi$
$234$	0	0
$235$	− 1.15980e11i	− 2.48072i
$236$	0	0
$237$	2.03370e10i	0.418716i
$238$	0	0
$239$	4.05955e10	0.804798	0.402399	−	0.915464i	$-0.368176\pi$
$239$	4.05955e10	0.804798	0.402399	+	0.915464i	$0.368176\pi$
$240$	0	0
$241$	3.45208e10	0.659181	0.329590	−	0.944124i	$-0.393089\pi$
$241$	3.45208e10	0.659181	0.329590	+	0.944124i	$0.393089\pi$
$242$	0	0
$243$	− 4.71541e10i	− 0.867544i
$244$	0	0
$245$	− 2.11671e10i	− 0.375331i
$246$	0	0
$247$	1.45409e10	0.248573
$248$	0	0
$249$	−6.13995e10	−1.01220
$250$	0	0
$251$	7.83857e10i	1.24654i	0.782008	+	0.623268i	$0.214196\pi$
$251$	7.83857e10i	1.24654i	−0.782008	+	0.623268i	$0.785804\pi$
$252$	0	0
$253$	− 2.12231e10i	− 0.325661i
$254$	0	0
$255$	3.59552e10	0.532513
$256$	0	0
$257$	1.06489e11	1.52267	0.761337	−	0.648356i	$-0.224543\pi$
$257$	1.06489e11	1.52267	0.761337	+	0.648356i	$0.224543\pi$
$258$	0	0
$259$	3.72988e9i	0.0515046i
$260$	0	0
$261$	1.08947e10i	0.145322i
$262$	0	0
$263$	6.39865e8	0.00824684	0.00412342	−	0.999991i	$-0.498687\pi$
$263$	6.39865e8	0.00824684	0.00412342	+	0.999991i	$0.498687\pi$
$264$	0	0
$265$	1.25452e11	1.56268
$266$	0	0
$267$	7.75302e10i	0.933620i
$268$	0	0
$269$	9.60820e10i	1.11881i	0.828894	+	0.559406i	$0.188971\pi$
$269$	9.60820e10i	1.11881i	−0.828894	+	0.559406i	$0.811029\pi$
$270$	0	0
$271$	9.78435e10	1.10197	0.550986	−	0.834515i	$-0.314252\pi$
$271$	9.78435e10	1.10197	0.550986	+	0.834515i	$0.314252\pi$
$272$	0	0
$273$	2.07945e10	0.226578
$274$	0	0
$275$	− 1.18238e11i	− 1.24669i
$276$	0	0
$277$	− 1.61396e11i	− 1.64716i	−0.567202	−	0.823579i	$-0.691974\pi$
$277$	− 1.61396e11i	− 1.64716i	0.567202	−	0.823579i	$-0.308026\pi$
$278$	0	0
$279$	−4.17728e10	−0.412738
$280$	0	0
$281$	−2.71451e10	−0.259724	−0.129862	−	0.991532i	$-0.541453\pi$
$281$	−2.71451e10	−0.259724	−0.129862	+	0.991532i	$0.541453\pi$
$282$	0	0
$283$	2.80153e10i	0.259631i	0.991538	+	0.129815i	$0.0414385\pi$
$283$	2.80153e10i	0.259631i	−0.991538	+	0.129815i	$0.958562\pi$
$284$	0	0
$285$	− 1.27069e11i	− 1.14087i
$286$	0	0
$287$	−1.01599e11	−0.883940
$288$	0	0
$289$	−9.93980e10	−0.838180
$290$	0	0
$291$	− 1.08813e11i	− 0.889534i
$292$	0	0
$293$	2.49001e10i	0.197377i	0.995118	+	0.0986884i	$0.0314647\pi$
$293$	2.49001e10i	0.197377i	−0.995118	+	0.0986884i	$0.968535\pi$
$294$	0	0
$295$	−1.08409e10	−0.0833421
$296$	0	0
$297$	7.36861e10	0.549518
$298$	0	0
$299$	2.51594e10i	0.182046i
$300$	0	0
$301$	2.76226e11i	1.93961i
$302$	0	0
$303$	−6.25239e10	−0.426142
$304$	0	0
$305$	−1.64971e11	−1.09159
$306$	0	0
$307$	1.71738e11i	1.10343i	0.834034	+	0.551713i	$0.186026\pi$
$307$	1.71738e11i	1.10343i	−0.834034	+	0.551713i	$0.813974\pi$
$308$	0	0
$309$	6.37223e10i	0.397629i
$310$	0	0
$311$	1.24783e11	0.756371	0.378185	−	0.925730i	$-0.376548\pi$
$311$	1.24783e11	0.756371	0.378185	+	0.925730i	$0.376548\pi$
$312$	0	0
$313$	−2.95278e10	−0.173893	−0.0869463	−	0.996213i	$-0.527711\pi$
$313$	−2.95278e10	−0.173893	−0.0869463	+	0.996213i	$0.527711\pi$
$314$	0	0
$315$	1.72538e11i	0.987388i
$316$	0	0
$317$	− 1.20582e11i	− 0.670678i	−0.942098	−	0.335339i	$-0.891149\pi$
$317$	− 1.20582e11i	− 0.670678i	0.942098	−	0.335339i	$-0.108851\pi$
$318$	0	0
$319$	−2.84735e10	−0.153951
$320$	0	0
$321$	−1.54887e11	−0.814223
$322$	0	0
$323$	− 6.78187e10i	− 0.346687i
$324$	0	0
$325$	1.40168e11i	0.696906i
$326$	0	0
$327$	−2.20844e11	−1.06812
$328$	0	0
$329$	−3.12845e11	−1.47214
$330$	0	0
$331$	− 4.18670e11i	− 1.91711i	−0.284913	−	0.958553i	$-0.591965\pi$
$331$	− 4.18670e11i	− 1.91711i	0.284913	−	0.958553i	$-0.408035\pi$
$332$	0	0
$333$	− 5.13179e9i	− 0.0228702i
$334$	0	0
$335$	−1.49721e11	−0.649504
$336$	0	0
$337$	2.95362e11	1.24744	0.623721	−	0.781647i	$-0.285620\pi$
$337$	2.95362e11	1.24744	0.623721	+	0.781647i	$0.285620\pi$
$338$	0	0
$339$	5.46092e10i	0.224578i
$340$	0	0
$341$	− 1.09174e11i	− 0.437245i
$342$	0	0
$343$	2.24073e11	0.874111
$344$	0	0
$345$	2.19862e11	0.835533
$346$	0	0
$347$	1.55047e10i	0.0574093i	0.999588	+	0.0287046i	$0.00913822\pi$
$347$	1.55047e10i	0.0574093i	−0.999588	+	0.0287046i	$0.990862\pi$
$348$	0	0
$349$	− 3.06969e11i	− 1.10759i	−0.832652	−	0.553797i	$-0.813178\pi$
$349$	− 3.06969e11i	− 1.10759i	0.832652	−	0.553797i	$-0.186822\pi$
$350$	0	0
$351$	−8.73531e10	−0.307182
$352$	0	0
$353$	2.99671e11	1.02721	0.513604	−	0.858028i	$-0.328310\pi$
$353$	2.99671e11	1.02721	0.513604	+	0.858028i	$0.328310\pi$
$354$	0	0
$355$	7.10124e11i	2.37305i
$356$	0	0
$357$	− 9.69856e10i	− 0.316010i
$358$	0	0
$359$	4.04368e11	1.28485	0.642424	−	0.766349i	$-0.277929\pi$
$359$	4.04368e11	1.28485	0.642424	+	0.766349i	$0.277929\pi$
$360$	0	0
$361$	8.30100e10	0.257246
$362$	0	0
$363$	− 1.73855e11i	− 0.525542i
$364$	0	0
$365$	2.36861e11i	0.698516i
$366$	0	0
$367$	−5.07965e11	−1.46163	−0.730813	−	0.682578i	$-0.760859\pi$
$367$	−5.07965e11	−1.46163	−0.730813	+	0.682578i	$0.760859\pi$
$368$	0	0
$369$	1.39786e11	0.392506
$370$	0	0
$371$	− 3.38395e11i	− 0.927345i
$372$	0	0
$373$	6.94617e10i	0.185804i	0.995675	+	0.0929022i	$0.0296144\pi$
$373$	6.94617e10i	0.185804i	−0.995675	+	0.0929022i	$0.970386\pi$
$374$	0	0
$375$	7.17953e11	1.87480
$376$	0	0
$377$	3.37546e10	0.0860591
$378$	0	0
$379$	− 4.45070e11i	− 1.10803i	−0.832507	−	0.554015i	$-0.813095\pi$
$379$	− 4.45070e11i	− 1.10803i	0.832507	−	0.554015i	$-0.186905\pi$
$380$	0	0
$381$	− 2.42466e11i	− 0.589507i
$382$	0	0
$383$	4.50375e11	1.06950	0.534749	−	0.845011i	$-0.320406\pi$
$383$	4.50375e11	1.06950	0.534749	+	0.845011i	$0.320406\pi$
$384$	0	0
$385$	−4.50932e11	−1.04601
$386$	0	0
$387$	− 3.80047e11i	− 0.861268i
$388$	0	0
$389$	5.76192e11i	1.27583i	0.770106	+	0.637916i	$0.220203\pi$
$389$	5.76192e11i	1.27583i	−0.770106	+	0.637916i	$0.779797\pi$
$390$	0	0
$391$	1.17344e11	0.253901
$392$	0	0
$393$	1.58505e11	0.335179
$394$	0	0
$395$	5.22807e11i	1.08057i
$396$	0	0
$397$	− 3.22788e11i	− 0.652169i	−0.945341	−	0.326084i	$-0.894271\pi$
$397$	− 3.22788e11i	− 0.652169i	0.945341	−	0.326084i	$-0.105729\pi$
$398$	0	0
$399$	−3.42756e11	−0.677030
$400$	0	0
$401$	1.47838e11	0.285520	0.142760	−	0.989757i	$-0.454402\pi$
$401$	1.47838e11	0.285520	0.142760	+	0.989757i	$0.454402\pi$
$402$	0	0
$403$	1.29423e11i	0.244421i
$404$	0	0
$405$	2.75949e11i	0.509661i
$406$	0	0
$407$	1.34120e10	0.0242281
$408$	0	0
$409$	−7.24078e11	−1.27947	−0.639735	−	0.768595i	$-0.720956\pi$
$409$	−7.24078e11	−1.27947	−0.639735	+	0.768595i	$0.720956\pi$
$410$	0	0
$411$	6.42153e11i	1.11007i
$412$	0	0
$413$	2.92422e10i	0.0494578i
$414$	0	0
$415$	−1.57841e12	−2.61218
$416$	0	0
$417$	−2.47081e11	−0.400153
$418$	0	0
$419$	− 6.10316e11i	− 0.967368i	−0.875243	−	0.483684i	$-0.839298\pi$
$419$	− 6.10316e11i	− 0.967368i	0.875243	−	0.483684i	$-0.160702\pi$
$420$	0	0
$421$	− 7.16040e11i	− 1.11088i	−0.831556	−	0.555441i	$-0.812550\pi$
$421$	− 7.16040e11i	− 1.11088i	0.831556	−	0.555441i	$-0.187450\pi$
$422$	0	0
$423$	4.30430e11	0.653689
$424$	0	0
$425$	6.53744e11	0.971981
$426$	0	0
$427$	4.44995e11i	0.647783i
$428$	0	0
$429$	− 7.47735e10i	− 0.106584i
$430$	0	0
$431$	−4.78823e11	−0.668385	−0.334193	−	0.942505i	$-0.608464\pi$
$431$	−4.78823e11	−0.668385	−0.334193	+	0.942505i	$0.608464\pi$
$432$	0	0
$433$	−3.13312e11	−0.428334	−0.214167	−	0.976797i	$-0.568704\pi$
$433$	−3.13312e11	−0.428334	−0.214167	+	0.976797i	$0.568704\pi$
$434$	0	0
$435$	− 2.94972e11i	− 0.394984i
$436$	0	0
$437$	− 4.14703e11i	− 0.543965i
$438$	0	0
$439$	−6.95985e11	−0.894354	−0.447177	−	0.894445i	$-0.647571\pi$
$439$	−6.95985e11	−0.894354	−0.447177	+	0.894445i	$0.647571\pi$
$440$	0	0
$441$	7.85563e10	0.0989025
$442$	0	0
$443$	3.55482e11i	0.438531i	0.975665	+	0.219266i	$0.0703661\pi$
$443$	3.55482e11i	0.438531i	−0.975665	+	0.219266i	$0.929634\pi$
$444$	0	0
$445$	1.99308e12i	2.40938i
$446$	0	0
$447$	3.86391e11	0.457765
$448$	0	0
$449$	−8.15188e11	−0.946562	−0.473281	−	0.880911i	$-0.656931\pi$
$449$	−8.15188e11	−0.946562	−0.473281	+	0.880911i	$0.656931\pi$
$450$	0	0
$451$	3.65334e11i	0.415811i
$452$	0	0
$453$	4.09589e11i	0.456990i
$454$	0	0
$455$	5.34568e11	0.584725
$456$	0	0
$457$	−5.94548e11	−0.637623	−0.318811	−	0.947818i	$-0.603284\pi$
$457$	−5.94548e11	−0.637623	−0.318811	+	0.947818i	$0.603284\pi$
$458$	0	0
$459$	4.07414e11i	0.428429i
$460$	0	0
$461$	− 1.42516e12i	− 1.46964i	−0.678265	−	0.734818i	$-0.737268\pi$
$461$	− 1.42516e12i	− 1.46964i	0.678265	−	0.734818i	$-0.262732\pi$
$462$	0	0
$463$	1.04996e12	1.06183	0.530917	−	0.847424i	$-0.321847\pi$
$463$	1.04996e12	1.06183	0.530917	+	0.847424i	$0.321847\pi$
$464$	0	0
$465$	1.13099e12	1.12182
$466$	0	0
$467$	− 1.15656e11i	− 0.112523i	−0.998416	−	0.0562616i	$-0.982082\pi$
$467$	− 1.15656e11i	− 0.112523i	0.998416	−	0.0562616i	$-0.0179181\pi$
$468$	0	0
$469$	4.03859e11i	0.385436i
$470$	0	0
$471$	7.63065e11	0.714443
$472$	0	0
$473$	9.93261e11	0.912406
$474$	0	0
$475$	− 2.31039e12i	− 2.08240i
$476$	0	0
$477$	4.65583e11i	0.411779i
$478$	0	0
$479$	−1.37302e12	−1.19170	−0.595849	−	0.803096i	$-0.703184\pi$
$479$	−1.37302e12	−1.19170	−0.595849	+	0.803096i	$0.703184\pi$
$480$	0	0
$481$	−1.58996e10	−0.0135436
$482$	0	0
$483$	− 5.93056e11i	− 0.495831i
$484$	0	0
$485$	− 2.79727e12i	− 2.29561i
$486$	0	0
$487$	1.53458e12	1.23626	0.618131	−	0.786075i	$-0.287890\pi$
$487$	1.53458e12	1.23626	0.618131	+	0.786075i	$0.287890\pi$
$488$	0	0
$489$	−7.74990e11	−0.612923
$490$	0	0
$491$	− 5.12475e11i	− 0.397929i	−0.980007	−	0.198965i	$-0.936242\pi$
$491$	− 5.12475e11i	− 0.397929i	0.980007	−	0.198965i	$-0.0637579\pi$
$492$	0	0
$493$	− 1.57431e11i	− 0.120027i
$494$	0	0
$495$	6.20418e11	0.464473
$496$	0	0
$497$	1.91549e12	1.40824
$498$	0	0
$499$	2.45745e12i	1.77432i	0.461463	+	0.887159i	$0.347325\pi$
$499$	2.45745e12i	1.77432i	−0.461463	+	0.887159i	$0.652675\pi$
$500$	0	0
$501$	1.01657e12i	0.720887i
$502$	0	0
$503$	2.11765e11	0.147502	0.0737512	−	0.997277i	$-0.476503\pi$
$503$	2.11765e11	0.147502	0.0737512	+	0.997277i	$0.476503\pi$
$504$	0	0
$505$	−1.60731e12	−1.09974
$506$	0	0
$507$	− 9.76913e11i	− 0.656629i
$508$	0	0
$509$	2.27707e12i	1.50365i	0.659363	+	0.751824i	$0.270826\pi$
$509$	2.27707e12i	1.50365i	−0.659363	+	0.751824i	$0.729174\pi$
$510$	0	0
$511$	6.38911e11	0.414521
$512$	0	0
$513$	1.43984e12	0.917881
$514$	0	0
$515$	1.63812e12i	1.02616i
$516$	0	0
$517$	1.12494e12i	0.692501i
$518$	0	0
$519$	−1.90637e12	−1.15333
$520$	0	0
$521$	3.15742e12	1.87742	0.938712	−	0.344703i	$-0.112020\pi$
$521$	3.15742e12	1.87742	0.938712	+	0.344703i	$0.112020\pi$
$522$	0	0
$523$	6.70464e11i	0.391848i	0.980619	+	0.195924i	$0.0627706\pi$
$523$	6.70464e11i	0.391848i	−0.980619	+	0.195924i	$0.937229\pi$
$524$	0	0
$525$	− 3.30403e12i	− 1.89814i
$526$	0	0
$527$	6.03629e11	0.340896
$528$	0	0
$529$	−1.08361e12	−0.601621
$530$	0	0
$531$	− 4.02330e10i	− 0.0219613i
$532$	0	0
$533$	− 4.33094e11i	− 0.232440i
$534$	0	0
$535$	−3.98172e12	−2.10125
$536$	0	0
$537$	−1.73506e12	−0.900387
$538$	0	0
$539$	2.05308e11i	0.104775i
$540$	0	0
$541$	− 6.87645e11i	− 0.345125i	−0.984999	−	0.172562i	$-0.944795\pi$
$541$	− 6.87645e11i	− 0.345125i	0.984999	−	0.172562i	$-0.0552047\pi$
$542$	0	0
$543$	1.03860e12	0.512685
$544$	0	0
$545$	−5.67726e12	−2.75648
$546$	0	0
$547$	− 2.73174e12i	− 1.30466i	−0.757936	−	0.652328i	$-0.773792\pi$
$547$	− 2.73174e12i	− 1.30466i	0.757936	−	0.652328i	$-0.226208\pi$
$548$	0	0
$549$	− 6.12249e11i	− 0.287642i
$550$	0	0
$551$	−5.56377e11	−0.257150
$552$	0	0
$553$	1.41022e12	0.641245
$554$	0	0
$555$	1.38942e11i	0.0621608i
$556$	0	0
$557$	3.59464e12i	1.58236i	0.611580	+	0.791182i	$0.290534\pi$
$557$	3.59464e12i	1.58236i	−0.611580	+	0.791182i	$0.709466\pi$
$558$	0	0
$559$	−1.17749e12	−0.510038
$560$	0	0
$561$	−3.48744e11	−0.148653
$562$	0	0
$563$	− 4.60263e11i	− 0.193072i	−0.995330	−	0.0965358i	$-0.969224\pi$
$563$	− 4.60263e11i	− 0.193072i	0.995330	−	0.0965358i	$-0.0307762\pi$
$564$	0	0
$565$	1.40385e12i	0.579565i
$566$	0	0
$567$	7.44347e11	0.302449
$568$	0	0
$569$	−1.91911e12	−0.767527	−0.383764	−	0.923431i	$-0.625372\pi$
$569$	−1.91911e12	−0.767527	−0.383764	+	0.923431i	$0.625372\pi$
$570$	0	0
$571$	2.01838e12i	0.794586i	0.917692	+	0.397293i	$0.130050\pi$
$571$	2.01838e12i	0.794586i	−0.917692	+	0.397293i	$0.869950\pi$
$572$	0	0
$573$	− 9.18701e11i	− 0.356023i
$574$	0	0
$575$	3.99757e12	1.52507
$576$	0	0
$577$	−6.91957e11	−0.259889	−0.129945	−	0.991521i	$-0.541480\pi$
$577$	−6.91957e11	−0.259889	−0.129945	+	0.991521i	$0.541480\pi$
$578$	0	0
$579$	− 8.49572e10i	− 0.0314157i
$580$	0	0
$581$	4.25760e12i	1.55015i
$582$	0	0
$583$	−1.21681e12	−0.436229
$584$	0	0
$585$	−7.35490e11	−0.259642
$586$	0	0
$587$	1.99542e12i	0.693686i	0.937923	+	0.346843i	$0.112746\pi$
$587$	1.99542e12i	0.693686i	−0.937923	+	0.346843i	$0.887254\pi$
$588$	0	0
$589$	− 2.13328e12i	− 0.730347i
$590$	0	0
$591$	−2.13430e11	−0.0719633
$592$	0	0
$593$	5.19417e12	1.72492	0.862462	−	0.506121i	$-0.168921\pi$
$593$	5.19417e12	1.72492	0.862462	+	0.506121i	$0.168921\pi$
$594$	0	0
$595$	− 2.49323e12i	− 0.815521i
$596$	0	0
$597$	− 2.94289e11i	− 0.0948177i
$598$	0	0
$599$	−4.47443e12	−1.42010	−0.710048	−	0.704154i	$-0.751327\pi$
$599$	−4.47443e12	−1.42010	−0.710048	+	0.704154i	$0.751327\pi$
$600$	0	0
$601$	3.10552e12	0.970955	0.485477	−	0.874249i	$-0.338646\pi$
$601$	3.10552e12	0.970955	0.485477	+	0.874249i	$0.338646\pi$
$602$	0	0
$603$	− 5.55652e11i	− 0.171149i
$604$	0	0
$605$	− 4.46932e12i	− 1.35626i
$606$	0	0
$607$	3.79043e12	1.13329	0.566643	−	0.823964i	$-0.308242\pi$
$607$	3.79043e12	1.13329	0.566643	+	0.823964i	$0.308242\pi$
$608$	0	0
$609$	−7.95660e11	−0.234396
$610$	0	0
$611$	− 1.33358e12i	− 0.387111i
$612$	0	0
$613$	− 6.09780e12i	− 1.74422i	−0.489310	−	0.872110i	$-0.662751\pi$
$613$	− 6.09780e12i	− 1.74422i	0.489310	−	0.872110i	$-0.337249\pi$
$614$	0	0
$615$	−3.78470e12	−1.06682
$616$	0	0
$617$	−2.07665e12	−0.576872	−0.288436	−	0.957499i	$-0.593135\pi$
$617$	−2.07665e12	−0.576872	−0.288436	+	0.957499i	$0.593135\pi$
$618$	0	0
$619$	6.03694e12i	1.65276i	0.563115	+	0.826379i	$0.309603\pi$
$619$	6.03694e12i	1.65276i	−0.563115	+	0.826379i	$0.690397\pi$
$620$	0	0
$621$	2.49129e12i	0.672221i
$622$	0	0
$623$	5.37615e12	1.42980
$624$	0	0
$625$	9.23927e12	2.42202
$626$	0	0
$627$	1.23249e12i	0.318479i
$628$	0	0
$629$	7.41558e10i	0.0188893i
$630$	0	0
$631$	7.01879e12	1.76250	0.881252	−	0.472648i	$-0.156702\pi$
$631$	7.01879e12	1.76250	0.881252	+	0.472648i	$0.156702\pi$
$632$	0	0
$633$	3.18095e12	0.787482
$634$	0	0
$635$	− 6.23312e12i	− 1.52133i
$636$	0	0
$637$	− 2.43388e11i	− 0.0585695i
$638$	0	0
$639$	−2.63544e12	−0.625316
$640$	0	0
$641$	−6.65994e12	−1.55815	−0.779075	−	0.626931i	$-0.784311\pi$
$641$	−6.65994e12	−1.55815	−0.779075	+	0.626931i	$0.784311\pi$
$642$	0	0
$643$	− 3.60863e12i	− 0.832516i	−0.909247	−	0.416258i	$-0.863341\pi$
$643$	− 3.60863e12i	− 0.832516i	0.909247	−	0.416258i	$-0.136659\pi$
$644$	0	0
$645$	1.02897e13i	2.34091i
$646$	0	0
$647$	−3.26245e12	−0.731938	−0.365969	−	0.930627i	$-0.619262\pi$
$647$	−3.26245e12	−0.731938	−0.365969	+	0.930627i	$0.619262\pi$
$648$	0	0
$649$	1.05150e11	0.0232652
$650$	0	0
$651$	− 3.05075e12i	− 0.665720i
$652$	0	0
$653$	4.35965e11i	0.0938302i	0.998899	+	0.0469151i	$0.0149390\pi$
$653$	4.35965e11i	0.0938302i	−0.998899	+	0.0469151i	$0.985061\pi$
$654$	0	0
$655$	4.07471e12	0.864990
$656$	0	0
$657$	−8.79050e11	−0.184064
$658$	0	0
$659$	2.27883e12i	0.470683i	0.971913	+	0.235341i	$0.0756208\pi$
$659$	2.27883e12i	0.470683i	−0.971913	+	0.235341i	$0.924379\pi$
$660$	0	0
$661$	− 6.28607e12i	− 1.28077i	−0.768052	−	0.640387i	$-0.778774\pi$
$661$	− 6.28607e12i	− 1.28077i	0.768052	−	0.640387i	$-0.221226\pi$
$662$	0	0
$663$	4.13427e11	0.0830975
$664$	0	0
$665$	−8.81130e12	−1.74720
$666$	0	0
$667$	− 9.62674e11i	− 0.188327i
$668$	0	0
$669$	− 4.94824e12i	− 0.955065i
$670$	0	0
$671$	1.60012e12	0.304721
$672$	0	0
$673$	−1.30391e12	−0.245007	−0.122504	−	0.992468i	$-0.539092\pi$
$673$	−1.30391e12	−0.245007	−0.122504	+	0.992468i	$0.539092\pi$
$674$	0	0
$675$	1.38795e13i	2.57339i
$676$	0	0
$677$	− 3.77662e12i	− 0.690962i	−0.938426	−	0.345481i	$-0.887716\pi$
$677$	− 3.77662e12i	− 0.690962i	0.938426	−	0.345481i	$-0.112284\pi$
$678$	0	0
$679$	−7.54538e12	−1.36228
$680$	0	0
$681$	5.69939e12	1.01547
$682$	0	0
$683$	− 8.09351e12i	− 1.42313i	−0.702622	−	0.711563i	$-0.747987\pi$
$683$	− 8.09351e12i	− 1.42313i	0.702622	−	0.711563i	$-0.252013\pi$
$684$	0	0
$685$	1.65079e13i	2.86474i
$686$	0	0
$687$	3.35251e11	0.0574202
$688$	0	0
$689$	1.44250e12	0.243853
$690$	0	0
$691$	− 1.04215e12i	− 0.173891i	−0.996213	−	0.0869455i	$-0.972289\pi$
$691$	− 1.04215e12i	− 0.173891i	0.996213	−	0.0869455i	$-0.0277106\pi$
$692$	0	0
$693$	− 1.67352e12i	− 0.275633i
$694$	0	0
$695$	−6.35174e12	−1.03267
$696$	0	0
$697$	−2.01995e12	−0.324185
$698$	0	0
$699$	4.62221e12i	0.732322i
$700$	0	0
$701$	2.19669e12i	0.343587i	0.985133	+	0.171794i	$0.0549562\pi$
$701$	2.19669e12i	0.343587i	−0.985133	+	0.171794i	$0.945044\pi$
$702$	0	0
$703$	2.62073e11	0.0404692
$704$	0	0
$705$	−1.16538e13	−1.77672
$706$	0	0
$707$	4.33557e12i	0.652618i
$708$	0	0
$709$	7.02990e11i	0.104482i	0.998635	+	0.0522410i	$0.0166364\pi$
$709$	7.02990e11i	0.104482i	−0.998635	+	0.0522410i	$0.983364\pi$
$710$	0	0
$711$	−1.94026e12	−0.284739
$712$	0	0
$713$	3.69112e12	0.534879
$714$	0	0
$715$	− 1.92222e12i	− 0.275058i
$716$	0	0
$717$	− 4.07909e12i	− 0.576404i
$718$	0	0
$719$	−1.18349e13	−1.65152	−0.825759	−	0.564022i	$-0.809253\pi$
$719$	−1.18349e13	−1.65152	−0.825759	+	0.564022i	$0.809253\pi$
$720$	0	0
$721$	4.41867e12	0.608952
$722$	0	0
$723$	− 3.46870e12i	− 0.472111i
$724$	0	0
$725$	− 5.36325e12i	− 0.720953i
$726$	0	0
$727$	4.66170e12	0.618927	0.309464	−	0.950911i	$-0.399850\pi$
$727$	4.66170e12	0.618927	0.309464	+	0.950911i	$0.399850\pi$
$728$	0	0
$729$	−6.84083e12	−0.897088
$730$	0	0
$731$	5.49179e12i	0.711354i
$732$	0	0
$733$	− 1.16839e13i	− 1.49493i	−0.664300	−	0.747466i	$-0.731270\pi$
$733$	− 1.16839e13i	− 1.49493i	0.664300	−	0.747466i	$-0.268730\pi$
$734$	0	0
$735$	−2.12690e12	−0.268815
$736$	0	0
$737$	1.45221e12	0.181311
$738$	0	0
$739$	− 1.25228e13i	− 1.54455i	−0.635290	−	0.772274i	$-0.719119\pi$
$739$	− 1.25228e13i	− 1.54455i	0.635290	−	0.772274i	$-0.280881\pi$
$740$	0	0
$741$	− 1.46109e12i	− 0.178031i
$742$	0	0
$743$	1.10289e13	1.32765	0.663823	−	0.747890i	$-0.268933\pi$
$743$	1.10289e13	1.32765	0.663823	+	0.747890i	$0.268933\pi$
$744$	0	0
$745$	9.93301e12	1.18135
$746$	0	0
$747$	− 5.85785e12i	− 0.688328i
$748$	0	0
$749$	1.07403e13i	1.24695i
$750$	0	0
$751$	6.45522e12	0.740511	0.370255	−	0.928930i	$-0.379270\pi$
$751$	6.45522e12	0.740511	0.370255	+	0.928930i	$0.379270\pi$
$752$	0	0
$753$	7.87631e12	0.892781
$754$	0	0
$755$	1.05294e13i	1.17935i
$756$	0	0
$757$	6.48010e12i	0.717217i	0.933488	+	0.358608i	$0.116749\pi$
$757$	6.48010e12i	0.717217i	−0.933488	+	0.358608i	$0.883251\pi$
$758$	0	0
$759$	−2.13253e12	−0.233242
$760$	0	0
$761$	2.62886e12	0.284142	0.142071	−	0.989856i	$-0.454624\pi$
$761$	2.62886e12	0.284142	0.142071	+	0.989856i	$0.454624\pi$
$762$	0	0
$763$	1.53139e13i	1.63578i
$764$	0	0
$765$	3.43032e12i	0.362125i
$766$	0	0
$767$	−1.24652e11	−0.0130053
$768$	0	0
$769$	−1.43517e13	−1.47991	−0.739955	−	0.672656i	$-0.765153\pi$
$769$	−1.43517e13	−1.47991	−0.739955	+	0.672656i	$0.765153\pi$
$770$	0	0
$771$	− 1.07002e13i	− 1.09055i
$772$	0	0
$773$	6.56407e12i	0.661250i	0.943762	+	0.330625i	$0.107260\pi$
$773$	6.56407e12i	0.661250i	−0.943762	+	0.330625i	$0.892740\pi$
$774$	0	0
$775$	2.05640e13	2.04762
$776$	0	0
$777$	3.74784e11	0.0368881
$778$	0	0
$779$	7.13870e12i	0.694545i
$780$	0	0
$781$	− 6.88778e12i	− 0.662444i
$782$	0	0
$783$	3.34238e12	0.317781
$784$	0	0
$785$	1.96162e13	1.84375
$786$	0	0
$787$	− 1.96637e13i	− 1.82717i	−0.406649	−	0.913584i	$-0.633303\pi$
$787$	− 1.96637e13i	− 1.82717i	0.406649	−	0.913584i	$-0.366697\pi$
$788$	0	0
$789$	− 6.42945e10i	− 0.00590646i
$790$	0	0
$791$	3.78675e12	0.343932
$792$	0	0
$793$	−1.89691e12	−0.170340
$794$	0	0
$795$	− 1.26056e13i	− 1.11921i
$796$	0	0
$797$	− 2.05859e12i	− 0.180721i	−0.995909	−	0.0903603i	$-0.971198\pi$
$797$	− 2.05859e12i	− 0.180721i	0.995909	−	0.0903603i	$-0.0288019\pi$
$798$	0	0
$799$	−6.21983e12	−0.539906
$800$	0	0
$801$	−7.39681e12	−0.634890
$802$	0	0
$803$	− 2.29741e12i	− 0.194993i
$804$	0	0
$805$	− 1.52458e13i	− 1.27958i
$806$	0	0
$807$	9.65446e12	0.801304
$808$	0	0
$809$	−9.44339e12	−0.775103	−0.387552	−	0.921848i	$-0.626679\pi$
$809$	−9.44339e12	−0.775103	−0.387552	+	0.921848i	$0.626679\pi$
$810$	0	0
$811$	3.40891e12i	0.276708i	0.990383	+	0.138354i	$0.0441812\pi$
$811$	3.40891e12i	0.276708i	−0.990383	+	0.138354i	$0.955819\pi$
$812$	0	0
$813$	− 9.83146e12i	− 0.789242i
$814$	0	0
$815$	−1.99228e13	−1.58176
$816$	0	0
$817$	1.94085e13	1.52403
$818$	0	0
$819$	1.98391e12i	0.154080i
$820$	0	0
$821$	− 1.44900e12i	− 0.111307i	−0.998450	−	0.0556535i	$-0.982276\pi$
$821$	− 1.44900e12i	− 0.111307i	0.998450	−	0.0556535i	$-0.0177242\pi$
$822$	0	0
$823$	−9.50599e12	−0.722267	−0.361134	−	0.932514i	$-0.617610\pi$
$823$	−9.50599e12	−0.722267	−0.361134	+	0.932514i	$0.617610\pi$
$824$	0	0
$825$	−1.18807e13	−0.892894
$826$	0	0
$827$	1.32590e13i	0.985683i	0.870119	+	0.492842i	$0.164042\pi$
$827$	1.32590e13i	0.985683i	−0.870119	+	0.492842i	$0.835958\pi$
$828$	0	0
$829$	1.63326e13i	1.20104i	0.799608	+	0.600522i	$0.205040\pi$
$829$	1.63326e13i	1.20104i	−0.799608	+	0.600522i	$0.794960\pi$
$830$	0	0
$831$	−1.62173e13	−1.17971
$832$	0	0
$833$	−1.13516e12	−0.0816873
$834$	0	0
$835$	2.61331e13i	1.86038i
$836$	0	0
$837$	1.28155e13i	0.902549i
$838$	0	0
$839$	−2.23005e12	−0.155377	−0.0776884	−	0.996978i	$-0.524754\pi$
$839$	−2.23005e12	−0.155377	−0.0776884	+	0.996978i	$0.524754\pi$
$840$	0	0
$841$	1.32156e13	0.910972
$842$	0	0
$843$	2.72758e12i	0.186017i
$844$	0	0
$845$	− 2.51137e13i	− 1.69455i
$846$	0	0
$847$	−1.20556e13	−0.804846
$848$	0	0
$849$	2.81501e12	0.185950
$850$	0	0
$851$	4.53454e11i	0.0296381i
$852$	0	0
$853$	2.83205e13i	1.83160i	0.401633	+	0.915801i	$0.368443\pi$
$853$	2.83205e13i	1.83160i	−0.401633	+	0.915801i	$0.631557\pi$
$854$	0	0
$855$	1.21231e13	0.775828
$856$	0	0
$857$	−6.76700e12	−0.428531	−0.214266	−	0.976775i	$-0.568736\pi$
$857$	−6.76700e12	−0.428531	−0.214266	+	0.976775i	$0.568736\pi$
$858$	0	0
$859$	− 2.30991e12i	− 0.144753i	−0.997377	−	0.0723763i	$-0.976942\pi$
$859$	− 2.30991e12i	− 0.144753i	0.997377	−	0.0723763i	$-0.0230582\pi$
$860$	0	0
$861$	1.02089e13i	0.633087i
$862$	0	0
$863$	1.05303e13	0.646237	0.323119	−	0.946358i	$-0.395269\pi$
$863$	1.05303e13	0.646237	0.323119	+	0.946358i	$0.395269\pi$
$864$	0	0
$865$	−4.90074e13	−2.97638
$866$	0	0
$867$	9.98766e12i	0.600313i
$868$	0	0
$869$	− 5.07091e12i	− 0.301646i
$870$	0	0
$871$	−1.72156e12	−0.101354
$872$	0	0
$873$	1.03814e13	0.604909
$874$	0	0
$875$	− 4.97848e13i	− 2.87118i
$876$	0	0
$877$	− 1.37761e13i	− 0.786373i	−0.919459	−	0.393186i	$-0.871373\pi$
$877$	− 1.37761e13i	− 0.786373i	0.919459	−	0.393186i	$-0.128627\pi$
$878$	0	0
$879$	2.50199e12	0.141363
$880$	0	0
$881$	−1.13133e13	−0.632698	−0.316349	−	0.948643i	$-0.602457\pi$
$881$	−1.13133e13	−0.632698	−0.316349	+	0.948643i	$0.602457\pi$
$882$	0	0
$883$	3.91577e12i	0.216768i	0.994109	+	0.108384i	$0.0345675\pi$
$883$	3.91577e12i	0.216768i	−0.994109	+	0.108384i	$0.965432\pi$
$884$	0	0
$885$	1.08930e12i	0.0596904i
$886$	0	0
$887$	1.62707e12	0.0882573	0.0441287	−	0.999026i	$-0.485949\pi$
$887$	1.62707e12	0.0882573	0.0441287	+	0.999026i	$0.485949\pi$
$888$	0	0
$889$	−1.68133e13	−0.902805
$890$	0	0
$891$	− 2.67654e12i	− 0.142274i
$892$	0	0
$893$	2.19815e13i	1.15671i
$894$	0	0
$895$	−4.46034e13	−2.32362
$896$	0	0
$897$	2.52806e12	0.130383
$898$	0	0
$899$	− 4.95211e12i	− 0.252855i
$900$	0	0
$901$	− 6.72780e12i	− 0.340104i
$902$	0	0
$903$	2.77556e13	1.38917
$904$	0	0
$905$	2.66995e13	1.32308
$906$	0	0
$907$	7.72659e12i	0.379101i	0.981871	+	0.189551i	$0.0607031\pi$
$907$	7.72659e12i	0.379101i	−0.981871	+	0.189551i	$0.939297\pi$
$908$	0	0
$909$	− 5.96513e12i	− 0.289789i
$910$	0	0
$911$	1.47051e13	0.707350	0.353675	−	0.935368i	$-0.384932\pi$
$911$	1.47051e13	0.707350	0.353675	+	0.935368i	$0.384932\pi$
$912$	0	0
$913$	1.53096e13	0.729198
$914$	0	0
$915$	1.65766e13i	0.781807i
$916$	0	0
$917$	− 1.09911e13i	− 0.513312i
$918$	0	0
$919$	3.87199e13	1.79066	0.895332	−	0.445399i	$-0.146938\pi$
$919$	3.87199e13	1.79066	0.895332	+	0.445399i	$0.146938\pi$
$920$	0	0
$921$	1.72565e13	0.790285
$922$	0	0
$923$	8.16529e12i	0.370309i
$924$	0	0
$925$	2.52628e12i	0.113460i
$926$	0	0
$927$	−6.07946e12	−0.270400
$928$	0	0
$929$	7.75866e12	0.341756	0.170878	−	0.985292i	$-0.445340\pi$
$929$	7.75866e12	0.341756	0.170878	+	0.985292i	$0.445340\pi$
$930$	0	0
$931$	4.01176e12i	0.175010i
$932$	0	0
$933$	− 1.25384e13i	− 0.541720i
$934$	0	0
$935$	−8.96521e12	−0.383626
$936$	0	0
$937$	−1.60112e13	−0.678573	−0.339286	−	0.940683i	$-0.610186\pi$
$937$	−1.60112e13	−0.678573	−0.339286	+	0.940683i	$0.610186\pi$
$938$	0	0
$939$	2.96699e12i	0.124543i
$940$	0	0
$941$	− 1.55404e13i	− 0.646115i	−0.946379	−	0.323057i	$-0.895289\pi$
$941$	− 1.55404e13i	− 0.646115i	0.946379	−	0.323057i	$-0.104711\pi$
$942$	0	0
$943$	−1.23518e13	−0.508659
$944$	0	0
$945$	5.29330e13	2.15915
$946$	0	0
$947$	− 1.38291e13i	− 0.558750i	−0.960182	−	0.279375i	$-0.909873\pi$
$947$	− 1.38291e13i	− 0.558750i	0.960182	−	0.279375i	$-0.0901273\pi$
$948$	0	0
$949$	2.72353e12i	0.109002i
$950$	0	0
$951$	−1.21162e13	−0.480346
$952$	0	0
$953$	−1.62441e12	−0.0637935	−0.0318967	−	0.999491i	$-0.510155\pi$
$953$	−1.62441e12	−0.0637935	−0.0318967	+	0.999491i	$0.510155\pi$
$954$	0	0
$955$	− 2.36172e13i	− 0.918784i
$956$	0	0
$957$	2.86105e12i	0.110261i
$958$	0	0
$959$	4.45285e13	1.70002
$960$	0	0
$961$	−7.45207e12	−0.281852
$962$	0	0
$963$	− 1.47771e13i	− 0.553696i
$964$	0	0
$965$	− 2.18401e12i	− 0.0810740i
$966$	0	0
$967$	−2.99781e13	−1.10252	−0.551259	−	0.834334i	$-0.685852\pi$
$967$	−2.99781e13	−1.10252	−0.551259	+	0.834334i	$0.685852\pi$
$968$	0	0
$969$	−6.81452e12	−0.248301
$970$	0	0
$971$	3.61921e13i	1.30655i	0.757119	+	0.653277i	$0.226606\pi$
$971$	3.61921e13i	1.30655i	−0.757119	+	0.653277i	$0.773394\pi$
$972$	0	0
$973$	1.71332e13i	0.612818i
$974$	0	0
$975$	1.40843e13	0.499131
$976$	0	0
$977$	2.29688e13	0.806515	0.403258	−	0.915087i	$-0.367878\pi$
$977$	2.29688e13	0.806515	0.403258	+	0.915087i	$0.367878\pi$
$978$	0	0
$979$	− 1.93317e13i	− 0.672586i
$980$	0	0
$981$	− 2.10697e13i	− 0.726353i
$982$	0	0
$983$	2.02509e13	0.691758	0.345879	−	0.938279i	$-0.387581\pi$
$983$	2.02509e13	0.691758	0.345879	+	0.938279i	$0.387581\pi$
$984$	0	0
$985$	−5.48667e12	−0.185715
$986$	0	0
$987$	3.14351e13i	1.05436i
$988$	0	0
$989$	3.35817e13i	1.11614i
$990$	0	0
$991$	1.63486e12	0.0538456	0.0269228	−	0.999638i	$-0.491429\pi$
$991$	1.63486e12	0.0538456	0.0269228	+	0.999638i	$0.491429\pi$
$992$	0	0
$993$	−4.20686e13	−1.37305
$994$	0	0
$995$	− 7.56533e12i	− 0.244694i
$996$	0	0
$997$	− 9.91675e12i	− 0.317864i	−0.987290	−	0.158932i	$-0.949195\pi$
$997$	− 9.91675e12i	− 0.317864i	0.987290	−	0.158932i	$-0.0508050\pi$
$998$	0	0
$999$	−1.57438e12	−0.0500110

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	32.10.b.a.17.3		8
3.2	odd	2		288.10.d.b.145.8		8
4.3	odd	2		8.10.b.a.5.1	✓	8
8.3	odd	2		8.10.b.a.5.2	yes	8
8.5	even	2	inner	32.10.b.a.17.6		8
12.11	even	2		72.10.d.b.37.8		8
16.3	odd	4		256.10.a.p.1.3		8
16.5	even	4		256.10.a.s.1.3		8
16.11	odd	4		256.10.a.p.1.6		8
16.13	even	4		256.10.a.s.1.6		8
24.5	odd	2		288.10.d.b.145.1		8
24.11	even	2		72.10.d.b.37.7		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8.10.b.a.5.1	✓	8	4.3	odd	2
8.10.b.a.5.2	yes	8	8.3	odd	2
32.10.b.a.17.3		8	1.1	even	1	trivial
32.10.b.a.17.6		8	8.5	even	2	inner
72.10.d.b.37.7		8	24.11	even	2
72.10.d.b.37.8		8	12.11	even	2
256.10.a.p.1.3		8	16.3	odd	4
256.10.a.p.1.6		8	16.11	odd	4
256.10.a.s.1.3		8	16.5	even	4
256.10.a.s.1.6		8	16.13	even	4
288.10.d.b.145.1		8	24.5	odd	2
288.10.d.b.145.8		8	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 32 = 2^{5} \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	32.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-100.481i q^{3} -2583.09i q^{5} -6967.65 q^{7} +9586.48 q^{9} +O(q^{10})\)	\(q-100.481i q^{3} -2583.09i q^{5} -6967.65 q^{7} +9586.48 q^{9} +25054.5i q^{11} -29701.4i q^{13} -259553. q^{15} -138527. q^{17} +489569. i q^{19} +700119. i q^{21} -847079. q^{23} -4.71924e6 q^{25} -2.94104e6i q^{27} +1.13646e6i q^{29} -4.35747e6 q^{31} +2.51751e6 q^{33} +1.79981e7i q^{35} -535315. i q^{37} -2.98444e6 q^{39} +1.45816e7 q^{41} -3.96441e7i q^{43} -2.47628e7i q^{45} +4.48997e7 q^{47} +8.19448e6 q^{49} +1.39194e7i q^{51} +4.85666e7i q^{53} +6.47180e7 q^{55} +4.91926e7 q^{57} -4.19685e6i q^{59} -6.38659e7i q^{61} -6.67952e7 q^{63} -7.67215e7 q^{65} -5.79621e7i q^{67} +8.51157e7i q^{69} -2.74912e8 q^{71} -9.16969e7 q^{73} +4.74196e8i q^{75} -1.74571e8i q^{77} -2.02396e8 q^{79} -1.06829e8 q^{81} -6.11053e8i q^{83} +3.57829e8i q^{85} +1.14193e8 q^{87} -7.71588e8 q^{89} +2.06949e8i q^{91} +4.37845e8i q^{93} +1.26460e9 q^{95} +1.08292e9 q^{97} +2.40184e8i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4800 q^{7} - 39368 q^{9}+O(q^{10}) \) 8 * q - 4800 * q^7 - 39368 * q^9	\( 8 q - 4800 q^{7} - 39368 q^{9} + 163136 q^{15} - 102000 q^{17} - 3412032 q^{23} - 2423384 q^{25} - 803584 q^{31} + 58272 q^{33} + 17590208 q^{39} - 2180784 q^{41} - 7432320 q^{47} + 24436680 q^{49} - 7056832 q^{55} + 134003744 q^{57} + 223198400 q^{63} - 146501760 q^{65} - 560234688 q^{71} - 523987120 q^{73} + 248943744 q^{79} + 231960296 q^{81} - 540527424 q^{87} + 744827856 q^{89} + 1465245504 q^{95} - 9932784 q^{97}+O(q^{100}) \) 8 * q - 4800 * q^7 - 39368 * q^9 + 163136 * q^15 - 102000 * q^17 - 3412032 * q^23 - 2423384 * q^25 - 803584 * q^31 + 58272 * q^33 + 17590208 * q^39 - 2180784 * q^41 - 7432320 * q^47 + 24436680 * q^49 - 7056832 * q^55 + 134003744 * q^57 + 223198400 * q^63 - 146501760 * q^65 - 560234688 * q^71 - 523987120 * q^73 + 248943744 * q^79 + 231960296 * q^81 - 540527424 * q^87 + 744827856 * q^89 + 1465245504 * q^95 - 9932784 * q^97

Introduction

L-functions

Modular forms

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Database

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