LMFDB - Embedded newform 24.8.f.a.11.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [24,8,Mod(11,24)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(24, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("24.11");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 24 = 2^{3} \cdot 3 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	24.f (of order $2$, degree $1$, 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:	$7.49724061162$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-2}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 2 $ x^2 + 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			11.2
Root			$1.41421i$ of defining polynomial
Character	$\chi$	$=$	24.11
Dual form			24.8.f.a.11.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+11.3137i q^{2} +(-43.0000 + 18.3848i) q^{3} -128.000 q^{4} +(-208.000 - 486.489i) q^{6} -1448.15i q^{8} +(1511.00 - 1581.09i) q^{9} +O(q^{10})$	\(q+11.3137i q^{2} +(-43.0000 + 18.3848i) q^{3} -128.000 q^{4} +(-208.000 - 486.489i) q^{6} -1448.15i q^{8} +(1511.00 - 1581.09i) q^{9} -511.945i q^{11} +(5504.00 - 2353.25i) q^{12} +16384.0 q^{16} -33901.5i q^{17} +(17888.0 + 17095.0i) q^{18} -59722.0 q^{19} +5792.00 q^{22} +(26624.0 + 62270.7i) q^{24} -78125.0 q^{25} +(-35905.0 + 95766.3i) q^{27} +185364. i q^{32} +(9412.00 + 22013.6i) q^{33} +383552. q^{34} +(-193408. + 202380. i) q^{36} -675677. i q^{38} -850237. i q^{41} -220510. q^{43} +65529.0i q^{44} +(-704512. + 301216. i) q^{48} +823543. q^{49} -883883. i q^{50} +(623272. + 1.45777e6i) q^{51} +(-1.08347e6 - 406219. i) q^{54} +(2.56805e6 - 1.09798e6i) q^{57} +2.98191e6i q^{59} -2.09715e6 q^{64} +(-249056. + 106485. i) q^{66} -3.85130e6 q^{67} +4.33940e6i q^{68} +(-2.28966e6 - 2.18816e6i) q^{72} -4.86561e6 q^{73} +(3.35938e6 - 1.43631e6i) q^{75} +7.64442e6 q^{76} +(-216727. - 4.77806e6i) q^{81} +9.61933e6 q^{82} -9.24221e6i q^{83} -2.49479e6i q^{86} -741376. q^{88} +1.12648e7i q^{89} +(-3.40787e6 - 7.97064e6i) q^{96} -9.93889e6 q^{97} +9.31733e6i q^{98} +(-809432. - 773549. i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 86 q^{3} - 256 q^{4} - 416 q^{6} + 3022 q^{9}+O(q^{10}) $ 2 * q - 86 * q^3 - 256 * q^4 - 416 * q^6 + 3022 * q^9	$ 2 q - 86 q^{3} - 256 q^{4} - 416 q^{6} + 3022 q^{9} + 11008 q^{12} + 32768 q^{16} + 35776 q^{18} - 119444 q^{19} + 11584 q^{22} + 53248 q^{24} - 156250 q^{25} - 71810 q^{27} + 18824 q^{33} + 767104 q^{34} - 386816 q^{36} - 441020 q^{43} - 1409024 q^{48} + 1647086 q^{49} + 1246544 q^{51} - 2166944 q^{54} + 5136092 q^{57} - 4194304 q^{64} - 498112 q^{66} - 7702604 q^{67} - 4579328 q^{72} - 9731228 q^{73} + 6718750 q^{75} + 15288832 q^{76} - 433454 q^{81} + 19238656 q^{82} - 1482752 q^{88} - 6815744 q^{96} - 19877780 q^{97} - 1618864 q^{99}+O(q^{100}) $ 2 * q - 86 * q^3 - 256 * q^4 - 416 * q^6 + 3022 * q^9 + 11008 * q^12 + 32768 * q^16 + 35776 * q^18 - 119444 * q^19 + 11584 * q^22 + 53248 * q^24 - 156250 * q^25 - 71810 * q^27 + 18824 * q^33 + 767104 * q^34 - 386816 * q^36 - 441020 * q^43 - 1409024 * q^48 + 1647086 * q^49 + 1246544 * q^51 - 2166944 * q^54 + 5136092 * q^57 - 4194304 * q^64 - 498112 * q^66 - 7702604 * q^67 - 4579328 * q^72 - 9731228 * q^73 + 6718750 * q^75 + 15288832 * q^76 - 433454 * q^81 + 19238656 * q^82 - 1482752 * q^88 - 6815744 * q^96 - 19877780 * q^97 - 1618864 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/24\mathbb{Z}\right)^\times$.

$n$	$7$	$13$	$17$
$\chi(n)$	$-1$	$-1$	$-1$

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 $

$2$			11.3137i			1.00000i
$2$			11.3137i			1.00000i
$3$	−43.0000	+	18.3848i	−0.919484	+	0.393128i
$3$	−43.0000	+	18.3848i	−0.919484	+	0.393128i
$4$	−128.000			−1.00000
$5$		0			0			−	1.00000i	$-0.5\pi$
$5$		0			0				1.00000i	$0.5\pi$
$6$	−208.000	−	486.489i	−0.393128	−	0.919484i
$7$		0			0		1.00000			$0$
$7$		0			0		−1.00000			$\pi$
$8$		−	1448.15i		−	1.00000i
$9$	1511.00	−	1581.09i	0.690901	−	0.722950i
$10$		0			0
$11$		−	511.945i		−	0.115971i	−0.998317	−	0.0579855i	$-0.981532\pi$
$11$		−	511.945i		−	0.115971i	0.998317	−	0.0579855i	$-0.0184677\pi$
$12$	5504.00	−	2353.25i	0.919484	−	0.393128i
$13$		0			0		1.00000			$0$
$13$		0			0		−1.00000			$\pi$
$14$		0			0
$15$		0			0
$16$	16384.0			1.00000
$17$		−	33901.5i		−	1.67359i	−0.547519	−	0.836793i	$-0.684428\pi$
$17$		−	33901.5i		−	1.67359i	0.547519	−	0.836793i	$-0.315572\pi$
$18$	17888.0	+	17095.0i	0.722950	+	0.690901i
$19$	−59722.0			−1.99755			−0.998773	−	0.0495250i	$-0.984229\pi$
$19$	−59722.0			−1.99755			−0.998773	+	0.0495250i	$0.984229\pi$
$20$		0			0
$21$		0			0
$22$	5792.00			0.115971
$23$		0			0			−	1.00000i	$-0.5\pi$
$23$		0			0				1.00000i	$0.5\pi$
$24$	26624.0	+	62270.7i	0.393128	+	0.919484i
$25$	−78125.0			−1.00000
$26$		0			0
$27$	−35905.0	+	95766.3i	−0.351060	+	0.936353i
$28$		0			0
$29$		0			0			−	1.00000i	$-0.5\pi$
$29$		0			0				1.00000i	$0.5\pi$
$30$		0			0
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$			185364.i			1.00000i
$33$	9412.00	+	22013.6i	0.0455914	+	0.106633i
$34$	383552.			1.67359
$35$		0			0
$36$	−193408.	+	202380.i	−0.690901	+	0.722950i
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$		−	675677.i		−	1.99755i
$39$		0			0
$40$		0			0
$41$		−	850237.i		−	1.92662i	−0.268390	−	0.963310i	$-0.586491\pi$
$41$		−	850237.i		−	1.92662i	0.268390	−	0.963310i	$-0.413509\pi$
$42$		0			0
$43$	−220510.			−0.422950			−0.211475	−	0.977383i	$-0.567827\pi$
$43$	−220510.			−0.422950			−0.211475	+	0.977383i	$0.567827\pi$
$44$			65529.0i			0.115971i
$45$		0			0
$46$		0			0
$47$		0			0			−	1.00000i	$-0.5\pi$
$47$		0			0				1.00000i	$0.5\pi$
$48$	−704512.	+	301216.i	−0.919484	+	0.393128i
$49$	823543.			1.00000
$50$		−	883883.i		−	1.00000i
$51$	623272.	+	1.45777e6i	0.657934	+	1.53884i
$52$		0			0
$53$		0			0			−	1.00000i	$-0.5\pi$
$53$		0			0				1.00000i	$0.5\pi$
$54$	−1.08347e6	−	406219.i	−0.936353	−	0.351060i
$55$		0			0
$56$		0			0
$57$	2.56805e6	−	1.09798e6i	1.83671	−	0.785291i
$58$		0			0
$59$			2.98191e6i			1.89022i	0.326750	+	0.945111i	$0.394047\pi$
$59$			2.98191e6i			1.89022i	−0.326750	+	0.945111i	$0.605953\pi$
$60$		0			0
$61$		0			0		1.00000			$0$
$61$		0			0		−1.00000			$\pi$
$62$		0			0
$63$		0			0
$64$	−2.09715e6			−1.00000
$65$		0			0
$66$	−249056.	+	106485.i	−0.106633	+	0.0455914i
$67$	−3.85130e6			−1.56439			−0.782196	−	0.623032i	$-0.785901\pi$
$67$	−3.85130e6			−1.56439			−0.782196	+	0.623032i	$0.785901\pi$
$68$			4.33940e6i			1.67359i
$69$		0			0
$70$		0			0
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$	−2.28966e6	−	2.18816e6i	−0.722950	−	0.690901i
$73$	−4.86561e6			−1.46389			−0.731944	−	0.681365i	$-0.761387\pi$
$73$	−4.86561e6			−1.46389			−0.731944	+	0.681365i	$0.761387\pi$
$74$		0			0
$75$	3.35938e6	−	1.43631e6i	0.919484	−	0.393128i
$76$	7.64442e6			1.99755
$77$		0			0
$78$		0			0
$79$		0			0		1.00000			$0$
$79$		0			0		−1.00000			$\pi$
$80$		0			0
$81$	−216727.	−	4.77806e6i	−0.0453122	−	0.998973i
$82$	9.61933e6			1.92662
$83$		−	9.24221e6i		−	1.77420i	−0.461578	−	0.887099i	$-0.652717\pi$
$83$		−	9.24221e6i		−	1.77420i	0.461578	−	0.887099i	$-0.347283\pi$
$84$		0			0
$85$		0			0
$86$		−	2.49479e6i		−	0.422950i
$87$		0			0
$88$	−741376.			−0.115971
$89$			1.12648e7i			1.69379i	0.531760	+	0.846895i	$0.321531\pi$
$89$			1.12648e7i			1.69379i	−0.531760	+	0.846895i	$0.678469\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$	−3.40787e6	−	7.97064e6i	−0.393128	−	0.919484i
$97$	−9.93889e6			−1.10570			−0.552849	−	0.833281i	$-0.686460\pi$
$97$	−9.93889e6			−1.10570			−0.552849	+	0.833281i	$0.686460\pi$
$98$			9.31733e6i			1.00000i
$99$	−809432.	−	773549.i	−0.0838411	−	0.0801244i
$100$	1.00000e7			1.00000
$101$		0			0			−	1.00000i	$-0.5\pi$
$101$		0			0				1.00000i	$0.5\pi$
$102$	−1.64927e7	+	7.05152e6i	−1.53884	+	0.657934i
$103$		0			0		1.00000			$0$
$103$		0			0		−1.00000			$\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	1.19039e7i		−	0.939389i	−0.882829	−	0.469695i	$-0.844364\pi$
$107$		−	1.19039e7i		−	0.939389i	0.882829	−	0.469695i	$-0.155636\pi$
$108$	4.59584e6	−	1.22581e7i	0.351060	−	0.936353i
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			2.17130e7i			1.41562i	0.706405	+	0.707808i	$0.250316\pi$
$113$			2.17130e7i			1.41562i	−0.706405	+	0.707808i	$0.749684\pi$
$114$	1.24222e7	+	2.90541e7i	0.785291	+	1.83671i
$115$		0			0
$116$		0			0
$117$		0			0
$118$	−3.37365e7			−1.89022
$119$		0			0
$120$		0			0
$121$	1.92251e7			0.986551
$122$		0			0
$123$	1.56314e7	+	3.65602e7i	0.757409	+	1.77150i
$124$		0			0
$125$		0			0
$126$		0			0
$127$		0			0		1.00000			$0$
$127$		0			0		−1.00000			$\pi$
$128$		−	2.37266e7i		−	1.00000i
$129$	9.48193e6	−	4.05403e6i	0.388895	−	0.166273i
$130$		0			0
$131$		−	5.13948e7i		−	1.99742i	−0.0507560	−	0.998711i	$-0.516163\pi$
$131$		−	5.13948e7i		−	1.99742i	0.0507560	−	0.998711i	$-0.483837\pi$
$132$	−1.20474e6	−	2.81775e6i	−0.0455914	−	0.106633i
$133$		0			0
$134$		−	4.35725e7i		−	1.56439i
$135$		0			0
$136$	−4.90947e7			−1.67359
$137$			1.45188e7i			0.482402i	0.970475	+	0.241201i	$0.0775413\pi$
$137$			1.45188e7i			0.482402i	−0.970475	+	0.241201i	$0.922459\pi$
$138$		0			0
$139$	5.34913e7			1.68939			0.844697	−	0.535244i	$-0.179781\pi$
$139$	5.34913e7			1.68939			0.844697	+	0.535244i	$0.179781\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	2.47562e7	−	2.59046e7i	0.690901	−	0.722950i
$145$		0			0
$146$		−	5.50481e7i		−	1.46389i
$147$	−3.54123e7	+	1.51407e7i	−0.919484	+	0.393128i
$148$		0			0
$149$		0			0			−	1.00000i	$-0.5\pi$
$149$		0			0				1.00000i	$0.5\pi$
$150$	1.62500e7	+	3.80070e7i	0.393128	+	0.919484i
$151$		0			0		1.00000			$0$
$151$		0			0		−1.00000			$\pi$
$152$			8.64867e7i			1.99755i
$153$	−5.36014e7	−	5.12252e7i	−1.20992	−	1.15628i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		1.00000			$0$
$157$		0			0		−1.00000			$\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$	5.40575e7	−	2.45199e6i	0.998973	−	0.0453122i
$163$	−5.76911e7			−1.04340			−0.521702	−	0.853128i	$-0.674703\pi$
$163$	−5.76911e7			−1.04340			−0.521702	+	0.853128i	$0.674703\pi$
$164$			1.08830e8i			1.92662i
$165$		0			0
$166$	1.04564e8			1.77420
$167$		0			0			−	1.00000i	$-0.5\pi$
$167$		0			0				1.00000i	$0.5\pi$
$168$		0			0
$169$	6.27485e7			1.00000
$170$		0			0
$171$	−9.02399e7	+	9.44259e7i	−1.38011	+	1.44412i
$172$	2.82253e7			0.422950
$173$		0			0			−	1.00000i	$-0.5\pi$
$173$		0			0				1.00000i	$0.5\pi$
$174$		0			0
$175$		0			0
$176$		−	8.38771e6i		−	0.115971i
$177$	−5.48218e7	−	1.28222e8i	−0.743099	−	1.73803i
$178$	−1.27447e8			−1.69379
$179$			6.70974e7i			0.874420i	0.899359	+	0.437210i	$0.144033\pi$
$179$			6.70974e7i			0.874420i	−0.899359	+	0.437210i	$0.855967\pi$
$180$		0			0
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−1.73557e7			−0.194087
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0			−	1.00000i	$-0.5\pi$
$191$		0			0				1.00000i	$0.5\pi$
$192$	9.01775e7	−	3.85557e7i	0.919484	−	0.393128i
$193$	2.25308e7			0.225593			0.112797	−	0.993618i	$-0.464019\pi$
$193$	2.25308e7			0.225593			0.112797	+	0.993618i	$0.464019\pi$
$194$		−	1.12446e8i		−	1.10570i
$195$		0			0
$196$	−1.05414e8			−1.00000
$197$		0			0			−	1.00000i	$-0.5\pi$
$197$		0			0				1.00000i	$0.5\pi$
$198$	8.75171e6	−	9.15768e6i	0.0801244	−	0.0838411i
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$			1.13137e8i			1.00000i
$201$	1.65606e8	−	7.08053e7i	1.43843	−	0.615007i
$202$		0			0
$203$		0			0
$204$	−7.97788e7	−	1.86594e8i	−0.657934	−	1.53884i
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$			3.05744e7i			0.231657i
$210$		0			0
$211$	1.00761e8			0.738420			0.369210	−	0.929346i	$-0.379628\pi$
$211$	1.00761e8			0.738420			0.369210	+	0.929346i	$0.379628\pi$
$212$		0			0
$213$		0			0
$214$	1.34677e8			0.939389
$215$		0			0
$216$	1.38684e8	+	5.19960e7i	0.936353	+	0.351060i
$217$		0			0
$218$		0			0
$219$	2.09221e8	−	8.94532e7i	1.34602	−	0.575495i
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		1.00000			$0$
$223$		0			0		−1.00000			$\pi$
$224$		0			0
$225$	−1.18047e8	+	1.23523e8i	−0.690901	+	0.722950i
$226$	−2.45655e8			−1.41562
$227$		−	2.15553e8i		−	1.22310i	−0.791204	−	0.611552i	$-0.790545\pi$
$227$		−	2.15553e8i		−	1.22310i	0.791204	−	0.611552i	$-0.209455\pi$
$228$	−3.28710e8	+	1.40541e8i	−1.83671	+	0.785291i
$229$		0			0		1.00000			$0$
$229$		0			0		−1.00000			$\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			3.72568e8i			1.92956i	0.263050	+	0.964782i	$0.415272\pi$
$233$			3.72568e8i			1.92956i	−0.263050	+	0.964782i	$0.584728\pi$
$234$		0			0
$235$		0			0
$236$		−	3.81685e8i		−	1.89022i
$237$		0			0
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		0			0
$241$	−2.67977e8			−1.23321			−0.616606	−	0.787272i	$-0.711493\pi$
$241$	−2.67977e8			−1.23321			−0.616606	+	0.787272i	$0.711493\pi$
$242$			2.17507e8i			0.986551i
$243$	9.71628e7	+	2.01472e8i	0.434388	+	0.900726i
$244$		0			0
$245$		0			0
$246$	−4.13631e8	+	1.76849e8i	−1.77150	+	0.757409i
$247$		0			0
$248$		0			0
$249$	1.69916e8	+	3.97415e8i	0.697487	+	1.63135i
$250$		0			0
$251$		−	1.17757e8i		−	0.470034i	−0.971991	−	0.235017i	$-0.924485\pi$
$251$		−	1.17757e8i		−	0.470034i	0.971991	−	0.235017i	$-0.0755146\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	2.68435e8			1.00000
$257$		−	3.14974e8i		−	1.15747i	−0.815517	−	0.578733i	$-0.803547\pi$
$257$		−	3.14974e8i		−	1.15747i	0.815517	−	0.578733i	$-0.196453\pi$
$258$	4.58661e7	+	1.07276e8i	0.166273	+	0.388895i
$259$		0			0
$260$		0			0
$261$		0			0
$262$	5.81466e8			1.99742
$263$		0			0			−	1.00000i	$-0.5\pi$
$263$		0			0				1.00000i	$0.5\pi$
$264$	3.18792e7	−	1.36300e7i	0.106633	−	0.0455914i
$265$		0			0
$266$		0			0
$267$	−2.07101e8	−	4.84388e8i	−0.665876	−	1.55741i
$268$	4.92967e8			1.56439
$269$		0			0			−	1.00000i	$-0.5\pi$
$269$		0			0				1.00000i	$0.5\pi$
$270$		0			0
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$		−	5.55443e8i		−	1.67359i
$273$		0			0
$274$	−1.64262e8			−0.482402
$275$			3.99957e7i			0.115971i
$276$		0			0
$277$		0			0		1.00000			$0$
$277$		0			0		−1.00000			$\pi$
$278$			6.05185e8i			1.68939i
$279$		0			0
$280$		0			0
$281$		−	5.04396e8i		−	1.35612i	−0.735005	−	0.678062i	$-0.762820\pi$
$281$		−	5.04396e8i		−	1.35612i	0.735005	−	0.678062i	$-0.237180\pi$
$282$		0			0
$283$	−7.33602e8			−1.92401			−0.962006	−	0.273029i	$-0.911975\pi$
$283$	−7.33602e8			−1.92401			−0.962006	+	0.273029i	$0.911975\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	2.93077e8	+	2.80085e8i	0.722950	+	0.690901i
$289$	−7.38975e8			−1.80089
$290$		0			0
$291$	4.27372e8	−	1.82724e8i	1.01667	−	0.434681i
$292$	6.22799e8			1.46389
$293$		0			0			−	1.00000i	$-0.5\pi$
$293$		0			0				1.00000i	$0.5\pi$
$294$	−1.71297e8	−	4.00645e8i	−0.393128	−	0.919484i
$295$		0			0
$296$		0			0
$297$	4.90271e7	+	1.83814e7i	0.108590	+	0.0407128i
$298$		0			0
$299$		0			0
$300$	−4.30000e8	+	1.83848e8i	−0.919484	+	0.393128i
$301$		0			0
$302$		0			0
$303$		0			0
$304$	−9.78485e8			−1.99755
$305$		0			0
$306$	5.79547e8	−	6.06431e8i	1.15628	−	1.20992i
$307$	1.42696e8			0.281466			0.140733	−	0.990048i	$-0.455054\pi$
$307$	1.42696e8			0.281466			0.140733	+	0.990048i	$0.455054\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0			−	1.00000i	$-0.5\pi$
$311$		0			0				1.00000i	$0.5\pi$
$312$		0			0
$313$	9.84035e8			1.81387			0.906933	−	0.421275i	$-0.138417\pi$
$313$	9.84035e8			1.81387			0.906933	+	0.421275i	$0.138417\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0			−	1.00000i	$-0.5\pi$
$317$		0			0				1.00000i	$0.5\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	2.18850e8	+	5.11867e8i	0.369300	+	0.863753i
$322$		0			0
$323$			2.02467e9i			3.34306i
$324$	2.77411e7	+	6.11591e8i	0.0453122	+	0.998973i
$325$		0			0
$326$		−	6.52700e8i		−	1.04340i
$327$		0			0
$328$	−1.23127e9			−1.92662
$329$		0			0
$330$		0			0
$331$	8.61506e8			1.30575			0.652876	−	0.757465i	$-0.273562\pi$
$331$	8.61506e8			1.30575			0.652876	+	0.757465i	$0.273562\pi$
$332$			1.18300e9i			1.77420i
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−5.51176e8			−0.784487			−0.392244	−	0.919861i	$-0.628301\pi$
$337$	−5.51176e8			−0.784487			−0.392244	+	0.919861i	$0.628301\pi$
$338$			7.09918e8i			1.00000i
$339$	−3.99189e8	−	9.33659e8i	−0.556518	−	1.30164i
$340$		0			0
$341$		0			0
$342$	−1.06831e9	−	1.02095e9i	−1.44412	−	1.38011i
$343$		0			0
$344$			3.19333e8i			0.422950i
$345$		0			0
$346$		0			0
$347$			6.60950e8i			0.849211i	0.905378	+	0.424606i	$0.139587\pi$
$347$			6.60950e8i			0.849211i	−0.905378	+	0.424606i	$0.860413\pi$
$348$		0			0
$349$		0			0		1.00000			$0$
$349$		0			0		−1.00000			$\pi$
$350$		0			0
$351$		0			0
$352$	9.48961e7			0.115971
$353$		−	1.62348e9i		−	1.96443i	−0.187771	−	0.982213i	$-0.560126\pi$
$353$		−	1.62348e9i		−	1.96443i	0.187771	−	0.982213i	$-0.439874\pi$
$354$	1.45067e9	−	6.20238e8i	1.73803	−	0.743099i
$355$		0			0
$356$		−	1.44190e9i		−	1.69379i
$357$		0			0
$358$	−7.59121e8			−0.874420
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$		0			0
$361$	2.67285e9			2.99019
$362$		0			0
$363$	−8.26679e8	+	3.53449e8i	−0.907117	+	0.387841i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		1.00000			$0$
$367$		0			0		−1.00000			$\pi$
$368$		0			0
$369$	−1.34430e9	−	1.28471e9i	−1.39285	−	1.33110i
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		1.00000			$0$
$373$		0			0		−1.00000			$\pi$
$374$		−	1.96358e8i		−	0.194087i
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	6.98234e7			0.0658816			0.0329408	−	0.999457i	$-0.489513\pi$
$379$	6.98234e7			0.0658816			0.0329408	+	0.999457i	$0.489513\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0			−	1.00000i	$-0.5\pi$
$383$		0			0				1.00000i	$0.5\pi$
$384$	4.36208e8	+	1.02024e9i	0.393128	+	0.919484i
$385$		0			0
$386$			2.54907e8i			0.225593i
$387$	−3.33191e8	+	3.48646e8i	−0.292216	+	0.305771i
$388$	1.27218e9			1.10570
$389$		0			0			−	1.00000i	$-0.5\pi$
$389$		0			0				1.00000i	$0.5\pi$
$390$		0			0
$391$		0			0
$392$		−	1.19262e9i		−	1.00000i
$393$	9.44882e8	+	2.20998e9i	0.785243	+	1.83660i
$394$		0			0
$395$		0			0
$396$	1.03607e8	+	9.90143e7i	0.0838411	+	0.0801244i
$397$		0			0		1.00000			$0$
$397$		0			0		−1.00000			$\pi$
$398$		0			0
$399$		0			0
$400$	−1.28000e9			−1.00000
$401$		−	1.27902e9i		−	0.990536i	−0.868740	−	0.495268i	$-0.835070\pi$
$401$		−	1.27902e9i		−	0.990536i	0.868740	−	0.495268i	$-0.164930\pi$
$402$	8.01071e8	+	1.87362e9i	0.615007	+	1.43843i
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$	2.11107e9	−	9.02594e8i	1.53884	−	0.657934i
$409$	−2.14006e9			−1.54666			−0.773330	−	0.634004i	$-0.781410\pi$
$409$	−2.14006e9			−1.54666			−0.773330	+	0.634004i	$0.781410\pi$
$410$		0			0
$411$	−2.66925e8	−	6.24309e8i	−0.189646	−	0.443561i
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−2.30012e9	+	9.83425e8i	−1.55337	+	0.664148i
$418$	−3.45910e8			−0.231657
$419$		−	3.00841e9i		−	1.99797i	−0.0450790	−	0.998983i	$-0.514354\pi$
$419$		−	3.00841e9i		−	1.99797i	0.0450790	−	0.998983i	$-0.485646\pi$
$420$		0			0
$421$		0			0		1.00000			$0$
$421$		0			0		−1.00000			$\pi$
$422$			1.13998e9i			0.738420i
$423$		0			0
$424$		0			0
$425$			2.64856e9i			1.67359i
$426$		0			0
$427$		0			0
$428$			1.52370e9i			0.939389i
$429$		0			0
$430$		0			0
$431$		0			0			−	1.00000i	$-0.5\pi$
$431$		0			0				1.00000i	$0.5\pi$
$432$	−5.88268e8	+	1.56904e9i	−0.351060	+	0.936353i
$433$	−3.31027e9			−1.95955			−0.979774	−	0.200109i	$-0.935870\pi$
$433$	−3.31027e9			−1.95955			−0.979774	+	0.200109i	$0.935870\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$	1.01205e9	+	2.36707e9i	0.575495	+	1.34602i
$439$		0			0		1.00000			$0$
$439$		0			0		−1.00000			$\pi$
$440$		0			0
$441$	1.24437e9	−	1.30210e9i	0.690901	−	0.722950i
$442$		0			0
$443$		−	1.65968e9i		−	0.907010i	−0.891254	−	0.453505i	$-0.850174\pi$
$443$		−	1.65968e9i		−	0.907010i	0.891254	−	0.453505i	$-0.149826\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$			3.09147e9i			1.61177i	0.592072	+	0.805885i	$0.298310\pi$
$449$			3.09147e9i			1.61177i	−0.592072	+	0.805885i	$0.701690\pi$
$450$	−1.39750e9	−	1.33555e9i	−0.722950	−	0.690901i
$451$	−4.35275e8			−0.223432
$452$		−	2.77927e9i		−	1.41562i
$453$		0			0
$454$	2.43870e9			1.22310
$455$		0			0
$456$	−1.59004e9	−	3.71893e9i	−0.785291	−	1.83671i
$457$	4.04270e9			1.98137			0.990684	−	0.136183i	$-0.0434834\pi$
$457$	4.04270e9			1.98137			0.990684	+	0.136183i	$0.0434834\pi$
$458$		0			0
$459$	3.24662e9	+	1.21723e9i	1.56707	+	0.587530i
$460$		0			0
$461$		0			0			−	1.00000i	$-0.5\pi$
$461$		0			0				1.00000i	$0.5\pi$
$462$		0			0
$463$		0			0		1.00000			$0$
$463$		0			0		−1.00000			$\pi$
$464$		0			0
$465$		0			0
$466$	−4.21512e9			−1.92956
$467$		−	2.69178e9i		−	1.22301i	−0.791241	−	0.611505i	$-0.790565\pi$
$467$		−	2.69178e9i		−	1.22301i	0.791241	−	0.611505i	$-0.209435\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$	4.31827e9			1.89022
$473$			1.12889e8i			0.0490499i
$474$		0			0
$475$	4.66578e9			1.99755
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$		0			0
$481$		0			0
$482$		−	3.03181e9i		−	1.23321i
$483$		0			0
$484$	−2.46081e9			−0.986551
$485$		0			0
$486$	−2.27939e9	+	1.09927e9i	−0.900726	+	0.434388i
$487$		0			0		1.00000			$0$
$487$		0			0		−1.00000			$\pi$
$488$		0			0
$489$	2.48072e9	−	1.06064e9i	0.959393	−	0.410191i
$490$		0			0
$491$			4.00291e9i			1.52613i	0.646323	+	0.763064i	$0.276306\pi$
$491$			4.00291e9i			1.52613i	−0.646323	+	0.763064i	$0.723694\pi$
$492$	−2.00082e9	−	4.67970e9i	−0.757409	−	1.77150i
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$	−4.49624e9	+	1.92238e9i	−1.63135	+	0.697487i
$499$	−4.38423e9			−1.57958			−0.789790	−	0.613378i	$-0.789810\pi$
$499$	−4.38423e9			−1.57958			−0.789790	+	0.613378i	$0.789810\pi$
$500$		0			0
$501$		0			0
$502$	1.33227e9			0.470034
$503$		0			0			−	1.00000i	$-0.5\pi$
$503$		0			0				1.00000i	$0.5\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−2.69819e9	+	1.15362e9i	−0.919484	+	0.393128i
$508$		0			0
$509$		0			0			−	1.00000i	$-0.5\pi$
$509$		0			0				1.00000i	$0.5\pi$
$510$		0			0
$511$		0			0
$512$			3.03700e9i			1.00000i
$513$	2.14432e9	−	5.71935e9i	0.701259	−	1.87041i
$514$	3.56352e9			1.15747
$515$		0			0
$516$	−1.21369e9	+	5.18915e8i	−0.388895	+	0.166273i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$			3.89730e9i			1.20735i	0.797232	+	0.603673i	$0.206297\pi$
$521$			3.89730e9i			1.20735i	−0.797232	+	0.603673i	$0.793703\pi$
$522$		0			0
$523$	−3.59001e9			−1.09734			−0.548668	−	0.836040i	$-0.684865\pi$
$523$	−3.59001e9			−1.09734			−0.548668	+	0.836040i	$0.684865\pi$
$524$			6.57854e9i			1.99742i
$525$		0			0
$526$		0			0
$527$		0			0
$528$	1.54206e8	+	3.60672e8i	0.0455914	+	0.106633i
$529$	−3.40483e9			−1.00000
$530$		0			0
$531$	4.71467e9	+	4.50567e9i	1.36654	+	1.30596i
$532$		0			0
$533$		0			0
$534$	5.48022e9	−	2.34308e9i	1.55741	−	0.665876i
$535$		0			0
$536$			5.57728e9i			1.56439i
$537$	−1.23357e9	−	2.88519e9i	−0.343759	−	0.804015i
$538$		0			0
$539$		−	4.21609e8i		−	0.115971i
$540$		0			0
$541$		0			0		1.00000			$0$
$541$		0			0		−1.00000			$\pi$
$542$		0			0
$543$		0			0
$544$	6.28412e9			1.67359
$545$		0			0
$546$		0			0
$547$	−2.21785e9			−0.579396			−0.289698	−	0.957118i	$-0.593555\pi$
$547$	−2.21785e9			−0.579396			−0.289698	+	0.957118i	$0.593555\pi$
$548$		−	1.85841e9i		−	0.482402i
$549$		0			0
$550$	−4.52500e8			−0.115971
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$	−6.84688e9			−1.68939
$557$		0			0			−	1.00000i	$-0.5\pi$
$557$		0			0				1.00000i	$0.5\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$	7.46296e8	−	3.19081e8i	0.178460	−	0.0763012i
$562$	5.70659e9			1.35612
$563$			6.60784e8i			0.156056i	0.996951	+	0.0780279i	$0.0248623\pi$
$563$			6.60784e8i			0.156056i	−0.996951	+	0.0780279i	$0.975138\pi$
$564$		0			0
$565$		0			0
$566$		−	8.29975e9i		−	1.92401i
$567$		0			0
$568$		0			0
$569$		−	2.74418e9i		−	0.624483i	−0.950003	−	0.312241i	$-0.898920\pi$
$569$		−	2.74418e9i		−	0.624483i	0.950003	−	0.312241i	$-0.101080\pi$
$570$		0			0
$571$	−1.82879e9			−0.411089			−0.205545	−	0.978648i	$-0.565897\pi$
$571$	−1.82879e9			−0.411089			−0.205545	+	0.978648i	$0.565897\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−3.16880e9	+	3.31579e9i	−0.690901	+	0.722950i
$577$	−6.48604e9			−1.40561			−0.702805	−	0.711383i	$-0.748069\pi$
$577$	−6.48604e9			−1.40561			−0.702805	+	0.711383i	$0.748069\pi$
$578$		−	8.36055e9i		−	1.80089i
$579$	−9.68824e8	+	4.14224e8i	−0.207429	+	0.0886870i
$580$		0			0
$581$		0			0
$582$	2.06729e9	+	4.83517e9i	0.434681	+	1.01667i
$583$		0			0
$584$			7.04616e9i			1.46389i
$585$		0			0
$586$		0			0
$587$		−	6.32738e9i		−	1.29119i	−0.763679	−	0.645596i	$-0.776609\pi$
$587$		−	6.32738e9i		−	1.29119i	0.763679	−	0.645596i	$-0.223391\pi$
$588$	4.53278e9	−	1.93800e9i	0.919484	−	0.393128i
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	8.95304e9i		−	1.76311i	−0.472082	−	0.881555i	$-0.656497\pi$
$593$		−	8.95304e9i		−	1.76311i	0.472082	−	0.881555i	$-0.343503\pi$
$594$	−2.07962e8	+	5.54678e8i	−0.0407128	+	0.108590i
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$	−2.08000e9	−	4.86489e9i	−0.393128	−	0.919484i
$601$	8.35888e9			1.57068			0.785339	−	0.619066i	$-0.212489\pi$
$601$	8.35888e9			1.57068			0.785339	+	0.619066i	$0.212489\pi$
$602$		0			0
$603$	−5.81932e9	+	6.08926e9i	−1.08084	+	1.13098i
$604$		0			0
$605$		0			0
$606$		0			0
$607$		0			0		1.00000			$0$
$607$		0			0		−1.00000			$\pi$
$608$		−	1.10703e10i		−	1.99755i
$609$		0			0
$610$		0			0
$611$		0			0
$612$	6.86098e9	+	6.55683e9i	1.20992	+	1.15628i
$613$		0			0		1.00000			$0$
$613$		0			0		−1.00000			$\pi$
$614$			1.61442e9i			0.281466i
$615$		0			0
$616$		0			0
$617$			2.01892e9i			0.346036i	0.984919	+	0.173018i	$0.0553519\pi$
$617$			2.01892e9i			0.346036i	−0.984919	+	0.173018i	$0.944648\pi$
$618$		0			0
$619$	7.66578e9			1.29909			0.649545	−	0.760323i	$-0.274959\pi$
$619$	7.66578e9			1.29909			0.649545	+	0.760323i	$0.274959\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	6.10352e9			1.00000
$626$			1.11331e10i			1.81387i
$627$	−5.62103e8	−	1.31470e9i	−0.0910710	−	0.213005i
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$		0			0
$633$	−4.33272e9	+	1.85247e9i	−0.678966	+	0.290294i
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$			1.12731e10i			1.69060i	0.534294	+	0.845299i	$0.320577\pi$
$641$			1.12731e10i			1.69060i	−0.534294	+	0.845299i	$0.679423\pi$
$642$	−5.79111e9	+	2.47601e9i	−0.863753	+	0.369300i
$643$	5.17655e9			0.767896			0.383948	−	0.923355i	$-0.374564\pi$
$643$	5.17655e9			0.767896			0.383948	+	0.923355i	$0.374564\pi$
$644$		0			0
$645$		0			0
$646$	−2.29065e10			−3.34306
$647$		0			0			−	1.00000i	$-0.5\pi$
$647$		0			0				1.00000i	$0.5\pi$
$648$	−6.91936e9	+	3.13854e8i	−0.998973	+	0.0453122i
$649$	1.52658e9			0.219211
$650$		0			0
$651$		0			0
$652$	7.38446e9			1.04340
$653$		0			0			−	1.00000i	$-0.5\pi$
$653$		0			0				1.00000i	$0.5\pi$
$654$		0			0
$655$		0			0
$656$		−	1.39303e10i		−	1.92662i
$657$	−7.35194e9	+	7.69298e9i	−1.01140	+	1.05832i
$658$		0			0
$659$			1.18365e10i			1.61111i	0.592520	+	0.805556i	$0.298133\pi$
$659$			1.18365e10i			1.61111i	−0.592520	+	0.805556i	$0.701867\pi$
$660$		0			0
$661$		0			0		1.00000			$0$
$661$		0			0		−1.00000			$\pi$
$662$			9.74683e9i			1.30575i
$663$		0			0
$664$	−1.33841e10			−1.77420
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	1.54576e10			1.95474			0.977369	−	0.211539i	$-0.0678477\pi$
$673$	1.54576e10			1.95474			0.977369	+	0.211539i	$0.0678477\pi$
$674$		−	6.23585e9i		−	0.784487i
$675$	2.80508e9	−	7.48174e9i	0.351060	−	0.936353i
$676$	−8.03181e9			−1.00000
$677$		0			0			−	1.00000i	$-0.5\pi$
$677$		0			0				1.00000i	$0.5\pi$
$678$	1.05632e10	−	4.51631e9i	1.30164	−	0.556518i
$679$		0			0
$680$		0			0
$681$	3.96289e9	+	9.26878e9i	0.480837	+	1.12463i
$682$		0			0
$683$		−	1.61382e10i		−	1.93813i	−0.246800	−	0.969066i	$-0.579379\pi$
$683$		−	1.61382e10i		−	1.93813i	0.246800	−	0.969066i	$-0.420621\pi$
$684$	1.15507e10	−	1.20865e10i	1.38011	−	1.44412i
$685$		0			0
$686$		0			0
$687$		0			0
$688$	−3.61284e9			−0.422950
$689$		0			0
$690$		0			0
$691$	1.60005e10			1.84484			0.922422	−	0.386184i	$-0.126207\pi$
$691$	1.60005e10			1.84484			0.922422	+	0.386184i	$0.126207\pi$
$692$		0			0
$693$		0			0
$694$	−7.47780e9			−0.849211
$695$		0			0
$696$		0			0
$697$	−2.88243e10			−3.22437
$698$		0			0
$699$	−6.84957e9	−	1.60204e10i	−0.758566	−	1.77420i
$700$		0			0
$701$		0			0			−	1.00000i	$-0.5\pi$
$701$		0			0				1.00000i	$0.5\pi$
$702$		0			0
$703$		0			0
$704$			1.07363e9i			0.115971i
$705$		0			0
$706$	1.83676e10			1.96443
$707$		0			0
$708$	7.01719e9	+	1.64124e10i	0.743099	+	1.73803i
$709$		0			0		1.00000			$0$
$709$		0			0		−1.00000			$\pi$
$710$		0			0
$711$		0			0
$712$	1.63132e10			1.69379
$713$		0			0
$714$		0			0
$715$		0			0
$716$		−	8.58847e9i		−	0.874420i
$717$		0			0
$718$		0			0
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$		0			0
$721$		0			0
$722$			3.02398e10i			2.99019i
$723$	1.15230e10	−	4.92669e9i	1.13392	−	0.484810i
$724$		0			0
$725$		0			0
$726$	−3.99882e9	−	9.35280e9i	−0.387841	−	0.907117i
$727$		0			0		1.00000			$0$
$727$		0			0		−1.00000			$\pi$
$728$		0			0
$729$	−7.88202e9	−	6.87698e9i	−0.753513	−	0.657433i
$730$		0			0
$731$			7.47563e9i			0.707843i
$732$		0			0
$733$		0			0		1.00000			$0$
$733$		0			0		−1.00000			$\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			1.97166e9i			0.181424i
$738$	1.45348e10	−	1.52090e10i	1.33110	−	1.39285i
$739$	−2.19394e10			−1.99972			−0.999860	−	0.0167159i	$-0.994679\pi$
$739$	−2.19394e10			−1.99972			−0.999860	+	0.0167159i	$0.994679\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0			−	1.00000i	$-0.5\pi$
$743$		0			0				1.00000i	$0.5\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−1.46128e10	−	1.39650e10i	−1.28266	−	1.22580i
$748$	2.22153e9			0.194087
$749$		0			0
$750$		0			0
$751$		0			0		1.00000			$0$
$751$		0			0		−1.00000			$\pi$
$752$		0			0
$753$	2.16494e9	+	5.06356e9i	0.184784	+	0.432189i
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0		1.00000			$0$
$757$		0			0		−1.00000			$\pi$
$758$			7.89962e8i			0.0658816i
$759$		0			0
$760$		0			0
$761$		−	2.41250e10i		−	1.98437i	−0.124789	−	0.992183i	$-0.539825\pi$
$761$		−	2.41250e10i		−	1.98437i	0.124789	−	0.992183i	$-0.460175\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	−1.15427e10	+	4.93513e9i	−0.919484	+	0.393128i
$769$	7.12479e9			0.564976			0.282488	−	0.959271i	$-0.408840\pi$
$769$	7.12479e9			0.564976			0.282488	+	0.959271i	$0.408840\pi$
$770$		0			0
$771$	5.79072e9	+	1.35439e10i	0.455032	+	1.06427i
$772$	−2.88394e9			−0.225593
$773$		0			0			−	1.00000i	$-0.5\pi$
$773$		0			0				1.00000i	$0.5\pi$
$774$	−3.94448e9	−	3.76962e9i	−0.305771	−	0.292216i
$775$		0			0
$776$			1.43931e10i			1.10570i
$777$		0			0
$778$		0			0
$779$			5.07778e10i			3.84851i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	1.34929e10			1.00000
$785$		0			0
$786$	−2.50030e10	+	1.06901e10i	−1.83660	+	0.785243i
$787$	−2.70219e10			−1.97608			−0.988041	−	0.154191i	$-0.950723\pi$
$787$	−2.70219e10			−1.97608			−0.988041	+	0.154191i	$0.950723\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$	−1.12022e9	+	1.17218e9i	−0.0801244	+	0.0838411i
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0			−	1.00000i	$-0.5\pi$
$797$		0			0				1.00000i	$0.5\pi$
$798$		0			0
$799$		0			0
$800$		−	1.44815e10i		−	1.00000i
$801$	1.78107e10	+	1.70212e10i	1.22452	+	1.17024i
$802$	1.44704e10			0.990536
$803$			2.49093e9i			0.169768i
$804$	−2.11976e10	+	9.06308e9i	−1.43843	+	0.615007i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	2.22486e10i		−	1.47735i	−0.674064	−	0.738673i	$-0.735453\pi$
$809$		−	2.22486e10i		−	1.47735i	0.674064	−	0.738673i	$-0.264547\pi$
$810$		0			0
$811$	2.79765e10			1.84171			0.920855	−	0.389906i	$-0.127492\pi$
$811$	2.79765e10			1.84171			0.920855	+	0.389906i	$0.127492\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$	1.02117e10	+	2.38840e10i	0.657934	+	1.53884i
$817$	1.31693e10			0.844861
$818$		−	2.42120e10i		−	1.54666i
$819$		0			0
$820$		0			0
$821$		0			0			−	1.00000i	$-0.5\pi$
$821$		0			0				1.00000i	$0.5\pi$
$822$	7.06325e9	−	3.01991e9i	0.443561	−	0.189646i
$823$		0			0		1.00000			$0$
$823$		0			0		−1.00000			$\pi$
$824$		0			0
$825$	−7.35312e8	−	1.71982e9i	−0.0455914	−	0.106633i
$826$		0			0
$827$		−	2.04957e10i		−	1.26006i	−0.776569	−	0.630032i	$-0.783042\pi$
$827$		−	2.04957e10i		−	1.26006i	0.776569	−	0.630032i	$-0.216958\pi$
$828$		0			0
$829$		0			0		1.00000			$0$
$829$		0			0		−1.00000			$\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	2.79194e10i		−	1.67359i
$834$	−1.11262e10	−	2.60229e10i	−0.664148	−	1.55337i
$835$		0			0
$836$		−	3.91352e9i		−	0.231657i
$837$		0			0
$838$	3.40363e10			1.99797
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	−1.72499e10			−1.00000
$842$		0			0
$843$	9.27321e9	+	2.16890e10i	0.533130	+	1.24693i
$844$	−1.28974e10			−0.738420
$845$		0			0
$846$		0			0
$847$		0			0
$848$		0			0
$849$	3.15449e10	−	1.34871e10i	1.76910	−	0.756383i
$850$	−2.99650e10			−1.67359
$851$		0			0
$852$		0			0
$853$		0			0		1.00000			$0$
$853$		0			0		−1.00000			$\pi$
$854$		0			0
$855$		0			0
$856$	−1.72387e10			−0.939389
$857$			1.31197e10i			0.712017i	0.934483	+	0.356008i	$0.115863\pi$
$857$			1.31197e10i			0.712017i	−0.934483	+	0.356008i	$0.884137\pi$
$858$		0			0
$859$	−1.95441e10			−1.05206			−0.526029	−	0.850467i	$-0.676320\pi$
$859$	−1.95441e10			−1.05206			−0.526029	+	0.850467i	$0.676320\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0			−	1.00000i	$-0.5\pi$
$863$		0			0				1.00000i	$0.5\pi$
$864$	−1.77516e10	−	6.65549e9i	−0.936353	−	0.351060i
$865$		0			0
$866$		−	3.74514e10i		−	1.95955i
$867$	3.17759e10	−	1.35859e10i	1.65589	−	0.707980i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	−1.50177e10	+	1.57143e10i	−0.763928	+	0.799364i
$874$		0			0
$875$		0			0
$876$	−2.67803e10	+	1.14500e10i	−1.34602	+	0.575495i
$877$		0			0		1.00000			$0$
$877$		0			0		−1.00000			$\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			2.24380e10i			1.10553i	0.833338	+	0.552763i	$0.186427\pi$
$881$			2.24380e10i			1.10553i	−0.833338	+	0.552763i	$0.813573\pi$
$882$	1.47315e10	+	1.40785e10i	0.722950	+	0.690901i
$883$	−9.55139e9			−0.466879			−0.233439	−	0.972371i	$-0.574998\pi$
$883$	−9.55139e9			−0.466879			−0.233439	+	0.972371i	$0.574998\pi$
$884$		0			0
$885$		0			0
$886$	1.87772e10			0.907010
$887$		0			0			−	1.00000i	$-0.5\pi$
$887$		0			0				1.00000i	$0.5\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	−2.44610e9	+	1.10952e8i	−0.115852	+	0.00525490i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	−3.49760e10			−1.61177
$899$		0			0
$900$	1.51100e10	−	1.58109e10i	0.690901	−	0.722950i
$901$		0			0
$902$		−	4.92457e9i		−	0.223432i
$903$		0			0
$904$	3.14438e10			1.41562
$905$		0			0
$906$		0			0
$907$	4.13378e10			1.83959			0.919797	−	0.392395i	$-0.128353\pi$
$907$	4.13378e10			1.83959			0.919797	+	0.392395i	$0.128353\pi$
$908$			2.75908e10i			1.22310i
$909$		0			0
$910$		0			0
$911$		0			0			−	1.00000i	$-0.5\pi$
$911$		0			0				1.00000i	$0.5\pi$
$912$	4.20749e10	−	1.79892e10i	1.83671	−	0.785291i
$913$	−4.73150e9			−0.205756
$914$			4.57380e10i			1.98137i
$915$		0			0
$916$		0			0
$917$		0			0
$918$	−1.37714e10	+	3.67314e10i	−0.587530	+	1.56707i
$919$		0			0		1.00000			$0$
$919$		0			0		−1.00000			$\pi$
$920$		0			0
$921$	−6.13592e9	+	2.62343e9i	−0.258804	+	0.110652i
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$			1.14207e10i			0.467346i	0.972315	+	0.233673i	$0.0750745\pi$
$929$			1.14207e10i			0.467346i	−0.972315	+	0.233673i	$0.924925\pi$
$930$		0			0
$931$	−4.91836e10			−1.99755
$932$		−	4.76886e10i		−	1.92956i
$933$		0			0
$934$	3.04540e10			1.22301
$935$		0			0
$936$		0			0
$937$	−4.18278e10			−1.66102			−0.830512	−	0.557000i	$-0.811952\pi$
$937$	−4.18278e10			−1.66102			−0.830512	+	0.557000i	$0.811952\pi$
$938$		0			0
$939$	−4.23135e10	+	1.80913e10i	−1.66782	+	0.713082i
$940$		0			0
$941$		0			0			−	1.00000i	$-0.5\pi$
$941$		0			0				1.00000i	$0.5\pi$
$942$		0			0
$943$		0			0
$944$			4.88556e10i			1.89022i
$945$		0			0
$946$	−1.27719e9			−0.0490499
$947$		−	4.76750e10i		−	1.82417i	−0.409998	−	0.912087i	$-0.634470\pi$
$947$		−	4.76750e10i		−	1.82417i	0.409998	−	0.912087i	$-0.365530\pi$
$948$		0			0
$949$		0			0
$950$			5.27873e10i			1.99755i
$951$		0			0
$952$		0			0
$953$			2.67564e10i			1.00139i	0.865624	+	0.500694i	$0.166922\pi$
$953$			2.67564e10i			1.00139i	−0.865624	+	0.500694i	$0.833078\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	2.75126e10			1.00000
$962$		0			0
$963$	−1.88211e10	−	1.79868e10i	−0.679131	−	0.649025i
$964$	3.43010e10			1.23321
$965$		0			0
$966$		0			0
$967$		0			0		1.00000			$0$
$967$		0			0		−1.00000			$\pi$
$968$		−	2.78409e10i		−	0.986551i
$969$	−3.72231e10	−	8.70607e10i	−1.31425	−	3.07389i
$970$		0			0
$971$		−	2.82185e10i		−	0.989161i	−0.869132	−	0.494581i	$-0.835322\pi$
$971$		−	2.82185e10i		−	0.989161i	0.869132	−	0.494581i	$-0.164678\pi$
$972$	−1.24368e10	−	2.57884e10i	−0.434388	−	0.900726i
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$			4.55914e10i			1.56405i	0.623245	+	0.782027i	$0.285814\pi$
$977$			4.55914e10i			1.56405i	−0.623245	+	0.782027i	$0.714186\pi$
$978$	1.19998e10	+	2.80661e10i	0.410191	+	0.959393i
$979$	5.76698e9			0.196430
$980$		0			0
$981$		0			0
$982$	−4.52878e10			−1.52613
$983$		0			0			−	1.00000i	$-0.5\pi$
$983$		0			0				1.00000i	$0.5\pi$
$984$	5.29448e10	−	2.26367e10i	1.77150	−	0.757409i
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		1.00000			$0$
$991$		0			0		−1.00000			$\pi$
$992$		0			0
$993$	−3.70448e10	+	1.58386e10i	−1.20062	+	0.513327i
$994$		0			0
$995$		0			0
$996$	−2.17492e10	−	5.08691e10i	−0.697487	−	1.63135i
$997$		0			0		1.00000			$0$
$997$		0			0		−1.00000			$\pi$
$998$		−	4.96019e10i		−	1.57958i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	24.8.f.a.11.2	yes	2
3.2	odd	2	inner	24.8.f.a.11.1	✓	2
4.3	odd	2		96.8.f.a.47.1		2
8.3	odd	2	CM	24.8.f.a.11.2	yes	2
8.5	even	2		96.8.f.a.47.1		2
12.11	even	2		96.8.f.a.47.2		2
24.5	odd	2		96.8.f.a.47.2		2
24.11	even	2	inner	24.8.f.a.11.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
24.8.f.a.11.1	✓	2	3.2	odd	2	inner
24.8.f.a.11.1	✓	2	24.11	even	2	inner
24.8.f.a.11.2	yes	2	1.1	even	1	trivial
24.8.f.a.11.2	yes	2	8.3	odd	2	CM
96.8.f.a.47.1		2	4.3	odd	2
96.8.f.a.47.1		2	8.5	even	2
96.8.f.a.47.2		2	12.11	even	2
96.8.f.a.47.2		2	24.5	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:	\( N \)	\(=\)	\( 24 = 2^{3} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	24.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+11.3137i q^{2} +(-43.0000 + 18.3848i) q^{3} -128.000 q^{4} +(-208.000 - 486.489i) q^{6} -1448.15i q^{8} +(1511.00 - 1581.09i) q^{9} +O(q^{10})\)	\(q+11.3137i q^{2} +(-43.0000 + 18.3848i) q^{3} -128.000 q^{4} +(-208.000 - 486.489i) q^{6} -1448.15i q^{8} +(1511.00 - 1581.09i) q^{9} -511.945i q^{11} +(5504.00 - 2353.25i) q^{12} +16384.0 q^{16} -33901.5i q^{17} +(17888.0 + 17095.0i) q^{18} -59722.0 q^{19} +5792.00 q^{22} +(26624.0 + 62270.7i) q^{24} -78125.0 q^{25} +(-35905.0 + 95766.3i) q^{27} +185364. i q^{32} +(9412.00 + 22013.6i) q^{33} +383552. q^{34} +(-193408. + 202380. i) q^{36} -675677. i q^{38} -850237. i q^{41} -220510. q^{43} +65529.0i q^{44} +(-704512. + 301216. i) q^{48} +823543. q^{49} -883883. i q^{50} +(623272. + 1.45777e6i) q^{51} +(-1.08347e6 - 406219. i) q^{54} +(2.56805e6 - 1.09798e6i) q^{57} +2.98191e6i q^{59} -2.09715e6 q^{64} +(-249056. + 106485. i) q^{66} -3.85130e6 q^{67} +4.33940e6i q^{68} +(-2.28966e6 - 2.18816e6i) q^{72} -4.86561e6 q^{73} +(3.35938e6 - 1.43631e6i) q^{75} +7.64442e6 q^{76} +(-216727. - 4.77806e6i) q^{81} +9.61933e6 q^{82} -9.24221e6i q^{83} -2.49479e6i q^{86} -741376. q^{88} +1.12648e7i q^{89} +(-3.40787e6 - 7.97064e6i) q^{96} -9.93889e6 q^{97} +9.31733e6i q^{98} +(-809432. - 773549. i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 86 q^{3} - 256 q^{4} - 416 q^{6} + 3022 q^{9}+O(q^{10}) \) 2 * q - 86 * q^3 - 256 * q^4 - 416 * q^6 + 3022 * q^9	\( 2 q - 86 q^{3} - 256 q^{4} - 416 q^{6} + 3022 q^{9} + 11008 q^{12} + 32768 q^{16} + 35776 q^{18} - 119444 q^{19} + 11584 q^{22} + 53248 q^{24} - 156250 q^{25} - 71810 q^{27} + 18824 q^{33} + 767104 q^{34} - 386816 q^{36} - 441020 q^{43} - 1409024 q^{48} + 1647086 q^{49} + 1246544 q^{51} - 2166944 q^{54} + 5136092 q^{57} - 4194304 q^{64} - 498112 q^{66} - 7702604 q^{67} - 4579328 q^{72} - 9731228 q^{73} + 6718750 q^{75} + 15288832 q^{76} - 433454 q^{81} + 19238656 q^{82} - 1482752 q^{88} - 6815744 q^{96} - 19877780 q^{97} - 1618864 q^{99}+O(q^{100}) \) 2 * q - 86 * q^3 - 256 * q^4 - 416 * q^6 + 3022 * q^9 + 11008 * q^12 + 32768 * q^16 + 35776 * q^18 - 119444 * q^19 + 11584 * q^22 + 53248 * q^24 - 156250 * q^25 - 71810 * q^27 + 18824 * q^33 + 767104 * q^34 - 386816 * q^36 - 441020 * q^43 - 1409024 * q^48 + 1647086 * q^49 + 1246544 * q^51 - 2166944 * q^54 + 5136092 * q^57 - 4194304 * q^64 - 498112 * q^66 - 7702604 * q^67 - 4579328 * q^72 - 9731228 * q^73 + 6718750 * q^75 + 15288832 * q^76 - 433454 * q^81 + 19238656 * q^82 - 1482752 * q^88 - 6815744 * q^96 - 19877780 * q^97 - 1618864 * q^99

Introduction

L-functions

Modular forms

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Database

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