LMFDB - Embedded newform 24.16.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [24,16,Mod(1,24)]

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([0, 0, 0]))

N = Newforms(chi, 16, names="a")

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

chi := DirichletCharacter("24.1");

S:= CuspForms(chi, 16);

N := Newforms(S);

Level:	$ N $	$=$	$ 24 = 2^{3} \cdot 3 $
Weight:	$ k $	$=$	$ 16 $
Character orbit:	$[\chi]$	$=$	24.a (trivial)

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:	yes
Analytic conductor:	$34.2464412240$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{22}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 22 $ x^2 - 22
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{7}\cdot 3\cdot 5 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-4.69042$ of defining polynomial
Character	$\chi$	$=$	24.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2187.00 q^{3} -180276. q^{5} +1.48794e6 q^{7} +4.78297e6 q^{9} +O(q^{10})$	$q+2187.00 q^{3} -180276. q^{5} +1.48794e6 q^{7} +4.78297e6 q^{9} +3.90021e7 q^{11} -2.07919e8 q^{13} -3.94264e8 q^{15} -2.64604e9 q^{17} +4.76142e9 q^{19} +3.25412e9 q^{21} -5.16387e9 q^{23} +1.98199e9 q^{25} +1.04604e10 q^{27} -1.10143e11 q^{29} -3.05870e10 q^{31} +8.52976e10 q^{33} -2.68240e11 q^{35} -7.68194e11 q^{37} -4.54720e11 q^{39} -9.15551e11 q^{41} -1.32094e12 q^{43} -8.62256e11 q^{45} -8.60974e11 q^{47} -2.53361e12 q^{49} -5.78690e12 q^{51} +9.28543e11 q^{53} -7.03115e12 q^{55} +1.04132e13 q^{57} -3.80695e13 q^{59} -1.83644e13 q^{61} +7.11675e12 q^{63} +3.74829e13 q^{65} +8.19085e13 q^{67} -1.12934e13 q^{69} +4.15622e13 q^{71} +1.55562e14 q^{73} +4.33461e12 q^{75} +5.80326e13 q^{77} -1.78006e14 q^{79} +2.28768e13 q^{81} +2.25716e13 q^{83} +4.77019e14 q^{85} -2.40882e14 q^{87} +5.76725e14 q^{89} -3.09371e14 q^{91} -6.68937e13 q^{93} -8.58371e14 q^{95} +4.91989e14 q^{97} +1.86546e14 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4374 q^{3} - 18340 q^{5} - 680400 q^{7} + 9565938 q^{9}+O(q^{10}) $ 2 * q + 4374 * q^3 - 18340 * q^5 - 680400 * q^7 + 9565938 * q^9	\( 2 q + 4374 q^{3} - 18340 q^{5} - 680400 q^{7} + 9565938 q^{9} - 77720632 q^{11} - 207881332 q^{13} - 40109580 q^{15} - 1673313692 q^{17} - 2240887112 q^{19} - 1488034800 q^{21} + 16655801552 q^{23} - 2312200850 q^{25} + 20920706406 q^{27} - 139493975892 q^{29} - 43133623424 q^{31} - 169975022184 q^{33} - 619372303200 q^{35} - 1457105544996 q^{37} - 454636473084 q^{39} - 1377923837772 q^{41} - 2763654497656 q^{43} - 87719651460 q^{45} - 7396441682400 q^{47} - 2579485205486 q^{49} - 3659537044404 q^{51} - 5824660261252 q^{53} - 25932807442960 q^{55} - 4900820113944 q^{57} - 22792784775448 q^{59} + 14623417395116 q^{61} - 3254332107600 q^{63} + 37489085612840 q^{65} + 68414350989784 q^{67} + 36426237994224 q^{69} + 64404223302256 q^{71} + 229480150537460 q^{73} - 5056783258950 q^{75} + 311126750827200 q^{77} - 8902155902944 q^{79} + 45753584909922 q^{81} - 239472633418184 q^{83} + 634539486862840 q^{85} - 305073325275804 q^{87} + 472605248938452 q^{89} - 309453044594400 q^{91} - 94333234428288 q^{93} - 19\!\cdots\!60 q^{95}+ \cdots - 371735373516408 q^{99}+O(q^{100}) \) 2 * q + 4374 * q^3 - 18340 * q^5 - 680400 * q^7 + 9565938 * q^9 - 77720632 * q^11 - 207881332 * q^13 - 40109580 * q^15 - 1673313692 * q^17 - 2240887112 * q^19 - 1488034800 * q^21 + 16655801552 * q^23 - 2312200850 * q^25 + 20920706406 * q^27 - 139493975892 * q^29 - 43133623424 * q^31 - 169975022184 * q^33 - 619372303200 * q^35 - 1457105544996 * q^37 - 454636473084 * q^39 - 1377923837772 * q^41 - 2763654497656 * q^43 - 87719651460 * q^45 - 7396441682400 * q^47 - 2579485205486 * q^49 - 3659537044404 * q^51 - 5824660261252 * q^53 - 25932807442960 * q^55 - 4900820113944 * q^57 - 22792784775448 * q^59 + 14623417395116 * q^61 - 3254332107600 * q^63 + 37489085612840 * q^65 + 68414350989784 * q^67 + 36426237994224 * q^69 + 64404223302256 * q^71 + 229480150537460 * q^73 - 5056783258950 * q^75 + 311126750827200 * q^77 - 8902155902944 * q^79 + 45753584909922 * q^81 - 239472633418184 * q^83 + 634539486862840 * q^85 - 305073325275804 * q^87 + 472605248938452 * q^89 - 309453044594400 * q^91 - 94333234428288 * q^93 - 1992299281656560 * q^95 + 2032136651194948 * q^97 - 371735373516408 * q^99

Display 100 coefficients 10 coefficients

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
$3$	2187.00	0.577350
$3$	2187.00	0.577350
$4$	0	0
$5$	−180276.	−1.03196	−0.515981	−	0.856600i	$-0.672573\pi$
$5$	−180276.	−1.03196	−0.515981	+	0.856600i	$0.672573\pi$
$6$	0	0
$7$	1.48794e6	0.682887	0.341444	−	0.939902i	$-0.389084\pi$
$7$	1.48794e6	0.682887	0.341444	+	0.939902i	$0.389084\pi$
$8$	0	0
$9$	4.78297e6	0.333333
$10$	0	0
$11$	3.90021e7	0.603452	0.301726	−	0.953395i	$-0.402437\pi$
$11$	3.90021e7	0.603452	0.301726	+	0.953395i	$0.402437\pi$
$12$	0	0
$13$	−2.07919e8	−0.919009	−0.459504	−	0.888175i	$-0.651973\pi$
$13$	−2.07919e8	−0.919009	−0.459504	+	0.888175i	$0.651973\pi$
$14$	0	0
$15$	−3.94264e8	−0.595804
$16$	0	0
$17$	−2.64604e9	−1.56398	−0.781988	−	0.623293i	$-0.785794\pi$
$17$	−2.64604e9	−1.56398	−0.781988	+	0.623293i	$0.785794\pi$
$18$	0	0
$19$	4.76142e9	1.22204	0.611019	−	0.791616i	$-0.290760\pi$
$19$	4.76142e9	1.22204	0.611019	+	0.791616i	$0.290760\pi$
$20$	0	0
$21$	3.25412e9	0.394265
$22$	0	0
$23$	−5.16387e9	−0.316240	−0.158120	−	0.987420i	$-0.550543\pi$
$23$	−5.16387e9	−0.316240	−0.158120	+	0.987420i	$0.550543\pi$
$24$	0	0
$25$	1.98199e9	0.0649459
$26$	0	0
$27$	1.04604e10	0.192450
$28$	0	0
$29$	−1.10143e11	−1.18569	−0.592845	−	0.805316i	$-0.701995\pi$
$29$	−1.10143e11	−1.18569	−0.592845	+	0.805316i	$0.701995\pi$
$30$	0	0
$31$	−3.05870e10	−0.199675	−0.0998375	−	0.995004i	$-0.531832\pi$
$31$	−3.05870e10	−0.199675	−0.0998375	+	0.995004i	$0.531832\pi$
$32$	0	0
$33$	8.52976e10	0.348403
$34$	0	0
$35$	−2.68240e11	−0.704714
$36$	0	0
$37$	−7.68194e11	−1.33033	−0.665163	−	0.746698i	$-0.731638\pi$
$37$	−7.68194e11	−1.33033	−0.665163	+	0.746698i	$0.731638\pi$
$38$	0	0
$39$	−4.54720e11	−0.530590
$40$	0	0
$41$	−9.15551e11	−0.734181	−0.367091	−	0.930185i	$-0.619646\pi$
$41$	−9.15551e11	−0.734181	−0.367091	+	0.930185i	$0.619646\pi$
$42$	0	0
$43$	−1.32094e12	−0.741091	−0.370545	−	0.928814i	$-0.620829\pi$
$43$	−1.32094e12	−0.741091	−0.370545	+	0.928814i	$0.620829\pi$
$44$	0	0
$45$	−8.62256e11	−0.343987
$46$	0	0
$47$	−8.60974e11	−0.247888	−0.123944	−	0.992289i	$-0.539554\pi$
$47$	−8.60974e11	−0.247888	−0.123944	+	0.992289i	$0.539554\pi$
$48$	0	0
$49$	−2.53361e12	−0.533665
$50$	0	0
$51$	−5.78690e12	−0.902962
$52$	0	0
$53$	9.28543e11	0.108576	0.0542879	−	0.998525i	$-0.482711\pi$
$53$	9.28543e11	0.108576	0.0542879	+	0.998525i	$0.482711\pi$
$54$	0	0
$55$	−7.03115e12	−0.622740
$56$	0	0
$57$	1.04132e13	0.705543
$58$	0	0
$59$	−3.80695e13	−1.99153	−0.995766	−	0.0919215i	$-0.970699\pi$
$59$	−3.80695e13	−1.99153	−0.995766	+	0.0919215i	$0.970699\pi$
$60$	0	0
$61$	−1.83644e13	−0.748175	−0.374087	−	0.927393i	$-0.622044\pi$
$61$	−1.83644e13	−0.748175	−0.374087	+	0.927393i	$0.622044\pi$
$62$	0	0
$63$	7.11675e12	0.227629
$64$	0	0
$65$	3.74829e13	0.948382
$66$	0	0
$67$	8.19085e13	1.65108	0.825540	−	0.564344i	$-0.190871\pi$
$67$	8.19085e13	1.65108	0.825540	+	0.564344i	$0.190871\pi$
$68$	0	0
$69$	−1.12934e13	−0.182581
$70$	0	0
$71$	4.15622e13	0.542327	0.271163	−	0.962533i	$-0.412592\pi$
$71$	4.15622e13	0.542327	0.271163	+	0.962533i	$0.412592\pi$
$72$	0	0
$73$	1.55562e14	1.64810	0.824048	−	0.566521i	$-0.191711\pi$
$73$	1.55562e14	1.64810	0.824048	+	0.566521i	$0.191711\pi$
$74$	0	0
$75$	4.33461e12	0.0374965
$76$	0	0
$77$	5.80326e13	0.412090
$78$	0	0
$79$	−1.78006e14	−1.04287	−0.521437	−	0.853290i	$-0.674604\pi$
$79$	−1.78006e14	−1.04287	−0.521437	+	0.853290i	$0.674604\pi$
$80$	0	0
$81$	2.28768e13	0.111111
$82$	0	0
$83$	2.25716e13	0.0913010	0.0456505	−	0.998957i	$-0.485464\pi$
$83$	2.25716e13	0.0913010	0.0456505	+	0.998957i	$0.485464\pi$
$84$	0	0
$85$	4.77019e14	1.61396
$86$	0	0
$87$	−2.40882e14	−0.684559
$88$	0	0
$89$	5.76725e14	1.38211	0.691057	−	0.722800i	$-0.257145\pi$
$89$	5.76725e14	1.38211	0.691057	+	0.722800i	$0.257145\pi$
$90$	0	0
$91$	−3.09371e14	−0.627580
$92$	0	0
$93$	−6.68937e13	−0.115282
$94$	0	0
$95$	−8.58371e14	−1.26110
$96$	0	0
$97$	4.91989e14	0.618255	0.309128	−	0.951021i	$-0.399963\pi$
$97$	4.91989e14	0.618255	0.309128	+	0.951021i	$0.399963\pi$
$98$	0	0
$99$	1.86546e14	0.201151
$100$	0	0
$101$	−1.44221e15	−1.33850	−0.669251	−	0.743036i	$-0.733385\pi$
$101$	−1.44221e15	−1.33850	−0.669251	+	0.743036i	$0.733385\pi$
$102$	0	0
$103$	−4.69202e14	−0.375907	−0.187954	−	0.982178i	$-0.560185\pi$
$103$	−4.69202e14	−0.375907	−0.187954	+	0.982178i	$0.560185\pi$
$104$	0	0
$105$	−5.86640e14	−0.406867
$106$	0	0
$107$	−2.01230e15	−1.21147	−0.605737	−	0.795665i	$-0.707122\pi$
$107$	−2.01230e15	−1.21147	−0.605737	+	0.795665i	$0.707122\pi$
$108$	0	0
$109$	2.49631e15	1.30798	0.653988	−	0.756505i	$-0.273095\pi$
$109$	2.49631e15	1.30798	0.653988	+	0.756505i	$0.273095\pi$
$110$	0	0
$111$	−1.68004e15	−0.768064
$112$	0	0
$113$	−1.14810e15	−0.459085	−0.229542	−	0.973299i	$-0.573723\pi$
$113$	−1.14810e15	−0.459085	−0.229542	+	0.973299i	$0.573723\pi$
$114$	0	0
$115$	9.30924e14	0.326348
$116$	0	0
$117$	−9.94472e14	−0.306336
$118$	0	0
$119$	−3.93714e15	−1.06802
$120$	0	0
$121$	−2.65609e15	−0.635846
$122$	0	0
$123$	−2.00231e15	−0.423880
$124$	0	0
$125$	5.14429e15	0.964940
$126$	0	0
$127$	1.00922e16	1.68057	0.840285	−	0.542145i	$-0.182388\pi$
$127$	1.00922e16	1.68057	0.840285	+	0.542145i	$0.182388\pi$
$128$	0	0
$129$	−2.88891e15	−0.427869
$130$	0	0
$131$	7.72987e15	1.02009	0.510044	−	0.860148i	$-0.329629\pi$
$131$	7.72987e15	1.02009	0.510044	+	0.860148i	$0.329629\pi$
$132$	0	0
$133$	7.08469e15	0.834514
$134$	0	0
$135$	−1.88575e15	−0.198601
$136$	0	0
$137$	−2.09348e15	−0.197454	−0.0987268	−	0.995115i	$-0.531477\pi$
$137$	−2.09348e15	−0.197454	−0.0987268	+	0.995115i	$0.531477\pi$
$138$	0	0
$139$	2.30677e16	1.95161	0.975806	−	0.218640i	$-0.0701619\pi$
$139$	2.30677e16	1.95161	0.975806	+	0.218640i	$0.0701619\pi$
$140$	0	0
$141$	−1.88295e15	−0.143118
$142$	0	0
$143$	−8.10929e15	−0.554578
$144$	0	0
$145$	1.98562e16	1.22359
$146$	0	0
$147$	−5.54100e15	−0.308112
$148$	0	0
$149$	−3.00896e16	−1.51189	−0.755943	−	0.654638i	$-0.772821\pi$
$149$	−3.00896e16	−1.51189	−0.755943	+	0.654638i	$0.772821\pi$
$150$	0	0
$151$	2.01711e16	0.917072	0.458536	−	0.888676i	$-0.348374\pi$
$151$	2.01711e16	0.917072	0.458536	+	0.888676i	$0.348374\pi$
$152$	0	0
$153$	−1.26559e16	−0.521325
$154$	0	0
$155$	5.51411e15	0.206057
$156$	0	0
$157$	−1.86362e15	−0.0632572	−0.0316286	−	0.999500i	$-0.510069\pi$
$157$	−1.86362e15	−0.0632572	−0.0316286	+	0.999500i	$0.510069\pi$
$158$	0	0
$159$	2.03072e15	0.0626863
$160$	0	0
$161$	−7.68351e15	−0.215956
$162$	0	0
$163$	7.03185e15	0.180162	0.0900808	−	0.995934i	$-0.471287\pi$
$163$	7.03185e15	0.180162	0.0900808	+	0.995934i	$0.471287\pi$
$164$	0	0
$165$	−1.53771e16	−0.359539
$166$	0	0
$167$	3.76392e15	0.0804020	0.0402010	−	0.999192i	$-0.487200\pi$
$167$	3.76392e15	0.0804020	0.0402010	+	0.999192i	$0.487200\pi$
$168$	0	0
$169$	−7.95546e15	−0.155423
$170$	0	0
$171$	2.27737e16	0.407346
$172$	0	0
$173$	−8.10801e16	−1.32913	−0.664567	−	0.747229i	$-0.731384\pi$
$173$	−8.10801e16	−1.32913	−0.664567	+	0.747229i	$0.731384\pi$
$174$	0	0
$175$	2.94908e15	0.0443507
$176$	0	0
$177$	−8.32581e16	−1.14981
$178$	0	0
$179$	−3.36381e16	−0.427005	−0.213503	−	0.976942i	$-0.568487\pi$
$179$	−3.36381e16	−0.427005	−0.213503	+	0.976942i	$0.568487\pi$
$180$	0	0
$181$	−7.25177e16	−0.846944	−0.423472	−	0.905909i	$-0.639189\pi$
$181$	−7.25177e16	−0.846944	−0.423472	+	0.905909i	$0.639189\pi$
$182$	0	0
$183$	−4.01629e16	−0.431959
$184$	0	0
$185$	1.38487e17	1.37285
$186$	0	0
$187$	−1.03201e17	−0.943784
$188$	0	0
$189$	1.55643e16	0.131422
$190$	0	0
$191$	−1.46831e17	−1.14569	−0.572845	−	0.819664i	$-0.694160\pi$
$191$	−1.46831e17	−1.14569	−0.572845	+	0.819664i	$0.694160\pi$
$192$	0	0
$193$	−1.95671e16	−0.141204	−0.0706019	−	0.997505i	$-0.522492\pi$
$193$	−1.95671e16	−0.141204	−0.0706019	+	0.997505i	$0.522492\pi$
$194$	0	0
$195$	8.19752e16	0.547549
$196$	0	0
$197$	1.99325e17	1.23329	0.616645	−	0.787241i	$-0.288491\pi$
$197$	1.99325e17	1.23329	0.616645	+	0.787241i	$0.288491\pi$
$198$	0	0
$199$	3.09007e17	1.77244	0.886219	−	0.463267i	$-0.153323\pi$
$199$	3.09007e17	1.77244	0.886219	+	0.463267i	$0.153323\pi$
$200$	0	0
$201$	1.79134e17	0.953251
$202$	0	0
$203$	−1.63886e17	−0.809693
$204$	0	0
$205$	1.65052e17	0.757647
$206$	0	0
$207$	−2.46986e16	−0.105413
$208$	0	0
$209$	1.85705e17	0.737441
$210$	0	0
$211$	−5.17876e17	−1.91473	−0.957364	−	0.288885i	$-0.906715\pi$
$211$	−5.17876e17	−1.91473	−0.957364	+	0.288885i	$0.906715\pi$
$212$	0	0
$213$	9.08965e16	0.313112
$214$	0	0
$215$	2.38135e17	0.764777
$216$	0	0
$217$	−4.55115e16	−0.136356
$218$	0	0
$219$	3.40214e17	0.951528
$220$	0	0
$221$	5.50164e17	1.43731
$222$	0	0
$223$	−1.24384e16	−0.0303722	−0.0151861	−	0.999885i	$-0.504834\pi$
$223$	−1.24384e16	−0.0303722	−0.0151861	+	0.999885i	$0.504834\pi$
$224$	0	0
$225$	9.47980e15	0.0216486
$226$	0	0
$227$	5.95457e16	0.127250	0.0636248	−	0.997974i	$-0.479734\pi$
$227$	5.95457e16	0.127250	0.0636248	+	0.997974i	$0.479734\pi$
$228$	0	0
$229$	−8.30572e17	−1.66192	−0.830962	−	0.556330i	$-0.812209\pi$
$229$	−8.30572e17	−1.66192	−0.830962	+	0.556330i	$0.812209\pi$
$230$	0	0
$231$	1.26917e17	0.237920
$232$	0	0
$233$	4.86376e17	0.854678	0.427339	−	0.904091i	$-0.359451\pi$
$233$	4.86376e17	0.854678	0.427339	+	0.904091i	$0.359451\pi$
$234$	0	0
$235$	1.55213e17	0.255811
$236$	0	0
$237$	−3.89299e17	−0.602103
$238$	0	0
$239$	−1.15538e18	−1.67780	−0.838901	−	0.544284i	$-0.816802\pi$
$239$	−1.15538e18	−1.67780	−0.838901	+	0.544284i	$0.816802\pi$
$240$	0	0
$241$	−7.02560e17	−0.958420	−0.479210	−	0.877700i	$-0.659077\pi$
$241$	−7.02560e17	−0.958420	−0.479210	+	0.877700i	$0.659077\pi$
$242$	0	0
$243$	5.00315e16	0.0641500
$244$	0	0
$245$	4.56749e17	0.550722
$246$	0	0
$247$	−9.89991e17	−1.12306
$248$	0	0
$249$	4.93640e16	0.0527127
$250$	0	0
$251$	−5.63469e16	−0.0566653	−0.0283327	−	0.999599i	$-0.509020\pi$
$251$	−5.63469e16	−0.0566653	−0.0283327	+	0.999599i	$0.509020\pi$
$252$	0	0
$253$	−2.01402e17	−0.190836
$254$	0	0
$255$	1.04324e18	0.931823
$256$	0	0
$257$	4.50968e17	0.379881	0.189940	−	0.981796i	$-0.439171\pi$
$257$	4.50968e17	0.379881	0.189940	+	0.981796i	$0.439171\pi$
$258$	0	0
$259$	−1.14302e18	−0.908463
$260$	0	0
$261$	−5.26810e17	−0.395230
$262$	0	0
$263$	−7.13174e17	−0.505275	−0.252637	−	0.967561i	$-0.581298\pi$
$263$	−7.13174e17	−0.505275	−0.252637	+	0.967561i	$0.581298\pi$
$264$	0	0
$265$	−1.67394e17	−0.112046
$266$	0	0
$267$	1.26130e18	0.797964
$268$	0	0
$269$	−2.32811e18	−1.39271	−0.696357	−	0.717695i	$-0.745197\pi$
$269$	−2.32811e18	−1.39271	−0.696357	+	0.717695i	$0.745197\pi$
$270$	0	0
$271$	2.29160e18	1.29679	0.648394	−	0.761305i	$-0.275441\pi$
$271$	2.29160e18	1.29679	0.648394	+	0.761305i	$0.275441\pi$
$272$	0	0
$273$	−6.76594e17	−0.362333
$274$	0	0
$275$	7.73018e16	0.0391917
$276$	0	0
$277$	−4.15077e18	−1.99311	−0.996554	−	0.0829450i	$-0.973567\pi$
$277$	−4.15077e18	−1.99311	−0.996554	+	0.0829450i	$0.973567\pi$
$278$	0	0
$279$	−1.46297e17	−0.0665583
$280$	0	0
$281$	2.67332e18	1.15280	0.576400	−	0.817168i	$-0.304457\pi$
$281$	2.67332e18	1.15280	0.576400	+	0.817168i	$0.304457\pi$
$282$	0	0
$283$	3.19921e18	1.30811	0.654055	−	0.756447i	$-0.273066\pi$
$283$	3.19921e18	1.30811	0.654055	+	0.756447i	$0.273066\pi$
$284$	0	0
$285$	−1.87726e18	−0.728094
$286$	0	0
$287$	−1.36228e18	−0.501363
$288$	0	0
$289$	4.13912e18	1.44602
$290$	0	0
$291$	1.07598e18	0.356950
$292$	0	0
$293$	2.61525e18	0.824149	0.412075	−	0.911150i	$-0.364804\pi$
$293$	2.61525e18	0.824149	0.412075	+	0.911150i	$0.364804\pi$
$294$	0	0
$295$	6.86304e18	2.05519
$296$	0	0
$297$	4.07976e17	0.116134
$298$	0	0
$299$	1.07367e18	0.290627
$300$	0	0
$301$	−1.96548e18	−0.506081
$302$	0	0
$303$	−3.15412e18	−0.772785
$304$	0	0
$305$	3.31067e18	0.772088
$306$	0	0
$307$	3.27984e18	0.728308	0.364154	−	0.931339i	$-0.381358\pi$
$307$	3.27984e18	0.728308	0.364154	+	0.931339i	$0.381358\pi$
$308$	0	0
$309$	−1.02615e18	−0.217030
$310$	0	0
$311$	1.63232e17	0.0328929	0.0164465	−	0.999865i	$-0.494765\pi$
$311$	1.63232e17	0.0328929	0.0164465	+	0.999865i	$0.494765\pi$
$312$	0	0
$313$	2.41673e18	0.464136	0.232068	−	0.972700i	$-0.425451\pi$
$313$	2.41673e18	0.464136	0.232068	+	0.972700i	$0.425451\pi$
$314$	0	0
$315$	−1.28298e18	−0.234905
$316$	0	0
$317$	−1.53593e18	−0.268180	−0.134090	−	0.990969i	$-0.542811\pi$
$317$	−1.53593e18	−0.268180	−0.134090	+	0.990969i	$0.542811\pi$
$318$	0	0
$319$	−4.29580e18	−0.715507
$320$	0	0
$321$	−4.40090e18	−0.699445
$322$	0	0
$323$	−1.25989e19	−1.91124
$324$	0	0
$325$	−4.12094e17	−0.0596858
$326$	0	0
$327$	5.45943e18	0.755161
$328$	0	0
$329$	−1.28107e18	−0.169280
$330$	0	0
$331$	1.02981e19	1.30032	0.650158	−	0.759799i	$-0.274703\pi$
$331$	1.02981e19	1.30032	0.650158	+	0.759799i	$0.274703\pi$
$332$	0	0
$333$	−3.67425e18	−0.443442
$334$	0	0
$335$	−1.47662e19	−1.70385
$336$	0	0
$337$	9.20546e18	1.01583	0.507915	−	0.861407i	$-0.330416\pi$
$337$	9.20546e18	1.01583	0.507915	+	0.861407i	$0.330416\pi$
$338$	0	0
$339$	−2.51090e18	−0.265053
$340$	0	0
$341$	−1.19296e18	−0.120494
$342$	0	0
$343$	−1.08339e19	−1.04732
$344$	0	0
$345$	2.03593e18	0.188417
$346$	0	0
$347$	−1.40238e19	−1.24279	−0.621393	−	0.783499i	$-0.713433\pi$
$347$	−1.40238e19	−1.24279	−0.621393	+	0.783499i	$0.713433\pi$
$348$	0	0
$349$	1.91289e18	0.162367	0.0811837	−	0.996699i	$-0.474130\pi$
$349$	1.91289e18	0.162367	0.0811837	+	0.996699i	$0.474130\pi$
$350$	0	0
$351$	−2.17491e18	−0.176863
$352$	0	0
$353$	2.19204e19	1.70820	0.854100	−	0.520108i	$-0.174108\pi$
$353$	2.19204e19	1.70820	0.854100	+	0.520108i	$0.174108\pi$
$354$	0	0
$355$	−7.49268e18	−0.559660
$356$	0	0
$357$	−8.61054e18	−0.616621
$358$	0	0
$359$	1.18890e19	0.816465	0.408232	−	0.912878i	$-0.366145\pi$
$359$	1.18890e19	0.816465	0.408232	+	0.912878i	$0.366145\pi$
$360$	0	0
$361$	7.48998e18	0.493375
$362$	0	0
$363$	−5.80886e18	−0.367106
$364$	0	0
$365$	−2.80442e19	−1.70077
$366$	0	0
$367$	−8.04460e18	−0.468284	−0.234142	−	0.972202i	$-0.575228\pi$
$367$	−8.04460e18	−0.468284	−0.234142	+	0.972202i	$0.575228\pi$
$368$	0	0
$369$	−4.37905e18	−0.244727
$370$	0	0
$371$	1.38161e18	0.0741450
$372$	0	0
$373$	6.28069e18	0.323736	0.161868	−	0.986812i	$-0.448248\pi$
$373$	6.28069e18	0.323736	0.161868	+	0.986812i	$0.448248\pi$
$374$	0	0
$375$	1.12506e19	0.557109
$376$	0	0
$377$	2.29008e19	1.08966
$378$	0	0
$379$	−1.77049e18	−0.0809652	−0.0404826	−	0.999180i	$-0.512890\pi$
$379$	−1.77049e18	−0.0809652	−0.0404826	+	0.999180i	$0.512890\pi$
$380$	0	0
$381$	2.20716e19	0.970277
$382$	0	0
$383$	3.93577e19	1.66356	0.831781	−	0.555104i	$-0.187321\pi$
$383$	3.93577e19	1.66356	0.831781	+	0.555104i	$0.187321\pi$
$384$	0	0
$385$	−1.04619e19	−0.425261
$386$	0	0
$387$	−6.31804e18	−0.247030
$388$	0	0
$389$	−8.94320e18	−0.336412	−0.168206	−	0.985752i	$-0.553797\pi$
$389$	−8.94320e18	−0.336412	−0.168206	+	0.985752i	$0.553797\pi$
$390$	0	0
$391$	1.36638e19	0.494592
$392$	0	0
$393$	1.69052e19	0.588948
$394$	0	0
$395$	3.20903e19	1.07621
$396$	0	0
$397$	−6.03814e19	−1.94973	−0.974864	−	0.222801i	$-0.928480\pi$
$397$	−6.03814e19	−1.94973	−0.974864	+	0.222801i	$0.928480\pi$
$398$	0	0
$399$	1.54942e19	0.481807
$400$	0	0
$401$	3.38522e19	1.01392	0.506961	−	0.861969i	$-0.330769\pi$
$401$	3.38522e19	1.01392	0.506961	+	0.861969i	$0.330769\pi$
$402$	0	0
$403$	6.35962e18	0.183503
$404$	0	0
$405$	−4.12415e18	−0.114662
$406$	0	0
$407$	−2.99612e19	−0.802788
$408$	0	0
$409$	6.86132e19	1.77208	0.886039	−	0.463610i	$-0.153446\pi$
$409$	6.86132e19	1.77208	0.886039	+	0.463610i	$0.153446\pi$
$410$	0	0
$411$	−4.57845e18	−0.114000
$412$	0	0
$413$	−5.66450e19	−1.35999
$414$	0	0
$415$	−4.06912e18	−0.0942192
$416$	0	0
$417$	5.04491e19	1.12676
$418$	0	0
$419$	2.41796e19	0.521007	0.260504	−	0.965473i	$-0.416111\pi$
$419$	2.41796e19	0.521007	0.260504	+	0.965473i	$0.416111\pi$
$420$	0	0
$421$	−4.62042e19	−0.960651	−0.480325	−	0.877090i	$-0.659481\pi$
$421$	−4.62042e19	−0.960651	−0.480325	+	0.877090i	$0.659481\pi$
$422$	0	0
$423$	−4.11801e18	−0.0826295
$424$	0	0
$425$	−5.24443e18	−0.101574
$426$	0	0
$427$	−2.73251e19	−0.510919
$428$	0	0
$429$	−1.77350e19	−0.320186
$430$	0	0
$431$	6.20343e19	1.08156	0.540782	−	0.841163i	$-0.318128\pi$
$431$	6.20343e19	1.08156	0.540782	+	0.841163i	$0.318128\pi$
$432$	0	0
$433$	−8.88817e19	−1.49676	−0.748382	−	0.663268i	$-0.769169\pi$
$433$	−8.88817e19	−1.49676	−0.748382	+	0.663268i	$0.769169\pi$
$434$	0	0
$435$	4.34254e19	0.706439
$436$	0	0
$437$	−2.45874e19	−0.386457
$438$	0	0
$439$	−8.31215e19	−1.26249	−0.631247	−	0.775581i	$-0.717457\pi$
$439$	−8.31215e19	−1.26249	−0.631247	+	0.775581i	$0.717457\pi$
$440$	0	0
$441$	−1.21182e19	−0.177888
$442$	0	0
$443$	9.77868e18	0.138756	0.0693781	−	0.997590i	$-0.477899\pi$
$443$	9.77868e18	0.138756	0.0693781	+	0.997590i	$0.477899\pi$
$444$	0	0
$445$	−1.03970e20	−1.42629
$446$	0	0
$447$	−6.58059e19	−0.872887
$448$	0	0
$449$	−1.32320e20	−1.69737	−0.848686	−	0.528896i	$-0.822606\pi$
$449$	−1.32320e20	−1.69737	−0.848686	+	0.528896i	$0.822606\pi$
$450$	0	0
$451$	−3.57084e19	−0.443043
$452$	0	0
$453$	4.41143e19	0.529472
$454$	0	0
$455$	5.57722e19	0.647638
$456$	0	0
$457$	5.82187e19	0.654170	0.327085	−	0.944995i	$-0.393934\pi$
$457$	5.82187e19	0.654170	0.327085	+	0.944995i	$0.393934\pi$
$458$	0	0
$459$	−2.76786e19	−0.300987
$460$	0	0
$461$	2.69152e19	0.283296	0.141648	−	0.989917i	$-0.454760\pi$
$461$	2.69152e19	0.283296	0.141648	+	0.989917i	$0.454760\pi$
$462$	0	0
$463$	−8.40379e18	−0.0856283	−0.0428142	−	0.999083i	$-0.513632\pi$
$463$	−8.40379e18	−0.0856283	−0.0428142	+	0.999083i	$0.513632\pi$
$464$	0	0
$465$	1.20594e19	0.118967
$466$	0	0
$467$	1.70165e20	1.62553	0.812763	−	0.582594i	$-0.197962\pi$
$467$	1.70165e20	1.62553	0.812763	+	0.582594i	$0.197962\pi$
$468$	0	0
$469$	1.21875e20	1.12750
$470$	0	0
$471$	−4.07574e18	−0.0365216
$472$	0	0
$473$	−5.15196e19	−0.447213
$474$	0	0
$475$	9.43709e18	0.0793662
$476$	0	0
$477$	4.44119e18	0.0361919
$478$	0	0
$479$	−2.27871e19	−0.179958	−0.0899792	−	0.995944i	$-0.528680\pi$
$479$	−2.27871e19	−0.179958	−0.0899792	+	0.995944i	$0.528680\pi$
$480$	0	0
$481$	1.59722e20	1.22258
$482$	0	0
$483$	−1.68038e19	−0.124682
$484$	0	0
$485$	−8.86941e19	−0.638016
$486$	0	0
$487$	−1.00043e20	−0.697785	−0.348892	−	0.937163i	$-0.613442\pi$
$487$	−1.00043e20	−0.697785	−0.348892	+	0.937163i	$0.613442\pi$
$488$	0	0
$489$	1.53786e19	0.104016
$490$	0	0
$491$	−1.86698e20	−1.22469	−0.612347	−	0.790589i	$-0.709774\pi$
$491$	−1.86698e20	−1.22469	−0.612347	+	0.790589i	$0.709774\pi$
$492$	0	0
$493$	2.91443e20	1.85439
$494$	0	0
$495$	−3.36298e19	−0.207580
$496$	0	0
$497$	6.18419e19	0.370348
$498$	0	0
$499$	−1.19986e19	−0.0697229	−0.0348615	−	0.999392i	$-0.511099\pi$
$499$	−1.19986e19	−0.0697229	−0.0348615	+	0.999392i	$0.511099\pi$
$500$	0	0
$501$	8.23170e18	0.0464201
$502$	0	0
$503$	−4.88293e19	−0.267252	−0.133626	−	0.991032i	$-0.542662\pi$
$503$	−4.88293e19	−0.267252	−0.133626	+	0.991032i	$0.542662\pi$
$504$	0	0
$505$	2.59997e20	1.38128
$506$	0	0
$507$	−1.73986e19	−0.0897334
$508$	0	0
$509$	4.07524e19	0.204066	0.102033	−	0.994781i	$-0.467465\pi$
$509$	4.07524e19	0.204066	0.102033	+	0.994781i	$0.467465\pi$
$510$	0	0
$511$	2.31466e20	1.12546
$512$	0	0
$513$	4.98061e19	0.235181
$514$	0	0
$515$	8.45861e19	0.387922
$516$	0	0
$517$	−3.35798e19	−0.149589
$518$	0	0
$519$	−1.77322e20	−0.767375
$520$	0	0
$521$	8.05648e19	0.338737	0.169368	−	0.985553i	$-0.445827\pi$
$521$	8.05648e19	0.338737	0.169368	+	0.985553i	$0.445827\pi$
$522$	0	0
$523$	5.97463e19	0.244089	0.122045	−	0.992525i	$-0.461055\pi$
$523$	5.97463e19	0.244089	0.122045	+	0.992525i	$0.461055\pi$
$524$	0	0
$525$	6.44963e18	0.0256059
$526$	0	0
$527$	8.09345e19	0.312287
$528$	0	0
$529$	−2.39970e20	−0.899992
$530$	0	0
$531$	−1.82085e20	−0.663844
$532$	0	0
$533$	1.90361e20	0.674719
$534$	0	0
$535$	3.62770e20	1.25020
$536$	0	0
$537$	−7.35665e19	−0.246532
$538$	0	0
$539$	−9.88159e19	−0.322041
$540$	0	0
$541$	3.73325e19	0.118334	0.0591668	−	0.998248i	$-0.481156\pi$
$541$	3.73325e19	0.118334	0.0591668	+	0.998248i	$0.481156\pi$
$542$	0	0
$543$	−1.58596e20	−0.488984
$544$	0	0
$545$	−4.50026e20	−1.34978
$546$	0	0
$547$	−3.54198e20	−1.03357	−0.516787	−	0.856114i	$-0.672872\pi$
$547$	−3.54198e20	−1.03357	−0.516787	+	0.856114i	$0.672872\pi$
$548$	0	0
$549$	−8.78364e19	−0.249392
$550$	0	0
$551$	−5.24436e20	−1.44896
$552$	0	0
$553$	−2.64862e20	−0.712165
$554$	0	0
$555$	3.02872e20	0.792613
$556$	0	0
$557$	−2.54417e20	−0.648084	−0.324042	−	0.946043i	$-0.605042\pi$
$557$	−2.54417e20	−0.648084	−0.324042	+	0.946043i	$0.605042\pi$
$558$	0	0
$559$	2.74650e20	0.681069
$560$	0	0
$561$	−2.25701e20	−0.544894
$562$	0	0
$563$	−2.20356e19	−0.0517979	−0.0258989	−	0.999665i	$-0.508245\pi$
$563$	−2.20356e19	−0.0517979	−0.0258989	+	0.999665i	$0.508245\pi$
$564$	0	0
$565$	2.06976e20	0.473758
$566$	0	0
$567$	3.40392e19	0.0758764
$568$	0	0
$569$	6.27873e19	0.136311	0.0681553	−	0.997675i	$-0.478289\pi$
$569$	6.27873e19	0.136311	0.0681553	+	0.997675i	$0.478289\pi$
$570$	0	0
$571$	−2.68918e20	−0.568656	−0.284328	−	0.958727i	$-0.591770\pi$
$571$	−2.68918e20	−0.568656	−0.284328	+	0.958727i	$0.591770\pi$
$572$	0	0
$573$	−3.21119e20	−0.661465
$574$	0	0
$575$	−1.02347e19	−0.0205385
$576$	0	0
$577$	−8.77749e19	−0.171614	−0.0858068	−	0.996312i	$-0.527347\pi$
$577$	−8.77749e19	−0.171614	−0.0858068	+	0.996312i	$0.527347\pi$
$578$	0	0
$579$	−4.27932e19	−0.0815241
$580$	0	0
$581$	3.35850e19	0.0623483
$582$	0	0
$583$	3.62151e19	0.0655203
$584$	0	0
$585$	1.79280e20	0.316127
$586$	0	0
$587$	7.86642e19	0.135205	0.0676023	−	0.997712i	$-0.478465\pi$
$587$	7.86642e19	0.135205	0.0676023	+	0.997712i	$0.478465\pi$
$588$	0	0
$589$	−1.45637e20	−0.244010
$590$	0	0
$591$	4.35924e20	0.712041
$592$	0	0
$593$	−5.19336e20	−0.827062	−0.413531	−	0.910490i	$-0.635705\pi$
$593$	−5.19336e20	−0.827062	−0.413531	+	0.910490i	$0.635705\pi$
$594$	0	0
$595$	7.09774e20	1.10216
$596$	0	0
$597$	6.75799e20	1.02332
$598$	0	0
$599$	8.64161e20	1.27613	0.638063	−	0.769984i	$-0.279736\pi$
$599$	8.64161e20	1.27613	0.638063	+	0.769984i	$0.279736\pi$
$600$	0	0
$601$	−1.03488e21	−1.49049	−0.745246	−	0.666789i	$-0.767668\pi$
$601$	−1.03488e21	−1.49049	−0.745246	+	0.666789i	$0.767668\pi$
$602$	0	0
$603$	3.91766e20	0.550360
$604$	0	0
$605$	4.78829e20	0.656169
$606$	0	0
$607$	−7.83953e20	−1.04803	−0.524016	−	0.851708i	$-0.675567\pi$
$607$	−7.83953e20	−1.04803	−0.524016	+	0.851708i	$0.675567\pi$
$608$	0	0
$609$	−3.58418e20	−0.467477
$610$	0	0
$611$	1.79013e20	0.227812
$612$	0	0
$613$	6.35634e20	0.789321	0.394660	−	0.918827i	$-0.370862\pi$
$613$	6.35634e20	0.789321	0.394660	+	0.918827i	$0.370862\pi$
$614$	0	0
$615$	3.60969e20	0.437428
$616$	0	0
$617$	6.35838e20	0.751982	0.375991	−	0.926623i	$-0.377302\pi$
$617$	6.35838e20	0.751982	0.375991	+	0.926623i	$0.377302\pi$
$618$	0	0
$619$	6.93312e20	0.800293	0.400146	−	0.916451i	$-0.368959\pi$
$619$	6.93312e20	0.800293	0.400146	+	0.916451i	$0.368959\pi$
$620$	0	0
$621$	−5.40159e19	−0.0608604
$622$	0	0
$623$	8.58131e20	0.943829
$624$	0	0
$625$	−9.87880e20	−1.06073
$626$	0	0
$627$	4.06137e20	0.425762
$628$	0	0
$629$	2.03267e21	2.08060
$630$	0	0
$631$	−5.31697e20	−0.531428	−0.265714	−	0.964052i	$-0.585608\pi$
$631$	−5.31697e20	−0.531428	−0.265714	+	0.964052i	$0.585608\pi$
$632$	0	0
$633$	−1.13259e21	−1.10547
$634$	0	0
$635$	−1.81938e21	−1.73428
$636$	0	0
$637$	5.26786e20	0.490443
$638$	0	0
$639$	1.98791e20	0.180776
$640$	0	0
$641$	9.53876e20	0.847338	0.423669	−	0.905817i	$-0.360742\pi$
$641$	9.53876e20	0.847338	0.423669	+	0.905817i	$0.360742\pi$
$642$	0	0
$643$	4.77350e20	0.414242	0.207121	−	0.978315i	$-0.433591\pi$
$643$	4.77350e20	0.414242	0.207121	+	0.978315i	$0.433591\pi$
$644$	0	0
$645$	5.20802e20	0.441544
$646$	0	0
$647$	−6.56151e19	−0.0543528	−0.0271764	−	0.999631i	$-0.508652\pi$
$647$	−6.56151e19	−0.0543528	−0.0271764	+	0.999631i	$0.508652\pi$
$648$	0	0
$649$	−1.48479e21	−1.20179
$650$	0	0
$651$	−9.95336e19	−0.0787249
$652$	0	0
$653$	1.02258e21	0.790405	0.395202	−	0.918594i	$-0.370674\pi$
$653$	1.02258e21	0.790405	0.395202	+	0.918594i	$0.370674\pi$
$654$	0	0
$655$	−1.39351e21	−1.05269
$656$	0	0
$657$	7.44049e20	0.549365
$658$	0	0
$659$	−1.53156e21	−1.10533	−0.552666	−	0.833403i	$-0.686390\pi$
$659$	−1.53156e21	−1.10533	−0.552666	+	0.833403i	$0.686390\pi$
$660$	0	0
$661$	−1.86754e21	−1.31752	−0.658762	−	0.752352i	$-0.728919\pi$
$661$	−1.86754e21	−1.31752	−0.658762	+	0.752352i	$0.728919\pi$
$662$	0	0
$663$	1.20321e21	0.829830
$664$	0	0
$665$	−1.27720e21	−0.861187
$666$	0	0
$667$	5.68764e20	0.374963
$668$	0	0
$669$	−2.72027e19	−0.0175354
$670$	0	0
$671$	−7.16250e20	−0.451487
$672$	0	0
$673$	−2.69039e20	−0.165845	−0.0829225	−	0.996556i	$-0.526425\pi$
$673$	−2.69039e20	−0.165845	−0.0829225	+	0.996556i	$0.526425\pi$
$674$	0	0
$675$	2.07323e19	0.0124988
$676$	0	0
$677$	6.96920e20	0.410930	0.205465	−	0.978664i	$-0.434129\pi$
$677$	6.96920e20	0.410930	0.205465	+	0.978664i	$0.434129\pi$
$678$	0	0
$679$	7.32049e20	0.422199
$680$	0	0
$681$	1.30226e20	0.0734676
$682$	0	0
$683$	6.53069e20	0.360416	0.180208	−	0.983629i	$-0.442323\pi$
$683$	6.53069e20	0.360416	0.180208	+	0.983629i	$0.442323\pi$
$684$	0	0
$685$	3.77405e20	0.203765
$686$	0	0
$687$	−1.81646e21	−0.959512
$688$	0	0
$689$	−1.93062e20	−0.0997821
$690$	0	0
$691$	−1.57344e21	−0.795728	−0.397864	−	0.917444i	$-0.630248\pi$
$691$	−1.57344e21	−0.795728	−0.397864	+	0.917444i	$0.630248\pi$
$692$	0	0
$693$	2.77568e20	0.137363
$694$	0	0
$695$	−4.15856e21	−2.01399
$696$	0	0
$697$	2.42259e21	1.14824
$698$	0	0
$699$	1.06370e21	0.493449
$700$	0	0
$701$	8.61982e20	0.391393	0.195696	−	0.980665i	$-0.437303\pi$
$701$	8.61982e20	0.391393	0.195696	+	0.980665i	$0.437303\pi$
$702$	0	0
$703$	−3.65769e21	−1.62571
$704$	0	0
$705$	3.39451e20	0.147693
$706$	0	0
$707$	−2.14592e21	−0.914047
$708$	0	0
$709$	−3.35234e20	−0.139798	−0.0698990	−	0.997554i	$-0.522268\pi$
$709$	−3.35234e20	−0.139798	−0.0698990	+	0.997554i	$0.522268\pi$
$710$	0	0
$711$	−8.51398e20	−0.347625
$712$	0	0
$713$	1.57947e20	0.0631452
$714$	0	0
$715$	1.46191e21	0.572303
$716$	0	0
$717$	−2.52682e21	−0.968680
$718$	0	0
$719$	8.77763e20	0.329542	0.164771	−	0.986332i	$-0.447311\pi$
$719$	8.77763e20	0.329542	0.164771	+	0.986332i	$0.447311\pi$
$720$	0	0
$721$	−6.98143e20	−0.256702
$722$	0	0
$723$	−1.53650e21	−0.553344
$724$	0	0
$725$	−2.18302e20	−0.0770057
$726$	0	0
$727$	2.41420e21	0.834189	0.417094	−	0.908863i	$-0.363048\pi$
$727$	2.41420e21	0.834189	0.417094	+	0.908863i	$0.363048\pi$
$728$	0	0
$729$	1.09419e20	0.0370370
$730$	0	0
$731$	3.49528e21	1.15905
$732$	0	0
$733$	4.91963e21	1.59828	0.799140	−	0.601145i	$-0.205289\pi$
$733$	4.91963e21	1.59828	0.799140	+	0.601145i	$0.205289\pi$
$734$	0	0
$735$	9.98911e20	0.317959
$736$	0	0
$737$	3.19460e21	0.996347
$738$	0	0
$739$	2.65946e21	0.812755	0.406377	−	0.913705i	$-0.366792\pi$
$739$	2.65946e21	0.812755	0.406377	+	0.913705i	$0.366792\pi$
$740$	0	0
$741$	−2.16511e21	−0.648401
$742$	0	0
$743$	4.83744e21	1.41971	0.709854	−	0.704349i	$-0.248761\pi$
$743$	4.83744e21	1.41971	0.709854	+	0.704349i	$0.248761\pi$
$744$	0	0
$745$	5.42444e21	1.56021
$746$	0	0
$747$	1.07959e20	0.0304337
$748$	0	0
$749$	−2.99417e21	−0.827301
$750$	0	0
$751$	4.22604e21	1.14455	0.572274	−	0.820063i	$-0.306061\pi$
$751$	4.22604e21	1.14455	0.572274	+	0.820063i	$0.306061\pi$
$752$	0	0
$753$	−1.23231e20	−0.0327157
$754$	0	0
$755$	−3.63638e21	−0.946383
$756$	0	0
$757$	−2.96399e21	−0.756235	−0.378118	−	0.925758i	$-0.623429\pi$
$757$	−2.96399e21	−0.756235	−0.378118	+	0.925758i	$0.623429\pi$
$758$	0	0
$759$	−4.40466e20	−0.110179
$760$	0	0
$761$	−7.04604e21	−1.72807	−0.864033	−	0.503435i	$-0.832069\pi$
$761$	−7.04604e21	−1.72807	−0.864033	+	0.503435i	$0.832069\pi$
$762$	0	0
$763$	3.71435e21	0.893201
$764$	0	0
$765$	2.28157e21	0.537988
$766$	0	0
$767$	7.91539e21	1.83024
$768$	0	0
$769$	−1.87704e21	−0.425623	−0.212811	−	0.977093i	$-0.568262\pi$
$769$	−1.87704e21	−0.425623	−0.212811	+	0.977093i	$0.568262\pi$
$770$	0	0
$771$	9.86267e20	0.219324
$772$	0	0
$773$	−1.89238e21	−0.412726	−0.206363	−	0.978476i	$-0.566163\pi$
$773$	−1.89238e21	−0.412726	−0.206363	+	0.978476i	$0.566163\pi$
$774$	0	0
$775$	−6.06231e19	−0.0129681
$776$	0	0
$777$	−2.49979e21	−0.524501
$778$	0	0
$779$	−4.35932e21	−0.897197
$780$	0	0
$781$	1.62101e21	0.327268
$782$	0	0
$783$	−1.15213e21	−0.228186
$784$	0	0
$785$	3.35967e20	0.0652791
$786$	0	0
$787$	−3.32042e21	−0.632968	−0.316484	−	0.948598i	$-0.602502\pi$
$787$	−3.32042e21	−0.632968	−0.316484	+	0.948598i	$0.602502\pi$
$788$	0	0
$789$	−1.55971e21	−0.291721
$790$	0	0
$791$	−1.70830e21	−0.313503
$792$	0	0
$793$	3.81831e21	0.687579
$794$	0	0
$795$	−3.66092e20	−0.0646899
$796$	0	0
$797$	−6.76651e21	−1.17335	−0.586675	−	0.809823i	$-0.699563\pi$
$797$	−6.76651e21	−1.17335	−0.586675	+	0.809823i	$0.699563\pi$
$798$	0	0
$799$	2.27818e21	0.387691
$800$	0	0
$801$	2.75846e21	0.460705
$802$	0	0
$803$	6.06725e21	0.994546
$804$	0	0
$805$	1.38516e21	0.222859
$806$	0	0
$807$	−5.09158e21	−0.804084
$808$	0	0
$809$	−2.69947e21	−0.418471	−0.209236	−	0.977865i	$-0.567098\pi$
$809$	−2.69947e21	−0.418471	−0.209236	+	0.977865i	$0.567098\pi$
$810$	0	0
$811$	3.26290e20	0.0496532	0.0248266	−	0.999692i	$-0.492097\pi$
$811$	3.26290e20	0.0496532	0.0248266	+	0.999692i	$0.492097\pi$
$812$	0	0
$813$	5.01173e21	0.748701
$814$	0	0
$815$	−1.26768e21	−0.185920
$816$	0	0
$817$	−6.28957e21	−0.905640
$818$	0	0
$819$	−1.47971e21	−0.209193
$820$	0	0
$821$	−8.38301e21	−1.16366	−0.581830	−	0.813310i	$-0.697663\pi$
$821$	−8.38301e21	−1.16366	−0.581830	+	0.813310i	$0.697663\pi$
$822$	0	0
$823$	−2.89689e21	−0.394851	−0.197426	−	0.980318i	$-0.563258\pi$
$823$	−2.89689e21	−0.394851	−0.197426	+	0.980318i	$0.563258\pi$
$824$	0	0
$825$	1.69059e20	0.0226273
$826$	0	0
$827$	−9.61354e21	−1.26355	−0.631774	−	0.775152i	$-0.717673\pi$
$827$	−9.61354e21	−1.26355	−0.631774	+	0.775152i	$0.717673\pi$
$828$	0	0
$829$	−8.25187e21	−1.06511	−0.532554	−	0.846396i	$-0.678768\pi$
$829$	−8.25187e21	−1.06511	−0.532554	+	0.846396i	$0.678768\pi$
$830$	0	0
$831$	−9.07774e21	−1.15072
$832$	0	0
$833$	6.70403e21	0.834639
$834$	0	0
$835$	−6.78546e20	−0.0829718
$836$	0	0
$837$	−3.19951e20	−0.0384275
$838$	0	0
$839$	1.55522e22	1.83475	0.917376	−	0.398023i	$-0.130304\pi$
$839$	1.55522e22	1.83475	0.917376	+	0.398023i	$0.130304\pi$
$840$	0	0
$841$	3.50226e21	0.405862
$842$	0	0
$843$	5.84656e21	0.665569
$844$	0	0
$845$	1.43418e21	0.160390
$846$	0	0
$847$	−3.95209e21	−0.434211
$848$	0	0
$849$	6.99667e21	0.755238
$850$	0	0
$851$	3.96686e21	0.420702
$852$	0	0
$853$	−1.58685e22	−1.65355	−0.826775	−	0.562533i	$-0.809827\pi$
$853$	−1.58685e22	−1.65355	−0.826775	+	0.562533i	$0.809827\pi$
$854$	0	0
$855$	−4.10556e21	−0.420365
$856$	0	0
$857$	2.84075e21	0.285810	0.142905	−	0.989736i	$-0.454356\pi$
$857$	2.84075e21	0.285810	0.142905	+	0.989736i	$0.454356\pi$
$858$	0	0
$859$	−1.71405e22	−1.69463	−0.847314	−	0.531092i	$-0.821782\pi$
$859$	−1.71405e22	−1.69463	−0.847314	+	0.531092i	$0.821782\pi$
$860$	0	0
$861$	−2.97931e21	−0.289462
$862$	0	0
$863$	1.14547e22	1.09371	0.546855	−	0.837227i	$-0.315825\pi$
$863$	1.14547e22	1.09371	0.546855	+	0.837227i	$0.315825\pi$
$864$	0	0
$865$	1.46168e22	1.37162
$866$	0	0
$867$	9.05226e21	0.834861
$868$	0	0
$869$	−6.94261e21	−0.629324
$870$	0	0
$871$	−1.70304e22	−1.51736
$872$	0	0
$873$	2.35317e21	0.206085
$874$	0	0
$875$	7.65438e21	0.658946
$876$	0	0
$877$	1.03589e22	0.876628	0.438314	−	0.898822i	$-0.355576\pi$
$877$	1.03589e22	0.876628	0.438314	+	0.898822i	$0.355576\pi$
$878$	0	0
$879$	5.71955e21	0.475823
$880$	0	0
$881$	6.85579e21	0.560710	0.280355	−	0.959896i	$-0.409548\pi$
$881$	6.85579e21	0.560710	0.280355	+	0.959896i	$0.409548\pi$
$882$	0	0
$883$	1.80330e21	0.144998	0.0724990	−	0.997368i	$-0.476903\pi$
$883$	1.80330e21	0.144998	0.0724990	+	0.997368i	$0.476903\pi$
$884$	0	0
$885$	1.50095e22	1.18656
$886$	0	0
$887$	−1.74255e22	−1.35444	−0.677218	−	0.735783i	$-0.736814\pi$
$887$	−1.74255e22	−1.35444	−0.677218	+	0.735783i	$0.736814\pi$
$888$	0	0
$889$	1.50165e22	1.14764
$890$	0	0
$891$	8.92243e20	0.0670502
$892$	0	0
$893$	−4.09946e21	−0.302929
$894$	0	0
$895$	6.06415e21	0.440653
$896$	0	0
$897$	2.34811e21	0.167794
$898$	0	0
$899$	3.36894e21	0.236753
$900$	0	0
$901$	−2.45697e21	−0.169810
$902$	0	0
$903$	−4.29851e21	−0.292186
$904$	0	0
$905$	1.30732e22	0.874014
$906$	0	0
$907$	2.57258e21	0.169167	0.0845833	−	0.996416i	$-0.473044\pi$
$907$	2.57258e21	0.169167	0.0845833	+	0.996416i	$0.473044\pi$
$908$	0	0
$909$	−6.89806e21	−0.446168
$910$	0	0
$911$	9.51786e21	0.605552	0.302776	−	0.953062i	$-0.402087\pi$
$911$	9.51786e21	0.605552	0.302776	+	0.953062i	$0.402087\pi$
$912$	0	0
$913$	8.80338e20	0.0550958
$914$	0	0
$915$	7.24043e21	0.445765
$916$	0	0
$917$	1.15016e22	0.696605
$918$	0	0
$919$	−1.52473e22	−0.908505	−0.454252	−	0.890873i	$-0.650094\pi$
$919$	−1.52473e22	−0.908505	−0.454252	+	0.890873i	$0.650094\pi$
$920$	0	0
$921$	7.17302e21	0.420489
$922$	0	0
$923$	−8.64158e21	−0.498403
$924$	0	0
$925$	−1.52255e21	−0.0863992
$926$	0	0
$927$	−2.24418e21	−0.125302
$928$	0	0
$929$	2.69216e22	1.47905	0.739525	−	0.673130i	$-0.235050\pi$
$929$	2.69216e22	1.47905	0.739525	+	0.673130i	$0.235050\pi$
$930$	0	0
$931$	−1.20636e22	−0.652158
$932$	0	0
$933$	3.56989e20	0.0189907
$934$	0	0
$935$	1.86047e22	0.973950
$936$	0	0
$937$	−4.04500e21	−0.208387	−0.104194	−	0.994557i	$-0.533226\pi$
$937$	−4.04500e21	−0.208387	−0.104194	+	0.994557i	$0.533226\pi$
$938$	0	0
$939$	5.28538e21	0.267969
$940$	0	0
$941$	−2.59999e22	−1.29733	−0.648663	−	0.761076i	$-0.724672\pi$
$941$	−2.59999e22	−1.29733	−0.648663	+	0.761076i	$0.724672\pi$
$942$	0	0
$943$	4.72779e21	0.232178
$944$	0	0
$945$	−2.80588e21	−0.135622
$946$	0	0
$947$	−2.85830e22	−1.35982	−0.679912	−	0.733294i	$-0.737982\pi$
$947$	−2.85830e22	−1.35982	−0.679912	+	0.733294i	$0.737982\pi$
$948$	0	0
$949$	−3.23444e22	−1.51461
$950$	0	0
$951$	−3.35908e21	−0.154834
$952$	0	0
$953$	−7.29856e21	−0.331162	−0.165581	−	0.986196i	$-0.552950\pi$
$953$	−7.29856e21	−0.331162	−0.165581	+	0.986196i	$0.552950\pi$
$954$	0	0
$955$	2.64701e22	1.18231
$956$	0	0
$957$	−9.39492e21	−0.413098
$958$	0	0
$959$	−3.11497e21	−0.134839
$960$	0	0
$961$	−2.25297e22	−0.960130
$962$	0	0
$963$	−9.62477e21	−0.403825
$964$	0	0
$965$	3.52749e21	0.145717
$966$	0	0
$967$	−4.15063e22	−1.68817	−0.844084	−	0.536211i	$-0.819855\pi$
$967$	−4.15063e22	−1.68817	−0.844084	+	0.536211i	$0.819855\pi$
$968$	0	0
$969$	−2.75538e22	−1.10345
$970$	0	0
$971$	−5.94891e21	−0.234581	−0.117291	−	0.993098i	$-0.537421\pi$
$971$	−5.94891e21	−0.234581	−0.117291	+	0.993098i	$0.537421\pi$
$972$	0	0
$973$	3.43233e22	1.33273
$974$	0	0
$975$	−9.01250e20	−0.0344596
$976$	0	0
$977$	−4.79538e22	−1.80557	−0.902783	−	0.430096i	$-0.858480\pi$
$977$	−4.79538e22	−1.80557	−0.902783	+	0.430096i	$0.858480\pi$
$978$	0	0
$979$	2.24935e22	0.834040
$980$	0	0
$981$	1.19398e22	0.435992
$982$	0	0
$983$	4.87969e22	1.75485	0.877426	−	0.479712i	$-0.159259\pi$
$983$	4.87969e22	1.75485	0.877426	+	0.479712i	$0.159259\pi$
$984$	0	0
$985$	−3.59336e22	−1.27271
$986$	0	0
$987$	−2.80171e21	−0.0977338
$988$	0	0
$989$	6.82119e21	0.234362
$990$	0	0
$991$	2.75510e21	0.0932362	0.0466181	−	0.998913i	$-0.485156\pi$
$991$	2.75510e21	0.0932362	0.0466181	+	0.998913i	$0.485156\pi$
$992$	0	0
$993$	2.25220e22	0.750738
$994$	0	0
$995$	−5.57067e22	−1.82909
$996$	0	0
$997$	1.10140e22	0.356232	0.178116	−	0.984009i	$-0.443000\pi$
$997$	1.10140e22	0.356232	0.178116	+	0.984009i	$0.443000\pi$
$998$	0	0
$999$	−8.03558e21	−0.256021

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	24.16.a.c.1.1	✓	2
3.2	odd	2		72.16.a.f.1.2		2
4.3	odd	2		48.16.a.h.1.1		2
12.11	even	2		144.16.a.u.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
24.16.a.c.1.1	✓	2	1.1	even	1	trivial
48.16.a.h.1.1		2	4.3	odd	2
72.16.a.f.1.2		2	3.2	odd	2
144.16.a.u.1.2		2	12.11	even	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:	\( k \)	\(=\)	\( 16 \)
Character orbit:	\([\chi]\)	\(=\)	24.a (trivial)

\(f(q)\)	\(=\)	\(q+2187.00 q^{3} -180276. q^{5} +1.48794e6 q^{7} +4.78297e6 q^{9} +O(q^{10})\)	\(q+2187.00 q^{3} -180276. q^{5} +1.48794e6 q^{7} +4.78297e6 q^{9} +3.90021e7 q^{11} -2.07919e8 q^{13} -3.94264e8 q^{15} -2.64604e9 q^{17} +4.76142e9 q^{19} +3.25412e9 q^{21} -5.16387e9 q^{23} +1.98199e9 q^{25} +1.04604e10 q^{27} -1.10143e11 q^{29} -3.05870e10 q^{31} +8.52976e10 q^{33} -2.68240e11 q^{35} -7.68194e11 q^{37} -4.54720e11 q^{39} -9.15551e11 q^{41} -1.32094e12 q^{43} -8.62256e11 q^{45} -8.60974e11 q^{47} -2.53361e12 q^{49} -5.78690e12 q^{51} +9.28543e11 q^{53} -7.03115e12 q^{55} +1.04132e13 q^{57} -3.80695e13 q^{59} -1.83644e13 q^{61} +7.11675e12 q^{63} +3.74829e13 q^{65} +8.19085e13 q^{67} -1.12934e13 q^{69} +4.15622e13 q^{71} +1.55562e14 q^{73} +4.33461e12 q^{75} +5.80326e13 q^{77} -1.78006e14 q^{79} +2.28768e13 q^{81} +2.25716e13 q^{83} +4.77019e14 q^{85} -2.40882e14 q^{87} +5.76725e14 q^{89} -3.09371e14 q^{91} -6.68937e13 q^{93} -8.58371e14 q^{95} +4.91989e14 q^{97} +1.86546e14 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4374 q^{3} - 18340 q^{5} - 680400 q^{7} + 9565938 q^{9}+O(q^{10}) \) 2 * q + 4374 * q^3 - 18340 * q^5 - 680400 * q^7 + 9565938 * q^9	\( 2 q + 4374 q^{3} - 18340 q^{5} - 680400 q^{7} + 9565938 q^{9} - 77720632 q^{11} - 207881332 q^{13} - 40109580 q^{15} - 1673313692 q^{17} - 2240887112 q^{19} - 1488034800 q^{21} + 16655801552 q^{23} - 2312200850 q^{25} + 20920706406 q^{27} - 139493975892 q^{29} - 43133623424 q^{31} - 169975022184 q^{33} - 619372303200 q^{35} - 1457105544996 q^{37} - 454636473084 q^{39} - 1377923837772 q^{41} - 2763654497656 q^{43} - 87719651460 q^{45} - 7396441682400 q^{47} - 2579485205486 q^{49} - 3659537044404 q^{51} - 5824660261252 q^{53} - 25932807442960 q^{55} - 4900820113944 q^{57} - 22792784775448 q^{59} + 14623417395116 q^{61} - 3254332107600 q^{63} + 37489085612840 q^{65} + 68414350989784 q^{67} + 36426237994224 q^{69} + 64404223302256 q^{71} + 229480150537460 q^{73} - 5056783258950 q^{75} + 311126750827200 q^{77} - 8902155902944 q^{79} + 45753584909922 q^{81} - 239472633418184 q^{83} + 634539486862840 q^{85} - 305073325275804 q^{87} + 472605248938452 q^{89} - 309453044594400 q^{91} - 94333234428288 q^{93} - 19\!\cdots\!60 q^{95}+ \cdots - 371735373516408 q^{99}+O(q^{100}) \) 2 * q + 4374 * q^3 - 18340 * q^5 - 680400 * q^7 + 9565938 * q^9 - 77720632 * q^11 - 207881332 * q^13 - 40109580 * q^15 - 1673313692 * q^17 - 2240887112 * q^19 - 1488034800 * q^21 + 16655801552 * q^23 - 2312200850 * q^25 + 20920706406 * q^27 - 139493975892 * q^29 - 43133623424 * q^31 - 169975022184 * q^33 - 619372303200 * q^35 - 1457105544996 * q^37 - 454636473084 * q^39 - 1377923837772 * q^41 - 2763654497656 * q^43 - 87719651460 * q^45 - 7396441682400 * q^47 - 2579485205486 * q^49 - 3659537044404 * q^51 - 5824660261252 * q^53 - 25932807442960 * q^55 - 4900820113944 * q^57 - 22792784775448 * q^59 + 14623417395116 * q^61 - 3254332107600 * q^63 + 37489085612840 * q^65 + 68414350989784 * q^67 + 36426237994224 * q^69 + 64404223302256 * q^71 + 229480150537460 * q^73 - 5056783258950 * q^75 + 311126750827200 * q^77 - 8902155902944 * q^79 + 45753584909922 * q^81 - 239472633418184 * q^83 + 634539486862840 * q^85 - 305073325275804 * q^87 + 472605248938452 * q^89 - 309453044594400 * q^91 - 94333234428288 * q^93 - 1992299281656560 * q^95 + 2032136651194948 * q^97 - 371735373516408 * q^99

Introduction

L-functions

Modular forms

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