LMFDB - Embedded newform 100.20.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [100,20,Mod(1,100)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(100, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 20, names="a")

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

chi := DirichletCharacter("100.1");

S:= CuspForms(chi, 20);

N := Newforms(S);

Level:	$ N $	$=$	$ 100 = 2^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 20 $
Character orbit:	$[\chi]$	$=$	100.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:	$228.816696556$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	$\mathbb{Q}[x]/(x^{3} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 1351720x + 139588750 $ x^3 - x^2 - 1351720*x + 139588750
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{11}\cdot 3^{3}\cdot 5 $
Twist minimal:	no (minimal twist has level 20)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1107.61$ of defining polynomial
Character	$\chi$	$=$	100.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-64511.3 q^{3} +1.51440e8 q^{7} +2.99945e9 q^{9} +O(q^{10})$	$q-64511.3 q^{3} +1.51440e8 q^{7} +2.99945e9 q^{9} +6.64602e8 q^{11} +6.93073e10 q^{13} +4.32704e10 q^{17} -1.65874e12 q^{19} -9.76959e12 q^{21} +3.51744e12 q^{23} -1.18519e14 q^{27} -1.49082e14 q^{29} +1.69561e14 q^{31} -4.28743e13 q^{33} -5.16643e14 q^{37} -4.47110e15 q^{39} -7.97319e14 q^{41} -1.73746e14 q^{43} +1.02870e16 q^{47} +1.15352e16 q^{49} -2.79143e15 q^{51} -2.47801e16 q^{53} +1.07007e17 q^{57} -8.13879e16 q^{59} -1.83613e16 q^{61} +4.54236e17 q^{63} -3.85335e16 q^{67} -2.26914e17 q^{69} -1.86393e17 q^{71} -4.58624e17 q^{73} +1.00647e17 q^{77} +2.19691e17 q^{79} +4.15968e18 q^{81} +1.99475e17 q^{83} +9.61749e18 q^{87} -3.46385e18 q^{89} +1.04959e19 q^{91} -1.09386e19 q^{93} -8.54697e17 q^{97} +1.99344e18 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 34086 q^{3} + 115130574 q^{7} + 3129228027 q^{9}+O(q^{10}) $ 3 * q - 34086 * q^3 + 115130574 * q^7 + 3129228027 * q^9	$ 3 q - 34086 q^{3} + 115130574 q^{7} + 3129228027 q^{9} + 6359213280 q^{11} + 33996539826 q^{13} + 60515110578 q^{17} - 1436218405908 q^{19} - 7119772762332 q^{21} - 1682066292342 q^{23} - 91316335875732 q^{27} - 181580995192278 q^{29} - 3566464773252 q^{31} + 90839870844480 q^{33} - 653714901466206 q^{37} - 43\!\cdots\!08 q^{39}+ \cdots + 37\!\cdots\!20 q^{99}+O(q^{100}) $ 3 * q - 34086 * q^3 + 115130574 * q^7 + 3129228027 * q^9 + 6359213280 * q^11 + 33996539826 * q^13 + 60515110578 * q^17 - 1436218405908 * q^19 - 7119772762332 * q^21 - 1682066292342 * q^23 - 91316335875732 * q^27 - 181580995192278 * q^29 - 3566464773252 * q^31 + 90839870844480 * q^33 - 653714901466206 * q^37 - 4339017394217508 * q^39 - 1257752912210622 * q^41 + 4361040178994610 * q^43 + 23174987171661798 * q^47 - 5454428919168921 * q^49 - 33371959424606244 * q^51 + 12180381712114362 * q^53 + 177419957801365416 * q^57 - 180023573560918116 * q^59 - 213491197168562514 * q^61 + 549307385783925246 * q^63 + 32494317103156134 * q^67 - 498099089220386004 * q^69 - 245957452394007276 * q^71 + 94437681636898506 * q^73 + 72972512123263680 * q^77 + 819623047726342536 * q^79 + 3159048884904329967 * q^81 - 2812828487653823982 * q^83 + 2510285506713629676 * q^87 + 4879439966436006078 * q^89 + 12213088562709029652 * q^91 - 16020238495381453176 * q^93 - 8189756396617107846 * q^97 + 3795512975671396320 * 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
$3$	−64511.3	−1.89227	−0.946137	−	0.323768i	$-0.895050\pi$
$3$	−64511.3	−1.89227	−0.946137	+	0.323768i	$0.895050\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.51440e8	1.41843	0.709217	−	0.704990i	$-0.249049\pi$
$7$	1.51440e8	1.41843	0.709217	+	0.704990i	$0.249049\pi$
$8$	0	0
$9$	2.99945e9	2.58070
$10$	0	0
$11$	6.64602e8	0.0849828	0.0424914	−	0.999097i	$-0.486470\pi$
$11$	6.64602e8	0.0849828	0.0424914	+	0.999097i	$0.486470\pi$
$12$	0	0
$13$	6.93073e10	1.81266	0.906331	−	0.422568i	$-0.138871\pi$
$13$	6.93073e10	1.81266	0.906331	+	0.422568i	$0.138871\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	4.32704e10	0.0884964	0.0442482	−	0.999021i	$-0.485911\pi$
$17$	4.32704e10	0.0884964	0.0442482	+	0.999021i	$0.485911\pi$
$18$	0	0
$19$	−1.65874e12	−1.17928	−0.589642	−	0.807665i	$-0.700731\pi$
$19$	−1.65874e12	−1.17928	−0.589642	+	0.807665i	$0.700731\pi$
$20$	0	0
$21$	−9.76959e12	−2.68407
$22$	0	0
$23$	3.51744e12	0.407204	0.203602	−	0.979054i	$-0.434735\pi$
$23$	3.51744e12	0.407204	0.203602	+	0.979054i	$0.434735\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−1.18519e14	−2.99111
$28$	0	0
$29$	−1.49082e14	−1.90829	−0.954146	−	0.299342i	$-0.903233\pi$
$29$	−1.49082e14	−1.90829	−0.954146	+	0.299342i	$0.903233\pi$
$30$	0	0
$31$	1.69561e14	1.15184	0.575918	−	0.817508i	$-0.304645\pi$
$31$	1.69561e14	1.15184	0.575918	+	0.817508i	$0.304645\pi$
$32$	0	0
$33$	−4.28743e13	−0.160811
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−5.16643e14	−0.653543	−0.326772	−	0.945103i	$-0.605961\pi$
$37$	−5.16643e14	−0.653543	−0.326772	+	0.945103i	$0.605961\pi$
$38$	0	0
$39$	−4.47110e15	−3.43005
$40$	0	0
$41$	−7.97319e14	−0.380352	−0.190176	−	0.981750i	$-0.560906\pi$
$41$	−7.97319e14	−0.380352	−0.190176	+	0.981750i	$0.560906\pi$
$42$	0	0
$43$	−1.73746e14	−0.0527188	−0.0263594	−	0.999653i	$-0.508391\pi$
$43$	−1.73746e14	−0.0527188	−0.0263594	+	0.999653i	$0.508391\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	1.02870e16	1.34078	0.670389	−	0.742009i	$-0.266127\pi$
$47$	1.02870e16	1.34078	0.670389	+	0.742009i	$0.266127\pi$
$48$	0	0
$49$	1.15352e16	1.01196
$50$	0	0
$51$	−2.79143e15	−0.167459
$52$	0	0
$53$	−2.47801e16	−1.03153	−0.515765	−	0.856730i	$-0.672492\pi$
$53$	−2.47801e16	−1.03153	−0.515765	+	0.856730i	$0.672492\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	1.07007e17	2.23153
$58$	0	0
$59$	−8.13879e16	−1.22311	−0.611555	−	0.791202i	$-0.709456\pi$
$59$	−8.13879e16	−1.22311	−0.611555	+	0.791202i	$0.709456\pi$
$60$	0	0
$61$	−1.83613e16	−0.201034	−0.100517	−	0.994935i	$-0.532050\pi$
$61$	−1.83613e16	−0.201034	−0.100517	+	0.994935i	$0.532050\pi$
$62$	0	0
$63$	4.54236e17	3.66055
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−3.85335e16	−0.173032	−0.0865162	−	0.996250i	$-0.527573\pi$
$67$	−3.85335e16	−0.173032	−0.0865162	+	0.996250i	$0.527573\pi$
$68$	0	0
$69$	−2.26914e17	−0.770541
$70$	0	0
$71$	−1.86393e17	−0.482476	−0.241238	−	0.970466i	$-0.577553\pi$
$71$	−1.86393e17	−0.482476	−0.241238	+	0.970466i	$0.577553\pi$
$72$	0	0
$73$	−4.58624e17	−0.911780	−0.455890	−	0.890036i	$-0.650679\pi$
$73$	−4.58624e17	−0.911780	−0.455890	+	0.890036i	$0.650679\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	1.00647e17	0.120543
$78$	0	0
$79$	2.19691e17	0.206232	0.103116	−	0.994669i	$-0.467119\pi$
$79$	2.19691e17	0.206232	0.103116	+	0.994669i	$0.467119\pi$
$80$	0	0
$81$	4.15968e18	3.07930
$82$	0	0
$83$	1.99475e17	0.117124	0.0585622	−	0.998284i	$-0.481348\pi$
$83$	1.99475e17	0.117124	0.0585622	+	0.998284i	$0.481348\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	9.61749e18	3.61101
$88$	0	0
$89$	−3.46385e18	−1.04798	−0.523991	−	0.851724i	$-0.675557\pi$
$89$	−3.46385e18	−1.04798	−0.523991	+	0.851724i	$0.675557\pi$
$90$	0	0
$91$	1.04959e19	2.57114
$92$	0	0
$93$	−1.09386e19	−2.17959
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−8.54697e17	−0.114151	−0.0570756	−	0.998370i	$-0.518178\pi$
$97$	−8.54697e17	−0.114151	−0.0570756	+	0.998370i	$0.518178\pi$
$98$	0	0
$99$	1.99344e18	0.219315
$100$	0	0
$101$	1.59591e19	1.45196	0.725980	−	0.687716i	$-0.241387\pi$
$101$	1.59591e19	1.45196	0.725980	+	0.687716i	$0.241387\pi$
$102$	0	0
$103$	2.57032e19	1.94104	0.970518	−	0.241028i	$-0.0774846\pi$
$103$	2.57032e19	1.94104	0.970518	+	0.241028i	$0.0774846\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−2.29590e19	−1.20728	−0.603640	−	0.797257i	$-0.706283\pi$
$107$	−2.29590e19	−1.20728	−0.603640	+	0.797257i	$0.706283\pi$
$108$	0	0
$109$	−2.98910e19	−1.31822	−0.659111	−	0.752046i	$-0.729067\pi$
$109$	−2.98910e19	−1.31822	−0.659111	+	0.752046i	$0.729067\pi$
$110$	0	0
$111$	3.33293e19	1.23668
$112$	0	0
$113$	1.51385e19	0.474065	0.237033	−	0.971502i	$-0.423825\pi$
$113$	1.51385e19	0.474065	0.237033	+	0.971502i	$0.423825\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	2.07883e20	4.67793
$118$	0	0
$119$	6.55286e18	0.125526
$120$	0	0
$121$	−6.07174e19	−0.992778
$122$	0	0
$123$	5.14361e19	0.719729
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−7.37167e19	−0.761080	−0.380540	−	0.924764i	$-0.624262\pi$
$127$	−7.37167e19	−0.761080	−0.380540	+	0.924764i	$0.624262\pi$
$128$	0	0
$129$	1.12086e19	0.0997584
$130$	0	0
$131$	1.20118e20	0.923703	0.461851	−	0.886957i	$-0.347185\pi$
$131$	1.20118e20	0.923703	0.461851	+	0.886957i	$0.347185\pi$
$132$	0	0
$133$	−2.51199e20	−1.67274
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−2.37840e20	−1.19520	−0.597598	−	0.801796i	$-0.703878\pi$
$137$	−2.37840e20	−1.19520	−0.597598	+	0.801796i	$0.703878\pi$
$138$	0	0
$139$	−4.71249e19	−0.206352	−0.103176	−	0.994663i	$-0.532901\pi$
$139$	−4.71249e19	−0.206352	−0.103176	+	0.994663i	$0.532901\pi$
$140$	0	0
$141$	−6.63625e20	−2.53712
$142$	0	0
$143$	4.60617e19	0.154045
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−7.44150e20	−1.91490
$148$	0	0
$149$	−2.12692e20	−0.481373	−0.240687	−	0.970603i	$-0.577373\pi$
$149$	−2.12692e20	−0.481373	−0.240687	+	0.970603i	$0.577373\pi$
$150$	0	0
$151$	−4.17246e20	−0.831976	−0.415988	−	0.909370i	$-0.636564\pi$
$151$	−4.17246e20	−0.831976	−0.415988	+	0.909370i	$0.636564\pi$
$152$	0	0
$153$	1.29787e20	0.228383
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−7.82726e20	−1.07786	−0.538931	−	0.842350i	$-0.681172\pi$
$157$	−7.82726e20	−1.07786	−0.538931	+	0.842350i	$0.681172\pi$
$158$	0	0
$159$	1.59860e21	1.95194
$160$	0	0
$161$	5.32680e20	0.577592
$162$	0	0
$163$	−6.38588e20	−0.615798	−0.307899	−	0.951419i	$-0.599626\pi$
$163$	−6.38588e20	−0.615798	−0.307899	+	0.951419i	$0.599626\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−1.22177e21	−0.935798	−0.467899	−	0.883782i	$-0.654989\pi$
$167$	−1.22177e21	−0.935798	−0.467899	+	0.883782i	$0.654989\pi$
$168$	0	0
$169$	3.34158e21	2.28575
$170$	0	0
$171$	−4.97529e21	−3.04338
$172$	0	0
$173$	−1.87793e21	−1.02859	−0.514295	−	0.857613i	$-0.671946\pi$
$173$	−1.87793e21	−1.02859	−0.514295	+	0.857613i	$0.671946\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	5.25044e21	2.31446
$178$	0	0
$179$	−1.91317e21	−0.757968	−0.378984	−	0.925403i	$-0.623726\pi$
$179$	−1.91317e21	−0.757968	−0.378984	+	0.925403i	$0.623726\pi$
$180$	0	0
$181$	3.38495e21	1.20672	0.603358	−	0.797470i	$-0.293829\pi$
$181$	3.38495e21	1.20672	0.603358	+	0.797470i	$0.293829\pi$
$182$	0	0
$183$	1.18451e21	0.380411
$184$	0	0
$185$	0	0
$186$	0	0
$187$	2.87576e19	0.00752067
$188$	0	0
$189$	−1.79485e22	−4.24270
$190$	0	0
$191$	4.28561e21	0.916634	0.458317	−	0.888789i	$-0.348452\pi$
$191$	4.28561e21	0.916634	0.458317	+	0.888789i	$0.348452\pi$
$192$	0	0
$193$	7.60745e21	1.47382	0.736911	−	0.675990i	$-0.236284\pi$
$193$	7.60745e21	1.47382	0.736911	+	0.675990i	$0.236284\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	2.43390e21	0.388038	0.194019	−	0.980998i	$-0.437848\pi$
$197$	2.43390e21	0.388038	0.194019	+	0.980998i	$0.437848\pi$
$198$	0	0
$199$	3.75758e21	0.544257	0.272129	−	0.962261i	$-0.412272\pi$
$199$	3.75758e21	0.544257	0.272129	+	0.962261i	$0.412272\pi$
$200$	0	0
$201$	2.48584e21	0.327425
$202$	0	0
$203$	−2.25770e22	−2.70679
$204$	0	0
$205$	0	0
$206$	0	0
$207$	1.05504e22	1.05087
$208$	0	0
$209$	−1.10240e21	−0.100219
$210$	0	0
$211$	5.14683e21	0.427422	0.213711	−	0.976897i	$-0.431445\pi$
$211$	5.14683e21	0.427422	0.213711	+	0.976897i	$0.431445\pi$
$212$	0	0
$213$	1.20245e22	0.912976
$214$	0	0
$215$	0	0
$216$	0	0
$217$	2.56783e22	1.63380
$218$	0	0
$219$	2.95865e22	1.72534
$220$	0	0
$221$	2.99895e21	0.160414
$222$	0	0
$223$	−1.01264e22	−0.497230	−0.248615	−	0.968602i	$-0.579975\pi$
$223$	−1.01264e22	−0.497230	−0.248615	+	0.968602i	$0.579975\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	1.16371e22	0.482615	0.241307	−	0.970449i	$-0.422424\pi$
$227$	1.16371e22	0.482615	0.241307	+	0.970449i	$0.422424\pi$
$228$	0	0
$229$	9.93369e21	0.379030	0.189515	−	0.981878i	$-0.439308\pi$
$229$	9.93369e21	0.379030	0.189515	+	0.981878i	$0.439308\pi$
$230$	0	0
$231$	−6.49289e21	−0.228099
$232$	0	0
$233$	−3.01584e22	−0.976174	−0.488087	−	0.872795i	$-0.662305\pi$
$233$	−3.01584e22	−0.976174	−0.488087	+	0.872795i	$0.662305\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−1.41726e22	−0.390247
$238$	0	0
$239$	−3.76780e22	−0.957873	−0.478936	−	0.877850i	$-0.658978\pi$
$239$	−3.76780e22	−0.957873	−0.478936	+	0.877850i	$0.658978\pi$
$240$	0	0
$241$	−4.44639e22	−1.04435	−0.522175	−	0.852838i	$-0.674879\pi$
$241$	−4.44639e22	−1.04435	−0.522175	+	0.852838i	$0.674879\pi$
$242$	0	0
$243$	−1.30596e23	−2.83577
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−1.14963e23	−2.13764
$248$	0	0
$249$	−1.28684e22	−0.221631
$250$	0	0
$251$	−7.27608e22	−1.16144	−0.580720	−	0.814103i	$-0.697229\pi$
$251$	−7.27608e22	−1.16144	−0.580720	+	0.814103i	$0.697229\pi$
$252$	0	0
$253$	2.33769e21	0.0346053
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−1.01400e23	−1.29322	−0.646612	−	0.762819i	$-0.723815\pi$
$257$	−1.01400e23	−1.29322	−0.646612	+	0.762819i	$0.723815\pi$
$258$	0	0
$259$	−7.82405e22	−0.927009
$260$	0	0
$261$	−4.47164e23	−4.92473
$262$	0	0
$263$	1.64829e23	1.68832	0.844160	−	0.536091i	$-0.180100\pi$
$263$	1.64829e23	1.68832	0.844160	+	0.536091i	$0.180100\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	2.23457e23	1.98307
$268$	0	0
$269$	−3.21036e22	−0.265403	−0.132702	−	0.991156i	$-0.542365\pi$
$269$	−3.21036e22	−0.265403	−0.132702	+	0.991156i	$0.542365\pi$
$270$	0	0
$271$	4.00913e21	0.0308917	0.0154458	−	0.999881i	$-0.495083\pi$
$271$	4.00913e21	0.0308917	0.0154458	+	0.999881i	$0.495083\pi$
$272$	0	0
$273$	−6.77104e23	−4.86531
$274$	0	0
$275$	0	0
$276$	0	0
$277$	3.16309e23	1.97949	0.989745	−	0.142846i	$-0.0456255\pi$
$277$	3.16309e23	1.97949	0.989745	+	0.142846i	$0.0456255\pi$
$278$	0	0
$279$	5.08590e23	2.97254
$280$	0	0
$281$	−1.54092e23	−0.841530	−0.420765	−	0.907170i	$-0.638238\pi$
$281$	−1.54092e23	−0.841530	−0.420765	+	0.907170i	$0.638238\pi$
$282$	0	0
$283$	2.19291e23	1.11956	0.559782	−	0.828640i	$-0.310885\pi$
$283$	2.19291e23	1.11956	0.559782	+	0.828640i	$0.310885\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−1.20746e23	−0.539504
$288$	0	0
$289$	−2.37200e23	−0.992168
$290$	0	0
$291$	5.51376e22	0.216005
$292$	0	0
$293$	6.15694e22	0.226007	0.113004	−	0.993595i	$-0.463953\pi$
$293$	6.15694e22	0.226007	0.113004	+	0.993595i	$0.463953\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−7.87680e22	−0.254193
$298$	0	0
$299$	2.43784e23	0.738123
$300$	0	0
$301$	−2.63122e22	−0.0747782
$302$	0	0
$303$	−1.02954e24	−2.74750
$304$	0	0
$305$	0	0
$306$	0	0
$307$	4.18978e22	0.0987135	0.0493567	−	0.998781i	$-0.484283\pi$
$307$	4.18978e22	0.0987135	0.0493567	+	0.998781i	$0.484283\pi$
$308$	0	0
$309$	−1.65815e24	−3.67297
$310$	0	0
$311$	−5.82239e23	−1.21305	−0.606523	−	0.795066i	$-0.707436\pi$
$311$	−5.82239e23	−1.21305	−0.606523	+	0.795066i	$0.707436\pi$
$312$	0	0
$313$	−5.98099e23	−1.17247	−0.586235	−	0.810141i	$-0.699391\pi$
$313$	−5.98099e23	−1.17247	−0.586235	+	0.810141i	$0.699391\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	3.64789e23	0.633839	0.316919	−	0.948453i	$-0.397352\pi$
$317$	3.64789e23	0.633839	0.316919	+	0.948453i	$0.397352\pi$
$318$	0	0
$319$	−9.90803e22	−0.162172
$320$	0	0
$321$	1.48112e24	2.28450
$322$	0	0
$323$	−7.17742e22	−0.104362
$324$	0	0
$325$	0	0
$326$	0	0
$327$	1.92831e24	2.49444
$328$	0	0
$329$	1.55786e24	1.90181
$330$	0	0
$331$	7.66370e23	0.883228	0.441614	−	0.897205i	$-0.354406\pi$
$331$	7.66370e23	0.883228	0.441614	+	0.897205i	$0.354406\pi$
$332$	0	0
$333$	−1.54964e24	−1.68660
$334$	0	0
$335$	0	0
$336$	0	0
$337$	1.23133e24	1.19644	0.598221	−	0.801331i	$-0.295874\pi$
$337$	1.23133e24	1.19644	0.598221	+	0.801331i	$0.295874\pi$
$338$	0	0
$339$	−9.76605e23	−0.897061
$340$	0	0
$341$	1.12691e23	0.0978861
$342$	0	0
$343$	2.06402e22	0.0169598
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−2.12707e23	−0.156549	−0.0782747	−	0.996932i	$-0.524941\pi$
$347$	−2.12707e23	−0.156549	−0.0782747	+	0.996932i	$0.524941\pi$
$348$	0	0
$349$	−5.87533e23	−0.409440	−0.204720	−	0.978821i	$-0.565628\pi$
$349$	−5.87533e23	−0.409440	−0.204720	+	0.978821i	$0.565628\pi$
$350$	0	0
$351$	−8.21424e24	−5.42188
$352$	0	0
$353$	6.25073e22	0.0390905	0.0195452	−	0.999809i	$-0.493778\pi$
$353$	6.25073e22	0.0390905	0.0195452	+	0.999809i	$0.493778\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−4.22734e23	−0.237530
$358$	0	0
$359$	2.59144e24	1.38084	0.690420	−	0.723409i	$-0.257426\pi$
$359$	2.59144e24	1.38084	0.690420	+	0.723409i	$0.257426\pi$
$360$	0	0
$361$	7.72992e23	0.390712
$362$	0	0
$363$	3.91696e24	1.87861
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−1.06341e24	−0.459595	−0.229797	−	0.973239i	$-0.573806\pi$
$367$	−1.06341e24	−0.459595	−0.229797	+	0.973239i	$0.573806\pi$
$368$	0	0
$369$	−2.39151e24	−0.981573
$370$	0	0
$371$	−3.75270e24	−1.46316
$372$	0	0
$373$	−2.05715e24	−0.762135	−0.381068	−	0.924547i	$-0.624444\pi$
$373$	−2.05715e24	−0.762135	−0.381068	+	0.924547i	$0.624444\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−1.03325e25	−3.45909
$378$	0	0
$379$	1.14682e24	0.365108	0.182554	−	0.983196i	$-0.441564\pi$
$379$	1.14682e24	0.365108	0.182554	+	0.983196i	$0.441564\pi$
$380$	0	0
$381$	4.75556e24	1.44017
$382$	0	0
$383$	−3.86076e24	−1.11246	−0.556231	−	0.831028i	$-0.687753\pi$
$383$	−3.86076e24	−1.11246	−0.556231	+	0.831028i	$0.687753\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−5.21143e23	−0.136051
$388$	0	0
$389$	−3.99976e24	−0.994288	−0.497144	−	0.867668i	$-0.665618\pi$
$389$	−3.99976e24	−0.994288	−0.497144	+	0.867668i	$0.665618\pi$
$390$	0	0
$391$	1.52201e23	0.0360361
$392$	0	0
$393$	−7.74899e24	−1.74790
$394$	0	0
$395$	0	0
$396$	0	0
$397$	5.09807e24	1.04447	0.522235	−	0.852802i	$-0.325099\pi$
$397$	5.09807e24	1.04447	0.522235	+	0.852802i	$0.325099\pi$
$398$	0	0
$399$	1.62052e25	3.16528
$400$	0	0
$401$	−7.17379e24	−1.33622	−0.668109	−	0.744064i	$-0.732896\pi$
$401$	−7.17379e24	−1.33622	−0.668109	+	0.744064i	$0.732896\pi$
$402$	0	0
$403$	1.17518e25	2.08789
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−3.43362e23	−0.0555399
$408$	0	0
$409$	−6.88509e24	−1.06301	−0.531506	−	0.847054i	$-0.678374\pi$
$409$	−6.88509e24	−1.06301	−0.531506	+	0.847054i	$0.678374\pi$
$410$	0	0
$411$	1.53433e25	2.26164
$412$	0	0
$413$	−1.23254e25	−1.73490
$414$	0	0
$415$	0	0
$416$	0	0
$417$	3.04009e24	0.390475
$418$	0	0
$419$	1.56620e24	0.192227	0.0961134	−	0.995370i	$-0.469359\pi$
$419$	1.56620e24	0.192227	0.0961134	+	0.995370i	$0.469359\pi$
$420$	0	0
$421$	−2.15126e23	−0.0252356	−0.0126178	−	0.999920i	$-0.504016\pi$
$421$	−2.15126e23	−0.0252356	−0.0126178	+	0.999920i	$0.504016\pi$
$422$	0	0
$423$	3.08552e25	3.46015
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−2.78063e24	−0.285153
$428$	0	0
$429$	−2.97150e24	−0.291495
$430$	0	0
$431$	1.32009e25	1.23900	0.619498	−	0.784998i	$-0.287336\pi$
$431$	1.32009e25	1.23900	0.619498	+	0.784998i	$0.287336\pi$
$432$	0	0
$433$	6.24562e24	0.560971	0.280486	−	0.959858i	$-0.409505\pi$
$433$	6.24562e24	0.560971	0.280486	+	0.959858i	$0.409505\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−5.83450e24	−0.480209
$438$	0	0
$439$	−3.75363e24	−0.295828	−0.147914	−	0.989000i	$-0.547256\pi$
$439$	−3.75363e24	−0.295828	−0.147914	+	0.989000i	$0.547256\pi$
$440$	0	0
$441$	3.45992e25	2.61155
$442$	0	0
$443$	1.76846e25	1.27867	0.639337	−	0.768927i	$-0.279209\pi$
$443$	1.76846e25	1.27867	0.639337	+	0.768927i	$0.279209\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	1.37210e25	0.910889
$448$	0	0
$449$	−2.51535e25	−1.60051	−0.800256	−	0.599659i	$-0.795303\pi$
$449$	−2.51535e25	−1.60051	−0.800256	+	0.599659i	$0.795303\pi$
$450$	0	0
$451$	−5.29899e23	−0.0323233
$452$	0	0
$453$	2.69171e25	1.57433
$454$	0	0
$455$	0	0
$456$	0	0
$457$	3.18165e25	1.71178	0.855892	−	0.517154i	$-0.173009\pi$
$457$	3.18165e25	1.71178	0.855892	+	0.517154i	$0.173009\pi$
$458$	0	0
$459$	−5.12837e24	−0.264703
$460$	0	0
$461$	6.61360e24	0.327551	0.163776	−	0.986498i	$-0.447633\pi$
$461$	6.61360e24	0.327551	0.163776	+	0.986498i	$0.447633\pi$
$462$	0	0
$463$	1.44650e25	0.687542	0.343771	−	0.939054i	$-0.388296\pi$
$463$	1.44650e25	0.687542	0.343771	+	0.939054i	$0.388296\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	4.46628e24	0.195630	0.0978151	−	0.995205i	$-0.468815\pi$
$467$	4.46628e24	0.195630	0.0978151	+	0.995205i	$0.468815\pi$
$468$	0	0
$469$	−5.83551e24	−0.245435
$470$	0	0
$471$	5.04947e25	2.03961
$472$	0	0
$473$	−1.15472e23	−0.00448019
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−7.43266e25	−2.66207
$478$	0	0
$479$	5.76346e25	1.98379	0.991895	−	0.127057i	$-0.0405530\pi$
$479$	5.76346e25	1.98379	0.991895	+	0.127057i	$0.0405530\pi$
$480$	0	0
$481$	−3.58071e25	−1.18465
$482$	0	0
$483$	−3.43639e25	−1.09296
$484$	0	0
$485$	0	0
$486$	0	0
$487$	4.77931e25	1.40553	0.702765	−	0.711422i	$-0.251948\pi$
$487$	4.77931e25	1.40553	0.702765	+	0.711422i	$0.251948\pi$
$488$	0	0
$489$	4.11961e25	1.16526
$490$	0	0
$491$	6.55305e24	0.178307	0.0891537	−	0.996018i	$-0.471584\pi$
$491$	6.55305e24	0.178307	0.0891537	+	0.996018i	$0.471584\pi$
$492$	0	0
$493$	−6.45084e24	−0.168877
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−2.82274e25	−0.684360
$498$	0	0
$499$	−5.70607e25	−1.33162	−0.665812	−	0.746120i	$-0.731915\pi$
$499$	−5.70607e25	−1.33162	−0.665812	+	0.746120i	$0.731915\pi$
$500$	0	0
$501$	7.88178e25	1.77078
$502$	0	0
$503$	−3.09183e25	−0.668837	−0.334418	−	0.942425i	$-0.608540\pi$
$503$	−3.09183e25	−0.668837	−0.334418	+	0.942425i	$0.608540\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−2.15570e26	−4.32526
$508$	0	0
$509$	1.18190e25	0.228434	0.114217	−	0.993456i	$-0.463564\pi$
$509$	1.18190e25	0.228434	0.114217	+	0.993456i	$0.463564\pi$
$510$	0	0
$511$	−6.94541e25	−1.29330
$512$	0	0
$513$	1.96592e26	3.52737
$514$	0	0
$515$	0	0
$516$	0	0
$517$	6.83673e24	0.113943
$518$	0	0
$519$	1.21148e26	1.94637
$520$	0	0
$521$	−7.37164e25	−1.14184	−0.570920	−	0.821005i	$-0.693413\pi$
$521$	−7.37164e25	−1.14184	−0.570920	+	0.821005i	$0.693413\pi$
$522$	0	0
$523$	4.21942e25	0.630212	0.315106	−	0.949056i	$-0.397960\pi$
$523$	4.21942e25	0.630212	0.315106	+	0.949056i	$0.397960\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	7.33697e24	0.101933
$528$	0	0
$529$	−6.22431e25	−0.834185
$530$	0	0
$531$	−2.44119e26	−3.15648
$532$	0	0
$533$	−5.52600e25	−0.689449
$534$	0	0
$535$	0	0
$536$	0	0
$537$	1.23421e26	1.43428
$538$	0	0
$539$	7.66631e24	0.0859989
$540$	0	0
$541$	1.30393e25	0.141214	0.0706072	−	0.997504i	$-0.477506\pi$
$541$	1.30393e25	0.141214	0.0706072	+	0.997504i	$0.477506\pi$
$542$	0	0
$543$	−2.18367e26	−2.28344
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−1.40434e26	−1.36960	−0.684798	−	0.728733i	$-0.740110\pi$
$547$	−1.40434e26	−1.36960	−0.684798	+	0.728733i	$0.740110\pi$
$548$	0	0
$549$	−5.50736e25	−0.518807
$550$	0	0
$551$	2.47288e26	2.25042
$552$	0	0
$553$	3.32700e25	0.292526
$554$	0	0
$555$	0	0
$556$	0	0
$557$	1.52983e26	1.25609	0.628044	−	0.778178i	$-0.283856\pi$
$557$	1.52983e26	1.25609	0.628044	+	0.778178i	$0.283856\pi$
$558$	0	0
$559$	−1.20419e25	−0.0955614
$560$	0	0
$561$	−1.85519e24	−0.0142312
$562$	0	0
$563$	1.08215e26	0.802522	0.401261	−	0.915964i	$-0.368572\pi$
$563$	1.08215e26	0.802522	0.401261	+	0.915964i	$0.368572\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	6.29943e26	4.36779
$568$	0	0
$569$	1.31768e26	0.883575	0.441788	−	0.897120i	$-0.354344\pi$
$569$	1.31768e26	0.883575	0.441788	+	0.897120i	$0.354344\pi$
$570$	0	0
$571$	1.30249e26	0.844753	0.422377	−	0.906420i	$-0.361196\pi$
$571$	1.30249e26	0.844753	0.422377	+	0.906420i	$0.361196\pi$
$572$	0	0
$573$	−2.76470e26	−1.73452
$574$	0	0
$575$	0	0
$576$	0	0
$577$	3.68374e24	0.0216331	0.0108165	−	0.999941i	$-0.496557\pi$
$577$	3.68374e24	0.0216331	0.0108165	+	0.999941i	$0.496557\pi$
$578$	0	0
$579$	−4.90767e26	−2.78887
$580$	0	0
$581$	3.02085e25	0.166133
$582$	0	0
$583$	−1.64689e25	−0.0876624
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−2.51071e26	−1.25238	−0.626188	−	0.779672i	$-0.715386\pi$
$587$	−2.51071e26	−1.25238	−0.626188	+	0.779672i	$0.715386\pi$
$588$	0	0
$589$	−2.81258e26	−1.35834
$590$	0	0
$591$	−1.57014e26	−0.734274
$592$	0	0
$593$	−1.31913e26	−0.597406	−0.298703	−	0.954346i	$-0.596554\pi$
$593$	−1.31913e26	−0.597406	−0.298703	+	0.954346i	$0.596554\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−2.42406e26	−1.02988
$598$	0	0
$599$	−3.67368e26	−1.51198	−0.755990	−	0.654583i	$-0.772844\pi$
$599$	−3.67368e26	−1.51198	−0.755990	+	0.654583i	$0.772844\pi$
$600$	0	0
$601$	−3.26911e26	−1.30353	−0.651767	−	0.758419i	$-0.725972\pi$
$601$	−3.26911e26	−1.30353	−0.651767	+	0.758419i	$0.725972\pi$
$602$	0	0
$603$	−1.15579e26	−0.446544
$604$	0	0
$605$	0	0
$606$	0	0
$607$	7.56333e25	0.274423	0.137212	−	0.990542i	$-0.456186\pi$
$607$	7.56333e25	0.274423	0.137212	+	0.990542i	$0.456186\pi$
$608$	0	0
$609$	1.45647e27	5.12198
$610$	0	0
$611$	7.12961e26	2.43038
$612$	0	0
$613$	3.13353e25	0.103552	0.0517761	−	0.998659i	$-0.483512\pi$
$613$	3.13353e25	0.103552	0.0517761	+	0.998659i	$0.483512\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	1.29644e26	0.402758	0.201379	−	0.979513i	$-0.435458\pi$
$617$	1.29644e26	0.402758	0.201379	+	0.979513i	$0.435458\pi$
$618$	0	0
$619$	8.76977e25	0.264197	0.132098	−	0.991237i	$-0.457829\pi$
$619$	8.76977e25	0.264197	0.132098	+	0.991237i	$0.457829\pi$
$620$	0	0
$621$	−4.16883e26	−1.21799
$622$	0	0
$623$	−5.24565e26	−1.48649
$624$	0	0
$625$	0	0
$626$	0	0
$627$	7.11173e25	0.189641
$628$	0	0
$629$	−2.23553e25	−0.0578363
$630$	0	0
$631$	−4.98110e26	−1.25039	−0.625196	−	0.780468i	$-0.714981\pi$
$631$	−4.98110e26	−1.25039	−0.625196	+	0.780468i	$0.714981\pi$
$632$	0	0
$633$	−3.32029e26	−0.808799
$634$	0	0
$635$	0	0
$636$	0	0
$637$	7.99472e26	1.83434
$638$	0	0
$639$	−5.59076e26	−1.24512
$640$	0	0
$641$	1.30342e26	0.281796	0.140898	−	0.990024i	$-0.455001\pi$
$641$	1.30342e26	0.281796	0.140898	+	0.990024i	$0.455001\pi$
$642$	0	0
$643$	−7.42903e26	−1.55929	−0.779647	−	0.626219i	$-0.784602\pi$
$643$	−7.42903e26	−1.55929	−0.779647	+	0.626219i	$0.784602\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	6.41134e26	1.26870	0.634348	−	0.773047i	$-0.281268\pi$
$647$	6.41134e26	1.26870	0.634348	+	0.773047i	$0.281268\pi$
$648$	0	0
$649$	−5.40906e25	−0.103943
$650$	0	0
$651$	−1.65654e27	−3.09160
$652$	0	0
$653$	3.48963e26	0.632563	0.316282	−	0.948665i	$-0.397566\pi$
$653$	3.48963e26	0.632563	0.316282	+	0.948665i	$0.397566\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−1.37562e27	−2.35303
$658$	0	0
$659$	−5.69000e26	−0.945586	−0.472793	−	0.881173i	$-0.656754\pi$
$659$	−5.69000e26	−0.945586	−0.472793	+	0.881173i	$0.656754\pi$
$660$	0	0
$661$	−1.01615e27	−1.64075	−0.820375	−	0.571826i	$-0.806235\pi$
$661$	−1.01615e27	−1.64075	−0.820375	+	0.571826i	$0.806235\pi$
$662$	0	0
$663$	−1.93466e26	−0.303547
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−5.24387e26	−0.777063
$668$	0	0
$669$	6.53265e26	0.940895
$670$	0	0
$671$	−1.22029e25	−0.0170844
$672$	0	0
$673$	9.90630e26	1.34824	0.674122	−	0.738620i	$-0.264522\pi$
$673$	9.90630e26	1.34824	0.674122	+	0.738620i	$0.264522\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−1.44432e27	−1.85811	−0.929053	−	0.369947i	$-0.879376\pi$
$677$	−1.44432e27	−1.85811	−0.929053	+	0.369947i	$0.879376\pi$
$678$	0	0
$679$	−1.29435e26	−0.161916
$680$	0	0
$681$	−7.50727e26	−0.913239
$682$	0	0
$683$	−1.46718e27	−1.73575	−0.867875	−	0.496783i	$-0.834514\pi$
$683$	−1.46718e27	−1.73575	−0.867875	+	0.496783i	$0.834514\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−6.40835e26	−0.717228
$688$	0	0
$689$	−1.71744e27	−1.86982
$690$	0	0
$691$	1.22743e27	1.30004	0.650020	−	0.759917i	$-0.274760\pi$
$691$	1.22743e27	1.30004	0.650020	+	0.759917i	$0.274760\pi$
$692$	0	0
$693$	3.01886e26	0.311084
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−3.45003e25	−0.0336598
$698$	0	0
$699$	1.94556e27	1.84719
$700$	0	0
$701$	−1.00489e27	−0.928533	−0.464266	−	0.885696i	$-0.653682\pi$
$701$	−1.00489e27	−0.928533	−0.464266	+	0.885696i	$0.653682\pi$
$702$	0	0
$703$	8.56976e26	0.770714
$704$	0	0
$705$	0	0
$706$	0	0
$707$	2.41684e27	2.05951
$708$	0	0
$709$	−5.79079e26	−0.480395	−0.240197	−	0.970724i	$-0.577212\pi$
$709$	−5.79079e26	−0.480395	−0.240197	+	0.970724i	$0.577212\pi$
$710$	0	0
$711$	6.58952e26	0.532222
$712$	0	0
$713$	5.96420e26	0.469031
$714$	0	0
$715$	0	0
$716$	0	0
$717$	2.43066e27	1.81256
$718$	0	0
$719$	1.47567e27	1.07168	0.535838	−	0.844321i	$-0.319996\pi$
$719$	1.47567e27	1.07168	0.535838	+	0.844321i	$0.319996\pi$
$720$	0	0
$721$	3.89249e27	2.75323
$722$	0	0
$723$	2.86843e27	1.97619
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−8.45090e26	−0.552492	−0.276246	−	0.961087i	$-0.589091\pi$
$727$	−8.45090e26	−0.552492	−0.276246	+	0.961087i	$0.589091\pi$
$728$	0	0
$729$	3.59030e27	2.28675
$730$	0	0
$731$	−7.51807e24	−0.00466542
$732$	0	0
$733$	−2.44378e27	−1.47766	−0.738831	−	0.673891i	$-0.764622\pi$
$733$	−2.44378e27	−1.47766	−0.738831	+	0.673891i	$0.764622\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−2.56094e25	−0.0147048
$738$	0	0
$739$	−1.49906e27	−0.838873	−0.419436	−	0.907785i	$-0.637772\pi$
$739$	−1.49906e27	−0.838873	−0.419436	+	0.907785i	$0.637772\pi$
$740$	0	0
$741$	7.41639e27	4.04501
$742$	0	0
$743$	−6.95305e26	−0.369642	−0.184821	−	0.982772i	$-0.559171\pi$
$743$	−6.95305e26	−0.369642	−0.184821	+	0.982772i	$0.559171\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	5.98315e26	0.302263
$748$	0	0
$749$	−3.47692e27	−1.71245
$750$	0	0
$751$	−8.33398e26	−0.400196	−0.200098	−	0.979776i	$-0.564126\pi$
$751$	−8.33398e26	−0.400196	−0.200098	+	0.979776i	$0.564126\pi$
$752$	0	0
$753$	4.69389e27	2.19776
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−9.69338e26	−0.431583	−0.215791	−	0.976439i	$-0.569233\pi$
$757$	−9.69338e26	−0.431583	−0.215791	+	0.976439i	$0.569233\pi$
$758$	0	0
$759$	−1.50808e26	−0.0654827
$760$	0	0
$761$	−2.50883e27	−1.06247	−0.531236	−	0.847224i	$-0.678272\pi$
$761$	−2.50883e27	−1.06247	−0.531236	+	0.847224i	$0.678272\pi$
$762$	0	0
$763$	−4.52669e27	−1.86981
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−5.64078e27	−2.21709
$768$	0	0
$769$	−2.94988e27	−1.13111	−0.565555	−	0.824711i	$-0.691338\pi$
$769$	−2.94988e27	−1.13111	−0.565555	+	0.824711i	$0.691338\pi$
$770$	0	0
$771$	6.54145e27	2.44713
$772$	0	0
$773$	4.55550e25	0.0166277	0.00831383	−	0.999965i	$-0.497354\pi$
$773$	4.55550e25	0.0166277	0.00831383	+	0.999965i	$0.497354\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	5.04739e27	1.75415
$778$	0	0
$779$	1.32254e27	0.448543
$780$	0	0
$781$	−1.23877e26	−0.0410021
$782$	0	0
$783$	1.76691e28	5.70792
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−2.51757e26	−0.0774856	−0.0387428	−	0.999249i	$-0.512335\pi$
$787$	−2.51757e26	−0.0774856	−0.0387428	+	0.999249i	$0.512335\pi$
$788$	0	0
$789$	−1.06333e28	−3.19476
$790$	0	0
$791$	2.29258e27	0.672431
$792$	0	0
$793$	−1.27257e27	−0.364406
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−3.04480e27	−0.831199	−0.415600	−	0.909548i	$-0.636428\pi$
$797$	−3.04480e27	−0.831199	−0.415600	+	0.909548i	$0.636428\pi$
$798$	0	0
$799$	4.45120e26	0.118654
$800$	0	0
$801$	−1.03896e28	−2.70453
$802$	0	0
$803$	−3.04803e26	−0.0774856
$804$	0	0
$805$	0	0
$806$	0	0
$807$	2.07104e27	0.502216
$808$	0	0
$809$	6.08654e27	1.44165	0.720825	−	0.693117i	$-0.243763\pi$
$809$	6.08654e27	1.44165	0.720825	+	0.693117i	$0.243763\pi$
$810$	0	0
$811$	5.89226e27	1.36328	0.681638	−	0.731689i	$-0.261268\pi$
$811$	5.89226e27	1.36328	0.681638	+	0.731689i	$0.261268\pi$
$812$	0	0
$813$	−2.58634e26	−0.0584555
$814$	0	0
$815$	0	0
$816$	0	0
$817$	2.88200e26	0.0621705
$818$	0	0
$819$	3.14819e28	6.63534
$820$	0	0
$821$	−3.76569e27	−0.775505	−0.387752	−	0.921764i	$-0.626748\pi$
$821$	−3.76569e27	−0.775505	−0.387752	+	0.921764i	$0.626748\pi$
$822$	0	0
$823$	−6.70942e27	−1.35016	−0.675082	−	0.737743i	$-0.735892\pi$
$823$	−6.70942e27	−1.35016	−0.675082	+	0.737743i	$0.735892\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	2.20966e27	0.424642	0.212321	−	0.977200i	$-0.431898\pi$
$827$	2.20966e27	0.424642	0.212321	+	0.977200i	$0.431898\pi$
$828$	0	0
$829$	3.05052e25	0.00572935	0.00286468	−	0.999996i	$-0.499088\pi$
$829$	3.05052e25	0.00572935	0.00286468	+	0.999996i	$0.499088\pi$
$830$	0	0
$831$	−2.04055e28	−3.74574
$832$	0	0
$833$	4.99132e26	0.0895545
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−2.00962e28	−3.44527
$838$	0	0
$839$	−1.07213e28	−1.79684	−0.898418	−	0.439141i	$-0.855283\pi$
$839$	−1.07213e28	−1.79684	−0.898418	+	0.439141i	$0.855283\pi$
$840$	0	0
$841$	1.61222e28	2.64158
$842$	0	0
$843$	9.94066e27	1.59240
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−9.19504e27	−1.40819
$848$	0	0
$849$	−1.41467e28	−2.11852
$850$	0	0
$851$	−1.81726e27	−0.266125
$852$	0	0
$853$	−5.76672e27	−0.825872	−0.412936	−	0.910760i	$-0.635497\pi$
$853$	−5.76672e27	−0.825872	−0.412936	+	0.910760i	$0.635497\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−1.88292e27	−0.257937	−0.128969	−	0.991649i	$-0.541167\pi$
$857$	−1.88292e27	−0.257937	−0.128969	+	0.991649i	$0.541167\pi$
$858$	0	0
$859$	4.48492e27	0.600924	0.300462	−	0.953794i	$-0.402859\pi$
$859$	4.48492e27	0.600924	0.300462	+	0.953794i	$0.402859\pi$
$860$	0	0
$861$	7.78948e27	1.02089
$862$	0	0
$863$	−1.35141e27	−0.173255	−0.0866275	−	0.996241i	$-0.527609\pi$
$863$	−1.35141e27	−0.173255	−0.0866275	+	0.996241i	$0.527609\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	1.53021e28	1.87745
$868$	0	0
$869$	1.46007e26	0.0175261
$870$	0	0
$871$	−2.67065e27	−0.313649
$872$	0	0
$873$	−2.56362e27	−0.294590
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−1.87978e27	−0.206828	−0.103414	−	0.994638i	$-0.532977\pi$
$877$	−1.87978e27	−0.206828	−0.103414	+	0.994638i	$0.532977\pi$
$878$	0	0
$879$	−3.97192e27	−0.427668
$880$	0	0
$881$	−1.96525e27	−0.207084	−0.103542	−	0.994625i	$-0.533018\pi$
$881$	−1.96525e27	−0.207084	−0.103542	+	0.994625i	$0.533018\pi$
$882$	0	0
$883$	−7.52802e27	−0.776344	−0.388172	−	0.921587i	$-0.626893\pi$
$883$	−7.52802e27	−0.776344	−0.388172	+	0.921587i	$0.626893\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	9.05381e27	0.894452	0.447226	−	0.894421i	$-0.352412\pi$
$887$	9.05381e27	0.894452	0.447226	+	0.894421i	$0.352412\pi$
$888$	0	0
$889$	−1.11637e28	−1.07954
$890$	0	0
$891$	2.76453e27	0.261688
$892$	0	0
$893$	−1.70634e28	−1.58116
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−1.57268e28	−1.39673
$898$	0	0
$899$	−2.52786e28	−2.19804
$900$	0	0
$901$	−1.07224e27	−0.0912868
$902$	0	0
$903$	1.69743e27	0.141501
$904$	0	0
$905$	0	0
$906$	0	0
$907$	1.23012e28	0.983281	0.491641	−	0.870798i	$-0.336397\pi$
$907$	1.23012e28	0.983281	0.491641	+	0.870798i	$0.336397\pi$
$908$	0	0
$909$	4.78684e28	3.74707
$910$	0	0
$911$	1.30146e28	0.997711	0.498855	−	0.866685i	$-0.333754\pi$
$911$	1.30146e28	0.997711	0.498855	+	0.866685i	$0.333754\pi$
$912$	0	0
$913$	1.32572e26	0.00995356
$914$	0	0
$915$	0	0
$916$	0	0
$917$	1.81907e28	1.31021
$918$	0	0
$919$	−1.12178e27	−0.0791429	−0.0395715	−	0.999217i	$-0.512599\pi$
$919$	−1.12178e27	−0.0791429	−0.0395715	+	0.999217i	$0.512599\pi$
$920$	0	0
$921$	−2.70288e27	−0.186793
$922$	0	0
$923$	−1.29184e28	−0.874566
$924$	0	0
$925$	0	0
$926$	0	0
$927$	7.70954e28	5.00923
$928$	0	0
$929$	−2.94158e28	−1.87254	−0.936270	−	0.351283i	$-0.885746\pi$
$929$	−2.94158e28	−1.87254	−0.936270	+	0.351283i	$0.885746\pi$
$930$	0	0
$931$	−1.91339e28	−1.19338
$932$	0	0
$933$	3.75610e28	2.29541
$934$	0	0
$935$	0	0
$936$	0	0
$937$	4.15594e27	0.243861	0.121931	−	0.992539i	$-0.461091\pi$
$937$	4.15594e27	0.243861	0.121931	+	0.992539i	$0.461091\pi$
$938$	0	0
$939$	3.85842e28	2.21863
$940$	0	0
$941$	−1.78354e27	−0.100503	−0.0502517	−	0.998737i	$-0.516002\pi$
$941$	−1.78354e27	−0.100503	−0.0502517	+	0.998737i	$0.516002\pi$
$942$	0	0
$943$	−2.80452e27	−0.154881
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−9.96565e27	−0.528665	−0.264333	−	0.964432i	$-0.585152\pi$
$947$	−9.96565e27	−0.528665	−0.264333	+	0.964432i	$0.585152\pi$
$948$	0	0
$949$	−3.17860e28	−1.65275
$950$	0	0
$951$	−2.35330e28	−1.19940
$952$	0	0
$953$	−9.27281e27	−0.463264	−0.231632	−	0.972803i	$-0.574407\pi$
$953$	−9.27281e27	−0.463264	−0.231632	+	0.972803i	$0.574407\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	6.39180e27	0.306874
$958$	0	0
$959$	−3.60185e28	−1.69531
$960$	0	0
$961$	7.08033e27	0.326724
$962$	0	0
$963$	−6.88644e28	−3.11562
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−2.78650e28	−1.21201	−0.606006	−	0.795460i	$-0.707229\pi$
$967$	−2.78650e28	−1.21201	−0.606006	+	0.795460i	$0.707229\pi$
$968$	0	0
$969$	4.63025e27	0.197482
$970$	0	0
$971$	3.65181e28	1.52730	0.763652	−	0.645628i	$-0.223404\pi$
$971$	3.65181e28	1.52730	0.763652	+	0.645628i	$0.223404\pi$
$972$	0	0
$973$	−7.13659e27	−0.292697
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−2.93762e28	−1.15877	−0.579385	−	0.815054i	$-0.696707\pi$
$977$	−2.93762e28	−1.15877	−0.579385	+	0.815054i	$0.696707\pi$
$978$	0	0
$979$	−2.30208e27	−0.0890604
$980$	0	0
$981$	−8.96564e28	−3.40193
$982$	0	0
$983$	1.89117e28	0.703838	0.351919	−	0.936030i	$-0.385529\pi$
$983$	1.89117e28	0.703838	0.351919	+	0.936030i	$0.385529\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−1.00499e29	−3.59874
$988$	0	0
$989$	−6.11142e26	−0.0214673
$990$	0	0
$991$	2.14871e28	0.740419	0.370209	−	0.928948i	$-0.379286\pi$
$991$	2.14871e28	0.740419	0.370209	+	0.928948i	$0.379286\pi$
$992$	0	0
$993$	−4.94396e28	−1.67131
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−1.45561e28	−0.473632	−0.236816	−	0.971554i	$-0.576104\pi$
$997$	−1.45561e28	−0.473632	−0.236816	+	0.971554i	$0.576104\pi$
$998$	0	0
$999$	6.12321e28	1.95482

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	100.20.a.b.1.1		3
5.2	odd	4		100.20.c.b.49.6		6
5.3	odd	4		100.20.c.b.49.1		6
5.4	even	2		20.20.a.a.1.3	✓	3
20.19	odd	2		80.20.a.e.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
20.20.a.a.1.3	✓	3	5.4	even	2
80.20.a.e.1.1		3	20.19	odd	2
100.20.a.b.1.1		3	1.1	even	1	trivial
100.20.c.b.49.1		6	5.3	odd	4
100.20.c.b.49.6		6	5.2	odd	4

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:	\( N \)	\(=\)	\( 100 = 2^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 20 \)
Character orbit:	\([\chi]\)	\(=\)	100.a (trivial)

\(f(q)\)	\(=\)	\(q-64511.3 q^{3} +1.51440e8 q^{7} +2.99945e9 q^{9} +O(q^{10})\)	\(q-64511.3 q^{3} +1.51440e8 q^{7} +2.99945e9 q^{9} +6.64602e8 q^{11} +6.93073e10 q^{13} +4.32704e10 q^{17} -1.65874e12 q^{19} -9.76959e12 q^{21} +3.51744e12 q^{23} -1.18519e14 q^{27} -1.49082e14 q^{29} +1.69561e14 q^{31} -4.28743e13 q^{33} -5.16643e14 q^{37} -4.47110e15 q^{39} -7.97319e14 q^{41} -1.73746e14 q^{43} +1.02870e16 q^{47} +1.15352e16 q^{49} -2.79143e15 q^{51} -2.47801e16 q^{53} +1.07007e17 q^{57} -8.13879e16 q^{59} -1.83613e16 q^{61} +4.54236e17 q^{63} -3.85335e16 q^{67} -2.26914e17 q^{69} -1.86393e17 q^{71} -4.58624e17 q^{73} +1.00647e17 q^{77} +2.19691e17 q^{79} +4.15968e18 q^{81} +1.99475e17 q^{83} +9.61749e18 q^{87} -3.46385e18 q^{89} +1.04959e19 q^{91} -1.09386e19 q^{93} -8.54697e17 q^{97} +1.99344e18 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 34086 q^{3} + 115130574 q^{7} + 3129228027 q^{9}+O(q^{10}) \) 3 * q - 34086 * q^3 + 115130574 * q^7 + 3129228027 * q^9	\( 3 q - 34086 q^{3} + 115130574 q^{7} + 3129228027 q^{9} + 6359213280 q^{11} + 33996539826 q^{13} + 60515110578 q^{17} - 1436218405908 q^{19} - 7119772762332 q^{21} - 1682066292342 q^{23} - 91316335875732 q^{27} - 181580995192278 q^{29} - 3566464773252 q^{31} + 90839870844480 q^{33} - 653714901466206 q^{37} - 43\!\cdots\!08 q^{39}+ \cdots + 37\!\cdots\!20 q^{99}+O(q^{100}) \) 3 * q - 34086 * q^3 + 115130574 * q^7 + 3129228027 * q^9 + 6359213280 * q^11 + 33996539826 * q^13 + 60515110578 * q^17 - 1436218405908 * q^19 - 7119772762332 * q^21 - 1682066292342 * q^23 - 91316335875732 * q^27 - 181580995192278 * q^29 - 3566464773252 * q^31 + 90839870844480 * q^33 - 653714901466206 * q^37 - 4339017394217508 * q^39 - 1257752912210622 * q^41 + 4361040178994610 * q^43 + 23174987171661798 * q^47 - 5454428919168921 * q^49 - 33371959424606244 * q^51 + 12180381712114362 * q^53 + 177419957801365416 * q^57 - 180023573560918116 * q^59 - 213491197168562514 * q^61 + 549307385783925246 * q^63 + 32494317103156134 * q^67 - 498099089220386004 * q^69 - 245957452394007276 * q^71 + 94437681636898506 * q^73 + 72972512123263680 * q^77 + 819623047726342536 * q^79 + 3159048884904329967 * q^81 - 2812828487653823982 * q^83 + 2510285506713629676 * q^87 + 4879439966436006078 * q^89 + 12213088562709029652 * q^91 - 16020238495381453176 * q^93 - 8189756396617107846 * q^97 + 3795512975671396320 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more