LMFDB - Embedded newform 1881.2.a.k.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1881,2,Mod(1,1881)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1881, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1881.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1881 = 3^{2} \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1881.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:	$15.0198606202$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.246832.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 5x^{3} + 6x^{2} + 7x - 2 $ x^5 - 2x^4 - 5x^3 + 6x^2 + 7x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 209)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.51908$ of defining polynomial
Character	$\chi$	$=$	1881.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.82669 q^{2} +1.33679 q^{4} -2.34577 q^{5} +1.69239 q^{7} +1.21147 q^{8} +4.28499 q^{10} -1.00000 q^{11} +4.11168 q^{13} -3.09147 q^{14} -4.88657 q^{16} +6.16499 q^{17} -1.00000 q^{19} -3.13581 q^{20} +1.82669 q^{22} -3.52199 q^{23} +0.502638 q^{25} -7.51076 q^{26} +2.26238 q^{28} +8.10336 q^{29} -2.30144 q^{31} +6.50330 q^{32} -11.2615 q^{34} -3.96996 q^{35} -6.56016 q^{37} +1.82669 q^{38} -2.84184 q^{40} -7.75013 q^{41} +7.75102 q^{43} -1.33679 q^{44} +6.43359 q^{46} +10.8969 q^{47} -4.13581 q^{49} -0.918163 q^{50} +5.49647 q^{52} -7.93511 q^{53} +2.34577 q^{55} +2.05029 q^{56} -14.8023 q^{58} -10.9247 q^{59} -4.51162 q^{61} +4.20401 q^{62} -2.10636 q^{64} -9.64506 q^{65} +14.7201 q^{67} +8.24132 q^{68} +7.25189 q^{70} -3.12026 q^{71} +11.5827 q^{73} +11.9834 q^{74} -1.33679 q^{76} -1.69239 q^{77} +4.96184 q^{79} +11.4628 q^{80} +14.1571 q^{82} +1.82905 q^{83} -14.4617 q^{85} -14.1587 q^{86} -1.21147 q^{88} +9.37496 q^{89} +6.95858 q^{91} -4.70818 q^{92} -19.9053 q^{94} +2.34577 q^{95} -10.9937 q^{97} +7.55484 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 2 q^{2} + 6 q^{4} + 5 q^{5} + 6 q^{7} - 6 q^{8} + 12 q^{10} - 5 q^{11} + 4 q^{13} + 14 q^{14} + 8 q^{16} + 4 q^{17} - 5 q^{19} + 8 q^{20} + 2 q^{22} - 3 q^{23} + 6 q^{25} + 6 q^{26} - 10 q^{28} - 10 q^{29}+ \cdots + 10 q^{98}+O(q^{100}) $ 5 * q - 2 * q^2 + 6 * q^4 + 5 * q^5 + 6 * q^7 - 6 * q^8 + 12 * q^10 - 5 * q^11 + 4 * q^13 + 14 * q^14 + 8 * q^16 + 4 * q^17 - 5 * q^19 + 8 * q^20 + 2 * q^22 - 3 * q^23 + 6 * q^25 + 6 * q^26 - 10 * q^28 - 10 * q^29 + 11 * q^31 - 14 * q^32 - 4 * q^34 + 8 * q^35 + q^37 + 2 * q^38 - 16 * q^40 - 2 * q^41 + 20 * q^43 - 6 * q^44 - 4 * q^46 + 20 * q^47 + 3 * q^49 + 32 * q^50 + 6 * q^52 + 14 * q^53 - 5 * q^55 + 38 * q^56 - 6 * q^58 - 3 * q^59 - 10 * q^61 + 6 * q^62 + 9 * q^67 - 24 * q^68 + 50 * q^70 - 23 * q^71 - 8 * q^74 - 6 * q^76 - 6 * q^77 + 44 * q^79 + 18 * q^80 - 30 * q^82 + 14 * q^83 - 12 * q^85 - 52 * q^86 + 6 * q^88 + 27 * q^89 + 24 * q^91 - 58 * q^92 - 8 * q^94 - 5 * q^95 + 15 * q^97 + 10 * q^98

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 $

$2$	−1.82669	−1.29166	−0.645832	−	0.763479i	$-0.723489\pi$
$2$	−1.82669	−1.29166	−0.645832	+	0.763479i	$0.723489\pi$
$3$	0	0
$3$	0	0
$4$	1.33679	0.668396
$5$	−2.34577	−1.04906	−0.524530	−	0.851392i	$-0.675759\pi$
$5$	−2.34577	−1.04906	−0.524530	+	0.851392i	$0.675759\pi$
$6$	0	0
$7$	1.69239	0.639664	0.319832	−	0.947474i	$-0.396374\pi$
$7$	1.69239	0.639664	0.319832	+	0.947474i	$0.396374\pi$
$8$	1.21147	0.428321
$9$	0	0
$10$	4.28499	1.35503
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	4.11168	1.14038	0.570188	−	0.821514i	$-0.306870\pi$
$13$	4.11168	1.14038	0.570188	+	0.821514i	$0.306870\pi$
$14$	−3.09147	−0.826231
$15$	0	0
$16$	−4.88657	−1.22164
$17$	6.16499	1.49523	0.747615	−	0.664132i	$-0.231199\pi$
$17$	6.16499	1.49523	0.747615	+	0.664132i	$0.231199\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	−3.13581	−0.701188
$21$	0	0
$22$	1.82669	0.389451
$23$	−3.52199	−0.734386	−0.367193	−	0.930145i	$-0.619681\pi$
$23$	−3.52199	−0.734386	−0.367193	+	0.930145i	$0.619681\pi$
$24$	0	0
$25$	0.502638	0.100528
$26$	−7.51076	−1.47298
$27$	0	0
$28$	2.26238	0.427549
$29$	8.10336	1.50476	0.752378	−	0.658731i	$-0.228906\pi$
$29$	8.10336	1.50476	0.752378	+	0.658731i	$0.228906\pi$
$30$	0	0
$31$	−2.30144	−0.413350	−0.206675	−	0.978410i	$-0.566264\pi$
$31$	−2.30144	−0.413350	−0.206675	+	0.978410i	$0.566264\pi$
$32$	6.50330	1.14963
$33$	0	0
$34$	−11.2615	−1.93134
$35$	−3.96996	−0.671046
$36$	0	0
$37$	−6.56016	−1.07848	−0.539242	−	0.842151i	$-0.681289\pi$
$37$	−6.56016	−1.07848	−0.539242	+	0.842151i	$0.681289\pi$
$38$	1.82669	0.296328
$39$	0	0
$40$	−2.84184	−0.449334
$41$	−7.75013	−1.21037	−0.605183	−	0.796086i	$-0.706900\pi$
$41$	−7.75013	−1.21037	−0.605183	+	0.796086i	$0.706900\pi$
$42$	0	0
$43$	7.75102	1.18202	0.591010	−	0.806664i	$-0.298729\pi$
$43$	7.75102	1.18202	0.591010	+	0.806664i	$0.298729\pi$
$44$	−1.33679	−0.201529
$45$	0	0
$46$	6.43359	0.948581
$47$	10.8969	1.58948	0.794742	−	0.606948i	$-0.207606\pi$
$47$	10.8969	1.58948	0.794742	+	0.606948i	$0.207606\pi$
$48$	0	0
$49$	−4.13581	−0.590830
$50$	−0.918163	−0.129848
$51$	0	0
$52$	5.49647	0.762223
$53$	−7.93511	−1.08997	−0.544986	−	0.838445i	$-0.683465\pi$
$53$	−7.93511	−1.08997	−0.544986	+	0.838445i	$0.683465\pi$
$54$	0	0
$55$	2.34577	0.316304
$56$	2.05029	0.273981
$57$	0	0
$58$	−14.8023	−1.94364
$59$	−10.9247	−1.42228	−0.711140	−	0.703051i	$-0.751821\pi$
$59$	−10.9247	−1.42228	−0.711140	+	0.703051i	$0.751821\pi$
$60$	0	0
$61$	−4.51162	−0.577653	−0.288827	−	0.957381i	$-0.593265\pi$
$61$	−4.51162	−0.577653	−0.288827	+	0.957381i	$0.593265\pi$
$62$	4.20401	0.533910
$63$	0	0
$64$	−2.10636	−0.263295
$65$	−9.64506	−1.19632
$66$	0	0
$67$	14.7201	1.79835	0.899173	−	0.437594i	$-0.144169\pi$
$67$	14.7201	1.79835	0.899173	+	0.437594i	$0.144169\pi$
$68$	8.24132	0.999407
$69$	0	0
$70$	7.25189	0.866766
$71$	−3.12026	−0.370307	−0.185153	−	0.982710i	$-0.559278\pi$
$71$	−3.12026	−0.370307	−0.185153	+	0.982710i	$0.559278\pi$
$72$	0	0
$73$	11.5827	1.35565	0.677824	−	0.735224i	$-0.262923\pi$
$73$	11.5827	1.35565	0.677824	+	0.735224i	$0.262923\pi$
$74$	11.9834	1.39304
$75$	0	0
$76$	−1.33679	−0.153341
$77$	−1.69239	−0.192866
$78$	0	0
$79$	4.96184	0.558250	0.279125	−	0.960255i	$-0.409956\pi$
$79$	4.96184	0.558250	0.279125	+	0.960255i	$0.409956\pi$
$80$	11.4628	1.28158
$81$	0	0
$82$	14.1571	1.56339
$83$	1.82905	0.200765	0.100382	−	0.994949i	$-0.467993\pi$
$83$	1.82905	0.200765	0.100382	+	0.994949i	$0.467993\pi$
$84$	0	0
$85$	−14.4617	−1.56859
$86$	−14.1587	−1.52677
$87$	0	0
$88$	−1.21147	−0.129144
$89$	9.37496	0.993743	0.496872	−	0.867824i	$-0.334482\pi$
$89$	9.37496	0.993743	0.496872	+	0.867824i	$0.334482\pi$
$90$	0	0
$91$	6.95858	0.729457
$92$	−4.70818	−0.490861
$93$	0	0
$94$	−19.9053	−2.05308
$95$	2.34577	0.240671
$96$	0	0
$97$	−10.9937	−1.11625	−0.558123	−	0.829758i	$-0.688478\pi$
$97$	−10.9937	−1.11625	−0.558123	+	0.829758i	$0.688478\pi$
$98$	7.55484	0.763154
$99$	0	0
$100$	0.671923	0.0671923
$101$	−2.30621	−0.229476	−0.114738	−	0.993396i	$-0.536603\pi$
$101$	−2.30621	−0.229476	−0.114738	+	0.993396i	$0.536603\pi$
$102$	0	0
$103$	6.12000	0.603021	0.301511	−	0.953463i	$-0.402509\pi$
$103$	6.12000	0.603021	0.301511	+	0.953463i	$0.402509\pi$
$104$	4.98119	0.488446
$105$	0	0
$106$	14.4950	1.40788
$107$	−0.422947	−0.0408878	−0.0204439	−	0.999791i	$-0.506508\pi$
$107$	−0.422947	−0.0408878	−0.0204439	+	0.999791i	$0.506508\pi$
$108$	0	0
$109$	14.0180	1.34268	0.671338	−	0.741151i	$-0.265720\pi$
$109$	14.0180	1.34268	0.671338	+	0.741151i	$0.265720\pi$
$110$	−4.28499	−0.408558
$111$	0	0
$112$	−8.26999	−0.781441
$113$	2.53638	0.238602	0.119301	−	0.992858i	$-0.461935\pi$
$113$	2.53638	0.238602	0.119301	+	0.992858i	$0.461935\pi$
$114$	0	0
$115$	8.26179	0.770416
$116$	10.8325	1.00577
$117$	0	0
$118$	19.9561	1.83711
$119$	10.4336	0.956445
$120$	0	0
$121$	1.00000	0.0909091
$122$	8.24132	0.746134
$123$	0	0
$124$	−3.07654	−0.276282
$125$	10.5498	0.943601
$126$	0	0
$127$	1.07275	0.0951914	0.0475957	−	0.998867i	$-0.484844\pi$
$127$	1.07275	0.0951914	0.0475957	+	0.998867i	$0.484844\pi$
$128$	−9.15893	−0.809543
$129$	0	0
$130$	17.6185	1.54525
$131$	7.82709	0.683856	0.341928	−	0.939726i	$-0.388920\pi$
$131$	7.82709	0.683856	0.341928	+	0.939726i	$0.388920\pi$
$132$	0	0
$133$	−1.69239	−0.146749
$134$	−26.8890	−2.32286
$135$	0	0
$136$	7.46873	0.640438
$137$	11.2473	0.960919	0.480460	−	0.877017i	$-0.340470\pi$
$137$	11.2473	0.960919	0.480460	+	0.877017i	$0.340470\pi$
$138$	0	0
$139$	−0.905926	−0.0768397	−0.0384198	−	0.999262i	$-0.512232\pi$
$139$	−0.905926	−0.0768397	−0.0384198	+	0.999262i	$0.512232\pi$
$140$	−5.30702	−0.448525
$141$	0	0
$142$	5.69975	0.478312
$143$	−4.11168	−0.343836
$144$	0	0
$145$	−19.0086	−1.57858
$146$	−21.1579	−1.75104
$147$	0	0
$148$	−8.76957	−0.720854
$149$	−10.5174	−0.861622	−0.430811	−	0.902442i	$-0.641772\pi$
$149$	−10.5174	−0.861622	−0.430811	+	0.902442i	$0.641772\pi$
$150$	0	0
$151$	13.2436	1.07775	0.538873	−	0.842387i	$-0.318850\pi$
$151$	13.2436	1.07775	0.538873	+	0.842387i	$0.318850\pi$
$152$	−1.21147	−0.0982635
$153$	0	0
$154$	3.09147	0.249118
$155$	5.39864	0.433629
$156$	0	0
$157$	1.99915	0.159549	0.0797747	−	0.996813i	$-0.474580\pi$
$157$	1.99915	0.159549	0.0797747	+	0.996813i	$0.474580\pi$
$158$	−9.06373	−0.721072
$159$	0	0
$160$	−15.2552	−1.20603
$161$	−5.96059	−0.469761
$162$	0	0
$163$	−18.7557	−1.46906	−0.734531	−	0.678575i	$-0.762598\pi$
$163$	−18.7557	−1.46906	−0.734531	+	0.678575i	$0.762598\pi$
$164$	−10.3603	−0.809005
$165$	0	0
$166$	−3.34111	−0.259320
$167$	6.31203	0.488440	0.244220	−	0.969720i	$-0.421468\pi$
$167$	6.31203	0.488440	0.244220	+	0.969720i	$0.421468\pi$
$168$	0	0
$169$	3.90593	0.300456
$170$	26.4170	2.02609
$171$	0	0
$172$	10.3615	0.790058
$173$	21.5269	1.63666	0.818328	−	0.574751i	$-0.194901\pi$
$173$	21.5269	1.63666	0.818328	+	0.574751i	$0.194901\pi$
$174$	0	0
$175$	0.850661	0.0643039
$176$	4.88657	0.368339
$177$	0	0
$178$	−17.1251	−1.28358
$179$	7.97018	0.595719	0.297860	−	0.954610i	$-0.403727\pi$
$179$	7.97018	0.595719	0.297860	+	0.954610i	$0.403727\pi$
$180$	0	0
$181$	−1.16955	−0.0869317	−0.0434659	−	0.999055i	$-0.513840\pi$
$181$	−1.16955	−0.0869317	−0.0434659	+	0.999055i	$0.513840\pi$
$182$	−12.7112	−0.942214
$183$	0	0
$184$	−4.26680	−0.314553
$185$	15.3886	1.13139
$186$	0	0
$187$	−6.16499	−0.450829
$188$	14.5670	1.06240
$189$	0	0
$190$	−4.28499	−0.310866
$191$	15.6673	1.13365	0.566824	−	0.823839i	$-0.308172\pi$
$191$	15.6673	1.13365	0.566824	+	0.823839i	$0.308172\pi$
$192$	0	0
$193$	21.6769	1.56034	0.780169	−	0.625568i	$-0.215133\pi$
$193$	21.6769	1.56034	0.780169	+	0.625568i	$0.215133\pi$
$194$	20.0821	1.44181
$195$	0	0
$196$	−5.52872	−0.394908
$197$	−4.58794	−0.326877	−0.163439	−	0.986554i	$-0.552259\pi$
$197$	−4.58794	−0.326877	−0.163439	+	0.986554i	$0.552259\pi$
$198$	0	0
$199$	13.0619	0.925937	0.462968	−	0.886375i	$-0.346784\pi$
$199$	13.0619	0.925937	0.462968	+	0.886375i	$0.346784\pi$
$200$	0.608933	0.0430580
$201$	0	0
$202$	4.21272	0.296406
$203$	13.7141	0.962539
$204$	0	0
$205$	18.1800	1.26975
$206$	−11.1793	−0.778901
$207$	0	0
$208$	−20.0920	−1.39313
$209$	1.00000	0.0691714
$210$	0	0
$211$	2.91188	0.200462	0.100231	−	0.994964i	$-0.468042\pi$
$211$	2.91188	0.200462	0.100231	+	0.994964i	$0.468042\pi$
$212$	−10.6076	−0.728533
$213$	0	0
$214$	0.772592	0.0528133
$215$	−18.1821	−1.24001
$216$	0	0
$217$	−3.89493	−0.264405
$218$	−25.6064	−1.73429
$219$	0	0
$220$	3.13581	0.211416
$221$	25.3485	1.70512
$222$	0	0
$223$	6.34161	0.424665	0.212333	−	0.977197i	$-0.431894\pi$
$223$	6.34161	0.424665	0.212333	+	0.977197i	$0.431894\pi$
$224$	11.0061	0.735378
$225$	0	0
$226$	−4.63317	−0.308194
$227$	2.00429	0.133029	0.0665147	−	0.997785i	$-0.478812\pi$
$227$	2.00429	0.133029	0.0665147	+	0.997785i	$0.478812\pi$
$228$	0	0
$229$	−20.9893	−1.38701	−0.693507	−	0.720450i	$-0.743935\pi$
$229$	−20.9893	−1.38701	−0.693507	+	0.720450i	$0.743935\pi$
$230$	−15.0917	−0.995118
$231$	0	0
$232$	9.81701	0.644518
$233$	17.6006	1.15305	0.576527	−	0.817078i	$-0.304407\pi$
$233$	17.6006	1.15305	0.576527	+	0.817078i	$0.304407\pi$
$234$	0	0
$235$	−25.5617	−1.66746
$236$	−14.6041	−0.950646
$237$	0	0
$238$	−19.0589	−1.23541
$239$	−26.6207	−1.72195	−0.860975	−	0.508647i	$-0.830146\pi$
$239$	−26.6207	−1.72195	−0.860975	+	0.508647i	$0.830146\pi$
$240$	0	0
$241$	13.1342	0.846049	0.423024	−	0.906118i	$-0.360968\pi$
$241$	13.1342	0.846049	0.423024	+	0.906118i	$0.360968\pi$
$242$	−1.82669	−0.117424
$243$	0	0
$244$	−6.03109	−0.386101
$245$	9.70166	0.619816
$246$	0	0
$247$	−4.11168	−0.261620
$248$	−2.78813	−0.177046
$249$	0	0
$250$	−19.2712	−1.21882
$251$	26.1636	1.65143	0.825715	−	0.564087i	$-0.190772\pi$
$251$	26.1636	1.65143	0.825715	+	0.564087i	$0.190772\pi$
$252$	0	0
$253$	3.52199	0.221426
$254$	−1.95959	−0.122955
$255$	0	0
$256$	20.9432	1.30895
$257$	−12.6117	−0.786697	−0.393349	−	0.919389i	$-0.628683\pi$
$257$	−12.6117	−0.786697	−0.393349	+	0.919389i	$0.628683\pi$
$258$	0	0
$259$	−11.1024	−0.689867
$260$	−12.8934	−0.799618
$261$	0	0
$262$	−14.2977	−0.883312
$263$	20.0550	1.23664	0.618322	−	0.785925i	$-0.287813\pi$
$263$	20.0550	1.23664	0.618322	+	0.785925i	$0.287813\pi$
$264$	0	0
$265$	18.6139	1.14345
$266$	3.09147	0.189550
$267$	0	0
$268$	19.6777	1.20201
$269$	20.1150	1.22644	0.613218	−	0.789914i	$-0.289875\pi$
$269$	20.1150	1.22644	0.613218	+	0.789914i	$0.289875\pi$
$270$	0	0
$271$	15.5586	0.945117	0.472558	−	0.881299i	$-0.343331\pi$
$271$	15.5586	0.945117	0.472558	+	0.881299i	$0.343331\pi$
$272$	−30.1257	−1.82664
$273$	0	0
$274$	−20.5453	−1.24119
$275$	−0.502638	−0.0303102
$276$	0	0
$277$	1.89211	0.113686	0.0568429	−	0.998383i	$-0.481897\pi$
$277$	1.89211	0.113686	0.0568429	+	0.998383i	$0.481897\pi$
$278$	1.65485	0.0992510
$279$	0	0
$280$	−4.80951	−0.287423
$281$	−4.59299	−0.273995	−0.136997	−	0.990571i	$-0.543745\pi$
$281$	−4.59299	−0.273995	−0.136997	+	0.990571i	$0.543745\pi$
$282$	0	0
$283$	−1.17798	−0.0700234	−0.0350117	−	0.999387i	$-0.511147\pi$
$283$	−1.17798	−0.0700234	−0.0350117	+	0.999387i	$0.511147\pi$
$284$	−4.17114	−0.247512
$285$	0	0
$286$	7.51076	0.444121
$287$	−13.1163	−0.774228
$288$	0	0
$289$	21.0071	1.23571
$290$	34.7229	2.03900
$291$	0	0
$292$	15.4836	0.906110
$293$	11.2606	0.657851	0.328925	−	0.944356i	$-0.393314\pi$
$293$	11.2606	0.657851	0.328925	+	0.944356i	$0.393314\pi$
$294$	0	0
$295$	25.6269	1.49206
$296$	−7.94745	−0.461936
$297$	0	0
$298$	19.2121	1.11293
$299$	−14.4813	−0.837476
$300$	0	0
$301$	13.1178	0.756096
$302$	−24.1919	−1.39209
$303$	0	0
$304$	4.88657	0.280264
$305$	10.5832	0.605993
$306$	0	0
$307$	4.35245	0.248407	0.124204	−	0.992257i	$-0.460362\pi$
$307$	4.35245	0.248407	0.124204	+	0.992257i	$0.460362\pi$
$308$	−2.26238	−0.128911
$309$	0	0
$310$	−9.86164	−0.560103
$311$	−7.61725	−0.431935	−0.215967	−	0.976401i	$-0.569291\pi$
$311$	−7.61725	−0.431935	−0.215967	+	0.976401i	$0.569291\pi$
$312$	0	0
$313$	−17.5654	−0.992855	−0.496427	−	0.868078i	$-0.665355\pi$
$313$	−17.5654	−0.992855	−0.496427	+	0.868078i	$0.665355\pi$
$314$	−3.65182	−0.206084
$315$	0	0
$316$	6.63295	0.373132
$317$	1.94192	0.109069	0.0545345	−	0.998512i	$-0.482633\pi$
$317$	1.94192	0.109069	0.0545345	+	0.998512i	$0.482633\pi$
$318$	0	0
$319$	−8.10336	−0.453701
$320$	4.94104	0.276213
$321$	0	0
$322$	10.8882	0.606773
$323$	−6.16499	−0.343029
$324$	0	0
$325$	2.06669	0.114639
$326$	34.2609	1.89754
$327$	0	0
$328$	−9.38907	−0.518425
$329$	18.4419	1.01674
$330$	0	0
$331$	13.3988	0.736462	0.368231	−	0.929734i	$-0.379964\pi$
$331$	13.3988	0.736462	0.368231	+	0.929734i	$0.379964\pi$
$332$	2.44506	0.134190
$333$	0	0
$334$	−11.5301	−0.630900
$335$	−34.5300	−1.88657
$336$	0	0
$337$	33.1631	1.80651	0.903254	−	0.429107i	$-0.141172\pi$
$337$	33.1631	1.80651	0.903254	+	0.429107i	$0.141172\pi$
$338$	−7.13491	−0.388088
$339$	0	0
$340$	−19.3322	−1.04844
$341$	2.30144	0.124630
$342$	0	0
$343$	−18.8462	−1.01760
$344$	9.39016	0.506283
$345$	0	0
$346$	−39.3229	−2.11401
$347$	−18.0268	−0.967730	−0.483865	−	0.875143i	$-0.660767\pi$
$347$	−18.0268	−0.967730	−0.483865	+	0.875143i	$0.660767\pi$
$348$	0	0
$349$	8.81411	0.471808	0.235904	−	0.971776i	$-0.424195\pi$
$349$	8.81411	0.471808	0.235904	+	0.971776i	$0.424195\pi$
$350$	−1.55389	−0.0830590
$351$	0	0
$352$	−6.50330	−0.346627
$353$	1.59249	0.0847599	0.0423799	−	0.999102i	$-0.486506\pi$
$353$	1.59249	0.0847599	0.0423799	+	0.999102i	$0.486506\pi$
$354$	0	0
$355$	7.31942	0.388474
$356$	12.5324	0.664214
$357$	0	0
$358$	−14.5590	−0.769469
$359$	−36.3774	−1.91993	−0.959963	−	0.280126i	$-0.909624\pi$
$359$	−36.3774	−1.91993	−0.959963	+	0.280126i	$0.909624\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	2.13640	0.112287
$363$	0	0
$364$	9.30218	0.487567
$365$	−27.1703	−1.42216
$366$	0	0
$367$	−5.00276	−0.261142	−0.130571	−	0.991439i	$-0.541681\pi$
$367$	−5.00276	−0.261142	−0.130571	+	0.991439i	$0.541681\pi$
$368$	17.2105	0.897158
$369$	0	0
$370$	−28.1102	−1.46138
$371$	−13.4293	−0.697216
$372$	0	0
$373$	−34.1313	−1.76725	−0.883625	−	0.468195i	$-0.844904\pi$
$373$	−34.1313	−1.76725	−0.883625	+	0.468195i	$0.844904\pi$
$374$	11.2615	0.582320
$375$	0	0
$376$	13.2014	0.680808
$377$	33.3185	1.71599
$378$	0	0
$379$	15.7749	0.810302	0.405151	−	0.914250i	$-0.367219\pi$
$379$	15.7749	0.810302	0.405151	+	0.914250i	$0.367219\pi$
$380$	3.13581	0.160864
$381$	0	0
$382$	−28.6193	−1.46429
$383$	22.4260	1.14592	0.572958	−	0.819585i	$-0.305796\pi$
$383$	22.4260	1.14592	0.572958	+	0.819585i	$0.305796\pi$
$384$	0	0
$385$	3.96996	0.202328
$386$	−39.5970	−2.01543
$387$	0	0
$388$	−14.6964	−0.746094
$389$	18.3488	0.930321	0.465160	−	0.885226i	$-0.345997\pi$
$389$	18.3488	0.930321	0.465160	+	0.885226i	$0.345997\pi$
$390$	0	0
$391$	−21.7131	−1.09808
$392$	−5.01042	−0.253065
$393$	0	0
$394$	8.38074	0.422216
$395$	−11.6393	−0.585638
$396$	0	0
$397$	−0.511483	−0.0256706	−0.0128353	−	0.999918i	$-0.504086\pi$
$397$	−0.511483	−0.0256706	−0.0128353	+	0.999918i	$0.504086\pi$
$398$	−23.8601	−1.19600
$399$	0	0
$400$	−2.45618	−0.122809
$401$	−18.9824	−0.947935	−0.473967	−	0.880542i	$-0.657179\pi$
$401$	−18.9824	−0.947935	−0.473967	+	0.880542i	$0.657179\pi$
$402$	0	0
$403$	−9.46277	−0.471374
$404$	−3.08292	−0.153381
$405$	0	0
$406$	−25.0513	−1.24328
$407$	6.56016	0.325175
$408$	0	0
$409$	1.10089	0.0544355	0.0272177	−	0.999630i	$-0.491335\pi$
$409$	1.10089	0.0544355	0.0272177	+	0.999630i	$0.491335\pi$
$410$	−33.2092	−1.64009
$411$	0	0
$412$	8.18117	0.403057
$413$	−18.4889	−0.909781
$414$	0	0
$415$	−4.29054	−0.210614
$416$	26.7395	1.31101
$417$	0	0
$418$	−1.82669	−0.0893463
$419$	18.7929	0.918094	0.459047	−	0.888412i	$-0.348191\pi$
$419$	18.7929	0.918094	0.459047	+	0.888412i	$0.348191\pi$
$420$	0	0
$421$	−24.7618	−1.20682	−0.603408	−	0.797433i	$-0.706191\pi$
$421$	−24.7618	−1.20682	−0.603408	+	0.797433i	$0.706191\pi$
$422$	−5.31910	−0.258930
$423$	0	0
$424$	−9.61318	−0.466857
$425$	3.09876	0.150312
$426$	0	0
$427$	−7.63542	−0.369504
$428$	−0.565392	−0.0273293
$429$	0	0
$430$	33.2131	1.60168
$431$	32.1985	1.55095	0.775474	−	0.631380i	$-0.217511\pi$
$431$	32.1985	1.55095	0.775474	+	0.631380i	$0.217511\pi$
$432$	0	0
$433$	−15.3042	−0.735475	−0.367737	−	0.929930i	$-0.619867\pi$
$433$	−15.3042	−0.735475	−0.367737	+	0.929930i	$0.619867\pi$
$434$	7.11483	0.341523
$435$	0	0
$436$	18.7391	0.897440
$437$	3.52199	0.168480
$438$	0	0
$439$	26.1812	1.24956	0.624781	−	0.780800i	$-0.285188\pi$
$439$	26.1812	1.24956	0.624781	+	0.780800i	$0.285188\pi$
$440$	2.84184	0.135479
$441$	0	0
$442$	−46.3038	−2.20245
$443$	−21.5579	−1.02425	−0.512123	−	0.858912i	$-0.671141\pi$
$443$	−21.5579	−1.02425	−0.512123	+	0.858912i	$0.671141\pi$
$444$	0	0
$445$	−21.9915	−1.04250
$446$	−11.5841	−0.548525
$447$	0	0
$448$	−3.56479	−0.168420
$449$	−11.7990	−0.556829	−0.278415	−	0.960461i	$-0.589809\pi$
$449$	−11.7990	−0.556829	−0.278415	+	0.960461i	$0.589809\pi$
$450$	0	0
$451$	7.75013	0.364939
$452$	3.39061	0.159481
$453$	0	0
$454$	−3.66122	−0.171829
$455$	−16.3232	−0.765245
$456$	0	0
$457$	−15.6863	−0.733773	−0.366886	−	0.930266i	$-0.619576\pi$
$457$	−15.6863	−0.733773	−0.366886	+	0.930266i	$0.619576\pi$
$458$	38.3410	1.79156
$459$	0	0
$460$	11.0443	0.514943
$461$	−27.2451	−1.26893	−0.634466	−	0.772951i	$-0.718780\pi$
$461$	−27.2451	−1.26893	−0.634466	+	0.772951i	$0.718780\pi$
$462$	0	0
$463$	−17.9364	−0.833573	−0.416787	−	0.909004i	$-0.636844\pi$
$463$	−17.9364	−0.833573	−0.416787	+	0.909004i	$0.636844\pi$
$464$	−39.5977	−1.83828
$465$	0	0
$466$	−32.1509	−1.48936
$467$	−29.0100	−1.34242	−0.671211	−	0.741266i	$-0.734226\pi$
$467$	−29.0100	−1.34242	−0.671211	+	0.741266i	$0.734226\pi$
$468$	0	0
$469$	24.9122	1.15034
$470$	46.6933	2.15380
$471$	0	0
$472$	−13.2350	−0.609191
$473$	−7.75102	−0.356392
$474$	0	0
$475$	−0.502638	−0.0230626
$476$	13.9475	0.639285
$477$	0	0
$478$	48.6277	2.22418
$479$	−34.0896	−1.55759	−0.778797	−	0.627276i	$-0.784170\pi$
$479$	−34.0896	−1.55759	−0.778797	+	0.627276i	$0.784170\pi$
$480$	0	0
$481$	−26.9733	−1.22988
$482$	−23.9921	−1.09281
$483$	0	0
$484$	1.33679	0.0607633
$485$	25.7888	1.17101
$486$	0	0
$487$	6.38349	0.289264	0.144632	−	0.989486i	$-0.453800\pi$
$487$	6.38349	0.289264	0.144632	+	0.989486i	$0.453800\pi$
$488$	−5.46570	−0.247421
$489$	0	0
$490$	−17.7219	−0.800594
$491$	31.4552	1.41956	0.709778	−	0.704426i	$-0.248795\pi$
$491$	31.4552	1.41956	0.709778	+	0.704426i	$0.248795\pi$
$492$	0	0
$493$	49.9572	2.24996
$494$	7.51076	0.337925
$495$	0	0
$496$	11.2461	0.504966
$497$	−5.28071	−0.236872
$498$	0	0
$499$	−31.9667	−1.43103	−0.715514	−	0.698599i	$-0.753807\pi$
$499$	−31.9667	−1.43103	−0.715514	+	0.698599i	$0.753807\pi$
$500$	14.1029	0.630699
$501$	0	0
$502$	−47.7927	−2.13309
$503$	34.0522	1.51831	0.759157	−	0.650907i	$-0.225611\pi$
$503$	34.0522	1.51831	0.759157	+	0.650907i	$0.225611\pi$
$504$	0	0
$505$	5.40983	0.240734
$506$	−6.43359	−0.286008
$507$	0	0
$508$	1.43405	0.0636256
$509$	−25.7904	−1.14314	−0.571569	−	0.820554i	$-0.693665\pi$
$509$	−25.7904	−1.14314	−0.571569	+	0.820554i	$0.693665\pi$
$510$	0	0
$511$	19.6024	0.867160
$512$	−19.9389	−0.881184
$513$	0	0
$514$	23.0377	1.01615
$515$	−14.3561	−0.632606
$516$	0	0
$517$	−10.8969	−0.479247
$518$	20.2806	0.891076
$519$	0	0
$520$	−11.6847	−0.512410
$521$	−31.3774	−1.37467	−0.687334	−	0.726342i	$-0.741219\pi$
$521$	−31.3774	−1.37467	−0.687334	+	0.726342i	$0.741219\pi$
$522$	0	0
$523$	−23.2388	−1.01616	−0.508082	−	0.861309i	$-0.669645\pi$
$523$	−23.2388	−1.01616	−0.508082	+	0.861309i	$0.669645\pi$
$524$	10.4632	0.457087
$525$	0	0
$526$	−36.6343	−1.59733
$527$	−14.1883	−0.618054
$528$	0	0
$529$	−10.5956	−0.460677
$530$	−34.0019	−1.47695
$531$	0	0
$532$	−2.26238	−0.0980865
$533$	−31.8661	−1.38027
$534$	0	0
$535$	0.992136	0.0428938
$536$	17.8330	0.770268
$537$	0	0
$538$	−36.7439	−1.58414
$539$	4.13581	0.178142
$540$	0	0
$541$	36.0085	1.54812	0.774062	−	0.633109i	$-0.218222\pi$
$541$	36.0085	1.54812	0.774062	+	0.633109i	$0.218222\pi$
$542$	−28.4207	−1.22077
$543$	0	0
$544$	40.0928	1.71896
$545$	−32.8829	−1.40855
$546$	0	0
$547$	−28.1855	−1.20513	−0.602563	−	0.798071i	$-0.705854\pi$
$547$	−28.1855	−1.20513	−0.602563	+	0.798071i	$0.705854\pi$
$548$	15.0353	0.642275
$549$	0	0
$550$	0.918163	0.0391506
$551$	−8.10336	−0.345215
$552$	0	0
$553$	8.39738	0.357093
$554$	−3.45629	−0.146844
$555$	0	0
$556$	−1.21104	−0.0513594
$557$	−21.6248	−0.916274	−0.458137	−	0.888882i	$-0.651483\pi$
$557$	−21.6248	−0.916274	−0.458137	+	0.888882i	$0.651483\pi$
$558$	0	0
$559$	31.8697	1.34795
$560$	19.3995	0.819779
$561$	0	0
$562$	8.38997	0.353909
$563$	−13.1791	−0.555434	−0.277717	−	0.960663i	$-0.589578\pi$
$563$	−13.1791	−0.555434	−0.277717	+	0.960663i	$0.589578\pi$
$564$	0	0
$565$	−5.94976	−0.250308
$566$	2.15179	0.0904467
$567$	0	0
$568$	−3.78011	−0.158610
$569$	−8.61143	−0.361010	−0.180505	−	0.983574i	$-0.557773\pi$
$569$	−8.61143	−0.361010	−0.180505	+	0.983574i	$0.557773\pi$
$570$	0	0
$571$	−17.6546	−0.738821	−0.369410	−	0.929266i	$-0.620440\pi$
$571$	−17.6546	−0.738821	−0.369410	+	0.929266i	$0.620440\pi$
$572$	−5.49647	−0.229819
$573$	0	0
$574$	23.9593	1.00004
$575$	−1.77029	−0.0738261
$576$	0	0
$577$	−31.0907	−1.29432	−0.647162	−	0.762352i	$-0.724044\pi$
$577$	−31.0907	−1.29432	−0.647162	+	0.762352i	$0.724044\pi$
$578$	−38.3735	−1.59613
$579$	0	0
$580$	−25.4106	−1.05512
$581$	3.09547	0.128422
$582$	0	0
$583$	7.93511	0.328639
$584$	14.0321	0.580652
$585$	0	0
$586$	−20.5696	−0.849722
$587$	26.7211	1.10290	0.551450	−	0.834208i	$-0.314075\pi$
$587$	26.7211	1.10290	0.551450	+	0.834208i	$0.314075\pi$
$588$	0	0
$589$	2.30144	0.0948290
$590$	−46.8124	−1.92724
$591$	0	0
$592$	32.0567	1.31752
$593$	24.2460	0.995666	0.497833	−	0.867273i	$-0.334129\pi$
$593$	24.2460	0.995666	0.497833	+	0.867273i	$0.334129\pi$
$594$	0	0
$595$	−24.4748	−1.00337
$596$	−14.0596	−0.575905
$597$	0	0
$598$	26.4529	1.08174
$599$	18.2095	0.744019	0.372009	−	0.928229i	$-0.378669\pi$
$599$	18.2095	0.744019	0.372009	+	0.928229i	$0.378669\pi$
$600$	0	0
$601$	37.9824	1.54934	0.774668	−	0.632368i	$-0.217917\pi$
$601$	37.9824	1.54934	0.774668	+	0.632368i	$0.217917\pi$
$602$	−23.9621	−0.976622
$603$	0	0
$604$	17.7039	0.720362
$605$	−2.34577	−0.0953691
$606$	0	0
$607$	−20.0130	−0.812301	−0.406151	−	0.913806i	$-0.633129\pi$
$607$	−20.0130	−0.812301	−0.406151	+	0.913806i	$0.633129\pi$
$608$	−6.50330	−0.263744
$609$	0	0
$610$	−19.3322	−0.782739
$611$	44.8048	1.81261
$612$	0	0
$613$	−42.1414	−1.70208	−0.851038	−	0.525105i	$-0.824026\pi$
$613$	−42.1414	−1.70208	−0.851038	+	0.525105i	$0.824026\pi$
$614$	−7.95057	−0.320859
$615$	0	0
$616$	−2.05029	−0.0826085
$617$	−16.9044	−0.680545	−0.340273	−	0.940327i	$-0.610519\pi$
$617$	−16.9044	−0.680545	−0.340273	+	0.940327i	$0.610519\pi$
$618$	0	0
$619$	5.31177	0.213498	0.106749	−	0.994286i	$-0.465956\pi$
$619$	5.31177	0.213498	0.106749	+	0.994286i	$0.465956\pi$
$620$	7.21686	0.289836
$621$	0	0
$622$	13.9144	0.557915
$623$	15.8661	0.635662
$624$	0	0
$625$	−27.2605	−1.09042
$626$	32.0865	1.28243
$627$	0	0
$628$	2.67245	0.106642
$629$	−40.4433	−1.61258
$630$	0	0
$631$	34.0425	1.35521	0.677604	−	0.735427i	$-0.263018\pi$
$631$	34.0425	1.35521	0.677604	+	0.735427i	$0.263018\pi$
$632$	6.01113	0.239110
$633$	0	0
$634$	−3.54728	−0.140880
$635$	−2.51643	−0.0998615
$636$	0	0
$637$	−17.0051	−0.673768
$638$	14.8023	0.586030
$639$	0	0
$640$	21.4847	0.849259
$641$	15.0405	0.594063	0.297031	−	0.954868i	$-0.404003\pi$
$641$	15.0405	0.594063	0.297031	+	0.954868i	$0.404003\pi$
$642$	0	0
$643$	32.3618	1.27622	0.638112	−	0.769943i	$-0.279715\pi$
$643$	32.3618	1.27622	0.638112	+	0.769943i	$0.279715\pi$
$644$	−7.96808	−0.313986
$645$	0	0
$646$	11.2615	0.443079
$647$	49.0042	1.92655	0.963276	−	0.268512i	$-0.0865319\pi$
$647$	49.0042	1.92655	0.963276	+	0.268512i	$0.0865319\pi$
$648$	0	0
$649$	10.9247	0.428833
$650$	−3.77520	−0.148075
$651$	0	0
$652$	−25.0725	−0.981916
$653$	−31.1992	−1.22092	−0.610460	−	0.792047i	$-0.709016\pi$
$653$	−31.1992	−1.22092	−0.610460	+	0.792047i	$0.709016\pi$
$654$	0	0
$655$	−18.3605	−0.717406
$656$	37.8715	1.47864
$657$	0	0
$658$	−33.6876	−1.31328
$659$	35.7237	1.39160	0.695799	−	0.718237i	$-0.255050\pi$
$659$	35.7237	1.39160	0.695799	+	0.718237i	$0.255050\pi$
$660$	0	0
$661$	33.8677	1.31730	0.658651	−	0.752449i	$-0.271128\pi$
$661$	33.8677	1.31730	0.658651	+	0.752449i	$0.271128\pi$
$662$	−24.4754	−0.951262
$663$	0	0
$664$	2.21585	0.0859916
$665$	3.96996	0.153949
$666$	0	0
$667$	−28.5400	−1.10507
$668$	8.43788	0.326471
$669$	0	0
$670$	63.0755	2.43682
$671$	4.51162	0.174169
$672$	0	0
$673$	21.1648	0.815842	0.407921	−	0.913017i	$-0.366254\pi$
$673$	21.1648	0.815842	0.407921	+	0.913017i	$0.366254\pi$
$674$	−60.5786	−2.33340
$675$	0	0
$676$	5.22141	0.200824
$677$	34.5239	1.32686	0.663431	−	0.748238i	$-0.269100\pi$
$677$	34.5239	1.32686	0.663431	+	0.748238i	$0.269100\pi$
$678$	0	0
$679$	−18.6057	−0.714022
$680$	−17.5199	−0.671858
$681$	0	0
$682$	−4.20401	−0.160980
$683$	−12.8022	−0.489862	−0.244931	−	0.969540i	$-0.578765\pi$
$683$	−12.8022	−0.489862	−0.244931	+	0.969540i	$0.578765\pi$
$684$	0	0
$685$	−26.3835	−1.00806
$686$	34.4261	1.31439
$687$	0	0
$688$	−37.8759	−1.44401
$689$	−32.6267	−1.24298
$690$	0	0
$691$	39.5406	1.50419	0.752097	−	0.659052i	$-0.229042\pi$
$691$	39.5406	1.50419	0.752097	+	0.659052i	$0.229042\pi$
$692$	28.7769	1.09394
$693$	0	0
$694$	32.9294	1.24998
$695$	2.12510	0.0806094
$696$	0	0
$697$	−47.7795	−1.80978
$698$	−16.1006	−0.609418
$699$	0	0
$700$	1.13716	0.0429805
$701$	−22.4546	−0.848097	−0.424049	−	0.905639i	$-0.639391\pi$
$701$	−22.4546	−0.848097	−0.424049	+	0.905639i	$0.639391\pi$
$702$	0	0
$703$	6.56016	0.247421
$704$	2.10636	0.0793865
$705$	0	0
$706$	−2.90899	−0.109481
$707$	−3.90301	−0.146788
$708$	0	0
$709$	−1.41244	−0.0530451	−0.0265226	−	0.999648i	$-0.508443\pi$
$709$	−1.41244	−0.0530451	−0.0265226	+	0.999648i	$0.508443\pi$
$710$	−13.3703	−0.501778
$711$	0	0
$712$	11.3575	0.425641
$713$	8.10564	0.303559
$714$	0	0
$715$	9.64506	0.360705
$716$	10.6545	0.398177
$717$	0	0
$718$	66.4502	2.47990
$719$	39.0879	1.45773	0.728867	−	0.684655i	$-0.240047\pi$
$719$	39.0879	1.45773	0.728867	+	0.684655i	$0.240047\pi$
$720$	0	0
$721$	10.3574	0.385731
$722$	−1.82669	−0.0679823
$723$	0	0
$724$	−1.56344	−0.0581049
$725$	4.07306	0.151270
$726$	0	0
$727$	9.42098	0.349405	0.174702	−	0.984621i	$-0.444104\pi$
$727$	9.42098	0.349405	0.174702	+	0.984621i	$0.444104\pi$
$728$	8.43013	0.312442
$729$	0	0
$730$	49.6316	1.83695
$731$	47.7850	1.76739
$732$	0	0
$733$	4.63361	0.171146	0.0855731	−	0.996332i	$-0.472728\pi$
$733$	4.63361	0.171146	0.0855731	+	0.996332i	$0.472728\pi$
$734$	9.13849	0.337308
$735$	0	0
$736$	−22.9046	−0.844274
$737$	−14.7201	−0.542222
$738$	0	0
$739$	−13.1654	−0.484298	−0.242149	−	0.970239i	$-0.577852\pi$
$739$	−13.1654	−0.484298	−0.242149	+	0.970239i	$0.577852\pi$
$740$	20.5714	0.756219
$741$	0	0
$742$	24.5312	0.900568
$743$	14.7689	0.541817	0.270909	−	0.962605i	$-0.412676\pi$
$743$	14.7689	0.541817	0.270909	+	0.962605i	$0.412676\pi$
$744$	0	0
$745$	24.6715	0.903894
$746$	62.3472	2.28269
$747$	0	0
$748$	−8.24132	−0.301332
$749$	−0.715792	−0.0261545
$750$	0	0
$751$	35.0415	1.27868	0.639341	−	0.768924i	$-0.279207\pi$
$751$	35.0415	1.27868	0.639341	+	0.768924i	$0.279207\pi$
$752$	−53.2487	−1.94178
$753$	0	0
$754$	−60.8625	−2.21648
$755$	−31.0664	−1.13062
$756$	0	0
$757$	−23.8005	−0.865044	−0.432522	−	0.901623i	$-0.642376\pi$
$757$	−23.8005	−0.865044	−0.432522	+	0.901623i	$0.642376\pi$
$758$	−28.8158	−1.04664
$759$	0	0
$760$	2.84184	0.103084
$761$	39.0282	1.41477	0.707386	−	0.706827i	$-0.249874\pi$
$761$	39.0282	1.41477	0.707386	+	0.706827i	$0.249874\pi$
$762$	0	0
$763$	23.7239	0.858862
$764$	20.9440	0.757726
$765$	0	0
$766$	−40.9654	−1.48014
$767$	−44.9190	−1.62193
$768$	0	0
$769$	9.84757	0.355113	0.177556	−	0.984111i	$-0.443181\pi$
$769$	9.84757	0.355113	0.177556	+	0.984111i	$0.443181\pi$
$770$	−7.25189	−0.261340
$771$	0	0
$772$	28.9776	1.04292
$773$	42.9837	1.54602	0.773008	−	0.634396i	$-0.218751\pi$
$773$	42.9837	1.54602	0.773008	+	0.634396i	$0.218751\pi$
$774$	0	0
$775$	−1.15679	−0.0415531
$776$	−13.3186	−0.478111
$777$	0	0
$778$	−33.5175	−1.20166
$779$	7.75013	0.277677
$780$	0	0
$781$	3.12026	0.111652
$782$	39.6630	1.41835
$783$	0	0
$784$	20.2099	0.721783
$785$	−4.68954	−0.167377
$786$	0	0
$787$	−25.8709	−0.922199	−0.461099	−	0.887349i	$-0.652545\pi$
$787$	−25.8709	−0.922199	−0.461099	+	0.887349i	$0.652545\pi$
$788$	−6.13313	−0.218484
$789$	0	0
$790$	21.2614	0.756448
$791$	4.29254	0.152625
$792$	0	0
$793$	−18.5503	−0.658741
$794$	0.934320	0.0331578
$795$	0	0
$796$	17.4611	0.618893
$797$	−3.62989	−0.128577	−0.0642886	−	0.997931i	$-0.520478\pi$
$797$	−3.62989	−0.128577	−0.0642886	+	0.997931i	$0.520478\pi$
$798$	0	0
$799$	67.1796	2.37664
$800$	3.26880	0.115570
$801$	0	0
$802$	34.6749	1.22441
$803$	−11.5827	−0.408743
$804$	0	0
$805$	13.9822	0.492807
$806$	17.2855	0.608857
$807$	0	0
$808$	−2.79391	−0.0982894
$809$	13.1327	0.461720	0.230860	−	0.972987i	$-0.425846\pi$
$809$	13.1327	0.461720	0.230860	+	0.972987i	$0.425846\pi$
$810$	0	0
$811$	9.40230	0.330160	0.165080	−	0.986280i	$-0.447212\pi$
$811$	9.40230	0.330160	0.165080	+	0.986280i	$0.447212\pi$
$812$	18.3329	0.643358
$813$	0	0
$814$	−11.9834	−0.420017
$815$	43.9966	1.54114
$816$	0	0
$817$	−7.75102	−0.271174
$818$	−2.01098	−0.0703124
$819$	0	0
$820$	24.3029	0.848695
$821$	21.1701	0.738843	0.369422	−	0.929262i	$-0.379556\pi$
$821$	21.1701	0.738843	0.369422	+	0.929262i	$0.379556\pi$
$822$	0	0
$823$	25.2277	0.879384	0.439692	−	0.898149i	$-0.355088\pi$
$823$	25.2277	0.879384	0.439692	+	0.898149i	$0.355088\pi$
$824$	7.41422	0.258286
$825$	0	0
$826$	33.7735	1.17513
$827$	39.4460	1.37167	0.685836	−	0.727756i	$-0.259437\pi$
$827$	39.4460	1.37167	0.685836	+	0.727756i	$0.259437\pi$
$828$	0	0
$829$	−36.5068	−1.26793	−0.633967	−	0.773360i	$-0.718574\pi$
$829$	−36.5068	−1.26793	−0.633967	+	0.773360i	$0.718574\pi$
$830$	7.83748	0.272043
$831$	0	0
$832$	−8.66069	−0.300255
$833$	−25.4972	−0.883427
$834$	0	0
$835$	−14.8066	−0.512403
$836$	1.33679	0.0462339
$837$	0	0
$838$	−34.3288	−1.18587
$839$	−42.4670	−1.46612	−0.733061	−	0.680163i	$-0.761909\pi$
$839$	−42.4670	−1.46612	−0.733061	+	0.680163i	$0.761909\pi$
$840$	0	0
$841$	36.6645	1.26429
$842$	45.2321	1.55880
$843$	0	0
$844$	3.89258	0.133988
$845$	−9.16241	−0.315196
$846$	0	0
$847$	1.69239	0.0581513
$848$	38.7755	1.33156
$849$	0	0
$850$	−5.66047	−0.194153
$851$	23.1048	0.792023
$852$	0	0
$853$	21.5088	0.736446	0.368223	−	0.929738i	$-0.379966\pi$
$853$	21.5088	0.736446	0.368223	+	0.929738i	$0.379966\pi$
$854$	13.9475	0.477275
$855$	0	0
$856$	−0.512389	−0.0175131
$857$	12.6370	0.431670	0.215835	−	0.976430i	$-0.430753\pi$
$857$	12.6370	0.431670	0.215835	+	0.976430i	$0.430753\pi$
$858$	0	0
$859$	−3.39720	−0.115911	−0.0579555	−	0.998319i	$-0.518458\pi$
$859$	−3.39720	−0.115911	−0.0579555	+	0.998319i	$0.518458\pi$
$860$	−24.3057	−0.828818
$861$	0	0
$862$	−58.8167	−2.00330
$863$	10.9308	0.372088	0.186044	−	0.982541i	$-0.440433\pi$
$863$	10.9308	0.372088	0.186044	+	0.982541i	$0.440433\pi$
$864$	0	0
$865$	−50.4971	−1.71695
$866$	27.9561	0.949987
$867$	0	0
$868$	−5.20672	−0.176727
$869$	−4.96184	−0.168319
$870$	0	0
$871$	60.5243	2.05079
$872$	16.9824	0.575096
$873$	0	0
$874$	−6.43359	−0.217619
$875$	17.8544	0.603588
$876$	0	0
$877$	31.7230	1.07121	0.535605	−	0.844468i	$-0.320083\pi$
$877$	31.7230	1.07121	0.535605	+	0.844468i	$0.320083\pi$
$878$	−47.8250	−1.61402
$879$	0	0
$880$	−11.4628	−0.386410
$881$	6.45839	0.217589	0.108794	−	0.994064i	$-0.465301\pi$
$881$	6.45839	0.217589	0.108794	+	0.994064i	$0.465301\pi$
$882$	0	0
$883$	−48.5242	−1.63297	−0.816485	−	0.577367i	$-0.804080\pi$
$883$	−48.5242	−1.63297	−0.816485	+	0.577367i	$0.804080\pi$
$884$	33.8857	1.13970
$885$	0	0
$886$	39.3796	1.32298
$887$	44.0097	1.47770	0.738851	−	0.673869i	$-0.235369\pi$
$887$	44.0097	1.47770	0.738851	+	0.673869i	$0.235369\pi$
$888$	0	0
$889$	1.81552	0.0608905
$890$	40.1716	1.34656
$891$	0	0
$892$	8.47742	0.283845
$893$	−10.8969	−0.364652
$894$	0	0
$895$	−18.6962	−0.624946
$896$	−15.5005	−0.517835
$897$	0	0
$898$	21.5531	0.719236
$899$	−18.6494	−0.621991
$900$	0	0
$901$	−48.9199	−1.62976
$902$	−14.1571	−0.471379
$903$	0	0
$904$	3.07275	0.102198
$905$	2.74349	0.0911966
$906$	0	0
$907$	−2.05084	−0.0680971	−0.0340485	−	0.999420i	$-0.510840\pi$
$907$	−2.05084	−0.0680971	−0.0340485	+	0.999420i	$0.510840\pi$
$908$	2.67932	0.0889164
$909$	0	0
$910$	29.8175	0.988439
$911$	16.8417	0.557990	0.278995	−	0.960293i	$-0.409999\pi$
$911$	16.8417	0.557990	0.278995	+	0.960293i	$0.409999\pi$
$912$	0	0
$913$	−1.82905	−0.0605328
$914$	28.6539	0.947788
$915$	0	0
$916$	−28.0584	−0.927075
$917$	13.2465	0.437438
$918$	0	0
$919$	8.01221	0.264299	0.132149	−	0.991230i	$-0.457812\pi$
$919$	8.01221	0.264299	0.132149	+	0.991230i	$0.457812\pi$
$920$	10.0089	0.329985
$921$	0	0
$922$	49.7683	1.63903
$923$	−12.8295	−0.422289
$924$	0	0
$925$	−3.29738	−0.108417
$926$	32.7641	1.07670
$927$	0	0
$928$	52.6986	1.72992
$929$	−16.7897	−0.550853	−0.275426	−	0.961322i	$-0.588819\pi$
$929$	−16.7897	−0.550853	−0.275426	+	0.961322i	$0.588819\pi$
$930$	0	0
$931$	4.13581	0.135546
$932$	23.5284	0.770698
$933$	0	0
$934$	52.9922	1.73396
$935$	14.4617	0.472947
$936$	0	0
$937$	21.8471	0.713713	0.356856	−	0.934159i	$-0.383849\pi$
$937$	21.8471	0.713713	0.356856	+	0.934159i	$0.383849\pi$
$938$	−45.5068	−1.48585
$939$	0	0
$940$	−34.1707	−1.11453
$941$	−17.8421	−0.581635	−0.290817	−	0.956779i	$-0.593927\pi$
$941$	−17.8421	−0.581635	−0.290817	+	0.956779i	$0.593927\pi$
$942$	0	0
$943$	27.2959	0.888877
$944$	53.3845	1.73752
$945$	0	0
$946$	14.1587	0.460339
$947$	−41.9736	−1.36396	−0.681980	−	0.731371i	$-0.738881\pi$
$947$	−41.9736	−1.36396	−0.681980	+	0.731371i	$0.738881\pi$
$948$	0	0
$949$	47.6242	1.54595
$950$	0.918163	0.0297892
$951$	0	0
$952$	12.6400	0.409665
$953$	8.06945	0.261395	0.130697	−	0.991422i	$-0.458278\pi$
$953$	8.06945	0.261395	0.130697	+	0.991422i	$0.458278\pi$
$954$	0	0
$955$	−36.7520	−1.18927
$956$	−35.5864	−1.15094
$957$	0	0
$958$	62.2711	2.01189
$959$	19.0348	0.614666
$960$	0	0
$961$	−25.7034	−0.829142
$962$	49.2718	1.58859
$963$	0	0
$964$	17.5577	0.565496
$965$	−50.8491	−1.63689
$966$	0	0
$967$	−3.86360	−0.124245	−0.0621225	−	0.998069i	$-0.519787\pi$
$967$	−3.86360	−0.124245	−0.0621225	+	0.998069i	$0.519787\pi$
$968$	1.21147	0.0389382
$969$	0	0
$970$	−47.1081	−1.51255
$971$	−42.7787	−1.37283	−0.686417	−	0.727208i	$-0.740818\pi$
$971$	−42.7787	−1.37283	−0.686417	+	0.727208i	$0.740818\pi$
$972$	0	0
$973$	−1.53318	−0.0491516
$974$	−11.6607	−0.373631
$975$	0	0
$976$	22.0463	0.705686
$977$	−54.5146	−1.74408	−0.872039	−	0.489437i	$-0.837202\pi$
$977$	−54.5146	−1.74408	−0.872039	+	0.489437i	$0.837202\pi$
$978$	0	0
$979$	−9.37496	−0.299625
$980$	12.9691	0.414283
$981$	0	0
$982$	−57.4590	−1.83359
$983$	−33.5422	−1.06983	−0.534916	−	0.844905i	$-0.679657\pi$
$983$	−33.5422	−1.06983	−0.534916	+	0.844905i	$0.679657\pi$
$984$	0	0
$985$	10.7623	0.342914
$986$	−91.2562	−2.90619
$987$	0	0
$988$	−5.49647	−0.174866
$989$	−27.2991	−0.868060
$990$	0	0
$991$	31.7767	1.00942	0.504710	−	0.863289i	$-0.331600\pi$
$991$	31.7767	1.00942	0.504710	+	0.863289i	$0.331600\pi$
$992$	−14.9669	−0.475200
$993$	0	0
$994$	9.64621	0.305959
$995$	−30.6403	−0.971363
$996$	0	0
$997$	−47.5668	−1.50646	−0.753228	−	0.657759i	$-0.771504\pi$
$997$	−47.5668	−1.50646	−0.753228	+	0.657759i	$0.771504\pi$
$998$	58.3933	1.84841
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1881.2.a.k.1.2		5
3.2	odd	2		209.2.a.c.1.4	✓	5
12.11	even	2		3344.2.a.t.1.4		5
15.14	odd	2		5225.2.a.h.1.2		5
33.32	even	2		2299.2.a.n.1.2		5
57.56	even	2		3971.2.a.h.1.2		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
209.2.a.c.1.4	✓	5	3.2	odd	2
1881.2.a.k.1.2		5	1.1	even	1	trivial
2299.2.a.n.1.2		5	33.32	even	2
3344.2.a.t.1.4		5	12.11	even	2
3971.2.a.h.1.2		5	57.56	even	2
5225.2.a.h.1.2		5	15.14	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1881 = 3^{2} \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1881.a (trivial)

\(f(q)\)	\(=\)	\(q-1.82669 q^{2} +1.33679 q^{4} -2.34577 q^{5} +1.69239 q^{7} +1.21147 q^{8} +4.28499 q^{10} -1.00000 q^{11} +4.11168 q^{13} -3.09147 q^{14} -4.88657 q^{16} +6.16499 q^{17} -1.00000 q^{19} -3.13581 q^{20} +1.82669 q^{22} -3.52199 q^{23} +0.502638 q^{25} -7.51076 q^{26} +2.26238 q^{28} +8.10336 q^{29} -2.30144 q^{31} +6.50330 q^{32} -11.2615 q^{34} -3.96996 q^{35} -6.56016 q^{37} +1.82669 q^{38} -2.84184 q^{40} -7.75013 q^{41} +7.75102 q^{43} -1.33679 q^{44} +6.43359 q^{46} +10.8969 q^{47} -4.13581 q^{49} -0.918163 q^{50} +5.49647 q^{52} -7.93511 q^{53} +2.34577 q^{55} +2.05029 q^{56} -14.8023 q^{58} -10.9247 q^{59} -4.51162 q^{61} +4.20401 q^{62} -2.10636 q^{64} -9.64506 q^{65} +14.7201 q^{67} +8.24132 q^{68} +7.25189 q^{70} -3.12026 q^{71} +11.5827 q^{73} +11.9834 q^{74} -1.33679 q^{76} -1.69239 q^{77} +4.96184 q^{79} +11.4628 q^{80} +14.1571 q^{82} +1.82905 q^{83} -14.4617 q^{85} -14.1587 q^{86} -1.21147 q^{88} +9.37496 q^{89} +6.95858 q^{91} -4.70818 q^{92} -19.9053 q^{94} +2.34577 q^{95} -10.9937 q^{97} +7.55484 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 2 q^{2} + 6 q^{4} + 5 q^{5} + 6 q^{7} - 6 q^{8} + 12 q^{10} - 5 q^{11} + 4 q^{13} + 14 q^{14} + 8 q^{16} + 4 q^{17} - 5 q^{19} + 8 q^{20} + 2 q^{22} - 3 q^{23} + 6 q^{25} + 6 q^{26} - 10 q^{28} - 10 q^{29}+ \cdots + 10 q^{98}+O(q^{100}) \) 5 * q - 2 * q^2 + 6 * q^4 + 5 * q^5 + 6 * q^7 - 6 * q^8 + 12 * q^10 - 5 * q^11 + 4 * q^13 + 14 * q^14 + 8 * q^16 + 4 * q^17 - 5 * q^19 + 8 * q^20 + 2 * q^22 - 3 * q^23 + 6 * q^25 + 6 * q^26 - 10 * q^28 - 10 * q^29 + 11 * q^31 - 14 * q^32 - 4 * q^34 + 8 * q^35 + q^37 + 2 * q^38 - 16 * q^40 - 2 * q^41 + 20 * q^43 - 6 * q^44 - 4 * q^46 + 20 * q^47 + 3 * q^49 + 32 * q^50 + 6 * q^52 + 14 * q^53 - 5 * q^55 + 38 * q^56 - 6 * q^58 - 3 * q^59 - 10 * q^61 + 6 * q^62 + 9 * q^67 - 24 * q^68 + 50 * q^70 - 23 * q^71 - 8 * q^74 - 6 * q^76 - 6 * q^77 + 44 * q^79 + 18 * q^80 - 30 * q^82 + 14 * q^83 - 12 * q^85 - 52 * q^86 + 6 * q^88 + 27 * q^89 + 24 * q^91 - 58 * q^92 - 8 * q^94 - 5 * q^95 + 15 * q^97 + 10 * q^98

Introduction

L-functions

Modular forms

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Database

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