LMFDB - Embedded newform 2031.2.a.e.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2031,2,Mod(1,2031)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2031, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("2031.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2031 = 3 \cdot 677 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2031.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:	$16.2176166505$
Analytic rank:	$1$
Dimension:	$7$
Coefficient field:	7.7.32354821.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 2x^{6} - 5x^{5} + 9x^{4} + 6x^{3} - 8x^{2} - x + 1 $ x^7 - 2x^6 - 5x^5 + 9x^4 + 6x^3 - 8*x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-1.08782$ of defining polynomial
Character	$\chi$	$=$	2031.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+0.455154 q^{2} -1.00000 q^{3} -1.79283 q^{4} -3.54297 q^{5} -0.455154 q^{6} +0.0807299 q^{7} -1.72633 q^{8} +1.00000 q^{9} -1.61260 q^{10} +4.14014 q^{11} +1.79283 q^{12} +4.01429 q^{13} +0.0367446 q^{14} +3.54297 q^{15} +2.79992 q^{16} -1.28738 q^{17} +0.455154 q^{18} -4.80382 q^{19} +6.35197 q^{20} -0.0807299 q^{21} +1.88440 q^{22} +4.07574 q^{23} +1.72633 q^{24} +7.55267 q^{25} +1.82712 q^{26} -1.00000 q^{27} -0.144735 q^{28} -1.26877 q^{29} +1.61260 q^{30} -1.33574 q^{31} +4.72705 q^{32} -4.14014 q^{33} -0.585956 q^{34} -0.286024 q^{35} -1.79283 q^{36} -2.69851 q^{37} -2.18648 q^{38} -4.01429 q^{39} +6.11633 q^{40} +4.57894 q^{41} -0.0367446 q^{42} -7.83553 q^{43} -7.42259 q^{44} -3.54297 q^{45} +1.85509 q^{46} +8.49584 q^{47} -2.79992 q^{48} -6.99348 q^{49} +3.43763 q^{50} +1.28738 q^{51} -7.19695 q^{52} -4.94876 q^{53} -0.455154 q^{54} -14.6684 q^{55} -0.139366 q^{56} +4.80382 q^{57} -0.577488 q^{58} +1.26205 q^{59} -6.35197 q^{60} +10.8384 q^{61} -0.607970 q^{62} +0.0807299 q^{63} -3.44831 q^{64} -14.2225 q^{65} -1.88440 q^{66} +6.74176 q^{67} +2.30806 q^{68} -4.07574 q^{69} -0.130185 q^{70} -10.7197 q^{71} -1.72633 q^{72} -14.2072 q^{73} -1.22824 q^{74} -7.55267 q^{75} +8.61245 q^{76} +0.334233 q^{77} -1.82712 q^{78} -9.57082 q^{79} -9.92006 q^{80} +1.00000 q^{81} +2.08413 q^{82} +3.90566 q^{83} +0.144735 q^{84} +4.56115 q^{85} -3.56638 q^{86} +1.26877 q^{87} -7.14723 q^{88} +0.470731 q^{89} -1.61260 q^{90} +0.324073 q^{91} -7.30713 q^{92} +1.33574 q^{93} +3.86692 q^{94} +17.0198 q^{95} -4.72705 q^{96} -11.7474 q^{97} -3.18311 q^{98} +4.14014 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - 2 q^{2} - 7 q^{3} + 2 q^{4} - 10 q^{5} + 2 q^{6} + 8 q^{7} - 3 q^{8} + 7 q^{9} - 8 q^{10} + 10 q^{11} - 2 q^{12} - 3 q^{13} - 3 q^{14} + 10 q^{15} - 12 q^{16} + 8 q^{17} - 2 q^{18} - 5 q^{19} + 4 q^{20}+ \cdots + 10 q^{99}+O(q^{100}) $ 7 * q - 2 * q^2 - 7 * q^3 + 2 * q^4 - 10 * q^5 + 2 * q^6 + 8 * q^7 - 3 * q^8 + 7 * q^9 - 8 * q^10 + 10 * q^11 - 2 * q^12 - 3 * q^13 - 3 * q^14 + 10 * q^15 - 12 * q^16 + 8 * q^17 - 2 * q^18 - 5 * q^19 + 4 * q^20 - 8 * q^21 - q^22 - q^23 + 3 * q^24 - q^25 + 10 * q^26 - 7 * q^27 + 10 * q^28 + 5 * q^29 + 8 * q^30 - 24 * q^31 + 12 * q^32 - 10 * q^33 - q^34 - 4 * q^35 + 2 * q^36 - 26 * q^37 + 11 * q^38 + 3 * q^39 + 15 * q^40 - 12 * q^41 + 3 * q^42 - 26 * q^43 + 16 * q^44 - 10 * q^45 + 16 * q^46 + 2 * q^47 + 12 * q^48 - 23 * q^49 + 23 * q^50 - 8 * q^51 - 27 * q^52 + 16 * q^53 + 2 * q^54 - 26 * q^55 + 4 * q^56 + 5 * q^57 - 14 * q^58 - 17 * q^59 - 4 * q^60 - 10 * q^61 - 3 * q^62 + 8 * q^63 - 15 * q^64 + q^66 + 20 * q^67 + 2 * q^68 + q^69 - 18 * q^70 - 5 * q^71 - 3 * q^72 - 13 * q^73 - 9 * q^74 + q^75 - 14 * q^76 + 17 * q^77 - 10 * q^78 - 32 * q^79 - 6 * q^80 + 7 * q^81 + 11 * q^83 - 10 * q^84 - 11 * q^85 - q^86 - 5 * q^87 - 14 * q^88 - 8 * q^89 - 8 * q^90 - 16 * q^91 - 32 * q^92 + 24 * q^93 - 50 * q^94 + 7 * q^95 - 12 * q^96 - 41 * q^97 + 10 * q^99

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 $

$2$	0.455154	0.321843	0.160921	−	0.986967i	$-0.448553\pi$
$2$	0.455154	0.321843	0.160921	+	0.986967i	$0.448553\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−1.79283	−0.896417
$5$	−3.54297	−1.58447	−0.792233	−	0.610219i	$-0.791082\pi$
$5$	−3.54297	−1.58447	−0.792233	+	0.610219i	$0.791082\pi$
$6$	−0.455154	−0.185816
$7$	0.0807299	0.0305130	0.0152565	−	0.999884i	$-0.495144\pi$
$7$	0.0807299	0.0305130	0.0152565	+	0.999884i	$0.495144\pi$
$8$	−1.72633	−0.610348
$9$	1.00000	0.333333
$10$	−1.61260	−0.509949
$11$	4.14014	1.24830	0.624150	−	0.781305i	$-0.285446\pi$
$11$	4.14014	1.24830	0.624150	+	0.781305i	$0.285446\pi$
$12$	1.79283	0.517547
$13$	4.01429	1.11336	0.556681	−	0.830726i	$-0.312074\pi$
$13$	4.01429	1.11336	0.556681	+	0.830726i	$0.312074\pi$
$14$	0.0367446	0.00982040
$15$	3.54297	0.914792
$16$	2.79992	0.699981
$17$	−1.28738	−0.312235	−0.156118	−	0.987738i	$-0.549898\pi$
$17$	−1.28738	−0.312235	−0.156118	+	0.987738i	$0.549898\pi$
$18$	0.455154	0.107281
$19$	−4.80382	−1.10207	−0.551035	−	0.834482i	$-0.685767\pi$
$19$	−4.80382	−1.10207	−0.551035	+	0.834482i	$0.685767\pi$
$20$	6.35197	1.42034
$21$	−0.0807299	−0.0176167
$22$	1.88440	0.401756
$23$	4.07574	0.849851	0.424925	−	0.905228i	$-0.360300\pi$
$23$	4.07574	0.849851	0.424925	+	0.905228i	$0.360300\pi$
$24$	1.72633	0.352385
$25$	7.55267	1.51053
$26$	1.82712	0.358328
$27$	−1.00000	−0.192450
$28$	−0.144735	−0.0273524
$29$	−1.26877	−0.235605	−0.117803	−	0.993037i	$-0.537585\pi$
$29$	−1.26877	−0.235605	−0.117803	+	0.993037i	$0.537585\pi$
$30$	1.61260	0.294419
$31$	−1.33574	−0.239907	−0.119953	−	0.992780i	$-0.538274\pi$
$31$	−1.33574	−0.239907	−0.119953	+	0.992780i	$0.538274\pi$
$32$	4.72705	0.835632
$33$	−4.14014	−0.720706
$34$	−0.585956	−0.100491
$35$	−0.286024	−0.0483469
$36$	−1.79283	−0.298806
$37$	−2.69851	−0.443632	−0.221816	−	0.975089i	$-0.571198\pi$
$37$	−2.69851	−0.443632	−0.221816	+	0.975089i	$0.571198\pi$
$38$	−2.18648	−0.354694
$39$	−4.01429	−0.642800
$40$	6.11633	0.967076
$41$	4.57894	0.715111	0.357555	−	0.933892i	$-0.383610\pi$
$41$	4.57894	0.715111	0.357555	+	0.933892i	$0.383610\pi$
$42$	−0.0367446	−0.00566981
$43$	−7.83553	−1.19491	−0.597454	−	0.801904i	$-0.703821\pi$
$43$	−7.83553	−1.19491	−0.597454	+	0.801904i	$0.703821\pi$
$44$	−7.42259	−1.11900
$45$	−3.54297	−0.528155
$46$	1.85509	0.273518
$47$	8.49584	1.23925	0.619623	−	0.784900i	$-0.287286\pi$
$47$	8.49584	1.23925	0.619623	+	0.784900i	$0.287286\pi$
$48$	−2.79992	−0.404134
$49$	−6.99348	−0.999069
$50$	3.43763	0.486154
$51$	1.28738	0.180269
$52$	−7.19695	−0.998037
$53$	−4.94876	−0.679765	−0.339882	−	0.940468i	$-0.610387\pi$
$53$	−4.94876	−0.679765	−0.339882	+	0.940468i	$0.610387\pi$
$54$	−0.455154	−0.0619387
$55$	−14.6684	−1.97789
$56$	−0.139366	−0.0186236
$57$	4.80382	0.636281
$58$	−0.577488	−0.0758279
$59$	1.26205	0.164306	0.0821528	−	0.996620i	$-0.473820\pi$
$59$	1.26205	0.164306	0.0821528	+	0.996620i	$0.473820\pi$
$60$	−6.35197	−0.820035
$61$	10.8384	1.38771	0.693857	−	0.720113i	$-0.255910\pi$
$61$	10.8384	1.38771	0.693857	+	0.720113i	$0.255910\pi$
$62$	−0.607970	−0.0772122
$63$	0.0807299	0.0101710
$64$	−3.44831	−0.431039
$65$	−14.2225	−1.76409
$66$	−1.88440	−0.231954
$67$	6.74176	0.823637	0.411819	−	0.911266i	$-0.364894\pi$
$67$	6.74176	0.823637	0.411819	+	0.911266i	$0.364894\pi$
$68$	2.30806	0.279893
$69$	−4.07574	−0.490662
$70$	−0.130185	−0.0155601
$71$	−10.7197	−1.27220	−0.636098	−	0.771608i	$-0.719453\pi$
$71$	−10.7197	−1.27220	−0.636098	+	0.771608i	$0.719453\pi$
$72$	−1.72633	−0.203449
$73$	−14.2072	−1.66282	−0.831411	−	0.555657i	$-0.812467\pi$
$73$	−14.2072	−1.66282	−0.831411	+	0.555657i	$0.812467\pi$
$74$	−1.22824	−0.142780
$75$	−7.55267	−0.872107
$76$	8.61245	0.987915
$77$	0.334233	0.0380894
$78$	−1.82712	−0.206881
$79$	−9.57082	−1.07680	−0.538400	−	0.842689i	$-0.680971\pi$
$79$	−9.57082	−1.07680	−0.538400	+	0.842689i	$0.680971\pi$
$80$	−9.92006	−1.10910
$81$	1.00000	0.111111
$82$	2.08413	0.230153
$83$	3.90566	0.428702	0.214351	−	0.976757i	$-0.431236\pi$
$83$	3.90566	0.428702	0.214351	+	0.976757i	$0.431236\pi$
$84$	0.144735	0.0157919
$85$	4.56115	0.494726
$86$	−3.56638	−0.384572
$87$	1.26877	0.136027
$88$	−7.14723	−0.761897
$89$	0.470731	0.0498974	0.0249487	−	0.999689i	$-0.492058\pi$
$89$	0.470731	0.0498974	0.0249487	+	0.999689i	$0.492058\pi$
$90$	−1.61260	−0.169983
$91$	0.324073	0.0339721
$92$	−7.30713	−0.761821
$93$	1.33574	0.138510
$94$	3.86692	0.398842
$95$	17.0198	1.74619
$96$	−4.72705	−0.482452
$97$	−11.7474	−1.19277	−0.596384	−	0.802699i	$-0.703396\pi$
$97$	−11.7474	−1.19277	−0.596384	+	0.802699i	$0.703396\pi$
$98$	−3.18311	−0.321543
$99$	4.14014	0.416100
$100$	−13.5407	−1.35407
$101$	−5.07845	−0.505325	−0.252662	−	0.967555i	$-0.581306\pi$
$101$	−5.07845	−0.505325	−0.252662	+	0.967555i	$0.581306\pi$
$102$	0.585956	0.0580183
$103$	−15.8224	−1.55902	−0.779512	−	0.626387i	$-0.784533\pi$
$103$	−15.8224	−1.55902	−0.779512	+	0.626387i	$0.784533\pi$
$104$	−6.92996	−0.679539
$105$	0.286024	0.0279131
$106$	−2.25245	−0.218777
$107$	−5.55150	−0.536684	−0.268342	−	0.963324i	$-0.586476\pi$
$107$	−5.55150	−0.536684	−0.268342	+	0.963324i	$0.586476\pi$
$108$	1.79283	0.172516
$109$	−2.06538	−0.197828	−0.0989140	−	0.995096i	$-0.531537\pi$
$109$	−2.06538	−0.197828	−0.0989140	+	0.995096i	$0.531537\pi$
$110$	−6.67639	−0.636569
$111$	2.69851	0.256131
$112$	0.226038	0.0213585
$113$	−2.85424	−0.268504	−0.134252	−	0.990947i	$-0.542863\pi$
$113$	−2.85424	−0.268504	−0.134252	+	0.990947i	$0.542863\pi$
$114$	2.18648	0.204782
$115$	−14.4402	−1.34656
$116$	2.27470	0.211201
$117$	4.01429	0.371121
$118$	0.574430	0.0528806
$119$	−0.103930	−0.00952725
$120$	−6.11633	−0.558342
$121$	6.14076	0.558251
$122$	4.93314	0.446626
$123$	−4.57894	−0.412869
$124$	2.39477	0.215056
$125$	−9.04403	−0.808922
$126$	0.0367446	0.00327347
$127$	−5.59703	−0.496656	−0.248328	−	0.968676i	$-0.579881\pi$
$127$	−5.59703	−0.496656	−0.248328	+	0.968676i	$0.579881\pi$
$128$	−11.0236	−0.974359
$129$	7.83553	0.689880
$130$	−6.47344	−0.567758
$131$	−1.92957	−0.168587	−0.0842936	−	0.996441i	$-0.526863\pi$
$131$	−1.92957	−0.168587	−0.0842936	+	0.996441i	$0.526863\pi$
$132$	7.42259	0.646053
$133$	−0.387812	−0.0336275
$134$	3.06854	0.265082
$135$	3.54297	0.304931
$136$	2.22244	0.190572
$137$	17.7316	1.51492	0.757458	−	0.652884i	$-0.226441\pi$
$137$	17.7316	1.51492	0.757458	+	0.652884i	$0.226441\pi$
$138$	−1.85509	−0.157916
$139$	3.42664	0.290644	0.145322	−	0.989384i	$-0.453578\pi$
$139$	3.42664	0.290644	0.145322	+	0.989384i	$0.453578\pi$
$140$	0.512794	0.0433390
$141$	−8.49584	−0.715479
$142$	−4.87913	−0.409447
$143$	16.6197	1.38981
$144$	2.79992	0.233327
$145$	4.49523	0.373309
$146$	−6.46645	−0.535168
$147$	6.99348	0.576813
$148$	4.83797	0.397679
$149$	−2.13254	−0.174704	−0.0873522	−	0.996177i	$-0.527841\pi$
$149$	−2.13254	−0.174704	−0.0873522	+	0.996177i	$0.527841\pi$
$150$	−3.43763	−0.280681
$151$	−12.1038	−0.984994	−0.492497	−	0.870314i	$-0.663916\pi$
$151$	−12.1038	−0.984994	−0.492497	+	0.870314i	$0.663916\pi$
$152$	8.29295	0.672647
$153$	−1.28738	−0.104078
$154$	0.152128	0.0122588
$155$	4.73250	0.380124
$156$	7.19695	0.576217
$157$	−1.04573	−0.0834582	−0.0417291	−	0.999129i	$-0.513287\pi$
$157$	−1.04573	−0.0834582	−0.0417291	+	0.999129i	$0.513287\pi$
$158$	−4.35620	−0.346561
$159$	4.94876	0.392462
$160$	−16.7478	−1.32403
$161$	0.329034	0.0259315
$162$	0.455154	0.0357603
$163$	17.8918	1.40139	0.700695	−	0.713461i	$-0.252873\pi$
$163$	17.8918	1.40139	0.700695	+	0.713461i	$0.252873\pi$
$164$	−8.20928	−0.641037
$165$	14.6684	1.14193
$166$	1.77768	0.137975
$167$	−21.9232	−1.69647	−0.848234	−	0.529622i	$-0.822334\pi$
$167$	−21.9232	−1.69647	−0.848234	+	0.529622i	$0.822334\pi$
$168$	0.139366	0.0107523
$169$	3.11449	0.239576
$170$	2.07603	0.159224
$171$	−4.80382	−0.367357
$172$	14.0478	1.07114
$173$	−8.75170	−0.665380	−0.332690	−	0.943036i	$-0.607956\pi$
$173$	−8.75170	−0.665380	−0.332690	+	0.943036i	$0.607956\pi$
$174$	0.577488	0.0437793
$175$	0.609726	0.0460910
$176$	11.5921	0.873786
$177$	−1.26205	−0.0948619
$178$	0.214255	0.0160591
$179$	17.7846	1.32928	0.664641	−	0.747163i	$-0.268585\pi$
$179$	17.7846	1.32928	0.664641	+	0.747163i	$0.268585\pi$
$180$	6.35197	0.473448
$181$	22.0033	1.63549	0.817747	−	0.575578i	$-0.195223\pi$
$181$	22.0033	1.63549	0.817747	+	0.575578i	$0.195223\pi$
$182$	0.147503	0.0109337
$183$	−10.8384	−0.801197
$184$	−7.03606	−0.518705
$185$	9.56074	0.702919
$186$	0.607970	0.0445785
$187$	−5.32993	−0.389763
$188$	−15.2316	−1.11088
$189$	−0.0807299	−0.00587224
$190$	7.74664	0.562000
$191$	12.3455	0.893292	0.446646	−	0.894711i	$-0.352618\pi$
$191$	12.3455	0.893292	0.446646	+	0.894711i	$0.352618\pi$
$192$	3.44831	0.248860
$193$	16.5406	1.19062	0.595309	−	0.803497i	$-0.297030\pi$
$193$	16.5406	1.19062	0.595309	+	0.803497i	$0.297030\pi$
$194$	−5.34688	−0.383884
$195$	14.2225	1.01850
$196$	12.5382	0.895583
$197$	6.13831	0.437337	0.218668	−	0.975799i	$-0.429829\pi$
$197$	6.13831	0.437337	0.218668	+	0.975799i	$0.429829\pi$
$198$	1.88440	0.133919
$199$	6.18107	0.438165	0.219082	−	0.975706i	$-0.429694\pi$
$199$	6.18107	0.438165	0.219082	+	0.975706i	$0.429694\pi$
$200$	−13.0384	−0.921951
$201$	−6.74176	−0.475527
$202$	−2.31148	−0.162635
$203$	−0.102428	−0.00718904
$204$	−2.30806	−0.161596
$205$	−16.2231	−1.13307
$206$	−7.20162	−0.501761
$207$	4.07574	0.283284
$208$	11.2397	0.779333
$209$	−19.8885	−1.37571
$210$	0.130185	0.00898363
$211$	−15.6291	−1.07595	−0.537974	−	0.842961i	$-0.680810\pi$
$211$	−15.6291	−1.07595	−0.537974	+	0.842961i	$0.680810\pi$
$212$	8.87231	0.609353
$213$	10.7197	0.734503
$214$	−2.52679	−0.172728
$215$	27.7611	1.89329
$216$	1.72633	0.117462
$217$	−0.107834	−0.00732028
$218$	−0.940069	−0.0636695
$219$	14.2072	0.960031
$220$	26.2980	1.77301
$221$	−5.16791	−0.347631
$222$	1.22824	0.0824339
$223$	−12.1765	−0.815401	−0.407701	−	0.913116i	$-0.633669\pi$
$223$	−12.1765	−0.815401	−0.407701	+	0.913116i	$0.633669\pi$
$224$	0.381614	0.0254977
$225$	7.55267	0.503511
$226$	−1.29912	−0.0864162
$227$	−12.8779	−0.854737	−0.427369	−	0.904078i	$-0.640559\pi$
$227$	−12.8779	−0.854737	−0.427369	+	0.904078i	$0.640559\pi$
$228$	−8.61245	−0.570373
$229$	−4.00438	−0.264617	−0.132308	−	0.991209i	$-0.542239\pi$
$229$	−4.00438	−0.264617	−0.132308	+	0.991209i	$0.542239\pi$
$230$	−6.57254	−0.433381
$231$	−0.334233	−0.0219909
$232$	2.19032	0.143801
$233$	5.37617	0.352204	0.176102	−	0.984372i	$-0.443651\pi$
$233$	5.37617	0.352204	0.176102	+	0.984372i	$0.443651\pi$
$234$	1.82712	0.119443
$235$	−30.1005	−1.96354
$236$	−2.26266	−0.147286
$237$	9.57082	0.621691
$238$	−0.0473042	−0.00306628
$239$	−0.517643	−0.0334835	−0.0167418	−	0.999860i	$-0.505329\pi$
$239$	−0.517643	−0.0334835	−0.0167418	+	0.999860i	$0.505329\pi$
$240$	9.92006	0.640337
$241$	−28.9814	−1.86686	−0.933428	−	0.358764i	$-0.883198\pi$
$241$	−28.9814	−1.86686	−0.933428	+	0.358764i	$0.883198\pi$
$242$	2.79500	0.179669
$243$	−1.00000	−0.0641500
$244$	−19.4314	−1.24397
$245$	24.7777	1.58299
$246$	−2.08413	−0.132879
$247$	−19.2839	−1.22700
$248$	2.30593	0.146427
$249$	−3.90566	−0.247511
$250$	−4.11643	−0.260346
$251$	−25.3020	−1.59705	−0.798524	−	0.601963i	$-0.794385\pi$
$251$	−25.3020	−1.59705	−0.798524	+	0.601963i	$0.794385\pi$
$252$	−0.144735	−0.00911747
$253$	16.8741	1.06087
$254$	−2.54751	−0.159845
$255$	−4.56115	−0.285630
$256$	1.87918	0.117448
$257$	−14.3829	−0.897179	−0.448589	−	0.893738i	$-0.648073\pi$
$257$	−14.3829	−0.897179	−0.448589	+	0.893738i	$0.648073\pi$
$258$	3.56638	0.222033
$259$	−0.217850	−0.0135366
$260$	25.4986	1.58136
$261$	−1.26877	−0.0785351
$262$	−0.878252	−0.0542586
$263$	11.9178	0.734883	0.367442	−	0.930047i	$-0.380234\pi$
$263$	11.9178	0.734883	0.367442	+	0.930047i	$0.380234\pi$
$264$	7.14723	0.439882
$265$	17.5333	1.07706
$266$	−0.176514	−0.0108228
$267$	−0.470731	−0.0288083
$268$	−12.0869	−0.738323
$269$	−8.57041	−0.522547	−0.261274	−	0.965265i	$-0.584142\pi$
$269$	−8.57041	−0.522547	−0.261274	+	0.965265i	$0.584142\pi$
$270$	1.61260	0.0981397
$271$	−23.5256	−1.42908	−0.714540	−	0.699595i	$-0.753364\pi$
$271$	−23.5256	−1.42908	−0.714540	+	0.699595i	$0.753364\pi$
$272$	−3.60456	−0.218559
$273$	−0.324073	−0.0196138
$274$	8.07064	0.487565
$275$	31.2691	1.88560
$276$	7.30713	0.439837
$277$	25.0321	1.50403	0.752016	−	0.659145i	$-0.229082\pi$
$277$	25.0321	1.50403	0.752016	+	0.659145i	$0.229082\pi$
$278$	1.55965	0.0935416
$279$	−1.33574	−0.0799689
$280$	0.493771	0.0295084
$281$	27.0697	1.61484	0.807421	−	0.589976i	$-0.200863\pi$
$281$	27.0697	1.61484	0.807421	+	0.589976i	$0.200863\pi$
$282$	−3.86692	−0.230272
$283$	7.42654	0.441462	0.220731	−	0.975335i	$-0.429156\pi$
$283$	7.42654	0.441462	0.220731	+	0.975335i	$0.429156\pi$
$284$	19.2187	1.14042
$285$	−17.0198	−1.00817
$286$	7.56453	0.447300
$287$	0.369658	0.0218202
$288$	4.72705	0.278544
$289$	−15.3427	−0.902509
$290$	2.04603	0.120147
$291$	11.7474	0.688645
$292$	25.4711	1.49058
$293$	−20.7262	−1.21084	−0.605418	−	0.795908i	$-0.706994\pi$
$293$	−20.7262	−1.21084	−0.605418	+	0.795908i	$0.706994\pi$
$294$	3.18311	0.185643
$295$	−4.47143	−0.260337
$296$	4.65850	0.270770
$297$	−4.14014	−0.240235
$298$	−0.970634	−0.0562273
$299$	16.3612	0.946192
$300$	13.5407	0.781771
$301$	−0.632562	−0.0364602
$302$	−5.50910	−0.317013
$303$	5.07845	0.291749
$304$	−13.4503	−0.771429
$305$	−38.4001	−2.19878
$306$	−0.585956	−0.0334969
$307$	2.95040	0.168388	0.0841942	−	0.996449i	$-0.473168\pi$
$307$	2.95040	0.168388	0.0841942	+	0.996449i	$0.473168\pi$
$308$	−0.599225	−0.0341440
$309$	15.8224	0.900103
$310$	2.15402	0.122340
$311$	9.73374	0.551950	0.275975	−	0.961165i	$-0.410999\pi$
$311$	9.73374	0.551950	0.275975	+	0.961165i	$0.410999\pi$
$312$	6.92996	0.392332
$313$	−7.87200	−0.444952	−0.222476	−	0.974938i	$-0.571414\pi$
$313$	−7.87200	−0.444952	−0.222476	+	0.974938i	$0.571414\pi$
$314$	−0.475968	−0.0268604
$315$	−0.286024	−0.0161156
$316$	17.1589	0.965263
$317$	−4.28906	−0.240898	−0.120449	−	0.992720i	$-0.538433\pi$
$317$	−4.28906	−0.240898	−0.120449	+	0.992720i	$0.538433\pi$
$318$	2.25245	0.126311
$319$	−5.25290	−0.294106
$320$	12.2173	0.682966
$321$	5.55150	0.309855
$322$	0.149761	0.00834588
$323$	6.18433	0.344106
$324$	−1.79283	−0.0996019
$325$	30.3186	1.68177
$326$	8.14351	0.451028
$327$	2.06538	0.114216
$328$	−7.90474	−0.436466
$329$	0.685869	0.0378132
$330$	6.67639	0.367523
$331$	5.23840	0.287928	0.143964	−	0.989583i	$-0.454015\pi$
$331$	5.23840	0.287928	0.143964	+	0.989583i	$0.454015\pi$
$332$	−7.00221	−0.384296
$333$	−2.69851	−0.147877
$334$	−9.97844	−0.545996
$335$	−23.8859	−1.30503
$336$	−0.226038	−0.0123314
$337$	11.6542	0.634842	0.317421	−	0.948285i	$-0.397183\pi$
$337$	11.6542	0.634842	0.317421	+	0.948285i	$0.397183\pi$
$338$	1.41757	0.0771059
$339$	2.85424	0.155021
$340$	−8.17739	−0.443481
$341$	−5.53017	−0.299475
$342$	−2.18648	−0.118231
$343$	−1.12969	−0.0609977
$344$	13.5267	0.729309
$345$	14.4402	0.777437
$346$	−3.98338	−0.214148
$347$	−4.51287	−0.242264	−0.121132	−	0.992636i	$-0.538652\pi$
$347$	−4.51287	−0.242264	−0.121132	+	0.992636i	$0.538652\pi$
$348$	−2.27470	−0.121937
$349$	−18.7079	−1.00141	−0.500706	−	0.865617i	$-0.666926\pi$
$349$	−18.7079	−1.00141	−0.500706	+	0.865617i	$0.666926\pi$
$350$	0.277520	0.0148340
$351$	−4.01429	−0.214267
$352$	19.5706	1.04312
$353$	0.0490583	0.00261111	0.00130555	−	0.999999i	$-0.499584\pi$
$353$	0.0490583	0.00261111	0.00130555	+	0.999999i	$0.499584\pi$
$354$	−0.574430	−0.0305306
$355$	37.9797	2.01575
$356$	−0.843942	−0.0447289
$357$	0.103930	0.00550056
$358$	8.09473	0.427820
$359$	−6.18415	−0.326387	−0.163193	−	0.986594i	$-0.552179\pi$
$359$	−6.18415	−0.326387	−0.163193	+	0.986594i	$0.552179\pi$
$360$	6.11633	0.322359
$361$	4.07665	0.214560
$362$	10.0149	0.526372
$363$	−6.14076	−0.322307
$364$	−0.581009	−0.0304532
$365$	50.3356	2.63469
$366$	−4.93314	−0.257859
$367$	−7.29147	−0.380612	−0.190306	−	0.981725i	$-0.560948\pi$
$367$	−7.29147	−0.380612	−0.190306	+	0.981725i	$0.560948\pi$
$368$	11.4118	0.594879
$369$	4.57894	0.238370
$370$	4.35161	0.226230
$371$	−0.399513	−0.0207417
$372$	−2.39477	−0.124163
$373$	−35.5605	−1.84125	−0.920626	−	0.390446i	$-0.872321\pi$
$373$	−35.5605	−1.84125	−0.920626	+	0.390446i	$0.872321\pi$
$374$	−2.42594	−0.125442
$375$	9.04403	0.467032
$376$	−14.6666	−0.756371
$377$	−5.09322	−0.262314
$378$	−0.0367446	−0.00188994
$379$	−11.8236	−0.607339	−0.303670	−	0.952777i	$-0.598212\pi$
$379$	−11.8236	−0.607339	−0.303670	+	0.952777i	$0.598212\pi$
$380$	−30.5137	−1.56532
$381$	5.59703	0.286745
$382$	5.61913	0.287500
$383$	−27.0166	−1.38048	−0.690241	−	0.723580i	$-0.742495\pi$
$383$	−27.0166	−1.38048	−0.690241	+	0.723580i	$0.742495\pi$
$384$	11.0236	0.562546
$385$	−1.18418	−0.0603514
$386$	7.52852	0.383192
$387$	−7.83553	−0.398302
$388$	21.0611	1.06922
$389$	−6.41962	−0.325488	−0.162744	−	0.986668i	$-0.552034\pi$
$389$	−6.41962	−0.325488	−0.162744	+	0.986668i	$0.552034\pi$
$390$	6.47344	0.327795
$391$	−5.24702	−0.265353
$392$	12.0730	0.609780
$393$	1.92957	0.0973339
$394$	2.79388	0.140754
$395$	33.9092	1.70615
$396$	−7.42259	−0.372999
$397$	−5.67043	−0.284591	−0.142295	−	0.989824i	$-0.545448\pi$
$397$	−5.67043	−0.284591	−0.142295	+	0.989824i	$0.545448\pi$
$398$	2.81334	0.141020
$399$	0.387812	0.0194149
$400$	21.1469	1.05734
$401$	−6.41640	−0.320420	−0.160210	−	0.987083i	$-0.551217\pi$
$401$	−6.41640	−0.320420	−0.160210	+	0.987083i	$0.551217\pi$
$402$	−3.06854	−0.153045
$403$	−5.36206	−0.267103
$404$	9.10482	0.452982
$405$	−3.54297	−0.176052
$406$	−0.0466206	−0.00231374
$407$	−11.1722	−0.553785
$408$	−2.22244	−0.110027
$409$	−33.3092	−1.64703	−0.823516	−	0.567293i	$-0.807991\pi$
$409$	−33.3092	−1.64703	−0.823516	+	0.567293i	$0.807991\pi$
$410$	−7.38400	−0.364670
$411$	−17.7316	−0.874637
$412$	28.3669	1.39754
$413$	0.101886	0.00501346
$414$	1.85509	0.0911728
$415$	−13.8377	−0.679264
$416$	18.9757	0.930361
$417$	−3.42664	−0.167803
$418$	−9.05233	−0.442764
$419$	−33.5439	−1.63873	−0.819364	−	0.573274i	$-0.805673\pi$
$419$	−33.5439	−1.63873	−0.819364	+	0.573274i	$0.805673\pi$
$420$	−0.512794	−0.0250218
$421$	−24.5206	−1.19506	−0.597531	−	0.801846i	$-0.703851\pi$
$421$	−24.5206	−1.19506	−0.597531	+	0.801846i	$0.703851\pi$
$422$	−7.11364	−0.346286
$423$	8.49584	0.413082
$424$	8.54317	0.414893
$425$	−9.72315	−0.471642
$426$	4.87913	0.236394
$427$	0.874982	0.0423433
$428$	9.95293	0.481093
$429$	−16.6197	−0.802407
$430$	12.6356	0.609342
$431$	36.2916	1.74811	0.874054	−	0.485830i	$-0.161482\pi$
$431$	36.2916	1.74811	0.874054	+	0.485830i	$0.161482\pi$
$432$	−2.79992	−0.134711
$433$	−37.7521	−1.81425	−0.907126	−	0.420860i	$-0.861728\pi$
$433$	−37.7521	−1.81425	−0.907126	+	0.420860i	$0.861728\pi$
$434$	−0.0490813	−0.00235598
$435$	−4.49523	−0.215530
$436$	3.70289	0.177336
$437$	−19.5791	−0.936596
$438$	6.46645	0.308979
$439$	−17.2107	−0.821421	−0.410710	−	0.911766i	$-0.634719\pi$
$439$	−17.2107	−0.821421	−0.410710	+	0.911766i	$0.634719\pi$
$440$	25.3225	1.20720
$441$	−6.99348	−0.333023
$442$	−2.35220	−0.111883
$443$	20.4563	0.971906	0.485953	−	0.873985i	$-0.338473\pi$
$443$	20.4563	0.971906	0.485953	+	0.873985i	$0.338473\pi$
$444$	−4.83797	−0.229600
$445$	−1.66779	−0.0790607
$446$	−5.54221	−0.262431
$447$	2.13254	0.100866
$448$	−0.278382	−0.0131523
$449$	−31.2593	−1.47522	−0.737608	−	0.675230i	$-0.764045\pi$
$449$	−31.2593	−1.47522	−0.737608	+	0.675230i	$0.764045\pi$
$450$	3.43763	0.162051
$451$	18.9575	0.892672
$452$	5.11718	0.240692
$453$	12.1038	0.568687
$454$	−5.86144	−0.275091
$455$	−1.14818	−0.0538276
$456$	−8.29295	−0.388353
$457$	5.18498	0.242543	0.121272	−	0.992619i	$-0.461303\pi$
$457$	5.18498	0.242543	0.121272	+	0.992619i	$0.461303\pi$
$458$	−1.82261	−0.0851650
$459$	1.28738	0.0600897
$460$	25.8890	1.20708
$461$	0.0383161	0.00178456	0.000892280	−	1.00000i	$-0.499716\pi$
$461$	0.0383161	0.00178456	0.000892280		1.00000i	$0.499716\pi$
$462$	−0.152128	−0.00707762
$463$	−2.86636	−0.133211	−0.0666054	−	0.997779i	$-0.521217\pi$
$463$	−2.86636	−0.133211	−0.0666054	+	0.997779i	$0.521217\pi$
$464$	−3.55247	−0.164919
$465$	−4.73250	−0.219465
$466$	2.44699	0.113354
$467$	8.97636	0.415376	0.207688	−	0.978195i	$-0.433406\pi$
$467$	8.97636	0.415376	0.207688	+	0.978195i	$0.433406\pi$
$468$	−7.19695	−0.332679
$469$	0.544262	0.0251317
$470$	−13.7004	−0.631952
$471$	1.04573	0.0481846
$472$	−2.17872	−0.100284
$473$	−32.4402	−1.49160
$474$	4.35620	0.200087
$475$	−36.2816	−1.66471
$476$	0.186329	0.00854039
$477$	−4.94876	−0.226588
$478$	−0.235608	−0.0107764
$479$	−4.62286	−0.211224	−0.105612	−	0.994407i	$-0.533680\pi$
$479$	−4.62286	−0.211224	−0.105612	+	0.994407i	$0.533680\pi$
$480$	16.7478	0.764430
$481$	−10.8326	−0.493923
$482$	−13.1910	−0.600834
$483$	−0.329034	−0.0149716
$484$	−11.0094	−0.500426
$485$	41.6207	1.88990
$486$	−0.455154	−0.0206462
$487$	−20.4183	−0.925243	−0.462622	−	0.886556i	$-0.653091\pi$
$487$	−20.4183	−0.925243	−0.462622	+	0.886556i	$0.653091\pi$
$488$	−18.7106	−0.846988
$489$	−17.8918	−0.809093
$490$	11.2777	0.509474
$491$	11.4612	0.517237	0.258618	−	0.965980i	$-0.416733\pi$
$491$	11.4612	0.517237	0.258618	+	0.965980i	$0.416733\pi$
$492$	8.20928	0.370103
$493$	1.63339	0.0735643
$494$	−8.77715	−0.394903
$495$	−14.6684	−0.659296
$496$	−3.73998	−0.167930
$497$	−0.865402	−0.0388186
$498$	−1.77768	−0.0796597
$499$	−42.3681	−1.89666	−0.948329	−	0.317288i	$-0.897228\pi$
$499$	−42.3681	−1.89666	−0.948329	+	0.317288i	$0.897228\pi$
$500$	16.2144	0.725132
$501$	21.9232	0.979456
$502$	−11.5163	−0.513998
$503$	37.6637	1.67934	0.839670	−	0.543097i	$-0.182748\pi$
$503$	37.6637	1.67934	0.839670	+	0.543097i	$0.182748\pi$
$504$	−0.139366	−0.00620786
$505$	17.9928	0.800670
$506$	7.68034	0.341433
$507$	−3.11449	−0.138319
$508$	10.0346	0.445211
$509$	12.2882	0.544664	0.272332	−	0.962203i	$-0.412205\pi$
$509$	12.2882	0.544664	0.272332	+	0.962203i	$0.412205\pi$
$510$	−2.07603	−0.0919281
$511$	−1.14694	−0.0507378
$512$	22.9025	1.01216
$513$	4.80382	0.212094
$514$	−6.54643	−0.288750
$515$	56.0583	2.47022
$516$	−14.0478	−0.618420
$517$	35.1740	1.54695
$518$	−0.0991555	−0.00435664
$519$	8.75170	0.384157
$520$	24.5527	1.07671
$521$	9.24249	0.404921	0.202461	−	0.979290i	$-0.435106\pi$
$521$	9.24249	0.404921	0.202461	+	0.979290i	$0.435106\pi$
$522$	−0.577488	−0.0252760
$523$	−30.7283	−1.34366	−0.671828	−	0.740707i	$-0.734490\pi$
$523$	−30.7283	−1.34366	−0.671828	+	0.740707i	$0.734490\pi$
$524$	3.45940	0.151124
$525$	−0.609726	−0.0266106
$526$	5.42444	0.236517
$527$	1.71961	0.0749073
$528$	−11.5921	−0.504480
$529$	−6.38834	−0.277754
$530$	7.98037	0.346645
$531$	1.26205	0.0547685
$532$	0.695282	0.0301443
$533$	18.3812	0.796177
$534$	−0.214255	−0.00927173
$535$	19.6688	0.850358
$536$	−11.6385	−0.502706
$537$	−17.7846	−0.767461
$538$	−3.90086	−0.168178
$539$	−28.9540	−1.24714
$540$	−6.35197	−0.273345
$541$	17.1608	0.737802	0.368901	−	0.929469i	$-0.379734\pi$
$541$	17.1608	0.737802	0.368901	+	0.929469i	$0.379734\pi$
$542$	−10.7078	−0.459939
$543$	−22.0033	−0.944253
$544$	−6.08551	−0.260914
$545$	7.31760	0.313452
$546$	−0.147503	−0.00631256
$547$	11.4319	0.488791	0.244396	−	0.969676i	$-0.421410\pi$
$547$	11.4319	0.488791	0.244396	+	0.969676i	$0.421410\pi$
$548$	−31.7899	−1.35800
$549$	10.8384	0.462571
$550$	14.2323	0.606866
$551$	6.09496	0.259654
$552$	7.03606	0.299474
$553$	−0.772651	−0.0328565
$554$	11.3935	0.484062
$555$	−9.56074	−0.405831
$556$	−6.14339	−0.260538
$557$	6.70736	0.284200	0.142100	−	0.989852i	$-0.454615\pi$
$557$	6.70736	0.284200	0.142100	+	0.989852i	$0.454615\pi$
$558$	−0.607970	−0.0257374
$559$	−31.4541	−1.33036
$560$	−0.800846	−0.0338419
$561$	5.32993	0.225030
$562$	12.3209	0.519725
$563$	40.4548	1.70497	0.852484	−	0.522754i	$-0.175095\pi$
$563$	40.4548	1.70497	0.852484	+	0.522754i	$0.175095\pi$
$564$	15.2316	0.641368
$565$	10.1125	0.425436
$566$	3.38022	0.142081
$567$	0.0807299	0.00339034
$568$	18.5057	0.776483
$569$	−12.4290	−0.521049	−0.260525	−	0.965467i	$-0.583896\pi$
$569$	−12.4290	−0.521049	−0.260525	+	0.965467i	$0.583896\pi$
$570$	−7.74664	−0.324471
$571$	42.6962	1.78678	0.893390	−	0.449281i	$-0.148320\pi$
$571$	42.6962	1.78678	0.893390	+	0.449281i	$0.148320\pi$
$572$	−29.7964	−1.24585
$573$	−12.3455	−0.515743
$574$	0.168251	0.00702267
$575$	30.7827	1.28373
$576$	−3.44831	−0.143680
$577$	−8.42846	−0.350881	−0.175441	−	0.984490i	$-0.556135\pi$
$577$	−8.42846	−0.350881	−0.175441	+	0.984490i	$0.556135\pi$
$578$	−6.98328	−0.290466
$579$	−16.5406	−0.687403
$580$	−8.05921	−0.334640
$581$	0.315304	0.0130810
$582$	5.34688	0.221635
$583$	−20.4886	−0.848550
$584$	24.5262	1.01490
$585$	−14.2225	−0.588028
$586$	−9.43361	−0.389699
$587$	−41.3993	−1.70873	−0.854367	−	0.519670i	$-0.826055\pi$
$587$	−41.3993	−1.70873	−0.854367	+	0.519670i	$0.826055\pi$
$588$	−12.5382	−0.517065
$589$	6.41667	0.264394
$590$	−2.03519	−0.0837875
$591$	−6.13831	−0.252497
$592$	−7.55561	−0.310534
$593$	28.3652	1.16482	0.582410	−	0.812895i	$-0.302110\pi$
$593$	28.3652	1.16482	0.582410	+	0.812895i	$0.302110\pi$
$594$	−1.88440	−0.0773180
$595$	0.368221	0.0150956
$596$	3.82329	0.156608
$597$	−6.18107	−0.252974
$598$	7.44687	0.304525
$599$	21.2329	0.867551	0.433776	−	0.901021i	$-0.357181\pi$
$599$	21.2329	0.867551	0.433776	+	0.901021i	$0.357181\pi$
$600$	13.0384	0.532289
$601$	−47.5531	−1.93973	−0.969865	−	0.243642i	$-0.921658\pi$
$601$	−47.5531	−1.93973	−0.969865	+	0.243642i	$0.921658\pi$
$602$	−0.287913	−0.0117345
$603$	6.74176	0.274546
$604$	21.7001	0.882966
$605$	−21.7566	−0.884530
$606$	2.31148	0.0938974
$607$	15.8930	0.645075	0.322538	−	0.946557i	$-0.395464\pi$
$607$	15.8930	0.645075	0.322538	+	0.946557i	$0.395464\pi$
$608$	−22.7079	−0.920926
$609$	0.102428	0.00415059
$610$	−17.4780	−0.707663
$611$	34.1047	1.37973
$612$	2.30806	0.0932977
$613$	40.6005	1.63984	0.819920	−	0.572479i	$-0.194018\pi$
$613$	40.6005	1.63984	0.819920	+	0.572479i	$0.194018\pi$
$614$	1.34289	0.0541946
$615$	16.2231	0.654177
$616$	−0.576995	−0.0232478
$617$	−15.2091	−0.612294	−0.306147	−	0.951984i	$-0.599040\pi$
$617$	−15.2091	−0.612294	−0.306147	+	0.951984i	$0.599040\pi$
$618$	7.20162	0.289692
$619$	−8.31316	−0.334134	−0.167067	−	0.985946i	$-0.553430\pi$
$619$	−8.31316	−0.334134	−0.167067	+	0.985946i	$0.553430\pi$
$620$	−8.48460	−0.340750
$621$	−4.07574	−0.163554
$622$	4.43036	0.177641
$623$	0.0380021	0.00152252
$624$	−11.2397	−0.449948
$625$	−5.72058	−0.228823
$626$	−3.58298	−0.143205
$627$	19.8885	0.794269
$628$	1.87482	0.0748134
$629$	3.47400	0.138517
$630$	−0.130185	−0.00518670
$631$	5.16487	0.205610	0.102805	−	0.994702i	$-0.467218\pi$
$631$	5.16487	0.205610	0.102805	+	0.994702i	$0.467218\pi$
$632$	16.5223	0.657224
$633$	15.6291	0.621199
$634$	−1.95218	−0.0775312
$635$	19.8301	0.786935
$636$	−8.87231	−0.351810
$637$	−28.0738	−1.11233
$638$	−2.39088	−0.0946559
$639$	−10.7197	−0.424065
$640$	39.0564	1.54384
$641$	−12.2716	−0.484701	−0.242350	−	0.970189i	$-0.577918\pi$
$641$	−12.2716	−0.484701	−0.242350	+	0.970189i	$0.577918\pi$
$642$	2.52679	0.0997245
$643$	−0.734498	−0.0289658	−0.0144829	−	0.999895i	$-0.504610\pi$
$643$	−0.734498	−0.0289658	−0.0144829	+	0.999895i	$0.504610\pi$
$644$	−0.589904	−0.0232455
$645$	−27.7611	−1.09309
$646$	2.81483	0.110748
$647$	13.0055	0.511301	0.255650	−	0.966769i	$-0.417710\pi$
$647$	13.0055	0.511301	0.255650	+	0.966769i	$0.417710\pi$
$648$	−1.72633	−0.0678165
$649$	5.22508	0.205103
$650$	13.7996	0.541266
$651$	0.107834	0.00422637
$652$	−32.0770	−1.25623
$653$	5.63620	0.220562	0.110281	−	0.993900i	$-0.464825\pi$
$653$	5.63620	0.220562	0.110281	+	0.993900i	$0.464825\pi$
$654$	0.940069	0.0367596
$655$	6.83641	0.267121
$656$	12.8207	0.500564
$657$	−14.2072	−0.554274
$658$	0.312176	0.0121699
$659$	17.2279	0.671103	0.335552	−	0.942022i	$-0.391077\pi$
$659$	17.2279	0.671103	0.335552	+	0.942022i	$0.391077\pi$
$660$	−26.2980	−1.02365
$661$	−10.0767	−0.391937	−0.195969	−	0.980610i	$-0.562785\pi$
$661$	−10.0767	−0.391937	−0.195969	+	0.980610i	$0.562785\pi$
$662$	2.38428	0.0926677
$663$	5.16791	0.200705
$664$	−6.74245	−0.261658
$665$	1.37401	0.0532817
$666$	−1.22824	−0.0475932
$667$	−5.17119	−0.200229
$668$	39.3046	1.52074
$669$	12.1765	0.470772
$670$	−10.8718	−0.420013
$671$	44.8724	1.73228
$672$	−0.381614	−0.0147211
$673$	17.1902	0.662633	0.331317	−	0.943520i	$-0.392507\pi$
$673$	17.1902	0.662633	0.331317	+	0.943520i	$0.392507\pi$
$674$	5.30444	0.204319
$675$	−7.55267	−0.290702
$676$	−5.58377	−0.214760
$677$	−1.00000	−0.0384331
$677$	−1.00000	−0.0384331
$678$	1.29912	0.0498924
$679$	−0.948367	−0.0363950
$680$	−7.87403	−0.301955
$681$	12.8779	0.493483
$682$	−2.51708	−0.0963840
$683$	−4.16118	−0.159223	−0.0796116	−	0.996826i	$-0.525368\pi$
$683$	−4.16118	−0.159223	−0.0796116	+	0.996826i	$0.525368\pi$
$684$	8.61245	0.329305
$685$	−62.8227	−2.40033
$686$	−0.514185	−0.0196317
$687$	4.00438	0.152777
$688$	−21.9389	−0.836412
$689$	−19.8657	−0.756825
$690$	6.57254	0.250212
$691$	48.6131	1.84933	0.924665	−	0.380783i	$-0.124345\pi$
$691$	48.6131	1.84933	0.924665	+	0.380783i	$0.124345\pi$
$692$	15.6904	0.596458
$693$	0.334233	0.0126965
$694$	−2.05405	−0.0779708
$695$	−12.1405	−0.460515
$696$	−2.19032	−0.0830238
$697$	−5.89484	−0.223283
$698$	−8.51500	−0.322298
$699$	−5.37617	−0.203345
$700$	−1.09314	−0.0413167
$701$	−0.107441	−0.00405798	−0.00202899	−	0.999998i	$-0.500646\pi$
$701$	−0.107441	−0.00405798	−0.00202899	+	0.999998i	$0.500646\pi$
$702$	−1.82712	−0.0689602
$703$	12.9631	0.488914
$704$	−14.2765	−0.538066
$705$	30.1005	1.13365
$706$	0.0223291	0.000840366 0
$707$	−0.409983	−0.0154190
$708$	2.26266	0.0850358
$709$	−22.2907	−0.837145	−0.418573	−	0.908183i	$-0.637469\pi$
$709$	−22.2907	−0.837145	−0.418573	+	0.908183i	$0.637469\pi$
$710$	17.2866	0.648755
$711$	−9.57082	−0.358934
$712$	−0.812635	−0.0304548
$713$	−5.44414	−0.203885
$714$	0.0473042	0.00177032
$715$	−58.8832	−2.20211
$716$	−31.8848	−1.19159
$717$	0.517643	0.0193317
$718$	−2.81474	−0.105045
$719$	26.6745	0.994791	0.497395	−	0.867524i	$-0.334290\pi$
$719$	26.6745	0.994791	0.497395	+	0.867524i	$0.334290\pi$
$720$	−9.92006	−0.369699
$721$	−1.27734	−0.0475706
$722$	1.85550	0.0690547
$723$	28.9814	1.07783
$724$	−39.4483	−1.46609
$725$	−9.58263	−0.355890
$726$	−2.79500	−0.103732
$727$	42.1822	1.56445	0.782225	−	0.622996i	$-0.214085\pi$
$727$	42.1822	1.56445	0.782225	+	0.622996i	$0.214085\pi$
$728$	−0.559455	−0.0207348
$729$	1.00000	0.0370370
$730$	22.9105	0.847955
$731$	10.0873	0.373092
$732$	19.4314	0.718206
$733$	1.84120	0.0680061	0.0340031	−	0.999422i	$-0.489174\pi$
$733$	1.84120	0.0680061	0.0340031	+	0.999422i	$0.489174\pi$
$734$	−3.31875	−0.122497
$735$	−24.7777	−0.913940
$736$	19.2662	0.710163
$737$	27.9118	1.02815
$738$	2.08413	0.0767177
$739$	−15.4533	−0.568459	−0.284230	−	0.958756i	$-0.591738\pi$
$739$	−15.4533	−0.568459	−0.284230	+	0.958756i	$0.591738\pi$
$740$	−17.1408	−0.630109
$741$	19.2839	0.708411
$742$	−0.181840	−0.00667556
$743$	−44.6296	−1.63730	−0.818650	−	0.574292i	$-0.805277\pi$
$743$	−44.6296	−1.63730	−0.818650	+	0.574292i	$0.805277\pi$
$744$	−2.30593	−0.0845394
$745$	7.55553	0.276813
$746$	−16.1855	−0.592594
$747$	3.90566	0.142901
$748$	9.55569	0.349390
$749$	−0.448172	−0.0163759
$750$	4.11643	0.150311
$751$	48.7546	1.77908	0.889540	−	0.456857i	$-0.151025\pi$
$751$	48.7546	1.77908	0.889540	+	0.456857i	$0.151025\pi$
$752$	23.7877	0.867449
$753$	25.3020	0.922056
$754$	−2.31820	−0.0844240
$755$	42.8835	1.56069
$756$	0.144735	0.00526397
$757$	−19.0533	−0.692503	−0.346252	−	0.938142i	$-0.612546\pi$
$757$	−19.0533	−0.692503	−0.346252	+	0.938142i	$0.612546\pi$
$758$	−5.38158	−0.195468
$759$	−16.8741	−0.612492
$760$	−29.3817	−1.06579
$761$	−4.57278	−0.165763	−0.0828816	−	0.996559i	$-0.526412\pi$
$761$	−4.57278	−0.165763	−0.0828816	+	0.996559i	$0.526412\pi$
$762$	2.54751	0.0922867
$763$	−0.166738	−0.00603633
$764$	−22.1335	−0.800763
$765$	4.56115	0.164909
$766$	−12.2967	−0.444298
$767$	5.06625	0.182932
$768$	−1.87918	−0.0678089
$769$	43.4106	1.56543	0.782713	−	0.622382i	$-0.213835\pi$
$769$	43.4106	1.56543	0.782713	+	0.622382i	$0.213835\pi$
$770$	−0.538985	−0.0194237
$771$	14.3829	0.517986
$772$	−29.6545	−1.06729
$773$	−9.24115	−0.332381	−0.166191	−	0.986094i	$-0.553147\pi$
$773$	−9.24115	−0.332381	−0.166191	+	0.986094i	$0.553147\pi$
$774$	−3.56638	−0.128191
$775$	−10.0884	−0.362387
$776$	20.2798	0.728004
$777$	0.217850	0.00781533
$778$	−2.92192	−0.104756
$779$	−21.9964	−0.788103
$780$	−25.4986	−0.912997
$781$	−44.3811	−1.58808
$782$	−2.38821	−0.0854021
$783$	1.26877	0.0453423
$784$	−19.5812	−0.699329
$785$	3.70499	0.132237
$786$	0.878252	0.0313262
$787$	28.9403	1.03161	0.515805	−	0.856706i	$-0.327493\pi$
$787$	28.9403	1.03161	0.515805	+	0.856706i	$0.327493\pi$
$788$	−11.0050	−0.392036
$789$	−11.9178	−0.424285
$790$	15.4339	0.549114
$791$	−0.230423	−0.00819289
$792$	−7.14723	−0.253966
$793$	43.5084	1.54503
$794$	−2.58092	−0.0915934
$795$	−17.5333	−0.621843
$796$	−11.0816	−0.392778
$797$	24.7971	0.878358	0.439179	−	0.898400i	$-0.355269\pi$
$797$	24.7971	0.878358	0.439179	+	0.898400i	$0.355269\pi$
$798$	0.176514	0.00624854
$799$	−10.9374	−0.386936
$800$	35.7018	1.26225
$801$	0.470731	0.0166325
$802$	−2.92045	−0.103125
$803$	−58.8197	−2.07570
$804$	12.0869	0.426271
$805$	−1.16576	−0.0410876
$806$	−2.44056	−0.0859652
$807$	8.57041	0.301693
$808$	8.76706	0.308424
$809$	−3.33378	−0.117209	−0.0586047	−	0.998281i	$-0.518665\pi$
$809$	−3.33378	−0.117209	−0.0586047	+	0.998281i	$0.518665\pi$
$810$	−1.61260	−0.0566610
$811$	17.9893	0.631691	0.315845	−	0.948811i	$-0.397712\pi$
$811$	17.9893	0.631691	0.315845	+	0.948811i	$0.397712\pi$
$812$	0.183636	0.00644438
$813$	23.5256	0.825079
$814$	−5.08507	−0.178232
$815$	−63.3900	−2.22046
$816$	3.60456	0.126185
$817$	37.6404	1.31687
$818$	−15.1608	−0.530085
$819$	0.324073	0.0113240
$820$	29.0853	1.01570
$821$	−39.5349	−1.37978	−0.689889	−	0.723916i	$-0.742341\pi$
$821$	−39.5349	−1.37978	−0.689889	+	0.723916i	$0.742341\pi$
$822$	−8.07064	−0.281496
$823$	54.2506	1.89106	0.945529	−	0.325539i	$-0.105546\pi$
$823$	54.2506	1.89106	0.945529	+	0.325539i	$0.105546\pi$
$824$	27.3146	0.951548
$825$	−31.2691	−1.08865
$826$	0.0463737	0.00161355
$827$	21.2331	0.738347	0.369174	−	0.929360i	$-0.379641\pi$
$827$	21.2331	0.738347	0.369174	+	0.929360i	$0.379641\pi$
$828$	−7.30713	−0.253940
$829$	−15.1398	−0.525828	−0.262914	−	0.964819i	$-0.584683\pi$
$829$	−15.1398	−0.525828	−0.262914	+	0.964819i	$0.584683\pi$
$830$	−6.29827	−0.218616
$831$	−25.0321	−0.868353
$832$	−13.8425	−0.479903
$833$	9.00327	0.311945
$834$	−1.55965	−0.0540062
$835$	77.6733	2.68799
$836$	35.6567	1.23321
$837$	1.33574	0.0461701
$838$	−15.2677	−0.527413
$839$	−8.13695	−0.280919	−0.140459	−	0.990086i	$-0.544858\pi$
$839$	−8.13695	−0.280919	−0.140459	+	0.990086i	$0.544858\pi$
$840$	−0.493771	−0.0170367
$841$	−27.3902	−0.944490
$842$	−11.1607	−0.384622
$843$	−27.0697	−0.932329
$844$	28.0203	0.964499
$845$	−11.0346	−0.379600
$846$	3.86692	0.132947
$847$	0.495743	0.0170339
$848$	−13.8562	−0.475822
$849$	−7.42654	−0.254878
$850$	−4.42553	−0.151795
$851$	−10.9984	−0.377021
$852$	−19.2187	−0.658421
$853$	28.0424	0.960154	0.480077	−	0.877226i	$-0.340609\pi$
$853$	28.0424	0.960154	0.480077	+	0.877226i	$0.340609\pi$
$854$	0.398252	0.0136279
$855$	17.0198	0.582065
$856$	9.58370	0.327564
$857$	−31.4085	−1.07289	−0.536447	−	0.843934i	$-0.680234\pi$
$857$	−31.4085	−1.07289	−0.536447	+	0.843934i	$0.680234\pi$
$858$	−7.56453	−0.258249
$859$	2.54996	0.0870034	0.0435017	−	0.999053i	$-0.486149\pi$
$859$	2.54996	0.0870034	0.0435017	+	0.999053i	$0.486149\pi$
$860$	−49.7710	−1.69718
$861$	−0.369658	−0.0125979
$862$	16.5183	0.562616
$863$	49.1718	1.67383	0.836914	−	0.547334i	$-0.184357\pi$
$863$	49.1718	1.67383	0.836914	+	0.547334i	$0.184357\pi$
$864$	−4.72705	−0.160817
$865$	31.0071	1.05427
$866$	−17.1830	−0.583904
$867$	15.3427	0.521064
$868$	0.193329	0.00656203
$869$	−39.6245	−1.34417
$870$	−2.04603	−0.0693668
$871$	27.0634	0.917007
$872$	3.56553	0.120744
$873$	−11.7474	−0.397589
$874$	−8.91152	−0.301437
$875$	−0.730123	−0.0246827
$876$	−25.4711	−0.860589
$877$	27.8456	0.940279	0.470140	−	0.882592i	$-0.344204\pi$
$877$	27.8456	0.940279	0.470140	+	0.882592i	$0.344204\pi$
$878$	−7.83351	−0.264368
$879$	20.7262	0.699077
$880$	−41.0704	−1.38448
$881$	−50.7571	−1.71005	−0.855024	−	0.518588i	$-0.826458\pi$
$881$	−50.7571	−1.71005	−0.855024	+	0.518588i	$0.826458\pi$
$882$	−3.18311	−0.107181
$883$	−9.45824	−0.318295	−0.159148	−	0.987255i	$-0.550875\pi$
$883$	−9.45824	−0.318295	−0.159148	+	0.987255i	$0.550875\pi$
$884$	9.26521	0.311623
$885$	4.47143	0.150305
$886$	9.31075	0.312801
$887$	−43.3314	−1.45493	−0.727463	−	0.686147i	$-0.759301\pi$
$887$	−43.3314	−1.45493	−0.727463	+	0.686147i	$0.759301\pi$
$888$	−4.65850	−0.156329
$889$	−0.451848	−0.0151545
$890$	−0.759101	−0.0254451
$891$	4.14014	0.138700
$892$	21.8305	0.730940
$893$	−40.8125	−1.36574
$894$	0.970634	0.0324629
$895$	−63.0103	−2.10620
$896$	−0.889935	−0.0297307
$897$	−16.3612	−0.546284
$898$	−14.2278	−0.474787
$899$	1.69476	0.0565233
$900$	−13.5407	−0.451356
$901$	6.37093	0.212247
$902$	8.62857	0.287300
$903$	0.632562	0.0210503
$904$	4.92735	0.163881
$905$	−77.9572	−2.59139
$906$	5.50910	0.183028
$907$	−23.0753	−0.766201	−0.383101	−	0.923707i	$-0.625144\pi$
$907$	−23.0753	−0.766201	−0.383101	+	0.923707i	$0.625144\pi$
$908$	23.0880	0.766201
$909$	−5.07845	−0.168442
$910$	−0.522600	−0.0173240
$911$	−37.8182	−1.25297	−0.626486	−	0.779433i	$-0.715507\pi$
$911$	−37.8182	−1.25297	−0.626486	+	0.779433i	$0.715507\pi$
$912$	13.4503	0.445385
$913$	16.1700	0.535149
$914$	2.35997	0.0780608
$915$	38.4001	1.26947
$916$	7.17919	0.237207
$917$	−0.155774	−0.00514411
$918$	0.585956	0.0193394
$919$	52.9813	1.74769	0.873846	−	0.486203i	$-0.161618\pi$
$919$	52.9813	1.74769	0.873846	+	0.486203i	$0.161618\pi$
$920$	24.9286	0.821870
$921$	−2.95040	−0.0972191
$922$	0.0174397	0.000574348 0
$923$	−43.0320	−1.41642
$924$	0.599225	0.0197130
$925$	−20.3809	−0.670120
$926$	−1.30464	−0.0428730
$927$	−15.8224	−0.519675
$928$	−5.99756	−0.196879
$929$	−34.4383	−1.12988	−0.564941	−	0.825131i	$-0.691101\pi$
$929$	−34.4383	−1.12988	−0.564941	+	0.825131i	$0.691101\pi$
$930$	−2.15402	−0.0706331
$931$	33.5954	1.10104
$932$	−9.63858	−0.315722
$933$	−9.73374	−0.318668
$934$	4.08563	0.133686
$935$	18.8838	0.617567
$936$	−6.92996	−0.226513
$937$	4.97427	0.162502	0.0812511	−	0.996694i	$-0.474108\pi$
$937$	4.97427	0.162502	0.0812511	+	0.996694i	$0.474108\pi$
$938$	0.247723	0.00808845
$939$	7.87200	0.256893
$940$	53.9653	1.76015
$941$	36.6470	1.19466	0.597329	−	0.801996i	$-0.296229\pi$
$941$	36.6470	1.19466	0.597329	+	0.801996i	$0.296229\pi$
$942$	0.475968	0.0155079
$943$	18.6626	0.607737
$944$	3.53366	0.115011
$945$	0.286024	0.00930436
$946$	−14.7653	−0.480061
$947$	8.10630	0.263419	0.131710	−	0.991288i	$-0.457953\pi$
$947$	8.10630	0.263419	0.131710	+	0.991288i	$0.457953\pi$
$948$	−17.1589	−0.557295
$949$	−57.0316	−1.85132
$950$	−16.5137	−0.535776
$951$	4.28906	0.139082
$952$	0.179417	0.00581494
$953$	−26.8792	−0.870702	−0.435351	−	0.900261i	$-0.643376\pi$
$953$	−26.8792	−0.870702	−0.435351	+	0.900261i	$0.643376\pi$
$954$	−2.25245	−0.0729258
$955$	−43.7399	−1.41539
$956$	0.928048	0.0300152
$957$	5.25290	0.169802
$958$	−2.10412	−0.0679809
$959$	1.43147	0.0462247
$960$	−12.2173	−0.394311
$961$	−29.2158	−0.942445
$962$	−4.93049	−0.158966
$963$	−5.55150	−0.178895
$964$	51.9589	1.67348
$965$	−58.6029	−1.88649
$966$	−0.149761	−0.00481849
$967$	−20.5250	−0.660039	−0.330020	−	0.943974i	$-0.607055\pi$
$967$	−20.5250	−0.660039	−0.330020	+	0.943974i	$0.607055\pi$
$968$	−10.6010	−0.340728
$969$	−6.18433	−0.198669
$970$	18.9439	0.608251
$971$	−15.3718	−0.493304	−0.246652	−	0.969104i	$-0.579330\pi$
$971$	−15.3718	−0.493304	−0.246652	+	0.969104i	$0.579330\pi$
$972$	1.79283	0.0575052
$973$	0.276632	0.00886842
$974$	−9.29350	−0.297783
$975$	−30.3186	−0.970971
$976$	30.3467	0.971373
$977$	7.41018	0.237073	0.118536	−	0.992950i	$-0.462180\pi$
$977$	7.41018	0.237073	0.118536	+	0.992950i	$0.462180\pi$
$978$	−8.14351	−0.260401
$979$	1.94889	0.0622869
$980$	−44.4224	−1.41902
$981$	−2.06538	−0.0659426
$982$	5.21662	0.166469
$983$	17.4618	0.556943	0.278472	−	0.960444i	$-0.410172\pi$
$983$	17.4618	0.556943	0.278472	+	0.960444i	$0.410172\pi$
$984$	7.90474	0.251994
$985$	−21.7479	−0.692945
$986$	0.743446	0.0236762
$987$	−0.685869	−0.0218314
$988$	34.5728	1.09991
$989$	−31.9356	−1.01549
$990$	−6.67639	−0.212190
$991$	9.83636	0.312462	0.156231	−	0.987721i	$-0.450066\pi$
$991$	9.83636	0.312462	0.156231	+	0.987721i	$0.450066\pi$
$992$	−6.31413	−0.200474
$993$	−5.23840	−0.166236
$994$	−0.393891	−0.0124935
$995$	−21.8994	−0.694257
$996$	7.00221	0.221873
$997$	−43.1148	−1.36546	−0.682730	−	0.730671i	$-0.739207\pi$
$997$	−43.1148	−1.36546	−0.682730	+	0.730671i	$0.739207\pi$
$998$	−19.2840	−0.610426
$999$	2.69851	0.0853770

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2031.2.a.e.1.5	✓	7
3.2	odd	2		6093.2.a.i.1.3		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2031.2.a.e.1.5	✓	7	1.1	even	1	trivial
6093.2.a.i.1.3		7	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2031 = 3 \cdot 677 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2031.a (trivial)

\(f(q)\)	\(=\)	\(q+0.455154 q^{2} -1.00000 q^{3} -1.79283 q^{4} -3.54297 q^{5} -0.455154 q^{6} +0.0807299 q^{7} -1.72633 q^{8} +1.00000 q^{9} -1.61260 q^{10} +4.14014 q^{11} +1.79283 q^{12} +4.01429 q^{13} +0.0367446 q^{14} +3.54297 q^{15} +2.79992 q^{16} -1.28738 q^{17} +0.455154 q^{18} -4.80382 q^{19} +6.35197 q^{20} -0.0807299 q^{21} +1.88440 q^{22} +4.07574 q^{23} +1.72633 q^{24} +7.55267 q^{25} +1.82712 q^{26} -1.00000 q^{27} -0.144735 q^{28} -1.26877 q^{29} +1.61260 q^{30} -1.33574 q^{31} +4.72705 q^{32} -4.14014 q^{33} -0.585956 q^{34} -0.286024 q^{35} -1.79283 q^{36} -2.69851 q^{37} -2.18648 q^{38} -4.01429 q^{39} +6.11633 q^{40} +4.57894 q^{41} -0.0367446 q^{42} -7.83553 q^{43} -7.42259 q^{44} -3.54297 q^{45} +1.85509 q^{46} +8.49584 q^{47} -2.79992 q^{48} -6.99348 q^{49} +3.43763 q^{50} +1.28738 q^{51} -7.19695 q^{52} -4.94876 q^{53} -0.455154 q^{54} -14.6684 q^{55} -0.139366 q^{56} +4.80382 q^{57} -0.577488 q^{58} +1.26205 q^{59} -6.35197 q^{60} +10.8384 q^{61} -0.607970 q^{62} +0.0807299 q^{63} -3.44831 q^{64} -14.2225 q^{65} -1.88440 q^{66} +6.74176 q^{67} +2.30806 q^{68} -4.07574 q^{69} -0.130185 q^{70} -10.7197 q^{71} -1.72633 q^{72} -14.2072 q^{73} -1.22824 q^{74} -7.55267 q^{75} +8.61245 q^{76} +0.334233 q^{77} -1.82712 q^{78} -9.57082 q^{79} -9.92006 q^{80} +1.00000 q^{81} +2.08413 q^{82} +3.90566 q^{83} +0.144735 q^{84} +4.56115 q^{85} -3.56638 q^{86} +1.26877 q^{87} -7.14723 q^{88} +0.470731 q^{89} -1.61260 q^{90} +0.324073 q^{91} -7.30713 q^{92} +1.33574 q^{93} +3.86692 q^{94} +17.0198 q^{95} -4.72705 q^{96} -11.7474 q^{97} -3.18311 q^{98} +4.14014 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 2 q^{2} - 7 q^{3} + 2 q^{4} - 10 q^{5} + 2 q^{6} + 8 q^{7} - 3 q^{8} + 7 q^{9} - 8 q^{10} + 10 q^{11} - 2 q^{12} - 3 q^{13} - 3 q^{14} + 10 q^{15} - 12 q^{16} + 8 q^{17} - 2 q^{18} - 5 q^{19} + 4 q^{20}+ \cdots + 10 q^{99}+O(q^{100}) \) 7 * q - 2 * q^2 - 7 * q^3 + 2 * q^4 - 10 * q^5 + 2 * q^6 + 8 * q^7 - 3 * q^8 + 7 * q^9 - 8 * q^10 + 10 * q^11 - 2 * q^12 - 3 * q^13 - 3 * q^14 + 10 * q^15 - 12 * q^16 + 8 * q^17 - 2 * q^18 - 5 * q^19 + 4 * q^20 - 8 * q^21 - q^22 - q^23 + 3 * q^24 - q^25 + 10 * q^26 - 7 * q^27 + 10 * q^28 + 5 * q^29 + 8 * q^30 - 24 * q^31 + 12 * q^32 - 10 * q^33 - q^34 - 4 * q^35 + 2 * q^36 - 26 * q^37 + 11 * q^38 + 3 * q^39 + 15 * q^40 - 12 * q^41 + 3 * q^42 - 26 * q^43 + 16 * q^44 - 10 * q^45 + 16 * q^46 + 2 * q^47 + 12 * q^48 - 23 * q^49 + 23 * q^50 - 8 * q^51 - 27 * q^52 + 16 * q^53 + 2 * q^54 - 26 * q^55 + 4 * q^56 + 5 * q^57 - 14 * q^58 - 17 * q^59 - 4 * q^60 - 10 * q^61 - 3 * q^62 + 8 * q^63 - 15 * q^64 + q^66 + 20 * q^67 + 2 * q^68 + q^69 - 18 * q^70 - 5 * q^71 - 3 * q^72 - 13 * q^73 - 9 * q^74 + q^75 - 14 * q^76 + 17 * q^77 - 10 * q^78 - 32 * q^79 - 6 * q^80 + 7 * q^81 + 11 * q^83 - 10 * q^84 - 11 * q^85 - q^86 - 5 * q^87 - 14 * q^88 - 8 * q^89 - 8 * q^90 - 16 * q^91 - 32 * q^92 + 24 * q^93 - 50 * q^94 + 7 * q^95 - 12 * q^96 - 41 * q^97 + 10 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more