LMFDB - Embedded newform 2009.2.a.u.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2009,2,Mod(1,2009)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2009, 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("2009.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2009 = 7^{2} \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2009.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:	$16.0419457661$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 2 x^{19} - 27 x^{18} + 52 x^{17} + 302 x^{16} - 552 x^{15} - 1820 x^{14} + 3090 x^{13} + \cdots + 49 $ x^20 - 2x^19 - 27x^18 + 52x^17 + 302x^16 - 552x^15 - 1820x^14 + 3090x^13 + 6449x^12 - 9852x^11 - 13797x^10 + 18080x^9 + 17721x^8 - 18446x^7 - 13352x^6 + 9524x^5 + 5690x^4 - 1972x^3 - 1179x^2 + 42*x + 49
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			1.3
Root			$2.27905$ of defining polynomial
Character	$\chi$	$=$	2009.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.27905 q^{2} -0.713649 q^{3} +3.19408 q^{4} -3.36532 q^{5} +1.62644 q^{6} -2.72136 q^{8} -2.49071 q^{9} +O(q^{10})$	\(q-2.27905 q^{2} -0.713649 q^{3} +3.19408 q^{4} -3.36532 q^{5} +1.62644 q^{6} -2.72136 q^{8} -2.49071 q^{9} +7.66975 q^{10} -2.45502 q^{11} -2.27945 q^{12} +1.53126 q^{13} +2.40166 q^{15} -0.186032 q^{16} -2.89641 q^{17} +5.67645 q^{18} +1.01605 q^{19} -10.7491 q^{20} +5.59512 q^{22} -4.93143 q^{23} +1.94209 q^{24} +6.32540 q^{25} -3.48983 q^{26} +3.91843 q^{27} -8.76332 q^{29} -5.47350 q^{30} -5.91571 q^{31} +5.86670 q^{32} +1.75202 q^{33} +6.60107 q^{34} -7.95550 q^{36} -8.45756 q^{37} -2.31564 q^{38} -1.09278 q^{39} +9.15826 q^{40} +1.00000 q^{41} -5.59089 q^{43} -7.84152 q^{44} +8.38203 q^{45} +11.2390 q^{46} +5.14085 q^{47} +0.132761 q^{48} -14.4159 q^{50} +2.06702 q^{51} +4.89097 q^{52} -7.81636 q^{53} -8.93031 q^{54} +8.26194 q^{55} -0.725106 q^{57} +19.9721 q^{58} -6.63151 q^{59} +7.67108 q^{60} +9.20226 q^{61} +13.4822 q^{62} -12.9984 q^{64} -5.15320 q^{65} -3.99295 q^{66} -9.54919 q^{67} -9.25136 q^{68} +3.51931 q^{69} -7.18144 q^{71} +6.77811 q^{72} -2.87408 q^{73} +19.2752 q^{74} -4.51412 q^{75} +3.24535 q^{76} +2.49051 q^{78} +3.80950 q^{79} +0.626057 q^{80} +4.67573 q^{81} -2.27905 q^{82} -14.4939 q^{83} +9.74737 q^{85} +12.7419 q^{86} +6.25393 q^{87} +6.68099 q^{88} -7.01102 q^{89} -19.1031 q^{90} -15.7513 q^{92} +4.22173 q^{93} -11.7163 q^{94} -3.41935 q^{95} -4.18676 q^{96} -5.72017 q^{97} +6.11473 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 2 q^{2} + 8 q^{3} + 18 q^{4} + 8 q^{5} + 12 q^{6} - 6 q^{8} + 16 q^{9}+O(q^{10}) $ 20 * q - 2 * q^2 + 8 * q^3 + 18 * q^4 + 8 * q^5 + 12 * q^6 - 6 * q^8 + 16 * q^9	$ 20 q - 2 q^{2} + 8 q^{3} + 18 q^{4} + 8 q^{5} + 12 q^{6} - 6 q^{8} + 16 q^{9} + 16 q^{10} + 6 q^{12} + 12 q^{13} + 14 q^{16} + 8 q^{17} - 18 q^{18} + 36 q^{19} + 24 q^{20} - 8 q^{22} - 12 q^{23} + 36 q^{24} + 20 q^{25} - 22 q^{26} + 32 q^{27} + 4 q^{29} + 28 q^{30} + 80 q^{31} + 6 q^{32} - 12 q^{33} + 48 q^{34} + 26 q^{36} + 4 q^{37} + 12 q^{38} - 28 q^{39} - 4 q^{40} + 20 q^{41} - 20 q^{44} + 40 q^{45} + 8 q^{46} + 32 q^{47} + 16 q^{48} + 6 q^{50} - 20 q^{51} + 36 q^{52} + 4 q^{53} - 50 q^{54} + 64 q^{55} - 4 q^{57} + 32 q^{59} + 20 q^{60} + 44 q^{61} - 8 q^{62} - 30 q^{64} - 8 q^{65} + 32 q^{66} - 4 q^{67} - 48 q^{68} - 24 q^{69} + 8 q^{71} - 8 q^{72} + 48 q^{73} - 38 q^{74} + 24 q^{75} + 84 q^{76} + 30 q^{78} - 4 q^{79} + 56 q^{80} - 2 q^{82} + 8 q^{83} - 12 q^{85} - 24 q^{86} + 40 q^{87} - 48 q^{88} + 20 q^{89} + 48 q^{90} - 50 q^{92} + 48 q^{93} + 26 q^{94} + 20 q^{95} + 70 q^{96} + 8 q^{97} - 8 q^{99}+O(q^{100}) $ 20 * q - 2 * q^2 + 8 * q^3 + 18 * q^4 + 8 * q^5 + 12 * q^6 - 6 * q^8 + 16 * q^9 + 16 * q^10 + 6 * q^12 + 12 * q^13 + 14 * q^16 + 8 * q^17 - 18 * q^18 + 36 * q^19 + 24 * q^20 - 8 * q^22 - 12 * q^23 + 36 * q^24 + 20 * q^25 - 22 * q^26 + 32 * q^27 + 4 * q^29 + 28 * q^30 + 80 * q^31 + 6 * q^32 - 12 * q^33 + 48 * q^34 + 26 * q^36 + 4 * q^37 + 12 * q^38 - 28 * q^39 - 4 * q^40 + 20 * q^41 - 20 * q^44 + 40 * q^45 + 8 * q^46 + 32 * q^47 + 16 * q^48 + 6 * q^50 - 20 * q^51 + 36 * q^52 + 4 * q^53 - 50 * q^54 + 64 * q^55 - 4 * q^57 + 32 * q^59 + 20 * q^60 + 44 * q^61 - 8 * q^62 - 30 * q^64 - 8 * q^65 + 32 * q^66 - 4 * q^67 - 48 * q^68 - 24 * q^69 + 8 * q^71 - 8 * q^72 + 48 * q^73 - 38 * q^74 + 24 * q^75 + 84 * q^76 + 30 * q^78 - 4 * q^79 + 56 * q^80 - 2 * q^82 + 8 * q^83 - 12 * q^85 - 24 * q^86 + 40 * q^87 - 48 * q^88 + 20 * q^89 + 48 * q^90 - 50 * q^92 + 48 * q^93 + 26 * q^94 + 20 * q^95 + 70 * q^96 + 8 * q^97 - 8 * 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 $

$2$	−2.27905	−1.61153	−0.805766	−	0.592234i	$-0.798246\pi$
$2$	−2.27905	−1.61153	−0.805766	+	0.592234i	$0.798246\pi$
$3$	−0.713649	−0.412025	−0.206013	−	0.978549i	$-0.566049\pi$
$3$	−0.713649	−0.412025	−0.206013	+	0.978549i	$0.566049\pi$
$4$	3.19408	1.59704
$5$	−3.36532	−1.50502	−0.752509	−	0.658582i	$-0.771157\pi$
$5$	−3.36532	−1.50502	−0.752509	+	0.658582i	$0.771157\pi$
$6$	1.62644	0.663992
$7$	0	0
$7$	0	0
$8$	−2.72136	−0.962146
$9$	−2.49071	−0.830235
$10$	7.66975	2.42539
$11$	−2.45502	−0.740216	−0.370108	−	0.928989i	$-0.620679\pi$
$11$	−2.45502	−0.740216	−0.370108	+	0.928989i	$0.620679\pi$
$12$	−2.27945	−0.658020
$13$	1.53126	0.424696	0.212348	−	0.977194i	$-0.431889\pi$
$13$	1.53126	0.424696	0.212348	+	0.977194i	$0.431889\pi$
$14$	0	0
$15$	2.40166	0.620106
$16$	−0.186032	−0.0465079
$17$	−2.89641	−0.702483	−0.351242	−	0.936285i	$-0.614240\pi$
$17$	−2.89641	−0.702483	−0.351242	+	0.936285i	$0.614240\pi$
$18$	5.67645	1.33795
$19$	1.01605	0.233099	0.116549	−	0.993185i	$-0.462817\pi$
$19$	1.01605	0.233099	0.116549	+	0.993185i	$0.462817\pi$
$20$	−10.7491	−2.40357
$21$	0	0
$22$	5.59512	1.19288
$23$	−4.93143	−1.02827	−0.514137	−	0.857708i	$-0.671888\pi$
$23$	−4.93143	−1.02827	−0.514137	+	0.857708i	$0.671888\pi$
$24$	1.94209	0.396428
$25$	6.32540	1.26508
$26$	−3.48983	−0.684412
$27$	3.91843	0.754103
$28$	0	0
$29$	−8.76332	−1.62731	−0.813654	−	0.581349i	$-0.802525\pi$
$29$	−8.76332	−1.62731	−0.813654	+	0.581349i	$0.802525\pi$
$30$	−5.47350	−0.999320
$31$	−5.91571	−1.06249	−0.531246	−	0.847218i	$-0.678276\pi$
$31$	−5.91571	−1.06249	−0.531246	+	0.847218i	$0.678276\pi$
$32$	5.86670	1.03710
$33$	1.75202	0.304988
$34$	6.60107	1.13207
$35$	0	0
$36$	−7.95550	−1.32592
$37$	−8.45756	−1.39041	−0.695207	−	0.718809i	$-0.744687\pi$
$37$	−8.45756	−1.39041	−0.695207	+	0.718809i	$0.744687\pi$
$38$	−2.31564	−0.375646
$39$	−1.09278	−0.174985
$40$	9.15826	1.44805
$41$	1.00000	0.156174
$41$	1.00000	0.156174
$42$	0	0
$43$	−5.59089	−0.852603	−0.426302	−	0.904581i	$-0.640184\pi$
$43$	−5.59089	−0.852603	−0.426302	+	0.904581i	$0.640184\pi$
$44$	−7.84152	−1.18215
$45$	8.38203	1.24952
$46$	11.2390	1.65710
$47$	5.14085	0.749871	0.374935	−	0.927051i	$-0.377665\pi$
$47$	5.14085	0.749871	0.374935	+	0.927051i	$0.377665\pi$
$48$	0.132761	0.0191624
$49$	0	0
$50$	−14.4159	−2.03872
$51$	2.06702	0.289441
$52$	4.89097	0.678256
$53$	−7.81636	−1.07366	−0.536830	−	0.843691i	$-0.680378\pi$
$53$	−7.81636	−1.07366	−0.536830	+	0.843691i	$0.680378\pi$
$54$	−8.93031	−1.21526
$55$	8.26194	1.11404
$56$	0	0
$57$	−0.725106	−0.0960426
$58$	19.9721	2.62246
$59$	−6.63151	−0.863349	−0.431674	−	0.902029i	$-0.642077\pi$
$59$	−6.63151	−0.863349	−0.431674	+	0.902029i	$0.642077\pi$
$60$	7.67108	0.990332
$61$	9.20226	1.17823	0.589114	−	0.808050i	$-0.299477\pi$
$61$	9.20226	1.17823	0.589114	+	0.808050i	$0.299477\pi$
$62$	13.4822	1.71224
$63$	0	0
$64$	−12.9984	−1.62480
$65$	−5.15320	−0.639176
$66$	−3.99295	−0.491498
$67$	−9.54919	−1.16662	−0.583310	−	0.812250i	$-0.698243\pi$
$67$	−9.54919	−1.16662	−0.583310	+	0.812250i	$0.698243\pi$
$68$	−9.25136	−1.12189
$69$	3.51931	0.423675
$70$	0	0
$71$	−7.18144	−0.852280	−0.426140	−	0.904657i	$-0.640127\pi$
$71$	−7.18144	−0.852280	−0.426140	+	0.904657i	$0.640127\pi$
$72$	6.77811	0.798808
$73$	−2.87408	−0.336385	−0.168193	−	0.985754i	$-0.553793\pi$
$73$	−2.87408	−0.336385	−0.168193	+	0.985754i	$0.553793\pi$
$74$	19.2752	2.24070
$75$	−4.51412	−0.521245
$76$	3.24535	0.372268
$77$	0	0
$78$	2.49051	0.281995
$79$	3.80950	0.428602	0.214301	−	0.976768i	$-0.431253\pi$
$79$	3.80950	0.428602	0.214301	+	0.976768i	$0.431253\pi$
$80$	0.626057	0.0699953
$81$	4.67573	0.519526
$82$	−2.27905	−0.251679
$83$	−14.4939	−1.59091	−0.795457	−	0.606010i	$-0.792769\pi$
$83$	−14.4939	−1.59091	−0.795457	+	0.606010i	$0.792769\pi$
$84$	0	0
$85$	9.74737	1.05725
$86$	12.7419	1.37400
$87$	6.25393	0.670492
$88$	6.68099	0.712196
$89$	−7.01102	−0.743167	−0.371583	−	0.928400i	$-0.621185\pi$
$89$	−7.01102	−0.743167	−0.371583	+	0.928400i	$0.621185\pi$
$90$	−19.1031	−2.01364
$91$	0	0
$92$	−15.7513	−1.64219
$93$	4.22173	0.437774
$94$	−11.7163	−1.20844
$95$	−3.41935	−0.350818
$96$	−4.18676	−0.427309
$97$	−5.72017	−0.580795	−0.290398	−	0.956906i	$-0.593788\pi$
$97$	−5.72017	−0.580795	−0.290398	+	0.956906i	$0.593788\pi$
$98$	0	0
$99$	6.11473	0.614554
$100$	20.2038	2.02038
$101$	3.87347	0.385425	0.192712	−	0.981255i	$-0.438272\pi$
$101$	3.87347	0.385425	0.192712	+	0.981255i	$0.438272\pi$
$102$	−4.71085	−0.466443
$103$	9.53019	0.939037	0.469519	−	0.882923i	$-0.344427\pi$
$103$	9.53019	0.939037	0.469519	+	0.882923i	$0.344427\pi$
$104$	−4.16712	−0.408620
$105$	0	0
$106$	17.8139	1.73024
$107$	3.75965	0.363459	0.181730	−	0.983349i	$-0.441830\pi$
$107$	3.75965	0.363459	0.181730	+	0.983349i	$0.441830\pi$
$108$	12.5158	1.20433
$109$	14.7130	1.40925	0.704624	−	0.709581i	$-0.251116\pi$
$109$	14.7130	1.40925	0.704624	+	0.709581i	$0.251116\pi$
$110$	−18.8294	−1.79531
$111$	6.03573	0.572886
$112$	0	0
$113$	3.37807	0.317782	0.158891	−	0.987296i	$-0.449208\pi$
$113$	3.37807	0.317782	0.158891	+	0.987296i	$0.449208\pi$
$114$	1.65255	0.154776
$115$	16.5958	1.54757
$116$	−27.9907	−2.59887
$117$	−3.81393	−0.352598
$118$	15.1135	1.39131
$119$	0	0
$120$	−6.53578	−0.596632
$121$	−4.97288	−0.452080
$122$	−20.9724	−1.89875
$123$	−0.713649	−0.0643475
$124$	−18.8952	−1.69684
$125$	−4.46042	−0.398952
$126$	0	0
$127$	−13.5562	−1.20292	−0.601460	−	0.798903i	$-0.705414\pi$
$127$	−13.5562	−1.20292	−0.601460	+	0.798903i	$0.705414\pi$
$128$	17.8907	1.58133
$129$	3.98993	0.351294
$130$	11.7444	1.03005
$131$	−6.65385	−0.581350	−0.290675	−	0.956822i	$-0.593880\pi$
$131$	−6.65385	−0.581350	−0.290675	+	0.956822i	$0.593880\pi$
$132$	5.59609	0.487077
$133$	0	0
$134$	21.7631	1.88004
$135$	−13.1868	−1.13494
$136$	7.88218	0.675892
$137$	10.8534	0.927272	0.463636	−	0.886026i	$-0.346545\pi$
$137$	10.8534	0.927272	0.463636	+	0.886026i	$0.346545\pi$
$138$	−8.02068	−0.682765
$139$	14.3569	1.21774	0.608870	−	0.793270i	$-0.291623\pi$
$139$	14.3569	1.21774	0.608870	+	0.793270i	$0.291623\pi$
$140$	0	0
$141$	−3.66876	−0.308966
$142$	16.3669	1.37348
$143$	−3.75928	−0.314367
$144$	0.463350	0.0386125
$145$	29.4914	2.44913
$146$	6.55017	0.542096
$147$	0	0
$148$	−27.0141	−2.22054
$149$	5.84740	0.479038	0.239519	−	0.970892i	$-0.423010\pi$
$149$	5.84740	0.479038	0.239519	+	0.970892i	$0.423010\pi$
$150$	10.2879	0.840004
$151$	−14.6829	−1.19488	−0.597439	−	0.801914i	$-0.703815\pi$
$151$	−14.6829	−1.19488	−0.597439	+	0.801914i	$0.703815\pi$
$152$	−2.76505	−0.224275
$153$	7.21411	0.583226
$154$	0	0
$155$	19.9083	1.59907
$156$	−3.49044	−0.279458
$157$	−22.5122	−1.79667	−0.898334	−	0.439313i	$-0.855222\pi$
$157$	−22.5122	−1.79667	−0.898334	+	0.439313i	$0.855222\pi$
$158$	−8.68204	−0.690706
$159$	5.57813	0.442375
$160$	−19.7433	−1.56085
$161$	0	0
$162$	−10.6562	−0.837233
$163$	−2.27657	−0.178315	−0.0891573	−	0.996018i	$-0.528417\pi$
$163$	−2.27657	−0.178315	−0.0891573	+	0.996018i	$0.528417\pi$
$164$	3.19408	0.249415
$165$	−5.89612	−0.459012
$166$	33.0324	2.56381
$167$	12.1188	0.937784	0.468892	−	0.883255i	$-0.344653\pi$
$167$	12.1188	0.937784	0.468892	+	0.883255i	$0.344653\pi$
$168$	0	0
$169$	−10.6552	−0.819633
$170$	−22.2148	−1.70379
$171$	−2.53069	−0.193527
$172$	−17.8577	−1.36164
$173$	−2.00653	−0.152554	−0.0762769	−	0.997087i	$-0.524303\pi$
$173$	−2.00653	−0.152554	−0.0762769	+	0.997087i	$0.524303\pi$
$174$	−14.2530	−1.08052
$175$	0	0
$176$	0.456712	0.0344259
$177$	4.73257	0.355721
$178$	15.9785	1.19764
$179$	−0.684662	−0.0511741	−0.0255870	−	0.999673i	$-0.508145\pi$
$179$	−0.684662	−0.0511741	−0.0255870	+	0.999673i	$0.508145\pi$
$180$	26.7728	1.99553
$181$	−19.2095	−1.42783	−0.713917	−	0.700230i	$-0.753081\pi$
$181$	−19.2095	−1.42783	−0.713917	+	0.700230i	$0.753081\pi$
$182$	0	0
$183$	−6.56718	−0.485460
$184$	13.4202	0.989349
$185$	28.4624	2.09260
$186$	−9.62155	−0.705486
$187$	7.11075	0.519990
$188$	16.4203	1.19757
$189$	0	0
$190$	7.79288	0.565355
$191$	−10.9931	−0.795435	−0.397718	−	0.917508i	$-0.630198\pi$
$191$	−10.9931	−0.795435	−0.397718	+	0.917508i	$0.630198\pi$
$192$	9.27632	0.669460
$193$	7.79901	0.561385	0.280693	−	0.959798i	$-0.409436\pi$
$193$	7.79901	0.561385	0.280693	+	0.959798i	$0.409436\pi$
$194$	13.0366	0.935970
$195$	3.67757	0.263356
$196$	0	0
$197$	24.8636	1.77146	0.885729	−	0.464203i	$-0.153659\pi$
$197$	24.8636	1.77146	0.885729	+	0.464203i	$0.153659\pi$
$198$	−13.9358	−0.990374
$199$	27.3911	1.94170	0.970852	−	0.239680i	$-0.0770425\pi$
$199$	27.3911	1.94170	0.970852	+	0.239680i	$0.0770425\pi$
$200$	−17.2137	−1.21719
$201$	6.81476	0.480676
$202$	−8.82785	−0.621125
$203$	0	0
$204$	6.60222	0.462248
$205$	−3.36532	−0.235044
$206$	−21.7198	−1.51329
$207$	12.2827	0.853709
$208$	−0.284864	−0.0197517
$209$	−2.49443	−0.172544
$210$	0	0
$211$	27.6154	1.90113	0.950563	−	0.310532i	$-0.100507\pi$
$211$	27.6154	1.90113	0.950563	+	0.310532i	$0.100507\pi$
$212$	−24.9660	−1.71467
$213$	5.12502	0.351161
$214$	−8.56844	−0.585726
$215$	18.8152	1.28318
$216$	−10.6635	−0.725557
$217$	0	0
$218$	−33.5316	−2.27105
$219$	2.05108	0.138599
$220$	26.3893	1.77916
$221$	−4.43517	−0.298342
$222$	−13.7557	−0.923224
$223$	17.7287	1.18720	0.593601	−	0.804759i	$-0.297706\pi$
$223$	17.7287	1.18720	0.593601	+	0.804759i	$0.297706\pi$
$224$	0	0
$225$	−15.7547	−1.05031
$226$	−7.69880	−0.512117
$227$	−10.6144	−0.704504	−0.352252	−	0.935905i	$-0.614584\pi$
$227$	−10.6144	−0.704504	−0.352252	+	0.935905i	$0.614584\pi$
$228$	−2.31604	−0.153384
$229$	26.6860	1.76346	0.881732	−	0.471751i	$-0.156378\pi$
$229$	26.6860	1.76346	0.881732	+	0.471751i	$0.156378\pi$
$230$	−37.8228	−2.49396
$231$	0	0
$232$	23.8482	1.56571
$233$	12.2730	0.804033	0.402017	−	0.915632i	$-0.368309\pi$
$233$	12.2730	0.804033	0.402017	+	0.915632i	$0.368309\pi$
$234$	8.69214	0.568223
$235$	−17.3006	−1.12857
$236$	−21.1815	−1.37880
$237$	−2.71864	−0.176595
$238$	0	0
$239$	17.2353	1.11486	0.557429	−	0.830225i	$-0.311788\pi$
$239$	17.2353	1.11486	0.557429	+	0.830225i	$0.311788\pi$
$240$	−0.446785	−0.0288398
$241$	10.5451	0.679272	0.339636	−	0.940557i	$-0.389696\pi$
$241$	10.5451	0.679272	0.339636	+	0.940557i	$0.389696\pi$
$242$	11.3334	0.728541
$243$	−15.0921	−0.968161
$244$	29.3927	1.88168
$245$	0	0
$246$	1.62644	0.103698
$247$	1.55585	0.0989962
$248$	16.0988	1.02227
$249$	10.3436	0.655496
$250$	10.1655	0.642924
$251$	21.3070	1.34489	0.672444	−	0.740148i	$-0.265245\pi$
$251$	21.3070	1.34489	0.672444	+	0.740148i	$0.265245\pi$
$252$	0	0
$253$	12.1068	0.761145
$254$	30.8953	1.93854
$255$	−6.95619	−0.435614
$256$	−14.7770	−0.923562
$257$	2.51146	0.156661	0.0783304	−	0.996927i	$-0.475041\pi$
$257$	2.51146	0.156661	0.0783304	+	0.996927i	$0.475041\pi$
$258$	−9.09326	−0.566122
$259$	0	0
$260$	−16.4597	−1.02079
$261$	21.8269	1.35105
$262$	15.1645	0.936864
$263$	−3.18543	−0.196422	−0.0982111	−	0.995166i	$-0.531312\pi$
$263$	−3.18543	−0.196422	−0.0982111	+	0.995166i	$0.531312\pi$
$264$	−4.76788	−0.293443
$265$	26.3046	1.61588
$266$	0	0
$267$	5.00341	0.306203
$268$	−30.5008	−1.86313
$269$	24.8332	1.51411	0.757054	−	0.653353i	$-0.226638\pi$
$269$	24.8332	1.51411	0.757054	+	0.653353i	$0.226638\pi$
$270$	30.0534	1.82899
$271$	29.4212	1.78721	0.893604	−	0.448857i	$-0.148169\pi$
$271$	29.4212	1.78721	0.893604	+	0.448857i	$0.148169\pi$
$272$	0.538825	0.0326710
$273$	0	0
$274$	−24.7355	−1.49433
$275$	−15.5290	−0.936434
$276$	11.2409	0.676624
$277$	−31.3205	−1.88187	−0.940934	−	0.338589i	$-0.890050\pi$
$277$	−31.3205	−1.88187	−0.940934	+	0.338589i	$0.890050\pi$
$278$	−32.7202	−1.96243
$279$	14.7343	0.882118
$280$	0	0
$281$	24.3540	1.45284	0.726419	−	0.687253i	$-0.241183\pi$
$281$	24.3540	1.45284	0.726419	+	0.687253i	$0.241183\pi$
$282$	8.36130	0.497908
$283$	3.38251	0.201069	0.100535	−	0.994934i	$-0.467945\pi$
$283$	3.38251	0.201069	0.100535	+	0.994934i	$0.467945\pi$
$284$	−22.9381	−1.36112
$285$	2.44021	0.144546
$286$	8.56760	0.506613
$287$	0	0
$288$	−14.6122	−0.861033
$289$	−8.61079	−0.506517
$290$	−67.2125	−3.94685
$291$	4.08219	0.239302
$292$	−9.18002	−0.537220
$293$	−13.1430	−0.767824	−0.383912	−	0.923370i	$-0.625423\pi$
$293$	−13.1430	−0.767824	−0.383912	+	0.923370i	$0.625423\pi$
$294$	0	0
$295$	22.3172	1.29936
$296$	23.0161	1.33778
$297$	−9.61984	−0.558199
$298$	−13.3265	−0.771985
$299$	−7.55131	−0.436704
$300$	−14.4184	−0.832448
$301$	0	0
$302$	33.4631	1.92558
$303$	−2.76430	−0.158805
$304$	−0.189018	−0.0108409
$305$	−30.9686	−1.77326
$306$	−16.4413	−0.939888
$307$	−15.5074	−0.885055	−0.442528	−	0.896755i	$-0.645918\pi$
$307$	−15.5074	−0.885055	−0.442528	+	0.896755i	$0.645918\pi$
$308$	0	0
$309$	−6.80121	−0.386907
$310$	−45.3720	−2.57695
$311$	−28.8024	−1.63323	−0.816616	−	0.577181i	$-0.804153\pi$
$311$	−28.8024	−1.63323	−0.816616	+	0.577181i	$0.804153\pi$
$312$	2.97386	0.168362
$313$	−24.1095	−1.36275	−0.681376	−	0.731933i	$-0.738618\pi$
$313$	−24.1095	−1.36275	−0.681376	+	0.731933i	$0.738618\pi$
$314$	51.3064	2.89539
$315$	0	0
$316$	12.1678	0.684493
$317$	−1.19150	−0.0669211	−0.0334605	−	0.999440i	$-0.510653\pi$
$317$	−1.19150	−0.0669211	−0.0334605	+	0.999440i	$0.510653\pi$
$318$	−12.7128	−0.712901
$319$	21.5141	1.20456
$320$	43.7440	2.44536
$321$	−2.68307	−0.149754
$322$	0	0
$323$	−2.94291	−0.163748
$324$	14.9346	0.829702
$325$	9.68586	0.537275
$326$	5.18841	0.287360
$327$	−10.4999	−0.580646
$328$	−2.72136	−0.150262
$329$	0	0
$330$	13.4376	0.739713
$331$	−15.6294	−0.859070	−0.429535	−	0.903050i	$-0.641323\pi$
$331$	−15.6294	−0.859070	−0.429535	+	0.903050i	$0.641323\pi$
$332$	−46.2947	−2.54075
$333$	21.0653	1.15437
$334$	−27.6195	−1.51127
$335$	32.1361	1.75578
$336$	0	0
$337$	−4.99767	−0.272240	−0.136120	−	0.990692i	$-0.543463\pi$
$337$	−4.99767	−0.272240	−0.136120	+	0.990692i	$0.543463\pi$
$338$	24.2838	1.32087
$339$	−2.41076	−0.130934
$340$	31.1338	1.68847
$341$	14.5232	0.786474
$342$	5.76758	0.311875
$343$	0	0
$344$	15.2148	0.820329
$345$	−11.8436	−0.637638
$346$	4.57299	0.245845
$347$	−22.2683	−1.19543	−0.597713	−	0.801710i	$-0.703924\pi$
$347$	−22.2683	−1.19543	−0.597713	+	0.801710i	$0.703924\pi$
$348$	19.9755	1.07080
$349$	16.2228	0.868386	0.434193	−	0.900820i	$-0.357034\pi$
$349$	16.2228	0.868386	0.434193	+	0.900820i	$0.357034\pi$
$350$	0	0
$351$	6.00016	0.320265
$352$	−14.4029	−0.767675
$353$	−5.59903	−0.298006	−0.149003	−	0.988837i	$-0.547606\pi$
$353$	−5.59903	−0.298006	−0.149003	+	0.988837i	$0.547606\pi$
$354$	−10.7858	−0.573257
$355$	24.1679	1.28270
$356$	−22.3937	−1.18687
$357$	0	0
$358$	1.56038	0.0824687
$359$	−9.55643	−0.504369	−0.252184	−	0.967679i	$-0.581149\pi$
$359$	−9.55643	−0.504369	−0.252184	+	0.967679i	$0.581149\pi$
$360$	−22.8105	−1.20222
$361$	−17.9676	−0.945665
$362$	43.7795	2.30100
$363$	3.54888	0.186268
$364$	0	0
$365$	9.67220	0.506266
$366$	14.9669	0.782334
$367$	−21.4264	−1.11845	−0.559225	−	0.829016i	$-0.688901\pi$
$367$	−21.4264	−1.11845	−0.559225	+	0.829016i	$0.688901\pi$
$368$	0.917402	0.0478229
$369$	−2.49071	−0.129661
$370$	−64.8673	−3.37229
$371$	0	0
$372$	13.4845	0.699141
$373$	18.4408	0.954827	0.477413	−	0.878679i	$-0.341574\pi$
$373$	18.4408	0.954827	0.477413	+	0.878679i	$0.341574\pi$
$374$	−16.2058	−0.837980
$375$	3.18317	0.164378
$376$	−13.9901	−0.721485
$377$	−13.4190	−0.691111
$378$	0	0
$379$	−7.15078	−0.367311	−0.183655	−	0.982991i	$-0.558793\pi$
$379$	−7.15078	−0.367311	−0.183655	+	0.982991i	$0.558793\pi$
$380$	−10.9217	−0.560270
$381$	9.67437	0.495633
$382$	25.0539	1.28187
$383$	−11.7979	−0.602844	−0.301422	−	0.953491i	$-0.597461\pi$
$383$	−11.7979	−0.602844	−0.301422	+	0.953491i	$0.597461\pi$
$384$	−12.7677	−0.651548
$385$	0	0
$386$	−17.7743	−0.904691
$387$	13.9253	0.707861
$388$	−18.2707	−0.927552
$389$	−9.03725	−0.458207	−0.229103	−	0.973402i	$-0.573579\pi$
$389$	−9.03725	−0.458207	−0.229103	+	0.973402i	$0.573579\pi$
$390$	−8.38138	−0.424408
$391$	14.2834	0.722345
$392$	0	0
$393$	4.74851	0.239531
$394$	−56.6654	−2.85476
$395$	−12.8202	−0.645054
$396$	19.5309	0.981466
$397$	−32.0399	−1.60804	−0.804019	−	0.594604i	$-0.797309\pi$
$397$	−32.0399	−1.60804	−0.804019	+	0.594604i	$0.797309\pi$
$398$	−62.4257	−3.12912
$399$	0	0
$400$	−1.17673	−0.0588363
$401$	31.5818	1.57712	0.788559	−	0.614959i	$-0.210828\pi$
$401$	31.5818	1.57712	0.788559	+	0.614959i	$0.210828\pi$
$402$	−15.5312	−0.774626
$403$	−9.05851	−0.451236
$404$	12.3722	0.615538
$405$	−15.7354	−0.781896
$406$	0	0
$407$	20.7635	1.02921
$408$	−5.62511	−0.278484
$409$	28.1870	1.39376	0.696880	−	0.717188i	$-0.254571\pi$
$409$	28.1870	1.39376	0.696880	+	0.717188i	$0.254571\pi$
$410$	7.66975	0.378782
$411$	−7.74554	−0.382059
$412$	30.4401	1.49968
$413$	0	0
$414$	−27.9930	−1.37578
$415$	48.7767	2.39435
$416$	8.98346	0.440450
$417$	−10.2458	−0.501740
$418$	5.68494	0.278060
$419$	−9.22594	−0.450717	−0.225358	−	0.974276i	$-0.572355\pi$
$419$	−9.22594	−0.450717	−0.225358	+	0.974276i	$0.572355\pi$
$420$	0	0
$421$	−11.6830	−0.569394	−0.284697	−	0.958617i	$-0.591893\pi$
$421$	−11.6830	−0.569394	−0.284697	+	0.958617i	$0.591893\pi$
$422$	−62.9370	−3.06373
$423$	−12.8044	−0.622569
$424$	21.2711	1.03302
$425$	−18.3210	−0.888698
$426$	−11.6802	−0.565907
$427$	0	0
$428$	12.0086	0.580458
$429$	2.68281	0.129527
$430$	−42.8807	−2.06789
$431$	5.44772	0.262407	0.131204	−	0.991355i	$-0.458116\pi$
$431$	5.44772	0.262407	0.131204	+	0.991355i	$0.458116\pi$
$432$	−0.728953	−0.0350718
$433$	1.24191	0.0596825	0.0298412	−	0.999555i	$-0.490500\pi$
$433$	1.24191	0.0596825	0.0298412	+	0.999555i	$0.490500\pi$
$434$	0	0
$435$	−21.0465	−1.00910
$436$	46.9944	2.25062
$437$	−5.01060	−0.239689
$438$	−4.67452	−0.223357
$439$	16.0697	0.766967	0.383483	−	0.923548i	$-0.374724\pi$
$439$	16.0697	0.766967	0.383483	+	0.923548i	$0.374724\pi$
$440$	−22.4837	−1.07187
$441$	0	0
$442$	10.1080	0.480788
$443$	−41.7940	−1.98569	−0.992846	−	0.119399i	$-0.961903\pi$
$443$	−41.7940	−1.98569	−0.992846	+	0.119399i	$0.961903\pi$
$444$	19.2786	0.914920
$445$	23.5944	1.11848
$446$	−40.4046	−1.91322
$447$	−4.17299	−0.197376
$448$	0	0
$449$	−26.4245	−1.24705	−0.623524	−	0.781804i	$-0.714300\pi$
$449$	−26.4245	−1.24705	−0.623524	+	0.781804i	$0.714300\pi$
$450$	35.9058	1.69262
$451$	−2.45502	−0.115602
$452$	10.7898	0.507511
$453$	10.4784	0.492320
$454$	24.1908	1.13533
$455$	0	0
$456$	1.97327	0.0924070
$457$	−8.57751	−0.401239	−0.200619	−	0.979669i	$-0.564295\pi$
$457$	−8.57751	−0.401239	−0.200619	+	0.979669i	$0.564295\pi$
$458$	−60.8189	−2.84188
$459$	−11.3494	−0.529745
$460$	53.0084	2.47153
$461$	−32.9161	−1.53305	−0.766527	−	0.642212i	$-0.778017\pi$
$461$	−32.9161	−1.53305	−0.766527	+	0.642212i	$0.778017\pi$
$462$	0	0
$463$	−37.2140	−1.72948	−0.864742	−	0.502217i	$-0.832518\pi$
$463$	−37.2140	−1.72948	−0.864742	+	0.502217i	$0.832518\pi$
$464$	1.63026	0.0756827
$465$	−14.2075	−0.658857
$466$	−27.9709	−1.29573
$467$	−2.89944	−0.134170	−0.0670851	−	0.997747i	$-0.521370\pi$
$467$	−2.89944	−0.134170	−0.0670851	+	0.997747i	$0.521370\pi$
$468$	−12.1820	−0.563112
$469$	0	0
$470$	39.4291	1.81873
$471$	16.0658	0.740272
$472$	18.0467	0.830668
$473$	13.7258	0.631111
$474$	6.19592	0.284588
$475$	6.42695	0.294889
$476$	0	0
$477$	19.4682	0.891390
$478$	−39.2801	−1.79663
$479$	−18.5894	−0.849371	−0.424685	−	0.905341i	$-0.639615\pi$
$479$	−18.5894	−0.849371	−0.424685	+	0.905341i	$0.639615\pi$
$480$	14.0898	0.643108
$481$	−12.9508	−0.590504
$482$	−24.0329	−1.09467
$483$	0	0
$484$	−15.8837	−0.721988
$485$	19.2502	0.874108
$486$	34.3957	1.56022
$487$	20.1238	0.911897	0.455948	−	0.890006i	$-0.349300\pi$
$487$	20.1238	0.911897	0.455948	+	0.890006i	$0.349300\pi$
$488$	−25.0427	−1.13363
$489$	1.62467	0.0734701
$490$	0	0
$491$	−11.6278	−0.524755	−0.262378	−	0.964965i	$-0.584507\pi$
$491$	−11.6278	−0.524755	−0.262378	+	0.964965i	$0.584507\pi$
$492$	−2.27945	−0.102765
$493$	25.3822	1.14316
$494$	−3.54586	−0.159536
$495$	−20.5781	−0.924915
$496$	1.10051	0.0494143
$497$	0	0
$498$	−23.5735	−1.05635
$499$	−11.2116	−0.501901	−0.250950	−	0.968000i	$-0.580743\pi$
$499$	−11.2116	−0.501901	−0.250950	+	0.968000i	$0.580743\pi$
$500$	−14.2469	−0.637141
$501$	−8.64860	−0.386391
$502$	−48.5598	−2.16733
$503$	14.7310	0.656823	0.328412	−	0.944535i	$-0.393487\pi$
$503$	14.7310	0.656823	0.328412	+	0.944535i	$0.393487\pi$
$504$	0	0
$505$	−13.0355	−0.580072
$506$	−27.5919	−1.22661
$507$	7.60409	0.337709
$508$	−43.2996	−1.92111
$509$	7.10443	0.314898	0.157449	−	0.987527i	$-0.449673\pi$
$509$	7.10443	0.314898	0.157449	+	0.987527i	$0.449673\pi$
$510$	15.8535	0.702006
$511$	0	0
$512$	−2.10391	−0.0929806
$513$	3.98134	0.175781
$514$	−5.72376	−0.252464
$515$	−32.0722	−1.41327
$516$	12.7442	0.561030
$517$	−12.6209	−0.555067
$518$	0	0
$519$	1.43196	0.0628560
$520$	14.0237	0.614980
$521$	6.47016	0.283463	0.141731	−	0.989905i	$-0.454733\pi$
$521$	6.47016	0.283463	0.141731	+	0.989905i	$0.454733\pi$
$522$	−49.7445	−2.17726
$523$	−8.56870	−0.374683	−0.187341	−	0.982295i	$-0.559987\pi$
$523$	−8.56870	−0.374683	−0.187341	+	0.982295i	$0.559987\pi$
$524$	−21.2529	−0.928438
$525$	0	0
$526$	7.25976	0.316541
$527$	17.1343	0.746383
$528$	−0.325932	−0.0141844
$529$	1.31897	0.0573464
$530$	−59.9495	−2.60404
$531$	16.5171	0.716783
$532$	0	0
$533$	1.53126	0.0663264
$534$	−11.4030	−0.493457
$535$	−12.6524	−0.547013
$536$	25.9868	1.12246
$537$	0.488608	0.0210850
$538$	−56.5962	−2.44003
$539$	0	0
$540$	−42.1196	−1.81254
$541$	25.7060	1.10519	0.552593	−	0.833451i	$-0.313638\pi$
$541$	25.7060	1.10519	0.552593	+	0.833451i	$0.313638\pi$
$542$	−67.0523	−2.88014
$543$	13.7089	0.588304
$544$	−16.9924	−0.728542
$545$	−49.5140	−2.12094
$546$	0	0
$547$	35.0020	1.49658	0.748290	−	0.663372i	$-0.230875\pi$
$547$	35.0020	1.49658	0.748290	+	0.663372i	$0.230875\pi$
$548$	34.6667	1.48089
$549$	−22.9201	−0.978207
$550$	35.3914	1.50909
$551$	−8.90401	−0.379324
$552$	−9.57730	−0.407637
$553$	0	0
$554$	71.3811	3.03269
$555$	−20.3122	−0.862204
$556$	45.8572	1.94478
$557$	−7.31991	−0.310155	−0.155077	−	0.987902i	$-0.549563\pi$
$557$	−7.31991	−0.310155	−0.155077	+	0.987902i	$0.549563\pi$
$558$	−33.5802	−1.42156
$559$	−8.56113	−0.362097
$560$	0	0
$561$	−5.07458	−0.214249
$562$	−55.5040	−2.34129
$563$	38.1709	1.60871	0.804356	−	0.594148i	$-0.202511\pi$
$563$	38.1709	1.60871	0.804356	+	0.594148i	$0.202511\pi$
$564$	−11.7183	−0.493430
$565$	−11.3683	−0.478268
$566$	−7.70892	−0.324030
$567$	0	0
$568$	19.5433	0.820018
$569$	−17.1259	−0.717957	−0.358979	−	0.933346i	$-0.616875\pi$
$569$	−17.1259	−0.717957	−0.358979	+	0.933346i	$0.616875\pi$
$570$	−5.56138	−0.232940
$571$	26.8558	1.12388	0.561941	−	0.827177i	$-0.310055\pi$
$571$	26.8558	1.12388	0.561941	+	0.827177i	$0.310055\pi$
$572$	−12.0074	−0.502056
$573$	7.84523	0.327739
$574$	0	0
$575$	−31.1933	−1.30085
$576$	32.3753	1.34897
$577$	11.4829	0.478040	0.239020	−	0.971015i	$-0.423174\pi$
$577$	11.4829	0.478040	0.239020	+	0.971015i	$0.423174\pi$
$578$	19.6244	0.816269
$579$	−5.56575	−0.231305
$580$	94.1978	3.91135
$581$	0	0
$582$	−9.30352	−0.385643
$583$	19.1893	0.794740
$584$	7.82140	0.323652
$585$	12.8351	0.530666
$586$	29.9536	1.23737
$587$	−22.5452	−0.930539	−0.465270	−	0.885169i	$-0.654043\pi$
$587$	−22.5452	−0.930539	−0.465270	+	0.885169i	$0.654043\pi$
$588$	0	0
$589$	−6.01068	−0.247666
$590$	−50.8620	−2.09395
$591$	−17.7439	−0.729885
$592$	1.57337	0.0646653
$593$	−19.8548	−0.815341	−0.407670	−	0.913129i	$-0.633659\pi$
$593$	−19.8548	−0.815341	−0.407670	+	0.913129i	$0.633659\pi$
$594$	21.9241	0.899557
$595$	0	0
$596$	18.6770	0.765042
$597$	−19.5476	−0.800031
$598$	17.2098	0.703762
$599$	1.35804	0.0554882	0.0277441	−	0.999615i	$-0.491168\pi$
$599$	1.35804	0.0554882	0.0277441	+	0.999615i	$0.491168\pi$
$600$	12.2845	0.501514
$601$	−41.5758	−1.69591	−0.847957	−	0.530065i	$-0.822167\pi$
$601$	−41.5758	−1.69591	−0.847957	+	0.530065i	$0.822167\pi$
$602$	0	0
$603$	23.7842	0.968568
$604$	−46.8983	−1.90827
$605$	16.7353	0.680388
$606$	6.29998	0.255919
$607$	−19.2495	−0.781315	−0.390657	−	0.920536i	$-0.627752\pi$
$607$	−19.2495	−0.781315	−0.390657	+	0.920536i	$0.627752\pi$
$608$	5.96088	0.241746
$609$	0	0
$610$	70.5790	2.85766
$611$	7.87200	0.318467
$612$	23.0424	0.931435
$613$	10.1832	0.411295	0.205647	−	0.978626i	$-0.434070\pi$
$613$	10.1832	0.411295	0.205647	+	0.978626i	$0.434070\pi$
$614$	35.3422	1.42630
$615$	2.40166	0.0968442
$616$	0	0
$617$	44.2234	1.78037	0.890184	−	0.455601i	$-0.150576\pi$
$617$	44.2234	1.78037	0.890184	+	0.455601i	$0.150576\pi$
$618$	15.5003	0.623513
$619$	30.6954	1.23375	0.616876	−	0.787061i	$-0.288398\pi$
$619$	30.6954	1.23375	0.616876	+	0.787061i	$0.288398\pi$
$620$	63.5885	2.55378
$621$	−19.3235	−0.775424
$622$	65.6421	2.63201
$623$	0	0
$624$	0.203293	0.00813821
$625$	−16.6163	−0.664651
$626$	54.9469	2.19612
$627$	1.78015	0.0710923
$628$	−71.9056	−2.86935
$629$	24.4966	0.976743
$630$	0	0
$631$	−36.5648	−1.45562	−0.727812	−	0.685777i	$-0.759463\pi$
$631$	−36.5648	−1.45562	−0.727812	+	0.685777i	$0.759463\pi$
$632$	−10.3670	−0.412378
$633$	−19.7077	−0.783312
$634$	2.71548	0.107846
$635$	45.6210	1.81042
$636$	17.8170	0.706489
$637$	0	0
$638$	−49.0318	−1.94119
$639$	17.8869	0.707593
$640$	−60.2081	−2.37993
$641$	−26.9317	−1.06374	−0.531868	−	0.846827i	$-0.678510\pi$
$641$	−26.9317	−1.06374	−0.531868	+	0.846827i	$0.678510\pi$
$642$	6.11485	0.241334
$643$	−28.1112	−1.10860	−0.554299	−	0.832318i	$-0.687014\pi$
$643$	−28.1112	−1.10860	−0.554299	+	0.832318i	$0.687014\pi$
$644$	0	0
$645$	−13.4274	−0.528704
$646$	6.70705	0.263885
$647$	39.1596	1.53952	0.769762	−	0.638331i	$-0.220375\pi$
$647$	39.1596	1.53952	0.769762	+	0.638331i	$0.220375\pi$
$648$	−12.7244	−0.499860
$649$	16.2805	0.639065
$650$	−22.0746	−0.865836
$651$	0	0
$652$	−7.27153	−0.284775
$653$	11.1357	0.435772	0.217886	−	0.975974i	$-0.430084\pi$
$653$	11.1357	0.435772	0.217886	+	0.975974i	$0.430084\pi$
$654$	23.9298	0.935730
$655$	22.3924	0.874942
$656$	−0.186032	−0.00726332
$657$	7.15848	0.279279
$658$	0	0
$659$	21.2444	0.827564	0.413782	−	0.910376i	$-0.364208\pi$
$659$	21.2444	0.827564	0.413782	+	0.910376i	$0.364208\pi$
$660$	−18.8327	−0.733060
$661$	−38.4871	−1.49697	−0.748487	−	0.663150i	$-0.769219\pi$
$661$	−38.4871	−1.49697	−0.748487	+	0.663150i	$0.769219\pi$
$662$	35.6202	1.38442
$663$	3.16515	0.122924
$664$	39.4432	1.53069
$665$	0	0
$666$	−48.0089	−1.86031
$667$	43.2157	1.67332
$668$	38.7085	1.49768
$669$	−12.6521	−0.489157
$670$	−73.2399	−2.82950
$671$	−22.5917	−0.872144
$672$	0	0
$673$	−36.0936	−1.39131	−0.695653	−	0.718378i	$-0.744885\pi$
$673$	−36.0936	−1.39131	−0.695653	+	0.718378i	$0.744885\pi$
$674$	11.3899	0.438724
$675$	24.7857	0.954001
$676$	−34.0336	−1.30899
$677$	−34.3742	−1.32111	−0.660554	−	0.750778i	$-0.729679\pi$
$677$	−34.3742	−1.32111	−0.660554	+	0.750778i	$0.729679\pi$
$678$	5.49424	0.211005
$679$	0	0
$680$	−26.5261	−1.01723
$681$	7.57496	0.290273
$682$	−33.0991	−1.26743
$683$	−21.7526	−0.832339	−0.416169	−	0.909287i	$-0.636628\pi$
$683$	−21.7526	−0.832339	−0.416169	+	0.909287i	$0.636628\pi$
$684$	−8.08322	−0.309070
$685$	−36.5253	−1.39556
$686$	0	0
$687$	−19.0445	−0.726591
$688$	1.04008	0.0396528
$689$	−11.9689	−0.455979
$690$	26.9922	1.02757
$691$	11.5953	0.441107	0.220553	−	0.975375i	$-0.429214\pi$
$691$	11.5953	0.441107	0.220553	+	0.975375i	$0.429214\pi$
$692$	−6.40902	−0.243634
$693$	0	0
$694$	50.7506	1.92647
$695$	−48.3158	−1.83272
$696$	−17.0192	−0.645111
$697$	−2.89641	−0.109709
$698$	−36.9726	−1.39943
$699$	−8.75863	−0.331282
$700$	0	0
$701$	−1.35962	−0.0513520	−0.0256760	−	0.999670i	$-0.508174\pi$
$701$	−1.35962	−0.0513520	−0.0256760	+	0.999670i	$0.508174\pi$
$702$	−13.6747	−0.516117
$703$	−8.59334	−0.324104
$704$	31.9114	1.20271
$705$	12.3466	0.464999
$706$	12.7605	0.480247
$707$	0	0
$708$	15.1162	0.568101
$709$	−23.7244	−0.890987	−0.445494	−	0.895285i	$-0.646972\pi$
$709$	−23.7244	−0.890987	−0.445494	+	0.895285i	$0.646972\pi$
$710$	−55.0798	−2.06711
$711$	−9.48833	−0.355840
$712$	19.0795	0.715035
$713$	29.1729	1.09253
$714$	0	0
$715$	12.6512	0.473128
$716$	−2.18686	−0.0817269
$717$	−12.2999	−0.459350
$718$	21.7796	0.812807
$719$	49.9170	1.86159	0.930794	−	0.365543i	$-0.119117\pi$
$719$	49.9170	1.86159	0.930794	+	0.365543i	$0.119117\pi$
$720$	−1.55932	−0.0581126
$721$	0	0
$722$	40.9492	1.52397
$723$	−7.52552	−0.279877
$724$	−61.3567	−2.28031
$725$	−55.4316	−2.05868
$726$	−8.08809	−0.300177
$727$	2.84194	0.105402	0.0527009	−	0.998610i	$-0.483217\pi$
$727$	2.84194	0.105402	0.0527009	+	0.998610i	$0.483217\pi$
$728$	0	0
$729$	−3.25672	−0.120619
$730$	−22.0435	−0.815865
$731$	16.1935	0.598940
$732$	−20.9761	−0.775298
$733$	−3.89743	−0.143955	−0.0719775	−	0.997406i	$-0.522931\pi$
$733$	−3.89743	−0.143955	−0.0719775	+	0.997406i	$0.522931\pi$
$734$	48.8319	1.80242
$735$	0	0
$736$	−28.9312	−1.06642
$737$	23.4435	0.863551
$738$	5.67645	0.208953
$739$	10.0006	0.367877	0.183939	−	0.982938i	$-0.441115\pi$
$739$	10.0006	0.367877	0.183939	+	0.982938i	$0.441115\pi$
$740$	90.9112	3.34196
$741$	−1.11033	−0.0407889
$742$	0	0
$743$	27.8643	1.02224	0.511121	−	0.859509i	$-0.329231\pi$
$743$	27.8643	1.02224	0.511121	+	0.859509i	$0.329231\pi$
$744$	−11.4889	−0.421202
$745$	−19.6784	−0.720961
$746$	−42.0275	−1.53873
$747$	36.1001	1.32083
$748$	22.7123	0.830443
$749$	0	0
$750$	−7.25461	−0.264901
$751$	−33.9019	−1.23710	−0.618549	−	0.785746i	$-0.712279\pi$
$751$	−33.9019	−1.23710	−0.618549	+	0.785746i	$0.712279\pi$
$752$	−0.956362	−0.0348749
$753$	−15.2057	−0.554127
$754$	30.5825	1.11375
$755$	49.4127	1.79831
$756$	0	0
$757$	−8.82791	−0.320856	−0.160428	−	0.987048i	$-0.551287\pi$
$757$	−8.82791	−0.320856	−0.160428	+	0.987048i	$0.551287\pi$
$758$	16.2970	0.591933
$759$	−8.63997	−0.313611
$760$	9.30529	0.337538
$761$	20.8344	0.755245	0.377622	−	0.925960i	$-0.376742\pi$
$761$	20.8344	0.755245	0.377622	+	0.925960i	$0.376742\pi$
$762$	−22.0484	−0.798729
$763$	0	0
$764$	−35.1129	−1.27034
$765$	−24.2778	−0.877767
$766$	26.8880	0.971503
$767$	−10.1546	−0.366661
$768$	10.5456	0.380531
$769$	27.4131	0.988542	0.494271	−	0.869308i	$-0.335435\pi$
$769$	27.4131	0.988542	0.494271	+	0.869308i	$0.335435\pi$
$770$	0	0
$771$	−1.79230	−0.0645482
$772$	24.9106	0.896553
$773$	26.5264	0.954089	0.477045	−	0.878879i	$-0.341708\pi$
$773$	26.5264	0.954089	0.477045	+	0.878879i	$0.341708\pi$
$774$	−31.7364	−1.14074
$775$	−37.4192	−1.34414
$776$	15.5666	0.558810
$777$	0	0
$778$	20.5964	0.738415
$779$	1.01605	0.0364039
$780$	11.7464	0.420590
$781$	17.6306	0.630872
$782$	−32.5527	−1.16408
$783$	−34.3385	−1.22716
$784$	0	0
$785$	75.7608	2.70402
$786$	−10.8221	−0.386012
$787$	−9.35752	−0.333560	−0.166780	−	0.985994i	$-0.553337\pi$
$787$	−9.35752	−0.333560	−0.166780	+	0.985994i	$0.553337\pi$
$788$	79.4162	2.82908
$789$	2.27328	0.0809309
$790$	29.2179	1.03953
$791$	0	0
$792$	−16.6404	−0.591291
$793$	14.0911	0.500389
$794$	73.0206	2.59141
$795$	−18.7722	−0.665782
$796$	87.4893	3.10097
$797$	−21.2446	−0.752523	−0.376261	−	0.926514i	$-0.622791\pi$
$797$	−21.2446	−0.752523	−0.376261	+	0.926514i	$0.622791\pi$
$798$	0	0
$799$	−14.8900	−0.526772
$800$	37.1092	1.31201
$801$	17.4624	0.617003
$802$	−71.9765	−2.54158
$803$	7.05592	0.248998
$804$	21.7669	0.767658
$805$	0	0
$806$	20.6448	0.727182
$807$	−17.7222	−0.623850
$808$	−10.5411	−0.370835
$809$	1.80730	0.0635413	0.0317706	−	0.999495i	$-0.489885\pi$
$809$	1.80730	0.0635413	0.0317706	+	0.999495i	$0.489885\pi$
$810$	35.8617	1.26005
$811$	11.0400	0.387667	0.193834	−	0.981034i	$-0.437908\pi$
$811$	11.0400	0.387667	0.193834	+	0.981034i	$0.437908\pi$
$812$	0	0
$813$	−20.9964	−0.736375
$814$	−47.3210	−1.65860
$815$	7.66139	0.268367
$816$	−0.384531	−0.0134613
$817$	−5.68065	−0.198741
$818$	−64.2397	−2.24609
$819$	0	0
$820$	−10.7491	−0.375375
$821$	−8.40206	−0.293234	−0.146617	−	0.989193i	$-0.546838\pi$
$821$	−8.40206	−0.293234	−0.146617	+	0.989193i	$0.546838\pi$
$822$	17.6525	0.615701
$823$	40.8634	1.42441	0.712204	−	0.701973i	$-0.247697\pi$
$823$	40.8634	1.42441	0.712204	+	0.701973i	$0.247697\pi$
$824$	−25.9351	−0.903491
$825$	11.0822	0.385834
$826$	0	0
$827$	−28.9216	−1.00570	−0.502851	−	0.864373i	$-0.667716\pi$
$827$	−28.9216	−1.00570	−0.502851	+	0.864373i	$0.667716\pi$
$828$	39.2320	1.36341
$829$	−38.3728	−1.33274	−0.666371	−	0.745620i	$-0.732153\pi$
$829$	−38.3728	−1.33274	−0.666371	+	0.745620i	$0.732153\pi$
$830$	−111.165	−3.85858
$831$	22.3519	0.775377
$832$	−19.9040	−0.690048
$833$	0	0
$834$	23.3507	0.808570
$835$	−40.7838	−1.41138
$836$	−7.96741	−0.275559
$837$	−23.1803	−0.801229
$838$	21.0264	0.726345
$839$	19.5341	0.674392	0.337196	−	0.941434i	$-0.390521\pi$
$839$	19.5341	0.674392	0.337196	+	0.941434i	$0.390521\pi$
$840$	0	0
$841$	47.7958	1.64813
$842$	26.6261	0.917598
$843$	−17.3802	−0.598605
$844$	88.2058	3.03617
$845$	35.8583	1.23356
$846$	29.1818	1.00329
$847$	0	0
$848$	1.45409	0.0499337
$849$	−2.41392	−0.0828457
$850$	41.7545	1.43217
$851$	41.7078	1.42973
$852$	16.3697	0.560817
$853$	−49.0610	−1.67982	−0.839908	−	0.542728i	$-0.817391\pi$
$853$	−49.0610	−1.67982	−0.839908	+	0.542728i	$0.817391\pi$
$854$	0	0
$855$	8.51660	0.291262
$856$	−10.2314	−0.349701
$857$	24.8455	0.848707	0.424354	−	0.905497i	$-0.360501\pi$
$857$	24.8455	0.848707	0.424354	+	0.905497i	$0.360501\pi$
$858$	−6.11426	−0.208737
$859$	21.6220	0.737733	0.368867	−	0.929482i	$-0.379746\pi$
$859$	21.6220	0.737733	0.368867	+	0.929482i	$0.379746\pi$
$860$	60.0971	2.04929
$861$	0	0
$862$	−12.4156	−0.422878
$863$	−25.2074	−0.858069	−0.429034	−	0.903288i	$-0.641146\pi$
$863$	−25.2074	−0.858069	−0.429034	+	0.903288i	$0.641146\pi$
$864$	22.9883	0.782077
$865$	6.75263	0.229596
$866$	−2.83038	−0.0961803
$867$	6.14508	0.208698
$868$	0	0
$869$	−9.35239	−0.317258
$870$	47.9661	1.62620
$871$	−14.6223	−0.495459
$872$	−40.0393	−1.35590
$873$	14.2473	0.482197
$874$	11.4194	0.386267
$875$	0	0
$876$	6.55131	0.221348
$877$	37.7456	1.27458	0.637289	−	0.770625i	$-0.280056\pi$
$877$	37.7456	1.27458	0.637289	+	0.770625i	$0.280056\pi$
$878$	−36.6238	−1.23599
$879$	9.37950	0.316363
$880$	−1.53698	−0.0518117
$881$	−34.3013	−1.15564	−0.577820	−	0.816164i	$-0.696096\pi$
$881$	−34.3013	−1.15564	−0.577820	+	0.816164i	$0.696096\pi$
$882$	0	0
$883$	−45.5946	−1.53438	−0.767191	−	0.641419i	$-0.778346\pi$
$883$	−45.5946	−1.53438	−0.767191	+	0.641419i	$0.778346\pi$
$884$	−14.1663	−0.476463
$885$	−15.9266	−0.535367
$886$	95.2507	3.20001
$887$	50.9019	1.70912	0.854559	−	0.519355i	$-0.173828\pi$
$887$	50.9019	1.70912	0.854559	+	0.519355i	$0.173828\pi$
$888$	−16.4254	−0.551200
$889$	0	0
$890$	−53.7728	−1.80247
$891$	−11.4790	−0.384562
$892$	56.6268	1.89601
$893$	5.22339	0.174794
$894$	9.51046	0.318077
$895$	2.30411	0.0770179
$896$	0	0
$897$	5.38898	0.179933
$898$	60.2227	2.00966
$899$	51.8412	1.72900
$900$	−50.3218	−1.67739
$901$	22.6394	0.754228
$902$	5.59512	0.186297
$903$	0	0
$904$	−9.19295	−0.305753
$905$	64.6463	2.14892
$906$	−23.8809	−0.793389
$907$	11.9891	0.398091	0.199046	−	0.979990i	$-0.436216\pi$
$907$	11.9891	0.398091	0.199046	+	0.979990i	$0.436216\pi$
$908$	−33.9033	−1.12512
$909$	−9.64768	−0.319993
$910$	0	0
$911$	42.1872	1.39772	0.698862	−	0.715257i	$-0.253690\pi$
$911$	42.1872	1.39772	0.698862	+	0.715257i	$0.253690\pi$
$912$	0.134893	0.00446674
$913$	35.5828	1.17762
$914$	19.5486	0.646610
$915$	22.1007	0.730626
$916$	85.2372	2.81632
$917$	0	0
$918$	25.8659	0.853701
$919$	56.9678	1.87919	0.939597	−	0.342283i	$-0.111200\pi$
$919$	56.9678	1.87919	0.939597	+	0.342283i	$0.111200\pi$
$920$	−45.1633	−1.48899
$921$	11.0669	0.364665
$922$	75.0174	2.47057
$923$	−10.9967	−0.361960
$924$	0	0
$925$	−53.4975	−1.75899
$926$	84.8127	2.78712
$927$	−23.7369	−0.779622
$928$	−51.4117	−1.68767
$929$	−1.84286	−0.0604623	−0.0302312	−	0.999543i	$-0.509624\pi$
$929$	−1.84286	−0.0604623	−0.0302312	+	0.999543i	$0.509624\pi$
$930$	32.3796	1.06177
$931$	0	0
$932$	39.2010	1.28407
$933$	20.5548	0.672933
$934$	6.60798	0.216220
$935$	−23.9300	−0.782594
$936$	10.3791	0.339251
$937$	−13.5370	−0.442235	−0.221117	−	0.975247i	$-0.570970\pi$
$937$	−13.5370	−0.442235	−0.221117	+	0.975247i	$0.570970\pi$
$938$	0	0
$939$	17.2057	0.561488
$940$	−55.2596	−1.80237
$941$	6.56336	0.213960	0.106980	−	0.994261i	$-0.465882\pi$
$941$	6.56336	0.213960	0.106980	+	0.994261i	$0.465882\pi$
$942$	−36.6148	−1.19297
$943$	−4.93143	−0.160589
$944$	1.23367	0.0401526
$945$	0	0
$946$	−31.2817	−1.01706
$947$	55.5125	1.80391	0.901956	−	0.431827i	$-0.142131\pi$
$947$	55.5125	1.80391	0.901956	+	0.431827i	$0.142131\pi$
$948$	−8.68355	−0.282028
$949$	−4.40097	−0.142862
$950$	−14.6474	−0.475223
$951$	0.850309	0.0275732
$952$	0	0
$953$	19.1455	0.620184	0.310092	−	0.950706i	$-0.399640\pi$
$953$	19.1455	0.620184	0.310092	+	0.950706i	$0.399640\pi$
$954$	−44.3691	−1.43650
$955$	36.9955	1.19714
$956$	55.0508	1.78047
$957$	−15.3535	−0.496309
$958$	42.3662	1.36879
$959$	0	0
$960$	−31.2178	−1.00755
$961$	3.99558	0.128890
$962$	29.5154	0.951616
$963$	−9.36418	−0.301757
$964$	33.6820	1.08482
$965$	−26.2462	−0.844895
$966$	0	0
$967$	−34.0203	−1.09402	−0.547010	−	0.837126i	$-0.684234\pi$
$967$	−34.0203	−1.09402	−0.547010	+	0.837126i	$0.684234\pi$
$968$	13.5330	0.434967
$969$	2.10020	0.0674683
$970$	−43.8722	−1.40865
$971$	−38.0921	−1.22243	−0.611217	−	0.791463i	$-0.709320\pi$
$971$	−38.0921	−1.22243	−0.611217	+	0.791463i	$0.709320\pi$
$972$	−48.2054	−1.54619
$973$	0	0
$974$	−45.8632	−1.46955
$975$	−6.91230	−0.221371
$976$	−1.71191	−0.0547970
$977$	−12.1362	−0.388271	−0.194135	−	0.980975i	$-0.562190\pi$
$977$	−12.1362	−0.388271	−0.194135	+	0.980975i	$0.562190\pi$
$978$	−3.70270	−0.118399
$979$	17.2122	0.550104
$980$	0	0
$981$	−36.6457	−1.17001
$982$	26.5004	0.845660
$983$	29.8678	0.952635	0.476318	−	0.879273i	$-0.341971\pi$
$983$	29.8678	0.952635	0.476318	+	0.879273i	$0.341971\pi$
$984$	1.94209	0.0619117
$985$	−83.6740	−2.66608
$986$	−57.8473	−1.84223
$987$	0	0
$988$	4.96949	0.158101
$989$	27.5711	0.876710
$990$	46.8985	1.49053
$991$	3.28825	0.104455	0.0522274	−	0.998635i	$-0.483368\pi$
$991$	3.28825	0.104455	0.0522274	+	0.998635i	$0.483368\pi$
$992$	−34.7056	−1.10191
$993$	11.1539	0.353959
$994$	0	0
$995$	−92.1799	−2.92230
$996$	33.0381	1.04685
$997$	−34.0598	−1.07868	−0.539342	−	0.842087i	$-0.681327\pi$
$997$	−34.0598	−1.07868	−0.539342	+	0.842087i	$0.681327\pi$
$998$	25.5519	0.808830
$999$	−33.1404	−1.04852

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2009.2.a.u.1.3	yes	20
7.6	odd	2		2009.2.a.t.1.3	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2009.2.a.t.1.3	✓	20	7.6	odd	2
2009.2.a.u.1.3	yes	20	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 2009 = 7^{2} \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2009.a (trivial)

\(f(q)\)	\(=\)	\(q-2.27905 q^{2} -0.713649 q^{3} +3.19408 q^{4} -3.36532 q^{5} +1.62644 q^{6} -2.72136 q^{8} -2.49071 q^{9} +O(q^{10})\)	\(q-2.27905 q^{2} -0.713649 q^{3} +3.19408 q^{4} -3.36532 q^{5} +1.62644 q^{6} -2.72136 q^{8} -2.49071 q^{9} +7.66975 q^{10} -2.45502 q^{11} -2.27945 q^{12} +1.53126 q^{13} +2.40166 q^{15} -0.186032 q^{16} -2.89641 q^{17} +5.67645 q^{18} +1.01605 q^{19} -10.7491 q^{20} +5.59512 q^{22} -4.93143 q^{23} +1.94209 q^{24} +6.32540 q^{25} -3.48983 q^{26} +3.91843 q^{27} -8.76332 q^{29} -5.47350 q^{30} -5.91571 q^{31} +5.86670 q^{32} +1.75202 q^{33} +6.60107 q^{34} -7.95550 q^{36} -8.45756 q^{37} -2.31564 q^{38} -1.09278 q^{39} +9.15826 q^{40} +1.00000 q^{41} -5.59089 q^{43} -7.84152 q^{44} +8.38203 q^{45} +11.2390 q^{46} +5.14085 q^{47} +0.132761 q^{48} -14.4159 q^{50} +2.06702 q^{51} +4.89097 q^{52} -7.81636 q^{53} -8.93031 q^{54} +8.26194 q^{55} -0.725106 q^{57} +19.9721 q^{58} -6.63151 q^{59} +7.67108 q^{60} +9.20226 q^{61} +13.4822 q^{62} -12.9984 q^{64} -5.15320 q^{65} -3.99295 q^{66} -9.54919 q^{67} -9.25136 q^{68} +3.51931 q^{69} -7.18144 q^{71} +6.77811 q^{72} -2.87408 q^{73} +19.2752 q^{74} -4.51412 q^{75} +3.24535 q^{76} +2.49051 q^{78} +3.80950 q^{79} +0.626057 q^{80} +4.67573 q^{81} -2.27905 q^{82} -14.4939 q^{83} +9.74737 q^{85} +12.7419 q^{86} +6.25393 q^{87} +6.68099 q^{88} -7.01102 q^{89} -19.1031 q^{90} -15.7513 q^{92} +4.22173 q^{93} -11.7163 q^{94} -3.41935 q^{95} -4.18676 q^{96} -5.72017 q^{97} +6.11473 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 2 q^{2} + 8 q^{3} + 18 q^{4} + 8 q^{5} + 12 q^{6} - 6 q^{8} + 16 q^{9}+O(q^{10}) \) 20 * q - 2 * q^2 + 8 * q^3 + 18 * q^4 + 8 * q^5 + 12 * q^6 - 6 * q^8 + 16 * q^9	\( 20 q - 2 q^{2} + 8 q^{3} + 18 q^{4} + 8 q^{5} + 12 q^{6} - 6 q^{8} + 16 q^{9} + 16 q^{10} + 6 q^{12} + 12 q^{13} + 14 q^{16} + 8 q^{17} - 18 q^{18} + 36 q^{19} + 24 q^{20} - 8 q^{22} - 12 q^{23} + 36 q^{24} + 20 q^{25} - 22 q^{26} + 32 q^{27} + 4 q^{29} + 28 q^{30} + 80 q^{31} + 6 q^{32} - 12 q^{33} + 48 q^{34} + 26 q^{36} + 4 q^{37} + 12 q^{38} - 28 q^{39} - 4 q^{40} + 20 q^{41} - 20 q^{44} + 40 q^{45} + 8 q^{46} + 32 q^{47} + 16 q^{48} + 6 q^{50} - 20 q^{51} + 36 q^{52} + 4 q^{53} - 50 q^{54} + 64 q^{55} - 4 q^{57} + 32 q^{59} + 20 q^{60} + 44 q^{61} - 8 q^{62} - 30 q^{64} - 8 q^{65} + 32 q^{66} - 4 q^{67} - 48 q^{68} - 24 q^{69} + 8 q^{71} - 8 q^{72} + 48 q^{73} - 38 q^{74} + 24 q^{75} + 84 q^{76} + 30 q^{78} - 4 q^{79} + 56 q^{80} - 2 q^{82} + 8 q^{83} - 12 q^{85} - 24 q^{86} + 40 q^{87} - 48 q^{88} + 20 q^{89} + 48 q^{90} - 50 q^{92} + 48 q^{93} + 26 q^{94} + 20 q^{95} + 70 q^{96} + 8 q^{97} - 8 q^{99}+O(q^{100}) \) 20 * q - 2 * q^2 + 8 * q^3 + 18 * q^4 + 8 * q^5 + 12 * q^6 - 6 * q^8 + 16 * q^9 + 16 * q^10 + 6 * q^12 + 12 * q^13 + 14 * q^16 + 8 * q^17 - 18 * q^18 + 36 * q^19 + 24 * q^20 - 8 * q^22 - 12 * q^23 + 36 * q^24 + 20 * q^25 - 22 * q^26 + 32 * q^27 + 4 * q^29 + 28 * q^30 + 80 * q^31 + 6 * q^32 - 12 * q^33 + 48 * q^34 + 26 * q^36 + 4 * q^37 + 12 * q^38 - 28 * q^39 - 4 * q^40 + 20 * q^41 - 20 * q^44 + 40 * q^45 + 8 * q^46 + 32 * q^47 + 16 * q^48 + 6 * q^50 - 20 * q^51 + 36 * q^52 + 4 * q^53 - 50 * q^54 + 64 * q^55 - 4 * q^57 + 32 * q^59 + 20 * q^60 + 44 * q^61 - 8 * q^62 - 30 * q^64 - 8 * q^65 + 32 * q^66 - 4 * q^67 - 48 * q^68 - 24 * q^69 + 8 * q^71 - 8 * q^72 + 48 * q^73 - 38 * q^74 + 24 * q^75 + 84 * q^76 + 30 * q^78 - 4 * q^79 + 56 * q^80 - 2 * q^82 + 8 * q^83 - 12 * q^85 - 24 * q^86 + 40 * q^87 - 48 * q^88 + 20 * q^89 + 48 * q^90 - 50 * q^92 + 48 * q^93 + 26 * q^94 + 20 * q^95 + 70 * q^96 + 8 * q^97 - 8 * q^99

Introduction

L-functions

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Database

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