LMFDB - Embedded newform 2009.2.a.u.1.4

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			1.4
Root			$2.27121$ 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)

$f(q)$	$=$	$q-2.27121 q^{2} +3.42520 q^{3} +3.15839 q^{4} -0.765314 q^{5} -7.77934 q^{6} -2.63095 q^{8} +8.73199 q^{9} +O(q^{10})$	\(q-2.27121 q^{2} +3.42520 q^{3} +3.15839 q^{4} -0.765314 q^{5} -7.77934 q^{6} -2.63095 q^{8} +8.73199 q^{9} +1.73819 q^{10} -1.96371 q^{11} +10.8181 q^{12} -3.97267 q^{13} -2.62135 q^{15} -0.341352 q^{16} -1.82855 q^{17} -19.8322 q^{18} +4.34193 q^{19} -2.41716 q^{20} +4.45999 q^{22} +2.97065 q^{23} -9.01151 q^{24} -4.41429 q^{25} +9.02277 q^{26} +19.6332 q^{27} -2.01383 q^{29} +5.95364 q^{30} +9.52999 q^{31} +6.03717 q^{32} -6.72609 q^{33} +4.15301 q^{34} +27.5790 q^{36} +11.3711 q^{37} -9.86144 q^{38} -13.6072 q^{39} +2.01350 q^{40} +1.00000 q^{41} -1.46685 q^{43} -6.20216 q^{44} -6.68271 q^{45} -6.74696 q^{46} +1.73730 q^{47} -1.16920 q^{48} +10.0258 q^{50} -6.26314 q^{51} -12.5473 q^{52} +5.14520 q^{53} -44.5911 q^{54} +1.50285 q^{55} +14.8720 q^{57} +4.57383 q^{58} -0.872037 q^{59} -8.27925 q^{60} +9.79505 q^{61} -21.6446 q^{62} -13.0290 q^{64} +3.04034 q^{65} +15.2764 q^{66} +4.01202 q^{67} -5.77527 q^{68} +10.1751 q^{69} +9.65178 q^{71} -22.9734 q^{72} +16.2960 q^{73} -25.8260 q^{74} -15.1198 q^{75} +13.7135 q^{76} +30.9048 q^{78} +5.15945 q^{79} +0.261241 q^{80} +41.0516 q^{81} -2.27121 q^{82} -15.1305 q^{83} +1.39941 q^{85} +3.33152 q^{86} -6.89777 q^{87} +5.16641 q^{88} +8.07545 q^{89} +15.1778 q^{90} +9.38246 q^{92} +32.6421 q^{93} -3.94578 q^{94} -3.32294 q^{95} +20.6785 q^{96} -12.7689 q^{97} -17.1471 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−2.27121	−1.60599	−0.802994	−	0.595988i	$-0.796761\pi$
$2$	−2.27121	−1.60599	−0.802994	+	0.595988i	$0.796761\pi$
$3$	3.42520	1.97754	0.988770	−	0.149448i	$-0.0477495\pi$
$3$	3.42520	1.97754	0.988770	+	0.149448i	$0.0477495\pi$
$4$	3.15839	1.57920
$5$	−0.765314	−0.342259	−0.171129	−	0.985249i	$-0.554742\pi$
$5$	−0.765314	−0.342259	−0.171129	+	0.985249i	$0.554742\pi$
$6$	−7.77934	−3.17590
$7$	0	0
$7$	0	0
$8$	−2.63095	−0.930180
$9$	8.73199	2.91066
$10$	1.73819	0.549663
$11$	−1.96371	−0.592080	−0.296040	−	0.955175i	$-0.595666\pi$
$11$	−1.96371	−0.592080	−0.296040	+	0.955175i	$0.595666\pi$
$12$	10.8181	3.12292
$13$	−3.97267	−1.10182	−0.550911	−	0.834564i	$-0.685720\pi$
$13$	−3.97267	−1.10182	−0.550911	+	0.834564i	$0.685720\pi$
$14$	0	0
$15$	−2.62135	−0.676830
$16$	−0.341352	−0.0853379
$17$	−1.82855	−0.443488	−0.221744	−	0.975105i	$-0.571175\pi$
$17$	−1.82855	−0.443488	−0.221744	+	0.975105i	$0.571175\pi$
$18$	−19.8322	−4.67449
$19$	4.34193	0.996108	0.498054	−	0.867146i	$-0.334048\pi$
$19$	4.34193	0.996108	0.498054	+	0.867146i	$0.334048\pi$
$20$	−2.41716	−0.540493
$21$	0	0
$22$	4.45999	0.950874
$23$	2.97065	0.619423	0.309711	−	0.950831i	$-0.399768\pi$
$23$	2.97065	0.619423	0.309711	+	0.950831i	$0.399768\pi$
$24$	−9.01151	−1.83947
$25$	−4.41429	−0.882859
$26$	9.02277	1.76951
$27$	19.6332	3.77841
$28$	0	0
$29$	−2.01383	−0.373959	−0.186980	−	0.982364i	$-0.559870\pi$
$29$	−2.01383	−0.373959	−0.186980	+	0.982364i	$0.559870\pi$
$30$	5.95364	1.08698
$31$	9.52999	1.71164	0.855818	−	0.517277i	$-0.173054\pi$
$31$	9.52999	1.71164	0.855818	+	0.517277i	$0.173054\pi$
$32$	6.03717	1.06723
$33$	−6.72609	−1.17086
$34$	4.15301	0.712236
$35$	0	0
$36$	27.5790	4.59650
$37$	11.3711	1.86939	0.934695	−	0.355450i	$-0.115672\pi$
$37$	11.3711	1.86939	0.934695	+	0.355450i	$0.115672\pi$
$38$	−9.86144	−1.59974
$39$	−13.6072	−2.17890
$40$	2.01350	0.318362
$41$	1.00000	0.156174
$41$	1.00000	0.156174
$42$	0	0
$43$	−1.46685	−0.223692	−0.111846	−	0.993726i	$-0.535676\pi$
$43$	−1.46685	−0.223692	−0.111846	+	0.993726i	$0.535676\pi$
$44$	−6.20216	−0.935011
$45$	−6.68271	−0.996199
$46$	−6.74696	−0.994785
$47$	1.73730	0.253412	0.126706	−	0.991940i	$-0.459560\pi$
$47$	1.73730	0.253412	0.126706	+	0.991940i	$0.459560\pi$
$48$	−1.16920	−0.168759
$49$	0	0
$50$	10.0258	1.41786
$51$	−6.26314	−0.877015
$52$	−12.5473	−1.73999
$53$	5.14520	0.706747	0.353374	−	0.935482i	$-0.385034\pi$
$53$	5.14520	0.706747	0.353374	+	0.935482i	$0.385034\pi$
$54$	−44.5911	−6.06808
$55$	1.50285	0.202645
$56$	0	0
$57$	14.8720	1.96984
$58$	4.57383	0.600574
$59$	−0.872037	−0.113530	−0.0567648	−	0.998388i	$-0.518079\pi$
$59$	−0.872037	−0.113530	−0.0567648	+	0.998388i	$0.518079\pi$
$60$	−8.27925	−1.06885
$61$	9.79505	1.25413	0.627064	−	0.778968i	$-0.284257\pi$
$61$	9.79505	1.25413	0.627064	+	0.778968i	$0.284257\pi$
$62$	−21.6446	−2.74887
$63$	0	0
$64$	−13.0290	−1.62862
$65$	3.04034	0.377108
$66$	15.2764	1.88039
$67$	4.01202	0.490146	0.245073	−	0.969505i	$-0.421188\pi$
$67$	4.01202	0.490146	0.245073	+	0.969505i	$0.421188\pi$
$68$	−5.77527	−0.700354
$69$	10.1751	1.22493
$70$	0	0
$71$	9.65178	1.14546	0.572728	−	0.819745i	$-0.305885\pi$
$71$	9.65178	1.14546	0.572728	+	0.819745i	$0.305885\pi$
$72$	−22.9734	−2.70744
$73$	16.2960	1.90731	0.953653	−	0.300907i	$-0.0972895\pi$
$73$	16.2960	1.90731	0.953653	+	0.300907i	$0.0972895\pi$
$74$	−25.8260	−3.00222
$75$	−15.1198	−1.74589
$76$	13.7135	1.57305
$77$	0	0
$78$	30.9048	3.49928
$79$	5.15945	0.580484	0.290242	−	0.956953i	$-0.406264\pi$
$79$	5.15945	0.580484	0.290242	+	0.956953i	$0.406264\pi$
$80$	0.261241	0.0292076
$81$	41.0516	4.56129
$82$	−2.27121	−0.250813
$83$	−15.1305	−1.66079	−0.830394	−	0.557176i	$-0.811885\pi$
$83$	−15.1305	−1.66079	−0.830394	+	0.557176i	$0.811885\pi$
$84$	0	0
$85$	1.39941	0.151788
$86$	3.33152	0.359247
$87$	−6.89777	−0.739519
$88$	5.16641	0.550741
$89$	8.07545	0.855996	0.427998	−	0.903780i	$-0.359219\pi$
$89$	8.07545	0.855996	0.427998	+	0.903780i	$0.359219\pi$
$90$	15.1778	1.59988
$91$	0	0
$92$	9.38246	0.978189
$93$	32.6421	3.38483
$94$	−3.94578	−0.406976
$95$	−3.32294	−0.340927
$96$	20.6785	2.11049
$97$	−12.7689	−1.29649	−0.648245	−	0.761432i	$-0.724497\pi$
$97$	−12.7689	−1.29649	−0.648245	+	0.761432i	$0.724497\pi$
$98$	0	0
$99$	−17.1471	−1.72335
$100$	−13.9421	−1.39421
$101$	−11.5893	−1.15318	−0.576591	−	0.817033i	$-0.695617\pi$
$101$	−11.5893	−1.15318	−0.576591	+	0.817033i	$0.695617\pi$
$102$	14.2249	1.40847
$103$	6.65054	0.655297	0.327649	−	0.944800i	$-0.393744\pi$
$103$	6.65054	0.655297	0.327649	+	0.944800i	$0.393744\pi$
$104$	10.4519	1.02489
$105$	0	0
$106$	−11.6858	−1.13503
$107$	−1.19895	−0.115907	−0.0579533	−	0.998319i	$-0.518457\pi$
$107$	−1.19895	−0.115907	−0.0579533	+	0.998319i	$0.518457\pi$
$108$	62.0093	5.96684
$109$	15.1852	1.45448	0.727240	−	0.686383i	$-0.240802\pi$
$109$	15.1852	1.45448	0.727240	+	0.686383i	$0.240802\pi$
$110$	−3.41329	−0.325445
$111$	38.9481	3.69679
$112$	0	0
$113$	−16.8541	−1.58550	−0.792749	−	0.609548i	$-0.791351\pi$
$113$	−16.8541	−1.58550	−0.792749	+	0.609548i	$0.791351\pi$
$114$	−33.7774	−3.16354
$115$	−2.27348	−0.212003
$116$	−6.36047	−0.590555
$117$	−34.6893	−3.20703
$118$	1.98058	0.182327
$119$	0	0
$120$	6.89664	0.629574
$121$	−7.14385	−0.649441
$122$	−22.2466	−2.01411
$123$	3.42520	0.308840
$124$	30.0994	2.70301
$125$	7.20489	0.644425
$126$	0	0
$127$	−13.4533	−1.19379	−0.596896	−	0.802319i	$-0.703599\pi$
$127$	−13.4533	−1.19379	−0.596896	+	0.802319i	$0.703599\pi$
$128$	17.5172	1.54832
$129$	−5.02424	−0.442360
$130$	−6.90525	−0.605631
$131$	−12.0746	−1.05497	−0.527483	−	0.849566i	$-0.676864\pi$
$131$	−12.0746	−1.05497	−0.527483	+	0.849566i	$0.676864\pi$
$132$	−21.2436	−1.84902
$133$	0	0
$134$	−9.11213	−0.787168
$135$	−15.0255	−1.29319
$136$	4.81081	0.412524
$137$	4.97408	0.424964	0.212482	−	0.977165i	$-0.431845\pi$
$137$	4.97408	0.424964	0.212482	+	0.977165i	$0.431845\pi$
$138$	−23.1097	−1.96723
$139$	5.50892	0.467260	0.233630	−	0.972326i	$-0.424939\pi$
$139$	5.50892	0.467260	0.233630	+	0.972326i	$0.424939\pi$
$140$	0	0
$141$	5.95061	0.501132
$142$	−21.9212	−1.83959
$143$	7.80117	0.652367
$144$	−2.98068	−0.248390
$145$	1.54121	0.127991
$146$	−37.0117	−3.06311
$147$	0	0
$148$	35.9142	2.95213
$149$	−16.0634	−1.31597	−0.657983	−	0.753033i	$-0.728590\pi$
$149$	−16.0634	−1.31597	−0.657983	+	0.753033i	$0.728590\pi$
$150$	34.3403	2.80387
$151$	−1.35595	−0.110345	−0.0551727	−	0.998477i	$-0.517571\pi$
$151$	−1.35595	−0.110345	−0.0551727	+	0.998477i	$0.517571\pi$
$152$	−11.4234	−0.926560
$153$	−15.9668	−1.29084
$154$	0	0
$155$	−7.29343	−0.585823
$156$	−42.9768	−3.44090
$157$	−0.159946	−0.0127651	−0.00638255	−	0.999980i	$-0.502032\pi$
$157$	−0.159946	−0.0127651	−0.00638255	+	0.999980i	$0.502032\pi$
$158$	−11.7182	−0.932249
$159$	17.6233	1.39762
$160$	−4.62033	−0.365269
$161$	0	0
$162$	−93.2368	−7.32537
$163$	−17.2564	−1.35162	−0.675812	−	0.737074i	$-0.736207\pi$
$163$	−17.2564	−1.35162	−0.675812	+	0.737074i	$0.736207\pi$
$164$	3.15839	0.246629
$165$	5.14757	0.400738
$166$	34.3645	2.66721
$167$	−6.83742	−0.529095	−0.264548	−	0.964373i	$-0.585223\pi$
$167$	−6.83742	−0.529095	−0.264548	+	0.964373i	$0.585223\pi$
$168$	0	0
$169$	2.78214	0.214011
$170$	−3.17836	−0.243769
$171$	37.9137	2.89933
$172$	−4.63288	−0.353254
$173$	11.0900	0.843155	0.421578	−	0.906792i	$-0.361477\pi$
$173$	11.0900	0.843155	0.421578	+	0.906792i	$0.361477\pi$
$174$	15.6663	1.18766
$175$	0	0
$176$	0.670315	0.0505269
$177$	−2.98690	−0.224509
$178$	−18.3410	−1.37472
$179$	−11.0385	−0.825055	−0.412527	−	0.910945i	$-0.635354\pi$
$179$	−11.0385	−0.825055	−0.412527	+	0.910945i	$0.635354\pi$
$180$	−21.1066	−1.57319
$181$	13.1597	0.978152	0.489076	−	0.872241i	$-0.337334\pi$
$181$	13.1597	0.978152	0.489076	+	0.872241i	$0.337334\pi$
$182$	0	0
$183$	33.5500	2.48009
$184$	−7.81561	−0.576175
$185$	−8.70243	−0.639815
$186$	−74.1370	−5.43599
$187$	3.59073	0.262581
$188$	5.48708	0.400187
$189$	0	0
$190$	7.54710	0.547524
$191$	−16.1263	−1.16686	−0.583429	−	0.812164i	$-0.698289\pi$
$191$	−16.1263	−1.16686	−0.583429	+	0.812164i	$0.698289\pi$
$192$	−44.6268	−3.22066
$193$	−6.85623	−0.493522	−0.246761	−	0.969076i	$-0.579366\pi$
$193$	−6.85623	−0.493522	−0.246761	+	0.969076i	$0.579366\pi$
$194$	29.0009	2.08215
$195$	10.4138	0.745746
$196$	0	0
$197$	3.21156	0.228814	0.114407	−	0.993434i	$-0.463503\pi$
$197$	3.21156	0.228814	0.114407	+	0.993434i	$0.463503\pi$
$198$	38.9446	2.76767
$199$	−6.40212	−0.453835	−0.226917	−	0.973914i	$-0.572865\pi$
$199$	−6.40212	−0.453835	−0.226917	+	0.973914i	$0.572865\pi$
$200$	11.6138	0.821218
$201$	13.7420	0.969283
$202$	26.3218	1.85200
$203$	0	0
$204$	−19.7814	−1.38498
$205$	−0.765314	−0.0534518
$206$	−15.1048	−1.05240
$207$	25.9396	1.80293
$208$	1.35608	0.0940272
$209$	−8.52629	−0.589776
$210$	0	0
$211$	6.52429	0.449151	0.224575	−	0.974457i	$-0.427901\pi$
$211$	6.52429	0.449151	0.224575	+	0.974457i	$0.427901\pi$
$212$	16.2505	1.11609
$213$	33.0593	2.26518
$214$	2.72306	0.186145
$215$	1.12260	0.0765606
$216$	−51.6539	−3.51460
$217$	0	0
$218$	−34.4888	−2.33588
$219$	55.8172	3.77177
$220$	4.74660	0.320016
$221$	7.26422	0.488645
$222$	−88.4593	−5.93700
$223$	−11.2474	−0.753183	−0.376592	−	0.926379i	$-0.622904\pi$
$223$	−11.2474	−0.753183	−0.376592	+	0.926379i	$0.622904\pi$
$224$	0	0
$225$	−38.5456	−2.56970
$226$	38.2791	2.54629
$227$	−23.1933	−1.53939	−0.769695	−	0.638411i	$-0.779592\pi$
$227$	−23.1933	−1.53939	−0.769695	+	0.638411i	$0.779592\pi$
$228$	46.9715	3.11077
$229$	−9.14138	−0.604080	−0.302040	−	0.953295i	$-0.597668\pi$
$229$	−9.14138	−0.604080	−0.302040	+	0.953295i	$0.597668\pi$
$230$	5.16354	0.340474
$231$	0	0
$232$	5.29828	0.347849
$233$	−9.78618	−0.641114	−0.320557	−	0.947229i	$-0.603870\pi$
$233$	−9.78618	−0.641114	−0.320557	+	0.947229i	$0.603870\pi$
$234$	78.7867	5.15045
$235$	−1.32958	−0.0867324
$236$	−2.75423	−0.179285
$237$	17.6721	1.14793
$238$	0	0
$239$	26.7245	1.72867	0.864333	−	0.502920i	$-0.167741\pi$
$239$	26.7245	1.72867	0.864333	+	0.502920i	$0.167741\pi$
$240$	0.894803	0.0577593
$241$	13.3286	0.858573	0.429286	−	0.903168i	$-0.358765\pi$
$241$	13.3286	0.858573	0.429286	+	0.903168i	$0.358765\pi$
$242$	16.2252	1.04299
$243$	81.7104	5.24172
$244$	30.9366	1.98051
$245$	0	0
$246$	−7.77934	−0.495993
$247$	−17.2491	−1.09753
$248$	−25.0729	−1.59213
$249$	−51.8250	−3.28427
$250$	−16.3638	−1.03494
$251$	2.88649	0.182194	0.0910969	−	0.995842i	$-0.470963\pi$
$251$	2.88649	0.182194	0.0910969	+	0.995842i	$0.470963\pi$
$252$	0	0
$253$	−5.83348	−0.366748
$254$	30.5554	1.91721
$255$	4.79327	0.300166
$256$	−13.7272	−0.857952
$257$	14.2134	0.886609	0.443305	−	0.896371i	$-0.353806\pi$
$257$	14.2134	0.886609	0.443305	+	0.896371i	$0.353806\pi$
$258$	11.4111	0.710425
$259$	0	0
$260$	9.60259	0.595527
$261$	−17.5848	−1.08847
$262$	27.4240	1.69426
$263$	18.0540	1.11325	0.556627	−	0.830762i	$-0.312095\pi$
$263$	18.0540	1.11325	0.556627	+	0.830762i	$0.312095\pi$
$264$	17.6960	1.08911
$265$	−3.93769	−0.241890
$266$	0	0
$267$	27.6600	1.69277
$268$	12.6715	0.774036
$269$	−12.0585	−0.735218	−0.367609	−	0.929980i	$-0.619823\pi$
$269$	−12.0585	−0.735218	−0.367609	+	0.929980i	$0.619823\pi$
$270$	34.1262	2.07685
$271$	19.9947	1.21459	0.607295	−	0.794476i	$-0.292254\pi$
$271$	19.9947	1.21459	0.607295	+	0.794476i	$0.292254\pi$
$272$	0.624178	0.0378463
$273$	0	0
$274$	−11.2972	−0.682487
$275$	8.66839	0.522724
$276$	32.1368	1.93441
$277$	−7.92347	−0.476075	−0.238038	−	0.971256i	$-0.576504\pi$
$277$	−7.92347	−0.476075	−0.238038	+	0.971256i	$0.576504\pi$
$278$	−12.5119	−0.750414
$279$	83.2157	4.98199
$280$	0	0
$281$	−19.8239	−1.18260	−0.591298	−	0.806453i	$-0.701384\pi$
$281$	−19.8239	−1.18260	−0.591298	+	0.806453i	$0.701384\pi$
$282$	−13.5151	−0.804811
$283$	−7.31457	−0.434806	−0.217403	−	0.976082i	$-0.569759\pi$
$283$	−7.31457	−0.434806	−0.217403	+	0.976082i	$0.569759\pi$
$284$	30.4841	1.80890
$285$	−11.3817	−0.674196
$286$	−17.7181	−1.04769
$287$	0	0
$288$	52.7165	3.10635
$289$	−13.6564	−0.803318
$290$	−3.50042	−0.205552
$291$	−43.7362	−2.56386
$292$	51.4692	3.01201
$293$	−3.79367	−0.221629	−0.110814	−	0.993841i	$-0.535346\pi$
$293$	−3.79367	−0.221629	−0.110814	+	0.993841i	$0.535346\pi$
$294$	0	0
$295$	0.667382	0.0388565
$296$	−29.9166	−1.73887
$297$	−38.5539	−2.23712
$298$	36.4833	2.11342
$299$	−11.8014	−0.682493
$300$	−47.7543	−2.75710
$301$	0	0
$302$	3.07964	0.177213
$303$	−39.6958	−2.28046
$304$	−1.48213	−0.0850058
$305$	−7.49629	−0.429236
$306$	36.2641	2.07308
$307$	18.8671	1.07680	0.538400	−	0.842689i	$-0.319029\pi$
$307$	18.8671	1.07680	0.538400	+	0.842689i	$0.319029\pi$
$308$	0	0
$309$	22.7794	1.29588
$310$	16.5649	0.940823
$311$	−12.3681	−0.701333	−0.350666	−	0.936500i	$-0.614045\pi$
$311$	−12.3681	−0.701333	−0.350666	+	0.936500i	$0.614045\pi$
$312$	35.7998	2.02676
$313$	−13.6577	−0.771977	−0.385989	−	0.922504i	$-0.626140\pi$
$313$	−13.6577	−0.771977	−0.385989	+	0.922504i	$0.626140\pi$
$314$	0.363271	0.0205006
$315$	0	0
$316$	16.2956	0.916697
$317$	13.1042	0.736008	0.368004	−	0.929824i	$-0.380041\pi$
$317$	13.1042	0.736008	0.368004	+	0.929824i	$0.380041\pi$
$318$	−40.0263	−2.24456
$319$	3.95458	0.221414
$320$	9.97126	0.557410
$321$	−4.10663	−0.229210
$322$	0	0
$323$	−7.93943	−0.441762
$324$	129.657	7.20317
$325$	17.5366	0.972753
$326$	39.1929	2.17069
$327$	52.0124	2.87629
$328$	−2.63095	−0.145270
$329$	0	0
$330$	−11.6912	−0.643580
$331$	15.5556	0.855011	0.427505	−	0.904013i	$-0.359392\pi$
$331$	15.5556	0.855011	0.427505	+	0.904013i	$0.359392\pi$
$332$	−47.7880	−2.62271
$333$	99.2919	5.44116
$334$	15.5292	0.849720
$335$	−3.07045	−0.167757
$336$	0	0
$337$	−9.01030	−0.490822	−0.245411	−	0.969419i	$-0.578923\pi$
$337$	−9.01030	−0.490822	−0.245411	+	0.969419i	$0.578923\pi$
$338$	−6.31882	−0.343698
$339$	−57.7285	−3.13538
$340$	4.41989	0.239702
$341$	−18.7141	−1.01343
$342$	−86.1099	−4.65629
$343$	0	0
$344$	3.85920	0.208074
$345$	−7.78711	−0.419244
$346$	−25.1877	−1.35410
$347$	−10.8836	−0.584263	−0.292132	−	0.956378i	$-0.594365\pi$
$347$	−10.8836	−0.584263	−0.292132	+	0.956378i	$0.594365\pi$
$348$	−21.7859	−1.16784
$349$	6.43965	0.344707	0.172353	−	0.985035i	$-0.444863\pi$
$349$	6.43965	0.344707	0.172353	+	0.985035i	$0.444863\pi$
$350$	0	0
$351$	−77.9963	−4.16313
$352$	−11.8553	−0.631887
$353$	25.8563	1.37619	0.688096	−	0.725620i	$-0.258447\pi$
$353$	25.8563	1.37619	0.688096	+	0.725620i	$0.258447\pi$
$354$	6.78388	0.360559
$355$	−7.38664	−0.392042
$356$	25.5054	1.35178
$357$	0	0
$358$	25.0707	1.32503
$359$	−23.6939	−1.25051	−0.625257	−	0.780419i	$-0.715006\pi$
$359$	−23.6939	−1.25051	−0.625257	+	0.780419i	$0.715006\pi$
$360$	17.5818	0.926645
$361$	−0.147608	−0.00776886
$362$	−29.8884	−1.57090
$363$	−24.4691	−1.28429
$364$	0	0
$365$	−12.4716	−0.652792
$366$	−76.1990	−3.98299
$367$	−9.40377	−0.490873	−0.245436	−	0.969413i	$-0.578931\pi$
$367$	−9.40377	−0.490873	−0.245436	+	0.969413i	$0.578931\pi$
$368$	−1.01404	−0.0528602
$369$	8.73199	0.454569
$370$	19.7650	1.02753
$371$	0	0
$372$	103.096	5.34530
$373$	26.1532	1.35416	0.677081	−	0.735909i	$-0.263245\pi$
$373$	26.1532	1.35416	0.677081	+	0.735909i	$0.263245\pi$
$374$	−8.15531	−0.421701
$375$	24.6782	1.27438
$376$	−4.57075	−0.235719
$377$	8.00030	0.412036
$378$	0	0
$379$	2.16545	0.111232	0.0556160	−	0.998452i	$-0.482288\pi$
$379$	2.16545	0.111232	0.0556160	+	0.998452i	$0.482288\pi$
$380$	−10.4951	−0.538390
$381$	−46.0804	−2.36077
$382$	36.6262	1.87396
$383$	−9.23159	−0.471712	−0.235856	−	0.971788i	$-0.575789\pi$
$383$	−9.23159	−0.471712	−0.235856	+	0.971788i	$0.575789\pi$
$384$	59.9998	3.06185
$385$	0	0
$386$	15.5719	0.792590
$387$	−12.8085	−0.651092
$388$	−40.3293	−2.04741
$389$	2.04695	0.103784	0.0518922	−	0.998653i	$-0.483475\pi$
$389$	2.04695	0.103784	0.0518922	+	0.998653i	$0.483475\pi$
$390$	−23.6519	−1.19766
$391$	−5.43197	−0.274706
$392$	0	0
$393$	−41.3580	−2.08624
$394$	−7.29411	−0.367472
$395$	−3.94860	−0.198676
$396$	−54.1572	−2.72150
$397$	17.4413	0.875356	0.437678	−	0.899132i	$-0.355801\pi$
$397$	17.4413	0.875356	0.437678	+	0.899132i	$0.355801\pi$
$398$	14.5406	0.728853
$399$	0	0
$400$	1.50683	0.0753413
$401$	10.9624	0.547437	0.273718	−	0.961810i	$-0.411746\pi$
$401$	10.9624	0.547437	0.273718	+	0.961810i	$0.411746\pi$
$402$	−31.2108	−1.55666
$403$	−37.8595	−1.88592
$404$	−36.6037	−1.82110
$405$	−31.4174	−1.56114
$406$	0	0
$407$	−22.3294	−1.10683
$408$	16.4780	0.815782
$409$	28.8759	1.42782	0.713910	−	0.700237i	$-0.246922\pi$
$409$	28.8759	1.42782	0.713910	+	0.700237i	$0.246922\pi$
$410$	1.73819	0.0858430
$411$	17.0372	0.840383
$412$	21.0050	1.03484
$413$	0	0
$414$	−58.9143	−2.89548
$415$	11.5796	0.568419
$416$	−23.9837	−1.17590
$417$	18.8691	0.924026
$418$	19.3650	0.947173
$419$	4.80062	0.234525	0.117263	−	0.993101i	$-0.462588\pi$
$419$	4.80062	0.234525	0.117263	+	0.993101i	$0.462588\pi$
$420$	0	0
$421$	22.3049	1.08707	0.543537	−	0.839385i	$-0.317085\pi$
$421$	22.3049	1.08707	0.543537	+	0.839385i	$0.317085\pi$
$422$	−14.8180	−0.721330
$423$	15.1701	0.737596
$424$	−13.5367	−0.657402
$425$	8.07175	0.391537
$426$	−75.0845	−3.63786
$427$	0	0
$428$	−3.78674	−0.183039
$429$	26.7206	1.29008
$430$	−2.54966	−0.122955
$431$	3.29229	0.158584	0.0792921	−	0.996851i	$-0.474734\pi$
$431$	3.29229	0.158584	0.0792921	+	0.996851i	$0.474734\pi$
$432$	−6.70182	−0.322442
$433$	3.35842	0.161395	0.0806976	−	0.996739i	$-0.474285\pi$
$433$	3.35842	0.161395	0.0806976	+	0.996739i	$0.474285\pi$
$434$	0	0
$435$	5.27896	0.253107
$436$	47.9609	2.29691
$437$	12.8984	0.617012
$438$	−126.772	−6.05742
$439$	−9.96180	−0.475451	−0.237725	−	0.971332i	$-0.576402\pi$
$439$	−9.96180	−0.475451	−0.237725	+	0.971332i	$0.576402\pi$
$440$	−3.95393	−0.188496
$441$	0	0
$442$	−16.4986	−0.784757
$443$	17.9113	0.850990	0.425495	−	0.904961i	$-0.360100\pi$
$443$	17.9113	0.850990	0.425495	+	0.904961i	$0.360100\pi$
$444$	123.013	5.83796
$445$	−6.18025	−0.292972
$446$	25.5453	1.20960
$447$	−55.0203	−2.60237
$448$	0	0
$449$	−9.24972	−0.436521	−0.218261	−	0.975890i	$-0.570038\pi$
$449$	−9.24972	−0.436521	−0.218261	+	0.975890i	$0.570038\pi$
$450$	87.5450	4.12691
$451$	−1.96371	−0.0924674
$452$	−53.2317	−2.50381
$453$	−4.64438	−0.218212
$454$	52.6767	2.47224
$455$	0	0
$456$	−39.1274	−1.83231
$457$	−20.5001	−0.958954	−0.479477	−	0.877554i	$-0.659174\pi$
$457$	−20.5001	−0.958954	−0.479477	+	0.877554i	$0.659174\pi$
$458$	20.7620	0.970144
$459$	−35.9002	−1.67568
$460$	−7.18053	−0.334794
$461$	−16.9991	−0.791728	−0.395864	−	0.918309i	$-0.629555\pi$
$461$	−16.9991	−0.791728	−0.395864	+	0.918309i	$0.629555\pi$
$462$	0	0
$463$	−13.9287	−0.647323	−0.323662	−	0.946173i	$-0.604914\pi$
$463$	−13.9287	−0.647323	−0.323662	+	0.946173i	$0.604914\pi$
$464$	0.687425	0.0319129
$465$	−24.9815	−1.15849
$466$	22.2265	1.02962
$467$	9.96343	0.461052	0.230526	−	0.973066i	$-0.425955\pi$
$467$	9.96343	0.461052	0.230526	+	0.973066i	$0.425955\pi$
$468$	−109.562	−5.06453
$469$	0	0
$470$	3.01976	0.139291
$471$	−0.547847	−0.0252435
$472$	2.29428	0.105603
$473$	2.88046	0.132444
$474$	−40.1371	−1.84356
$475$	−19.1666	−0.879423
$476$	0	0
$477$	44.9278	2.05710
$478$	−60.6970	−2.77622
$479$	29.8313	1.36303	0.681514	−	0.731805i	$-0.261322\pi$
$479$	29.8313	1.36303	0.681514	+	0.731805i	$0.261322\pi$
$480$	−15.8256	−0.722334
$481$	−45.1735	−2.05973
$482$	−30.2721	−1.37886
$483$	0	0
$484$	−22.5631	−1.02559
$485$	9.77225	0.443735
$486$	−185.581	−8.41814
$487$	−25.7515	−1.16691	−0.583455	−	0.812145i	$-0.698300\pi$
$487$	−25.7515	−1.16691	−0.583455	+	0.812145i	$0.698300\pi$
$488$	−25.7702	−1.16656
$489$	−59.1065	−2.67289
$490$	0	0
$491$	−37.3650	−1.68626	−0.843130	−	0.537710i	$-0.819290\pi$
$491$	−37.3650	−1.68626	−0.843130	+	0.537710i	$0.819290\pi$
$492$	10.8181	0.487718
$493$	3.68239	0.165846
$494$	39.1763	1.76262
$495$	13.1229	0.589830
$496$	−3.25308	−0.146067
$497$	0	0
$498$	117.705	5.27450
$499$	−24.3878	−1.09175	−0.545874	−	0.837867i	$-0.683802\pi$
$499$	−24.3878	−1.09175	−0.545874	+	0.837867i	$0.683802\pi$
$500$	22.7559	1.01767
$501$	−23.4195	−1.04631
$502$	−6.55583	−0.292601
$503$	26.7476	1.19262	0.596308	−	0.802755i	$-0.296634\pi$
$503$	26.7476	1.19262	0.596308	+	0.802755i	$0.296634\pi$
$504$	0	0
$505$	8.86948	0.394687
$506$	13.2491	0.588993
$507$	9.52938	0.423214
$508$	−42.4909	−1.88523
$509$	−9.18609	−0.407166	−0.203583	−	0.979058i	$-0.565259\pi$
$509$	−9.18609	−0.407166	−0.203583	+	0.979058i	$0.565259\pi$
$510$	−10.8865	−0.482063
$511$	0	0
$512$	−3.85696	−0.170455
$513$	85.2460	3.76370
$514$	−32.2817	−1.42388
$515$	−5.08975	−0.224281
$516$	−15.8685	−0.698573
$517$	−3.41156	−0.150040
$518$	0	0
$519$	37.9854	1.66737
$520$	−7.99898	−0.350778
$521$	43.2138	1.89323	0.946616	−	0.322364i	$-0.104478\pi$
$521$	43.2138	1.89323	0.946616	+	0.322364i	$0.104478\pi$
$522$	39.9386	1.74807
$523$	−4.02613	−0.176051	−0.0880253	−	0.996118i	$-0.528056\pi$
$523$	−4.02613	−0.176051	−0.0880253	+	0.996118i	$0.528056\pi$
$524$	−38.1364	−1.66600
$525$	0	0
$526$	−41.0043	−1.78787
$527$	−17.4260	−0.759090
$528$	2.29596	0.0999190
$529$	−14.1753	−0.616316
$530$	8.94332	0.388473
$531$	−7.61462	−0.330446
$532$	0	0
$533$	−3.97267	−0.172076
$534$	−62.8217	−2.71856
$535$	0.917571	0.0396700
$536$	−10.5554	−0.455924
$537$	−37.8090	−1.63158
$538$	27.3873	1.18075
$539$	0	0
$540$	−47.4565	−2.04220
$541$	−25.3081	−1.08808	−0.544041	−	0.839059i	$-0.683106\pi$
$541$	−25.3081	−1.08808	−0.544041	+	0.839059i	$0.683106\pi$
$542$	−45.4121	−1.95062
$543$	45.0746	1.93433
$544$	−11.0393	−0.473304
$545$	−11.6215	−0.497809
$546$	0	0
$547$	−30.5168	−1.30480	−0.652402	−	0.757873i	$-0.726239\pi$
$547$	−30.5168	−1.30480	−0.652402	+	0.757873i	$0.726239\pi$
$548$	15.7101	0.671101
$549$	85.5302	3.65034
$550$	−19.6877	−0.839487
$551$	−8.74393	−0.372504
$552$	−26.7700	−1.13941
$553$	0	0
$554$	17.9959	0.764570
$555$	−29.8075	−1.26526
$556$	17.3993	0.737895
$557$	−30.6528	−1.29880	−0.649400	−	0.760447i	$-0.724980\pi$
$557$	−30.6528	−1.29880	−0.649400	+	0.760447i	$0.724980\pi$
$558$	−189.000	−8.00102
$559$	5.82731	0.246469
$560$	0	0
$561$	12.2990	0.519263
$562$	45.0243	1.89923
$563$	−7.53563	−0.317589	−0.158794	−	0.987312i	$-0.550761\pi$
$563$	−7.53563	−0.317589	−0.158794	+	0.987312i	$0.550761\pi$
$564$	18.7944	0.791385
$565$	12.8987	0.542650
$566$	16.6129	0.698293
$567$	0	0
$568$	−25.3933	−1.06548
$569$	14.8804	0.623817	0.311909	−	0.950112i	$-0.399032\pi$
$569$	14.8804	0.623817	0.311909	+	0.950112i	$0.399032\pi$
$570$	25.8503	1.08275
$571$	−39.2393	−1.64211	−0.821057	−	0.570846i	$-0.806615\pi$
$571$	−39.2393	−1.64211	−0.821057	+	0.570846i	$0.806615\pi$
$572$	24.6392	1.03021
$573$	−55.2358	−2.30751
$574$	0	0
$575$	−13.1133	−0.546863
$576$	−113.769	−4.74037
$577$	28.5533	1.18869	0.594344	−	0.804211i	$-0.297412\pi$
$577$	28.5533	1.18869	0.594344	+	0.804211i	$0.297412\pi$
$578$	31.0166	1.29012
$579$	−23.4839	−0.975959
$580$	4.86775	0.202122
$581$	0	0
$582$	99.3340	4.11753
$583$	−10.1037	−0.418451
$584$	−42.8740	−1.77414
$585$	26.5482	1.09763
$586$	8.61622	0.355933
$587$	−5.78276	−0.238680	−0.119340	−	0.992853i	$-0.538078\pi$
$587$	−5.78276	−0.238680	−0.119340	+	0.992853i	$0.538078\pi$
$588$	0	0
$589$	41.3786	1.70497
$590$	−1.51576	−0.0624030
$591$	11.0002	0.452488
$592$	−3.88153	−0.159530
$593$	−5.29959	−0.217628	−0.108814	−	0.994062i	$-0.534705\pi$
$593$	−5.29959	−0.217628	−0.108814	+	0.994062i	$0.534705\pi$
$594$	87.5639	3.59279
$595$	0	0
$596$	−50.7345	−2.07817
$597$	−21.9285	−0.897476
$598$	26.8035	1.09608
$599$	11.2348	0.459040	0.229520	−	0.973304i	$-0.426284\pi$
$599$	11.2348	0.459040	0.229520	+	0.973304i	$0.426284\pi$
$600$	39.7795	1.62399
$601$	12.1098	0.493970	0.246985	−	0.969019i	$-0.420560\pi$
$601$	12.1098	0.493970	0.246985	+	0.969019i	$0.420560\pi$
$602$	0	0
$603$	35.0329	1.42665
$604$	−4.28261	−0.174257
$605$	5.46729	0.222277
$606$	90.1574	3.66240
$607$	−19.5998	−0.795530	−0.397765	−	0.917487i	$-0.630214\pi$
$607$	−19.5998	−0.795530	−0.397765	+	0.917487i	$0.630214\pi$
$608$	26.2130	1.06308
$609$	0	0
$610$	17.0256	0.689348
$611$	−6.90174	−0.279215
$612$	−50.4295	−2.03849
$613$	−9.28350	−0.374957	−0.187478	−	0.982269i	$-0.560031\pi$
$613$	−9.28350	−0.374957	−0.187478	+	0.982269i	$0.560031\pi$
$614$	−42.8511	−1.72933
$615$	−2.62135	−0.105703
$616$	0	0
$617$	−45.0121	−1.81212	−0.906059	−	0.423152i	$-0.860924\pi$
$617$	−45.0121	−1.81212	−0.906059	+	0.423152i	$0.860924\pi$
$618$	−51.7368	−2.08116
$619$	−25.9723	−1.04392	−0.521958	−	0.852971i	$-0.674798\pi$
$619$	−25.9723	−1.04392	−0.521958	+	0.852971i	$0.674798\pi$
$620$	−23.0355	−0.925128
$621$	58.3233	2.34043
$622$	28.0906	1.12633
$623$	0	0
$624$	4.64484	0.185942
$625$	16.5575	0.662299
$626$	31.0194	1.23979
$627$	−29.2043	−1.16631
$628$	−0.505172	−0.0201586
$629$	−20.7925	−0.829052
$630$	0	0
$631$	−38.9039	−1.54874	−0.774369	−	0.632734i	$-0.781933\pi$
$631$	−38.9039	−1.54874	−0.774369	+	0.632734i	$0.781933\pi$
$632$	−13.5742	−0.539954
$633$	22.3470	0.888213
$634$	−29.7625	−1.18202
$635$	10.2960	0.408586
$636$	55.6613	2.20712
$637$	0	0
$638$	−8.98168	−0.355588
$639$	84.2792	3.33404
$640$	−13.4061	−0.529924
$641$	47.7644	1.88658	0.943290	−	0.331970i	$-0.107713\pi$
$641$	47.7644	1.88658	0.943290	+	0.331970i	$0.107713\pi$
$642$	9.32702	0.368108
$643$	16.7350	0.659965	0.329982	−	0.943987i	$-0.392957\pi$
$643$	16.7350	0.659965	0.329982	+	0.943987i	$0.392957\pi$
$644$	0	0
$645$	3.84512	0.151402
$646$	18.0321	0.709464
$647$	−26.2471	−1.03188	−0.515939	−	0.856625i	$-0.672557\pi$
$647$	−26.2471	−1.03188	−0.515939	+	0.856625i	$0.672557\pi$
$648$	−108.005	−4.24282
$649$	1.71243	0.0672186
$650$	−39.8292	−1.56223
$651$	0	0
$652$	−54.5024	−2.13448
$653$	−26.9647	−1.05521	−0.527606	−	0.849489i	$-0.676910\pi$
$653$	−26.9647	−1.05521	−0.527606	+	0.849489i	$0.676910\pi$
$654$	−118.131	−4.61929
$655$	9.24088	0.361071
$656$	−0.341352	−0.0133275
$657$	142.297	5.55153
$658$	0	0
$659$	6.22414	0.242458	0.121229	−	0.992625i	$-0.461316\pi$
$659$	6.22414	0.242458	0.121229	+	0.992625i	$0.461316\pi$
$660$	16.2580	0.632843
$661$	−21.2430	−0.826255	−0.413128	−	0.910673i	$-0.635564\pi$
$661$	−21.2430	−0.826255	−0.413128	+	0.910673i	$0.635564\pi$
$662$	−35.3299	−1.37314
$663$	24.8814	0.966314
$664$	39.8075	1.54483
$665$	0	0
$666$	−225.513	−8.73844
$667$	−5.98238	−0.231639
$668$	−21.5952	−0.835545
$669$	−38.5247	−1.48945
$670$	6.97364	0.269415
$671$	−19.2346	−0.742544
$672$	0	0
$673$	12.2619	0.472661	0.236331	−	0.971673i	$-0.424055\pi$
$673$	12.2619	0.472661	0.236331	+	0.971673i	$0.424055\pi$
$674$	20.4643	0.788254
$675$	−86.6667	−3.33580
$676$	8.78708	0.337965
$677$	36.9880	1.42156	0.710781	−	0.703413i	$-0.248342\pi$
$677$	36.9880	1.42156	0.710781	+	0.703413i	$0.248342\pi$
$678$	131.114	5.03539
$679$	0	0
$680$	−3.68178	−0.141190
$681$	−79.4415	−3.04421
$682$	42.5037	1.62755
$683$	13.2643	0.507545	0.253772	−	0.967264i	$-0.418329\pi$
$683$	13.2643	0.507545	0.253772	+	0.967264i	$0.418329\pi$
$684$	119.746	4.57861
$685$	−3.80673	−0.145448
$686$	0	0
$687$	−31.3111	−1.19459
$688$	0.500711	0.0190894
$689$	−20.4402	−0.778709
$690$	17.6862	0.673300
$691$	43.4202	1.65178	0.825892	−	0.563829i	$-0.190672\pi$
$691$	43.4202	1.65178	0.825892	+	0.563829i	$0.190672\pi$
$692$	35.0265	1.33151
$693$	0	0
$694$	24.7190	0.938319
$695$	−4.21605	−0.159924
$696$	18.1477	0.687886
$697$	−1.82855	−0.0692612
$698$	−14.6258	−0.553595
$699$	−33.5196	−1.26783
$700$	0	0
$701$	−16.0091	−0.604654	−0.302327	−	0.953204i	$-0.597763\pi$
$701$	−16.0091	−0.604654	−0.302327	+	0.953204i	$0.597763\pi$
$702$	177.146	6.68594
$703$	49.3724	1.86211
$704$	25.5851	0.964276
$705$	−4.55408	−0.171517
$706$	−58.7250	−2.21015
$707$	0	0
$708$	−9.43380	−0.354544
$709$	−40.5577	−1.52318	−0.761588	−	0.648061i	$-0.775580\pi$
$709$	−40.5577	−1.52318	−0.761588	+	0.648061i	$0.775580\pi$
$710$	16.7766	0.629615
$711$	45.0523	1.68959
$712$	−21.2461	−0.796230
$713$	28.3102	1.06023
$714$	0	0
$715$	−5.97035	−0.223278
$716$	−34.8638	−1.30292
$717$	91.5368	3.41851
$718$	53.8137	2.00831
$719$	43.3925	1.61827	0.809134	−	0.587624i	$-0.199937\pi$
$719$	43.3925	1.61827	0.809134	+	0.587624i	$0.199937\pi$
$720$	2.28115	0.0850136
$721$	0	0
$722$	0.335250	0.0124767
$723$	45.6532	1.69786
$724$	41.5634	1.54469
$725$	8.88965	0.330153
$726$	55.5744	2.06256
$727$	13.7997	0.511803	0.255901	−	0.966703i	$-0.417628\pi$
$727$	13.7997	0.511803	0.255901	+	0.966703i	$0.417628\pi$
$728$	0	0
$729$	156.719	5.80442
$730$	28.3256	1.04838
$731$	2.68220	0.0992048
$732$	105.964	3.91654
$733$	−34.6584	−1.28014	−0.640069	−	0.768318i	$-0.721094\pi$
$733$	−34.6584	−1.28014	−0.640069	+	0.768318i	$0.721094\pi$
$734$	21.3579	0.788335
$735$	0	0
$736$	17.9343	0.661067
$737$	−7.87843	−0.290206
$738$	−19.8322	−0.730032
$739$	14.0191	0.515700	0.257850	−	0.966185i	$-0.416986\pi$
$739$	14.0191	0.515700	0.257850	+	0.966185i	$0.416986\pi$
$740$	−27.4857	−1.01039
$741$	−59.0816	−2.17042
$742$	0	0
$743$	1.45881	0.0535186	0.0267593	−	0.999642i	$-0.491481\pi$
$743$	1.45881	0.0535186	0.0267593	+	0.999642i	$0.491481\pi$
$744$	−85.8796	−3.14850
$745$	12.2935	0.450401
$746$	−59.3994	−2.17477
$747$	−132.119	−4.83399
$748$	11.3409	0.414666
$749$	0	0
$750$	−56.0493	−2.04663
$751$	48.2121	1.75929	0.879643	−	0.475634i	$-0.157781\pi$
$751$	48.2121	1.75929	0.879643	+	0.475634i	$0.157781\pi$
$752$	−0.593032	−0.0216256
$753$	9.88681	0.360295
$754$	−18.1703	−0.661725
$755$	1.03772	0.0377666
$756$	0	0
$757$	31.3313	1.13876	0.569378	−	0.822076i	$-0.307184\pi$
$757$	31.3313	1.13876	0.569378	+	0.822076i	$0.307184\pi$
$758$	−4.91820	−0.178637
$759$	−19.9808	−0.725259
$760$	8.74248	0.317123
$761$	−21.2772	−0.771296	−0.385648	−	0.922646i	$-0.626022\pi$
$761$	−21.2772	−0.771296	−0.385648	+	0.922646i	$0.626022\pi$
$762$	104.658	3.79137
$763$	0	0
$764$	−50.9332	−1.84270
$765$	12.2196	0.441802
$766$	20.9669	0.757564
$767$	3.46432	0.125089
$768$	−47.0185	−1.69663
$769$	−14.4294	−0.520338	−0.260169	−	0.965563i	$-0.583778\pi$
$769$	−14.4294	−0.520338	−0.260169	+	0.965563i	$0.583778\pi$
$770$	0	0
$771$	48.6838	1.75330
$772$	−21.6546	−0.779367
$773$	6.87039	0.247111	0.123555	−	0.992338i	$-0.460570\pi$
$773$	6.87039	0.247111	0.123555	+	0.992338i	$0.460570\pi$
$774$	29.0908	1.04565
$775$	−42.0682	−1.51113
$776$	33.5944	1.20597
$777$	0	0
$778$	−4.64905	−0.166677
$779$	4.34193	0.155566
$780$	32.8908	1.17768
$781$	−18.9533	−0.678202
$782$	12.3371	0.441175
$783$	−39.5379	−1.41297
$784$	0	0
$785$	0.122409	0.00436897
$786$	93.9327	3.35047
$787$	28.1854	1.00470	0.502351	−	0.864664i	$-0.332468\pi$
$787$	28.1854	1.00470	0.502351	+	0.864664i	$0.332468\pi$
$788$	10.1433	0.361342
$789$	61.8384	2.20150
$790$	8.96809	0.319070
$791$	0	0
$792$	45.1130	1.60302
$793$	−38.9125	−1.38182
$794$	−39.6129	−1.40581
$795$	−13.4874	−0.478348
$796$	−20.2204	−0.716693
$797$	−55.7654	−1.97531	−0.987655	−	0.156642i	$-0.949933\pi$
$797$	−55.7654	−1.97531	−0.987655	+	0.156642i	$0.949933\pi$
$798$	0	0
$799$	−3.17674	−0.112385
$800$	−26.6499	−0.942215
$801$	70.5147	2.49152
$802$	−24.8979	−0.879177
$803$	−32.0007	−1.12928
$804$	43.4025	1.53069
$805$	0	0
$806$	85.9869	3.02876
$807$	−41.3026	−1.45392
$808$	30.4909	1.07267
$809$	−26.7794	−0.941513	−0.470756	−	0.882263i	$-0.656019\pi$
$809$	−26.7794	−0.941513	−0.470756	+	0.882263i	$0.656019\pi$
$810$	71.3554	2.50717
$811$	18.1743	0.638185	0.319092	−	0.947724i	$-0.396622\pi$
$811$	18.1743	0.638185	0.319092	+	0.947724i	$0.396622\pi$
$812$	0	0
$813$	68.4858	2.40190
$814$	50.7148	1.77755
$815$	13.2065	0.462605
$816$	2.13793	0.0748426
$817$	−6.36896	−0.222822
$818$	−65.5832	−2.29306
$819$	0	0
$820$	−2.41716	−0.0844109
$821$	19.0490	0.664816	0.332408	−	0.943136i	$-0.392139\pi$
$821$	19.0490	0.664816	0.332408	+	0.943136i	$0.392139\pi$
$822$	−38.6950	−1.34964
$823$	−28.3743	−0.989067	−0.494533	−	0.869159i	$-0.664661\pi$
$823$	−28.3743	−0.989067	−0.494533	+	0.869159i	$0.664661\pi$
$824$	−17.4972	−0.609545
$825$	29.6910	1.03371
$826$	0	0
$827$	−36.8073	−1.27992	−0.639958	−	0.768410i	$-0.721048\pi$
$827$	−36.8073	−1.27992	−0.639958	+	0.768410i	$0.721048\pi$
$828$	81.9275	2.84718
$829$	−1.63939	−0.0569383	−0.0284692	−	0.999595i	$-0.509063\pi$
$829$	−1.63939	−0.0569383	−0.0284692	+	0.999595i	$0.509063\pi$
$830$	−26.2997	−0.912874
$831$	−27.1395	−0.941457
$832$	51.7599	1.79445
$833$	0	0
$834$	−42.8558	−1.48397
$835$	5.23277	0.181087
$836$	−26.9294	−0.931372
$837$	187.104	6.46726
$838$	−10.9032	−0.376645
$839$	20.6006	0.711212	0.355606	−	0.934636i	$-0.384275\pi$
$839$	20.6006	0.711212	0.355606	+	0.934636i	$0.384275\pi$
$840$	0	0
$841$	−24.9445	−0.860155
$842$	−50.6591	−1.74583
$843$	−67.9008	−2.33863
$844$	20.6062	0.709296
$845$	−2.12921	−0.0732470
$846$	−34.4545	−1.18457
$847$	0	0
$848$	−1.75632	−0.0603124
$849$	−25.0539	−0.859846
$850$	−18.3326	−0.628804
$851$	33.7794	1.15794
$852$	104.414	3.57717
$853$	−9.86458	−0.337757	−0.168878	−	0.985637i	$-0.554015\pi$
$853$	−9.86458	−0.337757	−0.168878	+	0.985637i	$0.554015\pi$
$854$	0	0
$855$	−29.0159	−0.992322
$856$	3.15437	0.107814
$857$	−1.21995	−0.0416727	−0.0208364	−	0.999783i	$-0.506633\pi$
$857$	−1.21995	−0.0416727	−0.0208364	+	0.999783i	$0.506633\pi$
$858$	−60.6880	−2.07185
$859$	−12.6095	−0.430231	−0.215115	−	0.976589i	$-0.569013\pi$
$859$	−12.6095	−0.430231	−0.215115	+	0.976589i	$0.569013\pi$
$860$	3.54560	0.120904
$861$	0	0
$862$	−7.47749	−0.254684
$863$	−10.9679	−0.373353	−0.186676	−	0.982421i	$-0.559772\pi$
$863$	−10.9679	−0.373353	−0.186676	+	0.982421i	$0.559772\pi$
$864$	118.529	4.03244
$865$	−8.48731	−0.288577
$866$	−7.62767	−0.259199
$867$	−46.7759	−1.58859
$868$	0	0
$869$	−10.1317	−0.343693
$870$	−11.9896	−0.406486
$871$	−15.9384	−0.540053
$872$	−39.9515	−1.35293
$873$	−111.498	−3.77364
$874$	−29.2949	−0.990913
$875$	0	0
$876$	176.292	5.95637
$877$	−3.78360	−0.127763	−0.0638816	−	0.997957i	$-0.520348\pi$
$877$	−3.78360	−0.127763	−0.0638816	+	0.997957i	$0.520348\pi$
$878$	22.6253	0.763568
$879$	−12.9941	−0.438279
$880$	−0.513002	−0.0172933
$881$	−21.7700	−0.733451	−0.366726	−	0.930329i	$-0.619521\pi$
$881$	−21.7700	−0.733451	−0.366726	+	0.930329i	$0.619521\pi$
$882$	0	0
$883$	−55.6895	−1.87410	−0.937050	−	0.349195i	$-0.886455\pi$
$883$	−55.6895	−1.87410	−0.937050	+	0.349195i	$0.886455\pi$
$884$	22.9433	0.771665
$885$	2.28592	0.0768402
$886$	−40.6802	−1.36668
$887$	1.97646	0.0663629	0.0331814	−	0.999449i	$-0.489436\pi$
$887$	1.97646	0.0663629	0.0331814	+	0.999449i	$0.489436\pi$
$888$	−102.470	−3.43868
$889$	0	0
$890$	14.0366	0.470510
$891$	−80.6134	−2.70065
$892$	−35.5238	−1.18942
$893$	7.54326	0.252426
$894$	124.963	4.17938
$895$	8.44790	0.282382
$896$	0	0
$897$	−40.4222	−1.34966
$898$	21.0081	0.701048
$899$	−19.1918	−0.640082
$900$	−121.742	−4.05806
$901$	−9.40824	−0.313434
$902$	4.45999	0.148502
$903$	0	0
$904$	44.3422	1.47480
$905$	−10.0713	−0.334781
$906$	10.5484	0.350446
$907$	−35.4440	−1.17690	−0.588450	−	0.808534i	$-0.700262\pi$
$907$	−35.4440	−1.17690	−0.588450	+	0.808534i	$0.700262\pi$
$908$	−73.2534	−2.43100
$909$	−101.198	−3.35652
$910$	0	0
$911$	−14.0067	−0.464063	−0.232032	−	0.972708i	$-0.574537\pi$
$911$	−14.0067	−0.464063	−0.232032	+	0.972708i	$0.574537\pi$
$912$	−5.07658	−0.168102
$913$	29.7119	0.983320
$914$	46.5600	1.54007
$915$	−25.6763	−0.848831
$916$	−28.8721	−0.953960
$917$	0	0
$918$	81.5369	2.69112
$919$	31.0357	1.02377	0.511887	−	0.859053i	$-0.328947\pi$
$919$	31.0357	1.02377	0.511887	+	0.859053i	$0.328947\pi$
$920$	5.98140	0.197201
$921$	64.6235	2.12942
$922$	38.6086	1.27151
$923$	−38.3434	−1.26209
$924$	0	0
$925$	−50.1952	−1.65041
$926$	31.6351	1.03959
$927$	58.0724	1.90735
$928$	−12.1579	−0.399101
$929$	34.4139	1.12908	0.564542	−	0.825404i	$-0.309053\pi$
$929$	34.4139	1.12908	0.564542	+	0.825404i	$0.309053\pi$
$930$	56.7381	1.86052
$931$	0	0
$932$	−30.9086	−1.01244
$933$	−42.3633	−1.38691
$934$	−22.6290	−0.740444
$935$	−2.74804	−0.0898705
$936$	91.2658	2.98311
$937$	−32.2511	−1.05360	−0.526798	−	0.849990i	$-0.676608\pi$
$937$	−32.2511	−1.05360	−0.526798	+	0.849990i	$0.676608\pi$
$938$	0	0
$939$	−46.7802	−1.52662
$940$	−4.19934	−0.136967
$941$	38.8631	1.26690	0.633451	−	0.773783i	$-0.281638\pi$
$941$	38.8631	1.26690	0.633451	+	0.773783i	$0.281638\pi$
$942$	1.24428	0.0405407
$943$	2.97065	0.0967376
$944$	0.297671	0.00968838
$945$	0	0
$946$	−6.54213	−0.212703
$947$	−15.2868	−0.496754	−0.248377	−	0.968663i	$-0.579897\pi$
$947$	−15.2868	−0.496754	−0.248377	+	0.968663i	$0.579897\pi$
$948$	55.8155	1.81280
$949$	−64.7388	−2.10151
$950$	43.5313	1.41234
$951$	44.8846	1.45548
$952$	0	0
$953$	−1.45687	−0.0471925	−0.0235963	−	0.999722i	$-0.507512\pi$
$953$	−1.45687	−0.0471925	−0.0235963	+	0.999722i	$0.507512\pi$
$954$	−102.040	−3.30368
$955$	12.3417	0.399367
$956$	84.4065	2.72990
$957$	13.5452	0.437855
$958$	−67.7532	−2.18901
$959$	0	0
$960$	34.1535	1.10230
$961$	59.8207	1.92970
$962$	102.598	3.30791
$963$	−10.4692	−0.337365
$964$	42.0970	1.35585
$965$	5.24716	0.168912
$966$	0	0
$967$	16.6851	0.536557	0.268279	−	0.963341i	$-0.413545\pi$
$967$	16.6851	0.536557	0.268279	+	0.963341i	$0.413545\pi$
$968$	18.7951	0.604097
$969$	−27.1941	−0.873601
$970$	−22.1948	−0.712633
$971$	46.0317	1.47723	0.738613	−	0.674129i	$-0.235481\pi$
$971$	46.0317	1.47723	0.738613	+	0.674129i	$0.235481\pi$
$972$	258.073	8.27770
$973$	0	0
$974$	58.4870	1.87404
$975$	60.0662	1.92366
$976$	−3.34356	−0.107025
$977$	17.5541	0.561604	0.280802	−	0.959766i	$-0.409400\pi$
$977$	17.5541	0.561604	0.280802	+	0.959766i	$0.409400\pi$
$978$	134.243	4.29263
$979$	−15.8578	−0.506819
$980$	0	0
$981$	132.597	4.23350
$982$	84.8638	2.70811
$983$	−12.7193	−0.405684	−0.202842	−	0.979212i	$-0.565018\pi$
$983$	−12.7193	−0.405684	−0.202842	+	0.979212i	$0.565018\pi$
$984$	−9.01151	−0.287277
$985$	−2.45785	−0.0783135
$986$	−8.36347	−0.266347
$987$	0	0
$988$	−54.4794	−1.73322
$989$	−4.35749	−0.138560
$990$	−29.8048	−0.947260
$991$	−20.6234	−0.655124	−0.327562	−	0.944830i	$-0.606227\pi$
$991$	−20.6234	−0.655124	−0.327562	+	0.944830i	$0.606227\pi$
$992$	57.5342	1.82671
$993$	53.2809	1.69082
$994$	0	0
$995$	4.89963	0.155329
$996$	−163.684	−5.18651
$997$	−35.5254	−1.12510	−0.562551	−	0.826763i	$-0.690180\pi$
$997$	−35.5254	−1.12510	−0.562551	+	0.826763i	$0.690180\pi$
$998$	55.3898	1.75333
$999$	223.250	7.06332

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2009.2.a.u.1.4	yes	20
7.6	odd	2		2009.2.a.t.1.4	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2009.2.a.t.1.4	✓	20	7.6	odd	2
2009.2.a.u.1.4	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)

\(f(q)\)	\(=\)	\(q-2.27121 q^{2} +3.42520 q^{3} +3.15839 q^{4} -0.765314 q^{5} -7.77934 q^{6} -2.63095 q^{8} +8.73199 q^{9} +O(q^{10})\)	\(q-2.27121 q^{2} +3.42520 q^{3} +3.15839 q^{4} -0.765314 q^{5} -7.77934 q^{6} -2.63095 q^{8} +8.73199 q^{9} +1.73819 q^{10} -1.96371 q^{11} +10.8181 q^{12} -3.97267 q^{13} -2.62135 q^{15} -0.341352 q^{16} -1.82855 q^{17} -19.8322 q^{18} +4.34193 q^{19} -2.41716 q^{20} +4.45999 q^{22} +2.97065 q^{23} -9.01151 q^{24} -4.41429 q^{25} +9.02277 q^{26} +19.6332 q^{27} -2.01383 q^{29} +5.95364 q^{30} +9.52999 q^{31} +6.03717 q^{32} -6.72609 q^{33} +4.15301 q^{34} +27.5790 q^{36} +11.3711 q^{37} -9.86144 q^{38} -13.6072 q^{39} +2.01350 q^{40} +1.00000 q^{41} -1.46685 q^{43} -6.20216 q^{44} -6.68271 q^{45} -6.74696 q^{46} +1.73730 q^{47} -1.16920 q^{48} +10.0258 q^{50} -6.26314 q^{51} -12.5473 q^{52} +5.14520 q^{53} -44.5911 q^{54} +1.50285 q^{55} +14.8720 q^{57} +4.57383 q^{58} -0.872037 q^{59} -8.27925 q^{60} +9.79505 q^{61} -21.6446 q^{62} -13.0290 q^{64} +3.04034 q^{65} +15.2764 q^{66} +4.01202 q^{67} -5.77527 q^{68} +10.1751 q^{69} +9.65178 q^{71} -22.9734 q^{72} +16.2960 q^{73} -25.8260 q^{74} -15.1198 q^{75} +13.7135 q^{76} +30.9048 q^{78} +5.15945 q^{79} +0.261241 q^{80} +41.0516 q^{81} -2.27121 q^{82} -15.1305 q^{83} +1.39941 q^{85} +3.33152 q^{86} -6.89777 q^{87} +5.16641 q^{88} +8.07545 q^{89} +15.1778 q^{90} +9.38246 q^{92} +32.6421 q^{93} -3.94578 q^{94} -3.32294 q^{95} +20.6785 q^{96} -12.7689 q^{97} -17.1471 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 2 q^{2} + 8 q^{3} + 18 q^{4} + 8 q^{5} + 12 q^{6} - 6 q^{8} + 16 q^{9}+O(q^{10}) \) 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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