LMFDB - Embedded newform 6422.2.a.bl.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6422,2,Mod(1,6422)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6422, 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("6422.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6422 = 2 \cdot 13^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6422.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:	$51.2799281781$
Analytic rank:	$1$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - x^{8} - 12x^{7} + 14x^{6} + 35x^{5} - 35x^{4} - 28x^{3} + 10x^{2} + 4x - 1 $ x^9 - x^8 - 12x^7 + 14x^6 + 35x^5 - 35x^4 - 28x^3 + 10x^2 + 4*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$0.219034$ of defining polynomial
Character	$\chi$	$=$	6422.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.28668 q^{3} +1.00000 q^{4} +4.29839 q^{5} +1.28668 q^{6} -4.70931 q^{7} +1.00000 q^{8} -1.34446 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +1.28668 q^{3} +1.00000 q^{4} +4.29839 q^{5} +1.28668 q^{6} -4.70931 q^{7} +1.00000 q^{8} -1.34446 q^{9} +4.29839 q^{10} -3.40286 q^{11} +1.28668 q^{12} -4.70931 q^{14} +5.53065 q^{15} +1.00000 q^{16} -2.91994 q^{17} -1.34446 q^{18} -1.00000 q^{19} +4.29839 q^{20} -6.05937 q^{21} -3.40286 q^{22} -5.88048 q^{23} +1.28668 q^{24} +13.4762 q^{25} -5.58992 q^{27} -4.70931 q^{28} -6.28981 q^{29} +5.53065 q^{30} -4.34971 q^{31} +1.00000 q^{32} -4.37839 q^{33} -2.91994 q^{34} -20.2425 q^{35} -1.34446 q^{36} -9.73303 q^{37} -1.00000 q^{38} +4.29839 q^{40} -1.32819 q^{41} -6.05937 q^{42} -8.24159 q^{43} -3.40286 q^{44} -5.77901 q^{45} -5.88048 q^{46} -1.55356 q^{47} +1.28668 q^{48} +15.1776 q^{49} +13.4762 q^{50} -3.75702 q^{51} +8.55208 q^{53} -5.58992 q^{54} -14.6268 q^{55} -4.70931 q^{56} -1.28668 q^{57} -6.28981 q^{58} -6.50672 q^{59} +5.53065 q^{60} +6.39817 q^{61} -4.34971 q^{62} +6.33147 q^{63} +1.00000 q^{64} -4.37839 q^{66} +1.20961 q^{67} -2.91994 q^{68} -7.56628 q^{69} -20.2425 q^{70} +7.74497 q^{71} -1.34446 q^{72} -0.805061 q^{73} -9.73303 q^{74} +17.3395 q^{75} -1.00000 q^{76} +16.0251 q^{77} +1.33306 q^{79} +4.29839 q^{80} -3.15905 q^{81} -1.32819 q^{82} -5.24803 q^{83} -6.05937 q^{84} -12.5510 q^{85} -8.24159 q^{86} -8.09297 q^{87} -3.40286 q^{88} +9.68229 q^{89} -5.77901 q^{90} -5.88048 q^{92} -5.59668 q^{93} -1.55356 q^{94} -4.29839 q^{95} +1.28668 q^{96} +6.38880 q^{97} +15.1776 q^{98} +4.57501 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 9 q^{2} + q^{3} + 9 q^{4} + q^{5} + q^{6} - 13 q^{7} + 9 q^{8} - 2 q^{9}+O(q^{10}) $ 9 * q + 9 * q^2 + q^3 + 9 * q^4 + q^5 + q^6 - 13 * q^7 + 9 * q^8 - 2 * q^9	$ 9 q + 9 q^{2} + q^{3} + 9 q^{4} + q^{5} + q^{6} - 13 q^{7} + 9 q^{8} - 2 q^{9} + q^{10} - 3 q^{11} + q^{12} - 13 q^{14} - 3 q^{15} + 9 q^{16} - 8 q^{17} - 2 q^{18} - 9 q^{19} + q^{20} - 24 q^{21} - 3 q^{22} - 10 q^{23} + q^{24} + 8 q^{25} + 10 q^{27} - 13 q^{28} - 20 q^{29} - 3 q^{30} - q^{31} + 9 q^{32} + 2 q^{33} - 8 q^{34} + 4 q^{35} - 2 q^{36} - 15 q^{37} - 9 q^{38} + q^{40} - 19 q^{41} - 24 q^{42} - 16 q^{43} - 3 q^{44} - 15 q^{45} - 10 q^{46} - 18 q^{47} + q^{48} + 18 q^{49} + 8 q^{50} - 11 q^{51} + 17 q^{53} + 10 q^{54} - 26 q^{55} - 13 q^{56} - q^{57} - 20 q^{58} - 24 q^{59} - 3 q^{60} + 6 q^{61} - q^{62} - q^{63} + 9 q^{64} + 2 q^{66} - 29 q^{67} - 8 q^{68} - 12 q^{69} + 4 q^{70} - 23 q^{71} - 2 q^{72} - 38 q^{73} - 15 q^{74} + 11 q^{75} - 9 q^{76} - 40 q^{77} - 20 q^{79} + q^{80} - 31 q^{81} - 19 q^{82} - 20 q^{83} - 24 q^{84} - 39 q^{85} - 16 q^{86} - 10 q^{87} - 3 q^{88} + 7 q^{89} - 15 q^{90} - 10 q^{92} - 11 q^{93} - 18 q^{94} - q^{95} + q^{96} - 28 q^{97} + 18 q^{98} - 25 q^{99}+O(q^{100}) $ 9 * q + 9 * q^2 + q^3 + 9 * q^4 + q^5 + q^6 - 13 * q^7 + 9 * q^8 - 2 * q^9 + q^10 - 3 * q^11 + q^12 - 13 * q^14 - 3 * q^15 + 9 * q^16 - 8 * q^17 - 2 * q^18 - 9 * q^19 + q^20 - 24 * q^21 - 3 * q^22 - 10 * q^23 + q^24 + 8 * q^25 + 10 * q^27 - 13 * q^28 - 20 * q^29 - 3 * q^30 - q^31 + 9 * q^32 + 2 * q^33 - 8 * q^34 + 4 * q^35 - 2 * q^36 - 15 * q^37 - 9 * q^38 + q^40 - 19 * q^41 - 24 * q^42 - 16 * q^43 - 3 * q^44 - 15 * q^45 - 10 * q^46 - 18 * q^47 + q^48 + 18 * q^49 + 8 * q^50 - 11 * q^51 + 17 * q^53 + 10 * q^54 - 26 * q^55 - 13 * q^56 - q^57 - 20 * q^58 - 24 * q^59 - 3 * q^60 + 6 * q^61 - q^62 - q^63 + 9 * q^64 + 2 * q^66 - 29 * q^67 - 8 * q^68 - 12 * q^69 + 4 * q^70 - 23 * q^71 - 2 * q^72 - 38 * q^73 - 15 * q^74 + 11 * q^75 - 9 * q^76 - 40 * q^77 - 20 * q^79 + q^80 - 31 * q^81 - 19 * q^82 - 20 * q^83 - 24 * q^84 - 39 * q^85 - 16 * q^86 - 10 * q^87 - 3 * q^88 + 7 * q^89 - 15 * q^90 - 10 * q^92 - 11 * q^93 - 18 * q^94 - q^95 + q^96 - 28 * q^97 + 18 * q^98 - 25 * 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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	1.28668	0.742864	0.371432	−	0.928460i	$-0.378867\pi$
$3$	1.28668	0.742864	0.371432	+	0.928460i	$0.378867\pi$
$4$	1.00000	0.500000
$5$	4.29839	1.92230	0.961150	−	0.276028i	$-0.0890184\pi$
$5$	4.29839	1.92230	0.961150	+	0.276028i	$0.0890184\pi$
$6$	1.28668	0.525284
$7$	−4.70931	−1.77995	−0.889976	−	0.456008i	$-0.849279\pi$
$7$	−4.70931	−1.77995	−0.889976	+	0.456008i	$0.849279\pi$
$8$	1.00000	0.353553
$9$	−1.34446	−0.448153
$10$	4.29839	1.35927
$11$	−3.40286	−1.02600	−0.513001	−	0.858388i	$-0.671466\pi$
$11$	−3.40286	−1.02600	−0.513001	+	0.858388i	$0.671466\pi$
$12$	1.28668	0.371432
$13$	0	0
$13$	0	0
$14$	−4.70931	−1.25862
$15$	5.53065	1.42801
$16$	1.00000	0.250000
$17$	−2.91994	−0.708189	−0.354094	−	0.935210i	$-0.615211\pi$
$17$	−2.91994	−0.708189	−0.354094	+	0.935210i	$0.615211\pi$
$18$	−1.34446	−0.316892
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	4.29839	0.961150
$21$	−6.05937	−1.32226
$22$	−3.40286	−0.725493
$23$	−5.88048	−1.22616	−0.613082	−	0.790019i	$-0.710071\pi$
$23$	−5.88048	−1.22616	−0.613082	+	0.790019i	$0.710071\pi$
$24$	1.28668	0.262642
$25$	13.4762	2.69523
$26$	0	0
$27$	−5.58992	−1.07578
$28$	−4.70931	−0.889976
$29$	−6.28981	−1.16799	−0.583995	−	0.811757i	$-0.698511\pi$
$29$	−6.28981	−1.16799	−0.583995	+	0.811757i	$0.698511\pi$
$30$	5.53065	1.00975
$31$	−4.34971	−0.781231	−0.390616	−	0.920554i	$-0.627738\pi$
$31$	−4.34971	−0.781231	−0.390616	+	0.920554i	$0.627738\pi$
$32$	1.00000	0.176777
$33$	−4.37839	−0.762180
$34$	−2.91994	−0.500765
$35$	−20.2425	−3.42160
$36$	−1.34446	−0.224077
$37$	−9.73303	−1.60010	−0.800050	−	0.599933i	$-0.795194\pi$
$37$	−9.73303	−1.60010	−0.800050	+	0.599933i	$0.795194\pi$
$38$	−1.00000	−0.162221
$39$	0	0
$40$	4.29839	0.679635
$41$	−1.32819	−0.207429	−0.103714	−	0.994607i	$-0.533073\pi$
$41$	−1.32819	−0.207429	−0.103714	+	0.994607i	$0.533073\pi$
$42$	−6.05937	−0.934981
$43$	−8.24159	−1.25683	−0.628415	−	0.777878i	$-0.716296\pi$
$43$	−8.24159	−1.25683	−0.628415	+	0.777878i	$0.716296\pi$
$44$	−3.40286	−0.513001
$45$	−5.77901	−0.861484
$46$	−5.88048	−0.867029
$47$	−1.55356	−0.226610	−0.113305	−	0.993560i	$-0.536144\pi$
$47$	−1.55356	−0.226610	−0.113305	+	0.993560i	$0.536144\pi$
$48$	1.28668	0.185716
$49$	15.1776	2.16823
$50$	13.4762	1.90582
$51$	−3.75702	−0.526088
$52$	0	0
$53$	8.55208	1.17472	0.587359	−	0.809327i	$-0.300168\pi$
$53$	8.55208	1.17472	0.587359	+	0.809327i	$0.300168\pi$
$54$	−5.58992	−0.760692
$55$	−14.6268	−1.97228
$56$	−4.70931	−0.629308
$57$	−1.28668	−0.170425
$58$	−6.28981	−0.825893
$59$	−6.50672	−0.847103	−0.423551	−	0.905872i	$-0.639217\pi$
$59$	−6.50672	−0.847103	−0.423551	+	0.905872i	$0.639217\pi$
$60$	5.53065	0.714003
$61$	6.39817	0.819202	0.409601	−	0.912265i	$-0.365668\pi$
$61$	6.39817	0.819202	0.409601	+	0.912265i	$0.365668\pi$
$62$	−4.34971	−0.552414
$63$	6.33147	0.797691
$64$	1.00000	0.125000
$65$	0	0
$66$	−4.37839	−0.538943
$67$	1.20961	0.147777	0.0738885	−	0.997267i	$-0.476459\pi$
$67$	1.20961	0.147777	0.0738885	+	0.997267i	$0.476459\pi$
$68$	−2.91994	−0.354094
$69$	−7.56628	−0.910873
$70$	−20.2425	−2.41944
$71$	7.74497	0.919159	0.459580	−	0.888137i	$-0.348000\pi$
$71$	7.74497	0.919159	0.459580	+	0.888137i	$0.348000\pi$
$72$	−1.34446	−0.158446
$73$	−0.805061	−0.0942252	−0.0471126	−	0.998890i	$-0.515002\pi$
$73$	−0.805061	−0.0942252	−0.0471126	+	0.998890i	$0.515002\pi$
$74$	−9.73303	−1.13144
$75$	17.3395	2.00219
$76$	−1.00000	−0.114708
$77$	16.0251	1.82623
$78$	0	0
$79$	1.33306	0.149981	0.0749903	−	0.997184i	$-0.476107\pi$
$79$	1.33306	0.149981	0.0749903	+	0.997184i	$0.476107\pi$
$80$	4.29839	0.480575
$81$	−3.15905	−0.351006
$82$	−1.32819	−0.146674
$83$	−5.24803	−0.576047	−0.288023	−	0.957623i	$-0.592998\pi$
$83$	−5.24803	−0.576047	−0.288023	+	0.957623i	$0.592998\pi$
$84$	−6.05937	−0.661131
$85$	−12.5510	−1.36135
$86$	−8.24159	−0.888713
$87$	−8.09297	−0.867657
$88$	−3.40286	−0.362747
$89$	9.68229	1.02632	0.513160	−	0.858293i	$-0.328475\pi$
$89$	9.68229	1.02632	0.513160	+	0.858293i	$0.328475\pi$
$90$	−5.77901	−0.609161
$91$	0	0
$92$	−5.88048	−0.613082
$93$	−5.59668	−0.580349
$94$	−1.55356	−0.160238
$95$	−4.29839	−0.441006
$96$	1.28668	0.131321
$97$	6.38880	0.648684	0.324342	−	0.945940i	$-0.394857\pi$
$97$	6.38880	0.648684	0.324342	+	0.945940i	$0.394857\pi$
$98$	15.1776	1.53317
$99$	4.57501	0.459806
$100$	13.4762	1.34762
$101$	15.1763	1.51010	0.755048	−	0.655669i	$-0.227613\pi$
$101$	15.1763	1.51010	0.755048	+	0.655669i	$0.227613\pi$
$102$	−3.75702	−0.372000
$103$	6.88840	0.678734	0.339367	−	0.940654i	$-0.389787\pi$
$103$	6.88840	0.678734	0.339367	+	0.940654i	$0.389787\pi$
$104$	0	0
$105$	−26.0455	−2.54178
$106$	8.55208	0.830651
$107$	8.48450	0.820227	0.410114	−	0.912034i	$-0.365489\pi$
$107$	8.48450	0.820227	0.410114	+	0.912034i	$0.365489\pi$
$108$	−5.58992	−0.537890
$109$	8.06230	0.772228	0.386114	−	0.922451i	$-0.373817\pi$
$109$	8.06230	0.772228	0.386114	+	0.922451i	$0.373817\pi$
$110$	−14.6268	−1.39461
$111$	−12.5233	−1.18866
$112$	−4.70931	−0.444988
$113$	15.6650	1.47364	0.736820	−	0.676089i	$-0.236326\pi$
$113$	15.6650	1.47364	0.736820	+	0.676089i	$0.236326\pi$
$114$	−1.28668	−0.120508
$115$	−25.2766	−2.35705
$116$	−6.28981	−0.583995
$117$	0	0
$118$	−6.50672	−0.598992
$119$	13.7509	1.26054
$120$	5.53065	0.504877
$121$	0.579488	0.0526807
$122$	6.39817	0.579264
$123$	−1.70896	−0.154092
$124$	−4.34971	−0.390616
$125$	36.4339	3.25875
$126$	6.33147	0.564053
$127$	−14.9710	−1.32847	−0.664233	−	0.747526i	$-0.731242\pi$
$127$	−14.9710	−1.32847	−0.664233	+	0.747526i	$0.731242\pi$
$128$	1.00000	0.0883883
$129$	−10.6043	−0.933654
$130$	0	0
$131$	−15.2834	−1.33532	−0.667659	−	0.744467i	$-0.732703\pi$
$131$	−15.2834	−1.33532	−0.667659	+	0.744467i	$0.732703\pi$
$132$	−4.37839	−0.381090
$133$	4.70931	0.408349
$134$	1.20961	0.104494
$135$	−24.0277	−2.06797
$136$	−2.91994	−0.250383
$137$	−16.3085	−1.39333	−0.696663	−	0.717399i	$-0.745333\pi$
$137$	−16.3085	−1.39333	−0.696663	+	0.717399i	$0.745333\pi$
$138$	−7.56628	−0.644085
$139$	13.8239	1.17253	0.586264	−	0.810120i	$-0.300598\pi$
$139$	13.8239	1.17253	0.586264	+	0.810120i	$0.300598\pi$
$140$	−20.2425	−1.71080
$141$	−1.99893	−0.168341
$142$	7.74497	0.649944
$143$	0	0
$144$	−1.34446	−0.112038
$145$	−27.0361	−2.24522
$146$	−0.805061	−0.0666273
$147$	19.5287	1.61070
$148$	−9.73303	−0.800050
$149$	−9.71911	−0.796221	−0.398110	−	0.917338i	$-0.630334\pi$
$149$	−9.71911	−0.796221	−0.398110	+	0.917338i	$0.630334\pi$
$150$	17.3395	1.41576
$151$	−4.30710	−0.350507	−0.175253	−	0.984523i	$-0.556074\pi$
$151$	−4.30710	−0.350507	−0.175253	+	0.984523i	$0.556074\pi$
$152$	−1.00000	−0.0811107
$153$	3.92574	0.317377
$154$	16.0251	1.29134
$155$	−18.6968	−1.50176
$156$	0	0
$157$	−4.27209	−0.340950	−0.170475	−	0.985362i	$-0.554530\pi$
$157$	−4.27209	−0.340950	−0.170475	+	0.985362i	$0.554530\pi$
$158$	1.33306	0.106052
$159$	11.0038	0.872656
$160$	4.29839	0.339818
$161$	27.6930	2.18251
$162$	−3.15905	−0.248199
$163$	17.6762	1.38450	0.692252	−	0.721656i	$-0.256619\pi$
$163$	17.6762	1.38450	0.692252	+	0.721656i	$0.256619\pi$
$164$	−1.32819	−0.103714
$165$	−18.8200	−1.46514
$166$	−5.24803	−0.407326
$167$	21.3276	1.65038	0.825188	−	0.564858i	$-0.191069\pi$
$167$	21.3276	1.65038	0.825188	+	0.564858i	$0.191069\pi$
$168$	−6.05937	−0.467490
$169$	0	0
$170$	−12.5510	−0.962620
$171$	1.34446	0.102813
$172$	−8.24159	−0.628415
$173$	−6.62670	−0.503818	−0.251909	−	0.967751i	$-0.581058\pi$
$173$	−6.62670	−0.503818	−0.251909	+	0.967751i	$0.581058\pi$
$174$	−8.09297	−0.613526
$175$	−63.4634	−4.79739
$176$	−3.40286	−0.256501
$177$	−8.37205	−0.629282
$178$	9.68229	0.725718
$179$	−12.6118	−0.942653	−0.471327	−	0.881959i	$-0.656225\pi$
$179$	−12.6118	−0.942653	−0.471327	+	0.881959i	$0.656225\pi$
$180$	−5.77901	−0.430742
$181$	−8.14571	−0.605466	−0.302733	−	0.953075i	$-0.597899\pi$
$181$	−8.14571	−0.605466	−0.302733	+	0.953075i	$0.597899\pi$
$182$	0	0
$183$	8.23239	0.608556
$184$	−5.88048	−0.433515
$185$	−41.8364	−3.07587
$186$	−5.59668	−0.410368
$187$	9.93615	0.726603
$188$	−1.55356	−0.113305
$189$	26.3247	1.91484
$190$	−4.29839	−0.311838
$191$	20.6625	1.49509	0.747543	−	0.664214i	$-0.231233\pi$
$191$	20.6625	1.49509	0.747543	+	0.664214i	$0.231233\pi$
$192$	1.28668	0.0928580
$193$	−18.9801	−1.36621	−0.683107	−	0.730318i	$-0.739372\pi$
$193$	−18.9801	−1.36621	−0.683107	+	0.730318i	$0.739372\pi$
$194$	6.38880	0.458689
$195$	0	0
$196$	15.1776	1.08411
$197$	−20.9968	−1.49596	−0.747980	−	0.663722i	$-0.768976\pi$
$197$	−20.9968	−1.49596	−0.747980	+	0.663722i	$0.768976\pi$
$198$	4.57501	0.325132
$199$	−19.1855	−1.36003	−0.680013	−	0.733200i	$-0.738026\pi$
$199$	−19.1855	−1.36003	−0.680013	+	0.733200i	$0.738026\pi$
$200$	13.4762	0.952909
$201$	1.55638	0.109778
$202$	15.1763	1.06780
$203$	29.6207	2.07896
$204$	−3.75702	−0.263044
$205$	−5.70909	−0.398740
$206$	6.88840	0.479937
$207$	7.90606	0.549509
$208$	0	0
$209$	3.40286	0.235381
$210$	−26.0455	−1.79731
$211$	8.51902	0.586474	0.293237	−	0.956040i	$-0.405268\pi$
$211$	8.51902	0.586474	0.293237	+	0.956040i	$0.405268\pi$
$212$	8.55208	0.587359
$213$	9.96529	0.682810
$214$	8.48450	0.579988
$215$	−35.4256	−2.41600
$216$	−5.58992	−0.380346
$217$	20.4841	1.39055
$218$	8.06230	0.546048
$219$	−1.03585	−0.0699965
$220$	−14.6268	−0.986142
$221$	0	0
$222$	−12.5233	−0.840508
$223$	−20.5059	−1.37318	−0.686588	−	0.727047i	$-0.740892\pi$
$223$	−20.5059	−1.37318	−0.686588	+	0.727047i	$0.740892\pi$
$224$	−4.70931	−0.314654
$225$	−18.1182	−1.20788
$226$	15.6650	1.04202
$227$	−8.09412	−0.537226	−0.268613	−	0.963248i	$-0.586565\pi$
$227$	−8.09412	−0.537226	−0.268613	+	0.963248i	$0.586565\pi$
$228$	−1.28668	−0.0852123
$229$	11.0960	0.733244	0.366622	−	0.930370i	$-0.380514\pi$
$229$	11.0960	0.733244	0.366622	+	0.930370i	$0.380514\pi$
$230$	−25.2766	−1.66669
$231$	20.6192	1.35664
$232$	−6.28981	−0.412947
$233$	0.692385	0.0453596	0.0226798	−	0.999743i	$-0.492780\pi$
$233$	0.692385	0.0453596	0.0226798	+	0.999743i	$0.492780\pi$
$234$	0	0
$235$	−6.67782	−0.435613
$236$	−6.50672	−0.423551
$237$	1.71521	0.111415
$238$	13.7509	0.891338
$239$	−5.05647	−0.327076	−0.163538	−	0.986537i	$-0.552291\pi$
$239$	−5.05647	−0.327076	−0.163538	+	0.986537i	$0.552291\pi$
$240$	5.53065	0.357002
$241$	1.17983	0.0759998	0.0379999	−	0.999278i	$-0.487901\pi$
$241$	1.17983	0.0759998	0.0379999	+	0.999278i	$0.487901\pi$
$242$	0.579488	0.0372509
$243$	12.7051	0.815031
$244$	6.39817	0.409601
$245$	65.2393	4.16798
$246$	−1.70896	−0.108959
$247$	0	0
$248$	−4.34971	−0.276207
$249$	−6.75253	−0.427924
$250$	36.4339	2.30428
$251$	16.1831	1.02147	0.510735	−	0.859738i	$-0.329373\pi$
$251$	16.1831	1.02147	0.510735	+	0.859738i	$0.329373\pi$
$252$	6.33147	0.398845
$253$	20.0105	1.25805
$254$	−14.9710	−0.939367
$255$	−16.1491	−1.01130
$256$	1.00000	0.0625000
$257$	14.5265	0.906141	0.453070	−	0.891475i	$-0.350329\pi$
$257$	14.5265	0.906141	0.453070	+	0.891475i	$0.350329\pi$
$258$	−10.6043	−0.660193
$259$	45.8359	2.84810
$260$	0	0
$261$	8.45640	0.523438
$262$	−15.2834	−0.944212
$263$	14.1077	0.869920	0.434960	−	0.900450i	$-0.356763\pi$
$263$	14.1077	0.869920	0.434960	+	0.900450i	$0.356763\pi$
$264$	−4.37839	−0.269471
$265$	36.7602	2.25816
$266$	4.70931	0.288746
$267$	12.4580	0.762417
$268$	1.20961	0.0738885
$269$	10.0839	0.614825	0.307413	−	0.951576i	$-0.400537\pi$
$269$	10.0839	0.614825	0.307413	+	0.951576i	$0.400537\pi$
$270$	−24.0277	−1.46228
$271$	−11.6395	−0.707052	−0.353526	−	0.935425i	$-0.615017\pi$
$271$	−11.6395	−0.707052	−0.353526	+	0.935425i	$0.615017\pi$
$272$	−2.91994	−0.177047
$273$	0	0
$274$	−16.3085	−0.985230
$275$	−45.8576	−2.76532
$276$	−7.56628	−0.455437
$277$	−0.529203	−0.0317967	−0.0158984	−	0.999874i	$-0.505061\pi$
$277$	−0.529203	−0.0317967	−0.0158984	+	0.999874i	$0.505061\pi$
$278$	13.8239	0.829102
$279$	5.84801	0.350111
$280$	−20.2425	−1.20972
$281$	−27.0663	−1.61464	−0.807320	−	0.590113i	$-0.799083\pi$
$281$	−27.0663	−1.61464	−0.807320	+	0.590113i	$0.799083\pi$
$282$	−1.99893	−0.119035
$283$	6.96429	0.413984	0.206992	−	0.978343i	$-0.433633\pi$
$283$	6.96429	0.413984	0.206992	+	0.978343i	$0.433633\pi$
$284$	7.74497	0.459580
$285$	−5.53065	−0.327607
$286$	0	0
$287$	6.25487	0.369214
$288$	−1.34446	−0.0792230
$289$	−8.47397	−0.498469
$290$	−27.0361	−1.58761
$291$	8.22032	0.481884
$292$	−0.805061	−0.0471126
$293$	−25.8717	−1.51144	−0.755719	−	0.654896i	$-0.772712\pi$
$293$	−25.8717	−1.51144	−0.755719	+	0.654896i	$0.772712\pi$
$294$	19.5287	1.13894
$295$	−27.9684	−1.62838
$296$	−9.73303	−0.565721
$297$	19.0217	1.10375
$298$	−9.71911	−0.563013
$299$	0	0
$300$	17.3395	1.00110
$301$	38.8122	2.23710
$302$	−4.30710	−0.247846
$303$	19.5270	1.12180
$304$	−1.00000	−0.0573539
$305$	27.5019	1.57475
$306$	3.92574	0.224419
$307$	−22.4406	−1.28075	−0.640377	−	0.768061i	$-0.721222\pi$
$307$	−22.4406	−1.28075	−0.640377	+	0.768061i	$0.721222\pi$
$308$	16.0251	0.913117
$309$	8.86315	0.504207
$310$	−18.6968	−1.06190
$311$	6.66590	0.377989	0.188994	−	0.981978i	$-0.439477\pi$
$311$	6.66590	0.377989	0.188994	+	0.981978i	$0.439477\pi$
$312$	0	0
$313$	−11.5401	−0.652283	−0.326142	−	0.945321i	$-0.605749\pi$
$313$	−11.5401	−0.652283	−0.326142	+	0.945321i	$0.605749\pi$
$314$	−4.27209	−0.241088
$315$	27.2152	1.53340
$316$	1.33306	0.0749903
$317$	−28.1127	−1.57897	−0.789484	−	0.613772i	$-0.789652\pi$
$317$	−28.1127	−1.57897	−0.789484	+	0.613772i	$0.789652\pi$
$318$	11.0038	0.617061
$319$	21.4034	1.19836
$320$	4.29839	0.240287
$321$	10.9168	0.609317
$322$	27.6930	1.54327
$323$	2.91994	0.162470
$324$	−3.15905	−0.175503
$325$	0	0
$326$	17.6762	0.978992
$327$	10.3736	0.573660
$328$	−1.32819	−0.0733372
$329$	7.31620	0.403355
$330$	−18.8200	−1.03601
$331$	−7.76517	−0.426813	−0.213406	−	0.976964i	$-0.568456\pi$
$331$	−7.76517	−0.426813	−0.213406	+	0.976964i	$0.568456\pi$
$332$	−5.24803	−0.288023
$333$	13.0857	0.717090
$334$	21.3276	1.16699
$335$	5.19936	0.284072
$336$	−6.05937	−0.330566
$337$	−31.9205	−1.73882	−0.869410	−	0.494091i	$-0.835501\pi$
$337$	−31.9205	−1.73882	−0.869410	+	0.494091i	$0.835501\pi$
$338$	0	0
$339$	20.1558	1.09471
$340$	−12.5510	−0.680675
$341$	14.8015	0.801545
$342$	1.34446	0.0727000
$343$	−38.5108	−2.07939
$344$	−8.24159	−0.444357
$345$	−32.5228	−1.75097
$346$	−6.62670	−0.356253
$347$	17.3924	0.933670	0.466835	−	0.884344i	$-0.345394\pi$
$347$	17.3924	0.933670	0.466835	+	0.884344i	$0.345394\pi$
$348$	−8.09297	−0.433829
$349$	−0.938581	−0.0502411	−0.0251205	−	0.999684i	$-0.507997\pi$
$349$	−0.938581	−0.0502411	−0.0251205	+	0.999684i	$0.507997\pi$
$350$	−63.4634	−3.39226
$351$	0	0
$352$	−3.40286	−0.181373
$353$	−9.63690	−0.512920	−0.256460	−	0.966555i	$-0.582556\pi$
$353$	−9.63690	−0.512920	−0.256460	+	0.966555i	$0.582556\pi$
$354$	−8.37205	−0.444970
$355$	33.2909	1.76690
$356$	9.68229	0.513160
$357$	17.6930	0.936411
$358$	−12.6118	−0.666557
$359$	−27.0024	−1.42513	−0.712567	−	0.701604i	$-0.752468\pi$
$359$	−27.0024	−1.42513	−0.712567	+	0.701604i	$0.752468\pi$
$360$	−5.77901	−0.304581
$361$	1.00000	0.0526316
$362$	−8.14571	−0.428129
$363$	0.745614	0.0391346
$364$	0	0
$365$	−3.46047	−0.181129
$366$	8.23239	0.430314
$367$	−0.507227	−0.0264770	−0.0132385	−	0.999912i	$-0.504214\pi$
$367$	−0.507227	−0.0264770	−0.0132385	+	0.999912i	$0.504214\pi$
$368$	−5.88048	−0.306541
$369$	1.78570	0.0929599
$370$	−41.8364	−2.17497
$371$	−40.2744	−2.09094
$372$	−5.59668	−0.290174
$373$	15.1224	0.783009	0.391505	−	0.920176i	$-0.371955\pi$
$373$	15.1224	0.783009	0.391505	+	0.920176i	$0.371955\pi$
$374$	9.93615	0.513786
$375$	46.8787	2.42080
$376$	−1.55356	−0.0801188
$377$	0	0
$378$	26.3247	1.35399
$379$	33.3864	1.71494	0.857472	−	0.514530i	$-0.172034\pi$
$379$	33.3864	1.71494	0.857472	+	0.514530i	$0.172034\pi$
$380$	−4.29839	−0.220503
$381$	−19.2629	−0.986869
$382$	20.6625	1.05719
$383$	−20.3998	−1.04238	−0.521191	−	0.853440i	$-0.674512\pi$
$383$	−20.3998	−1.04238	−0.521191	+	0.853440i	$0.674512\pi$
$384$	1.28668	0.0656605
$385$	68.8823	3.51057
$386$	−18.9801	−0.966060
$387$	11.0805	0.563252
$388$	6.38880	0.324342
$389$	30.0088	1.52151	0.760754	−	0.649041i	$-0.224830\pi$
$389$	30.0088	1.52151	0.760754	+	0.649041i	$0.224830\pi$
$390$	0	0
$391$	17.1706	0.868356
$392$	15.1776	0.766584
$393$	−19.6648	−0.991959
$394$	−20.9968	−1.05780
$395$	5.73000	0.288307
$396$	4.57501	0.229903
$397$	−12.6058	−0.632668	−0.316334	−	0.948648i	$-0.602452\pi$
$397$	−12.6058	−0.632668	−0.316334	+	0.948648i	$0.602452\pi$
$398$	−19.1855	−0.961684
$399$	6.05937	0.303348
$400$	13.4762	0.673808
$401$	−20.4344	−1.02044	−0.510222	−	0.860043i	$-0.670437\pi$
$401$	−20.4344	−1.02044	−0.510222	+	0.860043i	$0.670437\pi$
$402$	1.55638	0.0776249
$403$	0	0
$404$	15.1763	0.755048
$405$	−13.5788	−0.674738
$406$	29.6207	1.47005
$407$	33.1202	1.64171
$408$	−3.75702	−0.186000
$409$	8.77027	0.433662	0.216831	−	0.976209i	$-0.430428\pi$
$409$	8.77027	0.433662	0.216831	+	0.976209i	$0.430428\pi$
$410$	−5.70909	−0.281952
$411$	−20.9837	−1.03505
$412$	6.88840	0.339367
$413$	30.6422	1.50780
$414$	7.90606	0.388562
$415$	−22.5581	−1.10733
$416$	0	0
$417$	17.7869	0.871028
$418$	3.40286	0.166440
$419$	−24.2608	−1.18522	−0.592609	−	0.805490i	$-0.701902\pi$
$419$	−24.2608	−1.18522	−0.592609	+	0.805490i	$0.701902\pi$
$420$	−26.0455	−1.27089
$421$	−36.7007	−1.78868	−0.894341	−	0.447386i	$-0.852355\pi$
$421$	−36.7007	−1.78868	−0.894341	+	0.447386i	$0.852355\pi$
$422$	8.51902	0.414699
$423$	2.08870	0.101556
$424$	8.55208	0.415326
$425$	−39.3496	−1.90873
$426$	9.96529	0.482820
$427$	−30.1310	−1.45814
$428$	8.48450	0.410114
$429$	0	0
$430$	−35.4256	−1.70837
$431$	−14.1188	−0.680078	−0.340039	−	0.940411i	$-0.610440\pi$
$431$	−14.1188	−0.680078	−0.340039	+	0.940411i	$0.610440\pi$
$432$	−5.58992	−0.268945
$433$	33.7529	1.62206	0.811031	−	0.585004i	$-0.198907\pi$
$433$	33.7529	1.62206	0.811031	+	0.585004i	$0.198907\pi$
$434$	20.4841	0.983270
$435$	−34.7867	−1.66790
$436$	8.06230	0.386114
$437$	5.88048	0.281301
$438$	−1.03585	−0.0494950
$439$	20.5686	0.981688	0.490844	−	0.871247i	$-0.336689\pi$
$439$	20.5686	0.981688	0.490844	+	0.871247i	$0.336689\pi$
$440$	−14.6268	−0.697307
$441$	−20.4057	−0.971698
$442$	0	0
$443$	−26.0671	−1.23849	−0.619243	−	0.785199i	$-0.712561\pi$
$443$	−26.0671	−1.23849	−0.619243	+	0.785199i	$0.712561\pi$
$444$	−12.5233	−0.594329
$445$	41.6183	1.97290
$446$	−20.5059	−0.970982
$447$	−12.5054	−0.591484
$448$	−4.70931	−0.222494
$449$	−8.77925	−0.414319	−0.207159	−	0.978307i	$-0.566422\pi$
$449$	−8.77925	−0.414319	−0.207159	+	0.978307i	$0.566422\pi$
$450$	−18.1182	−0.854098
$451$	4.51966	0.212823
$452$	15.6650	0.736820
$453$	−5.54185	−0.260379
$454$	−8.09412	−0.379876
$455$	0	0
$456$	−1.28668	−0.0602542
$457$	−24.1874	−1.13144	−0.565720	−	0.824597i	$-0.691402\pi$
$457$	−24.1874	−1.13144	−0.565720	+	0.824597i	$0.691402\pi$
$458$	11.0960	0.518482
$459$	16.3222	0.761856
$460$	−25.2766	−1.17853
$461$	−23.3854	−1.08917	−0.544584	−	0.838706i	$-0.683312\pi$
$461$	−23.3854	−1.08917	−0.544584	+	0.838706i	$0.683312\pi$
$462$	20.6192	0.959292
$463$	22.9038	1.06443	0.532215	−	0.846609i	$-0.321360\pi$
$463$	22.9038	1.06443	0.532215	+	0.846609i	$0.321360\pi$
$464$	−6.28981	−0.291997
$465$	−24.0567	−1.11560
$466$	0.692385	0.0320741
$467$	−23.2869	−1.07759	−0.538794	−	0.842438i	$-0.681120\pi$
$467$	−23.2869	−1.07759	−0.538794	+	0.842438i	$0.681120\pi$
$468$	0	0
$469$	−5.69641	−0.263036
$470$	−6.67782	−0.308025
$471$	−5.49680	−0.253279
$472$	−6.50672	−0.299496
$473$	28.0450	1.28951
$474$	1.71521	0.0787824
$475$	−13.4762	−0.618329
$476$	13.7509	0.630271
$477$	−11.4979	−0.526453
$478$	−5.05647	−0.231277
$479$	25.3107	1.15648	0.578238	−	0.815868i	$-0.303741\pi$
$479$	25.3107	1.15648	0.578238	+	0.815868i	$0.303741\pi$
$480$	5.53065	0.252438
$481$	0	0
$482$	1.17983	0.0537400
$483$	35.6320	1.62131
$484$	0.579488	0.0263404
$485$	27.4615	1.24696
$486$	12.7051	0.576314
$487$	21.4003	0.969741	0.484870	−	0.874586i	$-0.338867\pi$
$487$	21.4003	0.969741	0.484870	+	0.874586i	$0.338867\pi$
$488$	6.39817	0.289632
$489$	22.7435	1.02850
$490$	65.2393	2.94721
$491$	36.3776	1.64170	0.820849	−	0.571146i	$-0.193501\pi$
$491$	36.3776	1.64170	0.820849	+	0.571146i	$0.193501\pi$
$492$	−1.70896	−0.0770458
$493$	18.3659	0.827157
$494$	0	0
$495$	19.6652	0.883885
$496$	−4.34971	−0.195308
$497$	−36.4735	−1.63606
$498$	−6.75253	−0.302588
$499$	1.49097	0.0667449	0.0333724	−	0.999443i	$-0.489375\pi$
$499$	1.49097	0.0667449	0.0333724	+	0.999443i	$0.489375\pi$
$500$	36.4339	1.62937
$501$	27.4417	1.22601
$502$	16.1831	0.722288
$503$	−14.6143	−0.651620	−0.325810	−	0.945435i	$-0.605637\pi$
$503$	−14.6143	−0.651620	−0.325810	+	0.945435i	$0.605637\pi$
$504$	6.33147	0.282026
$505$	65.2336	2.90286
$506$	20.0105	0.889574
$507$	0	0
$508$	−14.9710	−0.664233
$509$	29.9572	1.32783	0.663915	−	0.747808i	$-0.268894\pi$
$509$	29.9572	1.32783	0.663915	+	0.747808i	$0.268894\pi$
$510$	−16.1491	−0.715096
$511$	3.79128	0.167716
$512$	1.00000	0.0441942
$513$	5.58992	0.246801
$514$	14.5265	0.640738
$515$	29.6090	1.30473
$516$	−10.6043	−0.466827
$517$	5.28656	0.232503
$518$	45.8359	2.01391
$519$	−8.52643	−0.374269
$520$	0	0
$521$	−24.7061	−1.08239	−0.541197	−	0.840896i	$-0.682029\pi$
$521$	−24.7061	−1.08239	−0.541197	+	0.840896i	$0.682029\pi$
$522$	8.45640	0.370127
$523$	21.7094	0.949284	0.474642	−	0.880179i	$-0.342578\pi$
$523$	21.7094	0.949284	0.474642	+	0.880179i	$0.342578\pi$
$524$	−15.2834	−0.667659
$525$	−81.6570	−3.56381
$526$	14.1077	0.615126
$527$	12.7009	0.553259
$528$	−4.37839	−0.190545
$529$	11.5800	0.503479
$530$	36.7602	1.59676
$531$	8.74802	0.379632
$532$	4.70931	0.204174
$533$	0	0
$534$	12.4580	0.539110
$535$	36.4697	1.57672
$536$	1.20961	0.0522471
$537$	−16.2274	−0.700263
$538$	10.0839	0.434747
$539$	−51.6473	−2.22461
$540$	−24.0277	−1.03399
$541$	18.8539	0.810591	0.405295	−	0.914186i	$-0.367169\pi$
$541$	18.8539	0.810591	0.405295	+	0.914186i	$0.367169\pi$
$542$	−11.6395	−0.499961
$543$	−10.4809	−0.449779
$544$	−2.91994	−0.125191
$545$	34.6549	1.48445
$546$	0	0
$547$	−29.8279	−1.27535	−0.637674	−	0.770306i	$-0.720103\pi$
$547$	−29.8279	−1.27535	−0.637674	+	0.770306i	$0.720103\pi$
$548$	−16.3085	−0.696663
$549$	−8.60208	−0.367128
$550$	−45.8576	−1.95537
$551$	6.28981	0.267955
$552$	−7.56628	−0.322042
$553$	−6.27777	−0.266958
$554$	−0.529203	−0.0224837
$555$	−53.8300	−2.28496
$556$	13.8239	0.586264
$557$	−0.201031	−0.00851795	−0.00425898	−	0.999991i	$-0.501356\pi$
$557$	−0.201031	−0.00851795	−0.00425898	+	0.999991i	$0.501356\pi$
$558$	5.84801	0.247566
$559$	0	0
$560$	−20.2425	−0.855400
$561$	12.7846	0.539767
$562$	−27.0663	−1.14172
$563$	−23.9888	−1.01101	−0.505504	−	0.862824i	$-0.668693\pi$
$563$	−23.9888	−1.01101	−0.505504	+	0.862824i	$0.668693\pi$
$564$	−1.99893	−0.0841703
$565$	67.3344	2.83278
$566$	6.96429	0.292731
$567$	14.8770	0.624773
$568$	7.74497	0.324972
$569$	−43.5244	−1.82464	−0.912319	−	0.409480i	$-0.865710\pi$
$569$	−43.5244	−1.82464	−0.912319	+	0.409480i	$0.865710\pi$
$570$	−5.53065	−0.231653
$571$	3.42049	0.143143	0.0715715	−	0.997435i	$-0.477199\pi$
$571$	3.42049	0.143143	0.0715715	+	0.997435i	$0.477199\pi$
$572$	0	0
$573$	26.5860	1.11065
$574$	6.25487	0.261073
$575$	−79.2463	−3.30480
$576$	−1.34446	−0.0560191
$577$	−1.22794	−0.0511196	−0.0255598	−	0.999673i	$-0.508137\pi$
$577$	−1.22794	−0.0511196	−0.0255598	+	0.999673i	$0.508137\pi$
$578$	−8.47397	−0.352471
$579$	−24.4212	−1.01491
$580$	−27.0361	−1.12261
$581$	24.7146	1.02534
$582$	8.22032	0.340743
$583$	−29.1016	−1.20526
$584$	−0.805061	−0.0333136
$585$	0	0
$586$	−25.8717	−1.06875
$587$	5.81303	0.239929	0.119965	−	0.992778i	$-0.461722\pi$
$587$	5.81303	0.239929	0.119965	+	0.992778i	$0.461722\pi$
$588$	19.5287	0.805349
$589$	4.34971	0.179227
$590$	−27.9684	−1.15144
$591$	−27.0161	−1.11129
$592$	−9.73303	−0.400025
$593$	−18.3007	−0.751519	−0.375759	−	0.926717i	$-0.622618\pi$
$593$	−18.3007	−0.751519	−0.375759	+	0.926717i	$0.622618\pi$
$594$	19.0217	0.780472
$595$	59.1067	2.42314
$596$	−9.71911	−0.398110
$597$	−24.6856	−1.01032
$598$	0	0
$599$	−14.7650	−0.603283	−0.301642	−	0.953421i	$-0.597535\pi$
$599$	−14.7650	−0.603283	−0.301642	+	0.953421i	$0.597535\pi$
$600$	17.3395	0.707882
$601$	−8.43171	−0.343937	−0.171968	−	0.985102i	$-0.555013\pi$
$601$	−8.43171	−0.343937	−0.171968	+	0.985102i	$0.555013\pi$
$602$	38.8122	1.58187
$603$	−1.62627	−0.0662267
$604$	−4.30710	−0.175253
$605$	2.49087	0.101268
$606$	19.5270	0.793230
$607$	−3.09995	−0.125823	−0.0629114	−	0.998019i	$-0.520039\pi$
$607$	−3.09995	−0.125823	−0.0629114	+	0.998019i	$0.520039\pi$
$608$	−1.00000	−0.0405554
$609$	38.1123	1.54439
$610$	27.5019	1.11352
$611$	0	0
$612$	3.92574	0.158688
$613$	18.3764	0.742215	0.371107	−	0.928590i	$-0.378978\pi$
$613$	18.3764	0.742215	0.371107	+	0.928590i	$0.378978\pi$
$614$	−22.4406	−0.905630
$615$	−7.34577	−0.296210
$616$	16.0251	0.645671
$617$	35.9015	1.44534	0.722669	−	0.691194i	$-0.242915\pi$
$617$	35.9015	1.44534	0.722669	+	0.691194i	$0.242915\pi$
$618$	8.86315	0.356528
$619$	−36.2094	−1.45538	−0.727690	−	0.685906i	$-0.759406\pi$
$619$	−36.2094	−1.45538	−0.727690	+	0.685906i	$0.759406\pi$
$620$	−18.6968	−0.750880
$621$	32.8714	1.31908
$622$	6.66590	0.267278
$623$	−45.5969	−1.82680
$624$	0	0
$625$	89.2262	3.56905
$626$	−11.5401	−0.461234
$627$	4.37839	0.174856
$628$	−4.27209	−0.170475
$629$	28.4198	1.13317
$630$	27.2152	1.08428
$631$	−15.7203	−0.625814	−0.312907	−	0.949784i	$-0.601303\pi$
$631$	−15.7203	−0.625814	−0.312907	+	0.949784i	$0.601303\pi$
$632$	1.33306	0.0530261
$633$	10.9612	0.435670
$634$	−28.1127	−1.11650
$635$	−64.3514	−2.55371
$636$	11.0038	0.436328
$637$	0	0
$638$	21.4034	0.847368
$639$	−10.4128	−0.411924
$640$	4.29839	0.169909
$641$	21.8833	0.864338	0.432169	−	0.901793i	$-0.357748\pi$
$641$	21.8833	0.864338	0.432169	+	0.901793i	$0.357748\pi$
$642$	10.9168	0.430852
$643$	−42.4501	−1.67407	−0.837034	−	0.547151i	$-0.815712\pi$
$643$	−42.4501	−1.67407	−0.837034	+	0.547151i	$0.815712\pi$
$644$	27.6930	1.09126
$645$	−45.5813	−1.79476
$646$	2.91994	0.114883
$647$	−46.5162	−1.82874	−0.914370	−	0.404879i	$-0.867314\pi$
$647$	−46.5162	−1.82874	−0.914370	+	0.404879i	$0.867314\pi$
$648$	−3.15905	−0.124099
$649$	22.1415	0.869129
$650$	0	0
$651$	26.3565	1.03299
$652$	17.6762	0.692252
$653$	26.4119	1.03358	0.516789	−	0.856113i	$-0.327127\pi$
$653$	26.4119	1.03358	0.516789	+	0.856113i	$0.327127\pi$
$654$	10.3736	0.405639
$655$	−65.6941	−2.56688
$656$	−1.32819	−0.0518572
$657$	1.08237	0.0422273
$658$	7.31620	0.285215
$659$	−20.4766	−0.797656	−0.398828	−	0.917026i	$-0.630583\pi$
$659$	−20.4766	−0.797656	−0.398828	+	0.917026i	$0.630583\pi$
$660$	−18.8200	−0.732569
$661$	−9.07404	−0.352939	−0.176470	−	0.984306i	$-0.556468\pi$
$661$	−9.07404	−0.352939	−0.176470	+	0.984306i	$0.556468\pi$
$662$	−7.76517	−0.301802
$663$	0	0
$664$	−5.24803	−0.203663
$665$	20.2425	0.784969
$666$	13.0857	0.507059
$667$	36.9871	1.43215
$668$	21.3276	0.825188
$669$	−26.3845	−1.02008
$670$	5.19936	0.200869
$671$	−21.7721	−0.840503
$672$	−6.05937	−0.233745
$673$	−28.6258	−1.10344	−0.551722	−	0.834028i	$-0.686029\pi$
$673$	−28.6258	−1.10344	−0.551722	+	0.834028i	$0.686029\pi$
$674$	−31.9205	−1.22953
$675$	−75.3307	−2.89948
$676$	0	0
$677$	−17.5519	−0.674575	−0.337288	−	0.941402i	$-0.609509\pi$
$677$	−17.5519	−0.674575	−0.337288	+	0.941402i	$0.609509\pi$
$678$	20.1558	0.774080
$679$	−30.0868	−1.15463
$680$	−12.5510	−0.481310
$681$	−10.4145	−0.399086
$682$	14.8015	0.566778
$683$	3.55815	0.136149	0.0680744	−	0.997680i	$-0.478314\pi$
$683$	3.55815	0.136149	0.0680744	+	0.997680i	$0.478314\pi$
$684$	1.34446	0.0514067
$685$	−70.1001	−2.67839
$686$	−38.5108	−1.47035
$687$	14.2770	0.544700
$688$	−8.24159	−0.314208
$689$	0	0
$690$	−32.5228	−1.23812
$691$	−2.30143	−0.0875505	−0.0437753	−	0.999041i	$-0.513939\pi$
$691$	−2.30143	−0.0875505	−0.0437753	+	0.999041i	$0.513939\pi$
$692$	−6.62670	−0.251909
$693$	−21.5451	−0.818433
$694$	17.3924	0.660205
$695$	59.4205	2.25395
$696$	−8.09297	−0.306763
$697$	3.87824	0.146899
$698$	−0.938581	−0.0355258
$699$	0.890876	0.0336960
$700$	−63.4634	−2.39869
$701$	12.2974	0.464465	0.232233	−	0.972660i	$-0.425397\pi$
$701$	12.2974	0.464465	0.232233	+	0.972660i	$0.425397\pi$
$702$	0	0
$703$	9.73303	0.367088
$704$	−3.40286	−0.128250
$705$	−8.59220	−0.323601
$706$	−9.63690	−0.362689
$707$	−71.4698	−2.68790
$708$	−8.37205	−0.314641
$709$	1.74224	0.0654310	0.0327155	−	0.999465i	$-0.489584\pi$
$709$	1.74224	0.0654310	0.0327155	+	0.999465i	$0.489584\pi$
$710$	33.2909	1.24939
$711$	−1.79224	−0.0672142
$712$	9.68229	0.362859
$713$	25.5784	0.957918
$714$	17.6930	0.662143
$715$	0	0
$716$	−12.6118	−0.471327
$717$	−6.50605	−0.242973
$718$	−27.0024	−1.00772
$719$	−13.6096	−0.507553	−0.253776	−	0.967263i	$-0.581673\pi$
$719$	−13.6096	−0.507553	−0.253776	+	0.967263i	$0.581673\pi$
$720$	−5.77901	−0.215371
$721$	−32.4396	−1.20811
$722$	1.00000	0.0372161
$723$	1.51807	0.0564575
$724$	−8.14571	−0.302733
$725$	−84.7626	−3.14800
$726$	0.745614	0.0276723
$727$	−10.3595	−0.384212	−0.192106	−	0.981374i	$-0.561532\pi$
$727$	−10.3595	−0.384212	−0.192106	+	0.981374i	$0.561532\pi$
$728$	0	0
$729$	25.8245	0.956463
$730$	−3.46047	−0.128078
$731$	24.0649	0.890073
$732$	8.23239	0.304278
$733$	−28.4071	−1.04924	−0.524620	−	0.851337i	$-0.675793\pi$
$733$	−28.4071	−1.04924	−0.524620	+	0.851337i	$0.675793\pi$
$734$	−0.507227	−0.0187221
$735$	83.9419	3.09624
$736$	−5.88048	−0.216757
$737$	−4.11613	−0.151620
$738$	1.78570	0.0657326
$739$	28.6863	1.05524	0.527622	−	0.849480i	$-0.323084\pi$
$739$	28.6863	1.05524	0.527622	+	0.849480i	$0.323084\pi$
$740$	−41.8364	−1.53794
$741$	0	0
$742$	−40.2744	−1.47852
$743$	22.6706	0.831703	0.415851	−	0.909433i	$-0.363484\pi$
$743$	22.6706	0.831703	0.415851	+	0.909433i	$0.363484\pi$
$744$	−5.59668	−0.205184
$745$	−41.7765	−1.53057
$746$	15.1224	0.553671
$747$	7.05577	0.258157
$748$	9.93615	0.363302
$749$	−39.9561	−1.45997
$750$	46.8787	1.71177
$751$	12.0663	0.440307	0.220153	−	0.975465i	$-0.429344\pi$
$751$	12.0663	0.440307	0.220153	+	0.975465i	$0.429344\pi$
$752$	−1.55356	−0.0566526
$753$	20.8225	0.758813
$754$	0	0
$755$	−18.5136	−0.673778
$756$	26.3247	0.957419
$757$	−1.68333	−0.0611818	−0.0305909	−	0.999532i	$-0.509739\pi$
$757$	−1.68333	−0.0611818	−0.0305909	+	0.999532i	$0.509739\pi$
$758$	33.3864	1.21265
$759$	25.7470	0.934558
$760$	−4.29839	−0.155919
$761$	15.6109	0.565893	0.282947	−	0.959136i	$-0.408688\pi$
$761$	15.6109	0.565893	0.282947	+	0.959136i	$0.408688\pi$
$762$	−19.2629	−0.697822
$763$	−37.9679	−1.37453
$764$	20.6625	0.747543
$765$	16.8743	0.610093
$766$	−20.3998	−0.737075
$767$	0	0
$768$	1.28668	0.0464290
$769$	32.5034	1.17210	0.586052	−	0.810274i	$-0.300682\pi$
$769$	32.5034	1.17210	0.586052	+	0.810274i	$0.300682\pi$
$770$	68.8823	2.48235
$771$	18.6910	0.673140
$772$	−18.9801	−0.683107
$773$	−44.4591	−1.59908	−0.799541	−	0.600611i	$-0.794924\pi$
$773$	−44.4591	−1.59908	−0.799541	+	0.600611i	$0.794924\pi$
$774$	11.0805	0.398279
$775$	−58.6174	−2.10560
$776$	6.38880	0.229344
$777$	58.9760	2.11575
$778$	30.0088	1.07587
$779$	1.32819	0.0475875
$780$	0	0
$781$	−26.3551	−0.943059
$782$	17.1706	0.614020
$783$	35.1596	1.25650
$784$	15.1776	0.542057
$785$	−18.3631	−0.655407
$786$	−19.6648	−0.701421
$787$	17.4255	0.621153	0.310576	−	0.950548i	$-0.399478\pi$
$787$	17.4255	0.621153	0.310576	+	0.950548i	$0.399478\pi$
$788$	−20.9968	−0.747980
$789$	18.1521	0.646232
$790$	5.73000	0.203864
$791$	−73.7714	−2.62301
$792$	4.57501	0.162566
$793$	0	0
$794$	−12.6058	−0.447364
$795$	47.2985	1.67751
$796$	−19.1855	−0.680013
$797$	45.5179	1.61233	0.806164	−	0.591692i	$-0.201540\pi$
$797$	45.5179	1.61233	0.806164	+	0.591692i	$0.201540\pi$
$798$	6.05937	0.214499
$799$	4.53630	0.160483
$800$	13.4762	0.476454
$801$	−13.0174	−0.459949
$802$	−20.4344	−0.721563
$803$	2.73951	0.0966753
$804$	1.55638	0.0548891
$805$	119.035	4.19544
$806$	0	0
$807$	12.9747	0.456731
$808$	15.1763	0.533900
$809$	2.01677	0.0709059	0.0354529	−	0.999371i	$-0.488713\pi$
$809$	2.01677	0.0709059	0.0354529	+	0.999371i	$0.488713\pi$
$810$	−13.5788	−0.477112
$811$	0.125405	0.00440356	0.00220178	−	0.999998i	$-0.499299\pi$
$811$	0.125405	0.00440356	0.00220178	+	0.999998i	$0.499299\pi$
$812$	29.6207	1.03948
$813$	−14.9764	−0.525244
$814$	33.1202	1.16086
$815$	75.9790	2.66143
$816$	−3.75702	−0.131522
$817$	8.24159	0.288337
$818$	8.77027	0.306645
$819$	0	0
$820$	−5.70909	−0.199370
$821$	−19.1549	−0.668511	−0.334256	−	0.942482i	$-0.608485\pi$
$821$	−19.1549	−0.668511	−0.334256	+	0.942482i	$0.608485\pi$
$822$	−20.9837	−0.731892
$823$	7.94656	0.277000	0.138500	−	0.990362i	$-0.455772\pi$
$823$	7.94656	0.277000	0.138500	+	0.990362i	$0.455772\pi$
$824$	6.88840	0.239969
$825$	−59.0039	−2.05425
$826$	30.6422	1.06618
$827$	−21.9463	−0.763147	−0.381573	−	0.924339i	$-0.624618\pi$
$827$	−21.9463	−0.763147	−0.381573	+	0.924339i	$0.624618\pi$
$828$	7.90606	0.274755
$829$	28.1275	0.976908	0.488454	−	0.872590i	$-0.337561\pi$
$829$	28.1275	0.976908	0.488454	+	0.872590i	$0.337561\pi$
$830$	−22.5581	−0.783003
$831$	−0.680914	−0.0236207
$832$	0	0
$833$	−44.3176	−1.53552
$834$	17.7869	0.615910
$835$	91.6742	3.17252
$836$	3.40286	0.117691
$837$	24.3145	0.840433
$838$	−24.2608	−0.838075
$839$	−49.4222	−1.70624	−0.853122	−	0.521711i	$-0.825294\pi$
$839$	−49.4222	−1.70624	−0.853122	+	0.521711i	$0.825294\pi$
$840$	−26.0455	−0.898656
$841$	10.5618	0.364199
$842$	−36.7007	−1.26479
$843$	−34.8256	−1.19946
$844$	8.51902	0.293237
$845$	0	0
$846$	2.08870	0.0718110
$847$	−2.72899	−0.0937691
$848$	8.55208	0.293680
$849$	8.96080	0.307534
$850$	−39.3496	−1.34968
$851$	57.2349	1.96199
$852$	9.96529	0.341405
$853$	54.4215	1.86336	0.931679	−	0.363283i	$-0.118344\pi$
$853$	54.4215	1.86336	0.931679	+	0.363283i	$0.118344\pi$
$854$	−30.1310	−1.03106
$855$	5.77901	0.197638
$856$	8.48450	0.289994
$857$	−19.5715	−0.668550	−0.334275	−	0.942476i	$-0.608491\pi$
$857$	−19.5715	−0.668550	−0.334275	+	0.942476i	$0.608491\pi$
$858$	0	0
$859$	11.0903	0.378395	0.189198	−	0.981939i	$-0.439411\pi$
$859$	11.0903	0.378395	0.189198	+	0.981939i	$0.439411\pi$
$860$	−35.4256	−1.20800
$861$	8.04801	0.274275
$862$	−14.1188	−0.480887
$863$	11.0109	0.374815	0.187408	−	0.982282i	$-0.439991\pi$
$863$	11.0109	0.374815	0.187408	+	0.982282i	$0.439991\pi$
$864$	−5.58992	−0.190173
$865$	−28.4841	−0.968489
$866$	33.7529	1.14697
$867$	−10.9033	−0.370294
$868$	20.4841	0.695277
$869$	−4.53621	−0.153880
$870$	−34.7867	−1.17938
$871$	0	0
$872$	8.06230	0.273024
$873$	−8.58947	−0.290710
$874$	5.88048	0.198910
$875$	−171.578	−5.80041
$876$	−1.03585	−0.0349983
$877$	−10.2386	−0.345733	−0.172866	−	0.984945i	$-0.555303\pi$
$877$	−10.2386	−0.345733	−0.172866	+	0.984945i	$0.555303\pi$
$878$	20.5686	0.694158
$879$	−33.2885	−1.12279
$880$	−14.6268	−0.493071
$881$	16.3401	0.550512	0.275256	−	0.961371i	$-0.411237\pi$
$881$	16.3401	0.550512	0.275256	+	0.961371i	$0.411237\pi$
$882$	−20.4057	−0.687094
$883$	−21.9618	−0.739073	−0.369536	−	0.929216i	$-0.620483\pi$
$883$	−21.9618	−0.739073	−0.369536	+	0.929216i	$0.620483\pi$
$884$	0	0
$885$	−35.9864	−1.20967
$886$	−26.0671	−0.875742
$887$	−12.4864	−0.419251	−0.209626	−	0.977782i	$-0.567225\pi$
$887$	−12.4864	−0.419251	−0.209626	+	0.977782i	$0.567225\pi$
$888$	−12.5233	−0.420254
$889$	70.5033	2.36460
$890$	41.6183	1.39505
$891$	10.7498	0.360133
$892$	−20.5059	−0.686588
$893$	1.55356	0.0519880
$894$	−12.5054	−0.418242
$895$	−54.2106	−1.81206
$896$	−4.70931	−0.157327
$897$	0	0
$898$	−8.77925	−0.292967
$899$	27.3589	0.912470
$900$	−18.1182	−0.603939
$901$	−24.9715	−0.831922
$902$	4.51966	0.150488
$903$	49.9388	1.66186
$904$	15.6650	0.521011
$905$	−35.0134	−1.16389
$906$	−5.54185	−0.184116
$907$	35.3798	1.17477	0.587383	−	0.809309i	$-0.300158\pi$
$907$	35.3798	1.17477	0.587383	+	0.809309i	$0.300158\pi$
$908$	−8.09412	−0.268613
$909$	−20.4039	−0.676754
$910$	0	0
$911$	−33.5135	−1.11035	−0.555176	−	0.831733i	$-0.687349\pi$
$911$	−33.5135	−1.11035	−0.555176	+	0.831733i	$0.687349\pi$
$912$	−1.28668	−0.0426062
$913$	17.8584	0.591025
$914$	−24.1874	−0.800049
$915$	35.3860	1.16983
$916$	11.0960	0.366622
$917$	71.9743	2.37680
$918$	16.3222	0.538713
$919$	26.1949	0.864090	0.432045	−	0.901852i	$-0.357792\pi$
$919$	26.1949	0.864090	0.432045	+	0.901852i	$0.357792\pi$
$920$	−25.2766	−0.833345
$921$	−28.8739	−0.951426
$922$	−23.3854	−0.770158
$923$	0	0
$924$	20.6192	0.678322
$925$	−131.164	−4.31265
$926$	22.9038	0.752666
$927$	−9.26117	−0.304177
$928$	−6.28981	−0.206473
$929$	10.7983	0.354280	0.177140	−	0.984186i	$-0.443315\pi$
$929$	10.7983	0.354280	0.177140	+	0.984186i	$0.443315\pi$
$930$	−24.0567	−0.788851
$931$	−15.1776	−0.497426
$932$	0.692385	0.0226798
$933$	8.57687	0.280794
$934$	−23.2869	−0.761969
$935$	42.7095	1.39675
$936$	0	0
$937$	24.1210	0.787999	0.393999	−	0.919111i	$-0.371091\pi$
$937$	24.1210	0.787999	0.393999	+	0.919111i	$0.371091\pi$
$938$	−5.69641	−0.185995
$939$	−14.8484	−0.484558
$940$	−6.67782	−0.217806
$941$	−25.6886	−0.837426	−0.418713	−	0.908119i	$-0.637519\pi$
$941$	−25.6886	−0.837426	−0.418713	+	0.908119i	$0.637519\pi$
$942$	−5.49680	−0.179095
$943$	7.81041	0.254342
$944$	−6.50672	−0.211776
$945$	113.154	3.68089
$946$	28.0450	0.911822
$947$	10.3104	0.335044	0.167522	−	0.985868i	$-0.446423\pi$
$947$	10.3104	0.335044	0.167522	+	0.985868i	$0.446423\pi$
$948$	1.71521	0.0557076
$949$	0	0
$950$	−13.4762	−0.437225
$951$	−36.1720	−1.17296
$952$	13.7509	0.445669
$953$	2.81033	0.0910354	0.0455177	−	0.998964i	$-0.485506\pi$
$953$	2.81033	0.0910354	0.0455177	+	0.998964i	$0.485506\pi$
$954$	−11.4979	−0.372259
$955$	88.8155	2.87400
$956$	−5.05647	−0.163538
$957$	27.5393	0.890218
$958$	25.3107	0.817751
$959$	76.8016	2.48005
$960$	5.53065	0.178501
$961$	−12.0800	−0.389678
$962$	0	0
$963$	−11.4071	−0.367587
$964$	1.17983	0.0379999
$965$	−81.5837	−2.62627
$966$	35.6320	1.14644
$967$	−30.1037	−0.968069	−0.484034	−	0.875049i	$-0.660829\pi$
$967$	−30.1037	−0.968069	−0.484034	+	0.875049i	$0.660829\pi$
$968$	0.579488	0.0186254
$969$	3.75702	0.120693
$970$	27.4615	0.881737
$971$	−35.1916	−1.12935	−0.564676	−	0.825313i	$-0.690999\pi$
$971$	−35.1916	−1.12935	−0.564676	+	0.825313i	$0.690999\pi$
$972$	12.7051	0.407516
$973$	−65.1010	−2.08704
$974$	21.4003	0.685710
$975$	0	0
$976$	6.39817	0.204801
$977$	−19.3617	−0.619435	−0.309718	−	0.950829i	$-0.600235\pi$
$977$	−19.3617	−0.619435	−0.309718	+	0.950829i	$0.600235\pi$
$978$	22.7435	0.727258
$979$	−32.9475	−1.05301
$980$	65.2393	2.08399
$981$	−10.8394	−0.346076
$982$	36.3776	1.16086
$983$	18.4818	0.589478	0.294739	−	0.955578i	$-0.404767\pi$
$983$	18.4818	0.589478	0.294739	+	0.955578i	$0.404767\pi$
$984$	−1.70896	−0.0544796
$985$	−90.2524	−2.87568
$986$	18.3659	0.584888
$987$	9.41360	0.299638
$988$	0	0
$989$	48.4645	1.54108
$990$	19.6652	0.625001
$991$	−15.0294	−0.477425	−0.238712	−	0.971090i	$-0.576725\pi$
$991$	−15.0294	−0.477425	−0.238712	+	0.971090i	$0.576725\pi$
$992$	−4.34971	−0.138103
$993$	−9.99128	−0.317064
$994$	−36.4735	−1.15687
$995$	−82.4670	−2.61438
$996$	−6.75253	−0.213962
$997$	−10.5640	−0.334566	−0.167283	−	0.985909i	$-0.553499\pi$
$997$	−10.5640	−0.334566	−0.167283	+	0.985909i	$0.553499\pi$
$998$	1.49097	0.0471958
$999$	54.4069	1.72136

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6422.2.a.bl.1.7	yes	9
13.12	even	2		6422.2.a.bj.1.7	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6422.2.a.bj.1.7	✓	9	13.12	even	2
6422.2.a.bl.1.7	yes	9	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:	\( N \)	\(=\)	\( 6422 = 2 \cdot 13^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6422.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.28668 q^{3} +1.00000 q^{4} +4.29839 q^{5} +1.28668 q^{6} -4.70931 q^{7} +1.00000 q^{8} -1.34446 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +1.28668 q^{3} +1.00000 q^{4} +4.29839 q^{5} +1.28668 q^{6} -4.70931 q^{7} +1.00000 q^{8} -1.34446 q^{9} +4.29839 q^{10} -3.40286 q^{11} +1.28668 q^{12} -4.70931 q^{14} +5.53065 q^{15} +1.00000 q^{16} -2.91994 q^{17} -1.34446 q^{18} -1.00000 q^{19} +4.29839 q^{20} -6.05937 q^{21} -3.40286 q^{22} -5.88048 q^{23} +1.28668 q^{24} +13.4762 q^{25} -5.58992 q^{27} -4.70931 q^{28} -6.28981 q^{29} +5.53065 q^{30} -4.34971 q^{31} +1.00000 q^{32} -4.37839 q^{33} -2.91994 q^{34} -20.2425 q^{35} -1.34446 q^{36} -9.73303 q^{37} -1.00000 q^{38} +4.29839 q^{40} -1.32819 q^{41} -6.05937 q^{42} -8.24159 q^{43} -3.40286 q^{44} -5.77901 q^{45} -5.88048 q^{46} -1.55356 q^{47} +1.28668 q^{48} +15.1776 q^{49} +13.4762 q^{50} -3.75702 q^{51} +8.55208 q^{53} -5.58992 q^{54} -14.6268 q^{55} -4.70931 q^{56} -1.28668 q^{57} -6.28981 q^{58} -6.50672 q^{59} +5.53065 q^{60} +6.39817 q^{61} -4.34971 q^{62} +6.33147 q^{63} +1.00000 q^{64} -4.37839 q^{66} +1.20961 q^{67} -2.91994 q^{68} -7.56628 q^{69} -20.2425 q^{70} +7.74497 q^{71} -1.34446 q^{72} -0.805061 q^{73} -9.73303 q^{74} +17.3395 q^{75} -1.00000 q^{76} +16.0251 q^{77} +1.33306 q^{79} +4.29839 q^{80} -3.15905 q^{81} -1.32819 q^{82} -5.24803 q^{83} -6.05937 q^{84} -12.5510 q^{85} -8.24159 q^{86} -8.09297 q^{87} -3.40286 q^{88} +9.68229 q^{89} -5.77901 q^{90} -5.88048 q^{92} -5.59668 q^{93} -1.55356 q^{94} -4.29839 q^{95} +1.28668 q^{96} +6.38880 q^{97} +15.1776 q^{98} +4.57501 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 9 q^{2} + q^{3} + 9 q^{4} + q^{5} + q^{6} - 13 q^{7} + 9 q^{8} - 2 q^{9}+O(q^{10}) \) 9 * q + 9 * q^2 + q^3 + 9 * q^4 + q^5 + q^6 - 13 * q^7 + 9 * q^8 - 2 * q^9	\( 9 q + 9 q^{2} + q^{3} + 9 q^{4} + q^{5} + q^{6} - 13 q^{7} + 9 q^{8} - 2 q^{9} + q^{10} - 3 q^{11} + q^{12} - 13 q^{14} - 3 q^{15} + 9 q^{16} - 8 q^{17} - 2 q^{18} - 9 q^{19} + q^{20} - 24 q^{21} - 3 q^{22} - 10 q^{23} + q^{24} + 8 q^{25} + 10 q^{27} - 13 q^{28} - 20 q^{29} - 3 q^{30} - q^{31} + 9 q^{32} + 2 q^{33} - 8 q^{34} + 4 q^{35} - 2 q^{36} - 15 q^{37} - 9 q^{38} + q^{40} - 19 q^{41} - 24 q^{42} - 16 q^{43} - 3 q^{44} - 15 q^{45} - 10 q^{46} - 18 q^{47} + q^{48} + 18 q^{49} + 8 q^{50} - 11 q^{51} + 17 q^{53} + 10 q^{54} - 26 q^{55} - 13 q^{56} - q^{57} - 20 q^{58} - 24 q^{59} - 3 q^{60} + 6 q^{61} - q^{62} - q^{63} + 9 q^{64} + 2 q^{66} - 29 q^{67} - 8 q^{68} - 12 q^{69} + 4 q^{70} - 23 q^{71} - 2 q^{72} - 38 q^{73} - 15 q^{74} + 11 q^{75} - 9 q^{76} - 40 q^{77} - 20 q^{79} + q^{80} - 31 q^{81} - 19 q^{82} - 20 q^{83} - 24 q^{84} - 39 q^{85} - 16 q^{86} - 10 q^{87} - 3 q^{88} + 7 q^{89} - 15 q^{90} - 10 q^{92} - 11 q^{93} - 18 q^{94} - q^{95} + q^{96} - 28 q^{97} + 18 q^{98} - 25 q^{99}+O(q^{100}) \) 9 * q + 9 * q^2 + q^3 + 9 * q^4 + q^5 + q^6 - 13 * q^7 + 9 * q^8 - 2 * q^9 + q^10 - 3 * q^11 + q^12 - 13 * q^14 - 3 * q^15 + 9 * q^16 - 8 * q^17 - 2 * q^18 - 9 * q^19 + q^20 - 24 * q^21 - 3 * q^22 - 10 * q^23 + q^24 + 8 * q^25 + 10 * q^27 - 13 * q^28 - 20 * q^29 - 3 * q^30 - q^31 + 9 * q^32 + 2 * q^33 - 8 * q^34 + 4 * q^35 - 2 * q^36 - 15 * q^37 - 9 * q^38 + q^40 - 19 * q^41 - 24 * q^42 - 16 * q^43 - 3 * q^44 - 15 * q^45 - 10 * q^46 - 18 * q^47 + q^48 + 18 * q^49 + 8 * q^50 - 11 * q^51 + 17 * q^53 + 10 * q^54 - 26 * q^55 - 13 * q^56 - q^57 - 20 * q^58 - 24 * q^59 - 3 * q^60 + 6 * q^61 - q^62 - q^63 + 9 * q^64 + 2 * q^66 - 29 * q^67 - 8 * q^68 - 12 * q^69 + 4 * q^70 - 23 * q^71 - 2 * q^72 - 38 * q^73 - 15 * q^74 + 11 * q^75 - 9 * q^76 - 40 * q^77 - 20 * q^79 + q^80 - 31 * q^81 - 19 * q^82 - 20 * q^83 - 24 * q^84 - 39 * q^85 - 16 * q^86 - 10 * q^87 - 3 * q^88 + 7 * q^89 - 15 * q^90 - 10 * q^92 - 11 * q^93 - 18 * q^94 - q^95 + q^96 - 28 * q^97 + 18 * q^98 - 25 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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