LMFDB - Embedded newform 6422.2.a.bf.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:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 2x^{6} - 16x^{5} + 29x^{4} + 60x^{3} - 79x^{2} - 47x - 2 $ x^7 - 2x^6 - 16x^5 + 29x^4 + 60x^3 - 79x^2 - 47x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 494)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$3.31105$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} +3.31105 q^{3} +1.00000 q^{4} -2.06886 q^{5} +3.31105 q^{6} -3.81920 q^{7} +1.00000 q^{8} +7.96307 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +3.31105 q^{3} +1.00000 q^{4} -2.06886 q^{5} +3.31105 q^{6} -3.81920 q^{7} +1.00000 q^{8} +7.96307 q^{9} -2.06886 q^{10} +1.60687 q^{11} +3.31105 q^{12} -3.81920 q^{14} -6.85009 q^{15} +1.00000 q^{16} +7.51533 q^{17} +7.96307 q^{18} +1.00000 q^{19} -2.06886 q^{20} -12.6456 q^{21} +1.60687 q^{22} +2.12103 q^{23} +3.31105 q^{24} -0.719834 q^{25} +16.4330 q^{27} -3.81920 q^{28} -7.07077 q^{29} -6.85009 q^{30} +6.92729 q^{31} +1.00000 q^{32} +5.32042 q^{33} +7.51533 q^{34} +7.90139 q^{35} +7.96307 q^{36} +3.61366 q^{37} +1.00000 q^{38} -2.06886 q^{40} -1.62956 q^{41} -12.6456 q^{42} +0.236327 q^{43} +1.60687 q^{44} -16.4744 q^{45} +2.12103 q^{46} -9.28311 q^{47} +3.31105 q^{48} +7.58633 q^{49} -0.719834 q^{50} +24.8836 q^{51} -1.14260 q^{53} +16.4330 q^{54} -3.32438 q^{55} -3.81920 q^{56} +3.31105 q^{57} -7.07077 q^{58} +11.1509 q^{59} -6.85009 q^{60} +2.97140 q^{61} +6.92729 q^{62} -30.4126 q^{63} +1.00000 q^{64} +5.32042 q^{66} -5.51631 q^{67} +7.51533 q^{68} +7.02283 q^{69} +7.90139 q^{70} +2.43427 q^{71} +7.96307 q^{72} +13.9360 q^{73} +3.61366 q^{74} -2.38341 q^{75} +1.00000 q^{76} -6.13695 q^{77} -11.0307 q^{79} -2.06886 q^{80} +30.5212 q^{81} -1.62956 q^{82} +9.88549 q^{83} -12.6456 q^{84} -15.5481 q^{85} +0.236327 q^{86} -23.4117 q^{87} +1.60687 q^{88} +8.37645 q^{89} -16.4744 q^{90} +2.12103 q^{92} +22.9366 q^{93} -9.28311 q^{94} -2.06886 q^{95} +3.31105 q^{96} +0.865228 q^{97} +7.58633 q^{98} +12.7956 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 7 q^{2} + 2 q^{3} + 7 q^{4} + 2 q^{5} + 2 q^{6} + q^{7} + 7 q^{8} + 15 q^{9}+O(q^{10}) $ 7 * q + 7 * q^2 + 2 * q^3 + 7 * q^4 + 2 * q^5 + 2 * q^6 + q^7 + 7 * q^8 + 15 * q^9	\( 7 q + 7 q^{2} + 2 q^{3} + 7 q^{4} + 2 q^{5} + 2 q^{6} + q^{7} + 7 q^{8} + 15 q^{9} + 2 q^{10} + 5 q^{11} + 2 q^{12} + q^{14} + 13 q^{15} + 7 q^{16} + 16 q^{17} + 15 q^{18} + 7 q^{19} + 2 q^{20} - 3 q^{21} + 5 q^{22} + 3 q^{23} + 2 q^{24} + 7 q^{25} + 5 q^{27} + q^{28} - 7 q^{29} + 13 q^{30} + 11 q^{31} + 7 q^{32} + 6 q^{33} + 16 q^{34} + q^{35} + 15 q^{36} - 3 q^{37} + 7 q^{38} + 2 q^{40} + 15 q^{41} - 3 q^{42} + 23 q^{43} + 5 q^{44} - 38 q^{45} + 3 q^{46} - q^{47} + 2 q^{48} + 14 q^{49} + 7 q^{50} - 16 q^{51} + 21 q^{53} + 5 q^{54} + 8 q^{55} + q^{56} + 2 q^{57} - 7 q^{58} + 8 q^{59} + 13 q^{60} - 6 q^{61} + 11 q^{62} - 22 q^{63} + 7 q^{64} + 6 q^{66} + 28 q^{67} + 16 q^{68} + 48 q^{69} + q^{70} + 4 q^{71} + 15 q^{72} + 12 q^{73} - 3 q^{74} + 5 q^{75} + 7 q^{76} - 26 q^{77} - 4 q^{79} + 2 q^{80} + 51 q^{81} + 15 q^{82} + 13 q^{83} - 3 q^{84} - 39 q^{85} + 23 q^{86} - 16 q^{87} + 5 q^{88} - 15 q^{89} - 38 q^{90} + 3 q^{92} + 6 q^{93} - q^{94} + 2 q^{95} + 2 q^{96} - 37 q^{97} + 14 q^{98} - 15 q^{99}+O(q^{100}) \) 7 * q + 7 * q^2 + 2 * q^3 + 7 * q^4 + 2 * q^5 + 2 * q^6 + q^7 + 7 * q^8 + 15 * q^9 + 2 * q^10 + 5 * q^11 + 2 * q^12 + q^14 + 13 * q^15 + 7 * q^16 + 16 * q^17 + 15 * q^18 + 7 * q^19 + 2 * q^20 - 3 * q^21 + 5 * q^22 + 3 * q^23 + 2 * q^24 + 7 * q^25 + 5 * q^27 + q^28 - 7 * q^29 + 13 * q^30 + 11 * q^31 + 7 * q^32 + 6 * q^33 + 16 * q^34 + q^35 + 15 * q^36 - 3 * q^37 + 7 * q^38 + 2 * q^40 + 15 * q^41 - 3 * q^42 + 23 * q^43 + 5 * q^44 - 38 * q^45 + 3 * q^46 - q^47 + 2 * q^48 + 14 * q^49 + 7 * q^50 - 16 * q^51 + 21 * q^53 + 5 * q^54 + 8 * q^55 + q^56 + 2 * q^57 - 7 * q^58 + 8 * q^59 + 13 * q^60 - 6 * q^61 + 11 * q^62 - 22 * q^63 + 7 * q^64 + 6 * q^66 + 28 * q^67 + 16 * q^68 + 48 * q^69 + q^70 + 4 * q^71 + 15 * q^72 + 12 * q^73 - 3 * q^74 + 5 * q^75 + 7 * q^76 - 26 * q^77 - 4 * q^79 + 2 * q^80 + 51 * q^81 + 15 * q^82 + 13 * q^83 - 3 * q^84 - 39 * q^85 + 23 * q^86 - 16 * q^87 + 5 * q^88 - 15 * q^89 - 38 * q^90 + 3 * q^92 + 6 * q^93 - q^94 + 2 * q^95 + 2 * q^96 - 37 * q^97 + 14 * q^98 - 15 * 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
$3$	3.31105	1.91164	0.955818	−	0.293958i	$-0.0949725\pi$
$3$	3.31105	1.91164	0.955818	+	0.293958i	$0.0949725\pi$
$4$	1.00000	0.500000
$5$	−2.06886	−0.925221	−0.462610	−	0.886562i	$-0.653087\pi$
$5$	−2.06886	−0.925221	−0.462610	+	0.886562i	$0.653087\pi$
$6$	3.31105	1.35173
$7$	−3.81920	−1.44352	−0.721762	−	0.692141i	$-0.756667\pi$
$7$	−3.81920	−1.44352	−0.721762	+	0.692141i	$0.756667\pi$
$8$	1.00000	0.353553
$9$	7.96307	2.65436
$10$	−2.06886	−0.654230
$11$	1.60687	0.484489	0.242244	−	0.970215i	$-0.422116\pi$
$11$	1.60687	0.484489	0.242244	+	0.970215i	$0.422116\pi$
$12$	3.31105	0.955818
$13$	0	0
$13$	0	0
$14$	−3.81920	−1.02073
$15$	−6.85009	−1.76869
$16$	1.00000	0.250000
$17$	7.51533	1.82274	0.911368	−	0.411594i	$-0.135028\pi$
$17$	7.51533	1.82274	0.911368	+	0.411594i	$0.135028\pi$
$18$	7.96307	1.87691
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	−2.06886	−0.462610
$21$	−12.6456	−2.75949
$22$	1.60687	0.342585
$23$	2.12103	0.442265	0.221132	−	0.975244i	$-0.429025\pi$
$23$	2.12103	0.442265	0.221132	+	0.975244i	$0.429025\pi$
$24$	3.31105	0.675866
$25$	−0.719834	−0.143967
$26$	0	0
$27$	16.4330	3.16253
$28$	−3.81920	−0.721762
$29$	−7.07077	−1.31301	−0.656504	−	0.754322i	$-0.727966\pi$
$29$	−7.07077	−1.31301	−0.656504	+	0.754322i	$0.727966\pi$
$30$	−6.85009	−1.25065
$31$	6.92729	1.24418	0.622089	−	0.782947i	$-0.286284\pi$
$31$	6.92729	1.24418	0.622089	+	0.782947i	$0.286284\pi$
$32$	1.00000	0.176777
$33$	5.32042	0.926166
$34$	7.51533	1.28887
$35$	7.90139	1.33558
$36$	7.96307	1.32718
$37$	3.61366	0.594083	0.297041	−	0.954865i	$-0.404000\pi$
$37$	3.61366	0.594083	0.297041	+	0.954865i	$0.404000\pi$
$38$	1.00000	0.162221
$39$	0	0
$40$	−2.06886	−0.327115
$41$	−1.62956	−0.254494	−0.127247	−	0.991871i	$-0.540614\pi$
$41$	−1.62956	−0.254494	−0.127247	+	0.991871i	$0.540614\pi$
$42$	−12.6456	−1.95126
$43$	0.236327	0.0360395	0.0180197	−	0.999838i	$-0.494264\pi$
$43$	0.236327	0.0360395	0.0180197	+	0.999838i	$0.494264\pi$
$44$	1.60687	0.242244
$45$	−16.4744	−2.45586
$46$	2.12103	0.312728
$47$	−9.28311	−1.35408	−0.677040	−	0.735946i	$-0.736738\pi$
$47$	−9.28311	−1.35408	−0.677040	+	0.735946i	$0.736738\pi$
$48$	3.31105	0.477909
$49$	7.58633	1.08376
$50$	−0.719834	−0.101800
$51$	24.8836	3.48441
$52$	0	0
$53$	−1.14260	−0.156948	−0.0784739	−	0.996916i	$-0.525005\pi$
$53$	−1.14260	−0.156948	−0.0784739	+	0.996916i	$0.525005\pi$
$54$	16.4330	2.23624
$55$	−3.32438	−0.448259
$56$	−3.81920	−0.510363
$57$	3.31105	0.438560
$58$	−7.07077	−0.928437
$59$	11.1509	1.45172	0.725862	−	0.687841i	$-0.241441\pi$
$59$	11.1509	1.45172	0.725862	+	0.687841i	$0.241441\pi$
$60$	−6.85009	−0.884343
$61$	2.97140	0.380448	0.190224	−	0.981741i	$-0.439078\pi$
$61$	2.97140	0.380448	0.190224	+	0.981741i	$0.439078\pi$
$62$	6.92729	0.879766
$63$	−30.4126	−3.83162
$64$	1.00000	0.125000
$65$	0	0
$66$	5.32042	0.654898
$67$	−5.51631	−0.673925	−0.336962	−	0.941518i	$-0.609400\pi$
$67$	−5.51631	−0.673925	−0.336962	+	0.941518i	$0.609400\pi$
$68$	7.51533	0.911368
$69$	7.02283	0.845450
$70$	7.90139	0.944396
$71$	2.43427	0.288895	0.144447	−	0.989512i	$-0.453860\pi$
$71$	2.43427	0.288895	0.144447	+	0.989512i	$0.453860\pi$
$72$	7.96307	0.938456
$73$	13.9360	1.63109	0.815543	−	0.578697i	$-0.196439\pi$
$73$	13.9360	1.63109	0.815543	+	0.578697i	$0.196439\pi$
$74$	3.61366	0.420080
$75$	−2.38341	−0.275212
$76$	1.00000	0.114708
$77$	−6.13695	−0.699371
$78$	0	0
$79$	−11.0307	−1.24105	−0.620523	−	0.784188i	$-0.713080\pi$
$79$	−11.0307	−1.24105	−0.620523	+	0.784188i	$0.713080\pi$
$80$	−2.06886	−0.231305
$81$	30.5212	3.39125
$82$	−1.62956	−0.179955
$83$	9.88549	1.08507	0.542537	−	0.840032i	$-0.317464\pi$
$83$	9.88549	1.08507	0.542537	+	0.840032i	$0.317464\pi$
$84$	−12.6456	−1.37975
$85$	−15.5481	−1.68643
$86$	0.236327	0.0254838
$87$	−23.4117	−2.51000
$88$	1.60687	0.171293
$89$	8.37645	0.887902	0.443951	−	0.896051i	$-0.353576\pi$
$89$	8.37645	0.887902	0.443951	+	0.896051i	$0.353576\pi$
$90$	−16.4744	−1.73656
$91$	0	0
$92$	2.12103	0.221132
$93$	22.9366	2.37842
$94$	−9.28311	−0.957479
$95$	−2.06886	−0.212260
$96$	3.31105	0.337933
$97$	0.865228	0.0878506	0.0439253	−	0.999035i	$-0.486014\pi$
$97$	0.865228	0.0878506	0.0439253	+	0.999035i	$0.486014\pi$
$98$	7.58633	0.766335
$99$	12.7956	1.28600
$100$	−0.719834	−0.0719834
$101$	16.8788	1.67950	0.839751	−	0.542971i	$-0.182701\pi$
$101$	16.8788	1.67950	0.839751	+	0.542971i	$0.182701\pi$
$102$	24.8836	2.46385
$103$	8.56239	0.843677	0.421839	−	0.906671i	$-0.361385\pi$
$103$	8.56239	0.843677	0.421839	+	0.906671i	$0.361385\pi$
$104$	0	0
$105$	26.1619	2.55314
$106$	−1.14260	−0.110979
$107$	4.02119	0.388743	0.194372	−	0.980928i	$-0.437733\pi$
$107$	4.02119	0.388743	0.194372	+	0.980928i	$0.437733\pi$
$108$	16.4330	1.58126
$109$	7.97857	0.764209	0.382104	−	0.924119i	$-0.375199\pi$
$109$	7.97857	0.764209	0.382104	+	0.924119i	$0.375199\pi$
$110$	−3.32438	−0.316967
$111$	11.9650	1.13567
$112$	−3.81920	−0.360881
$113$	−16.2605	−1.52966	−0.764830	−	0.644232i	$-0.777177\pi$
$113$	−16.2605	−1.52966	−0.764830	+	0.644232i	$0.777177\pi$
$114$	3.31105	0.310108
$115$	−4.38810	−0.409193
$116$	−7.07077	−0.656504
$117$	0	0
$118$	11.1509	1.02652
$119$	−28.7026	−2.63116
$120$	−6.85009	−0.625325
$121$	−8.41798	−0.765271
$122$	2.97140	0.269018
$123$	−5.39556	−0.486501
$124$	6.92729	0.622089
$125$	11.8335	1.05842
$126$	−30.4126	−2.70937
$127$	1.89332	0.168005	0.0840025	−	0.996466i	$-0.473230\pi$
$127$	1.89332	0.168005	0.0840025	+	0.996466i	$0.473230\pi$
$128$	1.00000	0.0883883
$129$	0.782490	0.0688944
$130$	0	0
$131$	7.05307	0.616229	0.308115	−	0.951349i	$-0.400302\pi$
$131$	7.05307	0.616229	0.308115	+	0.951349i	$0.400302\pi$
$132$	5.32042	0.463083
$133$	−3.81920	−0.331167
$134$	−5.51631	−0.476537
$135$	−33.9975	−2.92603
$136$	7.51533	0.644434
$137$	0.297868	0.0254486	0.0127243	−	0.999919i	$-0.495950\pi$
$137$	0.297868	0.0254486	0.0127243	+	0.999919i	$0.495950\pi$
$138$	7.02283	0.597823
$139$	3.85600	0.327062	0.163531	−	0.986538i	$-0.447712\pi$
$139$	3.85600	0.327062	0.163531	+	0.986538i	$0.447712\pi$
$140$	7.90139	0.667789
$141$	−30.7368	−2.58851
$142$	2.43427	0.204280
$143$	0	0
$144$	7.96307	0.663589
$145$	14.6284	1.21482
$146$	13.9360	1.15335
$147$	25.1187	2.07176
$148$	3.61366	0.297041
$149$	3.49418	0.286255	0.143127	−	0.989704i	$-0.454284\pi$
$149$	3.49418	0.286255	0.143127	+	0.989704i	$0.454284\pi$
$150$	−2.38341	−0.194604
$151$	−6.85736	−0.558044	−0.279022	−	0.960285i	$-0.590010\pi$
$151$	−6.85736	−0.558044	−0.279022	+	0.960285i	$0.590010\pi$
$152$	1.00000	0.0811107
$153$	59.8451	4.83819
$154$	−6.13695	−0.494530
$155$	−14.3316	−1.15114
$156$	0	0
$157$	11.3773	0.908010	0.454005	−	0.890999i	$-0.349995\pi$
$157$	11.3773	0.908010	0.454005	+	0.890999i	$0.349995\pi$
$158$	−11.0307	−0.877552
$159$	−3.78320	−0.300027
$160$	−2.06886	−0.163557
$161$	−8.10064	−0.638420
$162$	30.5212	2.39797
$163$	−11.1303	−0.871789	−0.435895	−	0.899998i	$-0.643568\pi$
$163$	−11.1303	−0.871789	−0.435895	+	0.899998i	$0.643568\pi$
$164$	−1.62956	−0.127247
$165$	−11.0072	−0.856908
$166$	9.88549	0.767263
$167$	9.41798	0.728785	0.364393	−	0.931245i	$-0.381277\pi$
$167$	9.41798	0.728785	0.364393	+	0.931245i	$0.381277\pi$
$168$	−12.6456	−0.975628
$169$	0	0
$170$	−15.5481	−1.19249
$171$	7.96307	0.608951
$172$	0.236327	0.0180197
$173$	−13.6480	−1.03764	−0.518820	−	0.854884i	$-0.673628\pi$
$173$	−13.6480	−1.03764	−0.518820	+	0.854884i	$0.673628\pi$
$174$	−23.4117	−1.77484
$175$	2.74919	0.207819
$176$	1.60687	0.121122
$177$	36.9212	2.77517
$178$	8.37645	0.627842
$179$	−22.5738	−1.68724	−0.843621	−	0.536939i	$-0.819581\pi$
$179$	−22.5738	−1.68724	−0.843621	+	0.536939i	$0.819581\pi$
$180$	−16.4744	−1.22793
$181$	−5.59853	−0.416135	−0.208068	−	0.978114i	$-0.566717\pi$
$181$	−5.59853	−0.416135	−0.208068	+	0.978114i	$0.566717\pi$
$182$	0	0
$183$	9.83845	0.727279
$184$	2.12103	0.156364
$185$	−7.47615	−0.549658
$186$	22.9366	1.68179
$187$	12.0761	0.883094
$188$	−9.28311	−0.677040
$189$	−62.7609	−4.56518
$190$	−2.06886	−0.150091
$191$	1.33635	0.0966946	0.0483473	−	0.998831i	$-0.484605\pi$
$191$	1.33635	0.0966946	0.0483473	+	0.998831i	$0.484605\pi$
$192$	3.31105	0.238955
$193$	−17.4031	−1.25270	−0.626350	−	0.779542i	$-0.715452\pi$
$193$	−17.4031	−1.25270	−0.626350	+	0.779542i	$0.715452\pi$
$194$	0.865228	0.0621197
$195$	0	0
$196$	7.58633	0.541880
$197$	−13.0097	−0.926905	−0.463453	−	0.886122i	$-0.653390\pi$
$197$	−13.0097	−0.926905	−0.463453	+	0.886122i	$0.653390\pi$
$198$	12.7956	0.909343
$199$	−18.8544	−1.33655	−0.668276	−	0.743913i	$-0.732968\pi$
$199$	−18.8544	−1.33655	−0.668276	+	0.743913i	$0.732968\pi$
$200$	−0.719834	−0.0508999
$201$	−18.2648	−1.28830
$202$	16.8788	1.18759
$203$	27.0047	1.89536
$204$	24.8836	1.74220
$205$	3.37132	0.235464
$206$	8.56239	0.596570
$207$	16.8899	1.17393
$208$	0	0
$209$	1.60687	0.111149
$210$	26.1619	1.80534
$211$	9.77351	0.672836	0.336418	−	0.941713i	$-0.390784\pi$
$211$	9.77351	0.672836	0.336418	+	0.941713i	$0.390784\pi$
$212$	−1.14260	−0.0784739
$213$	8.06000	0.552262
$214$	4.02119	0.274883
$215$	−0.488926	−0.0333445
$216$	16.4330	1.11812
$217$	−26.4567	−1.79600
$218$	7.97857	0.540377
$219$	46.1428	3.11804
$220$	−3.32438	−0.224129
$221$	0	0
$222$	11.9650	0.803040
$223$	−7.13599	−0.477861	−0.238930	−	0.971037i	$-0.576797\pi$
$223$	−7.13599	−0.477861	−0.238930	+	0.971037i	$0.576797\pi$
$224$	−3.81920	−0.255181
$225$	−5.73208	−0.382139
$226$	−16.2605	−1.08163
$227$	19.7446	1.31049	0.655247	−	0.755415i	$-0.272565\pi$
$227$	19.7446	1.31049	0.655247	+	0.755415i	$0.272565\pi$
$228$	3.31105	0.219280
$229$	−16.5986	−1.09687	−0.548434	−	0.836194i	$-0.684776\pi$
$229$	−16.5986	−1.09687	−0.548434	+	0.836194i	$0.684776\pi$
$230$	−4.38810	−0.289343
$231$	−20.3198	−1.33694
$232$	−7.07077	−0.464219
$233$	−13.7098	−0.898157	−0.449079	−	0.893492i	$-0.648248\pi$
$233$	−13.7098	−0.898157	−0.449079	+	0.893492i	$0.648248\pi$
$234$	0	0
$235$	19.2054	1.25282
$236$	11.1509	0.725862
$237$	−36.5231	−2.37243
$238$	−28.7026	−1.86051
$239$	5.25402	0.339855	0.169927	−	0.985457i	$-0.445647\pi$
$239$	5.25402	0.339855	0.169927	+	0.985457i	$0.445647\pi$
$240$	−6.85009	−0.442171
$241$	−27.9251	−1.79881	−0.899406	−	0.437115i	$-0.856000\pi$
$241$	−27.9251	−1.79881	−0.899406	+	0.437115i	$0.856000\pi$
$242$	−8.41798	−0.541128
$243$	51.7584	3.32031
$244$	2.97140	0.190224
$245$	−15.6950	−1.00272
$246$	−5.39556	−0.344008
$247$	0	0
$248$	6.92729	0.439883
$249$	32.7314	2.07427
$250$	11.8335	0.748417
$251$	−11.2278	−0.708694	−0.354347	−	0.935114i	$-0.615297\pi$
$251$	−11.2278	−0.708694	−0.354347	+	0.935114i	$0.615297\pi$
$252$	−30.4126	−1.91581
$253$	3.40821	0.214272
$254$	1.89332	0.118797
$255$	−51.4807	−3.22385
$256$	1.00000	0.0625000
$257$	−10.1475	−0.632987	−0.316493	−	0.948595i	$-0.602505\pi$
$257$	−10.1475	−0.632987	−0.316493	+	0.948595i	$0.602505\pi$
$258$	0.782490	0.0487157
$259$	−13.8013	−0.857572
$260$	0	0
$261$	−56.3050	−3.48519
$262$	7.05307	0.435740
$263$	1.53544	0.0946794	0.0473397	−	0.998879i	$-0.484926\pi$
$263$	1.53544	0.0946794	0.0473397	+	0.998879i	$0.484926\pi$
$264$	5.32042	0.327449
$265$	2.36387	0.145211
$266$	−3.81920	−0.234170
$267$	27.7349	1.69735
$268$	−5.51631	−0.336962
$269$	24.3965	1.48748	0.743739	−	0.668470i	$-0.233050\pi$
$269$	24.3965	1.48748	0.743739	+	0.668470i	$0.233050\pi$
$270$	−33.9975	−2.06902
$271$	2.24257	0.136226	0.0681131	−	0.997678i	$-0.478302\pi$
$271$	2.24257	0.136226	0.0681131	+	0.997678i	$0.478302\pi$
$272$	7.51533	0.455684
$273$	0	0
$274$	0.297868	0.0179948
$275$	−1.15668	−0.0697503
$276$	7.02283	0.422725
$277$	21.2111	1.27445	0.637227	−	0.770676i	$-0.280081\pi$
$277$	21.2111	1.27445	0.637227	+	0.770676i	$0.280081\pi$
$278$	3.85600	0.231268
$279$	55.1624	3.30249
$280$	7.90139	0.472198
$281$	−26.3516	−1.57200	−0.786002	−	0.618224i	$-0.787853\pi$
$281$	−26.3516	−1.57200	−0.786002	+	0.618224i	$0.787853\pi$
$282$	−30.7368	−1.83035
$283$	−2.71491	−0.161385	−0.0806923	−	0.996739i	$-0.525713\pi$
$283$	−2.71491	−0.161385	−0.0806923	+	0.996739i	$0.525713\pi$
$284$	2.43427	0.144447
$285$	−6.85009	−0.405764
$286$	0	0
$287$	6.22362	0.367369
$288$	7.96307	0.469228
$289$	39.4802	2.32236
$290$	14.6284	0.859009
$291$	2.86481	0.167938
$292$	13.9360	0.815543
$293$	−4.37345	−0.255499	−0.127750	−	0.991806i	$-0.540775\pi$
$293$	−4.37345	−0.255499	−0.127750	+	0.991806i	$0.540775\pi$
$294$	25.1187	1.46495
$295$	−23.0696	−1.34316
$296$	3.61366	0.210040
$297$	26.4056	1.53221
$298$	3.49418	0.202413
$299$	0	0
$300$	−2.38341	−0.137606
$301$	−0.902580	−0.0520239
$302$	−6.85736	−0.394597
$303$	55.8866	3.21060
$304$	1.00000	0.0573539
$305$	−6.14739	−0.351999
$306$	59.8451	3.42111
$307$	20.4151	1.16515	0.582576	−	0.812776i	$-0.302045\pi$
$307$	20.4151	1.16515	0.582576	+	0.812776i	$0.302045\pi$
$308$	−6.13695	−0.349685
$309$	28.3505	1.61280
$310$	−14.3316	−0.813978
$311$	−4.38530	−0.248667	−0.124334	−	0.992240i	$-0.539679\pi$
$311$	−4.38530	−0.248667	−0.124334	+	0.992240i	$0.539679\pi$
$312$	0	0
$313$	−32.5863	−1.84189	−0.920943	−	0.389698i	$-0.872579\pi$
$313$	−32.5863	−1.84189	−0.920943	+	0.389698i	$0.872579\pi$
$314$	11.3773	0.642060
$315$	62.9193	3.54510
$316$	−11.0307	−0.620523
$317$	23.8392	1.33894	0.669470	−	0.742839i	$-0.266521\pi$
$317$	23.8392	1.33894	0.669470	+	0.742839i	$0.266521\pi$
$318$	−3.78320	−0.212151
$319$	−11.3618	−0.636138
$320$	−2.06886	−0.115653
$321$	13.3144	0.743136
$322$	−8.10064	−0.451431
$323$	7.51533	0.418164
$324$	30.5212	1.69562
$325$	0	0
$326$	−11.1303	−0.616448
$327$	26.4175	1.46089
$328$	−1.62956	−0.0899774
$329$	35.4541	1.95465
$330$	−11.0072	−0.605926
$331$	−24.1546	−1.32766	−0.663829	−	0.747884i	$-0.731070\pi$
$331$	−24.1546	−1.32766	−0.663829	+	0.747884i	$0.731070\pi$
$332$	9.88549	0.542537
$333$	28.7758	1.57691
$334$	9.41798	0.515329
$335$	11.4125	0.623529
$336$	−12.6456	−0.689873
$337$	−7.68980	−0.418890	−0.209445	−	0.977820i	$-0.567166\pi$
$337$	−7.68980	−0.418890	−0.209445	+	0.977820i	$0.567166\pi$
$338$	0	0
$339$	−53.8394	−2.92416
$340$	−15.5481	−0.843216
$341$	11.1312	0.602790
$342$	7.96307	0.430593
$343$	−2.23930	−0.120911
$344$	0.236327	0.0127419
$345$	−14.5292	−0.782228
$346$	−13.6480	−0.733722
$347$	−5.15461	−0.276714	−0.138357	−	0.990382i	$-0.544182\pi$
$347$	−5.15461	−0.276714	−0.138357	+	0.990382i	$0.544182\pi$
$348$	−23.4117	−1.25500
$349$	−10.5165	−0.562935	−0.281468	−	0.959571i	$-0.590821\pi$
$349$	−10.5165	−0.562935	−0.281468	+	0.959571i	$0.590821\pi$
$350$	2.74919	0.146951
$351$	0	0
$352$	1.60687	0.0856463
$353$	0.693757	0.0369250	0.0184625	−	0.999830i	$-0.494123\pi$
$353$	0.693757	0.0369250	0.0184625	+	0.999830i	$0.494123\pi$
$354$	36.9212	1.96234
$355$	−5.03616	−0.267292
$356$	8.37645	0.443951
$357$	−95.0357	−5.02982
$358$	−22.5738	−1.19306
$359$	5.93217	0.313088	0.156544	−	0.987671i	$-0.449965\pi$
$359$	5.93217	0.313088	0.156544	+	0.987671i	$0.449965\pi$
$360$	−16.4744	−0.868279
$361$	1.00000	0.0526316
$362$	−5.59853	−0.294252
$363$	−27.8724	−1.46292
$364$	0	0
$365$	−28.8316	−1.50911
$366$	9.83845	0.514264
$367$	−22.6867	−1.18424	−0.592119	−	0.805850i	$-0.701709\pi$
$367$	−22.6867	−1.18424	−0.592119	+	0.805850i	$0.701709\pi$
$368$	2.12103	0.110566
$369$	−12.9763	−0.675519
$370$	−7.47615	−0.388667
$371$	4.36381	0.226558
$372$	22.9366	1.18921
$373$	8.74445	0.452770	0.226385	−	0.974038i	$-0.427309\pi$
$373$	8.74445	0.452770	0.226385	+	0.974038i	$0.427309\pi$
$374$	12.0761	0.624442
$375$	39.1814	2.02332
$376$	−9.28311	−0.478740
$377$	0	0
$378$	−62.7609	−3.22807
$379$	−35.6911	−1.83333	−0.916665	−	0.399657i	$-0.869129\pi$
$379$	−35.6911	−1.83333	−0.916665	+	0.399657i	$0.869129\pi$
$380$	−2.06886	−0.106130
$381$	6.26888	0.321165
$382$	1.33635	0.0683734
$383$	8.19985	0.418993	0.209496	−	0.977809i	$-0.432818\pi$
$383$	8.19985	0.418993	0.209496	+	0.977809i	$0.432818\pi$
$384$	3.31105	0.168966
$385$	12.6965	0.647072
$386$	−17.4031	−0.885793
$387$	1.88189	0.0956616
$388$	0.865228	0.0439253
$389$	−19.4878	−0.988073	−0.494036	−	0.869441i	$-0.664479\pi$
$389$	−19.4878	−0.988073	−0.494036	+	0.869441i	$0.664479\pi$
$390$	0	0
$391$	15.9402	0.806132
$392$	7.58633	0.383167
$393$	23.3531	1.17801
$394$	−13.0097	−0.655421
$395$	22.8208	1.14824
$396$	12.7956	0.643002
$397$	−14.7711	−0.741342	−0.370671	−	0.928764i	$-0.620872\pi$
$397$	−14.7711	−0.741342	−0.370671	+	0.928764i	$0.620872\pi$
$398$	−18.8544	−0.945085
$399$	−12.6456	−0.633071
$400$	−0.719834	−0.0359917
$401$	19.7289	0.985217	0.492608	−	0.870251i	$-0.336044\pi$
$401$	19.7289	0.985217	0.492608	+	0.870251i	$0.336044\pi$
$402$	−18.2648	−0.910965
$403$	0	0
$404$	16.8788	0.839751
$405$	−63.1440	−3.13765
$406$	27.0047	1.34022
$407$	5.80668	0.287826
$408$	24.8836	1.23192
$409$	−18.0828	−0.894135	−0.447067	−	0.894500i	$-0.647532\pi$
$409$	−18.0828	−0.894135	−0.447067	+	0.894500i	$0.647532\pi$
$410$	3.37132	0.166498
$411$	0.986256	0.0486484
$412$	8.56239	0.421839
$413$	−42.5876	−2.09560
$414$	16.8899	0.830092
$415$	−20.4517	−1.00393
$416$	0	0
$417$	12.7674	0.625223
$418$	1.60687	0.0785944
$419$	−17.2375	−0.842105	−0.421053	−	0.907036i	$-0.638339\pi$
$419$	−17.2375	−0.842105	−0.421053	+	0.907036i	$0.638339\pi$
$420$	26.1619	1.27657
$421$	−16.6605	−0.811983	−0.405991	−	0.913877i	$-0.633074\pi$
$421$	−16.6605	−0.811983	−0.405991	+	0.913877i	$0.633074\pi$
$422$	9.77351	0.475767
$423$	−73.9220	−3.59421
$424$	−1.14260	−0.0554894
$425$	−5.40979	−0.262413
$426$	8.06000	0.390508
$427$	−11.3484	−0.549186
$428$	4.02119	0.194372
$429$	0	0
$430$	−0.488926	−0.0235781
$431$	−25.0673	−1.20745	−0.603724	−	0.797194i	$-0.706317\pi$
$431$	−25.0673	−1.20745	−0.603724	+	0.797194i	$0.706317\pi$
$432$	16.4330	0.790632
$433$	14.1339	0.679234	0.339617	−	0.940564i	$-0.389703\pi$
$433$	14.1339	0.679234	0.339617	+	0.940564i	$0.389703\pi$
$434$	−26.4567	−1.26996
$435$	48.4354	2.32230
$436$	7.97857	0.382104
$437$	2.12103	0.101463
$438$	46.1428	2.20479
$439$	23.2727	1.11075	0.555373	−	0.831601i	$-0.312575\pi$
$439$	23.2727	1.11075	0.555373	+	0.831601i	$0.312575\pi$
$440$	−3.32438	−0.158483
$441$	60.4104	2.87669
$442$	0	0
$443$	−5.00740	−0.237909	−0.118954	−	0.992900i	$-0.537954\pi$
$443$	−5.00740	−0.237909	−0.118954	+	0.992900i	$0.537954\pi$
$444$	11.9650	0.567835
$445$	−17.3297	−0.821505
$446$	−7.13599	−0.337899
$447$	11.5694	0.547215
$448$	−3.81920	−0.180440
$449$	22.4512	1.05954	0.529768	−	0.848143i	$-0.322279\pi$
$449$	22.4512	1.05954	0.529768	+	0.848143i	$0.322279\pi$
$450$	−5.73208	−0.270213
$451$	−2.61849	−0.123300
$452$	−16.2605	−0.764830
$453$	−22.7051	−1.06678
$454$	19.7446	0.926659
$455$	0	0
$456$	3.31105	0.155054
$457$	10.2779	0.480778	0.240389	−	0.970677i	$-0.422725\pi$
$457$	10.2779	0.480778	0.240389	+	0.970677i	$0.422725\pi$
$458$	−16.5986	−0.775603
$459$	123.499	5.76445
$460$	−4.38810	−0.204596
$461$	−3.07052	−0.143008	−0.0715042	−	0.997440i	$-0.522780\pi$
$461$	−3.07052	−0.143008	−0.0715042	+	0.997440i	$0.522780\pi$
$462$	−20.3198	−0.945361
$463$	−11.6200	−0.540025	−0.270012	−	0.962857i	$-0.587028\pi$
$463$	−11.6200	−0.540025	−0.270012	+	0.962857i	$0.587028\pi$
$464$	−7.07077	−0.328252
$465$	−47.4525	−2.20056
$466$	−13.7098	−0.635093
$467$	−2.07845	−0.0961793	−0.0480896	−	0.998843i	$-0.515313\pi$
$467$	−2.07845	−0.0961793	−0.0480896	+	0.998843i	$0.515313\pi$
$468$	0	0
$469$	21.0679	0.972827
$470$	19.2054	0.885880
$471$	37.6710	1.73579
$472$	11.1509	0.513262
$473$	0.379746	0.0174607
$474$	−36.5231	−1.67756
$475$	−0.719834	−0.0330282
$476$	−28.7026	−1.31558
$477$	−9.09858	−0.416595
$478$	5.25402	0.240313
$479$	7.16031	0.327163	0.163581	−	0.986530i	$-0.447695\pi$
$479$	7.16031	0.327163	0.163581	+	0.986530i	$0.447695\pi$
$480$	−6.85009	−0.312662
$481$	0	0
$482$	−27.9251	−1.27195
$483$	−26.8216	−1.22043
$484$	−8.41798	−0.382635
$485$	−1.79003	−0.0812812
$486$	51.7584	2.34781
$487$	−1.15351	−0.0522704	−0.0261352	−	0.999658i	$-0.508320\pi$
$487$	−1.15351	−0.0522704	−0.0261352	+	0.999658i	$0.508320\pi$
$488$	2.97140	0.134509
$489$	−36.8529	−1.66654
$490$	−15.6950	−0.709029
$491$	−11.6305	−0.524879	−0.262440	−	0.964948i	$-0.584527\pi$
$491$	−11.6305	−0.524879	−0.262440	+	0.964948i	$0.584527\pi$
$492$	−5.39556	−0.243250
$493$	−53.1392	−2.39327
$494$	0	0
$495$	−26.4722	−1.18984
$496$	6.92729	0.311044
$497$	−9.29698	−0.417027
$498$	32.7314	1.46673
$499$	−12.3876	−0.554545	−0.277273	−	0.960791i	$-0.589431\pi$
$499$	−12.3876	−0.554545	−0.277273	+	0.960791i	$0.589431\pi$
$500$	11.8335	0.529211
$501$	31.1834	1.39317
$502$	−11.2278	−0.501122
$503$	−13.2763	−0.591962	−0.295981	−	0.955194i	$-0.595646\pi$
$503$	−13.2763	−0.591962	−0.295981	+	0.955194i	$0.595646\pi$
$504$	−30.4126	−1.35468
$505$	−34.9198	−1.55391
$506$	3.40821	0.151513
$507$	0	0
$508$	1.89332	0.0840025
$509$	25.1619	1.11528	0.557640	−	0.830083i	$-0.311707\pi$
$509$	25.1619	1.11528	0.557640	+	0.830083i	$0.311707\pi$
$510$	−51.4807	−2.27960
$511$	−53.2244	−2.35451
$512$	1.00000	0.0441942
$513$	16.4330	0.725533
$514$	−10.1475	−0.447589
$515$	−17.7144	−0.780588
$516$	0.782490	0.0344472
$517$	−14.9167	−0.656036
$518$	−13.8013	−0.606395
$519$	−45.1893	−1.98359
$520$	0	0
$521$	16.4535	0.720841	0.360421	−	0.932790i	$-0.382633\pi$
$521$	16.4535	0.720841	0.360421	+	0.932790i	$0.382633\pi$
$522$	−56.3050	−2.46440
$523$	4.28254	0.187262	0.0936312	−	0.995607i	$-0.470153\pi$
$523$	4.28254	0.187262	0.0936312	+	0.995607i	$0.470153\pi$
$524$	7.05307	0.308115
$525$	9.10272	0.397275
$526$	1.53544	0.0669484
$527$	52.0608	2.26781
$528$	5.32042	0.231542
$529$	−18.5012	−0.804402
$530$	2.36387	0.102680
$531$	88.7953	3.85339
$532$	−3.81920	−0.165584
$533$	0	0
$534$	27.7349	1.20021
$535$	−8.31926	−0.359673
$536$	−5.51631	−0.238268
$537$	−74.7429	−3.22539
$538$	24.3965	1.05181
$539$	12.1902	0.525070
$540$	−33.9975	−1.46302
$541$	43.1628	1.85571	0.927857	−	0.372937i	$-0.121649\pi$
$541$	43.1628	1.85571	0.927857	+	0.372937i	$0.121649\pi$
$542$	2.24257	0.0963265
$543$	−18.5370	−0.795499
$544$	7.51533	0.322217
$545$	−16.5065	−0.707062
$546$	0	0
$547$	−9.95138	−0.425490	−0.212745	−	0.977108i	$-0.568240\pi$
$547$	−9.95138	−0.425490	−0.212745	+	0.977108i	$0.568240\pi$
$548$	0.297868	0.0127243
$549$	23.6614	1.00985
$550$	−1.15668	−0.0493209
$551$	−7.07077	−0.301225
$552$	7.02283	0.298912
$553$	42.1283	1.79148
$554$	21.2111	0.901175
$555$	−24.7539	−1.05075
$556$	3.85600	0.163531
$557$	−38.2539	−1.62087	−0.810435	−	0.585828i	$-0.800769\pi$
$557$	−38.2539	−1.62087	−0.810435	+	0.585828i	$0.800769\pi$
$558$	55.1624	2.33521
$559$	0	0
$560$	7.90139	0.333894
$561$	39.9847	1.68816
$562$	−26.3516	−1.11157
$563$	−6.52428	−0.274966	−0.137483	−	0.990504i	$-0.543901\pi$
$563$	−6.52428	−0.274966	−0.137483	+	0.990504i	$0.543901\pi$
$564$	−30.7368	−1.29425
$565$	33.6407	1.41527
$566$	−2.71491	−0.114116
$567$	−116.567	−4.89534
$568$	2.43427	0.102140
$569$	−22.1263	−0.927582	−0.463791	−	0.885945i	$-0.653511\pi$
$569$	−22.1263	−0.927582	−0.463791	+	0.885945i	$0.653511\pi$
$570$	−6.85009	−0.286919
$571$	37.8337	1.58329	0.791645	−	0.610981i	$-0.209225\pi$
$571$	37.8337	1.58329	0.791645	+	0.610981i	$0.209225\pi$
$572$	0	0
$573$	4.42471	0.184845
$574$	6.22362	0.259769
$575$	−1.52679	−0.0636714
$576$	7.96307	0.331794
$577$	17.1044	0.712065	0.356032	−	0.934474i	$-0.384129\pi$
$577$	17.1044	0.712065	0.356032	+	0.934474i	$0.384129\pi$
$578$	39.4802	1.64216
$579$	−57.6225	−2.39471
$580$	14.6284	0.607411
$581$	−37.7547	−1.56633
$582$	2.86481	0.118750
$583$	−1.83600	−0.0760394
$584$	13.9360	0.576676
$585$	0	0
$586$	−4.37345	−0.180665
$587$	−20.2336	−0.835129	−0.417565	−	0.908647i	$-0.637116\pi$
$587$	−20.2336	−0.835129	−0.417565	+	0.908647i	$0.637116\pi$
$588$	25.1187	1.03588
$589$	6.92729	0.285434
$590$	−23.0696	−0.949761
$591$	−43.0759	−1.77191
$592$	3.61366	0.148521
$593$	−47.1987	−1.93822	−0.969108	−	0.246636i	$-0.920675\pi$
$593$	−47.1987	−1.93822	−0.969108	+	0.246636i	$0.920675\pi$
$594$	26.4056	1.08343
$595$	59.3815	2.43440
$596$	3.49418	0.143127
$597$	−62.4279	−2.55500
$598$	0	0
$599$	18.0321	0.736770	0.368385	−	0.929673i	$-0.379911\pi$
$599$	18.0321	0.736770	0.368385	+	0.929673i	$0.379911\pi$
$600$	−2.38341	−0.0973022
$601$	−19.4644	−0.793970	−0.396985	−	0.917825i	$-0.629944\pi$
$601$	−19.4644	−0.793970	−0.396985	+	0.917825i	$0.629944\pi$
$602$	−0.902580	−0.0367864
$603$	−43.9268	−1.78884
$604$	−6.85736	−0.279022
$605$	17.4156	0.708044
$606$	55.8866	2.27024
$607$	27.5123	1.11669	0.558344	−	0.829610i	$-0.311437\pi$
$607$	27.5123	1.11669	0.558344	+	0.829610i	$0.311437\pi$
$608$	1.00000	0.0405554
$609$	89.4140	3.62324
$610$	−6.14739	−0.248901
$611$	0	0
$612$	59.8451	2.41909
$613$	−29.5648	−1.19411	−0.597055	−	0.802200i	$-0.703663\pi$
$613$	−29.5648	−1.19411	−0.597055	+	0.802200i	$0.703663\pi$
$614$	20.4151	0.823887
$615$	11.1626	0.450121
$616$	−6.13695	−0.247265
$617$	−35.8703	−1.44409	−0.722043	−	0.691848i	$-0.756797\pi$
$617$	−35.8703	−1.44409	−0.722043	+	0.691848i	$0.756797\pi$
$618$	28.3505	1.14042
$619$	−34.1128	−1.37111	−0.685555	−	0.728021i	$-0.740440\pi$
$619$	−34.1128	−1.37111	−0.685555	+	0.728021i	$0.740440\pi$
$620$	−14.3316	−0.575569
$621$	34.8548	1.39867
$622$	−4.38530	−0.175834
$623$	−31.9914	−1.28171
$624$	0	0
$625$	−20.8827	−0.835307
$626$	−32.5863	−1.30241
$627$	5.32042	0.212477
$628$	11.3773	0.454005
$629$	27.1579	1.08286
$630$	62.9193	2.50676
$631$	33.8741	1.34851	0.674254	−	0.738500i	$-0.264465\pi$
$631$	33.8741	1.34851	0.674254	+	0.738500i	$0.264465\pi$
$632$	−11.0307	−0.438776
$633$	32.3606	1.28622
$634$	23.8392	0.946774
$635$	−3.91701	−0.155442
$636$	−3.78320	−0.150014
$637$	0	0
$638$	−11.3618	−0.449817
$639$	19.3843	0.766830
$640$	−2.06886	−0.0817787
$641$	−16.2515	−0.641896	−0.320948	−	0.947097i	$-0.604001\pi$
$641$	−16.2515	−0.641896	−0.320948	+	0.947097i	$0.604001\pi$
$642$	13.3144	0.525476
$643$	−10.0186	−0.395096	−0.197548	−	0.980293i	$-0.563298\pi$
$643$	−10.0186	−0.395096	−0.197548	+	0.980293i	$0.563298\pi$
$644$	−8.10064	−0.319210
$645$	−1.61886	−0.0637425
$646$	7.51533	0.295687
$647$	6.28746	0.247185	0.123593	−	0.992333i	$-0.460558\pi$
$647$	6.28746	0.247185	0.123593	+	0.992333i	$0.460558\pi$
$648$	30.5212	1.19899
$649$	17.9180	0.703343
$650$	0	0
$651$	−87.5996	−3.43330
$652$	−11.1303	−0.435895
$653$	−36.2620	−1.41904	−0.709521	−	0.704684i	$-0.751089\pi$
$653$	−36.2620	−1.41904	−0.709521	+	0.704684i	$0.751089\pi$
$654$	26.4175	1.03300
$655$	−14.5918	−0.570148
$656$	−1.62956	−0.0636236
$657$	110.973	4.32948
$658$	35.4541	1.38214
$659$	−10.2406	−0.398916	−0.199458	−	0.979906i	$-0.563918\pi$
$659$	−10.2406	−0.398916	−0.199458	+	0.979906i	$0.563918\pi$
$660$	−11.0072	−0.428454
$661$	−6.24293	−0.242822	−0.121411	−	0.992602i	$-0.538742\pi$
$661$	−6.24293	−0.242822	−0.121411	+	0.992602i	$0.538742\pi$
$662$	−24.1546	−0.938796
$663$	0	0
$664$	9.88549	0.383631
$665$	7.90139	0.306403
$666$	28.7758	1.11504
$667$	−14.9973	−0.580698
$668$	9.41798	0.364393
$669$	−23.6276	−0.913497
$670$	11.4125	0.440902
$671$	4.77464	0.184323
$672$	−12.6456	−0.487814
$673$	26.1706	1.00880	0.504402	−	0.863469i	$-0.331713\pi$
$673$	26.1706	1.00880	0.504402	+	0.863469i	$0.331713\pi$
$674$	−7.68980	−0.296200
$675$	−11.8290	−0.455299
$676$	0	0
$677$	13.8537	0.532441	0.266220	−	0.963912i	$-0.414225\pi$
$677$	13.8537	0.532441	0.266220	+	0.963912i	$0.414225\pi$
$678$	−53.8394	−2.06769
$679$	−3.30448	−0.126814
$680$	−15.5481	−0.596244
$681$	65.3753	2.50519
$682$	11.1312	0.426237
$683$	−31.1542	−1.19208	−0.596042	−	0.802954i	$-0.703261\pi$
$683$	−31.1542	−1.19208	−0.596042	+	0.802954i	$0.703261\pi$
$684$	7.96307	0.304475
$685$	−0.616246	−0.0235455
$686$	−2.23930	−0.0854968
$687$	−54.9589	−2.09681
$688$	0.236327	0.00900987
$689$	0	0
$690$	−14.5292	−0.553118
$691$	15.2189	0.578954	0.289477	−	0.957185i	$-0.406519\pi$
$691$	15.2189	0.578954	0.289477	+	0.957185i	$0.406519\pi$
$692$	−13.6480	−0.518820
$693$	−48.8690	−1.85638
$694$	−5.15461	−0.195666
$695$	−7.97751	−0.302604
$696$	−23.4117	−0.887418
$697$	−12.2467	−0.463876
$698$	−10.5165	−0.398055
$699$	−45.3938	−1.71695
$700$	2.74919	0.103910
$701$	13.3257	0.503306	0.251653	−	0.967818i	$-0.419026\pi$
$701$	13.3257	0.503306	0.251653	+	0.967818i	$0.419026\pi$
$702$	0	0
$703$	3.61366	0.136292
$704$	1.60687	0.0605611
$705$	63.5901	2.39494
$706$	0.693757	0.0261099
$707$	−64.4636	−2.42440
$708$	36.9212	1.38758
$709$	−17.4913	−0.656899	−0.328450	−	0.944521i	$-0.606526\pi$
$709$	−17.4913	−0.656899	−0.328450	+	0.944521i	$0.606526\pi$
$710$	−5.03616	−0.189004
$711$	−87.8379	−3.29418
$712$	8.37645	0.313921
$713$	14.6930	0.550256
$714$	−95.0357	−3.55662
$715$	0	0
$716$	−22.5738	−0.843621
$717$	17.3963	0.649679
$718$	5.93217	0.221387
$719$	34.5785	1.28956	0.644781	−	0.764367i	$-0.276949\pi$
$719$	34.5785	1.28956	0.644781	+	0.764367i	$0.276949\pi$
$720$	−16.4744	−0.613966
$721$	−32.7015	−1.21787
$722$	1.00000	0.0372161
$723$	−92.4613	−3.43867
$724$	−5.59853	−0.208068
$725$	5.08978	0.189030
$726$	−27.8724	−1.03444
$727$	−31.7321	−1.17688	−0.588439	−	0.808541i	$-0.700257\pi$
$727$	−31.7321	−1.17688	−0.588439	+	0.808541i	$0.700257\pi$
$728$	0	0
$729$	79.8112	2.95597
$730$	−28.8316	−1.06710
$731$	1.77607	0.0656904
$732$	9.83845	0.363640
$733$	−35.2520	−1.30206	−0.651032	−	0.759051i	$-0.725663\pi$
$733$	−35.2520	−1.30206	−0.651032	+	0.759051i	$0.725663\pi$
$734$	−22.6867	−0.837383
$735$	−51.9670	−1.91683
$736$	2.12103	0.0781821
$737$	−8.86398	−0.326509
$738$	−12.9763	−0.477664
$739$	7.47892	0.275117	0.137558	−	0.990494i	$-0.456075\pi$
$739$	7.47892	0.275117	0.137558	+	0.990494i	$0.456075\pi$
$740$	−7.47615	−0.274829
$741$	0	0
$742$	4.36381	0.160201
$743$	−26.2537	−0.963155	−0.481578	−	0.876403i	$-0.659936\pi$
$743$	−26.2537	−0.963155	−0.481578	+	0.876403i	$0.659936\pi$
$744$	22.9366	0.840897
$745$	−7.22896	−0.264849
$746$	8.74445	0.320157
$747$	78.7188	2.88017
$748$	12.0761	0.441547
$749$	−15.3577	−0.561160
$750$	39.1814	1.43070
$751$	−14.5515	−0.530992	−0.265496	−	0.964112i	$-0.585536\pi$
$751$	−14.5515	−0.530992	−0.265496	+	0.964112i	$0.585536\pi$
$752$	−9.28311	−0.338520
$753$	−37.1759	−1.35477
$754$	0	0
$755$	14.1869	0.516314
$756$	−62.7609	−2.28259
$757$	23.6468	0.859458	0.429729	−	0.902958i	$-0.358609\pi$
$757$	23.6468	0.859458	0.429729	+	0.902958i	$0.358609\pi$
$758$	−35.6911	−1.29636
$759$	11.2848	0.409611
$760$	−2.06886	−0.0750453
$761$	−19.6856	−0.713604	−0.356802	−	0.934180i	$-0.616133\pi$
$761$	−19.6856	−0.713604	−0.356802	+	0.934180i	$0.616133\pi$
$762$	6.26888	0.227098
$763$	−30.4718	−1.10315
$764$	1.33635	0.0483473
$765$	−123.811	−4.47639
$766$	8.19985	0.296273
$767$	0	0
$768$	3.31105	0.119477
$769$	3.77183	0.136016	0.0680079	−	0.997685i	$-0.478336\pi$
$769$	3.77183	0.136016	0.0680079	+	0.997685i	$0.478336\pi$
$770$	12.6965	0.457549
$771$	−33.5991	−1.21004
$772$	−17.4031	−0.626350
$773$	−40.2642	−1.44820	−0.724101	−	0.689694i	$-0.757745\pi$
$773$	−40.2642	−1.44820	−0.724101	+	0.689694i	$0.757745\pi$
$774$	1.88189	0.0676430
$775$	−4.98650	−0.179120
$776$	0.865228	0.0310599
$777$	−45.6969	−1.63937
$778$	−19.4878	−0.698673
$779$	−1.62956	−0.0583850
$780$	0	0
$781$	3.91155	0.139966
$782$	15.9402	0.570021
$783$	−116.194	−4.15242
$784$	7.58633	0.270940
$785$	−23.5381	−0.840110
$786$	23.3531	0.832977
$787$	−25.1798	−0.897562	−0.448781	−	0.893642i	$-0.648142\pi$
$787$	−25.1798	−0.897562	−0.448781	+	0.893642i	$0.648142\pi$
$788$	−13.0097	−0.463453
$789$	5.08393	0.180993
$790$	22.8208	0.811929
$791$	62.1022	2.20810
$792$	12.7956	0.454671
$793$	0	0
$794$	−14.7711	−0.524208
$795$	7.82689	0.277591
$796$	−18.8544	−0.668276
$797$	18.9890	0.672624	0.336312	−	0.941751i	$-0.390820\pi$
$797$	18.9890	0.672624	0.336312	+	0.941751i	$0.390820\pi$
$798$	−12.6456	−0.447649
$799$	−69.7656	−2.46813
$800$	−0.719834	−0.0254500
$801$	66.7022	2.35681
$802$	19.7289	0.696653
$803$	22.3933	0.790242
$804$	−18.2648	−0.644150
$805$	16.7591	0.590679
$806$	0	0
$807$	80.7779	2.84352
$808$	16.8788	0.593794
$809$	14.9037	0.523987	0.261994	−	0.965070i	$-0.415620\pi$
$809$	14.9037	0.523987	0.261994	+	0.965070i	$0.415620\pi$
$810$	−63.1440	−2.21865
$811$	−13.3802	−0.469842	−0.234921	−	0.972014i	$-0.575483\pi$
$811$	−13.3802	−0.469842	−0.234921	+	0.972014i	$0.575483\pi$
$812$	27.0047	0.947680
$813$	7.42526	0.260415
$814$	5.80668	0.203524
$815$	23.0269	0.806597
$816$	24.8836	0.871102
$817$	0.236327	0.00826803
$818$	−18.0828	−0.632249
$819$	0	0
$820$	3.37132	0.117732
$821$	−26.8896	−0.938452	−0.469226	−	0.883078i	$-0.655467\pi$
$821$	−26.8896	−0.938452	−0.469226	+	0.883078i	$0.655467\pi$
$822$	0.986256	0.0343996
$823$	−39.1283	−1.36393	−0.681963	−	0.731386i	$-0.738874\pi$
$823$	−39.1283	−1.36393	−0.681963	+	0.731386i	$0.738874\pi$
$824$	8.56239	0.298285
$825$	−3.82982	−0.133337
$826$	−42.5876	−1.48181
$827$	19.9328	0.693131	0.346566	−	0.938026i	$-0.387348\pi$
$827$	19.9328	0.693131	0.346566	+	0.938026i	$0.387348\pi$
$828$	16.8899	0.586964
$829$	32.3529	1.12366	0.561832	−	0.827251i	$-0.310097\pi$
$829$	32.3529	1.12366	0.561832	+	0.827251i	$0.310097\pi$
$830$	−20.4517	−0.709887
$831$	70.2312	2.43629
$832$	0	0
$833$	57.0137	1.97541
$834$	12.7674	0.442100
$835$	−19.4844	−0.674287
$836$	1.60687	0.0555747
$837$	113.836	3.93474
$838$	−17.2375	−0.595458
$839$	1.74868	0.0603712	0.0301856	−	0.999544i	$-0.490390\pi$
$839$	1.74868	0.0603712	0.0301856	+	0.999544i	$0.490390\pi$
$840$	26.1619	0.902671
$841$	20.9958	0.723992
$842$	−16.6605	−0.574159
$843$	−87.2515	−3.00510
$844$	9.77351	0.336418
$845$	0	0
$846$	−73.9220	−2.54149
$847$	32.1500	1.10469
$848$	−1.14260	−0.0392369
$849$	−8.98920	−0.308509
$850$	−5.40979	−0.185554
$851$	7.66468	0.262742
$852$	8.06000	0.276131
$853$	55.6492	1.90539	0.952696	−	0.303924i	$-0.0982968\pi$
$853$	55.6492	1.90539	0.952696	+	0.303924i	$0.0982968\pi$
$854$	−11.3484	−0.388333
$855$	−16.4744	−0.563414
$856$	4.02119	0.137441
$857$	17.2627	0.589681	0.294841	−	0.955546i	$-0.404733\pi$
$857$	17.2627	0.589681	0.294841	+	0.955546i	$0.404733\pi$
$858$	0	0
$859$	−5.84042	−0.199273	−0.0996363	−	0.995024i	$-0.531768\pi$
$859$	−5.84042	−0.199273	−0.0996363	+	0.995024i	$0.531768\pi$
$860$	−0.488926	−0.0166722
$861$	20.6067	0.702276
$862$	−25.0673	−0.853794
$863$	57.0964	1.94358	0.971792	−	0.235841i	$-0.0757845\pi$
$863$	57.0964	1.94358	0.971792	+	0.235841i	$0.0757845\pi$
$864$	16.4330	0.559061
$865$	28.2358	0.960046
$866$	14.1339	0.480291
$867$	130.721	4.43951
$868$	−26.4567	−0.898000
$869$	−17.7248	−0.601273
$870$	48.4354	1.64211
$871$	0	0
$872$	7.97857	0.270189
$873$	6.88987	0.233187
$874$	2.12103	0.0717448
$875$	−45.1946	−1.52786
$876$	46.1428	1.55902
$877$	20.4286	0.689825	0.344913	−	0.938635i	$-0.387909\pi$
$877$	20.4286	0.689825	0.344913	+	0.938635i	$0.387909\pi$
$878$	23.2727	0.785416
$879$	−14.4807	−0.488422
$880$	−3.32438	−0.112065
$881$	27.0322	0.910737	0.455369	−	0.890303i	$-0.349507\pi$
$881$	27.0322	0.910737	0.455369	+	0.890303i	$0.349507\pi$
$882$	60.4104	2.03412
$883$	48.7246	1.63971	0.819856	−	0.572570i	$-0.194054\pi$
$883$	48.7246	1.63971	0.819856	+	0.572570i	$0.194054\pi$
$884$	0	0
$885$	−76.3847	−2.56764
$886$	−5.00740	−0.168227
$887$	25.6609	0.861608	0.430804	−	0.902446i	$-0.358230\pi$
$887$	25.6609	0.861608	0.430804	+	0.902446i	$0.358230\pi$
$888$	11.9650	0.401520
$889$	−7.23098	−0.242519
$890$	−17.3297	−0.580892
$891$	49.0435	1.64302
$892$	−7.13599	−0.238930
$893$	−9.28311	−0.310647
$894$	11.5694	0.386939
$895$	46.7019	1.56107
$896$	−3.81920	−0.127591
$897$	0	0
$898$	22.4512	0.749205
$899$	−48.9812	−1.63362
$900$	−5.73208	−0.191069
$901$	−8.58699	−0.286074
$902$	−2.61849	−0.0871860
$903$	−2.98849	−0.0994507
$904$	−16.2605	−0.540817
$905$	11.5825	0.385017
$906$	−22.7051	−0.754326
$907$	9.18550	0.304999	0.152500	−	0.988304i	$-0.451268\pi$
$907$	9.18550	0.304999	0.152500	+	0.988304i	$0.451268\pi$
$908$	19.7446	0.655247
$909$	134.407	4.45800
$910$	0	0
$911$	27.8204	0.921730	0.460865	−	0.887470i	$-0.347539\pi$
$911$	27.8204	0.921730	0.460865	+	0.887470i	$0.347539\pi$
$912$	3.31105	0.109640
$913$	15.8847	0.525706
$914$	10.2779	0.339961
$915$	−20.3543	−0.672894
$916$	−16.5986	−0.548434
$917$	−26.9371	−0.889542
$918$	123.499	4.07608
$919$	−22.7901	−0.751777	−0.375889	−	0.926665i	$-0.622663\pi$
$919$	−22.7901	−0.751777	−0.375889	+	0.926665i	$0.622663\pi$
$920$	−4.38810	−0.144671
$921$	67.5955	2.22735
$922$	−3.07052	−0.101122
$923$	0	0
$924$	−20.3198	−0.668471
$925$	−2.60124	−0.0855282
$926$	−11.6200	−0.381855
$927$	68.1829	2.23942
$928$	−7.07077	−0.232109
$929$	9.72061	0.318923	0.159461	−	0.987204i	$-0.449024\pi$
$929$	9.72061	0.318923	0.159461	+	0.987204i	$0.449024\pi$
$930$	−47.4525	−1.55603
$931$	7.58633	0.248632
$932$	−13.7098	−0.449079
$933$	−14.5200	−0.475362
$934$	−2.07845	−0.0680090
$935$	−24.9838	−0.817057
$936$	0	0
$937$	5.58279	0.182382	0.0911908	−	0.995833i	$-0.470933\pi$
$937$	5.58279	0.182382	0.0911908	+	0.995833i	$0.470933\pi$
$938$	21.0679	0.687892
$939$	−107.895	−3.52102
$940$	19.2054	0.626412
$941$	−53.2607	−1.73625	−0.868125	−	0.496345i	$-0.834675\pi$
$941$	−53.2607	−1.73625	−0.868125	+	0.496345i	$0.834675\pi$
$942$	37.6710	1.22739
$943$	−3.45634	−0.112554
$944$	11.1509	0.362931
$945$	129.843	4.22380
$946$	0.379746	0.0123466
$947$	−36.5415	−1.18744	−0.593720	−	0.804672i	$-0.702341\pi$
$947$	−36.5415	−1.18744	−0.593720	+	0.804672i	$0.702341\pi$
$948$	−36.5231	−1.18621
$949$	0	0
$950$	−0.719834	−0.0233545
$951$	78.9327	2.55957
$952$	−28.7026	−0.930256
$953$	−54.6812	−1.77130	−0.885648	−	0.464357i	$-0.846286\pi$
$953$	−54.6812	−1.77130	−0.885648	+	0.464357i	$0.846286\pi$
$954$	−9.09858	−0.294577
$955$	−2.76471	−0.0894639
$956$	5.25402	0.169927
$957$	−37.6195	−1.21606
$958$	7.16031	0.231339
$959$	−1.13762	−0.0367356
$960$	−6.85009	−0.221086
$961$	16.9873	0.547978
$962$	0	0
$963$	32.0210	1.03186
$964$	−27.9251	−0.899406
$965$	36.0044	1.15902
$966$	−26.8216	−0.862972
$967$	28.6607	0.921667	0.460834	−	0.887487i	$-0.347551\pi$
$967$	28.6607	0.921667	0.460834	+	0.887487i	$0.347551\pi$
$968$	−8.41798	−0.270564
$969$	24.8836	0.799378
$970$	−1.79003	−0.0574745
$971$	−37.5903	−1.20633	−0.603165	−	0.797617i	$-0.706094\pi$
$971$	−37.5903	−1.20633	−0.603165	+	0.797617i	$0.706094\pi$
$972$	51.7584	1.66015
$973$	−14.7269	−0.472121
$974$	−1.15351	−0.0369608
$975$	0	0
$976$	2.97140	0.0951121
$977$	31.3809	1.00397	0.501983	−	0.864878i	$-0.332604\pi$
$977$	31.3809	1.00397	0.501983	+	0.864878i	$0.332604\pi$
$978$	−36.8529	−1.17842
$979$	13.4598	0.430179
$980$	−15.6950	−0.501359
$981$	63.5339	2.02848
$982$	−11.6305	−0.371146
$983$	−0.958788	−0.0305806	−0.0152903	−	0.999883i	$-0.504867\pi$
$983$	−0.958788	−0.0305806	−0.0152903	+	0.999883i	$0.504867\pi$
$984$	−5.39556	−0.172004
$985$	26.9153	0.857592
$986$	−53.1392	−1.69230
$987$	117.390	3.73658
$988$	0	0
$989$	0.501256	0.0159390
$990$	−26.4722	−0.841343
$991$	53.5509	1.70110	0.850550	−	0.525895i	$-0.176269\pi$
$991$	53.5509	1.70110	0.850550	+	0.525895i	$0.176269\pi$
$992$	6.92729	0.219942
$993$	−79.9772	−2.53800
$994$	−9.29698	−0.294882
$995$	39.0070	1.23661
$996$	32.7314	1.03713
$997$	42.1502	1.33491	0.667455	−	0.744650i	$-0.267384\pi$
$997$	42.1502	1.33491	0.667455	+	0.744650i	$0.267384\pi$
$998$	−12.3876	−0.392123
$999$	59.3832	1.87880

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6422.2.a.bf.1.7		7
13.5	odd	4		494.2.d.c.77.7	✓	14
13.8	odd	4		494.2.d.c.77.14	yes	14
13.12	even	2		6422.2.a.be.1.7		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
494.2.d.c.77.7	✓	14	13.5	odd	4
494.2.d.c.77.14	yes	14	13.8	odd	4
6422.2.a.be.1.7		7	13.12	even	2
6422.2.a.bf.1.7		7	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +3.31105 q^{3} +1.00000 q^{4} -2.06886 q^{5} +3.31105 q^{6} -3.81920 q^{7} +1.00000 q^{8} +7.96307 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +3.31105 q^{3} +1.00000 q^{4} -2.06886 q^{5} +3.31105 q^{6} -3.81920 q^{7} +1.00000 q^{8} +7.96307 q^{9} -2.06886 q^{10} +1.60687 q^{11} +3.31105 q^{12} -3.81920 q^{14} -6.85009 q^{15} +1.00000 q^{16} +7.51533 q^{17} +7.96307 q^{18} +1.00000 q^{19} -2.06886 q^{20} -12.6456 q^{21} +1.60687 q^{22} +2.12103 q^{23} +3.31105 q^{24} -0.719834 q^{25} +16.4330 q^{27} -3.81920 q^{28} -7.07077 q^{29} -6.85009 q^{30} +6.92729 q^{31} +1.00000 q^{32} +5.32042 q^{33} +7.51533 q^{34} +7.90139 q^{35} +7.96307 q^{36} +3.61366 q^{37} +1.00000 q^{38} -2.06886 q^{40} -1.62956 q^{41} -12.6456 q^{42} +0.236327 q^{43} +1.60687 q^{44} -16.4744 q^{45} +2.12103 q^{46} -9.28311 q^{47} +3.31105 q^{48} +7.58633 q^{49} -0.719834 q^{50} +24.8836 q^{51} -1.14260 q^{53} +16.4330 q^{54} -3.32438 q^{55} -3.81920 q^{56} +3.31105 q^{57} -7.07077 q^{58} +11.1509 q^{59} -6.85009 q^{60} +2.97140 q^{61} +6.92729 q^{62} -30.4126 q^{63} +1.00000 q^{64} +5.32042 q^{66} -5.51631 q^{67} +7.51533 q^{68} +7.02283 q^{69} +7.90139 q^{70} +2.43427 q^{71} +7.96307 q^{72} +13.9360 q^{73} +3.61366 q^{74} -2.38341 q^{75} +1.00000 q^{76} -6.13695 q^{77} -11.0307 q^{79} -2.06886 q^{80} +30.5212 q^{81} -1.62956 q^{82} +9.88549 q^{83} -12.6456 q^{84} -15.5481 q^{85} +0.236327 q^{86} -23.4117 q^{87} +1.60687 q^{88} +8.37645 q^{89} -16.4744 q^{90} +2.12103 q^{92} +22.9366 q^{93} -9.28311 q^{94} -2.06886 q^{95} +3.31105 q^{96} +0.865228 q^{97} +7.58633 q^{98} +12.7956 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 7 q^{2} + 2 q^{3} + 7 q^{4} + 2 q^{5} + 2 q^{6} + q^{7} + 7 q^{8} + 15 q^{9}+O(q^{10}) \) 7 * q + 7 * q^2 + 2 * q^3 + 7 * q^4 + 2 * q^5 + 2 * q^6 + q^7 + 7 * q^8 + 15 * q^9	\( 7 q + 7 q^{2} + 2 q^{3} + 7 q^{4} + 2 q^{5} + 2 q^{6} + q^{7} + 7 q^{8} + 15 q^{9} + 2 q^{10} + 5 q^{11} + 2 q^{12} + q^{14} + 13 q^{15} + 7 q^{16} + 16 q^{17} + 15 q^{18} + 7 q^{19} + 2 q^{20} - 3 q^{21} + 5 q^{22} + 3 q^{23} + 2 q^{24} + 7 q^{25} + 5 q^{27} + q^{28} - 7 q^{29} + 13 q^{30} + 11 q^{31} + 7 q^{32} + 6 q^{33} + 16 q^{34} + q^{35} + 15 q^{36} - 3 q^{37} + 7 q^{38} + 2 q^{40} + 15 q^{41} - 3 q^{42} + 23 q^{43} + 5 q^{44} - 38 q^{45} + 3 q^{46} - q^{47} + 2 q^{48} + 14 q^{49} + 7 q^{50} - 16 q^{51} + 21 q^{53} + 5 q^{54} + 8 q^{55} + q^{56} + 2 q^{57} - 7 q^{58} + 8 q^{59} + 13 q^{60} - 6 q^{61} + 11 q^{62} - 22 q^{63} + 7 q^{64} + 6 q^{66} + 28 q^{67} + 16 q^{68} + 48 q^{69} + q^{70} + 4 q^{71} + 15 q^{72} + 12 q^{73} - 3 q^{74} + 5 q^{75} + 7 q^{76} - 26 q^{77} - 4 q^{79} + 2 q^{80} + 51 q^{81} + 15 q^{82} + 13 q^{83} - 3 q^{84} - 39 q^{85} + 23 q^{86} - 16 q^{87} + 5 q^{88} - 15 q^{89} - 38 q^{90} + 3 q^{92} + 6 q^{93} - q^{94} + 2 q^{95} + 2 q^{96} - 37 q^{97} + 14 q^{98} - 15 q^{99}+O(q^{100}) \) 7 * q + 7 * q^2 + 2 * q^3 + 7 * q^4 + 2 * q^5 + 2 * q^6 + q^7 + 7 * q^8 + 15 * q^9 + 2 * q^10 + 5 * q^11 + 2 * q^12 + q^14 + 13 * q^15 + 7 * q^16 + 16 * q^17 + 15 * q^18 + 7 * q^19 + 2 * q^20 - 3 * q^21 + 5 * q^22 + 3 * q^23 + 2 * q^24 + 7 * q^25 + 5 * q^27 + q^28 - 7 * q^29 + 13 * q^30 + 11 * q^31 + 7 * q^32 + 6 * q^33 + 16 * q^34 + q^35 + 15 * q^36 - 3 * q^37 + 7 * q^38 + 2 * q^40 + 15 * q^41 - 3 * q^42 + 23 * q^43 + 5 * q^44 - 38 * q^45 + 3 * q^46 - q^47 + 2 * q^48 + 14 * q^49 + 7 * q^50 - 16 * q^51 + 21 * q^53 + 5 * q^54 + 8 * q^55 + q^56 + 2 * q^57 - 7 * q^58 + 8 * q^59 + 13 * q^60 - 6 * q^61 + 11 * q^62 - 22 * q^63 + 7 * q^64 + 6 * q^66 + 28 * q^67 + 16 * q^68 + 48 * q^69 + q^70 + 4 * q^71 + 15 * q^72 + 12 * q^73 - 3 * q^74 + 5 * q^75 + 7 * q^76 - 26 * q^77 - 4 * q^79 + 2 * q^80 + 51 * q^81 + 15 * q^82 + 13 * q^83 - 3 * q^84 - 39 * q^85 + 23 * q^86 - 16 * q^87 + 5 * q^88 - 15 * q^89 - 38 * q^90 + 3 * q^92 + 6 * q^93 - q^94 + 2 * q^95 + 2 * q^96 - 37 * q^97 + 14 * q^98 - 15 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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