LMFDB - Embedded newform 2541.2.a.bc.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2541,2,Mod(1,2541)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2541, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("2541.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2541 = 3 \cdot 7 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2541.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:	$20.2899871536$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.56155$ of defining polynomial
Character	$\chi$	$=$	2541.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.56155 q^{2} +1.00000 q^{3} +4.56155 q^{4} +1.00000 q^{5} +2.56155 q^{6} +1.00000 q^{7} +6.56155 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+2.56155 q^{2} +1.00000 q^{3} +4.56155 q^{4} +1.00000 q^{5} +2.56155 q^{6} +1.00000 q^{7} +6.56155 q^{8} +1.00000 q^{9} +2.56155 q^{10} +4.56155 q^{12} +1.43845 q^{13} +2.56155 q^{14} +1.00000 q^{15} +7.68466 q^{16} -4.12311 q^{17} +2.56155 q^{18} +6.00000 q^{19} +4.56155 q^{20} +1.00000 q^{21} -4.00000 q^{23} +6.56155 q^{24} -4.00000 q^{25} +3.68466 q^{26} +1.00000 q^{27} +4.56155 q^{28} -5.68466 q^{29} +2.56155 q^{30} -8.24621 q^{31} +6.56155 q^{32} -10.5616 q^{34} +1.00000 q^{35} +4.56155 q^{36} +9.68466 q^{37} +15.3693 q^{38} +1.43845 q^{39} +6.56155 q^{40} -3.43845 q^{41} +2.56155 q^{42} +6.68466 q^{43} +1.00000 q^{45} -10.2462 q^{46} +7.56155 q^{47} +7.68466 q^{48} +1.00000 q^{49} -10.2462 q^{50} -4.12311 q^{51} +6.56155 q^{52} -12.8078 q^{53} +2.56155 q^{54} +6.56155 q^{56} +6.00000 q^{57} -14.5616 q^{58} -8.43845 q^{59} +4.56155 q^{60} +11.3693 q^{61} -21.1231 q^{62} +1.00000 q^{63} +1.43845 q^{64} +1.43845 q^{65} -11.8078 q^{67} -18.8078 q^{68} -4.00000 q^{69} +2.56155 q^{70} -5.12311 q^{71} +6.56155 q^{72} -1.12311 q^{73} +24.8078 q^{74} -4.00000 q^{75} +27.3693 q^{76} +3.68466 q^{78} -5.12311 q^{79} +7.68466 q^{80} +1.00000 q^{81} -8.80776 q^{82} +11.8078 q^{83} +4.56155 q^{84} -4.12311 q^{85} +17.1231 q^{86} -5.68466 q^{87} +5.24621 q^{89} +2.56155 q^{90} +1.43845 q^{91} -18.2462 q^{92} -8.24621 q^{93} +19.3693 q^{94} +6.00000 q^{95} +6.56155 q^{96} +13.6847 q^{97} +2.56155 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{2} + 2 q^{3} + 5 q^{4} + 2 q^{5} + q^{6} + 2 q^{7} + 9 q^{8} + 2 q^{9}+O(q^{10}) $ 2 * q + q^2 + 2 * q^3 + 5 * q^4 + 2 * q^5 + q^6 + 2 * q^7 + 9 * q^8 + 2 * q^9	$ 2 q + q^{2} + 2 q^{3} + 5 q^{4} + 2 q^{5} + q^{6} + 2 q^{7} + 9 q^{8} + 2 q^{9} + q^{10} + 5 q^{12} + 7 q^{13} + q^{14} + 2 q^{15} + 3 q^{16} + q^{18} + 12 q^{19} + 5 q^{20} + 2 q^{21} - 8 q^{23} + 9 q^{24} - 8 q^{25} - 5 q^{26} + 2 q^{27} + 5 q^{28} + q^{29} + q^{30} + 9 q^{32} - 17 q^{34} + 2 q^{35} + 5 q^{36} + 7 q^{37} + 6 q^{38} + 7 q^{39} + 9 q^{40} - 11 q^{41} + q^{42} + q^{43} + 2 q^{45} - 4 q^{46} + 11 q^{47} + 3 q^{48} + 2 q^{49} - 4 q^{50} + 9 q^{52} - 5 q^{53} + q^{54} + 9 q^{56} + 12 q^{57} - 25 q^{58} - 21 q^{59} + 5 q^{60} - 2 q^{61} - 34 q^{62} + 2 q^{63} + 7 q^{64} + 7 q^{65} - 3 q^{67} - 17 q^{68} - 8 q^{69} + q^{70} - 2 q^{71} + 9 q^{72} + 6 q^{73} + 29 q^{74} - 8 q^{75} + 30 q^{76} - 5 q^{78} - 2 q^{79} + 3 q^{80} + 2 q^{81} + 3 q^{82} + 3 q^{83} + 5 q^{84} + 26 q^{86} + q^{87} - 6 q^{89} + q^{90} + 7 q^{91} - 20 q^{92} + 14 q^{94} + 12 q^{95} + 9 q^{96} + 15 q^{97} + q^{98}+O(q^{100}) $ 2 * q + q^2 + 2 * q^3 + 5 * q^4 + 2 * q^5 + q^6 + 2 * q^7 + 9 * q^8 + 2 * q^9 + q^10 + 5 * q^12 + 7 * q^13 + q^14 + 2 * q^15 + 3 * q^16 + q^18 + 12 * q^19 + 5 * q^20 + 2 * q^21 - 8 * q^23 + 9 * q^24 - 8 * q^25 - 5 * q^26 + 2 * q^27 + 5 * q^28 + q^29 + q^30 + 9 * q^32 - 17 * q^34 + 2 * q^35 + 5 * q^36 + 7 * q^37 + 6 * q^38 + 7 * q^39 + 9 * q^40 - 11 * q^41 + q^42 + q^43 + 2 * q^45 - 4 * q^46 + 11 * q^47 + 3 * q^48 + 2 * q^49 - 4 * q^50 + 9 * q^52 - 5 * q^53 + q^54 + 9 * q^56 + 12 * q^57 - 25 * q^58 - 21 * q^59 + 5 * q^60 - 2 * q^61 - 34 * q^62 + 2 * q^63 + 7 * q^64 + 7 * q^65 - 3 * q^67 - 17 * q^68 - 8 * q^69 + q^70 - 2 * q^71 + 9 * q^72 + 6 * q^73 + 29 * q^74 - 8 * q^75 + 30 * q^76 - 5 * q^78 - 2 * q^79 + 3 * q^80 + 2 * q^81 + 3 * q^82 + 3 * q^83 + 5 * q^84 + 26 * q^86 + q^87 - 6 * q^89 + q^90 + 7 * q^91 - 20 * q^92 + 14 * q^94 + 12 * q^95 + 9 * q^96 + 15 * q^97 + q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	2.56155	1.81129	0.905646	−	0.424035i	$-0.139387\pi$
$2$	2.56155	1.81129	0.905646	+	0.424035i	$0.139387\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	4.56155	2.28078
$5$	1.00000	0.447214	0.223607	−	0.974679i	$-0.428217\pi$
$5$	1.00000	0.447214	0.223607	+	0.974679i	$0.428217\pi$
$6$	2.56155	1.04575
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	6.56155	2.31986
$9$	1.00000	0.333333
$10$	2.56155	0.810034
$11$	0	0
$11$	0	0
$12$	4.56155	1.31681
$13$	1.43845	0.398953	0.199477	−	0.979903i	$-0.436076\pi$
$13$	1.43845	0.398953	0.199477	+	0.979903i	$0.436076\pi$
$14$	2.56155	0.684604
$15$	1.00000	0.258199
$16$	7.68466	1.92116
$17$	−4.12311	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$17$	−4.12311	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$18$	2.56155	0.603764
$19$	6.00000	1.37649	0.688247	−	0.725476i	$-0.258380\pi$
$19$	6.00000	1.37649	0.688247	+	0.725476i	$0.258380\pi$
$20$	4.56155	1.01999
$21$	1.00000	0.218218
$22$	0	0
$23$	−4.00000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	6.56155	1.33937
$25$	−4.00000	−0.800000
$26$	3.68466	0.722621
$27$	1.00000	0.192450
$28$	4.56155	0.862052
$29$	−5.68466	−1.05561	−0.527807	−	0.849364i	$-0.676986\pi$
$29$	−5.68466	−1.05561	−0.527807	+	0.849364i	$0.676986\pi$
$30$	2.56155	0.467673
$31$	−8.24621	−1.48106	−0.740532	−	0.672022i	$-0.765426\pi$
$31$	−8.24621	−1.48106	−0.740532	+	0.672022i	$0.765426\pi$
$32$	6.56155	1.15993
$33$	0	0
$34$	−10.5616	−1.81129
$35$	1.00000	0.169031
$36$	4.56155	0.760259
$37$	9.68466	1.59215	0.796074	−	0.605199i	$-0.206907\pi$
$37$	9.68466	1.59215	0.796074	+	0.605199i	$0.206907\pi$
$38$	15.3693	2.49323
$39$	1.43845	0.230336
$40$	6.56155	1.03747
$41$	−3.43845	−0.536995	−0.268498	−	0.963280i	$-0.586527\pi$
$41$	−3.43845	−0.536995	−0.268498	+	0.963280i	$0.586527\pi$
$42$	2.56155	0.395256
$43$	6.68466	1.01940	0.509700	−	0.860352i	$-0.329756\pi$
$43$	6.68466	1.01940	0.509700	+	0.860352i	$0.329756\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	−10.2462	−1.51072
$47$	7.56155	1.10297	0.551483	−	0.834186i	$-0.314062\pi$
$47$	7.56155	1.10297	0.551483	+	0.834186i	$0.314062\pi$
$48$	7.68466	1.10918
$49$	1.00000	0.142857
$50$	−10.2462	−1.44903
$51$	−4.12311	−0.577350
$52$	6.56155	0.909924
$53$	−12.8078	−1.75928	−0.879641	−	0.475638i	$-0.842217\pi$
$53$	−12.8078	−1.75928	−0.879641	+	0.475638i	$0.842217\pi$
$54$	2.56155	0.348583
$55$	0	0
$56$	6.56155	0.876824
$57$	6.00000	0.794719
$58$	−14.5616	−1.91203
$59$	−8.43845	−1.09859	−0.549296	−	0.835628i	$-0.685104\pi$
$59$	−8.43845	−1.09859	−0.549296	+	0.835628i	$0.685104\pi$
$60$	4.56155	0.588894
$61$	11.3693	1.45569	0.727846	−	0.685741i	$-0.240522\pi$
$61$	11.3693	1.45569	0.727846	+	0.685741i	$0.240522\pi$
$62$	−21.1231	−2.68264
$63$	1.00000	0.125988
$64$	1.43845	0.179806
$65$	1.43845	0.178417
$66$	0	0
$67$	−11.8078	−1.44255	−0.721274	−	0.692650i	$-0.756443\pi$
$67$	−11.8078	−1.44255	−0.721274	+	0.692650i	$0.756443\pi$
$68$	−18.8078	−2.28078
$69$	−4.00000	−0.481543
$70$	2.56155	0.306164
$71$	−5.12311	−0.608001	−0.304000	−	0.952672i	$-0.598322\pi$
$71$	−5.12311	−0.608001	−0.304000	+	0.952672i	$0.598322\pi$
$72$	6.56155	0.773286
$73$	−1.12311	−0.131450	−0.0657248	−	0.997838i	$-0.520936\pi$
$73$	−1.12311	−0.131450	−0.0657248	+	0.997838i	$0.520936\pi$
$74$	24.8078	2.88384
$75$	−4.00000	−0.461880
$76$	27.3693	3.13948
$77$	0	0
$78$	3.68466	0.417205
$79$	−5.12311	−0.576394	−0.288197	−	0.957571i	$-0.593056\pi$
$79$	−5.12311	−0.576394	−0.288197	+	0.957571i	$0.593056\pi$
$80$	7.68466	0.859171
$81$	1.00000	0.111111
$82$	−8.80776	−0.972655
$83$	11.8078	1.29607	0.648035	−	0.761610i	$-0.275591\pi$
$83$	11.8078	1.29607	0.648035	+	0.761610i	$0.275591\pi$
$84$	4.56155	0.497706
$85$	−4.12311	−0.447214
$86$	17.1231	1.84643
$87$	−5.68466	−0.609459
$88$	0	0
$89$	5.24621	0.556097	0.278049	−	0.960567i	$-0.410312\pi$
$89$	5.24621	0.556097	0.278049	+	0.960567i	$0.410312\pi$
$90$	2.56155	0.270011
$91$	1.43845	0.150790
$92$	−18.2462	−1.90230
$93$	−8.24621	−0.855092
$94$	19.3693	1.99779
$95$	6.00000	0.615587
$96$	6.56155	0.669686
$97$	13.6847	1.38947	0.694733	−	0.719267i	$-0.255522\pi$
$97$	13.6847	1.38947	0.694733	+	0.719267i	$0.255522\pi$
$98$	2.56155	0.258756
$99$	0	0
$100$	−18.2462	−1.82462
$101$	0.438447	0.0436271	0.0218136	−	0.999762i	$-0.493056\pi$
$101$	0.438447	0.0436271	0.0218136	+	0.999762i	$0.493056\pi$
$102$	−10.5616	−1.04575
$103$	−8.00000	−0.788263	−0.394132	−	0.919054i	$-0.628955\pi$
$103$	−8.00000	−0.788263	−0.394132	+	0.919054i	$0.628955\pi$
$104$	9.43845	0.925516
$105$	1.00000	0.0975900
$106$	−32.8078	−3.18657
$107$	5.12311	0.495269	0.247635	−	0.968853i	$-0.420347\pi$
$107$	5.12311	0.495269	0.247635	+	0.968853i	$0.420347\pi$
$108$	4.56155	0.438936
$109$	12.1231	1.16118	0.580591	−	0.814195i	$-0.302821\pi$
$109$	12.1231	1.16118	0.580591	+	0.814195i	$0.302821\pi$
$110$	0	0
$111$	9.68466	0.919227
$112$	7.68466	0.726132
$113$	19.9309	1.87494	0.937469	−	0.348068i	$-0.113162\pi$
$113$	19.9309	1.87494	0.937469	+	0.348068i	$0.113162\pi$
$114$	15.3693	1.43947
$115$	−4.00000	−0.373002
$116$	−25.9309	−2.40762
$117$	1.43845	0.132984
$118$	−21.6155	−1.98987
$119$	−4.12311	−0.377964
$120$	6.56155	0.598985
$121$	0	0
$122$	29.1231	2.63668
$123$	−3.43845	−0.310034
$124$	−37.6155	−3.37797
$125$	−9.00000	−0.804984
$126$	2.56155	0.228201
$127$	−6.93087	−0.615015	−0.307508	−	0.951546i	$-0.599495\pi$
$127$	−6.93087	−0.615015	−0.307508	+	0.951546i	$0.599495\pi$
$128$	−9.43845	−0.834249
$129$	6.68466	0.588551
$130$	3.68466	0.323166
$131$	1.56155	0.136434	0.0682168	−	0.997671i	$-0.478269\pi$
$131$	1.56155	0.136434	0.0682168	+	0.997671i	$0.478269\pi$
$132$	0	0
$133$	6.00000	0.520266
$134$	−30.2462	−2.61287
$135$	1.00000	0.0860663
$136$	−27.0540	−2.31986
$137$	−14.0000	−1.19610	−0.598050	−	0.801459i	$-0.704058\pi$
$137$	−14.0000	−1.19610	−0.598050	+	0.801459i	$0.704058\pi$
$138$	−10.2462	−0.872215
$139$	−10.2462	−0.869072	−0.434536	−	0.900654i	$-0.643088\pi$
$139$	−10.2462	−0.869072	−0.434536	+	0.900654i	$0.643088\pi$
$140$	4.56155	0.385522
$141$	7.56155	0.636798
$142$	−13.1231	−1.10127
$143$	0	0
$144$	7.68466	0.640388
$145$	−5.68466	−0.472085
$146$	−2.87689	−0.238093
$147$	1.00000	0.0824786
$148$	44.1771	3.63133
$149$	−17.6847	−1.44878	−0.724392	−	0.689388i	$-0.757879\pi$
$149$	−17.6847	−1.44878	−0.724392	+	0.689388i	$0.757879\pi$
$150$	−10.2462	−0.836600
$151$	15.5616	1.26638	0.633191	−	0.773996i	$-0.281745\pi$
$151$	15.5616	1.26638	0.633191	+	0.773996i	$0.281745\pi$
$152$	39.3693	3.19327
$153$	−4.12311	−0.333333
$154$	0	0
$155$	−8.24621	−0.662352
$156$	6.56155	0.525345
$157$	19.3693	1.54584	0.772920	−	0.634504i	$-0.218795\pi$
$157$	19.3693	1.54584	0.772920	+	0.634504i	$0.218795\pi$
$158$	−13.1231	−1.04402
$159$	−12.8078	−1.01572
$160$	6.56155	0.518736
$161$	−4.00000	−0.315244
$162$	2.56155	0.201255
$163$	−7.36932	−0.577209	−0.288605	−	0.957448i	$-0.593191\pi$
$163$	−7.36932	−0.577209	−0.288605	+	0.957448i	$0.593191\pi$
$164$	−15.6847	−1.22477
$165$	0	0
$166$	30.2462	2.34756
$167$	7.80776	0.604183	0.302091	−	0.953279i	$-0.402315\pi$
$167$	7.80776	0.604183	0.302091	+	0.953279i	$0.402315\pi$
$168$	6.56155	0.506235
$169$	−10.9309	−0.840836
$170$	−10.5616	−0.810034
$171$	6.00000	0.458831
$172$	30.4924	2.32503
$173$	8.43845	0.641563	0.320782	−	0.947153i	$-0.396054\pi$
$173$	8.43845	0.641563	0.320782	+	0.947153i	$0.396054\pi$
$174$	−14.5616	−1.10391
$175$	−4.00000	−0.302372
$176$	0	0
$177$	−8.43845	−0.634273
$178$	13.4384	1.00725
$179$	−13.1231	−0.980867	−0.490433	−	0.871479i	$-0.663162\pi$
$179$	−13.1231	−0.980867	−0.490433	+	0.871479i	$0.663162\pi$
$180$	4.56155	0.339998
$181$	−20.8078	−1.54663	−0.773314	−	0.634023i	$-0.781403\pi$
$181$	−20.8078	−1.54663	−0.773314	+	0.634023i	$0.781403\pi$
$182$	3.68466	0.273125
$183$	11.3693	0.840444
$184$	−26.2462	−1.93490
$185$	9.68466	0.712030
$186$	−21.1231	−1.54882
$187$	0	0
$188$	34.4924	2.51562
$189$	1.00000	0.0727393
$190$	15.3693	1.11501
$191$	4.24621	0.307245	0.153623	−	0.988130i	$-0.450906\pi$
$191$	4.24621	0.307245	0.153623	+	0.988130i	$0.450906\pi$
$192$	1.43845	0.103811
$193$	1.24621	0.0897042	0.0448521	−	0.998994i	$-0.485718\pi$
$193$	1.24621	0.0897042	0.0448521	+	0.998994i	$0.485718\pi$
$194$	35.0540	2.51673
$195$	1.43845	0.103009
$196$	4.56155	0.325825
$197$	−18.1771	−1.29506	−0.647532	−	0.762039i	$-0.724199\pi$
$197$	−18.1771	−1.29506	−0.647532	+	0.762039i	$0.724199\pi$
$198$	0	0
$199$	1.12311	0.0796148	0.0398074	−	0.999207i	$-0.487326\pi$
$199$	1.12311	0.0796148	0.0398074	+	0.999207i	$0.487326\pi$
$200$	−26.2462	−1.85589
$201$	−11.8078	−0.832855
$202$	1.12311	0.0790214
$203$	−5.68466	−0.398985
$204$	−18.8078	−1.31681
$205$	−3.43845	−0.240152
$206$	−20.4924	−1.42777
$207$	−4.00000	−0.278019
$208$	11.0540	0.766455
$209$	0	0
$210$	2.56155	0.176764
$211$	−10.0540	−0.692144	−0.346072	−	0.938208i	$-0.612485\pi$
$211$	−10.0540	−0.692144	−0.346072	+	0.938208i	$0.612485\pi$
$212$	−58.4233	−4.01253
$213$	−5.12311	−0.351029
$214$	13.1231	0.897077
$215$	6.68466	0.455890
$216$	6.56155	0.446457
$217$	−8.24621	−0.559789
$218$	31.0540	2.10324
$219$	−1.12311	−0.0758924
$220$	0	0
$221$	−5.93087	−0.398953
$222$	24.8078	1.66499
$223$	24.0000	1.60716	0.803579	−	0.595198i	$-0.202926\pi$
$223$	24.0000	1.60716	0.803579	+	0.595198i	$0.202926\pi$
$224$	6.56155	0.438412
$225$	−4.00000	−0.266667
$226$	51.0540	3.39606
$227$	−6.68466	−0.443676	−0.221838	−	0.975083i	$-0.571206\pi$
$227$	−6.68466	−0.443676	−0.221838	+	0.975083i	$0.571206\pi$
$228$	27.3693	1.81258
$229$	−24.8078	−1.63934	−0.819672	−	0.572834i	$-0.805844\pi$
$229$	−24.8078	−1.63934	−0.819672	+	0.572834i	$0.805844\pi$
$230$	−10.2462	−0.675615
$231$	0	0
$232$	−37.3002	−2.44888
$233$	18.1771	1.19082	0.595410	−	0.803422i	$-0.296990\pi$
$233$	18.1771	1.19082	0.595410	+	0.803422i	$0.296990\pi$
$234$	3.68466	0.240874
$235$	7.56155	0.493261
$236$	−38.4924	−2.50564
$237$	−5.12311	−0.332781
$238$	−10.5616	−0.684604
$239$	5.36932	0.347312	0.173656	−	0.984806i	$-0.444442\pi$
$239$	5.36932	0.347312	0.173656	+	0.984806i	$0.444442\pi$
$240$	7.68466	0.496043
$241$	4.24621	0.273523	0.136761	−	0.990604i	$-0.456331\pi$
$241$	4.24621	0.273523	0.136761	+	0.990604i	$0.456331\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	51.8617	3.32011
$245$	1.00000	0.0638877
$246$	−8.80776	−0.561563
$247$	8.63068	0.549157
$248$	−54.1080	−3.43586
$249$	11.8078	0.748287
$250$	−23.0540	−1.45806
$251$	4.00000	0.252478	0.126239	−	0.992000i	$-0.459709\pi$
$251$	4.00000	0.252478	0.126239	+	0.992000i	$0.459709\pi$
$252$	4.56155	0.287351
$253$	0	0
$254$	−17.7538	−1.11397
$255$	−4.12311	−0.258199
$256$	−27.0540	−1.69087
$257$	23.0000	1.43470	0.717350	−	0.696713i	$-0.245355\pi$
$257$	23.0000	1.43470	0.717350	+	0.696713i	$0.245355\pi$
$258$	17.1231	1.06604
$259$	9.68466	0.601775
$260$	6.56155	0.406930
$261$	−5.68466	−0.351872
$262$	4.00000	0.247121
$263$	−18.7386	−1.15547	−0.577737	−	0.816223i	$-0.696064\pi$
$263$	−18.7386	−1.15547	−0.577737	+	0.816223i	$0.696064\pi$
$264$	0	0
$265$	−12.8078	−0.786775
$266$	15.3693	0.942353
$267$	5.24621	0.321063
$268$	−53.8617	−3.29013
$269$	−8.56155	−0.522007	−0.261004	−	0.965338i	$-0.584053\pi$
$269$	−8.56155	−0.522007	−0.261004	+	0.965338i	$0.584053\pi$
$270$	2.56155	0.155891
$271$	0	0		−	1.00000i	$-0.5\pi$
$271$	0	0			1.00000i	$0.5\pi$
$272$	−31.6847	−1.92116
$273$	1.43845	0.0870588
$274$	−35.8617	−2.16649
$275$	0	0
$276$	−18.2462	−1.09829
$277$	3.24621	0.195046	0.0975230	−	0.995233i	$-0.468908\pi$
$277$	3.24621	0.195046	0.0975230	+	0.995233i	$0.468908\pi$
$278$	−26.2462	−1.57414
$279$	−8.24621	−0.493688
$280$	6.56155	0.392128
$281$	29.1231	1.73734	0.868669	−	0.495392i	$-0.164976\pi$
$281$	29.1231	1.73734	0.868669	+	0.495392i	$0.164976\pi$
$282$	19.3693	1.15343
$283$	6.87689	0.408789	0.204394	−	0.978889i	$-0.434477\pi$
$283$	6.87689	0.408789	0.204394	+	0.978889i	$0.434477\pi$
$284$	−23.3693	−1.38671
$285$	6.00000	0.355409
$286$	0	0
$287$	−3.43845	−0.202965
$288$	6.56155	0.386643
$289$	0	0
$290$	−14.5616	−0.855084
$291$	13.6847	0.802209
$292$	−5.12311	−0.299807
$293$	2.36932	0.138417	0.0692085	−	0.997602i	$-0.477953\pi$
$293$	2.36932	0.138417	0.0692085	+	0.997602i	$0.477953\pi$
$294$	2.56155	0.149393
$295$	−8.43845	−0.491305
$296$	63.5464	3.69356
$297$	0	0
$298$	−45.3002	−2.62417
$299$	−5.75379	−0.332750
$300$	−18.2462	−1.05345
$301$	6.68466	0.385297
$302$	39.8617	2.29379
$303$	0.438447	0.0251881
$304$	46.1080	2.64447
$305$	11.3693	0.651005
$306$	−10.5616	−0.603764
$307$	31.1231	1.77629	0.888145	−	0.459564i	$-0.151994\pi$
$307$	31.1231	1.77629	0.888145	+	0.459564i	$0.151994\pi$
$308$	0	0
$309$	−8.00000	−0.455104
$310$	−21.1231	−1.19971
$311$	0.438447	0.0248621	0.0124310	−	0.999923i	$-0.496043\pi$
$311$	0.438447	0.0248621	0.0124310	+	0.999923i	$0.496043\pi$
$312$	9.43845	0.534347
$313$	6.56155	0.370881	0.185440	−	0.982656i	$-0.440629\pi$
$313$	6.56155	0.370881	0.185440	+	0.982656i	$0.440629\pi$
$314$	49.6155	2.79997
$315$	1.00000	0.0563436
$316$	−23.3693	−1.31463
$317$	−18.8769	−1.06023	−0.530116	−	0.847925i	$-0.677852\pi$
$317$	−18.8769	−1.06023	−0.530116	+	0.847925i	$0.677852\pi$
$318$	−32.8078	−1.83977
$319$	0	0
$320$	1.43845	0.0804116
$321$	5.12311	0.285944
$322$	−10.2462	−0.570999
$323$	−24.7386	−1.37649
$324$	4.56155	0.253420
$325$	−5.75379	−0.319163
$326$	−18.8769	−1.04549
$327$	12.1231	0.670409
$328$	−22.5616	−1.24575
$329$	7.56155	0.416882
$330$	0	0
$331$	−5.31534	−0.292158	−0.146079	−	0.989273i	$-0.546665\pi$
$331$	−5.31534	−0.292158	−0.146079	+	0.989273i	$0.546665\pi$
$332$	53.8617	2.95605
$333$	9.68466	0.530716
$334$	20.0000	1.09435
$335$	−11.8078	−0.645127
$336$	7.68466	0.419232
$337$	−18.1771	−0.990169	−0.495084	−	0.868845i	$-0.664863\pi$
$337$	−18.1771	−0.990169	−0.495084	+	0.868845i	$0.664863\pi$
$338$	−28.0000	−1.52300
$339$	19.9309	1.08250
$340$	−18.8078	−1.01999
$341$	0	0
$342$	15.3693	0.831077
$343$	1.00000	0.0539949
$344$	43.8617	2.36487
$345$	−4.00000	−0.215353
$346$	21.6155	1.16206
$347$	−14.0000	−0.751559	−0.375780	−	0.926709i	$-0.622625\pi$
$347$	−14.0000	−0.751559	−0.375780	+	0.926709i	$0.622625\pi$
$348$	−25.9309	−1.39004
$349$	29.0540	1.55522	0.777612	−	0.628745i	$-0.216431\pi$
$349$	29.0540	1.55522	0.777612	+	0.628745i	$0.216431\pi$
$350$	−10.2462	−0.547683
$351$	1.43845	0.0767786
$352$	0	0
$353$	−22.1771	−1.18037	−0.590183	−	0.807269i	$-0.700945\pi$
$353$	−22.1771	−1.18037	−0.590183	+	0.807269i	$0.700945\pi$
$354$	−21.6155	−1.14885
$355$	−5.12311	−0.271906
$356$	23.9309	1.26833
$357$	−4.12311	−0.218218
$358$	−33.6155	−1.77664
$359$	28.0000	1.47778	0.738892	−	0.673824i	$-0.235349\pi$
$359$	28.0000	1.47778	0.738892	+	0.673824i	$0.235349\pi$
$360$	6.56155	0.345824
$361$	17.0000	0.894737
$362$	−53.3002	−2.80140
$363$	0	0
$364$	6.56155	0.343919
$365$	−1.12311	−0.0587860
$366$	29.1231	1.52229
$367$	−18.4924	−0.965297	−0.482648	−	0.875814i	$-0.660325\pi$
$367$	−18.4924	−0.965297	−0.482648	+	0.875814i	$0.660325\pi$
$368$	−30.7386	−1.60236
$369$	−3.43845	−0.178998
$370$	24.8078	1.28969
$371$	−12.8078	−0.664946
$372$	−37.6155	−1.95027
$373$	−13.8078	−0.714939	−0.357469	−	0.933925i	$-0.616360\pi$
$373$	−13.8078	−0.714939	−0.357469	+	0.933925i	$0.616360\pi$
$374$	0	0
$375$	−9.00000	−0.464758
$376$	49.6155	2.55873
$377$	−8.17708	−0.421141
$378$	2.56155	0.131752
$379$	−10.4384	−0.536187	−0.268094	−	0.963393i	$-0.586394\pi$
$379$	−10.4384	−0.536187	−0.268094	+	0.963393i	$0.586394\pi$
$380$	27.3693	1.40402
$381$	−6.93087	−0.355079
$382$	10.8769	0.556510
$383$	21.5616	1.10174	0.550872	−	0.834590i	$-0.314295\pi$
$383$	21.5616	1.10174	0.550872	+	0.834590i	$0.314295\pi$
$384$	−9.43845	−0.481654
$385$	0	0
$386$	3.19224	0.162481
$387$	6.68466	0.339800
$388$	62.4233	3.16906
$389$	13.4384	0.681356	0.340678	−	0.940180i	$-0.389343\pi$
$389$	13.4384	0.681356	0.340678	+	0.940180i	$0.389343\pi$
$390$	3.68466	0.186580
$391$	16.4924	0.834058
$392$	6.56155	0.331408
$393$	1.56155	0.0787699
$394$	−46.5616	−2.34574
$395$	−5.12311	−0.257771
$396$	0	0
$397$	−25.6847	−1.28908	−0.644538	−	0.764572i	$-0.722950\pi$
$397$	−25.6847	−1.28908	−0.644538	+	0.764572i	$0.722950\pi$
$398$	2.87689	0.144206
$399$	6.00000	0.300376
$400$	−30.7386	−1.53693
$401$	−27.6847	−1.38251	−0.691253	−	0.722613i	$-0.742941\pi$
$401$	−27.6847	−1.38251	−0.691253	+	0.722613i	$0.742941\pi$
$402$	−30.2462	−1.50854
$403$	−11.8617	−0.590875
$404$	2.00000	0.0995037
$405$	1.00000	0.0496904
$406$	−14.5616	−0.722678
$407$	0	0
$408$	−27.0540	−1.33937
$409$	−19.3002	−0.954333	−0.477166	−	0.878813i	$-0.658336\pi$
$409$	−19.3002	−0.954333	−0.477166	+	0.878813i	$0.658336\pi$
$410$	−8.80776	−0.434984
$411$	−14.0000	−0.690569
$412$	−36.4924	−1.79785
$413$	−8.43845	−0.415229
$414$	−10.2462	−0.503574
$415$	11.8078	0.579620
$416$	9.43845	0.462758
$417$	−10.2462	−0.501759
$418$	0	0
$419$	32.3002	1.57797	0.788984	−	0.614414i	$-0.210608\pi$
$419$	32.3002	1.57797	0.788984	+	0.614414i	$0.210608\pi$
$420$	4.56155	0.222581
$421$	−7.00000	−0.341159	−0.170580	−	0.985344i	$-0.554564\pi$
$421$	−7.00000	−0.341159	−0.170580	+	0.985344i	$0.554564\pi$
$422$	−25.7538	−1.25367
$423$	7.56155	0.367655
$424$	−84.0388	−4.08129
$425$	16.4924	0.800000
$426$	−13.1231	−0.635817
$427$	11.3693	0.550200
$428$	23.3693	1.12960
$429$	0	0
$430$	17.1231	0.825749
$431$	35.3693	1.70368	0.851840	−	0.523802i	$-0.175487\pi$
$431$	35.3693	1.70368	0.851840	+	0.523802i	$0.175487\pi$
$432$	7.68466	0.369728
$433$	15.9309	0.765589	0.382794	−	0.923834i	$-0.374962\pi$
$433$	15.9309	0.765589	0.382794	+	0.923834i	$0.374962\pi$
$434$	−21.1231	−1.01394
$435$	−5.68466	−0.272559
$436$	55.3002	2.64840
$437$	−24.0000	−1.14808
$438$	−2.87689	−0.137463
$439$	7.12311	0.339967	0.169984	−	0.985447i	$-0.445629\pi$
$439$	7.12311	0.339967	0.169984	+	0.985447i	$0.445629\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	−15.1922	−0.722621
$443$	24.4924	1.16367	0.581835	−	0.813307i	$-0.302335\pi$
$443$	24.4924	1.16367	0.581835	+	0.813307i	$0.302335\pi$
$444$	44.1771	2.09655
$445$	5.24621	0.248694
$446$	61.4773	2.91103
$447$	−17.6847	−0.836456
$448$	1.43845	0.0679602
$449$	−9.05398	−0.427284	−0.213642	−	0.976912i	$-0.568533\pi$
$449$	−9.05398	−0.427284	−0.213642	+	0.976912i	$0.568533\pi$
$450$	−10.2462	−0.483011
$451$	0	0
$452$	90.9157	4.27632
$453$	15.5616	0.731146
$454$	−17.1231	−0.803627
$455$	1.43845	0.0674354
$456$	39.3693	1.84364
$457$	30.6155	1.43213	0.716067	−	0.698032i	$-0.245940\pi$
$457$	30.6155	1.43213	0.716067	+	0.698032i	$0.245940\pi$
$458$	−63.5464	−2.96933
$459$	−4.12311	−0.192450
$460$	−18.2462	−0.850734
$461$	14.3693	0.669246	0.334623	−	0.942352i	$-0.391391\pi$
$461$	14.3693	0.669246	0.334623	+	0.942352i	$0.391391\pi$
$462$	0	0
$463$	9.12311	0.423987	0.211993	−	0.977271i	$-0.432004\pi$
$463$	9.12311	0.423987	0.211993	+	0.977271i	$0.432004\pi$
$464$	−43.6847	−2.02801
$465$	−8.24621	−0.382409
$466$	46.5616	2.15692
$467$	−38.2462	−1.76982	−0.884912	−	0.465759i	$-0.845781\pi$
$467$	−38.2462	−1.76982	−0.884912	+	0.465759i	$0.845781\pi$
$468$	6.56155	0.303308
$469$	−11.8078	−0.545232
$470$	19.3693	0.893440
$471$	19.3693	0.892491
$472$	−55.3693	−2.54858
$473$	0	0
$474$	−13.1231	−0.602764
$475$	−24.0000	−1.10120
$476$	−18.8078	−0.862052
$477$	−12.8078	−0.586427
$478$	13.7538	0.629084
$479$	−17.3153	−0.791158	−0.395579	−	0.918432i	$-0.629456\pi$
$479$	−17.3153	−0.791158	−0.395579	+	0.918432i	$0.629456\pi$
$480$	6.56155	0.299493
$481$	13.9309	0.635193
$482$	10.8769	0.495429
$483$	−4.00000	−0.182006
$484$	0	0
$485$	13.6847	0.621388
$486$	2.56155	0.116194
$487$	23.5616	1.06768	0.533838	−	0.845587i	$-0.320749\pi$
$487$	23.5616	1.06768	0.533838	+	0.845587i	$0.320749\pi$
$488$	74.6004	3.37700
$489$	−7.36932	−0.333252
$490$	2.56155	0.115719
$491$	20.0000	0.902587	0.451294	−	0.892375i	$-0.350963\pi$
$491$	20.0000	0.902587	0.451294	+	0.892375i	$0.350963\pi$
$492$	−15.6847	−0.707119
$493$	23.4384	1.05561
$494$	22.1080	0.994684
$495$	0	0
$496$	−63.3693	−2.84537
$497$	−5.12311	−0.229803
$498$	30.2462	1.35537
$499$	30.5464	1.36744	0.683722	−	0.729742i	$-0.260360\pi$
$499$	30.5464	1.36744	0.683722	+	0.729742i	$0.260360\pi$
$500$	−41.0540	−1.83599
$501$	7.80776	0.348825
$502$	10.2462	0.457311
$503$	−7.31534	−0.326175	−0.163087	−	0.986612i	$-0.552145\pi$
$503$	−7.31534	−0.326175	−0.163087	+	0.986612i	$0.552145\pi$
$504$	6.56155	0.292275
$505$	0.438447	0.0195106
$506$	0	0
$507$	−10.9309	−0.485457
$508$	−31.6155	−1.40271
$509$	13.3153	0.590192	0.295096	−	0.955468i	$-0.404648\pi$
$509$	13.3153	0.590192	0.295096	+	0.955468i	$0.404648\pi$
$510$	−10.5616	−0.467673
$511$	−1.12311	−0.0496833
$512$	−50.4233	−2.22842
$513$	6.00000	0.264906
$514$	58.9157	2.59866
$515$	−8.00000	−0.352522
$516$	30.4924	1.34235
$517$	0	0
$518$	24.8078	1.08999
$519$	8.43845	0.370407
$520$	9.43845	0.413903
$521$	15.0691	0.660191	0.330095	−	0.943948i	$-0.392919\pi$
$521$	15.0691	0.660191	0.330095	+	0.943948i	$0.392919\pi$
$522$	−14.5616	−0.637342
$523$	−16.8769	−0.737975	−0.368988	−	0.929434i	$-0.620295\pi$
$523$	−16.8769	−0.737975	−0.368988	+	0.929434i	$0.620295\pi$
$524$	7.12311	0.311174
$525$	−4.00000	−0.174574
$526$	−48.0000	−2.09290
$527$	34.0000	1.48106
$528$	0	0
$529$	−7.00000	−0.304348
$530$	−32.8078	−1.42508
$531$	−8.43845	−0.366197
$532$	27.3693	1.18661
$533$	−4.94602	−0.214236
$534$	13.4384	0.581538
$535$	5.12311	0.221491
$536$	−77.4773	−3.34651
$537$	−13.1231	−0.566304
$538$	−21.9309	−0.945507
$539$	0	0
$540$	4.56155	0.196298
$541$	−16.9309	−0.727915	−0.363957	−	0.931416i	$-0.618575\pi$
$541$	−16.9309	−0.727915	−0.363957	+	0.931416i	$0.618575\pi$
$542$	0	0
$543$	−20.8078	−0.892947
$544$	−27.0540	−1.15993
$545$	12.1231	0.519297
$546$	3.68466	0.157689
$547$	3.36932	0.144062	0.0720308	−	0.997402i	$-0.477052\pi$
$547$	3.36932	0.144062	0.0720308	+	0.997402i	$0.477052\pi$
$548$	−63.8617	−2.72804
$549$	11.3693	0.485231
$550$	0	0
$551$	−34.1080	−1.45305
$552$	−26.2462	−1.11711
$553$	−5.12311	−0.217857
$554$	8.31534	0.353285
$555$	9.68466	0.411091
$556$	−46.7386	−1.98216
$557$	−9.50758	−0.402849	−0.201424	−	0.979504i	$-0.564557\pi$
$557$	−9.50758	−0.402849	−0.201424	+	0.979504i	$0.564557\pi$
$558$	−21.1231	−0.894212
$559$	9.61553	0.406694
$560$	7.68466	0.324736
$561$	0	0
$562$	74.6004	3.14683
$563$	−13.8078	−0.581928	−0.290964	−	0.956734i	$-0.593976\pi$
$563$	−13.8078	−0.581928	−0.290964	+	0.956734i	$0.593976\pi$
$564$	34.4924	1.45239
$565$	19.9309	0.838498
$566$	17.6155	0.740436
$567$	1.00000	0.0419961
$568$	−33.6155	−1.41048
$569$	−6.00000	−0.251533	−0.125767	−	0.992060i	$-0.540139\pi$
$569$	−6.00000	−0.251533	−0.125767	+	0.992060i	$0.540139\pi$
$570$	15.3693	0.643750
$571$	6.24621	0.261396	0.130698	−	0.991422i	$-0.458278\pi$
$571$	6.24621	0.261396	0.130698	+	0.991422i	$0.458278\pi$
$572$	0	0
$573$	4.24621	0.177388
$574$	−8.80776	−0.367629
$575$	16.0000	0.667246
$576$	1.43845	0.0599353
$577$	−7.19224	−0.299417	−0.149708	−	0.988730i	$-0.547833\pi$
$577$	−7.19224	−0.299417	−0.149708	+	0.988730i	$0.547833\pi$
$578$	0	0
$579$	1.24621	0.0517908
$580$	−25.9309	−1.07672
$581$	11.8078	0.489869
$582$	35.0540	1.45303
$583$	0	0
$584$	−7.36932	−0.304945
$585$	1.43845	0.0594725
$586$	6.06913	0.250713
$587$	−7.56155	−0.312099	−0.156049	−	0.987749i	$-0.549876\pi$
$587$	−7.56155	−0.312099	−0.156049	+	0.987749i	$0.549876\pi$
$588$	4.56155	0.188115
$589$	−49.4773	−2.03868
$590$	−21.6155	−0.889897
$591$	−18.1771	−0.747705
$592$	74.4233	3.05878
$593$	−19.6307	−0.806136	−0.403068	−	0.915170i	$-0.632056\pi$
$593$	−19.6307	−0.806136	−0.403068	+	0.915170i	$0.632056\pi$
$594$	0	0
$595$	−4.12311	−0.169031
$596$	−80.6695	−3.30435
$597$	1.12311	0.0459657
$598$	−14.7386	−0.602708
$599$	41.2311	1.68466	0.842328	−	0.538966i	$-0.181185\pi$
$599$	41.2311	1.68466	0.842328	+	0.538966i	$0.181185\pi$
$600$	−26.2462	−1.07150
$601$	−7.68466	−0.313464	−0.156732	−	0.987641i	$-0.550096\pi$
$601$	−7.68466	−0.313464	−0.156732	+	0.987641i	$0.550096\pi$
$602$	17.1231	0.697886
$603$	−11.8078	−0.480849
$604$	70.9848	2.88833
$605$	0	0
$606$	1.12311	0.0456230
$607$	2.24621	0.0911709	0.0455855	−	0.998960i	$-0.485485\pi$
$607$	2.24621	0.0911709	0.0455855	+	0.998960i	$0.485485\pi$
$608$	39.3693	1.59664
$609$	−5.68466	−0.230354
$610$	29.1231	1.17916
$611$	10.8769	0.440032
$612$	−18.8078	−0.760259
$613$	5.49242	0.221837	0.110918	−	0.993830i	$-0.464621\pi$
$613$	5.49242	0.221837	0.110918	+	0.993830i	$0.464621\pi$
$614$	79.7235	3.21738
$615$	−3.43845	−0.138652
$616$	0	0
$617$	−1.82292	−0.0733880	−0.0366940	−	0.999327i	$-0.511683\pi$
$617$	−1.82292	−0.0733880	−0.0366940	+	0.999327i	$0.511683\pi$
$618$	−20.4924	−0.824326
$619$	−13.5076	−0.542915	−0.271458	−	0.962450i	$-0.587506\pi$
$619$	−13.5076	−0.542915	−0.271458	+	0.962450i	$0.587506\pi$
$620$	−37.6155	−1.51068
$621$	−4.00000	−0.160514
$622$	1.12311	0.0450324
$623$	5.24621	0.210185
$624$	11.0540	0.442513
$625$	11.0000	0.440000
$626$	16.8078	0.671773
$627$	0	0
$628$	88.3542	3.52571
$629$	−39.9309	−1.59215
$630$	2.56155	0.102055
$631$	−19.0691	−0.759130	−0.379565	−	0.925165i	$-0.623926\pi$
$631$	−19.0691	−0.759130	−0.379565	+	0.925165i	$0.623926\pi$
$632$	−33.6155	−1.33715
$633$	−10.0540	−0.399610
$634$	−48.3542	−1.92039
$635$	−6.93087	−0.275043
$636$	−58.4233	−2.31663
$637$	1.43845	0.0569934
$638$	0	0
$639$	−5.12311	−0.202667
$640$	−9.43845	−0.373087
$641$	−2.31534	−0.0914505	−0.0457252	−	0.998954i	$-0.514560\pi$
$641$	−2.31534	−0.0914505	−0.0457252	+	0.998954i	$0.514560\pi$
$642$	13.1231	0.517928
$643$	38.7386	1.52770	0.763851	−	0.645392i	$-0.223306\pi$
$643$	38.7386	1.52770	0.763851	+	0.645392i	$0.223306\pi$
$644$	−18.2462	−0.719001
$645$	6.68466	0.263208
$646$	−63.3693	−2.49323
$647$	−14.9309	−0.586993	−0.293497	−	0.955960i	$-0.594819\pi$
$647$	−14.9309	−0.586993	−0.293497	+	0.955960i	$0.594819\pi$
$648$	6.56155	0.257762
$649$	0	0
$650$	−14.7386	−0.578097
$651$	−8.24621	−0.323195
$652$	−33.6155	−1.31649
$653$	−8.24621	−0.322699	−0.161350	−	0.986897i	$-0.551585\pi$
$653$	−8.24621	−0.322699	−0.161350	+	0.986897i	$0.551585\pi$
$654$	31.0540	1.21431
$655$	1.56155	0.0610149
$656$	−26.4233	−1.03166
$657$	−1.12311	−0.0438165
$658$	19.3693	0.755095
$659$	−10.0000	−0.389545	−0.194772	−	0.980848i	$-0.562397\pi$
$659$	−10.0000	−0.389545	−0.194772	+	0.980848i	$0.562397\pi$
$660$	0	0
$661$	30.4233	1.18333	0.591664	−	0.806184i	$-0.298471\pi$
$661$	30.4233	1.18333	0.591664	+	0.806184i	$0.298471\pi$
$662$	−13.6155	−0.529183
$663$	−5.93087	−0.230336
$664$	77.4773	3.00670
$665$	6.00000	0.232670
$666$	24.8078	0.961281
$667$	22.7386	0.880443
$668$	35.6155	1.37801
$669$	24.0000	0.927894
$670$	−30.2462	−1.16851
$671$	0	0
$672$	6.56155	0.253117
$673$	−0.930870	−0.0358824	−0.0179412	−	0.999839i	$-0.505711\pi$
$673$	−0.930870	−0.0358824	−0.0179412	+	0.999839i	$0.505711\pi$
$674$	−46.5616	−1.79348
$675$	−4.00000	−0.153960
$676$	−49.8617	−1.91776
$677$	34.8617	1.33985	0.669923	−	0.742431i	$-0.266327\pi$
$677$	34.8617	1.33985	0.669923	+	0.742431i	$0.266327\pi$
$678$	51.0540	1.96072
$679$	13.6847	0.525169
$680$	−27.0540	−1.03747
$681$	−6.68466	−0.256157
$682$	0	0
$683$	−27.6155	−1.05668	−0.528339	−	0.849033i	$-0.677185\pi$
$683$	−27.6155	−1.05668	−0.528339	+	0.849033i	$0.677185\pi$
$684$	27.3693	1.04649
$685$	−14.0000	−0.534913
$686$	2.56155	0.0978005
$687$	−24.8078	−0.946475
$688$	51.3693	1.95844
$689$	−18.4233	−0.701872
$690$	−10.2462	−0.390067
$691$	−38.1080	−1.44969	−0.724847	−	0.688909i	$-0.758090\pi$
$691$	−38.1080	−1.44969	−0.724847	+	0.688909i	$0.758090\pi$
$692$	38.4924	1.46326
$693$	0	0
$694$	−35.8617	−1.36129
$695$	−10.2462	−0.388661
$696$	−37.3002	−1.41386
$697$	14.1771	0.536995
$698$	74.4233	2.81696
$699$	18.1771	0.687520
$700$	−18.2462	−0.689642
$701$	−38.1771	−1.44193	−0.720964	−	0.692972i	$-0.756301\pi$
$701$	−38.1771	−1.44193	−0.720964	+	0.692972i	$0.756301\pi$
$702$	3.68466	0.139068
$703$	58.1080	2.19158
$704$	0	0
$705$	7.56155	0.284785
$706$	−56.8078	−2.13799
$707$	0.438447	0.0164895
$708$	−38.4924	−1.44663
$709$	−7.06913	−0.265487	−0.132743	−	0.991150i	$-0.542379\pi$
$709$	−7.06913	−0.265487	−0.132743	+	0.991150i	$0.542379\pi$
$710$	−13.1231	−0.492501
$711$	−5.12311	−0.192131
$712$	34.4233	1.29007
$713$	32.9848	1.23529
$714$	−10.5616	−0.395256
$715$	0	0
$716$	−59.8617	−2.23714
$717$	5.36932	0.200521
$718$	71.7235	2.67670
$719$	35.8617	1.33742	0.668709	−	0.743525i	$-0.266847\pi$
$719$	35.8617	1.33742	0.668709	+	0.743525i	$0.266847\pi$
$720$	7.68466	0.286390
$721$	−8.00000	−0.297936
$722$	43.5464	1.62063
$723$	4.24621	0.157918
$724$	−94.9157	−3.52751
$725$	22.7386	0.844492
$726$	0	0
$727$	33.2311	1.23247	0.616236	−	0.787562i	$-0.288657\pi$
$727$	33.2311	1.23247	0.616236	+	0.787562i	$0.288657\pi$
$728$	9.43845	0.349812
$729$	1.00000	0.0370370
$730$	−2.87689	−0.106479
$731$	−27.5616	−1.01940
$732$	51.8617	1.91687
$733$	−38.5616	−1.42430	−0.712152	−	0.702026i	$-0.752279\pi$
$733$	−38.5616	−1.42430	−0.712152	+	0.702026i	$0.752279\pi$
$734$	−47.3693	−1.74843
$735$	1.00000	0.0368856
$736$	−26.2462	−0.967448
$737$	0	0
$738$	−8.80776	−0.324218
$739$	49.6155	1.82514	0.912568	−	0.408924i	$-0.134096\pi$
$739$	49.6155	1.82514	0.912568	+	0.408924i	$0.134096\pi$
$740$	44.1771	1.62398
$741$	8.63068	0.317056
$742$	−32.8078	−1.20441
$743$	−41.1231	−1.50866	−0.754330	−	0.656495i	$-0.772038\pi$
$743$	−41.1231	−1.50866	−0.754330	+	0.656495i	$0.772038\pi$
$744$	−54.1080	−1.98369
$745$	−17.6847	−0.647916
$746$	−35.3693	−1.29496
$747$	11.8078	0.432023
$748$	0	0
$749$	5.12311	0.187194
$750$	−23.0540	−0.841812
$751$	8.43845	0.307923	0.153962	−	0.988077i	$-0.450797\pi$
$751$	8.43845	0.307923	0.153962	+	0.988077i	$0.450797\pi$
$752$	58.1080	2.11898
$753$	4.00000	0.145768
$754$	−20.9460	−0.762809
$755$	15.5616	0.566343
$756$	4.56155	0.165902
$757$	40.6155	1.47620	0.738098	−	0.674693i	$-0.235724\pi$
$757$	40.6155	1.47620	0.738098	+	0.674693i	$0.235724\pi$
$758$	−26.7386	−0.971191
$759$	0	0
$760$	39.3693	1.42808
$761$	51.9848	1.88445	0.942225	−	0.334982i	$-0.108730\pi$
$761$	51.9848	1.88445	0.942225	+	0.334982i	$0.108730\pi$
$762$	−17.7538	−0.643152
$763$	12.1231	0.438886
$764$	19.3693	0.700757
$765$	−4.12311	−0.149071
$766$	55.2311	1.99558
$767$	−12.1383	−0.438287
$768$	−27.0540	−0.976226
$769$	33.3002	1.20084	0.600418	−	0.799687i	$-0.295001\pi$
$769$	33.3002	1.20084	0.600418	+	0.799687i	$0.295001\pi$
$770$	0	0
$771$	23.0000	0.828325
$772$	5.68466	0.204595
$773$	−40.5464	−1.45835	−0.729176	−	0.684326i	$-0.760097\pi$
$773$	−40.5464	−1.45835	−0.729176	+	0.684326i	$0.760097\pi$
$774$	17.1231	0.615477
$775$	32.9848	1.18485
$776$	89.7926	3.22337
$777$	9.68466	0.347435
$778$	34.4233	1.23413
$779$	−20.6307	−0.739171
$780$	6.56155	0.234941
$781$	0	0
$782$	42.2462	1.51072
$783$	−5.68466	−0.203153
$784$	7.68466	0.274452
$785$	19.3693	0.691321
$786$	4.00000	0.142675
$787$	28.8769	1.02935	0.514675	−	0.857385i	$-0.327913\pi$
$787$	28.8769	1.02935	0.514675	+	0.857385i	$0.327913\pi$
$788$	−82.9157	−2.95375
$789$	−18.7386	−0.667113
$790$	−13.1231	−0.466899
$791$	19.9309	0.708660
$792$	0	0
$793$	16.3542	0.580753
$794$	−65.7926	−2.33489
$795$	−12.8078	−0.454245
$796$	5.12311	0.181584
$797$	−14.3002	−0.506539	−0.253269	−	0.967396i	$-0.581506\pi$
$797$	−14.3002	−0.506539	−0.253269	+	0.967396i	$0.581506\pi$
$798$	15.3693	0.544068
$799$	−31.1771	−1.10297
$800$	−26.2462	−0.927944
$801$	5.24621	0.185366
$802$	−70.9157	−2.50412
$803$	0	0
$804$	−53.8617	−1.89956
$805$	−4.00000	−0.140981
$806$	−30.3845	−1.07025
$807$	−8.56155	−0.301381
$808$	2.87689	0.101209
$809$	−11.8617	−0.417037	−0.208518	−	0.978018i	$-0.566864\pi$
$809$	−11.8617	−0.417037	−0.208518	+	0.978018i	$0.566864\pi$
$810$	2.56155	0.0900038
$811$	22.0000	0.772524	0.386262	−	0.922389i	$-0.373766\pi$
$811$	22.0000	0.772524	0.386262	+	0.922389i	$0.373766\pi$
$812$	−25.9309	−0.909995
$813$	0	0
$814$	0	0
$815$	−7.36932	−0.258136
$816$	−31.6847	−1.10918
$817$	40.1080	1.40320
$818$	−49.4384	−1.72857
$819$	1.43845	0.0502634
$820$	−15.6847	−0.547732
$821$	−34.4924	−1.20379	−0.601897	−	0.798574i	$-0.705588\pi$
$821$	−34.4924	−1.20379	−0.601897	+	0.798574i	$0.705588\pi$
$822$	−35.8617	−1.25082
$823$	−21.7538	−0.758289	−0.379145	−	0.925337i	$-0.623782\pi$
$823$	−21.7538	−0.758289	−0.379145	+	0.925337i	$0.623782\pi$
$824$	−52.4924	−1.82866
$825$	0	0
$826$	−21.6155	−0.752100
$827$	−30.8769	−1.07369	−0.536847	−	0.843679i	$-0.680385\pi$
$827$	−30.8769	−1.07369	−0.536847	+	0.843679i	$0.680385\pi$
$828$	−18.2462	−0.634100
$829$	27.1922	0.944425	0.472213	−	0.881485i	$-0.343455\pi$
$829$	27.1922	0.944425	0.472213	+	0.881485i	$0.343455\pi$
$830$	30.2462	1.04986
$831$	3.24621	0.112610
$832$	2.06913	0.0717342
$833$	−4.12311	−0.142857
$834$	−26.2462	−0.908832
$835$	7.80776	0.270199
$836$	0	0
$837$	−8.24621	−0.285031
$838$	82.7386	2.85816
$839$	19.9460	0.688613	0.344307	−	0.938857i	$-0.388114\pi$
$839$	19.9460	0.688613	0.344307	+	0.938857i	$0.388114\pi$
$840$	6.56155	0.226395
$841$	3.31534	0.114322
$842$	−17.9309	−0.617939
$843$	29.1231	1.00305
$844$	−45.8617	−1.57863
$845$	−10.9309	−0.376033
$846$	19.3693	0.665931
$847$	0	0
$848$	−98.4233	−3.37987
$849$	6.87689	0.236014
$850$	42.2462	1.44903
$851$	−38.7386	−1.32794
$852$	−23.3693	−0.800620
$853$	−33.7926	−1.15704	−0.578518	−	0.815669i	$-0.696369\pi$
$853$	−33.7926	−1.15704	−0.578518	+	0.815669i	$0.696369\pi$
$854$	29.1231	0.996572
$855$	6.00000	0.205196
$856$	33.6155	1.14896
$857$	−29.8078	−1.01821	−0.509107	−	0.860703i	$-0.670024\pi$
$857$	−29.8078	−1.01821	−0.509107	+	0.860703i	$0.670024\pi$
$858$	0	0
$859$	41.3693	1.41150	0.705751	−	0.708460i	$-0.250610\pi$
$859$	41.3693	1.41150	0.705751	+	0.708460i	$0.250610\pi$
$860$	30.4924	1.03978
$861$	−3.43845	−0.117182
$862$	90.6004	3.08586
$863$	19.7538	0.672427	0.336213	−	0.941786i	$-0.390854\pi$
$863$	19.7538	0.672427	0.336213	+	0.941786i	$0.390854\pi$
$864$	6.56155	0.223229
$865$	8.43845	0.286916
$866$	40.8078	1.38670
$867$	0	0
$868$	−37.6155	−1.27675
$869$	0	0
$870$	−14.5616	−0.493683
$871$	−16.9848	−0.575510
$872$	79.5464	2.69378
$873$	13.6847	0.463156
$874$	−61.4773	−2.07950
$875$	−9.00000	−0.304256
$876$	−5.12311	−0.173094
$877$	8.42329	0.284434	0.142217	−	0.989835i	$-0.454577\pi$
$877$	8.42329	0.284434	0.142217	+	0.989835i	$0.454577\pi$
$878$	18.2462	0.615780
$879$	2.36932	0.0799151
$880$	0	0
$881$	−47.9309	−1.61483	−0.807416	−	0.589983i	$-0.799135\pi$
$881$	−47.9309	−1.61483	−0.807416	+	0.589983i	$0.799135\pi$
$882$	2.56155	0.0862520
$883$	43.1771	1.45302	0.726512	−	0.687154i	$-0.241140\pi$
$883$	43.1771	1.45302	0.726512	+	0.687154i	$0.241140\pi$
$884$	−27.0540	−0.909924
$885$	−8.43845	−0.283655
$886$	62.7386	2.10775
$887$	27.8078	0.933693	0.466847	−	0.884338i	$-0.345390\pi$
$887$	27.8078	0.933693	0.466847	+	0.884338i	$0.345390\pi$
$888$	63.5464	2.13248
$889$	−6.93087	−0.232454
$890$	13.4384	0.450458
$891$	0	0
$892$	109.477	3.66557
$893$	45.3693	1.51823
$894$	−45.3002	−1.51506
$895$	−13.1231	−0.438657
$896$	−9.43845	−0.315316
$897$	−5.75379	−0.192113
$898$	−23.1922	−0.773935
$899$	46.8769	1.56343
$900$	−18.2462	−0.608207
$901$	52.8078	1.75928
$902$	0	0
$903$	6.68466	0.222452
$904$	130.777	4.34959
$905$	−20.8078	−0.691673
$906$	39.8617	1.32432
$907$	43.3693	1.44005	0.720027	−	0.693946i	$-0.244129\pi$
$907$	43.3693	1.44005	0.720027	+	0.693946i	$0.244129\pi$
$908$	−30.4924	−1.01193
$909$	0.438447	0.0145424
$910$	3.68466	0.122145
$911$	5.36932	0.177893	0.0889467	−	0.996036i	$-0.471650\pi$
$911$	5.36932	0.177893	0.0889467	+	0.996036i	$0.471650\pi$
$912$	46.1080	1.52679
$913$	0	0
$914$	78.4233	2.59401
$915$	11.3693	0.375858
$916$	−113.162	−3.73898
$917$	1.56155	0.0515670
$918$	−10.5616	−0.348583
$919$	−29.6155	−0.976926	−0.488463	−	0.872585i	$-0.662442\pi$
$919$	−29.6155	−0.976926	−0.488463	+	0.872585i	$0.662442\pi$
$920$	−26.2462	−0.865312
$921$	31.1231	1.02554
$922$	36.8078	1.21220
$923$	−7.36932	−0.242564
$924$	0	0
$925$	−38.7386	−1.27372
$926$	23.3693	0.767963
$927$	−8.00000	−0.262754
$928$	−37.3002	−1.22444
$929$	35.4924	1.16447	0.582234	−	0.813021i	$-0.302179\pi$
$929$	35.4924	1.16447	0.582234	+	0.813021i	$0.302179\pi$
$930$	−21.1231	−0.692654
$931$	6.00000	0.196642
$932$	82.9157	2.71599
$933$	0.438447	0.0143541
$934$	−97.9697	−3.20567
$935$	0	0
$936$	9.43845	0.308505
$937$	23.9309	0.781787	0.390894	−	0.920436i	$-0.372166\pi$
$937$	23.9309	0.781787	0.390894	+	0.920436i	$0.372166\pi$
$938$	−30.2462	−0.987574
$939$	6.56155	0.214128
$940$	34.4924	1.12502
$941$	−32.5616	−1.06148	−0.530738	−	0.847536i	$-0.678085\pi$
$941$	−32.5616	−1.06148	−0.530738	+	0.847536i	$0.678085\pi$
$942$	49.6155	1.61656
$943$	13.7538	0.447885
$944$	−64.8466	−2.11058
$945$	1.00000	0.0325300
$946$	0	0
$947$	2.00000	0.0649913	0.0324956	−	0.999472i	$-0.489654\pi$
$947$	2.00000	0.0649913	0.0324956	+	0.999472i	$0.489654\pi$
$948$	−23.3693	−0.759000
$949$	−1.61553	−0.0524423
$950$	−61.4773	−1.99459
$951$	−18.8769	−0.612125
$952$	−27.0540	−0.876824
$953$	34.6695	1.12306	0.561528	−	0.827458i	$-0.310214\pi$
$953$	34.6695	1.12306	0.561528	+	0.827458i	$0.310214\pi$
$954$	−32.8078	−1.06219
$955$	4.24621	0.137404
$956$	24.4924	0.792142
$957$	0	0
$958$	−44.3542	−1.43302
$959$	−14.0000	−0.452084
$960$	1.43845	0.0464257
$961$	37.0000	1.19355
$962$	35.6847	1.15052
$963$	5.12311	0.165090
$964$	19.3693	0.623844
$965$	1.24621	0.0401170
$966$	−10.2462	−0.329666
$967$	−34.3002	−1.10302	−0.551510	−	0.834168i	$-0.685948\pi$
$967$	−34.3002	−1.10302	−0.551510	+	0.834168i	$0.685948\pi$
$968$	0	0
$969$	−24.7386	−0.794719
$970$	35.0540	1.12552
$971$	28.6847	0.920534	0.460267	−	0.887780i	$-0.347754\pi$
$971$	28.6847	0.920534	0.460267	+	0.887780i	$0.347754\pi$
$972$	4.56155	0.146312
$973$	−10.2462	−0.328478
$974$	60.3542	1.93387
$975$	−5.75379	−0.184269
$976$	87.3693	2.79662
$977$	−20.6695	−0.661276	−0.330638	−	0.943758i	$-0.607264\pi$
$977$	−20.6695	−0.661276	−0.330638	+	0.943758i	$0.607264\pi$
$978$	−18.8769	−0.603617
$979$	0	0
$980$	4.56155	0.145713
$981$	12.1231	0.387061
$982$	51.2311	1.63485
$983$	−31.3693	−1.00053	−0.500263	−	0.865874i	$-0.666763\pi$
$983$	−31.3693	−1.00053	−0.500263	+	0.865874i	$0.666763\pi$
$984$	−22.5616	−0.719236
$985$	−18.1771	−0.579170
$986$	60.0388	1.91203
$987$	7.56155	0.240687
$988$	39.3693	1.25250
$989$	−26.7386	−0.850239
$990$	0	0
$991$	4.13826	0.131456	0.0657281	−	0.997838i	$-0.479063\pi$
$991$	4.13826	0.131456	0.0657281	+	0.997838i	$0.479063\pi$
$992$	−54.1080	−1.71793
$993$	−5.31534	−0.168677
$994$	−13.1231	−0.416240
$995$	1.12311	0.0356048
$996$	53.8617	1.70667
$997$	62.0388	1.96479	0.982395	−	0.186818i	$-0.0598174\pi$
$997$	62.0388	1.96479	0.982395	+	0.186818i	$0.0598174\pi$
$998$	78.2462	2.47684
$999$	9.68466	0.306409

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2541.2.a.bc.1.2	yes	2
3.2	odd	2		7623.2.a.be.1.1		2
11.10	odd	2		2541.2.a.u.1.1	✓	2
33.32	even	2		7623.2.a.bt.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2541.2.a.u.1.1	✓	2	11.10	odd	2
2541.2.a.bc.1.2	yes	2	1.1	even	1	trivial
7623.2.a.be.1.1		2	3.2	odd	2
7623.2.a.bt.1.2		2	33.32	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2541 = 3 \cdot 7 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2541.a (trivial)

\(f(q)\)	\(=\)	\(q+2.56155 q^{2} +1.00000 q^{3} +4.56155 q^{4} +1.00000 q^{5} +2.56155 q^{6} +1.00000 q^{7} +6.56155 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+2.56155 q^{2} +1.00000 q^{3} +4.56155 q^{4} +1.00000 q^{5} +2.56155 q^{6} +1.00000 q^{7} +6.56155 q^{8} +1.00000 q^{9} +2.56155 q^{10} +4.56155 q^{12} +1.43845 q^{13} +2.56155 q^{14} +1.00000 q^{15} +7.68466 q^{16} -4.12311 q^{17} +2.56155 q^{18} +6.00000 q^{19} +4.56155 q^{20} +1.00000 q^{21} -4.00000 q^{23} +6.56155 q^{24} -4.00000 q^{25} +3.68466 q^{26} +1.00000 q^{27} +4.56155 q^{28} -5.68466 q^{29} +2.56155 q^{30} -8.24621 q^{31} +6.56155 q^{32} -10.5616 q^{34} +1.00000 q^{35} +4.56155 q^{36} +9.68466 q^{37} +15.3693 q^{38} +1.43845 q^{39} +6.56155 q^{40} -3.43845 q^{41} +2.56155 q^{42} +6.68466 q^{43} +1.00000 q^{45} -10.2462 q^{46} +7.56155 q^{47} +7.68466 q^{48} +1.00000 q^{49} -10.2462 q^{50} -4.12311 q^{51} +6.56155 q^{52} -12.8078 q^{53} +2.56155 q^{54} +6.56155 q^{56} +6.00000 q^{57} -14.5616 q^{58} -8.43845 q^{59} +4.56155 q^{60} +11.3693 q^{61} -21.1231 q^{62} +1.00000 q^{63} +1.43845 q^{64} +1.43845 q^{65} -11.8078 q^{67} -18.8078 q^{68} -4.00000 q^{69} +2.56155 q^{70} -5.12311 q^{71} +6.56155 q^{72} -1.12311 q^{73} +24.8078 q^{74} -4.00000 q^{75} +27.3693 q^{76} +3.68466 q^{78} -5.12311 q^{79} +7.68466 q^{80} +1.00000 q^{81} -8.80776 q^{82} +11.8078 q^{83} +4.56155 q^{84} -4.12311 q^{85} +17.1231 q^{86} -5.68466 q^{87} +5.24621 q^{89} +2.56155 q^{90} +1.43845 q^{91} -18.2462 q^{92} -8.24621 q^{93} +19.3693 q^{94} +6.00000 q^{95} +6.56155 q^{96} +13.6847 q^{97} +2.56155 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{2} + 2 q^{3} + 5 q^{4} + 2 q^{5} + q^{6} + 2 q^{7} + 9 q^{8} + 2 q^{9}+O(q^{10}) \) 2 * q + q^2 + 2 * q^3 + 5 * q^4 + 2 * q^5 + q^6 + 2 * q^7 + 9 * q^8 + 2 * q^9	\( 2 q + q^{2} + 2 q^{3} + 5 q^{4} + 2 q^{5} + q^{6} + 2 q^{7} + 9 q^{8} + 2 q^{9} + q^{10} + 5 q^{12} + 7 q^{13} + q^{14} + 2 q^{15} + 3 q^{16} + q^{18} + 12 q^{19} + 5 q^{20} + 2 q^{21} - 8 q^{23} + 9 q^{24} - 8 q^{25} - 5 q^{26} + 2 q^{27} + 5 q^{28} + q^{29} + q^{30} + 9 q^{32} - 17 q^{34} + 2 q^{35} + 5 q^{36} + 7 q^{37} + 6 q^{38} + 7 q^{39} + 9 q^{40} - 11 q^{41} + q^{42} + q^{43} + 2 q^{45} - 4 q^{46} + 11 q^{47} + 3 q^{48} + 2 q^{49} - 4 q^{50} + 9 q^{52} - 5 q^{53} + q^{54} + 9 q^{56} + 12 q^{57} - 25 q^{58} - 21 q^{59} + 5 q^{60} - 2 q^{61} - 34 q^{62} + 2 q^{63} + 7 q^{64} + 7 q^{65} - 3 q^{67} - 17 q^{68} - 8 q^{69} + q^{70} - 2 q^{71} + 9 q^{72} + 6 q^{73} + 29 q^{74} - 8 q^{75} + 30 q^{76} - 5 q^{78} - 2 q^{79} + 3 q^{80} + 2 q^{81} + 3 q^{82} + 3 q^{83} + 5 q^{84} + 26 q^{86} + q^{87} - 6 q^{89} + q^{90} + 7 q^{91} - 20 q^{92} + 14 q^{94} + 12 q^{95} + 9 q^{96} + 15 q^{97} + q^{98}+O(q^{100}) \) 2 * q + q^2 + 2 * q^3 + 5 * q^4 + 2 * q^5 + q^6 + 2 * q^7 + 9 * q^8 + 2 * q^9 + q^10 + 5 * q^12 + 7 * q^13 + q^14 + 2 * q^15 + 3 * q^16 + q^18 + 12 * q^19 + 5 * q^20 + 2 * q^21 - 8 * q^23 + 9 * q^24 - 8 * q^25 - 5 * q^26 + 2 * q^27 + 5 * q^28 + q^29 + q^30 + 9 * q^32 - 17 * q^34 + 2 * q^35 + 5 * q^36 + 7 * q^37 + 6 * q^38 + 7 * q^39 + 9 * q^40 - 11 * q^41 + q^42 + q^43 + 2 * q^45 - 4 * q^46 + 11 * q^47 + 3 * q^48 + 2 * q^49 - 4 * q^50 + 9 * q^52 - 5 * q^53 + q^54 + 9 * q^56 + 12 * q^57 - 25 * q^58 - 21 * q^59 + 5 * q^60 - 2 * q^61 - 34 * q^62 + 2 * q^63 + 7 * q^64 + 7 * q^65 - 3 * q^67 - 17 * q^68 - 8 * q^69 + q^70 - 2 * q^71 + 9 * q^72 + 6 * q^73 + 29 * q^74 - 8 * q^75 + 30 * q^76 - 5 * q^78 - 2 * q^79 + 3 * q^80 + 2 * q^81 + 3 * q^82 + 3 * q^83 + 5 * q^84 + 26 * q^86 + q^87 - 6 * q^89 + q^90 + 7 * q^91 - 20 * q^92 + 14 * q^94 + 12 * q^95 + 9 * q^96 + 15 * q^97 + q^98

Introduction

L-functions

Modular forms

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