LMFDB - Embedded newform 3822.2.a.by.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3822,2,Mod(1,3822)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3822, 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("3822.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3822 = 2 \cdot 3 \cdot 7^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3822.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:	$30.5188236525$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.10304.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 7x^{2} + 8x + 8 $ x^4 - 2x^3 - 7x^2 + 8*x + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{29}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.16053$ of defining polynomial
Character	$\chi$	$=$	3822.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.16053 q^{5} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.16053 q^{5} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +1.16053 q^{10} +1.51929 q^{11} +1.00000 q^{12} -1.00000 q^{13} -1.16053 q^{15} +1.00000 q^{16} -5.84035 q^{17} -1.00000 q^{18} +1.45774 q^{19} -1.16053 q^{20} -1.51929 q^{22} +5.77880 q^{23} -1.00000 q^{24} -3.65317 q^{25} +1.00000 q^{26} +1.00000 q^{27} +3.59771 q^{29} +1.16053 q^{30} -7.29809 q^{31} -1.00000 q^{32} +1.51929 q^{33} +5.84035 q^{34} +1.00000 q^{36} +1.95648 q^{37} -1.45774 q^{38} -1.00000 q^{39} +1.16053 q^{40} +9.88388 q^{41} -1.77880 q^{43} +1.51929 q^{44} -1.16053 q^{45} -5.77880 q^{46} +11.5867 q^{47} +1.00000 q^{48} +3.65317 q^{50} -5.84035 q^{51} -1.00000 q^{52} +9.58667 q^{53} -1.00000 q^{54} -1.76319 q^{55} +1.45774 q^{57} -3.59771 q^{58} -0.0156151 q^{59} -1.16053 q^{60} -9.20988 q^{61} +7.29809 q^{62} +1.00000 q^{64} +1.16053 q^{65} -1.51929 q^{66} +4.07842 q^{67} -5.84035 q^{68} +5.77880 q^{69} -0.358761 q^{71} -1.00000 q^{72} -3.53491 q^{73} -1.95648 q^{74} -3.65317 q^{75} +1.45774 q^{76} +1.00000 q^{78} +10.2374 q^{79} -1.16053 q^{80} +1.00000 q^{81} -9.88388 q^{82} +7.33579 q^{83} +6.77792 q^{85} +1.77880 q^{86} +3.59771 q^{87} -1.51929 q^{88} +9.20406 q^{89} +1.16053 q^{90} +5.77880 q^{92} -7.29809 q^{93} -11.5867 q^{94} -1.69175 q^{95} -1.00000 q^{96} -5.19543 q^{97} +1.51929 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{2} + 4 q^{3} + 4 q^{4} + 6 q^{5} - 4 q^{6} - 4 q^{8} + 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^2 + 4 * q^3 + 4 * q^4 + 6 * q^5 - 4 * q^6 - 4 * q^8 + 4 * q^9	$ 4 q - 4 q^{2} + 4 q^{3} + 4 q^{4} + 6 q^{5} - 4 q^{6} - 4 q^{8} + 4 q^{9} - 6 q^{10} + 2 q^{11} + 4 q^{12} - 4 q^{13} + 6 q^{15} + 4 q^{16} + 2 q^{17} - 4 q^{18} + 2 q^{19} + 6 q^{20} - 2 q^{22} - 2 q^{23} - 4 q^{24} + 6 q^{25} + 4 q^{26} + 4 q^{27} + 6 q^{29} - 6 q^{30} - 4 q^{32} + 2 q^{33} - 2 q^{34} + 4 q^{36} + 6 q^{37} - 2 q^{38} - 4 q^{39} - 6 q^{40} + 16 q^{41} + 18 q^{43} + 2 q^{44} + 6 q^{45} + 2 q^{46} + 16 q^{47} + 4 q^{48} - 6 q^{50} + 2 q^{51} - 4 q^{52} + 8 q^{53} - 4 q^{54} + 2 q^{55} + 2 q^{57} - 6 q^{58} + 16 q^{59} + 6 q^{60} - 2 q^{61} + 4 q^{64} - 6 q^{65} - 2 q^{66} + 12 q^{67} + 2 q^{68} - 2 q^{69} - 8 q^{71} - 4 q^{72} + 6 q^{73} - 6 q^{74} + 6 q^{75} + 2 q^{76} + 4 q^{78} - 24 q^{79} + 6 q^{80} + 4 q^{81} - 16 q^{82} + 28 q^{83} + 38 q^{85} - 18 q^{86} + 6 q^{87} - 2 q^{88} + 28 q^{89} - 6 q^{90} - 2 q^{92} - 16 q^{94} + 22 q^{95} - 4 q^{96} - 4 q^{97} + 2 q^{99}+O(q^{100}) $ 4 * q - 4 * q^2 + 4 * q^3 + 4 * q^4 + 6 * q^5 - 4 * q^6 - 4 * q^8 + 4 * q^9 - 6 * q^10 + 2 * q^11 + 4 * q^12 - 4 * q^13 + 6 * q^15 + 4 * q^16 + 2 * q^17 - 4 * q^18 + 2 * q^19 + 6 * q^20 - 2 * q^22 - 2 * q^23 - 4 * q^24 + 6 * q^25 + 4 * q^26 + 4 * q^27 + 6 * q^29 - 6 * q^30 - 4 * q^32 + 2 * q^33 - 2 * q^34 + 4 * q^36 + 6 * q^37 - 2 * q^38 - 4 * q^39 - 6 * q^40 + 16 * q^41 + 18 * q^43 + 2 * q^44 + 6 * q^45 + 2 * q^46 + 16 * q^47 + 4 * q^48 - 6 * q^50 + 2 * q^51 - 4 * q^52 + 8 * q^53 - 4 * q^54 + 2 * q^55 + 2 * q^57 - 6 * q^58 + 16 * q^59 + 6 * q^60 - 2 * q^61 + 4 * q^64 - 6 * q^65 - 2 * q^66 + 12 * q^67 + 2 * q^68 - 2 * q^69 - 8 * q^71 - 4 * q^72 + 6 * q^73 - 6 * q^74 + 6 * q^75 + 2 * q^76 + 4 * q^78 - 24 * q^79 + 6 * q^80 + 4 * q^81 - 16 * q^82 + 28 * q^83 + 38 * q^85 - 18 * q^86 + 6 * q^87 - 2 * q^88 + 28 * q^89 - 6 * q^90 - 2 * q^92 - 16 * q^94 + 22 * q^95 - 4 * q^96 - 4 * q^97 + 2 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.00000	0.500000
$5$	−1.16053	−0.519005	−0.259503	−	0.965742i	$-0.583559\pi$
$5$	−1.16053	−0.519005	−0.259503	+	0.965742i	$0.583559\pi$
$6$	−1.00000	−0.408248
$7$	0	0
$7$	0	0
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	1.16053	0.366992
$11$	1.51929	0.458084	0.229042	−	0.973417i	$-0.426441\pi$
$11$	1.51929	0.458084	0.229042	+	0.973417i	$0.426441\pi$
$12$	1.00000	0.288675
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	−1.16053	−0.299648
$16$	1.00000	0.250000
$17$	−5.84035	−1.41649	−0.708247	−	0.705965i	$-0.750514\pi$
$17$	−5.84035	−1.41649	−0.708247	+	0.705965i	$0.750514\pi$
$18$	−1.00000	−0.235702
$19$	1.45774	0.334428	0.167214	−	0.985921i	$-0.446523\pi$
$19$	1.45774	0.334428	0.167214	+	0.985921i	$0.446523\pi$
$20$	−1.16053	−0.259503
$21$	0	0
$22$	−1.51929	−0.323914
$23$	5.77880	1.20496	0.602482	−	0.798133i	$-0.294179\pi$
$23$	5.77880	1.20496	0.602482	+	0.798133i	$0.294179\pi$
$24$	−1.00000	−0.204124
$25$	−3.65317	−0.730633
$26$	1.00000	0.196116
$27$	1.00000	0.192450
$28$	0	0
$29$	3.59771	0.668079	0.334039	−	0.942559i	$-0.391588\pi$
$29$	3.59771	0.668079	0.334039	+	0.942559i	$0.391588\pi$
$30$	1.16053	0.211883
$31$	−7.29809	−1.31078	−0.655388	−	0.755292i	$-0.727495\pi$
$31$	−7.29809	−1.31078	−0.655388	+	0.755292i	$0.727495\pi$
$32$	−1.00000	−0.176777
$33$	1.51929	0.264475
$34$	5.84035	1.00161
$35$	0	0
$36$	1.00000	0.166667
$37$	1.95648	0.321643	0.160821	−	0.986984i	$-0.448586\pi$
$37$	1.95648	0.321643	0.160821	+	0.986984i	$0.448586\pi$
$38$	−1.45774	−0.236476
$39$	−1.00000	−0.160128
$40$	1.16053	0.183496
$41$	9.88388	1.54360	0.771801	−	0.635864i	$-0.219356\pi$
$41$	9.88388	1.54360	0.771801	+	0.635864i	$0.219356\pi$
$42$	0	0
$43$	−1.77880	−0.271265	−0.135632	−	0.990759i	$-0.543307\pi$
$43$	−1.77880	−0.271265	−0.135632	+	0.990759i	$0.543307\pi$
$44$	1.51929	0.229042
$45$	−1.16053	−0.173002
$46$	−5.77880	−0.852038
$47$	11.5867	1.69009	0.845045	−	0.534695i	$-0.179574\pi$
$47$	11.5867	1.69009	0.845045	+	0.534695i	$0.179574\pi$
$48$	1.00000	0.144338
$49$	0	0
$50$	3.65317	0.516636
$51$	−5.84035	−0.817813
$52$	−1.00000	−0.138675
$53$	9.58667	1.31683	0.658415	−	0.752655i	$-0.271227\pi$
$53$	9.58667	1.31683	0.658415	+	0.752655i	$0.271227\pi$
$54$	−1.00000	−0.136083
$55$	−1.76319	−0.237748
$56$	0	0
$57$	1.45774	0.193082
$58$	−3.59771	−0.472403
$59$	−0.0156151	−0.00203291	−0.00101645	−	0.999999i	$-0.500324\pi$
$59$	−0.0156151	−0.00203291	−0.00101645	+	0.999999i	$0.500324\pi$
$60$	−1.16053	−0.149824
$61$	−9.20988	−1.17920	−0.589602	−	0.807694i	$-0.700716\pi$
$61$	−9.20988	−1.17920	−0.589602	+	0.807694i	$0.700716\pi$
$62$	7.29809	0.926859
$63$	0	0
$64$	1.00000	0.125000
$65$	1.16053	0.143946
$66$	−1.51929	−0.187012
$67$	4.07842	0.498259	0.249129	−	0.968470i	$-0.419856\pi$
$67$	4.07842	0.498259	0.249129	+	0.968470i	$0.419856\pi$
$68$	−5.84035	−0.708247
$69$	5.77880	0.695686
$70$	0	0
$71$	−0.358761	−0.0425771	−0.0212885	−	0.999773i	$-0.506777\pi$
$71$	−0.358761	−0.0425771	−0.0212885	+	0.999773i	$0.506777\pi$
$72$	−1.00000	−0.117851
$73$	−3.53491	−0.413730	−0.206865	−	0.978370i	$-0.566326\pi$
$73$	−3.53491	−0.413730	−0.206865	+	0.978370i	$0.566326\pi$
$74$	−1.95648	−0.227436
$75$	−3.65317	−0.421831
$76$	1.45774	0.167214
$77$	0	0
$78$	1.00000	0.113228
$79$	10.2374	1.15180	0.575900	−	0.817520i	$-0.304652\pi$
$79$	10.2374	1.15180	0.575900	+	0.817520i	$0.304652\pi$
$80$	−1.16053	−0.129751
$81$	1.00000	0.111111
$82$	−9.88388	−1.09149
$83$	7.33579	0.805208	0.402604	−	0.915374i	$-0.368105\pi$
$83$	7.33579	0.805208	0.402604	+	0.915374i	$0.368105\pi$
$84$	0	0
$85$	6.77792	0.735168
$86$	1.77880	0.191813
$87$	3.59771	0.385715
$88$	−1.51929	−0.161957
$89$	9.20406	0.975628	0.487814	−	0.872948i	$-0.337794\pi$
$89$	9.20406	0.975628	0.487814	+	0.872948i	$0.337794\pi$
$90$	1.16053	0.122331
$91$	0	0
$92$	5.77880	0.602482
$93$	−7.29809	−0.756777
$94$	−11.5867	−1.19507
$95$	−1.69175	−0.173570
$96$	−1.00000	−0.102062
$97$	−5.19543	−0.527516	−0.263758	−	0.964589i	$-0.584962\pi$
$97$	−5.19543	−0.527516	−0.263758	+	0.964589i	$0.584962\pi$
$98$	0	0
$99$	1.51929	0.152695
$100$	−3.65317	−0.365317
$101$	−12.0273	−1.19676	−0.598379	−	0.801213i	$-0.704188\pi$
$101$	−12.0273	−1.19676	−0.598379	+	0.801213i	$0.704188\pi$
$102$	5.84035	0.578281
$103$	7.81128	0.769669	0.384834	−	0.922986i	$-0.374259\pi$
$103$	7.81128	0.769669	0.384834	+	0.922986i	$0.374259\pi$
$104$	1.00000	0.0980581
$105$	0	0
$106$	−9.58667	−0.931140
$107$	−5.10480	−0.493500	−0.246750	−	0.969079i	$-0.579363\pi$
$107$	−5.10480	−0.493500	−0.246750	+	0.969079i	$0.579363\pi$
$108$	1.00000	0.0962250
$109$	6.20037	0.593888	0.296944	−	0.954895i	$-0.404033\pi$
$109$	6.20037	0.593888	0.296944	+	0.954895i	$0.404033\pi$
$110$	1.76319	0.168113
$111$	1.95648	0.185700
$112$	0	0
$113$	−17.0659	−1.60542	−0.802710	−	0.596369i	$-0.796609\pi$
$113$	−17.0659	−1.60542	−0.802710	+	0.596369i	$0.796609\pi$
$114$	−1.45774	−0.136530
$115$	−6.70648	−0.625382
$116$	3.59771	0.334039
$117$	−1.00000	−0.0924500
$118$	0.0156151	0.00143748
$119$	0	0
$120$	1.16053	0.105942
$121$	−8.69175	−0.790159
$122$	9.20988	0.833824
$123$	9.88388	0.891199
$124$	−7.29809	−0.655388
$125$	10.0423	0.898208
$126$	0	0
$127$	17.1651	1.52316	0.761578	−	0.648073i	$-0.224425\pi$
$127$	17.1651	1.52316	0.761578	+	0.648073i	$0.224425\pi$
$128$	−1.00000	−0.0883883
$129$	−1.77880	−0.156615
$130$	−1.16053	−0.101785
$131$	12.9127	1.12819	0.564093	−	0.825711i	$-0.309226\pi$
$131$	12.9127	1.12819	0.564093	+	0.825711i	$0.309226\pi$
$132$	1.51929	0.132237
$133$	0	0
$134$	−4.07842	−0.352322
$135$	−1.16053	−0.0998826
$136$	5.84035	0.500806
$137$	10.4348	0.891503	0.445751	−	0.895157i	$-0.352937\pi$
$137$	10.4348	0.891503	0.445751	+	0.895157i	$0.352937\pi$
$138$	−5.77880	−0.491924
$139$	20.7071	1.75635	0.878176	−	0.478337i	$-0.158760\pi$
$139$	20.7071	1.75635	0.878176	+	0.478337i	$0.158760\pi$
$140$	0	0
$141$	11.5867	0.975774
$142$	0.358761	0.0301065
$143$	−1.51929	−0.127050
$144$	1.00000	0.0833333
$145$	−4.17526	−0.346736
$146$	3.53491	0.292551
$147$	0	0
$148$	1.95648	0.160821
$149$	−2.50736	−0.205411	−0.102706	−	0.994712i	$-0.532750\pi$
$149$	−2.50736	−0.205411	−0.102706	+	0.994712i	$0.532750\pi$
$150$	3.65317	0.298280
$151$	1.67614	0.136402	0.0682010	−	0.997672i	$-0.478274\pi$
$151$	1.67614	0.136402	0.0682010	+	0.997672i	$0.478274\pi$
$152$	−1.45774	−0.118238
$153$	−5.84035	−0.472165
$154$	0	0
$155$	8.46967	0.680300
$156$	−1.00000	−0.0800641
$157$	−10.8551	−0.866330	−0.433165	−	0.901315i	$-0.642603\pi$
$157$	−10.8551	−0.866330	−0.433165	+	0.901315i	$0.642603\pi$
$158$	−10.2374	−0.814446
$159$	9.58667	0.760273
$160$	1.16053	0.0917481
$161$	0	0
$162$	−1.00000	−0.0785674
$163$	8.00699	0.627156	0.313578	−	0.949563i	$-0.398472\pi$
$163$	8.00699	0.627156	0.313578	+	0.949563i	$0.398472\pi$
$164$	9.88388	0.771801
$165$	−1.76319	−0.137264
$166$	−7.33579	−0.569368
$167$	0.339091	0.0262397	0.0131198	−	0.999914i	$-0.495824\pi$
$167$	0.339091	0.0262397	0.0131198	+	0.999914i	$0.495824\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	−6.77792	−0.519842
$171$	1.45774	0.111476
$172$	−1.77880	−0.135632
$173$	−2.22577	−0.169222	−0.0846112	−	0.996414i	$-0.526965\pi$
$173$	−2.22577	−0.169222	−0.0846112	+	0.996414i	$0.526965\pi$
$174$	−3.59771	−0.272742
$175$	0	0
$176$	1.51929	0.114521
$177$	−0.0156151	−0.00117370
$178$	−9.20406	−0.689873
$179$	−6.45280	−0.482305	−0.241152	−	0.970487i	$-0.577525\pi$
$179$	−6.45280	−0.482305	−0.241152	+	0.970487i	$0.577525\pi$
$180$	−1.16053	−0.0865009
$181$	0.0238555	0.00177317	0.000886584	−	1.00000i	$-0.499718\pi$
$181$	0.0238555	0.00177317	0.000886584		1.00000i	$0.499718\pi$
$182$	0	0
$183$	−9.20988	−0.680814
$184$	−5.77880	−0.426019
$185$	−2.27055	−0.166934
$186$	7.29809	0.535122
$187$	−8.87321	−0.648873
$188$	11.5867	0.845045
$189$	0	0
$190$	1.69175	0.122733
$191$	9.20164	0.665808	0.332904	−	0.942961i	$-0.391972\pi$
$191$	9.20164	0.665808	0.332904	+	0.942961i	$0.391972\pi$
$192$	1.00000	0.0721688
$193$	23.2522	1.67373	0.836863	−	0.547412i	$-0.184387\pi$
$193$	23.2522	1.67373	0.836863	+	0.547412i	$0.184387\pi$
$194$	5.19543	0.373010
$195$	1.16053	0.0831074
$196$	0	0
$197$	21.7521	1.54978	0.774888	−	0.632098i	$-0.217806\pi$
$197$	21.7521	1.54978	0.774888	+	0.632098i	$0.217806\pi$
$198$	−1.51929	−0.107971
$199$	−17.4902	−1.23985	−0.619924	−	0.784661i	$-0.712837\pi$
$199$	−17.4902	−1.23985	−0.619924	+	0.784661i	$0.712837\pi$
$200$	3.65317	0.258318
$201$	4.07842	0.287670
$202$	12.0273	0.846236
$203$	0	0
$204$	−5.84035	−0.408907
$205$	−11.4706	−0.801138
$206$	−7.81128	−0.544238
$207$	5.77880	0.401654
$208$	−1.00000	−0.0693375
$209$	2.21473	0.153196
$210$	0	0
$211$	4.87070	0.335313	0.167656	−	0.985845i	$-0.446380\pi$
$211$	4.87070	0.335313	0.167656	+	0.985845i	$0.446380\pi$
$212$	9.58667	0.658415
$213$	−0.358761	−0.0245819
$214$	5.10480	0.348957
$215$	2.06435	0.140788
$216$	−1.00000	−0.0680414
$217$	0	0
$218$	−6.20037	−0.419942
$219$	−3.53491	−0.238867
$220$	−1.76319	−0.118874
$221$	5.84035	0.392865
$222$	−1.95648	−0.131310
$223$	8.01165	0.536500	0.268250	−	0.963349i	$-0.413555\pi$
$223$	8.01165	0.536500	0.268250	+	0.963349i	$0.413555\pi$
$224$	0	0
$225$	−3.65317	−0.243544
$226$	17.0659	1.13520
$227$	23.8532	1.58319	0.791595	−	0.611046i	$-0.209251\pi$
$227$	23.8532	1.58319	0.791595	+	0.611046i	$0.209251\pi$
$228$	1.45774	0.0965411
$229$	−14.4935	−0.957759	−0.478879	−	0.877881i	$-0.658957\pi$
$229$	−14.4935	−0.957759	−0.478879	+	0.877881i	$0.658957\pi$
$230$	6.70648	0.442212
$231$	0	0
$232$	−3.59771	−0.236202
$233$	17.0723	1.11844	0.559222	−	0.829018i	$-0.311100\pi$
$233$	17.0723	1.11844	0.559222	+	0.829018i	$0.311100\pi$
$234$	1.00000	0.0653720
$235$	−13.4467	−0.877166
$236$	−0.0156151	−0.00101645
$237$	10.2374	0.664992
$238$	0	0
$239$	−19.8943	−1.28685	−0.643427	−	0.765507i	$-0.722488\pi$
$239$	−19.8943	−1.28685	−0.643427	+	0.765507i	$0.722488\pi$
$240$	−1.16053	−0.0749120
$241$	6.03374	0.388667	0.194334	−	0.980936i	$-0.437746\pi$
$241$	6.03374	0.388667	0.194334	+	0.980936i	$0.437746\pi$
$242$	8.69175	0.558727
$243$	1.00000	0.0641500
$244$	−9.20988	−0.589602
$245$	0	0
$246$	−9.88388	−0.630173
$247$	−1.45774	−0.0927537
$248$	7.29809	0.463429
$249$	7.33579	0.464887
$250$	−10.0423	−0.635129
$251$	−10.5817	−0.667913	−0.333957	−	0.942588i	$-0.608384\pi$
$251$	−10.5817	−0.667913	−0.333957	+	0.942588i	$0.608384\pi$
$252$	0	0
$253$	8.77969	0.551974
$254$	−17.1651	−1.07703
$255$	6.77792	0.424450
$256$	1.00000	0.0625000
$257$	2.27512	0.141918	0.0709592	−	0.997479i	$-0.477394\pi$
$257$	2.27512	0.141918	0.0709592	+	0.997479i	$0.477394\pi$
$258$	1.77880	0.110743
$259$	0	0
$260$	1.16053	0.0719731
$261$	3.59771	0.222693
$262$	−12.9127	−0.797748
$263$	14.9853	0.924031	0.462016	−	0.886872i	$-0.347126\pi$
$263$	14.9853	0.924031	0.462016	+	0.886872i	$0.347126\pi$
$264$	−1.51929	−0.0935060
$265$	−11.1256	−0.683442
$266$	0	0
$267$	9.20406	0.563279
$268$	4.07842	0.249129
$269$	4.34656	0.265014	0.132507	−	0.991182i	$-0.457697\pi$
$269$	4.34656	0.265014	0.132507	+	0.991182i	$0.457697\pi$
$270$	1.16053	0.0706277
$271$	5.32195	0.323285	0.161643	−	0.986849i	$-0.448321\pi$
$271$	5.32195	0.323285	0.161643	+	0.986849i	$0.448321\pi$
$272$	−5.84035	−0.354124
$273$	0	0
$274$	−10.4348	−0.630387
$275$	−5.55023	−0.334691
$276$	5.77880	0.347843
$277$	−22.0052	−1.32216	−0.661082	−	0.750314i	$-0.729902\pi$
$277$	−22.0052	−1.32216	−0.661082	+	0.750314i	$0.729902\pi$
$278$	−20.7071	−1.24193
$279$	−7.29809	−0.436925
$280$	0	0
$281$	21.6348	1.29062	0.645311	−	0.763920i	$-0.276728\pi$
$281$	21.6348	1.29062	0.645311	+	0.763920i	$0.276728\pi$
$282$	−11.5867	−0.689976
$283$	16.2530	0.966143	0.483072	−	0.875581i	$-0.339521\pi$
$283$	16.2530	0.966143	0.483072	+	0.875581i	$0.339521\pi$
$284$	−0.358761	−0.0212885
$285$	−1.69175	−0.100211
$286$	1.51929	0.0898376
$287$	0	0
$288$	−1.00000	−0.0589256
$289$	17.1097	1.00646
$290$	4.17526	0.245180
$291$	−5.19543	−0.304561
$292$	−3.53491	−0.206865
$293$	4.43020	0.258815	0.129407	−	0.991592i	$-0.458693\pi$
$293$	4.43020	0.258815	0.129407	+	0.991592i	$0.458693\pi$
$294$	0	0
$295$	0.0181218	0.00105509
$296$	−1.95648	−0.113718
$297$	1.51929	0.0881583
$298$	2.50736	0.145248
$299$	−5.77880	−0.334197
$300$	−3.65317	−0.210916
$301$	0	0
$302$	−1.67614	−0.0964508
$303$	−12.0273	−0.690949
$304$	1.45774	0.0836070
$305$	10.6884	0.612014
$306$	5.84035	0.333871
$307$	3.75611	0.214372	0.107186	−	0.994239i	$-0.465816\pi$
$307$	3.75611	0.214372	0.107186	+	0.994239i	$0.465816\pi$
$308$	0	0
$309$	7.81128	0.444368
$310$	−8.46967	−0.481045
$311$	−28.1348	−1.59538	−0.797688	−	0.603070i	$-0.793944\pi$
$311$	−28.1348	−1.59538	−0.797688	+	0.603070i	$0.793944\pi$
$312$	1.00000	0.0566139
$313$	−33.5490	−1.89630	−0.948150	−	0.317823i	$-0.897048\pi$
$313$	−33.5490	−1.89630	−0.948150	+	0.317823i	$0.897048\pi$
$314$	10.8551	0.612588
$315$	0	0
$316$	10.2374	0.575900
$317$	16.4714	0.925128	0.462564	−	0.886586i	$-0.346930\pi$
$317$	16.4714	0.925128	0.462564	+	0.886586i	$0.346930\pi$
$318$	−9.58667	−0.537594
$319$	5.46598	0.306036
$320$	−1.16053	−0.0648757
$321$	−5.10480	−0.284922
$322$	0	0
$323$	−8.51371	−0.473716
$324$	1.00000	0.0555556
$325$	3.65317	0.202641
$326$	−8.00699	−0.443466
$327$	6.20037	0.342881
$328$	−9.88388	−0.545746
$329$	0	0
$330$	1.76319	0.0970602
$331$	−1.98477	−0.109093	−0.0545465	−	0.998511i	$-0.517371\pi$
$331$	−1.98477	−0.109093	−0.0545465	+	0.998511i	$0.517371\pi$
$332$	7.33579	0.402604
$333$	1.95648	0.107214
$334$	−0.339091	−0.0185543
$335$	−4.73314	−0.258599
$336$	0	0
$337$	19.3707	1.05519	0.527594	−	0.849496i	$-0.323094\pi$
$337$	19.3707	1.05519	0.527594	+	0.849496i	$0.323094\pi$
$338$	−1.00000	−0.0543928
$339$	−17.0659	−0.926890
$340$	6.77792	0.367584
$341$	−11.0879	−0.600445
$342$	−1.45774	−0.0788255
$343$	0	0
$344$	1.77880	0.0959065
$345$	−6.70648	−0.361065
$346$	2.22577	0.119658
$347$	15.3921	0.826293	0.413146	−	0.910665i	$-0.364430\pi$
$347$	15.3921	0.826293	0.413146	+	0.910665i	$0.364430\pi$
$348$	3.59771	0.192858
$349$	−29.0585	−1.55546	−0.777732	−	0.628596i	$-0.783630\pi$
$349$	−29.0585	−1.55546	−0.777732	+	0.628596i	$0.783630\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	−1.51929	−0.0809785
$353$	−17.8955	−0.952483	−0.476242	−	0.879314i	$-0.658001\pi$
$353$	−17.8955	−0.952483	−0.476242	+	0.879314i	$0.658001\pi$
$354$	0.0156151	0.000829931 0
$355$	0.416353	0.0220977
$356$	9.20406	0.487814
$357$	0	0
$358$	6.45280	0.341041
$359$	−9.71445	−0.512709	−0.256354	−	0.966583i	$-0.582521\pi$
$359$	−9.71445	−0.512709	−0.256354	+	0.966583i	$0.582521\pi$
$360$	1.16053	0.0611654
$361$	−16.8750	−0.888158
$362$	−0.0238555	−0.00125382
$363$	−8.69175	−0.456199
$364$	0	0
$365$	4.10237	0.214728
$366$	9.20988	0.481408
$367$	−4.84530	−0.252922	−0.126461	−	0.991972i	$-0.540362\pi$
$367$	−4.84530	−0.252922	−0.126461	+	0.991972i	$0.540362\pi$
$368$	5.77880	0.301241
$369$	9.88388	0.514534
$370$	2.27055	0.118040
$371$	0	0
$372$	−7.29809	−0.378389
$373$	15.1001	0.781856	0.390928	−	0.920421i	$-0.372154\pi$
$373$	15.1001	0.781856	0.390928	+	0.920421i	$0.372154\pi$
$374$	8.87321	0.458823
$375$	10.0423	0.518581
$376$	−11.5867	−0.597537
$377$	−3.59771	−0.185292
$378$	0	0
$379$	7.30508	0.375237	0.187618	−	0.982242i	$-0.439923\pi$
$379$	7.30508	0.375237	0.187618	+	0.982242i	$0.439923\pi$
$380$	−1.69175	−0.0867850
$381$	17.1651	0.879395
$382$	−9.20164	−0.470797
$383$	−8.36800	−0.427585	−0.213792	−	0.976879i	$-0.568582\pi$
$383$	−8.36800	−0.427585	−0.213792	+	0.976879i	$0.568582\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	−23.2522	−1.18350
$387$	−1.77880	−0.0904216
$388$	−5.19543	−0.263758
$389$	11.8042	0.598497	0.299248	−	0.954175i	$-0.403264\pi$
$389$	11.8042	0.598497	0.299248	+	0.954175i	$0.403264\pi$
$390$	−1.16053	−0.0587658
$391$	−33.7503	−1.70682
$392$	0	0
$393$	12.9127	0.651358
$394$	−21.7521	−1.09586
$395$	−11.8809	−0.597791
$396$	1.51929	0.0763473
$397$	6.64950	0.333729	0.166864	−	0.985980i	$-0.446636\pi$
$397$	6.64950	0.333729	0.166864	+	0.985980i	$0.446636\pi$
$398$	17.4902	0.876706
$399$	0	0
$400$	−3.65317	−0.182658
$401$	−2.59619	−0.129647	−0.0648237	−	0.997897i	$-0.520649\pi$
$401$	−2.59619	−0.129647	−0.0648237	+	0.997897i	$0.520649\pi$
$402$	−4.07842	−0.203413
$403$	7.29809	0.363544
$404$	−12.0273	−0.598379
$405$	−1.16053	−0.0576673
$406$	0	0
$407$	2.97246	0.147339
$408$	5.84035	0.289141
$409$	−9.20824	−0.455318	−0.227659	−	0.973741i	$-0.573107\pi$
$409$	−9.20824	−0.455318	−0.227659	+	0.973741i	$0.573107\pi$
$410$	11.4706	0.566490
$411$	10.4348	0.514709
$412$	7.81128	0.384834
$413$	0	0
$414$	−5.77880	−0.284013
$415$	−8.51342	−0.417907
$416$	1.00000	0.0490290
$417$	20.7071	1.01403
$418$	−2.21473	−0.108326
$419$	10.5383	0.514830	0.257415	−	0.966301i	$-0.417129\pi$
$419$	10.5383	0.514830	0.257415	+	0.966301i	$0.417129\pi$
$420$	0	0
$421$	17.6400	0.859720	0.429860	−	0.902895i	$-0.358563\pi$
$421$	17.6400	0.859720	0.429860	+	0.902895i	$0.358563\pi$
$422$	−4.87070	−0.237102
$423$	11.5867	0.563363
$424$	−9.58667	−0.465570
$425$	21.3358	1.03494
$426$	0.358761	0.0173820
$427$	0	0
$428$	−5.10480	−0.246750
$429$	−1.51929	−0.0733521
$430$	−2.06435	−0.0995520
$431$	−9.42944	−0.454200	−0.227100	−	0.973871i	$-0.572924\pi$
$431$	−9.42944	−0.454200	−0.227100	+	0.973871i	$0.572924\pi$
$432$	1.00000	0.0481125
$433$	15.4316	0.741595	0.370798	−	0.928714i	$-0.379084\pi$
$433$	15.4316	0.741595	0.370798	+	0.928714i	$0.379084\pi$
$434$	0	0
$435$	−4.17526	−0.200188
$436$	6.20037	0.296944
$437$	8.42398	0.402974
$438$	3.53491	0.168904
$439$	−27.0938	−1.29311	−0.646557	−	0.762865i	$-0.723792\pi$
$439$	−27.0938	−1.29311	−0.646557	+	0.762865i	$0.723792\pi$
$440$	1.76319	0.0840566
$441$	0	0
$442$	−5.84035	−0.277797
$443$	34.8250	1.65458	0.827292	−	0.561771i	$-0.189880\pi$
$443$	34.8250	1.65458	0.827292	+	0.561771i	$0.189880\pi$
$444$	1.95648	0.0928502
$445$	−10.6816	−0.506356
$446$	−8.01165	−0.379363
$447$	−2.50736	−0.118594
$448$	0	0
$449$	−20.9556	−0.988954	−0.494477	−	0.869191i	$-0.664640\pi$
$449$	−20.9556	−0.988954	−0.494477	+	0.869191i	$0.664640\pi$
$450$	3.65317	0.172212
$451$	15.0165	0.707099
$452$	−17.0659	−0.802710
$453$	1.67614	0.0787518
$454$	−23.8532	−1.11948
$455$	0	0
$456$	−1.45774	−0.0682649
$457$	37.8171	1.76901	0.884505	−	0.466531i	$-0.154496\pi$
$457$	37.8171	1.76901	0.884505	+	0.466531i	$0.154496\pi$
$458$	14.4935	0.677238
$459$	−5.84035	−0.272604
$460$	−6.70648	−0.312691
$461$	24.0146	1.11847	0.559235	−	0.829009i	$-0.311095\pi$
$461$	24.0146	1.11847	0.559235	+	0.829009i	$0.311095\pi$
$462$	0	0
$463$	17.8594	0.829996	0.414998	−	0.909822i	$-0.363782\pi$
$463$	17.8594	0.829996	0.414998	+	0.909822i	$0.363782\pi$
$464$	3.59771	0.167020
$465$	8.46967	0.392771
$466$	−17.0723	−0.790860
$467$	36.2355	1.67678	0.838390	−	0.545070i	$-0.183497\pi$
$467$	36.2355	1.67678	0.838390	+	0.545070i	$0.183497\pi$
$468$	−1.00000	−0.0462250
$469$	0	0
$470$	13.4467	0.620250
$471$	−10.8551	−0.500176
$472$	0.0156151	0.000718741 0
$473$	−2.70252	−0.124262
$474$	−10.2374	−0.470221
$475$	−5.32536	−0.244344
$476$	0	0
$477$	9.58667	0.438944
$478$	19.8943	0.909943
$479$	23.8606	1.09022	0.545110	−	0.838364i	$-0.316488\pi$
$479$	23.8606	1.09022	0.545110	+	0.838364i	$0.316488\pi$
$480$	1.16053	0.0529708
$481$	−1.95648	−0.0892076
$482$	−6.03374	−0.274829
$483$	0	0
$484$	−8.69175	−0.395080
$485$	6.02946	0.273784
$486$	−1.00000	−0.0453609
$487$	−34.0551	−1.54318	−0.771592	−	0.636118i	$-0.780539\pi$
$487$	−34.0551	−1.54318	−0.771592	+	0.636118i	$0.780539\pi$
$488$	9.20988	0.416912
$489$	8.00699	0.362088
$490$	0	0
$491$	−5.02422	−0.226740	−0.113370	−	0.993553i	$-0.536165\pi$
$491$	−5.02422	−0.226740	−0.113370	+	0.993553i	$0.536165\pi$
$492$	9.88388	0.445600
$493$	−21.0119	−0.946330
$494$	1.45774	0.0655868
$495$	−1.76319	−0.0792493
$496$	−7.29809	−0.327694
$497$	0	0
$498$	−7.33579	−0.328725
$499$	24.1863	1.08273	0.541363	−	0.840789i	$-0.317908\pi$
$499$	24.1863	1.08273	0.541363	+	0.840789i	$0.317908\pi$
$500$	10.0423	0.449104
$501$	0.339091	0.0151495
$502$	10.5817	0.472286
$503$	−31.6915	−1.41305	−0.706527	−	0.707686i	$-0.749739\pi$
$503$	−31.6915	−1.41305	−0.706527	+	0.707686i	$0.749739\pi$
$504$	0	0
$505$	13.9580	0.621124
$506$	−8.77969	−0.390305
$507$	1.00000	0.0444116
$508$	17.1651	0.761578
$509$	36.1410	1.60192	0.800960	−	0.598717i	$-0.204323\pi$
$509$	36.1410	1.60192	0.800960	+	0.598717i	$0.204323\pi$
$510$	−6.77792	−0.300131
$511$	0	0
$512$	−1.00000	−0.0441942
$513$	1.45774	0.0643607
$514$	−2.27512	−0.100351
$515$	−9.06524	−0.399462
$516$	−1.77880	−0.0783074
$517$	17.6035	0.774203
$518$	0	0
$519$	−2.22577	−0.0977006
$520$	−1.16053	−0.0508927
$521$	11.5089	0.504212	0.252106	−	0.967700i	$-0.418877\pi$
$521$	11.5089	0.504212	0.252106	+	0.967700i	$0.418877\pi$
$522$	−3.59771	−0.157468
$523$	−12.6798	−0.554450	−0.277225	−	0.960805i	$-0.589415\pi$
$523$	−12.6798	−0.554450	−0.277225	+	0.960805i	$0.589415\pi$
$524$	12.9127	0.564093
$525$	0	0
$526$	−14.9853	−0.653389
$527$	42.6235	1.85671
$528$	1.51929	0.0661187
$529$	10.3945	0.451937
$530$	11.1256	0.483267
$531$	−0.0156151	−0.000677636 0
$532$	0	0
$533$	−9.88388	−0.428118
$534$	−9.20406	−0.398298
$535$	5.92429	0.256129
$536$	−4.07842	−0.176161
$537$	−6.45280	−0.278459
$538$	−4.34656	−0.187393
$539$	0	0
$540$	−1.16053	−0.0499413
$541$	−4.65014	−0.199925	−0.0999626	−	0.994991i	$-0.531872\pi$
$541$	−4.65014	−0.199925	−0.0999626	+	0.994991i	$0.531872\pi$
$542$	−5.32195	−0.228597
$543$	0.0238555	0.00102374
$544$	5.84035	0.250403
$545$	−7.19572	−0.308231
$546$	0	0
$547$	17.7916	0.760715	0.380357	−	0.924840i	$-0.375801\pi$
$547$	17.7916	0.760715	0.380357	+	0.924840i	$0.375801\pi$
$548$	10.4348	0.445751
$549$	−9.20988	−0.393068
$550$	5.55023	0.236663
$551$	5.24453	0.223424
$552$	−5.77880	−0.245962
$553$	0	0
$554$	22.0052	0.934911
$555$	−2.27055	−0.0963795
$556$	20.7071	0.878176
$557$	30.6373	1.29814	0.649072	−	0.760727i	$-0.275157\pi$
$557$	30.6373	1.29814	0.649072	+	0.760727i	$0.275157\pi$
$558$	7.29809	0.308953
$559$	1.77880	0.0752353
$560$	0	0
$561$	−8.87321	−0.374627
$562$	−21.6348	−0.912608
$563$	−18.3741	−0.774375	−0.387188	−	0.922001i	$-0.626553\pi$
$563$	−18.3741	−0.774375	−0.387188	+	0.922001i	$0.626553\pi$
$564$	11.5867	0.487887
$565$	19.8055	0.833222
$566$	−16.2530	−0.683166
$567$	0	0
$568$	0.358761	0.0150533
$569$	−17.9394	−0.752057	−0.376028	−	0.926608i	$-0.622710\pi$
$569$	−17.9394	−0.752057	−0.376028	+	0.926608i	$0.622710\pi$
$570$	1.69175	0.0708597
$571$	−18.6495	−0.780457	−0.390229	−	0.920718i	$-0.627604\pi$
$571$	−18.6495	−0.780457	−0.390229	+	0.920718i	$0.627604\pi$
$572$	−1.51929	−0.0635248
$573$	9.20164	0.384404
$574$	0	0
$575$	−21.1109	−0.880386
$576$	1.00000	0.0416667
$577$	−36.3523	−1.51337	−0.756683	−	0.653782i	$-0.773181\pi$
$577$	−36.3523	−1.51337	−0.756683	+	0.653782i	$0.773181\pi$
$578$	−17.1097	−0.711672
$579$	23.2522	0.966327
$580$	−4.17526	−0.173368
$581$	0	0
$582$	5.19543	0.215357
$583$	14.5650	0.603219
$584$	3.53491	0.146276
$585$	1.16053	0.0479821
$586$	−4.43020	−0.183010
$587$	−18.7622	−0.774398	−0.387199	−	0.921996i	$-0.626557\pi$
$587$	−18.7622	−0.774398	−0.387199	+	0.921996i	$0.626557\pi$
$588$	0	0
$589$	−10.6387	−0.438360
$590$	−0.0181218	−0.000746061 0
$591$	21.7521	0.894764
$592$	1.95648	0.0804106
$593$	−12.0975	−0.496783	−0.248391	−	0.968660i	$-0.579902\pi$
$593$	−12.0975	−0.496783	−0.248391	+	0.968660i	$0.579902\pi$
$594$	−1.51929	−0.0623373
$595$	0	0
$596$	−2.50736	−0.102706
$597$	−17.4902	−0.715827
$598$	5.77880	0.236313
$599$	−19.3009	−0.788613	−0.394307	−	0.918979i	$-0.629015\pi$
$599$	−19.3009	−0.788613	−0.394307	+	0.918979i	$0.629015\pi$
$600$	3.65317	0.149140
$601$	17.6854	0.721402	0.360701	−	0.932682i	$-0.382538\pi$
$601$	17.6854	0.721402	0.360701	+	0.932682i	$0.382538\pi$
$602$	0	0
$603$	4.07842	0.166086
$604$	1.67614	0.0682010
$605$	10.0870	0.410097
$606$	12.0273	0.488574
$607$	33.1514	1.34557	0.672786	−	0.739837i	$-0.265097\pi$
$607$	33.1514	1.34557	0.672786	+	0.739837i	$0.265097\pi$
$608$	−1.45774	−0.0591191
$609$	0	0
$610$	−10.6884	−0.432759
$611$	−11.5867	−0.468747
$612$	−5.84035	−0.236082
$613$	−28.6489	−1.15712	−0.578558	−	0.815641i	$-0.696384\pi$
$613$	−28.6489	−1.15712	−0.578558	+	0.815641i	$0.696384\pi$
$614$	−3.75611	−0.151584
$615$	−11.4706	−0.462537
$616$	0	0
$617$	−22.2380	−0.895270	−0.447635	−	0.894216i	$-0.647734\pi$
$617$	−22.2380	−0.895270	−0.447635	+	0.894216i	$0.647734\pi$
$618$	−7.81128	−0.314216
$619$	38.5778	1.55057	0.775286	−	0.631611i	$-0.217606\pi$
$619$	38.5778	1.55057	0.775286	+	0.631611i	$0.217606\pi$
$620$	8.46967	0.340150
$621$	5.77880	0.231895
$622$	28.1348	1.12810
$623$	0	0
$624$	−1.00000	−0.0400320
$625$	6.61146	0.264459
$626$	33.5490	1.34089
$627$	2.21473	0.0884478
$628$	−10.8551	−0.433165
$629$	−11.4265	−0.455605
$630$	0	0
$631$	43.5419	1.73337	0.866687	−	0.498852i	$-0.166245\pi$
$631$	43.5419	1.73337	0.866687	+	0.498852i	$0.166245\pi$
$632$	−10.2374	−0.407223
$633$	4.87070	0.193593
$634$	−16.4714	−0.654164
$635$	−19.9206	−0.790527
$636$	9.58667	0.380136
$637$	0	0
$638$	−5.46598	−0.216400
$639$	−0.358761	−0.0141924
$640$	1.16053	0.0458740
$641$	12.2184	0.482598	0.241299	−	0.970451i	$-0.422427\pi$
$641$	12.2184	0.482598	0.241299	+	0.970451i	$0.422427\pi$
$642$	5.10480	0.201471
$643$	−26.4417	−1.04276	−0.521380	−	0.853325i	$-0.674583\pi$
$643$	−26.4417	−1.04276	−0.521380	+	0.853325i	$0.674583\pi$
$644$	0	0
$645$	2.06435	0.0812839
$646$	8.51371	0.334967
$647$	8.39591	0.330077	0.165039	−	0.986287i	$-0.447225\pi$
$647$	8.39591	0.330077	0.165039	+	0.986287i	$0.447225\pi$
$648$	−1.00000	−0.0392837
$649$	−0.0237238	−0.000931242 0
$650$	−3.65317	−0.143289
$651$	0	0
$652$	8.00699	0.313578
$653$	−23.3820	−0.915007	−0.457503	−	0.889208i	$-0.651256\pi$
$653$	−23.3820	−0.915007	−0.457503	+	0.889208i	$0.651256\pi$
$654$	−6.20037	−0.242454
$655$	−14.9856	−0.585534
$656$	9.88388	0.385901
$657$	−3.53491	−0.137910
$658$	0	0
$659$	−45.7625	−1.78266	−0.891328	−	0.453359i	$-0.850225\pi$
$659$	−45.7625	−1.78266	−0.891328	+	0.453359i	$0.850225\pi$
$660$	−1.76319	−0.0686319
$661$	0.169046	0.00657512	0.00328756	−	0.999995i	$-0.498954\pi$
$661$	0.169046	0.00657512	0.00328756	+	0.999995i	$0.498954\pi$
$662$	1.98477	0.0771404
$663$	5.84035	0.226821
$664$	−7.33579	−0.284684
$665$	0	0
$666$	−1.95648	−0.0758119
$667$	20.7905	0.805010
$668$	0.339091	0.0131198
$669$	8.01165	0.309748
$670$	4.73314	0.182857
$671$	−13.9925	−0.540175
$672$	0	0
$673$	47.1905	1.81906	0.909530	−	0.415639i	$-0.136442\pi$
$673$	47.1905	1.81906	0.909530	+	0.415639i	$0.136442\pi$
$674$	−19.3707	−0.746131
$675$	−3.65317	−0.140610
$676$	1.00000	0.0384615
$677$	−32.7187	−1.25748	−0.628742	−	0.777614i	$-0.716430\pi$
$677$	−32.7187	−1.25748	−0.628742	+	0.777614i	$0.716430\pi$
$678$	17.0659	0.655410
$679$	0	0
$680$	−6.77792	−0.259921
$681$	23.8532	0.914055
$682$	11.0879	0.424579
$683$	−17.9035	−0.685060	−0.342530	−	0.939507i	$-0.611284\pi$
$683$	−17.9035	−0.685060	−0.342530	+	0.939507i	$0.611284\pi$
$684$	1.45774	0.0557380
$685$	−12.1099	−0.462695
$686$	0	0
$687$	−14.4935	−0.552962
$688$	−1.77880	−0.0678162
$689$	−9.58667	−0.365223
$690$	6.70648	0.255311
$691$	−7.03858	−0.267760	−0.133880	−	0.990998i	$-0.542744\pi$
$691$	−7.03858	−0.267760	−0.133880	+	0.990998i	$0.542744\pi$
$692$	−2.22577	−0.0846112
$693$	0	0
$694$	−15.3921	−0.584277
$695$	−24.0312	−0.911557
$696$	−3.59771	−0.136371
$697$	−57.7254	−2.18650
$698$	29.0585	1.09988
$699$	17.0723	0.645734
$700$	0	0
$701$	−36.7808	−1.38919	−0.694596	−	0.719400i	$-0.744417\pi$
$701$	−36.7808	−1.38919	−0.694596	+	0.719400i	$0.744417\pi$
$702$	1.00000	0.0377426
$703$	2.85203	0.107566
$704$	1.51929	0.0572605
$705$	−13.4467	−0.506432
$706$	17.8955	0.673507
$707$	0	0
$708$	−0.0156151	−0.000586850 0
$709$	−23.0021	−0.863863	−0.431932	−	0.901906i	$-0.642168\pi$
$709$	−23.0021	−0.863863	−0.431932	+	0.901906i	$0.642168\pi$
$710$	−0.416353	−0.0156255
$711$	10.2374	0.383933
$712$	−9.20406	−0.344937
$713$	−42.1742	−1.57944
$714$	0	0
$715$	1.76319	0.0659394
$716$	−6.45280	−0.241152
$717$	−19.8943	−0.742966
$718$	9.71445	0.362540
$719$	−4.66817	−0.174093	−0.0870467	−	0.996204i	$-0.527743\pi$
$719$	−4.66817	−0.174093	−0.0870467	+	0.996204i	$0.527743\pi$
$720$	−1.16053	−0.0432505
$721$	0	0
$722$	16.8750	0.628022
$723$	6.03374	0.224397
$724$	0.0238555	0.000886584 0
$725$	−13.1431	−0.488121
$726$	8.69175	0.322581
$727$	−25.6085	−0.949767	−0.474883	−	0.880049i	$-0.657510\pi$
$727$	−25.6085	−0.949767	−0.474883	+	0.880049i	$0.657510\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	−4.10237	−0.151836
$731$	10.3888	0.384245
$732$	−9.20988	−0.340407
$733$	−1.84962	−0.0683174	−0.0341587	−	0.999416i	$-0.510875\pi$
$733$	−1.84962	−0.0683174	−0.0341587	+	0.999416i	$0.510875\pi$
$734$	4.84530	0.178843
$735$	0	0
$736$	−5.77880	−0.213009
$737$	6.19631	0.228244
$738$	−9.88388	−0.363831
$739$	−23.9488	−0.880973	−0.440486	−	0.897759i	$-0.645194\pi$
$739$	−23.9488	−0.880973	−0.440486	+	0.897759i	$0.645194\pi$
$740$	−2.27055	−0.0834671
$741$	−1.45774	−0.0535514
$742$	0	0
$743$	−10.4754	−0.384305	−0.192153	−	0.981365i	$-0.561547\pi$
$743$	−10.4754	−0.384305	−0.192153	+	0.981365i	$0.561547\pi$
$744$	7.29809	0.267561
$745$	2.90988	0.106610
$746$	−15.1001	−0.552855
$747$	7.33579	0.268403
$748$	−8.87321	−0.324437
$749$	0	0
$750$	−10.0423	−0.366692
$751$	−23.5849	−0.860624	−0.430312	−	0.902680i	$-0.641596\pi$
$751$	−23.5849	−0.860624	−0.430312	+	0.902680i	$0.641596\pi$
$752$	11.5867	0.422522
$753$	−10.5817	−0.385620
$754$	3.59771	0.131021
$755$	−1.94521	−0.0707934
$756$	0	0
$757$	−18.6602	−0.678218	−0.339109	−	0.940747i	$-0.610126\pi$
$757$	−18.6602	−0.678218	−0.339109	+	0.940747i	$0.610126\pi$
$758$	−7.30508	−0.265332
$759$	8.77969	0.318682
$760$	1.69175	0.0613663
$761$	27.1786	0.985222	0.492611	−	0.870250i	$-0.336043\pi$
$761$	27.1786	0.985222	0.492611	+	0.870250i	$0.336043\pi$
$762$	−17.1651	−0.621826
$763$	0	0
$764$	9.20164	0.332904
$765$	6.77792	0.245056
$766$	8.36800	0.302348
$767$	0.0156151	0.000563827 0
$768$	1.00000	0.0360844
$769$	30.8339	1.11190	0.555949	−	0.831217i	$-0.312355\pi$
$769$	30.8339	1.11190	0.555949	+	0.831217i	$0.312355\pi$
$770$	0	0
$771$	2.27512	0.0819366
$772$	23.2522	0.836863
$773$	14.0245	0.504425	0.252213	−	0.967672i	$-0.418842\pi$
$773$	14.0245	0.504425	0.252213	+	0.967672i	$0.418842\pi$
$774$	1.77880	0.0639377
$775$	26.6612	0.957697
$776$	5.19543	0.186505
$777$	0	0
$778$	−11.8042	−0.423201
$779$	14.4081	0.516224
$780$	1.16053	0.0415537
$781$	−0.545062	−0.0195039
$782$	33.7503	1.20691
$783$	3.59771	0.128572
$784$	0	0
$785$	12.5977	0.449630
$786$	−12.9127	−0.460580
$787$	24.7524	0.882328	0.441164	−	0.897426i	$-0.354566\pi$
$787$	24.7524	0.882328	0.441164	+	0.897426i	$0.354566\pi$
$788$	21.7521	0.774888
$789$	14.9853	0.533490
$790$	11.8809	0.422702
$791$	0	0
$792$	−1.51929	−0.0539857
$793$	9.20988	0.327053
$794$	−6.64950	−0.235982
$795$	−11.1256	−0.394586
$796$	−17.4902	−0.619924
$797$	−17.5536	−0.621782	−0.310891	−	0.950446i	$-0.600627\pi$
$797$	−17.5536	−0.621782	−0.310891	+	0.950446i	$0.600627\pi$
$798$	0	0
$799$	−67.6703	−2.39400
$800$	3.65317	0.129159
$801$	9.20406	0.325209
$802$	2.59619	0.0916745
$803$	−5.37056	−0.189523
$804$	4.07842	0.143835
$805$	0	0
$806$	−7.29809	−0.257064
$807$	4.34656	0.153006
$808$	12.0273	0.423118
$809$	−0.153431	−0.00539434	−0.00269717	−	0.999996i	$-0.500859\pi$
$809$	−0.153431	−0.00539434	−0.00269717	+	0.999996i	$0.500859\pi$
$810$	1.16053	0.0407769
$811$	−41.7585	−1.46634	−0.733169	−	0.680046i	$-0.761960\pi$
$811$	−41.7585	−1.46634	−0.733169	+	0.680046i	$0.761960\pi$
$812$	0	0
$813$	5.32195	0.186649
$814$	−2.97246	−0.104185
$815$	−9.29236	−0.325497
$816$	−5.84035	−0.204453
$817$	−2.59303	−0.0907185
$818$	9.20824	0.321959
$819$	0	0
$820$	−11.4706	−0.400569
$821$	−50.4865	−1.76199	−0.880996	−	0.473124i	$-0.843126\pi$
$821$	−50.4865	−1.76199	−0.880996	+	0.473124i	$0.843126\pi$
$822$	−10.4348	−0.363954
$823$	−52.6062	−1.83374	−0.916868	−	0.399190i	$-0.869292\pi$
$823$	−52.6062	−1.83374	−0.916868	+	0.399190i	$0.869292\pi$
$824$	−7.81128	−0.272119
$825$	−5.55023	−0.193234
$826$	0	0
$827$	2.35509	0.0818946	0.0409473	−	0.999161i	$-0.486962\pi$
$827$	2.35509	0.0818946	0.0409473	+	0.999161i	$0.486962\pi$
$828$	5.77880	0.200827
$829$	−1.43169	−0.0497248	−0.0248624	−	0.999691i	$-0.507915\pi$
$829$	−1.43169	−0.0497248	−0.0248624	+	0.999691i	$0.507915\pi$
$830$	8.51342	0.295505
$831$	−22.0052	−0.763351
$832$	−1.00000	−0.0346688
$833$	0	0
$834$	−20.7071	−0.717028
$835$	−0.393526	−0.0136185
$836$	2.21473	0.0765981
$837$	−7.29809	−0.252259
$838$	−10.5383	−0.364040
$839$	22.5382	0.778105	0.389053	−	0.921216i	$-0.372802\pi$
$839$	22.5382	0.778105	0.389053	+	0.921216i	$0.372802\pi$
$840$	0	0
$841$	−16.0565	−0.553671
$842$	−17.6400	−0.607914
$843$	21.6348	0.745141
$844$	4.87070	0.167656
$845$	−1.16053	−0.0399235
$846$	−11.5867	−0.398358
$847$	0	0
$848$	9.58667	0.329208
$849$	16.2530	0.557803
$850$	−21.3358	−0.731812
$851$	11.3061	0.387567
$852$	−0.358761	−0.0122909
$853$	−4.09925	−0.140356	−0.0701779	−	0.997534i	$-0.522357\pi$
$853$	−4.09925	−0.140356	−0.0701779	+	0.997534i	$0.522357\pi$
$854$	0	0
$855$	−1.69175	−0.0578567
$856$	5.10480	0.174479
$857$	−11.8622	−0.405204	−0.202602	−	0.979261i	$-0.564940\pi$
$857$	−11.8622	−0.405204	−0.202602	+	0.979261i	$0.564940\pi$
$858$	1.51929	0.0518678
$859$	44.5980	1.52166	0.760831	−	0.648950i	$-0.224791\pi$
$859$	44.5980	1.52166	0.760831	+	0.648950i	$0.224791\pi$
$860$	2.06435	0.0703939
$861$	0	0
$862$	9.42944	0.321168
$863$	9.83022	0.334625	0.167312	−	0.985904i	$-0.446491\pi$
$863$	9.83022	0.334625	0.167312	+	0.985904i	$0.446491\pi$
$864$	−1.00000	−0.0340207
$865$	2.58308	0.0878273
$866$	−15.4316	−0.524387
$867$	17.1097	0.581077
$868$	0	0
$869$	15.5536	0.527621
$870$	4.17526	0.141555
$871$	−4.07842	−0.138192
$872$	−6.20037	−0.209971
$873$	−5.19543	−0.175839
$874$	−8.42398	−0.284945
$875$	0	0
$876$	−3.53491	−0.119433
$877$	−38.3415	−1.29470	−0.647350	−	0.762193i	$-0.724123\pi$
$877$	−38.3415	−1.29470	−0.647350	+	0.762193i	$0.724123\pi$
$878$	27.0938	0.914370
$879$	4.43020	0.149427
$880$	−1.76319	−0.0594370
$881$	−14.2467	−0.479983	−0.239992	−	0.970775i	$-0.577145\pi$
$881$	−14.2467	−0.479983	−0.239992	+	0.970775i	$0.577145\pi$
$882$	0	0
$883$	−54.1329	−1.82172	−0.910858	−	0.412720i	$-0.864579\pi$
$883$	−54.1329	−1.82172	−0.910858	+	0.412720i	$0.864579\pi$
$884$	5.84035	0.196432
$885$	0.0181218	0.000609156 0
$886$	−34.8250	−1.16997
$887$	8.85788	0.297419	0.148709	−	0.988881i	$-0.452488\pi$
$887$	8.85788	0.297419	0.148709	+	0.988881i	$0.452488\pi$
$888$	−1.95648	−0.0656550
$889$	0	0
$890$	10.6816	0.358048
$891$	1.51929	0.0508982
$892$	8.01165	0.268250
$893$	16.8903	0.565214
$894$	2.50736	0.0838588
$895$	7.48867	0.250319
$896$	0	0
$897$	−5.77880	−0.192949
$898$	20.9556	0.699296
$899$	−26.2565	−0.875702
$900$	−3.65317	−0.121772
$901$	−55.9896	−1.86528
$902$	−15.0165	−0.499995
$903$	0	0
$904$	17.0659	0.567602
$905$	−0.0276851	−0.000920284 0
$906$	−1.67614	−0.0556859
$907$	−13.1980	−0.438232	−0.219116	−	0.975699i	$-0.570317\pi$
$907$	−13.1980	−0.438232	−0.219116	+	0.975699i	$0.570317\pi$
$908$	23.8532	0.791595
$909$	−12.0273	−0.398919
$910$	0	0
$911$	−38.6427	−1.28029	−0.640146	−	0.768254i	$-0.721126\pi$
$911$	−38.6427	−1.28029	−0.640146	+	0.768254i	$0.721126\pi$
$912$	1.45774	0.0482705
$913$	11.1452	0.368853
$914$	−37.8171	−1.25088
$915$	10.6884	0.353346
$916$	−14.4935	−0.478879
$917$	0	0
$918$	5.84035	0.192760
$919$	−10.5294	−0.347334	−0.173667	−	0.984804i	$-0.555562\pi$
$919$	−10.5294	−0.347334	−0.173667	+	0.984804i	$0.555562\pi$
$920$	6.70648	0.221106
$921$	3.75611	0.123768
$922$	−24.0146	−0.790878
$923$	0.358761	0.0118088
$924$	0	0
$925$	−7.14733	−0.235003
$926$	−17.8594	−0.586896
$927$	7.81128	0.256556
$928$	−3.59771	−0.118101
$929$	33.7331	1.10675	0.553373	−	0.832933i	$-0.313340\pi$
$929$	33.7331	1.10675	0.553373	+	0.832933i	$0.313340\pi$
$930$	−8.46967	−0.277731
$931$	0	0
$932$	17.0723	0.559222
$933$	−28.1348	−0.921091
$934$	−36.2355	−1.18566
$935$	10.2976	0.336769
$936$	1.00000	0.0326860
$937$	6.05885	0.197934	0.0989669	−	0.995091i	$-0.468446\pi$
$937$	6.05885	0.197934	0.0989669	+	0.995091i	$0.468446\pi$
$938$	0	0
$939$	−33.5490	−1.09483
$940$	−13.4467	−0.438583
$941$	−48.1109	−1.56837	−0.784185	−	0.620527i	$-0.786919\pi$
$941$	−48.1109	−1.56837	−0.784185	+	0.620527i	$0.786919\pi$
$942$	10.8551	0.353678
$943$	57.1170	1.85998
$944$	−0.0156151	−0.000508227 0
$945$	0	0
$946$	2.70252	0.0878665
$947$	−46.8777	−1.52332	−0.761660	−	0.647977i	$-0.775615\pi$
$947$	−46.8777	−1.52332	−0.761660	+	0.647977i	$0.775615\pi$
$948$	10.2374	0.332496
$949$	3.53491	0.114748
$950$	5.32536	0.172778
$951$	16.4714	0.534123
$952$	0	0
$953$	−34.3401	−1.11238	−0.556192	−	0.831054i	$-0.687738\pi$
$953$	−34.3401	−1.11238	−0.556192	+	0.831054i	$0.687738\pi$
$954$	−9.58667	−0.310380
$955$	−10.6788	−0.345558
$956$	−19.8943	−0.643427
$957$	5.46598	0.176690
$958$	−23.8606	−0.770902
$959$	0	0
$960$	−1.16053	−0.0374560
$961$	22.2622	0.718134
$962$	1.95648	0.0630793
$963$	−5.10480	−0.164500
$964$	6.03374	0.194334
$965$	−26.9849	−0.868673
$966$	0	0
$967$	−11.9586	−0.384564	−0.192282	−	0.981340i	$-0.561589\pi$
$967$	−11.9586	−0.384564	−0.192282	+	0.981340i	$0.561589\pi$
$968$	8.69175	0.279363
$969$	−8.51371	−0.273500
$970$	−6.02946	−0.193594
$971$	−5.64697	−0.181220	−0.0906100	−	0.995886i	$-0.528882\pi$
$971$	−5.64697	−0.181220	−0.0906100	+	0.995886i	$0.528882\pi$
$972$	1.00000	0.0320750
$973$	0	0
$974$	34.0551	1.09120
$975$	3.65317	0.116995
$976$	−9.20988	−0.294801
$977$	54.2348	1.73512	0.867562	−	0.497329i	$-0.165686\pi$
$977$	54.2348	1.73512	0.867562	+	0.497329i	$0.165686\pi$
$978$	−8.00699	−0.256035
$979$	13.9837	0.446919
$980$	0	0
$981$	6.20037	0.197963
$982$	5.02422	0.160329
$983$	35.0175	1.11688	0.558442	−	0.829543i	$-0.311399\pi$
$983$	35.0175	1.11688	0.558442	+	0.829543i	$0.311399\pi$
$984$	−9.88388	−0.315087
$985$	−25.2440	−0.804342
$986$	21.0119	0.669156
$987$	0	0
$988$	−1.45774	−0.0463768
$989$	−10.2793	−0.326864
$990$	1.76319	0.0560377
$991$	−12.4835	−0.396552	−0.198276	−	0.980146i	$-0.563534\pi$
$991$	−12.4835	−0.396552	−0.198276	+	0.980146i	$0.563534\pi$
$992$	7.29809	0.231715
$993$	−1.98477	−0.0629848
$994$	0	0
$995$	20.2980	0.643488
$996$	7.33579	0.232443
$997$	−59.4282	−1.88211	−0.941054	−	0.338255i	$-0.890163\pi$
$997$	−59.4282	−1.88211	−0.941054	+	0.338255i	$0.890163\pi$
$998$	−24.1863	−0.765603
$999$	1.95648	0.0619001

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3822.2.a.by.1.1	yes	4
7.6	odd	2		3822.2.a.bx.1.4	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3822.2.a.bx.1.4	✓	4	7.6	odd	2
3822.2.a.by.1.1	yes	4	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:	\( N \)	\(=\)	\( 3822 = 2 \cdot 3 \cdot 7^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3822.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.16053 q^{5} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.16053 q^{5} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +1.16053 q^{10} +1.51929 q^{11} +1.00000 q^{12} -1.00000 q^{13} -1.16053 q^{15} +1.00000 q^{16} -5.84035 q^{17} -1.00000 q^{18} +1.45774 q^{19} -1.16053 q^{20} -1.51929 q^{22} +5.77880 q^{23} -1.00000 q^{24} -3.65317 q^{25} +1.00000 q^{26} +1.00000 q^{27} +3.59771 q^{29} +1.16053 q^{30} -7.29809 q^{31} -1.00000 q^{32} +1.51929 q^{33} +5.84035 q^{34} +1.00000 q^{36} +1.95648 q^{37} -1.45774 q^{38} -1.00000 q^{39} +1.16053 q^{40} +9.88388 q^{41} -1.77880 q^{43} +1.51929 q^{44} -1.16053 q^{45} -5.77880 q^{46} +11.5867 q^{47} +1.00000 q^{48} +3.65317 q^{50} -5.84035 q^{51} -1.00000 q^{52} +9.58667 q^{53} -1.00000 q^{54} -1.76319 q^{55} +1.45774 q^{57} -3.59771 q^{58} -0.0156151 q^{59} -1.16053 q^{60} -9.20988 q^{61} +7.29809 q^{62} +1.00000 q^{64} +1.16053 q^{65} -1.51929 q^{66} +4.07842 q^{67} -5.84035 q^{68} +5.77880 q^{69} -0.358761 q^{71} -1.00000 q^{72} -3.53491 q^{73} -1.95648 q^{74} -3.65317 q^{75} +1.45774 q^{76} +1.00000 q^{78} +10.2374 q^{79} -1.16053 q^{80} +1.00000 q^{81} -9.88388 q^{82} +7.33579 q^{83} +6.77792 q^{85} +1.77880 q^{86} +3.59771 q^{87} -1.51929 q^{88} +9.20406 q^{89} +1.16053 q^{90} +5.77880 q^{92} -7.29809 q^{93} -11.5867 q^{94} -1.69175 q^{95} -1.00000 q^{96} -5.19543 q^{97} +1.51929 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{2} + 4 q^{3} + 4 q^{4} + 6 q^{5} - 4 q^{6} - 4 q^{8} + 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^2 + 4 * q^3 + 4 * q^4 + 6 * q^5 - 4 * q^6 - 4 * q^8 + 4 * q^9	\( 4 q - 4 q^{2} + 4 q^{3} + 4 q^{4} + 6 q^{5} - 4 q^{6} - 4 q^{8} + 4 q^{9} - 6 q^{10} + 2 q^{11} + 4 q^{12} - 4 q^{13} + 6 q^{15} + 4 q^{16} + 2 q^{17} - 4 q^{18} + 2 q^{19} + 6 q^{20} - 2 q^{22} - 2 q^{23} - 4 q^{24} + 6 q^{25} + 4 q^{26} + 4 q^{27} + 6 q^{29} - 6 q^{30} - 4 q^{32} + 2 q^{33} - 2 q^{34} + 4 q^{36} + 6 q^{37} - 2 q^{38} - 4 q^{39} - 6 q^{40} + 16 q^{41} + 18 q^{43} + 2 q^{44} + 6 q^{45} + 2 q^{46} + 16 q^{47} + 4 q^{48} - 6 q^{50} + 2 q^{51} - 4 q^{52} + 8 q^{53} - 4 q^{54} + 2 q^{55} + 2 q^{57} - 6 q^{58} + 16 q^{59} + 6 q^{60} - 2 q^{61} + 4 q^{64} - 6 q^{65} - 2 q^{66} + 12 q^{67} + 2 q^{68} - 2 q^{69} - 8 q^{71} - 4 q^{72} + 6 q^{73} - 6 q^{74} + 6 q^{75} + 2 q^{76} + 4 q^{78} - 24 q^{79} + 6 q^{80} + 4 q^{81} - 16 q^{82} + 28 q^{83} + 38 q^{85} - 18 q^{86} + 6 q^{87} - 2 q^{88} + 28 q^{89} - 6 q^{90} - 2 q^{92} - 16 q^{94} + 22 q^{95} - 4 q^{96} - 4 q^{97} + 2 q^{99}+O(q^{100}) \) 4 * q - 4 * q^2 + 4 * q^3 + 4 * q^4 + 6 * q^5 - 4 * q^6 - 4 * q^8 + 4 * q^9 - 6 * q^10 + 2 * q^11 + 4 * q^12 - 4 * q^13 + 6 * q^15 + 4 * q^16 + 2 * q^17 - 4 * q^18 + 2 * q^19 + 6 * q^20 - 2 * q^22 - 2 * q^23 - 4 * q^24 + 6 * q^25 + 4 * q^26 + 4 * q^27 + 6 * q^29 - 6 * q^30 - 4 * q^32 + 2 * q^33 - 2 * q^34 + 4 * q^36 + 6 * q^37 - 2 * q^38 - 4 * q^39 - 6 * q^40 + 16 * q^41 + 18 * q^43 + 2 * q^44 + 6 * q^45 + 2 * q^46 + 16 * q^47 + 4 * q^48 - 6 * q^50 + 2 * q^51 - 4 * q^52 + 8 * q^53 - 4 * q^54 + 2 * q^55 + 2 * q^57 - 6 * q^58 + 16 * q^59 + 6 * q^60 - 2 * q^61 + 4 * q^64 - 6 * q^65 - 2 * q^66 + 12 * q^67 + 2 * q^68 - 2 * q^69 - 8 * q^71 - 4 * q^72 + 6 * q^73 - 6 * q^74 + 6 * q^75 + 2 * q^76 + 4 * q^78 - 24 * q^79 + 6 * q^80 + 4 * q^81 - 16 * q^82 + 28 * q^83 + 38 * q^85 - 18 * q^86 + 6 * q^87 - 2 * q^88 + 28 * q^89 - 6 * q^90 - 2 * q^92 - 16 * q^94 + 22 * q^95 - 4 * q^96 - 4 * q^97 + 2 * q^99

Introduction

L-functions

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Database

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