LMFDB - Embedded newform 6422.2.a.bc.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6422, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("6422.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6422 = 2 \cdot 13^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6422.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$51.2799281781$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.316645497.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 12x^{4} + 24x^{3} + 13x^{2} - 24x + 5 $ x^6 - 2x^5 - 12x^4 + 24x^3 + 13x^2 - 24*x + 5
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 494)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$3.23845$ of defining polynomial
Character	$\chi$	$=$	6422.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +3.23845 q^{3} +1.00000 q^{4} +0.399620 q^{5} -3.23845 q^{6} -1.11615 q^{7} -1.00000 q^{8} +7.48758 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +3.23845 q^{3} +1.00000 q^{4} +0.399620 q^{5} -3.23845 q^{6} -1.11615 q^{7} -1.00000 q^{8} +7.48758 q^{9} -0.399620 q^{10} +0.399620 q^{11} +3.23845 q^{12} +1.11615 q^{14} +1.29415 q^{15} +1.00000 q^{16} +5.00147 q^{17} -7.48758 q^{18} +1.00000 q^{19} +0.399620 q^{20} -3.61459 q^{21} -0.399620 q^{22} -0.891897 q^{23} -3.23845 q^{24} -4.84030 q^{25} +14.5328 q^{27} -1.11615 q^{28} +4.74618 q^{29} -1.29415 q^{30} +1.79858 q^{31} -1.00000 q^{32} +1.29415 q^{33} -5.00147 q^{34} -0.446034 q^{35} +7.48758 q^{36} +10.0733 q^{37} -1.00000 q^{38} -0.399620 q^{40} -8.00335 q^{41} +3.61459 q^{42} -0.468865 q^{43} +0.399620 q^{44} +2.99219 q^{45} +0.891897 q^{46} -2.60038 q^{47} +3.23845 q^{48} -5.75422 q^{49} +4.84030 q^{50} +16.1970 q^{51} +4.69976 q^{53} -14.5328 q^{54} +0.159696 q^{55} +1.11615 q^{56} +3.23845 q^{57} -4.74618 q^{58} -9.25676 q^{59} +1.29415 q^{60} +10.1317 q^{61} -1.79858 q^{62} -8.35723 q^{63} +1.00000 q^{64} -1.29415 q^{66} +7.22630 q^{67} +5.00147 q^{68} -2.88837 q^{69} +0.446034 q^{70} +4.51863 q^{71} -7.48758 q^{72} +3.67840 q^{73} -10.0733 q^{74} -15.6751 q^{75} +1.00000 q^{76} -0.446034 q^{77} +4.48160 q^{79} +0.399620 q^{80} +24.6012 q^{81} +8.00335 q^{82} +14.8329 q^{83} -3.61459 q^{84} +1.99869 q^{85} +0.468865 q^{86} +15.3703 q^{87} -0.399620 q^{88} -1.64679 q^{89} -2.99219 q^{90} -0.891897 q^{92} +5.82463 q^{93} +2.60038 q^{94} +0.399620 q^{95} -3.23845 q^{96} -18.1119 q^{97} +5.75422 q^{98} +2.99219 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{2} + 2 q^{3} + 6 q^{4} - 2 q^{5} - 2 q^{6} - q^{7} - 6 q^{8} + 10 q^{9}+O(q^{10}) $ 6 * q - 6 * q^2 + 2 * q^3 + 6 * q^4 - 2 * q^5 - 2 * q^6 - q^7 - 6 * q^8 + 10 * q^9	$ 6 q - 6 q^{2} + 2 q^{3} + 6 q^{4} - 2 q^{5} - 2 q^{6} - q^{7} - 6 q^{8} + 10 q^{9} + 2 q^{10} - 2 q^{11} + 2 q^{12} + q^{14} + 9 q^{15} + 6 q^{16} - 2 q^{17} - 10 q^{18} + 6 q^{19} - 2 q^{20} - q^{21} + 2 q^{22} + 8 q^{23} - 2 q^{24} + 16 q^{25} - 4 q^{27} - q^{28} + 20 q^{29} - 9 q^{30} - 3 q^{31} - 6 q^{32} + 9 q^{33} + 2 q^{34} + 7 q^{35} + 10 q^{36} + 9 q^{37} - 6 q^{38} + 2 q^{40} - 3 q^{41} + q^{42} + 13 q^{43} - 2 q^{44} - 38 q^{45} - 8 q^{46} - 20 q^{47} + 2 q^{48} - 7 q^{49} - 16 q^{50} + 2 q^{51} + 25 q^{53} + 4 q^{54} + 46 q^{55} + q^{56} + 2 q^{57} - 20 q^{58} + 9 q^{60} + 6 q^{61} + 3 q^{62} - 46 q^{63} + 6 q^{64} - 9 q^{66} + 32 q^{67} - 2 q^{68} - 29 q^{69} - 7 q^{70} - 39 q^{71} - 10 q^{72} - 7 q^{73} - 9 q^{74} - 15 q^{75} + 6 q^{76} + 7 q^{77} + 18 q^{79} - 2 q^{80} + 54 q^{81} + 3 q^{82} + 7 q^{83} - q^{84} + 2 q^{85} - 13 q^{86} + 12 q^{87} + 2 q^{88} - 9 q^{89} + 38 q^{90} + 8 q^{92} + 47 q^{93} + 20 q^{94} - 2 q^{95} - 2 q^{96} + 4 q^{97} + 7 q^{98} - 38 q^{99}+O(q^{100}) $ 6 * q - 6 * q^2 + 2 * q^3 + 6 * q^4 - 2 * q^5 - 2 * q^6 - q^7 - 6 * q^8 + 10 * q^9 + 2 * q^10 - 2 * q^11 + 2 * q^12 + q^14 + 9 * q^15 + 6 * q^16 - 2 * q^17 - 10 * q^18 + 6 * q^19 - 2 * q^20 - q^21 + 2 * q^22 + 8 * q^23 - 2 * q^24 + 16 * q^25 - 4 * q^27 - q^28 + 20 * q^29 - 9 * q^30 - 3 * q^31 - 6 * q^32 + 9 * q^33 + 2 * q^34 + 7 * q^35 + 10 * q^36 + 9 * q^37 - 6 * q^38 + 2 * q^40 - 3 * q^41 + q^42 + 13 * q^43 - 2 * q^44 - 38 * q^45 - 8 * q^46 - 20 * q^47 + 2 * q^48 - 7 * q^49 - 16 * q^50 + 2 * q^51 + 25 * q^53 + 4 * q^54 + 46 * q^55 + q^56 + 2 * q^57 - 20 * q^58 + 9 * q^60 + 6 * q^61 + 3 * q^62 - 46 * q^63 + 6 * q^64 - 9 * q^66 + 32 * q^67 - 2 * q^68 - 29 * q^69 - 7 * q^70 - 39 * q^71 - 10 * q^72 - 7 * q^73 - 9 * q^74 - 15 * q^75 + 6 * q^76 + 7 * q^77 + 18 * q^79 - 2 * q^80 + 54 * q^81 + 3 * q^82 + 7 * q^83 - q^84 + 2 * q^85 - 13 * q^86 + 12 * q^87 + 2 * q^88 - 9 * q^89 + 38 * q^90 + 8 * q^92 + 47 * q^93 + 20 * q^94 - 2 * q^95 - 2 * q^96 + 4 * q^97 + 7 * q^98 - 38 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	3.23845	1.86972	0.934861	−	0.355014i	$-0.115524\pi$
$3$	3.23845	1.86972	0.934861	+	0.355014i	$0.115524\pi$
$4$	1.00000	0.500000
$5$	0.399620	0.178715	0.0893577	−	0.996000i	$-0.471519\pi$
$5$	0.399620	0.178715	0.0893577	+	0.996000i	$0.471519\pi$
$6$	−3.23845	−1.32209
$7$	−1.11615	−0.421863	−0.210932	−	0.977501i	$-0.567650\pi$
$7$	−1.11615	−0.421863	−0.210932	+	0.977501i	$0.567650\pi$
$8$	−1.00000	−0.353553
$9$	7.48758	2.49586
$10$	−0.399620	−0.126371
$11$	0.399620	0.120490	0.0602450	−	0.998184i	$-0.480812\pi$
$11$	0.399620	0.120490	0.0602450	+	0.998184i	$0.480812\pi$
$12$	3.23845	0.934861
$13$	0	0
$13$	0	0
$14$	1.11615	0.298302
$15$	1.29415	0.334148
$16$	1.00000	0.250000
$17$	5.00147	1.21303	0.606517	−	0.795070i	$-0.292566\pi$
$17$	5.00147	1.21303	0.606517	+	0.795070i	$0.292566\pi$
$18$	−7.48758	−1.76484
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0.399620	0.0893577
$21$	−3.61459	−0.788767
$22$	−0.399620	−0.0851992
$23$	−0.891897	−0.185973	−0.0929867	−	0.995667i	$-0.529641\pi$
$23$	−0.891897	−0.185973	−0.0929867	+	0.995667i	$0.529641\pi$
$24$	−3.23845	−0.661047
$25$	−4.84030	−0.968061
$26$	0	0
$27$	14.5328	2.79685
$28$	−1.11615	−0.210932
$29$	4.74618	0.881343	0.440671	−	0.897668i	$-0.354740\pi$
$29$	4.74618	0.881343	0.440671	+	0.897668i	$0.354740\pi$
$30$	−1.29415	−0.236278
$31$	1.79858	0.323035	0.161518	−	0.986870i	$-0.448361\pi$
$31$	1.79858	0.323035	0.161518	+	0.986870i	$0.448361\pi$
$32$	−1.00000	−0.176777
$33$	1.29415	0.225283
$34$	−5.00147	−0.857745
$35$	−0.446034	−0.0753935
$36$	7.48758	1.24793
$37$	10.0733	1.65603	0.828016	−	0.560704i	$-0.189470\pi$
$37$	10.0733	1.65603	0.828016	+	0.560704i	$0.189470\pi$
$38$	−1.00000	−0.162221
$39$	0	0
$40$	−0.399620	−0.0631854
$41$	−8.00335	−1.24991	−0.624957	−	0.780660i	$-0.714883\pi$
$41$	−8.00335	−1.24991	−0.624957	+	0.780660i	$0.714883\pi$
$42$	3.61459	0.557743
$43$	−0.468865	−0.0715013	−0.0357506	−	0.999361i	$-0.511382\pi$
$43$	−0.468865	−0.0715013	−0.0357506	+	0.999361i	$0.511382\pi$
$44$	0.399620	0.0602450
$45$	2.99219	0.446049
$46$	0.891897	0.131503
$47$	−2.60038	−0.379304	−0.189652	−	0.981851i	$-0.560736\pi$
$47$	−2.60038	−0.379304	−0.189652	+	0.981851i	$0.560736\pi$
$48$	3.23845	0.467431
$49$	−5.75422	−0.822031
$50$	4.84030	0.684522
$51$	16.1970	2.26804
$52$	0	0
$53$	4.69976	0.645562	0.322781	−	0.946474i	$-0.395382\pi$
$53$	4.69976	0.645562	0.322781	+	0.946474i	$0.395382\pi$
$54$	−14.5328	−1.97767
$55$	0.159696	0.0215334
$56$	1.11615	0.149151
$57$	3.23845	0.428944
$58$	−4.74618	−0.623204
$59$	−9.25676	−1.20513	−0.602564	−	0.798071i	$-0.705854\pi$
$59$	−9.25676	−1.20513	−0.602564	+	0.798071i	$0.705854\pi$
$60$	1.29415	0.167074
$61$	10.1317	1.29724	0.648618	−	0.761114i	$-0.275347\pi$
$61$	10.1317	1.29724	0.648618	+	0.761114i	$0.275347\pi$
$62$	−1.79858	−0.228420
$63$	−8.35723	−1.05291
$64$	1.00000	0.125000
$65$	0	0
$66$	−1.29415	−0.159299
$67$	7.22630	0.882833	0.441416	−	0.897302i	$-0.354476\pi$
$67$	7.22630	0.882833	0.441416	+	0.897302i	$0.354476\pi$
$68$	5.00147	0.606517
$69$	−2.88837	−0.347719
$70$	0.446034	0.0533112
$71$	4.51863	0.536263	0.268131	−	0.963382i	$-0.413594\pi$
$71$	4.51863	0.536263	0.268131	+	0.963382i	$0.413594\pi$
$72$	−7.48758	−0.882420
$73$	3.67840	0.430524	0.215262	−	0.976556i	$-0.430939\pi$
$73$	3.67840	0.430524	0.215262	+	0.976556i	$0.430939\pi$
$74$	−10.0733	−1.17099
$75$	−15.6751	−1.81000
$76$	1.00000	0.114708
$77$	−0.446034	−0.0508303
$78$	0	0
$79$	4.48160	0.504220	0.252110	−	0.967699i	$-0.418876\pi$
$79$	4.48160	0.504220	0.252110	+	0.967699i	$0.418876\pi$
$80$	0.399620	0.0446789
$81$	24.6012	2.73346
$82$	8.00335	0.883822
$83$	14.8329	1.62812	0.814061	−	0.580779i	$-0.197252\pi$
$83$	14.8329	1.62812	0.814061	+	0.580779i	$0.197252\pi$
$84$	−3.61459	−0.394384
$85$	1.99869	0.216788
$86$	0.468865	0.0505591
$87$	15.3703	1.64787
$88$	−0.399620	−0.0425996
$89$	−1.64679	−0.174560	−0.0872799	−	0.996184i	$-0.527817\pi$
$89$	−1.64679	−0.174560	−0.0872799	+	0.996184i	$0.527817\pi$
$90$	−2.99219	−0.315404
$91$	0	0
$92$	−0.891897	−0.0929867
$93$	5.82463	0.603986
$94$	2.60038	0.268209
$95$	0.399620	0.0410001
$96$	−3.23845	−0.330523
$97$	−18.1119	−1.83898	−0.919490	−	0.393113i	$-0.871398\pi$
$97$	−18.1119	−1.83898	−0.919490	+	0.393113i	$0.871398\pi$
$98$	5.75422	0.581264
$99$	2.99219	0.300726
$100$	−4.84030	−0.484030
$101$	−5.90192	−0.587263	−0.293631	−	0.955919i	$-0.594864\pi$
$101$	−5.90192	−0.587263	−0.293631	+	0.955919i	$0.594864\pi$
$102$	−16.1970	−1.60374
$103$	−0.619175	−0.0610091	−0.0305046	−	0.999535i	$-0.509711\pi$
$103$	−0.619175	−0.0610091	−0.0305046	+	0.999535i	$0.509711\pi$
$104$	0	0
$105$	−1.44446	−0.140965
$106$	−4.69976	−0.456481
$107$	−7.98937	−0.772362	−0.386181	−	0.922423i	$-0.626206\pi$
$107$	−7.98937	−0.772362	−0.386181	+	0.922423i	$0.626206\pi$
$108$	14.5328	1.39842
$109$	14.9120	1.42831	0.714156	−	0.699987i	$-0.246811\pi$
$109$	14.9120	1.42831	0.714156	+	0.699987i	$0.246811\pi$
$110$	−0.159696	−0.0152264
$111$	32.6218	3.09632
$112$	−1.11615	−0.105466
$113$	14.3024	1.34546	0.672729	−	0.739889i	$-0.265122\pi$
$113$	14.3024	1.34546	0.672729	+	0.739889i	$0.265122\pi$
$114$	−3.23845	−0.303309
$115$	−0.356420	−0.0332363
$116$	4.74618	0.440671
$117$	0	0
$118$	9.25676	0.852154
$119$	−5.58237	−0.511735
$120$	−1.29415	−0.118139
$121$	−10.8403	−0.985482
$122$	−10.1317	−0.917285
$123$	−25.9185	−2.33699
$124$	1.79858	0.161518
$125$	−3.93238	−0.351723
$126$	8.35723	0.744521
$127$	−11.1889	−0.992855	−0.496428	−	0.868078i	$-0.665355\pi$
$127$	−11.1889	−0.992855	−0.496428	+	0.868078i	$0.665355\pi$
$128$	−1.00000	−0.0883883
$129$	−1.51840	−0.133688
$130$	0	0
$131$	16.3769	1.43086	0.715429	−	0.698685i	$-0.246231\pi$
$131$	16.3769	1.43086	0.715429	+	0.698685i	$0.246231\pi$
$132$	1.29415	0.112641
$133$	−1.11615	−0.0967821
$134$	−7.22630	−0.624257
$135$	5.80761	0.499839
$136$	−5.00147	−0.428873
$137$	9.80333	0.837555	0.418777	−	0.908089i	$-0.362459\pi$
$137$	9.80333	0.837555	0.418777	+	0.908089i	$0.362459\pi$
$138$	2.88837	0.245874
$139$	−10.8206	−0.917789	−0.458895	−	0.888491i	$-0.651754\pi$
$139$	−10.8206	−0.917789	−0.458895	+	0.888491i	$0.651754\pi$
$140$	−0.446034	−0.0376967
$141$	−8.42121	−0.709194
$142$	−4.51863	−0.379195
$143$	0	0
$144$	7.48758	0.623965
$145$	1.89667	0.157510
$146$	−3.67840	−0.304427
$147$	−18.6348	−1.53697
$148$	10.0733	0.828016
$149$	1.67429	0.137163	0.0685815	−	0.997646i	$-0.478153\pi$
$149$	1.67429	0.137163	0.0685815	+	0.997646i	$0.478153\pi$
$150$	15.6751	1.27987
$151$	1.76302	0.143472	0.0717361	−	0.997424i	$-0.477146\pi$
$151$	1.76302	0.143472	0.0717361	+	0.997424i	$0.477146\pi$
$152$	−1.00000	−0.0811107
$153$	37.4489	3.02757
$154$	0.446034	0.0359424
$155$	0.718750	0.0577314
$156$	0	0
$157$	−23.3468	−1.86328	−0.931640	−	0.363382i	$-0.881622\pi$
$157$	−23.3468	−1.86328	−0.931640	+	0.363382i	$0.881622\pi$
$158$	−4.48160	−0.356537
$159$	15.2200	1.20702
$160$	−0.399620	−0.0315927
$161$	0.995487	0.0784554
$162$	−24.6012	−1.93285
$163$	−5.78191	−0.452874	−0.226437	−	0.974026i	$-0.572708\pi$
$163$	−5.78191	−0.452874	−0.226437	+	0.974026i	$0.572708\pi$
$164$	−8.00335	−0.624957
$165$	0.517168	0.0402615
$166$	−14.8329	−1.15126
$167$	−16.6605	−1.28923	−0.644614	−	0.764509i	$-0.722982\pi$
$167$	−16.6605	−1.28923	−0.644614	+	0.764509i	$0.722982\pi$
$168$	3.61459	0.278871
$169$	0	0
$170$	−1.99869	−0.153292
$171$	7.48758	0.572590
$172$	−0.468865	−0.0357506
$173$	−17.5024	−1.33069	−0.665343	−	0.746538i	$-0.731715\pi$
$173$	−17.5024	−1.33069	−0.665343	+	0.746538i	$0.731715\pi$
$174$	−15.3703	−1.16522
$175$	5.40248	0.408389
$176$	0.399620	0.0301225
$177$	−29.9776	−2.25325
$178$	1.64679	0.123432
$179$	23.7353	1.77406	0.887030	−	0.461711i	$-0.152764\pi$
$179$	23.7353	1.77406	0.887030	+	0.461711i	$0.152764\pi$
$180$	2.99219	0.223024
$181$	8.88081	0.660106	0.330053	−	0.943962i	$-0.392933\pi$
$181$	8.88081	0.660106	0.330053	+	0.943962i	$0.392933\pi$
$182$	0	0
$183$	32.8112	2.42547
$184$	0.891897	0.0657515
$185$	4.02547	0.295959
$186$	−5.82463	−0.427082
$187$	1.99869	0.146158
$188$	−2.60038	−0.189652
$189$	−16.2208	−1.17989
$190$	−0.399620	−0.0289915
$191$	−13.8846	−1.00465	−0.502326	−	0.864678i	$-0.667522\pi$
$191$	−13.8846	−1.00465	−0.502326	+	0.864678i	$0.667522\pi$
$192$	3.23845	0.233715
$193$	12.2366	0.880807	0.440403	−	0.897800i	$-0.354835\pi$
$193$	12.2366	0.880807	0.440403	+	0.897800i	$0.354835\pi$
$194$	18.1119	1.30036
$195$	0	0
$196$	−5.75422	−0.411016
$197$	23.9726	1.70798	0.853989	−	0.520291i	$-0.174177\pi$
$197$	23.9726	1.70798	0.853989	+	0.520291i	$0.174177\pi$
$198$	−2.99219	−0.212645
$199$	23.8510	1.69075	0.845377	−	0.534170i	$-0.179376\pi$
$199$	23.8510	1.69075	0.845377	+	0.534170i	$0.179376\pi$
$200$	4.84030	0.342261
$201$	23.4020	1.65065
$202$	5.90192	0.415257
$203$	−5.29742	−0.371806
$204$	16.1970	1.13402
$205$	−3.19830	−0.223379
$206$	0.619175	0.0431400
$207$	−6.67815	−0.464164
$208$	0	0
$209$	0.399620	0.0276423
$210$	1.44446	0.0996772
$211$	−13.7885	−0.949242	−0.474621	−	0.880190i	$-0.657415\pi$
$211$	−13.7885	−0.949242	−0.474621	+	0.880190i	$0.657415\pi$
$212$	4.69976	0.322781
$213$	14.6334	1.00266
$214$	7.98937	0.546142
$215$	−0.187368	−0.0127784
$216$	−14.5328	−0.988834
$217$	−2.00748	−0.136277
$218$	−14.9120	−1.00997
$219$	11.9123	0.804961
$220$	0.159696	0.0107667
$221$	0	0
$222$	−32.6218	−2.18943
$223$	−11.4453	−0.766437	−0.383219	−	0.923658i	$-0.625184\pi$
$223$	−11.4453	−0.766437	−0.383219	+	0.923658i	$0.625184\pi$
$224$	1.11615	0.0745756
$225$	−36.2422	−2.41615
$226$	−14.3024	−0.951383
$227$	−8.28756	−0.550065	−0.275032	−	0.961435i	$-0.588689\pi$
$227$	−8.28756	−0.550065	−0.275032	+	0.961435i	$0.588689\pi$
$228$	3.23845	0.214472
$229$	11.7711	0.777853	0.388927	−	0.921269i	$-0.372846\pi$
$229$	11.7711	0.777853	0.388927	+	0.921269i	$0.372846\pi$
$230$	0.356420	0.0235016
$231$	−1.44446	−0.0950385
$232$	−4.74618	−0.311602
$233$	7.91735	0.518683	0.259341	−	0.965786i	$-0.416495\pi$
$233$	7.91735	0.518683	0.259341	+	0.965786i	$0.416495\pi$
$234$	0	0
$235$	−1.03916	−0.0677876
$236$	−9.25676	−0.602564
$237$	14.5135	0.942750
$238$	5.58237	0.361851
$239$	22.5966	1.46165	0.730827	−	0.682562i	$-0.239134\pi$
$239$	22.5966	1.46165	0.730827	+	0.682562i	$0.239134\pi$
$240$	1.29415	0.0835371
$241$	14.0142	0.902734	0.451367	−	0.892338i	$-0.350937\pi$
$241$	14.0142	0.902734	0.451367	+	0.892338i	$0.350937\pi$
$242$	10.8403	0.696841
$243$	36.0712	2.31397
$244$	10.1317	0.648618
$245$	−2.29950	−0.146910
$246$	25.9185	1.65250
$247$	0	0
$248$	−1.79858	−0.114210
$249$	48.0357	3.04414
$250$	3.93238	0.248706
$251$	2.20114	0.138935	0.0694674	−	0.997584i	$-0.477870\pi$
$251$	2.20114	0.138935	0.0694674	+	0.997584i	$0.477870\pi$
$252$	−8.35723	−0.526456
$253$	−0.356420	−0.0224079
$254$	11.1889	0.702055
$255$	6.47265	0.405333
$256$	1.00000	0.0625000
$257$	3.28665	0.205016	0.102508	−	0.994732i	$-0.467313\pi$
$257$	3.28665	0.205016	0.102508	+	0.994732i	$0.467313\pi$
$258$	1.51840	0.0945314
$259$	−11.2432	−0.698619
$260$	0	0
$261$	35.5374	2.19971
$262$	−16.3769	−1.01177
$263$	1.98736	0.122546	0.0612729	−	0.998121i	$-0.480484\pi$
$263$	1.98736	0.122546	0.0612729	+	0.998121i	$0.480484\pi$
$264$	−1.29415	−0.0796495
$265$	1.87812	0.115372
$266$	1.11615	0.0684353
$267$	−5.33307	−0.326378
$268$	7.22630	0.441416
$269$	−27.5466	−1.67955	−0.839774	−	0.542936i	$-0.817312\pi$
$269$	−27.5466	−1.67955	−0.839774	+	0.542936i	$0.817312\pi$
$270$	−5.80761	−0.353440
$271$	−1.18194	−0.0717980	−0.0358990	−	0.999355i	$-0.511429\pi$
$271$	−1.18194	−0.0717980	−0.0358990	+	0.999355i	$0.511429\pi$
$272$	5.00147	0.303259
$273$	0	0
$274$	−9.80333	−0.592241
$275$	−1.93428	−0.116642
$276$	−2.88837	−0.173859
$277$	5.01864	0.301541	0.150770	−	0.988569i	$-0.451825\pi$
$277$	5.01864	0.301541	0.150770	+	0.988569i	$0.451825\pi$
$278$	10.8206	0.648975
$279$	13.4670	0.806251
$280$	0.446034	0.0266556
$281$	25.7147	1.53401	0.767005	−	0.641642i	$-0.221746\pi$
$281$	25.7147	1.53401	0.767005	+	0.641642i	$0.221746\pi$
$282$	8.42121	0.501476
$283$	7.13053	0.423866	0.211933	−	0.977284i	$-0.432024\pi$
$283$	7.13053	0.423866	0.211933	+	0.977284i	$0.432024\pi$
$284$	4.51863	0.268131
$285$	1.29415	0.0766589
$286$	0	0
$287$	8.93290	0.527293
$288$	−7.48758	−0.441210
$289$	8.01470	0.471453
$290$	−1.89667	−0.111376
$291$	−58.6544	−3.43838
$292$	3.67840	0.215262
$293$	−13.9330	−0.813972	−0.406986	−	0.913434i	$-0.633420\pi$
$293$	−13.9330	−0.813972	−0.406986	+	0.913434i	$0.633420\pi$
$294$	18.6348	1.08680
$295$	−3.69919	−0.215375
$296$	−10.0733	−0.585496
$297$	5.80761	0.336992
$298$	−1.67429	−0.0969889
$299$	0	0
$300$	−15.6751	−0.905002
$301$	0.523322	0.0301638
$302$	−1.76302	−0.101450
$303$	−19.1131	−1.09802
$304$	1.00000	0.0573539
$305$	4.04885	0.231836
$306$	−37.4489	−2.14081
$307$	−25.7619	−1.47031	−0.735155	−	0.677899i	$-0.762891\pi$
$307$	−25.7619	−1.47031	−0.735155	+	0.677899i	$0.762891\pi$
$308$	−0.446034	−0.0254151
$309$	−2.00517	−0.114070
$310$	−0.718750	−0.0408222
$311$	−20.9127	−1.18585	−0.592926	−	0.805257i	$-0.702027\pi$
$311$	−20.9127	−1.18585	−0.592926	+	0.805257i	$0.702027\pi$
$312$	0	0
$313$	7.71160	0.435886	0.217943	−	0.975962i	$-0.430065\pi$
$313$	7.71160	0.435886	0.217943	+	0.975962i	$0.430065\pi$
$314$	23.3468	1.31754
$315$	−3.33972	−0.188172
$316$	4.48160	0.252110
$317$	−24.3225	−1.36609	−0.683045	−	0.730376i	$-0.739345\pi$
$317$	−24.3225	−1.36609	−0.683045	+	0.730376i	$0.739345\pi$
$318$	−15.2200	−0.853493
$319$	1.89667	0.106193
$320$	0.399620	0.0223394
$321$	−25.8732	−1.44410
$322$	−0.995487	−0.0554763
$323$	5.00147	0.278289
$324$	24.6012	1.36673
$325$	0	0
$326$	5.78191	0.320230
$327$	48.2919	2.67055
$328$	8.00335	0.441911
$329$	2.90240	0.160015
$330$	−0.517168	−0.0284692
$331$	−20.0672	−1.10299	−0.551497	−	0.834177i	$-0.685943\pi$
$331$	−20.0672	−1.10299	−0.551497	+	0.834177i	$0.685943\pi$
$332$	14.8329	0.814061
$333$	75.4243	4.13323
$334$	16.6605	0.911621
$335$	2.88777	0.157776
$336$	−3.61459	−0.197192
$337$	36.3220	1.97859	0.989293	−	0.145943i	$-0.0466217\pi$
$337$	36.3220	1.97859	0.989293	+	0.145943i	$0.0466217\pi$
$338$	0	0
$339$	46.3177	2.51563
$340$	1.99869	0.108394
$341$	0.718750	0.0389225
$342$	−7.48758	−0.404882
$343$	14.2356	0.768648
$344$	0.468865	0.0252795
$345$	−1.15425	−0.0621427
$346$	17.5024	0.940937
$347$	−10.8288	−0.581320	−0.290660	−	0.956826i	$-0.593875\pi$
$347$	−10.8288	−0.581320	−0.290660	+	0.956826i	$0.593875\pi$
$348$	15.3703	0.823933
$349$	6.97037	0.373115	0.186558	−	0.982444i	$-0.440267\pi$
$349$	6.97037	0.373115	0.186558	+	0.982444i	$0.440267\pi$
$350$	−5.40248	−0.288775
$351$	0	0
$352$	−0.399620	−0.0212998
$353$	−16.3065	−0.867908	−0.433954	−	0.900935i	$-0.642882\pi$
$353$	−16.3065	−0.867908	−0.433954	+	0.900935i	$0.642882\pi$
$354$	29.9776	1.59329
$355$	1.80573	0.0958384
$356$	−1.64679	−0.0872799
$357$	−18.0782	−0.956802
$358$	−23.7353	−1.25445
$359$	23.3696	1.23340	0.616701	−	0.787198i	$-0.288469\pi$
$359$	23.3696	1.23340	0.616701	+	0.787198i	$0.288469\pi$
$360$	−2.99219	−0.157702
$361$	1.00000	0.0526316
$362$	−8.88081	−0.466765
$363$	−35.1058	−1.84258
$364$	0	0
$365$	1.46996	0.0769413
$366$	−32.8112	−1.71507
$367$	28.3553	1.48013	0.740067	−	0.672533i	$-0.234794\pi$
$367$	28.3553	1.48013	0.740067	+	0.672533i	$0.234794\pi$
$368$	−0.891897	−0.0464933
$369$	−59.9258	−3.11961
$370$	−4.02547	−0.209274
$371$	−5.24562	−0.272339
$372$	5.82463	0.301993
$373$	12.8596	0.665846	0.332923	−	0.942954i	$-0.391965\pi$
$373$	12.8596	0.665846	0.332923	+	0.942954i	$0.391965\pi$
$374$	−1.99869	−0.103350
$375$	−12.7348	−0.657624
$376$	2.60038	0.134104
$377$	0	0
$378$	16.2208	0.834306
$379$	−22.1102	−1.13572	−0.567862	−	0.823124i	$-0.692229\pi$
$379$	−22.1102	−1.13572	−0.567862	+	0.823124i	$0.692229\pi$
$380$	0.399620	0.0205001
$381$	−36.2348	−1.85636
$382$	13.8846	0.710396
$383$	−4.96830	−0.253868	−0.126934	−	0.991911i	$-0.540514\pi$
$383$	−4.96830	−0.253868	−0.126934	+	0.991911i	$0.540514\pi$
$384$	−3.23845	−0.165262
$385$	−0.178244	−0.00908416
$386$	−12.2366	−0.622824
$387$	−3.51067	−0.178457
$388$	−18.1119	−0.919490
$389$	6.78314	0.343919	0.171959	−	0.985104i	$-0.444990\pi$
$389$	6.78314	0.343919	0.171959	+	0.985104i	$0.444990\pi$
$390$	0	0
$391$	−4.46080	−0.225592
$392$	5.75422	0.290632
$393$	53.0359	2.67531
$394$	−23.9726	−1.20772
$395$	1.79094	0.0901118
$396$	2.99219	0.150363
$397$	0.145988	0.00732695	0.00366348	−	0.999993i	$-0.498834\pi$
$397$	0.145988	0.00732695	0.00366348	+	0.999993i	$0.498834\pi$
$398$	−23.8510	−1.19554
$399$	−3.61459	−0.180956
$400$	−4.84030	−0.242015
$401$	17.2912	0.863483	0.431741	−	0.901997i	$-0.357899\pi$
$401$	17.2912	0.863483	0.431741	+	0.901997i	$0.357899\pi$
$402$	−23.4020	−1.16719
$403$	0	0
$404$	−5.90192	−0.293631
$405$	9.83111	0.488512
$406$	5.29742	0.262907
$407$	4.02547	0.199535
$408$	−16.1970	−0.801872
$409$	−8.53647	−0.422101	−0.211051	−	0.977475i	$-0.567688\pi$
$409$	−8.53647	−0.422101	−0.211051	+	0.977475i	$0.567688\pi$
$410$	3.19830	0.157953
$411$	31.7476	1.56599
$412$	−0.619175	−0.0305046
$413$	10.3319	0.508399
$414$	6.67815	0.328213
$415$	5.92752	0.290971
$416$	0	0
$417$	−35.0419	−1.71601
$418$	−0.399620	−0.0195460
$419$	−35.3620	−1.72755	−0.863774	−	0.503880i	$-0.831906\pi$
$419$	−35.3620	−1.72755	−0.863774	+	0.503880i	$0.831906\pi$
$420$	−1.44446	−0.0704824
$421$	16.7352	0.815622	0.407811	−	0.913066i	$-0.366292\pi$
$421$	16.7352	0.815622	0.407811	+	0.913066i	$0.366292\pi$
$422$	13.7885	0.671215
$423$	−19.4706	−0.946691
$424$	−4.69976	−0.228241
$425$	−24.2086	−1.17429
$426$	−14.6334	−0.708989
$427$	−11.3085	−0.547257
$428$	−7.98937	−0.386181
$429$	0	0
$430$	0.187368	0.00903568
$431$	1.74503	0.0840551	0.0420276	−	0.999116i	$-0.486618\pi$
$431$	1.74503	0.0840551	0.0420276	+	0.999116i	$0.486618\pi$
$432$	14.5328	0.699211
$433$	−16.6026	−0.797872	−0.398936	−	0.916979i	$-0.630620\pi$
$433$	−16.6026	−0.797872	−0.398936	+	0.916979i	$0.630620\pi$
$434$	2.00748	0.0963621
$435$	6.14227	0.294499
$436$	14.9120	0.714156
$437$	−0.891897	−0.0426652
$438$	−11.9123	−0.569193
$439$	36.2979	1.73240	0.866202	−	0.499695i	$-0.166554\pi$
$439$	36.2979	1.73240	0.866202	+	0.499695i	$0.166554\pi$
$440$	−0.159696	−0.00761321
$441$	−43.0852	−2.05168
$442$	0	0
$443$	−31.1649	−1.48069	−0.740344	−	0.672228i	$-0.765337\pi$
$443$	−31.1649	−1.48069	−0.740344	+	0.672228i	$0.765337\pi$
$444$	32.6218	1.54816
$445$	−0.658092	−0.0311965
$446$	11.4453	0.541953
$447$	5.42211	0.256457
$448$	−1.11615	−0.0527329
$449$	−0.495883	−0.0234021	−0.0117011	−	0.999932i	$-0.503725\pi$
$449$	−0.495883	−0.0234021	−0.0117011	+	0.999932i	$0.503725\pi$
$450$	36.2422	1.70847
$451$	−3.19830	−0.150602
$452$	14.3024	0.672729
$453$	5.70945	0.268253
$454$	8.28756	0.388954
$455$	0	0
$456$	−3.23845	−0.151655
$457$	20.2843	0.948861	0.474431	−	0.880293i	$-0.342654\pi$
$457$	20.2843	0.948861	0.474431	+	0.880293i	$0.342654\pi$
$458$	−11.7711	−0.550025
$459$	72.6855	3.39267
$460$	−0.356420	−0.0166182
$461$	−35.2203	−1.64037	−0.820186	−	0.572098i	$-0.806130\pi$
$461$	−35.2203	−1.64037	−0.820186	+	0.572098i	$0.806130\pi$
$462$	1.44446	0.0672024
$463$	14.6669	0.681628	0.340814	−	0.940131i	$-0.389297\pi$
$463$	14.6669	0.681628	0.340814	+	0.940131i	$0.389297\pi$
$464$	4.74618	0.220336
$465$	2.32764	0.107942
$466$	−7.91735	−0.366764
$467$	−41.3189	−1.91201	−0.956006	−	0.293349i	$-0.905230\pi$
$467$	−41.3189	−1.91201	−0.956006	+	0.293349i	$0.905230\pi$
$468$	0	0
$469$	−8.06560	−0.372435
$470$	1.03916	0.0479330
$471$	−75.6077	−3.48382
$472$	9.25676	0.426077
$473$	−0.187368	−0.00861519
$474$	−14.5135	−0.666625
$475$	−4.84030	−0.222088
$476$	−5.58237	−0.255867
$477$	35.1899	1.61123
$478$	−22.5966	−1.03355
$479$	−32.3574	−1.47845	−0.739225	−	0.673459i	$-0.764808\pi$
$479$	−32.3574	−1.47845	−0.739225	+	0.673459i	$0.764808\pi$
$480$	−1.29415	−0.0590696
$481$	0	0
$482$	−14.0142	−0.638329
$483$	3.22384	0.146690
$484$	−10.8403	−0.492741
$485$	−7.23786	−0.328654
$486$	−36.0712	−1.63622
$487$	27.2116	1.23307	0.616537	−	0.787326i	$-0.288535\pi$
$487$	27.2116	1.23307	0.616537	+	0.787326i	$0.288535\pi$
$488$	−10.1317	−0.458643
$489$	−18.7244	−0.846748
$490$	2.29950	0.103881
$491$	−24.2656	−1.09509	−0.547547	−	0.836775i	$-0.684438\pi$
$491$	−24.2656	−1.09509	−0.547547	+	0.836775i	$0.684438\pi$
$492$	−25.9185	−1.16850
$493$	23.7379	1.06910
$494$	0	0
$495$	1.19574	0.0537444
$496$	1.79858	0.0807588
$497$	−5.04345	−0.226230
$498$	−48.0357	−2.15253
$499$	−31.2924	−1.40084	−0.700420	−	0.713731i	$-0.747004\pi$
$499$	−31.2924	−1.40084	−0.700420	+	0.713731i	$0.747004\pi$
$500$	−3.93238	−0.175861
$501$	−53.9542	−2.41050
$502$	−2.20114	−0.0982418
$503$	−9.61992	−0.428931	−0.214465	−	0.976732i	$-0.568801\pi$
$503$	−9.61992	−0.428931	−0.214465	+	0.976732i	$0.568801\pi$
$504$	8.35723	0.372261
$505$	−2.35852	−0.104953
$506$	0.356420	0.0158448
$507$	0	0
$508$	−11.1889	−0.496428
$509$	7.11058	0.315171	0.157586	−	0.987505i	$-0.449629\pi$
$509$	7.11058	0.315171	0.157586	+	0.987505i	$0.449629\pi$
$510$	−6.47265	−0.286614
$511$	−4.10563	−0.181622
$512$	−1.00000	−0.0441942
$513$	14.5328	0.641640
$514$	−3.28665	−0.144968
$515$	−0.247435	−0.0109033
$516$	−1.51840	−0.0668438
$517$	−1.03916	−0.0457024
$518$	11.2432	0.493998
$519$	−56.6808	−2.48801
$520$	0	0
$521$	−4.60734	−0.201851	−0.100926	−	0.994894i	$-0.532180\pi$
$521$	−4.60734	−0.201851	−0.100926	+	0.994894i	$0.532180\pi$
$522$	−35.5374	−1.55543
$523$	−22.6558	−0.990668	−0.495334	−	0.868703i	$-0.664954\pi$
$523$	−22.6558	−0.990668	−0.495334	+	0.868703i	$0.664954\pi$
$524$	16.3769	0.715429
$525$	17.4957	0.763575
$526$	−1.98736	−0.0866530
$527$	8.99556	0.391853
$528$	1.29415	0.0563207
$529$	−22.2045	−0.965414
$530$	−1.87812	−0.0815803
$531$	−69.3108	−3.00783
$532$	−1.11615	−0.0483910
$533$	0	0
$534$	5.33307	0.230784
$535$	−3.19271	−0.138033
$536$	−7.22630	−0.312129
$537$	76.8657	3.31700
$538$	27.5466	1.18762
$539$	−2.29950	−0.0990465
$540$	5.80761	0.249920
$541$	16.3286	0.702022	0.351011	−	0.936371i	$-0.385838\pi$
$541$	16.3286	0.702022	0.351011	+	0.936371i	$0.385838\pi$
$542$	1.18194	0.0507688
$543$	28.7601	1.23421
$544$	−5.00147	−0.214436
$545$	5.95914	0.255261
$546$	0	0
$547$	7.11722	0.304310	0.152155	−	0.988357i	$-0.451379\pi$
$547$	7.11722	0.304310	0.152155	+	0.988357i	$0.451379\pi$
$548$	9.80333	0.418777
$549$	75.8623	3.23772
$550$	1.93428	0.0824780
$551$	4.74618	0.202194
$552$	2.88837	0.122937
$553$	−5.00212	−0.212712
$554$	−5.01864	−0.213222
$555$	13.0363	0.553360
$556$	−10.8206	−0.458895
$557$	−15.4875	−0.656227	−0.328113	−	0.944638i	$-0.606413\pi$
$557$	−15.4875	−0.656227	−0.328113	+	0.944638i	$0.606413\pi$
$558$	−13.4670	−0.570105
$559$	0	0
$560$	−0.446034	−0.0188484
$561$	6.47265	0.273276
$562$	−25.7147	−1.08471
$563$	−15.4877	−0.652727	−0.326363	−	0.945244i	$-0.605823\pi$
$563$	−15.4877	−0.652727	−0.326363	+	0.945244i	$0.605823\pi$
$564$	−8.42121	−0.354597
$565$	5.71553	0.240454
$566$	−7.13053	−0.299718
$567$	−27.4585	−1.15315
$568$	−4.51863	−0.189597
$569$	−29.8777	−1.25254	−0.626270	−	0.779606i	$-0.715419\pi$
$569$	−29.8777	−1.25254	−0.626270	+	0.779606i	$0.715419\pi$
$570$	−1.29415	−0.0542060
$571$	28.5689	1.19557	0.597786	−	0.801655i	$-0.296047\pi$
$571$	28.5689	1.19557	0.597786	+	0.801655i	$0.296047\pi$
$572$	0	0
$573$	−44.9645	−1.87842
$574$	−8.93290	−0.372852
$575$	4.31705	0.180034
$576$	7.48758	0.311983
$577$	−10.1612	−0.423016	−0.211508	−	0.977376i	$-0.567837\pi$
$577$	−10.1612	−0.423016	−0.211508	+	0.977376i	$0.567837\pi$
$578$	−8.01470	−0.333368
$579$	39.6275	1.64686
$580$	1.89667	0.0787548
$581$	−16.5557	−0.686845
$582$	58.6544	2.43130
$583$	1.87812	0.0777837
$584$	−3.67840	−0.152213
$585$	0	0
$586$	13.9330	0.575565
$587$	0.742345	0.0306399	0.0153199	−	0.999883i	$-0.495123\pi$
$587$	0.742345	0.0306399	0.0153199	+	0.999883i	$0.495123\pi$
$588$	−18.6348	−0.768485
$589$	1.79858	0.0741093
$590$	3.69919	0.152293
$591$	77.6342	3.19344
$592$	10.0733	0.414008
$593$	−25.1685	−1.03355	−0.516773	−	0.856122i	$-0.672867\pi$
$593$	−25.1685	−1.03355	−0.516773	+	0.856122i	$0.672867\pi$
$594$	−5.80761	−0.238289
$595$	−2.23083	−0.0914549
$596$	1.67429	0.0685815
$597$	77.2405	3.16124
$598$	0	0
$599$	32.6060	1.33225	0.666123	−	0.745842i	$-0.267953\pi$
$599$	32.6060	1.33225	0.666123	+	0.745842i	$0.267953\pi$
$600$	15.6751	0.639933
$601$	−4.44863	−0.181463	−0.0907317	−	0.995875i	$-0.528921\pi$
$601$	−4.44863	−0.181463	−0.0907317	+	0.995875i	$0.528921\pi$
$602$	−0.523322	−0.0213290
$603$	54.1075	2.20343
$604$	1.76302	0.0717361
$605$	−4.33200	−0.176121
$606$	19.1131	0.776416
$607$	−32.6694	−1.32601	−0.663004	−	0.748616i	$-0.730719\pi$
$607$	−32.6694	−1.32601	−0.663004	+	0.748616i	$0.730719\pi$
$608$	−1.00000	−0.0405554
$609$	−17.1555	−0.695174
$610$	−4.04885	−0.163933
$611$	0	0
$612$	37.4489	1.51378
$613$	24.6329	0.994912	0.497456	−	0.867489i	$-0.334268\pi$
$613$	24.6329	0.994912	0.497456	+	0.867489i	$0.334268\pi$
$614$	25.7619	1.03967
$615$	−10.3575	−0.417656
$616$	0.446034	0.0179712
$617$	−44.9737	−1.81057	−0.905287	−	0.424801i	$-0.860344\pi$
$617$	−44.9737	−1.81057	−0.905287	+	0.424801i	$0.860344\pi$
$618$	2.00517	0.0806598
$619$	−6.88164	−0.276597	−0.138298	−	0.990391i	$-0.544163\pi$
$619$	−6.88164	−0.276597	−0.138298	+	0.990391i	$0.544163\pi$
$620$	0.718750	0.0288657
$621$	−12.9618	−0.520139
$622$	20.9127	0.838524
$623$	1.83806	0.0736404
$624$	0	0
$625$	22.6301	0.905202
$626$	−7.71160	−0.308218
$627$	1.29415	0.0516834
$628$	−23.3468	−0.931640
$629$	50.3811	2.00882
$630$	3.33972	0.133057
$631$	40.1685	1.59908	0.799541	−	0.600611i	$-0.205076\pi$
$631$	40.1685	1.59908	0.799541	+	0.600611i	$0.205076\pi$
$632$	−4.48160	−0.178269
$633$	−44.6535	−1.77482
$634$	24.3225	0.965971
$635$	−4.47131	−0.177439
$636$	15.2200	0.603511
$637$	0	0
$638$	−1.89667	−0.0750897
$639$	33.8336	1.33844
$640$	−0.399620	−0.0157964
$641$	14.6092	0.577030	0.288515	−	0.957475i	$-0.406838\pi$
$641$	14.6092	0.577030	0.288515	+	0.957475i	$0.406838\pi$
$642$	25.8732	1.02113
$643$	−34.2756	−1.35170	−0.675849	−	0.737040i	$-0.736223\pi$
$643$	−34.2756	−1.35170	−0.675849	+	0.737040i	$0.736223\pi$
$644$	0.995487	0.0392277
$645$	−0.606782	−0.0238920
$646$	−5.00147	−0.196780
$647$	−20.0474	−0.788143	−0.394072	−	0.919080i	$-0.628934\pi$
$647$	−20.0474	−0.788143	−0.394072	+	0.919080i	$0.628934\pi$
$648$	−24.6012	−0.966425
$649$	−3.69919	−0.145206
$650$	0	0
$651$	−6.50113	−0.254799
$652$	−5.78191	−0.226437
$653$	−45.2639	−1.77131	−0.885656	−	0.464342i	$-0.846291\pi$
$653$	−45.2639	−1.77131	−0.885656	+	0.464342i	$0.846291\pi$
$654$	−48.2919	−1.88836
$655$	6.54454	0.255716
$656$	−8.00335	−0.312478
$657$	27.5423	1.07453
$658$	−2.90240	−0.113147
$659$	−44.1866	−1.72127	−0.860633	−	0.509226i	$-0.829932\pi$
$659$	−44.1866	−1.72127	−0.860633	+	0.509226i	$0.829932\pi$
$660$	0.517168	0.0201307
$661$	14.7839	0.575025	0.287513	−	0.957777i	$-0.407172\pi$
$661$	14.7839	0.575025	0.287513	+	0.957777i	$0.407172\pi$
$662$	20.0672	0.779935
$663$	0	0
$664$	−14.8329	−0.575628
$665$	−0.446034	−0.0172965
$666$	−75.4243	−2.92263
$667$	−4.23310	−0.163906
$668$	−16.6605	−0.644614
$669$	−37.0652	−1.43302
$670$	−2.88777	−0.111564
$671$	4.04885	0.156304
$672$	3.61459	0.139436
$673$	−16.5401	−0.637574	−0.318787	−	0.947826i	$-0.603275\pi$
$673$	−16.5401	−0.637574	−0.318787	+	0.947826i	$0.603275\pi$
$674$	−36.3220	−1.39907
$675$	−70.3433	−2.70752
$676$	0	0
$677$	28.9720	1.11349	0.556743	−	0.830685i	$-0.312051\pi$
$677$	28.9720	1.11349	0.556743	+	0.830685i	$0.312051\pi$
$678$	−46.3177	−1.77882
$679$	20.2155	0.775798
$680$	−1.99869	−0.0766461
$681$	−26.8389	−1.02847
$682$	−0.718750	−0.0275223
$683$	15.4444	0.590964	0.295482	−	0.955348i	$-0.404520\pi$
$683$	15.4444	0.590964	0.295482	+	0.955348i	$0.404520\pi$
$684$	7.48758	0.286295
$685$	3.91760	0.149684
$686$	−14.2356	−0.543516
$687$	38.1200	1.45437
$688$	−0.468865	−0.0178753
$689$	0	0
$690$	1.15425	0.0439415
$691$	−35.1469	−1.33705	−0.668526	−	0.743689i	$-0.733074\pi$
$691$	−35.1469	−1.33705	−0.668526	+	0.743689i	$0.733074\pi$
$692$	−17.5024	−0.665343
$693$	−3.33972	−0.126865
$694$	10.8288	0.411056
$695$	−4.32412	−0.164023
$696$	−15.3703	−0.582609
$697$	−40.0285	−1.51619
$698$	−6.97037	−0.263832
$699$	25.6400	0.969792
$700$	5.40248	0.204195
$701$	−8.05223	−0.304128	−0.152064	−	0.988371i	$-0.548592\pi$
$701$	−8.05223	−0.304128	−0.152064	+	0.988371i	$0.548592\pi$
$702$	0	0
$703$	10.0733	0.379920
$704$	0.399620	0.0150612
$705$	−3.36528	−0.126744
$706$	16.3065	0.613703
$707$	6.58740	0.247745
$708$	−29.9776	−1.12663
$709$	24.6296	0.924985	0.462492	−	0.886623i	$-0.346955\pi$
$709$	24.6296	0.924985	0.462492	+	0.886623i	$0.346955\pi$
$710$	−1.80573	−0.0677680
$711$	33.5564	1.25846
$712$	1.64679	0.0617162
$713$	−1.60415	−0.0600759
$714$	18.0782	0.676561
$715$	0	0
$716$	23.7353	0.887030
$717$	73.1782	2.73289
$718$	−23.3696	−0.872146
$719$	1.98652	0.0740847	0.0370423	−	0.999314i	$-0.488206\pi$
$719$	1.98652	0.0740847	0.0370423	+	0.999314i	$0.488206\pi$
$720$	2.99219	0.111512
$721$	0.691089	0.0257375
$722$	−1.00000	−0.0372161
$723$	45.3843	1.68786
$724$	8.88081	0.330053
$725$	−22.9729	−0.853194
$726$	35.1058	1.30290
$727$	7.53051	0.279291	0.139646	−	0.990202i	$-0.455404\pi$
$727$	7.53051	0.279291	0.139646	+	0.990202i	$0.455404\pi$
$728$	0	0
$729$	43.0115	1.59302
$730$	−1.46996	−0.0544057
$731$	−2.34502	−0.0867336
$732$	32.8112	1.21274
$733$	25.6844	0.948673	0.474337	−	0.880344i	$-0.342688\pi$
$733$	25.6844	0.948673	0.474337	+	0.880344i	$0.342688\pi$
$734$	−28.3553	−1.04661
$735$	−7.44683	−0.274680
$736$	0.891897	0.0328758
$737$	2.88777	0.106372
$738$	59.9258	2.20590
$739$	4.90257	0.180344	0.0901720	−	0.995926i	$-0.471258\pi$
$739$	4.90257	0.180344	0.0901720	+	0.995926i	$0.471258\pi$
$740$	4.02547	0.147979
$741$	0	0
$742$	5.24562	0.192573
$743$	30.9536	1.13558	0.567790	−	0.823174i	$-0.307799\pi$
$743$	30.9536	1.13558	0.567790	+	0.823174i	$0.307799\pi$
$744$	−5.82463	−0.213541
$745$	0.669079	0.0245132
$746$	−12.8596	−0.470824
$747$	111.063	4.06357
$748$	1.99869	0.0730792
$749$	8.91730	0.325831
$750$	12.7348	0.465010
$751$	−36.0547	−1.31566	−0.657828	−	0.753169i	$-0.728524\pi$
$751$	−36.0547	−1.31566	−0.657828	+	0.753169i	$0.728524\pi$
$752$	−2.60038	−0.0948261
$753$	7.12830	0.259770
$754$	0	0
$755$	0.704536	0.0256407
$756$	−16.2208	−0.589943
$757$	−42.7353	−1.55324	−0.776620	−	0.629969i	$-0.783067\pi$
$757$	−42.7353	−1.55324	−0.776620	+	0.629969i	$0.783067\pi$
$758$	22.1102	0.803078
$759$	−1.15425	−0.0418966
$760$	−0.399620	−0.0144957
$761$	−26.8380	−0.972878	−0.486439	−	0.873715i	$-0.661704\pi$
$761$	−26.8380	−0.972878	−0.486439	+	0.873715i	$0.661704\pi$
$762$	36.2348	1.31265
$763$	−16.6440	−0.602552
$764$	−13.8846	−0.502326
$765$	14.9653	0.541073
$766$	4.96830	0.179512
$767$	0	0
$768$	3.23845	0.116858
$769$	−34.2010	−1.23332	−0.616661	−	0.787229i	$-0.711515\pi$
$769$	−34.2010	−1.23332	−0.616661	+	0.787229i	$0.711515\pi$
$770$	0.178244	0.00642347
$771$	10.6437	0.383323
$772$	12.2366	0.440403
$773$	−2.58310	−0.0929075	−0.0464538	−	0.998920i	$-0.514792\pi$
$773$	−2.58310	−0.0929075	−0.0464538	+	0.998920i	$0.514792\pi$
$774$	3.51067	0.126188
$775$	−8.70569	−0.312718
$776$	18.1119	0.650178
$777$	−36.4106	−1.30622
$778$	−6.78314	−0.243187
$779$	−8.00335	−0.286750
$780$	0	0
$781$	1.80573	0.0646142
$782$	4.46080	0.159518
$783$	68.9754	2.46498
$784$	−5.75422	−0.205508
$785$	−9.32986	−0.332997
$786$	−53.0359	−1.89173
$787$	40.4174	1.44072	0.720362	−	0.693599i	$-0.243976\pi$
$787$	40.4174	1.44072	0.720362	+	0.693599i	$0.243976\pi$
$788$	23.9726	0.853989
$789$	6.43597	0.229127
$790$	−1.79094	−0.0637187
$791$	−15.9636	−0.567600
$792$	−2.99219	−0.106323
$793$	0	0
$794$	−0.145988	−0.00518094
$795$	6.08220	0.215713
$796$	23.8510	0.845377
$797$	−15.8791	−0.562465	−0.281233	−	0.959640i	$-0.590743\pi$
$797$	−15.8791	−0.562465	−0.281233	+	0.959640i	$0.590743\pi$
$798$	3.61459	0.127955
$799$	−13.0057	−0.460109
$800$	4.84030	0.171131
$801$	−12.3305	−0.435677
$802$	−17.2912	−0.610575
$803$	1.46996	0.0518738
$804$	23.4020	0.825326
$805$	0.397816	0.0140212
$806$	0	0
$807$	−89.2085	−3.14029
$808$	5.90192	0.207629
$809$	9.16436	0.322202	0.161101	−	0.986938i	$-0.448496\pi$
$809$	9.16436	0.322202	0.161101	+	0.986938i	$0.448496\pi$
$810$	−9.83111	−0.345430
$811$	−42.9912	−1.50963	−0.754813	−	0.655940i	$-0.772272\pi$
$811$	−42.9912	−1.50963	−0.754813	+	0.655940i	$0.772272\pi$
$812$	−5.29742	−0.185903
$813$	−3.82767	−0.134242
$814$	−4.02547	−0.141093
$815$	−2.31056	−0.0809355
$816$	16.1970	0.567009
$817$	−0.468865	−0.0164035
$818$	8.53647	0.298471
$819$	0	0
$820$	−3.19830	−0.111689
$821$	24.9986	0.872456	0.436228	−	0.899836i	$-0.356314\pi$
$821$	24.9986	0.872456	0.436228	+	0.899836i	$0.356314\pi$
$822$	−31.7476	−1.10733
$823$	54.8300	1.91126	0.955628	−	0.294578i	$-0.0951789\pi$
$823$	54.8300	1.91126	0.955628	+	0.294578i	$0.0951789\pi$
$824$	0.619175	0.0215700
$825$	−6.26408	−0.218087
$826$	−10.3319	−0.359493
$827$	3.23836	0.112609	0.0563043	−	0.998414i	$-0.482068\pi$
$827$	3.23836	0.112609	0.0563043	+	0.998414i	$0.482068\pi$
$828$	−6.67815	−0.232082
$829$	−2.58551	−0.0897986	−0.0448993	−	0.998992i	$-0.514297\pi$
$829$	−2.58551	−0.0897986	−0.0448993	+	0.998992i	$0.514297\pi$
$830$	−5.92752	−0.205747
$831$	16.2526	0.563798
$832$	0	0
$833$	−28.7796	−0.997152
$834$	35.0419	1.21340
$835$	−6.65786	−0.230405
$836$	0.399620	0.0138211
$837$	26.1385	0.903479
$838$	35.3620	1.22156
$839$	19.1170	0.659992	0.329996	−	0.943982i	$-0.392953\pi$
$839$	19.1170	0.659992	0.329996	+	0.943982i	$0.392953\pi$
$840$	1.44446	0.0498386
$841$	−6.47381	−0.223235
$842$	−16.7352	−0.576732
$843$	83.2758	2.86817
$844$	−13.7885	−0.474621
$845$	0	0
$846$	19.4706	0.669412
$847$	12.0994	0.415739
$848$	4.69976	0.161391
$849$	23.0919	0.792511
$850$	24.2086	0.830349
$851$	−8.98430	−0.307978
$852$	14.6334	0.501331
$853$	−42.8586	−1.46745	−0.733726	−	0.679446i	$-0.762220\pi$
$853$	−42.8586	−1.46745	−0.733726	+	0.679446i	$0.762220\pi$
$854$	11.3085	0.386969
$855$	2.99219	0.102331
$856$	7.98937	0.273071
$857$	3.25052	0.111036	0.0555179	−	0.998458i	$-0.482319\pi$
$857$	3.25052	0.111036	0.0555179	+	0.998458i	$0.482319\pi$
$858$	0	0
$859$	43.1504	1.47227	0.736136	−	0.676834i	$-0.236648\pi$
$859$	43.1504	1.47227	0.736136	+	0.676834i	$0.236648\pi$
$860$	−0.187368	−0.00638919
$861$	28.9288	0.985891
$862$	−1.74503	−0.0594360
$863$	19.3004	0.656992	0.328496	−	0.944505i	$-0.393458\pi$
$863$	19.3004	0.656992	0.328496	+	0.944505i	$0.393458\pi$
$864$	−14.5328	−0.494417
$865$	−6.99432	−0.237814
$866$	16.6026	0.564181
$867$	25.9552	0.881486
$868$	−2.00748	−0.0681383
$869$	1.79094	0.0607534
$870$	−6.14227	−0.208242
$871$	0	0
$872$	−14.9120	−0.504984
$873$	−135.614	−4.58984
$874$	0.891897	0.0301689
$875$	4.38911	0.148379
$876$	11.9123	0.402480
$877$	56.7479	1.91624	0.958120	−	0.286365i	$-0.0924471\pi$
$877$	56.7479	1.91624	0.958120	+	0.286365i	$0.0924471\pi$
$878$	−36.2979	−1.22499
$879$	−45.1213	−1.52190
$880$	0.159696	0.00538335
$881$	6.30025	0.212261	0.106130	−	0.994352i	$-0.466154\pi$
$881$	6.30025	0.212261	0.106130	+	0.994352i	$0.466154\pi$
$882$	43.0852	1.45075
$883$	46.2556	1.55662	0.778312	−	0.627878i	$-0.216076\pi$
$883$	46.2556	1.55662	0.778312	+	0.627878i	$0.216076\pi$
$884$	0	0
$885$	−11.9796	−0.402691
$886$	31.1649	1.04700
$887$	−41.1666	−1.38224	−0.691120	−	0.722740i	$-0.742883\pi$
$887$	−41.1666	−1.38224	−0.691120	+	0.722740i	$0.742883\pi$
$888$	−32.6218	−1.09471
$889$	12.4885	0.418849
$890$	0.658092	0.0220593
$891$	9.83111	0.329355
$892$	−11.4453	−0.383219
$893$	−2.60038	−0.0870184
$894$	−5.42211	−0.181342
$895$	9.48510	0.317052
$896$	1.11615	0.0372878
$897$	0	0
$898$	0.495883	0.0165478
$899$	8.53639	0.284705
$900$	−36.2422	−1.20807
$901$	23.5057	0.783089
$902$	3.19830	0.106492
$903$	1.69475	0.0563979
$904$	−14.3024	−0.475692
$905$	3.54895	0.117971
$906$	−5.70945	−0.189684
$907$	50.8975	1.69002	0.845011	−	0.534749i	$-0.179594\pi$
$907$	50.8975	1.69002	0.845011	+	0.534749i	$0.179594\pi$
$908$	−8.28756	−0.275032
$909$	−44.1911	−1.46573
$910$	0	0
$911$	−57.1753	−1.89430	−0.947151	−	0.320788i	$-0.896052\pi$
$911$	−57.1753	−1.89430	−0.947151	+	0.320788i	$0.896052\pi$
$912$	3.23845	0.107236
$913$	5.92752	0.196172
$914$	−20.2843	−0.670946
$915$	13.1120	0.433469
$916$	11.7711	0.388927
$917$	−18.2790	−0.603627
$918$	−72.6855	−2.39898
$919$	−11.1499	−0.367801	−0.183901	−	0.982945i	$-0.558872\pi$
$919$	−11.1499	−0.367801	−0.183901	+	0.982945i	$0.558872\pi$
$920$	0.356420	0.0117508
$921$	−83.4287	−2.74907
$922$	35.2203	1.15992
$923$	0	0
$924$	−1.44446	−0.0475193
$925$	−48.7576	−1.60314
$926$	−14.6669	−0.481984
$927$	−4.63613	−0.152270
$928$	−4.74618	−0.155801
$929$	41.4032	1.35839	0.679197	−	0.733956i	$-0.262328\pi$
$929$	41.4032	1.35839	0.679197	+	0.733956i	$0.262328\pi$
$930$	−2.32764	−0.0763262
$931$	−5.75422	−0.188587
$932$	7.91735	0.259341
$933$	−67.7249	−2.21721
$934$	41.3189	1.35200
$935$	0.798715	0.0261208
$936$	0	0
$937$	−4.10483	−0.134099	−0.0670495	−	0.997750i	$-0.521359\pi$
$937$	−4.10483	−0.134099	−0.0670495	+	0.997750i	$0.521359\pi$
$938$	8.06560	0.263351
$939$	24.9737	0.814985
$940$	−1.03916	−0.0338938
$941$	−13.7147	−0.447085	−0.223543	−	0.974694i	$-0.571762\pi$
$941$	−13.7147	−0.447085	−0.223543	+	0.974694i	$0.571762\pi$
$942$	75.6077	2.46343
$943$	7.13816	0.232451
$944$	−9.25676	−0.301282
$945$	−6.48214	−0.210864
$946$	0.187368	0.00609186
$947$	−25.5085	−0.828913	−0.414457	−	0.910069i	$-0.636028\pi$
$947$	−25.5085	−0.828913	−0.414457	+	0.910069i	$0.636028\pi$
$948$	14.5135	0.471375
$949$	0	0
$950$	4.84030	0.157040
$951$	−78.7674	−2.55421
$952$	5.58237	0.180926
$953$	19.4744	0.630837	0.315419	−	0.948953i	$-0.397855\pi$
$953$	19.4744	0.630837	0.315419	+	0.948953i	$0.397855\pi$
$954$	−35.1899	−1.13931
$955$	−5.54855	−0.179547
$956$	22.5966	0.730827
$957$	6.14227	0.198551
$958$	32.3574	1.04542
$959$	−10.9419	−0.353334
$960$	1.29415	0.0417685
$961$	−27.7651	−0.895648
$962$	0	0
$963$	−59.8211	−1.92771
$964$	14.0142	0.451367
$965$	4.88997	0.157414
$966$	−3.22384	−0.103725
$967$	0.0676905	0.00217678	0.00108839	−	0.999999i	$-0.499654\pi$
$967$	0.0676905	0.00217678	0.00108839	+	0.999999i	$0.499654\pi$
$968$	10.8403	0.348421
$969$	16.1970	0.520324
$970$	7.23786	0.232394
$971$	18.0001	0.577649	0.288825	−	0.957382i	$-0.406736\pi$
$971$	18.0001	0.577649	0.288825	+	0.957382i	$0.406736\pi$
$972$	36.0712	1.15699
$973$	12.0773	0.387182
$974$	−27.2116	−0.871916
$975$	0	0
$976$	10.1317	0.324309
$977$	−33.9688	−1.08676	−0.543379	−	0.839487i	$-0.682855\pi$
$977$	−33.9688	−1.08676	−0.543379	+	0.839487i	$0.682855\pi$
$978$	18.7244	0.598741
$979$	−0.658092	−0.0210327
$980$	−2.29950	−0.0734548
$981$	111.655	3.56487
$982$	24.2656	0.774348
$983$	23.0955	0.736632	0.368316	−	0.929701i	$-0.379935\pi$
$983$	23.0955	0.736632	0.368316	+	0.929701i	$0.379935\pi$
$984$	25.9185	0.826251
$985$	9.57993	0.305242
$986$	−23.7379	−0.755967
$987$	9.39930	0.299183
$988$	0	0
$989$	0.418180	0.0132973
$990$	−1.19574	−0.0380030
$991$	−7.97924	−0.253469	−0.126734	−	0.991937i	$-0.540450\pi$
$991$	−7.97924	−0.253469	−0.126734	+	0.991937i	$0.540450\pi$
$992$	−1.79858	−0.0571051
$993$	−64.9868	−2.06229
$994$	5.04345	0.159968
$995$	9.53135	0.302164
$996$	48.0357	1.52207
$997$	18.8207	0.596059	0.298029	−	0.954557i	$-0.403671\pi$
$997$	18.8207	0.596059	0.298029	+	0.954557i	$0.403671\pi$
$998$	31.2924	0.990544
$999$	146.393	4.63167

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6422.2.a.bc.1.6		6
13.4	even	6		494.2.g.f.419.1	yes	12
13.10	even	6		494.2.g.f.191.1	✓	12
13.12	even	2		6422.2.a.bd.1.6		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
494.2.g.f.191.1	✓	12	13.10	even	6
494.2.g.f.419.1	yes	12	13.4	even	6
6422.2.a.bc.1.6		6	1.1	even	1	trivial
6422.2.a.bd.1.6		6	13.12	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:	\( N \)	\(=\)	\( 6422 = 2 \cdot 13^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6422.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +3.23845 q^{3} +1.00000 q^{4} +0.399620 q^{5} -3.23845 q^{6} -1.11615 q^{7} -1.00000 q^{8} +7.48758 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +3.23845 q^{3} +1.00000 q^{4} +0.399620 q^{5} -3.23845 q^{6} -1.11615 q^{7} -1.00000 q^{8} +7.48758 q^{9} -0.399620 q^{10} +0.399620 q^{11} +3.23845 q^{12} +1.11615 q^{14} +1.29415 q^{15} +1.00000 q^{16} +5.00147 q^{17} -7.48758 q^{18} +1.00000 q^{19} +0.399620 q^{20} -3.61459 q^{21} -0.399620 q^{22} -0.891897 q^{23} -3.23845 q^{24} -4.84030 q^{25} +14.5328 q^{27} -1.11615 q^{28} +4.74618 q^{29} -1.29415 q^{30} +1.79858 q^{31} -1.00000 q^{32} +1.29415 q^{33} -5.00147 q^{34} -0.446034 q^{35} +7.48758 q^{36} +10.0733 q^{37} -1.00000 q^{38} -0.399620 q^{40} -8.00335 q^{41} +3.61459 q^{42} -0.468865 q^{43} +0.399620 q^{44} +2.99219 q^{45} +0.891897 q^{46} -2.60038 q^{47} +3.23845 q^{48} -5.75422 q^{49} +4.84030 q^{50} +16.1970 q^{51} +4.69976 q^{53} -14.5328 q^{54} +0.159696 q^{55} +1.11615 q^{56} +3.23845 q^{57} -4.74618 q^{58} -9.25676 q^{59} +1.29415 q^{60} +10.1317 q^{61} -1.79858 q^{62} -8.35723 q^{63} +1.00000 q^{64} -1.29415 q^{66} +7.22630 q^{67} +5.00147 q^{68} -2.88837 q^{69} +0.446034 q^{70} +4.51863 q^{71} -7.48758 q^{72} +3.67840 q^{73} -10.0733 q^{74} -15.6751 q^{75} +1.00000 q^{76} -0.446034 q^{77} +4.48160 q^{79} +0.399620 q^{80} +24.6012 q^{81} +8.00335 q^{82} +14.8329 q^{83} -3.61459 q^{84} +1.99869 q^{85} +0.468865 q^{86} +15.3703 q^{87} -0.399620 q^{88} -1.64679 q^{89} -2.99219 q^{90} -0.891897 q^{92} +5.82463 q^{93} +2.60038 q^{94} +0.399620 q^{95} -3.23845 q^{96} -18.1119 q^{97} +5.75422 q^{98} +2.99219 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{2} + 2 q^{3} + 6 q^{4} - 2 q^{5} - 2 q^{6} - q^{7} - 6 q^{8} + 10 q^{9}+O(q^{10}) \) 6 * q - 6 * q^2 + 2 * q^3 + 6 * q^4 - 2 * q^5 - 2 * q^6 - q^7 - 6 * q^8 + 10 * q^9	\( 6 q - 6 q^{2} + 2 q^{3} + 6 q^{4} - 2 q^{5} - 2 q^{6} - q^{7} - 6 q^{8} + 10 q^{9} + 2 q^{10} - 2 q^{11} + 2 q^{12} + q^{14} + 9 q^{15} + 6 q^{16} - 2 q^{17} - 10 q^{18} + 6 q^{19} - 2 q^{20} - q^{21} + 2 q^{22} + 8 q^{23} - 2 q^{24} + 16 q^{25} - 4 q^{27} - q^{28} + 20 q^{29} - 9 q^{30} - 3 q^{31} - 6 q^{32} + 9 q^{33} + 2 q^{34} + 7 q^{35} + 10 q^{36} + 9 q^{37} - 6 q^{38} + 2 q^{40} - 3 q^{41} + q^{42} + 13 q^{43} - 2 q^{44} - 38 q^{45} - 8 q^{46} - 20 q^{47} + 2 q^{48} - 7 q^{49} - 16 q^{50} + 2 q^{51} + 25 q^{53} + 4 q^{54} + 46 q^{55} + q^{56} + 2 q^{57} - 20 q^{58} + 9 q^{60} + 6 q^{61} + 3 q^{62} - 46 q^{63} + 6 q^{64} - 9 q^{66} + 32 q^{67} - 2 q^{68} - 29 q^{69} - 7 q^{70} - 39 q^{71} - 10 q^{72} - 7 q^{73} - 9 q^{74} - 15 q^{75} + 6 q^{76} + 7 q^{77} + 18 q^{79} - 2 q^{80} + 54 q^{81} + 3 q^{82} + 7 q^{83} - q^{84} + 2 q^{85} - 13 q^{86} + 12 q^{87} + 2 q^{88} - 9 q^{89} + 38 q^{90} + 8 q^{92} + 47 q^{93} + 20 q^{94} - 2 q^{95} - 2 q^{96} + 4 q^{97} + 7 q^{98} - 38 q^{99}+O(q^{100}) \) 6 * q - 6 * q^2 + 2 * q^3 + 6 * q^4 - 2 * q^5 - 2 * q^6 - q^7 - 6 * q^8 + 10 * q^9 + 2 * q^10 - 2 * q^11 + 2 * q^12 + q^14 + 9 * q^15 + 6 * q^16 - 2 * q^17 - 10 * q^18 + 6 * q^19 - 2 * q^20 - q^21 + 2 * q^22 + 8 * q^23 - 2 * q^24 + 16 * q^25 - 4 * q^27 - q^28 + 20 * q^29 - 9 * q^30 - 3 * q^31 - 6 * q^32 + 9 * q^33 + 2 * q^34 + 7 * q^35 + 10 * q^36 + 9 * q^37 - 6 * q^38 + 2 * q^40 - 3 * q^41 + q^42 + 13 * q^43 - 2 * q^44 - 38 * q^45 - 8 * q^46 - 20 * q^47 + 2 * q^48 - 7 * q^49 - 16 * q^50 + 2 * q^51 + 25 * q^53 + 4 * q^54 + 46 * q^55 + q^56 + 2 * q^57 - 20 * q^58 + 9 * q^60 + 6 * q^61 + 3 * q^62 - 46 * q^63 + 6 * q^64 - 9 * q^66 + 32 * q^67 - 2 * q^68 - 29 * q^69 - 7 * q^70 - 39 * q^71 - 10 * q^72 - 7 * q^73 - 9 * q^74 - 15 * q^75 + 6 * q^76 + 7 * q^77 + 18 * q^79 - 2 * q^80 + 54 * q^81 + 3 * q^82 + 7 * q^83 - q^84 + 2 * q^85 - 13 * q^86 + 12 * q^87 + 2 * q^88 - 9 * q^89 + 38 * q^90 + 8 * q^92 + 47 * q^93 + 20 * q^94 - 2 * q^95 - 2 * q^96 + 4 * q^97 + 7 * q^98 - 38 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

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