LMFDB - Embedded newform 4022.2.a.e.1.19

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4022,2,Mod(1,4022)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4022, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4022.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4022 = 2 \cdot 2011 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4022.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:	$32.1158316930$
Analytic rank:	$0$
Dimension:	$46$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.19
Character	$\chi$	$=$	4022.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -0.498264 q^{3} +1.00000 q^{4} +0.121384 q^{5} +0.498264 q^{6} -4.42409 q^{7} -1.00000 q^{8} -2.75173 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -0.498264 q^{3} +1.00000 q^{4} +0.121384 q^{5} +0.498264 q^{6} -4.42409 q^{7} -1.00000 q^{8} -2.75173 q^{9} -0.121384 q^{10} -2.32784 q^{11} -0.498264 q^{12} -1.79799 q^{13} +4.42409 q^{14} -0.0604811 q^{15} +1.00000 q^{16} +0.739422 q^{17} +2.75173 q^{18} -7.11297 q^{19} +0.121384 q^{20} +2.20437 q^{21} +2.32784 q^{22} -1.00494 q^{23} +0.498264 q^{24} -4.98527 q^{25} +1.79799 q^{26} +2.86588 q^{27} -4.42409 q^{28} -5.80422 q^{29} +0.0604811 q^{30} +2.80468 q^{31} -1.00000 q^{32} +1.15988 q^{33} -0.739422 q^{34} -0.537013 q^{35} -2.75173 q^{36} +4.95610 q^{37} +7.11297 q^{38} +0.895875 q^{39} -0.121384 q^{40} -0.415845 q^{41} -2.20437 q^{42} -11.3245 q^{43} -2.32784 q^{44} -0.334015 q^{45} +1.00494 q^{46} -10.4816 q^{47} -0.498264 q^{48} +12.5726 q^{49} +4.98527 q^{50} -0.368427 q^{51} -1.79799 q^{52} -0.00603078 q^{53} -2.86588 q^{54} -0.282562 q^{55} +4.42409 q^{56} +3.54414 q^{57} +5.80422 q^{58} -0.658526 q^{59} -0.0604811 q^{60} -1.30058 q^{61} -2.80468 q^{62} +12.1739 q^{63} +1.00000 q^{64} -0.218247 q^{65} -1.15988 q^{66} -4.65335 q^{67} +0.739422 q^{68} +0.500725 q^{69} +0.537013 q^{70} -5.25203 q^{71} +2.75173 q^{72} -14.8684 q^{73} -4.95610 q^{74} +2.48398 q^{75} -7.11297 q^{76} +10.2986 q^{77} -0.895875 q^{78} -2.10101 q^{79} +0.121384 q^{80} +6.82723 q^{81} +0.415845 q^{82} +14.2745 q^{83} +2.20437 q^{84} +0.0897538 q^{85} +11.3245 q^{86} +2.89203 q^{87} +2.32784 q^{88} +1.42660 q^{89} +0.334015 q^{90} +7.95448 q^{91} -1.00494 q^{92} -1.39747 q^{93} +10.4816 q^{94} -0.863398 q^{95} +0.498264 q^{96} +9.03301 q^{97} -12.5726 q^{98} +6.40560 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 46 q - 46 q^{2} + 8 q^{3} + 46 q^{4} + 14 q^{5} - 8 q^{6} + 28 q^{7} - 46 q^{8} + 58 q^{9}+O(q^{10}) $ 46 * q - 46 * q^2 + 8 * q^3 + 46 * q^4 + 14 * q^5 - 8 * q^6 + 28 * q^7 - 46 * q^8 + 58 * q^9	\( 46 q - 46 q^{2} + 8 q^{3} + 46 q^{4} + 14 q^{5} - 8 q^{6} + 28 q^{7} - 46 q^{8} + 58 q^{9} - 14 q^{10} - 6 q^{11} + 8 q^{12} + 37 q^{13} - 28 q^{14} + 9 q^{15} + 46 q^{16} + 6 q^{17} - 58 q^{18} + 18 q^{19} + 14 q^{20} + 19 q^{21} + 6 q^{22} - 4 q^{23} - 8 q^{24} + 86 q^{25} - 37 q^{26} + 32 q^{27} + 28 q^{28} + 15 q^{29} - 9 q^{30} + 18 q^{31} - 46 q^{32} + 37 q^{33} - 6 q^{34} - 2 q^{35} + 58 q^{36} + 74 q^{37} - 18 q^{38} - 3 q^{39} - 14 q^{40} - 18 q^{41} - 19 q^{42} + 25 q^{43} - 6 q^{44} + 94 q^{45} + 4 q^{46} + 18 q^{47} + 8 q^{48} + 92 q^{49} - 86 q^{50} - 10 q^{51} + 37 q^{52} + 17 q^{53} - 32 q^{54} + 37 q^{55} - 28 q^{56} + 43 q^{57} - 15 q^{58} - 24 q^{59} + 9 q^{60} + 46 q^{61} - 18 q^{62} + 80 q^{63} + 46 q^{64} + 24 q^{65} - 37 q^{66} + 61 q^{67} + 6 q^{68} + 59 q^{69} + 2 q^{70} - 8 q^{71} - 58 q^{72} + 101 q^{73} - 74 q^{74} + 34 q^{75} + 18 q^{76} + 40 q^{77} + 3 q^{78} + 9 q^{79} + 14 q^{80} + 58 q^{81} + 18 q^{82} + 18 q^{83} + 19 q^{84} + 60 q^{85} - 25 q^{86} + 20 q^{87} + 6 q^{88} - 25 q^{89} - 94 q^{90} + 51 q^{91} - 4 q^{92} + 63 q^{93} - 18 q^{94} - 31 q^{95} - 8 q^{96} + 76 q^{97} - 92 q^{98} + 21 q^{99}+O(q^{100}) \) 46 * q - 46 * q^2 + 8 * q^3 + 46 * q^4 + 14 * q^5 - 8 * q^6 + 28 * q^7 - 46 * q^8 + 58 * q^9 - 14 * q^10 - 6 * q^11 + 8 * q^12 + 37 * q^13 - 28 * q^14 + 9 * q^15 + 46 * q^16 + 6 * q^17 - 58 * q^18 + 18 * q^19 + 14 * q^20 + 19 * q^21 + 6 * q^22 - 4 * q^23 - 8 * q^24 + 86 * q^25 - 37 * q^26 + 32 * q^27 + 28 * q^28 + 15 * q^29 - 9 * q^30 + 18 * q^31 - 46 * q^32 + 37 * q^33 - 6 * q^34 - 2 * q^35 + 58 * q^36 + 74 * q^37 - 18 * q^38 - 3 * q^39 - 14 * q^40 - 18 * q^41 - 19 * q^42 + 25 * q^43 - 6 * q^44 + 94 * q^45 + 4 * q^46 + 18 * q^47 + 8 * q^48 + 92 * q^49 - 86 * q^50 - 10 * q^51 + 37 * q^52 + 17 * q^53 - 32 * q^54 + 37 * q^55 - 28 * q^56 + 43 * q^57 - 15 * q^58 - 24 * q^59 + 9 * q^60 + 46 * q^61 - 18 * q^62 + 80 * q^63 + 46 * q^64 + 24 * q^65 - 37 * q^66 + 61 * q^67 + 6 * q^68 + 59 * q^69 + 2 * q^70 - 8 * q^71 - 58 * q^72 + 101 * q^73 - 74 * q^74 + 34 * q^75 + 18 * q^76 + 40 * q^77 + 3 * q^78 + 9 * q^79 + 14 * q^80 + 58 * q^81 + 18 * q^82 + 18 * q^83 + 19 * q^84 + 60 * q^85 - 25 * q^86 + 20 * q^87 + 6 * q^88 - 25 * q^89 - 94 * q^90 + 51 * q^91 - 4 * q^92 + 63 * q^93 - 18 * q^94 - 31 * q^95 - 8 * q^96 + 76 * q^97 - 92 * q^98 + 21 * 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
$3$	−0.498264	−0.287673	−0.143836	−	0.989601i	$-0.545944\pi$
$3$	−0.498264	−0.287673	−0.143836	+	0.989601i	$0.545944\pi$
$4$	1.00000	0.500000
$5$	0.121384	0.0542844	0.0271422	−	0.999632i	$-0.491359\pi$
$5$	0.121384	0.0542844	0.0271422	+	0.999632i	$0.491359\pi$
$6$	0.498264	0.203415
$7$	−4.42409	−1.67215	−0.836075	−	0.548615i	$-0.815155\pi$
$7$	−4.42409	−1.67215	−0.836075	+	0.548615i	$0.815155\pi$
$8$	−1.00000	−0.353553
$9$	−2.75173	−0.917244
$10$	−0.121384	−0.0383849
$11$	−2.32784	−0.701871	−0.350935	−	0.936400i	$-0.614136\pi$
$11$	−2.32784	−0.701871	−0.350935	+	0.936400i	$0.614136\pi$
$12$	−0.498264	−0.143836
$13$	−1.79799	−0.498673	−0.249337	−	0.968417i	$-0.580213\pi$
$13$	−1.79799	−0.498673	−0.249337	+	0.968417i	$0.580213\pi$
$14$	4.42409	1.18239
$15$	−0.0604811	−0.0156162
$16$	1.00000	0.250000
$17$	0.739422	0.179336	0.0896681	−	0.995972i	$-0.471419\pi$
$17$	0.739422	0.179336	0.0896681	+	0.995972i	$0.471419\pi$
$18$	2.75173	0.648590
$19$	−7.11297	−1.63183	−0.815913	−	0.578174i	$-0.803765\pi$
$19$	−7.11297	−1.63183	−0.815913	+	0.578174i	$0.803765\pi$
$20$	0.121384	0.0271422
$21$	2.20437	0.481032
$22$	2.32784	0.496298
$23$	−1.00494	−0.209544	−0.104772	−	0.994496i	$-0.533411\pi$
$23$	−1.00494	−0.209544	−0.104772	+	0.994496i	$0.533411\pi$
$24$	0.498264	0.101708
$25$	−4.98527	−0.997053
$26$	1.79799	0.352615
$27$	2.86588	0.551539
$28$	−4.42409	−0.836075
$29$	−5.80422	−1.07782	−0.538908	−	0.842364i	$-0.681163\pi$
$29$	−5.80422	−1.07782	−0.538908	+	0.842364i	$0.681163\pi$
$30$	0.0604811	0.0110423
$31$	2.80468	0.503735	0.251868	−	0.967762i	$-0.418955\pi$
$31$	2.80468	0.503735	0.251868	+	0.967762i	$0.418955\pi$
$32$	−1.00000	−0.176777
$33$	1.15988	0.201909
$34$	−0.739422	−0.126810
$35$	−0.537013	−0.0907717
$36$	−2.75173	−0.458622
$37$	4.95610	0.814777	0.407389	−	0.913255i	$-0.366439\pi$
$37$	4.95610	0.814777	0.407389	+	0.913255i	$0.366439\pi$
$38$	7.11297	1.15388
$39$	0.895875	0.143455
$40$	−0.121384	−0.0191924
$41$	−0.415845	−0.0649440	−0.0324720	−	0.999473i	$-0.510338\pi$
$41$	−0.415845	−0.0649440	−0.0324720	+	0.999473i	$0.510338\pi$
$42$	−2.20437	−0.340141
$43$	−11.3245	−1.72697	−0.863484	−	0.504377i	$-0.831722\pi$
$43$	−11.3245	−1.72697	−0.863484	+	0.504377i	$0.831722\pi$
$44$	−2.32784	−0.350935
$45$	−0.334015	−0.0497921
$46$	1.00494	0.148170
$47$	−10.4816	−1.52891	−0.764453	−	0.644680i	$-0.776991\pi$
$47$	−10.4816	−1.52891	−0.764453	+	0.644680i	$0.776991\pi$
$48$	−0.498264	−0.0719182
$49$	12.5726	1.79609
$50$	4.98527	0.705023
$51$	−0.368427	−0.0515902
$52$	−1.79799	−0.249337
$53$	−0.00603078	−0.000828391 0	−0.000414195	−	1.00000i	$-0.500132\pi$
$53$	−0.00603078	−0.000828391 0	−0.000414195		1.00000i	$0.500132\pi$
$54$	−2.86588	−0.389997
$55$	−0.282562	−0.0381007
$56$	4.42409	0.591194
$57$	3.54414	0.469432
$58$	5.80422	0.762132
$59$	−0.658526	−0.0857328	−0.0428664	−	0.999081i	$-0.513649\pi$
$59$	−0.658526	−0.0857328	−0.0428664	+	0.999081i	$0.513649\pi$
$60$	−0.0604811	−0.00780808
$61$	−1.30058	−0.166522	−0.0832609	−	0.996528i	$-0.526533\pi$
$61$	−1.30058	−0.166522	−0.0832609	+	0.996528i	$0.526533\pi$
$62$	−2.80468	−0.356195
$63$	12.1739	1.53377
$64$	1.00000	0.125000
$65$	−0.218247	−0.0270702
$66$	−1.15988	−0.142771
$67$	−4.65335	−0.568498	−0.284249	−	0.958751i	$-0.591744\pi$
$67$	−4.65335	−0.568498	−0.284249	+	0.958751i	$0.591744\pi$
$68$	0.739422	0.0896681
$69$	0.500725	0.0602802
$70$	0.537013	0.0641853
$71$	−5.25203	−0.623301	−0.311650	−	0.950197i	$-0.600882\pi$
$71$	−5.25203	−0.623301	−0.311650	+	0.950197i	$0.600882\pi$
$72$	2.75173	0.324295
$73$	−14.8684	−1.74021	−0.870105	−	0.492867i	$-0.835949\pi$
$73$	−14.8684	−1.74021	−0.870105	+	0.492867i	$0.835949\pi$
$74$	−4.95610	−0.576135
$75$	2.48398	0.286825
$76$	−7.11297	−0.815913
$77$	10.2986	1.17363
$78$	−0.895875	−0.101438
$79$	−2.10101	−0.236382	−0.118191	−	0.992991i	$-0.537710\pi$
$79$	−2.10101	−0.236382	−0.118191	+	0.992991i	$0.537710\pi$
$80$	0.121384	0.0135711
$81$	6.82723	0.758581
$82$	0.415845	0.0459224
$83$	14.2745	1.56683	0.783417	−	0.621496i	$-0.213475\pi$
$83$	14.2745	1.56683	0.783417	+	0.621496i	$0.213475\pi$
$84$	2.20437	0.240516
$85$	0.0897538	0.00973516
$86$	11.3245	1.22115
$87$	2.89203	0.310059
$88$	2.32784	0.248149
$89$	1.42660	0.151220	0.0756099	−	0.997137i	$-0.475910\pi$
$89$	1.42660	0.151220	0.0756099	+	0.997137i	$0.475910\pi$
$90$	0.334015	0.0352083
$91$	7.95448	0.833856
$92$	−1.00494	−0.104772
$93$	−1.39747	−0.144911
$94$	10.4816	1.08110
$95$	−0.863398	−0.0885827
$96$	0.498264	0.0508539
$97$	9.03301	0.917163	0.458582	−	0.888652i	$-0.348358\pi$
$97$	9.03301	0.917163	0.458582	+	0.888652i	$0.348358\pi$
$98$	−12.5726	−1.27002
$99$	6.40560	0.643787
$100$	−4.98527	−0.498527
$101$	0.380575	0.0378686	0.0189343	−	0.999821i	$-0.493973\pi$
$101$	0.380575	0.0378686	0.0189343	+	0.999821i	$0.493973\pi$
$102$	0.368427	0.0364797
$103$	−2.11308	−0.208208	−0.104104	−	0.994566i	$-0.533197\pi$
$103$	−2.11308	−0.208208	−0.104104	+	0.994566i	$0.533197\pi$
$104$	1.79799	0.176308
$105$	0.267574	0.0261125
$106$	0.00603078	0.000585761 0
$107$	1.61586	0.156211	0.0781056	−	0.996945i	$-0.475113\pi$
$107$	1.61586	0.156211	0.0781056	+	0.996945i	$0.475113\pi$
$108$	2.86588	0.275770
$109$	−2.04762	−0.196126	−0.0980631	−	0.995180i	$-0.531265\pi$
$109$	−2.04762	−0.196126	−0.0980631	+	0.995180i	$0.531265\pi$
$110$	0.282562	0.0269412
$111$	−2.46944	−0.234389
$112$	−4.42409	−0.418037
$113$	10.2108	0.960555	0.480278	−	0.877117i	$-0.340536\pi$
$113$	10.2108	0.960555	0.480278	+	0.877117i	$0.340536\pi$
$114$	−3.54414	−0.331939
$115$	−0.121983	−0.0113750
$116$	−5.80422	−0.538908
$117$	4.94759	0.457405
$118$	0.658526	0.0606223
$119$	−3.27127	−0.299877
$120$	0.0604811	0.00552114
$121$	−5.58115	−0.507377
$122$	1.30058	0.117749
$123$	0.207200	0.0186826
$124$	2.80468	0.251868
$125$	−1.21205	−0.108409
$126$	−12.1739	−1.08454
$127$	17.0737	1.51505	0.757523	−	0.652808i	$-0.226409\pi$
$127$	17.0737	1.51505	0.757523	+	0.652808i	$0.226409\pi$
$128$	−1.00000	−0.0883883
$129$	5.64258	0.496802
$130$	0.218247	0.0191415
$131$	5.82775	0.509173	0.254586	−	0.967050i	$-0.418061\pi$
$131$	5.82775	0.509173	0.254586	+	0.967050i	$0.418061\pi$
$132$	1.15988	0.100955
$133$	31.4684	2.72866
$134$	4.65335	0.401989
$135$	0.347871	0.0299400
$136$	−0.739422	−0.0634049
$137$	12.5303	1.07054	0.535268	−	0.844682i	$-0.320211\pi$
$137$	12.5303	1.07054	0.535268	+	0.844682i	$0.320211\pi$
$138$	−0.500725	−0.0426245
$139$	−4.19061	−0.355443	−0.177721	−	0.984081i	$-0.556873\pi$
$139$	−4.19061	−0.355443	−0.177721	+	0.984081i	$0.556873\pi$
$140$	−0.537013	−0.0453858
$141$	5.22263	0.439824
$142$	5.25203	0.440740
$143$	4.18544	0.350004
$144$	−2.75173	−0.229311
$145$	−0.704538	−0.0585087
$146$	14.8684	1.23051
$147$	−6.26447	−0.516685
$148$	4.95610	0.407389
$149$	8.50006	0.696352	0.348176	−	0.937429i	$-0.386801\pi$
$149$	8.50006	0.696352	0.348176	+	0.937429i	$0.386801\pi$
$150$	−2.48398	−0.202816
$151$	−18.1596	−1.47781	−0.738905	−	0.673810i	$-0.764657\pi$
$151$	−18.1596	−1.47781	−0.738905	+	0.673810i	$0.764657\pi$
$152$	7.11297	0.576938
$153$	−2.03469	−0.164495
$154$	−10.2986	−0.829884
$155$	0.340442	0.0273450
$156$	0.895875	0.0717274
$157$	2.94342	0.234910	0.117455	−	0.993078i	$-0.462526\pi$
$157$	2.94342	0.234910	0.117455	+	0.993078i	$0.462526\pi$
$158$	2.10101	0.167147
$159$	0.00300492	0.000238306 0
$160$	−0.121384	−0.00959622
$161$	4.44594	0.350389
$162$	−6.82723	−0.536398
$163$	6.04077	0.473149	0.236575	−	0.971613i	$-0.423975\pi$
$163$	6.04077	0.473149	0.236575	+	0.971613i	$0.423975\pi$
$164$	−0.415845	−0.0324720
$165$	0.140790	0.0109605
$166$	−14.2745	−1.10792
$167$	−20.4539	−1.58277	−0.791386	−	0.611317i	$-0.790640\pi$
$167$	−20.4539	−1.58277	−0.791386	+	0.611317i	$0.790640\pi$
$168$	−2.20437	−0.170071
$169$	−9.76723	−0.751325
$170$	−0.0897538	−0.00688380
$171$	19.5730	1.49678
$172$	−11.3245	−0.863484
$173$	−10.9229	−0.830450	−0.415225	−	0.909719i	$-0.636297\pi$
$173$	−10.9229	−0.830450	−0.415225	+	0.909719i	$0.636297\pi$
$174$	−2.89203	−0.219245
$175$	22.0553	1.66722
$176$	−2.32784	−0.175468
$177$	0.328120	0.0246630
$178$	−1.42660	−0.106928
$179$	−12.3706	−0.924626	−0.462313	−	0.886717i	$-0.652980\pi$
$179$	−12.3706	−0.924626	−0.462313	+	0.886717i	$0.652980\pi$
$180$	−0.334015	−0.0248960
$181$	−3.38109	−0.251314	−0.125657	−	0.992074i	$-0.540104\pi$
$181$	−3.38109	−0.251314	−0.125657	+	0.992074i	$0.540104\pi$
$182$	−7.95448	−0.589625
$183$	0.648031	0.0479038
$184$	1.00494	0.0740850
$185$	0.601589	0.0442297
$186$	1.39747	0.102468
$187$	−1.72126	−0.125871
$188$	−10.4816	−0.764453
$189$	−12.6789	−0.922256
$190$	0.863398	0.0626375
$191$	10.3823	0.751239	0.375619	−	0.926774i	$-0.377430\pi$
$191$	10.3823	0.751239	0.375619	+	0.926774i	$0.377430\pi$
$192$	−0.498264	−0.0359591
$193$	11.0243	0.793544	0.396772	−	0.917917i	$-0.370130\pi$
$193$	11.0243	0.793544	0.396772	+	0.917917i	$0.370130\pi$
$194$	−9.03301	−0.648532
$195$	0.108745	0.00778736
$196$	12.5726	0.898043
$197$	0.569735	0.0405919	0.0202960	−	0.999794i	$-0.493539\pi$
$197$	0.569735	0.0405919	0.0202960	+	0.999794i	$0.493539\pi$
$198$	−6.40560	−0.455226
$199$	−13.1208	−0.930110	−0.465055	−	0.885282i	$-0.653965\pi$
$199$	−13.1208	−0.930110	−0.465055	+	0.885282i	$0.653965\pi$
$200$	4.98527	0.352512
$201$	2.31860	0.163541
$202$	−0.380575	−0.0267772
$203$	25.6784	1.80227
$204$	−0.368427	−0.0257951
$205$	−0.0504768	−0.00352545
$206$	2.11308	0.147225
$207$	2.76532	0.192203
$208$	−1.79799	−0.124668
$209$	16.5579	1.14533
$210$	−0.267574	−0.0184644
$211$	3.88386	0.267376	0.133688	−	0.991023i	$-0.457318\pi$
$211$	3.88386	0.267376	0.133688	+	0.991023i	$0.457318\pi$
$212$	−0.00603078	−0.000414195 0
$213$	2.61690	0.179307
$214$	−1.61586	−0.110458
$215$	−1.37461	−0.0937474
$216$	−2.86588	−0.194999
$217$	−12.4082	−0.842321
$218$	2.04762	0.138682
$219$	7.40837	0.500611
$220$	−0.282562	−0.0190503
$221$	−1.32947	−0.0894302
$222$	2.46944	0.165738
$223$	13.4557	0.901058	0.450529	−	0.892762i	$-0.351235\pi$
$223$	13.4557	0.901058	0.450529	+	0.892762i	$0.351235\pi$
$224$	4.42409	0.295597
$225$	13.7181	0.914541
$226$	−10.2108	−0.679215
$227$	−2.55028	−0.169268	−0.0846340	−	0.996412i	$-0.526972\pi$
$227$	−2.55028	−0.169268	−0.0846340	+	0.996412i	$0.526972\pi$
$228$	3.54414	0.234716
$229$	−0.476939	−0.0315170	−0.0157585	−	0.999876i	$-0.505016\pi$
$229$	−0.476939	−0.0315170	−0.0157585	+	0.999876i	$0.505016\pi$
$230$	0.121983	0.00804333
$231$	−5.13142	−0.337622
$232$	5.80422	0.381066
$233$	−14.6171	−0.957601	−0.478801	−	0.877924i	$-0.658928\pi$
$233$	−14.6171	−0.957601	−0.478801	+	0.877924i	$0.658928\pi$
$234$	−4.94759	−0.323434
$235$	−1.27230	−0.0829957
$236$	−0.658526	−0.0428664
$237$	1.04686	0.0680007
$238$	3.27127	0.212045
$239$	−16.4596	−1.06468	−0.532341	−	0.846530i	$-0.678687\pi$
$239$	−16.4596	−1.06468	−0.532341	+	0.846530i	$0.678687\pi$
$240$	−0.0604811	−0.00390404
$241$	−12.5624	−0.809218	−0.404609	−	0.914490i	$-0.632592\pi$
$241$	−12.5624	−0.809218	−0.404609	+	0.914490i	$0.632592\pi$
$242$	5.58115	0.358770
$243$	−11.9994	−0.769762
$244$	−1.30058	−0.0832609
$245$	1.52611	0.0974994
$246$	−0.207200	−0.0132106
$247$	12.7891	0.813748
$248$	−2.80468	−0.178097
$249$	−7.11249	−0.450736
$250$	1.21205	0.0766566
$251$	−8.95060	−0.564957	−0.282478	−	0.959274i	$-0.591157\pi$
$251$	−8.95060	−0.564957	−0.282478	+	0.959274i	$0.591157\pi$
$252$	12.1739	0.766885
$253$	2.33934	0.147073
$254$	−17.0737	−1.07130
$255$	−0.0447211	−0.00280054
$256$	1.00000	0.0625000
$257$	−3.15026	−0.196508	−0.0982538	−	0.995161i	$-0.531326\pi$
$257$	−3.15026	−0.196508	−0.0982538	+	0.995161i	$0.531326\pi$
$258$	−5.64258	−0.351292
$259$	−21.9262	−1.36243
$260$	−0.218247	−0.0135351
$261$	15.9717	0.988621
$262$	−5.82775	−0.360039
$263$	−22.6372	−1.39587	−0.697933	−	0.716163i	$-0.745897\pi$
$263$	−22.6372	−1.39587	−0.697933	+	0.716163i	$0.745897\pi$
$264$	−1.15988	−0.0713857
$265$	−0.000732038 0	−4.49687e−5 0
$266$	−31.4684	−1.92945
$267$	−0.710825	−0.0435018
$268$	−4.65335	−0.284249
$269$	−1.12543	−0.0686189	−0.0343095	−	0.999411i	$-0.510923\pi$
$269$	−1.12543	−0.0686189	−0.0343095	+	0.999411i	$0.510923\pi$
$270$	−0.347871	−0.0211708
$271$	20.3082	1.23363	0.616817	−	0.787106i	$-0.288422\pi$
$271$	20.3082	1.23363	0.616817	+	0.787106i	$0.288422\pi$
$272$	0.739422	0.0448340
$273$	−3.96343	−0.239878
$274$	−12.5303	−0.756983
$275$	11.6049	0.699803
$276$	0.500725	0.0301401
$277$	−9.85326	−0.592025	−0.296012	−	0.955184i	$-0.595657\pi$
$277$	−9.85326	−0.592025	−0.296012	+	0.955184i	$0.595657\pi$
$278$	4.19061	0.251336
$279$	−7.71773	−0.462048
$280$	0.537013	0.0320926
$281$	27.4801	1.63933	0.819663	−	0.572846i	$-0.194161\pi$
$281$	27.4801	1.63933	0.819663	+	0.572846i	$0.194161\pi$
$282$	−5.22263	−0.311003
$283$	−3.94891	−0.234738	−0.117369	−	0.993088i	$-0.537446\pi$
$283$	−3.94891	−0.234738	−0.117369	+	0.993088i	$0.537446\pi$
$284$	−5.25203	−0.311650
$285$	0.430200	0.0254829
$286$	−4.18544	−0.247490
$287$	1.83974	0.108596
$288$	2.75173	0.162147
$289$	−16.4533	−0.967839
$290$	0.704538	0.0413719
$291$	−4.50082	−0.263843
$292$	−14.8684	−0.870105
$293$	−3.12408	−0.182511	−0.0912553	−	0.995828i	$-0.529088\pi$
$293$	−3.12408	−0.182511	−0.0912553	+	0.995828i	$0.529088\pi$
$294$	6.26447	0.365351
$295$	−0.0799343	−0.00465396
$296$	−4.95610	−0.288067
$297$	−6.67132	−0.387109
$298$	−8.50006	−0.492395
$299$	1.80687	0.104494
$300$	2.48398	0.143413
$301$	50.1006	2.88775
$302$	18.1596	1.04497
$303$	−0.189627	−0.0108938
$304$	−7.11297	−0.407957
$305$	−0.157869	−0.00903954
$306$	2.03469	0.116316
$307$	−7.65897	−0.437120	−0.218560	−	0.975823i	$-0.570136\pi$
$307$	−7.65897	−0.437120	−0.218560	+	0.975823i	$0.570136\pi$
$308$	10.2986	0.586817
$309$	1.05287	0.0598957
$310$	−0.340442	−0.0193358
$311$	−25.8455	−1.46556	−0.732782	−	0.680463i	$-0.761779\pi$
$311$	−25.8455	−1.46556	−0.732782	+	0.680463i	$0.761779\pi$
$312$	−0.895875	−0.0507189
$313$	21.3604	1.20736	0.603680	−	0.797226i	$-0.293700\pi$
$313$	21.3604	1.20736	0.603680	+	0.797226i	$0.293700\pi$
$314$	−2.94342	−0.166107
$315$	1.47772	0.0832598
$316$	−2.10101	−0.118191
$317$	13.4217	0.753838	0.376919	−	0.926246i	$-0.376984\pi$
$317$	13.4217	0.753838	0.376919	+	0.926246i	$0.376984\pi$
$318$	−0.00300492	−0.000168507 0
$319$	13.5113	0.756488
$320$	0.121384	0.00678555
$321$	−0.805125	−0.0449377
$322$	−4.44594	−0.247763
$323$	−5.25948	−0.292646
$324$	6.82723	0.379291
$325$	8.96347	0.497204
$326$	−6.04077	−0.334567
$327$	1.02025	0.0564202
$328$	0.415845	0.0229612
$329$	46.3718	2.55656
$330$	−0.140790	−0.00775026
$331$	33.7717	1.85626	0.928131	−	0.372253i	$-0.121415\pi$
$331$	33.7717	1.85626	0.928131	+	0.372253i	$0.121415\pi$
$332$	14.2745	0.783417
$333$	−13.6379	−0.747350
$334$	20.4539	1.11919
$335$	−0.564841	−0.0308606
$336$	2.20437	0.120258
$337$	20.9826	1.14299	0.571497	−	0.820604i	$-0.306363\pi$
$337$	20.9826	1.14299	0.571497	+	0.820604i	$0.306363\pi$
$338$	9.76723	0.531267
$339$	−5.08769	−0.276326
$340$	0.0897538	0.00486758
$341$	−6.52885	−0.353557
$342$	−19.5730	−1.05839
$343$	−24.6537	−1.33117
$344$	11.3245	0.610575
$345$	0.0607798	0.00327227
$346$	10.9229	0.587217
$347$	5.95733	0.319806	0.159903	−	0.987133i	$-0.448882\pi$
$347$	5.95733	0.319806	0.159903	+	0.987133i	$0.448882\pi$
$348$	2.89203	0.155029
$349$	30.4328	1.62903	0.814516	−	0.580141i	$-0.197002\pi$
$349$	30.4328	1.62903	0.814516	+	0.580141i	$0.197002\pi$
$350$	−22.0553	−1.17890
$351$	−5.15283	−0.275038
$352$	2.32784	0.124074
$353$	1.57223	0.0836816	0.0418408	−	0.999124i	$-0.486678\pi$
$353$	1.57223	0.0836816	0.0418408	+	0.999124i	$0.486678\pi$
$354$	−0.328120	−0.0174394
$355$	−0.637510	−0.0338355
$356$	1.42660	0.0756099
$357$	1.62996	0.0862665
$358$	12.3706	0.653809
$359$	−21.4922	−1.13432	−0.567159	−	0.823609i	$-0.691957\pi$
$359$	−21.4922	−1.13432	−0.567159	+	0.823609i	$0.691957\pi$
$360$	0.334015	0.0176042
$361$	31.5943	1.66286
$362$	3.38109	0.177706
$363$	2.78089	0.145959
$364$	7.95448	0.416928
$365$	−1.80478	−0.0944663
$366$	−0.648031	−0.0338731
$367$	16.5830	0.865626	0.432813	−	0.901484i	$-0.357521\pi$
$367$	16.5830	0.865626	0.432813	+	0.901484i	$0.357521\pi$
$368$	−1.00494	−0.0523860
$369$	1.14429	0.0595696
$370$	−0.601589	−0.0312751
$371$	0.0266807	0.00138519
$372$	−1.39747	−0.0724555
$373$	−3.18625	−0.164978	−0.0824889	−	0.996592i	$-0.526287\pi$
$373$	−3.18625	−0.164978	−0.0824889	+	0.996592i	$0.526287\pi$
$374$	1.72126	0.0890041
$375$	0.603920	0.0311863
$376$	10.4816	0.540550
$377$	10.4359	0.537478
$378$	12.6789	0.652134
$379$	−9.92971	−0.510055	−0.255027	−	0.966934i	$-0.582084\pi$
$379$	−9.92971	−0.510055	−0.255027	+	0.966934i	$0.582084\pi$
$380$	−0.863398	−0.0442914
$381$	−8.50721	−0.435838
$382$	−10.3823	−0.531206
$383$	−5.75503	−0.294068	−0.147034	−	0.989131i	$-0.546973\pi$
$383$	−5.75503	−0.294068	−0.147034	+	0.989131i	$0.546973\pi$
$384$	0.498264	0.0254269
$385$	1.25008	0.0637100
$386$	−11.0243	−0.561120
$387$	31.1620	1.58405
$388$	9.03301	0.458582
$389$	−6.66472	−0.337915	−0.168957	−	0.985623i	$-0.554040\pi$
$389$	−6.66472	−0.337915	−0.168957	+	0.985623i	$0.554040\pi$
$390$	−0.108745	−0.00550649
$391$	−0.743074	−0.0375788
$392$	−12.5726	−0.635012
$393$	−2.90376	−0.146475
$394$	−0.569735	−0.0287028
$395$	−0.255028	−0.0128319
$396$	6.40560	0.321894
$397$	10.5735	0.530669	0.265334	−	0.964156i	$-0.414518\pi$
$397$	10.5735	0.530669	0.265334	+	0.964156i	$0.414518\pi$
$398$	13.1208	0.657687
$399$	−15.6796	−0.784961
$400$	−4.98527	−0.249263
$401$	4.27734	0.213600	0.106800	−	0.994281i	$-0.465939\pi$
$401$	4.27734	0.213600	0.106800	+	0.994281i	$0.465939\pi$
$402$	−2.31860	−0.115641
$403$	−5.04279	−0.251199
$404$	0.380575	0.0189343
$405$	0.828715	0.0411792
$406$	−25.6784	−1.27440
$407$	−11.5370	−0.571869
$408$	0.368427	0.0182399
$409$	26.1250	1.29180	0.645899	−	0.763423i	$-0.276483\pi$
$409$	26.1250	1.29180	0.645899	+	0.763423i	$0.276483\pi$
$410$	0.0504768	0.00249287
$411$	−6.24340	−0.307964
$412$	−2.11308	−0.104104
$413$	2.91338	0.143358
$414$	−2.76532	−0.135908
$415$	1.73270	0.0850547
$416$	1.79799	0.0881538
$417$	2.08803	0.102251
$418$	−16.5579	−0.809872
$419$	2.69334	0.131578	0.0657891	−	0.997834i	$-0.479044\pi$
$419$	2.69334	0.131578	0.0657891	+	0.997834i	$0.479044\pi$
$420$	0.267574	0.0130563
$421$	3.69940	0.180298	0.0901488	−	0.995928i	$-0.471266\pi$
$421$	3.69940	0.180298	0.0901488	+	0.995928i	$0.471266\pi$
$422$	−3.88386	−0.189063
$423$	28.8427	1.40238
$424$	0.00603078	0.000292880 0
$425$	−3.68622	−0.178808
$426$	−2.61690	−0.126789
$427$	5.75387	0.278449
$428$	1.61586	0.0781056
$429$	−2.08546	−0.100687
$430$	1.37461	0.0662894
$431$	13.6394	0.656987	0.328493	−	0.944506i	$-0.393459\pi$
$431$	13.6394	0.656987	0.328493	+	0.944506i	$0.393459\pi$
$432$	2.86588	0.137885
$433$	14.4233	0.693141	0.346570	−	0.938024i	$-0.387346\pi$
$433$	14.4233	0.693141	0.346570	+	0.938024i	$0.387346\pi$
$434$	12.4082	0.595611
$435$	0.351046	0.0168314
$436$	−2.04762	−0.0980631
$437$	7.14809	0.341940
$438$	−7.40837	−0.353985
$439$	6.21599	0.296673	0.148337	−	0.988937i	$-0.452608\pi$
$439$	6.21599	0.296673	0.148337	+	0.988937i	$0.452608\pi$
$440$	0.282562	0.0134706
$441$	−34.5964	−1.64745
$442$	1.32947	0.0632367
$443$	−8.25744	−0.392323	−0.196161	−	0.980572i	$-0.562848\pi$
$443$	−8.25744	−0.392323	−0.196161	+	0.980572i	$0.562848\pi$
$444$	−2.46944	−0.117195
$445$	0.173166	0.00820887
$446$	−13.4557	−0.637144
$447$	−4.23527	−0.200321
$448$	−4.42409	−0.209019
$449$	−20.6620	−0.975099	−0.487549	−	0.873095i	$-0.662109\pi$
$449$	−20.6620	−0.975099	−0.487549	+	0.873095i	$0.662109\pi$
$450$	−13.7181	−0.646678
$451$	0.968021	0.0455823
$452$	10.2108	0.480278
$453$	9.04829	0.425126
$454$	2.55028	0.119691
$455$	0.965544	0.0452654
$456$	−3.54414	−0.165969
$457$	5.46396	0.255593	0.127797	−	0.991800i	$-0.459210\pi$
$457$	5.46396	0.255593	0.127797	+	0.991800i	$0.459210\pi$
$458$	0.476939	0.0222859
$459$	2.11910	0.0989109
$460$	−0.121983	−0.00568749
$461$	−17.4900	−0.814592	−0.407296	−	0.913296i	$-0.633528\pi$
$461$	−17.4900	−0.814592	−0.407296	+	0.913296i	$0.633528\pi$
$462$	5.13142	0.238735
$463$	20.6181	0.958203	0.479101	−	0.877760i	$-0.340963\pi$
$463$	20.6181	0.958203	0.479101	+	0.877760i	$0.340963\pi$
$464$	−5.80422	−0.269454
$465$	−0.169630	−0.00786641
$466$	14.6171	0.677126
$467$	−30.7962	−1.42508	−0.712539	−	0.701632i	$-0.752455\pi$
$467$	−30.7962	−1.42508	−0.712539	+	0.701632i	$0.752455\pi$
$468$	4.94759	0.228703
$469$	20.5869	0.950613
$470$	1.27230	0.0586868
$471$	−1.46660	−0.0675773
$472$	0.658526	0.0303111
$473$	26.3616	1.21211
$474$	−1.04686	−0.0480837
$475$	35.4600	1.62702
$476$	−3.27127	−0.149938
$477$	0.0165951	0.000759837 0
$478$	16.4596	0.752843
$479$	−10.6221	−0.485336	−0.242668	−	0.970109i	$-0.578023\pi$
$479$	−10.6221	−0.485336	−0.242668	+	0.970109i	$0.578023\pi$
$480$	0.0604811	0.00276057
$481$	−8.91102	−0.406308
$482$	12.5624	0.572204
$483$	−2.21525	−0.100797
$484$	−5.58115	−0.253689
$485$	1.09646	0.0497877
$486$	11.9994	0.544304
$487$	8.33254	0.377583	0.188792	−	0.982017i	$-0.439543\pi$
$487$	8.33254	0.377583	0.188792	+	0.982017i	$0.439543\pi$
$488$	1.30058	0.0588743
$489$	−3.00990	−0.136112
$490$	−1.52611	−0.0689425
$491$	−27.6045	−1.24577	−0.622887	−	0.782312i	$-0.714040\pi$
$491$	−27.6045	−1.24577	−0.622887	+	0.782312i	$0.714040\pi$
$492$	0.207200	0.00934132
$493$	−4.29177	−0.193292
$494$	−12.7891	−0.575407
$495$	0.777535	0.0349476
$496$	2.80468	0.125934
$497$	23.2354	1.04225
$498$	7.11249	0.318718
$499$	10.0868	0.451548	0.225774	−	0.974180i	$-0.427509\pi$
$499$	10.0868	0.451548	0.225774	+	0.974180i	$0.427509\pi$
$500$	−1.21205	−0.0542044
$501$	10.1914	0.455320
$502$	8.95060	0.399485
$503$	−13.9066	−0.620065	−0.310033	−	0.950726i	$-0.600340\pi$
$503$	−13.9066	−0.620065	−0.310033	+	0.950726i	$0.600340\pi$
$504$	−12.1739	−0.542270
$505$	0.0461956	0.00205568
$506$	−2.33934	−0.103996
$507$	4.86666	0.216136
$508$	17.0737	0.757523
$509$	16.7969	0.744508	0.372254	−	0.928131i	$-0.378585\pi$
$509$	16.7969	0.744508	0.372254	+	0.928131i	$0.378585\pi$
$510$	0.0447211	0.00198028
$511$	65.7790	2.90989
$512$	−1.00000	−0.0441942
$513$	−20.3849	−0.900016
$514$	3.15026	0.138952
$515$	−0.256493	−0.0113024
$516$	5.64258	0.248401
$517$	24.3996	1.07309
$518$	21.9262	0.963383
$519$	5.44247	0.238898
$520$	0.218247	0.00957076
$521$	−31.3515	−1.37353	−0.686767	−	0.726877i	$-0.740971\pi$
$521$	−31.3515	−1.37353	−0.686767	+	0.726877i	$0.740971\pi$
$522$	−15.9717	−0.699061
$523$	−28.6805	−1.25411	−0.627055	−	0.778975i	$-0.715740\pi$
$523$	−28.6805	−1.25411	−0.627055	+	0.778975i	$0.715740\pi$
$524$	5.82775	0.254586
$525$	−10.9894	−0.479615
$526$	22.6372	0.987027
$527$	2.07384	0.0903380
$528$	1.15988	0.0504773
$529$	−21.9901	−0.956091
$530$	0.000732038 0	3.17977e−5 0
$531$	1.81209	0.0786380
$532$	31.4684	1.36433
$533$	0.747685	0.0323859
$534$	0.710825	0.0307604
$535$	0.196139	0.00847983
$536$	4.65335	0.200994
$537$	6.16385	0.265990
$538$	1.12543	0.0485209
$539$	−29.2670	−1.26062
$540$	0.347871	0.0149700
$541$	−34.4843	−1.48260	−0.741299	−	0.671175i	$-0.765790\pi$
$541$	−34.4843	−1.48260	−0.741299	+	0.671175i	$0.765790\pi$
$542$	−20.3082	−0.872311
$543$	1.68467	0.0722963
$544$	−0.739422	−0.0317025
$545$	−0.248547	−0.0106466
$546$	3.96343	0.169619
$547$	37.6121	1.60818	0.804089	−	0.594509i	$-0.202654\pi$
$547$	37.6121	1.60818	0.804089	+	0.594509i	$0.202654\pi$
$548$	12.5303	0.535268
$549$	3.57884	0.152741
$550$	−11.6049	−0.494835
$551$	41.2852	1.75881
$552$	−0.500725	−0.0213123
$553$	9.29506	0.395266
$554$	9.85326	0.418625
$555$	−0.299750	−0.0127237
$556$	−4.19061	−0.177721
$557$	−3.87680	−0.164265	−0.0821327	−	0.996621i	$-0.526173\pi$
$557$	−3.87680	−0.164265	−0.0821327	+	0.996621i	$0.526173\pi$
$558$	7.71773	0.326718
$559$	20.3613	0.861192
$560$	−0.537013	−0.0226929
$561$	0.857641	0.0362096
$562$	−27.4801	−1.15918
$563$	−9.62849	−0.405792	−0.202896	−	0.979200i	$-0.565035\pi$
$563$	−9.62849	−0.405792	−0.202896	+	0.979200i	$0.565035\pi$
$564$	5.22263	0.219912
$565$	1.23943	0.0521432
$566$	3.94891	0.165985
$567$	−30.2043	−1.26846
$568$	5.25203	0.220370
$569$	12.6805	0.531593	0.265796	−	0.964029i	$-0.414365\pi$
$569$	12.6805	0.531593	0.265796	+	0.964029i	$0.414365\pi$
$570$	−0.430200	−0.0180191
$571$	−19.7463	−0.826355	−0.413178	−	0.910650i	$-0.635581\pi$
$571$	−19.7463	−0.826355	−0.413178	+	0.910650i	$0.635581\pi$
$572$	4.18544	0.175002
$573$	−5.17314	−0.216111
$574$	−1.83974	−0.0767891
$575$	5.00988	0.208927
$576$	−2.75173	−0.114656
$577$	−4.78230	−0.199090	−0.0995450	−	0.995033i	$-0.531739\pi$
$577$	−4.78230	−0.199090	−0.0995450	+	0.995033i	$0.531739\pi$
$578$	16.4533	0.684365
$579$	−5.49299	−0.228281
$580$	−0.704538	−0.0292543
$581$	−63.1519	−2.61998
$582$	4.50082	0.186565
$583$	0.0140387	0.000581424 0
$584$	14.8684	0.615257
$585$	0.600557	0.0248300
$586$	3.12408	0.129054
$587$	−15.3346	−0.632927	−0.316463	−	0.948605i	$-0.602495\pi$
$587$	−15.3346	−0.632927	−0.316463	+	0.948605i	$0.602495\pi$
$588$	−6.26447	−0.258342
$589$	−19.9496	−0.822009
$590$	0.0799343	0.00329084
$591$	−0.283878	−0.0116772
$592$	4.95610	0.203694
$593$	−40.2505	−1.65289	−0.826444	−	0.563019i	$-0.809640\pi$
$593$	−40.2505	−1.65289	−0.826444	+	0.563019i	$0.809640\pi$
$594$	6.67132	0.273728
$595$	−0.397079	−0.0162786
$596$	8.50006	0.348176
$597$	6.53763	0.267567
$598$	−1.80687	−0.0738884
$599$	−5.29211	−0.216230	−0.108115	−	0.994138i	$-0.534481\pi$
$599$	−5.29211	−0.216230	−0.108115	+	0.994138i	$0.534481\pi$
$600$	−2.48398	−0.101408
$601$	−19.9001	−0.811742	−0.405871	−	0.913930i	$-0.633032\pi$
$601$	−19.9001	−0.811742	−0.405871	+	0.913930i	$0.633032\pi$
$602$	−50.1006	−2.04195
$603$	12.8048	0.521451
$604$	−18.1596	−0.738905
$605$	−0.677460	−0.0275427
$606$	0.189627	0.00770307
$607$	−15.6977	−0.637150	−0.318575	−	0.947898i	$-0.603204\pi$
$607$	−15.6977	−0.637150	−0.318575	+	0.947898i	$0.603204\pi$
$608$	7.11297	0.288469
$609$	−12.7946	−0.518465
$610$	0.157869	0.00639192
$611$	18.8459	0.762424
$612$	−2.03469	−0.0822476
$613$	3.00614	0.121417	0.0607084	−	0.998156i	$-0.480664\pi$
$613$	3.00614	0.121417	0.0607084	+	0.998156i	$0.480664\pi$
$614$	7.65897	0.309091
$615$	0.0251507	0.00101418
$616$	−10.2986	−0.414942
$617$	24.0509	0.968251	0.484126	−	0.874999i	$-0.339138\pi$
$617$	24.0509	0.968251	0.484126	+	0.874999i	$0.339138\pi$
$618$	−1.05287	−0.0423527
$619$	−12.2531	−0.492494	−0.246247	−	0.969207i	$-0.579197\pi$
$619$	−12.2531	−0.492494	−0.246247	+	0.969207i	$0.579197\pi$
$620$	0.340442	0.0136725
$621$	−2.88003	−0.115572
$622$	25.8455	1.03631
$623$	−6.31143	−0.252862
$624$	0.895875	0.0358637
$625$	24.7792	0.991168
$626$	−21.3604	−0.853733
$627$	−8.25019	−0.329481
$628$	2.94342	0.117455
$629$	3.66465	0.146119
$630$	−1.47772	−0.0588736
$631$	23.7089	0.943835	0.471918	−	0.881643i	$-0.343562\pi$
$631$	23.7089	0.943835	0.471918	+	0.881643i	$0.343562\pi$
$632$	2.10101	0.0835736
$633$	−1.93519	−0.0769168
$634$	−13.4217	−0.533044
$635$	2.07247	0.0822434
$636$	0.00300492	0.000119153 0
$637$	−22.6054	−0.895660
$638$	−13.5113	−0.534918
$639$	14.4522	0.571719
$640$	−0.121384	−0.00479811
$641$	−38.2267	−1.50986	−0.754932	−	0.655803i	$-0.772330\pi$
$641$	−38.2267	−1.50986	−0.754932	+	0.655803i	$0.772330\pi$
$642$	0.805125	0.0317758
$643$	−7.38082	−0.291071	−0.145536	−	0.989353i	$-0.546491\pi$
$643$	−7.38082	−0.291071	−0.145536	+	0.989353i	$0.546491\pi$
$644$	4.44594	0.175195
$645$	0.684917	0.0269686
$646$	5.25948	0.206932
$647$	−11.0553	−0.434627	−0.217313	−	0.976102i	$-0.569729\pi$
$647$	−11.0553	−0.434627	−0.217313	+	0.976102i	$0.569729\pi$
$648$	−6.82723	−0.268199
$649$	1.53295	0.0601734
$650$	−8.96347	−0.351576
$651$	6.18254	0.242313
$652$	6.04077	0.236575
$653$	42.1505	1.64948	0.824739	−	0.565514i	$-0.191322\pi$
$653$	42.1505	1.64948	0.824739	+	0.565514i	$0.191322\pi$
$654$	−1.02025	−0.0398951
$655$	0.707393	0.0276401
$656$	−0.415845	−0.0162360
$657$	40.9137	1.59620
$658$	−46.3718	−1.80776
$659$	1.57223	0.0612454	0.0306227	−	0.999531i	$-0.490251\pi$
$659$	1.57223	0.0612454	0.0306227	+	0.999531i	$0.490251\pi$
$660$	0.140790	0.00548026
$661$	10.5015	0.408461	0.204231	−	0.978923i	$-0.434531\pi$
$661$	10.5015	0.408461	0.204231	+	0.978923i	$0.434531\pi$
$662$	−33.7717	−1.31258
$663$	0.662429	0.0257266
$664$	−14.2745	−0.553960
$665$	3.81975	0.148124
$666$	13.6379	0.528456
$667$	5.83288	0.225850
$668$	−20.4539	−0.791386
$669$	−6.70447	−0.259210
$670$	0.564841	0.0218217
$671$	3.02754	0.116877
$672$	−2.20437	−0.0850353
$673$	2.12380	0.0818665	0.0409332	−	0.999162i	$-0.486967\pi$
$673$	2.12380	0.0818665	0.0409332	+	0.999162i	$0.486967\pi$
$674$	−20.9826	−0.808218
$675$	−14.2872	−0.549914
$676$	−9.76723	−0.375663
$677$	27.0289	1.03881	0.519403	−	0.854530i	$-0.326155\pi$
$677$	27.0289	1.03881	0.519403	+	0.854530i	$0.326155\pi$
$678$	5.08769	0.195392
$679$	−39.9629	−1.53363
$680$	−0.0897538	−0.00344190
$681$	1.27071	0.0486938
$682$	6.52885	0.250003
$683$	22.9301	0.877396	0.438698	−	0.898635i	$-0.355440\pi$
$683$	22.9301	0.877396	0.438698	+	0.898635i	$0.355440\pi$
$684$	19.5730	0.748392
$685$	1.52097	0.0581134
$686$	24.6537	0.941282
$687$	0.237641	0.00906659
$688$	−11.3245	−0.431742
$689$	0.0108433	0.000413096 0
$690$	−0.0607798	−0.00231385
$691$	−51.6623	−1.96533	−0.982664	−	0.185398i	$-0.940643\pi$
$691$	−51.6623	−1.96533	−0.982664	+	0.185398i	$0.940643\pi$
$692$	−10.9229	−0.415225
$693$	−28.3390	−1.07651
$694$	−5.95733	−0.226137
$695$	−0.508671	−0.0192950
$696$	−2.89203	−0.109622
$697$	−0.307485	−0.0116468
$698$	−30.4328	−1.15190
$699$	7.28320	0.275476
$700$	22.0553	0.833611
$701$	−26.2706	−0.992226	−0.496113	−	0.868258i	$-0.665240\pi$
$701$	−26.2706	−0.992226	−0.496113	+	0.868258i	$0.665240\pi$
$702$	5.15283	0.194481
$703$	−35.2525	−1.32958
$704$	−2.32784	−0.0877339
$705$	0.633941	0.0238756
$706$	−1.57223	−0.0591718
$707$	−1.68370	−0.0633221
$708$	0.328120	0.0123315
$709$	−50.5660	−1.89904	−0.949522	−	0.313700i	$-0.898431\pi$
$709$	−50.5660	−1.89904	−0.949522	+	0.313700i	$0.898431\pi$
$710$	0.637510	0.0239253
$711$	5.78142	0.216820
$712$	−1.42660	−0.0534642
$713$	−2.81853	−0.105555
$714$	−1.62996	−0.0609996
$715$	0.508044	0.0189998
$716$	−12.3706	−0.462313
$717$	8.20121	0.306280
$718$	21.4922	0.802083
$719$	−21.9633	−0.819093	−0.409546	−	0.912289i	$-0.634313\pi$
$719$	−21.9633	−0.819093	−0.409546	+	0.912289i	$0.634313\pi$
$720$	−0.334015	−0.0124480
$721$	9.34845	0.348154
$722$	−31.5943	−1.17582
$723$	6.25941	0.232790
$724$	−3.38109	−0.125657
$725$	28.9356	1.07464
$726$	−2.78089	−0.103208
$727$	42.7289	1.58473	0.792363	−	0.610050i	$-0.208851\pi$
$727$	42.7289	1.58473	0.792363	+	0.610050i	$0.208851\pi$
$728$	−7.95448	−0.294813
$729$	−14.5028	−0.537142
$730$	1.80478	0.0667977
$731$	−8.37357	−0.309708
$732$	0.648031	0.0239519
$733$	−18.0968	−0.668422	−0.334211	−	0.942498i	$-0.608470\pi$
$733$	−18.0968	−0.668422	−0.334211	+	0.942498i	$0.608470\pi$
$734$	−16.5830	−0.612090
$735$	−0.760404	−0.0280479
$736$	1.00494	0.0370425
$737$	10.8323	0.399012
$738$	−1.14429	−0.0421220
$739$	−4.36832	−0.160691	−0.0803455	−	0.996767i	$-0.525602\pi$
$739$	−4.36832	−0.160691	−0.0803455	+	0.996767i	$0.525602\pi$
$740$	0.601589	0.0221149
$741$	−6.37233	−0.234093
$742$	−0.0266807	−0.000979480 0
$743$	7.23682	0.265493	0.132747	−	0.991150i	$-0.457620\pi$
$743$	7.23682	0.265493	0.132747	+	0.991150i	$0.457620\pi$
$744$	1.39747	0.0512338
$745$	1.03177	0.0378011
$746$	3.18625	0.116657
$747$	−39.2797	−1.43717
$748$	−1.72126	−0.0629354
$749$	−7.14872	−0.261209
$750$	−0.603920	−0.0220520
$751$	−12.9753	−0.473477	−0.236738	−	0.971573i	$-0.576078\pi$
$751$	−12.9753	−0.473477	−0.236738	+	0.971573i	$0.576078\pi$
$752$	−10.4816	−0.382226
$753$	4.45976	0.162523
$754$	−10.4359	−0.380055
$755$	−2.20428	−0.0802221
$756$	−12.6789	−0.461128
$757$	−8.06482	−0.293121	−0.146560	−	0.989202i	$-0.546820\pi$
$757$	−8.06482	−0.293121	−0.146560	+	0.989202i	$0.546820\pi$
$758$	9.92971	0.360663
$759$	−1.16561	−0.0423089
$760$	0.863398	0.0313187
$761$	6.23470	0.226008	0.113004	−	0.993595i	$-0.463953\pi$
$761$	6.23470	0.226008	0.113004	+	0.993595i	$0.463953\pi$
$762$	8.50721	0.308184
$763$	9.05885	0.327953
$764$	10.3823	0.375619
$765$	−0.246978	−0.00892952
$766$	5.75503	0.207938
$767$	1.18403	0.0427527
$768$	−0.498264	−0.0179796
$769$	−35.2473	−1.27105	−0.635525	−	0.772080i	$-0.719216\pi$
$769$	−35.2473	−1.27105	−0.635525	+	0.772080i	$0.719216\pi$
$770$	−1.25008	−0.0450498
$771$	1.56966	0.0565299
$772$	11.0243	0.396772
$773$	−51.8611	−1.86531	−0.932657	−	0.360764i	$-0.882516\pi$
$773$	−51.8611	−1.86531	−0.932657	+	0.360764i	$0.882516\pi$
$774$	−31.1620	−1.12009
$775$	−13.9821	−0.502251
$776$	−9.03301	−0.324266
$777$	10.9251	0.391934
$778$	6.66472	0.238942
$779$	2.95789	0.105977
$780$	0.108745	0.00389368
$781$	12.2259	0.437477
$782$	0.743074	0.0265723
$783$	−16.6342	−0.594458
$784$	12.5726	0.449021
$785$	0.357283	0.0127520
$786$	2.90376	0.103574
$787$	13.7236	0.489194	0.244597	−	0.969625i	$-0.421344\pi$
$787$	13.7236	0.489194	0.244597	+	0.969625i	$0.421344\pi$
$788$	0.569735	0.0202960
$789$	11.2793	0.401553
$790$	0.255028	0.00907349
$791$	−45.1737	−1.60619
$792$	−6.40560	−0.227613
$793$	2.33843	0.0830400
$794$	−10.5735	−0.375240
$795$	0.000364748 0	1.29363e−5 0
$796$	−13.1208	−0.465055
$797$	27.9833	0.991219	0.495609	−	0.868546i	$-0.334945\pi$
$797$	27.9833	0.991219	0.495609	+	0.868546i	$0.334945\pi$
$798$	15.6796	0.555051
$799$	−7.75036	−0.274188
$800$	4.98527	0.176256
$801$	−3.92563	−0.138705
$802$	−4.27734	−0.151038
$803$	34.6112	1.22140
$804$	2.31860	0.0817707
$805$	0.539664	0.0190207
$806$	5.04279	0.177625
$807$	0.560763	0.0197398
$808$	−0.380575	−0.0133886
$809$	−24.5599	−0.863480	−0.431740	−	0.901998i	$-0.642100\pi$
$809$	−24.5599	−0.863480	−0.431740	+	0.901998i	$0.642100\pi$
$810$	−0.828715	−0.0291181
$811$	16.5338	0.580580	0.290290	−	0.956939i	$-0.406248\pi$
$811$	16.5338	0.580580	0.290290	+	0.956939i	$0.406248\pi$
$812$	25.6784	0.901136
$813$	−10.1188	−0.354883
$814$	11.5370	0.404372
$815$	0.733250	0.0256846
$816$	−0.368427	−0.0128975
$817$	80.5507	2.81811
$818$	−26.1250	−0.913439
$819$	−21.8886	−0.764850
$820$	−0.0504768	−0.00176272
$821$	−37.1319	−1.29591	−0.647956	−	0.761678i	$-0.724376\pi$
$821$	−37.1319	−1.29591	−0.647956	+	0.761678i	$0.724376\pi$
$822$	6.24340	0.217763
$823$	26.1870	0.912822	0.456411	−	0.889769i	$-0.349135\pi$
$823$	26.1870	0.912822	0.456411	+	0.889769i	$0.349135\pi$
$824$	2.11308	0.0736125
$825$	−5.78231	−0.201314
$826$	−2.91338	−0.101370
$827$	8.02784	0.279155	0.139578	−	0.990211i	$-0.455426\pi$
$827$	8.02784	0.279155	0.139578	+	0.990211i	$0.455426\pi$
$828$	2.76532	0.0961016
$829$	28.4558	0.988311	0.494156	−	0.869373i	$-0.335477\pi$
$829$	28.4558	0.988311	0.494156	+	0.869373i	$0.335477\pi$
$830$	−1.73270	−0.0601427
$831$	4.90952	0.170309
$832$	−1.79799	−0.0623342
$833$	9.29646	0.322103
$834$	−2.08803	−0.0723025
$835$	−2.48277	−0.0859198
$836$	16.5579	0.572666
$837$	8.03788	0.277830
$838$	−2.69334	−0.0930398
$839$	35.6003	1.22906	0.614529	−	0.788894i	$-0.289346\pi$
$839$	35.6003	1.22906	0.614529	+	0.788894i	$0.289346\pi$
$840$	−0.267574	−0.00923218
$841$	4.68899	0.161689
$842$	−3.69940	−0.127490
$843$	−13.6924	−0.471590
$844$	3.88386	0.133688
$845$	−1.18558	−0.0407852
$846$	−28.8427	−0.991632
$847$	24.6915	0.848411
$848$	−0.00603078	−0.000207098 0
$849$	1.96760	0.0675278
$850$	3.68622	0.126436
$851$	−4.98057	−0.170732
$852$	2.61690	0.0896534
$853$	−0.403519	−0.0138162	−0.00690811	−	0.999976i	$-0.502199\pi$
$853$	−0.403519	−0.0138162	−0.00690811	+	0.999976i	$0.502199\pi$
$854$	−5.75387	−0.196893
$855$	2.37584	0.0812520
$856$	−1.61586	−0.0552290
$857$	−9.23657	−0.315515	−0.157758	−	0.987478i	$-0.550426\pi$
$857$	−9.23657	−0.315515	−0.157758	+	0.987478i	$0.550426\pi$
$858$	2.08546	0.0711963
$859$	−36.0179	−1.22891	−0.614457	−	0.788950i	$-0.710625\pi$
$859$	−36.0179	−1.22891	−0.614457	+	0.788950i	$0.710625\pi$
$860$	−1.37461	−0.0468737
$861$	−0.916674	−0.0312402
$862$	−13.6394	−0.464560
$863$	−33.7190	−1.14781	−0.573905	−	0.818922i	$-0.694572\pi$
$863$	−33.7190	−1.14781	−0.573905	+	0.818922i	$0.694572\pi$
$864$	−2.86588	−0.0974993
$865$	−1.32586	−0.0450805
$866$	−14.4233	−0.490124
$867$	8.19806	0.278421
$868$	−12.4082	−0.421160
$869$	4.89082	0.165910
$870$	−0.351046	−0.0119016
$871$	8.36669	0.283495
$872$	2.04762	0.0693411
$873$	−24.8564	−0.841263
$874$	−7.14809	−0.241788
$875$	5.36221	0.181276
$876$	7.40837	0.250306
$877$	−5.08900	−0.171843	−0.0859216	−	0.996302i	$-0.527383\pi$
$877$	−5.08900	−0.171843	−0.0859216	+	0.996302i	$0.527383\pi$
$878$	−6.21599	−0.209780
$879$	1.55662	0.0525034
$880$	−0.282562	−0.00952516
$881$	27.5782	0.929132	0.464566	−	0.885539i	$-0.346210\pi$
$881$	27.5782	0.929132	0.464566	+	0.885539i	$0.346210\pi$
$882$	34.5964	1.16492
$883$	−46.0044	−1.54817	−0.774085	−	0.633081i	$-0.781790\pi$
$883$	−46.0044	−1.54817	−0.774085	+	0.633081i	$0.781790\pi$
$884$	−1.32947	−0.0447151
$885$	0.0398284	0.00133882
$886$	8.25744	0.277414
$887$	24.9236	0.836853	0.418426	−	0.908251i	$-0.362582\pi$
$887$	24.9236	0.836853	0.418426	+	0.908251i	$0.362582\pi$
$888$	2.46944	0.0828691
$889$	−75.5357	−2.53339
$890$	−0.173166	−0.00580455
$891$	−15.8927	−0.532426
$892$	13.4557	0.450529
$893$	74.5556	2.49491
$894$	4.23527	0.141649
$895$	−1.50159	−0.0501928
$896$	4.42409	0.147799
$897$	−0.900299	−0.0300601
$898$	20.6620	0.689499
$899$	−16.2790	−0.542934
$900$	13.7181	0.457271
$901$	−0.00445929	−0.000148560 0
$902$	−0.968021	−0.0322316
$903$	−24.9633	−0.830727
$904$	−10.2108	−0.339608
$905$	−0.410409	−0.0136425
$906$	−9.04829	−0.300609
$907$	20.3506	0.675732	0.337866	−	0.941194i	$-0.390295\pi$
$907$	20.3506	0.675732	0.337866	+	0.941194i	$0.390295\pi$
$908$	−2.55028	−0.0846340
$909$	−1.04724	−0.0347348
$910$	−0.965544	−0.0320075
$911$	−5.28911	−0.175236	−0.0876180	−	0.996154i	$-0.527925\pi$
$911$	−5.28911	−0.175236	−0.0876180	+	0.996154i	$0.527925\pi$
$912$	3.54414	0.117358
$913$	−33.2289	−1.09972
$914$	−5.46396	−0.180732
$915$	0.0786603	0.00260043
$916$	−0.476939	−0.0157585
$917$	−25.7825	−0.851413
$918$	−2.11910	−0.0699406
$919$	−33.4362	−1.10296	−0.551479	−	0.834189i	$-0.685936\pi$
$919$	−33.4362	−1.10296	−0.551479	+	0.834189i	$0.685936\pi$
$920$	0.121983	0.00402166
$921$	3.81619	0.125748
$922$	17.4900	0.576003
$923$	9.44310	0.310823
$924$	−5.13142	−0.168811
$925$	−24.7075	−0.812376
$926$	−20.6181	−0.677551
$927$	5.81462	0.190977
$928$	5.80422	0.190533
$929$	−30.2123	−0.991232	−0.495616	−	0.868542i	$-0.665058\pi$
$929$	−30.2123	−0.991232	−0.495616	+	0.868542i	$0.665058\pi$
$930$	0.169630	0.00556239
$931$	−89.4285	−2.93090
$932$	−14.6171	−0.478801
$933$	12.8779	0.421603
$934$	30.7962	1.00768
$935$	−0.208933	−0.00683283
$936$	−4.94759	−0.161717
$937$	35.4789	1.15905	0.579523	−	0.814956i	$-0.303239\pi$
$937$	35.4789	1.15905	0.579523	+	0.814956i	$0.303239\pi$
$938$	−20.5869	−0.672185
$939$	−10.6431	−0.347325
$940$	−1.27230	−0.0414979
$941$	−4.21119	−0.137281	−0.0686404	−	0.997641i	$-0.521866\pi$
$941$	−4.21119	−0.137281	−0.0686404	+	0.997641i	$0.521866\pi$
$942$	1.46660	0.0477844
$943$	0.417898	0.0136086
$944$	−0.658526	−0.0214332
$945$	−1.53901	−0.0500641
$946$	−26.3616	−0.857090
$947$	−57.0743	−1.85466	−0.927332	−	0.374239i	$-0.877904\pi$
$947$	−57.0743	−1.85466	−0.927332	+	0.374239i	$0.877904\pi$
$948$	1.04686	0.0340003
$949$	26.7332	0.867796
$950$	−35.4600	−1.15048
$951$	−6.68755	−0.216859
$952$	3.27127	0.106023
$953$	13.0640	0.423183	0.211592	−	0.977358i	$-0.432135\pi$
$953$	13.0640	0.423183	0.211592	+	0.977358i	$0.432135\pi$
$954$	−0.0165951	−0.000537286 0
$955$	1.26024	0.0407806
$956$	−16.4596	−0.532341
$957$	−6.73220	−0.217621
$958$	10.6221	0.343184
$959$	−55.4352	−1.79010
$960$	−0.0604811	−0.00195202
$961$	−23.1338	−0.746251
$962$	8.91102	0.287303
$963$	−4.44642	−0.143284
$964$	−12.5624	−0.404609
$965$	1.33817	0.0430771
$966$	2.21525	0.0712746
$967$	20.6882	0.665288	0.332644	−	0.943053i	$-0.392059\pi$
$967$	20.6882	0.665288	0.332644	+	0.943053i	$0.392059\pi$
$968$	5.58115	0.179385
$969$	2.62061	0.0841862
$970$	−1.09646	−0.0352052
$971$	−12.7975	−0.410692	−0.205346	−	0.978689i	$-0.565832\pi$
$971$	−12.7975	−0.410692	−0.205346	+	0.978689i	$0.565832\pi$
$972$	−11.9994	−0.384881
$973$	18.5396	0.594353
$974$	−8.33254	−0.266992
$975$	−4.46617	−0.143032
$976$	−1.30058	−0.0416304
$977$	−0.956086	−0.0305879	−0.0152940	−	0.999883i	$-0.504868\pi$
$977$	−0.956086	−0.0305879	−0.0152940	+	0.999883i	$0.504868\pi$
$978$	3.00990	0.0962459
$979$	−3.32091	−0.106137
$980$	1.52611	0.0487497
$981$	5.63450	0.179896
$982$	27.6045	0.880895
$983$	−16.6723	−0.531763	−0.265882	−	0.964006i	$-0.585663\pi$
$983$	−16.6723	−0.531763	−0.265882	+	0.964006i	$0.585663\pi$
$984$	−0.207200	−0.00660531
$985$	0.0691565	0.00220351
$986$	4.29177	0.136678
$987$	−23.1054	−0.735452
$988$	12.7891	0.406874
$989$	11.3804	0.361876
$990$	−0.777535	−0.0247117
$991$	30.3827	0.965138	0.482569	−	0.875858i	$-0.339704\pi$
$991$	30.3827	0.965138	0.482569	+	0.875858i	$0.339704\pi$
$992$	−2.80468	−0.0890487
$993$	−16.8272	−0.533996
$994$	−23.2354	−0.736984
$995$	−1.59265	−0.0504905
$996$	−7.11249	−0.225368
$997$	−2.94157	−0.0931603	−0.0465802	−	0.998915i	$-0.514832\pi$
$997$	−2.94157	−0.0931603	−0.0465802	+	0.998915i	$0.514832\pi$
$998$	−10.0868	−0.319293
$999$	14.2036	0.449382

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4022.2.a.e.1.19	✓	46

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4022.2.a.e.1.19	✓	46	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:	\( N \)	\(=\)	\( 4022 = 2 \cdot 2011 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4022.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -0.498264 q^{3} +1.00000 q^{4} +0.121384 q^{5} +0.498264 q^{6} -4.42409 q^{7} -1.00000 q^{8} -2.75173 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -0.498264 q^{3} +1.00000 q^{4} +0.121384 q^{5} +0.498264 q^{6} -4.42409 q^{7} -1.00000 q^{8} -2.75173 q^{9} -0.121384 q^{10} -2.32784 q^{11} -0.498264 q^{12} -1.79799 q^{13} +4.42409 q^{14} -0.0604811 q^{15} +1.00000 q^{16} +0.739422 q^{17} +2.75173 q^{18} -7.11297 q^{19} +0.121384 q^{20} +2.20437 q^{21} +2.32784 q^{22} -1.00494 q^{23} +0.498264 q^{24} -4.98527 q^{25} +1.79799 q^{26} +2.86588 q^{27} -4.42409 q^{28} -5.80422 q^{29} +0.0604811 q^{30} +2.80468 q^{31} -1.00000 q^{32} +1.15988 q^{33} -0.739422 q^{34} -0.537013 q^{35} -2.75173 q^{36} +4.95610 q^{37} +7.11297 q^{38} +0.895875 q^{39} -0.121384 q^{40} -0.415845 q^{41} -2.20437 q^{42} -11.3245 q^{43} -2.32784 q^{44} -0.334015 q^{45} +1.00494 q^{46} -10.4816 q^{47} -0.498264 q^{48} +12.5726 q^{49} +4.98527 q^{50} -0.368427 q^{51} -1.79799 q^{52} -0.00603078 q^{53} -2.86588 q^{54} -0.282562 q^{55} +4.42409 q^{56} +3.54414 q^{57} +5.80422 q^{58} -0.658526 q^{59} -0.0604811 q^{60} -1.30058 q^{61} -2.80468 q^{62} +12.1739 q^{63} +1.00000 q^{64} -0.218247 q^{65} -1.15988 q^{66} -4.65335 q^{67} +0.739422 q^{68} +0.500725 q^{69} +0.537013 q^{70} -5.25203 q^{71} +2.75173 q^{72} -14.8684 q^{73} -4.95610 q^{74} +2.48398 q^{75} -7.11297 q^{76} +10.2986 q^{77} -0.895875 q^{78} -2.10101 q^{79} +0.121384 q^{80} +6.82723 q^{81} +0.415845 q^{82} +14.2745 q^{83} +2.20437 q^{84} +0.0897538 q^{85} +11.3245 q^{86} +2.89203 q^{87} +2.32784 q^{88} +1.42660 q^{89} +0.334015 q^{90} +7.95448 q^{91} -1.00494 q^{92} -1.39747 q^{93} +10.4816 q^{94} -0.863398 q^{95} +0.498264 q^{96} +9.03301 q^{97} -12.5726 q^{98} +6.40560 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 46 q - 46 q^{2} + 8 q^{3} + 46 q^{4} + 14 q^{5} - 8 q^{6} + 28 q^{7} - 46 q^{8} + 58 q^{9}+O(q^{10}) \) 46 * q - 46 * q^2 + 8 * q^3 + 46 * q^4 + 14 * q^5 - 8 * q^6 + 28 * q^7 - 46 * q^8 + 58 * q^9	\( 46 q - 46 q^{2} + 8 q^{3} + 46 q^{4} + 14 q^{5} - 8 q^{6} + 28 q^{7} - 46 q^{8} + 58 q^{9} - 14 q^{10} - 6 q^{11} + 8 q^{12} + 37 q^{13} - 28 q^{14} + 9 q^{15} + 46 q^{16} + 6 q^{17} - 58 q^{18} + 18 q^{19} + 14 q^{20} + 19 q^{21} + 6 q^{22} - 4 q^{23} - 8 q^{24} + 86 q^{25} - 37 q^{26} + 32 q^{27} + 28 q^{28} + 15 q^{29} - 9 q^{30} + 18 q^{31} - 46 q^{32} + 37 q^{33} - 6 q^{34} - 2 q^{35} + 58 q^{36} + 74 q^{37} - 18 q^{38} - 3 q^{39} - 14 q^{40} - 18 q^{41} - 19 q^{42} + 25 q^{43} - 6 q^{44} + 94 q^{45} + 4 q^{46} + 18 q^{47} + 8 q^{48} + 92 q^{49} - 86 q^{50} - 10 q^{51} + 37 q^{52} + 17 q^{53} - 32 q^{54} + 37 q^{55} - 28 q^{56} + 43 q^{57} - 15 q^{58} - 24 q^{59} + 9 q^{60} + 46 q^{61} - 18 q^{62} + 80 q^{63} + 46 q^{64} + 24 q^{65} - 37 q^{66} + 61 q^{67} + 6 q^{68} + 59 q^{69} + 2 q^{70} - 8 q^{71} - 58 q^{72} + 101 q^{73} - 74 q^{74} + 34 q^{75} + 18 q^{76} + 40 q^{77} + 3 q^{78} + 9 q^{79} + 14 q^{80} + 58 q^{81} + 18 q^{82} + 18 q^{83} + 19 q^{84} + 60 q^{85} - 25 q^{86} + 20 q^{87} + 6 q^{88} - 25 q^{89} - 94 q^{90} + 51 q^{91} - 4 q^{92} + 63 q^{93} - 18 q^{94} - 31 q^{95} - 8 q^{96} + 76 q^{97} - 92 q^{98} + 21 q^{99}+O(q^{100}) \) 46 * q - 46 * q^2 + 8 * q^3 + 46 * q^4 + 14 * q^5 - 8 * q^6 + 28 * q^7 - 46 * q^8 + 58 * q^9 - 14 * q^10 - 6 * q^11 + 8 * q^12 + 37 * q^13 - 28 * q^14 + 9 * q^15 + 46 * q^16 + 6 * q^17 - 58 * q^18 + 18 * q^19 + 14 * q^20 + 19 * q^21 + 6 * q^22 - 4 * q^23 - 8 * q^24 + 86 * q^25 - 37 * q^26 + 32 * q^27 + 28 * q^28 + 15 * q^29 - 9 * q^30 + 18 * q^31 - 46 * q^32 + 37 * q^33 - 6 * q^34 - 2 * q^35 + 58 * q^36 + 74 * q^37 - 18 * q^38 - 3 * q^39 - 14 * q^40 - 18 * q^41 - 19 * q^42 + 25 * q^43 - 6 * q^44 + 94 * q^45 + 4 * q^46 + 18 * q^47 + 8 * q^48 + 92 * q^49 - 86 * q^50 - 10 * q^51 + 37 * q^52 + 17 * q^53 - 32 * q^54 + 37 * q^55 - 28 * q^56 + 43 * q^57 - 15 * q^58 - 24 * q^59 + 9 * q^60 + 46 * q^61 - 18 * q^62 + 80 * q^63 + 46 * q^64 + 24 * q^65 - 37 * q^66 + 61 * q^67 + 6 * q^68 + 59 * q^69 + 2 * q^70 - 8 * q^71 - 58 * q^72 + 101 * q^73 - 74 * q^74 + 34 * q^75 + 18 * q^76 + 40 * q^77 + 3 * q^78 + 9 * q^79 + 14 * q^80 + 58 * q^81 + 18 * q^82 + 18 * q^83 + 19 * q^84 + 60 * q^85 - 25 * q^86 + 20 * q^87 + 6 * q^88 - 25 * q^89 - 94 * q^90 + 51 * q^91 - 4 * q^92 + 63 * q^93 - 18 * q^94 - 31 * q^95 - 8 * q^96 + 76 * q^97 - 92 * q^98 + 21 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more