LMFDB - Embedded newform 8005.2.a.f.1.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8005,2,Mod(1,8005)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8005, 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("8005.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8005 = 5 \cdot 1601 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8005.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:	$63.9202468180$
Analytic rank:	$1$
Dimension:	$127$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
Character	$\chi$	$=$	8005.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.50760 q^{2} +2.92815 q^{3} +4.28808 q^{4} -1.00000 q^{5} -7.34263 q^{6} -1.39021 q^{7} -5.73759 q^{8} +5.57405 q^{9} +O(q^{10})$	\(q-2.50760 q^{2} +2.92815 q^{3} +4.28808 q^{4} -1.00000 q^{5} -7.34263 q^{6} -1.39021 q^{7} -5.73759 q^{8} +5.57405 q^{9} +2.50760 q^{10} +0.659694 q^{11} +12.5561 q^{12} +2.57624 q^{13} +3.48611 q^{14} -2.92815 q^{15} +5.81145 q^{16} +0.195545 q^{17} -13.9775 q^{18} +0.692907 q^{19} -4.28808 q^{20} -4.07075 q^{21} -1.65425 q^{22} -4.89742 q^{23} -16.8005 q^{24} +1.00000 q^{25} -6.46018 q^{26} +7.53719 q^{27} -5.96134 q^{28} -9.71809 q^{29} +7.34263 q^{30} -0.0735084 q^{31} -3.09763 q^{32} +1.93168 q^{33} -0.490349 q^{34} +1.39021 q^{35} +23.9019 q^{36} +5.59059 q^{37} -1.73754 q^{38} +7.54361 q^{39} +5.73759 q^{40} -1.90327 q^{41} +10.2078 q^{42} -3.86376 q^{43} +2.82882 q^{44} -5.57405 q^{45} +12.2808 q^{46} -11.0052 q^{47} +17.0168 q^{48} -5.06731 q^{49} -2.50760 q^{50} +0.572584 q^{51} +11.0471 q^{52} +5.08984 q^{53} -18.9003 q^{54} -0.659694 q^{55} +7.97648 q^{56} +2.02894 q^{57} +24.3691 q^{58} +2.27381 q^{59} -12.5561 q^{60} +11.8065 q^{61} +0.184330 q^{62} -7.74912 q^{63} -3.85527 q^{64} -2.57624 q^{65} -4.84389 q^{66} -6.73846 q^{67} +0.838511 q^{68} -14.3404 q^{69} -3.48611 q^{70} -2.05805 q^{71} -31.9816 q^{72} -3.30870 q^{73} -14.0190 q^{74} +2.92815 q^{75} +2.97124 q^{76} -0.917115 q^{77} -18.9164 q^{78} -12.7125 q^{79} -5.81145 q^{80} +5.34786 q^{81} +4.77266 q^{82} +4.48908 q^{83} -17.4557 q^{84} -0.195545 q^{85} +9.68878 q^{86} -28.4560 q^{87} -3.78505 q^{88} -16.4915 q^{89} +13.9775 q^{90} -3.58152 q^{91} -21.0005 q^{92} -0.215243 q^{93} +27.5968 q^{94} -0.692907 q^{95} -9.07031 q^{96} +3.83317 q^{97} +12.7068 q^{98} +3.67716 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 127 q - 6 q^{2} - 18 q^{3} + 114 q^{4} - 127 q^{5} - 20 q^{6} + 28 q^{7} - 18 q^{8} + 101 q^{9}+O(q^{10}) $ 127 * q - 6 * q^2 - 18 * q^3 + 114 * q^4 - 127 * q^5 - 20 * q^6 + 28 * q^7 - 18 * q^8 + 101 * q^9	\( 127 q - 6 q^{2} - 18 q^{3} + 114 q^{4} - 127 q^{5} - 20 q^{6} + 28 q^{7} - 18 q^{8} + 101 q^{9} + 6 q^{10} - 45 q^{11} - 30 q^{12} - 53 q^{14} + 18 q^{15} + 84 q^{16} - 36 q^{17} - 10 q^{18} - 49 q^{19} - 114 q^{20} - 48 q^{21} + 13 q^{22} - 29 q^{23} - 63 q^{24} + 127 q^{25} - 55 q^{26} - 75 q^{27} + 44 q^{28} - 45 q^{29} + 20 q^{30} - 49 q^{31} - 32 q^{32} - 8 q^{33} - 52 q^{34} - 28 q^{35} + 44 q^{36} + 36 q^{37} - 65 q^{38} - 52 q^{39} + 18 q^{40} - 66 q^{41} - 18 q^{42} - 5 q^{43} - 93 q^{44} - 101 q^{45} - 25 q^{46} - 32 q^{47} - 54 q^{48} + 77 q^{49} - 6 q^{50} - 102 q^{51} - 13 q^{52} - 67 q^{53} - 53 q^{54} + 45 q^{55} - 158 q^{56} + 16 q^{57} + 35 q^{58} - 213 q^{59} + 30 q^{60} - 62 q^{61} - 33 q^{62} + 59 q^{63} + 34 q^{64} - 60 q^{66} + 10 q^{67} - 94 q^{68} - 93 q^{69} + 53 q^{70} - 118 q^{71} - 24 q^{72} + 35 q^{73} - 107 q^{74} - 18 q^{75} - 98 q^{76} - 93 q^{77} + 21 q^{78} - 64 q^{79} - 84 q^{80} + 15 q^{81} + 15 q^{82} - 187 q^{83} - 118 q^{84} + 36 q^{85} - 126 q^{86} - 53 q^{87} + 15 q^{88} - 138 q^{89} + 10 q^{90} - 138 q^{91} - 86 q^{92} + 23 q^{93} - 60 q^{94} + 49 q^{95} - 92 q^{96} + 9 q^{97} - 67 q^{98} - 147 q^{99}+O(q^{100}) \) 127 * q - 6 * q^2 - 18 * q^3 + 114 * q^4 - 127 * q^5 - 20 * q^6 + 28 * q^7 - 18 * q^8 + 101 * q^9 + 6 * q^10 - 45 * q^11 - 30 * q^12 - 53 * q^14 + 18 * q^15 + 84 * q^16 - 36 * q^17 - 10 * q^18 - 49 * q^19 - 114 * q^20 - 48 * q^21 + 13 * q^22 - 29 * q^23 - 63 * q^24 + 127 * q^25 - 55 * q^26 - 75 * q^27 + 44 * q^28 - 45 * q^29 + 20 * q^30 - 49 * q^31 - 32 * q^32 - 8 * q^33 - 52 * q^34 - 28 * q^35 + 44 * q^36 + 36 * q^37 - 65 * q^38 - 52 * q^39 + 18 * q^40 - 66 * q^41 - 18 * q^42 - 5 * q^43 - 93 * q^44 - 101 * q^45 - 25 * q^46 - 32 * q^47 - 54 * q^48 + 77 * q^49 - 6 * q^50 - 102 * q^51 - 13 * q^52 - 67 * q^53 - 53 * q^54 + 45 * q^55 - 158 * q^56 + 16 * q^57 + 35 * q^58 - 213 * q^59 + 30 * q^60 - 62 * q^61 - 33 * q^62 + 59 * q^63 + 34 * q^64 - 60 * q^66 + 10 * q^67 - 94 * q^68 - 93 * q^69 + 53 * q^70 - 118 * q^71 - 24 * q^72 + 35 * q^73 - 107 * q^74 - 18 * q^75 - 98 * q^76 - 93 * q^77 + 21 * q^78 - 64 * q^79 - 84 * q^80 + 15 * q^81 + 15 * q^82 - 187 * q^83 - 118 * q^84 + 36 * q^85 - 126 * q^86 - 53 * q^87 + 15 * q^88 - 138 * q^89 + 10 * q^90 - 138 * q^91 - 86 * q^92 + 23 * q^93 - 60 * q^94 + 49 * q^95 - 92 * q^96 + 9 * q^97 - 67 * q^98 - 147 * 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 $

$2$	−2.50760	−1.77314	−0.886572	−	0.462591i	$-0.846920\pi$
$2$	−2.50760	−1.77314	−0.886572	+	0.462591i	$0.846920\pi$
$3$	2.92815	1.69057	0.845283	−	0.534318i	$-0.179432\pi$
$3$	2.92815	1.69057	0.845283	+	0.534318i	$0.179432\pi$
$4$	4.28808	2.14404
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−7.34263	−2.99762
$7$	−1.39021	−0.525451	−0.262726	−	0.964871i	$-0.584621\pi$
$7$	−1.39021	−0.525451	−0.262726	+	0.964871i	$0.584621\pi$
$8$	−5.73759	−2.02854
$9$	5.57405	1.85802
$10$	2.50760	0.792974
$11$	0.659694	0.198905	0.0994525	−	0.995042i	$-0.468291\pi$
$11$	0.659694	0.198905	0.0994525	+	0.995042i	$0.468291\pi$
$12$	12.5561	3.62464
$13$	2.57624	0.714520	0.357260	−	0.934005i	$-0.383711\pi$
$13$	2.57624	0.714520	0.357260	+	0.934005i	$0.383711\pi$
$14$	3.48611	0.931701
$15$	−2.92815	−0.756044
$16$	5.81145	1.45286
$17$	0.195545	0.0474266	0.0237133	−	0.999719i	$-0.492451\pi$
$17$	0.195545	0.0474266	0.0237133	+	0.999719i	$0.492451\pi$
$18$	−13.9775	−3.29453
$19$	0.692907	0.158964	0.0794819	−	0.996836i	$-0.474673\pi$
$19$	0.692907	0.158964	0.0794819	+	0.996836i	$0.474673\pi$
$20$	−4.28808	−0.958843
$21$	−4.07075	−0.888311
$22$	−1.65425	−0.352687
$23$	−4.89742	−1.02118	−0.510591	−	0.859824i	$-0.670573\pi$
$23$	−4.89742	−1.02118	−0.510591	+	0.859824i	$0.670573\pi$
$24$	−16.8005	−3.42939
$25$	1.00000	0.200000
$26$	−6.46018	−1.26695
$27$	7.53719	1.45053
$28$	−5.96134	−1.12659
$29$	−9.71809	−1.80460	−0.902302	−	0.431105i	$-0.858124\pi$
$29$	−9.71809	−1.80460	−0.902302	+	0.431105i	$0.858124\pi$
$30$	7.34263	1.34058
$31$	−0.0735084	−0.0132025	−0.00660125	−	0.999978i	$-0.502101\pi$
$31$	−0.0735084	−0.0132025	−0.00660125	+	0.999978i	$0.502101\pi$
$32$	−3.09763	−0.547588
$33$	1.93168	0.336262
$34$	−0.490349	−0.0840941
$35$	1.39021	0.234989
$36$	23.9019	3.98366
$37$	5.59059	0.919088	0.459544	−	0.888155i	$-0.348013\pi$
$37$	5.59059	0.919088	0.459544	+	0.888155i	$0.348013\pi$
$38$	−1.73754	−0.281866
$39$	7.54361	1.20794
$40$	5.73759	0.907193
$41$	−1.90327	−0.297242	−0.148621	−	0.988894i	$-0.547483\pi$
$41$	−1.90327	−0.297242	−0.148621	+	0.988894i	$0.547483\pi$
$42$	10.2078	1.57510
$43$	−3.86376	−0.589218	−0.294609	−	0.955618i	$-0.595189\pi$
$43$	−3.86376	−0.589218	−0.294609	+	0.955618i	$0.595189\pi$
$44$	2.82882	0.426460
$45$	−5.57405	−0.830930
$46$	12.2808	1.81070
$47$	−11.0052	−1.60528	−0.802640	−	0.596464i	$-0.796572\pi$
$47$	−11.0052	−1.60528	−0.802640	+	0.596464i	$0.796572\pi$
$48$	17.0168	2.45616
$49$	−5.06731	−0.723901
$50$	−2.50760	−0.354629
$51$	0.572584	0.0801778
$52$	11.0471	1.53196
$53$	5.08984	0.699144	0.349572	−	0.936910i	$-0.386327\pi$
$53$	5.08984	0.699144	0.349572	+	0.936910i	$0.386327\pi$
$54$	−18.9003	−2.57200
$55$	−0.659694	−0.0889531
$56$	7.97648	1.06590
$57$	2.02894	0.268739
$58$	24.3691	3.19982
$59$	2.27381	0.296025	0.148013	−	0.988985i	$-0.452712\pi$
$59$	2.27381	0.296025	0.148013	+	0.988985i	$0.452712\pi$
$60$	−12.5561	−1.62099
$61$	11.8065	1.51167	0.755835	−	0.654762i	$-0.227231\pi$
$61$	11.8065	1.51167	0.755835	+	0.654762i	$0.227231\pi$
$62$	0.184330	0.0234099
$63$	−7.74912	−0.976297
$64$	−3.85527	−0.481909
$65$	−2.57624	−0.319543
$66$	−4.84389	−0.596241
$67$	−6.73846	−0.823234	−0.411617	−	0.911357i	$-0.635036\pi$
$67$	−6.73846	−0.823234	−0.411617	+	0.911357i	$0.635036\pi$
$68$	0.838511	0.101684
$69$	−14.3404	−1.72638
$70$	−3.48611	−0.416669
$71$	−2.05805	−0.244245	−0.122123	−	0.992515i	$-0.538970\pi$
$71$	−2.05805	−0.244245	−0.122123	+	0.992515i	$0.538970\pi$
$72$	−31.9816	−3.76907
$73$	−3.30870	−0.387254	−0.193627	−	0.981075i	$-0.562025\pi$
$73$	−3.30870	−0.387254	−0.193627	+	0.981075i	$0.562025\pi$
$74$	−14.0190	−1.62968
$75$	2.92815	0.338113
$76$	2.97124	0.340825
$77$	−0.917115	−0.104515
$78$	−18.9164	−2.14186
$79$	−12.7125	−1.43027	−0.715136	−	0.698986i	$-0.753635\pi$
$79$	−12.7125	−1.43027	−0.715136	+	0.698986i	$0.753635\pi$
$80$	−5.81145	−0.649740
$81$	5.34786	0.594207
$82$	4.77266	0.527052
$83$	4.48908	0.492741	0.246371	−	0.969176i	$-0.420762\pi$
$83$	4.48908	0.492741	0.246371	+	0.969176i	$0.420762\pi$
$84$	−17.4557	−1.90457
$85$	−0.195545	−0.0212098
$86$	9.68878	1.04477
$87$	−28.4560	−3.05080
$88$	−3.78505	−0.403488
$89$	−16.4915	−1.74810	−0.874050	−	0.485835i	$-0.838515\pi$
$89$	−16.4915	−1.74810	−0.874050	+	0.485835i	$0.838515\pi$
$90$	13.9775	1.47336
$91$	−3.58152	−0.375445
$92$	−21.0005	−2.18945
$93$	−0.215243	−0.0223197
$94$	27.5968	2.84639
$95$	−0.692907	−0.0710908
$96$	−9.07031	−0.925735
$97$	3.83317	0.389199	0.194599	−	0.980883i	$-0.437659\pi$
$97$	3.83317	0.389199	0.194599	+	0.980883i	$0.437659\pi$
$98$	12.7068	1.28358
$99$	3.67716	0.369569
$100$	4.28808	0.428808
$101$	−2.88231	−0.286801	−0.143400	−	0.989665i	$-0.545804\pi$
$101$	−2.88231	−0.286801	−0.143400	+	0.989665i	$0.545804\pi$
$102$	−1.43581	−0.142167
$103$	−8.57423	−0.844844	−0.422422	−	0.906399i	$-0.638820\pi$
$103$	−8.57423	−0.844844	−0.422422	+	0.906399i	$0.638820\pi$
$104$	−14.7814	−1.44944
$105$	4.07075	0.397265
$106$	−12.7633	−1.23968
$107$	−18.0069	−1.74079	−0.870395	−	0.492353i	$-0.836137\pi$
$107$	−18.0069	−1.74079	−0.870395	+	0.492353i	$0.836137\pi$
$108$	32.3201	3.11000
$109$	8.76898	0.839916	0.419958	−	0.907544i	$-0.362045\pi$
$109$	8.76898	0.839916	0.419958	+	0.907544i	$0.362045\pi$
$110$	1.65425	0.157727
$111$	16.3701	1.55378
$112$	−8.07915	−0.763408
$113$	5.26173	0.494982	0.247491	−	0.968890i	$-0.420394\pi$
$113$	5.26173	0.494982	0.247491	+	0.968890i	$0.420394\pi$
$114$	−5.08777	−0.476513
$115$	4.89742	0.456687
$116$	−41.6719	−3.86914
$117$	14.3601	1.32759
$118$	−5.70182	−0.524895
$119$	−0.271849	−0.0249203
$120$	16.8005	1.53367
$121$	−10.5648	−0.960437
$122$	−29.6061	−2.68041
$123$	−5.57307	−0.502507
$124$	−0.315209	−0.0283066
$125$	−1.00000	−0.0894427
$126$	19.4317	1.73111
$127$	22.2612	1.97536	0.987681	−	0.156479i	$-0.0500144\pi$
$127$	22.2612	1.97536	0.987681	+	0.156479i	$0.0500144\pi$
$128$	15.8627	1.40208
$129$	−11.3137	−0.996112
$130$	6.46018	0.566596
$131$	−1.62628	−0.142089	−0.0710444	−	0.997473i	$-0.522633\pi$
$131$	−1.62628	−0.142089	−0.0710444	+	0.997473i	$0.522633\pi$
$132$	8.28319	0.720959
$133$	−0.963289	−0.0835278
$134$	16.8974	1.45971
$135$	−7.53719	−0.648698
$136$	−1.12196	−0.0962069
$137$	−4.65701	−0.397875	−0.198938	−	0.980012i	$-0.563749\pi$
$137$	−4.65701	−0.397875	−0.198938	+	0.980012i	$0.563749\pi$
$138$	35.9600	3.06111
$139$	−15.7059	−1.33216	−0.666080	−	0.745881i	$-0.732029\pi$
$139$	−15.7059	−1.33216	−0.666080	+	0.745881i	$0.732029\pi$
$140$	5.96134	0.503825
$141$	−32.2250	−2.71383
$142$	5.16077	0.433082
$143$	1.69953	0.142122
$144$	32.3933	2.69944
$145$	9.71809	0.807043
$146$	8.29691	0.686657
$147$	−14.8378	−1.22380
$148$	23.9729	1.97056
$149$	5.72490	0.469002	0.234501	−	0.972116i	$-0.424654\pi$
$149$	5.72490	0.469002	0.234501	+	0.972116i	$0.424654\pi$
$150$	−7.34263	−0.599524
$151$	3.74438	0.304713	0.152357	−	0.988326i	$-0.451314\pi$
$151$	3.74438	0.304713	0.152357	+	0.988326i	$0.451314\pi$
$152$	−3.97562	−0.322465
$153$	1.08998	0.0881193
$154$	2.29976	0.185320
$155$	0.0735084	0.00590433
$156$	32.3476	2.58988
$157$	−9.87249	−0.787910	−0.393955	−	0.919130i	$-0.628893\pi$
$157$	−9.87249	−0.787910	−0.393955	+	0.919130i	$0.628893\pi$
$158$	31.8780	2.53608
$159$	14.9038	1.18195
$160$	3.09763	0.244889
$161$	6.80846	0.536582
$162$	−13.4103	−1.05361
$163$	−4.34301	−0.340171	−0.170085	−	0.985429i	$-0.554404\pi$
$163$	−4.34301	−0.340171	−0.170085	+	0.985429i	$0.554404\pi$
$164$	−8.16139	−0.637297
$165$	−1.93168	−0.150381
$166$	−11.2568	−0.873701
$167$	13.2538	1.02561	0.512806	−	0.858505i	$-0.328606\pi$
$167$	13.2538	1.02561	0.512806	+	0.858505i	$0.328606\pi$
$168$	23.3563	1.80198
$169$	−6.36300	−0.489461
$170$	0.490349	0.0376080
$171$	3.86230	0.295357
$172$	−16.5681	−1.26331
$173$	19.8546	1.50952	0.754761	−	0.656000i	$-0.227753\pi$
$173$	19.8546	1.50952	0.754761	+	0.656000i	$0.227753\pi$
$174$	71.3564	5.40951
$175$	−1.39021	−0.105090
$176$	3.83377	0.288982
$177$	6.65806	0.500451
$178$	41.3543	3.09963
$179$	−6.04782	−0.452035	−0.226018	−	0.974123i	$-0.572571\pi$
$179$	−6.04782	−0.452035	−0.226018	+	0.974123i	$0.572571\pi$
$180$	−23.9019	−1.78155
$181$	12.9369	0.961594	0.480797	−	0.876832i	$-0.340347\pi$
$181$	12.9369	0.961594	0.480797	+	0.876832i	$0.340347\pi$
$182$	8.98104	0.665719
$183$	34.5712	2.55558
$184$	28.0994	2.07151
$185$	−5.59059	−0.411029
$186$	0.539745	0.0395760
$187$	0.129000	0.00943338
$188$	−47.1913	−3.44178
$189$	−10.4783	−0.762185
$190$	1.73754	0.126054
$191$	12.4536	0.901112	0.450556	−	0.892748i	$-0.351226\pi$
$191$	12.4536	0.901112	0.450556	+	0.892748i	$0.351226\pi$
$192$	−11.2888	−0.814699
$193$	20.4712	1.47355	0.736773	−	0.676140i	$-0.236349\pi$
$193$	20.4712	1.47355	0.736773	+	0.676140i	$0.236349\pi$
$194$	−9.61206	−0.690106
$195$	−7.54361	−0.540209
$196$	−21.7290	−1.55207
$197$	14.7692	1.05227	0.526133	−	0.850403i	$-0.323642\pi$
$197$	14.7692	1.05227	0.526133	+	0.850403i	$0.323642\pi$
$198$	−9.22087	−0.655299
$199$	−5.79731	−0.410960	−0.205480	−	0.978661i	$-0.565876\pi$
$199$	−5.79731	−0.410960	−0.205480	+	0.978661i	$0.565876\pi$
$200$	−5.73759	−0.405709
$201$	−19.7312	−1.39173
$202$	7.22770	0.508539
$203$	13.5102	0.948231
$204$	2.45528	0.171904
$205$	1.90327	0.132930
$206$	21.5008	1.49803
$207$	−27.2984	−1.89737
$208$	14.9717	1.03810
$209$	0.457107	0.0316187
$210$	−10.2078	−0.704407
$211$	1.84616	0.127095	0.0635473	−	0.997979i	$-0.479759\pi$
$211$	1.84616	0.127095	0.0635473	+	0.997979i	$0.479759\pi$
$212$	21.8256	1.49899
$213$	−6.02626	−0.412913
$214$	45.1541	3.08667
$215$	3.86376	0.263506
$216$	−43.2453	−2.94247
$217$	0.102192	0.00693727
$218$	−21.9891	−1.48929
$219$	−9.68836	−0.654679
$220$	−2.82882	−0.190719
$221$	0.503770	0.0338872
$222$	−41.0497	−2.75507
$223$	−2.41507	−0.161725	−0.0808625	−	0.996725i	$-0.525767\pi$
$223$	−2.41507	−0.161725	−0.0808625	+	0.996725i	$0.525767\pi$
$224$	4.30636	0.287731
$225$	5.57405	0.371603
$226$	−13.1943	−0.877674
$227$	−27.7092	−1.83913	−0.919563	−	0.392944i	$-0.871457\pi$
$227$	−27.7092	−1.83913	−0.919563	+	0.392944i	$0.871457\pi$
$228$	8.70023	0.576187
$229$	−15.5349	−1.02657	−0.513287	−	0.858217i	$-0.671572\pi$
$229$	−15.5349	−1.02657	−0.513287	+	0.858217i	$0.671572\pi$
$230$	−12.2808	−0.809771
$231$	−2.68545	−0.176690
$232$	55.7584	3.66072
$233$	−2.37207	−0.155400	−0.0776999	−	0.996977i	$-0.524758\pi$
$233$	−2.37207	−0.155400	−0.0776999	+	0.996977i	$0.524758\pi$
$234$	−36.0094	−2.35401
$235$	11.0052	0.717903
$236$	9.75029	0.634690
$237$	−37.2242	−2.41797
$238$	0.681689	0.0441874
$239$	−15.0207	−0.971606	−0.485803	−	0.874068i	$-0.661473\pi$
$239$	−15.0207	−0.971606	−0.485803	+	0.874068i	$0.661473\pi$
$240$	−17.0168	−1.09843
$241$	−1.34593	−0.0866992	−0.0433496	−	0.999060i	$-0.513803\pi$
$241$	−1.34593	−0.0866992	−0.0433496	+	0.999060i	$0.513803\pi$
$242$	26.4923	1.70299
$243$	−6.95224	−0.445986
$244$	50.6273	3.24108
$245$	5.06731	0.323738
$246$	13.9750	0.891017
$247$	1.78509	0.113583
$248$	0.421761	0.0267818
$249$	13.1447	0.833012
$250$	2.50760	0.158595
$251$	−18.7015	−1.18043	−0.590214	−	0.807247i	$-0.700957\pi$
$251$	−18.7015	−1.18043	−0.590214	+	0.807247i	$0.700957\pi$
$252$	−33.2288	−2.09322
$253$	−3.23080	−0.203118
$254$	−55.8223	−3.50260
$255$	−0.572584	−0.0358566
$256$	−32.0669	−2.00418
$257$	−7.75032	−0.483452	−0.241726	−	0.970345i	$-0.577713\pi$
$257$	−7.75032	−0.483452	−0.241726	+	0.970345i	$0.577713\pi$
$258$	28.3702	1.76625
$259$	−7.77212	−0.482936
$260$	−11.0471	−0.685112
$261$	−54.1691	−3.35298
$262$	4.07807	0.251944
$263$	−22.5575	−1.39096	−0.695478	−	0.718547i	$-0.744807\pi$
$263$	−22.5575	−1.39096	−0.695478	+	0.718547i	$0.744807\pi$
$264$	−11.0832	−0.682123
$265$	−5.08984	−0.312667
$266$	2.41555	0.148107
$267$	−48.2897	−2.95528
$268$	−28.8950	−1.76505
$269$	−10.4172	−0.635149	−0.317575	−	0.948233i	$-0.602868\pi$
$269$	−10.4172	−0.635149	−0.317575	+	0.948233i	$0.602868\pi$
$270$	18.9003	1.15023
$271$	13.5804	0.824949	0.412474	−	0.910969i	$-0.364665\pi$
$271$	13.5804	0.824949	0.412474	+	0.910969i	$0.364665\pi$
$272$	1.13640	0.0689042
$273$	−10.4872	−0.634716
$274$	11.6779	0.705490
$275$	0.659694	0.0397810
$276$	−61.4926	−3.70142
$277$	−13.4831	−0.810122	−0.405061	−	0.914290i	$-0.632750\pi$
$277$	−13.4831	−0.810122	−0.405061	+	0.914290i	$0.632750\pi$
$278$	39.3842	2.36211
$279$	−0.409739	−0.0245304
$280$	−7.97648	−0.476686
$281$	3.66590	0.218689	0.109345	−	0.994004i	$-0.465125\pi$
$281$	3.66590	0.218689	0.109345	+	0.994004i	$0.465125\pi$
$282$	80.8074	4.81201
$283$	−9.37233	−0.557127	−0.278564	−	0.960418i	$-0.589858\pi$
$283$	−9.37233	−0.557127	−0.278564	+	0.960418i	$0.589858\pi$
$284$	−8.82506	−0.523671
$285$	−2.02894	−0.120184
$286$	−4.26174	−0.252002
$287$	2.64596	0.156186
$288$	−17.2663	−1.01743
$289$	−16.9618	−0.997751
$290$	−24.3691	−1.43100
$291$	11.2241	0.657967
$292$	−14.1880	−0.830288
$293$	21.6768	1.26637	0.633185	−	0.774001i	$-0.281747\pi$
$293$	21.6768	1.26637	0.633185	+	0.774001i	$0.281747\pi$
$294$	37.2074	2.16998
$295$	−2.27381	−0.132387
$296$	−32.0765	−1.86441
$297$	4.97224	0.288518
$298$	−14.3558	−0.831609
$299$	−12.6169	−0.729655
$300$	12.5561	0.724928
$301$	5.37145	0.309605
$302$	−9.38941	−0.540300
$303$	−8.43984	−0.484856
$304$	4.02679	0.230953
$305$	−11.8065	−0.676039
$306$	−2.73323	−0.156248
$307$	13.6859	0.781093	0.390547	−	0.920583i	$-0.372286\pi$
$307$	13.6859	0.781093	0.390547	+	0.920583i	$0.372286\pi$
$308$	−3.93266	−0.224084
$309$	−25.1066	−1.42826
$310$	−0.184330	−0.0104692
$311$	4.84260	0.274599	0.137299	−	0.990530i	$-0.456158\pi$
$311$	4.84260	0.274599	0.137299	+	0.990530i	$0.456158\pi$
$312$	−43.2821	−2.45037
$313$	7.99713	0.452024	0.226012	−	0.974124i	$-0.427431\pi$
$313$	7.99713	0.452024	0.226012	+	0.974124i	$0.427431\pi$
$314$	24.7563	1.39708
$315$	7.74912	0.436613
$316$	−54.5123	−3.06656
$317$	5.04416	0.283308	0.141654	−	0.989916i	$-0.454758\pi$
$317$	5.04416	0.283308	0.141654	+	0.989916i	$0.454758\pi$
$318$	−37.3728	−2.09577
$319$	−6.41096	−0.358945
$320$	3.85527	0.215516
$321$	−52.7268	−2.94292
$322$	−17.0729	−0.951437
$323$	0.135494	0.00753911
$324$	22.9321	1.27400
$325$	2.57624	0.142904
$326$	10.8905	0.603172
$327$	25.6769	1.41993
$328$	10.9202	0.602968
$329$	15.2996	0.843496
$330$	4.84389	0.266647
$331$	23.6079	1.29761	0.648804	−	0.760956i	$-0.275270\pi$
$331$	23.6079	1.29761	0.648804	+	0.760956i	$0.275270\pi$
$332$	19.2495	1.05646
$333$	31.1622	1.70768
$334$	−33.2353	−1.81856
$335$	6.73846	0.368162
$336$	−23.6570	−1.29059
$337$	−5.32338	−0.289983	−0.144991	−	0.989433i	$-0.546315\pi$
$337$	−5.32338	−0.289983	−0.144991	+	0.989433i	$0.546315\pi$
$338$	15.9559	0.867885
$339$	15.4071	0.836800
$340$	−0.838511	−0.0454746
$341$	−0.0484930	−0.00262604
$342$	−9.68512	−0.523711
$343$	16.7761	0.905826
$344$	22.1687	1.19525
$345$	14.3404	0.772059
$346$	−49.7876	−2.67660
$347$	−11.0566	−0.593548	−0.296774	−	0.954948i	$-0.595911\pi$
$347$	−11.0566	−0.593548	−0.296774	+	0.954948i	$0.595911\pi$
$348$	−122.021	−6.54104
$349$	14.2411	0.762307	0.381154	−	0.924512i	$-0.375527\pi$
$349$	14.2411	0.762307	0.381154	+	0.924512i	$0.375527\pi$
$350$	3.48611	0.186340
$351$	19.4176	1.03643
$352$	−2.04349	−0.108918
$353$	−26.9348	−1.43360	−0.716799	−	0.697280i	$-0.754393\pi$
$353$	−26.9348	−1.43360	−0.716799	+	0.697280i	$0.754393\pi$
$354$	−16.6958	−0.887371
$355$	2.05805	0.109230
$356$	−70.7170	−3.74799
$357$	−0.796014	−0.0421295
$358$	15.1655	0.801523
$359$	−15.5632	−0.821396	−0.410698	−	0.911771i	$-0.634715\pi$
$359$	−15.5632	−0.821396	−0.410698	+	0.911771i	$0.634715\pi$
$360$	31.9816	1.68558
$361$	−18.5199	−0.974730
$362$	−32.4407	−1.70504
$363$	−30.9353	−1.62368
$364$	−15.3578	−0.804969
$365$	3.30870	0.173185
$366$	−86.6909	−4.53141
$367$	−8.91643	−0.465434	−0.232717	−	0.972545i	$-0.574762\pi$
$367$	−8.91643	−0.465434	−0.232717	+	0.972545i	$0.574762\pi$
$368$	−28.4611	−1.48364
$369$	−10.6089	−0.552280
$370$	14.0190	0.728813
$371$	−7.07597	−0.367366
$372$	−0.922980	−0.0478543
$373$	24.1120	1.24847	0.624236	−	0.781236i	$-0.285410\pi$
$373$	24.1120	1.24847	0.624236	+	0.781236i	$0.285410\pi$
$374$	−0.323480	−0.0167267
$375$	−2.92815	−0.151209
$376$	63.1435	3.25638
$377$	−25.0361	−1.28943
$378$	26.2754	1.35146
$379$	9.99907	0.513618	0.256809	−	0.966462i	$-0.417329\pi$
$379$	9.99907	0.513618	0.256809	+	0.966462i	$0.417329\pi$
$380$	−2.97124	−0.152421
$381$	65.1841	3.33948
$382$	−31.2287	−1.59780
$383$	10.3680	0.529783	0.264891	−	0.964278i	$-0.414664\pi$
$383$	10.3680	0.529783	0.264891	+	0.964278i	$0.414664\pi$
$384$	46.4485	2.37031
$385$	0.917115	0.0467405
$386$	−51.3336	−2.61281
$387$	−21.5368	−1.09478
$388$	16.4369	0.834457
$389$	−16.0414	−0.813330	−0.406665	−	0.913577i	$-0.633308\pi$
$389$	−16.0414	−0.813330	−0.406665	+	0.913577i	$0.633308\pi$
$390$	18.9164	0.957868
$391$	−0.957664	−0.0484312
$392$	29.0741	1.46846
$393$	−4.76199	−0.240211
$394$	−37.0354	−1.86582
$395$	12.7125	0.639637
$396$	15.7680	0.792370
$397$	32.3371	1.62295	0.811476	−	0.584386i	$-0.198665\pi$
$397$	32.3371	1.62295	0.811476	+	0.584386i	$0.198665\pi$
$398$	14.5374	0.728692
$399$	−2.82065	−0.141209
$400$	5.81145	0.290572
$401$	−27.9090	−1.39371	−0.696855	−	0.717212i	$-0.745418\pi$
$401$	−27.9090	−1.39371	−0.696855	+	0.717212i	$0.745418\pi$
$402$	49.4781	2.46774
$403$	−0.189375	−0.00943344
$404$	−12.3596	−0.614912
$405$	−5.34786	−0.265738
$406$	−33.8783	−1.68135
$407$	3.68808	0.182811
$408$	−3.28525	−0.162644
$409$	17.8652	0.883378	0.441689	−	0.897168i	$-0.354379\pi$
$409$	17.8652	0.883378	0.441689	+	0.897168i	$0.354379\pi$
$410$	−4.77266	−0.235705
$411$	−13.6364	−0.672635
$412$	−36.7669	−1.81138
$413$	−3.16109	−0.155547
$414$	68.4537	3.36432
$415$	−4.48908	−0.220360
$416$	−7.98023	−0.391263
$417$	−45.9893	−2.25210
$418$	−1.14624	−0.0560645
$419$	39.0451	1.90748	0.953738	−	0.300638i	$-0.0971997\pi$
$419$	39.0451	1.90748	0.953738	+	0.300638i	$0.0971997\pi$
$420$	17.4557	0.851751
$421$	−8.60614	−0.419438	−0.209719	−	0.977762i	$-0.567255\pi$
$421$	−8.60614	−0.419438	−0.209719	+	0.977762i	$0.567255\pi$
$422$	−4.62943	−0.225357
$423$	−61.3437	−2.98263
$424$	−29.2034	−1.41824
$425$	0.195545	0.00948531
$426$	15.1115	0.732154
$427$	−16.4136	−0.794309
$428$	−77.2149	−3.73232
$429$	4.97647	0.240266
$430$	−9.68878	−0.467234
$431$	−37.2574	−1.79463	−0.897313	−	0.441395i	$-0.854484\pi$
$431$	−37.2574	−1.79463	−0.897313	+	0.441395i	$0.854484\pi$
$432$	43.8020	2.10742
$433$	−7.92545	−0.380873	−0.190436	−	0.981700i	$-0.560990\pi$
$433$	−7.92545	−0.380873	−0.190436	+	0.981700i	$0.560990\pi$
$434$	−0.256258	−0.0123008
$435$	28.4560	1.36436
$436$	37.6020	1.80081
$437$	−3.39346	−0.162331
$438$	24.2946	1.16084
$439$	−6.98664	−0.333454	−0.166727	−	0.986003i	$-0.553320\pi$
$439$	−6.98664	−0.333454	−0.166727	+	0.986003i	$0.553320\pi$
$440$	3.78505	0.180445
$441$	−28.2454	−1.34502
$442$	−1.26325	−0.0600869
$443$	4.90116	0.232861	0.116430	−	0.993199i	$-0.462855\pi$
$443$	4.90116	0.232861	0.116430	+	0.993199i	$0.462855\pi$
$444$	70.1962	3.33136
$445$	16.4915	0.781774
$446$	6.05604	0.286762
$447$	16.7634	0.792880
$448$	5.35965	0.253220
$449$	−2.01985	−0.0953226	−0.0476613	−	0.998864i	$-0.515177\pi$
$449$	−2.01985	−0.0953226	−0.0476613	+	0.998864i	$0.515177\pi$
$450$	−13.9775	−0.658906
$451$	−1.25558	−0.0591229
$452$	22.5627	1.06126
$453$	10.9641	0.515138
$454$	69.4837	3.26103
$455$	3.58152	0.167904
$456$	−11.6412	−0.545149
$457$	−26.7521	−1.25141	−0.625705	−	0.780060i	$-0.715189\pi$
$457$	−26.7521	−1.25141	−0.625705	+	0.780060i	$0.715189\pi$
$458$	38.9553	1.82026
$459$	1.47386	0.0687938
$460$	21.0005	0.979154
$461$	19.8936	0.926537	0.463269	−	0.886218i	$-0.346677\pi$
$461$	19.8936	0.926537	0.463269	+	0.886218i	$0.346677\pi$
$462$	6.73404	0.313296
$463$	12.8020	0.594958	0.297479	−	0.954728i	$-0.403854\pi$
$463$	12.8020	0.594958	0.297479	+	0.954728i	$0.403854\pi$
$464$	−56.4762	−2.62184
$465$	0.215243	0.00998167
$466$	5.94822	0.275546
$467$	−21.9186	−1.01427	−0.507137	−	0.861866i	$-0.669296\pi$
$467$	−21.9186	−1.01427	−0.507137	+	0.861866i	$0.669296\pi$
$468$	61.5771	2.84640
$469$	9.36790	0.432570
$470$	−27.5968	−1.27294
$471$	−28.9081	−1.33201
$472$	−13.0462	−0.600500
$473$	−2.54890	−0.117198
$474$	93.3434	4.28741
$475$	0.692907	0.0317928
$476$	−1.16571	−0.0534302
$477$	28.3710	1.29902
$478$	37.6659	1.72280
$479$	−24.8247	−1.13427	−0.567136	−	0.823624i	$-0.691948\pi$
$479$	−24.8247	−1.13427	−0.567136	+	0.823624i	$0.691948\pi$
$480$	9.07031	0.414001
$481$	14.4027	0.656707
$482$	3.37507	0.153730
$483$	19.9362	0.907127
$484$	−45.3027	−2.05921
$485$	−3.83317	−0.174055
$486$	17.4335	0.790798
$487$	10.3509	0.469045	0.234522	−	0.972111i	$-0.424647\pi$
$487$	10.3509	0.469045	0.234522	+	0.972111i	$0.424647\pi$
$488$	−67.7410	−3.06649
$489$	−12.7170	−0.575081
$490$	−12.7068	−0.574034
$491$	−1.65391	−0.0746401	−0.0373200	−	0.999303i	$-0.511882\pi$
$491$	−1.65391	−0.0746401	−0.0373200	+	0.999303i	$0.511882\pi$
$492$	−23.8977	−1.07739
$493$	−1.90032	−0.0855861
$494$	−4.47631	−0.201399
$495$	−3.67716	−0.165276
$496$	−0.427190	−0.0191814
$497$	2.86112	0.128339
$498$	−32.9617	−1.47705
$499$	9.32296	0.417353	0.208676	−	0.977985i	$-0.433084\pi$
$499$	9.32296	0.417353	0.208676	+	0.977985i	$0.433084\pi$
$500$	−4.28808	−0.191769
$501$	38.8092	1.73387
$502$	46.8959	2.09307
$503$	−27.9332	−1.24548	−0.622739	−	0.782429i	$-0.713980\pi$
$503$	−27.9332	−1.24548	−0.622739	+	0.782429i	$0.713980\pi$
$504$	44.4613	1.98046
$505$	2.88231	0.128261
$506$	8.10156	0.360158
$507$	−18.6318	−0.827467
$508$	95.4577	4.23525
$509$	−1.78220	−0.0789948	−0.0394974	−	0.999220i	$-0.512576\pi$
$509$	−1.78220	−0.0789948	−0.0394974	+	0.999220i	$0.512576\pi$
$510$	1.43581	0.0635789
$511$	4.59980	0.203483
$512$	48.6857	2.15162
$513$	5.22258	0.230582
$514$	19.4347	0.857229
$515$	8.57423	0.377826
$516$	−48.5138	−2.13570
$517$	−7.26009	−0.319298
$518$	19.4894	0.856315
$519$	58.1373	2.55195
$520$	14.7814	0.648207
$521$	−4.03274	−0.176678	−0.0883388	−	0.996090i	$-0.528156\pi$
$521$	−4.03274	−0.176678	−0.0883388	+	0.996090i	$0.528156\pi$
$522$	135.835	5.94532
$523$	15.5949	0.681919	0.340959	−	0.940078i	$-0.389248\pi$
$523$	15.5949	0.681919	0.340959	+	0.940078i	$0.389248\pi$
$524$	−6.97362	−0.304644
$525$	−4.07075	−0.177662
$526$	56.5653	2.46636
$527$	−0.0143742	−0.000626149 0
$528$	11.2259	0.488543
$529$	0.984716	0.0428137
$530$	12.7633	0.554403
$531$	12.6743	0.550020
$532$	−4.13066	−0.179087
$533$	−4.90329	−0.212385
$534$	121.091	5.24014
$535$	18.0069	0.778505
$536$	38.6625	1.66997
$537$	−17.7089	−0.764196
$538$	26.1223	1.12621
$539$	−3.34287	−0.143988
$540$	−32.3201	−1.39083
$541$	−16.3061	−0.701053	−0.350526	−	0.936553i	$-0.613997\pi$
$541$	−16.3061	−0.701053	−0.350526	+	0.936553i	$0.613997\pi$
$542$	−34.0542	−1.46275
$543$	37.8812	1.62564
$544$	−0.605725	−0.0259702
$545$	−8.76898	−0.375622
$546$	26.2978	1.12544
$547$	11.7914	0.504164	0.252082	−	0.967706i	$-0.418885\pi$
$547$	11.7914	0.504164	0.252082	+	0.967706i	$0.418885\pi$
$548$	−19.9696	−0.853060
$549$	65.8101	2.80871
$550$	−1.65425	−0.0705375
$551$	−6.73374	−0.286867
$552$	82.2791	3.50203
$553$	17.6731	0.751538
$554$	33.8103	1.43646
$555$	−16.3701	−0.694871
$556$	−67.3482	−2.85620
$557$	17.7241	0.750994	0.375497	−	0.926824i	$-0.377472\pi$
$557$	17.7241	0.750994	0.375497	+	0.926824i	$0.377472\pi$
$558$	1.02746	0.0434960
$559$	−9.95397	−0.421008
$560$	8.07915	0.341407
$561$	0.377730	0.0159478
$562$	−9.19263	−0.387767
$563$	−21.7695	−0.917476	−0.458738	−	0.888572i	$-0.651698\pi$
$563$	−21.7695	−0.917476	−0.458738	+	0.888572i	$0.651698\pi$
$564$	−138.183	−5.81856
$565$	−5.26173	−0.221363
$566$	23.5021	0.987866
$567$	−7.43468	−0.312227
$568$	11.8082	0.495462
$569$	23.3970	0.980855	0.490428	−	0.871482i	$-0.336841\pi$
$569$	23.3970	0.980855	0.490428	+	0.871482i	$0.336841\pi$
$570$	5.08777	0.213103
$571$	25.6564	1.07369	0.536843	−	0.843682i	$-0.319617\pi$
$571$	25.6564	1.07369	0.536843	+	0.843682i	$0.319617\pi$
$572$	7.28771	0.304714
$573$	36.4660	1.52339
$574$	−6.63502	−0.276940
$575$	−4.89742	−0.204237
$576$	−21.4895	−0.895394
$577$	−39.7163	−1.65341	−0.826706	−	0.562635i	$-0.809788\pi$
$577$	−39.7163	−1.65341	−0.826706	+	0.562635i	$0.809788\pi$
$578$	42.5334	1.76916
$579$	59.9426	2.49113
$580$	41.6719	1.73033
$581$	−6.24079	−0.258911
$582$	−28.1455	−1.16667
$583$	3.35774	0.139063
$584$	18.9840	0.785562
$585$	−14.3601	−0.593716
$586$	−54.3567	−2.24546
$587$	−9.78048	−0.403684	−0.201842	−	0.979418i	$-0.564693\pi$
$587$	−9.78048	−0.403684	−0.201842	+	0.979418i	$0.564693\pi$
$588$	−63.6257	−2.62388
$589$	−0.0509345	−0.00209872
$590$	5.70182	0.234740
$591$	43.2465	1.77892
$592$	32.4894	1.33531
$593$	23.9175	0.982175	0.491088	−	0.871110i	$-0.336600\pi$
$593$	23.9175	0.982175	0.491088	+	0.871110i	$0.336600\pi$
$594$	−12.4684	−0.511585
$595$	0.271849	0.0111447
$596$	24.5488	1.00556
$597$	−16.9754	−0.694756
$598$	31.6382	1.29378
$599$	8.19785	0.334955	0.167478	−	0.985876i	$-0.446438\pi$
$599$	8.19785	0.334955	0.167478	+	0.985876i	$0.446438\pi$
$600$	−16.8005	−0.685878
$601$	−43.3259	−1.76730	−0.883650	−	0.468147i	$-0.844922\pi$
$601$	−43.3259	−1.76730	−0.883650	+	0.468147i	$0.844922\pi$
$602$	−13.4695	−0.548975
$603$	−37.5605	−1.52958
$604$	16.0562	0.653316
$605$	10.5648	0.429520
$606$	21.1638	0.859719
$607$	−0.0124872	−0.000506841 0	−0.000253420	−	1.00000i	$-0.500081\pi$
$607$	−0.0124872	−0.000506841 0	−0.000253420		1.00000i	$0.500081\pi$
$608$	−2.14637	−0.0870468
$609$	39.5599	1.60305
$610$	29.6061	1.19871
$611$	−28.3521	−1.14700
$612$	4.67390	0.188931
$613$	−3.41942	−0.138109	−0.0690545	−	0.997613i	$-0.521998\pi$
$613$	−3.41942	−0.138109	−0.0690545	+	0.997613i	$0.521998\pi$
$614$	−34.3187	−1.38499
$615$	5.57307	0.224728
$616$	5.26203	0.212013
$617$	40.9720	1.64947	0.824735	−	0.565520i	$-0.191324\pi$
$617$	40.9720	1.64947	0.824735	+	0.565520i	$0.191324\pi$
$618$	62.9574	2.53252
$619$	−32.2805	−1.29746	−0.648731	−	0.761017i	$-0.724700\pi$
$619$	−32.2805	−1.29746	−0.648731	+	0.761017i	$0.724700\pi$
$620$	0.315209	0.0126591
$621$	−36.9128	−1.48126
$622$	−12.1433	−0.486903
$623$	22.9268	0.918542
$624$	43.8393	1.75498
$625$	1.00000	0.0400000
$626$	−20.0536	−0.801504
$627$	1.33848	0.0534536
$628$	−42.3340	−1.68931
$629$	1.09321	0.0435892
$630$	−19.4317	−0.774178
$631$	−19.6981	−0.784168	−0.392084	−	0.919929i	$-0.628246\pi$
$631$	−19.6981	−0.784168	−0.392084	+	0.919929i	$0.628246\pi$
$632$	72.9393	2.90137
$633$	5.40581	0.214862
$634$	−12.6487	−0.502346
$635$	−22.2612	−0.883409
$636$	63.9087	2.53414
$637$	−13.0546	−0.517242
$638$	16.0761	0.636461
$639$	−11.4717	−0.453812
$640$	−15.8627	−0.627030
$641$	5.19266	0.205098	0.102549	−	0.994728i	$-0.467300\pi$
$641$	5.19266	0.205098	0.102549	+	0.994728i	$0.467300\pi$
$642$	132.218	5.21823
$643$	22.9081	0.903408	0.451704	−	0.892168i	$-0.350816\pi$
$643$	22.9081	0.903408	0.451704	+	0.892168i	$0.350816\pi$
$644$	29.1952	1.15045
$645$	11.3137	0.445475
$646$	−0.339766	−0.0133679
$647$	−19.1632	−0.753382	−0.376691	−	0.926339i	$-0.622938\pi$
$647$	−19.1632	−0.753382	−0.376691	+	0.926339i	$0.622938\pi$
$648$	−30.6839	−1.20538
$649$	1.50002	0.0588809
$650$	−6.46018	−0.253389
$651$	0.299234	0.0117279
$652$	−18.6232	−0.729339
$653$	−2.14361	−0.0838858	−0.0419429	−	0.999120i	$-0.513355\pi$
$653$	−2.14361	−0.0838858	−0.0419429	+	0.999120i	$0.513355\pi$
$654$	−64.3874	−2.51775
$655$	1.62628	0.0635441
$656$	−11.0608	−0.431851
$657$	−18.4429	−0.719524
$658$	−38.3654	−1.49564
$659$	−2.13009	−0.0829763	−0.0414882	−	0.999139i	$-0.513210\pi$
$659$	−2.13009	−0.0829763	−0.0414882	+	0.999139i	$0.513210\pi$
$660$	−8.28319	−0.322423
$661$	6.36378	0.247522	0.123761	−	0.992312i	$-0.460504\pi$
$661$	6.36378	0.247522	0.123761	+	0.992312i	$0.460504\pi$
$662$	−59.1992	−2.30084
$663$	1.47511	0.0572886
$664$	−25.7565	−0.999547
$665$	0.963289	0.0373548
$666$	−78.1425	−3.02796
$667$	47.5936	1.84283
$668$	56.8334	2.19895
$669$	−7.07168	−0.273407
$670$	−16.8974	−0.652803
$671$	7.78868	0.300679
$672$	12.6097	0.486429
$673$	−15.8226	−0.609917	−0.304959	−	0.952366i	$-0.598643\pi$
$673$	−15.8226	−0.609917	−0.304959	+	0.952366i	$0.598643\pi$
$674$	13.3489	0.514181
$675$	7.53719	0.290107
$676$	−27.2850	−1.04942
$677$	11.4554	0.440269	0.220134	−	0.975470i	$-0.429350\pi$
$677$	11.4554	0.440269	0.220134	+	0.975470i	$0.429350\pi$
$678$	−38.6350	−1.48377
$679$	−5.32892	−0.204505
$680$	1.12196	0.0430250
$681$	−81.1367	−3.10916
$682$	0.121601	0.00465635
$683$	−41.3267	−1.58132	−0.790662	−	0.612253i	$-0.790263\pi$
$683$	−41.3267	−1.58132	−0.790662	+	0.612253i	$0.790263\pi$
$684$	16.5618	0.633258
$685$	4.65701	0.177935
$686$	−42.0679	−1.60616
$687$	−45.4884	−1.73549
$688$	−22.4540	−0.856052
$689$	13.1126	0.499552
$690$	−35.9600	−1.36897
$691$	−16.1635	−0.614889	−0.307445	−	0.951566i	$-0.599474\pi$
$691$	−16.1635	−0.614889	−0.307445	+	0.951566i	$0.599474\pi$
$692$	85.1383	3.23647
$693$	−5.11204	−0.194190
$694$	27.7255	1.05245
$695$	15.7059	0.595760
$696$	163.269	6.18869
$697$	−0.372175	−0.0140971
$698$	−35.7110	−1.35168
$699$	−6.94578	−0.262714
$700$	−5.96134	−0.225318
$701$	−11.1603	−0.421521	−0.210760	−	0.977538i	$-0.567594\pi$
$701$	−11.1603	−0.421521	−0.210760	+	0.977538i	$0.567594\pi$
$702$	−48.6916	−1.83775
$703$	3.87376	0.146102
$704$	−2.54330	−0.0958541
$705$	32.2250	1.21366
$706$	67.5419	2.54197
$707$	4.00703	0.150700
$708$	28.5503	1.07299
$709$	48.4865	1.82095	0.910474	−	0.413567i	$-0.135717\pi$
$709$	48.4865	1.82095	0.910474	+	0.413567i	$0.135717\pi$
$710$	−5.16077	−0.193680
$711$	−70.8602	−2.65747
$712$	94.6217	3.54610
$713$	0.360001	0.0134822
$714$	1.99609	0.0747017
$715$	−1.69953	−0.0635587
$716$	−25.9335	−0.969181
$717$	−43.9827	−1.64256
$718$	39.0264	1.45645
$719$	−42.0501	−1.56820	−0.784101	−	0.620633i	$-0.786876\pi$
$719$	−42.0501	−1.56820	−0.784101	+	0.620633i	$0.786876\pi$
$720$	−32.3933	−1.20723
$721$	11.9200	0.443924
$722$	46.4405	1.72834
$723$	−3.94109	−0.146571
$724$	55.4745	2.06169
$725$	−9.71809	−0.360921
$726$	77.5735	2.87902
$727$	14.9101	0.552987	0.276493	−	0.961016i	$-0.410828\pi$
$727$	14.9101	0.552987	0.276493	+	0.961016i	$0.410828\pi$
$728$	20.5493	0.761608
$729$	−36.4008	−1.34818
$730$	−8.29691	−0.307082
$731$	−0.755538	−0.0279446
$732$	148.244	5.47926
$733$	−32.7273	−1.20881	−0.604405	−	0.796677i	$-0.706589\pi$
$733$	−32.7273	−1.20881	−0.604405	+	0.796677i	$0.706589\pi$
$734$	22.3589	0.825281
$735$	14.8378	0.547301
$736$	15.1704	0.559188
$737$	−4.44532	−0.163745
$738$	26.6030	0.979271
$739$	−51.0363	−1.87740	−0.938701	−	0.344733i	$-0.887970\pi$
$739$	−51.0363	−1.87740	−0.938701	+	0.344733i	$0.887970\pi$
$740$	−23.9729	−0.881261
$741$	5.22702	0.192019
$742$	17.7437	0.651393
$743$	10.9313	0.401031	0.200515	−	0.979691i	$-0.435738\pi$
$743$	10.9313	0.401031	0.200515	+	0.979691i	$0.435738\pi$
$744$	1.23498	0.0452765
$745$	−5.72490	−0.209744
$746$	−60.4634	−2.21372
$747$	25.0224	0.915521
$748$	0.553160	0.0202255
$749$	25.0334	0.914701
$750$	7.34263	0.268115
$751$	−14.0532	−0.512810	−0.256405	−	0.966569i	$-0.582538\pi$
$751$	−14.0532	−0.512810	−0.256405	+	0.966569i	$0.582538\pi$
$752$	−63.9564	−2.33225
$753$	−54.7607	−1.99559
$754$	62.7806	2.28634
$755$	−3.74438	−0.136272
$756$	−44.9318	−1.63415
$757$	50.8556	1.84838	0.924188	−	0.381937i	$-0.124743\pi$
$757$	50.8556	1.84838	0.924188	+	0.381937i	$0.124743\pi$
$758$	−25.0737	−0.910718
$759$	−9.46025	−0.343385
$760$	3.97562	0.144211
$761$	−5.53251	−0.200553	−0.100277	−	0.994960i	$-0.531973\pi$
$761$	−5.53251	−0.200553	−0.100277	+	0.994960i	$0.531973\pi$
$762$	−163.456	−5.92138
$763$	−12.1908	−0.441335
$764$	53.4020	1.93202
$765$	−1.08998	−0.0394081
$766$	−25.9990	−0.939381
$767$	5.85789	0.211516
$768$	−93.8967	−3.38821
$769$	34.8942	1.25832	0.629159	−	0.777276i	$-0.283399\pi$
$769$	34.8942	1.25832	0.629159	+	0.777276i	$0.283399\pi$
$770$	−2.29976	−0.0828776
$771$	−22.6941	−0.817307
$772$	87.7819	3.15934
$773$	47.0988	1.69403	0.847013	−	0.531571i	$-0.178398\pi$
$773$	47.0988	1.69403	0.847013	+	0.531571i	$0.178398\pi$
$774$	54.0057	1.94120
$775$	−0.0735084	−0.00264050
$776$	−21.9931	−0.789507
$777$	−22.7579	−0.816436
$778$	40.2254	1.44215
$779$	−1.31879	−0.0472507
$780$	−32.3476	−1.15823
$781$	−1.35768	−0.0485816
$782$	2.40144	0.0858754
$783$	−73.2471	−2.61764
$784$	−29.4484	−1.05173
$785$	9.87249	0.352364
$786$	11.9412	0.425928
$787$	−8.90141	−0.317301	−0.158651	−	0.987335i	$-0.550714\pi$
$787$	−8.90141	−0.317301	−0.158651	+	0.987335i	$0.550714\pi$
$788$	63.3317	2.25610
$789$	−66.0517	−2.35150
$790$	−31.8780	−1.13417
$791$	−7.31493	−0.260089
$792$	−21.0981	−0.749687
$793$	30.4164	1.08012
$794$	−81.0886	−2.87773
$795$	−14.9038	−0.528584
$796$	−24.8593	−0.881115
$797$	25.8719	0.916430	0.458215	−	0.888841i	$-0.348489\pi$
$797$	25.8719	0.916430	0.458215	+	0.888841i	$0.348489\pi$
$798$	7.07308	0.250384
$799$	−2.15202	−0.0761329
$800$	−3.09763	−0.109518
$801$	−91.9247	−3.24800
$802$	69.9848	2.47125
$803$	−2.18273	−0.0770268
$804$	−84.6090	−2.98393
$805$	−6.80846	−0.239967
$806$	0.474878	0.0167268
$807$	−30.5032	−1.07376
$808$	16.5375	0.581788
$809$	−23.0302	−0.809699	−0.404850	−	0.914383i	$-0.632676\pi$
$809$	−23.0302	−0.809699	−0.404850	+	0.914383i	$0.632676\pi$
$810$	13.4103	0.471191
$811$	29.3312	1.02996	0.514978	−	0.857203i	$-0.327800\pi$
$811$	29.3312	1.02996	0.514978	+	0.857203i	$0.327800\pi$
$812$	57.9329	2.03304
$813$	39.7653	1.39463
$814$	−9.24824	−0.324151
$815$	4.34301	0.152129
$816$	3.32754	0.116487
$817$	−2.67723	−0.0936644
$818$	−44.7989	−1.56636
$819$	−19.9636	−0.697584
$820$	8.16139	0.285008
$821$	35.1880	1.22807	0.614036	−	0.789278i	$-0.289545\pi$
$821$	35.1880	1.22807	0.614036	+	0.789278i	$0.289545\pi$
$822$	34.1947	1.19268
$823$	−47.3887	−1.65187	−0.825933	−	0.563769i	$-0.809351\pi$
$823$	−47.3887	−1.65187	−0.825933	+	0.563769i	$0.809351\pi$
$824$	49.1954	1.71380
$825$	1.93168	0.0672525
$826$	7.92675	0.275807
$827$	44.1065	1.53373	0.766866	−	0.641807i	$-0.221815\pi$
$827$	44.1065	1.53373	0.766866	+	0.641807i	$0.221815\pi$
$828$	−117.058	−4.06804
$829$	−17.2839	−0.600294	−0.300147	−	0.953893i	$-0.597036\pi$
$829$	−17.2839	−0.600294	−0.300147	+	0.953893i	$0.597036\pi$
$830$	11.2568	0.390731
$831$	−39.4805	−1.36957
$832$	−9.93210	−0.344333
$833$	−0.990885	−0.0343321
$834$	115.323	3.99330
$835$	−13.2538	−0.458668
$836$	1.96011	0.0677918
$837$	−0.554047	−0.0191507
$838$	−97.9096	−3.38223
$839$	−14.4853	−0.500090	−0.250045	−	0.968234i	$-0.580445\pi$
$839$	−14.4853	−0.500090	−0.250045	+	0.968234i	$0.580445\pi$
$840$	−23.3563	−0.805869
$841$	65.4412	2.25659
$842$	21.5808	0.743723
$843$	10.7343	0.369709
$844$	7.91645	0.272496
$845$	6.36300	0.218894
$846$	153.826	5.28864
$847$	14.6873	0.504663
$848$	29.5794	1.01576
$849$	−27.4436	−0.941861
$850$	−0.490349	−0.0168188
$851$	−27.3795	−0.938557
$852$	−25.8411	−0.885301
$853$	10.7825	0.369186	0.184593	−	0.982815i	$-0.440903\pi$
$853$	10.7825	0.369186	0.184593	+	0.982815i	$0.440903\pi$
$854$	41.1588	1.40842
$855$	−3.86230	−0.132088
$856$	103.316	3.53127
$857$	−6.31609	−0.215753	−0.107877	−	0.994164i	$-0.534405\pi$
$857$	−6.31609	−0.215753	−0.107877	+	0.994164i	$0.534405\pi$
$858$	−12.4790	−0.426026
$859$	−54.6649	−1.86514	−0.932572	−	0.360985i	$-0.882441\pi$
$859$	−54.6649	−1.86514	−0.932572	+	0.360985i	$0.882441\pi$
$860$	16.5681	0.564968
$861$	7.74776	0.264043
$862$	93.4268	3.18213
$863$	−9.69264	−0.329941	−0.164971	−	0.986298i	$-0.552753\pi$
$863$	−9.69264	−0.329941	−0.164971	+	0.986298i	$0.552753\pi$
$864$	−23.3474	−0.794295
$865$	−19.8546	−0.675078
$866$	19.8739	0.675342
$867$	−49.6665	−1.68676
$868$	0.438209	0.0148738
$869$	−8.38637	−0.284488
$870$	−71.3564	−2.41921
$871$	−17.3599	−0.588217
$872$	−50.3128	−1.70381
$873$	21.3662	0.723138
$874$	8.50945	0.287836
$875$	1.39021	0.0469978
$876$	−41.5444	−1.40366
$877$	49.4906	1.67118	0.835589	−	0.549355i	$-0.185127\pi$
$877$	49.4906	1.67118	0.835589	+	0.549355i	$0.185127\pi$
$878$	17.5197	0.591262
$879$	63.4727	2.14088
$880$	−3.83377	−0.129237
$881$	33.2987	1.12186	0.560930	−	0.827863i	$-0.310444\pi$
$881$	33.2987	1.12186	0.560930	+	0.827863i	$0.310444\pi$
$882$	70.8283	2.38491
$883$	39.2157	1.31971	0.659856	−	0.751392i	$-0.270617\pi$
$883$	39.2157	1.31971	0.659856	+	0.751392i	$0.270617\pi$
$884$	2.16020	0.0726555
$885$	−6.65806	−0.223808
$886$	−12.2902	−0.412896
$887$	−2.34488	−0.0787332	−0.0393666	−	0.999225i	$-0.512534\pi$
$887$	−2.34488	−0.0787332	−0.0393666	+	0.999225i	$0.512534\pi$
$888$	−93.9248	−3.15191
$889$	−30.9478	−1.03796
$890$	−41.3543	−1.38620
$891$	3.52795	0.118191
$892$	−10.3560	−0.346745
$893$	−7.62561	−0.255181
$894$	−42.0359	−1.40589
$895$	6.04782	0.202156
$896$	−22.0526	−0.736726
$897$	−36.9442	−1.23353
$898$	5.06498	0.169021
$899$	0.714361	0.0238253
$900$	23.9019	0.796731
$901$	0.995292	0.0331580
$902$	3.14849	0.104833
$903$	15.7284	0.523409
$904$	−30.1896	−1.00409
$905$	−12.9369	−0.430038
$906$	−27.4936	−0.913413
$907$	−14.4405	−0.479490	−0.239745	−	0.970836i	$-0.577064\pi$
$907$	−14.4405	−0.479490	−0.239745	+	0.970836i	$0.577064\pi$
$908$	−118.819	−3.94315
$909$	−16.0661	−0.532880
$910$	−8.98104	−0.297718
$911$	−45.7131	−1.51454	−0.757271	−	0.653101i	$-0.773468\pi$
$911$	−45.7131	−1.51454	−0.757271	+	0.653101i	$0.773468\pi$
$912$	11.7910	0.390441
$913$	2.96142	0.0980087
$914$	67.0836	2.21893
$915$	−34.5712	−1.14289
$916$	−66.6148	−2.20101
$917$	2.26088	0.0746608
$918$	−3.69585	−0.121981
$919$	23.1291	0.762959	0.381480	−	0.924377i	$-0.375415\pi$
$919$	23.1291	0.762959	0.381480	+	0.924377i	$0.375415\pi$
$920$	−28.0994	−0.926409
$921$	40.0742	1.32049
$922$	−49.8853	−1.64288
$923$	−5.30202	−0.174518
$924$	−11.5154	−0.378829
$925$	5.59059	0.183818
$926$	−32.1023	−1.05495
$927$	−47.7932	−1.56973
$928$	30.1030	0.988180
$929$	−43.7632	−1.43582	−0.717912	−	0.696133i	$-0.754902\pi$
$929$	−43.7632	−1.43582	−0.717912	+	0.696133i	$0.754902\pi$
$930$	−0.539745	−0.0176989
$931$	−3.51117	−0.115074
$932$	−10.1716	−0.333183
$933$	14.1799	0.464228
$934$	54.9633	1.79845
$935$	−0.129000	−0.00421874
$936$	−82.3922	−2.69307
$937$	−35.3184	−1.15380	−0.576900	−	0.816814i	$-0.695738\pi$
$937$	−35.3184	−1.15380	−0.576900	+	0.816814i	$0.695738\pi$
$938$	−23.4910	−0.767008
$939$	23.4168	0.764177
$940$	47.1913	1.53921
$941$	−40.0779	−1.30650	−0.653251	−	0.757141i	$-0.726595\pi$
$941$	−40.0779	−1.30650	−0.653251	+	0.757141i	$0.726595\pi$
$942$	72.4900	2.36185
$943$	9.32113	0.303538
$944$	13.2141	0.430084
$945$	10.4783	0.340859
$946$	6.39163	0.207810
$947$	−56.9788	−1.85156	−0.925780	−	0.378062i	$-0.876591\pi$
$947$	−56.9788	−1.85156	−0.925780	+	0.378062i	$0.876591\pi$
$948$	−159.620	−5.18422
$949$	−8.52400	−0.276701
$950$	−1.73754	−0.0563732
$951$	14.7700	0.478951
$952$	1.55976	0.0505520
$953$	16.3153	0.528503	0.264252	−	0.964454i	$-0.414875\pi$
$953$	16.3153	0.528503	0.264252	+	0.964454i	$0.414875\pi$
$954$	−71.1433	−2.30335
$955$	−12.4536	−0.402989
$956$	−64.4098	−2.08316
$957$	−18.7722	−0.606820
$958$	62.2506	2.01123
$959$	6.47424	0.209064
$960$	11.2888	0.364344
$961$	−30.9946	−0.999826
$962$	−36.1163	−1.16444
$963$	−100.371	−3.23442
$964$	−5.77146	−0.185886
$965$	−20.4712	−0.658990
$966$	−49.9920	−1.60847
$967$	40.9142	1.31571	0.657855	−	0.753144i	$-0.271464\pi$
$967$	40.9142	1.31571	0.657855	+	0.753144i	$0.271464\pi$
$968$	60.6165	1.94829
$969$	0.396747	0.0127454
$970$	9.61206	0.308625
$971$	−6.40885	−0.205670	−0.102835	−	0.994698i	$-0.532791\pi$
$971$	−6.40885	−0.205670	−0.102835	+	0.994698i	$0.532791\pi$
$972$	−29.8117	−0.956212
$973$	21.8346	0.699985
$974$	−25.9560	−0.831684
$975$	7.54361	0.241589
$976$	68.6130	2.19625
$977$	−11.6019	−0.371177	−0.185589	−	0.982628i	$-0.559419\pi$
$977$	−11.6019	−0.371177	−0.185589	+	0.982628i	$0.559419\pi$
$978$	31.8891	1.01970
$979$	−10.8794	−0.347706
$980$	21.7290	0.694107
$981$	48.8787	1.56058
$982$	4.14736	0.132348
$983$	−31.6383	−1.00910	−0.504552	−	0.863381i	$-0.668342\pi$
$983$	−31.6383	−1.00910	−0.504552	+	0.863381i	$0.668342\pi$
$984$	31.9760	1.01936
$985$	−14.7692	−0.470587
$986$	4.76525	0.151756
$987$	44.7996	1.42599
$988$	7.65462	0.243526
$989$	18.9225	0.601699
$990$	9.22087	0.293058
$991$	43.8980	1.39447	0.697234	−	0.716844i	$-0.254414\pi$
$991$	43.8980	1.39447	0.697234	+	0.716844i	$0.254414\pi$
$992$	0.227702	0.00722953
$993$	69.1274	2.19369
$994$	−7.17457	−0.227563
$995$	5.79731	0.183787
$996$	56.3655	1.78601
$997$	−32.6275	−1.03332	−0.516662	−	0.856189i	$-0.672826\pi$
$997$	−32.6275	−1.03332	−0.516662	+	0.856189i	$0.672826\pi$
$998$	−23.3783	−0.740026
$999$	42.1374	1.33317

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8005.2.a.f.1.11	✓	127

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8005.2.a.f.1.11	✓	127	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:	\( N \)	\(=\)	\( 8005 = 5 \cdot 1601 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8005.a (trivial)

\(f(q)\)	\(=\)	\(q-2.50760 q^{2} +2.92815 q^{3} +4.28808 q^{4} -1.00000 q^{5} -7.34263 q^{6} -1.39021 q^{7} -5.73759 q^{8} +5.57405 q^{9} +O(q^{10})\)	\(q-2.50760 q^{2} +2.92815 q^{3} +4.28808 q^{4} -1.00000 q^{5} -7.34263 q^{6} -1.39021 q^{7} -5.73759 q^{8} +5.57405 q^{9} +2.50760 q^{10} +0.659694 q^{11} +12.5561 q^{12} +2.57624 q^{13} +3.48611 q^{14} -2.92815 q^{15} +5.81145 q^{16} +0.195545 q^{17} -13.9775 q^{18} +0.692907 q^{19} -4.28808 q^{20} -4.07075 q^{21} -1.65425 q^{22} -4.89742 q^{23} -16.8005 q^{24} +1.00000 q^{25} -6.46018 q^{26} +7.53719 q^{27} -5.96134 q^{28} -9.71809 q^{29} +7.34263 q^{30} -0.0735084 q^{31} -3.09763 q^{32} +1.93168 q^{33} -0.490349 q^{34} +1.39021 q^{35} +23.9019 q^{36} +5.59059 q^{37} -1.73754 q^{38} +7.54361 q^{39} +5.73759 q^{40} -1.90327 q^{41} +10.2078 q^{42} -3.86376 q^{43} +2.82882 q^{44} -5.57405 q^{45} +12.2808 q^{46} -11.0052 q^{47} +17.0168 q^{48} -5.06731 q^{49} -2.50760 q^{50} +0.572584 q^{51} +11.0471 q^{52} +5.08984 q^{53} -18.9003 q^{54} -0.659694 q^{55} +7.97648 q^{56} +2.02894 q^{57} +24.3691 q^{58} +2.27381 q^{59} -12.5561 q^{60} +11.8065 q^{61} +0.184330 q^{62} -7.74912 q^{63} -3.85527 q^{64} -2.57624 q^{65} -4.84389 q^{66} -6.73846 q^{67} +0.838511 q^{68} -14.3404 q^{69} -3.48611 q^{70} -2.05805 q^{71} -31.9816 q^{72} -3.30870 q^{73} -14.0190 q^{74} +2.92815 q^{75} +2.97124 q^{76} -0.917115 q^{77} -18.9164 q^{78} -12.7125 q^{79} -5.81145 q^{80} +5.34786 q^{81} +4.77266 q^{82} +4.48908 q^{83} -17.4557 q^{84} -0.195545 q^{85} +9.68878 q^{86} -28.4560 q^{87} -3.78505 q^{88} -16.4915 q^{89} +13.9775 q^{90} -3.58152 q^{91} -21.0005 q^{92} -0.215243 q^{93} +27.5968 q^{94} -0.692907 q^{95} -9.07031 q^{96} +3.83317 q^{97} +12.7068 q^{98} +3.67716 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 127 q - 6 q^{2} - 18 q^{3} + 114 q^{4} - 127 q^{5} - 20 q^{6} + 28 q^{7} - 18 q^{8} + 101 q^{9}+O(q^{10}) \) 127 * q - 6 * q^2 - 18 * q^3 + 114 * q^4 - 127 * q^5 - 20 * q^6 + 28 * q^7 - 18 * q^8 + 101 * q^9	\( 127 q - 6 q^{2} - 18 q^{3} + 114 q^{4} - 127 q^{5} - 20 q^{6} + 28 q^{7} - 18 q^{8} + 101 q^{9} + 6 q^{10} - 45 q^{11} - 30 q^{12} - 53 q^{14} + 18 q^{15} + 84 q^{16} - 36 q^{17} - 10 q^{18} - 49 q^{19} - 114 q^{20} - 48 q^{21} + 13 q^{22} - 29 q^{23} - 63 q^{24} + 127 q^{25} - 55 q^{26} - 75 q^{27} + 44 q^{28} - 45 q^{29} + 20 q^{30} - 49 q^{31} - 32 q^{32} - 8 q^{33} - 52 q^{34} - 28 q^{35} + 44 q^{36} + 36 q^{37} - 65 q^{38} - 52 q^{39} + 18 q^{40} - 66 q^{41} - 18 q^{42} - 5 q^{43} - 93 q^{44} - 101 q^{45} - 25 q^{46} - 32 q^{47} - 54 q^{48} + 77 q^{49} - 6 q^{50} - 102 q^{51} - 13 q^{52} - 67 q^{53} - 53 q^{54} + 45 q^{55} - 158 q^{56} + 16 q^{57} + 35 q^{58} - 213 q^{59} + 30 q^{60} - 62 q^{61} - 33 q^{62} + 59 q^{63} + 34 q^{64} - 60 q^{66} + 10 q^{67} - 94 q^{68} - 93 q^{69} + 53 q^{70} - 118 q^{71} - 24 q^{72} + 35 q^{73} - 107 q^{74} - 18 q^{75} - 98 q^{76} - 93 q^{77} + 21 q^{78} - 64 q^{79} - 84 q^{80} + 15 q^{81} + 15 q^{82} - 187 q^{83} - 118 q^{84} + 36 q^{85} - 126 q^{86} - 53 q^{87} + 15 q^{88} - 138 q^{89} + 10 q^{90} - 138 q^{91} - 86 q^{92} + 23 q^{93} - 60 q^{94} + 49 q^{95} - 92 q^{96} + 9 q^{97} - 67 q^{98} - 147 q^{99}+O(q^{100}) \) 127 * q - 6 * q^2 - 18 * q^3 + 114 * q^4 - 127 * q^5 - 20 * q^6 + 28 * q^7 - 18 * q^8 + 101 * q^9 + 6 * q^10 - 45 * q^11 - 30 * q^12 - 53 * q^14 + 18 * q^15 + 84 * q^16 - 36 * q^17 - 10 * q^18 - 49 * q^19 - 114 * q^20 - 48 * q^21 + 13 * q^22 - 29 * q^23 - 63 * q^24 + 127 * q^25 - 55 * q^26 - 75 * q^27 + 44 * q^28 - 45 * q^29 + 20 * q^30 - 49 * q^31 - 32 * q^32 - 8 * q^33 - 52 * q^34 - 28 * q^35 + 44 * q^36 + 36 * q^37 - 65 * q^38 - 52 * q^39 + 18 * q^40 - 66 * q^41 - 18 * q^42 - 5 * q^43 - 93 * q^44 - 101 * q^45 - 25 * q^46 - 32 * q^47 - 54 * q^48 + 77 * q^49 - 6 * q^50 - 102 * q^51 - 13 * q^52 - 67 * q^53 - 53 * q^54 + 45 * q^55 - 158 * q^56 + 16 * q^57 + 35 * q^58 - 213 * q^59 + 30 * q^60 - 62 * q^61 - 33 * q^62 + 59 * q^63 + 34 * q^64 - 60 * q^66 + 10 * q^67 - 94 * q^68 - 93 * q^69 + 53 * q^70 - 118 * q^71 - 24 * q^72 + 35 * q^73 - 107 * q^74 - 18 * q^75 - 98 * q^76 - 93 * q^77 + 21 * q^78 - 64 * q^79 - 84 * q^80 + 15 * q^81 + 15 * q^82 - 187 * q^83 - 118 * q^84 + 36 * q^85 - 126 * q^86 - 53 * q^87 + 15 * q^88 - 138 * q^89 + 10 * q^90 - 138 * q^91 - 86 * q^92 + 23 * q^93 - 60 * q^94 + 49 * q^95 - 92 * q^96 + 9 * q^97 - 67 * q^98 - 147 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more