LMFDB - Embedded newform 6015.2.a.e.1.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6015,2,Mod(1,6015)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6015, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("6015.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6015 = 3 \cdot 5 \cdot 401 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6015.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:	$48.0300168158$
Analytic rank:	$0$
Dimension:	$31$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.15
Character	$\chi$	$=$	6015.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.0495237 q^{2} -1.00000 q^{3} -1.99755 q^{4} -1.00000 q^{5} -0.0495237 q^{6} -1.50762 q^{7} -0.197974 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+0.0495237 q^{2} -1.00000 q^{3} -1.99755 q^{4} -1.00000 q^{5} -0.0495237 q^{6} -1.50762 q^{7} -0.197974 q^{8} +1.00000 q^{9} -0.0495237 q^{10} +1.36628 q^{11} +1.99755 q^{12} -6.59677 q^{13} -0.0746630 q^{14} +1.00000 q^{15} +3.98529 q^{16} +2.05546 q^{17} +0.0495237 q^{18} +4.23470 q^{19} +1.99755 q^{20} +1.50762 q^{21} +0.0676634 q^{22} +1.56752 q^{23} +0.197974 q^{24} +1.00000 q^{25} -0.326697 q^{26} -1.00000 q^{27} +3.01154 q^{28} -6.29879 q^{29} +0.0495237 q^{30} -2.44850 q^{31} +0.593314 q^{32} -1.36628 q^{33} +0.101794 q^{34} +1.50762 q^{35} -1.99755 q^{36} -6.90433 q^{37} +0.209718 q^{38} +6.59677 q^{39} +0.197974 q^{40} -8.13111 q^{41} +0.0746630 q^{42} +4.49722 q^{43} -2.72921 q^{44} -1.00000 q^{45} +0.0776297 q^{46} +1.58039 q^{47} -3.98529 q^{48} -4.72708 q^{49} +0.0495237 q^{50} -2.05546 q^{51} +13.1774 q^{52} -14.0609 q^{53} -0.0495237 q^{54} -1.36628 q^{55} +0.298469 q^{56} -4.23470 q^{57} -0.311939 q^{58} -5.18085 q^{59} -1.99755 q^{60} -10.5297 q^{61} -0.121259 q^{62} -1.50762 q^{63} -7.94120 q^{64} +6.59677 q^{65} -0.0676634 q^{66} +6.50642 q^{67} -4.10587 q^{68} -1.56752 q^{69} +0.0746630 q^{70} -4.56999 q^{71} -0.197974 q^{72} -4.68254 q^{73} -0.341928 q^{74} -1.00000 q^{75} -8.45901 q^{76} -2.05983 q^{77} +0.326697 q^{78} +0.561877 q^{79} -3.98529 q^{80} +1.00000 q^{81} -0.402683 q^{82} +14.8970 q^{83} -3.01154 q^{84} -2.05546 q^{85} +0.222719 q^{86} +6.29879 q^{87} -0.270488 q^{88} -5.25035 q^{89} -0.0495237 q^{90} +9.94543 q^{91} -3.13121 q^{92} +2.44850 q^{93} +0.0782667 q^{94} -4.23470 q^{95} -0.593314 q^{96} +9.91016 q^{97} -0.234103 q^{98} +1.36628 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 31 q + 6 q^{2} - 31 q^{3} + 24 q^{4} - 31 q^{5} - 6 q^{6} - 4 q^{7} + 15 q^{8} + 31 q^{9}+O(q^{10}) $ 31 * q + 6 * q^2 - 31 * q^3 + 24 * q^4 - 31 * q^5 - 6 * q^6 - 4 * q^7 + 15 * q^8 + 31 * q^9	\( 31 q + 6 q^{2} - 31 q^{3} + 24 q^{4} - 31 q^{5} - 6 q^{6} - 4 q^{7} + 15 q^{8} + 31 q^{9} - 6 q^{10} - q^{11} - 24 q^{12} + 8 q^{13} + 8 q^{14} + 31 q^{15} + 10 q^{16} + 42 q^{17} + 6 q^{18} - 19 q^{19} - 24 q^{20} + 4 q^{21} + 17 q^{22} + 26 q^{23} - 15 q^{24} + 31 q^{25} + 4 q^{26} - 31 q^{27} - 16 q^{28} - 11 q^{29} + 6 q^{30} - 3 q^{31} + 41 q^{32} + q^{33} + 6 q^{34} + 4 q^{35} + 24 q^{36} + 10 q^{37} + 12 q^{38} - 8 q^{39} - 15 q^{40} + 27 q^{41} - 8 q^{42} - 37 q^{43} - 9 q^{44} - 31 q^{45} - 23 q^{46} + 48 q^{47} - 10 q^{48} + 31 q^{49} + 6 q^{50} - 42 q^{51} + 18 q^{52} + 35 q^{53} - 6 q^{54} + q^{55} + 7 q^{56} + 19 q^{57} - 26 q^{58} - 3 q^{59} + 24 q^{60} + 11 q^{61} + 49 q^{62} - 4 q^{63} - 27 q^{64} - 8 q^{65} - 17 q^{66} - 44 q^{67} + 72 q^{68} - 26 q^{69} - 8 q^{70} + 6 q^{71} + 15 q^{72} + 49 q^{73} - 6 q^{74} - 31 q^{75} - 36 q^{76} + 66 q^{77} - 4 q^{78} - 41 q^{79} - 10 q^{80} + 31 q^{81} + 9 q^{82} + 53 q^{83} + 16 q^{84} - 42 q^{85} + 3 q^{86} + 11 q^{87} + 21 q^{88} + 31 q^{89} - 6 q^{90} - 45 q^{91} + 45 q^{92} + 3 q^{93} - 4 q^{94} + 19 q^{95} - 41 q^{96} + 37 q^{97} + 41 q^{98} - q^{99}+O(q^{100}) \) 31 * q + 6 * q^2 - 31 * q^3 + 24 * q^4 - 31 * q^5 - 6 * q^6 - 4 * q^7 + 15 * q^8 + 31 * q^9 - 6 * q^10 - q^11 - 24 * q^12 + 8 * q^13 + 8 * q^14 + 31 * q^15 + 10 * q^16 + 42 * q^17 + 6 * q^18 - 19 * q^19 - 24 * q^20 + 4 * q^21 + 17 * q^22 + 26 * q^23 - 15 * q^24 + 31 * q^25 + 4 * q^26 - 31 * q^27 - 16 * q^28 - 11 * q^29 + 6 * q^30 - 3 * q^31 + 41 * q^32 + q^33 + 6 * q^34 + 4 * q^35 + 24 * q^36 + 10 * q^37 + 12 * q^38 - 8 * q^39 - 15 * q^40 + 27 * q^41 - 8 * q^42 - 37 * q^43 - 9 * q^44 - 31 * q^45 - 23 * q^46 + 48 * q^47 - 10 * q^48 + 31 * q^49 + 6 * q^50 - 42 * q^51 + 18 * q^52 + 35 * q^53 - 6 * q^54 + q^55 + 7 * q^56 + 19 * q^57 - 26 * q^58 - 3 * q^59 + 24 * q^60 + 11 * q^61 + 49 * q^62 - 4 * q^63 - 27 * q^64 - 8 * q^65 - 17 * q^66 - 44 * q^67 + 72 * q^68 - 26 * q^69 - 8 * q^70 + 6 * q^71 + 15 * q^72 + 49 * q^73 - 6 * q^74 - 31 * q^75 - 36 * q^76 + 66 * q^77 - 4 * q^78 - 41 * q^79 - 10 * q^80 + 31 * q^81 + 9 * q^82 + 53 * q^83 + 16 * q^84 - 42 * q^85 + 3 * q^86 + 11 * q^87 + 21 * q^88 + 31 * q^89 - 6 * q^90 - 45 * q^91 + 45 * q^92 + 3 * q^93 - 4 * q^94 + 19 * q^95 - 41 * q^96 + 37 * q^97 + 41 * q^98 - 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 $

$2$	0.0495237	0.0350186	0.0175093	−	0.999847i	$-0.494426\pi$
$2$	0.0495237	0.0350186	0.0175093	+	0.999847i	$0.494426\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−1.99755	−0.998774
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−0.0495237	−0.0202180
$7$	−1.50762	−0.569827	−0.284913	−	0.958553i	$-0.591965\pi$
$7$	−1.50762	−0.569827	−0.284913	+	0.958553i	$0.591965\pi$
$8$	−0.197974	−0.0699942
$9$	1.00000	0.333333
$10$	−0.0495237	−0.0156608
$11$	1.36628	0.411950	0.205975	−	0.978557i	$-0.433963\pi$
$11$	1.36628	0.411950	0.205975	+	0.978557i	$0.433963\pi$
$12$	1.99755	0.576642
$13$	−6.59677	−1.82962	−0.914808	−	0.403889i	$-0.867658\pi$
$13$	−6.59677	−1.82962	−0.914808	+	0.403889i	$0.867658\pi$
$14$	−0.0746630	−0.0199545
$15$	1.00000	0.258199
$16$	3.98529	0.996323
$17$	2.05546	0.498522	0.249261	−	0.968436i	$-0.419812\pi$
$17$	2.05546	0.498522	0.249261	+	0.968436i	$0.419812\pi$
$18$	0.0495237	0.0116729
$19$	4.23470	0.971506	0.485753	−	0.874096i	$-0.338545\pi$
$19$	4.23470	0.971506	0.485753	+	0.874096i	$0.338545\pi$
$20$	1.99755	0.446665
$21$	1.50762	0.328990
$22$	0.0676634	0.0144259
$23$	1.56752	0.326852	0.163426	−	0.986556i	$-0.447746\pi$
$23$	1.56752	0.326852	0.163426	+	0.986556i	$0.447746\pi$
$24$	0.197974	0.0404112
$25$	1.00000	0.200000
$26$	−0.326697	−0.0640705
$27$	−1.00000	−0.192450
$28$	3.01154	0.569128
$29$	−6.29879	−1.16966	−0.584828	−	0.811158i	$-0.698838\pi$
$29$	−6.29879	−1.16966	−0.584828	+	0.811158i	$0.698838\pi$
$30$	0.0495237	0.00904176
$31$	−2.44850	−0.439764	−0.219882	−	0.975526i	$-0.570567\pi$
$31$	−2.44850	−0.439764	−0.219882	+	0.975526i	$0.570567\pi$
$32$	0.593314	0.104884
$33$	−1.36628	−0.237839
$34$	0.101794	0.0174575
$35$	1.50762	0.254834
$36$	−1.99755	−0.332925
$37$	−6.90433	−1.13507	−0.567533	−	0.823351i	$-0.692102\pi$
$37$	−6.90433	−1.13507	−0.567533	+	0.823351i	$0.692102\pi$
$38$	0.209718	0.0340208
$39$	6.59677	1.05633
$40$	0.197974	0.0313024
$41$	−8.13111	−1.26987	−0.634933	−	0.772567i	$-0.718972\pi$
$41$	−8.13111	−1.26987	−0.634933	+	0.772567i	$0.718972\pi$
$42$	0.0746630	0.0115207
$43$	4.49722	0.685820	0.342910	−	0.939368i	$-0.388587\pi$
$43$	4.49722	0.685820	0.342910	+	0.939368i	$0.388587\pi$
$44$	−2.72921	−0.411445
$45$	−1.00000	−0.149071
$46$	0.0776297	0.0114459
$47$	1.58039	0.230523	0.115262	−	0.993335i	$-0.463229\pi$
$47$	1.58039	0.230523	0.115262	+	0.993335i	$0.463229\pi$
$48$	−3.98529	−0.575227
$49$	−4.72708	−0.675298
$50$	0.0495237	0.00700371
$51$	−2.05546	−0.287822
$52$	13.1774	1.82737
$53$	−14.0609	−1.93141	−0.965706	−	0.259639i	$-0.916396\pi$
$53$	−14.0609	−1.93141	−0.965706	+	0.259639i	$0.916396\pi$
$54$	−0.0495237	−0.00673933
$55$	−1.36628	−0.184230
$56$	0.298469	0.0398846
$57$	−4.23470	−0.560899
$58$	−0.311939	−0.0409597
$59$	−5.18085	−0.674490	−0.337245	−	0.941417i	$-0.609495\pi$
$59$	−5.18085	−0.674490	−0.337245	+	0.941417i	$0.609495\pi$
$60$	−1.99755	−0.257882
$61$	−10.5297	−1.34819	−0.674094	−	0.738646i	$-0.735466\pi$
$61$	−10.5297	−1.34819	−0.674094	+	0.738646i	$0.735466\pi$
$62$	−0.121259	−0.0153999
$63$	−1.50762	−0.189942
$64$	−7.94120	−0.992650
$65$	6.59677	0.818229
$66$	−0.0676634	−0.00832879
$67$	6.50642	0.794886	0.397443	−	0.917627i	$-0.369898\pi$
$67$	6.50642	0.794886	0.397443	+	0.917627i	$0.369898\pi$
$68$	−4.10587	−0.497910
$69$	−1.56752	−0.188708
$70$	0.0746630	0.00892393
$71$	−4.56999	−0.542358	−0.271179	−	0.962529i	$-0.587414\pi$
$71$	−4.56999	−0.542358	−0.271179	+	0.962529i	$0.587414\pi$
$72$	−0.197974	−0.0233314
$73$	−4.68254	−0.548050	−0.274025	−	0.961723i	$-0.588355\pi$
$73$	−4.68254	−0.548050	−0.274025	+	0.961723i	$0.588355\pi$
$74$	−0.341928	−0.0397484
$75$	−1.00000	−0.115470
$76$	−8.45901	−0.970315
$77$	−2.05983	−0.234740
$78$	0.326697	0.0369911
$79$	0.561877	0.0632161	0.0316080	−	0.999500i	$-0.489937\pi$
$79$	0.561877	0.0632161	0.0316080	+	0.999500i	$0.489937\pi$
$80$	−3.98529	−0.445569
$81$	1.00000	0.111111
$82$	−0.402683	−0.0444689
$83$	14.8970	1.63516	0.817579	−	0.575817i	$-0.195316\pi$
$83$	14.8970	1.63516	0.817579	+	0.575817i	$0.195316\pi$
$84$	−3.01154	−0.328586
$85$	−2.05546	−0.222946
$86$	0.222719	0.0240164
$87$	6.29879	0.675301
$88$	−0.270488	−0.0288341
$89$	−5.25035	−0.556536	−0.278268	−	0.960503i	$-0.589760\pi$
$89$	−5.25035	−0.556536	−0.278268	+	0.960503i	$0.589760\pi$
$90$	−0.0495237	−0.00522026
$91$	9.94543	1.04256
$92$	−3.13121	−0.326451
$93$	2.44850	0.253898
$94$	0.0782667	0.00807260
$95$	−4.23470	−0.434471
$96$	−0.593314	−0.0605548
$97$	9.91016	1.00622	0.503112	−	0.864221i	$-0.332188\pi$
$97$	9.91016	1.00622	0.503112	+	0.864221i	$0.332188\pi$
$98$	−0.234103	−0.0236480
$99$	1.36628	0.137317
$100$	−1.99755	−0.199755
$101$	−5.67995	−0.565177	−0.282588	−	0.959241i	$-0.591193\pi$
$101$	−5.67995	−0.565177	−0.282588	+	0.959241i	$0.591193\pi$
$102$	−0.101794	−0.0100791
$103$	15.4761	1.52491	0.762453	−	0.647044i	$-0.223995\pi$
$103$	15.4761	1.52491	0.762453	+	0.647044i	$0.223995\pi$
$104$	1.30599	0.128062
$105$	−1.50762	−0.147129
$106$	−0.696348	−0.0676353
$107$	12.3806	1.19687	0.598437	−	0.801170i	$-0.295789\pi$
$107$	12.3806	1.19687	0.598437	+	0.801170i	$0.295789\pi$
$108$	1.99755	0.192214
$109$	1.98009	0.189658	0.0948291	−	0.995494i	$-0.469770\pi$
$109$	1.98009	0.189658	0.0948291	+	0.995494i	$0.469770\pi$
$110$	−0.0676634	−0.00645146
$111$	6.90433	0.655330
$112$	−6.00830	−0.567731
$113$	−7.35089	−0.691514	−0.345757	−	0.938324i	$-0.612378\pi$
$113$	−7.35089	−0.691514	−0.345757	+	0.938324i	$0.612378\pi$
$114$	−0.209718	−0.0196419
$115$	−1.56752	−0.146172
$116$	12.5821	1.16822
$117$	−6.59677	−0.609872
$118$	−0.256575	−0.0236197
$119$	−3.09885	−0.284071
$120$	−0.197974	−0.0180724
$121$	−9.13327	−0.830297
$122$	−0.521469	−0.0472116
$123$	8.13111	0.733157
$124$	4.89100	0.439225
$125$	−1.00000	−0.0894427
$126$	−0.0746630	−0.00665151
$127$	10.2938	0.913426	0.456713	−	0.889614i	$-0.349027\pi$
$127$	10.2938	0.913426	0.456713	+	0.889614i	$0.349027\pi$
$128$	−1.57990	−0.139645
$129$	−4.49722	−0.395958
$130$	0.326697	0.0286532
$131$	−20.2544	−1.76963	−0.884816	−	0.465942i	$-0.845716\pi$
$131$	−20.2544	−1.76963	−0.884816	+	0.465942i	$0.845716\pi$
$132$	2.72921	0.237548
$133$	−6.38431	−0.553590
$134$	0.322222	0.0278358
$135$	1.00000	0.0860663
$136$	−0.406926	−0.0348936
$137$	2.37293	0.202733	0.101367	−	0.994849i	$-0.467679\pi$
$137$	2.37293	0.202733	0.101367	+	0.994849i	$0.467679\pi$
$138$	−0.0776297	−0.00660828
$139$	4.66917	0.396033	0.198017	−	0.980199i	$-0.436550\pi$
$139$	4.66917	0.396033	0.198017	+	0.980199i	$0.436550\pi$
$140$	−3.01154	−0.254522
$141$	−1.58039	−0.133093
$142$	−0.226323	−0.0189926
$143$	−9.01306	−0.753710
$144$	3.98529	0.332108
$145$	6.29879	0.523086
$146$	−0.231897	−0.0191919
$147$	4.72708	0.389883
$148$	13.7917	1.13367
$149$	−12.2928	−1.00707	−0.503533	−	0.863976i	$-0.667967\pi$
$149$	−12.2928	−1.00707	−0.503533	+	0.863976i	$0.667967\pi$
$150$	−0.0495237	−0.00404360
$151$	0.0739115	0.00601483	0.00300741	−	0.999995i	$-0.499043\pi$
$151$	0.0739115	0.00601483	0.00300741	+	0.999995i	$0.499043\pi$
$152$	−0.838358	−0.0679998
$153$	2.05546	0.166174
$154$	−0.102011	−0.00822026
$155$	2.44850	0.196668
$156$	−13.1774	−1.05503
$157$	15.1139	1.20622	0.603111	−	0.797657i	$-0.293928\pi$
$157$	15.1139	1.20622	0.603111	+	0.797657i	$0.293928\pi$
$158$	0.0278263	0.00221374
$159$	14.0609	1.11510
$160$	−0.593314	−0.0469056
$161$	−2.36323	−0.186249
$162$	0.0495237	0.00389095
$163$	−15.9428	−1.24874	−0.624370	−	0.781129i	$-0.714644\pi$
$163$	−15.9428	−1.24874	−0.624370	+	0.781129i	$0.714644\pi$
$164$	16.2423	1.26831
$165$	1.36628	0.106365
$166$	0.737755	0.0572609
$167$	18.1128	1.40161	0.700806	−	0.713352i	$-0.252824\pi$
$167$	18.1128	1.40161	0.700806	+	0.713352i	$0.252824\pi$
$168$	−0.298469	−0.0230274
$169$	30.5174	2.34749
$170$	−0.101794	−0.00780724
$171$	4.23470	0.323835
$172$	−8.98341	−0.684979
$173$	21.5389	1.63757	0.818786	−	0.574098i	$-0.194647\pi$
$173$	21.5389	1.63757	0.818786	+	0.574098i	$0.194647\pi$
$174$	0.311939	0.0236481
$175$	−1.50762	−0.113965
$176$	5.44503	0.410435
$177$	5.18085	0.389417
$178$	−0.260017	−0.0194891
$179$	12.5564	0.938513	0.469257	−	0.883062i	$-0.344522\pi$
$179$	12.5564	0.938513	0.469257	+	0.883062i	$0.344522\pi$
$180$	1.99755	0.148888
$181$	4.85742	0.361049	0.180525	−	0.983570i	$-0.442220\pi$
$181$	4.85742	0.361049	0.180525	+	0.983570i	$0.442220\pi$
$182$	0.492535	0.0365091
$183$	10.5297	0.778376
$184$	−0.310328	−0.0228777
$185$	6.90433	0.507617
$186$	0.121259	0.00889114
$187$	2.80834	0.205366
$188$	−3.15690	−0.230241
$189$	1.50762	0.109663
$190$	−0.209718	−0.0152145
$191$	−6.95203	−0.503031	−0.251515	−	0.967853i	$-0.580929\pi$
$191$	−6.95203	−0.503031	−0.251515	+	0.967853i	$0.580929\pi$
$192$	7.94120	0.573107
$193$	−17.9860	−1.29466	−0.647331	−	0.762209i	$-0.724115\pi$
$193$	−17.9860	−1.29466	−0.647331	+	0.762209i	$0.724115\pi$
$194$	0.490788	0.0352365
$195$	−6.59677	−0.472405
$196$	9.44257	0.674469
$197$	14.0754	1.00283	0.501416	−	0.865206i	$-0.332813\pi$
$197$	14.0754	1.00283	0.501416	+	0.865206i	$0.332813\pi$
$198$	0.0676634	0.00480863
$199$	−14.3799	−1.01936	−0.509682	−	0.860363i	$-0.670237\pi$
$199$	−14.3799	−1.01936	−0.509682	+	0.860363i	$0.670237\pi$
$200$	−0.197974	−0.0139988
$201$	−6.50642	−0.458927
$202$	−0.281293	−0.0197917
$203$	9.49617	0.666501
$204$	4.10587	0.287469
$205$	8.13111	0.567901
$206$	0.766435	0.0534000
$207$	1.56752	0.108951
$208$	−26.2901	−1.82289
$209$	5.78579	0.400212
$210$	−0.0746630	−0.00515223
$211$	8.67518	0.597224	0.298612	−	0.954375i	$-0.403476\pi$
$211$	8.67518	0.597224	0.298612	+	0.954375i	$0.403476\pi$
$212$	28.0873	1.92904
$213$	4.56999	0.313131
$214$	0.613132	0.0419128
$215$	−4.49722	−0.306708
$216$	0.197974	0.0134704
$217$	3.69141	0.250589
$218$	0.0980615	0.00664156
$219$	4.68254	0.316417
$220$	2.72921	0.184004
$221$	−13.5594	−0.912103
$222$	0.341928	0.0229487
$223$	−8.24649	−0.552226	−0.276113	−	0.961125i	$-0.589046\pi$
$223$	−8.24649	−0.552226	−0.276113	+	0.961125i	$0.589046\pi$
$224$	−0.894491	−0.0597657
$225$	1.00000	0.0666667
$226$	−0.364044	−0.0242158
$227$	−9.54107	−0.633263	−0.316631	−	0.948549i	$-0.602552\pi$
$227$	−9.54107	−0.633263	−0.316631	+	0.948549i	$0.602552\pi$
$228$	8.45901	0.560211
$229$	−13.3650	−0.883184	−0.441592	−	0.897216i	$-0.645586\pi$
$229$	−13.3650	−0.883184	−0.441592	+	0.897216i	$0.645586\pi$
$230$	−0.0776297	−0.00511875
$231$	2.05983	0.135527
$232$	1.24699	0.0818691
$233$	25.4997	1.67054	0.835272	−	0.549837i	$-0.185310\pi$
$233$	25.4997	1.67054	0.835272	+	0.549837i	$0.185310\pi$
$234$	−0.326697	−0.0213568
$235$	−1.58039	−0.103093
$236$	10.3490	0.673663
$237$	−0.561877	−0.0364978
$238$	−0.153467	−0.00994776
$239$	−18.2279	−1.17907	−0.589533	−	0.807744i	$-0.700688\pi$
$239$	−18.2279	−1.17907	−0.589533	+	0.807744i	$0.700688\pi$
$240$	3.98529	0.257249
$241$	11.4888	0.740060	0.370030	−	0.929020i	$-0.379347\pi$
$241$	11.4888	0.740060	0.370030	+	0.929020i	$0.379347\pi$
$242$	−0.452314	−0.0290758
$243$	−1.00000	−0.0641500
$244$	21.0335	1.34653
$245$	4.72708	0.302002
$246$	0.402683	0.0256741
$247$	−27.9353	−1.77748
$248$	0.484739	0.0307809
$249$	−14.8970	−0.944058
$250$	−0.0495237	−0.00313216
$251$	−8.45268	−0.533528	−0.266764	−	0.963762i	$-0.585954\pi$
$251$	−8.45268	−0.533528	−0.266764	+	0.963762i	$0.585954\pi$
$252$	3.01154	0.189709
$253$	2.14168	0.134646
$254$	0.509787	0.0319869
$255$	2.05546	0.128718
$256$	15.8042	0.987760
$257$	19.5102	1.21701	0.608507	−	0.793549i	$-0.291769\pi$
$257$	19.5102	1.21701	0.608507	+	0.793549i	$0.291769\pi$
$258$	−0.222719	−0.0138659
$259$	10.4091	0.646791
$260$	−13.1774	−0.817226
$261$	−6.29879	−0.389885
$262$	−1.00307	−0.0619700
$263$	−28.5496	−1.76044	−0.880222	−	0.474561i	$-0.842607\pi$
$263$	−28.5496	−1.76044	−0.880222	+	0.474561i	$0.842607\pi$
$264$	0.270488	0.0166474
$265$	14.0609	0.863753
$266$	−0.316175	−0.0193859
$267$	5.25035	0.321316
$268$	−12.9969	−0.793911
$269$	5.77788	0.352284	0.176142	−	0.984365i	$-0.443638\pi$
$269$	5.77788	0.352284	0.176142	+	0.984365i	$0.443638\pi$
$270$	0.0495237	0.00301392
$271$	4.90190	0.297769	0.148885	−	0.988855i	$-0.452432\pi$
$271$	4.90190	0.297769	0.148885	+	0.988855i	$0.452432\pi$
$272$	8.19159	0.496688
$273$	−9.94543	−0.601925
$274$	0.117516	0.00709942
$275$	1.36628	0.0823900
$276$	3.13121	0.188476
$277$	21.9801	1.32066	0.660329	−	0.750977i	$-0.270417\pi$
$277$	21.9801	1.32066	0.660329	+	0.750977i	$0.270417\pi$
$278$	0.231235	0.0138685
$279$	−2.44850	−0.146588
$280$	−0.298469	−0.0178369
$281$	8.65385	0.516245	0.258123	−	0.966112i	$-0.416896\pi$
$281$	8.65385	0.516245	0.258123	+	0.966112i	$0.416896\pi$
$282$	−0.0782667	−0.00466072
$283$	−19.6558	−1.16842	−0.584209	−	0.811603i	$-0.698595\pi$
$283$	−19.6558	−1.16842	−0.584209	+	0.811603i	$0.698595\pi$
$284$	9.12878	0.541693
$285$	4.23470	0.250842
$286$	−0.446360	−0.0263938
$287$	12.2586	0.723603
$288$	0.593314	0.0349613
$289$	−12.7751	−0.751476
$290$	0.311939	0.0183177
$291$	−9.91016	−0.580944
$292$	9.35359	0.547378
$293$	23.0574	1.34703	0.673513	−	0.739175i	$-0.264784\pi$
$293$	23.0574	1.34703	0.673513	+	0.739175i	$0.264784\pi$
$294$	0.234103	0.0136532
$295$	5.18085	0.301641
$296$	1.36688	0.0794480
$297$	−1.36628	−0.0792798
$298$	−0.608786	−0.0352660
$299$	−10.3406	−0.598013
$300$	1.99755	0.115328
$301$	−6.78010	−0.390798
$302$	0.00366037	0.000210631 0
$303$	5.67995	0.326305
$304$	16.8765	0.967933
$305$	10.5297	0.602928
$306$	0.101794	0.00581917
$307$	−26.5906	−1.51760	−0.758802	−	0.651321i	$-0.774215\pi$
$307$	−26.5906	−1.51760	−0.758802	+	0.651321i	$0.774215\pi$
$308$	4.11462	0.234452
$309$	−15.4761	−0.880405
$310$	0.121259	0.00688705
$311$	7.57838	0.429730	0.214865	−	0.976644i	$-0.431069\pi$
$311$	7.57838	0.429730	0.214865	+	0.976644i	$0.431069\pi$
$312$	−1.30599	−0.0739369
$313$	6.62822	0.374649	0.187324	−	0.982298i	$-0.440018\pi$
$313$	6.62822	0.374649	0.187324	+	0.982298i	$0.440018\pi$
$314$	0.748498	0.0422402
$315$	1.50762	0.0849447
$316$	−1.12238	−0.0631386
$317$	−10.2622	−0.576385	−0.288192	−	0.957573i	$-0.593054\pi$
$317$	−10.2622	−0.576385	−0.288192	+	0.957573i	$0.593054\pi$
$318$	0.696348	0.0390492
$319$	−8.60592	−0.481839
$320$	7.94120	0.443926
$321$	−12.3806	−0.691015
$322$	−0.117036	−0.00652217
$323$	8.70424	0.484317
$324$	−1.99755	−0.110975
$325$	−6.59677	−0.365923
$326$	−0.789550	−0.0437291
$327$	−1.98009	−0.109499
$328$	1.60974	0.0888832
$329$	−2.38262	−0.131358
$330$	0.0676634	0.00372475
$331$	4.01811	0.220855	0.110428	−	0.993884i	$-0.464778\pi$
$331$	4.01811	0.220855	0.110428	+	0.993884i	$0.464778\pi$
$332$	−29.7574	−1.63315
$333$	−6.90433	−0.378355
$334$	0.897015	0.0490825
$335$	−6.50642	−0.355484
$336$	6.00830	0.327780
$337$	16.6529	0.907143	0.453572	−	0.891220i	$-0.350150\pi$
$337$	16.6529	0.907143	0.453572	+	0.891220i	$0.350150\pi$
$338$	1.51134	0.0822059
$339$	7.35089	0.399246
$340$	4.10587	0.222672
$341$	−3.34535	−0.181161
$342$	0.209718	0.0113403
$343$	17.6800	0.954629
$344$	−0.890330	−0.0480034
$345$	1.56752	0.0843927
$346$	1.06669	0.0573455
$347$	4.26884	0.229163	0.114582	−	0.993414i	$-0.463447\pi$
$347$	4.26884	0.229163	0.114582	+	0.993414i	$0.463447\pi$
$348$	−12.5821	−0.674473
$349$	16.9096	0.905150	0.452575	−	0.891726i	$-0.350506\pi$
$349$	16.9096	0.905150	0.452575	+	0.891726i	$0.350506\pi$
$350$	−0.0746630	−0.00399090
$351$	6.59677	0.352110
$352$	0.810634	0.0432069
$353$	−16.4965	−0.878018	−0.439009	−	0.898483i	$-0.644670\pi$
$353$	−16.4965	−0.878018	−0.439009	+	0.898483i	$0.644670\pi$
$354$	0.256575	0.0136368
$355$	4.56999	0.242550
$356$	10.4878	0.555854
$357$	3.09885	0.164008
$358$	0.621842	0.0328654
$359$	33.3341	1.75931	0.879654	−	0.475615i	$-0.157774\pi$
$359$	33.3341	1.75931	0.879654	+	0.475615i	$0.157774\pi$
$360$	0.197974	0.0104341
$361$	−1.06734	−0.0561759
$362$	0.240558	0.0126434
$363$	9.13327	0.479372
$364$	−19.8665	−1.04129
$365$	4.68254	0.245095
$366$	0.521469	0.0272576
$367$	20.9367	1.09289	0.546443	−	0.837496i	$-0.315981\pi$
$367$	20.9367	1.09289	0.546443	+	0.837496i	$0.315981\pi$
$368$	6.24704	0.325650
$369$	−8.13111	−0.423289
$370$	0.341928	0.0177760
$371$	21.1985	1.10057
$372$	−4.89100	−0.253587
$373$	−19.5819	−1.01391	−0.506956	−	0.861972i	$-0.669229\pi$
$373$	−19.5819	−1.01391	−0.506956	+	0.861972i	$0.669229\pi$
$374$	0.139079	0.00719162
$375$	1.00000	0.0516398
$376$	−0.312875	−0.0161353
$377$	41.5517	2.14002
$378$	0.0746630	0.00384025
$379$	18.6552	0.958254	0.479127	−	0.877746i	$-0.340953\pi$
$379$	18.6552	0.958254	0.479127	+	0.877746i	$0.340953\pi$
$380$	8.45901	0.433938
$381$	−10.2938	−0.527366
$382$	−0.344290	−0.0176154
$383$	18.0356	0.921578	0.460789	−	0.887510i	$-0.347566\pi$
$383$	18.0356	0.921578	0.460789	+	0.887510i	$0.347566\pi$
$384$	1.57990	0.0806242
$385$	2.05983	0.104979
$386$	−0.890735	−0.0453372
$387$	4.49722	0.228607
$388$	−19.7960	−1.00499
$389$	9.97407	0.505705	0.252853	−	0.967505i	$-0.418631\pi$
$389$	9.97407	0.505705	0.252853	+	0.967505i	$0.418631\pi$
$390$	−0.326697	−0.0165429
$391$	3.22198	0.162943
$392$	0.935837	0.0472669
$393$	20.2544	1.02170
$394$	0.697067	0.0351177
$395$	−0.561877	−0.0282711
$396$	−2.72921	−0.137148
$397$	−36.6681	−1.84032	−0.920160	−	0.391543i	$-0.871942\pi$
$397$	−36.6681	−1.84032	−0.920160	+	0.391543i	$0.871942\pi$
$398$	−0.712147	−0.0356967
$399$	6.38431	0.319615
$400$	3.98529	0.199265
$401$	1.00000	0.0499376
$401$	1.00000	0.0499376
$402$	−0.322222	−0.0160710
$403$	16.1522	0.804599
$404$	11.3460	0.564484
$405$	−1.00000	−0.0496904
$406$	0.470286	0.0233399
$407$	−9.43327	−0.467590
$408$	0.406926	0.0201458
$409$	18.5077	0.915145	0.457572	−	0.889172i	$-0.348719\pi$
$409$	18.5077	0.915145	0.457572	+	0.889172i	$0.348719\pi$
$410$	0.402683	0.0198871
$411$	−2.37293	−0.117048
$412$	−30.9143	−1.52304
$413$	7.81076	0.384342
$414$	0.0776297	0.00381529
$415$	−14.8970	−0.731264
$416$	−3.91395	−0.191897
$417$	−4.66917	−0.228650
$418$	0.286534	0.0140148
$419$	28.6023	1.39731	0.698656	−	0.715458i	$-0.253782\pi$
$419$	28.6023	1.39731	0.698656	+	0.715458i	$0.253782\pi$
$420$	3.01154	0.146948
$421$	−15.1948	−0.740548	−0.370274	−	0.928923i	$-0.620736\pi$
$421$	−15.1948	−0.740548	−0.370274	+	0.928923i	$0.620736\pi$
$422$	0.429628	0.0209139
$423$	1.58039	0.0768411
$424$	2.78368	0.135188
$425$	2.05546	0.0997043
$426$	0.226323	0.0109654
$427$	15.8747	0.768233
$428$	−24.7308	−1.19541
$429$	9.01306	0.435155
$430$	−0.222719	−0.0107405
$431$	−19.8945	−0.958286	−0.479143	−	0.877737i	$-0.659053\pi$
$431$	−19.8945	−0.958286	−0.479143	+	0.877737i	$0.659053\pi$
$432$	−3.98529	−0.191742
$433$	31.9153	1.53375	0.766877	−	0.641794i	$-0.221810\pi$
$433$	31.9153	1.53375	0.766877	+	0.641794i	$0.221810\pi$
$434$	0.182812	0.00877528
$435$	−6.29879	−0.302004
$436$	−3.95532	−0.189426
$437$	6.63799	0.317538
$438$	0.231897	0.0110805
$439$	23.2525	1.10978	0.554890	−	0.831924i	$-0.312760\pi$
$439$	23.2525	1.10978	0.554890	+	0.831924i	$0.312760\pi$
$440$	0.270488	0.0128950
$441$	−4.72708	−0.225099
$442$	−0.671512	−0.0319405
$443$	27.7906	1.32037	0.660185	−	0.751103i	$-0.270478\pi$
$443$	27.7906	1.32037	0.660185	+	0.751103i	$0.270478\pi$
$444$	−13.7917	−0.654527
$445$	5.25035	0.248891
$446$	−0.408397	−0.0193382
$447$	12.2928	0.581430
$448$	11.9723	0.565638
$449$	23.6187	1.11464	0.557318	−	0.830299i	$-0.311830\pi$
$449$	23.6187	1.11464	0.557318	+	0.830299i	$0.311830\pi$
$450$	0.0495237	0.00233457
$451$	−11.1094	−0.523121
$452$	14.6838	0.690666
$453$	−0.0739115	−0.00347266
$454$	−0.472509	−0.0221760
$455$	−9.94543	−0.466249
$456$	0.838358	0.0392597
$457$	30.8588	1.44352	0.721758	−	0.692146i	$-0.243334\pi$
$457$	30.8588	1.44352	0.721758	+	0.692146i	$0.243334\pi$
$458$	−0.661885	−0.0309278
$459$	−2.05546	−0.0959405
$460$	3.13121	0.145993
$461$	−21.5333	−1.00290	−0.501452	−	0.865186i	$-0.667201\pi$
$461$	−21.5333	−1.00290	−0.501452	+	0.865186i	$0.667201\pi$
$462$	0.102011	0.00474597
$463$	14.0991	0.655241	0.327620	−	0.944809i	$-0.393753\pi$
$463$	14.0991	0.655241	0.327620	+	0.944809i	$0.393753\pi$
$464$	−25.1025	−1.16535
$465$	−2.44850	−0.113547
$466$	1.26284	0.0585001
$467$	−25.2081	−1.16649	−0.583247	−	0.812295i	$-0.698218\pi$
$467$	−25.2081	−1.16649	−0.583247	+	0.812295i	$0.698218\pi$
$468$	13.1774	0.609124
$469$	−9.80921	−0.452947
$470$	−0.0782667	−0.00361017
$471$	−15.1139	−0.696413
$472$	1.02567	0.0472104
$473$	6.14447	0.282523
$474$	−0.0278263	−0.00127810
$475$	4.23470	0.194301
$476$	6.19010	0.283723
$477$	−14.0609	−0.643804
$478$	−0.902714	−0.0412892
$479$	35.8424	1.63768	0.818840	−	0.574022i	$-0.194618\pi$
$479$	35.8424	1.63768	0.818840	+	0.574022i	$0.194618\pi$
$480$	0.593314	0.0270809
$481$	45.5463	2.07673
$482$	0.568970	0.0259159
$483$	2.36323	0.107531
$484$	18.2441	0.829279
$485$	−9.91016	−0.449997
$486$	−0.0495237	−0.00224644
$487$	−25.4352	−1.15258	−0.576289	−	0.817246i	$-0.695500\pi$
$487$	−25.4352	−1.15258	−0.576289	+	0.817246i	$0.695500\pi$
$488$	2.08460	0.0943653
$489$	15.9428	0.720961
$490$	0.234103	0.0105757
$491$	29.7507	1.34263	0.671316	−	0.741171i	$-0.265729\pi$
$491$	29.7507	1.34263	0.671316	+	0.741171i	$0.265729\pi$
$492$	−16.2423	−0.732258
$493$	−12.9469	−0.583098
$494$	−1.38346	−0.0622449
$495$	−1.36628	−0.0614098
$496$	−9.75799	−0.438147
$497$	6.88981	0.309050
$498$	−0.737755	−0.0330596
$499$	−7.15217	−0.320175	−0.160088	−	0.987103i	$-0.551178\pi$
$499$	−7.15217	−0.320175	−0.160088	+	0.987103i	$0.551178\pi$
$500$	1.99755	0.0893330
$501$	−18.1128	−0.809221
$502$	−0.418608	−0.0186834
$503$	5.04971	0.225155	0.112578	−	0.993643i	$-0.464089\pi$
$503$	5.04971	0.225155	0.112578	+	0.993643i	$0.464089\pi$
$504$	0.298469	0.0132949
$505$	5.67995	0.252755
$506$	0.106064	0.00471513
$507$	−30.5174	−1.35533
$508$	−20.5623	−0.912305
$509$	14.5723	0.645904	0.322952	−	0.946415i	$-0.395325\pi$
$509$	14.5723	0.645904	0.322952	+	0.946415i	$0.395325\pi$
$510$	0.101794	0.00450751
$511$	7.05949	0.312293
$512$	3.94249	0.174235
$513$	−4.23470	−0.186966
$514$	0.966218	0.0426181
$515$	−15.4761	−0.681959
$516$	8.98341	0.395473
$517$	2.15926	0.0949640
$518$	0.515498	0.0226497
$519$	−21.5389	−0.945453
$520$	−1.30599	−0.0572713
$521$	20.6140	0.903115	0.451558	−	0.892242i	$-0.350869\pi$
$521$	20.6140	0.903115	0.451558	+	0.892242i	$0.350869\pi$
$522$	−0.311939	−0.0136532
$523$	−9.79909	−0.428484	−0.214242	−	0.976781i	$-0.568728\pi$
$523$	−9.79909	−0.428484	−0.214242	+	0.976781i	$0.568728\pi$
$524$	40.4590	1.76746
$525$	1.50762	0.0657979
$526$	−1.41388	−0.0616483
$527$	−5.03279	−0.219232
$528$	−5.44503	−0.236965
$529$	−20.5429	−0.893168
$530$	0.696348	0.0302474
$531$	−5.18085	−0.224830
$532$	12.7530	0.552911
$533$	53.6391	2.32337
$534$	0.260017	0.0112520
$535$	−12.3806	−0.535258
$536$	−1.28810	−0.0556374
$537$	−12.5564	−0.541851
$538$	0.286142	0.0123365
$539$	−6.45853	−0.278189
$540$	−1.99755	−0.0859608
$541$	−4.84735	−0.208404	−0.104202	−	0.994556i	$-0.533229\pi$
$541$	−4.84735	−0.208404	−0.104202	+	0.994556i	$0.533229\pi$
$542$	0.242760	0.0104274
$543$	−4.85742	−0.208452
$544$	1.21953	0.0522869
$545$	−1.98009	−0.0848177
$546$	−0.492535	−0.0210785
$547$	−24.9615	−1.06728	−0.533638	−	0.845713i	$-0.679176\pi$
$547$	−24.9615	−1.06728	−0.533638	+	0.845713i	$0.679176\pi$
$548$	−4.74004	−0.202484
$549$	−10.5297	−0.449396
$550$	0.0676634	0.00288518
$551$	−26.6734	−1.13633
$552$	0.310328	0.0132085
$553$	−0.847097	−0.0360222
$554$	1.08854	0.0462476
$555$	−6.90433	−0.293073
$556$	−9.32688	−0.395548
$557$	−6.78053	−0.287300	−0.143650	−	0.989629i	$-0.545884\pi$
$557$	−6.78053	−0.287300	−0.143650	+	0.989629i	$0.545884\pi$
$558$	−0.121259	−0.00513330
$559$	−29.6671	−1.25479
$560$	6.00830	0.253897
$561$	−2.80834	−0.118568
$562$	0.428571	0.0180782
$563$	22.3189	0.940630	0.470315	−	0.882499i	$-0.344140\pi$
$563$	22.3189	0.940630	0.470315	+	0.882499i	$0.344140\pi$
$564$	3.15690	0.132929
$565$	7.35089	0.309254
$566$	−0.973430	−0.0409163
$567$	−1.50762	−0.0633141
$568$	0.904737	0.0379619
$569$	11.5536	0.484353	0.242177	−	0.970232i	$-0.422139\pi$
$569$	11.5536	0.484353	0.242177	+	0.970232i	$0.422139\pi$
$570$	0.209718	0.00878412
$571$	−9.90771	−0.414625	−0.207312	−	0.978275i	$-0.566472\pi$
$571$	−9.90771	−0.414625	−0.207312	+	0.978275i	$0.566472\pi$
$572$	18.0040	0.752786
$573$	6.95203	0.290425
$574$	0.607093	0.0253396
$575$	1.56752	0.0653703
$576$	−7.94120	−0.330883
$577$	17.4430	0.726162	0.363081	−	0.931758i	$-0.381725\pi$
$577$	17.4430	0.726162	0.363081	+	0.931758i	$0.381725\pi$
$578$	−0.632671	−0.0263156
$579$	17.9860	0.747473
$580$	−12.5821	−0.522444
$581$	−22.4590	−0.931756
$582$	−0.490788	−0.0203438
$583$	−19.2111	−0.795645
$584$	0.927019	0.0383603
$585$	6.59677	0.272743
$586$	1.14189	0.0471710
$587$	−3.35101	−0.138311	−0.0691555	−	0.997606i	$-0.522030\pi$
$587$	−3.35101	−0.138311	−0.0691555	+	0.997606i	$0.522030\pi$
$588$	−9.44257	−0.389405
$589$	−10.3687	−0.427233
$590$	0.256575	0.0105630
$591$	−14.0754	−0.578985
$592$	−27.5158	−1.13089
$593$	−14.4924	−0.595132	−0.297566	−	0.954701i	$-0.596175\pi$
$593$	−14.4924	−0.595132	−0.297566	+	0.954701i	$0.596175\pi$
$594$	−0.0676634	−0.00277626
$595$	3.09885	0.127040
$596$	24.5555	1.00583
$597$	14.3799	0.588530
$598$	−0.512106	−0.0209416
$599$	6.74572	0.275623	0.137811	−	0.990458i	$-0.455993\pi$
$599$	6.74572	0.275623	0.137811	+	0.990458i	$0.455993\pi$
$600$	0.197974	0.00808223
$601$	0.509088	0.0207662	0.0103831	−	0.999946i	$-0.496695\pi$
$601$	0.509088	0.0207662	0.0103831	+	0.999946i	$0.496695\pi$
$602$	−0.335776	−0.0136852
$603$	6.50642	0.264962
$604$	−0.147642	−0.00600745
$605$	9.13327	0.371320
$606$	0.281293	0.0114267
$607$	−17.0690	−0.692809	−0.346404	−	0.938085i	$-0.612597\pi$
$607$	−17.0690	−0.692809	−0.346404	+	0.938085i	$0.612597\pi$
$608$	2.51250	0.101895
$609$	−9.49617	−0.384804
$610$	0.521469	0.0211137
$611$	−10.4255	−0.421769
$612$	−4.10587	−0.165970
$613$	−25.8719	−1.04496	−0.522478	−	0.852653i	$-0.674992\pi$
$613$	−25.8719	−1.04496	−0.522478	+	0.852653i	$0.674992\pi$
$614$	−1.31686	−0.0531443
$615$	−8.13111	−0.327878
$616$	0.407793	0.0164304
$617$	9.56570	0.385100	0.192550	−	0.981287i	$-0.438324\pi$
$617$	9.56570	0.385100	0.192550	+	0.981287i	$0.438324\pi$
$618$	−0.766435	−0.0308305
$619$	6.46705	0.259933	0.129966	−	0.991518i	$-0.458513\pi$
$619$	6.46705	0.259933	0.129966	+	0.991518i	$0.458513\pi$
$620$	−4.89100	−0.196427
$621$	−1.56752	−0.0629026
$622$	0.375309	0.0150485
$623$	7.91553	0.317129
$624$	26.2901	1.05244
$625$	1.00000	0.0400000
$626$	0.328254	0.0131197
$627$	−5.78579	−0.231062
$628$	−30.1908	−1.20474
$629$	−14.1916	−0.565855
$630$	0.0746630	0.00297464
$631$	1.46057	0.0581444	0.0290722	−	0.999577i	$-0.490745\pi$
$631$	1.46057	0.0581444	0.0290722	+	0.999577i	$0.490745\pi$
$632$	−0.111237	−0.00442476
$633$	−8.67518	−0.344808
$634$	−0.508224	−0.0201842
$635$	−10.2938	−0.408496
$636$	−28.0873	−1.11373
$637$	31.1835	1.23554
$638$	−0.426198	−0.0168733
$639$	−4.56999	−0.180786
$640$	1.57990	0.0624512
$641$	6.76958	0.267382	0.133691	−	0.991023i	$-0.457317\pi$
$641$	6.76958	0.267382	0.133691	+	0.991023i	$0.457317\pi$
$642$	−0.613132	−0.0241984
$643$	45.2518	1.78455	0.892277	−	0.451487i	$-0.149106\pi$
$643$	45.2518	1.78455	0.892277	+	0.451487i	$0.149106\pi$
$644$	4.72067	0.186020
$645$	4.49722	0.177078
$646$	0.431066	0.0169601
$647$	−13.0346	−0.512442	−0.256221	−	0.966618i	$-0.582478\pi$
$647$	−13.0346	−0.512442	−0.256221	+	0.966618i	$0.582478\pi$
$648$	−0.197974	−0.00777713
$649$	−7.07851	−0.277856
$650$	−0.326697	−0.0128141
$651$	−3.69141	−0.144678
$652$	31.8466	1.24721
$653$	−2.75882	−0.107961	−0.0539804	−	0.998542i	$-0.517191\pi$
$653$	−2.75882	−0.107961	−0.0539804	+	0.998542i	$0.517191\pi$
$654$	−0.0980615	−0.00383451
$655$	20.2544	0.791403
$656$	−32.4048	−1.26520
$657$	−4.68254	−0.182683
$658$	−0.117996	−0.00459998
$659$	−33.6866	−1.31224	−0.656122	−	0.754655i	$-0.727804\pi$
$659$	−33.6866	−1.31224	−0.656122	+	0.754655i	$0.727804\pi$
$660$	−2.72921	−0.106235
$661$	17.3009	0.672929	0.336464	−	0.941696i	$-0.390769\pi$
$661$	17.3009	0.672929	0.336464	+	0.941696i	$0.390769\pi$
$662$	0.198992	0.00773403
$663$	13.5594	0.526603
$664$	−2.94921	−0.114452
$665$	6.38431	0.247573
$666$	−0.341928	−0.0132495
$667$	−9.87350	−0.382304
$668$	−36.1812	−1.39989
$669$	8.24649	0.318828
$670$	−0.322222	−0.0124485
$671$	−14.3865	−0.555385
$672$	0.894491	0.0345057
$673$	−44.5272	−1.71640	−0.858198	−	0.513318i	$-0.828416\pi$
$673$	−44.5272	−1.71640	−0.858198	+	0.513318i	$0.828416\pi$
$674$	0.824716	0.0317669
$675$	−1.00000	−0.0384900
$676$	−60.9600	−2.34462
$677$	−38.2413	−1.46973	−0.734866	−	0.678212i	$-0.762755\pi$
$677$	−38.2413	−1.46973	−0.734866	+	0.678212i	$0.762755\pi$
$678$	0.364044	0.0139810
$679$	−14.9408	−0.573374
$680$	0.406926	0.0156049
$681$	9.54107	0.365614
$682$	−0.165674	−0.00634399
$683$	−13.1080	−0.501564	−0.250782	−	0.968044i	$-0.580688\pi$
$683$	−13.1080	−0.501564	−0.250782	+	0.968044i	$0.580688\pi$
$684$	−8.45901	−0.323438
$685$	−2.37293	−0.0906650
$686$	0.875579	0.0334298
$687$	13.3650	0.509906
$688$	17.9227	0.683298
$689$	92.7565	3.53374
$690$	0.0776297	0.00295531
$691$	38.7951	1.47584	0.737918	−	0.674890i	$-0.235809\pi$
$691$	38.7951	1.47584	0.737918	+	0.674890i	$0.235809\pi$
$692$	−43.0250	−1.63556
$693$	−2.05983	−0.0782467
$694$	0.211409	0.00802497
$695$	−4.66917	−0.177112
$696$	−1.24699	−0.0472671
$697$	−16.7131	−0.633056
$698$	0.837426	0.0316970
$699$	−25.4997	−0.964489
$700$	3.01154	0.113826
$701$	−0.636906	−0.0240556	−0.0120278	−	0.999928i	$-0.503829\pi$
$701$	−0.636906	−0.0240556	−0.0120278	+	0.999928i	$0.503829\pi$
$702$	0.326697	0.0123304
$703$	−29.2378	−1.10272
$704$	−10.8499	−0.408922
$705$	1.58039	0.0595209
$706$	−0.816966	−0.0307469
$707$	8.56321	0.322053
$708$	−10.3490	−0.388939
$709$	40.2891	1.51309	0.756544	−	0.653943i	$-0.226886\pi$
$709$	40.2891	1.51309	0.756544	+	0.653943i	$0.226886\pi$
$710$	0.226323	0.00849376
$711$	0.561877	0.0210720
$712$	1.03943	0.0389543
$713$	−3.83809	−0.143738
$714$	0.153467	0.00574334
$715$	9.01306	0.337069
$716$	−25.0821	−0.937362
$717$	18.2279	0.680734
$718$	1.65083	0.0616084
$719$	25.5505	0.952874	0.476437	−	0.879209i	$-0.341928\pi$
$719$	25.5505	0.952874	0.476437	+	0.879209i	$0.341928\pi$
$720$	−3.98529	−0.148523
$721$	−23.3321	−0.868932
$722$	−0.0528588	−0.00196720
$723$	−11.4888	−0.427274
$724$	−9.70293	−0.360606
$725$	−6.29879	−0.233931
$726$	0.452314	0.0167869
$727$	−4.90791	−0.182024	−0.0910121	−	0.995850i	$-0.529010\pi$
$727$	−4.90791	−0.182024	−0.0910121	+	0.995850i	$0.529010\pi$
$728$	−1.96893	−0.0729734
$729$	1.00000	0.0370370
$730$	0.231897	0.00858289
$731$	9.24384	0.341896
$732$	−21.0335	−0.777422
$733$	−45.2868	−1.67271	−0.836353	−	0.548191i	$-0.815317\pi$
$733$	−45.2868	−1.67271	−0.836353	+	0.548191i	$0.815317\pi$
$734$	1.03686	0.0382713
$735$	−4.72708	−0.174361
$736$	0.930034	0.0342815
$737$	8.88961	0.327453
$738$	−0.402683	−0.0148230
$739$	−34.6521	−1.27470	−0.637350	−	0.770575i	$-0.719969\pi$
$739$	−34.6521	−1.27470	−0.637350	+	0.770575i	$0.719969\pi$
$740$	−13.7917	−0.506994
$741$	27.9353	1.02623
$742$	1.04983	0.0385404
$743$	−0.596137	−0.0218701	−0.0109351	−	0.999940i	$-0.503481\pi$
$743$	−0.596137	−0.0218701	−0.0109351	+	0.999940i	$0.503481\pi$
$744$	−0.484739	−0.0177714
$745$	12.2928	0.450373
$746$	−0.969769	−0.0355058
$747$	14.8970	0.545052
$748$	−5.60978	−0.205114
$749$	−18.6652	−0.682011
$750$	0.0495237	0.00180835
$751$	11.8602	0.432785	0.216393	−	0.976306i	$-0.430571\pi$
$751$	11.8602	0.432785	0.216393	+	0.976306i	$0.430571\pi$
$752$	6.29830	0.229676
$753$	8.45268	0.308033
$754$	2.05779	0.0749404
$755$	−0.0739115	−0.00268991
$756$	−3.01154	−0.109529
$757$	19.3769	0.704265	0.352133	−	0.935950i	$-0.385457\pi$
$757$	19.3769	0.704265	0.352133	+	0.935950i	$0.385457\pi$
$758$	0.923876	0.0335567
$759$	−2.14168	−0.0777381
$760$	0.838358	0.0304104
$761$	−28.1214	−1.01940	−0.509700	−	0.860352i	$-0.670244\pi$
$761$	−28.1214	−1.01940	−0.509700	+	0.860352i	$0.670244\pi$
$762$	−0.509787	−0.0184676
$763$	−2.98522	−0.108072
$764$	13.8870	0.502414
$765$	−2.05546	−0.0743152
$766$	0.893192	0.0322724
$767$	34.1769	1.23406
$768$	−15.8042	−0.570283
$769$	−47.3536	−1.70761	−0.853807	−	0.520590i	$-0.825712\pi$
$769$	−47.3536	−1.70761	−0.853807	+	0.520590i	$0.825712\pi$
$770$	0.102011	0.00367621
$771$	−19.5102	−0.702643
$772$	35.9279	1.29307
$773$	−18.0880	−0.650580	−0.325290	−	0.945614i	$-0.605462\pi$
$773$	−18.0880	−0.650580	−0.325290	+	0.945614i	$0.605462\pi$
$774$	0.222719	0.00800547
$775$	−2.44850	−0.0879528
$776$	−1.96195	−0.0704299
$777$	−10.4091	−0.373425
$778$	0.493953	0.0177091
$779$	−34.4328	−1.23368
$780$	13.1774	0.471825
$781$	−6.24390	−0.223424
$782$	0.159565	0.00570602
$783$	6.29879	0.225100
$784$	−18.8388	−0.672814
$785$	−15.1139	−0.539439
$786$	1.00307	0.0357784
$787$	43.9359	1.56615	0.783073	−	0.621930i	$-0.213651\pi$
$787$	43.9359	1.56615	0.783073	+	0.621930i	$0.213651\pi$
$788$	−28.1163	−1.00160
$789$	28.5496	1.01639
$790$	−0.0278263	−0.000990014 0
$791$	11.0824	0.394043
$792$	−0.270488	−0.00961137
$793$	69.4619	2.46666
$794$	−1.81594	−0.0644454
$795$	−14.0609	−0.498688
$796$	28.7245	1.01811
$797$	12.2379	0.433487	0.216744	−	0.976229i	$-0.430456\pi$
$797$	12.2379	0.433487	0.216744	+	0.976229i	$0.430456\pi$
$798$	0.316175	0.0111925
$799$	3.24842	0.114921
$800$	0.593314	0.0209768
$801$	−5.25035	−0.185512
$802$	0.0495237	0.00174874
$803$	−6.39767	−0.225769
$804$	12.9969	0.458365
$805$	2.36323	0.0832930
$806$	0.799918	0.0281759
$807$	−5.77788	−0.203391
$808$	1.12448	0.0395591
$809$	−55.9453	−1.96693	−0.983466	−	0.181094i	$-0.942036\pi$
$809$	−55.9453	−1.96693	−0.983466	+	0.181094i	$0.942036\pi$
$810$	−0.0495237	−0.00174009
$811$	−38.0642	−1.33662	−0.668308	−	0.743885i	$-0.732981\pi$
$811$	−38.0642	−1.33662	−0.668308	+	0.743885i	$0.732981\pi$
$812$	−18.9691	−0.665683
$813$	−4.90190	−0.171917
$814$	−0.467171	−0.0163743
$815$	15.9428	0.558454
$816$	−8.19159	−0.286763
$817$	19.0444	0.666278
$818$	0.916568	0.0320471
$819$	9.94543	0.347521
$820$	−16.2423	−0.567205
$821$	3.32791	0.116145	0.0580725	−	0.998312i	$-0.481505\pi$
$821$	3.32791	0.116145	0.0580725	+	0.998312i	$0.481505\pi$
$822$	−0.117516	−0.00409885
$823$	15.3975	0.536723	0.268361	−	0.963318i	$-0.413518\pi$
$823$	15.3975	0.536723	0.268361	+	0.963318i	$0.413518\pi$
$824$	−3.06386	−0.106735
$825$	−1.36628	−0.0475679
$826$	0.386818	0.0134591
$827$	1.73258	0.0602479	0.0301239	−	0.999546i	$-0.490410\pi$
$827$	1.73258	0.0602479	0.0301239	+	0.999546i	$0.490410\pi$
$828$	−3.13121	−0.108817
$829$	−20.6795	−0.718228	−0.359114	−	0.933294i	$-0.616921\pi$
$829$	−20.6795	−0.718228	−0.359114	+	0.933294i	$0.616921\pi$
$830$	−0.737755	−0.0256078
$831$	−21.9801	−0.762482
$832$	52.3863	1.81617
$833$	−9.71632	−0.336650
$834$	−0.231235	−0.00800700
$835$	−18.1128	−0.626820
$836$	−11.5574	−0.399721
$837$	2.44850	0.0846326
$838$	1.41649	0.0489319
$839$	17.4122	0.601137	0.300569	−	0.953760i	$-0.402824\pi$
$839$	17.4122	0.601137	0.300569	+	0.953760i	$0.402824\pi$
$840$	0.298469	0.0102982
$841$	10.6747	0.368093
$842$	−0.752502	−0.0259329
$843$	−8.65385	−0.298054
$844$	−17.3291	−0.596492
$845$	−30.5174	−1.04983
$846$	0.0782667	0.00269087
$847$	13.7695	0.473126
$848$	−56.0367	−1.92431
$849$	19.6558	0.674586
$850$	0.101794	0.00349150
$851$	−10.8227	−0.370998
$852$	−9.12878	−0.312747
$853$	−21.1540	−0.724299	−0.362150	−	0.932120i	$-0.617957\pi$
$853$	−21.1540	−0.724299	−0.362150	+	0.932120i	$0.617957\pi$
$854$	0.786177	0.0269024
$855$	−4.23470	−0.144824
$856$	−2.45102	−0.0837742
$857$	33.1262	1.13157	0.565785	−	0.824553i	$-0.308573\pi$
$857$	33.1262	1.13157	0.565785	+	0.824553i	$0.308573\pi$
$858$	0.446360	0.0152385
$859$	12.5343	0.427665	0.213832	−	0.976870i	$-0.431405\pi$
$859$	12.5343	0.427665	0.213832	+	0.976870i	$0.431405\pi$
$860$	8.98341	0.306332
$861$	−12.2586	−0.417773
$862$	−0.985252	−0.0335578
$863$	13.9891	0.476195	0.238098	−	0.971241i	$-0.423476\pi$
$863$	13.9891	0.476195	0.238098	+	0.971241i	$0.423476\pi$
$864$	−0.593314	−0.0201849
$865$	−21.5389	−0.732345
$866$	1.58057	0.0537099
$867$	12.7751	0.433865
$868$	−7.37377	−0.250282
$869$	0.767683	0.0260419
$870$	−0.311939	−0.0105757
$871$	−42.9214	−1.45434
$872$	−0.392005	−0.0132750
$873$	9.91016	0.335408
$874$	0.328738	0.0111197
$875$	1.50762	0.0509668
$876$	−9.35359	−0.316029
$877$	41.8145	1.41198	0.705988	−	0.708224i	$-0.250503\pi$
$877$	41.8145	1.41198	0.705988	+	0.708224i	$0.250503\pi$
$878$	1.15155	0.0388629
$879$	−23.0574	−0.777706
$880$	−5.44503	−0.183552
$881$	9.83749	0.331434	0.165717	−	0.986173i	$-0.447006\pi$
$881$	9.83749	0.331434	0.165717	+	0.986173i	$0.447006\pi$
$882$	−0.234103	−0.00788265
$883$	−53.4080	−1.79732	−0.898661	−	0.438644i	$-0.855459\pi$
$883$	−53.4080	−1.79732	−0.898661	+	0.438644i	$0.855459\pi$
$884$	27.0855	0.910985
$885$	−5.18085	−0.174153
$886$	1.37629	0.0462375
$887$	7.87584	0.264445	0.132222	−	0.991220i	$-0.457789\pi$
$887$	7.87584	0.264445	0.132222	+	0.991220i	$0.457789\pi$
$888$	−1.36688	−0.0458693
$889$	−15.5191	−0.520494
$890$	0.260017	0.00871579
$891$	1.36628	0.0457722
$892$	16.4728	0.551549
$893$	6.69246	0.223955
$894$	0.608786	0.0203608
$895$	−12.5564	−0.419716
$896$	2.38190	0.0795736
$897$	10.3406	0.345263
$898$	1.16969	0.0390330
$899$	15.4226	0.514372
$900$	−1.99755	−0.0665849
$901$	−28.9016	−0.962850
$902$	−0.550179	−0.0183189
$903$	6.78010	0.225627
$904$	1.45528	0.0484020
$905$	−4.85742	−0.161466
$906$	−0.00366037	−0.000121608 0
$907$	19.8319	0.658508	0.329254	−	0.944241i	$-0.393203\pi$
$907$	19.8319	0.658508	0.329254	+	0.944241i	$0.393203\pi$
$908$	19.0587	0.632486
$909$	−5.67995	−0.188392
$910$	−0.492535	−0.0163274
$911$	28.6048	0.947720	0.473860	−	0.880600i	$-0.342860\pi$
$911$	28.6048	0.947720	0.473860	+	0.880600i	$0.342860\pi$
$912$	−16.8765	−0.558837
$913$	20.3535	0.673603
$914$	1.52825	0.0505499
$915$	−10.5297	−0.348100
$916$	26.6972	0.882101
$917$	30.5359	1.00838
$918$	−0.101794	−0.00335970
$919$	−39.2967	−1.29628	−0.648139	−	0.761522i	$-0.724452\pi$
$919$	−39.2967	−1.29628	−0.648139	+	0.761522i	$0.724452\pi$
$920$	0.310328	0.0102312
$921$	26.5906	0.876189
$922$	−1.06641	−0.0351203
$923$	30.1472	0.992307
$924$	−4.11462	−0.135361
$925$	−6.90433	−0.227013
$926$	0.698240	0.0229456
$927$	15.4761	0.508302
$928$	−3.73715	−0.122678
$929$	34.7476	1.14003	0.570016	−	0.821634i	$-0.306937\pi$
$929$	34.7476	1.14003	0.570016	+	0.821634i	$0.306937\pi$
$930$	−0.121259	−0.00397624
$931$	−20.0178	−0.656056
$932$	−50.9369	−1.66850
$933$	−7.57838	−0.248105
$934$	−1.24840	−0.0408490
$935$	−2.80834	−0.0918424
$936$	1.30599	0.0426875
$937$	−14.5538	−0.475451	−0.237726	−	0.971332i	$-0.576402\pi$
$937$	−14.5538	−0.475451	−0.237726	+	0.971332i	$0.576402\pi$
$938$	−0.485789	−0.0158616
$939$	−6.62822	−0.216304
$940$	3.15690	0.102967
$941$	−4.14637	−0.135168	−0.0675839	−	0.997714i	$-0.521529\pi$
$941$	−4.14637	−0.135168	−0.0675839	+	0.997714i	$0.521529\pi$
$942$	−0.748498	−0.0243874
$943$	−12.7457	−0.415058
$944$	−20.6472	−0.672009
$945$	−1.50762	−0.0490429
$946$	0.304297	0.00989356
$947$	26.1181	0.848725	0.424362	−	0.905492i	$-0.360498\pi$
$947$	26.1181	0.848725	0.424362	+	0.905492i	$0.360498\pi$
$948$	1.12238	0.0364531
$949$	30.8897	1.00272
$950$	0.209718	0.00680415
$951$	10.2622	0.332776
$952$	0.613490	0.0198833
$953$	55.8917	1.81051	0.905255	−	0.424868i	$-0.139680\pi$
$953$	55.8917	1.81051	0.905255	+	0.424868i	$0.139680\pi$
$954$	−0.696348	−0.0225451
$955$	6.95203	0.224962
$956$	36.4111	1.17762
$957$	8.60592	0.278190
$958$	1.77505	0.0573492
$959$	−3.57748	−0.115523
$960$	−7.94120	−0.256301
$961$	−25.0048	−0.806608
$962$	2.25562	0.0727242
$963$	12.3806	0.398958
$964$	−22.9495	−0.739153
$965$	17.9860	0.578990
$966$	0.117036	0.00376557
$967$	−32.9636	−1.06004	−0.530018	−	0.847986i	$-0.677815\pi$
$967$	−32.9636	−1.06004	−0.530018	+	0.847986i	$0.677815\pi$
$968$	1.80815	0.0581160
$969$	−8.70424	−0.279620
$970$	−0.490788	−0.0157583
$971$	22.7607	0.730425	0.365212	−	0.930924i	$-0.380996\pi$
$971$	22.7607	0.730425	0.365212	+	0.930924i	$0.380996\pi$
$972$	1.99755	0.0640714
$973$	−7.03933	−0.225670
$974$	−1.25965	−0.0403616
$975$	6.59677	0.211266
$976$	−41.9638	−1.34323
$977$	7.80148	0.249591	0.124796	−	0.992182i	$-0.460172\pi$
$977$	7.80148	0.249591	0.124796	+	0.992182i	$0.460172\pi$
$978$	0.789550	0.0252470
$979$	−7.17347	−0.229265
$980$	−9.44257	−0.301632
$981$	1.98009	0.0632194
$982$	1.47337	0.0470170
$983$	18.5762	0.592489	0.296245	−	0.955112i	$-0.404266\pi$
$983$	18.5762	0.592489	0.296245	+	0.955112i	$0.404266\pi$
$984$	−1.60974	−0.0513168
$985$	−14.0754	−0.448480
$986$	−0.641178	−0.0204193
$987$	2.38262	0.0758398
$988$	55.8022	1.77530
$989$	7.04950	0.224161
$990$	−0.0676634	−0.00215049
$991$	−24.2779	−0.771212	−0.385606	−	0.922664i	$-0.626008\pi$
$991$	−24.2779	−0.771212	−0.385606	+	0.922664i	$0.626008\pi$
$992$	−1.45273	−0.0461242
$993$	−4.01811	−0.127511
$994$	0.341209	0.0108225
$995$	14.3799	0.455874
$996$	29.7574	0.942901
$997$	19.6813	0.623313	0.311657	−	0.950195i	$-0.399116\pi$
$997$	19.6813	0.623313	0.311657	+	0.950195i	$0.399116\pi$
$998$	−0.354202	−0.0112121
$999$	6.90433	0.218443

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6015.2.a.e.1.15	✓	31

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6015.2.a.e.1.15	✓	31	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:	\( N \)	\(=\)	\( 6015 = 3 \cdot 5 \cdot 401 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6015.a (trivial)

\(f(q)\)	\(=\)	\(q+0.0495237 q^{2} -1.00000 q^{3} -1.99755 q^{4} -1.00000 q^{5} -0.0495237 q^{6} -1.50762 q^{7} -0.197974 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+0.0495237 q^{2} -1.00000 q^{3} -1.99755 q^{4} -1.00000 q^{5} -0.0495237 q^{6} -1.50762 q^{7} -0.197974 q^{8} +1.00000 q^{9} -0.0495237 q^{10} +1.36628 q^{11} +1.99755 q^{12} -6.59677 q^{13} -0.0746630 q^{14} +1.00000 q^{15} +3.98529 q^{16} +2.05546 q^{17} +0.0495237 q^{18} +4.23470 q^{19} +1.99755 q^{20} +1.50762 q^{21} +0.0676634 q^{22} +1.56752 q^{23} +0.197974 q^{24} +1.00000 q^{25} -0.326697 q^{26} -1.00000 q^{27} +3.01154 q^{28} -6.29879 q^{29} +0.0495237 q^{30} -2.44850 q^{31} +0.593314 q^{32} -1.36628 q^{33} +0.101794 q^{34} +1.50762 q^{35} -1.99755 q^{36} -6.90433 q^{37} +0.209718 q^{38} +6.59677 q^{39} +0.197974 q^{40} -8.13111 q^{41} +0.0746630 q^{42} +4.49722 q^{43} -2.72921 q^{44} -1.00000 q^{45} +0.0776297 q^{46} +1.58039 q^{47} -3.98529 q^{48} -4.72708 q^{49} +0.0495237 q^{50} -2.05546 q^{51} +13.1774 q^{52} -14.0609 q^{53} -0.0495237 q^{54} -1.36628 q^{55} +0.298469 q^{56} -4.23470 q^{57} -0.311939 q^{58} -5.18085 q^{59} -1.99755 q^{60} -10.5297 q^{61} -0.121259 q^{62} -1.50762 q^{63} -7.94120 q^{64} +6.59677 q^{65} -0.0676634 q^{66} +6.50642 q^{67} -4.10587 q^{68} -1.56752 q^{69} +0.0746630 q^{70} -4.56999 q^{71} -0.197974 q^{72} -4.68254 q^{73} -0.341928 q^{74} -1.00000 q^{75} -8.45901 q^{76} -2.05983 q^{77} +0.326697 q^{78} +0.561877 q^{79} -3.98529 q^{80} +1.00000 q^{81} -0.402683 q^{82} +14.8970 q^{83} -3.01154 q^{84} -2.05546 q^{85} +0.222719 q^{86} +6.29879 q^{87} -0.270488 q^{88} -5.25035 q^{89} -0.0495237 q^{90} +9.94543 q^{91} -3.13121 q^{92} +2.44850 q^{93} +0.0782667 q^{94} -4.23470 q^{95} -0.593314 q^{96} +9.91016 q^{97} -0.234103 q^{98} +1.36628 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 31 q + 6 q^{2} - 31 q^{3} + 24 q^{4} - 31 q^{5} - 6 q^{6} - 4 q^{7} + 15 q^{8} + 31 q^{9}+O(q^{10}) \) 31 * q + 6 * q^2 - 31 * q^3 + 24 * q^4 - 31 * q^5 - 6 * q^6 - 4 * q^7 + 15 * q^8 + 31 * q^9	\( 31 q + 6 q^{2} - 31 q^{3} + 24 q^{4} - 31 q^{5} - 6 q^{6} - 4 q^{7} + 15 q^{8} + 31 q^{9} - 6 q^{10} - q^{11} - 24 q^{12} + 8 q^{13} + 8 q^{14} + 31 q^{15} + 10 q^{16} + 42 q^{17} + 6 q^{18} - 19 q^{19} - 24 q^{20} + 4 q^{21} + 17 q^{22} + 26 q^{23} - 15 q^{24} + 31 q^{25} + 4 q^{26} - 31 q^{27} - 16 q^{28} - 11 q^{29} + 6 q^{30} - 3 q^{31} + 41 q^{32} + q^{33} + 6 q^{34} + 4 q^{35} + 24 q^{36} + 10 q^{37} + 12 q^{38} - 8 q^{39} - 15 q^{40} + 27 q^{41} - 8 q^{42} - 37 q^{43} - 9 q^{44} - 31 q^{45} - 23 q^{46} + 48 q^{47} - 10 q^{48} + 31 q^{49} + 6 q^{50} - 42 q^{51} + 18 q^{52} + 35 q^{53} - 6 q^{54} + q^{55} + 7 q^{56} + 19 q^{57} - 26 q^{58} - 3 q^{59} + 24 q^{60} + 11 q^{61} + 49 q^{62} - 4 q^{63} - 27 q^{64} - 8 q^{65} - 17 q^{66} - 44 q^{67} + 72 q^{68} - 26 q^{69} - 8 q^{70} + 6 q^{71} + 15 q^{72} + 49 q^{73} - 6 q^{74} - 31 q^{75} - 36 q^{76} + 66 q^{77} - 4 q^{78} - 41 q^{79} - 10 q^{80} + 31 q^{81} + 9 q^{82} + 53 q^{83} + 16 q^{84} - 42 q^{85} + 3 q^{86} + 11 q^{87} + 21 q^{88} + 31 q^{89} - 6 q^{90} - 45 q^{91} + 45 q^{92} + 3 q^{93} - 4 q^{94} + 19 q^{95} - 41 q^{96} + 37 q^{97} + 41 q^{98} - q^{99}+O(q^{100}) \) 31 * q + 6 * q^2 - 31 * q^3 + 24 * q^4 - 31 * q^5 - 6 * q^6 - 4 * q^7 + 15 * q^8 + 31 * q^9 - 6 * q^10 - q^11 - 24 * q^12 + 8 * q^13 + 8 * q^14 + 31 * q^15 + 10 * q^16 + 42 * q^17 + 6 * q^18 - 19 * q^19 - 24 * q^20 + 4 * q^21 + 17 * q^22 + 26 * q^23 - 15 * q^24 + 31 * q^25 + 4 * q^26 - 31 * q^27 - 16 * q^28 - 11 * q^29 + 6 * q^30 - 3 * q^31 + 41 * q^32 + q^33 + 6 * q^34 + 4 * q^35 + 24 * q^36 + 10 * q^37 + 12 * q^38 - 8 * q^39 - 15 * q^40 + 27 * q^41 - 8 * q^42 - 37 * q^43 - 9 * q^44 - 31 * q^45 - 23 * q^46 + 48 * q^47 - 10 * q^48 + 31 * q^49 + 6 * q^50 - 42 * q^51 + 18 * q^52 + 35 * q^53 - 6 * q^54 + q^55 + 7 * q^56 + 19 * q^57 - 26 * q^58 - 3 * q^59 + 24 * q^60 + 11 * q^61 + 49 * q^62 - 4 * q^63 - 27 * q^64 - 8 * q^65 - 17 * q^66 - 44 * q^67 + 72 * q^68 - 26 * q^69 - 8 * q^70 + 6 * q^71 + 15 * q^72 + 49 * q^73 - 6 * q^74 - 31 * q^75 - 36 * q^76 + 66 * q^77 - 4 * q^78 - 41 * q^79 - 10 * q^80 + 31 * q^81 + 9 * q^82 + 53 * q^83 + 16 * q^84 - 42 * q^85 + 3 * q^86 + 11 * q^87 + 21 * q^88 + 31 * q^89 - 6 * q^90 - 45 * q^91 + 45 * q^92 + 3 * q^93 - 4 * q^94 + 19 * q^95 - 41 * q^96 + 37 * q^97 + 41 * q^98 - q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more