LMFDB - Embedded newform 4013.2.a.b.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4013,2,Mod(1,4013)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4013, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4013.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4013 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4013.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$32.0439663311$
Analytic rank:	$1$
Dimension:	$157$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Character	$\chi$	$=$	4013.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.54071 q^{2} +0.214630 q^{3} +4.45522 q^{4} +0.391559 q^{5} -0.545314 q^{6} +2.42547 q^{7} -6.23801 q^{8} -2.95393 q^{9} +O(q^{10})$	\(q-2.54071 q^{2} +0.214630 q^{3} +4.45522 q^{4} +0.391559 q^{5} -0.545314 q^{6} +2.42547 q^{7} -6.23801 q^{8} -2.95393 q^{9} -0.994840 q^{10} +0.791914 q^{11} +0.956226 q^{12} -0.224616 q^{13} -6.16242 q^{14} +0.0840406 q^{15} +6.93856 q^{16} +6.43585 q^{17} +7.50510 q^{18} -3.43556 q^{19} +1.74448 q^{20} +0.520579 q^{21} -2.01203 q^{22} -1.10992 q^{23} -1.33887 q^{24} -4.84668 q^{25} +0.570684 q^{26} -1.27789 q^{27} +10.8060 q^{28} -3.29791 q^{29} -0.213523 q^{30} -4.41775 q^{31} -5.15286 q^{32} +0.169969 q^{33} -16.3516 q^{34} +0.949715 q^{35} -13.1604 q^{36} -2.11515 q^{37} +8.72876 q^{38} -0.0482093 q^{39} -2.44255 q^{40} -0.162321 q^{41} -1.32264 q^{42} +6.24430 q^{43} +3.52815 q^{44} -1.15664 q^{45} +2.81999 q^{46} +10.5940 q^{47} +1.48923 q^{48} -1.11711 q^{49} +12.3140 q^{50} +1.38133 q^{51} -1.00071 q^{52} -9.12899 q^{53} +3.24676 q^{54} +0.310082 q^{55} -15.1301 q^{56} -0.737375 q^{57} +8.37904 q^{58} +1.94577 q^{59} +0.374419 q^{60} +3.62768 q^{61} +11.2242 q^{62} -7.16467 q^{63} -0.785184 q^{64} -0.0879504 q^{65} -0.431842 q^{66} +0.263649 q^{67} +28.6731 q^{68} -0.238223 q^{69} -2.41295 q^{70} -11.5350 q^{71} +18.4267 q^{72} +0.708976 q^{73} +5.37398 q^{74} -1.04024 q^{75} -15.3062 q^{76} +1.92076 q^{77} +0.122486 q^{78} -15.4186 q^{79} +2.71686 q^{80} +8.58753 q^{81} +0.412410 q^{82} +0.603639 q^{83} +2.31930 q^{84} +2.52002 q^{85} -15.8650 q^{86} -0.707832 q^{87} -4.93997 q^{88} -12.6317 q^{89} +2.93869 q^{90} -0.544798 q^{91} -4.94494 q^{92} -0.948183 q^{93} -26.9164 q^{94} -1.34522 q^{95} -1.10596 q^{96} -15.7793 q^{97} +2.83824 q^{98} -2.33926 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 157 q - 15 q^{2} - 51 q^{3} + 137 q^{4} - 13 q^{5} - 15 q^{6} - 49 q^{7} - 42 q^{8} + 144 q^{9}+O(q^{10}) $ 157 * q - 15 * q^2 - 51 * q^3 + 137 * q^4 - 13 * q^5 - 15 * q^6 - 49 * q^7 - 42 * q^8 + 144 * q^9	\( 157 q - 15 q^{2} - 51 q^{3} + 137 q^{4} - 13 q^{5} - 15 q^{6} - 49 q^{7} - 42 q^{8} + 144 q^{9} - 61 q^{10} - 27 q^{11} - 93 q^{12} - 97 q^{13} - 12 q^{14} - 36 q^{15} + 105 q^{16} - 45 q^{17} - 68 q^{18} - 128 q^{19} - 30 q^{20} - 26 q^{21} - 68 q^{22} - 41 q^{23} - 40 q^{24} + 102 q^{25} - 5 q^{26} - 189 q^{27} - 115 q^{28} - 26 q^{29} - 12 q^{30} - 88 q^{31} - 89 q^{32} - 52 q^{33} - 61 q^{34} - 87 q^{35} + 110 q^{36} - 62 q^{37} - 37 q^{38} - 20 q^{39} - 161 q^{40} - 34 q^{41} - 53 q^{42} - 254 q^{43} - 19 q^{44} - 46 q^{45} - 52 q^{46} - 76 q^{47} - 162 q^{48} + 96 q^{49} - 54 q^{50} - 76 q^{51} - 259 q^{52} - 48 q^{53} - 12 q^{54} - 194 q^{55} - 10 q^{56} - 30 q^{57} - 52 q^{58} - 64 q^{59} - 31 q^{60} - 107 q^{61} - 51 q^{62} - 106 q^{63} + 54 q^{64} - 17 q^{65} - 13 q^{66} - 193 q^{67} - 118 q^{68} - 55 q^{69} - 86 q^{70} - 11 q^{71} - 172 q^{72} - 173 q^{73} - 11 q^{74} - 209 q^{75} - 213 q^{76} - 84 q^{77} - 30 q^{78} - 111 q^{79} - 6 q^{80} + 157 q^{81} - 117 q^{82} - 154 q^{83} - 6 q^{84} - 91 q^{85} + 28 q^{86} - 165 q^{87} - 165 q^{88} - 32 q^{89} - 103 q^{90} - 200 q^{91} - 86 q^{92} - 39 q^{93} - 118 q^{94} - 22 q^{95} - 28 q^{96} - 151 q^{97} - 38 q^{98} - 91 q^{99}+O(q^{100}) \) 157 * q - 15 * q^2 - 51 * q^3 + 137 * q^4 - 13 * q^5 - 15 * q^6 - 49 * q^7 - 42 * q^8 + 144 * q^9 - 61 * q^10 - 27 * q^11 - 93 * q^12 - 97 * q^13 - 12 * q^14 - 36 * q^15 + 105 * q^16 - 45 * q^17 - 68 * q^18 - 128 * q^19 - 30 * q^20 - 26 * q^21 - 68 * q^22 - 41 * q^23 - 40 * q^24 + 102 * q^25 - 5 * q^26 - 189 * q^27 - 115 * q^28 - 26 * q^29 - 12 * q^30 - 88 * q^31 - 89 * q^32 - 52 * q^33 - 61 * q^34 - 87 * q^35 + 110 * q^36 - 62 * q^37 - 37 * q^38 - 20 * q^39 - 161 * q^40 - 34 * q^41 - 53 * q^42 - 254 * q^43 - 19 * q^44 - 46 * q^45 - 52 * q^46 - 76 * q^47 - 162 * q^48 + 96 * q^49 - 54 * q^50 - 76 * q^51 - 259 * q^52 - 48 * q^53 - 12 * q^54 - 194 * q^55 - 10 * q^56 - 30 * q^57 - 52 * q^58 - 64 * q^59 - 31 * q^60 - 107 * q^61 - 51 * q^62 - 106 * q^63 + 54 * q^64 - 17 * q^65 - 13 * q^66 - 193 * q^67 - 118 * q^68 - 55 * q^69 - 86 * q^70 - 11 * q^71 - 172 * q^72 - 173 * q^73 - 11 * q^74 - 209 * q^75 - 213 * q^76 - 84 * q^77 - 30 * q^78 - 111 * q^79 - 6 * q^80 + 157 * q^81 - 117 * q^82 - 154 * q^83 - 6 * q^84 - 91 * q^85 + 28 * q^86 - 165 * q^87 - 165 * q^88 - 32 * q^89 - 103 * q^90 - 200 * q^91 - 86 * q^92 - 39 * q^93 - 118 * q^94 - 22 * q^95 - 28 * q^96 - 151 * q^97 - 38 * q^98 - 91 * 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 $

$2$	−2.54071	−1.79656	−0.898278	−	0.439428i	$-0.855181\pi$
$2$	−2.54071	−1.79656	−0.898278	+	0.439428i	$0.855181\pi$
$3$	0.214630	0.123917	0.0619584	−	0.998079i	$-0.480265\pi$
$3$	0.214630	0.123917	0.0619584	+	0.998079i	$0.480265\pi$
$4$	4.45522	2.22761
$5$	0.391559	0.175111	0.0875554	−	0.996160i	$-0.472095\pi$
$5$	0.391559	0.175111	0.0875554	+	0.996160i	$0.472095\pi$
$6$	−0.545314	−0.222624
$7$	2.42547	0.916741	0.458370	−	0.888761i	$-0.348433\pi$
$7$	2.42547	0.916741	0.458370	+	0.888761i	$0.348433\pi$
$8$	−6.23801	−2.20547
$9$	−2.95393	−0.984645
$10$	−0.994840	−0.314596
$11$	0.791914	0.238771	0.119386	−	0.992848i	$-0.461908\pi$
$11$	0.791914	0.238771	0.119386	+	0.992848i	$0.461908\pi$
$12$	0.956226	0.276039
$13$	−0.224616	−0.0622971	−0.0311486	−	0.999515i	$-0.509917\pi$
$13$	−0.224616	−0.0622971	−0.0311486	+	0.999515i	$0.509917\pi$
$14$	−6.16242	−1.64698
$15$	0.0840406	0.0216992
$16$	6.93856	1.73464
$17$	6.43585	1.56092	0.780461	−	0.625204i	$-0.214984\pi$
$17$	6.43585	1.56092	0.780461	+	0.625204i	$0.214984\pi$
$18$	7.50510	1.76897
$19$	−3.43556	−0.788171	−0.394085	−	0.919074i	$-0.628939\pi$
$19$	−3.43556	−0.788171	−0.394085	+	0.919074i	$0.628939\pi$
$20$	1.74448	0.390079
$21$	0.520579	0.113600
$22$	−2.01203	−0.428966
$23$	−1.10992	−0.231434	−0.115717	−	0.993282i	$-0.536917\pi$
$23$	−1.10992	−0.231434	−0.115717	+	0.993282i	$0.536917\pi$
$24$	−1.33887	−0.273295
$25$	−4.84668	−0.969336
$26$	0.570684	0.111920
$27$	−1.27789	−0.245931
$28$	10.8060	2.04214
$29$	−3.29791	−0.612407	−0.306203	−	0.951966i	$-0.599059\pi$
$29$	−3.29791	−0.612407	−0.306203	+	0.951966i	$0.599059\pi$
$30$	−0.213523	−0.0389838
$31$	−4.41775	−0.793451	−0.396726	−	0.917937i	$-0.629854\pi$
$31$	−4.41775	−0.793451	−0.396726	+	0.917937i	$0.629854\pi$
$32$	−5.15286	−0.910905
$33$	0.169969	0.0295878
$34$	−16.3516	−2.80428
$35$	0.949715	0.160531
$36$	−13.1604	−2.19341
$37$	−2.11515	−0.347728	−0.173864	−	0.984770i	$-0.555625\pi$
$37$	−2.11515	−0.347728	−0.173864	+	0.984770i	$0.555625\pi$
$38$	8.72876	1.41599
$39$	−0.0482093	−0.00771967
$40$	−2.44255	−0.386202
$41$	−0.162321	−0.0253502	−0.0126751	−	0.999920i	$-0.504035\pi$
$41$	−0.162321	−0.0253502	−0.0126751	+	0.999920i	$0.504035\pi$
$42$	−1.32264	−0.204088
$43$	6.24430	0.952247	0.476124	−	0.879378i	$-0.342041\pi$
$43$	6.24430	0.952247	0.476124	+	0.879378i	$0.342041\pi$
$44$	3.52815	0.531889
$45$	−1.15664	−0.172422
$46$	2.81999	0.415785
$47$	10.5940	1.54530	0.772649	−	0.634834i	$-0.218931\pi$
$47$	10.5940	1.54530	0.772649	+	0.634834i	$0.218931\pi$
$48$	1.48923	0.214951
$49$	−1.11711	−0.159586
$50$	12.3140	1.74147
$51$	1.38133	0.193425
$52$	−1.00071	−0.138774
$53$	−9.12899	−1.25396	−0.626981	−	0.779034i	$-0.715710\pi$
$53$	−9.12899	−1.25396	−0.626981	+	0.779034i	$0.715710\pi$
$54$	3.24676	0.441829
$55$	0.310082	0.0418114
$56$	−15.1301	−2.02185
$57$	−0.737375	−0.0976677
$58$	8.37904	1.10022
$59$	1.94577	0.253318	0.126659	−	0.991946i	$-0.459575\pi$
$59$	1.94577	0.253318	0.126659	+	0.991946i	$0.459575\pi$
$60$	0.374419	0.0483373
$61$	3.62768	0.464476	0.232238	−	0.972659i	$-0.425395\pi$
$61$	3.62768	0.464476	0.232238	+	0.972659i	$0.425395\pi$
$62$	11.2242	1.42548
$63$	−7.16467	−0.902664
$64$	−0.785184	−0.0981480
$65$	−0.0879504	−0.0109089
$66$	−0.431842	−0.0531561
$67$	0.263649	0.0322098	0.0161049	−	0.999870i	$-0.494873\pi$
$67$	0.263649	0.0322098	0.0161049	+	0.999870i	$0.494873\pi$
$68$	28.6731	3.47713
$69$	−0.238223	−0.0286786
$70$	−2.41295	−0.288403
$71$	−11.5350	−1.36895	−0.684475	−	0.729037i	$-0.739968\pi$
$71$	−11.5350	−1.36895	−0.684475	+	0.729037i	$0.739968\pi$
$72$	18.4267	2.17161
$73$	0.708976	0.0829794	0.0414897	−	0.999139i	$-0.486790\pi$
$73$	0.708976	0.0829794	0.0414897	+	0.999139i	$0.486790\pi$
$74$	5.37398	0.624712
$75$	−1.04024	−0.120117
$76$	−15.3062	−1.75574
$77$	1.92076	0.218891
$78$	0.122486	0.0138688
$79$	−15.4186	−1.73473	−0.867365	−	0.497673i	$-0.834188\pi$
$79$	−15.4186	−1.73473	−0.867365	+	0.497673i	$0.834188\pi$
$80$	2.71686	0.303754
$81$	8.58753	0.954170
$82$	0.412410	0.0455431
$83$	0.603639	0.0662580	0.0331290	−	0.999451i	$-0.489453\pi$
$83$	0.603639	0.0662580	0.0331290	+	0.999451i	$0.489453\pi$
$84$	2.31930	0.253056
$85$	2.52002	0.273334
$86$	−15.8650	−1.71076
$87$	−0.707832	−0.0758875
$88$	−4.93997	−0.526603
$89$	−12.6317	−1.33896	−0.669480	−	0.742830i	$-0.733483\pi$
$89$	−12.6317	−1.33896	−0.669480	+	0.742830i	$0.733483\pi$
$90$	2.93869	0.309765
$91$	−0.544798	−0.0571103
$92$	−4.94494	−0.515546
$93$	−0.948183	−0.0983220
$94$	−26.9164	−2.77621
$95$	−1.34522	−0.138017
$96$	−1.10596	−0.112877
$97$	−15.7793	−1.60214	−0.801071	−	0.598570i	$-0.795736\pi$
$97$	−15.7793	−1.60214	−0.801071	+	0.598570i	$0.795736\pi$
$98$	2.83824	0.286706
$99$	−2.33926	−0.235105
$100$	−21.5930	−2.15930
$101$	16.3302	1.62492	0.812459	−	0.583019i	$-0.198129\pi$
$101$	16.3302	1.62492	0.812459	+	0.583019i	$0.198129\pi$
$102$	−3.50956	−0.347498
$103$	−19.7598	−1.94699	−0.973497	−	0.228700i	$-0.926552\pi$
$103$	−19.7598	−1.94699	−0.973497	+	0.228700i	$0.926552\pi$
$104$	1.40116	0.137395
$105$	0.203838	0.0198925
$106$	23.1941	2.25281
$107$	−9.32813	−0.901784	−0.450892	−	0.892578i	$-0.648894\pi$
$107$	−9.32813	−0.901784	−0.450892	+	0.892578i	$0.648894\pi$
$108$	−5.69331	−0.547839
$109$	−4.64506	−0.444916	−0.222458	−	0.974942i	$-0.571408\pi$
$109$	−4.64506	−0.444916	−0.222458	+	0.974942i	$0.571408\pi$
$110$	−0.787828	−0.0751165
$111$	−0.453974	−0.0430894
$112$	16.8293	1.59021
$113$	−4.56058	−0.429024	−0.214512	−	0.976721i	$-0.568816\pi$
$113$	−4.56058	−0.429024	−0.214512	+	0.976721i	$0.568816\pi$
$114$	1.87346	0.175465
$115$	−0.434600	−0.0405267
$116$	−14.6929	−1.36420
$117$	0.663499	0.0613406
$118$	−4.94365	−0.455100
$119$	15.6099	1.43096
$120$	−0.524246	−0.0478569
$121$	−10.3729	−0.942988
$122$	−9.21688	−0.834457
$123$	−0.0348390	−0.00314132
$124$	−19.6820	−1.76750
$125$	−3.85556	−0.344852
$126$	18.2034	1.62169
$127$	16.4541	1.46006	0.730031	−	0.683414i	$-0.239506\pi$
$127$	16.4541	1.46006	0.730031	+	0.683414i	$0.239506\pi$
$128$	12.3006	1.08723
$129$	1.34022	0.118000
$130$	0.223457	0.0195984
$131$	9.11083	0.796017	0.398008	−	0.917382i	$-0.369702\pi$
$131$	9.11083	0.796017	0.398008	+	0.917382i	$0.369702\pi$
$132$	0.757249	0.0659101
$133$	−8.33283	−0.722548
$134$	−0.669855	−0.0578667
$135$	−0.500372	−0.0430652
$136$	−40.1469	−3.44257
$137$	6.86436	0.586462	0.293231	−	0.956042i	$-0.405270\pi$
$137$	6.86436	0.586462	0.293231	+	0.956042i	$0.405270\pi$
$138$	0.605255	0.0515228
$139$	−0.598196	−0.0507383	−0.0253691	−	0.999678i	$-0.508076\pi$
$139$	−0.598196	−0.0507383	−0.0253691	+	0.999678i	$0.508076\pi$
$140$	4.23119	0.357601
$141$	2.27380	0.191488
$142$	29.3070	2.45939
$143$	−0.177876	−0.0148748
$144$	−20.4960	−1.70800
$145$	−1.29133	−0.107239
$146$	−1.80131	−0.149077
$147$	−0.239765	−0.0197755
$148$	−9.42344	−0.774602
$149$	18.3210	1.50091	0.750457	−	0.660920i	$-0.229834\pi$
$149$	18.3210	1.50091	0.750457	+	0.660920i	$0.229834\pi$
$150$	2.64296	0.215797
$151$	2.63555	0.214478	0.107239	−	0.994233i	$-0.465799\pi$
$151$	2.63555	0.214478	0.107239	+	0.994233i	$0.465799\pi$
$152$	21.4310	1.73829
$153$	−19.0111	−1.53695
$154$	−4.88011	−0.393250
$155$	−1.72981	−0.138942
$156$	−0.214783	−0.0171964
$157$	12.4239	0.991537	0.495768	−	0.868455i	$-0.334886\pi$
$157$	12.4239	0.991537	0.495768	+	0.868455i	$0.334886\pi$
$158$	39.1743	3.11654
$159$	−1.95936	−0.155387
$160$	−2.01765	−0.159509
$161$	−2.69208	−0.212165
$162$	−21.8184	−1.71422
$163$	−11.7392	−0.919482	−0.459741	−	0.888053i	$-0.652058\pi$
$163$	−11.7392	−0.919482	−0.459741	+	0.888053i	$0.652058\pi$
$164$	−0.723175	−0.0564705
$165$	0.0665529	0.00518114
$166$	−1.53367	−0.119036
$167$	0.413286	0.0319810	0.0159905	−	0.999872i	$-0.494910\pi$
$167$	0.413286	0.0319810	0.0159905	+	0.999872i	$0.494910\pi$
$168$	−3.24738	−0.250541
$169$	−12.9495	−0.996119
$170$	−6.40264	−0.491060
$171$	10.1484	0.776068
$172$	27.8198	2.12124
$173$	−24.4455	−1.85856	−0.929279	−	0.369380i	$-0.879570\pi$
$173$	−24.4455	−1.85856	−0.929279	+	0.369380i	$0.879570\pi$
$174$	1.79840	0.136336
$175$	−11.7555	−0.888630
$176$	5.49474	0.414182
$177$	0.417622	0.0313904
$178$	32.0936	2.40552
$179$	3.67319	0.274547	0.137274	−	0.990533i	$-0.456166\pi$
$179$	3.67319	0.274547	0.137274	+	0.990533i	$0.456166\pi$
$180$	−5.15309	−0.384089
$181$	13.6914	1.01768	0.508839	−	0.860862i	$-0.330075\pi$
$181$	13.6914	1.01768	0.508839	+	0.860862i	$0.330075\pi$
$182$	1.38417	0.102602
$183$	0.778609	0.0575565
$184$	6.92370	0.510422
$185$	−0.828205	−0.0608909
$186$	2.40906	0.176641
$187$	5.09664	0.372703
$188$	47.1987	3.44232
$189$	−3.09949	−0.225455
$190$	3.41783	0.247955
$191$	19.5937	1.41775	0.708874	−	0.705335i	$-0.249203\pi$
$191$	19.5937	1.41775	0.708874	+	0.705335i	$0.249203\pi$
$192$	−0.168524	−0.0121622
$193$	−3.03701	−0.218609	−0.109304	−	0.994008i	$-0.534862\pi$
$193$	−3.03701	−0.218609	−0.109304	+	0.994008i	$0.534862\pi$
$194$	40.0906	2.87834
$195$	−0.0188768	−0.00135180
$196$	−4.97695	−0.355497
$197$	−9.51284	−0.677761	−0.338881	−	0.940829i	$-0.610048\pi$
$197$	−9.51284	−0.677761	−0.338881	+	0.940829i	$0.610048\pi$
$198$	5.94339	0.422379
$199$	−4.02521	−0.285340	−0.142670	−	0.989770i	$-0.545569\pi$
$199$	−4.02521	−0.285340	−0.142670	+	0.989770i	$0.545569\pi$
$200$	30.2337	2.13784
$201$	0.0565870	0.00399134
$202$	−41.4904	−2.91925
$203$	−7.99898	−0.561418
$204$	6.15412	0.430875
$205$	−0.0635582	−0.00443910
$206$	50.2041	3.49788
$207$	3.27863	0.227881
$208$	−1.55851	−0.108063
$209$	−2.72067	−0.188192
$210$	−0.517893	−0.0357380
$211$	21.1559	1.45643	0.728215	−	0.685348i	$-0.240350\pi$
$211$	21.1559	1.45643	0.728215	+	0.685348i	$0.240350\pi$
$212$	−40.6717	−2.79334
$213$	−2.47576	−0.169636
$214$	23.7001	1.62011
$215$	2.44502	0.166749
$216$	7.97153	0.542394
$217$	−10.7151	−0.727389
$218$	11.8018	0.799316
$219$	0.152168	0.0102825
$220$	1.38148	0.0931395
$221$	−1.44559	−0.0972410
$222$	1.15342	0.0774124
$223$	−13.7676	−0.921945	−0.460972	−	0.887414i	$-0.652499\pi$
$223$	−13.7676	−0.921945	−0.460972	+	0.887414i	$0.652499\pi$
$224$	−12.4981	−0.835064
$225$	14.3168	0.954452
$226$	11.5871	0.770765
$227$	15.4187	1.02338	0.511688	−	0.859171i	$-0.329020\pi$
$227$	15.4187	1.02338	0.511688	+	0.859171i	$0.329020\pi$
$228$	−3.28517	−0.217566
$229$	3.55073	0.234639	0.117319	−	0.993094i	$-0.462570\pi$
$229$	3.55073	0.234639	0.117319	+	0.993094i	$0.462570\pi$
$230$	1.10419	0.0728084
$231$	0.412254	0.0271243
$232$	20.5724	1.35064
$233$	18.3224	1.20034	0.600172	−	0.799871i	$-0.295099\pi$
$233$	18.3224	1.20034	0.600172	+	0.799871i	$0.295099\pi$
$234$	−1.68576	−0.110202
$235$	4.14819	0.270598
$236$	8.66885	0.564294
$237$	−3.30930	−0.214962
$238$	−39.6604	−2.57080
$239$	−10.8038	−0.698840	−0.349420	−	0.936966i	$-0.613621\pi$
$239$	−10.8038	−0.698840	−0.349420	+	0.936966i	$0.613621\pi$
$240$	0.583120	0.0376403
$241$	−11.8780	−0.765129	−0.382564	−	0.923929i	$-0.624959\pi$
$241$	−11.8780	−0.765129	−0.382564	+	0.923929i	$0.624959\pi$
$242$	26.3545	1.69413
$243$	5.67683	0.364169
$244$	16.1621	1.03467
$245$	−0.437413	−0.0279453
$246$	0.0885158	0.00564356
$247$	0.771679	0.0491008
$248$	27.5580	1.74993
$249$	0.129559	0.00821049
$250$	9.79587	0.619545
$251$	−18.3915	−1.16086	−0.580431	−	0.814309i	$-0.697116\pi$
$251$	−18.3915	−1.16086	−0.580431	+	0.814309i	$0.697116\pi$
$252$	−31.9202	−2.01078
$253$	−0.878962	−0.0552599
$254$	−41.8051	−2.62308
$255$	0.540872	0.0338707
$256$	−29.6820	−1.85513
$257$	25.6024	1.59703	0.798516	−	0.601974i	$-0.205619\pi$
$257$	25.6024	1.59703	0.798516	+	0.601974i	$0.205619\pi$
$258$	−3.40511	−0.211993
$259$	−5.13022	−0.318776
$260$	−0.391838	−0.0243008
$261$	9.74181	0.603003
$262$	−23.1480	−1.43009
$263$	−21.3965	−1.31937	−0.659683	−	0.751544i	$-0.729309\pi$
$263$	−21.3965	−1.31937	−0.659683	+	0.751544i	$0.729309\pi$
$264$	−1.06027	−0.0652550
$265$	−3.57454	−0.219582
$266$	21.1713	1.29810
$267$	−2.71115	−0.165920
$268$	1.17461	0.0717509
$269$	−5.42192	−0.330580	−0.165290	−	0.986245i	$-0.552856\pi$
$269$	−5.42192	−0.330580	−0.165290	+	0.986245i	$0.552856\pi$
$270$	1.27130	0.0773689
$271$	−4.08136	−0.247925	−0.123962	−	0.992287i	$-0.539560\pi$
$271$	−4.08136	−0.247925	−0.123962	+	0.992287i	$0.539560\pi$
$272$	44.6555	2.70764
$273$	−0.116930	−0.00707694
$274$	−17.4404	−1.05361
$275$	−3.83816	−0.231449
$276$	−1.06134	−0.0638849
$277$	−23.1535	−1.39116	−0.695581	−	0.718448i	$-0.744853\pi$
$277$	−23.1535	−1.39116	−0.695581	+	0.718448i	$0.744853\pi$
$278$	1.51984	0.0911541
$279$	13.0497	0.781267
$280$	−5.92434	−0.354047
$281$	−29.0734	−1.73437	−0.867186	−	0.497984i	$-0.834074\pi$
$281$	−29.0734	−1.73437	−0.867186	+	0.497984i	$0.834074\pi$
$282$	−5.77707	−0.344020
$283$	−14.8876	−0.884977	−0.442488	−	0.896774i	$-0.645904\pi$
$283$	−14.8876	−0.884977	−0.442488	+	0.896774i	$0.645904\pi$
$284$	−51.3909	−3.04949
$285$	−0.288726	−0.0171027
$286$	0.451932	0.0267233
$287$	−0.393704	−0.0232396
$288$	15.2212	0.896918
$289$	24.4201	1.43648
$290$	3.28089	0.192661
$291$	−3.38671	−0.198532
$292$	3.15865	0.184846
$293$	20.2284	1.18176	0.590878	−	0.806761i	$-0.298782\pi$
$293$	20.2284	1.18176	0.590878	+	0.806761i	$0.298782\pi$
$294$	0.609173	0.0355277
$295$	0.761886	0.0443587
$296$	13.1943	0.766904
$297$	−1.01198	−0.0587212
$298$	−46.5483	−2.69647
$299$	0.249305	0.0144177
$300$	−4.63452	−0.267574
$301$	15.1454	0.872964
$302$	−6.69617	−0.385321
$303$	3.50496	0.201355
$304$	−23.8378	−1.36719
$305$	1.42045	0.0813348
$306$	48.3017	2.76122
$307$	−15.9731	−0.911634	−0.455817	−	0.890074i	$-0.650653\pi$
$307$	−15.9731	−0.911634	−0.455817	+	0.890074i	$0.650653\pi$
$308$	8.55742	0.487604
$309$	−4.24106	−0.241265
$310$	4.39495	0.249617
$311$	29.6882	1.68346	0.841732	−	0.539896i	$-0.181536\pi$
$311$	29.6882	1.68346	0.841732	+	0.539896i	$0.181536\pi$
$312$	0.300730	0.0170255
$313$	−16.8243	−0.950964	−0.475482	−	0.879725i	$-0.657726\pi$
$313$	−16.8243	−0.950964	−0.475482	+	0.879725i	$0.657726\pi$
$314$	−31.5656	−1.78135
$315$	−2.80540	−0.158066
$316$	−68.6934	−3.86430
$317$	−30.1809	−1.69513	−0.847563	−	0.530695i	$-0.821931\pi$
$317$	−30.1809	−1.69513	−0.847563	+	0.530695i	$0.821931\pi$
$318$	4.97816	0.279162
$319$	−2.61166	−0.146225
$320$	−0.307446	−0.0171868
$321$	−2.00210	−0.111746
$322$	6.83980	0.381167
$323$	−22.1107	−1.23027
$324$	38.2593	2.12552
$325$	1.08864	0.0603869
$326$	29.8258	1.65190
$327$	−0.996970	−0.0551326
$328$	1.01256	0.0559092
$329$	25.6955	1.41664
$330$	−0.169092	−0.00930820
$331$	3.75423	0.206351	0.103176	−	0.994663i	$-0.467100\pi$
$331$	3.75423	0.206351	0.103176	+	0.994663i	$0.467100\pi$
$332$	2.68935	0.147597
$333$	6.24800	0.342388
$334$	−1.05004	−0.0574557
$335$	0.103234	0.00564028
$336$	3.61207	0.197055
$337$	−3.40875	−0.185687	−0.0928433	−	0.995681i	$-0.529596\pi$
$337$	−3.40875	−0.185687	−0.0928433	+	0.995681i	$0.529596\pi$
$338$	32.9011	1.78958
$339$	−0.978840	−0.0531633
$340$	11.2272	0.608882
$341$	−3.49848	−0.189453
$342$	−25.7842	−1.39425
$343$	−19.6878	−1.06304
$344$	−38.9521	−2.10015
$345$	−0.0932784	−0.00502194
$346$	62.1090	3.33900
$347$	−3.89496	−0.209093	−0.104546	−	0.994520i	$-0.533339\pi$
$347$	−3.89496	−0.209093	−0.104546	+	0.994520i	$0.533339\pi$
$348$	−3.15355	−0.169048
$349$	20.9218	1.11992	0.559959	−	0.828520i	$-0.310817\pi$
$349$	20.9218	1.11992	0.559959	+	0.828520i	$0.310817\pi$
$350$	29.8673	1.59647
$351$	0.287035	0.0153208
$352$	−4.08062	−0.217498
$353$	35.9818	1.91512	0.957559	−	0.288238i	$-0.0930693\pi$
$353$	35.9818	1.91512	0.957559	+	0.288238i	$0.0930693\pi$
$354$	−1.06106	−0.0563946
$355$	−4.51663	−0.239718
$356$	−56.2771	−2.98268
$357$	3.35037	0.177320
$358$	−9.33252	−0.493239
$359$	17.9018	0.944821	0.472410	−	0.881379i	$-0.343384\pi$
$359$	17.9018	0.944821	0.472410	+	0.881379i	$0.343384\pi$
$360$	7.21514	0.380271
$361$	−7.19696	−0.378787
$362$	−34.7860	−1.82831
$363$	−2.22633	−0.116852
$364$	−2.42720	−0.127220
$365$	0.277606	0.0145306
$366$	−1.97822	−0.103403
$367$	−2.79620	−0.145960	−0.0729802	−	0.997333i	$-0.523251\pi$
$367$	−2.79620	−0.145960	−0.0729802	+	0.997333i	$0.523251\pi$
$368$	−7.70125	−0.401455
$369$	0.479485	0.0249610
$370$	2.10423	0.109394
$371$	−22.1421	−1.14956
$372$	−4.22437	−0.219023
$373$	−16.7394	−0.866734	−0.433367	−	0.901218i	$-0.642675\pi$
$373$	−16.7394	−0.866734	−0.433367	+	0.901218i	$0.642675\pi$
$374$	−12.9491	−0.669582
$375$	−0.827521	−0.0427330
$376$	−66.0857	−3.40811
$377$	0.740762	0.0381512
$378$	7.87492	0.405042
$379$	−12.6331	−0.648919	−0.324459	−	0.945900i	$-0.605182\pi$
$379$	−12.6331	−0.648919	−0.324459	+	0.945900i	$0.605182\pi$
$380$	−5.99327	−0.307448
$381$	3.53154	0.180926
$382$	−49.7819	−2.54706
$383$	16.3832	0.837143	0.418572	−	0.908184i	$-0.362531\pi$
$383$	16.3832	0.837143	0.418572	+	0.908184i	$0.362531\pi$
$384$	2.64009	0.134727
$385$	0.752093	0.0383302
$386$	7.71617	0.392743
$387$	−18.4453	−0.937625
$388$	−70.3001	−3.56895
$389$	−20.5042	−1.03961	−0.519803	−	0.854286i	$-0.673995\pi$
$389$	−20.5042	−1.03961	−0.519803	+	0.854286i	$0.673995\pi$
$390$	0.0479606	0.00242858
$391$	−7.14328	−0.361251
$392$	6.96852	0.351963
$393$	1.95546	0.0986399
$394$	24.1694	1.21764
$395$	−6.03731	−0.303770
$396$	−10.4219	−0.523722
$397$	−8.07921	−0.405484	−0.202742	−	0.979232i	$-0.564985\pi$
$397$	−8.07921	−0.405484	−0.202742	+	0.979232i	$0.564985\pi$
$398$	10.2269	0.512629
$399$	−1.78848	−0.0895359
$400$	−33.6290	−1.68145
$401$	10.0760	0.503174	0.251587	−	0.967835i	$-0.419048\pi$
$401$	10.0760	0.503174	0.251587	+	0.967835i	$0.419048\pi$
$402$	−0.143771	−0.00717066
$403$	0.992295	0.0494297
$404$	72.7547	3.61968
$405$	3.36253	0.167085
$406$	20.3231	1.00862
$407$	−1.67501	−0.0830274
$408$	−8.61675	−0.426592
$409$	13.9594	0.690250	0.345125	−	0.938557i	$-0.387837\pi$
$409$	13.9594	0.690250	0.345125	+	0.938557i	$0.387837\pi$
$410$	0.161483	0.00797509
$411$	1.47330	0.0726725
$412$	−88.0344	−4.33714
$413$	4.71941	0.232227
$414$	−8.33006	−0.409400
$415$	0.236361	0.0116025
$416$	1.15741	0.0567468
$417$	−0.128391	−0.00628733
$418$	6.91243	0.338098
$419$	−18.1639	−0.887364	−0.443682	−	0.896184i	$-0.646328\pi$
$419$	−18.1639	−0.887364	−0.443682	+	0.896184i	$0.646328\pi$
$420$	0.908142	0.0443128
$421$	1.47523	0.0718985	0.0359493	−	0.999354i	$-0.488555\pi$
$421$	1.47523	0.0718985	0.0359493	+	0.999354i	$0.488555\pi$
$422$	−53.7510	−2.61656
$423$	−31.2940	−1.52157
$424$	56.9467	2.76558
$425$	−31.1925	−1.51306
$426$	6.29018	0.304760
$427$	8.79881	0.425804
$428$	−41.5589	−2.00882
$429$	−0.0381776	−0.00184323
$430$	−6.21208	−0.299573
$431$	12.1438	0.584944	0.292472	−	0.956274i	$-0.405522\pi$
$431$	12.1438	0.584944	0.292472	+	0.956274i	$0.405522\pi$
$432$	−8.86675	−0.426602
$433$	8.55717	0.411231	0.205616	−	0.978633i	$-0.434080\pi$
$433$	8.55717	0.411231	0.205616	+	0.978633i	$0.434080\pi$
$434$	27.2240	1.30679
$435$	−0.277158	−0.0132887
$436$	−20.6948	−0.991099
$437$	3.81319	0.182410
$438$	−0.386615	−0.0184732
$439$	−1.14863	−0.0548211	−0.0274105	−	0.999624i	$-0.508726\pi$
$439$	−1.14863	−0.0548211	−0.0274105	+	0.999624i	$0.508726\pi$
$440$	−1.93429	−0.0922138
$441$	3.29986	0.157136
$442$	3.67283	0.174699
$443$	10.1519	0.482330	0.241165	−	0.970484i	$-0.422470\pi$
$443$	10.1519	0.482330	0.241165	+	0.970484i	$0.422470\pi$
$444$	−2.02256	−0.0959863
$445$	−4.94607	−0.234466
$446$	34.9794	1.65632
$447$	3.93224	0.185988
$448$	−1.90444	−0.0899763
$449$	−32.3454	−1.52647	−0.763237	−	0.646118i	$-0.776391\pi$
$449$	−32.3454	−1.52647	−0.763237	+	0.646118i	$0.776391\pi$
$450$	−36.3748	−1.71473
$451$	−0.128544	−0.00605290
$452$	−20.3184	−0.955698
$453$	0.565669	0.0265774
$454$	−39.1745	−1.83855
$455$	−0.213321	−0.0100006
$456$	4.59975	0.215403
$457$	−16.6101	−0.776988	−0.388494	−	0.921451i	$-0.627005\pi$
$457$	−16.6101	−0.776988	−0.388494	+	0.921451i	$0.627005\pi$
$458$	−9.02139	−0.421542
$459$	−8.22434	−0.383879
$460$	−1.93624	−0.0902776
$461$	−33.6349	−1.56653	−0.783267	−	0.621686i	$-0.786448\pi$
$461$	−33.6349	−1.56653	−0.783267	+	0.621686i	$0.786448\pi$
$462$	−1.04742	−0.0487303
$463$	−8.98704	−0.417663	−0.208832	−	0.977952i	$-0.566966\pi$
$463$	−8.98704	−0.417663	−0.208832	+	0.977952i	$0.566966\pi$
$464$	−22.8827	−1.06230
$465$	−0.371270	−0.0172172
$466$	−46.5521	−2.15648
$467$	−40.1338	−1.85717	−0.928585	−	0.371120i	$-0.878974\pi$
$467$	−40.1338	−1.85717	−0.928585	+	0.371120i	$0.878974\pi$
$468$	2.95604	0.136643
$469$	0.639471	0.0295280
$470$	−10.5394	−0.486144
$471$	2.66655	0.122868
$472$	−12.1378	−0.558686
$473$	4.94495	0.227369
$474$	8.40799	0.386192
$475$	16.6510	0.764002
$476$	69.5458	3.18762
$477$	26.9664	1.23471
$478$	27.4494	1.25550
$479$	12.9362	0.591069	0.295534	−	0.955332i	$-0.404502\pi$
$479$	12.9362	0.591069	0.295534	+	0.955332i	$0.404502\pi$
$480$	−0.433049	−0.0197659
$481$	0.475095	0.0216625
$482$	30.1786	1.37460
$483$	−0.577802	−0.0262909
$484$	−46.2134	−2.10061
$485$	−6.17852	−0.280552
$486$	−14.4232	−0.654249
$487$	5.58222	0.252955	0.126477	−	0.991970i	$-0.459633\pi$
$487$	5.58222	0.252955	0.126477	+	0.991970i	$0.459633\pi$
$488$	−22.6295	−1.02439
$489$	−2.51958	−0.113939
$490$	1.11134	0.0502053
$491$	−35.7859	−1.61500	−0.807498	−	0.589870i	$-0.799179\pi$
$491$	−35.7859	−1.61500	−0.807498	+	0.589870i	$0.799179\pi$
$492$	−0.155215	−0.00699764
$493$	−21.2248	−0.955919
$494$	−1.96062	−0.0882123
$495$	−0.915960	−0.0411694
$496$	−30.6528	−1.37635
$497$	−27.9777	−1.25497
$498$	−0.329173	−0.0147506
$499$	−7.96863	−0.356725	−0.178362	−	0.983965i	$-0.557080\pi$
$499$	−7.96863	−0.356725	−0.178362	+	0.983965i	$0.557080\pi$
$500$	−17.1774	−0.768196
$501$	0.0887037	0.00396299
$502$	46.7276	2.08555
$503$	24.0208	1.07103	0.535516	−	0.844525i	$-0.320117\pi$
$503$	24.0208	1.07103	0.535516	+	0.844525i	$0.320117\pi$
$504$	44.6933	1.99080
$505$	6.39425	0.284540
$506$	2.23319	0.0992774
$507$	−2.77937	−0.123436
$508$	73.3065	3.25245
$509$	−16.6381	−0.737471	−0.368735	−	0.929534i	$-0.620209\pi$
$509$	−16.6381	−0.737471	−0.368735	+	0.929534i	$0.620209\pi$
$510$	−1.37420	−0.0608506
$511$	1.71960	0.0760706
$512$	50.8122	2.24561
$513$	4.39028	0.193836
$514$	−65.0483	−2.86916
$515$	−7.73715	−0.340939
$516$	5.97096	0.262857
$517$	8.38956	0.368972
$518$	13.0344	0.572699
$519$	−5.24675	−0.230307
$520$	0.548636	0.0240593
$521$	32.4123	1.42001	0.710005	−	0.704196i	$-0.248693\pi$
$521$	32.4123	1.42001	0.710005	+	0.704196i	$0.248693\pi$
$522$	−24.7511	−1.08333
$523$	38.0044	1.66182	0.830908	−	0.556410i	$-0.187821\pi$
$523$	38.0044	1.66182	0.830908	+	0.556410i	$0.187821\pi$
$524$	40.5908	1.77322
$525$	−2.52308	−0.110116
$526$	54.3624	2.37031
$527$	−28.4320	−1.23852
$528$	1.17934	0.0513241
$529$	−21.7681	−0.946438
$530$	9.08188	0.394492
$531$	−5.74769	−0.249428
$532$	−37.1246	−1.60956
$533$	0.0364598	0.00157925
$534$	6.88826	0.298084
$535$	−3.65252	−0.157912
$536$	−1.64464	−0.0710378
$537$	0.788378	0.0340210
$538$	13.7755	0.593906
$539$	−0.884652	−0.0381046
$540$	−2.22927	−0.0959324
$541$	14.3213	0.615723	0.307861	−	0.951431i	$-0.400387\pi$
$541$	14.3213	0.615723	0.307861	+	0.951431i	$0.400387\pi$
$542$	10.3696	0.445411
$543$	2.93860	0.126107
$544$	−33.1630	−1.42185
$545$	−1.81882	−0.0779095
$546$	0.297086	0.0127141
$547$	−34.6558	−1.48178	−0.740888	−	0.671629i	$-0.765595\pi$
$547$	−34.6558	−1.48178	−0.740888	+	0.671629i	$0.765595\pi$
$548$	30.5823	1.30641
$549$	−10.7159	−0.457344
$550$	9.75165	0.415812
$551$	11.3302	0.482681
$552$	1.48604	0.0632499
$553$	−37.3974	−1.59030
$554$	58.8265	2.49930
$555$	−0.177758	−0.00754541
$556$	−2.66509	−0.113025
$557$	−37.0238	−1.56875	−0.784375	−	0.620287i	$-0.787016\pi$
$557$	−37.0238	−1.56875	−0.784375	+	0.620287i	$0.787016\pi$
$558$	−33.1556	−1.40359
$559$	−1.40257	−0.0593223
$560$	6.58965	0.278464
$561$	1.09389	0.0461842
$562$	73.8671	3.11590
$563$	17.6376	0.743338	0.371669	−	0.928365i	$-0.378786\pi$
$563$	17.6376	0.743338	0.371669	+	0.928365i	$0.378786\pi$
$564$	10.1303	0.426562
$565$	−1.78574	−0.0751266
$566$	37.8252	1.58991
$567$	20.8288	0.874726
$568$	71.9553	3.01918
$569$	10.4343	0.437427	0.218714	−	0.975789i	$-0.429814\pi$
$569$	10.4343	0.437427	0.218714	+	0.975789i	$0.429814\pi$
$570$	0.733570	0.0307259
$571$	−13.8316	−0.578836	−0.289418	−	0.957203i	$-0.593462\pi$
$571$	−13.8316	−0.578836	−0.289418	+	0.957203i	$0.593462\pi$
$572$	−0.792478	−0.0331352
$573$	4.20540	0.175683
$574$	1.00029	0.0417512
$575$	5.37943	0.224338
$576$	2.31938	0.0966409
$577$	21.0131	0.874786	0.437393	−	0.899271i	$-0.355902\pi$
$577$	21.0131	0.874786	0.437393	+	0.899271i	$0.355902\pi$
$578$	−62.0446	−2.58071
$579$	−0.651835	−0.0270893
$580$	−5.75315	−0.238887
$581$	1.46411	0.0607414
$582$	8.60466	0.356674
$583$	−7.22937	−0.299410
$584$	−4.42260	−0.183009
$585$	0.259800	0.0107414
$586$	−51.3946	−2.12309
$587$	−32.9654	−1.36063	−0.680314	−	0.732921i	$-0.738157\pi$
$587$	−32.9654	−1.36063	−0.680314	+	0.732921i	$0.738157\pi$
$588$	−1.06821	−0.0440520
$589$	15.1774	0.625375
$590$	−1.93573	−0.0796929
$591$	−2.04174	−0.0839861
$592$	−14.6761	−0.603183
$593$	−13.1061	−0.538201	−0.269100	−	0.963112i	$-0.586726\pi$
$593$	−13.1061	−0.538201	−0.269100	+	0.963112i	$0.586726\pi$
$594$	2.57116	0.105496
$595$	6.11222	0.250577
$596$	81.6240	3.34345
$597$	−0.863933	−0.0353584
$598$	−0.633414	−0.0259022
$599$	−9.25726	−0.378241	−0.189121	−	0.981954i	$-0.560564\pi$
$599$	−9.25726	−0.378241	−0.189121	+	0.981954i	$0.560564\pi$
$600$	6.48906	0.264915
$601$	−40.1553	−1.63797	−0.818985	−	0.573816i	$-0.805463\pi$
$601$	−40.1553	−1.63797	−0.818985	+	0.573816i	$0.805463\pi$
$602$	−38.4800	−1.56833
$603$	−0.778800	−0.0317152
$604$	11.7420	0.477773
$605$	−4.06160	−0.165127
$606$	−8.90510	−0.361745
$607$	−10.9632	−0.444984	−0.222492	−	0.974935i	$-0.571419\pi$
$607$	−10.9632	−0.444984	−0.222492	+	0.974935i	$0.571419\pi$
$608$	17.7029	0.717949
$609$	−1.71682	−0.0695692
$610$	−3.60896	−0.146122
$611$	−2.37958	−0.0962676
$612$	−84.6985	−3.42374
$613$	−30.0595	−1.21409	−0.607046	−	0.794666i	$-0.707646\pi$
$613$	−30.0595	−1.21409	−0.607046	+	0.794666i	$0.707646\pi$
$614$	40.5831	1.63780
$615$	−0.0136415	−0.000550079 0
$616$	−11.9817	−0.482758
$617$	−31.6991	−1.27616	−0.638080	−	0.769970i	$-0.720271\pi$
$617$	−31.6991	−1.27616	−0.638080	+	0.769970i	$0.720271\pi$
$618$	10.7753	0.433447
$619$	5.39509	0.216847	0.108424	−	0.994105i	$-0.465420\pi$
$619$	5.39509	0.216847	0.108424	+	0.994105i	$0.465420\pi$
$620$	−7.70669	−0.309508
$621$	1.41836	0.0569169
$622$	−75.4292	−3.02444
$623$	−30.6378	−1.22748
$624$	−0.334503	−0.0133908
$625$	22.7237	0.908949
$626$	42.7457	1.70846
$627$	−0.583937	−0.0233202
$628$	55.3513	2.20876
$629$	−13.6128	−0.542776
$630$	7.12770	0.283975
$631$	49.2824	1.96190	0.980950	−	0.194261i	$-0.0622310\pi$
$631$	49.2824	1.96190	0.980950	+	0.194261i	$0.0622310\pi$
$632$	96.1816	3.82590
$633$	4.54069	0.180476
$634$	76.6809	3.04539
$635$	6.44275	0.255673
$636$	−8.72937	−0.346142
$637$	0.250919	0.00994178
$638$	6.63548	0.262701
$639$	34.0735	1.34793
$640$	4.81643	0.190386
$641$	−49.0894	−1.93892	−0.969458	−	0.245259i	$-0.921127\pi$
$641$	−49.0894	−1.93892	−0.969458	+	0.245259i	$0.921127\pi$
$642$	5.08676	0.200758
$643$	−8.37116	−0.330126	−0.165063	−	0.986283i	$-0.552783\pi$
$643$	−8.37116	−0.330126	−0.165063	+	0.986283i	$0.552783\pi$
$644$	−11.9938	−0.472622
$645$	0.524775	0.0206630
$646$	56.1770	2.21025
$647$	−37.4216	−1.47120	−0.735598	−	0.677418i	$-0.763099\pi$
$647$	−37.4216	−1.47120	−0.735598	+	0.677418i	$0.763099\pi$
$648$	−53.5691	−2.10439
$649$	1.54089	0.0604851
$650$	−2.76592	−0.108488
$651$	−2.29979	−0.0901358
$652$	−52.3006	−2.04825
$653$	−31.0133	−1.21365	−0.606823	−	0.794837i	$-0.707556\pi$
$653$	−31.0133	−1.21365	−0.606823	+	0.794837i	$0.707556\pi$
$654$	2.53302	0.0990487
$655$	3.56743	0.139391
$656$	−1.12627	−0.0439735
$657$	−2.09427	−0.0817052
$658$	−65.2848	−2.54507
$659$	−16.9813	−0.661498	−0.330749	−	0.943719i	$-0.607301\pi$
$659$	−16.9813	−0.661498	−0.330749	+	0.943719i	$0.607301\pi$
$660$	0.296508	0.0115416
$661$	−36.4268	−1.41684	−0.708420	−	0.705791i	$-0.750592\pi$
$661$	−36.4268	−1.41684	−0.708420	+	0.705791i	$0.750592\pi$
$662$	−9.53843	−0.370722
$663$	−0.310268	−0.0120498
$664$	−3.76551	−0.146130
$665$	−3.26280	−0.126526
$666$	−15.8744	−0.615120
$667$	3.66042	0.141732
$668$	1.84128	0.0712413
$669$	−2.95494	−0.114245
$670$	−0.262288	−0.0101331
$671$	2.87281	0.110904
$672$	−2.68247	−0.103479
$673$	43.2227	1.66611	0.833056	−	0.553189i	$-0.186589\pi$
$673$	43.2227	1.66611	0.833056	+	0.553189i	$0.186589\pi$
$674$	8.66066	0.333596
$675$	6.19355	0.238390
$676$	−57.6931	−2.21897
$677$	21.0828	0.810279	0.405140	−	0.914255i	$-0.367223\pi$
$677$	21.0828	0.810279	0.405140	+	0.914255i	$0.367223\pi$
$678$	2.48695	0.0955108
$679$	−38.2721	−1.46875
$680$	−15.7199	−0.602831
$681$	3.30932	0.126814
$682$	8.88863	0.340363
$683$	−0.839371	−0.0321177	−0.0160588	−	0.999871i	$-0.505112\pi$
$683$	−0.839371	−0.0321177	−0.0160588	+	0.999871i	$0.505112\pi$
$684$	45.2134	1.72878
$685$	2.68781	0.102696
$686$	50.0210	1.90981
$687$	0.762095	0.0290757
$688$	43.3265	1.65181
$689$	2.05051	0.0781183
$690$	0.236994	0.00902219
$691$	50.4450	1.91902	0.959509	−	0.281678i	$-0.0908910\pi$
$691$	50.4450	1.91902	0.959509	+	0.281678i	$0.0908910\pi$
$692$	−108.910	−4.14014
$693$	−5.67381	−0.215530
$694$	9.89599	0.375647
$695$	−0.234229	−0.00888482
$696$	4.41546	0.167368
$697$	−1.04467	−0.0395697
$698$	−53.1563	−2.01199
$699$	3.93255	0.148743
$700$	−52.3732	−1.97952
$701$	−40.0417	−1.51235	−0.756176	−	0.654368i	$-0.772935\pi$
$701$	−40.0417	−1.51235	−0.756176	+	0.654368i	$0.772935\pi$
$702$	−0.729274	−0.0275247
$703$	7.26670	0.274069
$704$	−0.621798	−0.0234349
$705$	0.890328	0.0335317
$706$	−91.4194	−3.44062
$707$	39.6084	1.48963
$708$	1.86060	0.0699256
$709$	−6.04532	−0.227037	−0.113518	−	0.993536i	$-0.536212\pi$
$709$	−6.04532	−0.227037	−0.113518	+	0.993536i	$0.536212\pi$
$710$	11.4755	0.430666
$711$	45.5456	1.70809
$712$	78.7969	2.95304
$713$	4.90335	0.183632
$714$	−8.51232	−0.318566
$715$	−0.0696491	−0.00260473
$716$	16.3649	0.611584
$717$	−2.31882	−0.0865981
$718$	−45.4833	−1.69742
$719$	−24.3609	−0.908506	−0.454253	−	0.890873i	$-0.650094\pi$
$719$	−24.3609	−0.908506	−0.454253	+	0.890873i	$0.650094\pi$
$720$	−8.02542	−0.299090
$721$	−47.9268	−1.78489
$722$	18.2854	0.680512
$723$	−2.54938	−0.0948124
$724$	60.9984	2.26699
$725$	15.9839	0.593628
$726$	5.65647	0.209931
$727$	37.0159	1.37284	0.686422	−	0.727203i	$-0.259180\pi$
$727$	37.0159	1.37284	0.686422	+	0.727203i	$0.259180\pi$
$728$	3.39846	0.125955
$729$	−24.5442	−0.909043
$730$	−0.705318	−0.0261050
$731$	40.1874	1.48638
$732$	3.46888	0.128213
$733$	15.0482	0.555818	0.277909	−	0.960607i	$-0.410359\pi$
$733$	15.0482	0.555818	0.277909	+	0.960607i	$0.410359\pi$
$734$	7.10434	0.262226
$735$	−0.0938822	−0.00346290
$736$	5.71926	0.210815
$737$	0.208787	0.00769077
$738$	−1.21823	−0.0448438
$739$	−5.94043	−0.218522	−0.109261	−	0.994013i	$-0.534848\pi$
$739$	−5.94043	−0.218522	−0.109261	+	0.994013i	$0.534848\pi$
$740$	−3.68984	−0.135641
$741$	0.165626	0.00608442
$742$	56.2566	2.06525
$743$	−43.6775	−1.60237	−0.801186	−	0.598416i	$-0.795797\pi$
$743$	−43.6775	−1.60237	−0.801186	+	0.598416i	$0.795797\pi$
$744$	5.91478	0.216846
$745$	7.17375	0.262826
$746$	42.5300	1.55714
$747$	−1.78311	−0.0652406
$748$	22.7067	0.830238
$749$	−22.6251	−0.826703
$750$	2.10249	0.0767722
$751$	−33.4089	−1.21911	−0.609554	−	0.792745i	$-0.708651\pi$
$751$	−33.4089	−1.21911	−0.609554	+	0.792745i	$0.708651\pi$
$752$	73.5073	2.68053
$753$	−3.94738	−0.143851
$754$	−1.88206	−0.0685407
$755$	1.03197	0.0375574
$756$	−13.8089	−0.502226
$757$	24.3025	0.883289	0.441644	−	0.897190i	$-0.354395\pi$
$757$	24.3025	0.883289	0.441644	+	0.897190i	$0.354395\pi$
$758$	32.0971	1.16582
$759$	−0.188652	−0.00684763
$760$	8.39153	0.304393
$761$	3.45817	0.125359	0.0626793	−	0.998034i	$-0.480035\pi$
$761$	3.45817	0.125359	0.0626793	+	0.998034i	$0.480035\pi$
$762$	−8.97263	−0.325044
$763$	−11.2664	−0.407872
$764$	87.2942	3.15819
$765$	−7.44396	−0.269137
$766$	−41.6250	−1.50397
$767$	−0.437051	−0.0157810
$768$	−6.37067	−0.229882
$769$	20.0275	0.722209	0.361104	−	0.932525i	$-0.382400\pi$
$769$	20.0275	0.722209	0.361104	+	0.932525i	$0.382400\pi$
$770$	−1.91085	−0.0688623
$771$	5.49504	0.197899
$772$	−13.5306	−0.486975
$773$	−26.9954	−0.970956	−0.485478	−	0.874249i	$-0.661354\pi$
$773$	−26.9954	−0.970956	−0.485478	+	0.874249i	$0.661354\pi$
$774$	46.8641	1.68450
$775$	21.4114	0.769121
$776$	98.4313	3.53348
$777$	−1.10110	−0.0395018
$778$	52.0954	1.86771
$779$	0.557662	0.0199803
$780$	−0.0841004	−0.00301128
$781$	−9.13471	−0.326866
$782$	18.1490	0.649008
$783$	4.21438	0.150610
$784$	−7.75110	−0.276825
$785$	4.86470	0.173629
$786$	−4.96826	−0.177212
$787$	27.2286	0.970594	0.485297	−	0.874349i	$-0.338711\pi$
$787$	27.2286	0.970594	0.485297	+	0.874349i	$0.338711\pi$
$788$	−42.3818	−1.50979
$789$	−4.59234	−0.163492
$790$	15.3391	0.545739
$791$	−11.0615	−0.393303
$792$	14.5924	0.518517
$793$	−0.814833	−0.0289356
$794$	20.5270	0.728474
$795$	−0.767205	−0.0272100
$796$	−17.9332	−0.635626
$797$	12.3893	0.438850	0.219425	−	0.975629i	$-0.429582\pi$
$797$	12.3893	0.438850	0.219425	+	0.975629i	$0.429582\pi$
$798$	4.54401	0.160856
$799$	68.1815	2.41209
$800$	24.9743	0.882974
$801$	37.3133	1.31840
$802$	−25.6003	−0.903980
$803$	0.561448	0.0198131
$804$	0.252108	0.00889115
$805$	−1.05411	−0.0371524
$806$	−2.52114	−0.0888033
$807$	−1.16371	−0.0409645
$808$	−101.868	−3.58371
$809$	34.9082	1.22731	0.613653	−	0.789576i	$-0.289700\pi$
$809$	34.9082	1.22731	0.613653	+	0.789576i	$0.289700\pi$
$810$	−8.54322	−0.300178
$811$	−3.75993	−0.132029	−0.0660146	−	0.997819i	$-0.521028\pi$
$811$	−3.75993	−0.132029	−0.0660146	+	0.997819i	$0.521028\pi$
$812$	−35.6372	−1.25062
$813$	−0.875984	−0.0307221
$814$	4.25573	0.149163
$815$	−4.59658	−0.161011
$816$	9.58443	0.335522
$817$	−21.4527	−0.750533
$818$	−35.4669	−1.24007
$819$	1.60930	0.0562334
$820$	−0.283166	−0.00988858
$821$	40.9579	1.42944	0.714720	−	0.699411i	$-0.246554\pi$
$821$	40.9579	1.42944	0.714720	+	0.699411i	$0.246554\pi$
$822$	−3.74323	−0.130560
$823$	−6.11435	−0.213133	−0.106566	−	0.994306i	$-0.533986\pi$
$823$	−6.11435	−0.213133	−0.106566	+	0.994306i	$0.533986\pi$
$824$	123.262	4.29404
$825$	−0.823785	−0.0286805
$826$	−11.9907	−0.417209
$827$	15.6551	0.544381	0.272190	−	0.962243i	$-0.412252\pi$
$827$	15.6551	0.544381	0.272190	+	0.962243i	$0.412252\pi$
$828$	14.6070	0.507630
$829$	31.1912	1.08332	0.541658	−	0.840599i	$-0.317797\pi$
$829$	31.1912	1.08332	0.541658	+	0.840599i	$0.317797\pi$
$830$	−0.600524	−0.0208445
$831$	−4.96945	−0.172388
$832$	0.176365	0.00611434
$833$	−7.18952	−0.249102
$834$	0.326205	0.0112955
$835$	0.161826	0.00560022
$836$	−12.1212	−0.419219
$837$	5.64542	0.195134
$838$	46.1492	1.59420
$839$	32.7938	1.13217	0.566084	−	0.824347i	$-0.308458\pi$
$839$	32.7938	1.13217	0.566084	+	0.824347i	$0.308458\pi$
$840$	−1.27154	−0.0438724
$841$	−18.1238	−0.624958
$842$	−3.74815	−0.129170
$843$	−6.24003	−0.214918
$844$	94.2541	3.24436
$845$	−5.07052	−0.174431
$846$	79.5092	2.73358
$847$	−25.1591	−0.864476
$848$	−63.3420	−2.17517
$849$	−3.19533	−0.109664
$850$	79.2512	2.71829
$851$	2.34764	0.0804762
$852$	−11.0300	−0.377883
$853$	−23.3916	−0.800912	−0.400456	−	0.916316i	$-0.631148\pi$
$853$	−23.3916	−0.800912	−0.400456	+	0.916316i	$0.631148\pi$
$854$	−22.3553	−0.764981
$855$	3.97370	0.135898
$856$	58.1890	1.98886
$857$	−21.2928	−0.727348	−0.363674	−	0.931526i	$-0.618478\pi$
$857$	−21.2928	−0.727348	−0.363674	+	0.931526i	$0.618478\pi$
$858$	0.0969984	0.00331147
$859$	12.8661	0.438986	0.219493	−	0.975614i	$-0.429560\pi$
$859$	12.8661	0.438986	0.219493	+	0.975614i	$0.429560\pi$
$860$	10.8931	0.371451
$861$	−0.0845008	−0.00287978
$862$	−30.8538	−1.05089
$863$	3.38978	0.115389	0.0576947	−	0.998334i	$-0.481625\pi$
$863$	3.38978	0.115389	0.0576947	+	0.998334i	$0.481625\pi$
$864$	6.58481	0.224020
$865$	−9.57187	−0.325453
$866$	−21.7413	−0.738800
$867$	5.24130	0.178004
$868$	−47.7382	−1.62034
$869$	−12.2102	−0.414203
$870$	0.704179	0.0238739
$871$	−0.0592196	−0.00200658
$872$	28.9759	0.981249
$873$	46.6109	1.57754
$874$	−9.68823	−0.327709
$875$	−9.35154	−0.316140
$876$	0.677941	0.0229055
$877$	52.0578	1.75787	0.878934	−	0.476943i	$-0.158255\pi$
$877$	52.0578	1.75787	0.878934	+	0.476943i	$0.158255\pi$
$878$	2.91834	0.0984891
$879$	4.34163	0.146439
$880$	2.15152	0.0725277
$881$	37.0489	1.24821	0.624105	−	0.781341i	$-0.285464\pi$
$881$	37.0489	1.24821	0.624105	+	0.781341i	$0.285464\pi$
$882$	−8.38399	−0.282303
$883$	−7.16094	−0.240985	−0.120492	−	0.992714i	$-0.538447\pi$
$883$	−7.16094	−0.240985	−0.120492	+	0.992714i	$0.538447\pi$
$884$	−6.44043	−0.216615
$885$	0.163524	0.00549680
$886$	−25.7930	−0.866533
$887$	−18.6853	−0.627392	−0.313696	−	0.949523i	$-0.601567\pi$
$887$	−18.6853	−0.627392	−0.313696	+	0.949523i	$0.601567\pi$
$888$	2.83190	0.0950323
$889$	39.9088	1.33850
$890$	12.5665	0.421232
$891$	6.80058	0.227828
$892$	−61.3376	−2.05373
$893$	−36.3964	−1.21796
$894$	−9.99069	−0.334139
$895$	1.43827	0.0480761
$896$	29.8348	0.996711
$897$	0.0535085	0.00178660
$898$	82.1804	2.74240
$899$	14.5693	0.485915
$900$	63.7844	2.12615
$901$	−58.7528	−1.95734
$902$	0.326594	0.0108744
$903$	3.25065	0.108175
$904$	28.4490	0.946199
$905$	5.36102	0.178206
$906$	−1.43720	−0.0477478
$907$	−8.95329	−0.297289	−0.148645	−	0.988891i	$-0.547491\pi$
$907$	−8.95329	−0.297289	−0.148645	+	0.988891i	$0.547491\pi$
$908$	68.6938	2.27968
$909$	−48.2384	−1.59997
$910$	0.541987	0.0179667
$911$	46.5097	1.54093	0.770467	−	0.637479i	$-0.220023\pi$
$911$	46.5097	1.54093	0.770467	+	0.637479i	$0.220023\pi$
$912$	−5.11632	−0.169418
$913$	0.478030	0.0158205
$914$	42.2015	1.39590
$915$	0.304872	0.0100788
$916$	15.8193	0.522684
$917$	22.0980	0.729741
$918$	20.8957	0.689660
$919$	−21.0586	−0.694660	−0.347330	−	0.937743i	$-0.612912\pi$
$919$	−21.0586	−0.694660	−0.347330	+	0.937743i	$0.612912\pi$
$920$	2.71104	0.0893804
$921$	−3.42831	−0.112967
$922$	85.4566	2.81436
$923$	2.59093	0.0852816
$924$	1.83668	0.0604224
$925$	10.2514	0.337065
$926$	22.8335	0.750355
$927$	58.3692	1.91710
$928$	16.9937	0.557844
$929$	27.4628	0.901024	0.450512	−	0.892770i	$-0.351241\pi$
$929$	27.4628	0.901024	0.450512	+	0.892770i	$0.351241\pi$
$930$	0.943291	0.0309317
$931$	3.83788	0.125781
$932$	81.6306	2.67390
$933$	6.37199	0.208610
$934$	101.968	3.33651
$935$	1.99564	0.0652643
$936$	−4.13892	−0.135285
$937$	19.8094	0.647144	0.323572	−	0.946204i	$-0.395116\pi$
$937$	19.8094	0.647144	0.323572	+	0.946204i	$0.395116\pi$
$938$	−1.62471	−0.0530487
$939$	−3.61100	−0.117841
$940$	18.4811	0.602787
$941$	−26.2425	−0.855481	−0.427740	−	0.903902i	$-0.640690\pi$
$941$	−26.2425	−0.855481	−0.427740	+	0.903902i	$0.640690\pi$
$942$	−6.77494	−0.220739
$943$	0.180163	0.00586692
$944$	13.5009	0.439416
$945$	−1.21364	−0.0394796
$946$	−12.5637	−0.408481
$947$	23.2276	0.754797	0.377398	−	0.926051i	$-0.376819\pi$
$947$	23.2276	0.754797	0.377398	+	0.926051i	$0.376819\pi$
$948$	−14.7437	−0.478853
$949$	−0.159247	−0.00516938
$950$	−42.3055	−1.37257
$951$	−6.47773	−0.210055
$952$	−97.3750	−3.15594
$953$	27.4814	0.890211	0.445105	−	0.895478i	$-0.353166\pi$
$953$	27.4814	0.890211	0.445105	+	0.895478i	$0.353166\pi$
$954$	−68.5139	−2.21822
$955$	7.67209	0.248263
$956$	−48.1334	−1.55674
$957$	−0.560542	−0.0181197
$958$	−32.8671	−1.06189
$959$	16.6493	0.537634
$960$	−0.0659873	−0.00212973
$961$	−11.4835	−0.370435
$962$	−1.20708	−0.0389178
$963$	27.5547	0.887937
$964$	−52.9191	−1.70441
$965$	−1.18917	−0.0382807
$966$	1.46803	0.0472330
$967$	−33.6548	−1.08227	−0.541133	−	0.840937i	$-0.682004\pi$
$967$	−33.6548	−1.08227	−0.541133	+	0.840937i	$0.682004\pi$
$968$	64.7061	2.07973
$969$	−4.74563	−0.152452
$970$	15.6978	0.504027
$971$	4.64984	0.149220	0.0746102	−	0.997213i	$-0.476229\pi$
$971$	4.64984	0.149220	0.0746102	+	0.997213i	$0.476229\pi$
$972$	25.2915	0.811226
$973$	−1.45090	−0.0465139
$974$	−14.1828	−0.454447
$975$	0.233655	0.00748296
$976$	25.1708	0.805699
$977$	17.9513	0.574314	0.287157	−	0.957884i	$-0.407290\pi$
$977$	17.9513	0.574314	0.287157	+	0.957884i	$0.407290\pi$
$978$	6.40153	0.204698
$979$	−10.0032	−0.319705
$980$	−1.94877	−0.0622513
$981$	13.7212	0.438084
$982$	90.9217	2.90143
$983$	21.5277	0.686626	0.343313	−	0.939221i	$-0.388451\pi$
$983$	21.5277	0.686626	0.343313	+	0.939221i	$0.388451\pi$
$984$	0.217326	0.00692810
$985$	−3.72484	−0.118683
$986$	53.9262	1.71736
$987$	5.51503	0.175545
$988$	3.43800	0.109377
$989$	−6.93068	−0.220383
$990$	2.32719	0.0739630
$991$	−57.7511	−1.83452	−0.917262	−	0.398284i	$-0.869606\pi$
$991$	−57.7511	−1.83452	−0.917262	+	0.398284i	$0.869606\pi$
$992$	22.7640	0.722759
$993$	0.805773	0.0255704
$994$	71.0833	2.25463
$995$	−1.57611	−0.0499661
$996$	0.577215	0.0182898
$997$	−8.28309	−0.262328	−0.131164	−	0.991361i	$-0.541871\pi$
$997$	−8.28309	−0.262328	−0.131164	+	0.991361i	$0.541871\pi$
$998$	20.2460	0.640876
$999$	2.70293	0.0855170

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4013.2.a.b.1.9	✓	157

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4013.2.a.b.1.9	✓	157	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:	\( N \)	\(=\)	\( 4013 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4013.a (trivial)

\(f(q)\)	\(=\)	\(q-2.54071 q^{2} +0.214630 q^{3} +4.45522 q^{4} +0.391559 q^{5} -0.545314 q^{6} +2.42547 q^{7} -6.23801 q^{8} -2.95393 q^{9} +O(q^{10})\)	\(q-2.54071 q^{2} +0.214630 q^{3} +4.45522 q^{4} +0.391559 q^{5} -0.545314 q^{6} +2.42547 q^{7} -6.23801 q^{8} -2.95393 q^{9} -0.994840 q^{10} +0.791914 q^{11} +0.956226 q^{12} -0.224616 q^{13} -6.16242 q^{14} +0.0840406 q^{15} +6.93856 q^{16} +6.43585 q^{17} +7.50510 q^{18} -3.43556 q^{19} +1.74448 q^{20} +0.520579 q^{21} -2.01203 q^{22} -1.10992 q^{23} -1.33887 q^{24} -4.84668 q^{25} +0.570684 q^{26} -1.27789 q^{27} +10.8060 q^{28} -3.29791 q^{29} -0.213523 q^{30} -4.41775 q^{31} -5.15286 q^{32} +0.169969 q^{33} -16.3516 q^{34} +0.949715 q^{35} -13.1604 q^{36} -2.11515 q^{37} +8.72876 q^{38} -0.0482093 q^{39} -2.44255 q^{40} -0.162321 q^{41} -1.32264 q^{42} +6.24430 q^{43} +3.52815 q^{44} -1.15664 q^{45} +2.81999 q^{46} +10.5940 q^{47} +1.48923 q^{48} -1.11711 q^{49} +12.3140 q^{50} +1.38133 q^{51} -1.00071 q^{52} -9.12899 q^{53} +3.24676 q^{54} +0.310082 q^{55} -15.1301 q^{56} -0.737375 q^{57} +8.37904 q^{58} +1.94577 q^{59} +0.374419 q^{60} +3.62768 q^{61} +11.2242 q^{62} -7.16467 q^{63} -0.785184 q^{64} -0.0879504 q^{65} -0.431842 q^{66} +0.263649 q^{67} +28.6731 q^{68} -0.238223 q^{69} -2.41295 q^{70} -11.5350 q^{71} +18.4267 q^{72} +0.708976 q^{73} +5.37398 q^{74} -1.04024 q^{75} -15.3062 q^{76} +1.92076 q^{77} +0.122486 q^{78} -15.4186 q^{79} +2.71686 q^{80} +8.58753 q^{81} +0.412410 q^{82} +0.603639 q^{83} +2.31930 q^{84} +2.52002 q^{85} -15.8650 q^{86} -0.707832 q^{87} -4.93997 q^{88} -12.6317 q^{89} +2.93869 q^{90} -0.544798 q^{91} -4.94494 q^{92} -0.948183 q^{93} -26.9164 q^{94} -1.34522 q^{95} -1.10596 q^{96} -15.7793 q^{97} +2.83824 q^{98} -2.33926 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 157 q - 15 q^{2} - 51 q^{3} + 137 q^{4} - 13 q^{5} - 15 q^{6} - 49 q^{7} - 42 q^{8} + 144 q^{9}+O(q^{10}) \) 157 * q - 15 * q^2 - 51 * q^3 + 137 * q^4 - 13 * q^5 - 15 * q^6 - 49 * q^7 - 42 * q^8 + 144 * q^9	\( 157 q - 15 q^{2} - 51 q^{3} + 137 q^{4} - 13 q^{5} - 15 q^{6} - 49 q^{7} - 42 q^{8} + 144 q^{9} - 61 q^{10} - 27 q^{11} - 93 q^{12} - 97 q^{13} - 12 q^{14} - 36 q^{15} + 105 q^{16} - 45 q^{17} - 68 q^{18} - 128 q^{19} - 30 q^{20} - 26 q^{21} - 68 q^{22} - 41 q^{23} - 40 q^{24} + 102 q^{25} - 5 q^{26} - 189 q^{27} - 115 q^{28} - 26 q^{29} - 12 q^{30} - 88 q^{31} - 89 q^{32} - 52 q^{33} - 61 q^{34} - 87 q^{35} + 110 q^{36} - 62 q^{37} - 37 q^{38} - 20 q^{39} - 161 q^{40} - 34 q^{41} - 53 q^{42} - 254 q^{43} - 19 q^{44} - 46 q^{45} - 52 q^{46} - 76 q^{47} - 162 q^{48} + 96 q^{49} - 54 q^{50} - 76 q^{51} - 259 q^{52} - 48 q^{53} - 12 q^{54} - 194 q^{55} - 10 q^{56} - 30 q^{57} - 52 q^{58} - 64 q^{59} - 31 q^{60} - 107 q^{61} - 51 q^{62} - 106 q^{63} + 54 q^{64} - 17 q^{65} - 13 q^{66} - 193 q^{67} - 118 q^{68} - 55 q^{69} - 86 q^{70} - 11 q^{71} - 172 q^{72} - 173 q^{73} - 11 q^{74} - 209 q^{75} - 213 q^{76} - 84 q^{77} - 30 q^{78} - 111 q^{79} - 6 q^{80} + 157 q^{81} - 117 q^{82} - 154 q^{83} - 6 q^{84} - 91 q^{85} + 28 q^{86} - 165 q^{87} - 165 q^{88} - 32 q^{89} - 103 q^{90} - 200 q^{91} - 86 q^{92} - 39 q^{93} - 118 q^{94} - 22 q^{95} - 28 q^{96} - 151 q^{97} - 38 q^{98} - 91 q^{99}+O(q^{100}) \) 157 * q - 15 * q^2 - 51 * q^3 + 137 * q^4 - 13 * q^5 - 15 * q^6 - 49 * q^7 - 42 * q^8 + 144 * q^9 - 61 * q^10 - 27 * q^11 - 93 * q^12 - 97 * q^13 - 12 * q^14 - 36 * q^15 + 105 * q^16 - 45 * q^17 - 68 * q^18 - 128 * q^19 - 30 * q^20 - 26 * q^21 - 68 * q^22 - 41 * q^23 - 40 * q^24 + 102 * q^25 - 5 * q^26 - 189 * q^27 - 115 * q^28 - 26 * q^29 - 12 * q^30 - 88 * q^31 - 89 * q^32 - 52 * q^33 - 61 * q^34 - 87 * q^35 + 110 * q^36 - 62 * q^37 - 37 * q^38 - 20 * q^39 - 161 * q^40 - 34 * q^41 - 53 * q^42 - 254 * q^43 - 19 * q^44 - 46 * q^45 - 52 * q^46 - 76 * q^47 - 162 * q^48 + 96 * q^49 - 54 * q^50 - 76 * q^51 - 259 * q^52 - 48 * q^53 - 12 * q^54 - 194 * q^55 - 10 * q^56 - 30 * q^57 - 52 * q^58 - 64 * q^59 - 31 * q^60 - 107 * q^61 - 51 * q^62 - 106 * q^63 + 54 * q^64 - 17 * q^65 - 13 * q^66 - 193 * q^67 - 118 * q^68 - 55 * q^69 - 86 * q^70 - 11 * q^71 - 172 * q^72 - 173 * q^73 - 11 * q^74 - 209 * q^75 - 213 * q^76 - 84 * q^77 - 30 * q^78 - 111 * q^79 - 6 * q^80 + 157 * q^81 - 117 * q^82 - 154 * q^83 - 6 * q^84 - 91 * q^85 + 28 * q^86 - 165 * q^87 - 165 * q^88 - 32 * q^89 - 103 * q^90 - 200 * q^91 - 86 * q^92 - 39 * q^93 - 118 * q^94 - 22 * q^95 - 28 * q^96 - 151 * q^97 - 38 * q^98 - 91 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more