LMFDB - Embedded newform 309.2.a.c.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [309,2,Mod(1,309)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(309, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("309.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 309 = 3 \cdot 103 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	309.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:	$2.46737742246$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	5.5.81509.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 5x^{3} + 3x^{2} + 5x - 2 $ x^5 - x^4 - 5x^3 + 3x^2 + 5*x - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$2.26835$ of defining polynomial
Character	$\chi$	$=$	309.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.721159 q^{2} -1.00000 q^{3} -1.47993 q^{4} -1.11830 q^{5} -0.721159 q^{6} +2.30134 q^{7} -2.50958 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+0.721159 q^{2} -1.00000 q^{3} -1.47993 q^{4} -1.11830 q^{5} -0.721159 q^{6} +2.30134 q^{7} -2.50958 q^{8} +1.00000 q^{9} -0.806475 q^{10} -6.53671 q^{11} +1.47993 q^{12} -6.26120 q^{13} +1.65963 q^{14} +1.11830 q^{15} +1.15005 q^{16} +1.56636 q^{17} +0.721159 q^{18} +0.107816 q^{19} +1.65501 q^{20} -2.30134 q^{21} -4.71401 q^{22} -4.56636 q^{23} +2.50958 q^{24} -3.74940 q^{25} -4.51532 q^{26} -1.00000 q^{27} -3.40582 q^{28} +1.74366 q^{29} +0.806475 q^{30} +1.65501 q^{31} +5.84854 q^{32} +6.53671 q^{33} +1.12959 q^{34} -2.57360 q^{35} -1.47993 q^{36} +9.42134 q^{37} +0.0777522 q^{38} +6.26120 q^{39} +2.80647 q^{40} +2.71974 q^{41} -1.65963 q^{42} +10.2076 q^{43} +9.67386 q^{44} -1.11830 q^{45} -3.29307 q^{46} -4.29085 q^{47} -1.15005 q^{48} -1.70383 q^{49} -2.70391 q^{50} -1.56636 q^{51} +9.26613 q^{52} -9.38524 q^{53} -0.721159 q^{54} +7.31002 q^{55} -5.77541 q^{56} -0.107816 q^{57} +1.25746 q^{58} -7.34442 q^{59} -1.65501 q^{60} +3.40792 q^{61} +1.19353 q^{62} +2.30134 q^{63} +1.91762 q^{64} +7.00192 q^{65} +4.71401 q^{66} -9.46340 q^{67} -2.31810 q^{68} +4.56636 q^{69} -1.85597 q^{70} -5.38301 q^{71} -2.50958 q^{72} +7.50829 q^{73} +6.79429 q^{74} +3.74940 q^{75} -0.159560 q^{76} -15.0432 q^{77} +4.51532 q^{78} +1.72076 q^{79} -1.28610 q^{80} +1.00000 q^{81} +1.96137 q^{82} -13.2641 q^{83} +3.40582 q^{84} -1.75167 q^{85} +7.36133 q^{86} -1.74366 q^{87} +16.4044 q^{88} +14.2632 q^{89} -0.806475 q^{90} -14.4092 q^{91} +6.75789 q^{92} -1.65501 q^{93} -3.09439 q^{94} -0.120571 q^{95} -5.84854 q^{96} -13.1050 q^{97} -1.22873 q^{98} -6.53671 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 2 q^{2} - 5 q^{3} + 2 q^{4} - 5 q^{5} + 2 q^{6} - 2 q^{7} - 6 q^{8} + 5 q^{9}+O(q^{10}) $ 5 * q - 2 * q^2 - 5 * q^3 + 2 * q^4 - 5 * q^5 + 2 * q^6 - 2 * q^7 - 6 * q^8 + 5 * q^9	\( 5 q - 2 q^{2} - 5 q^{3} + 2 q^{4} - 5 q^{5} + 2 q^{6} - 2 q^{7} - 6 q^{8} + 5 q^{9} - q^{10} - 12 q^{11} - 2 q^{12} + q^{13} - 8 q^{14} + 5 q^{15} - 4 q^{16} - 10 q^{17} - 2 q^{18} - 16 q^{19} - 13 q^{20} + 2 q^{21} + 4 q^{22} - 5 q^{23} + 6 q^{24} + 12 q^{25} - 10 q^{26} - 5 q^{27} + 7 q^{28} - 16 q^{29} + q^{30} - 13 q^{31} + 11 q^{32} + 12 q^{33} - 2 q^{34} - 22 q^{35} + 2 q^{36} + 4 q^{37} + 21 q^{38} - q^{39} + 11 q^{40} - 20 q^{41} + 8 q^{42} + 7 q^{43} + 2 q^{44} - 5 q^{45} + 8 q^{46} + 8 q^{47} + 4 q^{48} + 23 q^{49} + 13 q^{50} + 10 q^{51} + 31 q^{52} - 8 q^{53} + 2 q^{54} - 6 q^{55} + 5 q^{56} + 16 q^{57} + 20 q^{58} - 19 q^{59} + 13 q^{60} - 19 q^{61} + 9 q^{62} - 2 q^{63} - 16 q^{64} + q^{65} - 4 q^{66} + 11 q^{67} + 15 q^{68} + 5 q^{69} + 44 q^{70} - 10 q^{71} - 6 q^{72} + 20 q^{73} + 44 q^{74} - 12 q^{75} - 8 q^{76} + 8 q^{77} + 10 q^{78} - 14 q^{79} + 45 q^{80} + 5 q^{81} - 3 q^{82} - 11 q^{83} - 7 q^{84} + 12 q^{85} - 11 q^{86} + 16 q^{87} + 30 q^{88} + 22 q^{89} - q^{90} - 42 q^{91} - 21 q^{92} + 13 q^{93} - 6 q^{94} - 10 q^{95} - 11 q^{96} + 7 q^{97} - 11 q^{98} - 12 q^{99}+O(q^{100}) \) 5 * q - 2 * q^2 - 5 * q^3 + 2 * q^4 - 5 * q^5 + 2 * q^6 - 2 * q^7 - 6 * q^8 + 5 * q^9 - q^10 - 12 * q^11 - 2 * q^12 + q^13 - 8 * q^14 + 5 * q^15 - 4 * q^16 - 10 * q^17 - 2 * q^18 - 16 * q^19 - 13 * q^20 + 2 * q^21 + 4 * q^22 - 5 * q^23 + 6 * q^24 + 12 * q^25 - 10 * q^26 - 5 * q^27 + 7 * q^28 - 16 * q^29 + q^30 - 13 * q^31 + 11 * q^32 + 12 * q^33 - 2 * q^34 - 22 * q^35 + 2 * q^36 + 4 * q^37 + 21 * q^38 - q^39 + 11 * q^40 - 20 * q^41 + 8 * q^42 + 7 * q^43 + 2 * q^44 - 5 * q^45 + 8 * q^46 + 8 * q^47 + 4 * q^48 + 23 * q^49 + 13 * q^50 + 10 * q^51 + 31 * q^52 - 8 * q^53 + 2 * q^54 - 6 * q^55 + 5 * q^56 + 16 * q^57 + 20 * q^58 - 19 * q^59 + 13 * q^60 - 19 * q^61 + 9 * q^62 - 2 * q^63 - 16 * q^64 + q^65 - 4 * q^66 + 11 * q^67 + 15 * q^68 + 5 * q^69 + 44 * q^70 - 10 * q^71 - 6 * q^72 + 20 * q^73 + 44 * q^74 - 12 * q^75 - 8 * q^76 + 8 * q^77 + 10 * q^78 - 14 * q^79 + 45 * q^80 + 5 * q^81 - 3 * q^82 - 11 * q^83 - 7 * q^84 + 12 * q^85 - 11 * q^86 + 16 * q^87 + 30 * q^88 + 22 * q^89 - q^90 - 42 * q^91 - 21 * q^92 + 13 * q^93 - 6 * q^94 - 10 * q^95 - 11 * q^96 + 7 * q^97 - 11 * q^98 - 12 * 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 $

$2$	0.721159	0.509937	0.254968	−	0.966949i	$-0.417935\pi$
$2$	0.721159	0.509937	0.254968	+	0.966949i	$0.417935\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−1.47993	−0.739965
$5$	−1.11830	−0.500120	−0.250060	−	0.968230i	$-0.580450\pi$
$5$	−1.11830	−0.500120	−0.250060	+	0.968230i	$0.580450\pi$
$6$	−0.721159	−0.294412
$7$	2.30134	0.869825	0.434913	−	0.900473i	$-0.356779\pi$
$7$	2.30134	0.869825	0.434913	+	0.900473i	$0.356779\pi$
$8$	−2.50958	−0.887272
$9$	1.00000	0.333333
$10$	−0.806475	−0.255030
$11$	−6.53671	−1.97089	−0.985446	−	0.169991i	$-0.945626\pi$
$11$	−6.53671	−1.97089	−0.985446	+	0.169991i	$0.945626\pi$
$12$	1.47993	0.427219
$13$	−6.26120	−1.73654	−0.868272	−	0.496088i	$-0.834769\pi$
$13$	−6.26120	−1.73654	−0.868272	+	0.496088i	$0.834769\pi$
$14$	1.65963	0.443556
$15$	1.11830	0.288745
$16$	1.15005	0.287513
$17$	1.56636	0.379898	0.189949	−	0.981794i	$-0.439168\pi$
$17$	1.56636	0.379898	0.189949	+	0.981794i	$0.439168\pi$
$18$	0.721159	0.169979
$19$	0.107816	0.0247346	0.0123673	−	0.999924i	$-0.496063\pi$
$19$	0.107816	0.0247346	0.0123673	+	0.999924i	$0.496063\pi$
$20$	1.65501	0.370071
$21$	−2.30134	−0.502194
$22$	−4.71401	−1.00503
$23$	−4.56636	−0.952152	−0.476076	−	0.879404i	$-0.657941\pi$
$23$	−4.56636	−0.952152	−0.476076	+	0.879404i	$0.657941\pi$
$24$	2.50958	0.512266
$25$	−3.74940	−0.749880
$26$	−4.51532	−0.885527
$27$	−1.00000	−0.192450
$28$	−3.40582	−0.643640
$29$	1.74366	0.323789	0.161895	−	0.986808i	$-0.448240\pi$
$29$	1.74366	0.323789	0.161895	+	0.986808i	$0.448240\pi$
$30$	0.806475	0.147241
$31$	1.65501	0.297249	0.148624	−	0.988894i	$-0.452516\pi$
$31$	1.65501	0.297249	0.148624	+	0.988894i	$0.452516\pi$
$32$	5.84854	1.03388
$33$	6.53671	1.13789
$34$	1.12959	0.193724
$35$	−2.57360	−0.435017
$36$	−1.47993	−0.246655
$37$	9.42134	1.54886	0.774430	−	0.632660i	$-0.218037\pi$
$37$	9.42134	1.54886	0.774430	+	0.632660i	$0.218037\pi$
$38$	0.0777522	0.0126131
$39$	6.26120	1.00259
$40$	2.80647	0.443743
$41$	2.71974	0.424753	0.212376	−	0.977188i	$-0.431880\pi$
$41$	2.71974	0.424753	0.212376	+	0.977188i	$0.431880\pi$
$42$	−1.65963	−0.256087
$43$	10.2076	1.55665	0.778325	−	0.627862i	$-0.216070\pi$
$43$	10.2076	1.55665	0.778325	+	0.627862i	$0.216070\pi$
$44$	9.67386	1.45839
$45$	−1.11830	−0.166707
$46$	−3.29307	−0.485537
$47$	−4.29085	−0.625885	−0.312943	−	0.949772i	$-0.601315\pi$
$47$	−4.29085	−0.625885	−0.312943	+	0.949772i	$0.601315\pi$
$48$	−1.15005	−0.165995
$49$	−1.70383	−0.243404
$50$	−2.70391	−0.382391
$51$	−1.56636	−0.219334
$52$	9.26613	1.28498
$53$	−9.38524	−1.28916	−0.644581	−	0.764536i	$-0.722968\pi$
$53$	−9.38524	−1.28916	−0.644581	+	0.764536i	$0.722968\pi$
$54$	−0.721159	−0.0981373
$55$	7.31002	0.985683
$56$	−5.77541	−0.771771
$57$	−0.107816	−0.0142805
$58$	1.25746	0.165112
$59$	−7.34442	−0.956162	−0.478081	−	0.878316i	$-0.658668\pi$
$59$	−7.34442	−0.956162	−0.478081	+	0.878316i	$0.658668\pi$
$60$	−1.65501	−0.213661
$61$	3.40792	0.436339	0.218169	−	0.975911i	$-0.429991\pi$
$61$	3.40792	0.436339	0.218169	+	0.975911i	$0.429991\pi$
$62$	1.19353	0.151578
$63$	2.30134	0.289942
$64$	1.91762	0.239703
$65$	7.00192	0.868481
$66$	4.71401	0.580254
$67$	−9.46340	−1.15614	−0.578069	−	0.815988i	$-0.696194\pi$
$67$	−9.46340	−1.15614	−0.578069	+	0.815988i	$0.696194\pi$
$68$	−2.31810	−0.281111
$69$	4.56636	0.549725
$70$	−1.85597	−0.221831
$71$	−5.38301	−0.638846	−0.319423	−	0.947612i	$-0.603489\pi$
$71$	−5.38301	−0.638846	−0.319423	+	0.947612i	$0.603489\pi$
$72$	−2.50958	−0.295757
$73$	7.50829	0.878779	0.439390	−	0.898297i	$-0.355195\pi$
$73$	7.50829	0.878779	0.439390	+	0.898297i	$0.355195\pi$
$74$	6.79429	0.789820
$75$	3.74940	0.432943
$76$	−0.159560	−0.0183027
$77$	−15.0432	−1.71433
$78$	4.51532	0.511260
$79$	1.72076	0.193601	0.0968006	−	0.995304i	$-0.469139\pi$
$79$	1.72076	0.193601	0.0968006	+	0.995304i	$0.469139\pi$
$80$	−1.28610	−0.143791
$81$	1.00000	0.111111
$82$	1.96137	0.216597
$83$	−13.2641	−1.45593	−0.727964	−	0.685615i	$-0.759533\pi$
$83$	−13.2641	−1.45593	−0.727964	+	0.685615i	$0.759533\pi$
$84$	3.40582	0.371606
$85$	−1.75167	−0.189995
$86$	7.36133	0.793792
$87$	−1.74366	−0.186940
$88$	16.4044	1.74872
$89$	14.2632	1.51190	0.755949	−	0.654631i	$-0.227176\pi$
$89$	14.2632	1.51190	0.755949	+	0.654631i	$0.227176\pi$
$90$	−0.806475	−0.0850099
$91$	−14.4092	−1.51049
$92$	6.75789	0.704559
$93$	−1.65501	−0.171617
$94$	−3.09439	−0.319162
$95$	−0.120571	−0.0123703
$96$	−5.84854	−0.596914
$97$	−13.1050	−1.33061	−0.665305	−	0.746572i	$-0.731698\pi$
$97$	−13.1050	−1.33061	−0.665305	+	0.746572i	$0.731698\pi$
$98$	−1.22873	−0.124121
$99$	−6.53671	−0.656964
$100$	5.54884	0.554884
$101$	−3.11163	−0.309619	−0.154810	−	0.987944i	$-0.549476\pi$
$101$	−3.11163	−0.309619	−0.154810	+	0.987944i	$0.549476\pi$
$102$	−1.12959	−0.111847
$103$	−1.00000	−0.0985329
$103$	−1.00000	−0.0985329
$104$	15.7130	1.54079
$105$	2.57360	0.251157
$106$	−6.76825	−0.657391
$107$	1.98275	0.191680	0.0958400	−	0.995397i	$-0.469446\pi$
$107$	1.98275	0.191680	0.0958400	+	0.995397i	$0.469446\pi$
$108$	1.47993	0.142406
$109$	−6.95986	−0.666634	−0.333317	−	0.942815i	$-0.608168\pi$
$109$	−6.95986	−0.666634	−0.333317	+	0.942815i	$0.608168\pi$
$110$	5.27169	0.502636
$111$	−9.42134	−0.894234
$112$	2.64666	0.250086
$113$	−0.258040	−0.0242743	−0.0121372	−	0.999926i	$-0.503863\pi$
$113$	−0.258040	−0.0242743	−0.0121372	+	0.999926i	$0.503863\pi$
$114$	−0.0777522	−0.00728216
$115$	5.10657	0.476191
$116$	−2.58049	−0.239593
$117$	−6.26120	−0.578848
$118$	−5.29650	−0.487582
$119$	3.60473	0.330445
$120$	−2.80647	−0.256195
$121$	31.7285	2.88441
$122$	2.45765	0.222505
$123$	−2.71974	−0.245231
$124$	−2.44930	−0.219953
$125$	9.78448	0.875150
$126$	1.65963	0.147852
$127$	−2.42244	−0.214957	−0.107479	−	0.994207i	$-0.534278\pi$
$127$	−2.42244	−0.214957	−0.107479	+	0.994207i	$0.534278\pi$
$128$	−10.3142	−0.911651
$129$	−10.2076	−0.898732
$130$	5.04950	0.442870
$131$	−14.6844	−1.28298	−0.641489	−	0.767132i	$-0.721683\pi$
$131$	−14.6844	−1.28298	−0.641489	+	0.767132i	$0.721683\pi$
$132$	−9.67386	−0.842002
$133$	0.248120	0.0215148
$134$	−6.82462	−0.589557
$135$	1.11830	0.0962482
$136$	−3.93091	−0.337073
$137$	−2.44481	−0.208874	−0.104437	−	0.994532i	$-0.533304\pi$
$137$	−2.44481	−0.208874	−0.104437	+	0.994532i	$0.533304\pi$
$138$	3.29307	0.280325
$139$	−16.9230	−1.43539	−0.717694	−	0.696359i	$-0.754802\pi$
$139$	−16.9230	−1.43539	−0.717694	+	0.696359i	$0.754802\pi$
$140$	3.80874	0.321897
$141$	4.29085	0.361355
$142$	−3.88201	−0.325771
$143$	40.9276	3.42254
$144$	1.15005	0.0958375
$145$	−1.94994	−0.161934
$146$	5.41467	0.448122
$147$	1.70383	0.140529
$148$	−13.9429	−1.14610
$149$	−13.1835	−1.08003	−0.540016	−	0.841655i	$-0.681582\pi$
$149$	−13.1835	−1.08003	−0.540016	+	0.841655i	$0.681582\pi$
$150$	2.70391	0.220774
$151$	6.63290	0.539778	0.269889	−	0.962891i	$-0.413013\pi$
$151$	6.63290	0.539778	0.269889	+	0.962891i	$0.413013\pi$
$152$	−0.270572	−0.0219463
$153$	1.56636	0.126633
$154$	−10.8485	−0.874200
$155$	−1.85080	−0.148660
$156$	−9.26613	−0.741884
$157$	1.24910	0.0996893	0.0498447	−	0.998757i	$-0.484127\pi$
$157$	1.24910	0.0996893	0.0498447	+	0.998757i	$0.484127\pi$
$158$	1.24095	0.0987243
$159$	9.38524	0.744298
$160$	−6.54044	−0.517067
$161$	−10.5088	−0.828206
$162$	0.721159	0.0566596
$163$	5.20447	0.407646	0.203823	−	0.979008i	$-0.434663\pi$
$163$	5.20447	0.407646	0.203823	+	0.979008i	$0.434663\pi$
$164$	−4.02503	−0.314302
$165$	−7.31002	−0.569084
$166$	−9.56556	−0.742431
$167$	−18.8427	−1.45809	−0.729045	−	0.684465i	$-0.760036\pi$
$167$	−18.8427	−1.45809	−0.729045	+	0.684465i	$0.760036\pi$
$168$	5.77541	0.445582
$169$	26.2026	2.01559
$170$	−1.26323	−0.0968853
$171$	0.107816	0.00824487
$172$	−15.1066	−1.15187
$173$	21.9035	1.66529	0.832644	−	0.553808i	$-0.186826\pi$
$173$	21.9035	1.66529	0.832644	+	0.553808i	$0.186826\pi$
$174$	−1.25746	−0.0953275
$175$	−8.62864	−0.652264
$176$	−7.51754	−0.566656
$177$	7.34442	0.552041
$178$	10.2860	0.770972
$179$	12.6825	0.947938	0.473969	−	0.880542i	$-0.342821\pi$
$179$	12.6825	0.947938	0.473969	+	0.880542i	$0.342821\pi$
$180$	1.65501	0.123357
$181$	−24.8125	−1.84430	−0.922148	−	0.386837i	$-0.873568\pi$
$181$	−24.8125	−1.84430	−0.922148	+	0.386837i	$0.873568\pi$
$182$	−10.3913	−0.770254
$183$	−3.40792	−0.251920
$184$	11.4597	0.844817
$185$	−10.5359	−0.774616
$186$	−1.19353	−0.0875135
$187$	−10.2388	−0.748738
$188$	6.35016	0.463133
$189$	−2.30134	−0.167398
$190$	−0.0869505	−0.00630806
$191$	3.14222	0.227363	0.113681	−	0.993517i	$-0.463736\pi$
$191$	3.14222	0.227363	0.113681	+	0.993517i	$0.463736\pi$
$192$	−1.91762	−0.138393
$193$	26.4653	1.90502	0.952508	−	0.304514i	$-0.0984941\pi$
$193$	26.4653	1.90502	0.952508	+	0.304514i	$0.0984941\pi$
$194$	−9.45078	−0.678527
$195$	−7.00192	−0.501418
$196$	2.52155	0.180111
$197$	25.2892	1.80178	0.900889	−	0.434050i	$-0.142916\pi$
$197$	25.2892	1.80178	0.900889	+	0.434050i	$0.142916\pi$
$198$	−4.71401	−0.335010
$199$	−23.3831	−1.65759	−0.828793	−	0.559556i	$-0.810972\pi$
$199$	−23.3831	−1.65759	−0.828793	+	0.559556i	$0.810972\pi$
$200$	9.40943	0.665347
$201$	9.46340	0.667497
$202$	−2.24398	−0.157886
$203$	4.01275	0.281640
$204$	2.31810	0.162300
$205$	−3.04150	−0.212427
$206$	−0.721159	−0.0502455
$207$	−4.56636	−0.317384
$208$	−7.20069	−0.499278
$209$	−0.704759	−0.0487492
$210$	1.85597	0.128074
$211$	−14.9056	−1.02614	−0.513072	−	0.858345i	$-0.671493\pi$
$211$	−14.9056	−1.02614	−0.513072	+	0.858345i	$0.671493\pi$
$212$	13.8895	0.953935
$213$	5.38301	0.368838
$214$	1.42988	0.0977446
$215$	−11.4152	−0.778512
$216$	2.50958	0.170755
$217$	3.80874	0.258554
$218$	−5.01917	−0.339941
$219$	−7.50829	−0.507363
$220$	−10.8183	−0.729371
$221$	−9.80729	−0.659710
$222$	−6.79429	−0.456003
$223$	−12.5391	−0.839679	−0.419839	−	0.907598i	$-0.637914\pi$
$223$	−12.5391	−0.839679	−0.419839	+	0.907598i	$0.637914\pi$
$224$	13.4595	0.899299
$225$	−3.74940	−0.249960
$226$	−0.186088	−0.0123784
$227$	−3.68398	−0.244515	−0.122257	−	0.992498i	$-0.539013\pi$
$227$	−3.68398	−0.244515	−0.122257	+	0.992498i	$0.539013\pi$
$228$	0.159560	0.0105671
$229$	−5.28992	−0.349568	−0.174784	−	0.984607i	$-0.555923\pi$
$229$	−5.28992	−0.349568	−0.174784	+	0.984607i	$0.555923\pi$
$230$	3.68265	0.242827
$231$	15.0432	0.989769
$232$	−4.37586	−0.287289
$233$	−20.2838	−1.32884	−0.664419	−	0.747360i	$-0.731321\pi$
$233$	−20.2838	−1.32884	−0.664419	+	0.747360i	$0.731321\pi$
$234$	−4.51532	−0.295176
$235$	4.79847	0.313018
$236$	10.8692	0.707526
$237$	−1.72076	−0.111776
$238$	2.59958	0.168506
$239$	−3.78055	−0.244543	−0.122272	−	0.992497i	$-0.539018\pi$
$239$	−3.78055	−0.244543	−0.122272	+	0.992497i	$0.539018\pi$
$240$	1.28610	0.0830177
$241$	4.27938	0.275659	0.137829	−	0.990456i	$-0.455987\pi$
$241$	4.27938	0.275659	0.137829	+	0.990456i	$0.455987\pi$
$242$	22.8813	1.47087
$243$	−1.00000	−0.0641500
$244$	−5.04348	−0.322875
$245$	1.90540	0.121731
$246$	−1.96137	−0.125052
$247$	−0.675055	−0.0429527
$248$	−4.15338	−0.263740
$249$	13.2641	0.840581
$250$	7.05617	0.446271
$251$	−13.9080	−0.877865	−0.438932	−	0.898520i	$-0.644643\pi$
$251$	−13.9080	−0.877865	−0.438932	+	0.898520i	$0.644643\pi$
$252$	−3.40582	−0.214547
$253$	29.8490	1.87659
$254$	−1.74697	−0.109614
$255$	1.75167	0.109694
$256$	−11.2734	−0.704587
$257$	6.44851	0.402247	0.201123	−	0.979566i	$-0.435541\pi$
$257$	6.44851	0.402247	0.201123	+	0.979566i	$0.435541\pi$
$258$	−7.36133	−0.458296
$259$	21.6817	1.34724
$260$	−10.3623	−0.642645
$261$	1.74366	0.107930
$262$	−10.5898	−0.654237
$263$	17.2921	1.06627	0.533137	−	0.846029i	$-0.321013\pi$
$263$	17.2921	1.06627	0.533137	+	0.846029i	$0.321013\pi$
$264$	−16.4044	−1.00962
$265$	10.4955	0.644736
$266$	0.178934	0.0109712
$267$	−14.2632	−0.872894
$268$	14.0052	0.855502
$269$	−4.47726	−0.272983	−0.136492	−	0.990641i	$-0.543583\pi$
$269$	−4.47726	−0.272983	−0.136492	+	0.990641i	$0.543583\pi$
$270$	0.806475	0.0490805
$271$	−15.2377	−0.925627	−0.462813	−	0.886456i	$-0.653160\pi$
$271$	−15.2377	−0.925627	−0.462813	+	0.886456i	$0.653160\pi$
$272$	1.80139	0.109225
$273$	14.4092	0.872082
$274$	−1.76309	−0.106512
$275$	24.5087	1.47793
$276$	−6.75789	−0.406777
$277$	12.9288	0.776817	0.388409	−	0.921487i	$-0.373025\pi$
$277$	12.9288	0.776817	0.388409	+	0.921487i	$0.373025\pi$
$278$	−12.2042	−0.731957
$279$	1.65501	0.0990828
$280$	6.45866	0.385978
$281$	−0.339583	−0.0202578	−0.0101289	−	0.999949i	$-0.503224\pi$
$281$	−0.339583	−0.0202578	−0.0101289	+	0.999949i	$0.503224\pi$
$282$	3.09439	0.184268
$283$	−17.8272	−1.05972	−0.529860	−	0.848085i	$-0.677756\pi$
$283$	−17.8272	−1.05972	−0.529860	+	0.848085i	$0.677756\pi$
$284$	7.96648	0.472723
$285$	0.120571	0.00714198
$286$	29.5153	1.74528
$287$	6.25906	0.369461
$288$	5.84854	0.344628
$289$	−14.5465	−0.855677
$290$	−1.40622	−0.0825759
$291$	13.1050	0.768228
$292$	−11.1117	−0.650266
$293$	23.4041	1.36728	0.683641	−	0.729818i	$-0.260395\pi$
$293$	23.4041	1.36728	0.683641	+	0.729818i	$0.260395\pi$
$294$	1.22873	0.0716611
$295$	8.21329	0.478196
$296$	−23.6436	−1.37426
$297$	6.53671	0.379298
$298$	−9.50737	−0.550747
$299$	28.5909	1.65345
$300$	−5.54884	−0.320363
$301$	23.4912	1.35401
$302$	4.78338	0.275253
$303$	3.11163	0.178759
$304$	0.123993	0.00711151
$305$	−3.81108	−0.218222
$306$	1.12959	0.0645746
$307$	28.4741	1.62511	0.812553	−	0.582888i	$-0.198077\pi$
$307$	28.4741	1.62511	0.812553	+	0.582888i	$0.198077\pi$
$308$	22.2629	1.26854
$309$	1.00000	0.0568880
$310$	−1.33472	−0.0758072
$311$	9.42222	0.534285	0.267143	−	0.963657i	$-0.413921\pi$
$311$	9.42222	0.534285	0.267143	+	0.963657i	$0.413921\pi$
$312$	−15.7130	−0.889574
$313$	−18.7320	−1.05880	−0.529399	−	0.848373i	$-0.677582\pi$
$313$	−18.7320	−1.05880	−0.529399	+	0.848373i	$0.677582\pi$
$314$	0.900803	0.0508352
$315$	−2.57360	−0.145006
$316$	−2.54661	−0.143258
$317$	−8.57707	−0.481736	−0.240868	−	0.970558i	$-0.577432\pi$
$317$	−8.57707	−0.481736	−0.240868	+	0.970558i	$0.577432\pi$
$318$	6.76825	0.379545
$319$	−11.3978	−0.638154
$320$	−2.14449	−0.119880
$321$	−1.98275	−0.110667
$322$	−7.57848	−0.422332
$323$	0.168878	0.00939663
$324$	−1.47993	−0.0822183
$325$	23.4757	1.30220
$326$	3.75325	0.207873
$327$	6.95986	0.384881
$328$	−6.82542	−0.376871
$329$	−9.87472	−0.544411
$330$	−5.27169	−0.290197
$331$	−24.3674	−1.33935	−0.669675	−	0.742654i	$-0.733567\pi$
$331$	−24.3674	−1.33935	−0.669675	+	0.742654i	$0.733567\pi$
$332$	19.6300	1.07734
$333$	9.42134	0.516286
$334$	−13.5886	−0.743534
$335$	10.5830	0.578209
$336$	−2.64666	−0.144387
$337$	16.0738	0.875595	0.437797	−	0.899074i	$-0.355759\pi$
$337$	16.0738	0.875595	0.437797	+	0.899074i	$0.355759\pi$
$338$	18.8963	1.02782
$339$	0.258040	0.0140148
$340$	2.59234	0.140589
$341$	−10.8183	−0.585845
$342$	0.0777522	0.00420436
$343$	−20.0305	−1.08154
$344$	−25.6169	−1.38117
$345$	−5.10657	−0.274929
$346$	15.7959	0.849192
$347$	31.3152	1.68109	0.840545	−	0.541741i	$-0.182235\pi$
$347$	31.3152	1.68109	0.840545	+	0.541741i	$0.182235\pi$
$348$	2.58049	0.138329
$349$	8.22265	0.440148	0.220074	−	0.975483i	$-0.429370\pi$
$349$	8.22265	0.440148	0.220074	+	0.975483i	$0.429370\pi$
$350$	−6.22263	−0.332613
$351$	6.26120	0.334198
$352$	−38.2302	−2.03767
$353$	−11.0229	−0.586689	−0.293345	−	0.956007i	$-0.594768\pi$
$353$	−11.0229	−0.586689	−0.293345	+	0.956007i	$0.594768\pi$
$354$	5.29650	0.281506
$355$	6.01984	0.319500
$356$	−21.1085	−1.11875
$357$	−3.60473	−0.190782
$358$	9.14613	0.483388
$359$	16.5665	0.874346	0.437173	−	0.899377i	$-0.355980\pi$
$359$	16.5665	0.874346	0.437173	+	0.899377i	$0.355980\pi$
$360$	2.80647	0.147914
$361$	−18.9884	−0.999388
$362$	−17.8937	−0.940474
$363$	−31.7285	−1.66532
$364$	21.3245	1.11771
$365$	−8.39655	−0.439495
$366$	−2.45765	−0.128463
$367$	35.3414	1.84481	0.922403	−	0.386229i	$-0.126223\pi$
$367$	35.3414	1.84481	0.922403	+	0.386229i	$0.126223\pi$
$368$	−5.25154	−0.273756
$369$	2.71974	0.141584
$370$	−7.59807	−0.395005
$371$	−21.5986	−1.12135
$372$	2.44930	0.126990
$373$	−16.7963	−0.869682	−0.434841	−	0.900507i	$-0.643195\pi$
$373$	−16.7963	−0.869682	−0.434841	+	0.900507i	$0.643195\pi$
$374$	−7.38383	−0.381809
$375$	−9.78448	−0.505268
$376$	10.7683	0.555330
$377$	−10.9174	−0.562275
$378$	−1.65963	−0.0853623
$379$	2.41241	0.123917	0.0619585	−	0.998079i	$-0.480265\pi$
$379$	2.41241	0.123917	0.0619585	+	0.998079i	$0.480265\pi$
$380$	0.178436	0.00915357
$381$	2.42244	0.124106
$382$	2.26604	0.115941
$383$	13.5743	0.693613	0.346806	−	0.937937i	$-0.387266\pi$
$383$	13.5743	0.693613	0.346806	+	0.937937i	$0.387266\pi$
$384$	10.3142	0.526342
$385$	16.8228	0.857372
$386$	19.0857	0.971437
$387$	10.2076	0.518883
$388$	19.3945	0.984604
$389$	24.4284	1.23857	0.619285	−	0.785166i	$-0.287422\pi$
$389$	24.4284	1.23857	0.619285	+	0.785166i	$0.287422\pi$
$390$	−5.04950	−0.255691
$391$	−7.15256	−0.361721
$392$	4.27590	0.215966
$393$	14.6844	0.740728
$394$	18.2375	0.918792
$395$	−1.92434	−0.0968239
$396$	9.67386	0.486130
$397$	−8.69080	−0.436179	−0.218089	−	0.975929i	$-0.569982\pi$
$397$	−8.69080	−0.436179	−0.218089	+	0.975929i	$0.569982\pi$
$398$	−16.8630	−0.845263
$399$	−0.248120	−0.0124216
$400$	−4.31200	−0.215600
$401$	20.0888	1.00318	0.501592	−	0.865104i	$-0.332748\pi$
$401$	20.0888	1.00318	0.501592	+	0.865104i	$0.332748\pi$
$402$	6.82462	0.340381
$403$	−10.3623	−0.516185
$404$	4.60500	0.229107
$405$	−1.11830	−0.0555689
$406$	2.89383	0.143619
$407$	−61.5846	−3.05263
$408$	3.93091	0.194609
$409$	16.0770	0.794958	0.397479	−	0.917611i	$-0.369885\pi$
$409$	16.0770	0.794958	0.397479	+	0.917611i	$0.369885\pi$
$410$	−2.19340	−0.108325
$411$	2.44481	0.120593
$412$	1.47993	0.0729109
$413$	−16.9020	−0.831694
$414$	−3.29307	−0.161846
$415$	14.8333	0.728139
$416$	−36.6188	−1.79539
$417$	16.9230	0.828721
$418$	−0.508243	−0.0248590
$419$	−9.54089	−0.466103	−0.233051	−	0.972464i	$-0.574871\pi$
$419$	−9.54089	−0.466103	−0.233051	+	0.972464i	$0.574871\pi$
$420$	−3.80874	−0.185848
$421$	7.17416	0.349647	0.174824	−	0.984600i	$-0.444064\pi$
$421$	7.17416	0.349647	0.174824	+	0.984600i	$0.444064\pi$
$422$	−10.7493	−0.523269
$423$	−4.29085	−0.208628
$424$	23.5530	1.14384
$425$	−5.87291	−0.284878
$426$	3.88201	0.188084
$427$	7.84278	0.379539
$428$	−2.93434	−0.141836
$429$	−40.9276	−1.97600
$430$	−8.23220	−0.396992
$431$	−17.2481	−0.830814	−0.415407	−	0.909636i	$-0.636361\pi$
$431$	−17.2481	−0.830814	−0.415407	+	0.909636i	$0.636361\pi$
$432$	−1.15005	−0.0553318
$433$	−24.4219	−1.17364	−0.586822	−	0.809716i	$-0.699621\pi$
$433$	−24.4219	−1.17364	−0.586822	+	0.809716i	$0.699621\pi$
$434$	2.74671	0.131846
$435$	1.94994	0.0934925
$436$	10.3001	0.493285
$437$	−0.492325	−0.0235511
$438$	−5.41467	−0.258723
$439$	−33.1495	−1.58214	−0.791069	−	0.611727i	$-0.790475\pi$
$439$	−33.1495	−1.58214	−0.791069	+	0.611727i	$0.790475\pi$
$440$	−18.3451	−0.874568
$441$	−1.70383	−0.0811347
$442$	−7.07262	−0.336410
$443$	23.8526	1.13327	0.566636	−	0.823968i	$-0.308245\pi$
$443$	23.8526	1.13327	0.566636	+	0.823968i	$0.308245\pi$
$444$	13.9429	0.661702
$445$	−15.9506	−0.756131
$446$	−9.04267	−0.428183
$447$	13.1835	0.623556
$448$	4.41311	0.208500
$449$	−41.8737	−1.97614	−0.988071	−	0.154000i	$-0.950784\pi$
$449$	−41.8737	−1.97614	−0.988071	+	0.154000i	$0.950784\pi$
$450$	−2.70391	−0.127464
$451$	−17.7782	−0.837141
$452$	0.381881	0.0179621
$453$	−6.63290	−0.311641
$454$	−2.65674	−0.124687
$455$	16.1138	0.755427
$456$	0.270572	0.0126707
$457$	15.7815	0.738229	0.369115	−	0.929384i	$-0.379661\pi$
$457$	15.7815	0.738229	0.369115	+	0.929384i	$0.379661\pi$
$458$	−3.81488	−0.178257
$459$	−1.56636	−0.0731114
$460$	−7.55737	−0.352364
$461$	−8.65453	−0.403082	−0.201541	−	0.979480i	$-0.564595\pi$
$461$	−8.65453	−0.403082	−0.201541	+	0.979480i	$0.564595\pi$
$462$	10.8485	0.504720
$463$	26.0368	1.21003	0.605015	−	0.796214i	$-0.293167\pi$
$463$	26.0368	1.21003	0.605015	+	0.796214i	$0.293167\pi$
$464$	2.00530	0.0930935
$465$	1.85080	0.0858289
$466$	−14.6279	−0.677623
$467$	−21.5493	−0.997181	−0.498590	−	0.866838i	$-0.666149\pi$
$467$	−21.5493	−0.997181	−0.498590	+	0.866838i	$0.666149\pi$
$468$	9.26613	0.428327
$469$	−21.7785	−1.00564
$470$	3.46046	0.159619
$471$	−1.24910	−0.0575557
$472$	18.4314	0.848376
$473$	−66.7243	−3.06799
$474$	−1.24095	−0.0569985
$475$	−0.404244	−0.0185480
$476$	−5.33474	−0.244518
$477$	−9.38524	−0.429721
$478$	−2.72638	−0.124702
$479$	−14.2993	−0.653352	−0.326676	−	0.945136i	$-0.605929\pi$
$479$	−14.2993	−0.653352	−0.326676	+	0.945136i	$0.605929\pi$
$480$	6.54044	0.298529
$481$	−58.9889	−2.68966
$482$	3.08611	0.140569
$483$	10.5088	0.478165
$484$	−46.9560	−2.13436
$485$	14.6553	0.665465
$486$	−0.721159	−0.0327124
$487$	38.8667	1.76122	0.880609	−	0.473843i	$-0.157134\pi$
$487$	38.8667	1.76122	0.880609	+	0.473843i	$0.157134\pi$
$488$	−8.55245	−0.387151
$489$	−5.20447	−0.235354
$490$	1.37409	0.0620753
$491$	−6.04142	−0.272645	−0.136323	−	0.990664i	$-0.543528\pi$
$491$	−6.04142	−0.272645	−0.136323	+	0.990664i	$0.543528\pi$
$492$	4.02503	0.181462
$493$	2.73120	0.123007
$494$	−0.486822	−0.0219032
$495$	7.31002	0.328561
$496$	1.90334	0.0854627
$497$	−12.3881	−0.555684
$498$	9.56556	0.428643
$499$	−4.08964	−0.183077	−0.0915387	−	0.995802i	$-0.529179\pi$
$499$	−4.08964	−0.183077	−0.0915387	+	0.995802i	$0.529179\pi$
$500$	−14.4803	−0.647580
$501$	18.8427	0.841829
$502$	−10.0299	−0.447655
$503$	29.7182	1.32507	0.662534	−	0.749032i	$-0.269481\pi$
$503$	29.7182	1.32507	0.662534	+	0.749032i	$0.269481\pi$
$504$	−5.77541	−0.257257
$505$	3.47975	0.154847
$506$	21.5258	0.956941
$507$	−26.2026	−1.16370
$508$	3.58505	0.159061
$509$	21.1968	0.939531	0.469766	−	0.882791i	$-0.344338\pi$
$509$	21.1968	0.939531	0.469766	+	0.882791i	$0.344338\pi$
$510$	1.26323	0.0559367
$511$	17.2791	0.764384
$512$	12.4984	0.552357
$513$	−0.107816	−0.00476018
$514$	4.65040	0.205120
$515$	1.11830	0.0492783
$516$	15.1066	0.665030
$517$	28.0481	1.23355
$518$	15.6360	0.687005
$519$	−21.9035	−0.961455
$520$	−17.5719	−0.770579
$521$	−0.737101	−0.0322930	−0.0161465	−	0.999870i	$-0.505140\pi$
$521$	−0.737101	−0.0322930	−0.0161465	+	0.999870i	$0.505140\pi$
$522$	1.25746	0.0550374
$523$	19.4201	0.849182	0.424591	−	0.905385i	$-0.360418\pi$
$523$	19.4201	0.849182	0.424591	+	0.905385i	$0.360418\pi$
$524$	21.7318	0.949358
$525$	8.62864	0.376585
$526$	12.4703	0.543732
$527$	2.59234	0.112924
$528$	7.51754	0.327159
$529$	−2.14835	−0.0934067
$530$	7.56896	0.328775
$531$	−7.34442	−0.318721
$532$	−0.367201	−0.0159202
$533$	−17.0289	−0.737602
$534$	−10.2860	−0.445121
$535$	−2.21732	−0.0958631
$536$	23.7492	1.02581
$537$	−12.6825	−0.547292
$538$	−3.22882	−0.139204
$539$	11.1374	0.479723
$540$	−1.65501	−0.0712203
$541$	27.8845	1.19885	0.599424	−	0.800432i	$-0.295396\pi$
$541$	27.8845	1.19885	0.599424	+	0.800432i	$0.295396\pi$
$542$	−10.9888	−0.472011
$543$	24.8125	1.06480
$544$	9.16091	0.392771
$545$	7.78323	0.333397
$546$	10.3913	0.444706
$547$	5.88042	0.251429	0.125714	−	0.992066i	$-0.459878\pi$
$547$	5.88042	0.251429	0.125714	+	0.992066i	$0.459878\pi$
$548$	3.61814	0.154559
$549$	3.40792	0.145446
$550$	17.6747	0.753651
$551$	0.187994	0.00800880
$552$	−11.4597	−0.487756
$553$	3.96007	0.168399
$554$	9.32374	0.396128
$555$	10.5359	0.447225
$556$	25.0448	1.06214
$557$	1.34576	0.0570217	0.0285108	−	0.999593i	$-0.490923\pi$
$557$	1.34576	0.0570217	0.0285108	+	0.999593i	$0.490923\pi$
$558$	1.19353	0.0505260
$559$	−63.9120	−2.70319
$560$	−2.95977	−0.125073
$561$	10.2388	0.432284
$562$	−0.244893	−0.0103302
$563$	−23.4867	−0.989847	−0.494924	−	0.868936i	$-0.664804\pi$
$563$	−23.4867	−0.989847	−0.494924	+	0.868936i	$0.664804\pi$
$564$	−6.35016	−0.267390
$565$	0.288567	0.0121401
$566$	−12.8563	−0.540390
$567$	2.30134	0.0966472
$568$	13.5091	0.566830
$569$	25.5274	1.07016	0.535082	−	0.844800i	$-0.320281\pi$
$569$	25.5274	1.07016	0.535082	+	0.844800i	$0.320281\pi$
$570$	0.0869505	0.00364196
$571$	−17.1221	−0.716539	−0.358270	−	0.933618i	$-0.616633\pi$
$571$	−17.1221	−0.716539	−0.358270	+	0.933618i	$0.616633\pi$
$572$	−60.5700	−2.53256
$573$	−3.14222	−0.131268
$574$	4.51378	0.188401
$575$	17.1211	0.713999
$576$	1.91762	0.0799010
$577$	−21.7723	−0.906394	−0.453197	−	0.891410i	$-0.649717\pi$
$577$	−21.7723	−0.906394	−0.453197	+	0.891410i	$0.649717\pi$
$578$	−10.4904	−0.436341
$579$	−26.4653	−1.09986
$580$	2.88577	0.119825
$581$	−30.5253	−1.26640
$582$	9.45078	0.391747
$583$	61.3486	2.54080
$584$	−18.8427	−0.779716
$585$	7.00192	0.289494
$586$	16.8781	0.697227
$587$	−10.4325	−0.430596	−0.215298	−	0.976548i	$-0.569072\pi$
$587$	−10.4325	−0.430596	−0.215298	+	0.976548i	$0.569072\pi$
$588$	−2.52155	−0.103987
$589$	0.178436	0.00735232
$590$	5.92309	0.243850
$591$	−25.2892	−1.04026
$592$	10.8350	0.445316
$593$	−24.0466	−0.987476	−0.493738	−	0.869611i	$-0.664370\pi$
$593$	−24.0466	−0.987476	−0.493738	+	0.869611i	$0.664370\pi$
$594$	4.71401	0.193418
$595$	−4.03118	−0.165262
$596$	19.5106	0.799185
$597$	23.3831	0.957007
$598$	20.6186	0.843157
$599$	5.13468	0.209797	0.104899	−	0.994483i	$-0.466548\pi$
$599$	5.13468	0.209797	0.104899	+	0.994483i	$0.466548\pi$
$600$	−9.40943	−0.384138
$601$	−19.0315	−0.776310	−0.388155	−	0.921594i	$-0.626887\pi$
$601$	−19.0315	−0.776310	−0.388155	+	0.921594i	$0.626887\pi$
$602$	16.9409	0.690461
$603$	−9.46340	−0.385380
$604$	−9.81623	−0.399417
$605$	−35.4821	−1.44255
$606$	2.24398	0.0911556
$607$	−6.99132	−0.283769	−0.141884	−	0.989883i	$-0.545316\pi$
$607$	−6.99132	−0.283769	−0.141884	+	0.989883i	$0.545316\pi$
$608$	0.630563	0.0255727
$609$	−4.01275	−0.162605
$610$	−2.74840	−0.111279
$611$	26.8659	1.08688
$612$	−2.31810	−0.0937037
$613$	−19.1524	−0.773557	−0.386779	−	0.922173i	$-0.626412\pi$
$613$	−19.1524	−0.773557	−0.386779	+	0.922173i	$0.626412\pi$
$614$	20.5344	0.828700
$615$	3.04150	0.122645
$616$	37.7521	1.52108
$617$	−29.5185	−1.18837	−0.594184	−	0.804329i	$-0.702525\pi$
$617$	−29.5185	−1.18837	−0.594184	+	0.804329i	$0.702525\pi$
$618$	0.721159	0.0290093
$619$	5.04887	0.202931	0.101466	−	0.994839i	$-0.467647\pi$
$619$	5.04887	0.202931	0.101466	+	0.994839i	$0.467647\pi$
$620$	2.73906	0.110003
$621$	4.56636	0.183242
$622$	6.79492	0.272452
$623$	32.8245	1.31509
$624$	7.20069	0.288258
$625$	7.80498	0.312199
$626$	−13.5088	−0.539920
$627$	0.704759	0.0281454
$628$	−1.84859	−0.0737666
$629$	14.7572	0.588409
$630$	−1.85597	−0.0739437
$631$	2.51522	0.100129	0.0500647	−	0.998746i	$-0.484057\pi$
$631$	2.51522	0.100129	0.0500647	+	0.998746i	$0.484057\pi$
$632$	−4.31840	−0.171777
$633$	14.9056	0.592445
$634$	−6.18543	−0.245655
$635$	2.70903	0.107504
$636$	−13.8895	−0.550754
$637$	10.6680	0.422682
$638$	−8.21962	−0.325418
$639$	−5.38301	−0.212949
$640$	11.5344	0.455935
$641$	17.5796	0.694351	0.347176	−	0.937800i	$-0.387141\pi$
$641$	17.5796	0.694351	0.347176	+	0.937800i	$0.387141\pi$
$642$	−1.42988	−0.0564329
$643$	6.14801	0.242454	0.121227	−	0.992625i	$-0.461317\pi$
$643$	6.14801	0.242454	0.121227	+	0.992625i	$0.461317\pi$
$644$	15.5522	0.612843
$645$	11.4152	0.449474
$646$	0.121788	0.00479168
$647$	−0.760173	−0.0298855	−0.0149427	−	0.999888i	$-0.504757\pi$
$647$	−0.760173	−0.0298855	−0.0149427	+	0.999888i	$0.504757\pi$
$648$	−2.50958	−0.0985857
$649$	48.0083	1.88449
$650$	16.9297	0.664039
$651$	−3.80874	−0.149276
$652$	−7.70225	−0.301643
$653$	38.6070	1.51081	0.755404	−	0.655259i	$-0.227441\pi$
$653$	38.6070	1.51081	0.755404	+	0.655259i	$0.227441\pi$
$654$	5.01917	0.196265
$655$	16.4216	0.641643
$656$	3.12784	0.122122
$657$	7.50829	0.292926
$658$	−7.12124	−0.277615
$659$	−37.7487	−1.47048	−0.735240	−	0.677806i	$-0.762931\pi$
$659$	−37.7487	−1.47048	−0.735240	+	0.677806i	$0.762931\pi$
$660$	10.8183	0.421102
$661$	−29.2907	−1.13927	−0.569637	−	0.821896i	$-0.692916\pi$
$661$	−29.2907	−1.13927	−0.569637	+	0.821896i	$0.692916\pi$
$662$	−17.5727	−0.682984
$663$	9.80729	0.380884
$664$	33.2875	1.29180
$665$	−0.277474	−0.0107600
$666$	6.79429	0.263273
$667$	−7.96218	−0.308297
$668$	27.8858	1.07894
$669$	12.5391	0.484789
$670$	7.63199	0.294850
$671$	−22.2765	−0.859977
$672$	−13.4595	−0.519210
$673$	−48.6893	−1.87683	−0.938417	−	0.345505i	$-0.887708\pi$
$673$	−48.6893	−1.87683	−0.938417	+	0.345505i	$0.887708\pi$
$674$	11.5918	0.446498
$675$	3.74940	0.144314
$676$	−38.7780	−1.49146
$677$	5.30463	0.203873	0.101937	−	0.994791i	$-0.467496\pi$
$677$	5.30463	0.203873	0.101937	+	0.994791i	$0.467496\pi$
$678$	0.186088	0.00714665
$679$	−30.1590	−1.15740
$680$	4.39595	0.168577
$681$	3.68398	0.141171
$682$	−7.80173	−0.298744
$683$	7.37546	0.282214	0.141107	−	0.989994i	$-0.454934\pi$
$683$	7.37546	0.282214	0.141107	+	0.989994i	$0.454934\pi$
$684$	−0.159560	−0.00610091
$685$	2.73403	0.104462
$686$	−14.4452	−0.551519
$687$	5.28992	0.201823
$688$	11.7393	0.447556
$689$	58.7629	2.23869
$690$	−3.68265	−0.140196
$691$	6.00330	0.228376	0.114188	−	0.993459i	$-0.463573\pi$
$691$	6.00330	0.228376	0.114188	+	0.993459i	$0.463573\pi$
$692$	−32.4156	−1.23226
$693$	−15.0432	−0.571444
$694$	22.5833	0.857249
$695$	18.9250	0.717867
$696$	4.37586	0.165866
$697$	4.26010	0.161363
$698$	5.92984	0.224448
$699$	20.2838	0.767205
$700$	12.7698	0.482652
$701$	23.1097	0.872842	0.436421	−	0.899743i	$-0.356246\pi$
$701$	23.1097	0.872842	0.436421	+	0.899743i	$0.356246\pi$
$702$	4.51532	0.170420
$703$	1.01577	0.0383104
$704$	−12.5349	−0.472429
$705$	−4.79847	−0.180721
$706$	−7.94926	−0.299174
$707$	−7.16093	−0.269315
$708$	−10.8692	−0.408491
$709$	−41.4529	−1.55680	−0.778398	−	0.627771i	$-0.783967\pi$
$709$	−41.4529	−1.55680	−0.778398	+	0.627771i	$0.783967\pi$
$710$	4.34126	0.162925
$711$	1.72076	0.0645337
$712$	−35.7947	−1.34146
$713$	−7.55737	−0.283026
$714$	−2.59958	−0.0972870
$715$	−45.7695	−1.71168
$716$	−18.7693	−0.701441
$717$	3.78055	0.141187
$718$	11.9471	0.445861
$719$	31.1845	1.16299	0.581493	−	0.813552i	$-0.302469\pi$
$719$	31.1845	1.16299	0.581493	+	0.813552i	$0.302469\pi$
$720$	−1.28610	−0.0479303
$721$	−2.30134	−0.0857064
$722$	−13.6936	−0.509625
$723$	−4.27938	−0.159152
$724$	36.7207	1.36471
$725$	−6.53767	−0.242803
$726$	−22.8813	−0.849206
$727$	−20.0686	−0.744302	−0.372151	−	0.928172i	$-0.621380\pi$
$727$	−20.0686	−0.744302	−0.372151	+	0.928172i	$0.621380\pi$
$728$	36.1610	1.34021
$729$	1.00000	0.0370370
$730$	−6.05525	−0.224115
$731$	15.9888	0.591368
$732$	5.04348	0.186412
$733$	−11.8663	−0.438292	−0.219146	−	0.975692i	$-0.570327\pi$
$733$	−11.8663	−0.438292	−0.219146	+	0.975692i	$0.570327\pi$
$734$	25.4868	0.940734
$735$	−1.90540	−0.0702817
$736$	−26.7065	−0.984415
$737$	61.8595	2.27862
$738$	1.96137	0.0721990
$739$	−31.2347	−1.14899	−0.574493	−	0.818510i	$-0.694801\pi$
$739$	−31.2347	−1.14899	−0.574493	+	0.818510i	$0.694801\pi$
$740$	15.5924	0.573189
$741$	0.675055	0.0247988
$742$	−15.5761	−0.571815
$743$	24.3051	0.891668	0.445834	−	0.895116i	$-0.352907\pi$
$743$	24.3051	0.891668	0.445834	+	0.895116i	$0.352907\pi$
$744$	4.15338	0.152270
$745$	14.7431	0.540146
$746$	−12.1128	−0.443483
$747$	−13.2641	−0.485309
$748$	15.1528	0.554040
$749$	4.56299	0.166728
$750$	−7.05617	−0.257655
$751$	−22.2722	−0.812725	−0.406362	−	0.913712i	$-0.633203\pi$
$751$	−22.2722	−0.812725	−0.406362	+	0.913712i	$0.633203\pi$
$752$	−4.93470	−0.179950
$753$	13.9080	0.506835
$754$	−7.87318	−0.286724
$755$	−7.41760	−0.269954
$756$	3.40582	0.123869
$757$	4.21399	0.153160	0.0765801	−	0.997063i	$-0.475600\pi$
$757$	4.21399	0.153160	0.0765801	+	0.997063i	$0.475600\pi$
$758$	1.73973	0.0631898
$759$	−29.8490	−1.08345
$760$	0.302582	0.0109758
$761$	42.3489	1.53514	0.767572	−	0.640962i	$-0.221464\pi$
$761$	42.3489	1.53514	0.767572	+	0.640962i	$0.221464\pi$
$762$	1.74697	0.0632859
$763$	−16.0170	−0.579855
$764$	−4.65026	−0.168241
$765$	−1.75167	−0.0633316
$766$	9.78921	0.353698
$767$	45.9849	1.66042
$768$	11.2734	0.406794
$769$	6.79565	0.245057	0.122529	−	0.992465i	$-0.460900\pi$
$769$	6.79565	0.245057	0.122529	+	0.992465i	$0.460900\pi$
$770$	12.1320	0.437205
$771$	−6.44851	−0.232237
$772$	−39.1668	−1.40964
$773$	−15.6926	−0.564424	−0.282212	−	0.959352i	$-0.591068\pi$
$773$	−15.6926	−0.564424	−0.282212	+	0.959352i	$0.591068\pi$
$774$	7.36133	0.264597
$775$	−6.20529	−0.222901
$776$	32.8881	1.18061
$777$	−21.6817	−0.777828
$778$	17.6168	0.631592
$779$	0.293231	0.0105061
$780$	10.3623	0.371032
$781$	35.1872	1.25910
$782$	−5.15814	−0.184455
$783$	−1.74366	−0.0623133
$784$	−1.95949	−0.0699818
$785$	−1.39688	−0.0498567
$786$	10.5898	0.377724
$787$	−33.5058	−1.19435	−0.597176	−	0.802110i	$-0.703711\pi$
$787$	−33.5058	−1.19435	−0.597176	+	0.802110i	$0.703711\pi$
$788$	−37.4262	−1.33325
$789$	−17.2921	−0.615614
$790$	−1.38775	−0.0493740
$791$	−0.593837	−0.0211144
$792$	16.4044	0.582905
$793$	−21.3376	−0.757722
$794$	−6.26745	−0.222423
$795$	−10.4955	−0.372239
$796$	34.6054	1.22655
$797$	41.0704	1.45479	0.727394	−	0.686220i	$-0.240731\pi$
$797$	41.0704	1.45479	0.727394	+	0.686220i	$0.240731\pi$
$798$	−0.178934	−0.00633421
$799$	−6.72102	−0.237773
$800$	−21.9285	−0.775289
$801$	14.2632	0.503966
$802$	14.4872	0.511561
$803$	−49.0795	−1.73198
$804$	−14.0052	−0.493924
$805$	11.7520	0.414203
$806$	−7.47290	−0.263222
$807$	4.47726	0.157607
$808$	7.80891	0.274716
$809$	−22.5830	−0.793974	−0.396987	−	0.917824i	$-0.629944\pi$
$809$	−22.5830	−0.793974	−0.396987	+	0.917824i	$0.629944\pi$
$810$	−0.806475	−0.0283366
$811$	−0.961009	−0.0337456	−0.0168728	−	0.999858i	$-0.505371\pi$
$811$	−0.961009	−0.0337456	−0.0168728	+	0.999858i	$0.505371\pi$
$812$	−5.93859	−0.208404
$813$	15.2377	0.534411
$814$	−44.4123	−1.55665
$815$	−5.82018	−0.203872
$816$	−1.80139	−0.0630614
$817$	1.10054	0.0385031
$818$	11.5941	0.405378
$819$	−14.4092	−0.503497
$820$	4.50120	0.157189
$821$	15.0680	0.525878	0.262939	−	0.964813i	$-0.415308\pi$
$821$	15.0680	0.525878	0.262939	+	0.964813i	$0.415308\pi$
$822$	1.76309	0.0614950
$823$	23.3504	0.813944	0.406972	−	0.913441i	$-0.366585\pi$
$823$	23.3504	0.813944	0.406972	+	0.913441i	$0.366585\pi$
$824$	2.50958	0.0874255
$825$	−24.5087	−0.853284
$826$	−12.1890	−0.424111
$827$	5.54270	0.192739	0.0963693	−	0.995346i	$-0.469277\pi$
$827$	5.54270	0.192739	0.0963693	+	0.995346i	$0.469277\pi$
$828$	6.75789	0.234853
$829$	−28.6416	−0.994763	−0.497382	−	0.867532i	$-0.665705\pi$
$829$	−28.6416	−0.994763	−0.497382	+	0.867532i	$0.665705\pi$
$830$	10.6972	0.371305
$831$	−12.9288	−0.448496
$832$	−12.0066	−0.416255
$833$	−2.66881	−0.0924688
$834$	12.2042	0.422595
$835$	21.0718	0.729221
$836$	1.04299	0.0360727
$837$	−1.65501	−0.0572055
$838$	−6.88050	−0.237683
$839$	−34.9588	−1.20691	−0.603457	−	0.797396i	$-0.706210\pi$
$839$	−34.9588	−1.20691	−0.603457	+	0.797396i	$0.706210\pi$
$840$	−6.45866	−0.222845
$841$	−25.9597	−0.895160
$842$	5.17371	0.178298
$843$	0.339583	0.0116958
$844$	22.0593	0.759311
$845$	−29.3025	−1.00804
$846$	−3.09439	−0.106387
$847$	73.0182	2.50893
$848$	−10.7935	−0.370650
$849$	17.8272	0.611830
$850$	−4.23530	−0.145270
$851$	−43.0212	−1.47475
$852$	−7.96648	−0.272927
$853$	35.2279	1.20618	0.603090	−	0.797673i	$-0.293936\pi$
$853$	35.2279	1.20618	0.603090	+	0.797673i	$0.293936\pi$
$854$	5.65589	0.193541
$855$	−0.120571	−0.00412343
$856$	−4.97588	−0.170072
$857$	−14.3825	−0.491297	−0.245648	−	0.969359i	$-0.579001\pi$
$857$	−14.3825	−0.491297	−0.245648	+	0.969359i	$0.579001\pi$
$858$	−29.5153	−1.00764
$859$	1.08267	0.0369403	0.0184702	−	0.999829i	$-0.494120\pi$
$859$	1.08267	0.0369403	0.0184702	+	0.999829i	$0.494120\pi$
$860$	16.8937	0.576071
$861$	−6.25906	−0.213308
$862$	−12.4387	−0.423662
$863$	31.4884	1.07188	0.535939	−	0.844257i	$-0.319958\pi$
$863$	31.4884	1.07188	0.535939	+	0.844257i	$0.319958\pi$
$864$	−5.84854	−0.198971
$865$	−24.4947	−0.832845
$866$	−17.6121	−0.598484
$867$	14.5465	0.494026
$868$	−5.63667	−0.191321
$869$	−11.2481	−0.381567
$870$	1.40622	0.0476752
$871$	59.2523	2.00769
$872$	17.4663	0.591485
$873$	−13.1050	−0.443537
$874$	−0.355045	−0.0120096
$875$	22.5174	0.761228
$876$	11.1117	0.375431
$877$	−35.0887	−1.18486	−0.592431	−	0.805621i	$-0.701832\pi$
$877$	−35.0887	−1.18486	−0.592431	+	0.805621i	$0.701832\pi$
$878$	−23.9060	−0.806790
$879$	−23.4041	−0.789401
$880$	8.40689	0.283396
$881$	−48.1542	−1.62236	−0.811179	−	0.584798i	$-0.801174\pi$
$881$	−48.1542	−1.62236	−0.811179	+	0.584798i	$0.801174\pi$
$882$	−1.22873	−0.0413736
$883$	15.9025	0.535163	0.267581	−	0.963535i	$-0.413776\pi$
$883$	15.9025	0.535163	0.267581	+	0.963535i	$0.413776\pi$
$884$	14.5141	0.488162
$885$	−8.21329	−0.276087
$886$	17.2015	0.577897
$887$	−26.8586	−0.901824	−0.450912	−	0.892568i	$-0.648901\pi$
$887$	−26.8586	−0.901824	−0.450912	+	0.892568i	$0.648901\pi$
$888$	23.6436	0.793429
$889$	−5.57487	−0.186975
$890$	−11.5029	−0.385579
$891$	−6.53671	−0.218988
$892$	18.5570	0.621333
$893$	−0.462621	−0.0154810
$894$	9.50737	0.317974
$895$	−14.1829	−0.474083
$896$	−23.7364	−0.792977
$897$	−28.5909	−0.954622
$898$	−30.1976	−1.00771
$899$	2.88577	0.0962459
$900$	5.54884	0.184961
$901$	−14.7007	−0.489750
$902$	−12.8209	−0.426889
$903$	−23.4912	−0.781740
$904$	0.647572	0.0215379
$905$	27.7479	0.922370
$906$	−4.78338	−0.158917
$907$	16.7277	0.555433	0.277717	−	0.960663i	$-0.410422\pi$
$907$	16.7277	0.555433	0.277717	+	0.960663i	$0.410422\pi$
$908$	5.45204	0.180932
$909$	−3.11163	−0.103206
$910$	11.6206	0.385220
$911$	30.1621	0.999316	0.499658	−	0.866223i	$-0.333459\pi$
$911$	30.1621	0.999316	0.499658	+	0.866223i	$0.333459\pi$
$912$	−0.123993	−0.00410583
$913$	86.7038	2.86948
$914$	11.3810	0.376450
$915$	3.81108	0.125991
$916$	7.82871	0.258668
$917$	−33.7937	−1.11597
$918$	−1.12959	−0.0372822
$919$	21.3411	0.703977	0.351988	−	0.936004i	$-0.385506\pi$
$919$	21.3411	0.703977	0.351988	+	0.936004i	$0.385506\pi$
$920$	−12.8154	−0.422510
$921$	−28.4741	−0.938255
$922$	−6.24129	−0.205546
$923$	33.7041	1.10938
$924$	−22.2629	−0.732394
$925$	−35.3244	−1.16146
$926$	18.7766	0.617039
$927$	−1.00000	−0.0328443
$928$	10.1979	0.334761
$929$	21.2673	0.697756	0.348878	−	0.937168i	$-0.386563\pi$
$929$	21.2673	0.697756	0.348878	+	0.937168i	$0.386563\pi$
$930$	1.33472	0.0437673
$931$	−0.183699	−0.00602050
$932$	30.0186	0.983294
$933$	−9.42222	−0.308470
$934$	−15.5404	−0.508499
$935$	11.4501	0.374459
$936$	15.7130	0.513596
$937$	44.6237	1.45779	0.728896	−	0.684624i	$-0.240034\pi$
$937$	44.6237	1.45779	0.728896	+	0.684624i	$0.240034\pi$
$938$	−15.7058	−0.512812
$939$	18.7320	0.611297
$940$	−7.10140	−0.231622
$941$	32.1991	1.04966	0.524831	−	0.851207i	$-0.324129\pi$
$941$	32.1991	1.04966	0.524831	+	0.851207i	$0.324129\pi$
$942$	−0.900803	−0.0293497
$943$	−12.4193	−0.404429
$944$	−8.44645	−0.274909
$945$	2.57360	0.0837191
$946$	−48.1188	−1.56448
$947$	24.0908	0.782846	0.391423	−	0.920211i	$-0.371983\pi$
$947$	24.0908	0.782846	0.391423	+	0.920211i	$0.371983\pi$
$948$	2.54661	0.0827101
$949$	−47.0109	−1.52604
$950$	−0.291524	−0.00945829
$951$	8.57707	0.278130
$952$	−9.04637	−0.293194
$953$	−38.3273	−1.24154	−0.620772	−	0.783991i	$-0.713181\pi$
$953$	−38.3273	−1.24154	−0.620772	+	0.783991i	$0.713181\pi$
$954$	−6.76825	−0.219130
$955$	−3.51395	−0.113709
$956$	5.59495	0.180954
$957$	11.3978	0.368438
$958$	−10.3121	−0.333168
$959$	−5.62633	−0.181684
$960$	2.14449	0.0692130
$961$	−28.2609	−0.911643
$962$	−42.5404	−1.37156
$963$	1.98275	0.0638933
$964$	−6.33317	−0.203978
$965$	−29.5963	−0.952737
$966$	7.57848	0.243834
$967$	3.33790	0.107340	0.0536698	−	0.998559i	$-0.482908\pi$
$967$	3.33790	0.107340	0.0536698	+	0.998559i	$0.482908\pi$
$968$	−79.6254	−2.55926
$969$	−0.168878	−0.00542515
$970$	10.5688	0.339345
$971$	−2.64019	−0.0847279	−0.0423639	−	0.999102i	$-0.513489\pi$
$971$	−2.64019	−0.0847279	−0.0423639	+	0.999102i	$0.513489\pi$
$972$	1.47993	0.0474688
$973$	−38.9455	−1.24854
$974$	28.0291	0.898110
$975$	−23.4757	−0.751825
$976$	3.91927	0.125453
$977$	−28.8418	−0.922730	−0.461365	−	0.887211i	$-0.652640\pi$
$977$	−28.8418	−0.922730	−0.461365	+	0.887211i	$0.652640\pi$
$978$	−3.75325	−0.120016
$979$	−93.2344	−2.97978
$980$	−2.81985	−0.0900769
$981$	−6.95986	−0.222211
$982$	−4.35682	−0.139032
$983$	−29.1578	−0.929991	−0.464995	−	0.885313i	$-0.653944\pi$
$983$	−29.1578	−0.929991	−0.464995	+	0.885313i	$0.653944\pi$
$984$	6.82542	0.217587
$985$	−28.2809	−0.901106
$986$	1.96963	0.0627258
$987$	9.87472	0.314316
$988$	0.999034	0.0317835
$989$	−46.6117	−1.48217
$990$	5.27169	0.167545
$991$	23.4052	0.743490	0.371745	−	0.928335i	$-0.378760\pi$
$991$	23.4052	0.743490	0.371745	+	0.928335i	$0.378760\pi$
$992$	9.67938	0.307321
$993$	24.3674	0.773275
$994$	−8.93382	−0.283364
$995$	26.1494	0.828992
$996$	−19.6300	−0.622000
$997$	28.5746	0.904968	0.452484	−	0.891773i	$-0.350538\pi$
$997$	28.5746	0.904968	0.452484	+	0.891773i	$0.350538\pi$
$998$	−2.94928	−0.0933579
$999$	−9.42134	−0.298078

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	309.2.a.c.1.4	✓	5
3.2	odd	2		927.2.a.e.1.2		5
4.3	odd	2		4944.2.a.bb.1.3		5
5.4	even	2		7725.2.a.t.1.2		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
309.2.a.c.1.4	✓	5	1.1	even	1	trivial
927.2.a.e.1.2		5	3.2	odd	2
4944.2.a.bb.1.3		5	4.3	odd	2
7725.2.a.t.1.2		5	5.4	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 309 = 3 \cdot 103 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	309.a (trivial)

\(f(q)\)	\(=\)	\(q+0.721159 q^{2} -1.00000 q^{3} -1.47993 q^{4} -1.11830 q^{5} -0.721159 q^{6} +2.30134 q^{7} -2.50958 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+0.721159 q^{2} -1.00000 q^{3} -1.47993 q^{4} -1.11830 q^{5} -0.721159 q^{6} +2.30134 q^{7} -2.50958 q^{8} +1.00000 q^{9} -0.806475 q^{10} -6.53671 q^{11} +1.47993 q^{12} -6.26120 q^{13} +1.65963 q^{14} +1.11830 q^{15} +1.15005 q^{16} +1.56636 q^{17} +0.721159 q^{18} +0.107816 q^{19} +1.65501 q^{20} -2.30134 q^{21} -4.71401 q^{22} -4.56636 q^{23} +2.50958 q^{24} -3.74940 q^{25} -4.51532 q^{26} -1.00000 q^{27} -3.40582 q^{28} +1.74366 q^{29} +0.806475 q^{30} +1.65501 q^{31} +5.84854 q^{32} +6.53671 q^{33} +1.12959 q^{34} -2.57360 q^{35} -1.47993 q^{36} +9.42134 q^{37} +0.0777522 q^{38} +6.26120 q^{39} +2.80647 q^{40} +2.71974 q^{41} -1.65963 q^{42} +10.2076 q^{43} +9.67386 q^{44} -1.11830 q^{45} -3.29307 q^{46} -4.29085 q^{47} -1.15005 q^{48} -1.70383 q^{49} -2.70391 q^{50} -1.56636 q^{51} +9.26613 q^{52} -9.38524 q^{53} -0.721159 q^{54} +7.31002 q^{55} -5.77541 q^{56} -0.107816 q^{57} +1.25746 q^{58} -7.34442 q^{59} -1.65501 q^{60} +3.40792 q^{61} +1.19353 q^{62} +2.30134 q^{63} +1.91762 q^{64} +7.00192 q^{65} +4.71401 q^{66} -9.46340 q^{67} -2.31810 q^{68} +4.56636 q^{69} -1.85597 q^{70} -5.38301 q^{71} -2.50958 q^{72} +7.50829 q^{73} +6.79429 q^{74} +3.74940 q^{75} -0.159560 q^{76} -15.0432 q^{77} +4.51532 q^{78} +1.72076 q^{79} -1.28610 q^{80} +1.00000 q^{81} +1.96137 q^{82} -13.2641 q^{83} +3.40582 q^{84} -1.75167 q^{85} +7.36133 q^{86} -1.74366 q^{87} +16.4044 q^{88} +14.2632 q^{89} -0.806475 q^{90} -14.4092 q^{91} +6.75789 q^{92} -1.65501 q^{93} -3.09439 q^{94} -0.120571 q^{95} -5.84854 q^{96} -13.1050 q^{97} -1.22873 q^{98} -6.53671 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 2 q^{2} - 5 q^{3} + 2 q^{4} - 5 q^{5} + 2 q^{6} - 2 q^{7} - 6 q^{8} + 5 q^{9}+O(q^{10}) \) 5 * q - 2 * q^2 - 5 * q^3 + 2 * q^4 - 5 * q^5 + 2 * q^6 - 2 * q^7 - 6 * q^8 + 5 * q^9	\( 5 q - 2 q^{2} - 5 q^{3} + 2 q^{4} - 5 q^{5} + 2 q^{6} - 2 q^{7} - 6 q^{8} + 5 q^{9} - q^{10} - 12 q^{11} - 2 q^{12} + q^{13} - 8 q^{14} + 5 q^{15} - 4 q^{16} - 10 q^{17} - 2 q^{18} - 16 q^{19} - 13 q^{20} + 2 q^{21} + 4 q^{22} - 5 q^{23} + 6 q^{24} + 12 q^{25} - 10 q^{26} - 5 q^{27} + 7 q^{28} - 16 q^{29} + q^{30} - 13 q^{31} + 11 q^{32} + 12 q^{33} - 2 q^{34} - 22 q^{35} + 2 q^{36} + 4 q^{37} + 21 q^{38} - q^{39} + 11 q^{40} - 20 q^{41} + 8 q^{42} + 7 q^{43} + 2 q^{44} - 5 q^{45} + 8 q^{46} + 8 q^{47} + 4 q^{48} + 23 q^{49} + 13 q^{50} + 10 q^{51} + 31 q^{52} - 8 q^{53} + 2 q^{54} - 6 q^{55} + 5 q^{56} + 16 q^{57} + 20 q^{58} - 19 q^{59} + 13 q^{60} - 19 q^{61} + 9 q^{62} - 2 q^{63} - 16 q^{64} + q^{65} - 4 q^{66} + 11 q^{67} + 15 q^{68} + 5 q^{69} + 44 q^{70} - 10 q^{71} - 6 q^{72} + 20 q^{73} + 44 q^{74} - 12 q^{75} - 8 q^{76} + 8 q^{77} + 10 q^{78} - 14 q^{79} + 45 q^{80} + 5 q^{81} - 3 q^{82} - 11 q^{83} - 7 q^{84} + 12 q^{85} - 11 q^{86} + 16 q^{87} + 30 q^{88} + 22 q^{89} - q^{90} - 42 q^{91} - 21 q^{92} + 13 q^{93} - 6 q^{94} - 10 q^{95} - 11 q^{96} + 7 q^{97} - 11 q^{98} - 12 q^{99}+O(q^{100}) \) 5 * q - 2 * q^2 - 5 * q^3 + 2 * q^4 - 5 * q^5 + 2 * q^6 - 2 * q^7 - 6 * q^8 + 5 * q^9 - q^10 - 12 * q^11 - 2 * q^12 + q^13 - 8 * q^14 + 5 * q^15 - 4 * q^16 - 10 * q^17 - 2 * q^18 - 16 * q^19 - 13 * q^20 + 2 * q^21 + 4 * q^22 - 5 * q^23 + 6 * q^24 + 12 * q^25 - 10 * q^26 - 5 * q^27 + 7 * q^28 - 16 * q^29 + q^30 - 13 * q^31 + 11 * q^32 + 12 * q^33 - 2 * q^34 - 22 * q^35 + 2 * q^36 + 4 * q^37 + 21 * q^38 - q^39 + 11 * q^40 - 20 * q^41 + 8 * q^42 + 7 * q^43 + 2 * q^44 - 5 * q^45 + 8 * q^46 + 8 * q^47 + 4 * q^48 + 23 * q^49 + 13 * q^50 + 10 * q^51 + 31 * q^52 - 8 * q^53 + 2 * q^54 - 6 * q^55 + 5 * q^56 + 16 * q^57 + 20 * q^58 - 19 * q^59 + 13 * q^60 - 19 * q^61 + 9 * q^62 - 2 * q^63 - 16 * q^64 + q^65 - 4 * q^66 + 11 * q^67 + 15 * q^68 + 5 * q^69 + 44 * q^70 - 10 * q^71 - 6 * q^72 + 20 * q^73 + 44 * q^74 - 12 * q^75 - 8 * q^76 + 8 * q^77 + 10 * q^78 - 14 * q^79 + 45 * q^80 + 5 * q^81 - 3 * q^82 - 11 * q^83 - 7 * q^84 + 12 * q^85 - 11 * q^86 + 16 * q^87 + 30 * q^88 + 22 * q^89 - q^90 - 42 * q^91 - 21 * q^92 + 13 * q^93 - 6 * q^94 - 10 * q^95 - 11 * q^96 + 7 * q^97 - 11 * q^98 - 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more