LMFDB - Embedded newform 6042.2.a.bh.1.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6042,2,Mod(1,6042)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6042, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("6042.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6042 = 2 \cdot 3 \cdot 19 \cdot 53 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6042.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$48.2456129013$
Analytic rank:	$0$
Dimension:	$13$
Coefficient field:	$\mathbb{Q}[x]/(x^{13} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{13} - 3 x^{12} - 40 x^{11} + 123 x^{10} + 537 x^{9} - 1707 x^{8} - 2914 x^{7} + 9639 x^{6} + \cdots + 1200 $ x^13 - 3x^12 - 40x^11 + 123x^10 + 537x^9 - 1707x^8 - 2914x^7 + 9639x^6 + 6938x^5 - 22200x^4 - 9466x^3 + 16812x^2 + 9304x + 1200
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
Root			$3.50660$ of defining polynomial
Character	$\chi$	$=$	6042.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +3.50660 q^{5} +1.00000 q^{6} -4.14603 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +3.50660 q^{5} +1.00000 q^{6} -4.14603 q^{7} +1.00000 q^{8} +1.00000 q^{9} +3.50660 q^{10} +1.87582 q^{11} +1.00000 q^{12} +6.85620 q^{13} -4.14603 q^{14} +3.50660 q^{15} +1.00000 q^{16} +0.160684 q^{17} +1.00000 q^{18} +1.00000 q^{19} +3.50660 q^{20} -4.14603 q^{21} +1.87582 q^{22} -6.46183 q^{23} +1.00000 q^{24} +7.29628 q^{25} +6.85620 q^{26} +1.00000 q^{27} -4.14603 q^{28} -7.03603 q^{29} +3.50660 q^{30} +1.88027 q^{31} +1.00000 q^{32} +1.87582 q^{33} +0.160684 q^{34} -14.5385 q^{35} +1.00000 q^{36} +5.70041 q^{37} +1.00000 q^{38} +6.85620 q^{39} +3.50660 q^{40} +1.04550 q^{41} -4.14603 q^{42} +7.82947 q^{43} +1.87582 q^{44} +3.50660 q^{45} -6.46183 q^{46} +2.58896 q^{47} +1.00000 q^{48} +10.1896 q^{49} +7.29628 q^{50} +0.160684 q^{51} +6.85620 q^{52} -1.00000 q^{53} +1.00000 q^{54} +6.57777 q^{55} -4.14603 q^{56} +1.00000 q^{57} -7.03603 q^{58} -2.76143 q^{59} +3.50660 q^{60} +12.8453 q^{61} +1.88027 q^{62} -4.14603 q^{63} +1.00000 q^{64} +24.0420 q^{65} +1.87582 q^{66} +1.88407 q^{67} +0.160684 q^{68} -6.46183 q^{69} -14.5385 q^{70} +0.685532 q^{71} +1.00000 q^{72} -10.9852 q^{73} +5.70041 q^{74} +7.29628 q^{75} +1.00000 q^{76} -7.77722 q^{77} +6.85620 q^{78} +3.64982 q^{79} +3.50660 q^{80} +1.00000 q^{81} +1.04550 q^{82} +1.89763 q^{83} -4.14603 q^{84} +0.563456 q^{85} +7.82947 q^{86} -7.03603 q^{87} +1.87582 q^{88} +4.08516 q^{89} +3.50660 q^{90} -28.4260 q^{91} -6.46183 q^{92} +1.88027 q^{93} +2.58896 q^{94} +3.50660 q^{95} +1.00000 q^{96} +1.44939 q^{97} +10.1896 q^{98} +1.87582 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 13 q + 13 q^{2} + 13 q^{3} + 13 q^{4} + 3 q^{5} + 13 q^{6} + 12 q^{7} + 13 q^{8} + 13 q^{9}+O(q^{10}) $ 13 * q + 13 * q^2 + 13 * q^3 + 13 * q^4 + 3 * q^5 + 13 * q^6 + 12 * q^7 + 13 * q^8 + 13 * q^9	\( 13 q + 13 q^{2} + 13 q^{3} + 13 q^{4} + 3 q^{5} + 13 q^{6} + 12 q^{7} + 13 q^{8} + 13 q^{9} + 3 q^{10} + 4 q^{11} + 13 q^{12} + 18 q^{13} + 12 q^{14} + 3 q^{15} + 13 q^{16} + 3 q^{17} + 13 q^{18} + 13 q^{19} + 3 q^{20} + 12 q^{21} + 4 q^{22} + 8 q^{23} + 13 q^{24} + 24 q^{25} + 18 q^{26} + 13 q^{27} + 12 q^{28} + 11 q^{29} + 3 q^{30} + 2 q^{31} + 13 q^{32} + 4 q^{33} + 3 q^{34} - 6 q^{35} + 13 q^{36} + 26 q^{37} + 13 q^{38} + 18 q^{39} + 3 q^{40} + 3 q^{41} + 12 q^{42} + 24 q^{43} + 4 q^{44} + 3 q^{45} + 8 q^{46} + 4 q^{47} + 13 q^{48} + 41 q^{49} + 24 q^{50} + 3 q^{51} + 18 q^{52} - 13 q^{53} + 13 q^{54} + 23 q^{55} + 12 q^{56} + 13 q^{57} + 11 q^{58} + 8 q^{59} + 3 q^{60} + 29 q^{61} + 2 q^{62} + 12 q^{63} + 13 q^{64} + 13 q^{65} + 4 q^{66} - 5 q^{67} + 3 q^{68} + 8 q^{69} - 6 q^{70} + 29 q^{71} + 13 q^{72} + 15 q^{73} + 26 q^{74} + 24 q^{75} + 13 q^{76} + 3 q^{77} + 18 q^{78} + 15 q^{79} + 3 q^{80} + 13 q^{81} + 3 q^{82} + 4 q^{83} + 12 q^{84} - q^{85} + 24 q^{86} + 11 q^{87} + 4 q^{88} + 3 q^{89} + 3 q^{90} - 29 q^{91} + 8 q^{92} + 2 q^{93} + 4 q^{94} + 3 q^{95} + 13 q^{96} + 50 q^{97} + 41 q^{98} + 4 q^{99}+O(q^{100}) \) 13 * q + 13 * q^2 + 13 * q^3 + 13 * q^4 + 3 * q^5 + 13 * q^6 + 12 * q^7 + 13 * q^8 + 13 * q^9 + 3 * q^10 + 4 * q^11 + 13 * q^12 + 18 * q^13 + 12 * q^14 + 3 * q^15 + 13 * q^16 + 3 * q^17 + 13 * q^18 + 13 * q^19 + 3 * q^20 + 12 * q^21 + 4 * q^22 + 8 * q^23 + 13 * q^24 + 24 * q^25 + 18 * q^26 + 13 * q^27 + 12 * q^28 + 11 * q^29 + 3 * q^30 + 2 * q^31 + 13 * q^32 + 4 * q^33 + 3 * q^34 - 6 * q^35 + 13 * q^36 + 26 * q^37 + 13 * q^38 + 18 * q^39 + 3 * q^40 + 3 * q^41 + 12 * q^42 + 24 * q^43 + 4 * q^44 + 3 * q^45 + 8 * q^46 + 4 * q^47 + 13 * q^48 + 41 * q^49 + 24 * q^50 + 3 * q^51 + 18 * q^52 - 13 * q^53 + 13 * q^54 + 23 * q^55 + 12 * q^56 + 13 * q^57 + 11 * q^58 + 8 * q^59 + 3 * q^60 + 29 * q^61 + 2 * q^62 + 12 * q^63 + 13 * q^64 + 13 * q^65 + 4 * q^66 - 5 * q^67 + 3 * q^68 + 8 * q^69 - 6 * q^70 + 29 * q^71 + 13 * q^72 + 15 * q^73 + 26 * q^74 + 24 * q^75 + 13 * q^76 + 3 * q^77 + 18 * q^78 + 15 * q^79 + 3 * q^80 + 13 * q^81 + 3 * q^82 + 4 * q^83 + 12 * q^84 - q^85 + 24 * q^86 + 11 * q^87 + 4 * q^88 + 3 * q^89 + 3 * q^90 - 29 * q^91 + 8 * q^92 + 2 * q^93 + 4 * q^94 + 3 * q^95 + 13 * q^96 + 50 * q^97 + 41 * q^98 + 4 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.00000	0.500000
$5$	3.50660	1.56820	0.784101	−	0.620634i	$-0.213125\pi$
$5$	3.50660	1.56820	0.784101	+	0.620634i	$0.213125\pi$
$6$	1.00000	0.408248
$7$	−4.14603	−1.56705	−0.783527	−	0.621358i	$-0.786581\pi$
$7$	−4.14603	−1.56705	−0.783527	+	0.621358i	$0.786581\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	3.50660	1.10889
$11$	1.87582	0.565582	0.282791	−	0.959182i	$-0.408740\pi$
$11$	1.87582	0.565582	0.282791	+	0.959182i	$0.408740\pi$
$12$	1.00000	0.288675
$13$	6.85620	1.90157	0.950783	−	0.309856i	$-0.100281\pi$
$13$	6.85620	1.90157	0.950783	+	0.309856i	$0.100281\pi$
$14$	−4.14603	−1.10807
$15$	3.50660	0.905401
$16$	1.00000	0.250000
$17$	0.160684	0.0389716	0.0194858	−	0.999810i	$-0.493797\pi$
$17$	0.160684	0.0389716	0.0194858	+	0.999810i	$0.493797\pi$
$18$	1.00000	0.235702
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	3.50660	0.784101
$21$	−4.14603	−0.904739
$22$	1.87582	0.399927
$23$	−6.46183	−1.34739	−0.673693	−	0.739011i	$-0.735293\pi$
$23$	−6.46183	−1.34739	−0.673693	+	0.739011i	$0.735293\pi$
$24$	1.00000	0.204124
$25$	7.29628	1.45926
$26$	6.85620	1.34461
$27$	1.00000	0.192450
$28$	−4.14603	−0.783527
$29$	−7.03603	−1.30656	−0.653279	−	0.757117i	$-0.726607\pi$
$29$	−7.03603	−1.30656	−0.653279	+	0.757117i	$0.726607\pi$
$30$	3.50660	0.640216
$31$	1.88027	0.337707	0.168854	−	0.985641i	$-0.445993\pi$
$31$	1.88027	0.337707	0.168854	+	0.985641i	$0.445993\pi$
$32$	1.00000	0.176777
$33$	1.87582	0.326539
$34$	0.160684	0.0275571
$35$	−14.5385	−2.45746
$36$	1.00000	0.166667
$37$	5.70041	0.937141	0.468571	−	0.883426i	$-0.344769\pi$
$37$	5.70041	0.937141	0.468571	+	0.883426i	$0.344769\pi$
$38$	1.00000	0.162221
$39$	6.85620	1.09787
$40$	3.50660	0.554443
$41$	1.04550	0.163280	0.0816401	−	0.996662i	$-0.473984\pi$
$41$	1.04550	0.163280	0.0816401	+	0.996662i	$0.473984\pi$
$42$	−4.14603	−0.639747
$43$	7.82947	1.19398	0.596991	−	0.802248i	$-0.296363\pi$
$43$	7.82947	1.19398	0.596991	+	0.802248i	$0.296363\pi$
$44$	1.87582	0.282791
$45$	3.50660	0.522734
$46$	−6.46183	−0.952746
$47$	2.58896	0.377638	0.188819	−	0.982012i	$-0.439534\pi$
$47$	2.58896	0.377638	0.188819	+	0.982012i	$0.439534\pi$
$48$	1.00000	0.144338
$49$	10.1896	1.45566
$50$	7.29628	1.03185
$51$	0.160684	0.0225003
$52$	6.85620	0.950783
$53$	−1.00000	−0.137361
$53$	−1.00000	−0.137361
$54$	1.00000	0.136083
$55$	6.57777	0.886946
$56$	−4.14603	−0.554037
$57$	1.00000	0.132453
$58$	−7.03603	−0.923877
$59$	−2.76143	−0.359507	−0.179754	−	0.983712i	$-0.557530\pi$
$59$	−2.76143	−0.359507	−0.179754	+	0.983712i	$0.557530\pi$
$60$	3.50660	0.452701
$61$	12.8453	1.64467	0.822334	−	0.569005i	$-0.192672\pi$
$61$	12.8453	1.64467	0.822334	+	0.569005i	$0.192672\pi$
$62$	1.88027	0.238795
$63$	−4.14603	−0.522351
$64$	1.00000	0.125000
$65$	24.0420	2.98204
$66$	1.87582	0.230898
$67$	1.88407	0.230175	0.115088	−	0.993355i	$-0.463285\pi$
$67$	1.88407	0.230175	0.115088	+	0.993355i	$0.463285\pi$
$68$	0.160684	0.0194858
$69$	−6.46183	−0.777914
$70$	−14.5385	−1.73768
$71$	0.685532	0.0813577	0.0406789	−	0.999172i	$-0.487048\pi$
$71$	0.685532	0.0813577	0.0406789	+	0.999172i	$0.487048\pi$
$72$	1.00000	0.117851
$73$	−10.9852	−1.28572	−0.642862	−	0.765982i	$-0.722253\pi$
$73$	−10.9852	−1.28572	−0.642862	+	0.765982i	$0.722253\pi$
$74$	5.70041	0.662659
$75$	7.29628	0.842502
$76$	1.00000	0.114708
$77$	−7.77722	−0.886297
$78$	6.85620	0.776311
$79$	3.64982	0.410636	0.205318	−	0.978695i	$-0.434177\pi$
$79$	3.64982	0.410636	0.205318	+	0.978695i	$0.434177\pi$
$80$	3.50660	0.392050
$81$	1.00000	0.111111
$82$	1.04550	0.115457
$83$	1.89763	0.208292	0.104146	−	0.994562i	$-0.466789\pi$
$83$	1.89763	0.208292	0.104146	+	0.994562i	$0.466789\pi$
$84$	−4.14603	−0.452369
$85$	0.563456	0.0611153
$86$	7.82947	0.844273
$87$	−7.03603	−0.754342
$88$	1.87582	0.199963
$89$	4.08516	0.433026	0.216513	−	0.976280i	$-0.430532\pi$
$89$	4.08516	0.433026	0.216513	+	0.976280i	$0.430532\pi$
$90$	3.50660	0.369629
$91$	−28.4260	−2.97986
$92$	−6.46183	−0.673693
$93$	1.88027	0.194975
$94$	2.58896	0.267030
$95$	3.50660	0.359770
$96$	1.00000	0.102062
$97$	1.44939	0.147163	0.0735815	−	0.997289i	$-0.476557\pi$
$97$	1.44939	0.147163	0.0735815	+	0.997289i	$0.476557\pi$
$98$	10.1896	1.02930
$99$	1.87582	0.188527
$100$	7.29628	0.729628
$101$	−4.98376	−0.495902	−0.247951	−	0.968773i	$-0.579757\pi$
$101$	−4.98376	−0.495902	−0.247951	+	0.968773i	$0.579757\pi$
$102$	0.160684	0.0159101
$103$	−14.5901	−1.43760	−0.718802	−	0.695215i	$-0.755309\pi$
$103$	−14.5901	−1.43760	−0.718802	+	0.695215i	$0.755309\pi$
$104$	6.85620	0.672305
$105$	−14.5385	−1.41881
$106$	−1.00000	−0.0971286
$107$	−16.4245	−1.58781	−0.793906	−	0.608040i	$-0.791956\pi$
$107$	−16.4245	−1.58781	−0.793906	+	0.608040i	$0.791956\pi$
$108$	1.00000	0.0962250
$109$	−6.62512	−0.634572	−0.317286	−	0.948330i	$-0.602771\pi$
$109$	−6.62512	−0.634572	−0.317286	+	0.948330i	$0.602771\pi$
$110$	6.57777	0.627166
$111$	5.70041	0.541059
$112$	−4.14603	−0.391763
$113$	13.9304	1.31046	0.655231	−	0.755428i	$-0.272571\pi$
$113$	13.9304	1.31046	0.655231	+	0.755428i	$0.272571\pi$
$114$	1.00000	0.0936586
$115$	−22.6591	−2.11297
$116$	−7.03603	−0.653279
$117$	6.85620	0.633856
$118$	−2.76143	−0.254210
$119$	−0.666201	−0.0610706
$120$	3.50660	0.320108
$121$	−7.48129	−0.680117
$122$	12.8453	1.16296
$123$	1.04550	0.0942699
$124$	1.88027	0.168854
$125$	8.05214	0.720205
$126$	−4.14603	−0.369358
$127$	19.9198	1.76760	0.883798	−	0.467869i	$-0.154978\pi$
$127$	19.9198	1.76760	0.883798	+	0.467869i	$0.154978\pi$
$128$	1.00000	0.0883883
$129$	7.82947	0.689346
$130$	24.0420	2.10862
$131$	8.71021	0.761015	0.380508	−	0.924778i	$-0.375749\pi$
$131$	8.71021	0.761015	0.380508	+	0.924778i	$0.375749\pi$
$132$	1.87582	0.163269
$133$	−4.14603	−0.359507
$134$	1.88407	0.162759
$135$	3.50660	0.301800
$136$	0.160684	0.0137785
$137$	−9.62749	−0.822532	−0.411266	−	0.911515i	$-0.634913\pi$
$137$	−9.62749	−0.822532	−0.411266	+	0.911515i	$0.634913\pi$
$138$	−6.46183	−0.550068
$139$	1.46891	0.124591	0.0622955	−	0.998058i	$-0.480158\pi$
$139$	1.46891	0.124591	0.0622955	+	0.998058i	$0.480158\pi$
$140$	−14.5385	−1.22873
$141$	2.58896	0.218029
$142$	0.685532	0.0575286
$143$	12.8610	1.07549
$144$	1.00000	0.0833333
$145$	−24.6726	−2.04895
$146$	−10.9852	−0.909144
$147$	10.1896	0.840423
$148$	5.70041	0.468571
$149$	−3.89058	−0.318729	−0.159364	−	0.987220i	$-0.550944\pi$
$149$	−3.89058	−0.318729	−0.159364	+	0.987220i	$0.550944\pi$
$150$	7.29628	0.595739
$151$	−20.4546	−1.66457	−0.832287	−	0.554345i	$-0.812969\pi$
$151$	−20.4546	−1.66457	−0.832287	+	0.554345i	$0.812969\pi$
$152$	1.00000	0.0811107
$153$	0.160684	0.0129905
$154$	−7.77722	−0.626706
$155$	6.59338	0.529593
$156$	6.85620	0.548935
$157$	8.60438	0.686704	0.343352	−	0.939207i	$-0.388438\pi$
$157$	8.60438	0.686704	0.343352	+	0.939207i	$0.388438\pi$
$158$	3.64982	0.290364
$159$	−1.00000	−0.0793052
$160$	3.50660	0.277221
$161$	26.7910	2.11143
$162$	1.00000	0.0785674
$163$	−8.49489	−0.665371	−0.332686	−	0.943038i	$-0.607955\pi$
$163$	−8.49489	−0.665371	−0.332686	+	0.943038i	$0.607955\pi$
$164$	1.04550	0.0816401
$165$	6.57777	0.512079
$166$	1.89763	0.147285
$167$	21.8706	1.69240	0.846200	−	0.532865i	$-0.178885\pi$
$167$	21.8706	1.69240	0.846200	+	0.532865i	$0.178885\pi$
$168$	−4.14603	−0.319873
$169$	34.0074	2.61596
$170$	0.563456	0.0432151
$171$	1.00000	0.0764719
$172$	7.82947	0.596991
$173$	−4.56370	−0.346972	−0.173486	−	0.984836i	$-0.555503\pi$
$173$	−4.56370	−0.346972	−0.173486	+	0.984836i	$0.555503\pi$
$174$	−7.03603	−0.533400
$175$	−30.2506	−2.28673
$176$	1.87582	0.141395
$177$	−2.76143	−0.207562
$178$	4.08516	0.306196
$179$	−22.6443	−1.69251	−0.846256	−	0.532777i	$-0.821149\pi$
$179$	−22.6443	−1.69251	−0.846256	+	0.532777i	$0.821149\pi$
$180$	3.50660	0.261367
$181$	4.18383	0.310982	0.155491	−	0.987837i	$-0.450304\pi$
$181$	4.18383	0.310982	0.155491	+	0.987837i	$0.450304\pi$
$182$	−28.4260	−2.10708
$183$	12.8453	0.949549
$184$	−6.46183	−0.476373
$185$	19.9891	1.46963
$186$	1.88027	0.137868
$187$	0.301415	0.0220416
$188$	2.58896	0.188819
$189$	−4.14603	−0.301580
$190$	3.50660	0.254396
$191$	3.35058	0.242440	0.121220	−	0.992626i	$-0.461319\pi$
$191$	3.35058	0.242440	0.121220	+	0.992626i	$0.461319\pi$
$192$	1.00000	0.0721688
$193$	−11.2723	−0.811399	−0.405700	−	0.914006i	$-0.632972\pi$
$193$	−11.2723	−0.811399	−0.405700	+	0.914006i	$0.632972\pi$
$194$	1.44939	0.104060
$195$	24.0420	1.72168
$196$	10.1896	0.727828
$197$	3.04684	0.217078	0.108539	−	0.994092i	$-0.465383\pi$
$197$	3.04684	0.217078	0.108539	+	0.994092i	$0.465383\pi$
$198$	1.87582	0.133309
$199$	−7.28348	−0.516312	−0.258156	−	0.966103i	$-0.583115\pi$
$199$	−7.28348	−0.516312	−0.258156	+	0.966103i	$0.583115\pi$
$200$	7.29628	0.515925
$201$	1.88407	0.132892
$202$	−4.98376	−0.350656
$203$	29.1716	2.04745
$204$	0.160684	0.0112501
$205$	3.66617	0.256056
$206$	−14.5901	−1.01654
$207$	−6.46183	−0.449129
$208$	6.85620	0.475392
$209$	1.87582	0.129753
$210$	−14.5385	−1.00325
$211$	17.6367	1.21416	0.607079	−	0.794642i	$-0.292341\pi$
$211$	17.6367	1.21416	0.607079	+	0.794642i	$0.292341\pi$
$212$	−1.00000	−0.0686803
$213$	0.685532	0.0469719
$214$	−16.4245	−1.12275
$215$	27.4549	1.87241
$216$	1.00000	0.0680414
$217$	−7.79568	−0.529205
$218$	−6.62512	−0.448710
$219$	−10.9852	−0.742313
$220$	6.57777	0.443473
$221$	1.10168	0.0741071
$222$	5.70041	0.382586
$223$	13.6624	0.914900	0.457450	−	0.889235i	$-0.348763\pi$
$223$	13.6624	0.914900	0.457450	+	0.889235i	$0.348763\pi$
$224$	−4.14603	−0.277018
$225$	7.29628	0.486419
$226$	13.9304	0.926637
$227$	−22.1629	−1.47100	−0.735502	−	0.677523i	$-0.763053\pi$
$227$	−22.1629	−1.47100	−0.735502	+	0.677523i	$0.763053\pi$
$228$	1.00000	0.0662266
$229$	2.21110	0.146114	0.0730569	−	0.997328i	$-0.476725\pi$
$229$	2.21110	0.146114	0.0730569	+	0.997328i	$0.476725\pi$
$230$	−22.6591	−1.49410
$231$	−7.77722	−0.511704
$232$	−7.03603	−0.461938
$233$	−2.48380	−0.162719	−0.0813595	−	0.996685i	$-0.525926\pi$
$233$	−2.48380	−0.162719	−0.0813595	+	0.996685i	$0.525926\pi$
$234$	6.85620	0.448204
$235$	9.07844	0.592212
$236$	−2.76143	−0.179754
$237$	3.64982	0.237081
$238$	−0.666201	−0.0431834
$239$	−11.2741	−0.729260	−0.364630	−	0.931152i	$-0.618805\pi$
$239$	−11.2741	−0.729260	−0.364630	+	0.931152i	$0.618805\pi$
$240$	3.50660	0.226350
$241$	18.5719	1.19632	0.598160	−	0.801377i	$-0.295899\pi$
$241$	18.5719	1.19632	0.598160	+	0.801377i	$0.295899\pi$
$242$	−7.48129	−0.480916
$243$	1.00000	0.0641500
$244$	12.8453	0.822334
$245$	35.7309	2.28276
$246$	1.04550	0.0666589
$247$	6.85620	0.436249
$248$	1.88027	0.119398
$249$	1.89763	0.120257
$250$	8.05214	0.509262
$251$	−17.7086	−1.11776	−0.558878	−	0.829250i	$-0.688768\pi$
$251$	−17.7086	−1.11776	−0.558878	+	0.829250i	$0.688768\pi$
$252$	−4.14603	−0.261176
$253$	−12.1213	−0.762057
$254$	19.9198	1.24988
$255$	0.563456	0.0352850
$256$	1.00000	0.0625000
$257$	−8.32063	−0.519026	−0.259513	−	0.965740i	$-0.583562\pi$
$257$	−8.32063	−0.519026	−0.259513	+	0.965740i	$0.583562\pi$
$258$	7.82947	0.487441
$259$	−23.6341	−1.46855
$260$	24.0420	1.49102
$261$	−7.03603	−0.435520
$262$	8.71021	0.538119
$263$	−19.7117	−1.21548	−0.607739	−	0.794137i	$-0.707923\pi$
$263$	−19.7117	−1.21548	−0.607739	+	0.794137i	$0.707923\pi$
$264$	1.87582	0.115449
$265$	−3.50660	−0.215409
$266$	−4.14603	−0.254210
$267$	4.08516	0.250008
$268$	1.88407	0.115088
$269$	7.62818	0.465098	0.232549	−	0.972585i	$-0.425293\pi$
$269$	7.62818	0.465098	0.232549	+	0.972585i	$0.425293\pi$
$270$	3.50660	0.213405
$271$	−20.1260	−1.22257	−0.611284	−	0.791412i	$-0.709347\pi$
$271$	−20.1260	−1.22257	−0.611284	+	0.791412i	$0.709347\pi$
$272$	0.160684	0.00974290
$273$	−28.4260	−1.72042
$274$	−9.62749	−0.581618
$275$	13.6865	0.825328
$276$	−6.46183	−0.388957
$277$	19.7368	1.18587	0.592934	−	0.805251i	$-0.297969\pi$
$277$	19.7368	1.18587	0.592934	+	0.805251i	$0.297969\pi$
$278$	1.46891	0.0880991
$279$	1.88027	0.112569
$280$	−14.5385	−0.868842
$281$	28.0426	1.67288	0.836440	−	0.548059i	$-0.184633\pi$
$281$	28.0426	1.67288	0.836440	+	0.548059i	$0.184633\pi$
$282$	2.58896	0.154170
$283$	9.44679	0.561553	0.280777	−	0.959773i	$-0.409408\pi$
$283$	9.44679	0.561553	0.280777	+	0.959773i	$0.409408\pi$
$284$	0.685532	0.0406789
$285$	3.50660	0.207713
$286$	12.8610	0.760487
$287$	−4.33469	−0.255869
$288$	1.00000	0.0589256
$289$	−16.9742	−0.998481
$290$	−24.6726	−1.44882
$291$	1.44939	0.0849646
$292$	−10.9852	−0.642862
$293$	6.08366	0.355411	0.177706	−	0.984084i	$-0.443132\pi$
$293$	6.08366	0.355411	0.177706	+	0.984084i	$0.443132\pi$
$294$	10.1896	0.594269
$295$	−9.68323	−0.563780
$296$	5.70041	0.331330
$297$	1.87582	0.108846
$298$	−3.89058	−0.225375
$299$	−44.3036	−2.56214
$300$	7.29628	0.421251
$301$	−32.4612	−1.87103
$302$	−20.4546	−1.17703
$303$	−4.98376	−0.286309
$304$	1.00000	0.0573539
$305$	45.0433	2.57917
$306$	0.160684	0.00918570
$307$	−25.4356	−1.45169	−0.725844	−	0.687860i	$-0.758550\pi$
$307$	−25.4356	−1.45169	−0.725844	+	0.687860i	$0.758550\pi$
$308$	−7.77722	−0.443148
$309$	−14.5901	−0.830001
$310$	6.59338	0.374479
$311$	25.2517	1.43190	0.715948	−	0.698154i	$-0.245995\pi$
$311$	25.2517	1.43190	0.715948	+	0.698154i	$0.245995\pi$
$312$	6.85620	0.388156
$313$	−10.9429	−0.618530	−0.309265	−	0.950976i	$-0.600083\pi$
$313$	−10.9429	−0.618530	−0.309265	+	0.950976i	$0.600083\pi$
$314$	8.60438	0.485573
$315$	−14.5385	−0.819152
$316$	3.64982	0.205318
$317$	−6.09834	−0.342517	−0.171258	−	0.985226i	$-0.554783\pi$
$317$	−6.09834	−0.342517	−0.171258	+	0.985226i	$0.554783\pi$
$318$	−1.00000	−0.0560772
$319$	−13.1984	−0.738966
$320$	3.50660	0.196025
$321$	−16.4245	−0.916724
$322$	26.7910	1.49300
$323$	0.160684	0.00894070
$324$	1.00000	0.0555556
$325$	50.0247	2.77487
$326$	−8.49489	−0.470488
$327$	−6.62512	−0.366370
$328$	1.04550	0.0577283
$329$	−10.7339	−0.591779
$330$	6.57777	0.362094
$331$	8.58797	0.472038	0.236019	−	0.971748i	$-0.424157\pi$
$331$	8.58797	0.472038	0.236019	+	0.971748i	$0.424157\pi$
$332$	1.89763	0.104146
$333$	5.70041	0.312380
$334$	21.8706	1.19671
$335$	6.60668	0.360961
$336$	−4.14603	−0.226185
$337$	13.9300	0.758815	0.379407	−	0.925230i	$-0.376128\pi$
$337$	13.9300	0.758815	0.379407	+	0.925230i	$0.376128\pi$
$338$	34.0074	1.84976
$339$	13.9304	0.756596
$340$	0.563456	0.0305577
$341$	3.52706	0.191001
$342$	1.00000	0.0540738
$343$	−13.2241	−0.714037
$344$	7.82947	0.422137
$345$	−22.6591	−1.21993
$346$	−4.56370	−0.245346
$347$	35.4976	1.90561	0.952806	−	0.303580i	$-0.0981821\pi$
$347$	35.4976	1.90561	0.952806	+	0.303580i	$0.0981821\pi$
$348$	−7.03603	−0.377171
$349$	−1.03977	−0.0556576	−0.0278288	−	0.999613i	$-0.508859\pi$
$349$	−1.03977	−0.0556576	−0.0278288	+	0.999613i	$0.508859\pi$
$350$	−30.2506	−1.61696
$351$	6.85620	0.365957
$352$	1.87582	0.0999817
$353$	7.14551	0.380317	0.190158	−	0.981753i	$-0.439100\pi$
$353$	7.14551	0.380317	0.190158	+	0.981753i	$0.439100\pi$
$354$	−2.76143	−0.146768
$355$	2.40389	0.127585
$356$	4.08516	0.216513
$357$	−0.666201	−0.0352591
$358$	−22.6443	−1.19679
$359$	1.00046	0.0528025	0.0264012	−	0.999651i	$-0.491595\pi$
$359$	1.00046	0.0528025	0.0264012	+	0.999651i	$0.491595\pi$
$360$	3.50660	0.184814
$361$	1.00000	0.0526316
$362$	4.18383	0.219897
$363$	−7.48129	−0.392666
$364$	−28.4260	−1.48993
$365$	−38.5209	−2.01627
$366$	12.8453	0.671433
$367$	−24.0160	−1.25362	−0.626812	−	0.779171i	$-0.715641\pi$
$367$	−24.0160	−1.25362	−0.626812	+	0.779171i	$0.715641\pi$
$368$	−6.46183	−0.336846
$369$	1.04550	0.0544267
$370$	19.9891	1.03918
$371$	4.14603	0.215251
$372$	1.88027	0.0974877
$373$	17.7378	0.918426	0.459213	−	0.888326i	$-0.348131\pi$
$373$	17.7378	0.918426	0.459213	+	0.888326i	$0.348131\pi$
$374$	0.301415	0.0155858
$375$	8.05214	0.415811
$376$	2.58896	0.133515
$377$	−48.2404	−2.48451
$378$	−4.14603	−0.213249
$379$	−11.9893	−0.615848	−0.307924	−	0.951411i	$-0.599634\pi$
$379$	−11.9893	−0.615848	−0.307924	+	0.951411i	$0.599634\pi$
$380$	3.50660	0.179885
$381$	19.9198	1.02052
$382$	3.35058	0.171431
$383$	−21.6370	−1.10560	−0.552799	−	0.833315i	$-0.686440\pi$
$383$	−21.6370	−1.10560	−0.552799	+	0.833315i	$0.686440\pi$
$384$	1.00000	0.0510310
$385$	−27.2716	−1.38989
$386$	−11.2723	−0.573746
$387$	7.82947	0.397994
$388$	1.44939	0.0735815
$389$	7.90562	0.400831	0.200415	−	0.979711i	$-0.435771\pi$
$389$	7.90562	0.400831	0.200415	+	0.979711i	$0.435771\pi$
$390$	24.0420	1.21741
$391$	−1.03831	−0.0525098
$392$	10.1896	0.514652
$393$	8.71021	0.439372
$394$	3.04684	0.153497
$395$	12.7985	0.643960
$396$	1.87582	0.0942636
$397$	13.1820	0.661587	0.330793	−	0.943703i	$-0.392684\pi$
$397$	13.1820	0.661587	0.330793	+	0.943703i	$0.392684\pi$
$398$	−7.28348	−0.365088
$399$	−4.14603	−0.207561
$400$	7.29628	0.364814
$401$	−15.8831	−0.793162	−0.396581	−	0.918000i	$-0.629803\pi$
$401$	−15.8831	−0.793162	−0.396581	+	0.918000i	$0.629803\pi$
$402$	1.88407	0.0939687
$403$	12.8915	0.642173
$404$	−4.98376	−0.247951
$405$	3.50660	0.174245
$406$	29.1716	1.44776
$407$	10.6930	0.530030
$408$	0.160684	0.00795505
$409$	8.46143	0.418391	0.209195	−	0.977874i	$-0.432916\pi$
$409$	8.46143	0.418391	0.209195	+	0.977874i	$0.432916\pi$
$410$	3.66617	0.181059
$411$	−9.62749	−0.474889
$412$	−14.5901	−0.718802
$413$	11.4490	0.563367
$414$	−6.46183	−0.317582
$415$	6.65424	0.326644
$416$	6.85620	0.336153
$417$	1.46891	0.0719326
$418$	1.87582	0.0917495
$419$	−20.1955	−0.986614	−0.493307	−	0.869855i	$-0.664212\pi$
$419$	−20.1955	−0.986614	−0.493307	+	0.869855i	$0.664212\pi$
$420$	−14.5385	−0.709406
$421$	6.43295	0.313523	0.156762	−	0.987636i	$-0.449895\pi$
$421$	6.43295	0.313523	0.156762	+	0.987636i	$0.449895\pi$
$422$	17.6367	0.858539
$423$	2.58896	0.125879
$424$	−1.00000	−0.0485643
$425$	1.17240	0.0568695
$426$	0.685532	0.0332141
$427$	−53.2569	−2.57728
$428$	−16.4245	−0.793906
$429$	12.8610	0.620935
$430$	27.4549	1.32399
$431$	−6.82924	−0.328953	−0.164476	−	0.986381i	$-0.552593\pi$
$431$	−6.82924	−0.328953	−0.164476	+	0.986381i	$0.552593\pi$
$432$	1.00000	0.0481125
$433$	−29.8446	−1.43424	−0.717120	−	0.696950i	$-0.754540\pi$
$433$	−29.8446	−1.43424	−0.717120	+	0.696950i	$0.754540\pi$
$434$	−7.79568	−0.374205
$435$	−24.6726	−1.18296
$436$	−6.62512	−0.317286
$437$	−6.46183	−0.309111
$438$	−10.9852	−0.524895
$439$	−32.8706	−1.56883	−0.784415	−	0.620237i	$-0.787036\pi$
$439$	−32.8706	−1.56883	−0.784415	+	0.620237i	$0.787036\pi$
$440$	6.57777	0.313583
$441$	10.1896	0.485219
$442$	1.10168	0.0524016
$443$	−14.7994	−0.703140	−0.351570	−	0.936162i	$-0.614352\pi$
$443$	−14.7994	−0.703140	−0.351570	+	0.936162i	$0.614352\pi$
$444$	5.70041	0.270529
$445$	14.3250	0.679072
$446$	13.6624	0.646932
$447$	−3.89058	−0.184018
$448$	−4.14603	−0.195882
$449$	−31.8370	−1.50248	−0.751240	−	0.660029i	$-0.770544\pi$
$449$	−31.8370	−1.50248	−0.751240	+	0.660029i	$0.770544\pi$
$450$	7.29628	0.343950
$451$	1.96118	0.0923483
$452$	13.9304	0.655231
$453$	−20.4546	−0.961042
$454$	−22.1629	−1.04016
$455$	−99.6788	−4.67302
$456$	1.00000	0.0468293
$457$	−4.84983	−0.226866	−0.113433	−	0.993546i	$-0.536185\pi$
$457$	−4.84983	−0.226866	−0.113433	+	0.993546i	$0.536185\pi$
$458$	2.21110	0.103318
$459$	0.160684	0.00750009
$460$	−22.6591	−1.05649
$461$	−28.0412	−1.30601	−0.653005	−	0.757354i	$-0.726492\pi$
$461$	−28.0412	−1.30601	−0.653005	+	0.757354i	$0.726492\pi$
$462$	−7.77722	−0.361829
$463$	19.4605	0.904408	0.452204	−	0.891915i	$-0.350638\pi$
$463$	19.4605	0.904408	0.452204	+	0.891915i	$0.350638\pi$
$464$	−7.03603	−0.326640
$465$	6.59338	0.305761
$466$	−2.48380	−0.115060
$467$	0.689300	0.0318970	0.0159485	−	0.999873i	$-0.494923\pi$
$467$	0.689300	0.0318970	0.0159485	+	0.999873i	$0.494923\pi$
$468$	6.85620	0.316928
$469$	−7.81140	−0.360697
$470$	9.07844	0.418757
$471$	8.60438	0.396469
$472$	−2.76143	−0.127105
$473$	14.6867	0.675295
$474$	3.64982	0.167642
$475$	7.29628	0.334776
$476$	−0.666201	−0.0305353
$477$	−1.00000	−0.0457869
$478$	−11.2741	−0.515665
$479$	39.3072	1.79599	0.897995	−	0.440006i	$-0.145024\pi$
$479$	39.3072	1.79599	0.897995	+	0.440006i	$0.145024\pi$
$480$	3.50660	0.160054
$481$	39.0831	1.78204
$482$	18.5719	0.845926
$483$	26.7910	1.21903
$484$	−7.48129	−0.340059
$485$	5.08243	0.230781
$486$	1.00000	0.0453609
$487$	−5.75810	−0.260925	−0.130462	−	0.991453i	$-0.541646\pi$
$487$	−5.75810	−0.260925	−0.130462	+	0.991453i	$0.541646\pi$
$488$	12.8453	0.581478
$489$	−8.49489	−0.384152
$490$	35.7309	1.61416
$491$	11.2108	0.505937	0.252968	−	0.967475i	$-0.418593\pi$
$491$	11.2108	0.505937	0.252968	+	0.967475i	$0.418593\pi$
$492$	1.04550	0.0471349
$493$	−1.13058	−0.0509187
$494$	6.85620	0.308475
$495$	6.57777	0.295649
$496$	1.88027	0.0844268
$497$	−2.84224	−0.127492
$498$	1.89763	0.0850349
$499$	−22.1508	−0.991608	−0.495804	−	0.868435i	$-0.665127\pi$
$499$	−22.1508	−0.991608	−0.495804	+	0.868435i	$0.665127\pi$
$500$	8.05214	0.360103
$501$	21.8706	0.977108
$502$	−17.7086	−0.790372
$503$	3.73328	0.166459	0.0832294	−	0.996530i	$-0.473477\pi$
$503$	3.73328	0.166459	0.0832294	+	0.996530i	$0.473477\pi$
$504$	−4.14603	−0.184679
$505$	−17.4761	−0.777675
$506$	−12.1213	−0.538856
$507$	34.0074	1.51032
$508$	19.9198	0.883798
$509$	−24.7159	−1.09551	−0.547757	−	0.836638i	$-0.684518\pi$
$509$	−24.7159	−1.09551	−0.547757	+	0.836638i	$0.684518\pi$
$510$	0.563456	0.0249502
$511$	45.5451	2.01480
$512$	1.00000	0.0441942
$513$	1.00000	0.0441511
$514$	−8.32063	−0.367007
$515$	−51.1617	−2.25445
$516$	7.82947	0.344673
$517$	4.85642	0.213585
$518$	−23.6341	−1.03842
$519$	−4.56370	−0.200324
$520$	24.0420	1.05431
$521$	2.04084	0.0894108	0.0447054	−	0.999000i	$-0.485765\pi$
$521$	2.04084	0.0894108	0.0447054	+	0.999000i	$0.485765\pi$
$522$	−7.03603	−0.307959
$523$	−43.9179	−1.92040	−0.960198	−	0.279320i	$-0.909891\pi$
$523$	−43.9179	−1.92040	−0.960198	+	0.279320i	$0.909891\pi$
$524$	8.71021	0.380508
$525$	−30.2506	−1.32024
$526$	−19.7117	−0.859473
$527$	0.302130	0.0131610
$528$	1.87582	0.0816347
$529$	18.7553	0.815448
$530$	−3.50660	−0.152317
$531$	−2.76143	−0.119836
$532$	−4.14603	−0.179753
$533$	7.16818	0.310488
$534$	4.08516	0.176782
$535$	−57.5941	−2.49001
$536$	1.88407	0.0813793
$537$	−22.6443	−0.977172
$538$	7.62818	0.328874
$539$	19.1139	0.823292
$540$	3.50660	0.150900
$541$	2.24526	0.0965314	0.0482657	−	0.998835i	$-0.484631\pi$
$541$	2.24526	0.0965314	0.0482657	+	0.998835i	$0.484631\pi$
$542$	−20.1260	−0.864486
$543$	4.18383	0.179545
$544$	0.160684	0.00688927
$545$	−23.2317	−0.995136
$546$	−28.4260	−1.21652
$547$	22.3710	0.956517	0.478258	−	0.878219i	$-0.341268\pi$
$547$	22.3710	0.956517	0.478258	+	0.878219i	$0.341268\pi$
$548$	−9.62749	−0.411266
$549$	12.8453	0.548223
$550$	13.6865	0.583595
$551$	−7.03603	−0.299745
$552$	−6.46183	−0.275034
$553$	−15.1323	−0.643489
$554$	19.7368	0.838535
$555$	19.9891	0.848489
$556$	1.46891	0.0622955
$557$	−8.10278	−0.343326	−0.171663	−	0.985156i	$-0.554914\pi$
$557$	−8.10278	−0.343326	−0.171663	+	0.985156i	$0.554914\pi$
$558$	1.88027	0.0795984
$559$	53.6804	2.27044
$560$	−14.5385	−0.614364
$561$	0.301415	0.0127257
$562$	28.0426	1.18290
$563$	−29.6336	−1.24891	−0.624453	−	0.781063i	$-0.714678\pi$
$563$	−29.6336	−1.24891	−0.624453	+	0.781063i	$0.714678\pi$
$564$	2.58896	0.109015
$565$	48.8484	2.05507
$566$	9.44679	0.397078
$567$	−4.14603	−0.174117
$568$	0.685532	0.0287643
$569$	−11.2871	−0.473179	−0.236590	−	0.971610i	$-0.576030\pi$
$569$	−11.2871	−0.473179	−0.236590	+	0.971610i	$0.576030\pi$
$570$	3.50660	0.146876
$571$	−30.5931	−1.28028	−0.640140	−	0.768258i	$-0.721124\pi$
$571$	−30.5931	−1.28028	−0.640140	+	0.768258i	$0.721124\pi$
$572$	12.8610	0.537746
$573$	3.35058	0.139973
$574$	−4.33469	−0.180927
$575$	−47.1473	−1.96618
$576$	1.00000	0.0416667
$577$	−2.95098	−0.122851	−0.0614254	−	0.998112i	$-0.519565\pi$
$577$	−2.95098	−0.122851	−0.0614254	+	0.998112i	$0.519565\pi$
$578$	−16.9742	−0.706033
$579$	−11.2723	−0.468462
$580$	−24.6726	−1.02447
$581$	−7.86764	−0.326405
$582$	1.44939	0.0600791
$583$	−1.87582	−0.0776886
$584$	−10.9852	−0.454572
$585$	24.0420	0.994013
$586$	6.08366	0.251314
$587$	8.17323	0.337345	0.168673	−	0.985672i	$-0.446052\pi$
$587$	8.17323	0.337345	0.168673	+	0.985672i	$0.446052\pi$
$588$	10.1896	0.420212
$589$	1.88027	0.0774754
$590$	−9.68323	−0.398652
$591$	3.04684	0.125330
$592$	5.70041	0.234285
$593$	5.44237	0.223491	0.111746	−	0.993737i	$-0.464356\pi$
$593$	5.44237	0.223491	0.111746	+	0.993737i	$0.464356\pi$
$594$	1.87582	0.0769659
$595$	−2.33611	−0.0957710
$596$	−3.89058	−0.159364
$597$	−7.28348	−0.298093
$598$	−44.3036	−1.81171
$599$	−38.3078	−1.56522	−0.782608	−	0.622515i	$-0.786111\pi$
$599$	−38.3078	−1.56522	−0.782608	+	0.622515i	$0.786111\pi$
$600$	7.29628	0.297869
$601$	30.9576	1.26279	0.631394	−	0.775462i	$-0.282483\pi$
$601$	30.9576	1.26279	0.631394	+	0.775462i	$0.282483\pi$
$602$	−32.4612	−1.32302
$603$	1.88407	0.0767251
$604$	−20.4546	−0.832287
$605$	−26.2339	−1.06656
$606$	−4.98376	−0.202451
$607$	−30.8316	−1.25141	−0.625707	−	0.780058i	$-0.715190\pi$
$607$	−30.8316	−1.25141	−0.625707	+	0.780058i	$0.715190\pi$
$608$	1.00000	0.0405554
$609$	29.1716	1.18209
$610$	45.0433	1.82375
$611$	17.7504	0.718104
$612$	0.160684	0.00649527
$613$	33.3223	1.34588	0.672938	−	0.739699i	$-0.265032\pi$
$613$	33.3223	1.34588	0.672938	+	0.739699i	$0.265032\pi$
$614$	−25.4356	−1.02650
$615$	3.66617	0.147834
$616$	−7.77722	−0.313353
$617$	−12.3074	−0.495477	−0.247739	−	0.968827i	$-0.579687\pi$
$617$	−12.3074	−0.495477	−0.247739	+	0.968827i	$0.579687\pi$
$618$	−14.5901	−0.586899
$619$	−30.2966	−1.21772	−0.608862	−	0.793276i	$-0.708374\pi$
$619$	−30.2966	−1.21772	−0.608862	+	0.793276i	$0.708374\pi$
$620$	6.59338	0.264796
$621$	−6.46183	−0.259305
$622$	25.2517	1.01250
$623$	−16.9372	−0.678575
$624$	6.85620	0.274468
$625$	−8.24572	−0.329829
$626$	−10.9429	−0.437367
$627$	1.87582	0.0749131
$628$	8.60438	0.343352
$629$	0.915965	0.0365219
$630$	−14.5385	−0.579228
$631$	40.6121	1.61674	0.808372	−	0.588673i	$-0.200349\pi$
$631$	40.6121	1.61674	0.808372	+	0.588673i	$0.200349\pi$
$632$	3.64982	0.145182
$633$	17.6367	0.700994
$634$	−6.09834	−0.242196
$635$	69.8508	2.77195
$636$	−1.00000	−0.0396526
$637$	69.8618	2.76803
$638$	−13.1984	−0.522528
$639$	0.685532	0.0271192
$640$	3.50660	0.138611
$641$	−45.8159	−1.80962	−0.904809	−	0.425818i	$-0.859986\pi$
$641$	−45.8159	−1.80962	−0.904809	+	0.425818i	$0.859986\pi$
$642$	−16.4245	−0.648222
$643$	−47.0008	−1.85353	−0.926765	−	0.375643i	$-0.877422\pi$
$643$	−47.0008	−1.85353	−0.926765	+	0.375643i	$0.877422\pi$
$644$	26.7910	1.05571
$645$	27.4549	1.08103
$646$	0.160684	0.00632203
$647$	0.141424	0.00555997	0.00277998	−	0.999996i	$-0.499115\pi$
$647$	0.141424	0.00555997	0.00277998	+	0.999996i	$0.499115\pi$
$648$	1.00000	0.0392837
$649$	−5.17995	−0.203331
$650$	50.0247	1.96213
$651$	−7.79568	−0.305537
$652$	−8.49489	−0.332686
$653$	19.3048	0.755456	0.377728	−	0.925917i	$-0.376706\pi$
$653$	19.3048	0.755456	0.377728	+	0.925917i	$0.376706\pi$
$654$	−6.62512	−0.259063
$655$	30.5433	1.19342
$656$	1.04550	0.0408200
$657$	−10.9852	−0.428575
$658$	−10.7339	−0.418451
$659$	24.5811	0.957543	0.478772	−	0.877940i	$-0.341082\pi$
$659$	24.5811	0.957543	0.478772	+	0.877940i	$0.341082\pi$
$660$	6.57777	0.256039
$661$	25.7073	0.999897	0.499949	−	0.866055i	$-0.333352\pi$
$661$	25.7073	0.999897	0.499949	+	0.866055i	$0.333352\pi$
$662$	8.58797	0.333781
$663$	1.10168	0.0427858
$664$	1.89763	0.0736423
$665$	−14.5385	−0.563779
$666$	5.70041	0.220886
$667$	45.4657	1.76044
$668$	21.8706	0.846200
$669$	13.6624	0.528217
$670$	6.60668	0.255238
$671$	24.0954	0.930194
$672$	−4.14603	−0.159937
$673$	32.4241	1.24986	0.624928	−	0.780682i	$-0.285128\pi$
$673$	32.4241	1.24986	0.624928	+	0.780682i	$0.285128\pi$
$674$	13.9300	0.536563
$675$	7.29628	0.280834
$676$	34.0074	1.30798
$677$	6.25592	0.240434	0.120217	−	0.992748i	$-0.461641\pi$
$677$	6.25592	0.240434	0.120217	+	0.992748i	$0.461641\pi$
$678$	13.9304	0.534994
$679$	−6.00921	−0.230612
$680$	0.563456	0.0216075
$681$	−22.1629	−0.849284
$682$	3.52706	0.135058
$683$	2.04903	0.0784039	0.0392020	−	0.999231i	$-0.487518\pi$
$683$	2.04903	0.0784039	0.0392020	+	0.999231i	$0.487518\pi$
$684$	1.00000	0.0382360
$685$	−33.7598	−1.28990
$686$	−13.2241	−0.504900
$687$	2.21110	0.0843589
$688$	7.82947	0.298496
$689$	−6.85620	−0.261200
$690$	−22.6591	−0.862617
$691$	43.1665	1.64213	0.821065	−	0.570835i	$-0.193380\pi$
$691$	43.1665	1.64213	0.821065	+	0.570835i	$0.193380\pi$
$692$	−4.56370	−0.173486
$693$	−7.77722	−0.295432
$694$	35.4976	1.34747
$695$	5.15087	0.195384
$696$	−7.03603	−0.266700
$697$	0.167996	0.00636329
$698$	−1.03977	−0.0393558
$699$	−2.48380	−0.0939459
$700$	−30.2506	−1.14337
$701$	−39.5074	−1.49217	−0.746086	−	0.665849i	$-0.768069\pi$
$701$	−39.5074	−1.49217	−0.746086	+	0.665849i	$0.768069\pi$
$702$	6.85620	0.258770
$703$	5.70041	0.214995
$704$	1.87582	0.0706977
$705$	9.07844	0.341914
$706$	7.14551	0.268925
$707$	20.6628	0.777105
$708$	−2.76143	−0.103781
$709$	28.2457	1.06079	0.530395	−	0.847751i	$-0.322044\pi$
$709$	28.2457	1.06079	0.530395	+	0.847751i	$0.322044\pi$
$710$	2.40389	0.0902164
$711$	3.64982	0.136879
$712$	4.08516	0.153098
$713$	−12.1500	−0.455022
$714$	−0.666201	−0.0249320
$715$	45.0985	1.68659
$716$	−22.6443	−0.846256
$717$	−11.2741	−0.421038
$718$	1.00046	0.0373370
$719$	27.6400	1.03080	0.515399	−	0.856950i	$-0.327644\pi$
$719$	27.6400	1.03080	0.515399	+	0.856950i	$0.327644\pi$
$720$	3.50660	0.130683
$721$	60.4910	2.25280
$722$	1.00000	0.0372161
$723$	18.5719	0.690695
$724$	4.18383	0.155491
$725$	−51.3369	−1.90660
$726$	−7.48129	−0.277657
$727$	−28.7469	−1.06616	−0.533082	−	0.846064i	$-0.678966\pi$
$727$	−28.7469	−1.06616	−0.533082	+	0.846064i	$0.678966\pi$
$728$	−28.4260	−1.05354
$729$	1.00000	0.0370370
$730$	−38.5209	−1.42572
$731$	1.25807	0.0465314
$732$	12.8453	0.474775
$733$	−27.8033	−1.02694	−0.513470	−	0.858108i	$-0.671640\pi$
$733$	−27.8033	−1.02694	−0.513470	+	0.858108i	$0.671640\pi$
$734$	−24.0160	−0.886446
$735$	35.7309	1.31795
$736$	−6.46183	−0.238186
$737$	3.53417	0.130183
$738$	1.04550	0.0384855
$739$	−8.91246	−0.327850	−0.163925	−	0.986473i	$-0.552416\pi$
$739$	−8.91246	−0.327850	−0.163925	+	0.986473i	$0.552416\pi$
$740$	19.9891	0.734813
$741$	6.85620	0.251869
$742$	4.14603	0.152206
$743$	−22.0795	−0.810019	−0.405009	−	0.914313i	$-0.632732\pi$
$743$	−22.0795	−0.810019	−0.405009	+	0.914313i	$0.632732\pi$
$744$	1.88027	0.0689342
$745$	−13.6427	−0.499831
$746$	17.7378	0.649425
$747$	1.89763	0.0694307
$748$	0.301415	0.0110208
$749$	68.0964	2.48819
$750$	8.05214	0.294023
$751$	−24.5291	−0.895080	−0.447540	−	0.894264i	$-0.647700\pi$
$751$	−24.5291	−0.895080	−0.447540	+	0.894264i	$0.647700\pi$
$752$	2.58896	0.0944095
$753$	−17.7086	−0.645336
$754$	−48.2404	−1.75681
$755$	−71.7263	−2.61039
$756$	−4.14603	−0.150790
$757$	19.7216	0.716794	0.358397	−	0.933569i	$-0.383323\pi$
$757$	19.7216	0.716794	0.358397	+	0.933569i	$0.383323\pi$
$758$	−11.9893	−0.435470
$759$	−12.1213	−0.439974
$760$	3.50660	0.127198
$761$	−33.5320	−1.21554	−0.607768	−	0.794115i	$-0.707935\pi$
$761$	−33.5320	−1.21554	−0.607768	+	0.794115i	$0.707935\pi$
$762$	19.9198	0.721618
$763$	27.4680	0.994407
$764$	3.35058	0.121220
$765$	0.563456	0.0203718
$766$	−21.6370	−0.781776
$767$	−18.9329	−0.683627
$768$	1.00000	0.0360844
$769$	−50.5821	−1.82404	−0.912018	−	0.410149i	$-0.865477\pi$
$769$	−50.5821	−1.82404	−0.912018	+	0.410149i	$0.865477\pi$
$770$	−27.2716	−0.982802
$771$	−8.32063	−0.299660
$772$	−11.2723	−0.405700
$773$	−4.76090	−0.171238	−0.0856189	−	0.996328i	$-0.527287\pi$
$773$	−4.76090	−0.171238	−0.0856189	+	0.996328i	$0.527287\pi$
$774$	7.82947	0.281424
$775$	13.7190	0.492801
$776$	1.44939	0.0520300
$777$	−23.6341	−0.847868
$778$	7.90562	0.283430
$779$	1.04550	0.0374590
$780$	24.0420	0.860841
$781$	1.28594	0.0460144
$782$	−1.03831	−0.0371300
$783$	−7.03603	−0.251447
$784$	10.1896	0.363914
$785$	30.1722	1.07689
$786$	8.71021	0.310683
$787$	25.8811	0.922560	0.461280	−	0.887255i	$-0.347390\pi$
$787$	25.8811	0.922560	0.461280	+	0.887255i	$0.347390\pi$
$788$	3.04684	0.108539
$789$	−19.7117	−0.701756
$790$	12.7985	0.455349
$791$	−57.7559	−2.05356
$792$	1.87582	0.0666545
$793$	88.0697	3.12745
$794$	13.1820	0.467812
$795$	−3.50660	−0.124366
$796$	−7.28348	−0.258156
$797$	16.2599	0.575955	0.287978	−	0.957637i	$-0.407017\pi$
$797$	16.2599	0.575955	0.287978	+	0.957637i	$0.407017\pi$
$798$	−4.14603	−0.146768
$799$	0.416004	0.0147172
$800$	7.29628	0.257962
$801$	4.08516	0.144342
$802$	−15.8831	−0.560850
$803$	−20.6063	−0.727182
$804$	1.88407	0.0664459
$805$	93.9454	3.31114
$806$	12.8915	0.454085
$807$	7.62818	0.268525
$808$	−4.98376	−0.175328
$809$	56.3128	1.97985	0.989926	−	0.141589i	$-0.0452210\pi$
$809$	56.3128	1.97985	0.989926	+	0.141589i	$0.0452210\pi$
$810$	3.50660	0.123210
$811$	−7.68665	−0.269915	−0.134957	−	0.990851i	$-0.543090\pi$
$811$	−7.68665	−0.269915	−0.134957	+	0.990851i	$0.543090\pi$
$812$	29.1716	1.02372
$813$	−20.1260	−0.705850
$814$	10.6930	0.374788
$815$	−29.7882	−1.04344
$816$	0.160684	0.00562507
$817$	7.82947	0.273918
$818$	8.46143	0.295847
$819$	−28.4260	−0.993286
$820$	3.66617	0.128028
$821$	27.9558	0.975665	0.487833	−	0.872937i	$-0.337788\pi$
$821$	27.9558	0.975665	0.487833	+	0.872937i	$0.337788\pi$
$822$	−9.62749	−0.335797
$823$	−49.3374	−1.71979	−0.859897	−	0.510468i	$-0.829472\pi$
$823$	−49.3374	−1.71979	−0.859897	+	0.510468i	$0.829472\pi$
$824$	−14.5901	−0.508270
$825$	13.6865	0.476504
$826$	11.4490	0.398360
$827$	22.9117	0.796719	0.398360	−	0.917229i	$-0.369580\pi$
$827$	22.9117	0.796719	0.398360	+	0.917229i	$0.369580\pi$
$828$	−6.46183	−0.224564
$829$	−28.9997	−1.00720	−0.503600	−	0.863937i	$-0.667992\pi$
$829$	−28.9997	−1.00720	−0.503600	+	0.863937i	$0.667992\pi$
$830$	6.65424	0.230972
$831$	19.7368	0.684661
$832$	6.85620	0.237696
$833$	1.63730	0.0567292
$834$	1.46891	0.0508640
$835$	76.6917	2.65402
$836$	1.87582	0.0648767
$837$	1.88027	0.0649918
$838$	−20.1955	−0.697642
$839$	−28.7277	−0.991790	−0.495895	−	0.868382i	$-0.665160\pi$
$839$	−28.7277	−0.991790	−0.495895	+	0.868382i	$0.665160\pi$
$840$	−14.5385	−0.501626
$841$	20.5058	0.707096
$842$	6.43295	0.221694
$843$	28.0426	0.965838
$844$	17.6367	0.607079
$845$	119.251	4.10235
$846$	2.58896	0.0890101
$847$	31.0177	1.06578
$848$	−1.00000	−0.0343401
$849$	9.44679	0.324213
$850$	1.17240	0.0402128
$851$	−36.8351	−1.26269
$852$	0.685532	0.0234860
$853$	22.1003	0.756700	0.378350	−	0.925663i	$-0.376492\pi$
$853$	22.1003	0.756700	0.378350	+	0.925663i	$0.376492\pi$
$854$	−53.2569	−1.82241
$855$	3.50660	0.119923
$856$	−16.4245	−0.561377
$857$	−34.0328	−1.16254	−0.581270	−	0.813711i	$-0.697444\pi$
$857$	−34.0328	−1.16254	−0.581270	+	0.813711i	$0.697444\pi$
$858$	12.8610	0.439068
$859$	25.5239	0.870865	0.435432	−	0.900221i	$-0.356596\pi$
$859$	25.5239	0.870865	0.435432	+	0.900221i	$0.356596\pi$
$860$	27.4549	0.936203
$861$	−4.33469	−0.147726
$862$	−6.82924	−0.232605
$863$	−57.9071	−1.97118	−0.985590	−	0.169150i	$-0.945898\pi$
$863$	−57.9071	−1.97118	−0.985590	+	0.169150i	$0.945898\pi$
$864$	1.00000	0.0340207
$865$	−16.0031	−0.544121
$866$	−29.8446	−1.01416
$867$	−16.9742	−0.576473
$868$	−7.79568	−0.264603
$869$	6.84641	0.232248
$870$	−24.6726	−0.836479
$871$	12.9175	0.437694
$872$	−6.62512	−0.224355
$873$	1.44939	0.0490544
$874$	−6.46183	−0.218575
$875$	−33.3844	−1.12860
$876$	−10.9852	−0.371156
$877$	5.34380	0.180447	0.0902236	−	0.995922i	$-0.471242\pi$
$877$	5.34380	0.180447	0.0902236	+	0.995922i	$0.471242\pi$
$878$	−32.8706	−1.10933
$879$	6.08366	0.205197
$880$	6.57777	0.221737
$881$	48.6709	1.63976	0.819882	−	0.572532i	$-0.194039\pi$
$881$	48.6709	1.63976	0.819882	+	0.572532i	$0.194039\pi$
$882$	10.1896	0.343101
$883$	15.9460	0.536624	0.268312	−	0.963332i	$-0.413534\pi$
$883$	15.9460	0.536624	0.268312	+	0.963332i	$0.413534\pi$
$884$	1.10168	0.0370536
$885$	−9.68323	−0.325498
$886$	−14.7994	−0.497195
$887$	29.4952	0.990353	0.495177	−	0.868792i	$-0.335103\pi$
$887$	29.4952	0.990353	0.495177	+	0.868792i	$0.335103\pi$
$888$	5.70041	0.191293
$889$	−82.5881	−2.76992
$890$	14.3250	0.480176
$891$	1.87582	0.0628424
$892$	13.6624	0.457450
$893$	2.58896	0.0866361
$894$	−3.89058	−0.130120
$895$	−79.4045	−2.65420
$896$	−4.14603	−0.138509
$897$	−44.3036	−1.47925
$898$	−31.8370	−1.06241
$899$	−13.2297	−0.441234
$900$	7.29628	0.243209
$901$	−0.160684	−0.00535316
$902$	1.96118	0.0653001
$903$	−32.4612	−1.08024
$904$	13.9304	0.463318
$905$	14.6710	0.487682
$906$	−20.4546	−0.679560
$907$	18.0375	0.598925	0.299463	−	0.954108i	$-0.403193\pi$
$907$	18.0375	0.598925	0.299463	+	0.954108i	$0.403193\pi$
$908$	−22.1629	−0.735502
$909$	−4.98376	−0.165301
$910$	−99.6788	−3.30432
$911$	44.0642	1.45991	0.729957	−	0.683493i	$-0.239540\pi$
$911$	44.0642	1.45991	0.729957	+	0.683493i	$0.239540\pi$
$912$	1.00000	0.0331133
$913$	3.55962	0.117806
$914$	−4.84983	−0.160418
$915$	45.0433	1.48908
$916$	2.21110	0.0730569
$917$	−36.1128	−1.19255
$918$	0.160684	0.00530336
$919$	13.7867	0.454781	0.227391	−	0.973804i	$-0.426981\pi$
$919$	13.7867	0.454781	0.227391	+	0.973804i	$0.426981\pi$
$920$	−22.6591	−0.747048
$921$	−25.4356	−0.838132
$922$	−28.0412	−0.923489
$923$	4.70014	0.154707
$924$	−7.77722	−0.255852
$925$	41.5918	1.36753
$926$	19.4605	0.639513
$927$	−14.5901	−0.479201
$928$	−7.03603	−0.230969
$929$	−46.2664	−1.51795	−0.758976	−	0.651118i	$-0.774300\pi$
$929$	−46.2664	−1.51795	−0.758976	+	0.651118i	$0.774300\pi$
$930$	6.59338	0.216205
$931$	10.1896	0.333950
$932$	−2.48380	−0.0813595
$933$	25.2517	0.826705
$934$	0.689300	0.0225546
$935$	1.05694	0.0345657
$936$	6.85620	0.224102
$937$	45.5724	1.48879	0.744393	−	0.667742i	$-0.232739\pi$
$937$	45.5724	1.48879	0.744393	+	0.667742i	$0.232739\pi$
$938$	−7.81140	−0.255051
$939$	−10.9429	−0.357108
$940$	9.07844	0.296106
$941$	−25.9771	−0.846829	−0.423414	−	0.905936i	$-0.639169\pi$
$941$	−25.9771	−0.846829	−0.423414	+	0.905936i	$0.639169\pi$
$942$	8.60438	0.280346
$943$	−6.75587	−0.220001
$944$	−2.76143	−0.0898768
$945$	−14.5385	−0.472937
$946$	14.6867	0.477506
$947$	40.3224	1.31030	0.655151	−	0.755498i	$-0.272605\pi$
$947$	40.3224	1.31030	0.655151	+	0.755498i	$0.272605\pi$
$948$	3.64982	0.118540
$949$	−75.3169	−2.44489
$950$	7.29628	0.236723
$951$	−6.09834	−0.197752
$952$	−0.666201	−0.0215917
$953$	−30.0963	−0.974915	−0.487458	−	0.873147i	$-0.662076\pi$
$953$	−30.0963	−0.974915	−0.487458	+	0.873147i	$0.662076\pi$
$954$	−1.00000	−0.0323762
$955$	11.7492	0.380194
$956$	−11.2741	−0.364630
$957$	−13.1984	−0.426642
$958$	39.3072	1.26996
$959$	39.9159	1.28895
$960$	3.50660	0.113175
$961$	−27.4646	−0.885954
$962$	39.0831	1.26009
$963$	−16.4245	−0.529271
$964$	18.5719	0.598160
$965$	−39.5276	−1.27244
$966$	26.7910	0.861986
$967$	−49.2836	−1.58485	−0.792426	−	0.609967i	$-0.791182\pi$
$967$	−49.2836	−1.58485	−0.792426	+	0.609967i	$0.791182\pi$
$968$	−7.48129	−0.240458
$969$	0.160684	0.00516192
$970$	5.08243	0.163187
$971$	−32.3905	−1.03946	−0.519731	−	0.854330i	$-0.673968\pi$
$971$	−32.3905	−1.03946	−0.519731	+	0.854330i	$0.673968\pi$
$972$	1.00000	0.0320750
$973$	−6.09013	−0.195241
$974$	−5.75810	−0.184502
$975$	50.0247	1.60207
$976$	12.8453	0.411167
$977$	49.3528	1.57893	0.789467	−	0.613793i	$-0.210357\pi$
$977$	49.3528	1.57893	0.789467	+	0.613793i	$0.210357\pi$
$978$	−8.49489	−0.271637
$979$	7.66303	0.244912
$980$	35.7309	1.14138
$981$	−6.62512	−0.211524
$982$	11.2108	0.357751
$983$	−21.3973	−0.682468	−0.341234	−	0.939978i	$-0.610845\pi$
$983$	−21.3973	−0.682468	−0.341234	+	0.939978i	$0.610845\pi$
$984$	1.04550	0.0333294
$985$	10.6841	0.340422
$986$	−1.13058	−0.0360050
$987$	−10.7339	−0.341664
$988$	6.85620	0.218125
$989$	−50.5927	−1.60876
$990$	6.57777	0.209055
$991$	25.8731	0.821886	0.410943	−	0.911661i	$-0.365199\pi$
$991$	25.8731	0.821886	0.410943	+	0.911661i	$0.365199\pi$
$992$	1.88027	0.0596988
$993$	8.58797	0.272531
$994$	−2.84224	−0.0901504
$995$	−25.5403	−0.809682
$996$	1.89763	0.0601287
$997$	−26.0638	−0.825450	−0.412725	−	0.910856i	$-0.635423\pi$
$997$	−26.0638	−0.825450	−0.412725	+	0.910856i	$0.635423\pi$
$998$	−22.1508	−0.701173
$999$	5.70041	0.180353

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6042.2.a.bh.1.12	✓	13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6042.2.a.bh.1.12	✓	13	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:	\( N \)	\(=\)	\( 6042 = 2 \cdot 3 \cdot 19 \cdot 53 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6042.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +3.50660 q^{5} +1.00000 q^{6} -4.14603 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +3.50660 q^{5} +1.00000 q^{6} -4.14603 q^{7} +1.00000 q^{8} +1.00000 q^{9} +3.50660 q^{10} +1.87582 q^{11} +1.00000 q^{12} +6.85620 q^{13} -4.14603 q^{14} +3.50660 q^{15} +1.00000 q^{16} +0.160684 q^{17} +1.00000 q^{18} +1.00000 q^{19} +3.50660 q^{20} -4.14603 q^{21} +1.87582 q^{22} -6.46183 q^{23} +1.00000 q^{24} +7.29628 q^{25} +6.85620 q^{26} +1.00000 q^{27} -4.14603 q^{28} -7.03603 q^{29} +3.50660 q^{30} +1.88027 q^{31} +1.00000 q^{32} +1.87582 q^{33} +0.160684 q^{34} -14.5385 q^{35} +1.00000 q^{36} +5.70041 q^{37} +1.00000 q^{38} +6.85620 q^{39} +3.50660 q^{40} +1.04550 q^{41} -4.14603 q^{42} +7.82947 q^{43} +1.87582 q^{44} +3.50660 q^{45} -6.46183 q^{46} +2.58896 q^{47} +1.00000 q^{48} +10.1896 q^{49} +7.29628 q^{50} +0.160684 q^{51} +6.85620 q^{52} -1.00000 q^{53} +1.00000 q^{54} +6.57777 q^{55} -4.14603 q^{56} +1.00000 q^{57} -7.03603 q^{58} -2.76143 q^{59} +3.50660 q^{60} +12.8453 q^{61} +1.88027 q^{62} -4.14603 q^{63} +1.00000 q^{64} +24.0420 q^{65} +1.87582 q^{66} +1.88407 q^{67} +0.160684 q^{68} -6.46183 q^{69} -14.5385 q^{70} +0.685532 q^{71} +1.00000 q^{72} -10.9852 q^{73} +5.70041 q^{74} +7.29628 q^{75} +1.00000 q^{76} -7.77722 q^{77} +6.85620 q^{78} +3.64982 q^{79} +3.50660 q^{80} +1.00000 q^{81} +1.04550 q^{82} +1.89763 q^{83} -4.14603 q^{84} +0.563456 q^{85} +7.82947 q^{86} -7.03603 q^{87} +1.87582 q^{88} +4.08516 q^{89} +3.50660 q^{90} -28.4260 q^{91} -6.46183 q^{92} +1.88027 q^{93} +2.58896 q^{94} +3.50660 q^{95} +1.00000 q^{96} +1.44939 q^{97} +10.1896 q^{98} +1.87582 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q + 13 q^{2} + 13 q^{3} + 13 q^{4} + 3 q^{5} + 13 q^{6} + 12 q^{7} + 13 q^{8} + 13 q^{9}+O(q^{10}) \) 13 * q + 13 * q^2 + 13 * q^3 + 13 * q^4 + 3 * q^5 + 13 * q^6 + 12 * q^7 + 13 * q^8 + 13 * q^9	\( 13 q + 13 q^{2} + 13 q^{3} + 13 q^{4} + 3 q^{5} + 13 q^{6} + 12 q^{7} + 13 q^{8} + 13 q^{9} + 3 q^{10} + 4 q^{11} + 13 q^{12} + 18 q^{13} + 12 q^{14} + 3 q^{15} + 13 q^{16} + 3 q^{17} + 13 q^{18} + 13 q^{19} + 3 q^{20} + 12 q^{21} + 4 q^{22} + 8 q^{23} + 13 q^{24} + 24 q^{25} + 18 q^{26} + 13 q^{27} + 12 q^{28} + 11 q^{29} + 3 q^{30} + 2 q^{31} + 13 q^{32} + 4 q^{33} + 3 q^{34} - 6 q^{35} + 13 q^{36} + 26 q^{37} + 13 q^{38} + 18 q^{39} + 3 q^{40} + 3 q^{41} + 12 q^{42} + 24 q^{43} + 4 q^{44} + 3 q^{45} + 8 q^{46} + 4 q^{47} + 13 q^{48} + 41 q^{49} + 24 q^{50} + 3 q^{51} + 18 q^{52} - 13 q^{53} + 13 q^{54} + 23 q^{55} + 12 q^{56} + 13 q^{57} + 11 q^{58} + 8 q^{59} + 3 q^{60} + 29 q^{61} + 2 q^{62} + 12 q^{63} + 13 q^{64} + 13 q^{65} + 4 q^{66} - 5 q^{67} + 3 q^{68} + 8 q^{69} - 6 q^{70} + 29 q^{71} + 13 q^{72} + 15 q^{73} + 26 q^{74} + 24 q^{75} + 13 q^{76} + 3 q^{77} + 18 q^{78} + 15 q^{79} + 3 q^{80} + 13 q^{81} + 3 q^{82} + 4 q^{83} + 12 q^{84} - q^{85} + 24 q^{86} + 11 q^{87} + 4 q^{88} + 3 q^{89} + 3 q^{90} - 29 q^{91} + 8 q^{92} + 2 q^{93} + 4 q^{94} + 3 q^{95} + 13 q^{96} + 50 q^{97} + 41 q^{98} + 4 q^{99}+O(q^{100}) \) 13 * q + 13 * q^2 + 13 * q^3 + 13 * q^4 + 3 * q^5 + 13 * q^6 + 12 * q^7 + 13 * q^8 + 13 * q^9 + 3 * q^10 + 4 * q^11 + 13 * q^12 + 18 * q^13 + 12 * q^14 + 3 * q^15 + 13 * q^16 + 3 * q^17 + 13 * q^18 + 13 * q^19 + 3 * q^20 + 12 * q^21 + 4 * q^22 + 8 * q^23 + 13 * q^24 + 24 * q^25 + 18 * q^26 + 13 * q^27 + 12 * q^28 + 11 * q^29 + 3 * q^30 + 2 * q^31 + 13 * q^32 + 4 * q^33 + 3 * q^34 - 6 * q^35 + 13 * q^36 + 26 * q^37 + 13 * q^38 + 18 * q^39 + 3 * q^40 + 3 * q^41 + 12 * q^42 + 24 * q^43 + 4 * q^44 + 3 * q^45 + 8 * q^46 + 4 * q^47 + 13 * q^48 + 41 * q^49 + 24 * q^50 + 3 * q^51 + 18 * q^52 - 13 * q^53 + 13 * q^54 + 23 * q^55 + 12 * q^56 + 13 * q^57 + 11 * q^58 + 8 * q^59 + 3 * q^60 + 29 * q^61 + 2 * q^62 + 12 * q^63 + 13 * q^64 + 13 * q^65 + 4 * q^66 - 5 * q^67 + 3 * q^68 + 8 * q^69 - 6 * q^70 + 29 * q^71 + 13 * q^72 + 15 * q^73 + 26 * q^74 + 24 * q^75 + 13 * q^76 + 3 * q^77 + 18 * q^78 + 15 * q^79 + 3 * q^80 + 13 * q^81 + 3 * q^82 + 4 * q^83 + 12 * q^84 - q^85 + 24 * q^86 + 11 * q^87 + 4 * q^88 + 3 * q^89 + 3 * q^90 - 29 * q^91 + 8 * q^92 + 2 * q^93 + 4 * q^94 + 3 * q^95 + 13 * q^96 + 50 * q^97 + 41 * q^98 + 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more