LMFDB - Embedded newform 6042.2.a.bc.1.4

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:	$1$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 3x^{8} - 20x^{7} + 69x^{6} + 27x^{5} - 185x^{4} + 8x^{3} + 109x^{2} - 8x - 14 $ x^9 - 3x^8 - 20x^7 + 69x^6 + 27x^5 - 185x^4 + 8x^3 + 109x^2 - 8x - 14
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$0.801114$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -0.801114 q^{5} -1.00000 q^{6} +1.86180 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -0.801114 q^{5} -1.00000 q^{6} +1.86180 q^{7} +1.00000 q^{8} +1.00000 q^{9} -0.801114 q^{10} +0.608374 q^{11} -1.00000 q^{12} +2.19376 q^{13} +1.86180 q^{14} +0.801114 q^{15} +1.00000 q^{16} -6.27980 q^{17} +1.00000 q^{18} +1.00000 q^{19} -0.801114 q^{20} -1.86180 q^{21} +0.608374 q^{22} -2.21526 q^{23} -1.00000 q^{24} -4.35822 q^{25} +2.19376 q^{26} -1.00000 q^{27} +1.86180 q^{28} -9.20293 q^{29} +0.801114 q^{30} -2.59670 q^{31} +1.00000 q^{32} -0.608374 q^{33} -6.27980 q^{34} -1.49152 q^{35} +1.00000 q^{36} +9.47660 q^{37} +1.00000 q^{38} -2.19376 q^{39} -0.801114 q^{40} -3.73764 q^{41} -1.86180 q^{42} -9.90815 q^{43} +0.608374 q^{44} -0.801114 q^{45} -2.21526 q^{46} -2.40508 q^{47} -1.00000 q^{48} -3.53369 q^{49} -4.35822 q^{50} +6.27980 q^{51} +2.19376 q^{52} -1.00000 q^{53} -1.00000 q^{54} -0.487377 q^{55} +1.86180 q^{56} -1.00000 q^{57} -9.20293 q^{58} -10.9486 q^{59} +0.801114 q^{60} +6.77548 q^{61} -2.59670 q^{62} +1.86180 q^{63} +1.00000 q^{64} -1.75745 q^{65} -0.608374 q^{66} -4.07906 q^{67} -6.27980 q^{68} +2.21526 q^{69} -1.49152 q^{70} +11.8355 q^{71} +1.00000 q^{72} +11.9482 q^{73} +9.47660 q^{74} +4.35822 q^{75} +1.00000 q^{76} +1.13267 q^{77} -2.19376 q^{78} +7.44078 q^{79} -0.801114 q^{80} +1.00000 q^{81} -3.73764 q^{82} -5.55706 q^{83} -1.86180 q^{84} +5.03084 q^{85} -9.90815 q^{86} +9.20293 q^{87} +0.608374 q^{88} +11.1485 q^{89} -0.801114 q^{90} +4.08434 q^{91} -2.21526 q^{92} +2.59670 q^{93} -2.40508 q^{94} -0.801114 q^{95} -1.00000 q^{96} -7.38789 q^{97} -3.53369 q^{98} +0.608374 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 9 q^{2} - 9 q^{3} + 9 q^{4} - 3 q^{5} - 9 q^{6} - 7 q^{7} + 9 q^{8} + 9 q^{9}+O(q^{10}) $ 9 * q + 9 * q^2 - 9 * q^3 + 9 * q^4 - 3 * q^5 - 9 * q^6 - 7 * q^7 + 9 * q^8 + 9 * q^9	\( 9 q + 9 q^{2} - 9 q^{3} + 9 q^{4} - 3 q^{5} - 9 q^{6} - 7 q^{7} + 9 q^{8} + 9 q^{9} - 3 q^{10} + 4 q^{11} - 9 q^{12} - 5 q^{13} - 7 q^{14} + 3 q^{15} + 9 q^{16} - 28 q^{17} + 9 q^{18} + 9 q^{19} - 3 q^{20} + 7 q^{21} + 4 q^{22} - 10 q^{23} - 9 q^{24} + 4 q^{25} - 5 q^{26} - 9 q^{27} - 7 q^{28} + 3 q^{30} + 5 q^{31} + 9 q^{32} - 4 q^{33} - 28 q^{34} - 10 q^{35} + 9 q^{36} - 25 q^{37} + 9 q^{38} + 5 q^{39} - 3 q^{40} - 7 q^{41} + 7 q^{42} - 16 q^{43} + 4 q^{44} - 3 q^{45} - 10 q^{46} - 9 q^{47} - 9 q^{48} + 44 q^{49} + 4 q^{50} + 28 q^{51} - 5 q^{52} - 9 q^{53} - 9 q^{54} - 31 q^{55} - 7 q^{56} - 9 q^{57} - 3 q^{59} + 3 q^{60} - 16 q^{61} + 5 q^{62} - 7 q^{63} + 9 q^{64} - 33 q^{65} - 4 q^{66} - 13 q^{67} - 28 q^{68} + 10 q^{69} - 10 q^{70} - 4 q^{71} + 9 q^{72} - 29 q^{73} - 25 q^{74} - 4 q^{75} + 9 q^{76} - 33 q^{77} + 5 q^{78} + 13 q^{79} - 3 q^{80} + 9 q^{81} - 7 q^{82} - 35 q^{83} + 7 q^{84} + 3 q^{85} - 16 q^{86} + 4 q^{88} - 19 q^{89} - 3 q^{90} - 10 q^{92} - 5 q^{93} - 9 q^{94} - 3 q^{95} - 9 q^{96} - 12 q^{97} + 44 q^{98} + 4 q^{99}+O(q^{100}) \) 9 * q + 9 * q^2 - 9 * q^3 + 9 * q^4 - 3 * q^5 - 9 * q^6 - 7 * q^7 + 9 * q^8 + 9 * q^9 - 3 * q^10 + 4 * q^11 - 9 * q^12 - 5 * q^13 - 7 * q^14 + 3 * q^15 + 9 * q^16 - 28 * q^17 + 9 * q^18 + 9 * q^19 - 3 * q^20 + 7 * q^21 + 4 * q^22 - 10 * q^23 - 9 * q^24 + 4 * q^25 - 5 * q^26 - 9 * q^27 - 7 * q^28 + 3 * q^30 + 5 * q^31 + 9 * q^32 - 4 * q^33 - 28 * q^34 - 10 * q^35 + 9 * q^36 - 25 * q^37 + 9 * q^38 + 5 * q^39 - 3 * q^40 - 7 * q^41 + 7 * q^42 - 16 * q^43 + 4 * q^44 - 3 * q^45 - 10 * q^46 - 9 * q^47 - 9 * q^48 + 44 * q^49 + 4 * q^50 + 28 * q^51 - 5 * q^52 - 9 * q^53 - 9 * q^54 - 31 * q^55 - 7 * q^56 - 9 * q^57 - 3 * q^59 + 3 * q^60 - 16 * q^61 + 5 * q^62 - 7 * q^63 + 9 * q^64 - 33 * q^65 - 4 * q^66 - 13 * q^67 - 28 * q^68 + 10 * q^69 - 10 * q^70 - 4 * q^71 + 9 * q^72 - 29 * q^73 - 25 * q^74 - 4 * q^75 + 9 * q^76 - 33 * q^77 + 5 * q^78 + 13 * q^79 - 3 * q^80 + 9 * q^81 - 7 * q^82 - 35 * q^83 + 7 * q^84 + 3 * q^85 - 16 * q^86 + 4 * q^88 - 19 * q^89 - 3 * q^90 - 10 * q^92 - 5 * q^93 - 9 * q^94 - 3 * q^95 - 9 * q^96 - 12 * q^97 + 44 * 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
$5$	−0.801114	−0.358269	−0.179135	−	0.983825i	$-0.557330\pi$
$5$	−0.801114	−0.358269	−0.179135	+	0.983825i	$0.557330\pi$
$6$	−1.00000	−0.408248
$7$	1.86180	0.703695	0.351848	−	0.936057i	$-0.385554\pi$
$7$	1.86180	0.703695	0.351848	+	0.936057i	$0.385554\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	−0.801114	−0.253335
$11$	0.608374	0.183432	0.0917158	−	0.995785i	$-0.470765\pi$
$11$	0.608374	0.183432	0.0917158	+	0.995785i	$0.470765\pi$
$12$	−1.00000	−0.288675
$13$	2.19376	0.608438	0.304219	−	0.952602i	$-0.401605\pi$
$13$	2.19376	0.608438	0.304219	+	0.952602i	$0.401605\pi$
$14$	1.86180	0.497588
$15$	0.801114	0.206847
$16$	1.00000	0.250000
$17$	−6.27980	−1.52308	−0.761538	−	0.648121i	$-0.775555\pi$
$17$	−6.27980	−1.52308	−0.761538	+	0.648121i	$0.775555\pi$
$18$	1.00000	0.235702
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	−0.801114	−0.179135
$21$	−1.86180	−0.406279
$22$	0.608374	0.129706
$23$	−2.21526	−0.461913	−0.230957	−	0.972964i	$-0.574186\pi$
$23$	−2.21526	−0.461913	−0.230957	+	0.972964i	$0.574186\pi$
$24$	−1.00000	−0.204124
$25$	−4.35822	−0.871643
$26$	2.19376	0.430231
$27$	−1.00000	−0.192450
$28$	1.86180	0.351848
$29$	−9.20293	−1.70894	−0.854470	−	0.519500i	$-0.826118\pi$
$29$	−9.20293	−1.70894	−0.854470	+	0.519500i	$0.826118\pi$
$30$	0.801114	0.146263
$31$	−2.59670	−0.466381	−0.233191	−	0.972431i	$-0.574917\pi$
$31$	−2.59670	−0.466381	−0.233191	+	0.972431i	$0.574917\pi$
$32$	1.00000	0.176777
$33$	−0.608374	−0.105904
$34$	−6.27980	−1.07698
$35$	−1.49152	−0.252112
$36$	1.00000	0.166667
$37$	9.47660	1.55794	0.778972	−	0.627059i	$-0.215741\pi$
$37$	9.47660	1.55794	0.778972	+	0.627059i	$0.215741\pi$
$38$	1.00000	0.162221
$39$	−2.19376	−0.351282
$40$	−0.801114	−0.126667
$41$	−3.73764	−0.583722	−0.291861	−	0.956461i	$-0.594274\pi$
$41$	−3.73764	−0.583722	−0.291861	+	0.956461i	$0.594274\pi$
$42$	−1.86180	−0.287282
$43$	−9.90815	−1.51098	−0.755489	−	0.655161i	$-0.772601\pi$
$43$	−9.90815	−1.51098	−0.755489	+	0.655161i	$0.772601\pi$
$44$	0.608374	0.0917158
$45$	−0.801114	−0.119423
$46$	−2.21526	−0.326622
$47$	−2.40508	−0.350816	−0.175408	−	0.984496i	$-0.556125\pi$
$47$	−2.40508	−0.350816	−0.175408	+	0.984496i	$0.556125\pi$
$48$	−1.00000	−0.144338
$49$	−3.53369	−0.504813
$50$	−4.35822	−0.616345
$51$	6.27980	0.879348
$52$	2.19376	0.304219
$53$	−1.00000	−0.137361
$53$	−1.00000	−0.137361
$54$	−1.00000	−0.136083
$55$	−0.487377	−0.0657179
$56$	1.86180	0.248794
$57$	−1.00000	−0.132453
$58$	−9.20293	−1.20840
$59$	−10.9486	−1.42538	−0.712692	−	0.701477i	$-0.752524\pi$
$59$	−10.9486	−1.42538	−0.712692	+	0.701477i	$0.752524\pi$
$60$	0.801114	0.103423
$61$	6.77548	0.867511	0.433755	−	0.901031i	$-0.357188\pi$
$61$	6.77548	0.867511	0.433755	+	0.901031i	$0.357188\pi$
$62$	−2.59670	−0.329781
$63$	1.86180	0.234565
$64$	1.00000	0.125000
$65$	−1.75745	−0.217985
$66$	−0.608374	−0.0748856
$67$	−4.07906	−0.498337	−0.249168	−	0.968460i	$-0.580157\pi$
$67$	−4.07906	−0.498337	−0.249168	+	0.968460i	$0.580157\pi$
$68$	−6.27980	−0.761538
$69$	2.21526	0.266686
$70$	−1.49152	−0.178270
$71$	11.8355	1.40461	0.702306	−	0.711875i	$-0.252154\pi$
$71$	11.8355	1.40461	0.702306	+	0.711875i	$0.252154\pi$
$72$	1.00000	0.117851
$73$	11.9482	1.39844	0.699218	−	0.714908i	$-0.253532\pi$
$73$	11.9482	1.39844	0.699218	+	0.714908i	$0.253532\pi$
$74$	9.47660	1.10163
$75$	4.35822	0.503243
$76$	1.00000	0.114708
$77$	1.13267	0.129080
$78$	−2.19376	−0.248394
$79$	7.44078	0.837153	0.418577	−	0.908181i	$-0.362529\pi$
$79$	7.44078	0.837153	0.418577	+	0.908181i	$0.362529\pi$
$80$	−0.801114	−0.0895673
$81$	1.00000	0.111111
$82$	−3.73764	−0.412754
$83$	−5.55706	−0.609967	−0.304983	−	0.952358i	$-0.598651\pi$
$83$	−5.55706	−0.609967	−0.304983	+	0.952358i	$0.598651\pi$
$84$	−1.86180	−0.203139
$85$	5.03084	0.545671
$86$	−9.90815	−1.06842
$87$	9.20293	0.986657
$88$	0.608374	0.0648528
$89$	11.1485	1.18174	0.590870	−	0.806767i	$-0.298785\pi$
$89$	11.1485	1.18174	0.590870	+	0.806767i	$0.298785\pi$
$90$	−0.801114	−0.0844448
$91$	4.08434	0.428155
$92$	−2.21526	−0.230957
$93$	2.59670	0.269265
$94$	−2.40508	−0.248065
$95$	−0.801114	−0.0821926
$96$	−1.00000	−0.102062
$97$	−7.38789	−0.750126	−0.375063	−	0.926999i	$-0.622379\pi$
$97$	−7.38789	−0.750126	−0.375063	+	0.926999i	$0.622379\pi$
$98$	−3.53369	−0.356957
$99$	0.608374	0.0611438
$100$	−4.35822	−0.435822
$101$	−17.2350	−1.71495	−0.857473	−	0.514530i	$-0.827967\pi$
$101$	−17.2350	−1.71495	−0.857473	+	0.514530i	$0.827967\pi$
$102$	6.27980	0.621793
$103$	−5.61064	−0.552833	−0.276416	−	0.961038i	$-0.589147\pi$
$103$	−5.61064	−0.552833	−0.276416	+	0.961038i	$0.589147\pi$
$104$	2.19376	0.215115
$105$	1.49152	0.145557
$106$	−1.00000	−0.0971286
$107$	−3.51716	−0.340016	−0.170008	−	0.985443i	$-0.554379\pi$
$107$	−3.51716	−0.340016	−0.170008	+	0.985443i	$0.554379\pi$
$108$	−1.00000	−0.0962250
$109$	6.70565	0.642284	0.321142	−	0.947031i	$-0.395933\pi$
$109$	6.70565	0.642284	0.321142	+	0.947031i	$0.395933\pi$
$110$	−0.487377	−0.0464695
$111$	−9.47660	−0.899479
$112$	1.86180	0.175924
$113$	3.41480	0.321238	0.160619	−	0.987017i	$-0.448651\pi$
$113$	3.41480	0.321238	0.160619	+	0.987017i	$0.448651\pi$
$114$	−1.00000	−0.0936586
$115$	1.77467	0.165489
$116$	−9.20293	−0.854470
$117$	2.19376	0.202813
$118$	−10.9486	−1.00790
$119$	−11.6917	−1.07178
$120$	0.801114	0.0731314
$121$	−10.6299	−0.966353
$122$	6.77548	0.613423
$123$	3.73764	0.337012
$124$	−2.59670	−0.233191
$125$	7.49700	0.670552
$126$	1.86180	0.165863
$127$	6.22160	0.552078	0.276039	−	0.961146i	$-0.410978\pi$
$127$	6.22160	0.552078	0.276039	+	0.961146i	$0.410978\pi$
$128$	1.00000	0.0883883
$129$	9.90815	0.872364
$130$	−1.75745	−0.154138
$131$	−15.6900	−1.37084	−0.685419	−	0.728149i	$-0.740381\pi$
$131$	−15.6900	−1.37084	−0.685419	+	0.728149i	$0.740381\pi$
$132$	−0.608374	−0.0529521
$133$	1.86180	0.161439
$134$	−4.07906	−0.352377
$135$	0.801114	0.0689489
$136$	−6.27980	−0.538488
$137$	−12.2650	−1.04787	−0.523935	−	0.851758i	$-0.675536\pi$
$137$	−12.2650	−1.04787	−0.523935	+	0.851758i	$0.675536\pi$
$138$	2.21526	0.188575
$139$	−21.9206	−1.85928	−0.929640	−	0.368468i	$-0.879882\pi$
$139$	−21.9206	−1.85928	−0.929640	+	0.368468i	$0.879882\pi$
$140$	−1.49152	−0.126056
$141$	2.40508	0.202544
$142$	11.8355	0.993211
$143$	1.33462	0.111607
$144$	1.00000	0.0833333
$145$	7.37259	0.612261
$146$	11.9482	0.988844
$147$	3.53369	0.291454
$148$	9.47660	0.778972
$149$	7.32858	0.600380	0.300190	−	0.953879i	$-0.402950\pi$
$149$	7.32858	0.600380	0.300190	+	0.953879i	$0.402950\pi$
$150$	4.35822	0.355847
$151$	−11.7373	−0.955170	−0.477585	−	0.878586i	$-0.658488\pi$
$151$	−11.7373	−0.955170	−0.477585	+	0.878586i	$0.658488\pi$
$152$	1.00000	0.0811107
$153$	−6.27980	−0.507692
$154$	1.13267	0.0912732
$155$	2.08025	0.167090
$156$	−2.19376	−0.175641
$157$	15.9464	1.27266	0.636331	−	0.771416i	$-0.280451\pi$
$157$	15.9464	1.27266	0.636331	+	0.771416i	$0.280451\pi$
$158$	7.44078	0.591957
$159$	1.00000	0.0793052
$160$	−0.801114	−0.0633336
$161$	−4.12437	−0.325046
$162$	1.00000	0.0785674
$163$	16.0201	1.25479	0.627396	−	0.778700i	$-0.284121\pi$
$163$	16.0201	1.25479	0.627396	+	0.778700i	$0.284121\pi$
$164$	−3.73764	−0.291861
$165$	0.487377	0.0379422
$166$	−5.55706	−0.431312
$167$	12.0520	0.932610	0.466305	−	0.884624i	$-0.345585\pi$
$167$	12.0520	0.932610	0.466305	+	0.884624i	$0.345585\pi$
$168$	−1.86180	−0.143641
$169$	−8.18744	−0.629803
$170$	5.03084	0.385848
$171$	1.00000	0.0764719
$172$	−9.90815	−0.755489
$173$	−17.8116	−1.35419	−0.677095	−	0.735895i	$-0.736762\pi$
$173$	−17.8116	−1.35419	−0.677095	+	0.735895i	$0.736762\pi$
$174$	9.20293	0.697672
$175$	−8.11414	−0.613371
$176$	0.608374	0.0458579
$177$	10.9486	0.822946
$178$	11.1485	0.835616
$179$	−5.26998	−0.393897	−0.196948	−	0.980414i	$-0.563103\pi$
$179$	−5.26998	−0.393897	−0.196948	+	0.980414i	$0.563103\pi$
$180$	−0.801114	−0.0597115
$181$	1.79726	0.133590	0.0667948	−	0.997767i	$-0.478723\pi$
$181$	1.79726	0.133590	0.0667948	+	0.997767i	$0.478723\pi$
$182$	4.08434	0.302751
$183$	−6.77548	−0.500858
$184$	−2.21526	−0.163311
$185$	−7.59184	−0.558163
$186$	2.59670	0.190399
$187$	−3.82046	−0.279380
$188$	−2.40508	−0.175408
$189$	−1.86180	−0.135426
$190$	−0.801114	−0.0581189
$191$	18.7443	1.35629	0.678145	−	0.734928i	$-0.262784\pi$
$191$	18.7443	1.35629	0.678145	+	0.734928i	$0.262784\pi$
$192$	−1.00000	−0.0721688
$193$	−17.1121	−1.23176	−0.615878	−	0.787841i	$-0.711199\pi$
$193$	−17.1121	−1.23176	−0.615878	+	0.787841i	$0.711199\pi$
$194$	−7.38789	−0.530419
$195$	1.75745	0.125853
$196$	−3.53369	−0.252407
$197$	1.06610	0.0759562	0.0379781	−	0.999279i	$-0.487908\pi$
$197$	1.06610	0.0759562	0.0379781	+	0.999279i	$0.487908\pi$
$198$	0.608374	0.0432352
$199$	−19.0891	−1.35319	−0.676595	−	0.736355i	$-0.736545\pi$
$199$	−19.0891	−1.35319	−0.676595	+	0.736355i	$0.736545\pi$
$200$	−4.35822	−0.308172
$201$	4.07906	0.287715
$202$	−17.2350	−1.21265
$203$	−17.1340	−1.20257
$204$	6.27980	0.439674
$205$	2.99428	0.209129
$206$	−5.61064	−0.390912
$207$	−2.21526	−0.153971
$208$	2.19376	0.152110
$209$	0.608374	0.0420821
$210$	1.49152	0.102924
$211$	14.5530	1.00187	0.500935	−	0.865485i	$-0.332990\pi$
$211$	14.5530	1.00187	0.500935	+	0.865485i	$0.332990\pi$
$212$	−1.00000	−0.0686803
$213$	−11.8355	−0.810953
$214$	−3.51716	−0.240428
$215$	7.93756	0.541337
$216$	−1.00000	−0.0680414
$217$	−4.83454	−0.328190
$218$	6.70565	0.454164
$219$	−11.9482	−0.807388
$220$	−0.487377	−0.0328589
$221$	−13.7763	−0.926697
$222$	−9.47660	−0.636028
$223$	−6.22895	−0.417121	−0.208561	−	0.978009i	$-0.566878\pi$
$223$	−6.22895	−0.417121	−0.208561	+	0.978009i	$0.566878\pi$
$224$	1.86180	0.124397
$225$	−4.35822	−0.290548
$226$	3.41480	0.227149
$227$	−3.62648	−0.240698	−0.120349	−	0.992732i	$-0.538401\pi$
$227$	−3.62648	−0.240698	−0.120349	+	0.992732i	$0.538401\pi$
$228$	−1.00000	−0.0662266
$229$	29.6404	1.95869	0.979347	−	0.202186i	$-0.0648045\pi$
$229$	29.6404	1.95869	0.979347	+	0.202186i	$0.0648045\pi$
$230$	1.77467	0.117019
$231$	−1.13267	−0.0745243
$232$	−9.20293	−0.604202
$233$	−8.86532	−0.580787	−0.290393	−	0.956907i	$-0.593786\pi$
$233$	−8.86532	−0.580787	−0.290393	+	0.956907i	$0.593786\pi$
$234$	2.19376	0.143410
$235$	1.92674	0.125687
$236$	−10.9486	−0.712692
$237$	−7.44078	−0.483331
$238$	−11.6917	−0.757863
$239$	4.83429	0.312705	0.156352	−	0.987701i	$-0.450026\pi$
$239$	4.83429	0.312705	0.156352	+	0.987701i	$0.450026\pi$
$240$	0.801114	0.0517117
$241$	−22.2892	−1.43577	−0.717886	−	0.696161i	$-0.754890\pi$
$241$	−22.2892	−1.43577	−0.717886	+	0.696161i	$0.754890\pi$
$242$	−10.6299	−0.683315
$243$	−1.00000	−0.0641500
$244$	6.77548	0.433755
$245$	2.83089	0.180859
$246$	3.73764	0.238303
$247$	2.19376	0.139585
$248$	−2.59670	−0.164891
$249$	5.55706	0.352165
$250$	7.49700	0.474152
$251$	13.3260	0.841132	0.420566	−	0.907262i	$-0.361831\pi$
$251$	13.3260	0.841132	0.420566	+	0.907262i	$0.361831\pi$
$252$	1.86180	0.117283
$253$	−1.34770	−0.0847294
$254$	6.22160	0.390378
$255$	−5.03084	−0.315043
$256$	1.00000	0.0625000
$257$	−16.2716	−1.01499	−0.507497	−	0.861654i	$-0.669429\pi$
$257$	−16.2716	−1.01499	−0.507497	+	0.861654i	$0.669429\pi$
$258$	9.90815	0.616854
$259$	17.6436	1.09632
$260$	−1.75745	−0.108992
$261$	−9.20293	−0.569647
$262$	−15.6900	−0.969329
$263$	−11.3721	−0.701236	−0.350618	−	0.936519i	$-0.614028\pi$
$263$	−11.3721	−0.701236	−0.350618	+	0.936519i	$0.614028\pi$
$264$	−0.608374	−0.0374428
$265$	0.801114	0.0492120
$266$	1.86180	0.114154
$267$	−11.1485	−0.682278
$268$	−4.07906	−0.249168
$269$	−10.8453	−0.661252	−0.330626	−	0.943762i	$-0.607260\pi$
$269$	−10.8453	−0.661252	−0.330626	+	0.943762i	$0.607260\pi$
$270$	0.801114	0.0487543
$271$	−13.6218	−0.827465	−0.413733	−	0.910398i	$-0.635775\pi$
$271$	−13.6218	−0.827465	−0.413733	+	0.910398i	$0.635775\pi$
$272$	−6.27980	−0.380769
$273$	−4.08434	−0.247195
$274$	−12.2650	−0.740956
$275$	−2.65142	−0.159887
$276$	2.21526	0.133343
$277$	−20.8786	−1.25447	−0.627237	−	0.778828i	$-0.715814\pi$
$277$	−20.8786	−1.25447	−0.627237	+	0.778828i	$0.715814\pi$
$278$	−21.9206	−1.31471
$279$	−2.59670	−0.155460
$280$	−1.49152	−0.0891351
$281$	2.47912	0.147892	0.0739460	−	0.997262i	$-0.476441\pi$
$281$	2.47912	0.147892	0.0739460	+	0.997262i	$0.476441\pi$
$282$	2.40508	0.143220
$283$	−29.7745	−1.76991	−0.884954	−	0.465678i	$-0.845810\pi$
$283$	−29.7745	−1.76991	−0.884954	+	0.465678i	$0.845810\pi$
$284$	11.8355	0.702306
$285$	0.801114	0.0474539
$286$	1.33462	0.0789179
$287$	−6.95875	−0.410762
$288$	1.00000	0.0589256
$289$	22.4359	1.31976
$290$	7.37259	0.432934
$291$	7.38789	0.433086
$292$	11.9482	0.699218
$293$	24.7469	1.44573	0.722864	−	0.690990i	$-0.242825\pi$
$293$	24.7469	1.44573	0.722864	+	0.690990i	$0.242825\pi$
$294$	3.53369	0.206089
$295$	8.77107	0.510671
$296$	9.47660	0.550816
$297$	−0.608374	−0.0353014
$298$	7.32858	0.424533
$299$	−4.85973	−0.281046
$300$	4.35822	0.251622
$301$	−18.4470	−1.06327
$302$	−11.7373	−0.675407
$303$	17.2350	0.990124
$304$	1.00000	0.0573539
$305$	−5.42793	−0.310802
$306$	−6.27980	−0.358992
$307$	−6.49156	−0.370493	−0.185246	−	0.982692i	$-0.559308\pi$
$307$	−6.49156	−0.370493	−0.185246	+	0.982692i	$0.559308\pi$
$308$	1.13267	0.0645399
$309$	5.61064	0.319178
$310$	2.08025	0.118150
$311$	−1.40965	−0.0799337	−0.0399668	−	0.999201i	$-0.512725\pi$
$311$	−1.40965	−0.0799337	−0.0399668	+	0.999201i	$0.512725\pi$
$312$	−2.19376	−0.124197
$313$	27.5317	1.55618	0.778091	−	0.628152i	$-0.216188\pi$
$313$	27.5317	1.55618	0.778091	+	0.628152i	$0.216188\pi$
$314$	15.9464	0.899908
$315$	−1.49152	−0.0840374
$316$	7.44078	0.418577
$317$	−18.4901	−1.03851	−0.519254	−	0.854620i	$-0.673790\pi$
$317$	−18.4901	−1.03851	−0.519254	+	0.854620i	$0.673790\pi$
$318$	1.00000	0.0560772
$319$	−5.59882	−0.313474
$320$	−0.801114	−0.0447836
$321$	3.51716	0.196309
$322$	−4.12437	−0.229842
$323$	−6.27980	−0.349417
$324$	1.00000	0.0555556
$325$	−9.56086	−0.530341
$326$	16.0201	0.887272
$327$	−6.70565	−0.370823
$328$	−3.73764	−0.206377
$329$	−4.47778	−0.246868
$330$	0.487377	0.0268292
$331$	−14.8955	−0.818731	−0.409366	−	0.912370i	$-0.634250\pi$
$331$	−14.8955	−0.818731	−0.409366	+	0.912370i	$0.634250\pi$
$332$	−5.55706	−0.304983
$333$	9.47660	0.519315
$334$	12.0520	0.659455
$335$	3.26780	0.178539
$336$	−1.86180	−0.101570
$337$	−4.34139	−0.236490	−0.118245	−	0.992984i	$-0.537727\pi$
$337$	−4.34139	−0.236490	−0.118245	+	0.992984i	$0.537727\pi$
$338$	−8.18744	−0.445338
$339$	−3.41480	−0.185467
$340$	5.03084	0.272835
$341$	−1.57976	−0.0855490
$342$	1.00000	0.0540738
$343$	−19.6117	−1.05893
$344$	−9.90815	−0.534212
$345$	−1.77467	−0.0955452
$346$	−17.8116	−0.957557
$347$	24.8424	1.33361	0.666805	−	0.745232i	$-0.267661\pi$
$347$	24.8424	1.33361	0.666805	+	0.745232i	$0.267661\pi$
$348$	9.20293	0.493329
$349$	32.8347	1.75760	0.878800	−	0.477190i	$-0.158345\pi$
$349$	32.8347	1.75760	0.878800	+	0.477190i	$0.158345\pi$
$350$	−8.11414	−0.433719
$351$	−2.19376	−0.117094
$352$	0.608374	0.0324264
$353$	−15.3970	−0.819499	−0.409750	−	0.912198i	$-0.634384\pi$
$353$	−15.3970	−0.819499	−0.409750	+	0.912198i	$0.634384\pi$
$354$	10.9486	0.581911
$355$	−9.48156	−0.503229
$356$	11.1485	0.590870
$357$	11.6917	0.618793
$358$	−5.26998	−0.278527
$359$	−29.5164	−1.55781	−0.778907	−	0.627139i	$-0.784226\pi$
$359$	−29.5164	−1.55781	−0.778907	+	0.627139i	$0.784226\pi$
$360$	−0.801114	−0.0422224
$361$	1.00000	0.0526316
$362$	1.79726	0.0944621
$363$	10.6299	0.557924
$364$	4.08434	0.214077
$365$	−9.57191	−0.501017
$366$	−6.77548	−0.354160
$367$	19.6874	1.02767	0.513836	−	0.857888i	$-0.328224\pi$
$367$	19.6874	1.02767	0.513836	+	0.857888i	$0.328224\pi$
$368$	−2.21526	−0.115478
$369$	−3.73764	−0.194574
$370$	−7.59184	−0.394681
$371$	−1.86180	−0.0966600
$372$	2.59670	0.134633
$373$	−4.03472	−0.208910	−0.104455	−	0.994530i	$-0.533310\pi$
$373$	−4.03472	−0.208910	−0.104455	+	0.994530i	$0.533310\pi$
$374$	−3.82046	−0.197552
$375$	−7.49700	−0.387143
$376$	−2.40508	−0.124032
$377$	−20.1890	−1.03978
$378$	−1.86180	−0.0957608
$379$	15.9704	0.820343	0.410172	−	0.912008i	$-0.365469\pi$
$379$	15.9704	0.820343	0.410172	+	0.912008i	$0.365469\pi$
$380$	−0.801114	−0.0410963
$381$	−6.22160	−0.318742
$382$	18.7443	0.959041
$383$	12.1134	0.618964	0.309482	−	0.950905i	$-0.399844\pi$
$383$	12.1134	0.618964	0.309482	+	0.950905i	$0.399844\pi$
$384$	−1.00000	−0.0510310
$385$	−0.907399	−0.0462453
$386$	−17.1121	−0.870983
$387$	−9.90815	−0.503659
$388$	−7.38789	−0.375063
$389$	−22.6756	−1.14970	−0.574849	−	0.818259i	$-0.694939\pi$
$389$	−22.6756	−1.14970	−0.574849	+	0.818259i	$0.694939\pi$
$390$	1.75745	0.0889918
$391$	13.9114	0.703529
$392$	−3.53369	−0.178478
$393$	15.6900	0.791454
$394$	1.06610	0.0537091
$395$	−5.96091	−0.299926
$396$	0.608374	0.0305719
$397$	5.61485	0.281801	0.140901	−	0.990024i	$-0.455000\pi$
$397$	5.61485	0.281801	0.140901	+	0.990024i	$0.455000\pi$
$398$	−19.0891	−0.956850
$399$	−1.86180	−0.0932067
$400$	−4.35822	−0.217911
$401$	21.3010	1.06372	0.531859	−	0.846833i	$-0.321493\pi$
$401$	21.3010	1.06372	0.531859	+	0.846833i	$0.321493\pi$
$402$	4.07906	0.203445
$403$	−5.69653	−0.283764
$404$	−17.2350	−0.857473
$405$	−0.801114	−0.0398077
$406$	−17.1340	−0.850348
$407$	5.76532	0.285776
$408$	6.27980	0.310896
$409$	−29.1568	−1.44171	−0.720856	−	0.693085i	$-0.756251\pi$
$409$	−29.1568	−1.44171	−0.720856	+	0.693085i	$0.756251\pi$
$410$	2.99428	0.147877
$411$	12.2650	0.604988
$412$	−5.61064	−0.276416
$413$	−20.3841	−1.00304
$414$	−2.21526	−0.108874
$415$	4.45184	0.218532
$416$	2.19376	0.107558
$417$	21.9206	1.07346
$418$	0.608374	0.0297565
$419$	16.0205	0.782652	0.391326	−	0.920252i	$-0.372017\pi$
$419$	16.0205	0.782652	0.391326	+	0.920252i	$0.372017\pi$
$420$	1.49152	0.0727785
$421$	−7.11913	−0.346965	−0.173483	−	0.984837i	$-0.555502\pi$
$421$	−7.11913	−0.346965	−0.173483	+	0.984837i	$0.555502\pi$
$422$	14.5530	0.708429
$423$	−2.40508	−0.116939
$424$	−1.00000	−0.0485643
$425$	27.3687	1.32758
$426$	−11.8355	−0.573431
$427$	12.6146	0.610463
$428$	−3.51716	−0.170008
$429$	−1.33462	−0.0644362
$430$	7.93756	0.382783
$431$	22.4869	1.08316	0.541578	−	0.840650i	$-0.317827\pi$
$431$	22.4869	1.08316	0.541578	+	0.840650i	$0.317827\pi$
$432$	−1.00000	−0.0481125
$433$	−12.7120	−0.610899	−0.305450	−	0.952208i	$-0.598807\pi$
$433$	−12.7120	−0.610899	−0.305450	+	0.952208i	$0.598807\pi$
$434$	−4.83454	−0.232066
$435$	−7.37259	−0.353489
$436$	6.70565	0.321142
$437$	−2.21526	−0.105970
$438$	−11.9482	−0.570909
$439$	37.1170	1.77150	0.885749	−	0.464165i	$-0.153646\pi$
$439$	37.1170	1.77150	0.885749	+	0.464165i	$0.153646\pi$
$440$	−0.487377	−0.0232348
$441$	−3.53369	−0.168271
$442$	−13.7763	−0.655274
$443$	−11.8005	−0.560660	−0.280330	−	0.959904i	$-0.590444\pi$
$443$	−11.8005	−0.560660	−0.280330	+	0.959904i	$0.590444\pi$
$444$	−9.47660	−0.449740
$445$	−8.93123	−0.423381
$446$	−6.22895	−0.294949
$447$	−7.32858	−0.346630
$448$	1.86180	0.0879619
$449$	−8.08335	−0.381477	−0.190738	−	0.981641i	$-0.561088\pi$
$449$	−8.08335	−0.381477	−0.190738	+	0.981641i	$0.561088\pi$
$450$	−4.35822	−0.205448
$451$	−2.27388	−0.107073
$452$	3.41480	0.160619
$453$	11.7373	0.551468
$454$	−3.62648	−0.170199
$455$	−3.27202	−0.153395
$456$	−1.00000	−0.0468293
$457$	−23.9362	−1.11969	−0.559844	−	0.828598i	$-0.689139\pi$
$457$	−23.9362	−1.11969	−0.559844	+	0.828598i	$0.689139\pi$
$458$	29.6404	1.38501
$459$	6.27980	0.293116
$460$	1.77467	0.0827446
$461$	−24.9424	−1.16168	−0.580841	−	0.814017i	$-0.697276\pi$
$461$	−24.9424	−1.16168	−0.580841	+	0.814017i	$0.697276\pi$
$462$	−1.13267	−0.0526966
$463$	29.2588	1.35977	0.679885	−	0.733319i	$-0.262030\pi$
$463$	29.2588	1.35977	0.679885	+	0.733319i	$0.262030\pi$
$464$	−9.20293	−0.427235
$465$	−2.08025	−0.0964695
$466$	−8.86532	−0.410678
$467$	28.8523	1.33513	0.667563	−	0.744553i	$-0.267337\pi$
$467$	28.8523	1.33513	0.667563	+	0.744553i	$0.267337\pi$
$468$	2.19376	0.101406
$469$	−7.59441	−0.350677
$470$	1.92674	0.0888739
$471$	−15.9464	−0.734772
$472$	−10.9486	−0.503950
$473$	−6.02786	−0.277161
$474$	−7.44078	−0.341766
$475$	−4.35822	−0.199969
$476$	−11.6917	−0.535890
$477$	−1.00000	−0.0457869
$478$	4.83429	0.221115
$479$	12.6718	0.578989	0.289494	−	0.957180i	$-0.406513\pi$
$479$	12.6718	0.578989	0.289494	+	0.957180i	$0.406513\pi$
$480$	0.801114	0.0365657
$481$	20.7893	0.947913
$482$	−22.2892	−1.01524
$483$	4.12437	0.187665
$484$	−10.6299	−0.483176
$485$	5.91854	0.268747
$486$	−1.00000	−0.0453609
$487$	34.9779	1.58500	0.792500	−	0.609872i	$-0.208779\pi$
$487$	34.9779	1.58500	0.792500	+	0.609872i	$0.208779\pi$
$488$	6.77548	0.306711
$489$	−16.0201	−0.724455
$490$	2.83089	0.127887
$491$	−5.40344	−0.243854	−0.121927	−	0.992539i	$-0.538907\pi$
$491$	−5.40344	−0.243854	−0.121927	+	0.992539i	$0.538907\pi$
$492$	3.73764	0.168506
$493$	57.7925	2.60285
$494$	2.19376	0.0987017
$495$	−0.487377	−0.0219060
$496$	−2.59670	−0.116595
$497$	22.0353	0.988419
$498$	5.55706	0.249018
$499$	−22.8202	−1.02157	−0.510786	−	0.859708i	$-0.670646\pi$
$499$	−22.8202	−1.02157	−0.510786	+	0.859708i	$0.670646\pi$
$500$	7.49700	0.335276
$501$	−12.0520	−0.538443
$502$	13.3260	0.594770
$503$	9.52932	0.424891	0.212446	−	0.977173i	$-0.431857\pi$
$503$	9.52932	0.424891	0.212446	+	0.977173i	$0.431857\pi$
$504$	1.86180	0.0829313
$505$	13.8072	0.614412
$506$	−1.34770	−0.0599128
$507$	8.18744	0.363617
$508$	6.22160	0.276039
$509$	−19.9645	−0.884909	−0.442455	−	0.896791i	$-0.645892\pi$
$509$	−19.9645	−0.884909	−0.442455	+	0.896791i	$0.645892\pi$
$510$	−5.03084	−0.222769
$511$	22.2453	0.984073
$512$	1.00000	0.0441942
$513$	−1.00000	−0.0441511
$514$	−16.2716	−0.717709
$515$	4.49476	0.198063
$516$	9.90815	0.436182
$517$	−1.46318	−0.0643508
$518$	17.6436	0.775214
$519$	17.8116	0.781842
$520$	−1.75745	−0.0770692
$521$	−15.5844	−0.682763	−0.341382	−	0.939925i	$-0.610895\pi$
$521$	−15.5844	−0.682763	−0.341382	+	0.939925i	$0.610895\pi$
$522$	−9.20293	−0.402801
$523$	−11.1729	−0.488557	−0.244279	−	0.969705i	$-0.578551\pi$
$523$	−11.1729	−0.488557	−0.244279	+	0.969705i	$0.578551\pi$
$524$	−15.6900	−0.685419
$525$	8.11414	0.354130
$526$	−11.3721	−0.495849
$527$	16.3068	0.710334
$528$	−0.608374	−0.0264761
$529$	−18.0926	−0.786636
$530$	0.801114	0.0347982
$531$	−10.9486	−0.475128
$532$	1.86180	0.0807194
$533$	−8.19947	−0.355159
$534$	−11.1485	−0.482443
$535$	2.81764	0.121817
$536$	−4.07906	−0.176189
$537$	5.26998	0.227417
$538$	−10.8453	−0.467576
$539$	−2.14981	−0.0925987
$540$	0.801114	0.0344745
$541$	−28.2979	−1.21662	−0.608310	−	0.793699i	$-0.708152\pi$
$541$	−28.2979	−1.21662	−0.608310	+	0.793699i	$0.708152\pi$
$542$	−13.6218	−0.585106
$543$	−1.79726	−0.0771280
$544$	−6.27980	−0.269244
$545$	−5.37199	−0.230111
$546$	−4.08434	−0.174794
$547$	22.3733	0.956613	0.478306	−	0.878193i	$-0.341251\pi$
$547$	22.3733	0.956613	0.478306	+	0.878193i	$0.341251\pi$
$548$	−12.2650	−0.523935
$549$	6.77548	0.289170
$550$	−2.65142	−0.113057
$551$	−9.20293	−0.392058
$552$	2.21526	0.0942876
$553$	13.8533	0.589101
$554$	−20.8786	−0.887047
$555$	7.59184	0.322256
$556$	−21.9206	−0.929640
$557$	−2.06179	−0.0873606	−0.0436803	−	0.999046i	$-0.513908\pi$
$557$	−2.06179	−0.0873606	−0.0436803	+	0.999046i	$0.513908\pi$
$558$	−2.59670	−0.109927
$559$	−21.7360	−0.919337
$560$	−1.49152	−0.0630281
$561$	3.82046	0.161300
$562$	2.47912	0.104576
$563$	−21.0368	−0.886595	−0.443297	−	0.896375i	$-0.646191\pi$
$563$	−21.0368	−0.886595	−0.443297	+	0.896375i	$0.646191\pi$
$564$	2.40508	0.101272
$565$	−2.73565	−0.115090
$566$	−29.7745	−1.25151
$567$	1.86180	0.0781883
$568$	11.8355	0.496605
$569$	13.5468	0.567911	0.283955	−	0.958837i	$-0.408353\pi$
$569$	13.5468	0.567911	0.283955	+	0.958837i	$0.408353\pi$
$570$	0.801114	0.0335550
$571$	41.8391	1.75091	0.875456	−	0.483299i	$-0.160561\pi$
$571$	41.8391	1.75091	0.875456	+	0.483299i	$0.160561\pi$
$572$	1.33462	0.0558034
$573$	−18.7443	−0.783054
$574$	−6.95875	−0.290453
$575$	9.65457	0.402623
$576$	1.00000	0.0416667
$577$	−29.1289	−1.21265	−0.606325	−	0.795217i	$-0.707357\pi$
$577$	−29.1289	−1.21265	−0.606325	+	0.795217i	$0.707357\pi$
$578$	22.4359	0.933210
$579$	17.1121	0.711155
$580$	7.37259	0.306130
$581$	−10.3462	−0.429231
$582$	7.38789	0.306238
$583$	−0.608374	−0.0251963
$584$	11.9482	0.494422
$585$	−1.75745	−0.0726615
$586$	24.7469	1.02228
$587$	17.4623	0.720745	0.360372	−	0.932809i	$-0.382650\pi$
$587$	17.4623	0.720745	0.360372	+	0.932809i	$0.382650\pi$
$588$	3.53369	0.145727
$589$	−2.59670	−0.106995
$590$	8.77107	0.361099
$591$	−1.06610	−0.0438533
$592$	9.47660	0.389486
$593$	23.0486	0.946493	0.473246	−	0.880930i	$-0.343082\pi$
$593$	23.0486	0.946493	0.473246	+	0.880930i	$0.343082\pi$
$594$	−0.608374	−0.0249619
$595$	9.36642	0.383986
$596$	7.32858	0.300190
$597$	19.0891	0.781265
$598$	−4.85973	−0.198729
$599$	8.69537	0.355283	0.177642	−	0.984095i	$-0.443153\pi$
$599$	8.69537	0.355283	0.177642	+	0.984095i	$0.443153\pi$
$600$	4.35822	0.177923
$601$	−4.05347	−0.165344	−0.0826722	−	0.996577i	$-0.526345\pi$
$601$	−4.05347	−0.165344	−0.0826722	+	0.996577i	$0.526345\pi$
$602$	−18.4470	−0.751844
$603$	−4.07906	−0.166112
$604$	−11.7373	−0.477585
$605$	8.51575	0.346214
$606$	17.2350	0.700124
$607$	−7.21533	−0.292861	−0.146431	−	0.989221i	$-0.546779\pi$
$607$	−7.21533	−0.292861	−0.146431	+	0.989221i	$0.546779\pi$
$608$	1.00000	0.0405554
$609$	17.1340	0.694306
$610$	−5.42793	−0.219770
$611$	−5.27615	−0.213450
$612$	−6.27980	−0.253846
$613$	5.41647	0.218769	0.109385	−	0.993999i	$-0.465112\pi$
$613$	5.41647	0.218769	0.109385	+	0.993999i	$0.465112\pi$
$614$	−6.49156	−0.261978
$615$	−2.99428	−0.120741
$616$	1.13267	0.0456366
$617$	−29.5279	−1.18875	−0.594374	−	0.804189i	$-0.702600\pi$
$617$	−29.5279	−1.18875	−0.594374	+	0.804189i	$0.702600\pi$
$618$	5.61064	0.225693
$619$	11.7860	0.473718	0.236859	−	0.971544i	$-0.423882\pi$
$619$	11.7860	0.473718	0.236859	+	0.971544i	$0.423882\pi$
$620$	2.08025	0.0835450
$621$	2.21526	0.0888952
$622$	−1.40965	−0.0565216
$623$	20.7563	0.831585
$624$	−2.19376	−0.0878205
$625$	15.7851	0.631405
$626$	27.5317	1.10039
$627$	−0.608374	−0.0242961
$628$	15.9464	0.636331
$629$	−59.5112	−2.37287
$630$	−1.49152	−0.0594234
$631$	−1.18002	−0.0469758	−0.0234879	−	0.999724i	$-0.507477\pi$
$631$	−1.18002	−0.0469758	−0.0234879	+	0.999724i	$0.507477\pi$
$632$	7.44078	0.295978
$633$	−14.5530	−0.578430
$634$	−18.4901	−0.734336
$635$	−4.98421	−0.197792
$636$	1.00000	0.0396526
$637$	−7.75206	−0.307148
$638$	−5.59882	−0.221659
$639$	11.8355	0.468204
$640$	−0.801114	−0.0316668
$641$	29.2430	1.15503	0.577514	−	0.816381i	$-0.304023\pi$
$641$	29.2430	1.15503	0.577514	+	0.816381i	$0.304023\pi$
$642$	3.51716	0.138811
$643$	−25.1189	−0.990591	−0.495296	−	0.868725i	$-0.664940\pi$
$643$	−25.1189	−0.990591	−0.495296	+	0.868725i	$0.664940\pi$
$644$	−4.12437	−0.162523
$645$	−7.93756	−0.312541
$646$	−6.27980	−0.247075
$647$	−27.7512	−1.09101	−0.545505	−	0.838107i	$-0.683662\pi$
$647$	−27.7512	−1.09101	−0.545505	+	0.838107i	$0.683662\pi$
$648$	1.00000	0.0392837
$649$	−6.66083	−0.261461
$650$	−9.56086	−0.375008
$651$	4.83454	0.189481
$652$	16.0201	0.627396
$653$	41.5846	1.62733	0.813665	−	0.581334i	$-0.197469\pi$
$653$	41.5846	1.62733	0.813665	+	0.581334i	$0.197469\pi$
$654$	−6.70565	−0.262212
$655$	12.5694	0.491129
$656$	−3.73764	−0.145930
$657$	11.9482	0.466146
$658$	−4.47778	−0.174562
$659$	29.4605	1.14762	0.573809	−	0.818989i	$-0.305465\pi$
$659$	29.4605	1.14762	0.573809	+	0.818989i	$0.305465\pi$
$660$	0.487377	0.0189711
$661$	29.5437	1.14912	0.574559	−	0.818463i	$-0.305174\pi$
$661$	29.5437	1.14912	0.574559	+	0.818463i	$0.305174\pi$
$662$	−14.8955	−0.578930
$663$	13.7763	0.535029
$664$	−5.55706	−0.215656
$665$	−1.49152	−0.0578385
$666$	9.47660	0.367211
$667$	20.3869	0.789382
$668$	12.0520	0.466305
$669$	6.22895	0.240825
$670$	3.26780	0.126246
$671$	4.12202	0.159129
$672$	−1.86180	−0.0718206
$673$	−48.3669	−1.86441	−0.932203	−	0.361937i	$-0.882116\pi$
$673$	−48.3669	−1.86441	−0.932203	+	0.361937i	$0.882116\pi$
$674$	−4.34139	−0.167224
$675$	4.35822	0.167748
$676$	−8.18744	−0.314901
$677$	−6.81508	−0.261925	−0.130962	−	0.991387i	$-0.541807\pi$
$677$	−6.81508	−0.261925	−0.130962	+	0.991387i	$0.541807\pi$
$678$	−3.41480	−0.131145
$679$	−13.7548	−0.527860
$680$	5.03084	0.192924
$681$	3.62648	0.138967
$682$	−1.57976	−0.0604923
$683$	−8.48939	−0.324838	−0.162419	−	0.986722i	$-0.551930\pi$
$683$	−8.48939	−0.324838	−0.162419	+	0.986722i	$0.551930\pi$
$684$	1.00000	0.0382360
$685$	9.82567	0.375419
$686$	−19.6117	−0.748776
$687$	−29.6404	−1.13085
$688$	−9.90815	−0.377745
$689$	−2.19376	−0.0835754
$690$	−1.77467	−0.0675607
$691$	40.4179	1.53757	0.768784	−	0.639509i	$-0.220862\pi$
$691$	40.4179	1.53757	0.768784	+	0.639509i	$0.220862\pi$
$692$	−17.8116	−0.677095
$693$	1.13267	0.0430266
$694$	24.8424	0.943004
$695$	17.5609	0.666123
$696$	9.20293	0.348836
$697$	23.4716	0.889052
$698$	32.8347	1.24281
$699$	8.86532	0.335317
$700$	−8.11414	−0.306686
$701$	30.0159	1.13368	0.566842	−	0.823827i	$-0.308165\pi$
$701$	30.0159	1.13368	0.566842	+	0.823827i	$0.308165\pi$
$702$	−2.19376	−0.0827979
$703$	9.47660	0.357417
$704$	0.608374	0.0229289
$705$	−1.92674	−0.0725652
$706$	−15.3970	−0.579474
$707$	−32.0881	−1.20680
$708$	10.9486	0.411473
$709$	−12.4732	−0.468441	−0.234221	−	0.972183i	$-0.575254\pi$
$709$	−12.4732	−0.468441	−0.234221	+	0.972183i	$0.575254\pi$
$710$	−9.48156	−0.355837
$711$	7.44078	0.279051
$712$	11.1485	0.417808
$713$	5.75236	0.215428
$714$	11.6917	0.437553
$715$	−1.06918	−0.0399852
$716$	−5.26998	−0.196948
$717$	−4.83429	−0.180540
$718$	−29.5164	−1.10154
$719$	−37.0088	−1.38019	−0.690097	−	0.723717i	$-0.742432\pi$
$719$	−37.0088	−1.38019	−0.690097	+	0.723717i	$0.742432\pi$
$720$	−0.801114	−0.0298558
$721$	−10.4459	−0.389026
$722$	1.00000	0.0372161
$723$	22.2892	0.828943
$724$	1.79726	0.0667948
$725$	40.1083	1.48959
$726$	10.6299	0.394512
$727$	37.7958	1.40177	0.700884	−	0.713275i	$-0.252789\pi$
$727$	37.7958	1.40177	0.700884	+	0.713275i	$0.252789\pi$
$728$	4.08434	0.151376
$729$	1.00000	0.0370370
$730$	−9.57191	−0.354272
$731$	62.2212	2.30133
$732$	−6.77548	−0.250429
$733$	24.5720	0.907587	0.453794	−	0.891107i	$-0.350070\pi$
$733$	24.5720	0.907587	0.453794	+	0.891107i	$0.350070\pi$
$734$	19.6874	0.726674
$735$	−2.83089	−0.104419
$736$	−2.21526	−0.0816555
$737$	−2.48159	−0.0914107
$738$	−3.73764	−0.137585
$739$	−28.2400	−1.03882	−0.519412	−	0.854524i	$-0.673849\pi$
$739$	−28.2400	−1.03882	−0.519412	+	0.854524i	$0.673849\pi$
$740$	−7.59184	−0.279082
$741$	−2.19376	−0.0805896
$742$	−1.86180	−0.0683489
$743$	17.5703	0.644592	0.322296	−	0.946639i	$-0.395545\pi$
$743$	17.5703	0.644592	0.322296	+	0.946639i	$0.395545\pi$
$744$	2.59670	0.0951997
$745$	−5.87103	−0.215098
$746$	−4.03472	−0.147722
$747$	−5.55706	−0.203322
$748$	−3.82046	−0.139690
$749$	−6.54825	−0.239268
$750$	−7.49700	−0.273752
$751$	41.9189	1.52964	0.764822	−	0.644242i	$-0.222827\pi$
$751$	41.9189	1.52964	0.764822	+	0.644242i	$0.222827\pi$
$752$	−2.40508	−0.0877041
$753$	−13.3260	−0.485628
$754$	−20.1890	−0.735239
$755$	9.40294	0.342208
$756$	−1.86180	−0.0677131
$757$	−26.1592	−0.950771	−0.475386	−	0.879778i	$-0.657691\pi$
$757$	−26.1592	−0.950771	−0.475386	+	0.879778i	$0.657691\pi$
$758$	15.9704	0.580070
$759$	1.34770	0.0489186
$760$	−0.801114	−0.0290595
$761$	−9.23991	−0.334947	−0.167473	−	0.985877i	$-0.553561\pi$
$761$	−9.23991	−0.334947	−0.167473	+	0.985877i	$0.553561\pi$
$762$	−6.22160	−0.225385
$763$	12.4846	0.451972
$764$	18.7443	0.678145
$765$	5.03084	0.181890
$766$	12.1134	0.437673
$767$	−24.0185	−0.867259
$768$	−1.00000	−0.0360844
$769$	11.3372	0.408830	0.204415	−	0.978884i	$-0.434471\pi$
$769$	11.3372	0.408830	0.204415	+	0.978884i	$0.434471\pi$
$770$	−0.907399	−0.0327004
$771$	16.2716	0.586007
$772$	−17.1121	−0.615878
$773$	−16.7854	−0.603727	−0.301864	−	0.953351i	$-0.597609\pi$
$773$	−16.7854	−0.603727	−0.301864	+	0.953351i	$0.597609\pi$
$774$	−9.90815	−0.356141
$775$	11.3170	0.406518
$776$	−7.38789	−0.265210
$777$	−17.6436	−0.632959
$778$	−22.6756	−0.812960
$779$	−3.73764	−0.133915
$780$	1.75745	0.0629267
$781$	7.20039	0.257650
$782$	13.9114	0.497470
$783$	9.20293	0.328886
$784$	−3.53369	−0.126203
$785$	−12.7749	−0.455956
$786$	15.6900	0.559642
$787$	−7.32702	−0.261180	−0.130590	−	0.991436i	$-0.541687\pi$
$787$	−7.32702	−0.261180	−0.130590	+	0.991436i	$0.541687\pi$
$788$	1.06610	0.0379781
$789$	11.3721	0.404859
$790$	−5.96091	−0.212080
$791$	6.35769	0.226053
$792$	0.608374	0.0216176
$793$	14.8637	0.527827
$794$	5.61485	0.199264
$795$	−0.801114	−0.0284126
$796$	−19.0891	−0.676595
$797$	−16.8445	−0.596662	−0.298331	−	0.954462i	$-0.596430\pi$
$797$	−16.8445	−0.596662	−0.298331	+	0.954462i	$0.596430\pi$
$798$	−1.86180	−0.0659071
$799$	15.1034	0.534320
$800$	−4.35822	−0.154086
$801$	11.1485	0.393913
$802$	21.3010	0.752163
$803$	7.26900	0.256517
$804$	4.07906	0.143857
$805$	3.30409	0.116454
$806$	−5.69653	−0.200652
$807$	10.8453	0.381774
$808$	−17.2350	−0.606325
$809$	−0.241883	−0.00850416	−0.00425208	−	0.999991i	$-0.501353\pi$
$809$	−0.241883	−0.00850416	−0.00425208	+	0.999991i	$0.501353\pi$
$810$	−0.801114	−0.0281483
$811$	−38.9313	−1.36706	−0.683532	−	0.729920i	$-0.739557\pi$
$811$	−38.9313	−1.36706	−0.683532	+	0.729920i	$0.739557\pi$
$812$	−17.1340	−0.601287
$813$	13.6218	0.477737
$814$	5.76532	0.202074
$815$	−12.8339	−0.449553
$816$	6.27980	0.219837
$817$	−9.90815	−0.346642
$818$	−29.1568	−1.01944
$819$	4.08434	0.142718
$820$	2.99428	0.104565
$821$	29.2073	1.01934	0.509670	−	0.860370i	$-0.329767\pi$
$821$	29.2073	1.01934	0.509670	+	0.860370i	$0.329767\pi$
$822$	12.2650	0.427791
$823$	20.9373	0.729828	0.364914	−	0.931041i	$-0.381098\pi$
$823$	20.9373	0.729828	0.364914	+	0.931041i	$0.381098\pi$
$824$	−5.61064	−0.195456
$825$	2.65142	0.0923107
$826$	−20.3841	−0.709254
$827$	−12.4404	−0.432594	−0.216297	−	0.976328i	$-0.569398\pi$
$827$	−12.4404	−0.432594	−0.216297	+	0.976328i	$0.569398\pi$
$828$	−2.21526	−0.0769855
$829$	−17.2559	−0.599321	−0.299660	−	0.954046i	$-0.596873\pi$
$829$	−17.2559	−0.599321	−0.299660	+	0.954046i	$0.596873\pi$
$830$	4.45184	0.154526
$831$	20.8786	0.724271
$832$	2.19376	0.0760548
$833$	22.1909	0.768869
$834$	21.9206	0.759048
$835$	−9.65501	−0.334125
$836$	0.608374	0.0210410
$837$	2.59670	0.0897551
$838$	16.0205	0.553418
$839$	−51.1860	−1.76714	−0.883568	−	0.468303i	$-0.844865\pi$
$839$	−51.1860	−1.76714	−0.883568	+	0.468303i	$0.844865\pi$
$840$	1.49152	0.0514622
$841$	55.6939	1.92048
$842$	−7.11913	−0.245341
$843$	−2.47912	−0.0853855
$844$	14.5530	0.500935
$845$	6.55907	0.225639
$846$	−2.40508	−0.0826882
$847$	−19.7907	−0.680018
$848$	−1.00000	−0.0343401
$849$	29.7745	1.02186
$850$	27.3687	0.938740
$851$	−20.9931	−0.719635
$852$	−11.8355	−0.405477
$853$	−7.02147	−0.240411	−0.120205	−	0.992749i	$-0.538355\pi$
$853$	−7.02147	−0.240411	−0.120205	+	0.992749i	$0.538355\pi$
$854$	12.6146	0.431663
$855$	−0.801114	−0.0273975
$856$	−3.51716	−0.120214
$857$	−43.8544	−1.49804	−0.749020	−	0.662548i	$-0.769475\pi$
$857$	−43.8544	−1.49804	−0.749020	+	0.662548i	$0.769475\pi$
$858$	−1.33462	−0.0455633
$859$	−14.1258	−0.481965	−0.240982	−	0.970529i	$-0.577470\pi$
$859$	−14.1258	−0.481965	−0.240982	+	0.970529i	$0.577470\pi$
$860$	7.93756	0.270668
$861$	6.95875	0.237154
$862$	22.4869	0.765907
$863$	46.5103	1.58323	0.791615	−	0.611020i	$-0.209241\pi$
$863$	46.5103	1.58323	0.791615	+	0.611020i	$0.209241\pi$
$864$	−1.00000	−0.0340207
$865$	14.2691	0.485165
$866$	−12.7120	−0.431971
$867$	−22.4359	−0.761963
$868$	−4.83454	−0.164095
$869$	4.52677	0.153560
$870$	−7.37259	−0.249954
$871$	−8.94847	−0.303207
$872$	6.70565	0.227082
$873$	−7.38789	−0.250042
$874$	−2.21526	−0.0749322
$875$	13.9579	0.471864
$876$	−11.9482	−0.403694
$877$	−26.6605	−0.900262	−0.450131	−	0.892962i	$-0.648623\pi$
$877$	−26.6605	−0.900262	−0.450131	+	0.892962i	$0.648623\pi$
$878$	37.1170	1.25264
$879$	−24.7469	−0.834691
$880$	−0.487377	−0.0164295
$881$	12.8058	0.431439	0.215720	−	0.976455i	$-0.430790\pi$
$881$	12.8058	0.431439	0.215720	+	0.976455i	$0.430790\pi$
$882$	−3.53369	−0.118986
$883$	3.25383	0.109500	0.0547500	−	0.998500i	$-0.482564\pi$
$883$	3.25383	0.109500	0.0547500	+	0.998500i	$0.482564\pi$
$884$	−13.7763	−0.463349
$885$	−8.77107	−0.294836
$886$	−11.8005	−0.396446
$887$	−42.1111	−1.41395	−0.706976	−	0.707237i	$-0.749941\pi$
$887$	−42.1111	−1.41395	−0.706976	+	0.707237i	$0.749941\pi$
$888$	−9.47660	−0.318014
$889$	11.5834	0.388494
$890$	−8.93123	−0.299376
$891$	0.608374	0.0203813
$892$	−6.22895	−0.208561
$893$	−2.40508	−0.0804828
$894$	−7.32858	−0.245104
$895$	4.22186	0.141121
$896$	1.86180	0.0621984
$897$	4.85973	0.162262
$898$	−8.08335	−0.269745
$899$	23.8973	0.797018
$900$	−4.35822	−0.145274
$901$	6.27980	0.209211
$902$	−2.27388	−0.0757120
$903$	18.4470	0.613878
$904$	3.41480	0.113575
$905$	−1.43981	−0.0478610
$906$	11.7373	0.389947
$907$	20.1653	0.669578	0.334789	−	0.942293i	$-0.391335\pi$
$907$	20.1653	0.669578	0.334789	+	0.942293i	$0.391335\pi$
$908$	−3.62648	−0.120349
$909$	−17.2350	−0.571648
$910$	−3.27202	−0.108466
$911$	−16.7318	−0.554348	−0.277174	−	0.960820i	$-0.589398\pi$
$911$	−16.7318	−0.554348	−0.277174	+	0.960820i	$0.589398\pi$
$912$	−1.00000	−0.0331133
$913$	−3.38077	−0.111887
$914$	−23.9362	−0.791739
$915$	5.42793	0.179442
$916$	29.6404	0.979347
$917$	−29.2116	−0.964652
$918$	6.27980	0.207264
$919$	−5.50337	−0.181540	−0.0907698	−	0.995872i	$-0.528933\pi$
$919$	−5.50337	−0.181540	−0.0907698	+	0.995872i	$0.528933\pi$
$920$	1.77467	0.0585093
$921$	6.49156	0.213904
$922$	−24.9424	−0.821433
$923$	25.9641	0.854620
$924$	−1.13267	−0.0372621
$925$	−41.3011	−1.35797
$926$	29.2588	0.961503
$927$	−5.61064	−0.184278
$928$	−9.20293	−0.302101
$929$	22.5151	0.738697	0.369349	−	0.929291i	$-0.379581\pi$
$929$	22.5151	0.738697	0.369349	+	0.929291i	$0.379581\pi$
$930$	−2.08025	−0.0682142
$931$	−3.53369	−0.115812
$932$	−8.86532	−0.290393
$933$	1.40965	0.0461497
$934$	28.8523	0.944077
$935$	3.06063	0.100093
$936$	2.19376	0.0717051
$937$	18.3555	0.599648	0.299824	−	0.953994i	$-0.403072\pi$
$937$	18.3555	0.599648	0.299824	+	0.953994i	$0.403072\pi$
$938$	−7.59441	−0.247966
$939$	−27.5317	−0.898462
$940$	1.92674	0.0628433
$941$	33.3132	1.08598	0.542990	−	0.839739i	$-0.317292\pi$
$941$	33.3132	1.08598	0.542990	+	0.839739i	$0.317292\pi$
$942$	−15.9464	−0.519562
$943$	8.27984	0.269629
$944$	−10.9486	−0.356346
$945$	1.49152	0.0485190
$946$	−6.02786	−0.195982
$947$	28.3204	0.920291	0.460145	−	0.887844i	$-0.347797\pi$
$947$	28.3204	0.920291	0.460145	+	0.887844i	$0.347797\pi$
$948$	−7.44078	−0.241665
$949$	26.2115	0.850862
$950$	−4.35822	−0.141399
$951$	18.4901	0.599583
$952$	−11.6917	−0.378932
$953$	29.4578	0.954233	0.477116	−	0.878840i	$-0.341682\pi$
$953$	29.4578	0.954233	0.477116	+	0.878840i	$0.341682\pi$
$954$	−1.00000	−0.0323762
$955$	−15.0163	−0.485917
$956$	4.83429	0.156352
$957$	5.59882	0.180984
$958$	12.6718	0.409407
$959$	−22.8350	−0.737381
$960$	0.801114	0.0258558
$961$	−24.2571	−0.782488
$962$	20.7893	0.670275
$963$	−3.51716	−0.113339
$964$	−22.2892	−0.717886
$965$	13.7088	0.441300
$966$	4.12437	0.132699
$967$	36.3465	1.16882	0.584412	−	0.811457i	$-0.301325\pi$
$967$	36.3465	1.16882	0.584412	+	0.811457i	$0.301325\pi$
$968$	−10.6299	−0.341657
$969$	6.27980	0.201736
$970$	5.91854	0.190033
$971$	49.6442	1.59316	0.796578	−	0.604535i	$-0.206641\pi$
$971$	49.6442	1.59316	0.796578	+	0.604535i	$0.206641\pi$
$972$	−1.00000	−0.0320750
$973$	−40.8118	−1.30837
$974$	34.9779	1.12076
$975$	9.56086	0.306193
$976$	6.77548	0.216878
$977$	22.3081	0.713700	0.356850	−	0.934162i	$-0.383851\pi$
$977$	22.3081	0.713700	0.356850	+	0.934162i	$0.383851\pi$
$978$	−16.0201	−0.512267
$979$	6.78246	0.216768
$980$	2.83089	0.0904295
$981$	6.70565	0.214095
$982$	−5.40344	−0.172431
$983$	−55.7668	−1.77869	−0.889343	−	0.457241i	$-0.848838\pi$
$983$	−55.7668	−1.77869	−0.889343	+	0.457241i	$0.848838\pi$
$984$	3.73764	0.119152
$985$	−0.854064	−0.0272127
$986$	57.7925	1.84049
$987$	4.47778	0.142529
$988$	2.19376	0.0697926
$989$	21.9491	0.697941
$990$	−0.487377	−0.0154898
$991$	25.1651	0.799395	0.399697	−	0.916647i	$-0.369115\pi$
$991$	25.1651	0.799395	0.399697	+	0.916647i	$0.369115\pi$
$992$	−2.59670	−0.0824453
$993$	14.8955	0.472695
$994$	22.0353	0.698918
$995$	15.2925	0.484806
$996$	5.55706	0.176082
$997$	52.1313	1.65101	0.825507	−	0.564392i	$-0.190889\pi$
$997$	52.1313	1.65101	0.825507	+	0.564392i	$0.190889\pi$
$998$	−22.8202	−0.722361
$999$	−9.47660	−0.299826

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6042.2.a.bc.1.4	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6042.2.a.bc.1.4	✓	9	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -0.801114 q^{5} -1.00000 q^{6} +1.86180 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -0.801114 q^{5} -1.00000 q^{6} +1.86180 q^{7} +1.00000 q^{8} +1.00000 q^{9} -0.801114 q^{10} +0.608374 q^{11} -1.00000 q^{12} +2.19376 q^{13} +1.86180 q^{14} +0.801114 q^{15} +1.00000 q^{16} -6.27980 q^{17} +1.00000 q^{18} +1.00000 q^{19} -0.801114 q^{20} -1.86180 q^{21} +0.608374 q^{22} -2.21526 q^{23} -1.00000 q^{24} -4.35822 q^{25} +2.19376 q^{26} -1.00000 q^{27} +1.86180 q^{28} -9.20293 q^{29} +0.801114 q^{30} -2.59670 q^{31} +1.00000 q^{32} -0.608374 q^{33} -6.27980 q^{34} -1.49152 q^{35} +1.00000 q^{36} +9.47660 q^{37} +1.00000 q^{38} -2.19376 q^{39} -0.801114 q^{40} -3.73764 q^{41} -1.86180 q^{42} -9.90815 q^{43} +0.608374 q^{44} -0.801114 q^{45} -2.21526 q^{46} -2.40508 q^{47} -1.00000 q^{48} -3.53369 q^{49} -4.35822 q^{50} +6.27980 q^{51} +2.19376 q^{52} -1.00000 q^{53} -1.00000 q^{54} -0.487377 q^{55} +1.86180 q^{56} -1.00000 q^{57} -9.20293 q^{58} -10.9486 q^{59} +0.801114 q^{60} +6.77548 q^{61} -2.59670 q^{62} +1.86180 q^{63} +1.00000 q^{64} -1.75745 q^{65} -0.608374 q^{66} -4.07906 q^{67} -6.27980 q^{68} +2.21526 q^{69} -1.49152 q^{70} +11.8355 q^{71} +1.00000 q^{72} +11.9482 q^{73} +9.47660 q^{74} +4.35822 q^{75} +1.00000 q^{76} +1.13267 q^{77} -2.19376 q^{78} +7.44078 q^{79} -0.801114 q^{80} +1.00000 q^{81} -3.73764 q^{82} -5.55706 q^{83} -1.86180 q^{84} +5.03084 q^{85} -9.90815 q^{86} +9.20293 q^{87} +0.608374 q^{88} +11.1485 q^{89} -0.801114 q^{90} +4.08434 q^{91} -2.21526 q^{92} +2.59670 q^{93} -2.40508 q^{94} -0.801114 q^{95} -1.00000 q^{96} -7.38789 q^{97} -3.53369 q^{98} +0.608374 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 9 q^{2} - 9 q^{3} + 9 q^{4} - 3 q^{5} - 9 q^{6} - 7 q^{7} + 9 q^{8} + 9 q^{9}+O(q^{10}) \) 9 * q + 9 * q^2 - 9 * q^3 + 9 * q^4 - 3 * q^5 - 9 * q^6 - 7 * q^7 + 9 * q^8 + 9 * q^9	\( 9 q + 9 q^{2} - 9 q^{3} + 9 q^{4} - 3 q^{5} - 9 q^{6} - 7 q^{7} + 9 q^{8} + 9 q^{9} - 3 q^{10} + 4 q^{11} - 9 q^{12} - 5 q^{13} - 7 q^{14} + 3 q^{15} + 9 q^{16} - 28 q^{17} + 9 q^{18} + 9 q^{19} - 3 q^{20} + 7 q^{21} + 4 q^{22} - 10 q^{23} - 9 q^{24} + 4 q^{25} - 5 q^{26} - 9 q^{27} - 7 q^{28} + 3 q^{30} + 5 q^{31} + 9 q^{32} - 4 q^{33} - 28 q^{34} - 10 q^{35} + 9 q^{36} - 25 q^{37} + 9 q^{38} + 5 q^{39} - 3 q^{40} - 7 q^{41} + 7 q^{42} - 16 q^{43} + 4 q^{44} - 3 q^{45} - 10 q^{46} - 9 q^{47} - 9 q^{48} + 44 q^{49} + 4 q^{50} + 28 q^{51} - 5 q^{52} - 9 q^{53} - 9 q^{54} - 31 q^{55} - 7 q^{56} - 9 q^{57} - 3 q^{59} + 3 q^{60} - 16 q^{61} + 5 q^{62} - 7 q^{63} + 9 q^{64} - 33 q^{65} - 4 q^{66} - 13 q^{67} - 28 q^{68} + 10 q^{69} - 10 q^{70} - 4 q^{71} + 9 q^{72} - 29 q^{73} - 25 q^{74} - 4 q^{75} + 9 q^{76} - 33 q^{77} + 5 q^{78} + 13 q^{79} - 3 q^{80} + 9 q^{81} - 7 q^{82} - 35 q^{83} + 7 q^{84} + 3 q^{85} - 16 q^{86} + 4 q^{88} - 19 q^{89} - 3 q^{90} - 10 q^{92} - 5 q^{93} - 9 q^{94} - 3 q^{95} - 9 q^{96} - 12 q^{97} + 44 q^{98} + 4 q^{99}+O(q^{100}) \) 9 * q + 9 * q^2 - 9 * q^3 + 9 * q^4 - 3 * q^5 - 9 * q^6 - 7 * q^7 + 9 * q^8 + 9 * q^9 - 3 * q^10 + 4 * q^11 - 9 * q^12 - 5 * q^13 - 7 * q^14 + 3 * q^15 + 9 * q^16 - 28 * q^17 + 9 * q^18 + 9 * q^19 - 3 * q^20 + 7 * q^21 + 4 * q^22 - 10 * q^23 - 9 * q^24 + 4 * q^25 - 5 * q^26 - 9 * q^27 - 7 * q^28 + 3 * q^30 + 5 * q^31 + 9 * q^32 - 4 * q^33 - 28 * q^34 - 10 * q^35 + 9 * q^36 - 25 * q^37 + 9 * q^38 + 5 * q^39 - 3 * q^40 - 7 * q^41 + 7 * q^42 - 16 * q^43 + 4 * q^44 - 3 * q^45 - 10 * q^46 - 9 * q^47 - 9 * q^48 + 44 * q^49 + 4 * q^50 + 28 * q^51 - 5 * q^52 - 9 * q^53 - 9 * q^54 - 31 * q^55 - 7 * q^56 - 9 * q^57 - 3 * q^59 + 3 * q^60 - 16 * q^61 + 5 * q^62 - 7 * q^63 + 9 * q^64 - 33 * q^65 - 4 * q^66 - 13 * q^67 - 28 * q^68 + 10 * q^69 - 10 * q^70 - 4 * q^71 + 9 * q^72 - 29 * q^73 - 25 * q^74 - 4 * q^75 + 9 * q^76 - 33 * q^77 + 5 * q^78 + 13 * q^79 - 3 * q^80 + 9 * q^81 - 7 * q^82 - 35 * q^83 + 7 * q^84 + 3 * q^85 - 16 * q^86 + 4 * q^88 - 19 * q^89 - 3 * q^90 - 10 * q^92 - 5 * q^93 - 9 * q^94 - 3 * q^95 - 9 * q^96 - 12 * q^97 + 44 * q^98 + 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more