LMFDB - Embedded newform 1690.2.a.t.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1690,2,Mod(1,1690)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1690, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1690.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1690 = 2 \cdot 5 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1690.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:	$13.4947179416$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.4752.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 3x^{2} + 4x + 1 $ x^4 - 2x^3 - 3x^2 + 4*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 130)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.21969$ of defining polynomial
Character	$\chi$	$=$	1690.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -0.600196 q^{3} +1.00000 q^{4} -1.00000 q^{5} +0.600196 q^{6} -1.43937 q^{7} -1.00000 q^{8} -2.63977 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -0.600196 q^{3} +1.00000 q^{4} -1.00000 q^{5} +0.600196 q^{6} -1.43937 q^{7} -1.00000 q^{8} -2.63977 q^{9} +1.00000 q^{10} -2.77162 q^{11} -0.600196 q^{12} +1.43937 q^{14} +0.600196 q^{15} +1.00000 q^{16} -4.50367 q^{17} +2.63977 q^{18} -4.33225 q^{19} -1.00000 q^{20} +0.863906 q^{21} +2.77162 q^{22} +4.43937 q^{23} +0.600196 q^{24} +1.00000 q^{25} +3.38496 q^{27} -1.43937 q^{28} +7.86488 q^{29} -0.600196 q^{30} -4.16082 q^{31} -1.00000 q^{32} +1.66351 q^{33} +4.50367 q^{34} +1.43937 q^{35} -2.63977 q^{36} +0.575468 q^{37} +4.33225 q^{38} +1.00000 q^{40} -4.22512 q^{41} -0.863906 q^{42} +2.61504 q^{43} -2.77162 q^{44} +2.63977 q^{45} -4.43937 q^{46} +12.7684 q^{47} -0.600196 q^{48} -4.92820 q^{49} -1.00000 q^{50} +2.70308 q^{51} +9.57123 q^{53} -3.38496 q^{54} +2.77162 q^{55} +1.43937 q^{56} +2.60020 q^{57} -7.86488 q^{58} +3.46410 q^{59} +0.600196 q^{60} -12.5037 q^{61} +4.16082 q^{62} +3.79961 q^{63} +1.00000 q^{64} -1.66351 q^{66} -5.32899 q^{67} -4.50367 q^{68} -2.66449 q^{69} -1.43937 q^{70} +6.00000 q^{71} +2.63977 q^{72} +1.66351 q^{73} -0.575468 q^{74} -0.600196 q^{75} -4.33225 q^{76} +3.98940 q^{77} +12.7288 q^{79} -1.00000 q^{80} +5.88766 q^{81} +4.22512 q^{82} -10.0469 q^{83} +0.863906 q^{84} +4.50367 q^{85} -2.61504 q^{86} -4.72047 q^{87} +2.77162 q^{88} -5.95717 q^{89} -2.63977 q^{90} +4.43937 q^{92} +2.49731 q^{93} -12.7684 q^{94} +4.33225 q^{95} +0.600196 q^{96} +1.58633 q^{97} +4.92820 q^{98} +7.31643 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{2} + 2 q^{3} + 4 q^{4} - 4 q^{5} - 2 q^{6} - 4 q^{8} + 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^2 + 2 * q^3 + 4 * q^4 - 4 * q^5 - 2 * q^6 - 4 * q^8 + 4 * q^9	$ 4 q - 4 q^{2} + 2 q^{3} + 4 q^{4} - 4 q^{5} - 2 q^{6} - 4 q^{8} + 4 q^{9} + 4 q^{10} + 6 q^{11} + 2 q^{12} - 2 q^{15} + 4 q^{16} + 6 q^{17} - 4 q^{18} - 6 q^{19} - 4 q^{20} - 6 q^{21} - 6 q^{22} + 12 q^{23} - 2 q^{24} + 4 q^{25} + 20 q^{27} + 2 q^{30} - 18 q^{31} - 4 q^{32} + 6 q^{33} - 6 q^{34} + 4 q^{36} + 6 q^{37} + 6 q^{38} + 4 q^{40} + 6 q^{42} + 4 q^{43} + 6 q^{44} - 4 q^{45} - 12 q^{46} + 2 q^{48} + 8 q^{49} - 4 q^{50} + 30 q^{53} - 20 q^{54} - 6 q^{55} + 6 q^{57} - 2 q^{60} - 26 q^{61} + 18 q^{62} + 24 q^{63} + 4 q^{64} - 6 q^{66} + 24 q^{67} + 6 q^{68} + 12 q^{69} + 24 q^{71} - 4 q^{72} + 6 q^{73} - 6 q^{74} + 2 q^{75} - 6 q^{76} + 30 q^{77} + 10 q^{79} - 4 q^{80} + 28 q^{81} + 18 q^{83} - 6 q^{84} - 6 q^{85} - 4 q^{86} - 48 q^{87} - 6 q^{88} + 4 q^{90} + 12 q^{92} + 12 q^{93} + 6 q^{95} - 2 q^{96} - 18 q^{97} - 8 q^{98} + 42 q^{99}+O(q^{100}) $ 4 * q - 4 * q^2 + 2 * q^3 + 4 * q^4 - 4 * q^5 - 2 * q^6 - 4 * q^8 + 4 * q^9 + 4 * q^10 + 6 * q^11 + 2 * q^12 - 2 * q^15 + 4 * q^16 + 6 * q^17 - 4 * q^18 - 6 * q^19 - 4 * q^20 - 6 * q^21 - 6 * q^22 + 12 * q^23 - 2 * q^24 + 4 * q^25 + 20 * q^27 + 2 * q^30 - 18 * q^31 - 4 * q^32 + 6 * q^33 - 6 * q^34 + 4 * q^36 + 6 * q^37 + 6 * q^38 + 4 * q^40 + 6 * q^42 + 4 * q^43 + 6 * q^44 - 4 * q^45 - 12 * q^46 + 2 * q^48 + 8 * q^49 - 4 * q^50 + 30 * q^53 - 20 * q^54 - 6 * q^55 + 6 * q^57 - 2 * q^60 - 26 * q^61 + 18 * q^62 + 24 * q^63 + 4 * q^64 - 6 * q^66 + 24 * q^67 + 6 * q^68 + 12 * q^69 + 24 * q^71 - 4 * q^72 + 6 * q^73 - 6 * q^74 + 2 * q^75 - 6 * q^76 + 30 * q^77 + 10 * q^79 - 4 * q^80 + 28 * q^81 + 18 * q^83 - 6 * q^84 - 6 * q^85 - 4 * q^86 - 48 * q^87 - 6 * q^88 + 4 * q^90 + 12 * q^92 + 12 * q^93 + 6 * q^95 - 2 * q^96 - 18 * q^97 - 8 * q^98 + 42 * 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
$3$	−0.600196	−0.346523	−0.173262	−	0.984876i	$-0.555431\pi$
$3$	−0.600196	−0.346523	−0.173262	+	0.984876i	$0.555431\pi$
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0.600196	0.245029
$7$	−1.43937	−0.544032	−0.272016	−	0.962293i	$-0.587690\pi$
$7$	−1.43937	−0.544032	−0.272016	+	0.962293i	$0.587690\pi$
$8$	−1.00000	−0.353553
$9$	−2.63977	−0.879922
$10$	1.00000	0.316228
$11$	−2.77162	−0.835675	−0.417837	−	0.908522i	$-0.637212\pi$
$11$	−2.77162	−0.835675	−0.417837	+	0.908522i	$0.637212\pi$
$12$	−0.600196	−0.173262
$13$	0	0
$13$	0	0
$14$	1.43937	0.384689
$15$	0.600196	0.154970
$16$	1.00000	0.250000
$17$	−4.50367	−1.09230	−0.546150	−	0.837687i	$-0.683907\pi$
$17$	−4.50367	−1.09230	−0.546150	+	0.837687i	$0.683907\pi$
$18$	2.63977	0.622199
$19$	−4.33225	−0.993886	−0.496943	−	0.867783i	$-0.665544\pi$
$19$	−4.33225	−0.993886	−0.496943	+	0.867783i	$0.665544\pi$
$20$	−1.00000	−0.223607
$21$	0.863906	0.188520
$22$	2.77162	0.590911
$23$	4.43937	0.925673	0.462837	−	0.886444i	$-0.346832\pi$
$23$	4.43937	0.925673	0.462837	+	0.886444i	$0.346832\pi$
$24$	0.600196	0.122514
$25$	1.00000	0.200000
$26$	0	0
$27$	3.38496	0.651436
$28$	−1.43937	−0.272016
$29$	7.86488	1.46047	0.730236	−	0.683195i	$-0.239410\pi$
$29$	7.86488	1.46047	0.730236	+	0.683195i	$0.239410\pi$
$30$	−0.600196	−0.109580
$31$	−4.16082	−0.747306	−0.373653	−	0.927569i	$-0.621895\pi$
$31$	−4.16082	−0.747306	−0.373653	+	0.927569i	$0.621895\pi$
$32$	−1.00000	−0.176777
$33$	1.66351	0.289581
$34$	4.50367	0.772373
$35$	1.43937	0.243299
$36$	−2.63977	−0.439961
$37$	0.575468	0.0946063	0.0473032	−	0.998881i	$-0.484937\pi$
$37$	0.575468	0.0946063	0.0473032	+	0.998881i	$0.484937\pi$
$38$	4.33225	0.702783
$39$	0	0
$40$	1.00000	0.158114
$41$	−4.22512	−0.659853	−0.329926	−	0.944007i	$-0.607024\pi$
$41$	−4.22512	−0.659853	−0.329926	+	0.944007i	$0.607024\pi$
$42$	−0.863906	−0.133304
$43$	2.61504	0.398789	0.199395	−	0.979919i	$-0.436102\pi$
$43$	2.61504	0.398789	0.199395	+	0.979919i	$0.436102\pi$
$44$	−2.77162	−0.417837
$45$	2.63977	0.393513
$46$	−4.43937	−0.654550
$47$	12.7684	1.86246	0.931228	−	0.364436i	$-0.118738\pi$
$47$	12.7684	1.86246	0.931228	+	0.364436i	$0.118738\pi$
$48$	−0.600196	−0.0866308
$49$	−4.92820	−0.704029
$50$	−1.00000	−0.141421
$51$	2.70308	0.378507
$52$	0	0
$53$	9.57123	1.31471	0.657355	−	0.753581i	$-0.271675\pi$
$53$	9.57123	1.31471	0.657355	+	0.753581i	$0.271675\pi$
$54$	−3.38496	−0.460635
$55$	2.77162	0.373725
$56$	1.43937	0.192344
$57$	2.60020	0.344404
$58$	−7.86488	−1.03271
$59$	3.46410	0.450988	0.225494	−	0.974245i	$-0.427600\pi$
$59$	3.46410	0.450988	0.225494	+	0.974245i	$0.427600\pi$
$60$	0.600196	0.0774849
$61$	−12.5037	−1.60093	−0.800466	−	0.599379i	$-0.795414\pi$
$61$	−12.5037	−1.60093	−0.800466	+	0.599379i	$0.795414\pi$
$62$	4.16082	0.528425
$63$	3.79961	0.478706
$64$	1.00000	0.125000
$65$	0	0
$66$	−1.66351	−0.204764
$67$	−5.32899	−0.651039	−0.325520	−	0.945535i	$-0.605539\pi$
$67$	−5.32899	−0.651039	−0.325520	+	0.945535i	$0.605539\pi$
$68$	−4.50367	−0.546150
$69$	−2.66449	−0.320767
$70$	−1.43937	−0.172038
$71$	6.00000	0.712069	0.356034	−	0.934473i	$-0.384129\pi$
$71$	6.00000	0.712069	0.356034	+	0.934473i	$0.384129\pi$
$72$	2.63977	0.311099
$73$	1.66351	0.194700	0.0973498	−	0.995250i	$-0.468963\pi$
$73$	1.66351	0.194700	0.0973498	+	0.995250i	$0.468963\pi$
$74$	−0.575468	−0.0668968
$75$	−0.600196	−0.0693046
$76$	−4.33225	−0.496943
$77$	3.98940	0.454634
$78$	0	0
$79$	12.7288	1.43210	0.716050	−	0.698049i	$-0.245948\pi$
$79$	12.7288	1.43210	0.716050	+	0.698049i	$0.245948\pi$
$80$	−1.00000	−0.111803
$81$	5.88766	0.654184
$82$	4.22512	0.466586
$83$	−10.0469	−1.10279	−0.551396	−	0.834244i	$-0.685905\pi$
$83$	−10.0469	−1.10279	−0.551396	+	0.834244i	$0.685905\pi$
$84$	0.863906	0.0942599
$85$	4.50367	0.488492
$86$	−2.61504	−0.281987
$87$	−4.72047	−0.506087
$88$	2.77162	0.295456
$89$	−5.95717	−0.631459	−0.315729	−	0.948849i	$-0.602249\pi$
$89$	−5.95717	−0.631459	−0.315729	+	0.948849i	$0.602249\pi$
$90$	−2.63977	−0.278256
$91$	0	0
$92$	4.43937	0.462837
$93$	2.49731	0.258959
$94$	−12.7684	−1.31696
$95$	4.33225	0.444479
$96$	0.600196	0.0612572
$97$	1.58633	0.161068	0.0805338	−	0.996752i	$-0.474337\pi$
$97$	1.58633	0.161068	0.0805338	+	0.996752i	$0.474337\pi$
$98$	4.92820	0.497824
$99$	7.31643	0.735329
$100$	1.00000	0.100000
$101$	13.7535	1.36853	0.684263	−	0.729235i	$-0.260124\pi$
$101$	13.7535	1.36853	0.684263	+	0.729235i	$0.260124\pi$
$102$	−2.70308	−0.267645
$103$	5.73205	0.564796	0.282398	−	0.959297i	$-0.408870\pi$
$103$	5.73205	0.564796	0.282398	+	0.959297i	$0.408870\pi$
$104$	0	0
$105$	−0.863906	−0.0843086
$106$	−9.57123	−0.929640
$107$	4.67933	0.452368	0.226184	−	0.974085i	$-0.427375\pi$
$107$	4.67933	0.452368	0.226184	+	0.974085i	$0.427375\pi$
$108$	3.38496	0.325718
$109$	−2.32164	−0.222373	−0.111187	−	0.993800i	$-0.535465\pi$
$109$	−2.32164	−0.222373	−0.111187	+	0.993800i	$0.535465\pi$
$110$	−2.77162	−0.264264
$111$	−0.345393	−0.0327833
$112$	−1.43937	−0.136008
$113$	7.73629	0.727769	0.363884	−	0.931444i	$-0.381450\pi$
$113$	7.73629	0.727769	0.363884	+	0.931444i	$0.381450\pi$
$114$	−2.60020	−0.243531
$115$	−4.43937	−0.413974
$116$	7.86488	0.730236
$117$	0	0
$118$	−3.46410	−0.318896
$119$	6.48247	0.594247
$120$	−0.600196	−0.0547901
$121$	−3.31812	−0.301647
$122$	12.5037	1.13203
$123$	2.53590	0.228654
$124$	−4.16082	−0.373653
$125$	−1.00000	−0.0894427
$126$	−3.79961	−0.338496
$127$	−16.1244	−1.43081	−0.715403	−	0.698712i	$-0.753757\pi$
$127$	−16.1244	−1.43081	−0.715403	+	0.698712i	$0.753757\pi$
$128$	−1.00000	−0.0883883
$129$	−1.56953	−0.138190
$130$	0	0
$131$	−19.7609	−1.72651	−0.863257	−	0.504764i	$-0.831579\pi$
$131$	−19.7609	−1.72651	−0.863257	+	0.504764i	$0.831579\pi$
$132$	1.66351	0.144790
$133$	6.23572	0.540706
$134$	5.32899	0.460354
$135$	−3.38496	−0.291331
$136$	4.50367	0.386187
$137$	17.2647	1.47502	0.737511	−	0.675335i	$-0.236001\pi$
$137$	17.2647	1.47502	0.737511	+	0.675335i	$0.236001\pi$
$138$	2.66449	0.226817
$139$	2.94657	0.249925	0.124962	−	0.992161i	$-0.460119\pi$
$139$	2.94657	0.249925	0.124962	+	0.992161i	$0.460119\pi$
$140$	1.43937	0.121649
$141$	−7.66351	−0.645384
$142$	−6.00000	−0.503509
$143$	0	0
$144$	−2.63977	−0.219980
$145$	−7.86488	−0.653143
$146$	−1.66351	−0.137673
$147$	2.95789	0.243962
$148$	0.575468	0.0473032
$149$	−7.06528	−0.578810	−0.289405	−	0.957207i	$-0.593457\pi$
$149$	−7.06528	−0.578810	−0.289405	+	0.957207i	$0.593457\pi$
$150$	0.600196	0.0490058
$151$	15.8327	1.28844	0.644222	−	0.764839i	$-0.277181\pi$
$151$	15.8327	1.28844	0.644222	+	0.764839i	$0.277181\pi$
$152$	4.33225	0.351392
$153$	11.8886	0.961139
$154$	−3.98940	−0.321475
$155$	4.16082	0.334205
$156$	0	0
$157$	12.6280	1.00783	0.503913	−	0.863754i	$-0.331893\pi$
$157$	12.6280	1.00783	0.503913	+	0.863754i	$0.331893\pi$
$158$	−12.7288	−1.01265
$159$	−5.74461	−0.455577
$160$	1.00000	0.0790569
$161$	−6.38992	−0.503596
$162$	−5.88766	−0.462578
$163$	18.3686	1.43874	0.719368	−	0.694629i	$-0.244431\pi$
$163$	18.3686	1.43874	0.719368	+	0.694629i	$0.244431\pi$
$164$	−4.22512	−0.329926
$165$	−1.66351	−0.129504
$166$	10.0469	0.779792
$167$	21.6085	1.67212	0.836059	−	0.548640i	$-0.184854\pi$
$167$	21.6085	1.67212	0.836059	+	0.548640i	$0.184854\pi$
$168$	−0.863906	−0.0666518
$169$	0	0
$170$	−4.50367	−0.345416
$171$	11.4361	0.874541
$172$	2.61504	0.199395
$173$	3.98940	0.303308	0.151654	−	0.988434i	$-0.451540\pi$
$173$	3.98940	0.303308	0.151654	+	0.988434i	$0.451540\pi$
$174$	4.72047	0.357858
$175$	−1.43937	−0.108806
$176$	−2.77162	−0.208919
$177$	−2.07914	−0.156278
$178$	5.95717	0.446509
$179$	12.4799	0.932793	0.466397	−	0.884576i	$-0.345552\pi$
$179$	12.4799	0.932793	0.466397	+	0.884576i	$0.345552\pi$
$180$	2.63977	0.196756
$181$	−19.4319	−1.44436	−0.722180	−	0.691705i	$-0.756860\pi$
$181$	−19.4319	−1.44436	−0.722180	+	0.691705i	$0.756860\pi$
$182$	0	0
$183$	7.50465	0.554760
$184$	−4.43937	−0.327275
$185$	−0.575468	−0.0423092
$186$	−2.49731	−0.183111
$187$	12.4825	0.912808
$188$	12.7684	0.931228
$189$	−4.87223	−0.354402
$190$	−4.33225	−0.314294
$191$	−18.3215	−1.32570	−0.662848	−	0.748754i	$-0.730653\pi$
$191$	−18.3215	−1.32570	−0.662848	+	0.748754i	$0.730653\pi$
$192$	−0.600196	−0.0433154
$193$	4.74363	0.341454	0.170727	−	0.985318i	$-0.445388\pi$
$193$	4.74363	0.341454	0.170727	+	0.985318i	$0.445388\pi$
$194$	−1.58633	−0.113892
$195$	0	0
$196$	−4.92820	−0.352015
$197$	10.8006	0.769510	0.384755	−	0.923019i	$-0.374286\pi$
$197$	10.8006	0.769510	0.384755	+	0.923019i	$0.374286\pi$
$198$	−7.31643	−0.519956
$199$	11.0073	0.780290	0.390145	−	0.920753i	$-0.372425\pi$
$199$	11.0073	0.780290	0.390145	+	0.920753i	$0.372425\pi$
$200$	−1.00000	−0.0707107
$201$	3.19843	0.225600
$202$	−13.7535	−0.967694
$203$	−11.3205	−0.794544
$204$	2.70308	0.189254
$205$	4.22512	0.295095
$206$	−5.73205	−0.399371
$207$	−11.7189	−0.814520
$208$	0	0
$209$	12.0073	0.830565
$210$	0.863906	0.0596152
$211$	26.2582	1.80769	0.903843	−	0.427863i	$-0.140734\pi$
$211$	26.2582	1.80769	0.903843	+	0.427863i	$0.140734\pi$
$212$	9.57123	0.657355
$213$	−3.60117	−0.246748
$214$	−4.67933	−0.319873
$215$	−2.61504	−0.178344
$216$	−3.38496	−0.230318
$217$	5.98898	0.406558
$218$	2.32164	0.157242
$219$	−0.998434	−0.0674679
$220$	2.77162	0.186863
$221$	0	0
$222$	0.345393	0.0231813
$223$	22.2042	1.48690	0.743452	−	0.668789i	$-0.233187\pi$
$223$	22.2042	1.48690	0.743452	+	0.668789i	$0.233187\pi$
$224$	1.43937	0.0961722
$225$	−2.63977	−0.175984
$226$	−7.73629	−0.514610
$227$	2.07914	0.137997	0.0689986	−	0.997617i	$-0.478020\pi$
$227$	2.07914	0.137997	0.0689986	+	0.997617i	$0.478020\pi$
$228$	2.60020	0.172202
$229$	4.67933	0.309219	0.154610	−	0.987976i	$-0.450588\pi$
$229$	4.67933	0.309219	0.154610	+	0.987976i	$0.450588\pi$
$230$	4.43937	0.292724
$231$	−2.39442	−0.157541
$232$	−7.86488	−0.516355
$233$	−0.611060	−0.0400319	−0.0200160	−	0.999800i	$-0.506372\pi$
$233$	−0.611060	−0.0400319	−0.0200160	+	0.999800i	$0.506372\pi$
$234$	0	0
$235$	−12.7684	−0.832916
$236$	3.46410	0.225494
$237$	−7.63977	−0.496256
$238$	−6.48247	−0.420196
$239$	25.6481	1.65904	0.829518	−	0.558479i	$-0.188615\pi$
$239$	25.6481	1.65904	0.829518	+	0.558479i	$0.188615\pi$
$240$	0.600196	0.0387425
$241$	−27.0931	−1.74522	−0.872610	−	0.488417i	$-0.837574\pi$
$241$	−27.0931	−1.74522	−0.872610	+	0.488417i	$0.837574\pi$
$242$	3.31812	0.213297
$243$	−13.6886	−0.878126
$244$	−12.5037	−0.800466
$245$	4.92820	0.314851
$246$	−2.53590	−0.161683
$247$	0	0
$248$	4.16082	0.264212
$249$	6.03011	0.382143
$250$	1.00000	0.0632456
$251$	1.50367	0.0949109	0.0474554	−	0.998873i	$-0.484889\pi$
$251$	1.50367	0.0949109	0.0474554	+	0.998873i	$0.484889\pi$
$252$	3.79961	0.239353
$253$	−12.3043	−0.773562
$254$	16.1244	1.01173
$255$	−2.70308	−0.169274
$256$	1.00000	0.0625000
$257$	−20.3448	−1.26907	−0.634537	−	0.772892i	$-0.718809\pi$
$257$	−20.3448	−1.26907	−0.634537	+	0.772892i	$0.718809\pi$
$258$	1.56953	0.0977149
$259$	−0.828313	−0.0514689
$260$	0	0
$261$	−20.7614	−1.28510
$262$	19.7609	1.22083
$263$	−17.1293	−1.05624	−0.528119	−	0.849170i	$-0.677103\pi$
$263$	−17.1293	−1.05624	−0.528119	+	0.849170i	$0.677103\pi$
$264$	−1.66351	−0.102382
$265$	−9.57123	−0.587956
$266$	−6.23572	−0.382337
$267$	3.57547	0.218815
$268$	−5.32899	−0.325520
$269$	19.3824	1.18177	0.590883	−	0.806757i	$-0.298779\pi$
$269$	19.3824	1.18177	0.590883	+	0.806757i	$0.298779\pi$
$270$	3.38496	0.206002
$271$	−18.3711	−1.11596	−0.557982	−	0.829853i	$-0.688424\pi$
$271$	−18.3711	−1.11596	−0.557982	+	0.829853i	$0.688424\pi$
$272$	−4.50367	−0.273075
$273$	0	0
$274$	−17.2647	−1.04300
$275$	−2.77162	−0.167135
$276$	−2.66449	−0.160384
$277$	−27.2342	−1.63634	−0.818171	−	0.574975i	$-0.805012\pi$
$277$	−27.2342	−1.63634	−0.818171	+	0.574975i	$0.805012\pi$
$278$	−2.94657	−0.176723
$279$	10.9836	0.657570
$280$	−1.43937	−0.0860190
$281$	−30.0815	−1.79451	−0.897257	−	0.441509i	$-0.854443\pi$
$281$	−30.0815	−1.79451	−0.897257	+	0.441509i	$0.854443\pi$
$282$	7.66351	0.456356
$283$	−25.4788	−1.51456	−0.757278	−	0.653092i	$-0.773471\pi$
$283$	−25.4788	−1.51456	−0.757278	+	0.653092i	$0.773471\pi$
$284$	6.00000	0.356034
$285$	−2.60020	−0.154022
$286$	0	0
$287$	6.08153	0.358981
$288$	2.63977	0.155550
$289$	3.28305	0.193121
$290$	7.86488	0.461842
$291$	−0.952110	−0.0558137
$292$	1.66351	0.0973498
$293$	16.5779	0.968489	0.484244	−	0.874933i	$-0.339095\pi$
$293$	16.5779	0.968489	0.484244	+	0.874933i	$0.339095\pi$
$294$	−2.95789	−0.172507
$295$	−3.46410	−0.201688
$296$	−0.575468	−0.0334484
$297$	−9.38183	−0.544389
$298$	7.06528	0.409280
$299$	0	0
$300$	−0.600196	−0.0346523
$301$	−3.76402	−0.216954
$302$	−15.8327	−0.911067
$303$	−8.25480	−0.474226
$304$	−4.33225	−0.248471
$305$	12.5037	0.715958
$306$	−11.8886	−0.679628
$307$	16.2722	0.928703	0.464351	−	0.885651i	$-0.346287\pi$
$307$	16.2722	0.928703	0.464351	+	0.885651i	$0.346287\pi$
$308$	3.98940	0.227317
$309$	−3.44035	−0.195715
$310$	−4.16082	−0.236319
$311$	16.1053	0.913246	0.456623	−	0.889660i	$-0.349059\pi$
$311$	16.1053	0.913246	0.456623	+	0.889660i	$0.349059\pi$
$312$	0	0
$313$	−10.2162	−0.577454	−0.288727	−	0.957411i	$-0.593232\pi$
$313$	−10.2162	−0.577454	−0.288727	+	0.957411i	$0.593232\pi$
$314$	−12.6280	−0.712641
$315$	−3.79961	−0.214084
$316$	12.7288	0.716050
$317$	0.204368	0.0114785	0.00573923	−	0.999984i	$-0.498173\pi$
$317$	0.204368	0.0114785	0.00573923	+	0.999984i	$0.498173\pi$
$318$	5.74461	0.322142
$319$	−21.7985	−1.22048
$320$	−1.00000	−0.0559017
$321$	−2.80852	−0.156756
$322$	6.38992	0.356096
$323$	19.5110	1.08562
$324$	5.88766	0.327092
$325$	0	0
$326$	−18.3686	−1.01734
$327$	1.39344	0.0770574
$328$	4.22512	0.233293
$329$	−18.3784	−1.01324
$330$	1.66351	0.0915735
$331$	19.0653	1.04792	0.523961	−	0.851742i	$-0.324454\pi$
$331$	19.0653	1.04792	0.523961	+	0.851742i	$0.324454\pi$
$332$	−10.0469	−0.551396
$333$	−1.51910	−0.0832462
$334$	−21.6085	−1.18237
$335$	5.32899	0.291154
$336$	0.863906	0.0471299
$337$	−19.8609	−1.08189	−0.540946	−	0.841057i	$-0.681934\pi$
$337$	−19.8609	−1.08189	−0.540946	+	0.841057i	$0.681934\pi$
$338$	0	0
$339$	−4.64329	−0.252189
$340$	4.50367	0.244246
$341$	11.5322	0.624505
$342$	−11.4361	−0.618394
$343$	17.1691	0.927047
$344$	−2.61504	−0.140993
$345$	2.66449	0.143451
$346$	−3.98940	−0.214471
$347$	17.0504	0.915315	0.457658	−	0.889129i	$-0.348689\pi$
$347$	17.0504	0.915315	0.457658	+	0.889129i	$0.348689\pi$
$348$	−4.72047	−0.253044
$349$	21.2370	1.13679	0.568394	−	0.822756i	$-0.307565\pi$
$349$	21.2370	1.13679	0.568394	+	0.822756i	$0.307565\pi$
$350$	1.43937	0.0769378
$351$	0	0
$352$	2.77162	0.147728
$353$	−16.0168	−0.852488	−0.426244	−	0.904608i	$-0.640164\pi$
$353$	−16.0168	−0.852488	−0.426244	+	0.904608i	$0.640164\pi$
$354$	2.07914	0.110505
$355$	−6.00000	−0.318447
$356$	−5.95717	−0.315729
$357$	−3.89075	−0.205920
$358$	−12.4799	−0.659584
$359$	−24.2487	−1.27980	−0.639899	−	0.768459i	$-0.721024\pi$
$359$	−24.2487	−1.27980	−0.639899	+	0.768459i	$0.721024\pi$
$360$	−2.63977	−0.139128
$361$	−0.231640	−0.0121916
$362$	19.4319	1.02132
$363$	1.99152	0.104528
$364$	0	0
$365$	−1.66351	−0.0870723
$366$	−7.50465	−0.392274
$367$	−29.7707	−1.55402	−0.777010	−	0.629488i	$-0.783265\pi$
$367$	−29.7707	−1.55402	−0.777010	+	0.629488i	$0.783265\pi$
$368$	4.43937	0.231418
$369$	11.1533	0.580619
$370$	0.575468	0.0299171
$371$	−13.7766	−0.715244
$372$	2.49731	0.129479
$373$	28.3858	1.46976	0.734880	−	0.678197i	$-0.237238\pi$
$373$	28.3858	1.46976	0.734880	+	0.678197i	$0.237238\pi$
$374$	−12.4825	−0.645453
$375$	0.600196	0.0309940
$376$	−12.7684	−0.658478
$377$	0	0
$378$	4.87223	0.250600
$379$	28.4138	1.45952	0.729759	−	0.683705i	$-0.239632\pi$
$379$	28.4138	1.45952	0.729759	+	0.683705i	$0.239632\pi$
$380$	4.33225	0.222240
$381$	9.67777	0.495807
$382$	18.3215	0.937409
$383$	−9.39726	−0.480178	−0.240089	−	0.970751i	$-0.577177\pi$
$383$	−9.39726	−0.480178	−0.240089	+	0.970751i	$0.577177\pi$
$384$	0.600196	0.0306286
$385$	−3.98940	−0.203319
$386$	−4.74363	−0.241445
$387$	−6.90308	−0.350903
$388$	1.58633	0.0805338
$389$	−11.6461	−0.590482	−0.295241	−	0.955423i	$-0.595400\pi$
$389$	−11.6461	−0.590482	−0.295241	+	0.955423i	$0.595400\pi$
$390$	0	0
$391$	−19.9935	−1.01111
$392$	4.92820	0.248912
$393$	11.8604	0.598277
$394$	−10.8006	−0.544126
$395$	−12.7288	−0.640455
$396$	7.31643	0.367664
$397$	0.399804	0.0200656	0.0100328	−	0.999950i	$-0.496806\pi$
$397$	0.399804	0.0200656	0.0100328	+	0.999950i	$0.496806\pi$
$398$	−11.0073	−0.551748
$399$	−3.74265	−0.187367
$400$	1.00000	0.0500000
$401$	8.50775	0.424857	0.212428	−	0.977177i	$-0.431863\pi$
$401$	8.50775	0.424857	0.212428	+	0.977177i	$0.431863\pi$
$402$	−3.19843	−0.159523
$403$	0	0
$404$	13.7535	0.684263
$405$	−5.88766	−0.292560
$406$	11.3205	0.561827
$407$	−1.59498	−0.0790601
$408$	−2.70308	−0.133823
$409$	−14.0131	−0.692906	−0.346453	−	0.938067i	$-0.612614\pi$
$409$	−14.0131	−0.692906	−0.346453	+	0.938067i	$0.612614\pi$
$410$	−4.22512	−0.208664
$411$	−10.3622	−0.511129
$412$	5.73205	0.282398
$413$	−4.98614	−0.245352
$414$	11.7189	0.575953
$415$	10.0469	0.493183
$416$	0	0
$417$	−1.76852	−0.0866047
$418$	−12.0073	−0.587298
$419$	−24.1954	−1.18202	−0.591012	−	0.806663i	$-0.701272\pi$
$419$	−24.1954	−1.18202	−0.591012	+	0.806663i	$0.701272\pi$
$420$	−0.863906	−0.0421543
$421$	−35.6392	−1.73695	−0.868474	−	0.495735i	$-0.834899\pi$
$421$	−35.6392	−1.73695	−0.868474	+	0.495735i	$0.834899\pi$
$422$	−26.2582	−1.27823
$423$	−33.7055	−1.63882
$424$	−9.57123	−0.464820
$425$	−4.50367	−0.218460
$426$	3.60117	0.174477
$427$	17.9975	0.870958
$428$	4.67933	0.226184
$429$	0	0
$430$	2.61504	0.126108
$431$	−4.96043	−0.238936	−0.119468	−	0.992838i	$-0.538119\pi$
$431$	−4.96043	−0.238936	−0.119468	+	0.992838i	$0.538119\pi$
$432$	3.38496	0.162859
$433$	4.21621	0.202618	0.101309	−	0.994855i	$-0.467697\pi$
$433$	4.21621	0.202618	0.101309	+	0.994855i	$0.467697\pi$
$434$	−5.98898	−0.287480
$435$	4.72047	0.226329
$436$	−2.32164	−0.111187
$437$	−19.2325	−0.920013
$438$	0.998434	0.0477070
$439$	−14.7500	−0.703979	−0.351989	−	0.936004i	$-0.614495\pi$
$439$	−14.7500	−0.703979	−0.351989	+	0.936004i	$0.614495\pi$
$440$	−2.77162	−0.132132
$441$	13.0093	0.619490
$442$	0	0
$443$	−12.2374	−0.581417	−0.290709	−	0.956812i	$-0.593891\pi$
$443$	−12.2374	−0.581417	−0.290709	+	0.956812i	$0.593891\pi$
$444$	−0.345393	−0.0163916
$445$	5.95717	0.282397
$446$	−22.2042	−1.05140
$447$	4.24055	0.200571
$448$	−1.43937	−0.0680040
$449$	19.0714	0.900034	0.450017	−	0.893020i	$-0.351418\pi$
$449$	19.0714	0.900034	0.450017	+	0.893020i	$0.351418\pi$
$450$	2.63977	0.124440
$451$	11.7104	0.551422
$452$	7.73629	0.363884
$453$	−9.50269	−0.446475
$454$	−2.07914	−0.0975788
$455$	0	0
$456$	−2.60020	−0.121765
$457$	−16.1138	−0.753770	−0.376885	−	0.926260i	$-0.623005\pi$
$457$	−16.1138	−0.753770	−0.376885	+	0.926260i	$0.623005\pi$
$458$	−4.67933	−0.218651
$459$	−15.2448	−0.711564
$460$	−4.43937	−0.206987
$461$	4.84466	0.225638	0.112819	−	0.993616i	$-0.464012\pi$
$461$	4.84466	0.225638	0.112819	+	0.993616i	$0.464012\pi$
$462$	2.39442	0.111398
$463$	8.23164	0.382557	0.191278	−	0.981536i	$-0.438737\pi$
$463$	8.23164	0.382557	0.191278	+	0.981536i	$0.438737\pi$
$464$	7.86488	0.365118
$465$	−2.49731	−0.115810
$466$	0.611060	0.0283068
$467$	33.4873	1.54961	0.774803	−	0.632203i	$-0.217849\pi$
$467$	33.4873	1.54961	0.774803	+	0.632203i	$0.217849\pi$
$468$	0	0
$469$	7.67040	0.354186
$470$	12.7684	0.588961
$471$	−7.57929	−0.349235
$472$	−3.46410	−0.159448
$473$	−7.24789	−0.333258
$474$	7.63977	0.350906
$475$	−4.33225	−0.198777
$476$	6.48247	0.297123
$477$	−25.2658	−1.15684
$478$	−25.6481	−1.17312
$479$	8.05343	0.367971	0.183985	−	0.982929i	$-0.441100\pi$
$479$	8.05343	0.367971	0.183985	+	0.982929i	$0.441100\pi$
$480$	−0.600196	−0.0273951
$481$	0	0
$482$	27.0931	1.23406
$483$	3.83520	0.174508
$484$	−3.31812	−0.150824
$485$	−1.58633	−0.0720317
$486$	13.6886	0.620929
$487$	−1.62352	−0.0735685	−0.0367842	−	0.999323i	$-0.511711\pi$
$487$	−1.62352	−0.0735685	−0.0367842	+	0.999323i	$0.511711\pi$
$488$	12.5037	0.566015
$489$	−11.0247	−0.498555
$490$	−4.92820	−0.222634
$491$	10.3207	0.465765	0.232883	−	0.972505i	$-0.425184\pi$
$491$	10.3207	0.465765	0.232883	+	0.972505i	$0.425184\pi$
$492$	2.53590	0.114327
$493$	−35.4209	−1.59527
$494$	0	0
$495$	−7.31643	−0.328849
$496$	−4.16082	−0.186826
$497$	−8.63624	−0.387388
$498$	−6.03011	−0.270216
$499$	13.9807	0.625860	0.312930	−	0.949776i	$-0.398689\pi$
$499$	13.9807	0.625860	0.312930	+	0.949776i	$0.398689\pi$
$500$	−1.00000	−0.0447214
$501$	−12.9693	−0.579427
$502$	−1.50367	−0.0671121
$503$	23.7849	1.06052	0.530258	−	0.847836i	$-0.322095\pi$
$503$	23.7849	1.06052	0.530258	+	0.847836i	$0.322095\pi$
$504$	−3.79961	−0.169248
$505$	−13.7535	−0.612024
$506$	12.3043	0.546991
$507$	0	0
$508$	−16.1244	−0.715403
$509$	25.6371	1.13634	0.568171	−	0.822910i	$-0.307651\pi$
$509$	25.6371	1.13634	0.568171	+	0.822910i	$0.307651\pi$
$510$	2.70308	0.119695
$511$	−2.39442	−0.105923
$512$	−1.00000	−0.0441942
$513$	−14.6645	−0.647453
$514$	20.3448	0.897371
$515$	−5.73205	−0.252584
$516$	−1.56953	−0.0690949
$517$	−35.3890	−1.55641
$518$	0.828313	0.0363940
$519$	−2.39442	−0.105103
$520$	0	0
$521$	−8.53476	−0.373915	−0.186957	−	0.982368i	$-0.559863\pi$
$521$	−8.53476	−0.373915	−0.186957	+	0.982368i	$0.559863\pi$
$522$	20.7614	0.908704
$523$	−12.6877	−0.554792	−0.277396	−	0.960756i	$-0.589471\pi$
$523$	−12.6877	−0.554792	−0.277396	+	0.960756i	$0.589471\pi$
$524$	−19.7609	−0.863257
$525$	0.863906	0.0377039
$526$	17.1293	0.746873
$527$	18.7390	0.816283
$528$	1.66351	0.0723952
$529$	−3.29196	−0.143129
$530$	9.57123	0.415748
$531$	−9.14441	−0.396834
$532$	6.23572	0.270353
$533$	0	0
$534$	−3.57547	−0.154726
$535$	−4.67933	−0.202305
$536$	5.32899	0.230177
$537$	−7.49040	−0.323234
$538$	−19.3824	−0.835635
$539$	13.6591	0.588339
$540$	−3.38496	−0.145666
$541$	−42.0507	−1.80790	−0.903951	−	0.427636i	$-0.859347\pi$
$541$	−42.0507	−1.80790	−0.903951	+	0.427636i	$0.859347\pi$
$542$	18.3711	0.789106
$543$	11.6629	0.500504
$544$	4.50367	0.193093
$545$	2.32164	0.0994483
$546$	0	0
$547$	24.6297	1.05309	0.526545	−	0.850147i	$-0.323487\pi$
$547$	24.6297	1.05309	0.526545	+	0.850147i	$0.323487\pi$
$548$	17.2647	0.737511
$549$	33.0068	1.40869
$550$	2.77162	0.118182
$551$	−34.0726	−1.45154
$552$	2.66449	0.113408
$553$	−18.3215	−0.779109
$554$	27.2342	1.15707
$555$	0.345393	0.0146611
$556$	2.94657	0.124962
$557$	7.90488	0.334941	0.167470	−	0.985877i	$-0.446440\pi$
$557$	7.90488	0.334941	0.167470	+	0.985877i	$0.446440\pi$
$558$	−10.9836	−0.464973
$559$	0	0
$560$	1.43937	0.0608246
$561$	−7.49192	−0.316309
$562$	30.0815	1.26891
$563$	13.2711	0.559308	0.279654	−	0.960101i	$-0.409780\pi$
$563$	13.2711	0.559308	0.279654	+	0.960101i	$0.409780\pi$
$564$	−7.66351	−0.322692
$565$	−7.73629	−0.325468
$566$	25.4788	1.07095
$567$	−8.47454	−0.355897
$568$	−6.00000	−0.251754
$569$	42.2645	1.77182	0.885911	−	0.463856i	$-0.153534\pi$
$569$	42.2645	1.77182	0.885911	+	0.463856i	$0.153534\pi$
$570$	2.60020	0.108910
$571$	−1.64965	−0.0690358	−0.0345179	−	0.999404i	$-0.510990\pi$
$571$	−1.64965	−0.0690358	−0.0345179	+	0.999404i	$0.510990\pi$
$572$	0	0
$573$	10.9965	0.459384
$574$	−6.08153	−0.253838
$575$	4.43937	0.185135
$576$	−2.63977	−0.109990
$577$	28.4651	1.18502	0.592508	−	0.805564i	$-0.298138\pi$
$577$	28.4651	1.18502	0.592508	+	0.805564i	$0.298138\pi$
$578$	−3.28305	−0.136557
$579$	−2.84711	−0.118322
$580$	−7.86488	−0.326572
$581$	14.4613	0.599954
$582$	0.952110	0.0394662
$583$	−26.5278	−1.09867
$584$	−1.66351	−0.0688367
$585$	0	0
$586$	−16.5779	−0.684825
$587$	−22.1696	−0.915036	−0.457518	−	0.889200i	$-0.651262\pi$
$587$	−22.1696	−0.915036	−0.457518	+	0.889200i	$0.651262\pi$
$588$	2.95789	0.121981
$589$	18.0257	0.742736
$590$	3.46410	0.142615
$591$	−6.48247	−0.266653
$592$	0.575468	0.0236516
$593$	−9.68683	−0.397791	−0.198895	−	0.980021i	$-0.563735\pi$
$593$	−9.68683	−0.397791	−0.198895	+	0.980021i	$0.563735\pi$
$594$	9.38183	0.384941
$595$	−6.48247	−0.265755
$596$	−7.06528	−0.289405
$597$	−6.60656	−0.270388
$598$	0	0
$599$	8.33012	0.340360	0.170180	−	0.985413i	$-0.445565\pi$
$599$	8.33012	0.340360	0.170180	+	0.985413i	$0.445565\pi$
$600$	0.600196	0.0245029
$601$	−12.2023	−0.497744	−0.248872	−	0.968536i	$-0.580060\pi$
$601$	−12.2023	−0.497744	−0.248872	+	0.968536i	$0.580060\pi$
$602$	3.76402	0.153410
$603$	14.0673	0.572864
$604$	15.8327	0.644222
$605$	3.31812	0.134901
$606$	8.25480	0.335328
$607$	−9.83892	−0.399349	−0.199675	−	0.979862i	$-0.563989\pi$
$607$	−9.83892	−0.399349	−0.199675	+	0.979862i	$0.563989\pi$
$608$	4.33225	0.175696
$609$	6.79452	0.275328
$610$	−12.5037	−0.506259
$611$	0	0
$612$	11.8886	0.480570
$613$	−15.2117	−0.614395	−0.307198	−	0.951646i	$-0.599391\pi$
$613$	−15.2117	−0.614395	−0.307198	+	0.951646i	$0.599391\pi$
$614$	−16.2722	−0.656692
$615$	−2.53590	−0.102257
$616$	−3.98940	−0.160737
$617$	−24.4714	−0.985183	−0.492592	−	0.870261i	$-0.663950\pi$
$617$	−24.4714	−0.985183	−0.492592	+	0.870261i	$0.663950\pi$
$618$	3.44035	0.138391
$619$	42.4157	1.70483	0.852416	−	0.522864i	$-0.175136\pi$
$619$	42.4157	1.70483	0.852416	+	0.522864i	$0.175136\pi$
$620$	4.16082	0.167103
$621$	15.0271	0.603017
$622$	−16.1053	−0.645763
$623$	8.57459	0.343534
$624$	0	0
$625$	1.00000	0.0400000
$626$	10.2162	0.408322
$627$	−7.20676	−0.287810
$628$	12.6280	0.503913
$629$	−2.59172	−0.103339
$630$	3.79961	0.151380
$631$	0.0282505	0.00112464	0.000562318	−	1.00000i	$-0.499821\pi$
$631$	0.0282505	0.00112464	0.000562318		1.00000i	$0.499821\pi$
$632$	−12.7288	−0.506324
$633$	−15.7600	−0.626405
$634$	−0.204368	−0.00811650
$635$	16.1244	0.639876
$636$	−5.74461	−0.227789
$637$	0	0
$638$	21.7985	0.863010
$639$	−15.8386	−0.626565
$640$	1.00000	0.0395285
$641$	16.2093	0.640229	0.320114	−	0.947379i	$-0.396279\pi$
$641$	16.2093	0.640229	0.320114	+	0.947379i	$0.396279\pi$
$642$	2.80852	0.110843
$643$	18.2117	0.718200	0.359100	−	0.933299i	$-0.383084\pi$
$643$	18.2117	0.718200	0.359100	+	0.933299i	$0.383084\pi$
$644$	−6.38992	−0.251798
$645$	1.56953	0.0618003
$646$	−19.5110	−0.767651
$647$	−7.12279	−0.280026	−0.140013	−	0.990150i	$-0.544714\pi$
$647$	−7.12279	−0.280026	−0.140013	+	0.990150i	$0.544714\pi$
$648$	−5.88766	−0.231289
$649$	−9.60117	−0.376879
$650$	0	0
$651$	−3.59456	−0.140882
$652$	18.3686	0.719368
$653$	−16.0709	−0.628904	−0.314452	−	0.949273i	$-0.601821\pi$
$653$	−16.0709	−0.628904	−0.314452	+	0.949273i	$0.601821\pi$
$654$	−1.39344	−0.0544878
$655$	19.7609	0.772121
$656$	−4.22512	−0.164963
$657$	−4.39129	−0.171320
$658$	18.3784	0.716466
$659$	45.1977	1.76065	0.880326	−	0.474369i	$-0.157324\pi$
$659$	45.1977	1.76065	0.880326	+	0.474369i	$0.157324\pi$
$660$	−1.66351	−0.0647522
$661$	−8.81445	−0.342842	−0.171421	−	0.985198i	$-0.554836\pi$
$661$	−8.81445	−0.342842	−0.171421	+	0.985198i	$0.554836\pi$
$662$	−19.0653	−0.740993
$663$	0	0
$664$	10.0469	0.389896
$665$	−6.23572	−0.241811
$666$	1.51910	0.0588639
$667$	34.9152	1.35192
$668$	21.6085	0.836059
$669$	−13.3269	−0.515247
$670$	−5.32899	−0.205877
$671$	34.6554	1.33786
$672$	−0.863906	−0.0333259
$673$	41.4398	1.59739	0.798693	−	0.601739i	$-0.205525\pi$
$673$	41.4398	1.59739	0.798693	+	0.601739i	$0.205525\pi$
$674$	19.8609	0.765014
$675$	3.38496	0.130287
$676$	0	0
$677$	−18.8510	−0.724504	−0.362252	−	0.932080i	$-0.617992\pi$
$677$	−18.8510	−0.724504	−0.362252	+	0.932080i	$0.617992\pi$
$678$	4.64329	0.178324
$679$	−2.28333	−0.0876260
$680$	−4.50367	−0.172708
$681$	−1.24789	−0.0478193
$682$	−11.5322	−0.441591
$683$	−22.6664	−0.867308	−0.433654	−	0.901080i	$-0.642776\pi$
$683$	−22.6664	−0.867308	−0.433654	+	0.901080i	$0.642776\pi$
$684$	11.4361	0.437271
$685$	−17.2647	−0.659650
$686$	−17.1691	−0.655521
$687$	−2.80852	−0.107152
$688$	2.61504	0.0996974
$689$	0	0
$690$	−2.66449	−0.101436
$691$	−31.8393	−1.21122	−0.605612	−	0.795760i	$-0.707072\pi$
$691$	−31.8393	−1.21122	−0.605612	+	0.795760i	$0.707072\pi$
$692$	3.98940	0.151654
$693$	−10.5311	−0.400042
$694$	−17.0504	−0.647226
$695$	−2.94657	−0.111770
$696$	4.72047	0.178929
$697$	19.0285	0.720758
$698$	−21.2370	−0.803831
$699$	0.366756	0.0138720
$700$	−1.43937	−0.0544032
$701$	−0.611060	−0.0230794	−0.0115397	−	0.999933i	$-0.503673\pi$
$701$	−0.611060	−0.0230794	−0.0115397	+	0.999933i	$0.503673\pi$
$702$	0	0
$703$	−2.49307	−0.0940279
$704$	−2.77162	−0.104459
$705$	7.66351	0.288625
$706$	16.0168	0.602800
$707$	−19.7965	−0.744522
$708$	−2.07914	−0.0781388
$709$	−12.9642	−0.486883	−0.243441	−	0.969916i	$-0.578276\pi$
$709$	−12.9642	−0.486883	−0.243441	+	0.969916i	$0.578276\pi$
$710$	6.00000	0.225176
$711$	−33.6010	−1.26014
$712$	5.95717	0.223254
$713$	−18.4714	−0.691761
$714$	3.89075	0.145608
$715$	0	0
$716$	12.4799	0.466397
$717$	−15.3939	−0.574895
$718$	24.2487	0.904954
$719$	35.7232	1.33225	0.666126	−	0.745839i	$-0.267951\pi$
$719$	35.7232	1.33225	0.666126	+	0.745839i	$0.267951\pi$
$720$	2.63977	0.0983782
$721$	−8.25056	−0.307267
$722$	0.231640	0.00862076
$723$	16.2612	0.604759
$724$	−19.4319	−0.722180
$725$	7.86488	0.292094
$726$	−1.99152	−0.0739123
$727$	51.7313	1.91861	0.959304	−	0.282375i	$-0.0911224\pi$
$727$	51.7313	1.91861	0.959304	+	0.282375i	$0.0911224\pi$
$728$	0	0
$729$	−9.44711	−0.349893
$730$	1.66351	0.0615694
$731$	−11.7773	−0.435598
$732$	7.50465	0.277380
$733$	26.4319	0.976284	0.488142	−	0.872764i	$-0.337675\pi$
$733$	26.4319	0.976284	0.488142	+	0.872764i	$0.337675\pi$
$734$	29.7707	1.09886
$735$	−2.95789	−0.109103
$736$	−4.43937	−0.163637
$737$	14.7699	0.544057
$738$	−11.1533	−0.410559
$739$	−27.4102	−1.00830	−0.504151	−	0.863615i	$-0.668195\pi$
$739$	−27.4102	−1.00830	−0.504151	+	0.863615i	$0.668195\pi$
$740$	−0.575468	−0.0211546
$741$	0	0
$742$	13.7766	0.505754
$743$	42.1819	1.54750	0.773751	−	0.633489i	$-0.218378\pi$
$743$	42.1819	1.54750	0.773751	+	0.633489i	$0.218378\pi$
$744$	−2.49731	−0.0915557
$745$	7.06528	0.258852
$746$	−28.3858	−1.03928
$747$	26.5215	0.970370
$748$	12.4825	0.456404
$749$	−6.73531	−0.246103
$750$	−0.600196	−0.0219160
$751$	19.3270	0.705253	0.352627	−	0.935764i	$-0.385289\pi$
$751$	19.3270	0.705253	0.352627	+	0.935764i	$0.385289\pi$
$752$	12.7684	0.465614
$753$	−0.902497	−0.0328888
$754$	0	0
$755$	−15.8327	−0.576209
$756$	−4.87223	−0.177201
$757$	−20.3026	−0.737909	−0.368955	−	0.929447i	$-0.620284\pi$
$757$	−20.3026	−0.737909	−0.368955	+	0.929447i	$0.620284\pi$
$758$	−28.4138	−1.03203
$759$	7.38496	0.268057
$760$	−4.33225	−0.157147
$761$	44.2696	1.60477	0.802386	−	0.596806i	$-0.203564\pi$
$761$	44.2696	1.60477	0.802386	+	0.596806i	$0.203564\pi$
$762$	−9.67777	−0.350589
$763$	3.34171	0.120978
$764$	−18.3215	−0.662848
$765$	−11.8886	−0.429834
$766$	9.39726	0.339537
$767$	0	0
$768$	−0.600196	−0.0216577
$769$	−3.74011	−0.134872	−0.0674359	−	0.997724i	$-0.521482\pi$
$769$	−3.74011	−0.134872	−0.0674359	+	0.997724i	$0.521482\pi$
$770$	3.98940	0.143768
$771$	12.2109	0.439764
$772$	4.74363	0.170727
$773$	4.48090	0.161167	0.0805834	−	0.996748i	$-0.474322\pi$
$773$	4.48090	0.161167	0.0805834	+	0.996748i	$0.474322\pi$
$774$	6.90308	0.248126
$775$	−4.16082	−0.149461
$776$	−1.58633	−0.0569460
$777$	0.497150	0.0178352
$778$	11.6461	0.417534
$779$	18.3043	0.655818
$780$	0	0
$781$	−16.6297	−0.595058
$782$	19.9935	0.714965
$783$	26.6223	0.951405
$784$	−4.92820	−0.176007
$785$	−12.6280	−0.450714
$786$	−11.8604	−0.423046
$787$	32.0964	1.14411	0.572056	−	0.820215i	$-0.306146\pi$
$787$	32.0964	1.14411	0.572056	+	0.820215i	$0.306146\pi$
$788$	10.8006	0.384755
$789$	10.2809	0.366011
$790$	12.7288	0.452870
$791$	−11.1354	−0.395930
$792$	−7.31643	−0.259978
$793$	0	0
$794$	−0.399804	−0.0141885
$795$	5.74461	0.203740
$796$	11.0073	0.390145
$797$	22.2166	0.786954	0.393477	−	0.919335i	$-0.371272\pi$
$797$	22.2166	0.786954	0.393477	+	0.919335i	$0.371272\pi$
$798$	3.74265	0.132488
$799$	−57.5045	−2.03436
$800$	−1.00000	−0.0353553
$801$	15.7255	0.555634
$802$	−8.50775	−0.300419
$803$	−4.61063	−0.162706
$804$	3.19843	0.112800
$805$	6.38992	0.225215
$806$	0	0
$807$	−11.6332	−0.409510
$808$	−13.7535	−0.483847
$809$	−10.5482	−0.370855	−0.185427	−	0.982658i	$-0.559367\pi$
$809$	−10.5482	−0.370855	−0.185427	+	0.982658i	$0.559367\pi$
$810$	5.88766	0.206871
$811$	41.6434	1.46230	0.731149	−	0.682218i	$-0.238984\pi$
$811$	41.6434	1.46230	0.731149	+	0.682218i	$0.238984\pi$
$812$	−11.3205	−0.397272
$813$	11.0263	0.386708
$814$	1.59498	0.0559040
$815$	−18.3686	−0.643422
$816$	2.70308	0.0946269
$817$	−11.3290	−0.396351
$818$	14.0131	0.489958
$819$	0	0
$820$	4.22512	0.147548
$821$	−23.2280	−0.810661	−0.405331	−	0.914170i	$-0.632844\pi$
$821$	−23.2280	−0.810661	−0.405331	+	0.914170i	$0.632844\pi$
$822$	10.3622	0.361423
$823$	−0.998854	−0.0348179	−0.0174089	−	0.999848i	$-0.505542\pi$
$823$	−0.998854	−0.0348179	−0.0174089	+	0.999848i	$0.505542\pi$
$824$	−5.73205	−0.199685
$825$	1.66351	0.0579161
$826$	4.98614	0.173490
$827$	29.8030	1.03635	0.518175	−	0.855274i	$-0.326611\pi$
$827$	29.8030	1.03635	0.518175	+	0.855274i	$0.326611\pi$
$828$	−11.7189	−0.407260
$829$	23.7015	0.823188	0.411594	−	0.911367i	$-0.364972\pi$
$829$	23.7015	0.823188	0.411594	+	0.911367i	$0.364972\pi$
$830$	−10.0469	−0.348733
$831$	16.3458	0.567030
$832$	0	0
$833$	22.1950	0.769011
$834$	1.76852	0.0612387
$835$	−21.6085	−0.747794
$836$	12.0073	0.415283
$837$	−14.0842	−0.486822
$838$	24.1954	0.835817
$839$	24.5941	0.849083	0.424541	−	0.905409i	$-0.360435\pi$
$839$	24.5941	0.849083	0.424541	+	0.905409i	$0.360435\pi$
$840$	0.863906	0.0298076
$841$	32.8564	1.13298
$842$	35.6392	1.22821
$843$	18.0548	0.621840
$844$	26.2582	0.903843
$845$	0	0
$846$	33.7055	1.15882
$847$	4.77602	0.164106
$848$	9.57123	0.328677
$849$	15.2923	0.524829
$850$	4.50367	0.154475
$851$	2.55472	0.0875746
$852$	−3.60117	−0.123374
$853$	41.7449	1.42932	0.714659	−	0.699473i	$-0.246582\pi$
$853$	41.7449	1.42932	0.714659	+	0.699473i	$0.246582\pi$
$854$	−17.9975	−0.615860
$855$	−11.4361	−0.391107
$856$	−4.67933	−0.159936
$857$	−18.7543	−0.640636	−0.320318	−	0.947310i	$-0.603790\pi$
$857$	−18.7543	−0.640636	−0.320318	+	0.947310i	$0.603790\pi$
$858$	0	0
$859$	13.8357	0.472066	0.236033	−	0.971745i	$-0.424153\pi$
$859$	13.8357	0.472066	0.236033	+	0.971745i	$0.424153\pi$
$860$	−2.61504	−0.0891720
$861$	−3.65011	−0.124395
$862$	4.96043	0.168953
$863$	−8.00891	−0.272626	−0.136313	−	0.990666i	$-0.543525\pi$
$863$	−8.00891	−0.272626	−0.136313	+	0.990666i	$0.543525\pi$
$864$	−3.38496	−0.115159
$865$	−3.98940	−0.135644
$866$	−4.21621	−0.143273
$867$	−1.97047	−0.0669208
$868$	5.98898	0.203279
$869$	−35.2794	−1.19677
$870$	−4.72047	−0.160039
$871$	0	0
$872$	2.32164	0.0786208
$873$	−4.18755	−0.141727
$874$	19.2325	0.650548
$875$	1.43937	0.0486597
$876$	−0.998434	−0.0337340
$877$	37.5316	1.26735	0.633677	−	0.773598i	$-0.281545\pi$
$877$	37.5316	1.26735	0.633677	+	0.773598i	$0.281545\pi$
$878$	14.7500	0.497788
$879$	−9.94996	−0.335604
$880$	2.77162	0.0934313
$881$	12.1162	0.408204	0.204102	−	0.978950i	$-0.434573\pi$
$881$	12.1162	0.408204	0.204102	+	0.978950i	$0.434573\pi$
$882$	−13.0093	−0.438046
$883$	−44.4026	−1.49427	−0.747133	−	0.664675i	$-0.768570\pi$
$883$	−44.4026	−1.49427	−0.747133	+	0.664675i	$0.768570\pi$
$884$	0	0
$885$	2.07914	0.0698895
$886$	12.2374	0.411124
$887$	−6.51427	−0.218728	−0.109364	−	0.994002i	$-0.534881\pi$
$887$	−6.51427	−0.218728	−0.109364	+	0.994002i	$0.534881\pi$
$888$	0.345393	0.0115906
$889$	23.2090	0.778404
$890$	−5.95717	−0.199685
$891$	−16.3183	−0.546685
$892$	22.2042	0.743452
$893$	−55.3157	−1.85107
$894$	−4.24055	−0.141825
$895$	−12.4799	−0.417158
$896$	1.43937	0.0480861
$897$	0	0
$898$	−19.0714	−0.636420
$899$	−32.7244	−1.09142
$900$	−2.63977	−0.0879922
$901$	−43.1057	−1.43606
$902$	−11.7104	−0.389915
$903$	2.25915	0.0751797
$904$	−7.73629	−0.257305
$905$	19.4319	0.645937
$906$	9.50269	0.315706
$907$	−5.93766	−0.197157	−0.0985784	−	0.995129i	$-0.531430\pi$
$907$	−5.93766	−0.197157	−0.0985784	+	0.995129i	$0.531430\pi$
$908$	2.07914	0.0689986
$909$	−36.3061	−1.20420
$910$	0	0
$911$	48.7860	1.61635	0.808177	−	0.588940i	$-0.200455\pi$
$911$	48.7860	1.61635	0.808177	+	0.588940i	$0.200455\pi$
$912$	2.60020	0.0861011
$913$	27.8462	0.921575
$914$	16.1138	0.532996
$915$	−7.50465	−0.248096
$916$	4.67933	0.154610
$917$	28.4433	0.939279
$918$	15.2448	0.503152
$919$	−29.7128	−0.980135	−0.490068	−	0.871684i	$-0.663028\pi$
$919$	−29.7128	−0.980135	−0.490068	+	0.871684i	$0.663028\pi$
$920$	4.43937	0.146362
$921$	−9.76650	−0.321817
$922$	−4.84466	−0.159550
$923$	0	0
$924$	−2.39442	−0.0787706
$925$	0.575468	0.0189213
$926$	−8.23164	−0.270508
$927$	−15.1313	−0.496976
$928$	−7.86488	−0.258177
$929$	18.0147	0.591043	0.295521	−	0.955336i	$-0.404507\pi$
$929$	18.0147	0.591043	0.295521	+	0.955336i	$0.404507\pi$
$930$	2.49731	0.0818899
$931$	21.3502	0.699724
$932$	−0.611060	−0.0200160
$933$	−9.66632	−0.316461
$934$	−33.4873	−1.09574
$935$	−12.4825	−0.408220
$936$	0	0
$937$	28.9796	0.946723	0.473361	−	0.880868i	$-0.343040\pi$
$937$	28.9796	0.946723	0.473361	+	0.880868i	$0.343040\pi$
$938$	−7.67040	−0.250448
$939$	6.13173	0.200101
$940$	−12.7684	−0.416458
$941$	4.61504	0.150446	0.0752230	−	0.997167i	$-0.476033\pi$
$941$	4.61504	0.150446	0.0752230	+	0.997167i	$0.476033\pi$
$942$	7.57929	0.246947
$943$	−18.7569	−0.610808
$944$	3.46410	0.112747
$945$	4.87223	0.158494
$946$	7.24789	0.235649
$947$	10.6006	0.344474	0.172237	−	0.985056i	$-0.444901\pi$
$947$	10.6006	0.344474	0.172237	+	0.985056i	$0.444901\pi$
$948$	−7.63977	−0.248128
$949$	0	0
$950$	4.33225	0.140557
$951$	−0.122661	−0.00397755
$952$	−6.48247	−0.210098
$953$	28.7411	0.931015	0.465508	−	0.885044i	$-0.345872\pi$
$953$	28.7411	0.931015	0.465508	+	0.885044i	$0.345872\pi$
$954$	25.2658	0.818010
$955$	18.3215	0.592869
$956$	25.6481	0.829518
$957$	13.0833	0.422925
$958$	−8.05343	−0.260195
$959$	−24.8503	−0.802459
$960$	0.600196	0.0193712
$961$	−13.6876	−0.441534
$962$	0	0
$963$	−12.3523	−0.398049
$964$	−27.0931	−0.872610
$965$	−4.74363	−0.152703
$966$	−3.83520	−0.123396
$967$	−8.08648	−0.260044	−0.130022	−	0.991511i	$-0.541505\pi$
$967$	−8.08648	−0.260044	−0.130022	+	0.991511i	$0.541505\pi$
$968$	3.31812	0.106648
$969$	−11.7104	−0.376193
$970$	1.58633	0.0509341
$971$	0.605581	0.0194340	0.00971701	−	0.999953i	$-0.496907\pi$
$971$	0.605581	0.0194340	0.00971701	+	0.999953i	$0.496907\pi$
$972$	−13.6886	−0.439063
$973$	−4.24121	−0.135967
$974$	1.62352	0.0520208
$975$	0	0
$976$	−12.5037	−0.400233
$977$	−49.0432	−1.56903	−0.784516	−	0.620109i	$-0.787089\pi$
$977$	−49.0432	−1.56903	−0.784516	+	0.620109i	$0.787089\pi$
$978$	11.0247	0.352532
$979$	16.5110	0.527694
$980$	4.92820	0.157426
$981$	6.12859	0.195671
$982$	−10.3207	−0.329346
$983$	−30.7684	−0.981358	−0.490679	−	0.871340i	$-0.663251\pi$
$983$	−30.7684	−0.981358	−0.490679	+	0.871340i	$0.663251\pi$
$984$	−2.53590	−0.0808415
$985$	−10.8006	−0.344135
$986$	35.4209	1.12803
$987$	11.0307	0.351110
$988$	0	0
$989$	11.6091	0.369149
$990$	7.31643	0.232531
$991$	15.4887	0.492015	0.246007	−	0.969268i	$-0.420881\pi$
$991$	15.4887	0.492015	0.246007	+	0.969268i	$0.420881\pi$
$992$	4.16082	0.132106
$993$	−11.4429	−0.363129
$994$	8.63624	0.273925
$995$	−11.0073	−0.348956
$996$	6.03011	0.191071
$997$	−7.82221	−0.247732	−0.123866	−	0.992299i	$-0.539529\pi$
$997$	−7.82221	−0.247732	−0.123866	+	0.992299i	$0.539529\pi$
$998$	−13.9807	−0.442550
$999$	1.94794	0.0616300

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1690.2.a.t.1.2		4
5.4	even	2		8450.2.a.cm.1.3		4
13.2	odd	12		1690.2.l.j.1161.2		8
13.3	even	3		1690.2.e.t.191.3		8
13.4	even	6		1690.2.e.s.991.3		8
13.5	odd	4		1690.2.d.k.1351.6		8
13.6	odd	12		130.2.l.b.101.4	✓	8
13.7	odd	12		1690.2.l.j.361.2		8
13.8	odd	4		1690.2.d.k.1351.2		8
13.9	even	3		1690.2.e.t.991.3		8
13.10	even	6		1690.2.e.s.191.3		8
13.11	odd	12		130.2.l.b.121.4	yes	8
13.12	even	2		1690.2.a.u.1.2		4
39.11	even	12		1170.2.bs.g.901.2		8
39.32	even	12		1170.2.bs.g.361.2		8
52.11	even	12		1040.2.da.d.641.2		8
52.19	even	12		1040.2.da.d.881.2		8
65.19	odd	12		650.2.m.c.101.1		8
65.24	odd	12		650.2.m.c.251.1		8
65.32	even	12		650.2.n.e.49.3		8
65.37	even	12		650.2.n.d.199.2		8
65.58	even	12		650.2.n.d.49.2		8
65.63	even	12		650.2.n.e.199.3		8
65.64	even	2		8450.2.a.ci.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
130.2.l.b.101.4	✓	8	13.6	odd	12
130.2.l.b.121.4	yes	8	13.11	odd	12
650.2.m.c.101.1		8	65.19	odd	12
650.2.m.c.251.1		8	65.24	odd	12
650.2.n.d.49.2		8	65.58	even	12
650.2.n.d.199.2		8	65.37	even	12
650.2.n.e.49.3		8	65.32	even	12
650.2.n.e.199.3		8	65.63	even	12
1040.2.da.d.641.2		8	52.11	even	12
1040.2.da.d.881.2		8	52.19	even	12
1170.2.bs.g.361.2		8	39.32	even	12
1170.2.bs.g.901.2		8	39.11	even	12
1690.2.a.t.1.2		4	1.1	even	1	trivial
1690.2.a.u.1.2		4	13.12	even	2
1690.2.d.k.1351.2		8	13.8	odd	4
1690.2.d.k.1351.6		8	13.5	odd	4
1690.2.e.s.191.3		8	13.10	even	6
1690.2.e.s.991.3		8	13.4	even	6
1690.2.e.t.191.3		8	13.3	even	3
1690.2.e.t.991.3		8	13.9	even	3
1690.2.l.j.361.2		8	13.7	odd	12
1690.2.l.j.1161.2		8	13.2	odd	12
8450.2.a.ci.1.3		4	65.64	even	2
8450.2.a.cm.1.3		4	5.4	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1690 = 2 \cdot 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1690.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -0.600196 q^{3} +1.00000 q^{4} -1.00000 q^{5} +0.600196 q^{6} -1.43937 q^{7} -1.00000 q^{8} -2.63977 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -0.600196 q^{3} +1.00000 q^{4} -1.00000 q^{5} +0.600196 q^{6} -1.43937 q^{7} -1.00000 q^{8} -2.63977 q^{9} +1.00000 q^{10} -2.77162 q^{11} -0.600196 q^{12} +1.43937 q^{14} +0.600196 q^{15} +1.00000 q^{16} -4.50367 q^{17} +2.63977 q^{18} -4.33225 q^{19} -1.00000 q^{20} +0.863906 q^{21} +2.77162 q^{22} +4.43937 q^{23} +0.600196 q^{24} +1.00000 q^{25} +3.38496 q^{27} -1.43937 q^{28} +7.86488 q^{29} -0.600196 q^{30} -4.16082 q^{31} -1.00000 q^{32} +1.66351 q^{33} +4.50367 q^{34} +1.43937 q^{35} -2.63977 q^{36} +0.575468 q^{37} +4.33225 q^{38} +1.00000 q^{40} -4.22512 q^{41} -0.863906 q^{42} +2.61504 q^{43} -2.77162 q^{44} +2.63977 q^{45} -4.43937 q^{46} +12.7684 q^{47} -0.600196 q^{48} -4.92820 q^{49} -1.00000 q^{50} +2.70308 q^{51} +9.57123 q^{53} -3.38496 q^{54} +2.77162 q^{55} +1.43937 q^{56} +2.60020 q^{57} -7.86488 q^{58} +3.46410 q^{59} +0.600196 q^{60} -12.5037 q^{61} +4.16082 q^{62} +3.79961 q^{63} +1.00000 q^{64} -1.66351 q^{66} -5.32899 q^{67} -4.50367 q^{68} -2.66449 q^{69} -1.43937 q^{70} +6.00000 q^{71} +2.63977 q^{72} +1.66351 q^{73} -0.575468 q^{74} -0.600196 q^{75} -4.33225 q^{76} +3.98940 q^{77} +12.7288 q^{79} -1.00000 q^{80} +5.88766 q^{81} +4.22512 q^{82} -10.0469 q^{83} +0.863906 q^{84} +4.50367 q^{85} -2.61504 q^{86} -4.72047 q^{87} +2.77162 q^{88} -5.95717 q^{89} -2.63977 q^{90} +4.43937 q^{92} +2.49731 q^{93} -12.7684 q^{94} +4.33225 q^{95} +0.600196 q^{96} +1.58633 q^{97} +4.92820 q^{98} +7.31643 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{2} + 2 q^{3} + 4 q^{4} - 4 q^{5} - 2 q^{6} - 4 q^{8} + 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^2 + 2 * q^3 + 4 * q^4 - 4 * q^5 - 2 * q^6 - 4 * q^8 + 4 * q^9	\( 4 q - 4 q^{2} + 2 q^{3} + 4 q^{4} - 4 q^{5} - 2 q^{6} - 4 q^{8} + 4 q^{9} + 4 q^{10} + 6 q^{11} + 2 q^{12} - 2 q^{15} + 4 q^{16} + 6 q^{17} - 4 q^{18} - 6 q^{19} - 4 q^{20} - 6 q^{21} - 6 q^{22} + 12 q^{23} - 2 q^{24} + 4 q^{25} + 20 q^{27} + 2 q^{30} - 18 q^{31} - 4 q^{32} + 6 q^{33} - 6 q^{34} + 4 q^{36} + 6 q^{37} + 6 q^{38} + 4 q^{40} + 6 q^{42} + 4 q^{43} + 6 q^{44} - 4 q^{45} - 12 q^{46} + 2 q^{48} + 8 q^{49} - 4 q^{50} + 30 q^{53} - 20 q^{54} - 6 q^{55} + 6 q^{57} - 2 q^{60} - 26 q^{61} + 18 q^{62} + 24 q^{63} + 4 q^{64} - 6 q^{66} + 24 q^{67} + 6 q^{68} + 12 q^{69} + 24 q^{71} - 4 q^{72} + 6 q^{73} - 6 q^{74} + 2 q^{75} - 6 q^{76} + 30 q^{77} + 10 q^{79} - 4 q^{80} + 28 q^{81} + 18 q^{83} - 6 q^{84} - 6 q^{85} - 4 q^{86} - 48 q^{87} - 6 q^{88} + 4 q^{90} + 12 q^{92} + 12 q^{93} + 6 q^{95} - 2 q^{96} - 18 q^{97} - 8 q^{98} + 42 q^{99}+O(q^{100}) \) 4 * q - 4 * q^2 + 2 * q^3 + 4 * q^4 - 4 * q^5 - 2 * q^6 - 4 * q^8 + 4 * q^9 + 4 * q^10 + 6 * q^11 + 2 * q^12 - 2 * q^15 + 4 * q^16 + 6 * q^17 - 4 * q^18 - 6 * q^19 - 4 * q^20 - 6 * q^21 - 6 * q^22 + 12 * q^23 - 2 * q^24 + 4 * q^25 + 20 * q^27 + 2 * q^30 - 18 * q^31 - 4 * q^32 + 6 * q^33 - 6 * q^34 + 4 * q^36 + 6 * q^37 + 6 * q^38 + 4 * q^40 + 6 * q^42 + 4 * q^43 + 6 * q^44 - 4 * q^45 - 12 * q^46 + 2 * q^48 + 8 * q^49 - 4 * q^50 + 30 * q^53 - 20 * q^54 - 6 * q^55 + 6 * q^57 - 2 * q^60 - 26 * q^61 + 18 * q^62 + 24 * q^63 + 4 * q^64 - 6 * q^66 + 24 * q^67 + 6 * q^68 + 12 * q^69 + 24 * q^71 - 4 * q^72 + 6 * q^73 - 6 * q^74 + 2 * q^75 - 6 * q^76 + 30 * q^77 + 10 * q^79 - 4 * q^80 + 28 * q^81 + 18 * q^83 - 6 * q^84 - 6 * q^85 - 4 * q^86 - 48 * q^87 - 6 * q^88 + 4 * q^90 + 12 * q^92 + 12 * q^93 + 6 * q^95 - 2 * q^96 - 18 * q^97 - 8 * q^98 + 42 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more