LMFDB - Embedded newform 6422.2.a.bn.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6422,2,Mod(1,6422)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6422, 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("6422.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6422 = 2 \cdot 13^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6422.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:	$51.2799281781$
Analytic rank:	$0$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - 4 x^{13} - 22 x^{12} + 98 x^{11} + 164 x^{10} - 912 x^{9} - 374 x^{8} + 3996 x^{7} - 817 x^{6} + \cdots + 358 $ x^14 - 4x^13 - 22x^12 + 98x^11 + 164x^10 - 912x^9 - 374x^8 + 3996x^7 - 817x^6 - 8194x^5 + 4418x^4 + 6430x^3 - 4327x^2 - 922*x + 358
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 494)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$1.30931$ of defining polynomial
Character	$\chi$	$=$	6422.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.30931 q^{3} +1.00000 q^{4} -4.16024 q^{5} +1.30931 q^{6} -4.89391 q^{7} +1.00000 q^{8} -1.28570 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +1.30931 q^{3} +1.00000 q^{4} -4.16024 q^{5} +1.30931 q^{6} -4.89391 q^{7} +1.00000 q^{8} -1.28570 q^{9} -4.16024 q^{10} -5.16660 q^{11} +1.30931 q^{12} -4.89391 q^{14} -5.44704 q^{15} +1.00000 q^{16} -0.826597 q^{17} -1.28570 q^{18} +1.00000 q^{19} -4.16024 q^{20} -6.40765 q^{21} -5.16660 q^{22} +0.110001 q^{23} +1.30931 q^{24} +12.3076 q^{25} -5.61132 q^{27} -4.89391 q^{28} -7.43442 q^{29} -5.44704 q^{30} -3.63471 q^{31} +1.00000 q^{32} -6.76469 q^{33} -0.826597 q^{34} +20.3598 q^{35} -1.28570 q^{36} +4.61991 q^{37} +1.00000 q^{38} -4.16024 q^{40} -1.49469 q^{41} -6.40765 q^{42} -3.12919 q^{43} -5.16660 q^{44} +5.34884 q^{45} +0.110001 q^{46} +3.15036 q^{47} +1.30931 q^{48} +16.9503 q^{49} +12.3076 q^{50} -1.08227 q^{51} +2.28874 q^{53} -5.61132 q^{54} +21.4943 q^{55} -4.89391 q^{56} +1.30931 q^{57} -7.43442 q^{58} +8.01071 q^{59} -5.44704 q^{60} -7.50720 q^{61} -3.63471 q^{62} +6.29212 q^{63} +1.00000 q^{64} -6.76469 q^{66} +4.05575 q^{67} -0.826597 q^{68} +0.144025 q^{69} +20.3598 q^{70} -6.62868 q^{71} -1.28570 q^{72} +4.15960 q^{73} +4.61991 q^{74} +16.1144 q^{75} +1.00000 q^{76} +25.2849 q^{77} +10.4525 q^{79} -4.16024 q^{80} -3.48985 q^{81} -1.49469 q^{82} -18.0017 q^{83} -6.40765 q^{84} +3.43884 q^{85} -3.12919 q^{86} -9.73397 q^{87} -5.16660 q^{88} +15.3778 q^{89} +5.34884 q^{90} +0.110001 q^{92} -4.75897 q^{93} +3.15036 q^{94} -4.16024 q^{95} +1.30931 q^{96} +1.69504 q^{97} +16.9503 q^{98} +6.64272 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q + 14 q^{2} + 4 q^{3} + 14 q^{4} - 2 q^{5} + 4 q^{6} + 2 q^{7} + 14 q^{8} + 18 q^{9}+O(q^{10}) $ 14 * q + 14 * q^2 + 4 * q^3 + 14 * q^4 - 2 * q^5 + 4 * q^6 + 2 * q^7 + 14 * q^8 + 18 * q^9	\( 14 q + 14 q^{2} + 4 q^{3} + 14 q^{4} - 2 q^{5} + 4 q^{6} + 2 q^{7} + 14 q^{8} + 18 q^{9} - 2 q^{10} + 10 q^{11} + 4 q^{12} + 2 q^{14} - 4 q^{15} + 14 q^{16} + 2 q^{17} + 18 q^{18} + 14 q^{19} - 2 q^{20} - 18 q^{21} + 10 q^{22} + 12 q^{23} + 4 q^{24} + 44 q^{25} + 10 q^{27} + 2 q^{28} + 4 q^{29} - 4 q^{30} + 4 q^{31} + 14 q^{32} + 12 q^{33} + 2 q^{34} + 14 q^{35} + 18 q^{36} + 18 q^{37} + 14 q^{38} - 2 q^{40} + 6 q^{41} - 18 q^{42} + 28 q^{43} + 10 q^{44} + 8 q^{45} + 12 q^{46} - 20 q^{47} + 4 q^{48} + 28 q^{49} + 44 q^{50} + 10 q^{51} + 12 q^{53} + 10 q^{54} - 2 q^{55} + 2 q^{56} + 4 q^{57} + 4 q^{58} + 16 q^{59} - 4 q^{60} + 30 q^{61} + 4 q^{62} + 28 q^{63} + 14 q^{64} + 12 q^{66} + 2 q^{67} + 2 q^{68} - 42 q^{69} + 14 q^{70} + 44 q^{71} + 18 q^{72} - 36 q^{73} + 18 q^{74} + 46 q^{75} + 14 q^{76} + 68 q^{77} + 34 q^{79} - 2 q^{80} - 6 q^{81} + 6 q^{82} + 2 q^{83} - 18 q^{84} - 30 q^{85} + 28 q^{86} + 52 q^{87} + 10 q^{88} + 42 q^{89} + 8 q^{90} + 12 q^{92} - 12 q^{93} - 20 q^{94} - 2 q^{95} + 4 q^{96} + 40 q^{97} + 28 q^{98} + 12 q^{99}+O(q^{100}) \) 14 * q + 14 * q^2 + 4 * q^3 + 14 * q^4 - 2 * q^5 + 4 * q^6 + 2 * q^7 + 14 * q^8 + 18 * q^9 - 2 * q^10 + 10 * q^11 + 4 * q^12 + 2 * q^14 - 4 * q^15 + 14 * q^16 + 2 * q^17 + 18 * q^18 + 14 * q^19 - 2 * q^20 - 18 * q^21 + 10 * q^22 + 12 * q^23 + 4 * q^24 + 44 * q^25 + 10 * q^27 + 2 * q^28 + 4 * q^29 - 4 * q^30 + 4 * q^31 + 14 * q^32 + 12 * q^33 + 2 * q^34 + 14 * q^35 + 18 * q^36 + 18 * q^37 + 14 * q^38 - 2 * q^40 + 6 * q^41 - 18 * q^42 + 28 * q^43 + 10 * q^44 + 8 * q^45 + 12 * q^46 - 20 * q^47 + 4 * q^48 + 28 * q^49 + 44 * q^50 + 10 * q^51 + 12 * q^53 + 10 * q^54 - 2 * q^55 + 2 * q^56 + 4 * q^57 + 4 * q^58 + 16 * q^59 - 4 * q^60 + 30 * q^61 + 4 * q^62 + 28 * q^63 + 14 * q^64 + 12 * q^66 + 2 * q^67 + 2 * q^68 - 42 * q^69 + 14 * q^70 + 44 * q^71 + 18 * q^72 - 36 * q^73 + 18 * q^74 + 46 * q^75 + 14 * q^76 + 68 * q^77 + 34 * q^79 - 2 * q^80 - 6 * q^81 + 6 * q^82 + 2 * q^83 - 18 * q^84 - 30 * q^85 + 28 * q^86 + 52 * q^87 + 10 * q^88 + 42 * q^89 + 8 * q^90 + 12 * q^92 - 12 * q^93 - 20 * q^94 - 2 * q^95 + 4 * q^96 + 40 * q^97 + 28 * q^98 + 12 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	1.30931	0.755931	0.377965	−	0.925820i	$-0.376624\pi$
$3$	1.30931	0.755931	0.377965	+	0.925820i	$0.376624\pi$
$4$	1.00000	0.500000
$5$	−4.16024	−1.86051	−0.930257	−	0.366909i	$-0.880416\pi$
$5$	−4.16024	−1.86051	−0.930257	+	0.366909i	$0.880416\pi$
$6$	1.30931	0.534524
$7$	−4.89391	−1.84972	−0.924862	−	0.380304i	$-0.875819\pi$
$7$	−4.89391	−1.84972	−0.924862	+	0.380304i	$0.875819\pi$
$8$	1.00000	0.353553
$9$	−1.28570	−0.428568
$10$	−4.16024	−1.31558
$11$	−5.16660	−1.55779	−0.778894	−	0.627155i	$-0.784219\pi$
$11$	−5.16660	−1.55779	−0.778894	+	0.627155i	$0.784219\pi$
$12$	1.30931	0.377965
$13$	0	0
$13$	0	0
$14$	−4.89391	−1.30795
$15$	−5.44704	−1.40642
$16$	1.00000	0.250000
$17$	−0.826597	−0.200479	−0.100240	−	0.994963i	$-0.531961\pi$
$17$	−0.826597	−0.200479	−0.100240	+	0.994963i	$0.531961\pi$
$18$	−1.28570	−0.303044
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	−4.16024	−0.930257
$21$	−6.40765	−1.39826
$22$	−5.16660	−1.10152
$23$	0.110001	0.0229368	0.0114684	−	0.999934i	$-0.496349\pi$
$23$	0.110001	0.0229368	0.0114684	+	0.999934i	$0.496349\pi$
$24$	1.30931	0.267262
$25$	12.3076	2.46151
$26$	0	0
$27$	−5.61132	−1.07990
$28$	−4.89391	−0.924862
$29$	−7.43442	−1.38054	−0.690268	−	0.723553i	$-0.742508\pi$
$29$	−7.43442	−1.38054	−0.690268	+	0.723553i	$0.742508\pi$
$30$	−5.44704	−0.994489
$31$	−3.63471	−0.652813	−0.326407	−	0.945229i	$-0.605838\pi$
$31$	−3.63471	−0.652813	−0.326407	+	0.945229i	$0.605838\pi$
$32$	1.00000	0.176777
$33$	−6.76469	−1.17758
$34$	−0.826597	−0.141760
$35$	20.3598	3.44144
$36$	−1.28570	−0.214284
$37$	4.61991	0.759509	0.379755	−	0.925087i	$-0.376008\pi$
$37$	4.61991	0.759509	0.379755	+	0.925087i	$0.376008\pi$
$38$	1.00000	0.162221
$39$	0	0
$40$	−4.16024	−0.657791
$41$	−1.49469	−0.233432	−0.116716	−	0.993165i	$-0.537237\pi$
$41$	−1.49469	−0.233432	−0.116716	+	0.993165i	$0.537237\pi$
$42$	−6.40765	−0.988721
$43$	−3.12919	−0.477197	−0.238598	−	0.971118i	$-0.576688\pi$
$43$	−3.12919	−0.477197	−0.238598	+	0.971118i	$0.576688\pi$
$44$	−5.16660	−0.778894
$45$	5.34884	0.797357
$46$	0.110001	0.0162187
$47$	3.15036	0.459527	0.229763	−	0.973247i	$-0.426205\pi$
$47$	3.15036	0.459527	0.229763	+	0.973247i	$0.426205\pi$
$48$	1.30931	0.188983
$49$	16.9503	2.42148
$50$	12.3076	1.74055
$51$	−1.08227	−0.151548
$52$	0	0
$53$	2.28874	0.314383	0.157192	−	0.987568i	$-0.449756\pi$
$53$	2.28874	0.314383	0.157192	+	0.987568i	$0.449756\pi$
$54$	−5.61132	−0.763604
$55$	21.4943	2.89829
$56$	−4.89391	−0.653976
$57$	1.30931	0.173422
$58$	−7.43442	−0.976187
$59$	8.01071	1.04291	0.521453	−	0.853280i	$-0.325390\pi$
$59$	8.01071	1.04291	0.521453	+	0.853280i	$0.325390\pi$
$60$	−5.44704	−0.703210
$61$	−7.50720	−0.961199	−0.480599	−	0.876940i	$-0.659581\pi$
$61$	−7.50720	−0.961199	−0.480599	+	0.876940i	$0.659581\pi$
$62$	−3.63471	−0.461609
$63$	6.29212	0.792733
$64$	1.00000	0.125000
$65$	0	0
$66$	−6.76469	−0.832675
$67$	4.05575	0.495489	0.247744	−	0.968825i	$-0.420311\pi$
$67$	4.05575	0.495489	0.247744	+	0.968825i	$0.420311\pi$
$68$	−0.826597	−0.100240
$69$	0.144025	0.0173386
$70$	20.3598	2.43346
$71$	−6.62868	−0.786680	−0.393340	−	0.919393i	$-0.628680\pi$
$71$	−6.62868	−0.786680	−0.393340	+	0.919393i	$0.628680\pi$
$72$	−1.28570	−0.151522
$73$	4.15960	0.486844	0.243422	−	0.969920i	$-0.421730\pi$
$73$	4.15960	0.486844	0.243422	+	0.969920i	$0.421730\pi$
$74$	4.61991	0.537054
$75$	16.1144	1.86073
$76$	1.00000	0.114708
$77$	25.2849	2.88148
$78$	0	0
$79$	10.4525	1.17600	0.588000	−	0.808861i	$-0.299915\pi$
$79$	10.4525	1.17600	0.588000	+	0.808861i	$0.299915\pi$
$80$	−4.16024	−0.465128
$81$	−3.48985	−0.387761
$82$	−1.49469	−0.165061
$83$	−18.0017	−1.97594	−0.987972	−	0.154636i	$-0.950580\pi$
$83$	−18.0017	−1.97594	−0.987972	+	0.154636i	$0.950580\pi$
$84$	−6.40765	−0.699131
$85$	3.43884	0.372994
$86$	−3.12919	−0.337429
$87$	−9.73397	−1.04359
$88$	−5.16660	−0.550762
$89$	15.3778	1.63004	0.815019	−	0.579434i	$-0.196726\pi$
$89$	15.3778	1.63004	0.815019	+	0.579434i	$0.196726\pi$
$90$	5.34884	0.563817
$91$	0	0
$92$	0.110001	0.0114684
$93$	−4.75897	−0.493482
$94$	3.15036	0.324934
$95$	−4.16024	−0.426831
$96$	1.30931	0.133631
$97$	1.69504	0.172106	0.0860528	−	0.996291i	$-0.472575\pi$
$97$	1.69504	0.172106	0.0860528	+	0.996291i	$0.472575\pi$
$98$	16.9503	1.71224
$99$	6.64272	0.667619
$100$	12.3076	1.23076
$101$	−7.06917	−0.703409	−0.351704	−	0.936111i	$-0.614398\pi$
$101$	−7.06917	−0.703409	−0.351704	+	0.936111i	$0.614398\pi$
$102$	−1.08227	−0.107161
$103$	2.09095	0.206027	0.103014	−	0.994680i	$-0.467152\pi$
$103$	2.09095	0.206027	0.103014	+	0.994680i	$0.467152\pi$
$104$	0	0
$105$	26.6573	2.60149
$106$	2.28874	0.222302
$107$	11.2185	1.08453	0.542266	−	0.840207i	$-0.317566\pi$
$107$	11.2185	1.08453	0.542266	+	0.840207i	$0.317566\pi$
$108$	−5.61132	−0.539950
$109$	−13.7272	−1.31483	−0.657413	−	0.753530i	$-0.728349\pi$
$109$	−13.7272	−1.31483	−0.657413	+	0.753530i	$0.728349\pi$
$110$	21.4943	2.04940
$111$	6.04891	0.574137
$112$	−4.89391	−0.462431
$113$	−12.8772	−1.21138	−0.605691	−	0.795700i	$-0.707103\pi$
$113$	−12.8772	−1.21138	−0.605691	+	0.795700i	$0.707103\pi$
$114$	1.30931	0.122628
$115$	−0.457630	−0.0426742
$116$	−7.43442	−0.690268
$117$	0	0
$118$	8.01071	0.737446
$119$	4.04529	0.370831
$120$	−5.44704	−0.497245
$121$	15.6938	1.42671
$122$	−7.50720	−0.679670
$123$	−1.95702	−0.176458
$124$	−3.63471	−0.326407
$125$	−30.4012	−2.71916
$126$	6.29212	0.560547
$127$	11.8558	1.05203	0.526016	−	0.850475i	$-0.323685\pi$
$127$	11.8558	1.05203	0.526016	+	0.850475i	$0.323685\pi$
$128$	1.00000	0.0883883
$129$	−4.09708	−0.360728
$130$	0	0
$131$	2.10549	0.183958	0.0919788	−	0.995761i	$-0.470681\pi$
$131$	2.10549	0.183958	0.0919788	+	0.995761i	$0.470681\pi$
$132$	−6.76469	−0.588790
$133$	−4.89391	−0.424356
$134$	4.05575	0.350364
$135$	23.3444	2.00917
$136$	−0.826597	−0.0708801
$137$	−15.8660	−1.35552	−0.677761	−	0.735282i	$-0.737050\pi$
$137$	−15.8660	−1.35552	−0.677761	+	0.735282i	$0.737050\pi$
$138$	0.144025	0.0122603
$139$	−16.8909	−1.43267	−0.716334	−	0.697758i	$-0.754181\pi$
$139$	−16.8909	−1.43267	−0.716334	+	0.697758i	$0.754181\pi$
$140$	20.3598	1.72072
$141$	4.12479	0.347370
$142$	−6.62868	−0.556266
$143$	0	0
$144$	−1.28570	−0.107142
$145$	30.9289	2.56851
$146$	4.15960	0.344251
$147$	22.1932	1.83047
$148$	4.61991	0.379755
$149$	10.8934	0.892426	0.446213	−	0.894927i	$-0.352772\pi$
$149$	10.8934	0.892426	0.446213	+	0.894927i	$0.352772\pi$
$150$	16.1144	1.31574
$151$	−11.3121	−0.920567	−0.460284	−	0.887772i	$-0.652252\pi$
$151$	−11.3121	−0.920567	−0.460284	+	0.887772i	$0.652252\pi$
$152$	1.00000	0.0811107
$153$	1.06276	0.0859190
$154$	25.2849	2.03751
$155$	15.1212	1.21457
$156$	0	0
$157$	−1.93140	−0.154143	−0.0770714	−	0.997026i	$-0.524557\pi$
$157$	−1.93140	−0.154143	−0.0770714	+	0.997026i	$0.524557\pi$
$158$	10.4525	0.831558
$159$	2.99668	0.237652
$160$	−4.16024	−0.328895
$161$	−0.538334	−0.0424267
$162$	−3.48985	−0.274188
$163$	−0.630891	−0.0494152	−0.0247076	−	0.999695i	$-0.507865\pi$
$163$	−0.630891	−0.0494152	−0.0247076	+	0.999695i	$0.507865\pi$
$164$	−1.49469	−0.116716
$165$	28.1427	2.19091
$166$	−18.0017	−1.39720
$167$	−0.526696	−0.0407570	−0.0203785	−	0.999792i	$-0.506487\pi$
$167$	−0.526696	−0.0407570	−0.0203785	+	0.999792i	$0.506487\pi$
$168$	−6.40765	−0.494361
$169$	0	0
$170$	3.43884	0.263747
$171$	−1.28570	−0.0983203
$172$	−3.12919	−0.238598
$173$	−12.6199	−0.959472	−0.479736	−	0.877413i	$-0.659268\pi$
$173$	−12.6199	−0.959472	−0.479736	+	0.877413i	$0.659268\pi$
$174$	−9.73397	−0.737930
$175$	−60.2320	−4.55311
$176$	−5.16660	−0.389447
$177$	10.4885	0.788365
$178$	15.3778	1.15261
$179$	11.0041	0.822488	0.411244	−	0.911525i	$-0.365094\pi$
$179$	11.0041	0.822488	0.411244	+	0.911525i	$0.365094\pi$
$180$	5.34884	0.398679
$181$	−8.54868	−0.635419	−0.317709	−	0.948188i	$-0.602914\pi$
$181$	−8.54868	−0.635419	−0.317709	+	0.948188i	$0.602914\pi$
$182$	0	0
$183$	−9.82926	−0.726600
$184$	0.110001	0.00810937
$185$	−19.2199	−1.41308
$186$	−4.75897	−0.348944
$187$	4.27069	0.312304
$188$	3.15036	0.229763
$189$	27.4613	1.99751
$190$	−4.16024	−0.301815
$191$	−2.11168	−0.152796	−0.0763978	−	0.997077i	$-0.524342\pi$
$191$	−2.11168	−0.152796	−0.0763978	+	0.997077i	$0.524342\pi$
$192$	1.30931	0.0944914
$193$	16.2511	1.16978	0.584889	−	0.811113i	$-0.301138\pi$
$193$	16.2511	1.16978	0.584889	+	0.811113i	$0.301138\pi$
$194$	1.69504	0.121697
$195$	0	0
$196$	16.9503	1.21074
$197$	−12.9063	−0.919534	−0.459767	−	0.888040i	$-0.652067\pi$
$197$	−12.9063	−0.919534	−0.459767	+	0.888040i	$0.652067\pi$
$198$	6.64272	0.472078
$199$	13.2604	0.940006	0.470003	−	0.882665i	$-0.344253\pi$
$199$	13.2604	0.940006	0.470003	+	0.882665i	$0.344253\pi$
$200$	12.3076	0.870276
$201$	5.31024	0.374555
$202$	−7.06917	−0.497385
$203$	36.3834	2.55361
$204$	−1.08227	−0.0757742
$205$	6.21828	0.434303
$206$	2.09095	0.145683
$207$	−0.141429	−0.00982998
$208$	0	0
$209$	−5.16660	−0.357381
$210$	26.6573	1.83953
$211$	0.616577	0.0424469	0.0212234	−	0.999775i	$-0.493244\pi$
$211$	0.616577	0.0424469	0.0212234	+	0.999775i	$0.493244\pi$
$212$	2.28874	0.157192
$213$	−8.67900	−0.594675
$214$	11.2185	0.766881
$215$	13.0182	0.887831
$216$	−5.61132	−0.381802
$217$	17.7879	1.20752
$218$	−13.7272	−0.929723
$219$	5.44621	0.368021
$220$	21.4943	1.44914
$221$	0	0
$222$	6.04891	0.405976
$223$	15.4713	1.03603	0.518016	−	0.855371i	$-0.326671\pi$
$223$	15.4713	1.03603	0.518016	+	0.855371i	$0.326671\pi$
$224$	−4.89391	−0.326988
$225$	−15.8239	−1.05493
$226$	−12.8772	−0.856577
$227$	−5.01952	−0.333157	−0.166578	−	0.986028i	$-0.553272\pi$
$227$	−5.01952	−0.333157	−0.166578	+	0.986028i	$0.553272\pi$
$228$	1.30931	0.0867112
$229$	−14.5207	−0.959557	−0.479779	−	0.877390i	$-0.659283\pi$
$229$	−14.5207	−0.959557	−0.479779	+	0.877390i	$0.659283\pi$
$230$	−0.457630	−0.0301752
$231$	33.1057	2.17820
$232$	−7.43442	−0.488094
$233$	−5.28417	−0.346177	−0.173089	−	0.984906i	$-0.555375\pi$
$233$	−5.28417	−0.346177	−0.173089	+	0.984906i	$0.555375\pi$
$234$	0	0
$235$	−13.1062	−0.854955
$236$	8.01071	0.521453
$237$	13.6856	0.888975
$238$	4.04529	0.262217
$239$	30.3753	1.96481	0.982407	−	0.186751i	$-0.0597957\pi$
$239$	30.3753	1.96481	0.982407	+	0.186751i	$0.0597957\pi$
$240$	−5.44704	−0.351605
$241$	21.5093	1.38554	0.692769	−	0.721160i	$-0.256391\pi$
$241$	21.5093	1.38554	0.692769	+	0.721160i	$0.256391\pi$
$242$	15.6938	1.00883
$243$	12.2647	0.786779
$244$	−7.50720	−0.480599
$245$	−70.5173	−4.50519
$246$	−1.95702	−0.124775
$247$	0	0
$248$	−3.63471	−0.230804
$249$	−23.5698	−1.49368
$250$	−30.4012	−1.92274
$251$	15.6493	0.987774	0.493887	−	0.869526i	$-0.335576\pi$
$251$	15.6493	0.987774	0.493887	+	0.869526i	$0.335576\pi$
$252$	6.29212	0.396366
$253$	−0.568331	−0.0357307
$254$	11.8558	0.743899
$255$	4.50251	0.281958
$256$	1.00000	0.0625000
$257$	−12.9068	−0.805103	−0.402551	−	0.915397i	$-0.631877\pi$
$257$	−12.9068	−0.805103	−0.402551	+	0.915397i	$0.631877\pi$
$258$	−4.09708	−0.255073
$259$	−22.6094	−1.40488
$260$	0	0
$261$	9.55847	0.591654
$262$	2.10549	0.130078
$263$	−19.5814	−1.20744	−0.603721	−	0.797196i	$-0.706316\pi$
$263$	−19.5814	−1.20744	−0.603721	+	0.797196i	$0.706316\pi$
$264$	−6.76469	−0.416338
$265$	−9.52171	−0.584914
$266$	−4.89391	−0.300065
$267$	20.1343	1.23220
$268$	4.05575	0.247744
$269$	−18.8216	−1.14757	−0.573786	−	0.819005i	$-0.694526\pi$
$269$	−18.8216	−1.14757	−0.573786	+	0.819005i	$0.694526\pi$
$270$	23.3444	1.42070
$271$	1.15476	0.0701464	0.0350732	−	0.999385i	$-0.488834\pi$
$271$	1.15476	0.0701464	0.0350732	+	0.999385i	$0.488834\pi$
$272$	−0.826597	−0.0501198
$273$	0	0
$274$	−15.8660	−0.958499
$275$	−63.5882	−3.83451
$276$	0.144025	0.00866931
$277$	15.1452	0.909985	0.454993	−	0.890495i	$-0.349642\pi$
$277$	15.1452	0.909985	0.454993	+	0.890495i	$0.349642\pi$
$278$	−16.8909	−1.01305
$279$	4.67316	0.279775
$280$	20.3598	1.21673
$281$	−9.05085	−0.539929	−0.269964	−	0.962870i	$-0.587012\pi$
$281$	−9.05085	−0.539929	−0.269964	+	0.962870i	$0.587012\pi$
$282$	4.12479	0.245628
$283$	−0.322507	−0.0191711	−0.00958553	−	0.999954i	$-0.503051\pi$
$283$	−0.322507	−0.0191711	−0.00958553	+	0.999954i	$0.503051\pi$
$284$	−6.62868	−0.393340
$285$	−5.44704	−0.322655
$286$	0	0
$287$	7.31489	0.431785
$288$	−1.28570	−0.0757609
$289$	−16.3167	−0.959808
$290$	30.9289	1.81621
$291$	2.21934	0.130100
$292$	4.15960	0.243422
$293$	0.577328	0.0337279	0.0168639	−	0.999858i	$-0.494632\pi$
$293$	0.577328	0.0337279	0.0168639	+	0.999858i	$0.494632\pi$
$294$	22.1932	1.29434
$295$	−33.3265	−1.94034
$296$	4.61991	0.268527
$297$	28.9915	1.68225
$298$	10.8934	0.631041
$299$	0	0
$300$	16.1144	0.930366
$301$	15.3140	0.882682
$302$	−11.3121	−0.650939
$303$	−9.25574	−0.531729
$304$	1.00000	0.0573539
$305$	31.2317	1.78832
$306$	1.06276	0.0607539
$307$	17.0687	0.974163	0.487082	−	0.873356i	$-0.338061\pi$
$307$	17.0687	0.974163	0.487082	+	0.873356i	$0.338061\pi$
$308$	25.2849	1.44074
$309$	2.73770	0.155742
$310$	15.1212	0.858829
$311$	0.0907973	0.00514864	0.00257432	−	0.999997i	$-0.499181\pi$
$311$	0.0907973	0.00514864	0.00257432	+	0.999997i	$0.499181\pi$
$312$	0	0
$313$	−22.9924	−1.29960	−0.649802	−	0.760103i	$-0.725148\pi$
$313$	−22.9924	−1.29960	−0.649802	+	0.760103i	$0.725148\pi$
$314$	−1.93140	−0.108995
$315$	−26.1767	−1.47489
$316$	10.4525	0.588000
$317$	−11.7090	−0.657641	−0.328820	−	0.944393i	$-0.606651\pi$
$317$	−11.7090	−0.657641	−0.328820	+	0.944393i	$0.606651\pi$
$318$	2.99668	0.168045
$319$	38.4107	2.15058
$320$	−4.16024	−0.232564
$321$	14.6885	0.819832
$322$	−0.538334	−0.0300002
$323$	−0.826597	−0.0459931
$324$	−3.48985	−0.193880
$325$	0	0
$326$	−0.630891	−0.0349418
$327$	−17.9732	−0.993918
$328$	−1.49469	−0.0825307
$329$	−15.4175	−0.849997
$330$	28.1427	1.54920
$331$	9.30885	0.511661	0.255830	−	0.966722i	$-0.417651\pi$
$331$	9.30885	0.511661	0.255830	+	0.966722i	$0.417651\pi$
$332$	−18.0017	−0.987972
$333$	−5.93985	−0.325502
$334$	−0.526696	−0.0288195
$335$	−16.8729	−0.921864
$336$	−6.40765	−0.349566
$337$	10.7917	0.587864	0.293932	−	0.955826i	$-0.405036\pi$
$337$	10.7917	0.587864	0.293932	+	0.955826i	$0.405036\pi$
$338$	0	0
$339$	−16.8602	−0.915722
$340$	3.43884	0.186497
$341$	18.7791	1.01695
$342$	−1.28570	−0.0695230
$343$	−48.6960	−2.62934
$344$	−3.12919	−0.168714
$345$	−0.599180	−0.0322587
$346$	−12.6199	−0.678449
$347$	10.3261	0.554336	0.277168	−	0.960821i	$-0.410604\pi$
$347$	10.3261	0.554336	0.277168	+	0.960821i	$0.410604\pi$
$348$	−9.73397	−0.521795
$349$	23.7338	1.27044	0.635220	−	0.772331i	$-0.280909\pi$
$349$	23.7338	1.27044	0.635220	+	0.772331i	$0.280909\pi$
$350$	−60.2320	−3.21954
$351$	0	0
$352$	−5.16660	−0.275381
$353$	27.3851	1.45756	0.728780	−	0.684748i	$-0.240088\pi$
$353$	27.3851	1.45756	0.728780	+	0.684748i	$0.240088\pi$
$354$	10.4885	0.557458
$355$	27.5769	1.46363
$356$	15.3778	0.815019
$357$	5.29654	0.280323
$358$	11.0041	0.581587
$359$	−6.71446	−0.354376	−0.177188	−	0.984177i	$-0.556700\pi$
$359$	−6.71446	−0.354376	−0.177188	+	0.984177i	$0.556700\pi$
$360$	5.34884	0.281908
$361$	1.00000	0.0526316
$362$	−8.54868	−0.449309
$363$	20.5480	1.07849
$364$	0	0
$365$	−17.3049	−0.905781
$366$	−9.82926	−0.513784
$367$	−37.2169	−1.94271	−0.971353	−	0.237640i	$-0.923626\pi$
$367$	−37.2169	−1.94271	−0.971353	+	0.237640i	$0.923626\pi$
$368$	0.110001	0.00573419
$369$	1.92174	0.100042
$370$	−19.2199	−0.999197
$371$	−11.2009	−0.581522
$372$	−4.75897	−0.246741
$373$	−0.358547	−0.0185649	−0.00928243	−	0.999957i	$-0.502955\pi$
$373$	−0.358547	−0.0185649	−0.00928243	+	0.999957i	$0.502955\pi$
$374$	4.27069	0.220832
$375$	−39.8046	−2.05550
$376$	3.15036	0.162467
$377$	0	0
$378$	27.4613	1.41246
$379$	1.19632	0.0614506	0.0307253	−	0.999528i	$-0.490218\pi$
$379$	1.19632	0.0614506	0.0307253	+	0.999528i	$0.490218\pi$
$380$	−4.16024	−0.213416
$381$	15.5229	0.795264
$382$	−2.11168	−0.108043
$383$	3.39507	0.173480	0.0867401	−	0.996231i	$-0.472355\pi$
$383$	3.39507	0.173480	0.0867401	+	0.996231i	$0.472355\pi$
$384$	1.30931	0.0668155
$385$	−105.191	−5.36103
$386$	16.2511	0.827159
$387$	4.02321	0.204511
$388$	1.69504	0.0860528
$389$	−19.9437	−1.01119	−0.505594	−	0.862772i	$-0.668727\pi$
$389$	−19.9437	−1.01119	−0.505594	+	0.862772i	$0.668727\pi$
$390$	0	0
$391$	−0.0909264	−0.00459834
$392$	16.9503	0.856121
$393$	2.75674	0.139059
$394$	−12.9063	−0.650209
$395$	−43.4850	−2.18797
$396$	6.64272	0.333809
$397$	6.81432	0.342001	0.171000	−	0.985271i	$-0.445300\pi$
$397$	6.81432	0.342001	0.171000	+	0.985271i	$0.445300\pi$
$398$	13.2604	0.664685
$399$	−6.40765	−0.320784
$400$	12.3076	0.615378
$401$	7.50198	0.374631	0.187316	−	0.982300i	$-0.440021\pi$
$401$	7.50198	0.374631	0.187316	+	0.982300i	$0.440021\pi$
$402$	5.31024	0.264851
$403$	0	0
$404$	−7.06917	−0.351704
$405$	14.5186	0.721434
$406$	36.3834	1.80568
$407$	−23.8693	−1.18316
$408$	−1.08227	−0.0535804
$409$	15.2222	0.752690	0.376345	−	0.926480i	$-0.377181\pi$
$409$	15.2222	0.752690	0.376345	+	0.926480i	$0.377181\pi$
$410$	6.21828	0.307099
$411$	−20.7735	−1.02468
$412$	2.09095	0.103014
$413$	−39.2037	−1.92909
$414$	−0.141429	−0.00695084
$415$	74.8913	3.67627
$416$	0	0
$417$	−22.1154	−1.08300
$418$	−5.16660	−0.252707
$419$	17.1300	0.836853	0.418427	−	0.908251i	$-0.362582\pi$
$419$	17.1300	0.836853	0.418427	+	0.908251i	$0.362582\pi$
$420$	26.6573	1.30074
$421$	−3.36243	−0.163875	−0.0819375	−	0.996637i	$-0.526111\pi$
$421$	−3.36243	−0.163875	−0.0819375	+	0.996637i	$0.526111\pi$
$422$	0.616577	0.0300145
$423$	−4.05043	−0.196939
$424$	2.28874	0.111151
$425$	−10.1734	−0.493482
$426$	−8.67900	−0.420499
$427$	36.7395	1.77795
$428$	11.2185	0.542266
$429$	0	0
$430$	13.0182	0.627791
$431$	25.1374	1.21082	0.605412	−	0.795912i	$-0.293008\pi$
$431$	25.1374	1.21082	0.605412	+	0.795912i	$0.293008\pi$
$432$	−5.61132	−0.269975
$433$	12.2856	0.590406	0.295203	−	0.955435i	$-0.404613\pi$
$433$	12.2856	0.590406	0.295203	+	0.955435i	$0.404613\pi$
$434$	17.7879	0.853848
$435$	40.4956	1.94161
$436$	−13.7272	−0.657413
$437$	0.110001	0.00526206
$438$	5.44621	0.260230
$439$	30.7772	1.46891	0.734457	−	0.678655i	$-0.237437\pi$
$439$	30.7772	1.46891	0.734457	+	0.678655i	$0.237437\pi$
$440$	21.4943	1.02470
$441$	−21.7931	−1.03777
$442$	0	0
$443$	2.04824	0.0973150	0.0486575	−	0.998816i	$-0.484506\pi$
$443$	2.04824	0.0973150	0.0486575	+	0.998816i	$0.484506\pi$
$444$	6.04891	0.287068
$445$	−63.9751	−3.03271
$446$	15.4713	0.732585
$447$	14.2629	0.674613
$448$	−4.89391	−0.231215
$449$	4.73169	0.223302	0.111651	−	0.993747i	$-0.464386\pi$
$449$	4.73169	0.223302	0.111651	+	0.993747i	$0.464386\pi$
$450$	−15.8239	−0.745945
$451$	7.72249	0.363638
$452$	−12.8772	−0.605691
$453$	−14.8111	−0.695885
$454$	−5.01952	−0.235578
$455$	0	0
$456$	1.30931	0.0613141
$457$	−21.5996	−1.01039	−0.505193	−	0.863006i	$-0.668579\pi$
$457$	−21.5996	−1.01039	−0.505193	+	0.863006i	$0.668579\pi$
$458$	−14.5207	−0.678509
$459$	4.63830	0.216497
$460$	−0.457630	−0.0213371
$461$	32.7873	1.52706	0.763528	−	0.645775i	$-0.223466\pi$
$461$	32.7873	1.52706	0.763528	+	0.645775i	$0.223466\pi$
$462$	33.1057	1.54022
$463$	1.39591	0.0648736	0.0324368	−	0.999474i	$-0.489673\pi$
$463$	1.39591	0.0648736	0.0324368	+	0.999474i	$0.489673\pi$
$464$	−7.43442	−0.345134
$465$	19.7984	0.918129
$466$	−5.28417	−0.244784
$467$	−9.84054	−0.455366	−0.227683	−	0.973735i	$-0.573115\pi$
$467$	−9.84054	−0.455366	−0.227683	+	0.973735i	$0.573115\pi$
$468$	0	0
$469$	−19.8485	−0.916517
$470$	−13.1062	−0.604545
$471$	−2.52881	−0.116521
$472$	8.01071	0.368723
$473$	16.1673	0.743372
$474$	13.6856	0.628601
$475$	12.3076	0.564709
$476$	4.04529	0.185415
$477$	−2.94265	−0.134735
$478$	30.3753	1.38933
$479$	−21.9271	−1.00187	−0.500937	−	0.865484i	$-0.667011\pi$
$479$	−21.9271	−1.00187	−0.500937	+	0.865484i	$0.667011\pi$
$480$	−5.44704	−0.248622
$481$	0	0
$482$	21.5093	0.979723
$483$	−0.704847	−0.0320716
$484$	15.6938	0.713353
$485$	−7.05178	−0.320205
$486$	12.2647	0.556337
$487$	−12.0251	−0.544909	−0.272455	−	0.962169i	$-0.587835\pi$
$487$	−12.0251	−0.544909	−0.272455	+	0.962169i	$0.587835\pi$
$488$	−7.50720	−0.339835
$489$	−0.826032	−0.0373545
$490$	−70.5173	−3.18565
$491$	−3.37909	−0.152496	−0.0762482	−	0.997089i	$-0.524294\pi$
$491$	−3.37909	−0.152496	−0.0762482	+	0.997089i	$0.524294\pi$
$492$	−1.95702	−0.0882292
$493$	6.14527	0.276769
$494$	0	0
$495$	−27.6353	−1.24211
$496$	−3.63471	−0.163203
$497$	32.4401	1.45514
$498$	−23.5698	−1.05619
$499$	−33.9601	−1.52026	−0.760132	−	0.649769i	$-0.774866\pi$
$499$	−33.9601	−1.52026	−0.760132	+	0.649769i	$0.774866\pi$
$500$	−30.4012	−1.35958
$501$	−0.689609	−0.0308094
$502$	15.6493	0.698462
$503$	10.3815	0.462887	0.231444	−	0.972848i	$-0.425655\pi$
$503$	10.3815	0.462887	0.231444	+	0.972848i	$0.425655\pi$
$504$	6.29212	0.280273
$505$	29.4094	1.30870
$506$	−0.568331	−0.0252654
$507$	0	0
$508$	11.8558	0.526016
$509$	−25.5045	−1.13047	−0.565235	−	0.824930i	$-0.691214\pi$
$509$	−25.5045	−1.13047	−0.565235	+	0.824930i	$0.691214\pi$
$510$	4.50251	0.199374
$511$	−20.3567	−0.900527
$512$	1.00000	0.0441942
$513$	−5.61132	−0.247746
$514$	−12.9068	−0.569294
$515$	−8.69883	−0.383316
$516$	−4.09708	−0.180364
$517$	−16.2766	−0.715845
$518$	−22.6094	−0.993402
$519$	−16.5234	−0.725295
$520$	0	0
$521$	−25.1613	−1.10234	−0.551168	−	0.834394i	$-0.685818\pi$
$521$	−25.1613	−1.10234	−0.551168	+	0.834394i	$0.685818\pi$
$522$	9.55847	0.418363
$523$	17.0491	0.745503	0.372752	−	0.927931i	$-0.378414\pi$
$523$	17.0491	0.745503	0.372752	+	0.927931i	$0.378414\pi$
$524$	2.10549	0.0919788
$525$	−78.8625	−3.44184
$526$	−19.5814	−0.853791
$527$	3.00444	0.130875
$528$	−6.76469	−0.294395
$529$	−22.9879	−0.999474
$530$	−9.52171	−0.413597
$531$	−10.2994	−0.446957
$532$	−4.89391	−0.212178
$533$	0	0
$534$	20.1343	0.871295
$535$	−46.6716	−2.01779
$536$	4.05575	0.175182
$537$	14.4078	0.621744
$538$	−18.8216	−0.811456
$539$	−87.5756	−3.77215
$540$	23.3444	1.00458
$541$	19.0495	0.819001	0.409501	−	0.912310i	$-0.365703\pi$
$541$	19.0495	0.819001	0.409501	+	0.912310i	$0.365703\pi$
$542$	1.15476	0.0496010
$543$	−11.1929	−0.480333
$544$	−0.826597	−0.0354400
$545$	57.1083	2.44625
$546$	0	0
$547$	33.0788	1.41435	0.707173	−	0.707040i	$-0.249970\pi$
$547$	33.0788	1.41435	0.707173	+	0.707040i	$0.249970\pi$
$548$	−15.8660	−0.677761
$549$	9.65205	0.411939
$550$	−63.5882	−2.71141
$551$	−7.43442	−0.316717
$552$	0.144025	0.00613013
$553$	−51.1537	−2.17528
$554$	15.1452	0.643457
$555$	−25.1649	−1.06819
$556$	−16.8909	−0.716334
$557$	−10.8409	−0.459346	−0.229673	−	0.973268i	$-0.573766\pi$
$557$	−10.8409	−0.459346	−0.229673	+	0.973268i	$0.573766\pi$
$558$	4.67316	0.197831
$559$	0	0
$560$	20.3598	0.860359
$561$	5.59167	0.236080
$562$	−9.05085	−0.381787
$563$	−16.7081	−0.704163	−0.352081	−	0.935969i	$-0.614526\pi$
$563$	−16.7081	−0.704163	−0.352081	+	0.935969i	$0.614526\pi$
$564$	4.12479	0.173685
$565$	53.5721	2.25379
$566$	−0.322507	−0.0135560
$567$	17.0790	0.717250
$568$	−6.62868	−0.278133
$569$	24.6474	1.03327	0.516637	−	0.856205i	$-0.327184\pi$
$569$	24.6474	1.03327	0.516637	+	0.856205i	$0.327184\pi$
$570$	−5.44704	−0.228151
$571$	22.0543	0.922944	0.461472	−	0.887155i	$-0.347321\pi$
$571$	22.0543	0.922944	0.461472	+	0.887155i	$0.347321\pi$
$572$	0	0
$573$	−2.76484	−0.115503
$574$	7.31489	0.305318
$575$	1.35384	0.0564591
$576$	−1.28570	−0.0535710
$577$	8.66164	0.360589	0.180294	−	0.983613i	$-0.442295\pi$
$577$	8.66164	0.360589	0.180294	+	0.983613i	$0.442295\pi$
$578$	−16.3167	−0.678687
$579$	21.2777	0.884272
$580$	30.9289	1.28425
$581$	88.0986	3.65495
$582$	2.21934	0.0919946
$583$	−11.8250	−0.489742
$584$	4.15960	0.172126
$585$	0	0
$586$	0.577328	0.0238492
$587$	39.3935	1.62595	0.812973	−	0.582302i	$-0.197848\pi$
$587$	39.3935	1.62595	0.812973	+	0.582302i	$0.197848\pi$
$588$	22.1932	0.915234
$589$	−3.63471	−0.149766
$590$	−33.3265	−1.37203
$591$	−16.8983	−0.695104
$592$	4.61991	0.189877
$593$	8.58130	0.352392	0.176196	−	0.984355i	$-0.443621\pi$
$593$	8.58130	0.352392	0.176196	+	0.984355i	$0.443621\pi$
$594$	28.9915	1.18953
$595$	−16.8293	−0.689936
$596$	10.8934	0.446213
$597$	17.3620	0.710580
$598$	0	0
$599$	−45.1931	−1.84654	−0.923270	−	0.384153i	$-0.874494\pi$
$599$	−45.1931	−1.84654	−0.923270	+	0.384153i	$0.874494\pi$
$600$	16.1144	0.657868
$601$	33.5447	1.36832	0.684158	−	0.729334i	$-0.260170\pi$
$601$	33.5447	1.36832	0.684158	+	0.729334i	$0.260170\pi$
$602$	15.3140	0.624150
$603$	−5.21450	−0.212351
$604$	−11.3121	−0.460284
$605$	−65.2898	−2.65441
$606$	−9.25574	−0.375989
$607$	6.59597	0.267722	0.133861	−	0.991000i	$-0.457262\pi$
$607$	6.59597	0.267722	0.133861	+	0.991000i	$0.457262\pi$
$608$	1.00000	0.0405554
$609$	47.6371	1.93035
$610$	31.2317	1.26454
$611$	0	0
$612$	1.06276	0.0429595
$613$	1.54817	0.0625298	0.0312649	−	0.999511i	$-0.490046\pi$
$613$	1.54817	0.0625298	0.0312649	+	0.999511i	$0.490046\pi$
$614$	17.0687	0.688837
$615$	8.14166	0.328303
$616$	25.2849	1.01876
$617$	−13.5808	−0.546742	−0.273371	−	0.961909i	$-0.588139\pi$
$617$	−13.5808	−0.546742	−0.273371	+	0.961909i	$0.588139\pi$
$618$	2.73770	0.110126
$619$	−11.9135	−0.478844	−0.239422	−	0.970916i	$-0.576958\pi$
$619$	−11.9135	−0.478844	−0.239422	+	0.970916i	$0.576958\pi$
$620$	15.1212	0.607284
$621$	−0.617250	−0.0247694
$622$	0.0907973	0.00364064
$623$	−75.2573	−3.01512
$624$	0	0
$625$	64.9382	2.59753
$626$	−22.9924	−0.918959
$627$	−6.76469	−0.270156
$628$	−1.93140	−0.0770714
$629$	−3.81881	−0.152266
$630$	−26.1767	−1.04290
$631$	−29.3904	−1.17001	−0.585007	−	0.811028i	$-0.698908\pi$
$631$	−29.3904	−1.17001	−0.585007	+	0.811028i	$0.698908\pi$
$632$	10.4525	0.415779
$633$	0.807291	0.0320869
$634$	−11.7090	−0.465022
$635$	−49.3229	−1.95732
$636$	2.99668	0.118826
$637$	0	0
$638$	38.4107	1.52069
$639$	8.52253	0.337146
$640$	−4.16024	−0.164448
$641$	44.8234	1.77042	0.885209	−	0.465194i	$-0.154016\pi$
$641$	44.8234	1.77042	0.885209	+	0.465194i	$0.154016\pi$
$642$	14.6885	0.579709
$643$	−18.0858	−0.713233	−0.356617	−	0.934251i	$-0.616070\pi$
$643$	−18.0858	−0.713233	−0.356617	+	0.934251i	$0.616070\pi$
$644$	−0.538334	−0.0212133
$645$	17.0448	0.671139
$646$	−0.826597	−0.0325220
$647$	−20.5682	−0.808621	−0.404311	−	0.914622i	$-0.632489\pi$
$647$	−20.5682	−0.808621	−0.404311	+	0.914622i	$0.632489\pi$
$648$	−3.48985	−0.137094
$649$	−41.3882	−1.62463
$650$	0	0
$651$	23.2899	0.912804
$652$	−0.630891	−0.0247076
$653$	26.0552	1.01962	0.509809	−	0.860288i	$-0.329716\pi$
$653$	26.0552	1.01962	0.509809	+	0.860288i	$0.329716\pi$
$654$	−17.9732	−0.702806
$655$	−8.75934	−0.342256
$656$	−1.49469	−0.0583580
$657$	−5.34802	−0.208646
$658$	−15.4175	−0.601038
$659$	−23.1716	−0.902638	−0.451319	−	0.892363i	$-0.649046\pi$
$659$	−23.1716	−0.902638	−0.451319	+	0.892363i	$0.649046\pi$
$660$	28.1427	1.09545
$661$	−43.4812	−1.69122	−0.845611	−	0.533800i	$-0.820763\pi$
$661$	−43.4812	−1.69122	−0.845611	+	0.533800i	$0.820763\pi$
$662$	9.30885	0.361799
$663$	0	0
$664$	−18.0017	−0.698601
$665$	20.3598	0.789519
$666$	−5.93985	−0.230164
$667$	−0.817793	−0.0316651
$668$	−0.526696	−0.0203785
$669$	20.2567	0.783169
$670$	−16.8729	−0.651856
$671$	38.7867	1.49734
$672$	−6.40765	−0.247180
$673$	5.02781	0.193808	0.0969038	−	0.995294i	$-0.469106\pi$
$673$	5.02781	0.193808	0.0969038	+	0.995294i	$0.469106\pi$
$674$	10.7917	0.415682
$675$	−69.0616	−2.65818
$676$	0	0
$677$	−20.9768	−0.806203	−0.403101	−	0.915155i	$-0.632068\pi$
$677$	−20.9768	−0.806203	−0.403101	+	0.915155i	$0.632068\pi$
$678$	−16.8602	−0.647513
$679$	−8.29539	−0.318348
$680$	3.43884	0.131873
$681$	−6.57211	−0.251844
$682$	18.7791	0.719089
$683$	43.2325	1.65425	0.827123	−	0.562021i	$-0.189976\pi$
$683$	43.2325	1.65425	0.827123	+	0.562021i	$0.189976\pi$
$684$	−1.28570	−0.0491602
$685$	66.0062	2.52197
$686$	−48.6960	−1.85922
$687$	−19.0122	−0.725359
$688$	−3.12919	−0.119299
$689$	0	0
$690$	−0.599180	−0.0228104
$691$	−2.52944	−0.0962243	−0.0481122	−	0.998842i	$-0.515320\pi$
$691$	−2.52944	−0.0962243	−0.0481122	+	0.998842i	$0.515320\pi$
$692$	−12.6199	−0.479736
$693$	−32.5089	−1.23491
$694$	10.3261	0.391975
$695$	70.2701	2.66550
$696$	−9.73397	−0.368965
$697$	1.23551	0.0467982
$698$	23.7338	0.898337
$699$	−6.91862	−0.261686
$700$	−60.2320	−2.27656
$701$	36.0943	1.36326	0.681631	−	0.731696i	$-0.261271\pi$
$701$	36.0943	1.36326	0.681631	+	0.731696i	$0.261271\pi$
$702$	0	0
$703$	4.61991	0.174243
$704$	−5.16660	−0.194724
$705$	−17.1601	−0.646287
$706$	27.3851	1.03065
$707$	34.5959	1.30111
$708$	10.4885	0.394183
$709$	−29.6557	−1.11374	−0.556871	−	0.830599i	$-0.687998\pi$
$709$	−29.6557	−1.11374	−0.556871	+	0.830599i	$0.687998\pi$
$710$	27.5769	1.03494
$711$	−13.4389	−0.503997
$712$	15.3778	0.576306
$713$	−0.399821	−0.0149734
$714$	5.29654	0.198218
$715$	0	0
$716$	11.0041	0.411244
$717$	39.7707	1.48526
$718$	−6.71446	−0.250581
$719$	−47.6635	−1.77755	−0.888775	−	0.458344i	$-0.848443\pi$
$719$	−47.6635	−1.77755	−0.888775	+	0.458344i	$0.848443\pi$
$720$	5.34884	0.199339
$721$	−10.2329	−0.381093
$722$	1.00000	0.0372161
$723$	28.1624	1.04737
$724$	−8.54868	−0.317709
$725$	−91.4995	−3.39821
$726$	20.5480	0.762608
$727$	−19.6899	−0.730259	−0.365129	−	0.930957i	$-0.618975\pi$
$727$	−19.6899	−0.730259	−0.365129	+	0.930957i	$0.618975\pi$
$728$	0	0
$729$	26.5278	0.982511
$730$	−17.3049	−0.640484
$731$	2.58658	0.0956680
$732$	−9.82926	−0.363300
$733$	27.8818	1.02984	0.514918	−	0.857239i	$-0.327822\pi$
$733$	27.8818	1.02984	0.514918	+	0.857239i	$0.327822\pi$
$734$	−37.2169	−1.37370
$735$	−92.3291	−3.40561
$736$	0.110001	0.00405469
$737$	−20.9544	−0.771867
$738$	1.92174	0.0707401
$739$	5.60156	0.206057	0.103028	−	0.994678i	$-0.467147\pi$
$739$	5.60156	0.206057	0.103028	+	0.994678i	$0.467147\pi$
$740$	−19.2199	−0.706539
$741$	0	0
$742$	−11.2009	−0.411198
$743$	−25.7763	−0.945642	−0.472821	−	0.881159i	$-0.656764\pi$
$743$	−25.7763	−0.945642	−0.472821	+	0.881159i	$0.656764\pi$
$744$	−4.75897	−0.174472
$745$	−45.3193	−1.66037
$746$	−0.358547	−0.0131273
$747$	23.1449	0.846827
$748$	4.27069	0.156152
$749$	−54.9023	−2.00609
$750$	−39.8046	−1.45346
$751$	−15.5784	−0.568466	−0.284233	−	0.958755i	$-0.591739\pi$
$751$	−15.5784	−0.568466	−0.284233	+	0.958755i	$0.591739\pi$
$752$	3.15036	0.114882
$753$	20.4898	0.746689
$754$	0	0
$755$	47.0611	1.71273
$756$	27.4613	0.998757
$757$	−11.4780	−0.417177	−0.208588	−	0.978004i	$-0.566887\pi$
$757$	−11.4780	−0.417177	−0.208588	+	0.978004i	$0.566887\pi$
$758$	1.19632	0.0434521
$759$	−0.744122	−0.0270099
$760$	−4.16024	−0.150908
$761$	−41.8703	−1.51780	−0.758899	−	0.651209i	$-0.774262\pi$
$761$	−41.8703	−1.51780	−0.758899	+	0.651209i	$0.774262\pi$
$762$	15.5229	0.562337
$763$	67.1796	2.43206
$764$	−2.11168	−0.0763978
$765$	−4.42133	−0.159853
$766$	3.39507	0.122669
$767$	0	0
$768$	1.30931	0.0472457
$769$	46.6813	1.68337	0.841685	−	0.539969i	$-0.181564\pi$
$769$	46.6813	1.68337	0.841685	+	0.539969i	$0.181564\pi$
$770$	−105.191	−3.79082
$771$	−16.8990	−0.608602
$772$	16.2511	0.584889
$773$	7.89148	0.283837	0.141918	−	0.989878i	$-0.454673\pi$
$773$	7.89148	0.283837	0.141918	+	0.989878i	$0.454673\pi$
$774$	4.02321	0.144611
$775$	−44.7344	−1.60691
$776$	1.69504	0.0608485
$777$	−29.6028	−1.06199
$778$	−19.9437	−0.715018
$779$	−1.49469	−0.0535530
$780$	0	0
$781$	34.2477	1.22548
$782$	−0.0909264	−0.00325152
$783$	41.7169	1.49084
$784$	16.9503	0.605369
$785$	8.03509	0.286785
$786$	2.75674	0.0983298
$787$	−12.5671	−0.447967	−0.223984	−	0.974593i	$-0.571906\pi$
$787$	−12.5671	−0.447967	−0.223984	+	0.974593i	$0.571906\pi$
$788$	−12.9063	−0.459767
$789$	−25.6382	−0.912743
$790$	−43.4850	−1.54713
$791$	63.0197	2.24072
$792$	6.64272	0.236039
$793$	0	0
$794$	6.81432	0.241831
$795$	−12.4669	−0.442155
$796$	13.2604	0.470003
$797$	−45.6927	−1.61852	−0.809259	−	0.587452i	$-0.800131\pi$
$797$	−45.6927	−1.61852	−0.809259	+	0.587452i	$0.800131\pi$
$798$	−6.40765	−0.226828
$799$	−2.60407	−0.0921255
$800$	12.3076	0.435138
$801$	−19.7713	−0.698583
$802$	7.50198	0.264904
$803$	−21.4910	−0.758401
$804$	5.31024	0.187278
$805$	2.23960	0.0789354
$806$	0	0
$807$	−24.6433	−0.867485
$808$	−7.06917	−0.248693
$809$	45.3882	1.59576	0.797882	−	0.602814i	$-0.205954\pi$
$809$	45.3882	1.59576	0.797882	+	0.602814i	$0.205954\pi$
$810$	14.5186	0.510131
$811$	−21.0904	−0.740585	−0.370293	−	0.928915i	$-0.620743\pi$
$811$	−21.0904	−0.740585	−0.370293	+	0.928915i	$0.620743\pi$
$812$	36.3834	1.27681
$813$	1.51193	0.0530258
$814$	−23.8693	−0.836617
$815$	2.62465	0.0919376
$816$	−1.08227	−0.0378871
$817$	−3.12919	−0.109476
$818$	15.2222	0.532232
$819$	0	0
$820$	6.21828	0.217152
$821$	−44.3538	−1.54796	−0.773979	−	0.633211i	$-0.781736\pi$
$821$	−44.3538	−1.54796	−0.773979	+	0.633211i	$0.781736\pi$
$822$	−20.7735	−0.724559
$823$	3.97523	0.138568	0.0692840	−	0.997597i	$-0.477929\pi$
$823$	3.97523	0.138568	0.0692840	+	0.997597i	$0.477929\pi$
$824$	2.09095	0.0728416
$825$	−83.2568	−2.89863
$826$	−39.2037	−1.36407
$827$	−6.92877	−0.240937	−0.120468	−	0.992717i	$-0.538440\pi$
$827$	−6.92877	−0.240937	−0.120468	+	0.992717i	$0.538440\pi$
$828$	−0.141429	−0.00491499
$829$	42.2582	1.46769	0.733843	−	0.679319i	$-0.237725\pi$
$829$	42.2582	1.46769	0.733843	+	0.679319i	$0.237725\pi$
$830$	74.8913	2.59952
$831$	19.8297	0.687886
$832$	0	0
$833$	−14.0111	−0.485455
$834$	−22.1154	−0.765795
$835$	2.19118	0.0758289
$836$	−5.16660	−0.178691
$837$	20.3955	0.704972
$838$	17.1300	0.591745
$839$	49.1714	1.69759	0.848793	−	0.528725i	$-0.177330\pi$
$839$	49.1714	1.69759	0.848793	+	0.528725i	$0.177330\pi$
$840$	26.6573	0.919765
$841$	26.2706	0.905882
$842$	−3.36243	−0.115877
$843$	−11.8504	−0.408149
$844$	0.616577	0.0212234
$845$	0	0
$846$	−4.05043	−0.139257
$847$	−76.8038	−2.63901
$848$	2.28874	0.0785958
$849$	−0.422262	−0.0144920
$850$	−10.1734	−0.348944
$851$	0.508195	0.0174207
$852$	−8.67900	−0.297338
$853$	−17.1488	−0.587165	−0.293582	−	0.955934i	$-0.594847\pi$
$853$	−17.1488	−0.587165	−0.293582	+	0.955934i	$0.594847\pi$
$854$	36.7395	1.25720
$855$	5.34884	0.182926
$856$	11.2185	0.383440
$857$	35.0972	1.19890	0.599448	−	0.800414i	$-0.295387\pi$
$857$	35.0972	1.19890	0.599448	+	0.800414i	$0.295387\pi$
$858$	0	0
$859$	3.19450	0.108995	0.0544974	−	0.998514i	$-0.482644\pi$
$859$	3.19450	0.108995	0.0544974	+	0.998514i	$0.482644\pi$
$860$	13.0182	0.443915
$861$	9.57747	0.326399
$862$	25.1374	0.856182
$863$	18.6884	0.636160	0.318080	−	0.948064i	$-0.396962\pi$
$863$	18.6884	0.636160	0.318080	+	0.948064i	$0.396962\pi$
$864$	−5.61132	−0.190901
$865$	52.5017	1.78511
$866$	12.2856	0.417480
$867$	−21.3637	−0.725549
$868$	17.7879	0.603762
$869$	−54.0040	−1.83196
$870$	40.4956	1.37293
$871$	0	0
$872$	−13.7272	−0.464861
$873$	−2.17933	−0.0737590
$874$	0.110001	0.00372084
$875$	148.780	5.02970
$876$	5.44621	0.184010
$877$	−49.4188	−1.66876	−0.834378	−	0.551193i	$-0.814173\pi$
$877$	−49.4188	−1.66876	−0.834378	+	0.551193i	$0.814173\pi$
$878$	30.7772	1.03868
$879$	0.755902	0.0254959
$880$	21.4943	0.724572
$881$	23.4293	0.789354	0.394677	−	0.918820i	$-0.370857\pi$
$881$	23.4293	0.789354	0.394677	+	0.918820i	$0.370857\pi$
$882$	−21.7931	−0.733813
$883$	9.60315	0.323172	0.161586	−	0.986859i	$-0.448339\pi$
$883$	9.60315	0.323172	0.161586	+	0.986859i	$0.448339\pi$
$884$	0	0
$885$	−43.6347	−1.46676
$886$	2.04824	0.0688121
$887$	24.7744	0.831844	0.415922	−	0.909400i	$-0.363459\pi$
$887$	24.7744	0.831844	0.415922	+	0.909400i	$0.363459\pi$
$888$	6.04891	0.202988
$889$	−58.0212	−1.94597
$890$	−63.9751	−2.14445
$891$	18.0307	0.604050
$892$	15.4713	0.518016
$893$	3.15036	0.105423
$894$	14.2629	0.477023
$895$	−45.7798	−1.53025
$896$	−4.89391	−0.163494
$897$	0	0
$898$	4.73169	0.157899
$899$	27.0220	0.901233
$900$	−15.8239	−0.527463
$901$	−1.89187	−0.0630272
$902$	7.72249	0.257131
$903$	20.0507	0.667246
$904$	−12.8772	−0.428288
$905$	35.5645	1.18220
$906$	−14.8111	−0.492065
$907$	8.66811	0.287820	0.143910	−	0.989591i	$-0.454032\pi$
$907$	8.66811	0.287820	0.143910	+	0.989591i	$0.454032\pi$
$908$	−5.01952	−0.166578
$909$	9.08887	0.301459
$910$	0	0
$911$	36.5544	1.21110	0.605551	−	0.795806i	$-0.292953\pi$
$911$	36.5544	1.21110	0.605551	+	0.795806i	$0.292953\pi$
$912$	1.30931	0.0433556
$913$	93.0076	3.07810
$914$	−21.5996	−0.714451
$915$	40.8920	1.35185
$916$	−14.5207	−0.479779
$917$	−10.3041	−0.340271
$918$	4.63830	0.153087
$919$	57.7683	1.90560	0.952801	−	0.303596i	$-0.0981874\pi$
$919$	57.7683	1.90560	0.952801	+	0.303596i	$0.0981874\pi$
$920$	−0.457630	−0.0150876
$921$	22.3483	0.736400
$922$	32.7873	1.07979
$923$	0	0
$924$	33.1057	1.08910
$925$	56.8599	1.86954
$926$	1.39591	0.0458726
$927$	−2.68834	−0.0882967
$928$	−7.43442	−0.244047
$929$	6.84579	0.224603	0.112302	−	0.993674i	$-0.464178\pi$
$929$	6.84579	0.224603	0.112302	+	0.993674i	$0.464178\pi$
$930$	19.7984	0.649216
$931$	16.9503	0.555525
$932$	−5.28417	−0.173089
$933$	0.118882	0.00389202
$934$	−9.84054	−0.321992
$935$	−17.7671	−0.581046
$936$	0	0
$937$	−7.00877	−0.228966	−0.114483	−	0.993425i	$-0.536521\pi$
$937$	−7.00877	−0.228966	−0.114483	+	0.993425i	$0.536521\pi$
$938$	−19.8485	−0.648075
$939$	−30.1041	−0.982412
$940$	−13.1062	−0.427478
$941$	54.8521	1.78813	0.894064	−	0.447939i	$-0.147842\pi$
$941$	54.8521	1.78813	0.894064	+	0.447939i	$0.147842\pi$
$942$	−2.52881	−0.0823930
$943$	−0.164418	−0.00535418
$944$	8.01071	0.260727
$945$	−114.245	−3.71640
$946$	16.1673	0.525643
$947$	20.5812	0.668800	0.334400	−	0.942431i	$-0.391466\pi$
$947$	20.5812	0.668800	0.334400	+	0.942431i	$0.391466\pi$
$948$	13.6856	0.444488
$949$	0	0
$950$	12.3076	0.399310
$951$	−15.3307	−0.497131
$952$	4.04529	0.131109
$953$	−9.82558	−0.318282	−0.159141	−	0.987256i	$-0.550872\pi$
$953$	−9.82558	−0.318282	−0.159141	+	0.987256i	$0.550872\pi$
$954$	−2.94265	−0.0952718
$955$	8.78508	0.284278
$956$	30.3753	0.982407
$957$	50.2915	1.62569
$958$	−21.9271	−0.708432
$959$	77.6466	2.50734
$960$	−5.44704	−0.175803
$961$	−17.7889	−0.573835
$962$	0	0
$963$	−14.4237	−0.464796
$964$	21.5093	0.692769
$965$	−67.6083	−2.17639
$966$	−0.704847	−0.0226781
$967$	39.4345	1.26813	0.634063	−	0.773281i	$-0.281386\pi$
$967$	39.4345	1.26813	0.634063	+	0.773281i	$0.281386\pi$
$968$	15.6938	0.504417
$969$	−1.08227	−0.0347676
$970$	−7.05178	−0.226419
$971$	−18.8417	−0.604658	−0.302329	−	0.953204i	$-0.597764\pi$
$971$	−18.8417	−0.604658	−0.302329	+	0.953204i	$0.597764\pi$
$972$	12.2647	0.393389
$973$	82.6625	2.65004
$974$	−12.0251	−0.385309
$975$	0	0
$976$	−7.50720	−0.240300
$977$	−15.2214	−0.486977	−0.243489	−	0.969904i	$-0.578292\pi$
$977$	−15.2214	−0.486977	−0.243489	+	0.969904i	$0.578292\pi$
$978$	−0.826032	−0.0264136
$979$	−79.4507	−2.53926
$980$	−70.5173	−2.25259
$981$	17.6491	0.563493
$982$	−3.37909	−0.107831
$983$	−19.9421	−0.636055	−0.318028	−	0.948081i	$-0.603020\pi$
$983$	−19.9421	−0.636055	−0.318028	+	0.948081i	$0.603020\pi$
$984$	−1.95702	−0.0623875
$985$	53.6931	1.71081
$986$	6.14527	0.195705
$987$	−20.1864	−0.642539
$988$	0	0
$989$	−0.344213	−0.0109454
$990$	−27.6353	−0.878307
$991$	−27.9661	−0.888372	−0.444186	−	0.895934i	$-0.646507\pi$
$991$	−27.9661	−0.888372	−0.444186	+	0.895934i	$0.646507\pi$
$992$	−3.63471	−0.115402
$993$	12.1882	0.386780
$994$	32.4401	1.02894
$995$	−55.1665	−1.74889
$996$	−23.5698	−0.746838
$997$	−40.5267	−1.28349	−0.641747	−	0.766917i	$-0.721790\pi$
$997$	−40.5267	−1.28349	−0.641747	+	0.766917i	$0.721790\pi$
$998$	−33.9601	−1.07499
$999$	−25.9238	−0.820193

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6422.2.a.bn.1.8		14
13.6	odd	12		494.2.m.b.153.4	✓	28
13.11	odd	12		494.2.m.b.381.4	yes	28
13.12	even	2		6422.2.a.bm.1.8		14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
494.2.m.b.153.4	✓	28	13.6	odd	12
494.2.m.b.381.4	yes	28	13.11	odd	12
6422.2.a.bm.1.8		14	13.12	even	2
6422.2.a.bn.1.8		14	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:	\( N \)	\(=\)	\( 6422 = 2 \cdot 13^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6422.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.30931 q^{3} +1.00000 q^{4} -4.16024 q^{5} +1.30931 q^{6} -4.89391 q^{7} +1.00000 q^{8} -1.28570 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +1.30931 q^{3} +1.00000 q^{4} -4.16024 q^{5} +1.30931 q^{6} -4.89391 q^{7} +1.00000 q^{8} -1.28570 q^{9} -4.16024 q^{10} -5.16660 q^{11} +1.30931 q^{12} -4.89391 q^{14} -5.44704 q^{15} +1.00000 q^{16} -0.826597 q^{17} -1.28570 q^{18} +1.00000 q^{19} -4.16024 q^{20} -6.40765 q^{21} -5.16660 q^{22} +0.110001 q^{23} +1.30931 q^{24} +12.3076 q^{25} -5.61132 q^{27} -4.89391 q^{28} -7.43442 q^{29} -5.44704 q^{30} -3.63471 q^{31} +1.00000 q^{32} -6.76469 q^{33} -0.826597 q^{34} +20.3598 q^{35} -1.28570 q^{36} +4.61991 q^{37} +1.00000 q^{38} -4.16024 q^{40} -1.49469 q^{41} -6.40765 q^{42} -3.12919 q^{43} -5.16660 q^{44} +5.34884 q^{45} +0.110001 q^{46} +3.15036 q^{47} +1.30931 q^{48} +16.9503 q^{49} +12.3076 q^{50} -1.08227 q^{51} +2.28874 q^{53} -5.61132 q^{54} +21.4943 q^{55} -4.89391 q^{56} +1.30931 q^{57} -7.43442 q^{58} +8.01071 q^{59} -5.44704 q^{60} -7.50720 q^{61} -3.63471 q^{62} +6.29212 q^{63} +1.00000 q^{64} -6.76469 q^{66} +4.05575 q^{67} -0.826597 q^{68} +0.144025 q^{69} +20.3598 q^{70} -6.62868 q^{71} -1.28570 q^{72} +4.15960 q^{73} +4.61991 q^{74} +16.1144 q^{75} +1.00000 q^{76} +25.2849 q^{77} +10.4525 q^{79} -4.16024 q^{80} -3.48985 q^{81} -1.49469 q^{82} -18.0017 q^{83} -6.40765 q^{84} +3.43884 q^{85} -3.12919 q^{86} -9.73397 q^{87} -5.16660 q^{88} +15.3778 q^{89} +5.34884 q^{90} +0.110001 q^{92} -4.75897 q^{93} +3.15036 q^{94} -4.16024 q^{95} +1.30931 q^{96} +1.69504 q^{97} +16.9503 q^{98} +6.64272 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q + 14 q^{2} + 4 q^{3} + 14 q^{4} - 2 q^{5} + 4 q^{6} + 2 q^{7} + 14 q^{8} + 18 q^{9}+O(q^{10}) \) 14 * q + 14 * q^2 + 4 * q^3 + 14 * q^4 - 2 * q^5 + 4 * q^6 + 2 * q^7 + 14 * q^8 + 18 * q^9	\( 14 q + 14 q^{2} + 4 q^{3} + 14 q^{4} - 2 q^{5} + 4 q^{6} + 2 q^{7} + 14 q^{8} + 18 q^{9} - 2 q^{10} + 10 q^{11} + 4 q^{12} + 2 q^{14} - 4 q^{15} + 14 q^{16} + 2 q^{17} + 18 q^{18} + 14 q^{19} - 2 q^{20} - 18 q^{21} + 10 q^{22} + 12 q^{23} + 4 q^{24} + 44 q^{25} + 10 q^{27} + 2 q^{28} + 4 q^{29} - 4 q^{30} + 4 q^{31} + 14 q^{32} + 12 q^{33} + 2 q^{34} + 14 q^{35} + 18 q^{36} + 18 q^{37} + 14 q^{38} - 2 q^{40} + 6 q^{41} - 18 q^{42} + 28 q^{43} + 10 q^{44} + 8 q^{45} + 12 q^{46} - 20 q^{47} + 4 q^{48} + 28 q^{49} + 44 q^{50} + 10 q^{51} + 12 q^{53} + 10 q^{54} - 2 q^{55} + 2 q^{56} + 4 q^{57} + 4 q^{58} + 16 q^{59} - 4 q^{60} + 30 q^{61} + 4 q^{62} + 28 q^{63} + 14 q^{64} + 12 q^{66} + 2 q^{67} + 2 q^{68} - 42 q^{69} + 14 q^{70} + 44 q^{71} + 18 q^{72} - 36 q^{73} + 18 q^{74} + 46 q^{75} + 14 q^{76} + 68 q^{77} + 34 q^{79} - 2 q^{80} - 6 q^{81} + 6 q^{82} + 2 q^{83} - 18 q^{84} - 30 q^{85} + 28 q^{86} + 52 q^{87} + 10 q^{88} + 42 q^{89} + 8 q^{90} + 12 q^{92} - 12 q^{93} - 20 q^{94} - 2 q^{95} + 4 q^{96} + 40 q^{97} + 28 q^{98} + 12 q^{99}+O(q^{100}) \) 14 * q + 14 * q^2 + 4 * q^3 + 14 * q^4 - 2 * q^5 + 4 * q^6 + 2 * q^7 + 14 * q^8 + 18 * q^9 - 2 * q^10 + 10 * q^11 + 4 * q^12 + 2 * q^14 - 4 * q^15 + 14 * q^16 + 2 * q^17 + 18 * q^18 + 14 * q^19 - 2 * q^20 - 18 * q^21 + 10 * q^22 + 12 * q^23 + 4 * q^24 + 44 * q^25 + 10 * q^27 + 2 * q^28 + 4 * q^29 - 4 * q^30 + 4 * q^31 + 14 * q^32 + 12 * q^33 + 2 * q^34 + 14 * q^35 + 18 * q^36 + 18 * q^37 + 14 * q^38 - 2 * q^40 + 6 * q^41 - 18 * q^42 + 28 * q^43 + 10 * q^44 + 8 * q^45 + 12 * q^46 - 20 * q^47 + 4 * q^48 + 28 * q^49 + 44 * q^50 + 10 * q^51 + 12 * q^53 + 10 * q^54 - 2 * q^55 + 2 * q^56 + 4 * q^57 + 4 * q^58 + 16 * q^59 - 4 * q^60 + 30 * q^61 + 4 * q^62 + 28 * q^63 + 14 * q^64 + 12 * q^66 + 2 * q^67 + 2 * q^68 - 42 * q^69 + 14 * q^70 + 44 * q^71 + 18 * q^72 - 36 * q^73 + 18 * q^74 + 46 * q^75 + 14 * q^76 + 68 * q^77 + 34 * q^79 - 2 * q^80 - 6 * q^81 + 6 * q^82 + 2 * q^83 - 18 * q^84 - 30 * q^85 + 28 * q^86 + 52 * q^87 + 10 * q^88 + 42 * q^89 + 8 * q^90 + 12 * q^92 - 12 * q^93 - 20 * q^94 - 2 * q^95 + 4 * q^96 + 40 * q^97 + 28 * q^98 + 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

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