LMFDB - Embedded newform 8470.2.a.co.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8470,2,Mod(1,8470)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8470, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8470.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8470 = 2 \cdot 5 \cdot 7 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8470.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:	$67.6332905120$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.4400.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 7x^{2} + 11 $ x^4 - 7*x^2 + 11
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 770)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.54336$ of defining polynomial
Character	$\chi$	$=$	8470.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +0.543362 q^{3} +1.00000 q^{4} +1.00000 q^{5} -0.543362 q^{6} -1.00000 q^{7} -1.00000 q^{8} -2.70476 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +0.543362 q^{3} +1.00000 q^{4} +1.00000 q^{5} -0.543362 q^{6} -1.00000 q^{7} -1.00000 q^{8} -2.70476 q^{9} -1.00000 q^{10} +0.543362 q^{12} -1.65861 q^{13} +1.00000 q^{14} +0.543362 q^{15} +1.00000 q^{16} +2.04615 q^{17} +2.70476 q^{18} +3.70476 q^{19} +1.00000 q^{20} -0.543362 q^{21} -4.99442 q^{23} -0.543362 q^{24} +1.00000 q^{25} +1.65861 q^{26} -3.09975 q^{27} -1.00000 q^{28} +7.13632 q^{29} -0.543362 q^{30} -10.1809 q^{31} -1.00000 q^{32} -2.04615 q^{34} -1.00000 q^{35} -2.70476 q^{36} +9.13632 q^{37} -3.70476 q^{38} -0.901224 q^{39} -1.00000 q^{40} +5.04615 q^{41} +0.543362 q^{42} +5.78688 q^{43} -2.70476 q^{45} +4.99442 q^{46} -4.56444 q^{47} +0.543362 q^{48} +1.00000 q^{49} -1.00000 q^{50} +1.11180 q^{51} -1.65861 q^{52} +8.88262 q^{53} +3.09975 q^{54} +1.00000 q^{56} +2.01302 q^{57} -7.13632 q^{58} -14.0846 q^{59} +0.543362 q^{60} -3.30516 q^{61} +10.1809 q^{62} +2.70476 q^{63} +1.00000 q^{64} -1.65861 q^{65} -3.18992 q^{67} +2.04615 q^{68} -2.71378 q^{69} +1.00000 q^{70} -7.80795 q^{71} +2.70476 q^{72} -10.9046 q^{73} -9.13632 q^{74} +0.543362 q^{75} +3.70476 q^{76} +0.901224 q^{78} +11.2054 q^{79} +1.00000 q^{80} +6.42999 q^{81} -5.04615 q^{82} +4.07812 q^{83} -0.543362 q^{84} +2.04615 q^{85} -5.78688 q^{86} +3.87760 q^{87} -3.32679 q^{89} +2.70476 q^{90} +1.65861 q^{91} -4.99442 q^{92} -5.53191 q^{93} +4.56444 q^{94} +3.70476 q^{95} -0.543362 q^{96} +9.19639 q^{97} -1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{2} - 4 q^{3} + 4 q^{4} + 4 q^{5} + 4 q^{6} - 4 q^{7} - 4 q^{8} + 6 q^{9}+O(q^{10}) $ 4 * q - 4 * q^2 - 4 * q^3 + 4 * q^4 + 4 * q^5 + 4 * q^6 - 4 * q^7 - 4 * q^8 + 6 * q^9	$ 4 q - 4 q^{2} - 4 q^{3} + 4 q^{4} + 4 q^{5} + 4 q^{6} - 4 q^{7} - 4 q^{8} + 6 q^{9} - 4 q^{10} - 4 q^{12} + 14 q^{13} + 4 q^{14} - 4 q^{15} + 4 q^{16} + 12 q^{17} - 6 q^{18} - 2 q^{19} + 4 q^{20} + 4 q^{21} + 4 q^{24} + 4 q^{25} - 14 q^{26} - 22 q^{27} - 4 q^{28} + 10 q^{29} + 4 q^{30} - 18 q^{31} - 4 q^{32} - 12 q^{34} - 4 q^{35} + 6 q^{36} + 18 q^{37} + 2 q^{38} - 30 q^{39} - 4 q^{40} + 24 q^{41} - 4 q^{42} + 10 q^{43} + 6 q^{45} - 8 q^{47} - 4 q^{48} + 4 q^{49} - 4 q^{50} + 14 q^{52} + 20 q^{53} + 22 q^{54} + 4 q^{56} + 30 q^{57} - 10 q^{58} - 14 q^{59} - 4 q^{60} + 14 q^{61} + 18 q^{62} - 6 q^{63} + 4 q^{64} + 14 q^{65} + 12 q^{68} - 4 q^{69} + 4 q^{70} - 14 q^{71} - 6 q^{72} + 30 q^{73} - 18 q^{74} - 4 q^{75} - 2 q^{76} + 30 q^{78} + 8 q^{79} + 4 q^{80} + 16 q^{81} - 24 q^{82} + 8 q^{83} + 4 q^{84} + 12 q^{85} - 10 q^{86} + 2 q^{87} - 4 q^{89} - 6 q^{90} - 14 q^{91} - 2 q^{93} + 8 q^{94} - 2 q^{95} + 4 q^{96} - 10 q^{97} - 4 q^{98}+O(q^{100}) $ 4 * q - 4 * q^2 - 4 * q^3 + 4 * q^4 + 4 * q^5 + 4 * q^6 - 4 * q^7 - 4 * q^8 + 6 * q^9 - 4 * q^10 - 4 * q^12 + 14 * q^13 + 4 * q^14 - 4 * q^15 + 4 * q^16 + 12 * q^17 - 6 * q^18 - 2 * q^19 + 4 * q^20 + 4 * q^21 + 4 * q^24 + 4 * q^25 - 14 * q^26 - 22 * q^27 - 4 * q^28 + 10 * q^29 + 4 * q^30 - 18 * q^31 - 4 * q^32 - 12 * q^34 - 4 * q^35 + 6 * q^36 + 18 * q^37 + 2 * q^38 - 30 * q^39 - 4 * q^40 + 24 * q^41 - 4 * q^42 + 10 * q^43 + 6 * q^45 - 8 * q^47 - 4 * q^48 + 4 * q^49 - 4 * q^50 + 14 * q^52 + 20 * q^53 + 22 * q^54 + 4 * q^56 + 30 * q^57 - 10 * q^58 - 14 * q^59 - 4 * q^60 + 14 * q^61 + 18 * q^62 - 6 * q^63 + 4 * q^64 + 14 * q^65 + 12 * q^68 - 4 * q^69 + 4 * q^70 - 14 * q^71 - 6 * q^72 + 30 * q^73 - 18 * q^74 - 4 * q^75 - 2 * q^76 + 30 * q^78 + 8 * q^79 + 4 * q^80 + 16 * q^81 - 24 * q^82 + 8 * q^83 + 4 * q^84 + 12 * q^85 - 10 * q^86 + 2 * q^87 - 4 * q^89 - 6 * q^90 - 14 * q^91 - 2 * q^93 + 8 * q^94 - 2 * q^95 + 4 * q^96 - 10 * q^97 - 4 * q^98

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
$3$	0.543362	0.313710	0.156855	−	0.987622i	$-0.449864\pi$
$3$	0.543362	0.313710	0.156855	+	0.987622i	$0.449864\pi$
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	−0.543362	−0.221827
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−1.00000	−0.353553
$9$	−2.70476	−0.901586
$10$	−1.00000	−0.316228
$11$	0	0
$11$	0	0
$12$	0.543362	0.156855
$13$	−1.65861	−0.460015	−0.230008	−	0.973189i	$-0.573875\pi$
$13$	−1.65861	−0.460015	−0.230008	+	0.973189i	$0.573875\pi$
$14$	1.00000	0.267261
$15$	0.543362	0.140295
$16$	1.00000	0.250000
$17$	2.04615	0.496264	0.248132	−	0.968726i	$-0.420183\pi$
$17$	2.04615	0.496264	0.248132	+	0.968726i	$0.420183\pi$
$18$	2.70476	0.637518
$19$	3.70476	0.849930	0.424965	−	0.905210i	$-0.360286\pi$
$19$	3.70476	0.849930	0.424965	+	0.905210i	$0.360286\pi$
$20$	1.00000	0.223607
$21$	−0.543362	−0.118571
$22$	0	0
$23$	−4.99442	−1.04141	−0.520705	−	0.853737i	$-0.674331\pi$
$23$	−4.99442	−1.04141	−0.520705	+	0.853737i	$0.674331\pi$
$24$	−0.543362	−0.110913
$25$	1.00000	0.200000
$26$	1.65861	0.325280
$27$	−3.09975	−0.596547
$28$	−1.00000	−0.188982
$29$	7.13632	1.32518	0.662591	−	0.748982i	$-0.269457\pi$
$29$	7.13632	1.32518	0.662591	+	0.748982i	$0.269457\pi$
$30$	−0.543362	−0.0992039
$31$	−10.1809	−1.82854	−0.914271	−	0.405102i	$-0.867236\pi$
$31$	−10.1809	−1.82854	−0.914271	+	0.405102i	$0.867236\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−2.04615	−0.350912
$35$	−1.00000	−0.169031
$36$	−2.70476	−0.450793
$37$	9.13632	1.50200	0.751001	−	0.660301i	$-0.229571\pi$
$37$	9.13632	1.50200	0.751001	+	0.660301i	$0.229571\pi$
$38$	−3.70476	−0.600991
$39$	−0.901224	−0.144311
$40$	−1.00000	−0.158114
$41$	5.04615	0.788076	0.394038	−	0.919094i	$-0.371078\pi$
$41$	5.04615	0.788076	0.394038	+	0.919094i	$0.371078\pi$
$42$	0.543362	0.0838426
$43$	5.78688	0.882491	0.441245	−	0.897387i	$-0.354537\pi$
$43$	5.78688	0.882491	0.441245	+	0.897387i	$0.354537\pi$
$44$	0	0
$45$	−2.70476	−0.403201
$46$	4.99442	0.736388
$47$	−4.56444	−0.665791	−0.332896	−	0.942964i	$-0.608026\pi$
$47$	−4.56444	−0.665791	−0.332896	+	0.942964i	$0.608026\pi$
$48$	0.543362	0.0784275
$49$	1.00000	0.142857
$50$	−1.00000	−0.141421
$51$	1.11180	0.155683
$52$	−1.65861	−0.230008
$53$	8.88262	1.22012	0.610061	−	0.792354i	$-0.291145\pi$
$53$	8.88262	1.22012	0.610061	+	0.792354i	$0.291145\pi$
$54$	3.09975	0.421822
$55$	0	0
$56$	1.00000	0.133631
$57$	2.01302	0.266632
$58$	−7.13632	−0.937045
$59$	−14.0846	−1.83366	−0.916829	−	0.399280i	$-0.869260\pi$
$59$	−14.0846	−1.83366	−0.916829	+	0.399280i	$0.869260\pi$
$60$	0.543362	0.0701477
$61$	−3.30516	−0.423183	−0.211591	−	0.977358i	$-0.567865\pi$
$61$	−3.30516	−0.423183	−0.211591	+	0.977358i	$0.567865\pi$
$62$	10.1809	1.29298
$63$	2.70476	0.340767
$64$	1.00000	0.125000
$65$	−1.65861	−0.205725
$66$	0	0
$67$	−3.18992	−0.389711	−0.194855	−	0.980832i	$-0.562424\pi$
$67$	−3.18992	−0.389711	−0.194855	+	0.980832i	$0.562424\pi$
$68$	2.04615	0.248132
$69$	−2.71378	−0.326701
$70$	1.00000	0.119523
$71$	−7.80795	−0.926633	−0.463317	−	0.886193i	$-0.653341\pi$
$71$	−7.80795	−0.926633	−0.463317	+	0.886193i	$0.653341\pi$
$72$	2.70476	0.318759
$73$	−10.9046	−1.27629	−0.638143	−	0.769918i	$-0.720297\pi$
$73$	−10.9046	−1.27629	−0.638143	+	0.769918i	$0.720297\pi$
$74$	−9.13632	−1.06208
$75$	0.543362	0.0627420
$76$	3.70476	0.424965
$77$	0	0
$78$	0.901224	0.102044
$79$	11.2054	1.26071	0.630354	−	0.776308i	$-0.282910\pi$
$79$	11.2054	1.26071	0.630354	+	0.776308i	$0.282910\pi$
$80$	1.00000	0.111803
$81$	6.42999	0.714443
$82$	−5.04615	−0.557254
$83$	4.07812	0.447632	0.223816	−	0.974631i	$-0.428149\pi$
$83$	4.07812	0.447632	0.223816	+	0.974631i	$0.428149\pi$
$84$	−0.543362	−0.0592856
$85$	2.04615	0.221936
$86$	−5.78688	−0.624015
$87$	3.87760	0.415723
$88$	0	0
$89$	−3.32679	−0.352639	−0.176320	−	0.984333i	$-0.556419\pi$
$89$	−3.32679	−0.352639	−0.176320	+	0.984333i	$0.556419\pi$
$90$	2.70476	0.285107
$91$	1.65861	0.173869
$92$	−4.99442	−0.520705
$93$	−5.53191	−0.573632
$94$	4.56444	0.470786
$95$	3.70476	0.380100
$96$	−0.543362	−0.0554566
$97$	9.19639	0.933752	0.466876	−	0.884323i	$-0.345379\pi$
$97$	9.19639	0.933752	0.466876	+	0.884323i	$0.345379\pi$
$98$	−1.00000	−0.101015
$99$	0	0
$100$	1.00000	0.100000
$101$	2.73328	0.271972	0.135986	−	0.990711i	$-0.456580\pi$
$101$	2.73328	0.271972	0.135986	+	0.990711i	$0.456580\pi$
$102$	−1.11180	−0.110085
$103$	−1.54736	−0.152466	−0.0762332	−	0.997090i	$-0.524289\pi$
$103$	−1.54736	−0.152466	−0.0762332	+	0.997090i	$0.524289\pi$
$104$	1.65861	0.162640
$105$	−0.543362	−0.0530267
$106$	−8.88262	−0.862757
$107$	12.2089	1.18028	0.590138	−	0.807303i	$-0.299073\pi$
$107$	12.2089	1.18028	0.590138	+	0.807303i	$0.299073\pi$
$108$	−3.09975	−0.298273
$109$	−1.32376	−0.126794	−0.0633968	−	0.997988i	$-0.520193\pi$
$109$	−1.32376	−0.126794	−0.0633968	+	0.997988i	$0.520193\pi$
$110$	0	0
$111$	4.96433	0.471193
$112$	−1.00000	−0.0944911
$113$	3.45851	0.325349	0.162675	−	0.986680i	$-0.447988\pi$
$113$	3.45851	0.325349	0.162675	+	0.986680i	$0.447988\pi$
$114$	−2.01302	−0.188537
$115$	−4.99442	−0.465732
$116$	7.13632	0.662591
$117$	4.48613	0.414743
$118$	14.0846	1.29659
$119$	−2.04615	−0.187570
$120$	−0.543362	−0.0496019
$121$	0	0
$122$	3.30516	0.299236
$123$	2.74189	0.247228
$124$	−10.1809	−0.914271
$125$	1.00000	0.0894427
$126$	−2.70476	−0.240959
$127$	−18.2239	−1.61711	−0.808557	−	0.588418i	$-0.799751\pi$
$127$	−18.2239	−1.61711	−0.808557	+	0.588418i	$0.799751\pi$
$128$	−1.00000	−0.0883883
$129$	3.14437	0.276846
$130$	1.65861	0.145470
$131$	−9.47001	−0.827398	−0.413699	−	0.910414i	$-0.635763\pi$
$131$	−9.47001	−0.827398	−0.413699	+	0.910414i	$0.635763\pi$
$132$	0	0
$133$	−3.70476	−0.321243
$134$	3.18992	0.275567
$135$	−3.09975	−0.266784
$136$	−2.04615	−0.175456
$137$	11.4807	0.980866	0.490433	−	0.871479i	$-0.336839\pi$
$137$	11.4807	0.980866	0.490433	+	0.871479i	$0.336839\pi$
$138$	2.71378	0.231012
$139$	−4.86312	−0.412485	−0.206242	−	0.978501i	$-0.566124\pi$
$139$	−4.86312	−0.412485	−0.206242	+	0.978501i	$0.566124\pi$
$140$	−1.00000	−0.0845154
$141$	−2.48014	−0.208866
$142$	7.80795	0.655229
$143$	0	0
$144$	−2.70476	−0.225396
$145$	7.13632	0.592639
$146$	10.9046	0.902471
$147$	0.543362	0.0448157
$148$	9.13632	0.751001
$149$	7.60901	0.623355	0.311677	−	0.950188i	$-0.399109\pi$
$149$	7.60901	0.623355	0.311677	+	0.950188i	$0.399109\pi$
$150$	−0.543362	−0.0443653
$151$	10.8640	0.884102	0.442051	−	0.896990i	$-0.354251\pi$
$151$	10.8640	0.884102	0.442051	+	0.896990i	$0.354251\pi$
$152$	−3.70476	−0.300496
$153$	−5.53434	−0.447425
$154$	0	0
$155$	−10.1809	−0.817749
$156$	−0.901224	−0.0721557
$157$	9.67811	0.772397	0.386199	−	0.922416i	$-0.373788\pi$
$157$	9.67811	0.772397	0.386199	+	0.922416i	$0.373788\pi$
$158$	−11.2054	−0.891455
$159$	4.82648	0.382765
$160$	−1.00000	−0.0790569
$161$	4.99442	0.393616
$162$	−6.42999	−0.505188
$163$	16.3076	1.27731	0.638656	−	0.769492i	$-0.279491\pi$
$163$	16.3076	1.27731	0.638656	+	0.769492i	$0.279491\pi$
$164$	5.04615	0.394038
$165$	0	0
$166$	−4.07812	−0.316523
$167$	−9.49351	−0.734630	−0.367315	−	0.930097i	$-0.619723\pi$
$167$	−9.49351	−0.734630	−0.367315	+	0.930097i	$0.619723\pi$
$168$	0.543362	0.0419213
$169$	−10.2490	−0.788386
$170$	−2.04615	−0.156933
$171$	−10.0205	−0.766285
$172$	5.78688	0.441245
$173$	1.44017	0.109494	0.0547470	−	0.998500i	$-0.482565\pi$
$173$	1.44017	0.109494	0.0547470	+	0.998500i	$0.482565\pi$
$174$	−3.87760	−0.293960
$175$	−1.00000	−0.0755929
$176$	0	0
$177$	−7.65303	−0.575237
$178$	3.32679	0.249354
$179$	−9.52847	−0.712191	−0.356095	−	0.934450i	$-0.615892\pi$
$179$	−9.52847	−0.712191	−0.356095	+	0.934450i	$0.615892\pi$
$180$	−2.70476	−0.201601
$181$	−11.5431	−0.857992	−0.428996	−	0.903306i	$-0.641133\pi$
$181$	−11.5431	−0.857992	−0.428996	+	0.903306i	$0.641133\pi$
$182$	−1.65861	−0.122944
$183$	−1.79590	−0.132757
$184$	4.99442	0.368194
$185$	9.13632	0.671716
$186$	5.53191	0.405619
$187$	0	0
$188$	−4.56444	−0.332896
$189$	3.09975	0.225474
$190$	−3.70476	−0.268771
$191$	25.8516	1.87055	0.935276	−	0.353918i	$-0.115151\pi$
$191$	25.8516	1.87055	0.935276	+	0.353918i	$0.115151\pi$
$192$	0.543362	0.0392138
$193$	25.6289	1.84481	0.922403	−	0.386230i	$-0.126223\pi$
$193$	25.6289	1.84481	0.922403	+	0.386230i	$0.126223\pi$
$194$	−9.19639	−0.660263
$195$	−0.901224	−0.0645380
$196$	1.00000	0.0714286
$197$	10.5223	0.749682	0.374841	−	0.927089i	$-0.377697\pi$
$197$	10.5223	0.749682	0.374841	+	0.927089i	$0.377697\pi$
$198$	0	0
$199$	18.3423	1.30025	0.650125	−	0.759827i	$-0.274716\pi$
$199$	18.3423	1.30025	0.650125	+	0.759827i	$0.274716\pi$
$200$	−1.00000	−0.0707107
$201$	−1.73328	−0.122256
$202$	−2.73328	−0.192313
$203$	−7.13632	−0.500871
$204$	1.11180	0.0778416
$205$	5.04615	0.352438
$206$	1.54736	0.107810
$207$	13.5087	0.938920
$208$	−1.65861	−0.115004
$209$	0	0
$210$	0.543362	0.0374955
$211$	−23.6304	−1.62679	−0.813393	−	0.581715i	$-0.802382\pi$
$211$	−23.6304	−1.62679	−0.813393	+	0.581715i	$0.802382\pi$
$212$	8.88262	0.610061
$213$	−4.24254	−0.290694
$214$	−12.2089	−0.834581
$215$	5.78688	0.394662
$216$	3.09975	0.210911
$217$	10.1809	0.691124
$218$	1.32376	0.0896566
$219$	−5.92514	−0.400384
$220$	0	0
$221$	−3.39376	−0.228289
$222$	−4.96433	−0.333184
$223$	2.76854	0.185395	0.0926974	−	0.995694i	$-0.470451\pi$
$223$	2.76854	0.185395	0.0926974	+	0.995694i	$0.470451\pi$
$224$	1.00000	0.0668153
$225$	−2.70476	−0.180317
$226$	−3.45851	−0.230057
$227$	1.59296	0.105728	0.0528642	−	0.998602i	$-0.483165\pi$
$227$	1.59296	0.105728	0.0528642	+	0.998602i	$0.483165\pi$
$228$	2.01302	0.133316
$229$	23.5747	1.55786	0.778929	−	0.627113i	$-0.215763\pi$
$229$	23.5747	1.55786	0.778929	+	0.627113i	$0.215763\pi$
$230$	4.99442	0.329323
$231$	0	0
$232$	−7.13632	−0.468522
$233$	3.72239	0.243862	0.121931	−	0.992539i	$-0.461091\pi$
$233$	3.72239	0.243862	0.121931	+	0.992539i	$0.461091\pi$
$234$	−4.48613	−0.293268
$235$	−4.56444	−0.297751
$236$	−14.0846	−0.916829
$237$	6.08860	0.395497
$238$	2.04615	0.132632
$239$	15.0088	0.970836	0.485418	−	0.874282i	$-0.338667\pi$
$239$	15.0088	0.970836	0.485418	+	0.874282i	$0.338667\pi$
$240$	0.543362	0.0350739
$241$	23.4709	1.51189	0.755947	−	0.654633i	$-0.227177\pi$
$241$	23.4709	1.51189	0.755947	+	0.654633i	$0.227177\pi$
$242$	0	0
$243$	12.7931	0.820675
$244$	−3.30516	−0.211591
$245$	1.00000	0.0638877
$246$	−2.74189	−0.174816
$247$	−6.14474	−0.390980
$248$	10.1809	0.646488
$249$	2.21589	0.140427
$250$	−1.00000	−0.0632456
$251$	23.8705	1.50669	0.753346	−	0.657624i	$-0.228438\pi$
$251$	23.8705	1.50669	0.753346	+	0.657624i	$0.228438\pi$
$252$	2.70476	0.170384
$253$	0	0
$254$	18.2239	1.14347
$255$	1.11180	0.0696236
$256$	1.00000	0.0625000
$257$	3.13935	0.195827	0.0979136	−	0.995195i	$-0.468783\pi$
$257$	3.13935	0.195827	0.0979136	+	0.995195i	$0.468783\pi$
$258$	−3.14437	−0.195760
$259$	−9.13632	−0.567703
$260$	−1.65861	−0.102862
$261$	−19.3020	−1.19476
$262$	9.47001	0.585059
$263$	18.0997	1.11608	0.558039	−	0.829815i	$-0.311554\pi$
$263$	18.0997	1.11608	0.558039	+	0.829815i	$0.311554\pi$
$264$	0	0
$265$	8.88262	0.545655
$266$	3.70476	0.227153
$267$	−1.80765	−0.110627
$268$	−3.18992	−0.194855
$269$	−5.65771	−0.344957	−0.172478	−	0.985013i	$-0.555177\pi$
$269$	−5.65771	−0.344957	−0.172478	+	0.985013i	$0.555177\pi$
$270$	3.09975	0.188645
$271$	31.7788	1.93042	0.965211	−	0.261472i	$-0.0842080\pi$
$271$	31.7788	1.93042	0.965211	+	0.261472i	$0.0842080\pi$
$272$	2.04615	0.124066
$273$	0.901224	0.0545446
$274$	−11.4807	−0.693577
$275$	0	0
$276$	−2.71378	−0.163350
$277$	−2.40678	−0.144610	−0.0723048	−	0.997383i	$-0.523035\pi$
$277$	−2.40678	−0.144610	−0.0723048	+	0.997383i	$0.523035\pi$
$278$	4.86312	0.291671
$279$	27.5369	1.64859
$280$	1.00000	0.0597614
$281$	11.2228	0.669495	0.334748	−	0.942308i	$-0.391349\pi$
$281$	11.2228	0.669495	0.334748	+	0.942308i	$0.391349\pi$
$282$	2.48014	0.147690
$283$	31.2559	1.85797	0.928986	−	0.370116i	$-0.120682\pi$
$283$	31.2559	1.85797	0.928986	+	0.370116i	$0.120682\pi$
$284$	−7.80795	−0.463317
$285$	2.01302	0.119241
$286$	0	0
$287$	−5.04615	−0.297865
$288$	2.70476	0.159379
$289$	−12.8133	−0.753722
$290$	−7.13632	−0.419059
$291$	4.99697	0.292928
$292$	−10.9046	−0.638143
$293$	2.27230	0.132749	0.0663745	−	0.997795i	$-0.478857\pi$
$293$	2.27230	0.132749	0.0663745	+	0.997795i	$0.478857\pi$
$294$	−0.543362	−0.0316895
$295$	−14.0846	−0.820037
$296$	−9.13632	−0.531038
$297$	0	0
$298$	−7.60901	−0.440778
$299$	8.28379	0.479064
$300$	0.543362	0.0313710
$301$	−5.78688	−0.333550
$302$	−10.8640	−0.625154
$303$	1.48516	0.0853202
$304$	3.70476	0.212482
$305$	−3.30516	−0.189253
$306$	5.53434	0.316377
$307$	5.55583	0.317088	0.158544	−	0.987352i	$-0.449320\pi$
$307$	5.55583	0.317088	0.158544	+	0.987352i	$0.449320\pi$
$308$	0	0
$309$	−0.840779	−0.0478302
$310$	10.1809	0.578236
$311$	−20.3446	−1.15364	−0.576818	−	0.816873i	$-0.695706\pi$
$311$	−20.3446	−1.15364	−0.576818	+	0.816873i	$0.695706\pi$
$312$	0.901224	0.0510218
$313$	28.0901	1.58775	0.793873	−	0.608083i	$-0.208061\pi$
$313$	28.0901	1.58775	0.793873	+	0.608083i	$0.208061\pi$
$314$	−9.67811	−0.546167
$315$	2.70476	0.152396
$316$	11.2054	0.630354
$317$	7.61646	0.427783	0.213892	−	0.976857i	$-0.431386\pi$
$317$	7.61646	0.427783	0.213892	+	0.976857i	$0.431386\pi$
$318$	−4.82648	−0.270656
$319$	0	0
$320$	1.00000	0.0559017
$321$	6.63383	0.370264
$322$	−4.99442	−0.278328
$323$	7.58049	0.421790
$324$	6.42999	0.357222
$325$	−1.65861	−0.0920030
$326$	−16.3076	−0.903197
$327$	−0.719283	−0.0397764
$328$	−5.04615	−0.278627
$329$	4.56444	0.251645
$330$	0	0
$331$	9.86518	0.542239	0.271120	−	0.962546i	$-0.412606\pi$
$331$	9.86518	0.542239	0.271120	+	0.962546i	$0.412606\pi$
$332$	4.07812	0.223816
$333$	−24.7115	−1.35418
$334$	9.49351	0.519462
$335$	−3.18992	−0.174284
$336$	−0.543362	−0.0296428
$337$	−22.8921	−1.24701	−0.623507	−	0.781818i	$-0.714293\pi$
$337$	−22.8921	−1.24701	−0.623507	+	0.781818i	$0.714293\pi$
$338$	10.2490	0.557473
$339$	1.87922	0.102065
$340$	2.04615	0.110968
$341$	0	0
$342$	10.0205	0.541845
$343$	−1.00000	−0.0539949
$344$	−5.78688	−0.312008
$345$	−2.71378	−0.146105
$346$	−1.44017	−0.0774239
$347$	−8.07715	−0.433604	−0.216802	−	0.976216i	$-0.569563\pi$
$347$	−8.07715	−0.433604	−0.216802	+	0.976216i	$0.569563\pi$
$348$	3.87760	0.207861
$349$	16.3618	0.875827	0.437913	−	0.899017i	$-0.355718\pi$
$349$	16.3618	0.875827	0.437913	+	0.899017i	$0.355718\pi$
$350$	1.00000	0.0534522
$351$	5.14127	0.274421
$352$	0	0
$353$	−18.5642	−0.988072	−0.494036	−	0.869442i	$-0.664479\pi$
$353$	−18.5642	−0.988072	−0.494036	+	0.869442i	$0.664479\pi$
$354$	7.65303	0.406754
$355$	−7.80795	−0.414403
$356$	−3.32679	−0.176320
$357$	−1.11180	−0.0588427
$358$	9.52847	0.503595
$359$	4.59048	0.242276	0.121138	−	0.992636i	$-0.461346\pi$
$359$	4.59048	0.242276	0.121138	+	0.992636i	$0.461346\pi$
$360$	2.70476	0.142553
$361$	−5.27477	−0.277619
$362$	11.5431	0.606692
$363$	0	0
$364$	1.65861	0.0869347
$365$	−10.9046	−0.570773
$366$	1.79590	0.0938732
$367$	12.7694	0.666559	0.333280	−	0.942828i	$-0.391845\pi$
$367$	12.7694	0.666559	0.333280	+	0.942828i	$0.391845\pi$
$368$	−4.99442	−0.260352
$369$	−13.6486	−0.710518
$370$	−9.13632	−0.474975
$371$	−8.88262	−0.461163
$372$	−5.53191	−0.286816
$373$	−22.4340	−1.16159	−0.580795	−	0.814050i	$-0.697258\pi$
$373$	−22.4340	−1.16159	−0.580795	+	0.814050i	$0.697258\pi$
$374$	0	0
$375$	0.543362	0.0280591
$376$	4.56444	0.235393
$377$	−11.8364	−0.609603
$378$	−3.09975	−0.159434
$379$	−30.1347	−1.54791	−0.773957	−	0.633238i	$-0.781725\pi$
$379$	−30.1347	−1.54791	−0.773957	+	0.633238i	$0.781725\pi$
$380$	3.70476	0.190050
$381$	−9.90220	−0.507305
$382$	−25.8516	−1.32268
$383$	27.9554	1.42845	0.714226	−	0.699915i	$-0.246779\pi$
$383$	27.9554	1.42845	0.714226	+	0.699915i	$0.246779\pi$
$384$	−0.543362	−0.0277283
$385$	0	0
$386$	−25.6289	−1.30447
$387$	−15.6521	−0.795641
$388$	9.19639	0.466876
$389$	4.57649	0.232037	0.116019	−	0.993247i	$-0.462987\pi$
$389$	4.57649	0.232037	0.116019	+	0.993247i	$0.462987\pi$
$390$	0.901224	0.0456353
$391$	−10.2193	−0.516814
$392$	−1.00000	−0.0505076
$393$	−5.14564	−0.259563
$394$	−10.5223	−0.530105
$395$	11.2054	0.563806
$396$	0	0
$397$	−37.9390	−1.90411	−0.952053	−	0.305933i	$-0.901032\pi$
$397$	−37.9390	−1.90411	−0.952053	+	0.305933i	$0.901032\pi$
$398$	−18.3423	−0.919416
$399$	−2.01302	−0.100777
$400$	1.00000	0.0500000
$401$	15.9557	0.796791	0.398395	−	0.917214i	$-0.369567\pi$
$401$	15.9557	0.796791	0.398395	+	0.917214i	$0.369567\pi$
$402$	1.73328	0.0864482
$403$	16.8861	0.841157
$404$	2.73328	0.135986
$405$	6.42999	0.319509
$406$	7.13632	0.354170
$407$	0	0
$408$	−1.11180	−0.0550423
$409$	−24.2684	−1.19999	−0.599997	−	0.800002i	$-0.704832\pi$
$409$	−24.2684	−1.19999	−0.599997	+	0.800002i	$0.704832\pi$
$410$	−5.04615	−0.249212
$411$	6.23820	0.307708
$412$	−1.54736	−0.0762332
$413$	14.0846	0.693058
$414$	−13.5087	−0.663917
$415$	4.07812	0.200187
$416$	1.65861	0.0813199
$417$	−2.64244	−0.129401
$418$	0	0
$419$	−25.6838	−1.25473	−0.627367	−	0.778724i	$-0.715867\pi$
$419$	−25.6838	−1.25473	−0.627367	+	0.778724i	$0.715867\pi$
$420$	−0.543362	−0.0265133
$421$	−9.36362	−0.456355	−0.228178	−	0.973620i	$-0.573277\pi$
$421$	−9.36362	−0.456355	−0.228178	+	0.973620i	$0.573277\pi$
$422$	23.6304	1.15031
$423$	12.3457	0.600268
$424$	−8.88262	−0.431378
$425$	2.04615	0.0992528
$426$	4.24254	0.205552
$427$	3.30516	0.159948
$428$	12.2089	0.590138
$429$	0	0
$430$	−5.78688	−0.279068
$431$	16.9902	0.818387	0.409194	−	0.912448i	$-0.365810\pi$
$431$	16.9902	0.818387	0.409194	+	0.912448i	$0.365810\pi$
$432$	−3.09975	−0.149137
$433$	6.70663	0.322300	0.161150	−	0.986930i	$-0.448480\pi$
$433$	6.70663	0.322300	0.161150	+	0.986930i	$0.448480\pi$
$434$	−10.1809	−0.488699
$435$	3.87760	0.185917
$436$	−1.32376	−0.0633968
$437$	−18.5031	−0.885125
$438$	5.92514	0.283114
$439$	12.8474	0.613171	0.306586	−	0.951843i	$-0.400813\pi$
$439$	12.8474	0.613171	0.306586	+	0.951843i	$0.400813\pi$
$440$	0	0
$441$	−2.70476	−0.128798
$442$	3.39376	0.161425
$443$	0.678365	0.0322301	0.0161151	−	0.999870i	$-0.494870\pi$
$443$	0.678365	0.0322301	0.0161151	+	0.999870i	$0.494870\pi$
$444$	4.96433	0.235597
$445$	−3.32679	−0.157705
$446$	−2.76854	−0.131094
$447$	4.13445	0.195553
$448$	−1.00000	−0.0472456
$449$	7.59525	0.358442	0.179221	−	0.983809i	$-0.442642\pi$
$449$	7.59525	0.358442	0.179221	+	0.983809i	$0.442642\pi$
$450$	2.70476	0.127504
$451$	0	0
$452$	3.45851	0.162675
$453$	5.90310	0.277352
$454$	−1.59296	−0.0747612
$455$	1.65861	0.0777567
$456$	−2.01302	−0.0942685
$457$	−33.0502	−1.54602	−0.773012	−	0.634391i	$-0.781251\pi$
$457$	−33.0502	−1.54602	−0.773012	+	0.634391i	$0.781251\pi$
$458$	−23.5747	−1.10157
$459$	−6.34255	−0.296045
$460$	−4.99442	−0.232866
$461$	20.2500	0.943136	0.471568	−	0.881830i	$-0.343688\pi$
$461$	20.2500	0.943136	0.471568	+	0.881830i	$0.343688\pi$
$462$	0	0
$463$	39.3213	1.82742	0.913708	−	0.406371i	$-0.133206\pi$
$463$	39.3213	1.82742	0.913708	+	0.406371i	$0.133206\pi$
$464$	7.13632	0.331295
$465$	−5.53191	−0.256536
$466$	−3.72239	−0.172436
$467$	−18.8167	−0.870734	−0.435367	−	0.900253i	$-0.643381\pi$
$467$	−18.8167	−0.870734	−0.435367	+	0.900253i	$0.643381\pi$
$468$	4.48613	0.207372
$469$	3.18992	0.147297
$470$	4.56444	0.210542
$471$	5.25872	0.242309
$472$	14.0846	0.648296
$473$	0	0
$474$	−6.08860	−0.279658
$475$	3.70476	0.169986
$476$	−2.04615	−0.0937851
$477$	−24.0253	−1.10005
$478$	−15.0088	−0.686485
$479$	−7.56631	−0.345713	−0.172857	−	0.984947i	$-0.555300\pi$
$479$	−7.56631	−0.345713	−0.172857	+	0.984947i	$0.555300\pi$
$480$	−0.543362	−0.0248010
$481$	−15.1536	−0.690943
$482$	−23.4709	−1.06907
$483$	2.71378	0.123481
$484$	0	0
$485$	9.19639	0.417587
$486$	−12.7931	−0.580305
$487$	7.92350	0.359048	0.179524	−	0.983754i	$-0.442544\pi$
$487$	7.92350	0.359048	0.179524	+	0.983754i	$0.442544\pi$
$488$	3.30516	0.149618
$489$	8.86095	0.400706
$490$	−1.00000	−0.0451754
$491$	−15.1101	−0.681909	−0.340954	−	0.940080i	$-0.610750\pi$
$491$	−15.1101	−0.681909	−0.340954	+	0.940080i	$0.610750\pi$
$492$	2.74189	0.123614
$493$	14.6020	0.657640
$494$	6.14474	0.276465
$495$	0	0
$496$	−10.1809	−0.457136
$497$	7.80795	0.350235
$498$	−2.21589	−0.0992966
$499$	−5.96549	−0.267052	−0.133526	−	0.991045i	$-0.542630\pi$
$499$	−5.96549	−0.267052	−0.133526	+	0.991045i	$0.542630\pi$
$500$	1.00000	0.0447214
$501$	−5.15841	−0.230461
$502$	−23.8705	−1.06539
$503$	−25.5217	−1.13795	−0.568977	−	0.822353i	$-0.692661\pi$
$503$	−25.5217	−1.13795	−0.568977	+	0.822353i	$0.692661\pi$
$504$	−2.70476	−0.120479
$505$	2.73328	0.121629
$506$	0	0
$507$	−5.56893	−0.247325
$508$	−18.2239	−0.808557
$509$	−27.7639	−1.23061	−0.615306	−	0.788289i	$-0.710967\pi$
$509$	−27.7639	−1.23061	−0.615306	+	0.788289i	$0.710967\pi$
$510$	−1.11180	−0.0492313
$511$	10.9046	0.482391
$512$	−1.00000	−0.0441942
$513$	−11.4838	−0.507023
$514$	−3.13935	−0.138471
$515$	−1.54736	−0.0681850
$516$	3.14437	0.138423
$517$	0	0
$518$	9.13632	0.401427
$519$	0.782532	0.0343494
$520$	1.65861	0.0727348
$521$	23.8699	1.04576	0.522880	−	0.852407i	$-0.324858\pi$
$521$	23.8699	1.04576	0.522880	+	0.852407i	$0.324858\pi$
$522$	19.3020	0.844826
$523$	2.38789	0.104415	0.0522075	−	0.998636i	$-0.483374\pi$
$523$	2.38789	0.104415	0.0522075	+	0.998636i	$0.483374\pi$
$524$	−9.47001	−0.413699
$525$	−0.543362	−0.0237143
$526$	−18.0997	−0.789187
$527$	−20.8316	−0.907440
$528$	0	0
$529$	1.94427	0.0845336
$530$	−8.88262	−0.385837
$531$	38.0954	1.65320
$532$	−3.70476	−0.160622
$533$	−8.36958	−0.362527
$534$	1.80765	0.0782248
$535$	12.2089	0.527835
$536$	3.18992	0.137783
$537$	−5.17741	−0.223422
$538$	5.65771	0.243921
$539$	0	0
$540$	−3.09975	−0.133392
$541$	24.8483	1.06831	0.534155	−	0.845386i	$-0.320630\pi$
$541$	24.8483	1.06831	0.534155	+	0.845386i	$0.320630\pi$
$542$	−31.7788	−1.36501
$543$	−6.27208	−0.269161
$544$	−2.04615	−0.0877280
$545$	−1.32376	−0.0567038
$546$	−0.901224	−0.0385688
$547$	33.1504	1.41741	0.708704	−	0.705506i	$-0.249280\pi$
$547$	33.1504	1.41741	0.708704	+	0.705506i	$0.249280\pi$
$548$	11.4807	0.490433
$549$	8.93967	0.381536
$550$	0	0
$551$	26.4383	1.12631
$552$	2.71378	0.115506
$553$	−11.2054	−0.476503
$554$	2.40678	0.102254
$555$	4.96433	0.210724
$556$	−4.86312	−0.206242
$557$	28.0256	1.18748	0.593741	−	0.804657i	$-0.297651\pi$
$557$	28.0256	1.18748	0.593741	+	0.804657i	$0.297651\pi$
$558$	−27.5369	−1.16573
$559$	−9.59816	−0.405959
$560$	−1.00000	−0.0422577
$561$	0	0
$562$	−11.2228	−0.473405
$563$	3.17711	0.133899	0.0669495	−	0.997756i	$-0.478673\pi$
$563$	3.17711	0.133899	0.0669495	+	0.997756i	$0.478673\pi$
$564$	−2.48014	−0.104433
$565$	3.45851	0.145501
$566$	−31.2559	−1.31378
$567$	−6.42999	−0.270034
$568$	7.80795	0.327614
$569$	18.7562	0.786302	0.393151	−	0.919474i	$-0.371385\pi$
$569$	18.7562	0.786302	0.393151	+	0.919474i	$0.371385\pi$
$570$	−2.01302	−0.0843163
$571$	−20.5329	−0.859276	−0.429638	−	0.903001i	$-0.641359\pi$
$571$	−20.5329	−0.859276	−0.429638	+	0.903001i	$0.641359\pi$
$572$	0	0
$573$	14.0468	0.586811
$574$	5.04615	0.210622
$575$	−4.99442	−0.208282
$576$	−2.70476	−0.112698
$577$	16.7729	0.698264	0.349132	−	0.937074i	$-0.386476\pi$
$577$	16.7729	0.698264	0.349132	+	0.937074i	$0.386476\pi$
$578$	12.8133	0.532962
$579$	13.9257	0.578734
$580$	7.13632	0.296320
$581$	−4.07812	−0.169189
$582$	−4.99697	−0.207131
$583$	0	0
$584$	10.9046	0.451235
$585$	4.48613	0.185479
$586$	−2.27230	−0.0938677
$587$	25.0595	1.03432	0.517158	−	0.855890i	$-0.326990\pi$
$587$	25.0595	1.03432	0.517158	+	0.855890i	$0.326990\pi$
$588$	0.543362	0.0224079
$589$	−37.7178	−1.55413
$590$	14.0846	0.579854
$591$	5.71741	0.235183
$592$	9.13632	0.375500
$593$	44.2586	1.81748	0.908741	−	0.417360i	$-0.137044\pi$
$593$	44.2586	1.81748	0.908741	+	0.417360i	$0.137044\pi$
$594$	0	0
$595$	−2.04615	−0.0838840
$596$	7.60901	0.311677
$597$	9.96650	0.407902
$598$	−8.28379	−0.338749
$599$	2.39133	0.0977072	0.0488536	−	0.998806i	$-0.484443\pi$
$599$	2.39133	0.0977072	0.0488536	+	0.998806i	$0.484443\pi$
$600$	−0.543362	−0.0221827
$601$	−34.6639	−1.41397	−0.706984	−	0.707229i	$-0.749945\pi$
$601$	−34.6639	−1.41397	−0.706984	+	0.707229i	$0.749945\pi$
$602$	5.78688	0.235856
$603$	8.62796	0.351358
$604$	10.8640	0.442051
$605$	0	0
$606$	−1.48516	−0.0603305
$607$	27.2676	1.10676	0.553379	−	0.832929i	$-0.313338\pi$
$607$	27.2676	1.10676	0.553379	+	0.832929i	$0.313338\pi$
$608$	−3.70476	−0.150248
$609$	−3.87760	−0.157128
$610$	3.30516	0.133822
$611$	7.57061	0.306274
$612$	−5.53434	−0.223712
$613$	0.453748	0.0183267	0.00916336	−	0.999958i	$-0.497083\pi$
$613$	0.453748	0.0183267	0.00916336	+	0.999958i	$0.497083\pi$
$614$	−5.55583	−0.224215
$615$	2.74189	0.110564
$616$	0	0
$617$	35.9399	1.44688	0.723442	−	0.690385i	$-0.242559\pi$
$617$	35.9399	1.44688	0.723442	+	0.690385i	$0.242559\pi$
$618$	0.840779	0.0338211
$619$	−44.4805	−1.78782	−0.893912	−	0.448243i	$-0.852050\pi$
$619$	−44.4805	−1.78782	−0.893912	+	0.448243i	$0.852050\pi$
$620$	−10.1809	−0.408875
$621$	15.4815	0.621249
$622$	20.3446	0.815743
$623$	3.32679	0.133285
$624$	−0.901224	−0.0360778
$625$	1.00000	0.0400000
$626$	−28.0901	−1.12271
$627$	0	0
$628$	9.67811	0.386199
$629$	18.6943	0.745390
$630$	−2.70476	−0.107760
$631$	19.3612	0.770756	0.385378	−	0.922759i	$-0.374071\pi$
$631$	19.3612	0.770756	0.385378	+	0.922759i	$0.374071\pi$
$632$	−11.2054	−0.445728
$633$	−12.8399	−0.510339
$634$	−7.61646	−0.302488
$635$	−18.2239	−0.723195
$636$	4.82648	0.191382
$637$	−1.65861	−0.0657164
$638$	0	0
$639$	21.1186	0.835440
$640$	−1.00000	−0.0395285
$641$	24.0572	0.950203	0.475102	−	0.879931i	$-0.342411\pi$
$641$	24.0572	0.950203	0.475102	+	0.879931i	$0.342411\pi$
$642$	−6.63383	−0.261816
$643$	−14.4357	−0.569286	−0.284643	−	0.958634i	$-0.591875\pi$
$643$	−14.4357	−0.569286	−0.284643	+	0.958634i	$0.591875\pi$
$644$	4.99442	0.196808
$645$	3.14437	0.123809
$646$	−7.58049	−0.298250
$647$	−14.6590	−0.576306	−0.288153	−	0.957584i	$-0.593041\pi$
$647$	−14.6590	−0.576306	−0.288153	+	0.957584i	$0.593041\pi$
$648$	−6.42999	−0.252594
$649$	0	0
$650$	1.65861	0.0650560
$651$	5.53191	0.216813
$652$	16.3076	0.638656
$653$	24.1399	0.944668	0.472334	−	0.881420i	$-0.343412\pi$
$653$	24.1399	0.944668	0.472334	+	0.881420i	$0.343412\pi$
$654$	0.719283	0.0281262
$655$	−9.47001	−0.370024
$656$	5.04615	0.197019
$657$	29.4943	1.15068
$658$	−4.56444	−0.177940
$659$	−25.1109	−0.978182	−0.489091	−	0.872233i	$-0.662671\pi$
$659$	−25.1109	−0.978182	−0.489091	+	0.872233i	$0.662671\pi$
$660$	0	0
$661$	−29.7048	−1.15538	−0.577692	−	0.816255i	$-0.696047\pi$
$661$	−29.7048	−1.15538	−0.577692	+	0.816255i	$0.696047\pi$
$662$	−9.86518	−0.383421
$663$	−1.84404	−0.0716166
$664$	−4.07812	−0.158262
$665$	−3.70476	−0.143664
$666$	24.7115	0.957552
$667$	−35.6418	−1.38006
$668$	−9.49351	−0.367315
$669$	1.50432	0.0581602
$670$	3.18992	0.123237
$671$	0	0
$672$	0.543362	0.0209606
$673$	−49.0571	−1.89101	−0.945505	−	0.325607i	$-0.894431\pi$
$673$	−49.0571	−1.89101	−0.945505	+	0.325607i	$0.894431\pi$
$674$	22.8921	0.881772
$675$	−3.09975	−0.119309
$676$	−10.2490	−0.394193
$677$	27.6246	1.06170	0.530849	−	0.847466i	$-0.321873\pi$
$677$	27.6246	1.06170	0.530849	+	0.847466i	$0.321873\pi$
$678$	−1.87922	−0.0721711
$679$	−9.19639	−0.352925
$680$	−2.04615	−0.0784663
$681$	0.865553	0.0331681
$682$	0	0
$683$	6.50934	0.249073	0.124536	−	0.992215i	$-0.460256\pi$
$683$	6.50934	0.249073	0.124536	+	0.992215i	$0.460256\pi$
$684$	−10.0205	−0.383142
$685$	11.4807	0.438657
$686$	1.00000	0.0381802
$687$	12.8096	0.488716
$688$	5.78688	0.220623
$689$	−14.7328	−0.561275
$690$	2.71378	0.103312
$691$	19.8453	0.754949	0.377475	−	0.926020i	$-0.376793\pi$
$691$	19.8453	0.754949	0.377475	+	0.926020i	$0.376793\pi$
$692$	1.44017	0.0547470
$693$	0	0
$694$	8.07715	0.306604
$695$	−4.86312	−0.184469
$696$	−3.87760	−0.146980
$697$	10.3252	0.391094
$698$	−16.3618	−0.619303
$699$	2.02260	0.0765019
$700$	−1.00000	−0.0377964
$701$	−9.40172	−0.355098	−0.177549	−	0.984112i	$-0.556817\pi$
$701$	−9.40172	−0.355098	−0.177549	+	0.984112i	$0.556817\pi$
$702$	−5.14127	−0.194045
$703$	33.8479	1.27660
$704$	0	0
$705$	−2.48014	−0.0934075
$706$	18.5642	0.698672
$707$	−2.73328	−0.102796
$708$	−7.65303	−0.287619
$709$	−26.9787	−1.01321	−0.506603	−	0.862180i	$-0.669099\pi$
$709$	−26.9787	−1.01321	−0.506603	+	0.862180i	$0.669099\pi$
$710$	7.80795	0.293027
$711$	−30.3079	−1.13664
$712$	3.32679	0.124677
$713$	50.8477	1.90426
$714$	1.11180	0.0416081
$715$	0	0
$716$	−9.52847	−0.356095
$717$	8.15519	0.304561
$718$	−4.59048	−0.171315
$719$	−23.7054	−0.884061	−0.442030	−	0.897000i	$-0.645742\pi$
$719$	−23.7054	−0.884061	−0.442030	+	0.897000i	$0.645742\pi$
$720$	−2.70476	−0.100800
$721$	1.54736	0.0576269
$722$	5.27477	0.196307
$723$	12.7532	0.474296
$724$	−11.5431	−0.428996
$725$	7.13632	0.265036
$726$	0	0
$727$	−42.5192	−1.57695	−0.788475	−	0.615066i	$-0.789129\pi$
$727$	−42.5192	−1.57695	−0.788475	+	0.615066i	$0.789129\pi$
$728$	−1.65861	−0.0614721
$729$	−12.3387	−0.456989
$730$	10.9046	0.403597
$731$	11.8408	0.437949
$732$	−1.79590	−0.0663784
$733$	−4.52655	−0.167192	−0.0835959	−	0.996500i	$-0.526641\pi$
$733$	−4.52655	−0.167192	−0.0835959	+	0.996500i	$0.526641\pi$
$734$	−12.7694	−0.471328
$735$	0.543362	0.0200422
$736$	4.99442	0.184097
$737$	0	0
$738$	13.6486	0.502412
$739$	31.4047	1.15524	0.577619	−	0.816306i	$-0.303982\pi$
$739$	31.4047	1.15524	0.577619	+	0.816306i	$0.303982\pi$
$740$	9.13632	0.335858
$741$	−3.33882	−0.122655
$742$	8.88262	0.326091
$743$	9.01673	0.330792	0.165396	−	0.986227i	$-0.447110\pi$
$743$	9.01673	0.330792	0.165396	+	0.986227i	$0.447110\pi$
$744$	5.53191	0.202810
$745$	7.60901	0.278773
$746$	22.4340	0.821368
$747$	−11.0303	−0.403578
$748$	0	0
$749$	−12.2089	−0.446102
$750$	−0.543362	−0.0198408
$751$	8.52599	0.311118	0.155559	−	0.987827i	$-0.450282\pi$
$751$	8.52599	0.311118	0.155559	+	0.987827i	$0.450282\pi$
$752$	−4.56444	−0.166448
$753$	12.9703	0.472665
$754$	11.8364	0.431055
$755$	10.8640	0.395382
$756$	3.09975	0.112737
$757$	31.4517	1.14313	0.571566	−	0.820556i	$-0.306336\pi$
$757$	31.4517	1.14313	0.571566	+	0.820556i	$0.306336\pi$
$758$	30.1347	1.09454
$759$	0	0
$760$	−3.70476	−0.134386
$761$	49.4950	1.79419	0.897096	−	0.441835i	$-0.145672\pi$
$761$	49.4950	1.79419	0.897096	+	0.441835i	$0.145672\pi$
$762$	9.90220	0.358719
$763$	1.32376	0.0479235
$764$	25.8516	0.935276
$765$	−5.53434	−0.200094
$766$	−27.9554	−1.01007
$767$	23.3608	0.843510
$768$	0.543362	0.0196069
$769$	−32.2787	−1.16400	−0.582000	−	0.813189i	$-0.697730\pi$
$769$	−32.2787	−1.16400	−0.582000	+	0.813189i	$0.697730\pi$
$770$	0	0
$771$	1.70580	0.0614330
$772$	25.6289	0.922403
$773$	5.16410	0.185740	0.0928699	−	0.995678i	$-0.470396\pi$
$773$	5.16410	0.185740	0.0928699	+	0.995678i	$0.470396\pi$
$774$	15.6521	0.562603
$775$	−10.1809	−0.365709
$776$	−9.19639	−0.330131
$777$	−4.96433	−0.178094
$778$	−4.57649	−0.164075
$779$	18.6948	0.669809
$780$	−0.901224	−0.0322690
$781$	0	0
$782$	10.2193	0.365443
$783$	−22.1208	−0.790533
$784$	1.00000	0.0357143
$785$	9.67811	0.345426
$786$	5.14564	0.183539
$787$	−14.6338	−0.521638	−0.260819	−	0.965388i	$-0.583993\pi$
$787$	−14.6338	−0.521638	−0.260819	+	0.965388i	$0.583993\pi$
$788$	10.5223	0.374841
$789$	9.83471	0.350125
$790$	−11.2054	−0.398671
$791$	−3.45851	−0.122970
$792$	0	0
$793$	5.48197	0.194671
$794$	37.9390	1.34641
$795$	4.82648	0.171178
$796$	18.3423	0.650125
$797$	29.1014	1.03082	0.515412	−	0.856942i	$-0.327639\pi$
$797$	29.1014	1.03082	0.515412	+	0.856942i	$0.327639\pi$
$798$	2.01302	0.0712603
$799$	−9.33952	−0.330408
$800$	−1.00000	−0.0353553
$801$	8.99817	0.317935
$802$	−15.9557	−0.563416
$803$	0	0
$804$	−1.73328	−0.0611281
$805$	4.99442	0.176030
$806$	−16.8861	−0.594788
$807$	−3.07418	−0.108216
$808$	−2.73328	−0.0961565
$809$	−29.3769	−1.03284	−0.516418	−	0.856337i	$-0.672735\pi$
$809$	−29.3769	−1.03284	−0.516418	+	0.856337i	$0.672735\pi$
$810$	−6.42999	−0.225927
$811$	27.8818	0.979064	0.489532	−	0.871985i	$-0.337168\pi$
$811$	27.8818	0.979064	0.489532	+	0.871985i	$0.337168\pi$
$812$	−7.13632	−0.250436
$813$	17.2674	0.605593
$814$	0	0
$815$	16.3076	0.571232
$816$	1.11180	0.0389208
$817$	21.4390	0.750055
$818$	24.2684	0.848524
$819$	−4.48613	−0.156758
$820$	5.04615	0.176219
$821$	5.70412	0.199075	0.0995375	−	0.995034i	$-0.468264\pi$
$821$	5.70412	0.199075	0.0995375	+	0.995034i	$0.468264\pi$
$822$	−6.23820	−0.217582
$823$	−11.2756	−0.393044	−0.196522	−	0.980499i	$-0.562965\pi$
$823$	−11.2756	−0.393044	−0.196522	+	0.980499i	$0.562965\pi$
$824$	1.54736	0.0539050
$825$	0	0
$826$	−14.0846	−0.490066
$827$	10.9311	0.380110	0.190055	−	0.981773i	$-0.439133\pi$
$827$	10.9311	0.380110	0.190055	+	0.981773i	$0.439133\pi$
$828$	13.5087	0.469460
$829$	0.763376	0.0265131	0.0132566	−	0.999912i	$-0.495780\pi$
$829$	0.763376	0.0265131	0.0132566	+	0.999912i	$0.495780\pi$
$830$	−4.07812	−0.141554
$831$	−1.30776	−0.0453655
$832$	−1.65861	−0.0575019
$833$	2.04615	0.0708949
$834$	2.64244	0.0915001
$835$	−9.49351	−0.328536
$836$	0	0
$837$	31.5582	1.09081
$838$	25.6838	0.887231
$839$	−27.3649	−0.944740	−0.472370	−	0.881400i	$-0.656601\pi$
$839$	−27.3649	−0.944740	−0.472370	+	0.881400i	$0.656601\pi$
$840$	0.543362	0.0187478
$841$	21.9271	0.756106
$842$	9.36362	0.322692
$843$	6.09803	0.210027
$844$	−23.6304	−0.813393
$845$	−10.2490	−0.352577
$846$	−12.3457	−0.424454
$847$	0	0
$848$	8.88262	0.305031
$849$	16.9833	0.582864
$850$	−2.04615	−0.0701824
$851$	−45.6307	−1.56420
$852$	−4.24254	−0.145347
$853$	36.3022	1.24296	0.621482	−	0.783428i	$-0.286531\pi$
$853$	36.3022	1.24296	0.621482	+	0.783428i	$0.286531\pi$
$854$	−3.30516	−0.113100
$855$	−10.0205	−0.342693
$856$	−12.2089	−0.417290
$857$	30.3248	1.03587	0.517937	−	0.855419i	$-0.326700\pi$
$857$	30.3248	1.03587	0.517937	+	0.855419i	$0.326700\pi$
$858$	0	0
$859$	−45.7487	−1.56093	−0.780463	−	0.625202i	$-0.785017\pi$
$859$	−45.7487	−1.56093	−0.780463	+	0.625202i	$0.785017\pi$
$860$	5.78688	0.197331
$861$	−2.74189	−0.0934432
$862$	−16.9902	−0.578687
$863$	−4.79812	−0.163330	−0.0816649	−	0.996660i	$-0.526024\pi$
$863$	−4.79812	−0.163330	−0.0816649	+	0.996660i	$0.526024\pi$
$864$	3.09975	0.105456
$865$	1.44017	0.0489672
$866$	−6.70663	−0.227901
$867$	−6.96224	−0.236450
$868$	10.1809	0.345562
$869$	0	0
$870$	−3.87760	−0.131463
$871$	5.29082	0.179273
$872$	1.32376	0.0448283
$873$	−24.8740	−0.841858
$874$	18.5031	0.625878
$875$	−1.00000	−0.0338062
$876$	−5.92514	−0.200192
$877$	32.5055	1.09763	0.548816	−	0.835943i	$-0.315079\pi$
$877$	32.5055	1.09763	0.548816	+	0.835943i	$0.315079\pi$
$878$	−12.8474	−0.433578
$879$	1.23468	0.0416447
$880$	0	0
$881$	−45.5066	−1.53316	−0.766578	−	0.642151i	$-0.778042\pi$
$881$	−45.5066	−1.53316	−0.766578	+	0.642151i	$0.778042\pi$
$882$	2.70476	0.0910739
$883$	−16.9399	−0.570071	−0.285036	−	0.958517i	$-0.592005\pi$
$883$	−16.9399	−0.570071	−0.285036	+	0.958517i	$0.592005\pi$
$884$	−3.39376	−0.114145
$885$	−7.65303	−0.257254
$886$	−0.678365	−0.0227901
$887$	−1.79960	−0.0604248	−0.0302124	−	0.999544i	$-0.509618\pi$
$887$	−1.79960	−0.0604248	−0.0302124	+	0.999544i	$0.509618\pi$
$888$	−4.96433	−0.166592
$889$	18.2239	0.611211
$890$	3.32679	0.111514
$891$	0	0
$892$	2.76854	0.0926974
$893$	−16.9101	−0.565876
$894$	−4.13445	−0.138277
$895$	−9.52847	−0.318501
$896$	1.00000	0.0334077
$897$	4.50110	0.150287
$898$	−7.59525	−0.253457
$899$	−72.6541	−2.42315
$900$	−2.70476	−0.0901586
$901$	18.1752	0.605503
$902$	0	0
$903$	−3.14437	−0.104638
$904$	−3.45851	−0.115028
$905$	−11.5431	−0.383706
$906$	−5.90310	−0.196117
$907$	18.0878	0.600595	0.300297	−	0.953846i	$-0.402914\pi$
$907$	18.0878	0.600595	0.300297	+	0.953846i	$0.402914\pi$
$908$	1.59296	0.0528642
$909$	−7.39286	−0.245206
$910$	−1.65861	−0.0549823
$911$	20.4160	0.676412	0.338206	−	0.941072i	$-0.390180\pi$
$911$	20.4160	0.676412	0.338206	+	0.941072i	$0.390180\pi$
$912$	2.01302	0.0666579
$913$	0	0
$914$	33.0502	1.09320
$915$	−1.79590	−0.0593706
$916$	23.5747	0.778929
$917$	9.47001	0.312727
$918$	6.34255	0.209335
$919$	41.4420	1.36704	0.683522	−	0.729930i	$-0.260447\pi$
$919$	41.4420	1.36704	0.683522	+	0.729930i	$0.260447\pi$
$920$	4.99442	0.164661
$921$	3.01883	0.0994737
$922$	−20.2500	−0.666898
$923$	12.9503	0.426265
$924$	0	0
$925$	9.13632	0.300400
$926$	−39.3213	−1.29218
$927$	4.18525	0.137461
$928$	−7.13632	−0.234261
$929$	−45.4529	−1.49126	−0.745630	−	0.666360i	$-0.767851\pi$
$929$	−45.4529	−1.49126	−0.745630	+	0.666360i	$0.767851\pi$
$930$	5.53191	0.181399
$931$	3.70476	0.121419
$932$	3.72239	0.121931
$933$	−11.0545	−0.361907
$934$	18.8167	0.615702
$935$	0	0
$936$	−4.48613	−0.146634
$937$	−24.2812	−0.793233	−0.396616	−	0.917984i	$-0.629816\pi$
$937$	−24.2812	−0.793233	−0.396616	+	0.917984i	$0.629816\pi$
$938$	−3.18992	−0.104155
$939$	15.2631	0.498092
$940$	−4.56444	−0.148875
$941$	−28.6929	−0.935361	−0.467680	−	0.883898i	$-0.654910\pi$
$941$	−28.6929	−0.935361	−0.467680	+	0.883898i	$0.654910\pi$
$942$	−5.25872	−0.171338
$943$	−25.2026	−0.820710
$944$	−14.0846	−0.458414
$945$	3.09975	0.100835
$946$	0	0
$947$	50.0233	1.62554	0.812770	−	0.582585i	$-0.197959\pi$
$947$	50.0233	1.62554	0.812770	+	0.582585i	$0.197959\pi$
$948$	6.08860	0.197748
$949$	18.0865	0.587111
$950$	−3.70476	−0.120198
$951$	4.13849	0.134200
$952$	2.04615	0.0663161
$953$	59.7306	1.93486	0.967431	−	0.253133i	$-0.0814612\pi$
$953$	59.7306	1.93486	0.967431	+	0.253133i	$0.0814612\pi$
$954$	24.0253	0.777849
$955$	25.8516	0.836537
$956$	15.0088	0.485418
$957$	0	0
$958$	7.56631	0.244456
$959$	−11.4807	−0.370732
$960$	0.543362	0.0175369
$961$	72.6506	2.34357
$962$	15.1536	0.488571
$963$	−33.0220	−1.06412
$964$	23.4709	0.755947
$965$	25.6289	0.825022
$966$	−2.71378	−0.0873144
$967$	−14.2933	−0.459640	−0.229820	−	0.973233i	$-0.573814\pi$
$967$	−14.2933	−0.459640	−0.229820	+	0.973233i	$0.573814\pi$
$968$	0	0
$969$	4.11895	0.132320
$970$	−9.19639	−0.295278
$971$	17.7082	0.568283	0.284142	−	0.958782i	$-0.408291\pi$
$971$	17.7082	0.568283	0.284142	+	0.958782i	$0.408291\pi$
$972$	12.7931	0.410337
$973$	4.86312	0.155905
$974$	−7.92350	−0.253885
$975$	−0.901224	−0.0288623
$976$	−3.30516	−0.105796
$977$	−24.9354	−0.797755	−0.398878	−	0.917004i	$-0.630600\pi$
$977$	−24.9354	−0.797755	−0.398878	+	0.917004i	$0.630600\pi$
$978$	−8.86095	−0.283342
$979$	0	0
$980$	1.00000	0.0319438
$981$	3.58046	0.114315
$982$	15.1101	0.482182
$983$	−50.1552	−1.59970	−0.799851	−	0.600199i	$-0.795088\pi$
$983$	−50.1552	−1.59970	−0.799851	+	0.600199i	$0.795088\pi$
$984$	−2.74189	−0.0874081
$985$	10.5223	0.335268
$986$	−14.6020	−0.465022
$987$	2.48014	0.0789437
$988$	−6.14474	−0.195490
$989$	−28.9021	−0.919034
$990$	0	0
$991$	−5.56911	−0.176909	−0.0884543	−	0.996080i	$-0.528193\pi$
$991$	−5.56911	−0.176909	−0.0884543	+	0.996080i	$0.528193\pi$
$992$	10.1809	0.323244
$993$	5.36036	0.170106
$994$	−7.80795	−0.247653
$995$	18.3423	0.581490
$996$	2.21589	0.0702133
$997$	−12.2104	−0.386708	−0.193354	−	0.981129i	$-0.561937\pi$
$997$	−12.2104	−0.386708	−0.193354	+	0.981129i	$0.561937\pi$
$998$	5.96549	0.188834
$999$	−28.3203	−0.896014

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8470.2.a.co.1.3		4
11.5	even	5		770.2.n.f.421.2	✓	8
11.9	even	5		770.2.n.f.631.2	yes	8
11.10	odd	2		8470.2.a.cs.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
770.2.n.f.421.2	✓	8	11.5	even	5
770.2.n.f.631.2	yes	8	11.9	even	5
8470.2.a.co.1.3		4	1.1	even	1	trivial
8470.2.a.cs.1.3		4	11.10	odd	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:	\( N \)	\(=\)	\( 8470 = 2 \cdot 5 \cdot 7 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8470.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +0.543362 q^{3} +1.00000 q^{4} +1.00000 q^{5} -0.543362 q^{6} -1.00000 q^{7} -1.00000 q^{8} -2.70476 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +0.543362 q^{3} +1.00000 q^{4} +1.00000 q^{5} -0.543362 q^{6} -1.00000 q^{7} -1.00000 q^{8} -2.70476 q^{9} -1.00000 q^{10} +0.543362 q^{12} -1.65861 q^{13} +1.00000 q^{14} +0.543362 q^{15} +1.00000 q^{16} +2.04615 q^{17} +2.70476 q^{18} +3.70476 q^{19} +1.00000 q^{20} -0.543362 q^{21} -4.99442 q^{23} -0.543362 q^{24} +1.00000 q^{25} +1.65861 q^{26} -3.09975 q^{27} -1.00000 q^{28} +7.13632 q^{29} -0.543362 q^{30} -10.1809 q^{31} -1.00000 q^{32} -2.04615 q^{34} -1.00000 q^{35} -2.70476 q^{36} +9.13632 q^{37} -3.70476 q^{38} -0.901224 q^{39} -1.00000 q^{40} +5.04615 q^{41} +0.543362 q^{42} +5.78688 q^{43} -2.70476 q^{45} +4.99442 q^{46} -4.56444 q^{47} +0.543362 q^{48} +1.00000 q^{49} -1.00000 q^{50} +1.11180 q^{51} -1.65861 q^{52} +8.88262 q^{53} +3.09975 q^{54} +1.00000 q^{56} +2.01302 q^{57} -7.13632 q^{58} -14.0846 q^{59} +0.543362 q^{60} -3.30516 q^{61} +10.1809 q^{62} +2.70476 q^{63} +1.00000 q^{64} -1.65861 q^{65} -3.18992 q^{67} +2.04615 q^{68} -2.71378 q^{69} +1.00000 q^{70} -7.80795 q^{71} +2.70476 q^{72} -10.9046 q^{73} -9.13632 q^{74} +0.543362 q^{75} +3.70476 q^{76} +0.901224 q^{78} +11.2054 q^{79} +1.00000 q^{80} +6.42999 q^{81} -5.04615 q^{82} +4.07812 q^{83} -0.543362 q^{84} +2.04615 q^{85} -5.78688 q^{86} +3.87760 q^{87} -3.32679 q^{89} +2.70476 q^{90} +1.65861 q^{91} -4.99442 q^{92} -5.53191 q^{93} +4.56444 q^{94} +3.70476 q^{95} -0.543362 q^{96} +9.19639 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{2} - 4 q^{3} + 4 q^{4} + 4 q^{5} + 4 q^{6} - 4 q^{7} - 4 q^{8} + 6 q^{9}+O(q^{10}) \) 4 * q - 4 * q^2 - 4 * q^3 + 4 * q^4 + 4 * q^5 + 4 * q^6 - 4 * q^7 - 4 * q^8 + 6 * q^9	\( 4 q - 4 q^{2} - 4 q^{3} + 4 q^{4} + 4 q^{5} + 4 q^{6} - 4 q^{7} - 4 q^{8} + 6 q^{9} - 4 q^{10} - 4 q^{12} + 14 q^{13} + 4 q^{14} - 4 q^{15} + 4 q^{16} + 12 q^{17} - 6 q^{18} - 2 q^{19} + 4 q^{20} + 4 q^{21} + 4 q^{24} + 4 q^{25} - 14 q^{26} - 22 q^{27} - 4 q^{28} + 10 q^{29} + 4 q^{30} - 18 q^{31} - 4 q^{32} - 12 q^{34} - 4 q^{35} + 6 q^{36} + 18 q^{37} + 2 q^{38} - 30 q^{39} - 4 q^{40} + 24 q^{41} - 4 q^{42} + 10 q^{43} + 6 q^{45} - 8 q^{47} - 4 q^{48} + 4 q^{49} - 4 q^{50} + 14 q^{52} + 20 q^{53} + 22 q^{54} + 4 q^{56} + 30 q^{57} - 10 q^{58} - 14 q^{59} - 4 q^{60} + 14 q^{61} + 18 q^{62} - 6 q^{63} + 4 q^{64} + 14 q^{65} + 12 q^{68} - 4 q^{69} + 4 q^{70} - 14 q^{71} - 6 q^{72} + 30 q^{73} - 18 q^{74} - 4 q^{75} - 2 q^{76} + 30 q^{78} + 8 q^{79} + 4 q^{80} + 16 q^{81} - 24 q^{82} + 8 q^{83} + 4 q^{84} + 12 q^{85} - 10 q^{86} + 2 q^{87} - 4 q^{89} - 6 q^{90} - 14 q^{91} - 2 q^{93} + 8 q^{94} - 2 q^{95} + 4 q^{96} - 10 q^{97} - 4 q^{98}+O(q^{100}) \) 4 * q - 4 * q^2 - 4 * q^3 + 4 * q^4 + 4 * q^5 + 4 * q^6 - 4 * q^7 - 4 * q^8 + 6 * q^9 - 4 * q^10 - 4 * q^12 + 14 * q^13 + 4 * q^14 - 4 * q^15 + 4 * q^16 + 12 * q^17 - 6 * q^18 - 2 * q^19 + 4 * q^20 + 4 * q^21 + 4 * q^24 + 4 * q^25 - 14 * q^26 - 22 * q^27 - 4 * q^28 + 10 * q^29 + 4 * q^30 - 18 * q^31 - 4 * q^32 - 12 * q^34 - 4 * q^35 + 6 * q^36 + 18 * q^37 + 2 * q^38 - 30 * q^39 - 4 * q^40 + 24 * q^41 - 4 * q^42 + 10 * q^43 + 6 * q^45 - 8 * q^47 - 4 * q^48 + 4 * q^49 - 4 * q^50 + 14 * q^52 + 20 * q^53 + 22 * q^54 + 4 * q^56 + 30 * q^57 - 10 * q^58 - 14 * q^59 - 4 * q^60 + 14 * q^61 + 18 * q^62 - 6 * q^63 + 4 * q^64 + 14 * q^65 + 12 * q^68 - 4 * q^69 + 4 * q^70 - 14 * q^71 - 6 * q^72 + 30 * q^73 - 18 * q^74 - 4 * q^75 - 2 * q^76 + 30 * q^78 + 8 * q^79 + 4 * q^80 + 16 * q^81 - 24 * q^82 + 8 * q^83 + 4 * q^84 + 12 * q^85 - 10 * q^86 + 2 * q^87 - 4 * q^89 - 6 * q^90 - 14 * q^91 - 2 * q^93 + 8 * q^94 - 2 * q^95 + 4 * q^96 - 10 * q^97 - 4 * q^98

Introduction

L-functions

Modular forms

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