LMFDB - Embedded newform 7569.2.a.bl.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7569,2,Mod(1,7569)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7569, 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("7569.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7569 = 3^{2} \cdot 29^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7569.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:	$60.4387692899$
Analytic rank:	$0$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - x^{8} - 14x^{7} + 9x^{6} + 70x^{5} - 23x^{4} - 141x^{3} + 14x^{2} + 84x - 7 $ x^9 - x^8 - 14x^7 + 9x^6 + 70x^5 - 23x^4 - 141x^3 + 14x^2 + 84*x - 7
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 87)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.07337$ of defining polynomial
Character	$\chi$	$=$	7569.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.07337 q^{2} +2.29887 q^{4} -1.89543 q^{5} -4.71271 q^{7} -0.619659 q^{8} +O(q^{10})$	$q-2.07337 q^{2} +2.29887 q^{4} -1.89543 q^{5} -4.71271 q^{7} -0.619659 q^{8} +3.92993 q^{10} -3.74001 q^{11} -1.12526 q^{13} +9.77119 q^{14} -3.31295 q^{16} +5.60261 q^{17} -1.93415 q^{19} -4.35734 q^{20} +7.75444 q^{22} +4.32591 q^{23} -1.40734 q^{25} +2.33307 q^{26} -10.8339 q^{28} +0.971138 q^{31} +8.10829 q^{32} -11.6163 q^{34} +8.93262 q^{35} -9.96009 q^{37} +4.01020 q^{38} +1.17452 q^{40} -1.59051 q^{41} +4.53504 q^{43} -8.59779 q^{44} -8.96921 q^{46} -5.86127 q^{47} +15.2096 q^{49} +2.91793 q^{50} -2.58681 q^{52} -5.15887 q^{53} +7.08895 q^{55} +2.92027 q^{56} -5.71853 q^{59} -10.4423 q^{61} -2.01353 q^{62} -10.1856 q^{64} +2.13285 q^{65} -3.96861 q^{67} +12.8796 q^{68} -18.5206 q^{70} +0.299065 q^{71} -15.1069 q^{73} +20.6510 q^{74} -4.44634 q^{76} +17.6256 q^{77} -13.8124 q^{79} +6.27947 q^{80} +3.29773 q^{82} -2.91826 q^{83} -10.6194 q^{85} -9.40282 q^{86} +2.31753 q^{88} -7.28497 q^{89} +5.30301 q^{91} +9.94468 q^{92} +12.1526 q^{94} +3.66605 q^{95} -12.8154 q^{97} -31.5352 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + q^{2} + 11 q^{4} + 5 q^{7} + 12 q^{8}+O(q^{10}) $ 9 * q + q^2 + 11 * q^4 + 5 * q^7 + 12 * q^8	$ 9 q + q^{2} + 11 q^{4} + 5 q^{7} + 12 q^{8} - 4 q^{10} + 3 q^{11} + 5 q^{13} + 15 q^{14} - 5 q^{16} + 16 q^{17} + q^{19} - 8 q^{20} + 24 q^{22} + 10 q^{23} + 9 q^{25} + 12 q^{26} - 24 q^{28} + 4 q^{31} + 25 q^{32} + 24 q^{34} + 44 q^{35} - 25 q^{37} + 10 q^{38} - 5 q^{40} + 34 q^{41} - 12 q^{43} - 23 q^{44} - 6 q^{46} + 8 q^{47} + 26 q^{49} + 27 q^{50} - 23 q^{52} + 32 q^{53} + 5 q^{55} - 14 q^{56} - 10 q^{59} - 51 q^{61} - 8 q^{62} - 8 q^{64} + 11 q^{65} + 7 q^{67} + 11 q^{68} + 14 q^{70} - 7 q^{71} - 17 q^{73} + 62 q^{74} + 6 q^{76} + 64 q^{77} - 13 q^{79} - 54 q^{80} + 37 q^{82} + 31 q^{83} - 42 q^{85} + 70 q^{86} - 29 q^{88} - 32 q^{89} + 45 q^{91} - 9 q^{92} + 38 q^{94} - 20 q^{95} - 16 q^{97} - 12 q^{98}+O(q^{100}) $ 9 * q + q^2 + 11 * q^4 + 5 * q^7 + 12 * q^8 - 4 * q^10 + 3 * q^11 + 5 * q^13 + 15 * q^14 - 5 * q^16 + 16 * q^17 + q^19 - 8 * q^20 + 24 * q^22 + 10 * q^23 + 9 * q^25 + 12 * q^26 - 24 * q^28 + 4 * q^31 + 25 * q^32 + 24 * q^34 + 44 * q^35 - 25 * q^37 + 10 * q^38 - 5 * q^40 + 34 * q^41 - 12 * q^43 - 23 * q^44 - 6 * q^46 + 8 * q^47 + 26 * q^49 + 27 * q^50 - 23 * q^52 + 32 * q^53 + 5 * q^55 - 14 * q^56 - 10 * q^59 - 51 * q^61 - 8 * q^62 - 8 * q^64 + 11 * q^65 + 7 * q^67 + 11 * q^68 + 14 * q^70 - 7 * q^71 - 17 * q^73 + 62 * q^74 + 6 * q^76 + 64 * q^77 - 13 * q^79 - 54 * q^80 + 37 * q^82 + 31 * q^83 - 42 * q^85 + 70 * q^86 - 29 * q^88 - 32 * q^89 + 45 * q^91 - 9 * q^92 + 38 * q^94 - 20 * q^95 - 16 * q^97 - 12 * 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 $

$2$	−2.07337	−1.46609	−0.733047	−	0.680178i	$-0.761903\pi$
$2$	−2.07337	−1.46609	−0.733047	+	0.680178i	$0.761903\pi$
$3$	0	0
$3$	0	0
$4$	2.29887	1.14943
$5$	−1.89543	−0.847663	−0.423832	−	0.905741i	$-0.639315\pi$
$5$	−1.89543	−0.847663	−0.423832	+	0.905741i	$0.639315\pi$
$6$	0	0
$7$	−4.71271	−1.78124	−0.890619	−	0.454751i	$-0.849728\pi$
$7$	−4.71271	−1.78124	−0.890619	+	0.454751i	$0.849728\pi$
$8$	−0.619659	−0.219082
$9$	0	0
$10$	3.92993	1.24275
$11$	−3.74001	−1.12766	−0.563828	−	0.825892i	$-0.690672\pi$
$11$	−3.74001	−1.12766	−0.563828	+	0.825892i	$0.690672\pi$
$12$	0	0
$13$	−1.12526	−0.312090	−0.156045	−	0.987750i	$-0.549874\pi$
$13$	−1.12526	−0.312090	−0.156045	+	0.987750i	$0.549874\pi$
$14$	9.77119	2.61146
$15$	0	0
$16$	−3.31295	−0.828237
$17$	5.60261	1.35883	0.679416	−	0.733753i	$-0.262233\pi$
$17$	5.60261	1.35883	0.679416	+	0.733753i	$0.262233\pi$
$18$	0	0
$19$	−1.93415	−0.443724	−0.221862	−	0.975078i	$-0.571213\pi$
$19$	−1.93415	−0.443724	−0.221862	+	0.975078i	$0.571213\pi$
$20$	−4.35734	−0.974332
$21$	0	0
$22$	7.75444	1.65325
$23$	4.32591	0.902014	0.451007	−	0.892520i	$-0.351065\pi$
$23$	4.32591	0.902014	0.451007	+	0.892520i	$0.351065\pi$
$24$	0	0
$25$	−1.40734	−0.281467
$26$	2.33307	0.457553
$27$	0	0
$28$	−10.8339	−2.04741
$29$	0	0
$29$	0	0
$30$	0	0
$31$	0.971138	0.174422	0.0872108	−	0.996190i	$-0.472205\pi$
$31$	0.971138	0.174422	0.0872108	+	0.996190i	$0.472205\pi$
$32$	8.10829	1.43336
$33$	0	0
$34$	−11.6163	−1.99218
$35$	8.93262	1.50989
$36$	0	0
$37$	−9.96009	−1.63743	−0.818714	−	0.574201i	$-0.805313\pi$
$37$	−9.96009	−1.63743	−0.818714	+	0.574201i	$0.805313\pi$
$38$	4.01020	0.650541
$39$	0	0
$40$	1.17452	0.185708
$41$	−1.59051	−0.248397	−0.124198	−	0.992257i	$-0.539636\pi$
$41$	−1.59051	−0.248397	−0.124198	+	0.992257i	$0.539636\pi$
$42$	0	0
$43$	4.53504	0.691587	0.345793	−	0.938311i	$-0.387610\pi$
$43$	4.53504	0.691587	0.345793	+	0.938311i	$0.387610\pi$
$44$	−8.59779	−1.29617
$45$	0	0
$46$	−8.96921	−1.32244
$47$	−5.86127	−0.854954	−0.427477	−	0.904026i	$-0.640598\pi$
$47$	−5.86127	−0.854954	−0.427477	+	0.904026i	$0.640598\pi$
$48$	0	0
$49$	15.2096	2.17281
$50$	2.91793	0.412657
$51$	0	0
$52$	−2.58681	−0.358726
$53$	−5.15887	−0.708625	−0.354312	−	0.935127i	$-0.615285\pi$
$53$	−5.15887	−0.708625	−0.354312	+	0.935127i	$0.615285\pi$
$54$	0	0
$55$	7.08895	0.955873
$56$	2.92027	0.390238
$57$	0	0
$58$	0	0
$59$	−5.71853	−0.744490	−0.372245	−	0.928135i	$-0.621412\pi$
$59$	−5.71853	−0.744490	−0.372245	+	0.928135i	$0.621412\pi$
$60$	0	0
$61$	−10.4423	−1.33700	−0.668502	−	0.743710i	$-0.733064\pi$
$61$	−10.4423	−1.33700	−0.668502	+	0.743710i	$0.733064\pi$
$62$	−2.01353	−0.255719
$63$	0	0
$64$	−10.1856	−1.27320
$65$	2.13285	0.264547
$66$	0	0
$67$	−3.96861	−0.484843	−0.242421	−	0.970171i	$-0.577942\pi$
$67$	−3.96861	−0.484843	−0.242421	+	0.970171i	$0.577942\pi$
$68$	12.8796	1.56189
$69$	0	0
$70$	−18.5206	−2.21364
$71$	0.299065	0.0354925	0.0177462	−	0.999843i	$-0.494351\pi$
$71$	0.299065	0.0354925	0.0177462	+	0.999843i	$0.494351\pi$
$72$	0	0
$73$	−15.1069	−1.76812	−0.884062	−	0.467370i	$-0.845202\pi$
$73$	−15.1069	−1.76812	−0.884062	+	0.467370i	$0.845202\pi$
$74$	20.6510	2.40062
$75$	0	0
$76$	−4.44634	−0.510031
$77$	17.6256	2.00862
$78$	0	0
$79$	−13.8124	−1.55401	−0.777007	−	0.629492i	$-0.783263\pi$
$79$	−13.8124	−1.55401	−0.777007	+	0.629492i	$0.783263\pi$
$80$	6.27947	0.702066
$81$	0	0
$82$	3.29773	0.364173
$83$	−2.91826	−0.320321	−0.160160	−	0.987091i	$-0.551201\pi$
$83$	−2.91826	−0.320321	−0.160160	+	0.987091i	$0.551201\pi$
$84$	0	0
$85$	−10.6194	−1.15183
$86$	−9.40282	−1.01393
$87$	0	0
$88$	2.31753	0.247050
$89$	−7.28497	−0.772205	−0.386102	−	0.922456i	$-0.626179\pi$
$89$	−7.28497	−0.772205	−0.386102	+	0.922456i	$0.626179\pi$
$90$	0	0
$91$	5.30301	0.555906
$92$	9.94468	1.03680
$93$	0	0
$94$	12.1526	1.25344
$95$	3.66605	0.376128
$96$	0	0
$97$	−12.8154	−1.30121	−0.650605	−	0.759416i	$-0.725485\pi$
$97$	−12.8154	−1.30121	−0.650605	+	0.759416i	$0.725485\pi$
$98$	−31.5352	−3.18554
$99$	0	0
$100$	−3.23527	−0.323527
$101$	9.45277	0.940586	0.470293	−	0.882510i	$-0.344148\pi$
$101$	9.45277	0.940586	0.470293	+	0.882510i	$0.344148\pi$
$102$	0	0
$103$	5.55324	0.547177	0.273588	−	0.961847i	$-0.411789\pi$
$103$	5.55324	0.547177	0.273588	+	0.961847i	$0.411789\pi$
$104$	0.697275	0.0683734
$105$	0	0
$106$	10.6962	1.03891
$107$	−6.29711	−0.608765	−0.304382	−	0.952550i	$-0.598450\pi$
$107$	−6.29711	−0.608765	−0.304382	+	0.952550i	$0.598450\pi$
$108$	0	0
$109$	−16.3981	−1.57065	−0.785324	−	0.619084i	$-0.787504\pi$
$109$	−16.3981	−1.57065	−0.785324	+	0.619084i	$0.787504\pi$
$110$	−14.6980	−1.40140
$111$	0	0
$112$	15.6130	1.47529
$113$	−2.74678	−0.258395	−0.129198	−	0.991619i	$-0.541240\pi$
$113$	−2.74678	−0.258395	−0.129198	+	0.991619i	$0.541240\pi$
$114$	0	0
$115$	−8.19947	−0.764604
$116$	0	0
$117$	0	0
$118$	11.8566	1.09149
$119$	−26.4035	−2.42040
$120$	0	0
$121$	2.98771	0.271610
$122$	21.6508	1.96017
$123$	0	0
$124$	2.23252	0.200486
$125$	12.1447	1.08625
$126$	0	0
$127$	−12.4117	−1.10136	−0.550681	−	0.834716i	$-0.685632\pi$
$127$	−12.4117	−1.10136	−0.550681	+	0.834716i	$0.685632\pi$
$128$	4.90192	0.433273
$129$	0	0
$130$	−4.42218	−0.387851
$131$	−18.9093	−1.65211	−0.826057	−	0.563587i	$-0.809421\pi$
$131$	−18.9093	−1.65211	−0.826057	+	0.563587i	$0.809421\pi$
$132$	0	0
$133$	9.11507	0.790377
$134$	8.22840	0.710825
$135$	0	0
$136$	−3.47170	−0.297696
$137$	−8.82449	−0.753927	−0.376963	−	0.926228i	$-0.623032\pi$
$137$	−8.82449	−0.753927	−0.376963	+	0.926228i	$0.623032\pi$
$138$	0	0
$139$	10.3454	0.877485	0.438743	−	0.898613i	$-0.355424\pi$
$139$	10.3454	0.877485	0.438743	+	0.898613i	$0.355424\pi$
$140$	20.5349	1.73552
$141$	0	0
$142$	−0.620073	−0.0520353
$143$	4.20847	0.351930
$144$	0	0
$145$	0	0
$146$	31.3221	2.59224
$147$	0	0
$148$	−22.8969	−1.88211
$149$	−12.1003	−0.991298	−0.495649	−	0.868523i	$-0.665070\pi$
$149$	−12.1003	−0.991298	−0.495649	+	0.868523i	$0.665070\pi$
$150$	0	0
$151$	9.54059	0.776402	0.388201	−	0.921575i	$-0.373097\pi$
$151$	9.54059	0.776402	0.388201	+	0.921575i	$0.373097\pi$
$152$	1.19851	0.0972121
$153$	0	0
$154$	−36.5444	−2.94483
$155$	−1.84073	−0.147851
$156$	0	0
$157$	0.889508	0.0709905	0.0354952	−	0.999370i	$-0.488699\pi$
$157$	0.889508	0.0709905	0.0354952	+	0.999370i	$0.488699\pi$
$158$	28.6382	2.27833
$159$	0	0
$160$	−15.3687	−1.21500
$161$	−20.3868	−1.60670
$162$	0	0
$163$	7.71251	0.604090	0.302045	−	0.953294i	$-0.402331\pi$
$163$	7.71251	0.604090	0.302045	+	0.953294i	$0.402331\pi$
$164$	−3.65638	−0.285515
$165$	0	0
$166$	6.05064	0.469621
$167$	−8.38877	−0.649143	−0.324571	−	0.945861i	$-0.605220\pi$
$167$	−8.38877	−0.649143	−0.324571	+	0.945861i	$0.605220\pi$
$168$	0	0
$169$	−11.7338	−0.902600
$170$	22.0179	1.68869
$171$	0	0
$172$	10.4254	0.794933
$173$	3.52185	0.267761	0.133881	−	0.990997i	$-0.457256\pi$
$173$	3.52185	0.267761	0.133881	+	0.990997i	$0.457256\pi$
$174$	0	0
$175$	6.63236	0.501360
$176$	12.3905	0.933967
$177$	0	0
$178$	15.1044	1.13213
$179$	−7.13537	−0.533323	−0.266661	−	0.963790i	$-0.585921\pi$
$179$	−7.13537	−0.533323	−0.266661	+	0.963790i	$0.585921\pi$
$180$	0	0
$181$	21.0805	1.56690	0.783452	−	0.621452i	$-0.213457\pi$
$181$	21.0805	1.56690	0.783452	+	0.621452i	$0.213457\pi$
$182$	−10.9951	−0.815011
$183$	0	0
$184$	−2.68059	−0.197615
$185$	18.8787	1.38799
$186$	0	0
$187$	−20.9538	−1.53230
$188$	−13.4743	−0.982712
$189$	0	0
$190$	−7.60107	−0.551440
$191$	13.8150	0.999620	0.499810	−	0.866135i	$-0.333403\pi$
$191$	13.8150	0.999620	0.499810	+	0.866135i	$0.333403\pi$
$192$	0	0
$193$	4.23298	0.304697	0.152348	−	0.988327i	$-0.451316\pi$
$193$	4.23298	0.304697	0.152348	+	0.988327i	$0.451316\pi$
$194$	26.5711	1.90770
$195$	0	0
$196$	34.9649	2.49749
$197$	−4.61478	−0.328789	−0.164395	−	0.986395i	$-0.552567\pi$
$197$	−4.61478	−0.328789	−0.164395	+	0.986395i	$0.552567\pi$
$198$	0	0
$199$	−3.82599	−0.271217	−0.135609	−	0.990762i	$-0.543299\pi$
$199$	−3.82599	−0.271217	−0.135609	+	0.990762i	$0.543299\pi$
$200$	0.872067	0.0616645
$201$	0	0
$202$	−19.5991	−1.37899
$203$	0	0
$204$	0	0
$205$	3.01471	0.210557
$206$	−11.5139	−0.802213
$207$	0	0
$208$	3.72792	0.258484
$209$	7.23374	0.500368
$210$	0	0
$211$	−9.98022	−0.687067	−0.343533	−	0.939140i	$-0.611624\pi$
$211$	−9.98022	−0.687067	−0.343533	+	0.939140i	$0.611624\pi$
$212$	−11.8595	−0.814516
$213$	0	0
$214$	13.0562	0.892507
$215$	−8.59586	−0.586233
$216$	0	0
$217$	−4.57669	−0.310686
$218$	33.9992	2.30272
$219$	0	0
$220$	16.2965	1.09871
$221$	−6.30437	−0.424078
$222$	0	0
$223$	7.69270	0.515141	0.257571	−	0.966259i	$-0.417078\pi$
$223$	7.69270	0.515141	0.257571	+	0.966259i	$0.417078\pi$
$224$	−38.2120	−2.55315
$225$	0	0
$226$	5.69509	0.378832
$227$	−6.60154	−0.438159	−0.219080	−	0.975707i	$-0.570305\pi$
$227$	−6.60154	−0.438159	−0.219080	+	0.975707i	$0.570305\pi$
$228$	0	0
$229$	−2.32866	−0.153882	−0.0769412	−	0.997036i	$-0.524515\pi$
$229$	−2.32866	−0.153882	−0.0769412	+	0.997036i	$0.524515\pi$
$230$	17.0005	1.12098
$231$	0	0
$232$	0	0
$233$	−1.74843	−0.114544	−0.0572719	−	0.998359i	$-0.518240\pi$
$233$	−1.74843	−0.114544	−0.0572719	+	0.998359i	$0.518240\pi$
$234$	0	0
$235$	11.1096	0.724713
$236$	−13.1461	−0.855741
$237$	0	0
$238$	54.7442	3.54854
$239$	22.8194	1.47607	0.738033	−	0.674765i	$-0.235755\pi$
$239$	22.8194	1.47607	0.738033	+	0.674765i	$0.235755\pi$
$240$	0	0
$241$	11.1763	0.719926	0.359963	−	0.932967i	$-0.382789\pi$
$241$	11.1763	0.719926	0.359963	+	0.932967i	$0.382789\pi$
$242$	−6.19463	−0.398206
$243$	0	0
$244$	−24.0055	−1.53680
$245$	−28.8288	−1.84181
$246$	0	0
$247$	2.17641	0.138482
$248$	−0.601774	−0.0382127
$249$	0	0
$250$	−25.1804	−1.59255
$251$	−20.7323	−1.30861	−0.654304	−	0.756231i	$-0.727038\pi$
$251$	−20.7323	−1.30861	−0.654304	+	0.756231i	$0.727038\pi$
$252$	0	0
$253$	−16.1790	−1.01716
$254$	25.7341	1.61470
$255$	0	0
$256$	10.2077	0.637980
$257$	−2.36599	−0.147586	−0.0737932	−	0.997274i	$-0.523510\pi$
$257$	−2.36599	−0.147586	−0.0737932	+	0.997274i	$0.523510\pi$
$258$	0	0
$259$	46.9390	2.91665
$260$	4.90313	0.304079
$261$	0	0
$262$	39.2060	2.42215
$263$	24.4606	1.50830	0.754152	−	0.656699i	$-0.228048\pi$
$263$	24.4606	1.50830	0.754152	+	0.656699i	$0.228048\pi$
$264$	0	0
$265$	9.77828	0.600675
$266$	−18.8989	−1.15877
$267$	0	0
$268$	−9.12330	−0.557294
$269$	−6.48855	−0.395614	−0.197807	−	0.980241i	$-0.563382\pi$
$269$	−6.48855	−0.395614	−0.197807	+	0.980241i	$0.563382\pi$
$270$	0	0
$271$	−0.479077	−0.0291019	−0.0145509	−	0.999894i	$-0.504632\pi$
$271$	−0.479077	−0.0291019	−0.0145509	+	0.999894i	$0.504632\pi$
$272$	−18.5611	−1.12543
$273$	0	0
$274$	18.2964	1.10533
$275$	5.26345	0.317398
$276$	0	0
$277$	4.16222	0.250084	0.125042	−	0.992151i	$-0.460094\pi$
$277$	4.16222	0.250084	0.125042	+	0.992151i	$0.460094\pi$
$278$	−21.4498	−1.28648
$279$	0	0
$280$	−5.53518	−0.330790
$281$	9.31820	0.555877	0.277939	−	0.960599i	$-0.410349\pi$
$281$	9.31820	0.555877	0.277939	+	0.960599i	$0.410349\pi$
$282$	0	0
$283$	9.78406	0.581602	0.290801	−	0.956784i	$-0.406078\pi$
$283$	9.78406	0.581602	0.290801	+	0.956784i	$0.406078\pi$
$284$	0.687510	0.0407962
$285$	0	0
$286$	−8.72573	−0.515963
$287$	7.49563	0.442453
$288$	0	0
$289$	14.3892	0.846424
$290$	0	0
$291$	0	0
$292$	−34.7286	−2.03234
$293$	11.4460	0.668685	0.334342	−	0.942452i	$-0.391486\pi$
$293$	11.4460	0.668685	0.334342	+	0.942452i	$0.391486\pi$
$294$	0	0
$295$	10.8391	0.631076
$296$	6.17185	0.358732
$297$	0	0
$298$	25.0885	1.45334
$299$	−4.86776	−0.281510
$300$	0	0
$301$	−21.3723	−1.23188
$302$	−19.7812	−1.13828
$303$	0	0
$304$	6.40773	0.367508
$305$	19.7927	1.13333
$306$	0	0
$307$	−20.1309	−1.14893	−0.574465	−	0.818529i	$-0.694790\pi$
$307$	−20.1309	−1.14893	−0.574465	+	0.818529i	$0.694790\pi$
$308$	40.5189	2.30878
$309$	0	0
$310$	3.81651	0.216763
$311$	−1.77641	−0.100731	−0.0503655	−	0.998731i	$-0.516039\pi$
$311$	−1.77641	−0.100731	−0.0503655	+	0.998731i	$0.516039\pi$
$312$	0	0
$313$	−14.9122	−0.842889	−0.421444	−	0.906854i	$-0.638477\pi$
$313$	−14.9122	−0.842889	−0.421444	+	0.906854i	$0.638477\pi$
$314$	−1.84428	−0.104079
$315$	0	0
$316$	−31.7528	−1.78623
$317$	11.4211	0.641471	0.320736	−	0.947169i	$-0.396070\pi$
$317$	11.4211	0.641471	0.320736	+	0.947169i	$0.396070\pi$
$318$	0	0
$319$	0	0
$320$	19.3061	1.07924
$321$	0	0
$322$	42.2693	2.35558
$323$	−10.8363	−0.602946
$324$	0	0
$325$	1.58361	0.0878430
$326$	−15.9909	−0.885654
$327$	0	0
$328$	0.985576	0.0544193
$329$	27.6225	1.52288
$330$	0	0
$331$	9.83216	0.540424	0.270212	−	0.962801i	$-0.412906\pi$
$331$	9.83216	0.540424	0.270212	+	0.962801i	$0.412906\pi$
$332$	−6.70869	−0.368187
$333$	0	0
$334$	17.3930	0.951704
$335$	7.52223	0.410983
$336$	0	0
$337$	−15.0044	−0.817344	−0.408672	−	0.912681i	$-0.634008\pi$
$337$	−15.0044	−0.817344	−0.408672	+	0.912681i	$0.634008\pi$
$338$	24.3285	1.32330
$339$	0	0
$340$	−24.4125	−1.32395
$341$	−3.63207	−0.196688
$342$	0	0
$343$	−38.6896	−2.08904
$344$	−2.81018	−0.151515
$345$	0	0
$346$	−7.30210	−0.392563
$347$	−33.4677	−1.79664	−0.898320	−	0.439341i	$-0.855212\pi$
$347$	−33.4677	−1.79664	−0.898320	+	0.439341i	$0.855212\pi$
$348$	0	0
$349$	27.7844	1.48727	0.743633	−	0.668588i	$-0.233101\pi$
$349$	27.7844	1.48727	0.743633	+	0.668588i	$0.233101\pi$
$350$	−13.7513	−0.735040
$351$	0	0
$352$	−30.3251	−1.61633
$353$	−13.7386	−0.731232	−0.365616	−	0.930766i	$-0.619142\pi$
$353$	−13.7386	−0.731232	−0.365616	+	0.930766i	$0.619142\pi$
$354$	0	0
$355$	−0.566858	−0.0300857
$356$	−16.7472	−0.887597
$357$	0	0
$358$	14.7943	0.781901
$359$	24.6004	1.29836	0.649179	−	0.760635i	$-0.275112\pi$
$359$	24.6004	1.29836	0.649179	+	0.760635i	$0.275112\pi$
$360$	0	0
$361$	−15.2591	−0.803109
$362$	−43.7078	−2.29723
$363$	0	0
$364$	12.1909	0.638977
$365$	28.6340	1.49877
$366$	0	0
$367$	−4.43348	−0.231426	−0.115713	−	0.993283i	$-0.536915\pi$
$367$	−4.43348	−0.231426	−0.115713	+	0.993283i	$0.536915\pi$
$368$	−14.3315	−0.747082
$369$	0	0
$370$	−39.1425	−2.03492
$371$	24.3122	1.26223
$372$	0	0
$373$	24.3764	1.26216	0.631080	−	0.775718i	$-0.282612\pi$
$373$	24.3764	1.26216	0.631080	+	0.775718i	$0.282612\pi$
$374$	43.4451	2.24649
$375$	0	0
$376$	3.63199	0.187305
$377$	0	0
$378$	0	0
$379$	13.5054	0.693727	0.346864	−	0.937916i	$-0.387247\pi$
$379$	13.5054	0.693727	0.346864	+	0.937916i	$0.387247\pi$
$380$	8.42774	0.432334
$381$	0	0
$382$	−28.6437	−1.46554
$383$	−24.7226	−1.26327	−0.631634	−	0.775267i	$-0.717615\pi$
$383$	−24.7226	−1.26327	−0.631634	+	0.775267i	$0.717615\pi$
$384$	0	0
$385$	−33.4081	−1.70264
$386$	−8.77654	−0.446714
$387$	0	0
$388$	−29.4610	−1.49565
$389$	−30.2883	−1.53568	−0.767839	−	0.640643i	$-0.778668\pi$
$389$	−30.2883	−1.53568	−0.767839	+	0.640643i	$0.778668\pi$
$390$	0	0
$391$	24.2364	1.22569
$392$	−9.42478	−0.476023
$393$	0	0
$394$	9.56814	0.482036
$395$	26.1804	1.31728
$396$	0	0
$397$	−13.5986	−0.682494	−0.341247	−	0.939974i	$-0.610849\pi$
$397$	−13.5986	−0.682494	−0.341247	+	0.939974i	$0.610849\pi$
$398$	7.93270	0.397630
$399$	0	0
$400$	4.66243	0.233121
$401$	14.9067	0.744407	0.372203	−	0.928151i	$-0.378602\pi$
$401$	14.9067	0.744407	0.372203	+	0.928151i	$0.378602\pi$
$402$	0	0
$403$	−1.09278	−0.0544352
$404$	21.7306	1.08114
$405$	0	0
$406$	0	0
$407$	37.2509	1.84646
$408$	0	0
$409$	−20.6030	−1.01875	−0.509377	−	0.860544i	$-0.670124\pi$
$409$	−20.6030	−1.01875	−0.509377	+	0.860544i	$0.670124\pi$
$410$	−6.25062	−0.308696
$411$	0	0
$412$	12.7661	0.628943
$413$	26.9498	1.32611
$414$	0	0
$415$	5.53137	0.271524
$416$	−9.12390	−0.447336
$417$	0	0
$418$	−14.9982	−0.733587
$419$	5.35781	0.261746	0.130873	−	0.991399i	$-0.458222\pi$
$419$	5.35781	0.261746	0.130873	+	0.991399i	$0.458222\pi$
$420$	0	0
$421$	1.91744	0.0934504	0.0467252	−	0.998908i	$-0.485121\pi$
$421$	1.91744	0.0934504	0.0467252	+	0.998908i	$0.485121\pi$
$422$	20.6927	1.00730
$423$	0	0
$424$	3.19674	0.155247
$425$	−7.88475	−0.382466
$426$	0	0
$427$	49.2117	2.38152
$428$	−14.4762	−0.699734
$429$	0	0
$430$	17.8224	0.859473
$431$	20.2439	0.975116	0.487558	−	0.873091i	$-0.337888\pi$
$431$	20.2439	0.975116	0.487558	+	0.873091i	$0.337888\pi$
$432$	0	0
$433$	−18.1995	−0.874614	−0.437307	−	0.899312i	$-0.644068\pi$
$433$	−18.1995	−0.874614	−0.437307	+	0.899312i	$0.644068\pi$
$434$	9.48918	0.455495
$435$	0	0
$436$	−37.6969	−1.80536
$437$	−8.36694	−0.400245
$438$	0	0
$439$	−0.392037	−0.0187109	−0.00935545	−	0.999956i	$-0.502978\pi$
$439$	−0.392037	−0.0187109	−0.00935545	+	0.999956i	$0.502978\pi$
$440$	−4.39273	−0.209415
$441$	0	0
$442$	13.0713	0.621738
$443$	−5.45469	−0.259160	−0.129580	−	0.991569i	$-0.541363\pi$
$443$	−5.45469	−0.259160	−0.129580	+	0.991569i	$0.541363\pi$
$444$	0	0
$445$	13.8082	0.654570
$446$	−15.9498	−0.755246
$447$	0	0
$448$	48.0017	2.26787
$449$	−2.77692	−0.131051	−0.0655254	−	0.997851i	$-0.520872\pi$
$449$	−2.77692	−0.131051	−0.0655254	+	0.997851i	$0.520872\pi$
$450$	0	0
$451$	5.94855	0.280106
$452$	−6.31448	−0.297008
$453$	0	0
$454$	13.6874	0.642383
$455$	−10.0515	−0.471221
$456$	0	0
$457$	−36.4271	−1.70399	−0.851994	−	0.523552i	$-0.824607\pi$
$457$	−36.4271	−1.70399	−0.851994	+	0.523552i	$0.824607\pi$
$458$	4.82818	0.225606
$459$	0	0
$460$	−18.8495	−0.878861
$461$	−25.3558	−1.18094	−0.590470	−	0.807060i	$-0.701057\pi$
$461$	−25.3558	−1.18094	−0.590470	+	0.807060i	$0.701057\pi$
$462$	0	0
$463$	28.4073	1.32020	0.660099	−	0.751179i	$-0.270514\pi$
$463$	28.4073	1.32020	0.660099	+	0.751179i	$0.270514\pi$
$464$	0	0
$465$	0	0
$466$	3.62515	0.167932
$467$	13.2737	0.614234	0.307117	−	0.951672i	$-0.400636\pi$
$467$	13.2737	0.614234	0.307117	+	0.951672i	$0.400636\pi$
$468$	0	0
$469$	18.7029	0.863620
$470$	−23.0344	−1.06250
$471$	0	0
$472$	3.54354	0.163105
$473$	−16.9611	−0.779873
$474$	0	0
$475$	2.72199	0.124894
$476$	−60.6980	−2.78209
$477$	0	0
$478$	−47.3131	−2.16405
$479$	−26.2597	−1.19984	−0.599918	−	0.800062i	$-0.704800\pi$
$479$	−26.2597	−1.19984	−0.599918	+	0.800062i	$0.704800\pi$
$480$	0	0
$481$	11.2077	0.511025
$482$	−23.1725	−1.05548
$483$	0	0
$484$	6.86834	0.312197
$485$	24.2908	1.10299
$486$	0	0
$487$	9.23188	0.418337	0.209168	−	0.977880i	$-0.432924\pi$
$487$	9.23188	0.418337	0.209168	+	0.977880i	$0.432924\pi$
$488$	6.47068	0.292914
$489$	0	0
$490$	59.7729	2.70026
$491$	12.1181	0.546883	0.273442	−	0.961889i	$-0.411838\pi$
$491$	12.1181	0.546883	0.273442	+	0.961889i	$0.411838\pi$
$492$	0	0
$493$	0	0
$494$	−4.51251	−0.203027
$495$	0	0
$496$	−3.21733	−0.144462
$497$	−1.40941	−0.0632205
$498$	0	0
$499$	−29.5908	−1.32466	−0.662332	−	0.749210i	$-0.730433\pi$
$499$	−29.5908	−1.32466	−0.662332	+	0.749210i	$0.730433\pi$
$500$	27.9190	1.24857
$501$	0	0
$502$	42.9857	1.91854
$503$	8.97720	0.400274	0.200137	−	0.979768i	$-0.435861\pi$
$503$	8.97720	0.400274	0.200137	+	0.979768i	$0.435861\pi$
$504$	0	0
$505$	−17.9171	−0.797300
$506$	33.5450	1.49126
$507$	0	0
$508$	−28.5328	−1.26594
$509$	−29.6633	−1.31480	−0.657402	−	0.753540i	$-0.728345\pi$
$509$	−29.6633	−1.31480	−0.657402	+	0.753540i	$0.728345\pi$
$510$	0	0
$511$	71.1942	3.14945
$512$	−30.9681	−1.36861
$513$	0	0
$514$	4.90557	0.216376
$515$	−10.5258	−0.463822
$516$	0	0
$517$	21.9212	0.964095
$518$	−97.3220	−4.27608
$519$	0	0
$520$	−1.32164	−0.0579576
$521$	23.8067	1.04299	0.521496	−	0.853254i	$-0.325374\pi$
$521$	23.8067	1.04299	0.521496	+	0.853254i	$0.325374\pi$
$522$	0	0
$523$	9.45221	0.413316	0.206658	−	0.978413i	$-0.433741\pi$
$523$	9.45221	0.413316	0.206658	+	0.978413i	$0.433741\pi$
$524$	−43.4699	−1.89899
$525$	0	0
$526$	−50.7159	−2.21132
$527$	5.44091	0.237010
$528$	0	0
$529$	−4.28652	−0.186370
$530$	−20.2740	−0.880646
$531$	0	0
$532$	20.9543	0.908485
$533$	1.78974	0.0775221
$534$	0	0
$535$	11.9358	0.516028
$536$	2.45918	0.106221
$537$	0	0
$538$	13.4532	0.580008
$539$	−56.8843	−2.45018
$540$	0	0
$541$	−22.9959	−0.988670	−0.494335	−	0.869272i	$-0.664588\pi$
$541$	−22.9959	−0.988670	−0.494335	+	0.869272i	$0.664588\pi$
$542$	0.993304	0.0426661
$543$	0	0
$544$	45.4275	1.94769
$545$	31.0814	1.33138
$546$	0	0
$547$	4.50082	0.192441	0.0962206	−	0.995360i	$-0.469325\pi$
$547$	4.50082	0.192441	0.0962206	+	0.995360i	$0.469325\pi$
$548$	−20.2863	−0.866588
$549$	0	0
$550$	−10.9131	−0.465336
$551$	0	0
$552$	0	0
$553$	65.0938	2.76807
$554$	−8.62983	−0.366646
$555$	0	0
$556$	23.7827	1.00861
$557$	13.8370	0.586292	0.293146	−	0.956068i	$-0.405298\pi$
$557$	13.8370	0.586292	0.293146	+	0.956068i	$0.405298\pi$
$558$	0	0
$559$	−5.10308	−0.215837
$560$	−29.5933	−1.25055
$561$	0	0
$562$	−19.3201	−0.814968
$563$	29.6777	1.25077	0.625384	−	0.780317i	$-0.284942\pi$
$563$	29.6777	1.25077	0.625384	+	0.780317i	$0.284942\pi$
$564$	0	0
$565$	5.20634	0.219032
$566$	−20.2860	−0.852684
$567$	0	0
$568$	−0.185318	−0.00777578
$569$	−10.5370	−0.441734	−0.220867	−	0.975304i	$-0.570889\pi$
$569$	−10.5370	−0.441734	−0.220867	+	0.975304i	$0.570889\pi$
$570$	0	0
$571$	−34.9153	−1.46116	−0.730581	−	0.682826i	$-0.760751\pi$
$571$	−34.9153	−1.46116	−0.730581	+	0.682826i	$0.760751\pi$
$572$	9.67472	0.404520
$573$	0	0
$574$	−15.5412	−0.648678
$575$	−6.08800	−0.253887
$576$	0	0
$577$	−8.34697	−0.347489	−0.173744	−	0.984791i	$-0.555587\pi$
$577$	−8.34697	−0.347489	−0.173744	+	0.984791i	$0.555587\pi$
$578$	−29.8341	−1.24094
$579$	0	0
$580$	0	0
$581$	13.7529	0.570568
$582$	0	0
$583$	19.2942	0.799086
$584$	9.36109	0.387365
$585$	0	0
$586$	−23.7319	−0.980355
$587$	16.8741	0.696468	0.348234	−	0.937408i	$-0.386782\pi$
$587$	16.8741	0.696468	0.348234	+	0.937408i	$0.386782\pi$
$588$	0	0
$589$	−1.87832	−0.0773950
$590$	−22.4735	−0.925218
$591$	0	0
$592$	32.9973	1.35618
$593$	28.1078	1.15425	0.577125	−	0.816656i	$-0.304175\pi$
$593$	28.1078	1.15425	0.577125	+	0.816656i	$0.304175\pi$
$594$	0	0
$595$	50.0460	2.05169
$596$	−27.8170	−1.13943
$597$	0	0
$598$	10.0927	0.412720
$599$	18.0476	0.737403	0.368702	−	0.929548i	$-0.379802\pi$
$599$	18.0476	0.737403	0.368702	+	0.929548i	$0.379802\pi$
$600$	0	0
$601$	−15.9159	−0.649225	−0.324612	−	0.945847i	$-0.605234\pi$
$601$	−15.9159	−0.649225	−0.324612	+	0.945847i	$0.605234\pi$
$602$	44.3127	1.80605
$603$	0	0
$604$	21.9325	0.892422
$605$	−5.66300	−0.230234
$606$	0	0
$607$	−35.4959	−1.44074	−0.720368	−	0.693592i	$-0.756027\pi$
$607$	−35.4959	−1.44074	−0.720368	+	0.693592i	$0.756027\pi$
$608$	−15.6826	−0.636014
$609$	0	0
$610$	−41.0377	−1.66157
$611$	6.59543	0.266823
$612$	0	0
$613$	−28.3858	−1.14649	−0.573246	−	0.819383i	$-0.694316\pi$
$613$	−28.3858	−1.14649	−0.573246	+	0.819383i	$0.694316\pi$
$614$	41.7388	1.68444
$615$	0	0
$616$	−10.9219	−0.440054
$617$	−30.7863	−1.23941	−0.619705	−	0.784835i	$-0.712748\pi$
$617$	−30.7863	−1.23941	−0.619705	+	0.784835i	$0.712748\pi$
$618$	0	0
$619$	1.91267	0.0768768	0.0384384	−	0.999261i	$-0.487762\pi$
$619$	1.91267	0.0768768	0.0384384	+	0.999261i	$0.487762\pi$
$620$	−4.23158	−0.169945
$621$	0	0
$622$	3.68315	0.147681
$623$	34.3319	1.37548
$624$	0	0
$625$	−15.9827	−0.639309
$626$	30.9186	1.23575
$627$	0	0
$628$	2.04486	0.0815988
$629$	−55.8025	−2.22499
$630$	0	0
$631$	6.41366	0.255324	0.127662	−	0.991818i	$-0.459253\pi$
$631$	6.41366	0.255324	0.127662	+	0.991818i	$0.459253\pi$
$632$	8.55896	0.340457
$633$	0	0
$634$	−23.6801	−0.940458
$635$	23.5256	0.933583
$636$	0	0
$637$	−17.1147	−0.678111
$638$	0	0
$639$	0	0
$640$	−9.29126	−0.367269
$641$	28.8063	1.13778	0.568891	−	0.822413i	$-0.307373\pi$
$641$	28.8063	1.13778	0.568891	+	0.822413i	$0.307373\pi$
$642$	0	0
$643$	3.06597	0.120910	0.0604551	−	0.998171i	$-0.480745\pi$
$643$	3.06597	0.120910	0.0604551	+	0.998171i	$0.480745\pi$
$644$	−46.8664	−1.84679
$645$	0	0
$646$	22.4676	0.883976
$647$	45.6599	1.79508	0.897538	−	0.440938i	$-0.145354\pi$
$647$	45.6599	1.79508	0.897538	+	0.440938i	$0.145354\pi$
$648$	0	0
$649$	21.3874	0.839529
$650$	−3.28342	−0.128786
$651$	0	0
$652$	17.7300	0.694361
$653$	12.8121	0.501377	0.250689	−	0.968068i	$-0.419343\pi$
$653$	12.8121	0.501377	0.250689	+	0.968068i	$0.419343\pi$
$654$	0	0
$655$	35.8413	1.40044
$656$	5.26929	0.205731
$657$	0	0
$658$	−57.2716	−2.23268
$659$	−25.2362	−0.983064	−0.491532	−	0.870860i	$-0.663563\pi$
$659$	−25.2362	−0.983064	−0.491532	+	0.870860i	$0.663563\pi$
$660$	0	0
$661$	−16.2771	−0.633106	−0.316553	−	0.948575i	$-0.602526\pi$
$661$	−16.2771	−0.633106	−0.316553	+	0.948575i	$0.602526\pi$
$662$	−20.3857	−0.792313
$663$	0	0
$664$	1.80833	0.0701767
$665$	−17.2770	−0.669974
$666$	0	0
$667$	0	0
$668$	−19.2847	−0.746146
$669$	0	0
$670$	−15.5964	−0.602540
$671$	39.0545	1.50768
$672$	0	0
$673$	−42.5390	−1.63976	−0.819879	−	0.572536i	$-0.805960\pi$
$673$	−42.5390	−1.63976	−0.819879	+	0.572536i	$0.805960\pi$
$674$	31.1098	1.19830
$675$	0	0
$676$	−26.9744	−1.03748
$677$	−20.8334	−0.800693	−0.400347	−	0.916364i	$-0.631110\pi$
$677$	−20.8334	−0.800693	−0.400347	+	0.916364i	$0.631110\pi$
$678$	0	0
$679$	60.3954	2.31776
$680$	6.58038	0.252346
$681$	0	0
$682$	7.53063	0.288363
$683$	−34.2477	−1.31045	−0.655226	−	0.755433i	$-0.727427\pi$
$683$	−34.2477	−1.31045	−0.655226	+	0.755433i	$0.727427\pi$
$684$	0	0
$685$	16.7262	0.639076
$686$	80.2180	3.06274
$687$	0	0
$688$	−15.0244	−0.572798
$689$	5.80505	0.221155
$690$	0	0
$691$	−9.12186	−0.347012	−0.173506	−	0.984833i	$-0.555510\pi$
$691$	−9.12186	−0.347012	−0.173506	+	0.984833i	$0.555510\pi$
$692$	8.09626	0.307774
$693$	0	0
$694$	69.3910	2.63404
$695$	−19.6090	−0.743812
$696$	0	0
$697$	−8.91103	−0.337529
$698$	−57.6074	−2.18047
$699$	0	0
$700$	15.2469	0.576279
$701$	42.8359	1.61789	0.808944	−	0.587885i	$-0.200039\pi$
$701$	42.8359	1.61789	0.808944	+	0.587885i	$0.200039\pi$
$702$	0	0
$703$	19.2643	0.726566
$704$	38.0942	1.43573
$705$	0	0
$706$	28.4852	1.07206
$707$	−44.5482	−1.67541
$708$	0	0
$709$	11.0057	0.413327	0.206663	−	0.978412i	$-0.433740\pi$
$709$	11.0057	0.413327	0.206663	+	0.978412i	$0.433740\pi$
$710$	1.17531	0.0441084
$711$	0	0
$712$	4.51419	0.169176
$713$	4.20106	0.157331
$714$	0	0
$715$	−7.97688	−0.298318
$716$	−16.4033	−0.613019
$717$	0	0
$718$	−51.0057	−1.90352
$719$	−18.1735	−0.677756	−0.338878	−	0.940830i	$-0.610047\pi$
$719$	−18.1735	−0.677756	−0.338878	+	0.940830i	$0.610047\pi$
$720$	0	0
$721$	−26.1708	−0.974651
$722$	31.6377	1.17743
$723$	0	0
$724$	48.4613	1.80105
$725$	0	0
$726$	0	0
$727$	−41.2879	−1.53128	−0.765641	−	0.643268i	$-0.777578\pi$
$727$	−41.2879	−1.53128	−0.765641	+	0.643268i	$0.777578\pi$
$728$	−3.28605	−0.121789
$729$	0	0
$730$	−59.3689	−2.19734
$731$	25.4080	0.939750
$732$	0	0
$733$	4.54792	0.167981	0.0839906	−	0.996467i	$-0.473233\pi$
$733$	4.54792	0.167981	0.0839906	+	0.996467i	$0.473233\pi$
$734$	9.19225	0.339292
$735$	0	0
$736$	35.0757	1.29291
$737$	14.8427	0.546736
$738$	0	0
$739$	47.3495	1.74178	0.870889	−	0.491480i	$-0.163544\pi$
$739$	47.3495	1.74178	0.870889	+	0.491480i	$0.163544\pi$
$740$	43.3995	1.59540
$741$	0	0
$742$	−50.4083	−1.85055
$743$	13.5622	0.497550	0.248775	−	0.968561i	$-0.419972\pi$
$743$	13.5622	0.497550	0.248775	+	0.968561i	$0.419972\pi$
$744$	0	0
$745$	22.9354	0.840286
$746$	−50.5412	−1.85045
$747$	0	0
$748$	−48.1700	−1.76127
$749$	29.6765	1.08435
$750$	0	0
$751$	28.5839	1.04304	0.521521	−	0.853239i	$-0.325365\pi$
$751$	28.5839	1.04304	0.521521	+	0.853239i	$0.325365\pi$
$752$	19.4181	0.708105
$753$	0	0
$754$	0	0
$755$	−18.0835	−0.658128
$756$	0	0
$757$	7.44366	0.270545	0.135272	−	0.990808i	$-0.456809\pi$
$757$	7.44366	0.270545	0.135272	+	0.990808i	$0.456809\pi$
$758$	−28.0018	−1.01707
$759$	0	0
$760$	−2.27170	−0.0824031
$761$	−44.1546	−1.60060	−0.800302	−	0.599597i	$-0.795327\pi$
$761$	−44.1546	−1.60060	−0.800302	+	0.599597i	$0.795327\pi$
$762$	0	0
$763$	77.2793	2.79770
$764$	31.7589	1.14900
$765$	0	0
$766$	51.2592	1.85207
$767$	6.43481	0.232348
$768$	0	0
$769$	50.0146	1.80357	0.901786	−	0.432183i	$-0.142256\pi$
$769$	50.0146	1.80357	0.901786	+	0.432183i	$0.142256\pi$
$770$	69.2675	2.49623
$771$	0	0
$772$	9.73106	0.350228
$773$	32.9745	1.18601	0.593004	−	0.805199i	$-0.297942\pi$
$773$	32.9745	1.18601	0.593004	+	0.805199i	$0.297942\pi$
$774$	0	0
$775$	−1.36672	−0.0490939
$776$	7.94119	0.285072
$777$	0	0
$778$	62.7988	2.25145
$779$	3.07629	0.110219
$780$	0	0
$781$	−1.11851	−0.0400233
$782$	−50.2510	−1.79697
$783$	0	0
$784$	−50.3888	−1.79960
$785$	−1.68600	−0.0601760
$786$	0	0
$787$	−22.3174	−0.795529	−0.397765	−	0.917488i	$-0.630214\pi$
$787$	−22.3174	−0.795529	−0.397765	+	0.917488i	$0.630214\pi$
$788$	−10.6087	−0.377921
$789$	0	0
$790$	−54.2818	−1.93126
$791$	12.9448	0.460264
$792$	0	0
$793$	11.7503	0.417265
$794$	28.1949	1.00060
$795$	0	0
$796$	−8.79544	−0.311746
$797$	−4.41478	−0.156380	−0.0781898	−	0.996938i	$-0.524914\pi$
$797$	−4.41478	−0.156380	−0.0781898	+	0.996938i	$0.524914\pi$
$798$	0	0
$799$	−32.8384	−1.16174
$800$	−11.4111	−0.403443
$801$	0	0
$802$	−30.9072	−1.09137
$803$	56.4999	1.99384
$804$	0	0
$805$	38.6417	1.36194
$806$	2.26574	0.0798072
$807$	0	0
$808$	−5.85749	−0.206066
$809$	−11.2673	−0.396138	−0.198069	−	0.980188i	$-0.563467\pi$
$809$	−11.2673	−0.396138	−0.198069	+	0.980188i	$0.563467\pi$
$810$	0	0
$811$	8.67128	0.304490	0.152245	−	0.988343i	$-0.451350\pi$
$811$	8.67128	0.304490	0.152245	+	0.988343i	$0.451350\pi$
$812$	0	0
$813$	0	0
$814$	−77.2349	−2.70708
$815$	−14.6185	−0.512065
$816$	0	0
$817$	−8.77143	−0.306874
$818$	42.7177	1.49359
$819$	0	0
$820$	6.93042	0.242021
$821$	28.9281	1.00960	0.504800	−	0.863237i	$-0.331566\pi$
$821$	28.9281	1.00960	0.504800	+	0.863237i	$0.331566\pi$
$822$	0	0
$823$	21.9692	0.765799	0.382900	−	0.923790i	$-0.374926\pi$
$823$	21.9692	0.765799	0.382900	+	0.923790i	$0.374926\pi$
$824$	−3.44111	−0.119877
$825$	0	0
$826$	−55.8769	−1.94421
$827$	−5.80098	−0.201720	−0.100860	−	0.994901i	$-0.532159\pi$
$827$	−5.80098	−0.201720	−0.100860	+	0.994901i	$0.532159\pi$
$828$	0	0
$829$	18.6693	0.648412	0.324206	−	0.945986i	$-0.394903\pi$
$829$	18.6693	0.648412	0.324206	+	0.945986i	$0.394903\pi$
$830$	−11.4686	−0.398080
$831$	0	0
$832$	11.4614	0.397352
$833$	85.2136	2.95248
$834$	0	0
$835$	15.9003	0.550254
$836$	16.6294	0.575139
$837$	0	0
$838$	−11.1087	−0.383745
$839$	44.1447	1.52404	0.762022	−	0.647551i	$-0.224207\pi$
$839$	44.1447	1.52404	0.762022	+	0.647551i	$0.224207\pi$
$840$	0	0
$841$	0	0
$842$	−3.97557	−0.137007
$843$	0	0
$844$	−22.9432	−0.789737
$845$	22.2406	0.765101
$846$	0	0
$847$	−14.0802	−0.483801
$848$	17.0911	0.586909
$849$	0	0
$850$	16.3480	0.560732
$851$	−43.0864	−1.47698
$852$	0	0
$853$	−52.7487	−1.80608	−0.903041	−	0.429555i	$-0.858670\pi$
$853$	−52.7487	−1.80608	−0.903041	+	0.429555i	$0.858670\pi$
$854$	−102.034	−3.49153
$855$	0	0
$856$	3.90206	0.133370
$857$	−51.1931	−1.74872	−0.874362	−	0.485274i	$-0.838720\pi$
$857$	−51.1931	−1.74872	−0.874362	+	0.485274i	$0.838720\pi$
$858$	0	0
$859$	−36.4802	−1.24469	−0.622344	−	0.782744i	$-0.713819\pi$
$859$	−36.4802	−1.24469	−0.622344	+	0.782744i	$0.713819\pi$
$860$	−19.7607	−0.673835
$861$	0	0
$862$	−41.9732	−1.42961
$863$	41.9778	1.42894	0.714470	−	0.699666i	$-0.246668\pi$
$863$	41.9778	1.42894	0.714470	+	0.699666i	$0.246668\pi$
$864$	0	0
$865$	−6.67543	−0.226971
$866$	37.7344	1.28227
$867$	0	0
$868$	−10.5212	−0.357113
$869$	51.6585	1.75239
$870$	0	0
$871$	4.46570	0.151315
$872$	10.1612	0.344102
$873$	0	0
$874$	17.3478	0.586797
$875$	−57.2343	−1.93487
$876$	0	0
$877$	−45.6889	−1.54281	−0.771403	−	0.636347i	$-0.780445\pi$
$877$	−45.6889	−1.54281	−0.771403	+	0.636347i	$0.780445\pi$
$878$	0.812838	0.0274319
$879$	0	0
$880$	−23.4853	−0.791690
$881$	43.5459	1.46710	0.733549	−	0.679636i	$-0.237862\pi$
$881$	43.5459	1.46710	0.733549	+	0.679636i	$0.237862\pi$
$882$	0	0
$883$	26.3374	0.886325	0.443162	−	0.896441i	$-0.353857\pi$
$883$	26.3374	0.886325	0.443162	+	0.896441i	$0.353857\pi$
$884$	−14.4929	−0.487449
$885$	0	0
$886$	11.3096	0.379953
$887$	29.6661	0.996091	0.498046	−	0.867151i	$-0.334051\pi$
$887$	29.6661	0.996091	0.498046	+	0.867151i	$0.334051\pi$
$888$	0	0
$889$	58.4928	1.96179
$890$	−28.6294	−0.959661
$891$	0	0
$892$	17.6845	0.592120
$893$	11.3366	0.379363
$894$	0	0
$895$	13.5246	0.452078
$896$	−23.1013	−0.771761
$897$	0	0
$898$	5.75758	0.192133
$899$	0	0
$900$	0	0
$901$	−28.9031	−0.962902
$902$	−12.3335	−0.410662
$903$	0	0
$904$	1.70207	0.0566099
$905$	−39.9567	−1.32821
$906$	0	0
$907$	−8.07782	−0.268220	−0.134110	−	0.990966i	$-0.542817\pi$
$907$	−8.07782	−0.268220	−0.134110	+	0.990966i	$0.542817\pi$
$908$	−15.1760	−0.503635
$909$	0	0
$910$	20.8405	0.690855
$911$	48.9849	1.62294	0.811471	−	0.584392i	$-0.198667\pi$
$911$	48.9849	1.62294	0.811471	+	0.584392i	$0.198667\pi$
$912$	0	0
$913$	10.9143	0.361212
$914$	75.5269	2.49821
$915$	0	0
$916$	−5.35328	−0.176877
$917$	89.1140	2.94281
$918$	0	0
$919$	43.7574	1.44342	0.721712	−	0.692194i	$-0.243356\pi$
$919$	43.7574	1.44342	0.721712	+	0.692194i	$0.243356\pi$
$920$	5.08087	0.167511
$921$	0	0
$922$	52.5721	1.73137
$923$	−0.336525	−0.0110768
$924$	0	0
$925$	14.0172	0.460882
$926$	−58.8988	−1.93554
$927$	0	0
$928$	0	0
$929$	−28.4491	−0.933386	−0.466693	−	0.884419i	$-0.654555\pi$
$929$	−28.4491	−0.933386	−0.466693	+	0.884419i	$0.654555\pi$
$930$	0	0
$931$	−29.4177	−0.964125
$932$	−4.01942	−0.131660
$933$	0	0
$934$	−27.5213	−0.900525
$935$	39.7166	1.29887
$936$	0	0
$937$	6.02644	0.196875	0.0984377	−	0.995143i	$-0.468615\pi$
$937$	6.02644	0.196875	0.0984377	+	0.995143i	$0.468615\pi$
$938$	−38.7780	−1.26615
$939$	0	0
$940$	25.5396	0.833009
$941$	37.8665	1.23441	0.617206	−	0.786801i	$-0.288264\pi$
$941$	37.8665	1.23441	0.617206	+	0.786801i	$0.288264\pi$
$942$	0	0
$943$	−6.88042	−0.224057
$944$	18.9452	0.616614
$945$	0	0
$946$	35.1667	1.14337
$947$	−12.9249	−0.420002	−0.210001	−	0.977701i	$-0.567347\pi$
$947$	−12.9249	−0.420002	−0.210001	+	0.977701i	$0.567347\pi$
$948$	0	0
$949$	16.9991	0.551814
$950$	−5.64370	−0.183106
$951$	0	0
$952$	16.3611	0.530267
$953$	6.55592	0.212367	0.106184	−	0.994347i	$-0.466137\pi$
$953$	6.55592	0.212367	0.106184	+	0.994347i	$0.466137\pi$
$954$	0	0
$955$	−26.1854	−0.847341
$956$	52.4588	1.69664
$957$	0	0
$958$	54.4461	1.75907
$959$	41.5873	1.34292
$960$	0	0
$961$	−30.0569	−0.969577
$962$	−23.2376	−0.749211
$963$	0	0
$964$	25.6927	0.827506
$965$	−8.02333	−0.258280
$966$	0	0
$967$	−34.1333	−1.09765	−0.548826	−	0.835937i	$-0.684925\pi$
$967$	−34.1333	−1.09765	−0.548826	+	0.835937i	$0.684925\pi$
$968$	−1.85136	−0.0595049
$969$	0	0
$970$	−50.3638	−1.61708
$971$	29.7514	0.954768	0.477384	−	0.878695i	$-0.341585\pi$
$971$	29.7514	0.954768	0.477384	+	0.878695i	$0.341585\pi$
$972$	0	0
$973$	−48.7549	−1.56301
$974$	−19.1411	−0.613321
$975$	0	0
$976$	34.5949	1.10736
$977$	32.5600	1.04169	0.520844	−	0.853652i	$-0.325617\pi$
$977$	32.5600	1.04169	0.520844	+	0.853652i	$0.325617\pi$
$978$	0	0
$979$	27.2459	0.870782
$980$	−66.2736	−2.11703
$981$	0	0
$982$	−25.1254	−0.801783
$983$	−12.2384	−0.390344	−0.195172	−	0.980769i	$-0.562526\pi$
$983$	−12.2384	−0.390344	−0.195172	+	0.980769i	$0.562526\pi$
$984$	0	0
$985$	8.74700	0.278702
$986$	0	0
$987$	0	0
$988$	5.00328	0.159175
$989$	19.6182	0.623821
$990$	0	0
$991$	24.6058	0.781628	0.390814	−	0.920470i	$-0.372194\pi$
$991$	24.6058	0.781628	0.390814	+	0.920470i	$0.372194\pi$
$992$	7.87427	0.250008
$993$	0	0
$994$	2.92222	0.0926873
$995$	7.25191	0.229901
$996$	0	0
$997$	−9.16138	−0.290144	−0.145072	−	0.989421i	$-0.546341\pi$
$997$	−9.16138	−0.290144	−0.145072	+	0.989421i	$0.546341\pi$
$998$	61.3526	1.94208
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7569.2.a.bl.1.1		9
3.2	odd	2		2523.2.a.p.1.9		9
29.7	even	7		261.2.k.b.136.3		18
29.25	even	7		261.2.k.b.190.3		18
29.28	even	2		7569.2.a.bk.1.9		9
87.65	odd	14		87.2.g.b.49.1	yes	18
87.83	odd	14		87.2.g.b.16.1	✓	18
87.86	odd	2		2523.2.a.q.1.1		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
87.2.g.b.16.1	✓	18	87.83	odd	14
87.2.g.b.49.1	yes	18	87.65	odd	14
261.2.k.b.136.3		18	29.7	even	7
261.2.k.b.190.3		18	29.25	even	7
2523.2.a.p.1.9		9	3.2	odd	2
2523.2.a.q.1.1		9	87.86	odd	2
7569.2.a.bk.1.9		9	29.28	even	2
7569.2.a.bl.1.1		9	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 7569 = 3^{2} \cdot 29^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7569.a (trivial)

\(f(q)\)	\(=\)	\(q-2.07337 q^{2} +2.29887 q^{4} -1.89543 q^{5} -4.71271 q^{7} -0.619659 q^{8} +O(q^{10})\)	\(q-2.07337 q^{2} +2.29887 q^{4} -1.89543 q^{5} -4.71271 q^{7} -0.619659 q^{8} +3.92993 q^{10} -3.74001 q^{11} -1.12526 q^{13} +9.77119 q^{14} -3.31295 q^{16} +5.60261 q^{17} -1.93415 q^{19} -4.35734 q^{20} +7.75444 q^{22} +4.32591 q^{23} -1.40734 q^{25} +2.33307 q^{26} -10.8339 q^{28} +0.971138 q^{31} +8.10829 q^{32} -11.6163 q^{34} +8.93262 q^{35} -9.96009 q^{37} +4.01020 q^{38} +1.17452 q^{40} -1.59051 q^{41} +4.53504 q^{43} -8.59779 q^{44} -8.96921 q^{46} -5.86127 q^{47} +15.2096 q^{49} +2.91793 q^{50} -2.58681 q^{52} -5.15887 q^{53} +7.08895 q^{55} +2.92027 q^{56} -5.71853 q^{59} -10.4423 q^{61} -2.01353 q^{62} -10.1856 q^{64} +2.13285 q^{65} -3.96861 q^{67} +12.8796 q^{68} -18.5206 q^{70} +0.299065 q^{71} -15.1069 q^{73} +20.6510 q^{74} -4.44634 q^{76} +17.6256 q^{77} -13.8124 q^{79} +6.27947 q^{80} +3.29773 q^{82} -2.91826 q^{83} -10.6194 q^{85} -9.40282 q^{86} +2.31753 q^{88} -7.28497 q^{89} +5.30301 q^{91} +9.94468 q^{92} +12.1526 q^{94} +3.66605 q^{95} -12.8154 q^{97} -31.5352 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + q^{2} + 11 q^{4} + 5 q^{7} + 12 q^{8}+O(q^{10}) \) 9 * q + q^2 + 11 * q^4 + 5 * q^7 + 12 * q^8	\( 9 q + q^{2} + 11 q^{4} + 5 q^{7} + 12 q^{8} - 4 q^{10} + 3 q^{11} + 5 q^{13} + 15 q^{14} - 5 q^{16} + 16 q^{17} + q^{19} - 8 q^{20} + 24 q^{22} + 10 q^{23} + 9 q^{25} + 12 q^{26} - 24 q^{28} + 4 q^{31} + 25 q^{32} + 24 q^{34} + 44 q^{35} - 25 q^{37} + 10 q^{38} - 5 q^{40} + 34 q^{41} - 12 q^{43} - 23 q^{44} - 6 q^{46} + 8 q^{47} + 26 q^{49} + 27 q^{50} - 23 q^{52} + 32 q^{53} + 5 q^{55} - 14 q^{56} - 10 q^{59} - 51 q^{61} - 8 q^{62} - 8 q^{64} + 11 q^{65} + 7 q^{67} + 11 q^{68} + 14 q^{70} - 7 q^{71} - 17 q^{73} + 62 q^{74} + 6 q^{76} + 64 q^{77} - 13 q^{79} - 54 q^{80} + 37 q^{82} + 31 q^{83} - 42 q^{85} + 70 q^{86} - 29 q^{88} - 32 q^{89} + 45 q^{91} - 9 q^{92} + 38 q^{94} - 20 q^{95} - 16 q^{97} - 12 q^{98}+O(q^{100}) \) 9 * q + q^2 + 11 * q^4 + 5 * q^7 + 12 * q^8 - 4 * q^10 + 3 * q^11 + 5 * q^13 + 15 * q^14 - 5 * q^16 + 16 * q^17 + q^19 - 8 * q^20 + 24 * q^22 + 10 * q^23 + 9 * q^25 + 12 * q^26 - 24 * q^28 + 4 * q^31 + 25 * q^32 + 24 * q^34 + 44 * q^35 - 25 * q^37 + 10 * q^38 - 5 * q^40 + 34 * q^41 - 12 * q^43 - 23 * q^44 - 6 * q^46 + 8 * q^47 + 26 * q^49 + 27 * q^50 - 23 * q^52 + 32 * q^53 + 5 * q^55 - 14 * q^56 - 10 * q^59 - 51 * q^61 - 8 * q^62 - 8 * q^64 + 11 * q^65 + 7 * q^67 + 11 * q^68 + 14 * q^70 - 7 * q^71 - 17 * q^73 + 62 * q^74 + 6 * q^76 + 64 * q^77 - 13 * q^79 - 54 * q^80 + 37 * q^82 + 31 * q^83 - 42 * q^85 + 70 * q^86 - 29 * q^88 - 32 * q^89 + 45 * q^91 - 9 * q^92 + 38 * q^94 - 20 * q^95 - 16 * q^97 - 12 * q^98

Introduction

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