LMFDB - Embedded newform 7569.2.a.bj.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} - 4x^{8} - 6x^{7} + 33x^{6} + 6x^{5} - 90x^{4} + 21x^{3} + 84x^{2} - 36x - 8 $ x^9 - 4x^8 - 6x^7 + 33x^6 + 6x^5 - 90x^4 + 21x^3 + 84x^2 - 36x - 8
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			$-1.77801$ 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)

$f(q)$	$=$	$q-2.77801 q^{2} +5.71733 q^{4} +2.53766 q^{5} +0.846706 q^{7} -10.3268 q^{8} -7.04965 q^{10} -1.25795 q^{11} -2.22142 q^{13} -2.35216 q^{14} +17.2532 q^{16} -3.75587 q^{17} -2.95482 q^{19} +14.5087 q^{20} +3.49460 q^{22} -6.41819 q^{23} +1.43972 q^{25} +6.17112 q^{26} +4.84090 q^{28} +0.575935 q^{31} -27.2761 q^{32} +10.4338 q^{34} +2.14865 q^{35} -4.01493 q^{37} +8.20851 q^{38} -26.2059 q^{40} +9.29991 q^{41} +9.54895 q^{43} -7.19214 q^{44} +17.8298 q^{46} +2.81581 q^{47} -6.28309 q^{49} -3.99957 q^{50} -12.7006 q^{52} -3.94794 q^{53} -3.19226 q^{55} -8.74375 q^{56} -2.24563 q^{59} -4.76610 q^{61} -1.59995 q^{62} +41.2667 q^{64} -5.63721 q^{65} -0.143981 q^{67} -21.4736 q^{68} -5.96898 q^{70} +5.11884 q^{71} +1.37989 q^{73} +11.1535 q^{74} -16.8937 q^{76} -1.06512 q^{77} +0.269362 q^{79} +43.7829 q^{80} -25.8352 q^{82} +6.83476 q^{83} -9.53112 q^{85} -26.5271 q^{86} +12.9906 q^{88} +4.61019 q^{89} -1.88089 q^{91} -36.6949 q^{92} -7.82234 q^{94} -7.49833 q^{95} -4.61991 q^{97} +17.4545 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - 5 q^{2} + 11 q^{4} + 4 q^{5} + 5 q^{7} - 24 q^{8} + q^{11} + q^{13} - 9 q^{14} + 35 q^{16} - 2 q^{17} + 9 q^{19} + 18 q^{20} - 4 q^{22} + 4 q^{23} + q^{25} + 8 q^{26} + 40 q^{28} + 8 q^{31} - 43 q^{32}+ \cdots + 30 q^{98}+O(q^{100}) $ 9 * q - 5 * q^2 + 11 * q^4 + 4 * q^5 + 5 * q^7 - 24 * q^8 + q^11 + q^13 - 9 * q^14 + 35 * q^16 - 2 * q^17 + 9 * q^19 + 18 * q^20 - 4 * q^22 + 4 * q^23 + q^25 + 8 * q^26 + 40 * q^28 + 8 * q^31 - 43 * q^32 - 4 * q^34 - 22 * q^35 + 27 * q^37 + 30 * q^38 - 29 * q^40 - 12 * q^41 + 16 * q^43 - 37 * q^44 - 22 * q^46 + 8 * q^47 - 6 * q^49 + 7 * q^50 + 33 * q^52 + 8 * q^53 + 9 * q^55 - 40 * q^56 + 16 * q^59 + 21 * q^61 + 32 * q^62 + 36 * q^64 + 31 * q^65 + 3 * q^67 - 33 * q^68 - 6 * q^70 + 33 * q^71 + 3 * q^73 - 28 * q^74 - 26 * q^76 - 24 * q^77 + 3 * q^79 + 64 * q^80 + 13 * q^82 - 13 * q^83 + 6 * q^85 - 58 * q^86 + 27 * q^88 - 6 * q^89 + q^91 + 29 * q^92 - 18 * q^94 - 48 * q^95 + 4 * q^97 + 30 * q^98

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 $

$2$	−2.77801	−1.96435	−0.982175	−	0.187971i	$-0.939809\pi$
$2$	−2.77801	−1.96435	−0.982175	+	0.187971i	$0.939809\pi$
$3$	0	0
$3$	0	0
$4$	5.71733	2.85867
$5$	2.53766	1.13488	0.567438	−	0.823416i	$-0.307935\pi$
$5$	2.53766	1.13488	0.567438	+	0.823416i	$0.307935\pi$
$6$	0	0
$7$	0.846706	0.320025	0.160012	−	0.987115i	$-0.448847\pi$
$7$	0.846706	0.320025	0.160012	+	0.987115i	$0.448847\pi$
$8$	−10.3268	−3.65107
$9$	0	0
$10$	−7.04965	−2.22929
$11$	−1.25795	−0.379287	−0.189644	−	0.981853i	$-0.560733\pi$
$11$	−1.25795	−0.379287	−0.189644	+	0.981853i	$0.560733\pi$
$12$	0	0
$13$	−2.22142	−0.616111	−0.308055	−	0.951368i	$-0.599678\pi$
$13$	−2.22142	−0.616111	−0.308055	+	0.951368i	$0.599678\pi$
$14$	−2.35216	−0.628640
$15$	0	0
$16$	17.2532	4.31331
$17$	−3.75587	−0.910932	−0.455466	−	0.890253i	$-0.650527\pi$
$17$	−3.75587	−0.910932	−0.455466	+	0.890253i	$0.650527\pi$
$18$	0	0
$19$	−2.95482	−0.677882	−0.338941	−	0.940808i	$-0.610069\pi$
$19$	−2.95482	−0.677882	−0.338941	+	0.940808i	$0.610069\pi$
$20$	14.5087	3.24423
$21$	0	0
$22$	3.49460	0.745052
$23$	−6.41819	−1.33828	−0.669142	−	0.743134i	$-0.733338\pi$
$23$	−6.41819	−1.33828	−0.669142	+	0.743134i	$0.733338\pi$
$24$	0	0
$25$	1.43972	0.287945
$26$	6.17112	1.21026
$27$	0	0
$28$	4.84090	0.914844
$29$	0	0
$29$	0	0
$30$	0	0
$31$	0.575935	0.103441	0.0517205	−	0.998662i	$-0.483529\pi$
$31$	0.575935	0.103441	0.0517205	+	0.998662i	$0.483529\pi$
$32$	−27.2761	−4.82178
$33$	0	0
$34$	10.4338	1.78939
$35$	2.14865	0.363189
$36$	0	0
$37$	−4.01493	−0.660051	−0.330026	−	0.943972i	$-0.607057\pi$
$37$	−4.01493	−0.660051	−0.330026	+	0.943972i	$0.607057\pi$
$38$	8.20851	1.33160
$39$	0	0
$40$	−26.2059	−4.14351
$41$	9.29991	1.45240	0.726201	−	0.687482i	$-0.241284\pi$
$41$	9.29991	1.45240	0.726201	+	0.687482i	$0.241284\pi$
$42$	0	0
$43$	9.54895	1.45620	0.728101	−	0.685470i	$-0.240403\pi$
$43$	9.54895	1.45620	0.728101	+	0.685470i	$0.240403\pi$
$44$	−7.19214	−1.08426
$45$	0	0
$46$	17.8298	2.62886
$47$	2.81581	0.410728	0.205364	−	0.978686i	$-0.434162\pi$
$47$	2.81581	0.410728	0.205364	+	0.978686i	$0.434162\pi$
$48$	0	0
$49$	−6.28309	−0.897584
$50$	−3.99957	−0.565624
$51$	0	0
$52$	−12.7006	−1.76126
$53$	−3.94794	−0.542291	−0.271145	−	0.962538i	$-0.587402\pi$
$53$	−3.94794	−0.542291	−0.271145	+	0.962538i	$0.587402\pi$
$54$	0	0
$55$	−3.19226	−0.430444
$56$	−8.74375	−1.16843
$57$	0	0
$58$	0	0
$59$	−2.24563	−0.292356	−0.146178	−	0.989258i	$-0.546697\pi$
$59$	−2.24563	−0.292356	−0.146178	+	0.989258i	$0.546697\pi$
$60$	0	0
$61$	−4.76610	−0.610237	−0.305119	−	0.952314i	$-0.598696\pi$
$61$	−4.76610	−0.610237	−0.305119	+	0.952314i	$0.598696\pi$
$62$	−1.59995	−0.203194
$63$	0	0
$64$	41.2667	5.15834
$65$	−5.63721	−0.699209
$66$	0	0
$67$	−0.143981	−0.0175900	−0.00879502	−	0.999961i	$-0.502800\pi$
$67$	−0.143981	−0.0175900	−0.00879502	+	0.999961i	$0.502800\pi$
$68$	−21.4736	−2.60405
$69$	0	0
$70$	−5.96898	−0.713429
$71$	5.11884	0.607495	0.303747	−	0.952753i	$-0.401762\pi$
$71$	5.11884	0.607495	0.303747	+	0.952753i	$0.401762\pi$
$72$	0	0
$73$	1.37989	0.161504	0.0807519	−	0.996734i	$-0.474268\pi$
$73$	1.37989	0.161504	0.0807519	+	0.996734i	$0.474268\pi$
$74$	11.1535	1.29657
$75$	0	0
$76$	−16.8937	−1.93784
$77$	−1.06512	−0.121381
$78$	0	0
$79$	0.269362	0.0303056	0.0151528	−	0.999885i	$-0.495177\pi$
$79$	0.269362	0.0303056	0.0151528	+	0.999885i	$0.495177\pi$
$80$	43.7829	4.89507
$81$	0	0
$82$	−25.8352	−2.85303
$83$	6.83476	0.750213	0.375106	−	0.926982i	$-0.377606\pi$
$83$	6.83476	0.750213	0.375106	+	0.926982i	$0.377606\pi$
$84$	0	0
$85$	−9.53112	−1.03380
$86$	−26.5271	−2.86049
$87$	0	0
$88$	12.9906	1.38480
$89$	4.61019	0.488679	0.244339	−	0.969690i	$-0.421429\pi$
$89$	4.61019	0.488679	0.244339	+	0.969690i	$0.421429\pi$
$90$	0	0
$91$	−1.88089	−0.197171
$92$	−36.6949	−3.82571
$93$	0	0
$94$	−7.82234	−0.806813
$95$	−7.49833	−0.769312
$96$	0	0
$97$	−4.61991	−0.469081	−0.234540	−	0.972106i	$-0.575358\pi$
$97$	−4.61991	−0.469081	−0.234540	+	0.972106i	$0.575358\pi$
$98$	17.4545	1.76317
$99$	0	0
$100$	8.23138	0.823138
$101$	9.11390	0.906867	0.453433	−	0.891290i	$-0.350199\pi$
$101$	9.11390	0.906867	0.453433	+	0.891290i	$0.350199\pi$
$102$	0	0
$103$	−14.4058	−1.41945	−0.709724	−	0.704480i	$-0.751180\pi$
$103$	−14.4058	−1.41945	−0.709724	+	0.704480i	$0.751180\pi$
$104$	22.9401	2.24946
$105$	0	0
$106$	10.9674	1.06525
$107$	1.83345	0.177246	0.0886230	−	0.996065i	$-0.471753\pi$
$107$	1.83345	0.177246	0.0886230	+	0.996065i	$0.471753\pi$
$108$	0	0
$109$	14.3457	1.37407	0.687033	−	0.726626i	$-0.258913\pi$
$109$	14.3457	1.37407	0.687033	+	0.726626i	$0.258913\pi$
$110$	8.86812	0.845542
$111$	0	0
$112$	14.6084	1.38037
$113$	11.7792	1.10810	0.554049	−	0.832484i	$-0.313082\pi$
$113$	11.7792	1.10810	0.554049	+	0.832484i	$0.313082\pi$
$114$	0	0
$115$	−16.2872	−1.51879
$116$	0	0
$117$	0	0
$118$	6.23838	0.574290
$119$	−3.18012	−0.291521
$120$	0	0
$121$	−9.41755	−0.856141
$122$	13.2403	1.19872
$123$	0	0
$124$	3.29282	0.295704
$125$	−9.03477	−0.808095
$126$	0	0
$127$	12.0293	1.06743	0.533713	−	0.845666i	$-0.320796\pi$
$127$	12.0293	1.06743	0.533713	+	0.845666i	$0.320796\pi$
$128$	−60.0872	−5.31100
$129$	0	0
$130$	15.6602	1.37349
$131$	2.38388	0.208280	0.104140	−	0.994563i	$-0.466791\pi$
$131$	2.38388	0.208280	0.104140	+	0.994563i	$0.466791\pi$
$132$	0	0
$133$	−2.50186	−0.216939
$134$	0.399980	0.0345530
$135$	0	0
$136$	38.7861	3.32588
$137$	13.5702	1.15938	0.579689	−	0.814838i	$-0.303174\pi$
$137$	13.5702	1.15938	0.579689	+	0.814838i	$0.303174\pi$
$138$	0	0
$139$	19.9821	1.69486	0.847430	−	0.530907i	$-0.178148\pi$
$139$	19.9821	1.69486	0.847430	+	0.530907i	$0.178148\pi$
$140$	12.2846	1.03824
$141$	0	0
$142$	−14.2202	−1.19333
$143$	2.79444	0.233683
$144$	0	0
$145$	0	0
$146$	−3.83335	−0.317250
$147$	0	0
$148$	−22.9547	−1.88687
$149$	16.1338	1.32173	0.660864	−	0.750505i	$-0.270190\pi$
$149$	16.1338	1.32173	0.660864	+	0.750505i	$0.270190\pi$
$150$	0	0
$151$	2.80144	0.227978	0.113989	−	0.993482i	$-0.463637\pi$
$151$	2.80144	0.227978	0.113989	+	0.993482i	$0.463637\pi$
$152$	30.5138	2.47499
$153$	0	0
$154$	2.95890	0.238435
$155$	1.46153	0.117393
$156$	0	0
$157$	8.49632	0.678080	0.339040	−	0.940772i	$-0.389898\pi$
$157$	8.49632	0.678080	0.339040	+	0.940772i	$0.389898\pi$
$158$	−0.748291	−0.0595309
$159$	0	0
$160$	−69.2175	−5.47212
$161$	−5.43432	−0.428284
$162$	0	0
$163$	−20.7440	−1.62479	−0.812396	−	0.583106i	$-0.801837\pi$
$163$	−20.7440	−1.62479	−0.812396	+	0.583106i	$0.801837\pi$
$164$	53.1707	4.15193
$165$	0	0
$166$	−18.9870	−1.47368
$167$	−9.79965	−0.758320	−0.379160	−	0.925331i	$-0.623787\pi$
$167$	−9.79965	−0.758320	−0.379160	+	0.925331i	$0.623787\pi$
$168$	0	0
$169$	−8.06530	−0.620408
$170$	26.4775	2.03073
$171$	0	0
$172$	54.5945	4.16280
$173$	20.1246	1.53005	0.765023	−	0.644003i	$-0.222728\pi$
$173$	20.1246	1.53005	0.765023	+	0.644003i	$0.222728\pi$
$174$	0	0
$175$	1.21902	0.0921494
$176$	−21.7038	−1.63598
$177$	0	0
$178$	−12.8071	−0.959936
$179$	24.5921	1.83810	0.919049	−	0.394143i	$-0.128958\pi$
$179$	24.5921	1.83810	0.919049	+	0.394143i	$0.128958\pi$
$180$	0	0
$181$	19.4105	1.44277	0.721384	−	0.692535i	$-0.243506\pi$
$181$	19.4105	1.44277	0.721384	+	0.692535i	$0.243506\pi$
$182$	5.22512	0.387312
$183$	0	0
$184$	66.2793	4.88617
$185$	−10.1885	−0.749077
$186$	0	0
$187$	4.72471	0.345505
$188$	16.0989	1.17413
$189$	0	0
$190$	20.8304	1.51120
$191$	0.923161	0.0667976	0.0333988	−	0.999442i	$-0.489367\pi$
$191$	0.923161	0.0667976	0.0333988	+	0.999442i	$0.489367\pi$
$192$	0	0
$193$	9.74495	0.701457	0.350729	−	0.936477i	$-0.385934\pi$
$193$	9.74495	0.701457	0.350729	+	0.936477i	$0.385934\pi$
$194$	12.8341	0.921438
$195$	0	0
$196$	−35.9225	−2.56589
$197$	7.47012	0.532224	0.266112	−	0.963942i	$-0.414261\pi$
$197$	7.47012	0.532224	0.266112	+	0.963942i	$0.414261\pi$
$198$	0	0
$199$	25.9244	1.83773	0.918864	−	0.394573i	$-0.129108\pi$
$199$	25.9244	1.83773	0.918864	+	0.394573i	$0.129108\pi$
$200$	−14.8677	−1.05131
$201$	0	0
$202$	−25.3185	−1.78140
$203$	0	0
$204$	0	0
$205$	23.6000	1.64830
$206$	40.0195	2.78829
$207$	0	0
$208$	−38.3267	−2.65748
$209$	3.71702	0.257112
$210$	0	0
$211$	19.2611	1.32599	0.662996	−	0.748623i	$-0.269285\pi$
$211$	19.2611	1.32599	0.662996	+	0.748623i	$0.269285\pi$
$212$	−22.5717	−1.55023
$213$	0	0
$214$	−5.09333	−0.348173
$215$	24.2320	1.65261
$216$	0	0
$217$	0.487648	0.0331037
$218$	−39.8524	−2.69914
$219$	0	0
$220$	−18.2512	−1.23050
$221$	8.34335	0.561235
$222$	0	0
$223$	−0.788104	−0.0527753	−0.0263877	−	0.999652i	$-0.508400\pi$
$223$	−0.788104	−0.0527753	−0.0263877	+	0.999652i	$0.508400\pi$
$224$	−23.0948	−1.54309
$225$	0	0
$226$	−32.7229	−2.17669
$227$	18.8209	1.24919	0.624595	−	0.780949i	$-0.285264\pi$
$227$	18.8209	1.24919	0.624595	+	0.780949i	$0.285264\pi$
$228$	0	0
$229$	26.9061	1.77800	0.889002	−	0.457903i	$-0.151399\pi$
$229$	26.9061	1.77800	0.889002	+	0.457903i	$0.151399\pi$
$230$	45.2459	2.98343
$231$	0	0
$232$	0	0
$233$	13.1181	0.859393	0.429696	−	0.902973i	$-0.358621\pi$
$233$	13.1181	0.859393	0.429696	+	0.902973i	$0.358621\pi$
$234$	0	0
$235$	7.14557	0.466126
$236$	−12.8390	−0.835749
$237$	0	0
$238$	8.83439	0.572648
$239$	−4.04911	−0.261915	−0.130958	−	0.991388i	$-0.541805\pi$
$239$	−4.04911	−0.261915	−0.130958	+	0.991388i	$0.541805\pi$
$240$	0	0
$241$	4.02740	0.259428	0.129714	−	0.991551i	$-0.458594\pi$
$241$	4.02740	0.259428	0.129714	+	0.991551i	$0.458594\pi$
$242$	26.1621	1.68176
$243$	0	0
$244$	−27.2494	−1.74446
$245$	−15.9444	−1.01865
$246$	0	0
$247$	6.56389	0.417650
$248$	−5.94756	−0.377671
$249$	0	0
$250$	25.0987	1.58738
$251$	−30.3241	−1.91404	−0.957019	−	0.290027i	$-0.906336\pi$
$251$	−30.3241	−1.91404	−0.957019	+	0.290027i	$0.906336\pi$
$252$	0	0
$253$	8.07378	0.507594
$254$	−33.4174	−2.09680
$255$	0	0
$256$	84.3892	5.27433
$257$	8.14701	0.508196	0.254098	−	0.967178i	$-0.418221\pi$
$257$	8.14701	0.508196	0.254098	+	0.967178i	$0.418221\pi$
$258$	0	0
$259$	−3.39947	−0.211233
$260$	−32.2298	−1.99881
$261$	0	0
$262$	−6.62243	−0.409135
$263$	5.67062	0.349666	0.174833	−	0.984598i	$-0.444062\pi$
$263$	5.67062	0.349666	0.174833	+	0.984598i	$0.444062\pi$
$264$	0	0
$265$	−10.0185	−0.615433
$266$	6.95019	0.426144
$267$	0	0
$268$	−0.823186	−0.0502841
$269$	−20.5529	−1.25313	−0.626565	−	0.779369i	$-0.715540\pi$
$269$	−20.5529	−1.25313	−0.626565	+	0.779369i	$0.715540\pi$
$270$	0	0
$271$	9.59174	0.582657	0.291328	−	0.956623i	$-0.405903\pi$
$271$	9.59174	0.582657	0.291328	+	0.956623i	$0.405903\pi$
$272$	−64.8009	−3.92913
$273$	0	0
$274$	−37.6980	−2.27742
$275$	−1.81110	−0.109214
$276$	0	0
$277$	−25.8418	−1.55269	−0.776343	−	0.630311i	$-0.782928\pi$
$277$	−25.8418	−1.55269	−0.776343	+	0.630311i	$0.782928\pi$
$278$	−55.5105	−3.32930
$279$	0	0
$280$	−22.1887	−1.32603
$281$	18.3009	1.09174	0.545870	−	0.837870i	$-0.316199\pi$
$281$	18.3009	1.09174	0.545870	+	0.837870i	$0.316199\pi$
$282$	0	0
$283$	−13.8005	−0.820355	−0.410177	−	0.912006i	$-0.634533\pi$
$283$	−13.8005	−0.820355	−0.410177	+	0.912006i	$0.634533\pi$
$284$	29.2661	1.73663
$285$	0	0
$286$	−7.76298	−0.459035
$287$	7.87429	0.464805
$288$	0	0
$289$	−2.89345	−0.170203
$290$	0	0
$291$	0	0
$292$	7.88929	0.461686
$293$	13.2248	0.772600	0.386300	−	0.922373i	$-0.373753\pi$
$293$	13.2248	0.772600	0.386300	+	0.922373i	$0.373753\pi$
$294$	0	0
$295$	−5.69865	−0.331788
$296$	41.4614	2.40989
$297$	0	0
$298$	−44.8197	−2.59634
$299$	14.2575	0.824531
$300$	0	0
$301$	8.08515	0.466020
$302$	−7.78241	−0.447828
$303$	0	0
$304$	−50.9802	−2.92391
$305$	−12.0948	−0.692544
$306$	0	0
$307$	23.3280	1.33140	0.665698	−	0.746221i	$-0.268134\pi$
$307$	23.3280	1.33140	0.665698	+	0.746221i	$0.268134\pi$
$308$	−6.08962	−0.346989
$309$	0	0
$310$	−4.06014	−0.230601
$311$	5.68433	0.322329	0.161164	−	0.986928i	$-0.448475\pi$
$311$	5.68433	0.322329	0.161164	+	0.986928i	$0.448475\pi$
$312$	0	0
$313$	3.99178	0.225629	0.112814	−	0.993616i	$-0.464013\pi$
$313$	3.99178	0.225629	0.112814	+	0.993616i	$0.464013\pi$
$314$	−23.6029	−1.33199
$315$	0	0
$316$	1.54004	0.0866337
$317$	−12.2855	−0.690020	−0.345010	−	0.938599i	$-0.612125\pi$
$317$	−12.2855	−0.690020	−0.345010	+	0.938599i	$0.612125\pi$
$318$	0	0
$319$	0	0
$320$	104.721	5.85408
$321$	0	0
$322$	15.0966	0.841300
$323$	11.0979	0.617504
$324$	0	0
$325$	−3.19823	−0.177406
$326$	57.6269	3.19166
$327$	0	0
$328$	−96.0382	−5.30282
$329$	2.38416	0.131443
$330$	0	0
$331$	−15.4931	−0.851579	−0.425790	−	0.904822i	$-0.640004\pi$
$331$	−15.4931	−0.851579	−0.425790	+	0.904822i	$0.640004\pi$
$332$	39.0766	2.14461
$333$	0	0
$334$	27.2235	1.48960
$335$	−0.365374	−0.0199625
$336$	0	0
$337$	−0.0322530	−0.00175693	−0.000878465	−	1.00000i	$-0.500280\pi$
$337$	−0.0322530	−0.00175693	−0.000878465		1.00000i	$0.500280\pi$
$338$	22.4055	1.21870
$339$	0	0
$340$	−54.4926	−2.95528
$341$	−0.724500	−0.0392339
$342$	0	0
$343$	−11.2469	−0.607274
$344$	−98.6100	−5.31669
$345$	0	0
$346$	−55.9063	−3.00554
$347$	4.44819	0.238791	0.119396	−	0.992847i	$-0.461904\pi$
$347$	4.44819	0.238791	0.119396	+	0.992847i	$0.461904\pi$
$348$	0	0
$349$	−17.0567	−0.913024	−0.456512	−	0.889717i	$-0.650901\pi$
$349$	−17.0567	−0.913024	−0.456512	+	0.889717i	$0.650901\pi$
$350$	−3.38646	−0.181014
$351$	0	0
$352$	34.3120	1.82884
$353$	8.76371	0.466445	0.233222	−	0.972423i	$-0.425073\pi$
$353$	8.76371	0.466445	0.233222	+	0.972423i	$0.425073\pi$
$354$	0	0
$355$	12.9899	0.689432
$356$	26.3580	1.39697
$357$	0	0
$358$	−68.3170	−3.61067
$359$	−2.25261	−0.118888	−0.0594441	−	0.998232i	$-0.518933\pi$
$359$	−2.25261	−0.118888	−0.0594441	+	0.998232i	$0.518933\pi$
$360$	0	0
$361$	−10.2691	−0.540476
$362$	−53.9224	−2.83410
$363$	0	0
$364$	−10.7537	−0.563645
$365$	3.50169	0.183287
$366$	0	0
$367$	8.93678	0.466496	0.233248	−	0.972417i	$-0.425065\pi$
$367$	8.93678	0.466496	0.233248	+	0.972417i	$0.425065\pi$
$368$	−110.735	−5.77244
$369$	0	0
$370$	28.3039	1.47145
$371$	−3.34274	−0.173547
$372$	0	0
$373$	−4.78101	−0.247551	−0.123776	−	0.992310i	$-0.539500\pi$
$373$	−4.78101	−0.247551	−0.123776	+	0.992310i	$0.539500\pi$
$374$	−13.1253	−0.678692
$375$	0	0
$376$	−29.0783	−1.49960
$377$	0	0
$378$	0	0
$379$	34.2900	1.76136	0.880681	−	0.473711i	$-0.157086\pi$
$379$	34.2900	1.76136	0.880681	+	0.473711i	$0.157086\pi$
$380$	−42.8704	−2.19921
$381$	0	0
$382$	−2.56455	−0.131214
$383$	−23.8006	−1.21615	−0.608076	−	0.793879i	$-0.708058\pi$
$383$	−23.8006	−1.21615	−0.608076	+	0.793879i	$0.708058\pi$
$384$	0	0
$385$	−2.70290	−0.137753
$386$	−27.0716	−1.37791
$387$	0	0
$388$	−26.4136	−1.34095
$389$	−18.5239	−0.939198	−0.469599	−	0.882880i	$-0.655601\pi$
$389$	−18.5239	−0.939198	−0.469599	+	0.882880i	$0.655601\pi$
$390$	0	0
$391$	24.1059	1.21909
$392$	64.8841	3.27714
$393$	0	0
$394$	−20.7521	−1.04547
$395$	0.683551	0.0343932
$396$	0	0
$397$	−10.8488	−0.544485	−0.272242	−	0.962229i	$-0.587765\pi$
$397$	−10.8488	−0.544485	−0.272242	+	0.962229i	$0.587765\pi$
$398$	−72.0181	−3.60994
$399$	0	0
$400$	24.8399	1.24199
$401$	−28.7000	−1.43321	−0.716604	−	0.697480i	$-0.754305\pi$
$401$	−28.7000	−1.43321	−0.716604	+	0.697480i	$0.754305\pi$
$402$	0	0
$403$	−1.27939	−0.0637311
$404$	52.1072	2.59243
$405$	0	0
$406$	0	0
$407$	5.05060	0.250349
$408$	0	0
$409$	−25.1951	−1.24582	−0.622910	−	0.782294i	$-0.714050\pi$
$409$	−25.1951	−1.24582	−0.622910	+	0.782294i	$0.714050\pi$
$410$	−65.5611	−3.23783
$411$	0	0
$412$	−82.3629	−4.05773
$413$	−1.90139	−0.0935612
$414$	0	0
$415$	17.3443	0.851399
$416$	60.5916	2.97075
$417$	0	0
$418$	−10.3259	−0.505057
$419$	−12.9466	−0.632482	−0.316241	−	0.948679i	$-0.602421\pi$
$419$	−12.9466	−0.632482	−0.316241	+	0.948679i	$0.602421\pi$
$420$	0	0
$421$	19.5788	0.954213	0.477107	−	0.878845i	$-0.341686\pi$
$421$	19.5788	0.954213	0.477107	+	0.878845i	$0.341686\pi$
$422$	−53.5076	−2.60471
$423$	0	0
$424$	40.7695	1.97994
$425$	−5.40741	−0.262298
$426$	0	0
$427$	−4.03549	−0.195291
$428$	10.4824	0.506688
$429$	0	0
$430$	−67.3167	−3.24630
$431$	3.60547	0.173669	0.0868346	−	0.996223i	$-0.472325\pi$
$431$	3.60547	0.173669	0.0868346	+	0.996223i	$0.472325\pi$
$432$	0	0
$433$	−4.80667	−0.230994	−0.115497	−	0.993308i	$-0.536846\pi$
$433$	−4.80667	−0.230994	−0.115497	+	0.993308i	$0.536846\pi$
$434$	−1.35469	−0.0650272
$435$	0	0
$436$	82.0190	3.92800
$437$	18.9646	0.907199
$438$	0	0
$439$	7.54657	0.360178	0.180089	−	0.983650i	$-0.442361\pi$
$439$	7.54657	0.360178	0.180089	+	0.983650i	$0.442361\pi$
$440$	32.9658	1.57158
$441$	0	0
$442$	−23.1779	−1.10246
$443$	−22.8176	−1.08410	−0.542050	−	0.840347i	$-0.682352\pi$
$443$	−22.8176	−1.08410	−0.542050	+	0.840347i	$0.682352\pi$
$444$	0	0
$445$	11.6991	0.554590
$446$	2.18936	0.103669
$447$	0	0
$448$	34.9408	1.65080
$449$	13.0855	0.617544	0.308772	−	0.951136i	$-0.400082\pi$
$449$	13.0855	0.617544	0.308772	+	0.951136i	$0.400082\pi$
$450$	0	0
$451$	−11.6989	−0.550877
$452$	67.3459	3.16768
$453$	0	0
$454$	−52.2847	−2.45384
$455$	−4.77306	−0.223764
$456$	0	0
$457$	−31.3399	−1.46602	−0.733009	−	0.680218i	$-0.761885\pi$
$457$	−31.3399	−1.46602	−0.733009	+	0.680218i	$0.761885\pi$
$458$	−74.7453	−3.49262
$459$	0	0
$460$	−93.1193	−4.34171
$461$	−38.1212	−1.77548	−0.887740	−	0.460346i	$-0.847725\pi$
$461$	−38.1212	−1.77548	−0.887740	+	0.460346i	$0.847725\pi$
$462$	0	0
$463$	39.1593	1.81989	0.909944	−	0.414732i	$-0.136125\pi$
$463$	39.1593	1.81989	0.909944	+	0.414732i	$0.136125\pi$
$464$	0	0
$465$	0	0
$466$	−36.4421	−1.68815
$467$	−14.0227	−0.648891	−0.324445	−	0.945904i	$-0.605178\pi$
$467$	−14.0227	−0.648891	−0.324445	+	0.945904i	$0.605178\pi$
$468$	0	0
$469$	−0.121909	−0.00562925
$470$	−19.8505	−0.915633
$471$	0	0
$472$	23.1902	1.06741
$473$	−12.0121	−0.552318
$474$	0	0
$475$	−4.25412	−0.195192
$476$	−18.1818	−0.833361
$477$	0	0
$478$	11.2485	0.514493
$479$	41.8152	1.91059	0.955294	−	0.295659i	$-0.0955391\pi$
$479$	41.8152	1.91059	0.955294	+	0.295659i	$0.0955391\pi$
$480$	0	0
$481$	8.91885	0.406665
$482$	−11.1882	−0.509607
$483$	0	0
$484$	−53.8433	−2.44742
$485$	−11.7238	−0.532349
$486$	0	0
$487$	14.1076	0.639277	0.319639	−	0.947540i	$-0.396438\pi$
$487$	14.1076	0.639277	0.319639	+	0.947540i	$0.396438\pi$
$488$	49.2185	2.22802
$489$	0	0
$490$	44.2936	2.00098
$491$	−26.5746	−1.19929	−0.599647	−	0.800265i	$-0.704692\pi$
$491$	−26.5746	−1.19929	−0.599647	+	0.800265i	$0.704692\pi$
$492$	0	0
$493$	0	0
$494$	−18.2345	−0.820411
$495$	0	0
$496$	9.93675	0.446173
$497$	4.33415	0.194413
$498$	0	0
$499$	19.1146	0.855687	0.427844	−	0.903853i	$-0.359273\pi$
$499$	19.1146	0.855687	0.427844	+	0.903853i	$0.359273\pi$
$500$	−51.6548	−2.31007
$501$	0	0
$502$	84.2405	3.75984
$503$	−16.1585	−0.720471	−0.360235	−	0.932861i	$-0.617304\pi$
$503$	−16.1585	−0.720471	−0.360235	+	0.932861i	$0.617304\pi$
$504$	0	0
$505$	23.1280	1.02918
$506$	−22.4290	−0.997092
$507$	0	0
$508$	68.7754	3.05141
$509$	25.1057	1.11279	0.556394	−	0.830918i	$-0.312184\pi$
$509$	25.1057	1.11279	0.556394	+	0.830918i	$0.312184\pi$
$510$	0	0
$511$	1.16836	0.0516852
$512$	−114.260	−5.04961
$513$	0	0
$514$	−22.6325	−0.998275
$515$	−36.5571	−1.61090
$516$	0	0
$517$	−3.54215	−0.155784
$518$	9.44375	0.414935
$519$	0	0
$520$	58.2142	2.55286
$521$	−2.82291	−0.123674	−0.0618370	−	0.998086i	$-0.519696\pi$
$521$	−2.82291	−0.123674	−0.0618370	+	0.998086i	$0.519696\pi$
$522$	0	0
$523$	30.2113	1.32105	0.660523	−	0.750805i	$-0.270334\pi$
$523$	30.2113	1.32105	0.660523	+	0.750805i	$0.270334\pi$
$524$	13.6294	0.595404
$525$	0	0
$526$	−15.7530	−0.686865
$527$	−2.16314	−0.0942278
$528$	0	0
$529$	18.1931	0.791006
$530$	27.8316	1.20893
$531$	0	0
$532$	−14.3040	−0.620156
$533$	−20.6590	−0.894840
$534$	0	0
$535$	4.65267	0.201152
$536$	1.48686	0.0642225
$537$	0	0
$538$	57.0961	2.46159
$539$	7.90383	0.340442
$540$	0	0
$541$	13.4843	0.579736	0.289868	−	0.957067i	$-0.406389\pi$
$541$	13.4843	0.579736	0.289868	+	0.957067i	$0.406389\pi$
$542$	−26.6459	−1.14454
$543$	0	0
$544$	102.445	4.39231
$545$	36.4044	1.55940
$546$	0	0
$547$	23.3404	0.997962	0.498981	−	0.866613i	$-0.333708\pi$
$547$	23.3404	0.997962	0.498981	+	0.866613i	$0.333708\pi$
$548$	77.5852	3.31427
$549$	0	0
$550$	5.03126	0.214534
$551$	0	0
$552$	0	0
$553$	0.228071	0.00969855
$554$	71.7889	3.05002
$555$	0	0
$556$	114.244	4.84504
$557$	−3.33567	−0.141337	−0.0706685	−	0.997500i	$-0.522513\pi$
$557$	−3.33567	−0.141337	−0.0706685	+	0.997500i	$0.522513\pi$
$558$	0	0
$559$	−21.2122	−0.897181
$560$	37.0712	1.56654
$561$	0	0
$562$	−50.8401	−2.14456
$563$	−7.82895	−0.329951	−0.164976	−	0.986298i	$-0.552755\pi$
$563$	−7.82895	−0.329951	−0.164976	+	0.986298i	$0.552755\pi$
$564$	0	0
$565$	29.8917	1.25755
$566$	38.3379	1.61146
$567$	0	0
$568$	−52.8612	−2.21801
$569$	−0.170857	−0.00716268	−0.00358134	−	0.999994i	$-0.501140\pi$
$569$	−0.170857	−0.00716268	−0.00358134	+	0.999994i	$0.501140\pi$
$570$	0	0
$571$	26.7532	1.11959	0.559793	−	0.828633i	$-0.310881\pi$
$571$	26.7532	1.11959	0.559793	+	0.828633i	$0.310881\pi$
$572$	15.9767	0.668021
$573$	0	0
$574$	−21.8748	−0.913039
$575$	−9.24042	−0.385352
$576$	0	0
$577$	−39.1710	−1.63071	−0.815356	−	0.578960i	$-0.803459\pi$
$577$	−39.1710	−1.63071	−0.815356	+	0.578960i	$0.803459\pi$
$578$	8.03804	0.334338
$579$	0	0
$580$	0	0
$581$	5.78703	0.240087
$582$	0	0
$583$	4.96632	0.205684
$584$	−14.2498	−0.589662
$585$	0	0
$586$	−36.7386	−1.51766
$587$	2.09173	0.0863350	0.0431675	−	0.999068i	$-0.486255\pi$
$587$	2.09173	0.0863350	0.0431675	+	0.999068i	$0.486255\pi$
$588$	0	0
$589$	−1.70178	−0.0701208
$590$	15.8309	0.651748
$591$	0	0
$592$	−69.2706	−2.84701
$593$	−21.6103	−0.887431	−0.443715	−	0.896168i	$-0.646340\pi$
$593$	−21.6103	−0.887431	−0.443715	+	0.896168i	$0.646340\pi$
$594$	0	0
$595$	−8.07006	−0.330840
$596$	92.2421	3.77838
$597$	0	0
$598$	−39.6074	−1.61967
$599$	9.93168	0.405798	0.202899	−	0.979200i	$-0.434964\pi$
$599$	9.93168	0.405798	0.202899	+	0.979200i	$0.434964\pi$
$600$	0	0
$601$	15.5063	0.632515	0.316258	−	0.948673i	$-0.397574\pi$
$601$	15.5063	0.632515	0.316258	+	0.948673i	$0.397574\pi$
$602$	−22.4606	−0.915427
$603$	0	0
$604$	16.0167	0.651712
$605$	−23.8986	−0.971615
$606$	0	0
$607$	0.403185	0.0163648	0.00818240	−	0.999967i	$-0.497395\pi$
$607$	0.403185	0.0163648	0.00818240	+	0.999967i	$0.497395\pi$
$608$	80.5959	3.26859
$609$	0	0
$610$	33.5993	1.36040
$611$	−6.25509	−0.253054
$612$	0	0
$613$	−14.7943	−0.597535	−0.298768	−	0.954326i	$-0.596576\pi$
$613$	−14.7943	−0.597535	−0.298768	+	0.954326i	$0.596576\pi$
$614$	−64.8053	−2.61533
$615$	0	0
$616$	10.9992	0.443171
$617$	−0.597950	−0.0240726	−0.0120363	−	0.999928i	$-0.503831\pi$
$617$	−0.597950	−0.0240726	−0.0120363	+	0.999928i	$0.503831\pi$
$618$	0	0
$619$	5.29901	0.212985	0.106493	−	0.994313i	$-0.466038\pi$
$619$	5.29901	0.212985	0.106493	+	0.994313i	$0.466038\pi$
$620$	8.35605	0.335587
$621$	0	0
$622$	−15.7911	−0.633166
$623$	3.90347	0.156389
$624$	0	0
$625$	−30.1258	−1.20503
$626$	−11.0892	−0.443214
$627$	0	0
$628$	48.5763	1.93841
$629$	15.0796	0.601262
$630$	0	0
$631$	−26.5484	−1.05687	−0.528437	−	0.848972i	$-0.677222\pi$
$631$	−26.5484	−1.05687	−0.528437	+	0.848972i	$0.677222\pi$
$632$	−2.78165	−0.110648
$633$	0	0
$634$	34.1291	1.35544
$635$	30.5262	1.21140
$636$	0	0
$637$	13.9574	0.553011
$638$	0	0
$639$	0	0
$640$	−152.481	−6.02733
$641$	−5.25020	−0.207370	−0.103685	−	0.994610i	$-0.533063\pi$
$641$	−5.25020	−0.207370	−0.103685	+	0.994610i	$0.533063\pi$
$642$	0	0
$643$	17.9092	0.706270	0.353135	−	0.935572i	$-0.385116\pi$
$643$	17.9092	0.706270	0.353135	+	0.935572i	$0.385116\pi$
$644$	−31.0698	−1.22432
$645$	0	0
$646$	−30.8301	−1.21299
$647$	35.0126	1.37649	0.688244	−	0.725479i	$-0.258382\pi$
$647$	35.0126	1.37649	0.688244	+	0.725479i	$0.258382\pi$
$648$	0	0
$649$	2.82490	0.110887
$650$	8.88471	0.348487
$651$	0	0
$652$	−118.600	−4.64474
$653$	−18.5363	−0.725381	−0.362691	−	0.931910i	$-0.618142\pi$
$653$	−18.5363	−0.725381	−0.362691	+	0.931910i	$0.618142\pi$
$654$	0	0
$655$	6.04947	0.236372
$656$	160.454	6.26466
$657$	0	0
$658$	−6.62322	−0.258200
$659$	1.30699	0.0509132	0.0254566	−	0.999676i	$-0.491896\pi$
$659$	1.30699	0.0509132	0.0254566	+	0.999676i	$0.491896\pi$
$660$	0	0
$661$	34.1343	1.32767	0.663836	−	0.747878i	$-0.268927\pi$
$661$	34.1343	1.32767	0.663836	+	0.747878i	$0.268927\pi$
$662$	43.0401	1.67280
$663$	0	0
$664$	−70.5811	−2.73908
$665$	−6.34888	−0.246199
$666$	0	0
$667$	0	0
$668$	−56.0279	−2.16778
$669$	0	0
$670$	1.01501	0.0392134
$671$	5.99553	0.231455
$672$	0	0
$673$	−19.0077	−0.732693	−0.366347	−	0.930478i	$-0.619392\pi$
$673$	−19.0077	−0.732693	−0.366347	+	0.930478i	$0.619392\pi$
$674$	0.0895990	0.00345122
$675$	0	0
$676$	−46.1120	−1.77354
$677$	−23.6929	−0.910593	−0.455297	−	0.890340i	$-0.650467\pi$
$677$	−23.6929	−0.910593	−0.455297	+	0.890340i	$0.650467\pi$
$678$	0	0
$679$	−3.91170	−0.150117
$680$	98.4259	3.77446
$681$	0	0
$682$	2.01267	0.0770690
$683$	−14.5206	−0.555614	−0.277807	−	0.960637i	$-0.589608\pi$
$683$	−14.5206	−0.555614	−0.277807	+	0.960637i	$0.589608\pi$
$684$	0	0
$685$	34.4365	1.31575
$686$	31.2439	1.19290
$687$	0	0
$688$	164.750	6.28105
$689$	8.77002	0.334111
$690$	0	0
$691$	−12.5999	−0.479322	−0.239661	−	0.970857i	$-0.577036\pi$
$691$	−12.5999	−0.479322	−0.239661	+	0.970857i	$0.577036\pi$
$692$	115.059	4.37389
$693$	0	0
$694$	−12.3571	−0.469069
$695$	50.7078	1.92346
$696$	0	0
$697$	−34.9292	−1.32304
$698$	47.3836	1.79350
$699$	0	0
$700$	6.96956	0.263425
$701$	−36.3398	−1.37254	−0.686268	−	0.727349i	$-0.740752\pi$
$701$	−36.3398	−1.37254	−0.686268	+	0.727349i	$0.740752\pi$
$702$	0	0
$703$	11.8634	0.447437
$704$	−51.9116	−1.95649
$705$	0	0
$706$	−24.3457	−0.916261
$707$	7.71679	0.290220
$708$	0	0
$709$	−20.1461	−0.756602	−0.378301	−	0.925683i	$-0.623492\pi$
$709$	−20.1461	−0.756602	−0.378301	+	0.925683i	$0.623492\pi$
$710$	−36.0860	−1.35428
$711$	0	0
$712$	−47.6084	−1.78420
$713$	−3.69646	−0.138434
$714$	0	0
$715$	7.09134	0.265201
$716$	140.601	5.25451
$717$	0	0
$718$	6.25777	0.233538
$719$	19.5168	0.727853	0.363926	−	0.931428i	$-0.381436\pi$
$719$	19.5168	0.727853	0.363926	+	0.931428i	$0.381436\pi$
$720$	0	0
$721$	−12.1975	−0.454259
$722$	28.5275	1.06168
$723$	0	0
$724$	110.976	4.12439
$725$	0	0
$726$	0	0
$727$	−0.984358	−0.0365078	−0.0182539	−	0.999833i	$-0.505811\pi$
$727$	−0.984358	−0.0365078	−0.0182539	+	0.999833i	$0.505811\pi$
$728$	19.4235	0.719884
$729$	0	0
$730$	−9.72773	−0.360040
$731$	−35.8646	−1.32650
$732$	0	0
$733$	−32.9945	−1.21868	−0.609340	−	0.792909i	$-0.708565\pi$
$733$	−32.9945	−1.21868	−0.609340	+	0.792909i	$0.708565\pi$
$734$	−24.8265	−0.916361
$735$	0	0
$736$	175.063	6.45291
$737$	0.181121	0.00667167
$738$	0	0
$739$	21.7629	0.800560	0.400280	−	0.916393i	$-0.368913\pi$
$739$	21.7629	0.800560	0.400280	+	0.916393i	$0.368913\pi$
$740$	−58.2513	−2.14136
$741$	0	0
$742$	9.28617	0.340906
$743$	17.1512	0.629216	0.314608	−	0.949222i	$-0.398127\pi$
$743$	17.1512	0.629216	0.314608	+	0.949222i	$0.398127\pi$
$744$	0	0
$745$	40.9420	1.50000
$746$	13.2817	0.486277
$747$	0	0
$748$	27.0127	0.987683
$749$	1.55239	0.0567231
$750$	0	0
$751$	37.6068	1.37229	0.686146	−	0.727464i	$-0.259301\pi$
$751$	37.6068	1.37229	0.686146	+	0.727464i	$0.259301\pi$
$752$	48.5818	1.77160
$753$	0	0
$754$	0	0
$755$	7.10910	0.258726
$756$	0	0
$757$	24.7412	0.899235	0.449617	−	0.893221i	$-0.351560\pi$
$757$	24.7412	0.899235	0.449617	+	0.893221i	$0.351560\pi$
$758$	−95.2581	−3.45993
$759$	0	0
$760$	77.4336	2.80881
$761$	23.6385	0.856895	0.428448	−	0.903567i	$-0.359061\pi$
$761$	23.6385	0.856895	0.428448	+	0.903567i	$0.359061\pi$
$762$	0	0
$763$	12.1466	0.439735
$764$	5.27802	0.190952
$765$	0	0
$766$	66.1182	2.38895
$767$	4.98849	0.180124
$768$	0	0
$769$	−2.07725	−0.0749074	−0.0374537	−	0.999298i	$-0.511925\pi$
$769$	−2.07725	−0.0749074	−0.0374537	+	0.999298i	$0.511925\pi$
$770$	7.50869	0.270594
$771$	0	0
$772$	55.7151	2.00523
$773$	−12.9678	−0.466419	−0.233209	−	0.972427i	$-0.574923\pi$
$773$	−12.9678	−0.466419	−0.233209	+	0.972427i	$0.574923\pi$
$774$	0	0
$775$	0.829188	0.0297853
$776$	47.7088	1.71265
$777$	0	0
$778$	51.4595	1.84491
$779$	−27.4795	−0.984557
$780$	0	0
$781$	−6.43926	−0.230415
$782$	−66.9663	−2.39471
$783$	0	0
$784$	−108.404	−3.87156
$785$	21.5608	0.769537
$786$	0	0
$787$	−6.87306	−0.244998	−0.122499	−	0.992469i	$-0.539091\pi$
$787$	−6.87306	−0.244998	−0.122499	+	0.992469i	$0.539091\pi$
$788$	42.7092	1.52145
$789$	0	0
$790$	−1.89891	−0.0675602
$791$	9.97356	0.354619
$792$	0	0
$793$	10.5875	0.375974
$794$	30.1380	1.06956
$795$	0	0
$796$	148.218	5.25346
$797$	13.4431	0.476179	0.238090	−	0.971243i	$-0.423479\pi$
$797$	13.4431	0.476179	0.238090	+	0.971243i	$0.423479\pi$
$798$	0	0
$799$	−10.5758	−0.374145
$800$	−39.2700	−1.38840
$801$	0	0
$802$	79.7288	2.81532
$803$	−1.73584	−0.0612563
$804$	0	0
$805$	−13.7905	−0.486050
$806$	3.55417	0.125190
$807$	0	0
$808$	−94.1173	−3.31104
$809$	−35.8188	−1.25932	−0.629661	−	0.776870i	$-0.716806\pi$
$809$	−35.8188	−1.25932	−0.629661	+	0.776870i	$0.716806\pi$
$810$	0	0
$811$	−42.0554	−1.47677	−0.738383	−	0.674382i	$-0.764410\pi$
$811$	−42.0554	−1.47677	−0.738383	+	0.674382i	$0.764410\pi$
$812$	0	0
$813$	0	0
$814$	−14.0306	−0.491773
$815$	−52.6412	−1.84394
$816$	0	0
$817$	−28.2154	−0.987132
$818$	69.9923	2.44722
$819$	0	0
$820$	134.929	4.71193
$821$	18.7528	0.654476	0.327238	−	0.944942i	$-0.393882\pi$
$821$	18.7528	0.654476	0.327238	+	0.944942i	$0.393882\pi$
$822$	0	0
$823$	−19.9500	−0.695414	−0.347707	−	0.937603i	$-0.613040\pi$
$823$	−19.9500	−0.695414	−0.347707	+	0.937603i	$0.613040\pi$
$824$	148.766	5.18251
$825$	0	0
$826$	5.28208	0.183787
$827$	−14.8030	−0.514752	−0.257376	−	0.966311i	$-0.582858\pi$
$827$	−14.8030	−0.514752	−0.257376	+	0.966311i	$0.582858\pi$
$828$	0	0
$829$	−33.7345	−1.17165	−0.585824	−	0.810438i	$-0.699229\pi$
$829$	−33.7345	−1.17165	−0.585824	+	0.810438i	$0.699229\pi$
$830$	−48.1826	−1.67244
$831$	0	0
$832$	−91.6707	−3.17811
$833$	23.5985	0.817638
$834$	0	0
$835$	−24.8682	−0.860599
$836$	21.2515	0.734997
$837$	0	0
$838$	35.9657	1.24242
$839$	−18.8511	−0.650813	−0.325407	−	0.945574i	$-0.605501\pi$
$839$	−18.8511	−0.650813	−0.325407	+	0.945574i	$0.605501\pi$
$840$	0	0
$841$	0	0
$842$	−54.3901	−1.87441
$843$	0	0
$844$	110.122	3.79057
$845$	−20.4670	−0.704086
$846$	0	0
$847$	−7.97390	−0.273986
$848$	−68.1147	−2.33907
$849$	0	0
$850$	15.0218	0.515245
$851$	25.7686	0.883336
$852$	0	0
$853$	50.4469	1.72727	0.863634	−	0.504119i	$-0.168183\pi$
$853$	50.4469	1.72727	0.863634	+	0.504119i	$0.168183\pi$
$854$	11.2106	0.383620
$855$	0	0
$856$	−18.9336	−0.647138
$857$	−56.8554	−1.94214	−0.971072	−	0.238787i	$-0.923250\pi$
$857$	−56.8554	−1.94214	−0.971072	+	0.238787i	$0.923250\pi$
$858$	0	0
$859$	−17.5756	−0.599673	−0.299837	−	0.953991i	$-0.596932\pi$
$859$	−17.5756	−0.599673	−0.299837	+	0.953991i	$0.596932\pi$
$860$	138.542	4.72426
$861$	0	0
$862$	−10.0160	−0.341147
$863$	31.3953	1.06871	0.534354	−	0.845261i	$-0.320555\pi$
$863$	31.3953	1.06871	0.534354	+	0.845261i	$0.320555\pi$
$864$	0	0
$865$	51.0694	1.73641
$866$	13.3530	0.453753
$867$	0	0
$868$	2.78805	0.0946325
$869$	−0.338845	−0.0114945
$870$	0	0
$871$	0.319841	0.0108374
$872$	−148.145	−5.01681
$873$	0	0
$874$	−52.6838	−1.78206
$875$	−7.64980	−0.258610
$876$	0	0
$877$	29.1626	0.984750	0.492375	−	0.870383i	$-0.336129\pi$
$877$	29.1626	0.984750	0.492375	+	0.870383i	$0.336129\pi$
$878$	−20.9644	−0.707515
$879$	0	0
$880$	−55.0768	−1.85664
$881$	6.19983	0.208878	0.104439	−	0.994531i	$-0.466695\pi$
$881$	6.19983	0.208878	0.104439	+	0.994531i	$0.466695\pi$
$882$	0	0
$883$	−52.2570	−1.75859	−0.879293	−	0.476280i	$-0.841985\pi$
$883$	−52.2570	−1.75859	−0.879293	+	0.476280i	$0.841985\pi$
$884$	47.7017	1.60438
$885$	0	0
$886$	63.3876	2.12955
$887$	26.4724	0.888855	0.444428	−	0.895815i	$-0.353407\pi$
$887$	26.4724	0.888855	0.444428	+	0.895815i	$0.353407\pi$
$888$	0	0
$889$	10.1853	0.341603
$890$	−32.5002	−1.08941
$891$	0	0
$892$	−4.50585	−0.150867
$893$	−8.32020	−0.278425
$894$	0	0
$895$	62.4064	2.08601
$896$	−50.8762	−1.69965
$897$	0	0
$898$	−36.3517	−1.21307
$899$	0	0
$900$	0	0
$901$	14.8279	0.493990
$902$	32.4995	1.08212
$903$	0	0
$904$	−121.642	−4.04575
$905$	49.2572	1.63736
$906$	0	0
$907$	−19.5034	−0.647598	−0.323799	−	0.946126i	$-0.604960\pi$
$907$	−19.5034	−0.647598	−0.323799	+	0.946126i	$0.604960\pi$
$908$	107.606	3.57102
$909$	0	0
$910$	13.2596	0.439551
$911$	−26.8766	−0.890461	−0.445231	−	0.895416i	$-0.646878\pi$
$911$	−26.8766	−0.890461	−0.445231	+	0.895416i	$0.646878\pi$
$912$	0	0
$913$	−8.59781	−0.284546
$914$	87.0625	2.87977
$915$	0	0
$916$	153.831	5.08272
$917$	2.01844	0.0666549
$918$	0	0
$919$	−23.5501	−0.776847	−0.388424	−	0.921481i	$-0.626980\pi$
$919$	−23.5501	−0.776847	−0.388424	+	0.921481i	$0.626980\pi$
$920$	168.194	5.54520
$921$	0	0
$922$	105.901	3.48766
$923$	−11.3711	−0.374284
$924$	0	0
$925$	−5.78040	−0.190058
$926$	−108.785	−3.57489
$927$	0	0
$928$	0	0
$929$	26.2377	0.860831	0.430416	−	0.902631i	$-0.358367\pi$
$929$	26.2377	0.860831	0.430416	+	0.902631i	$0.358367\pi$
$930$	0	0
$931$	18.5654	0.608456
$932$	75.0003	2.45672
$933$	0	0
$934$	38.9550	1.27465
$935$	11.9897	0.392105
$936$	0	0
$937$	43.7650	1.42974	0.714870	−	0.699257i	$-0.246486\pi$
$937$	43.7650	1.42974	0.714870	+	0.699257i	$0.246486\pi$
$938$	0.338665	0.0110578
$939$	0	0
$940$	40.8536	1.33250
$941$	−15.3106	−0.499111	−0.249555	−	0.968361i	$-0.580284\pi$
$941$	−15.3106	−0.499111	−0.249555	+	0.968361i	$0.580284\pi$
$942$	0	0
$943$	−59.6886	−1.94373
$944$	−38.7444	−1.26102
$945$	0	0
$946$	33.3698	1.08495
$947$	12.1417	0.394552	0.197276	−	0.980348i	$-0.436791\pi$
$947$	12.1417	0.394552	0.197276	+	0.980348i	$0.436791\pi$
$948$	0	0
$949$	−3.06531	−0.0995042
$950$	11.8180	0.383426
$951$	0	0
$952$	32.8404	1.06436
$953$	14.2993	0.463199	0.231600	−	0.972811i	$-0.425604\pi$
$953$	14.2993	0.463199	0.231600	+	0.972811i	$0.425604\pi$
$954$	0	0
$955$	2.34267	0.0758070
$956$	−23.1501	−0.748729
$957$	0	0
$958$	−116.163	−3.75306
$959$	11.4899	0.371029
$960$	0	0
$961$	−30.6683	−0.989300
$962$	−24.7766	−0.798831
$963$	0	0
$964$	23.0260	0.741618
$965$	24.7294	0.796067
$966$	0	0
$967$	−12.1295	−0.390059	−0.195029	−	0.980797i	$-0.562480\pi$
$967$	−12.1295	−0.390059	−0.195029	+	0.980797i	$0.562480\pi$
$968$	97.2531	3.12583
$969$	0	0
$970$	32.5687	1.04572
$971$	52.2721	1.67749	0.838745	−	0.544524i	$-0.183290\pi$
$971$	52.2721	1.67749	0.838745	+	0.544524i	$0.183290\pi$
$972$	0	0
$973$	16.9190	0.542397
$974$	−39.1911	−1.25576
$975$	0	0
$976$	−82.2307	−2.63214
$977$	−44.6255	−1.42770	−0.713849	−	0.700300i	$-0.753050\pi$
$977$	−44.6255	−1.42770	−0.713849	+	0.700300i	$0.753050\pi$
$978$	0	0
$979$	−5.79940	−0.185350
$980$	−91.1592	−2.91197
$981$	0	0
$982$	73.8244	2.35583
$983$	−5.21076	−0.166197	−0.0830987	−	0.996541i	$-0.526482\pi$
$983$	−5.21076	−0.166197	−0.0830987	+	0.996541i	$0.526482\pi$
$984$	0	0
$985$	18.9566	0.604008
$986$	0	0
$987$	0	0
$988$	37.5279	1.19392
$989$	−61.2870	−1.94881
$990$	0	0
$991$	38.3071	1.21687	0.608433	−	0.793606i	$-0.291799\pi$
$991$	38.3071	1.21687	0.608433	+	0.793606i	$0.291799\pi$
$992$	−15.7093	−0.498770
$993$	0	0
$994$	−12.0403	−0.381896
$995$	65.7872	2.08560
$996$	0	0
$997$	11.4438	0.362428	0.181214	−	0.983444i	$-0.441997\pi$
$997$	11.4438	0.362428	0.181214	+	0.983444i	$0.441997\pi$
$998$	−53.1005	−1.68087
$999$	0	0

Twists

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

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-2.77801 q^{2} +5.71733 q^{4} +2.53766 q^{5} +0.846706 q^{7} -10.3268 q^{8} -7.04965 q^{10} -1.25795 q^{11} -2.22142 q^{13} -2.35216 q^{14} +17.2532 q^{16} -3.75587 q^{17} -2.95482 q^{19} +14.5087 q^{20} +3.49460 q^{22} -6.41819 q^{23} +1.43972 q^{25} +6.17112 q^{26} +4.84090 q^{28} +0.575935 q^{31} -27.2761 q^{32} +10.4338 q^{34} +2.14865 q^{35} -4.01493 q^{37} +8.20851 q^{38} -26.2059 q^{40} +9.29991 q^{41} +9.54895 q^{43} -7.19214 q^{44} +17.8298 q^{46} +2.81581 q^{47} -6.28309 q^{49} -3.99957 q^{50} -12.7006 q^{52} -3.94794 q^{53} -3.19226 q^{55} -8.74375 q^{56} -2.24563 q^{59} -4.76610 q^{61} -1.59995 q^{62} +41.2667 q^{64} -5.63721 q^{65} -0.143981 q^{67} -21.4736 q^{68} -5.96898 q^{70} +5.11884 q^{71} +1.37989 q^{73} +11.1535 q^{74} -16.8937 q^{76} -1.06512 q^{77} +0.269362 q^{79} +43.7829 q^{80} -25.8352 q^{82} +6.83476 q^{83} -9.53112 q^{85} -26.5271 q^{86} +12.9906 q^{88} +4.61019 q^{89} -1.88089 q^{91} -36.6949 q^{92} -7.82234 q^{94} -7.49833 q^{95} -4.61991 q^{97} +17.4545 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - 5 q^{2} + 11 q^{4} + 4 q^{5} + 5 q^{7} - 24 q^{8} + q^{11} + q^{13} - 9 q^{14} + 35 q^{16} - 2 q^{17} + 9 q^{19} + 18 q^{20} - 4 q^{22} + 4 q^{23} + q^{25} + 8 q^{26} + 40 q^{28} + 8 q^{31} - 43 q^{32}+ \cdots + 30 q^{98}+O(q^{100}) \) 9 * q - 5 * q^2 + 11 * q^4 + 4 * q^5 + 5 * q^7 - 24 * q^8 + q^11 + q^13 - 9 * q^14 + 35 * q^16 - 2 * q^17 + 9 * q^19 + 18 * q^20 - 4 * q^22 + 4 * q^23 + q^25 + 8 * q^26 + 40 * q^28 + 8 * q^31 - 43 * q^32 - 4 * q^34 - 22 * q^35 + 27 * q^37 + 30 * q^38 - 29 * q^40 - 12 * q^41 + 16 * q^43 - 37 * q^44 - 22 * q^46 + 8 * q^47 - 6 * q^49 + 7 * q^50 + 33 * q^52 + 8 * q^53 + 9 * q^55 - 40 * q^56 + 16 * q^59 + 21 * q^61 + 32 * q^62 + 36 * q^64 + 31 * q^65 + 3 * q^67 - 33 * q^68 - 6 * q^70 + 33 * q^71 + 3 * q^73 - 28 * q^74 - 26 * q^76 - 24 * q^77 + 3 * q^79 + 64 * q^80 + 13 * q^82 - 13 * q^83 + 6 * q^85 - 58 * q^86 + 27 * q^88 - 6 * q^89 + q^91 + 29 * q^92 - 18 * q^94 - 48 * q^95 + 4 * q^97 + 30 * q^98

Introduction

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