LMFDB - Embedded newform 7569.2.a.bj.1.3

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			1.3
Root			$-1.53422$ 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)

$f(q)$	$=$	$q-2.53422 q^{2} +4.42225 q^{4} -1.08952 q^{5} +1.73057 q^{7} -6.13851 q^{8} +2.76107 q^{10} -6.29513 q^{11} -1.27948 q^{13} -4.38564 q^{14} +6.71181 q^{16} +5.08291 q^{17} -6.52684 q^{19} -4.81811 q^{20} +15.9532 q^{22} +4.52415 q^{23} -3.81296 q^{25} +3.24249 q^{26} +7.65302 q^{28} -6.85487 q^{31} -4.73216 q^{32} -12.8812 q^{34} -1.88548 q^{35} +6.17424 q^{37} +16.5404 q^{38} +6.68800 q^{40} -4.04550 q^{41} +1.83124 q^{43} -27.8387 q^{44} -11.4652 q^{46} -1.51649 q^{47} -4.00512 q^{49} +9.66286 q^{50} -5.65820 q^{52} +3.77409 q^{53} +6.85864 q^{55} -10.6231 q^{56} +8.72197 q^{59} -3.08764 q^{61} +17.3717 q^{62} -1.43131 q^{64} +1.39402 q^{65} -2.39689 q^{67} +22.4779 q^{68} +4.77822 q^{70} -9.34384 q^{71} -1.79781 q^{73} -15.6469 q^{74} -28.8633 q^{76} -10.8942 q^{77} -2.86562 q^{79} -7.31262 q^{80} +10.2522 q^{82} -6.59352 q^{83} -5.53791 q^{85} -4.64077 q^{86} +38.6427 q^{88} -3.34552 q^{89} -2.21424 q^{91} +20.0069 q^{92} +3.84311 q^{94} +7.11109 q^{95} +2.26753 q^{97} +10.1498 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−2.53422	−1.79196	−0.895981	−	0.444093i	$-0.853526\pi$
$2$	−2.53422	−1.79196	−0.895981	+	0.444093i	$0.853526\pi$
$3$	0	0
$3$	0	0
$4$	4.42225	2.21113
$5$	−1.08952	−0.487246	−0.243623	−	0.969870i	$-0.578336\pi$
$5$	−1.08952	−0.487246	−0.243623	+	0.969870i	$0.578336\pi$
$6$	0	0
$7$	1.73057	0.654094	0.327047	−	0.945008i	$-0.393946\pi$
$7$	1.73057	0.654094	0.327047	+	0.945008i	$0.393946\pi$
$8$	−6.13851	−2.17029
$9$	0	0
$10$	2.76107	0.873126
$11$	−6.29513	−1.89805	−0.949027	−	0.315195i	$-0.897930\pi$
$11$	−6.29513	−1.89805	−0.949027	+	0.315195i	$0.897930\pi$
$12$	0	0
$13$	−1.27948	−0.354865	−0.177432	−	0.984133i	$-0.556779\pi$
$13$	−1.27948	−0.354865	−0.177432	+	0.984133i	$0.556779\pi$
$14$	−4.38564	−1.17211
$15$	0	0
$16$	6.71181	1.67795
$17$	5.08291	1.23279	0.616393	−	0.787439i	$-0.288593\pi$
$17$	5.08291	1.23279	0.616393	+	0.787439i	$0.288593\pi$
$18$	0	0
$19$	−6.52684	−1.49736	−0.748680	−	0.662931i	$-0.769312\pi$
$19$	−6.52684	−1.49736	−0.748680	+	0.662931i	$0.769312\pi$
$20$	−4.81811	−1.07736
$21$	0	0
$22$	15.9532	3.40124
$23$	4.52415	0.943351	0.471675	−	0.881772i	$-0.343649\pi$
$23$	4.52415	0.943351	0.471675	+	0.881772i	$0.343649\pi$
$24$	0	0
$25$	−3.81296	−0.762591
$26$	3.24249	0.635904
$27$	0	0
$28$	7.65302	1.44629
$29$	0	0
$29$	0	0
$30$	0	0
$31$	−6.85487	−1.23117	−0.615586	−	0.788070i	$-0.711081\pi$
$31$	−6.85487	−1.23117	−0.615586	+	0.788070i	$0.711081\pi$
$32$	−4.73216	−0.836535
$33$	0	0
$34$	−12.8812	−2.20911
$35$	−1.88548	−0.318705
$36$	0	0
$37$	6.17424	1.01504	0.507519	−	0.861640i	$-0.330563\pi$
$37$	6.17424	1.01504	0.507519	+	0.861640i	$0.330563\pi$
$38$	16.5404	2.68321
$39$	0	0
$40$	6.68800	1.05747
$41$	−4.04550	−0.631801	−0.315901	−	0.948792i	$-0.602307\pi$
$41$	−4.04550	−0.631801	−0.315901	+	0.948792i	$0.602307\pi$
$42$	0	0
$43$	1.83124	0.279262	0.139631	−	0.990204i	$-0.455408\pi$
$43$	1.83124	0.279262	0.139631	+	0.990204i	$0.455408\pi$
$44$	−27.8387	−4.19684
$45$	0	0
$46$	−11.4652	−1.69045
$47$	−1.51649	−0.221203	−0.110601	−	0.993865i	$-0.535278\pi$
$47$	−1.51649	−0.221203	−0.110601	+	0.993865i	$0.535278\pi$
$48$	0	0
$49$	−4.00512	−0.572160
$50$	9.66286	1.36653
$51$	0	0
$52$	−5.65820	−0.784651
$53$	3.77409	0.518411	0.259206	−	0.965822i	$-0.416539\pi$
$53$	3.77409	0.518411	0.259206	+	0.965822i	$0.416539\pi$
$54$	0	0
$55$	6.85864	0.924819
$56$	−10.6231	−1.41958
$57$	0	0
$58$	0	0
$59$	8.72197	1.13550	0.567752	−	0.823200i	$-0.307813\pi$
$59$	8.72197	1.13550	0.567752	+	0.823200i	$0.307813\pi$
$60$	0	0
$61$	−3.08764	−0.395332	−0.197666	−	0.980269i	$-0.563336\pi$
$61$	−3.08764	−0.395332	−0.197666	+	0.980269i	$0.563336\pi$
$62$	17.3717	2.20621
$63$	0	0
$64$	−1.43131	−0.178914
$65$	1.39402	0.172906
$66$	0	0
$67$	−2.39689	−0.292826	−0.146413	−	0.989224i	$-0.546773\pi$
$67$	−2.39689	−0.292826	−0.146413	+	0.989224i	$0.546773\pi$
$68$	22.4779	2.72585
$69$	0	0
$70$	4.77822	0.571107
$71$	−9.34384	−1.10891	−0.554455	−	0.832214i	$-0.687073\pi$
$71$	−9.34384	−1.10891	−0.554455	+	0.832214i	$0.687073\pi$
$72$	0	0
$73$	−1.79781	−0.210417	−0.105209	−	0.994450i	$-0.533551\pi$
$73$	−1.79781	−0.210417	−0.105209	+	0.994450i	$0.533551\pi$
$74$	−15.6469	−1.81891
$75$	0	0
$76$	−28.8633	−3.31085
$77$	−10.8942	−1.24151
$78$	0	0
$79$	−2.86562	−0.322407	−0.161204	−	0.986921i	$-0.551538\pi$
$79$	−2.86562	−0.322407	−0.161204	+	0.986921i	$0.551538\pi$
$80$	−7.31262	−0.817576
$81$	0	0
$82$	10.2522	1.13216
$83$	−6.59352	−0.723733	−0.361866	−	0.932230i	$-0.617860\pi$
$83$	−6.59352	−0.723733	−0.361866	+	0.932230i	$0.617860\pi$
$84$	0	0
$85$	−5.53791	−0.600670
$86$	−4.64077	−0.500427
$87$	0	0
$88$	38.6427	4.11933
$89$	−3.34552	−0.354624	−0.177312	−	0.984155i	$-0.556740\pi$
$89$	−3.34552	−0.354624	−0.177312	+	0.984155i	$0.556740\pi$
$90$	0	0
$91$	−2.21424	−0.232115
$92$	20.0069	2.08587
$93$	0	0
$94$	3.84311	0.396387
$95$	7.11109	0.729583
$96$	0	0
$97$	2.26753	0.230233	0.115116	−	0.993352i	$-0.463276\pi$
$97$	2.26753	0.230233	0.115116	+	0.993352i	$0.463276\pi$
$98$	10.1498	1.02529
$99$	0	0
$100$	−16.8619	−1.68619
$101$	−12.5571	−1.24948	−0.624741	−	0.780832i	$-0.714796\pi$
$101$	−12.5571	−1.24948	−0.624741	+	0.780832i	$0.714796\pi$
$102$	0	0
$103$	2.77536	0.273464	0.136732	−	0.990608i	$-0.456340\pi$
$103$	2.77536	0.273464	0.136732	+	0.990608i	$0.456340\pi$
$104$	7.85412	0.770160
$105$	0	0
$106$	−9.56436	−0.928973
$107$	−9.95779	−0.962656	−0.481328	−	0.876540i	$-0.659845\pi$
$107$	−9.95779	−0.962656	−0.481328	+	0.876540i	$0.659845\pi$
$108$	0	0
$109$	−4.72545	−0.452616	−0.226308	−	0.974056i	$-0.572666\pi$
$109$	−4.72545	−0.452616	−0.226308	+	0.974056i	$0.572666\pi$
$110$	−17.3813	−1.65724
$111$	0	0
$112$	11.6153	1.09754
$113$	0.548457	0.0515945	0.0257973	−	0.999667i	$-0.491788\pi$
$113$	0.548457	0.0515945	0.0257973	+	0.999667i	$0.491788\pi$
$114$	0	0
$115$	−4.92913	−0.459644
$116$	0	0
$117$	0	0
$118$	−22.1034	−2.03478
$119$	8.79634	0.806359
$120$	0	0
$121$	28.6287	2.60261
$122$	7.82476	0.708420
$123$	0	0
$124$	−30.3140	−2.72228
$125$	9.60185	0.858816
$126$	0	0
$127$	14.2467	1.26419	0.632096	−	0.774890i	$-0.282195\pi$
$127$	14.2467	1.26419	0.632096	+	0.774890i	$0.282195\pi$
$128$	13.0916	1.15714
$129$	0	0
$130$	−3.53274	−0.309842
$131$	−6.83805	−0.597443	−0.298722	−	0.954340i	$-0.596560\pi$
$131$	−6.83805	−0.597443	−0.298722	+	0.954340i	$0.596560\pi$
$132$	0	0
$133$	−11.2952	−0.979415
$134$	6.07423	0.524733
$135$	0	0
$136$	−31.2015	−2.67551
$137$	−13.4742	−1.15118	−0.575588	−	0.817740i	$-0.695227\pi$
$137$	−13.4742	−1.15118	−0.575588	+	0.817740i	$0.695227\pi$
$138$	0	0
$139$	11.7820	0.999334	0.499667	−	0.866218i	$-0.333456\pi$
$139$	11.7820	0.999334	0.499667	+	0.866218i	$0.333456\pi$
$140$	−8.33808	−0.704697
$141$	0	0
$142$	23.6793	1.98712
$143$	8.05451	0.673552
$144$	0	0
$145$	0	0
$146$	4.55603	0.377060
$147$	0	0
$148$	27.3040	2.24438
$149$	−0.761963	−0.0624224	−0.0312112	−	0.999513i	$-0.509936\pi$
$149$	−0.761963	−0.0624224	−0.0312112	+	0.999513i	$0.509936\pi$
$150$	0	0
$151$	18.4366	1.50035	0.750176	−	0.661238i	$-0.229969\pi$
$151$	18.4366	1.50035	0.750176	+	0.661238i	$0.229969\pi$
$152$	40.0651	3.24971
$153$	0	0
$154$	27.6082	2.22473
$155$	7.46849	0.599883
$156$	0	0
$157$	10.9681	0.875353	0.437676	−	0.899133i	$-0.355802\pi$
$157$	10.9681	0.875353	0.437676	+	0.899133i	$0.355802\pi$
$158$	7.26210	0.577741
$159$	0	0
$160$	5.15576	0.407598
$161$	7.82937	0.617041
$162$	0	0
$163$	23.8504	1.86811	0.934054	−	0.357133i	$-0.116246\pi$
$163$	23.8504	1.86811	0.934054	+	0.357133i	$0.116246\pi$
$164$	−17.8902	−1.39699
$165$	0	0
$166$	16.7094	1.29690
$167$	14.3878	1.11336	0.556679	−	0.830728i	$-0.312075\pi$
$167$	14.3878	1.11336	0.556679	+	0.830728i	$0.312075\pi$
$168$	0	0
$169$	−11.3629	−0.874071
$170$	14.0342	1.07638
$171$	0	0
$172$	8.09822	0.617484
$173$	3.94904	0.300240	0.150120	−	0.988668i	$-0.452034\pi$
$173$	3.94904	0.300240	0.150120	+	0.988668i	$0.452034\pi$
$174$	0	0
$175$	−6.59859	−0.498807
$176$	−42.2517	−3.18484
$177$	0	0
$178$	8.47826	0.635473
$179$	−16.4137	−1.22682	−0.613410	−	0.789765i	$-0.710203\pi$
$179$	−16.4137	−1.22682	−0.613410	+	0.789765i	$0.710203\pi$
$180$	0	0
$181$	−16.7532	−1.24525	−0.622626	−	0.782519i	$-0.713934\pi$
$181$	−16.7532	−1.24525	−0.622626	+	0.782519i	$0.713934\pi$
$182$	5.61135	0.415941
$183$	0	0
$184$	−27.7716	−2.04735
$185$	−6.72693	−0.494574
$186$	0	0
$187$	−31.9976	−2.33989
$188$	−6.70630	−0.489107
$189$	0	0
$190$	−18.0211	−1.30738
$191$	2.95004	0.213457	0.106729	−	0.994288i	$-0.465962\pi$
$191$	2.95004	0.213457	0.106729	+	0.994288i	$0.465962\pi$
$192$	0	0
$193$	−4.88334	−0.351511	−0.175755	−	0.984434i	$-0.556237\pi$
$193$	−4.88334	−0.351511	−0.175755	+	0.984434i	$0.556237\pi$
$194$	−5.74640	−0.412568
$195$	0	0
$196$	−17.7117	−1.26512
$197$	2.51855	0.179439	0.0897195	−	0.995967i	$-0.471403\pi$
$197$	2.51855	0.179439	0.0897195	+	0.995967i	$0.471403\pi$
$198$	0	0
$199$	25.5517	1.81131	0.905655	−	0.424015i	$-0.139380\pi$
$199$	25.5517	1.81131	0.905655	+	0.424015i	$0.139380\pi$
$200$	23.4059	1.65505
$201$	0	0
$202$	31.8225	2.23903
$203$	0	0
$204$	0	0
$205$	4.40764	0.307843
$206$	−7.03336	−0.490038
$207$	0	0
$208$	−8.58765	−0.595446
$209$	41.0873	2.84207
$210$	0	0
$211$	−1.84511	−0.127023	−0.0635114	−	0.997981i	$-0.520230\pi$
$211$	−1.84511	−0.127023	−0.0635114	+	0.997981i	$0.520230\pi$
$212$	16.6900	1.14627
$213$	0	0
$214$	25.2352	1.72504
$215$	−1.99517	−0.136069
$216$	0	0
$217$	−11.8628	−0.805303
$218$	11.9753	0.811070
$219$	0	0
$220$	30.3306	2.04489
$221$	−6.50349	−0.437472
$222$	0	0
$223$	−24.0492	−1.61045	−0.805225	−	0.592969i	$-0.797956\pi$
$223$	−24.0492	−1.61045	−0.805225	+	0.592969i	$0.797956\pi$
$224$	−8.18933	−0.547173
$225$	0	0
$226$	−1.38991	−0.0924554
$227$	22.6399	1.50267	0.751333	−	0.659924i	$-0.229411\pi$
$227$	22.6399	1.50267	0.751333	+	0.659924i	$0.229411\pi$
$228$	0	0
$229$	−13.8021	−0.912070	−0.456035	−	0.889962i	$-0.650731\pi$
$229$	−13.8021	−0.912070	−0.456035	+	0.889962i	$0.650731\pi$
$230$	12.4915	0.823664
$231$	0	0
$232$	0	0
$233$	6.69816	0.438811	0.219405	−	0.975634i	$-0.429588\pi$
$233$	6.69816	0.438811	0.219405	+	0.975634i	$0.429588\pi$
$234$	0	0
$235$	1.65224	0.107780
$236$	38.5708	2.51074
$237$	0	0
$238$	−22.2918	−1.44496
$239$	−12.1625	−0.786726	−0.393363	−	0.919383i	$-0.628688\pi$
$239$	−12.1625	−0.786726	−0.393363	+	0.919383i	$0.628688\pi$
$240$	0	0
$241$	2.65618	0.171099	0.0855497	−	0.996334i	$-0.472735\pi$
$241$	2.65618	0.171099	0.0855497	+	0.996334i	$0.472735\pi$
$242$	−72.5513	−4.66377
$243$	0	0
$244$	−13.6543	−0.874129
$245$	4.36364	0.278783
$246$	0	0
$247$	8.35098	0.531360
$248$	42.0787	2.67200
$249$	0	0
$250$	−24.3332	−1.53896
$251$	22.7654	1.43694	0.718471	−	0.695557i	$-0.244843\pi$
$251$	22.7654	1.43694	0.718471	+	0.695557i	$0.244843\pi$
$252$	0	0
$253$	−28.4801	−1.79053
$254$	−36.1043	−2.26538
$255$	0	0
$256$	−30.3142	−1.89464
$257$	−7.70838	−0.480836	−0.240418	−	0.970669i	$-0.577284\pi$
$257$	−7.70838	−0.480836	−0.240418	+	0.970669i	$0.577284\pi$
$258$	0	0
$259$	10.6850	0.663931
$260$	6.16469	0.382318
$261$	0	0
$262$	17.3291	1.07060
$263$	0.769558	0.0474530	0.0237265	−	0.999718i	$-0.492447\pi$
$263$	0.769558	0.0474530	0.0237265	+	0.999718i	$0.492447\pi$
$264$	0	0
$265$	−4.11193	−0.252594
$266$	28.6244	1.75507
$267$	0	0
$268$	−10.5996	−0.647476
$269$	−15.3466	−0.935696	−0.467848	−	0.883809i	$-0.654971\pi$
$269$	−15.3466	−0.935696	−0.467848	+	0.883809i	$0.654971\pi$
$270$	0	0
$271$	15.7301	0.955538	0.477769	−	0.878485i	$-0.341446\pi$
$271$	15.7301	0.955538	0.477769	+	0.878485i	$0.341446\pi$
$272$	34.1155	2.06856
$273$	0	0
$274$	34.1465	2.06286
$275$	24.0031	1.44744
$276$	0	0
$277$	−22.6255	−1.35944	−0.679718	−	0.733473i	$-0.737898\pi$
$277$	−22.6255	−1.35944	−0.679718	+	0.733473i	$0.737898\pi$
$278$	−29.8581	−1.79077
$279$	0	0
$280$	11.5741	0.691682
$281$	22.0455	1.31512	0.657561	−	0.753401i	$-0.271588\pi$
$281$	22.0455	1.31512	0.657561	+	0.753401i	$0.271588\pi$
$282$	0	0
$283$	5.18145	0.308005	0.154003	−	0.988070i	$-0.450784\pi$
$283$	5.18145	0.308005	0.154003	+	0.988070i	$0.450784\pi$
$284$	−41.3208	−2.45194
$285$	0	0
$286$	−20.4119	−1.20698
$287$	−7.00103	−0.413258
$288$	0	0
$289$	8.83596	0.519762
$290$	0	0
$291$	0	0
$292$	−7.95035	−0.465259
$293$	−3.76172	−0.219762	−0.109881	−	0.993945i	$-0.535047\pi$
$293$	−3.76172	−0.219762	−0.109881	+	0.993945i	$0.535047\pi$
$294$	0	0
$295$	−9.50272	−0.553270
$296$	−37.9006	−2.20293
$297$	0	0
$298$	1.93098	0.111859
$299$	−5.78857	−0.334762
$300$	0	0
$301$	3.16910	0.182664
$302$	−46.7224	−2.68857
$303$	0	0
$304$	−43.8069	−2.51250
$305$	3.36403	0.192624
$306$	0	0
$307$	−4.15007	−0.236857	−0.118428	−	0.992963i	$-0.537786\pi$
$307$	−4.15007	−0.236857	−0.118428	+	0.992963i	$0.537786\pi$
$308$	−48.1768	−2.74513
$309$	0	0
$310$	−18.9268	−1.07497
$311$	8.20010	0.464985	0.232492	−	0.972598i	$-0.425312\pi$
$311$	8.20010	0.464985	0.232492	+	0.972598i	$0.425312\pi$
$312$	0	0
$313$	−5.81526	−0.328698	−0.164349	−	0.986402i	$-0.552552\pi$
$313$	−5.81526	−0.328698	−0.164349	+	0.986402i	$0.552552\pi$
$314$	−27.7956	−1.56860
$315$	0	0
$316$	−12.6725	−0.712883
$317$	11.0951	0.623161	0.311581	−	0.950220i	$-0.399142\pi$
$317$	11.0951	0.623161	0.311581	+	0.950220i	$0.399142\pi$
$318$	0	0
$319$	0	0
$320$	1.55944	0.0871752
$321$	0	0
$322$	−19.8413	−1.10571
$323$	−33.1753	−1.84593
$324$	0	0
$325$	4.87861	0.270617
$326$	−60.4421	−3.34758
$327$	0	0
$328$	24.8334	1.37119
$329$	−2.62439	−0.144687
$330$	0	0
$331$	16.7926	0.923002	0.461501	−	0.887140i	$-0.347311\pi$
$331$	16.7926	0.923002	0.461501	+	0.887140i	$0.347311\pi$
$332$	−29.1582	−1.60026
$333$	0	0
$334$	−36.4617	−1.99509
$335$	2.61144	0.142678
$336$	0	0
$337$	21.0778	1.14818	0.574091	−	0.818791i	$-0.305355\pi$
$337$	21.0778	1.14818	0.574091	+	0.818791i	$0.305355\pi$
$338$	28.7961	1.56630
$339$	0	0
$340$	−24.4900	−1.32816
$341$	43.1523	2.33683
$342$	0	0
$343$	−19.0452	−1.02834
$344$	−11.2411	−0.606080
$345$	0	0
$346$	−10.0077	−0.538018
$347$	31.0309	1.66583	0.832914	−	0.553402i	$-0.186671\pi$
$347$	31.0309	1.66583	0.832914	+	0.553402i	$0.186671\pi$
$348$	0	0
$349$	−35.7539	−1.91386	−0.956931	−	0.290315i	$-0.906240\pi$
$349$	−35.7539	−1.91386	−0.956931	+	0.290315i	$0.906240\pi$
$350$	16.7223	0.893843
$351$	0	0
$352$	29.7895	1.58779
$353$	6.02826	0.320852	0.160426	−	0.987048i	$-0.448713\pi$
$353$	6.02826	0.320852	0.160426	+	0.987048i	$0.448713\pi$
$354$	0	0
$355$	10.1803	0.540312
$356$	−14.7947	−0.784118
$357$	0	0
$358$	41.5959	2.19841
$359$	5.71397	0.301572	0.150786	−	0.988566i	$-0.451820\pi$
$359$	5.71397	0.301572	0.150786	+	0.988566i	$0.451820\pi$
$360$	0	0
$361$	23.5997	1.24209
$362$	42.4561	2.23145
$363$	0	0
$364$	−9.79191	−0.513236
$365$	1.95874	0.102525
$366$	0	0
$367$	27.3435	1.42732	0.713658	−	0.700494i	$-0.247037\pi$
$367$	27.3435	1.42732	0.713658	+	0.700494i	$0.247037\pi$
$368$	30.3652	1.58290
$369$	0	0
$370$	17.0475	0.886257
$371$	6.53133	0.339090
$372$	0	0
$373$	−12.6216	−0.653522	−0.326761	−	0.945107i	$-0.605957\pi$
$373$	−12.6216	−0.653522	−0.326761	+	0.945107i	$0.605957\pi$
$374$	81.0888	4.19300
$375$	0	0
$376$	9.30899	0.480074
$377$	0	0
$378$	0	0
$379$	−3.30059	−0.169540	−0.0847700	−	0.996401i	$-0.527016\pi$
$379$	−3.30059	−0.169540	−0.0847700	+	0.996401i	$0.527016\pi$
$380$	31.4471	1.61320
$381$	0	0
$382$	−7.47603	−0.382507
$383$	−9.63804	−0.492481	−0.246240	−	0.969209i	$-0.579195\pi$
$383$	−9.63804	−0.492481	−0.246240	+	0.969209i	$0.579195\pi$
$384$	0	0
$385$	11.8694	0.604919
$386$	12.3754	0.629893
$387$	0	0
$388$	10.0276	0.509073
$389$	−10.7987	−0.547518	−0.273759	−	0.961798i	$-0.588267\pi$
$389$	−10.7987	−0.547518	−0.273759	+	0.961798i	$0.588267\pi$
$390$	0	0
$391$	22.9958	1.16295
$392$	24.5855	1.24175
$393$	0	0
$394$	−6.38254	−0.321548
$395$	3.12213	0.157092
$396$	0	0
$397$	29.7323	1.49222	0.746111	−	0.665822i	$-0.231919\pi$
$397$	29.7323	1.49222	0.746111	+	0.665822i	$0.231919\pi$
$398$	−64.7535	−3.24580
$399$	0	0
$400$	−25.5918	−1.27959
$401$	−11.4349	−0.571030	−0.285515	−	0.958374i	$-0.592165\pi$
$401$	−11.4349	−0.571030	−0.285515	+	0.958374i	$0.592165\pi$
$402$	0	0
$403$	8.77069	0.436899
$404$	−55.5309	−2.76276
$405$	0	0
$406$	0	0
$407$	−38.8677	−1.92660
$408$	0	0
$409$	29.9586	1.48136	0.740679	−	0.671859i	$-0.234504\pi$
$409$	29.9586	1.48136	0.740679	+	0.671859i	$0.234504\pi$
$410$	−11.1699	−0.551642
$411$	0	0
$412$	12.2733	0.604664
$413$	15.0940	0.742727
$414$	0	0
$415$	7.18374	0.352636
$416$	6.05471	0.296857
$417$	0	0
$418$	−104.124	−5.09288
$419$	20.3911	0.996168	0.498084	−	0.867129i	$-0.334037\pi$
$419$	20.3911	0.996168	0.498084	+	0.867129i	$0.334037\pi$
$420$	0	0
$421$	−22.9880	−1.12037	−0.560183	−	0.828369i	$-0.689269\pi$
$421$	−22.9880	−1.12037	−0.560183	+	0.828369i	$0.689269\pi$
$422$	4.67591	0.227620
$423$	0	0
$424$	−23.1673	−1.12510
$425$	−19.3809	−0.940112
$426$	0	0
$427$	−5.34339	−0.258585
$428$	−44.0359	−2.12855
$429$	0	0
$430$	5.05619	0.243831
$431$	4.04708	0.194941	0.0974704	−	0.995238i	$-0.468925\pi$
$431$	4.04708	0.194941	0.0974704	+	0.995238i	$0.468925\pi$
$432$	0	0
$433$	2.62955	0.126368	0.0631841	−	0.998002i	$-0.479874\pi$
$433$	2.62955	0.126368	0.0631841	+	0.998002i	$0.479874\pi$
$434$	30.0630	1.44307
$435$	0	0
$436$	−20.8971	−1.00079
$437$	−29.5284	−1.41254
$438$	0	0
$439$	−26.0095	−1.24137	−0.620683	−	0.784061i	$-0.713145\pi$
$439$	−26.0095	−1.24137	−0.620683	+	0.784061i	$0.713145\pi$
$440$	−42.1018	−2.00713
$441$	0	0
$442$	16.4813	0.783934
$443$	−18.8152	−0.893935	−0.446968	−	0.894550i	$-0.647496\pi$
$443$	−18.8152	−0.893935	−0.446968	+	0.894550i	$0.647496\pi$
$444$	0	0
$445$	3.64499	0.172789
$446$	60.9458	2.88587
$447$	0	0
$448$	−2.47699	−0.117027
$449$	−12.3698	−0.583767	−0.291883	−	0.956454i	$-0.594282\pi$
$449$	−12.3698	−0.583767	−0.291883	+	0.956454i	$0.594282\pi$
$450$	0	0
$451$	25.4670	1.19919
$452$	2.42542	0.114082
$453$	0	0
$454$	−57.3745	−2.69272
$455$	2.41244	0.113097
$456$	0	0
$457$	−20.1273	−0.941514	−0.470757	−	0.882263i	$-0.656019\pi$
$457$	−20.1273	−0.941514	−0.470757	+	0.882263i	$0.656019\pi$
$458$	34.9776	1.63440
$459$	0	0
$460$	−21.7979	−1.01633
$461$	−19.7785	−0.921175	−0.460587	−	0.887614i	$-0.652361\pi$
$461$	−19.7785	−0.921175	−0.460587	+	0.887614i	$0.652361\pi$
$462$	0	0
$463$	2.22268	0.103297	0.0516484	−	0.998665i	$-0.483552\pi$
$463$	2.22268	0.103297	0.0516484	+	0.998665i	$0.483552\pi$
$464$	0	0
$465$	0	0
$466$	−16.9746	−0.786332
$467$	−4.63380	−0.214427	−0.107213	−	0.994236i	$-0.534193\pi$
$467$	−4.63380	−0.214427	−0.107213	+	0.994236i	$0.534193\pi$
$468$	0	0
$469$	−4.14798	−0.191536
$470$	−4.18713	−0.193138
$471$	0	0
$472$	−53.5399	−2.46437
$473$	−11.5279	−0.530054
$474$	0	0
$475$	24.8866	1.14187
$476$	38.8996	1.78296
$477$	0	0
$478$	30.8224	1.40978
$479$	9.50600	0.434340	0.217170	−	0.976134i	$-0.430317\pi$
$479$	9.50600	0.434340	0.217170	+	0.976134i	$0.430317\pi$
$480$	0	0
$481$	−7.89983	−0.360201
$482$	−6.73132	−0.306603
$483$	0	0
$484$	126.603	5.75469
$485$	−2.47051	−0.112180
$486$	0	0
$487$	7.59754	0.344278	0.172139	−	0.985073i	$-0.444932\pi$
$487$	7.59754	0.344278	0.172139	+	0.985073i	$0.444932\pi$
$488$	18.9535	0.857986
$489$	0	0
$490$	−11.0584	−0.499568
$491$	22.5226	1.01643	0.508216	−	0.861230i	$-0.330305\pi$
$491$	22.5226	1.01643	0.508216	+	0.861230i	$0.330305\pi$
$492$	0	0
$493$	0	0
$494$	−21.1632	−0.952177
$495$	0	0
$496$	−46.0086	−2.06585
$497$	−16.1702	−0.725332
$498$	0	0
$499$	−28.7796	−1.28835	−0.644176	−	0.764878i	$-0.722799\pi$
$499$	−28.7796	−1.28835	−0.644176	+	0.764878i	$0.722799\pi$
$500$	42.4618	1.89895
$501$	0	0
$502$	−57.6925	−2.57494
$503$	5.00532	0.223176	0.111588	−	0.993755i	$-0.464406\pi$
$503$	5.00532	0.223176	0.111588	+	0.993755i	$0.464406\pi$
$504$	0	0
$505$	13.6812	0.608805
$506$	72.1748	3.20856
$507$	0	0
$508$	63.0026	2.79529
$509$	−4.98133	−0.220794	−0.110397	−	0.993888i	$-0.535212\pi$
$509$	−4.98133	−0.220794	−0.110397	+	0.993888i	$0.535212\pi$
$510$	0	0
$511$	−3.11123	−0.137633
$512$	50.6397	2.23798
$513$	0	0
$514$	19.5347	0.861639
$515$	−3.02380	−0.133244
$516$	0	0
$517$	9.54650	0.419855
$518$	−27.0780	−1.18974
$519$	0	0
$520$	−8.55718	−0.375257
$521$	−4.54404	−0.199078	−0.0995389	−	0.995034i	$-0.531737\pi$
$521$	−4.54404	−0.199078	−0.0995389	+	0.995034i	$0.531737\pi$
$522$	0	0
$523$	−3.08967	−0.135102	−0.0675510	−	0.997716i	$-0.521519\pi$
$523$	−3.08967	−0.135102	−0.0675510	+	0.997716i	$0.521519\pi$
$524$	−30.2396	−1.32102
$525$	0	0
$526$	−1.95023	−0.0850339
$527$	−34.8427	−1.51777
$528$	0	0
$529$	−2.53205	−0.110089
$530$	10.4205	0.452638
$531$	0	0
$532$	−49.9501	−2.16561
$533$	5.17615	0.224204
$534$	0	0
$535$	10.8492	0.469050
$536$	14.7133	0.635518
$537$	0	0
$538$	38.8915	1.67673
$539$	25.2128	1.08599
$540$	0	0
$541$	21.4608	0.922670	0.461335	−	0.887226i	$-0.347371\pi$
$541$	21.4608	0.922670	0.461335	+	0.887226i	$0.347371\pi$
$542$	−39.8636	−1.71229
$543$	0	0
$544$	−24.0531	−1.03127
$545$	5.14845	0.220535
$546$	0	0
$547$	−14.5151	−0.620619	−0.310309	−	0.950636i	$-0.600433\pi$
$547$	−14.5151	−0.620619	−0.310309	+	0.950636i	$0.600433\pi$
$548$	−59.5862	−2.54540
$549$	0	0
$550$	−60.8290	−2.59376
$551$	0	0
$552$	0	0
$553$	−4.95916	−0.210885
$554$	57.3380	2.43606
$555$	0	0
$556$	52.1029	2.20965
$557$	−24.1558	−1.02351	−0.511757	−	0.859130i	$-0.671005\pi$
$557$	−24.1558	−1.02351	−0.511757	+	0.859130i	$0.671005\pi$
$558$	0	0
$559$	−2.34305	−0.0991003
$560$	−12.6550	−0.534772
$561$	0	0
$562$	−55.8680	−2.35665
$563$	7.54185	0.317851	0.158926	−	0.987291i	$-0.449197\pi$
$563$	7.54185	0.317851	0.158926	+	0.987291i	$0.449197\pi$
$564$	0	0
$565$	−0.597553	−0.0251392
$566$	−13.1309	−0.551933
$567$	0	0
$568$	57.3572	2.40666
$569$	20.0507	0.840568	0.420284	−	0.907393i	$-0.361930\pi$
$569$	20.0507	0.840568	0.420284	+	0.907393i	$0.361930\pi$
$570$	0	0
$571$	−4.55564	−0.190648	−0.0953239	−	0.995446i	$-0.530389\pi$
$571$	−4.55564	−0.190648	−0.0953239	+	0.995446i	$0.530389\pi$
$572$	35.6191	1.48931
$573$	0	0
$574$	17.7421	0.740542
$575$	−17.2504	−0.719391
$576$	0	0
$577$	19.6367	0.817485	0.408742	−	0.912650i	$-0.365967\pi$
$577$	19.6367	0.817485	0.408742	+	0.912650i	$0.365967\pi$
$578$	−22.3922	−0.931394
$579$	0	0
$580$	0	0
$581$	−11.4106	−0.473390
$582$	0	0
$583$	−23.7584	−0.983972
$584$	11.0358	0.456667
$585$	0	0
$586$	9.53300	0.393805
$587$	40.1991	1.65919	0.829596	−	0.558364i	$-0.188571\pi$
$587$	40.1991	1.65919	0.829596	+	0.558364i	$0.188571\pi$
$588$	0	0
$589$	44.7407	1.84351
$590$	24.0819	0.991438
$591$	0	0
$592$	41.4403	1.70319
$593$	−45.9625	−1.88746	−0.943728	−	0.330724i	$-0.892707\pi$
$593$	−45.9625	−1.88746	−0.943728	+	0.330724i	$0.892707\pi$
$594$	0	0
$595$	−9.58374	−0.392895
$596$	−3.36959	−0.138024
$597$	0	0
$598$	14.6695	0.599880
$599$	10.0554	0.410853	0.205426	−	0.978673i	$-0.434142\pi$
$599$	10.0554	0.410853	0.205426	+	0.978673i	$0.434142\pi$
$600$	0	0
$601$	−1.22427	−0.0499390	−0.0249695	−	0.999688i	$-0.507949\pi$
$601$	−1.22427	−0.0499390	−0.0249695	+	0.999688i	$0.507949\pi$
$602$	−8.03118	−0.327327
$603$	0	0
$604$	81.5315	3.31747
$605$	−31.1914	−1.26811
$606$	0	0
$607$	30.7767	1.24919	0.624593	−	0.780950i	$-0.285265\pi$
$607$	30.7767	1.24919	0.624593	+	0.780950i	$0.285265\pi$
$608$	30.8860	1.25259
$609$	0	0
$610$	−8.52519	−0.345175
$611$	1.94032	0.0784970
$612$	0	0
$613$	−7.90846	−0.319420	−0.159710	−	0.987164i	$-0.551056\pi$
$613$	−7.90846	−0.319420	−0.159710	+	0.987164i	$0.551056\pi$
$614$	10.5172	0.424438
$615$	0	0
$616$	66.8740	2.69443
$617$	12.8000	0.515308	0.257654	−	0.966237i	$-0.417051\pi$
$617$	12.8000	0.515308	0.257654	+	0.966237i	$0.417051\pi$
$618$	0	0
$619$	−42.4563	−1.70646	−0.853230	−	0.521534i	$-0.825360\pi$
$619$	−42.4563	−1.70646	−0.853230	+	0.521534i	$0.825360\pi$
$620$	33.0275	1.32642
$621$	0	0
$622$	−20.7808	−0.833235
$623$	−5.78965	−0.231958
$624$	0	0
$625$	8.60343	0.344137
$626$	14.7371	0.589014
$627$	0	0
$628$	48.5039	1.93552
$629$	31.3831	1.25133
$630$	0	0
$631$	44.0652	1.75421	0.877104	−	0.480300i	$-0.159472\pi$
$631$	44.0652	1.75421	0.877104	+	0.480300i	$0.159472\pi$
$632$	17.5906	0.699718
$633$	0	0
$634$	−28.1173	−1.11668
$635$	−15.5220	−0.615972
$636$	0	0
$637$	5.12449	0.203040
$638$	0	0
$639$	0	0
$640$	−14.2635	−0.563813
$641$	40.3894	1.59528	0.797642	−	0.603131i	$-0.206080\pi$
$641$	40.3894	1.59528	0.797642	+	0.603131i	$0.206080\pi$
$642$	0	0
$643$	7.69076	0.303294	0.151647	−	0.988435i	$-0.451542\pi$
$643$	7.69076	0.303294	0.151647	+	0.988435i	$0.451542\pi$
$644$	34.6234	1.36435
$645$	0	0
$646$	84.0735	3.30783
$647$	24.5624	0.965649	0.482824	−	0.875717i	$-0.339611\pi$
$647$	24.5624	0.965649	0.482824	+	0.875717i	$0.339611\pi$
$648$	0	0
$649$	−54.9060	−2.15525
$650$	−12.3635	−0.484935
$651$	0	0
$652$	105.472	4.13062
$653$	−26.6658	−1.04351	−0.521756	−	0.853095i	$-0.674723\pi$
$653$	−26.6658	−1.04351	−0.521756	+	0.853095i	$0.674723\pi$
$654$	0	0
$655$	7.45016	0.291102
$656$	−27.1526	−1.06013
$657$	0	0
$658$	6.65078	0.259274
$659$	12.2372	0.476692	0.238346	−	0.971180i	$-0.423395\pi$
$659$	12.2372	0.476692	0.238346	+	0.971180i	$0.423395\pi$
$660$	0	0
$661$	40.4225	1.57225	0.786126	−	0.618067i	$-0.212084\pi$
$661$	40.4225	1.57225	0.786126	+	0.618067i	$0.212084\pi$
$662$	−42.5560	−1.65398
$663$	0	0
$664$	40.4744	1.57071
$665$	12.3063	0.477216
$666$	0	0
$667$	0	0
$668$	63.6263	2.46178
$669$	0	0
$670$	−6.61796	−0.255674
$671$	19.4371	0.750362
$672$	0	0
$673$	48.1741	1.85698	0.928488	−	0.371362i	$-0.121109\pi$
$673$	48.1741	1.85698	0.928488	+	0.371362i	$0.121109\pi$
$674$	−53.4158	−2.05750
$675$	0	0
$676$	−50.2497	−1.93268
$677$	−2.26329	−0.0869852	−0.0434926	−	0.999054i	$-0.513848\pi$
$677$	−2.26329	−0.0869852	−0.0434926	+	0.999054i	$0.513848\pi$
$678$	0	0
$679$	3.92412	0.150594
$680$	33.9945	1.30363
$681$	0	0
$682$	−109.357	−4.18751
$683$	27.9688	1.07020	0.535099	−	0.844789i	$-0.320274\pi$
$683$	27.9688	1.07020	0.535099	+	0.844789i	$0.320274\pi$
$684$	0	0
$685$	14.6803	0.560906
$686$	48.2645	1.84275
$687$	0	0
$688$	12.2910	0.468589
$689$	−4.82889	−0.183966
$690$	0	0
$691$	−11.9344	−0.454006	−0.227003	−	0.973894i	$-0.572893\pi$
$691$	−11.9344	−0.454006	−0.227003	+	0.973894i	$0.572893\pi$
$692$	17.4636	0.663868
$693$	0	0
$694$	−78.6391	−2.98510
$695$	−12.8366	−0.486921
$696$	0	0
$697$	−20.5629	−0.778876
$698$	90.6081	3.42957
$699$	0	0
$700$	−29.1806	−1.10292
$701$	3.43744	0.129830	0.0649152	−	0.997891i	$-0.479322\pi$
$701$	3.43744	0.129830	0.0649152	+	0.997891i	$0.479322\pi$
$702$	0	0
$703$	−40.2983	−1.51988
$704$	9.01030	0.339589
$705$	0	0
$706$	−15.2769	−0.574954
$707$	−21.7310	−0.817280
$708$	0	0
$709$	−26.5487	−0.997058	−0.498529	−	0.866873i	$-0.666126\pi$
$709$	−26.5487	−0.997058	−0.498529	+	0.866873i	$0.666126\pi$
$710$	−25.7990	−0.968218
$711$	0	0
$712$	20.5365	0.769637
$713$	−31.0125	−1.16143
$714$	0	0
$715$	−8.77551	−0.328186
$716$	−72.5857	−2.71265
$717$	0	0
$718$	−14.4804	−0.540405
$719$	31.1862	1.16305	0.581524	−	0.813530i	$-0.302457\pi$
$719$	31.1862	1.16305	0.581524	+	0.813530i	$0.302457\pi$
$720$	0	0
$721$	4.80296	0.178872
$722$	−59.8067	−2.22578
$723$	0	0
$724$	−74.0867	−2.75341
$725$	0	0
$726$	0	0
$727$	17.1516	0.636119	0.318060	−	0.948071i	$-0.396969\pi$
$727$	17.1516	0.636119	0.318060	+	0.948071i	$0.396969\pi$
$728$	13.5921	0.503757
$729$	0	0
$730$	−4.96386	−0.183721
$731$	9.30805	0.344271
$732$	0	0
$733$	21.3935	0.790186	0.395093	−	0.918641i	$-0.370712\pi$
$733$	21.3935	0.790186	0.395093	+	0.918641i	$0.370712\pi$
$734$	−69.2942	−2.55770
$735$	0	0
$736$	−21.4090	−0.789146
$737$	15.0887	0.555800
$738$	0	0
$739$	−17.3296	−0.637478	−0.318739	−	0.947842i	$-0.603259\pi$
$739$	−17.3296	−0.637478	−0.318739	+	0.947842i	$0.603259\pi$
$740$	−29.7482	−1.09356
$741$	0	0
$742$	−16.5518	−0.607636
$743$	52.1369	1.91272	0.956358	−	0.292197i	$-0.0943862\pi$
$743$	52.1369	1.91272	0.956358	+	0.292197i	$0.0943862\pi$
$744$	0	0
$745$	0.830170	0.0304151
$746$	31.9859	1.17109
$747$	0	0
$748$	−141.501	−5.17380
$749$	−17.2327	−0.629668
$750$	0	0
$751$	23.9867	0.875288	0.437644	−	0.899148i	$-0.355813\pi$
$751$	23.9867	0.875288	0.437644	+	0.899148i	$0.355813\pi$
$752$	−10.1784	−0.371168
$753$	0	0
$754$	0	0
$755$	−20.0870	−0.731041
$756$	0	0
$757$	47.4501	1.72460	0.862301	−	0.506395i	$-0.169022\pi$
$757$	47.4501	1.72460	0.862301	+	0.506395i	$0.169022\pi$
$758$	8.36441	0.303809
$759$	0	0
$760$	−43.6515	−1.58341
$761$	−38.7137	−1.40337	−0.701684	−	0.712488i	$-0.747568\pi$
$761$	−38.7137	−1.40337	−0.701684	+	0.712488i	$0.747568\pi$
$762$	0	0
$763$	−8.17772	−0.296053
$764$	13.0458	0.471981
$765$	0	0
$766$	24.4249	0.882507
$767$	−11.1596	−0.402950
$768$	0	0
$769$	42.9530	1.54893	0.774463	−	0.632619i	$-0.218020\pi$
$769$	42.9530	1.54893	0.774463	+	0.632619i	$0.218020\pi$
$770$	−30.0795	−1.08399
$771$	0	0
$772$	−21.5954	−0.777234
$773$	−28.2216	−1.01506	−0.507531	−	0.861634i	$-0.669442\pi$
$773$	−28.2216	−1.01506	−0.507531	+	0.861634i	$0.669442\pi$
$774$	0	0
$775$	26.1373	0.938881
$776$	−13.9192	−0.499672
$777$	0	0
$778$	27.3664	0.981132
$779$	26.4044	0.946034
$780$	0	0
$781$	58.8207	2.10477
$782$	−58.2765	−2.08396
$783$	0	0
$784$	−26.8816	−0.960058
$785$	−11.9500	−0.426512
$786$	0	0
$787$	13.3827	0.477042	0.238521	−	0.971137i	$-0.423337\pi$
$787$	13.3827	0.477042	0.238521	+	0.971137i	$0.423337\pi$
$788$	11.1377	0.396762
$789$	0	0
$790$	−7.91216	−0.281502
$791$	0.949145	0.0337477
$792$	0	0
$793$	3.95059	0.140289
$794$	−75.3481	−2.67400
$795$	0	0
$796$	112.996	4.00503
$797$	29.7230	1.05284	0.526421	−	0.850224i	$-0.323534\pi$
$797$	29.7230	1.05284	0.526421	+	0.850224i	$0.323534\pi$
$798$	0	0
$799$	−7.70818	−0.272696
$800$	18.0435	0.637934
$801$	0	0
$802$	28.9784	1.02326
$803$	11.3174	0.399383
$804$	0	0
$805$	−8.53021	−0.300651
$806$	−22.2268	−0.782907
$807$	0	0
$808$	77.0822	2.71174
$809$	46.8802	1.64822	0.824111	−	0.566429i	$-0.191675\pi$
$809$	46.8802	1.64822	0.824111	+	0.566429i	$0.191675\pi$
$810$	0	0
$811$	5.86820	0.206060	0.103030	−	0.994678i	$-0.467146\pi$
$811$	5.86820	0.206060	0.103030	+	0.994678i	$0.467146\pi$
$812$	0	0
$813$	0	0
$814$	98.4990	3.45239
$815$	−25.9854	−0.910228
$816$	0	0
$817$	−11.9522	−0.418156
$818$	−75.9216	−2.65454
$819$	0	0
$820$	19.4917	0.680679
$821$	22.3264	0.779198	0.389599	−	0.920985i	$-0.372614\pi$
$821$	22.3264	0.779198	0.389599	+	0.920985i	$0.372614\pi$
$822$	0	0
$823$	−18.4220	−0.642152	−0.321076	−	0.947053i	$-0.604045\pi$
$823$	−18.4220	−0.642152	−0.321076	+	0.947053i	$0.604045\pi$
$824$	−17.0366	−0.593497
$825$	0	0
$826$	−38.2514	−1.33094
$827$	−28.3783	−0.986809	−0.493404	−	0.869800i	$-0.664248\pi$
$827$	−28.3783	−0.986809	−0.493404	+	0.869800i	$0.664248\pi$
$828$	0	0
$829$	−9.92857	−0.344833	−0.172417	−	0.985024i	$-0.555158\pi$
$829$	−9.92857	−0.344833	−0.172417	+	0.985024i	$0.555158\pi$
$830$	−18.2051	−0.631910
$831$	0	0
$832$	1.83134	0.0634903
$833$	−20.3577	−0.705352
$834$	0	0
$835$	−15.6757	−0.542479
$836$	181.699	6.28418
$837$	0	0
$838$	−51.6753	−1.78509
$839$	−52.9639	−1.82852	−0.914259	−	0.405130i	$-0.867226\pi$
$839$	−52.9639	−1.82852	−0.914259	+	0.405130i	$0.867226\pi$
$840$	0	0
$841$	0	0
$842$	58.2565	2.00765
$843$	0	0
$844$	−8.15955	−0.280863
$845$	12.3801	0.425888
$846$	0	0
$847$	49.5440	1.70235
$848$	25.3310	0.869870
$849$	0	0
$850$	49.1154	1.68464
$851$	27.9332	0.957538
$852$	0	0
$853$	44.7737	1.53302	0.766512	−	0.642230i	$-0.221991\pi$
$853$	44.7737	1.53302	0.766512	+	0.642230i	$0.221991\pi$
$854$	13.5413	0.463374
$855$	0	0
$856$	61.1260	2.08924
$857$	−0.211298	−0.00721779	−0.00360889	−	0.999993i	$-0.501149\pi$
$857$	−0.211298	−0.00721779	−0.00360889	+	0.999993i	$0.501149\pi$
$858$	0	0
$859$	56.0665	1.91296	0.956482	−	0.291790i	$-0.0942508\pi$
$859$	56.0665	1.91296	0.956482	+	0.291790i	$0.0942508\pi$
$860$	−8.82314	−0.300866
$861$	0	0
$862$	−10.2562	−0.349326
$863$	−26.3407	−0.896647	−0.448323	−	0.893871i	$-0.647979\pi$
$863$	−26.3407	−0.896647	−0.448323	+	0.893871i	$0.647979\pi$
$864$	0	0
$865$	−4.30253	−0.146291
$866$	−6.66385	−0.226447
$867$	0	0
$868$	−52.4605	−1.78063
$869$	18.0394	0.611946
$870$	0	0
$871$	3.06678	0.103914
$872$	29.0072	0.982308
$873$	0	0
$874$	74.8314	2.53121
$875$	16.6167	0.561746
$876$	0	0
$877$	33.6648	1.13678	0.568390	−	0.822759i	$-0.307566\pi$
$877$	33.6648	1.13678	0.568390	+	0.822759i	$0.307566\pi$
$878$	65.9138	2.22448
$879$	0	0
$880$	46.0339	1.55180
$881$	16.4282	0.553481	0.276741	−	0.960945i	$-0.410746\pi$
$881$	16.4282	0.553481	0.276741	+	0.960945i	$0.410746\pi$
$882$	0	0
$883$	4.29020	0.144377	0.0721884	−	0.997391i	$-0.477002\pi$
$883$	4.29020	0.144377	0.0721884	+	0.997391i	$0.477002\pi$
$884$	−28.7601	−0.967306
$885$	0	0
$886$	47.6817	1.60190
$887$	39.5279	1.32722	0.663608	−	0.748081i	$-0.269024\pi$
$887$	39.5279	1.32722	0.663608	+	0.748081i	$0.269024\pi$
$888$	0	0
$889$	24.6550	0.826901
$890$	−9.23719	−0.309631
$891$	0	0
$892$	−106.351	−3.56091
$893$	9.89789	0.331220
$894$	0	0
$895$	17.8830	0.597763
$896$	22.6559	0.756880
$897$	0	0
$898$	31.3477	1.04609
$899$	0	0
$900$	0	0
$901$	19.1834	0.639090
$902$	−64.5388	−2.14891
$903$	0	0
$904$	−3.36671	−0.111975
$905$	18.2528	0.606744
$906$	0	0
$907$	27.1253	0.900681	0.450340	−	0.892857i	$-0.351303\pi$
$907$	27.1253	0.900681	0.450340	+	0.892857i	$0.351303\pi$
$908$	100.119	3.32258
$909$	0	0
$910$	−6.11365	−0.202666
$911$	12.8679	0.426333	0.213166	−	0.977016i	$-0.431622\pi$
$911$	12.8679	0.426333	0.213166	+	0.977016i	$0.431622\pi$
$912$	0	0
$913$	41.5071	1.37368
$914$	51.0069	1.68716
$915$	0	0
$916$	−61.0365	−2.01670
$917$	−11.8337	−0.390784
$918$	0	0
$919$	−23.3758	−0.771097	−0.385548	−	0.922688i	$-0.625988\pi$
$919$	−23.3758	−0.771097	−0.385548	+	0.922688i	$0.625988\pi$
$920$	30.2575	0.997561
$921$	0	0
$922$	50.1229	1.65071
$923$	11.9553	0.393513
$924$	0	0
$925$	−23.5421	−0.774060
$926$	−5.63276	−0.185104
$927$	0	0
$928$	0	0
$929$	6.68335	0.219274	0.109637	−	0.993972i	$-0.465031\pi$
$929$	6.68335	0.219274	0.109637	+	0.993972i	$0.465031\pi$
$930$	0	0
$931$	26.1408	0.856731
$932$	29.6209	0.970266
$933$	0	0
$934$	11.7430	0.384244
$935$	34.8618	1.14010
$936$	0	0
$937$	−35.4245	−1.15727	−0.578633	−	0.815588i	$-0.696414\pi$
$937$	−35.4245	−1.15727	−0.578633	+	0.815588i	$0.696414\pi$
$938$	10.5119	0.343225
$939$	0	0
$940$	7.30661	0.238315
$941$	−54.0181	−1.76094	−0.880470	−	0.474103i	$-0.842773\pi$
$941$	−54.0181	−1.76094	−0.880470	+	0.474103i	$0.842773\pi$
$942$	0	0
$943$	−18.3025	−0.596010
$944$	58.5402	1.90532
$945$	0	0
$946$	29.2142	0.949837
$947$	46.9883	1.52691	0.763457	−	0.645859i	$-0.223501\pi$
$947$	46.9883	1.52691	0.763457	+	0.645859i	$0.223501\pi$
$948$	0	0
$949$	2.30026	0.0746696
$950$	−63.0680	−2.04619
$951$	0	0
$952$	−53.9964	−1.75003
$953$	39.4554	1.27809	0.639043	−	0.769171i	$-0.279330\pi$
$953$	39.4554	1.27809	0.639043	+	0.769171i	$0.279330\pi$
$954$	0	0
$955$	−3.21411	−0.104006
$956$	−53.7856	−1.73955
$957$	0	0
$958$	−24.0903	−0.778321
$959$	−23.3180	−0.752978
$960$	0	0
$961$	15.9893	0.515783
$962$	20.0199	0.645467
$963$	0	0
$964$	11.7463	0.378322
$965$	5.32047	0.171272
$966$	0	0
$967$	−19.1270	−0.615082	−0.307541	−	0.951535i	$-0.599506\pi$
$967$	−19.1270	−0.615082	−0.307541	+	0.951535i	$0.599506\pi$
$968$	−175.737	−5.64842
$969$	0	0
$970$	6.26079	0.201022
$971$	−18.6748	−0.599304	−0.299652	−	0.954049i	$-0.596871\pi$
$971$	−18.6748	−0.599304	−0.299652	+	0.954049i	$0.596871\pi$
$972$	0	0
$973$	20.3895	0.653659
$974$	−19.2538	−0.616932
$975$	0	0
$976$	−20.7237	−0.663349
$977$	−54.0174	−1.72817	−0.864085	−	0.503345i	$-0.832102\pi$
$977$	−54.0174	−1.72817	−0.864085	+	0.503345i	$0.832102\pi$
$978$	0	0
$979$	21.0605	0.673095
$980$	19.2971	0.616424
$981$	0	0
$982$	−57.0772	−1.82141
$983$	53.7012	1.71280	0.856401	−	0.516311i	$-0.172695\pi$
$983$	53.7012	1.71280	0.856401	+	0.516311i	$0.172695\pi$
$984$	0	0
$985$	−2.74399	−0.0874309
$986$	0	0
$987$	0	0
$988$	36.9302	1.17490
$989$	8.28483	0.263442
$990$	0	0
$991$	−3.10936	−0.0987721	−0.0493861	−	0.998780i	$-0.515726\pi$
$991$	−3.10936	−0.0987721	−0.0493861	+	0.998780i	$0.515726\pi$
$992$	32.4383	1.02992
$993$	0	0
$994$	40.9787	1.29977
$995$	−27.8389	−0.882553
$996$	0	0
$997$	−37.8192	−1.19774	−0.598872	−	0.800844i	$-0.704384\pi$
$997$	−37.8192	−1.19774	−0.598872	+	0.800844i	$0.704384\pi$
$998$	72.9337	2.30868
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7569.2.a.bj.1.3		9
3.2	odd	2		2523.2.a.r.1.7		9
29.23	even	7		261.2.k.c.181.3		18
29.24	even	7		261.2.k.c.199.3		18
29.28	even	2		7569.2.a.bm.1.7		9
87.23	odd	14		87.2.g.a.7.1	✓	18
87.53	odd	14		87.2.g.a.25.1	yes	18
87.86	odd	2		2523.2.a.o.1.3		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
87.2.g.a.7.1	✓	18	87.23	odd	14
87.2.g.a.25.1	yes	18	87.53	odd	14
261.2.k.c.181.3		18	29.23	even	7
261.2.k.c.199.3		18	29.24	even	7
2523.2.a.o.1.3		9	87.86	odd	2
2523.2.a.r.1.7		9	3.2	odd	2
7569.2.a.bj.1.3		9	1.1	even	1	trivial
7569.2.a.bm.1.7		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)

\(f(q)\)	\(=\)	\(q-2.53422 q^{2} +4.42225 q^{4} -1.08952 q^{5} +1.73057 q^{7} -6.13851 q^{8} +2.76107 q^{10} -6.29513 q^{11} -1.27948 q^{13} -4.38564 q^{14} +6.71181 q^{16} +5.08291 q^{17} -6.52684 q^{19} -4.81811 q^{20} +15.9532 q^{22} +4.52415 q^{23} -3.81296 q^{25} +3.24249 q^{26} +7.65302 q^{28} -6.85487 q^{31} -4.73216 q^{32} -12.8812 q^{34} -1.88548 q^{35} +6.17424 q^{37} +16.5404 q^{38} +6.68800 q^{40} -4.04550 q^{41} +1.83124 q^{43} -27.8387 q^{44} -11.4652 q^{46} -1.51649 q^{47} -4.00512 q^{49} +9.66286 q^{50} -5.65820 q^{52} +3.77409 q^{53} +6.85864 q^{55} -10.6231 q^{56} +8.72197 q^{59} -3.08764 q^{61} +17.3717 q^{62} -1.43131 q^{64} +1.39402 q^{65} -2.39689 q^{67} +22.4779 q^{68} +4.77822 q^{70} -9.34384 q^{71} -1.79781 q^{73} -15.6469 q^{74} -28.8633 q^{76} -10.8942 q^{77} -2.86562 q^{79} -7.31262 q^{80} +10.2522 q^{82} -6.59352 q^{83} -5.53791 q^{85} -4.64077 q^{86} +38.6427 q^{88} -3.34552 q^{89} -2.21424 q^{91} +20.0069 q^{92} +3.84311 q^{94} +7.11109 q^{95} +2.26753 q^{97} +10.1498 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - 5 q^{2} + 11 q^{4} + 4 q^{5} + 5 q^{7} - 24 q^{8} + q^{11} + q^{13} - 9 q^{14} + 35 q^{16} - 2 q^{17} + 9 q^{19} + 18 q^{20} - 4 q^{22} + 4 q^{23} + q^{25} + 8 q^{26} + 40 q^{28} + 8 q^{31} - 43 q^{32}+ \cdots + 30 q^{98}+O(q^{100}) \) 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

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more