LMFDB - Embedded newform 7569.2.a.bg.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:	$1$
Dimension:	$8$
Coefficient field:	8.8.14884000000.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 11x^{6} + 36x^{4} - 31x^{2} + 1 $ x^8 - 11x^6 + 36x^4 - 31*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.13370$ 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)

$f(q)$	$=$	\(q-1.13370 q^{2} -0.714715 q^{4} -0.407162 q^{5} +2.15643 q^{7} +3.07768 q^{8} +0.461601 q^{10} -3.55259 q^{11} -0.0966811 q^{13} -2.44476 q^{14} -2.05975 q^{16} -0.587012 q^{17} -3.61803 q^{19} +0.291004 q^{20} +4.02758 q^{22} +8.26725 q^{23} -4.83422 q^{25} +0.109608 q^{26} -1.54124 q^{28} +1.67779 q^{31} -3.82022 q^{32} +0.665498 q^{34} -0.878017 q^{35} -0.0122881 q^{37} +4.10178 q^{38} -1.25311 q^{40} +9.57247 q^{41} +3.47689 q^{43} +2.53909 q^{44} -9.37262 q^{46} -3.31333 q^{47} -2.34980 q^{49} +5.48057 q^{50} +0.0690994 q^{52} +6.88596 q^{53} +1.44648 q^{55} +6.63682 q^{56} -6.45137 q^{59} -9.67779 q^{61} -1.90211 q^{62} +8.45050 q^{64} +0.0393648 q^{65} +8.02282 q^{67} +0.419546 q^{68} +0.995411 q^{70} -9.30889 q^{71} -9.28647 q^{73} +0.0139310 q^{74} +2.58586 q^{76} -7.66092 q^{77} -11.0410 q^{79} +0.838652 q^{80} -10.8523 q^{82} +12.8784 q^{83} +0.239009 q^{85} -3.94177 q^{86} -10.9337 q^{88} -2.23999 q^{89} -0.208486 q^{91} -5.90873 q^{92} +3.75634 q^{94} +1.47312 q^{95} +5.72809 q^{97} +2.66397 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 6 q^{4} + 12 q^{10} + 2 q^{13} - 2 q^{16} - 20 q^{19} - 14 q^{22} + 2 q^{25} - 20 q^{28} - 10 q^{31} - 36 q^{34} - 18 q^{37} + 10 q^{40} - 28 q^{43} - 26 q^{46} + 4 q^{49} + 44 q^{52} - 14 q^{55}+ \cdots + 6 q^{97}+O(q^{100}) $ 8 * q + 6 * q^4 + 12 * q^10 + 2 * q^13 - 2 * q^16 - 20 * q^19 - 14 * q^22 + 2 * q^25 - 20 * q^28 - 10 * q^31 - 36 * q^34 - 18 * q^37 + 10 * q^40 - 28 * q^43 - 26 * q^46 + 4 * q^49 + 44 * q^52 - 14 * q^55 - 54 * q^61 - 44 * q^64 + 38 * q^67 - 70 * q^70 + 16 * q^73 - 20 * q^76 - 14 * q^79 + 2 * q^82 - 56 * q^85 - 60 * q^88 - 30 * q^91 + 52 * q^94 + 6 * q^97

Coefficient data

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

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

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

$2$	−1.13370	−0.801650	−0.400825	−	0.916155i	$-0.631276\pi$
$2$	−1.13370	−0.801650	−0.400825	+	0.916155i	$0.631276\pi$
$3$	0	0
$3$	0	0
$4$	−0.714715	−0.357358
$5$	−0.407162	−0.182088	−0.0910441	−	0.995847i	$-0.529020\pi$
$5$	−0.407162	−0.182088	−0.0910441	+	0.995847i	$0.529020\pi$
$6$	0	0
$7$	2.15643	0.815055	0.407528	−	0.913193i	$-0.366391\pi$
$7$	2.15643	0.815055	0.407528	+	0.913193i	$0.366391\pi$
$8$	3.07768	1.08813
$9$	0	0
$10$	0.461601	0.145971
$11$	−3.55259	−1.07115	−0.535573	−	0.844489i	$-0.679904\pi$
$11$	−3.55259	−1.07115	−0.535573	+	0.844489i	$0.679904\pi$
$12$	0	0
$13$	−0.0966811	−0.0268145	−0.0134073	−	0.999910i	$-0.504268\pi$
$13$	−0.0966811	−0.0268145	−0.0134073	+	0.999910i	$0.504268\pi$
$14$	−2.44476	−0.653389
$15$	0	0
$16$	−2.05975	−0.514938
$17$	−0.587012	−0.142371	−0.0711857	−	0.997463i	$-0.522678\pi$
$17$	−0.587012	−0.142371	−0.0711857	+	0.997463i	$0.522678\pi$
$18$	0	0
$19$	−3.61803	−0.830034	−0.415017	−	0.909814i	$-0.636224\pi$
$19$	−3.61803	−0.830034	−0.415017	+	0.909814i	$0.636224\pi$
$20$	0.291004	0.0650706
$21$	0	0
$22$	4.02758	0.858683
$23$	8.26725	1.72384	0.861921	−	0.507043i	$-0.169261\pi$
$23$	8.26725	1.72384	0.861921	+	0.507043i	$0.169261\pi$
$24$	0	0
$25$	−4.83422	−0.966844
$26$	0.109608	0.0214958
$27$	0	0
$28$	−1.54124	−0.291266
$29$	0	0
$29$	0	0
$30$	0	0
$31$	1.67779	0.301339	0.150670	−	0.988584i	$-0.451857\pi$
$31$	1.67779	0.301339	0.150670	+	0.988584i	$0.451857\pi$
$32$	−3.82022	−0.675325
$33$	0	0
$34$	0.665498	0.114132
$35$	−0.878017	−0.148412
$36$	0	0
$37$	−0.0122881	−0.00202014	−0.00101007	−	0.999999i	$-0.500322\pi$
$37$	−0.0122881	−0.00202014	−0.00101007	+	0.999999i	$0.500322\pi$
$38$	4.10178	0.665397
$39$	0	0
$40$	−1.25311	−0.198135
$41$	9.57247	1.49497	0.747484	−	0.664279i	$-0.231261\pi$
$41$	9.57247	1.49497	0.747484	+	0.664279i	$0.231261\pi$
$42$	0	0
$43$	3.47689	0.530221	0.265111	−	0.964218i	$-0.414591\pi$
$43$	3.47689	0.530221	0.265111	+	0.964218i	$0.414591\pi$
$44$	2.53909	0.382782
$45$	0	0
$46$	−9.37262	−1.38192
$47$	−3.31333	−0.483299	−0.241650	−	0.970364i	$-0.577688\pi$
$47$	−3.31333	−0.483299	−0.241650	+	0.970364i	$0.577688\pi$
$48$	0	0
$49$	−2.34980	−0.335685
$50$	5.48057	0.775070
$51$	0	0
$52$	0.0690994	0.00958237
$53$	6.88596	0.945859	0.472929	−	0.881100i	$-0.343197\pi$
$53$	6.88596	0.945859	0.472929	+	0.881100i	$0.343197\pi$
$54$	0	0
$55$	1.44648	0.195043
$56$	6.63682	0.886882
$57$	0	0
$58$	0	0
$59$	−6.45137	−0.839897	−0.419949	−	0.907548i	$-0.637952\pi$
$59$	−6.45137	−0.839897	−0.419949	+	0.907548i	$0.637952\pi$
$60$	0	0
$61$	−9.67779	−1.23911	−0.619557	−	0.784952i	$-0.712688\pi$
$61$	−9.67779	−1.23911	−0.619557	+	0.784952i	$0.712688\pi$
$62$	−1.90211	−0.241569
$63$	0	0
$64$	8.45050	1.05631
$65$	0.0393648	0.00488261
$66$	0	0
$67$	8.02282	0.980144	0.490072	−	0.871682i	$-0.336971\pi$
$67$	8.02282	0.980144	0.490072	+	0.871682i	$0.336971\pi$
$68$	0.419546	0.0508775
$69$	0	0
$70$	0.995411	0.118974
$71$	−9.30889	−1.10476	−0.552381	−	0.833592i	$-0.686281\pi$
$71$	−9.30889	−1.10476	−0.552381	+	0.833592i	$0.686281\pi$
$72$	0	0
$73$	−9.28647	−1.08690	−0.543450	−	0.839442i	$-0.682882\pi$
$73$	−9.28647	−1.08690	−0.543450	+	0.839442i	$0.682882\pi$
$74$	0.0139310	0.00161945
$75$	0	0
$76$	2.58586	0.296619
$77$	−7.66092	−0.873042
$78$	0	0
$79$	−11.0410	−1.24220	−0.621102	−	0.783730i	$-0.713315\pi$
$79$	−11.0410	−1.24220	−0.621102	+	0.783730i	$0.713315\pi$
$80$	0.838652	0.0937641
$81$	0	0
$82$	−10.8523	−1.19844
$83$	12.8784	1.41359	0.706795	−	0.707418i	$-0.250140\pi$
$83$	12.8784	1.41359	0.706795	+	0.707418i	$0.250140\pi$
$84$	0	0
$85$	0.239009	0.0259241
$86$	−3.94177	−0.425052
$87$	0	0
$88$	−10.9337	−1.16554
$89$	−2.23999	−0.237438	−0.118719	−	0.992928i	$-0.537879\pi$
$89$	−2.23999	−0.237438	−0.118719	+	0.992928i	$0.537879\pi$
$90$	0	0
$91$	−0.208486	−0.0218553
$92$	−5.90873	−0.616028
$93$	0	0
$94$	3.75634	0.387437
$95$	1.47312	0.151139
$96$	0	0
$97$	5.72809	0.581599	0.290800	−	0.956784i	$-0.406079\pi$
$97$	5.72809	0.581599	0.290800	+	0.956784i	$0.406079\pi$
$98$	2.66397	0.269102
$99$	0	0
$100$	3.45509	0.345509
$101$	15.9995	1.59201	0.796006	−	0.605289i	$-0.206942\pi$
$101$	15.9995	1.59201	0.796006	+	0.605289i	$0.206942\pi$
$102$	0	0
$103$	−11.2267	−1.10620	−0.553101	−	0.833114i	$-0.686556\pi$
$103$	−11.2267	−1.10620	−0.553101	+	0.833114i	$0.686556\pi$
$104$	−0.297554	−0.0291776
$105$	0	0
$106$	−7.80664	−0.758248
$107$	−10.5022	−1.01529	−0.507645	−	0.861566i	$-0.669484\pi$
$107$	−10.5022	−1.01529	−0.507645	+	0.861566i	$0.669484\pi$
$108$	0	0
$109$	3.87115	0.370789	0.185394	−	0.982664i	$-0.440644\pi$
$109$	3.87115	0.370789	0.185394	+	0.982664i	$0.440644\pi$
$110$	−1.63988	−0.156356
$111$	0	0
$112$	−4.44172	−0.419703
$113$	−11.5563	−1.08712	−0.543561	−	0.839370i	$-0.682924\pi$
$113$	−11.5563	−1.08712	−0.543561	+	0.839370i	$0.682924\pi$
$114$	0	0
$115$	−3.36611	−0.313891
$116$	0	0
$117$	0	0
$118$	7.31395	0.673304
$119$	−1.26585	−0.116041
$120$	0	0
$121$	1.62087	0.147352
$122$	10.9717	0.993335
$123$	0	0
$124$	−1.19914	−0.107686
$125$	4.00412	0.358139
$126$	0	0
$127$	9.69886	0.860634	0.430317	−	0.902678i	$-0.358402\pi$
$127$	9.69886	0.860634	0.430317	+	0.902678i	$0.358402\pi$
$128$	−1.93993	−0.171467
$129$	0	0
$130$	−0.0446281	−0.00391414
$131$	17.3488	1.51577	0.757887	−	0.652386i	$-0.226232\pi$
$131$	17.3488	1.51577	0.757887	+	0.652386i	$0.226232\pi$
$132$	0	0
$133$	−7.80205	−0.676523
$134$	−9.09551	−0.785732
$135$	0	0
$136$	−1.80664	−0.154918
$137$	−20.7061	−1.76904	−0.884520	−	0.466502i	$-0.845514\pi$
$137$	−20.7061	−1.76904	−0.884520	+	0.466502i	$0.845514\pi$
$138$	0	0
$139$	16.3187	1.38413	0.692067	−	0.721833i	$-0.256700\pi$
$139$	16.3187	1.38413	0.692067	+	0.721833i	$0.256700\pi$
$140$	0.627532	0.0530361
$141$	0	0
$142$	10.5535	0.885632
$143$	0.343468	0.0287222
$144$	0	0
$145$	0	0
$146$	10.5281	0.871313
$147$	0	0
$148$	0.00878246	0.000721914 0
$149$	5.71650	0.468314	0.234157	−	0.972199i	$-0.424767\pi$
$149$	5.71650	0.468314	0.234157	+	0.972199i	$0.424767\pi$
$150$	0	0
$151$	−0.741109	−0.0603106	−0.0301553	−	0.999545i	$-0.509600\pi$
$151$	−0.741109	−0.0603106	−0.0301553	+	0.999545i	$0.509600\pi$
$152$	−11.1352	−0.903181
$153$	0	0
$154$	8.68521	0.699874
$155$	−0.683130	−0.0548703
$156$	0	0
$157$	−20.1518	−1.60829	−0.804146	−	0.594432i	$-0.797377\pi$
$157$	−20.1518	−1.60829	−0.804146	+	0.594432i	$0.797377\pi$
$158$	12.5172	0.995813
$159$	0	0
$160$	1.55545	0.122969
$161$	17.8278	1.40503
$162$	0	0
$163$	−23.1453	−1.81288	−0.906441	−	0.422332i	$-0.861212\pi$
$163$	−23.1453	−1.81288	−0.906441	+	0.422332i	$0.861212\pi$
$164$	−6.84159	−0.534238
$165$	0	0
$166$	−14.6003	−1.13320
$167$	15.7922	1.22204	0.611020	−	0.791615i	$-0.290759\pi$
$167$	15.7922	1.22204	0.611020	+	0.791615i	$0.290759\pi$
$168$	0	0
$169$	−12.9907	−0.999281
$170$	−0.270965	−0.0207821
$171$	0	0
$172$	−2.48499	−0.189479
$173$	17.6792	1.34412	0.672062	−	0.740495i	$-0.265409\pi$
$173$	17.6792	1.34412	0.672062	+	0.740495i	$0.265409\pi$
$174$	0	0
$175$	−10.4247	−0.788031
$176$	7.31745	0.551573
$177$	0	0
$178$	2.53948	0.190342
$179$	−22.1429	−1.65504	−0.827521	−	0.561435i	$-0.810249\pi$
$179$	−22.1429	−1.65504	−0.827521	+	0.561435i	$0.810249\pi$
$180$	0	0
$181$	24.1746	1.79688	0.898441	−	0.439095i	$-0.144701\pi$
$181$	24.1746	1.79688	0.898441	+	0.439095i	$0.144701\pi$
$182$	0.236362	0.0175203
$183$	0	0
$184$	25.4440	1.87576
$185$	0.00500322	0.000367844 0
$186$	0	0
$187$	2.08541	0.152500
$188$	2.36809	0.172711
$189$	0	0
$190$	−1.67009	−0.121161
$191$	6.35069	0.459520	0.229760	−	0.973247i	$-0.426206\pi$
$191$	6.35069	0.459520	0.229760	+	0.973247i	$0.426206\pi$
$192$	0	0
$193$	−17.9490	−1.29200	−0.646000	−	0.763338i	$-0.723559\pi$
$193$	−17.9490	−1.29200	−0.646000	+	0.763338i	$0.723559\pi$
$194$	−6.49395	−0.466239
$195$	0	0
$196$	1.67943	0.119960
$197$	−4.15154	−0.295785	−0.147893	−	0.989003i	$-0.547249\pi$
$197$	−4.15154	−0.295785	−0.147893	+	0.989003i	$0.547249\pi$
$198$	0	0
$199$	−7.04095	−0.499120	−0.249560	−	0.968359i	$-0.580286\pi$
$199$	−7.04095	−0.499120	−0.249560	+	0.968359i	$0.580286\pi$
$200$	−14.8782	−1.05205
$201$	0	0
$202$	−18.1387	−1.27624
$203$	0	0
$204$	0	0
$205$	−3.89754	−0.272216
$206$	12.7278	0.886786
$207$	0	0
$208$	0.199139	0.0138078
$209$	12.8534	0.889087
$210$	0	0
$211$	−1.57816	−0.108645	−0.0543227	−	0.998523i	$-0.517300\pi$
$211$	−1.57816	−0.108645	−0.0543227	+	0.998523i	$0.517300\pi$
$212$	−4.92150	−0.338010
$213$	0	0
$214$	11.9064	0.813907
$215$	−1.41566	−0.0965471
$216$	0	0
$217$	3.61803	0.245608
$218$	−4.38874	−0.297243
$219$	0	0
$220$	−1.03382	−0.0697000
$221$	0.0567530	0.00381762
$222$	0	0
$223$	23.1323	1.54905	0.774527	−	0.632541i	$-0.217988\pi$
$223$	23.1323	1.54905	0.774527	+	0.632541i	$0.217988\pi$
$224$	−8.23804	−0.550427
$225$	0	0
$226$	13.1014	0.871490
$227$	14.4670	0.960208	0.480104	−	0.877212i	$-0.340599\pi$
$227$	14.4670	0.960208	0.480104	+	0.877212i	$0.340599\pi$
$228$	0	0
$229$	−6.77849	−0.447935	−0.223968	−	0.974597i	$-0.571901\pi$
$229$	−6.77849	−0.447935	−0.223968	+	0.974597i	$0.571901\pi$
$230$	3.81617	0.251631
$231$	0	0
$232$	0	0
$233$	−12.4345	−0.814614	−0.407307	−	0.913291i	$-0.633532\pi$
$233$	−12.4345	−0.814614	−0.407307	+	0.913291i	$0.633532\pi$
$234$	0	0
$235$	1.34906	0.0880031
$236$	4.61089	0.300144
$237$	0	0
$238$	1.43510	0.0930239
$239$	−0.266676	−0.0172498	−0.00862491	−	0.999963i	$-0.502745\pi$
$239$	−0.266676	−0.0172498	−0.00862491	+	0.999963i	$0.502745\pi$
$240$	0	0
$241$	23.7084	1.52719	0.763595	−	0.645695i	$-0.223432\pi$
$241$	23.7084	1.52719	0.763595	+	0.645695i	$0.223432\pi$
$242$	−1.83759	−0.118125
$243$	0	0
$244$	6.91686	0.442807
$245$	0.956746	0.0611243
$246$	0	0
$247$	0.349795	0.0222570
$248$	5.16369	0.327895
$249$	0	0
$250$	−4.53948	−0.287102
$251$	−26.7465	−1.68822	−0.844111	−	0.536169i	$-0.819871\pi$
$251$	−26.7465	−1.68822	−0.844111	+	0.536169i	$0.819871\pi$
$252$	0	0
$253$	−29.3701	−1.84648
$254$	−10.9956	−0.689927
$255$	0	0
$256$	−14.7017	−0.918856
$257$	15.4693	0.964946	0.482473	−	0.875911i	$-0.339739\pi$
$257$	15.4693	0.964946	0.482473	+	0.875911i	$0.339739\pi$
$258$	0	0
$259$	−0.0264984	−0.00164653
$260$	−0.0281346	−0.00174484
$261$	0	0
$262$	−19.6684	−1.21512
$263$	5.72078	0.352759	0.176379	−	0.984322i	$-0.443561\pi$
$263$	5.72078	0.352759	0.176379	+	0.984322i	$0.443561\pi$
$264$	0	0
$265$	−2.80370	−0.172230
$266$	8.84521	0.542335
$267$	0	0
$268$	−5.73403	−0.350262
$269$	10.2232	0.623318	0.311659	−	0.950194i	$-0.399115\pi$
$269$	10.2232	0.623318	0.311659	+	0.950194i	$0.399115\pi$
$270$	0	0
$271$	14.6298	0.888694	0.444347	−	0.895855i	$-0.353436\pi$
$271$	14.6298	0.888694	0.444347	+	0.895855i	$0.353436\pi$
$272$	1.20910	0.0733124
$273$	0	0
$274$	23.4746	1.41815
$275$	17.1740	1.03563
$276$	0	0
$277$	−7.41063	−0.445262	−0.222631	−	0.974903i	$-0.571464\pi$
$277$	−7.41063	−0.445262	−0.222631	+	0.974903i	$0.571464\pi$
$278$	−18.5006	−1.10959
$279$	0	0
$280$	−2.70226	−0.161491
$281$	29.9112	1.78435	0.892176	−	0.451687i	$-0.149178\pi$
$281$	29.9112	1.78435	0.892176	+	0.451687i	$0.149178\pi$
$282$	0	0
$283$	−13.9339	−0.828285	−0.414142	−	0.910212i	$-0.635918\pi$
$283$	−13.9339	−0.828285	−0.414142	+	0.910212i	$0.635918\pi$
$284$	6.65320	0.394795
$285$	0	0
$286$	−0.389391	−0.0230252
$287$	20.6424	1.21848
$288$	0	0
$289$	−16.6554	−0.979730
$290$	0	0
$291$	0	0
$292$	6.63718	0.388412
$293$	−17.2303	−1.00660	−0.503302	−	0.864110i	$-0.667882\pi$
$293$	−17.2303	−1.00660	−0.503302	+	0.864110i	$0.667882\pi$
$294$	0	0
$295$	2.62675	0.152935
$296$	−0.0378187	−0.00219817
$297$	0	0
$298$	−6.48081	−0.375423
$299$	−0.799287	−0.0462240
$300$	0	0
$301$	7.49769	0.432160
$302$	0.840198	0.0483480
$303$	0	0
$304$	7.45225	0.427416
$305$	3.94042	0.225628
$306$	0	0
$307$	−24.7246	−1.41111	−0.705553	−	0.708657i	$-0.749301\pi$
$307$	−24.7246	−1.41111	−0.705553	+	0.708657i	$0.749301\pi$
$308$	5.47537	0.311988
$309$	0	0
$310$	0.774467	0.0439868
$311$	17.4125	0.987374	0.493687	−	0.869640i	$-0.335649\pi$
$311$	17.4125	0.987374	0.493687	+	0.869640i	$0.335649\pi$
$312$	0	0
$313$	9.29106	0.525162	0.262581	−	0.964910i	$-0.415426\pi$
$313$	9.29106	0.525162	0.262581	+	0.964910i	$0.415426\pi$
$314$	22.8462	1.28929
$315$	0	0
$316$	7.89114	0.443911
$317$	4.37858	0.245926	0.122963	−	0.992411i	$-0.460760\pi$
$317$	4.37858	0.245926	0.122963	+	0.992411i	$0.460760\pi$
$318$	0	0
$319$	0	0
$320$	−3.44072	−0.192342
$321$	0	0
$322$	−20.2114	−1.12634
$323$	2.12383	0.118173
$324$	0	0
$325$	0.467378	0.0259254
$326$	26.2400	1.45330
$327$	0	0
$328$	29.4610	1.62671
$329$	−7.14498	−0.393915
$330$	0	0
$331$	−30.0440	−1.65137	−0.825683	−	0.564135i	$-0.809210\pi$
$331$	−30.0440	−1.65137	−0.825683	+	0.564135i	$0.809210\pi$
$332$	−9.20440	−0.505157
$333$	0	0
$334$	−17.9037	−0.979649
$335$	−3.26659	−0.178473
$336$	0	0
$337$	−12.9613	−0.706048	−0.353024	−	0.935614i	$-0.614847\pi$
$337$	−12.9613	−0.706048	−0.353024	+	0.935614i	$0.614847\pi$
$338$	14.7276	0.801073
$339$	0	0
$340$	−0.170823	−0.00926419
$341$	−5.96048	−0.322778
$342$	0	0
$343$	−20.1622	−1.08866
$344$	10.7008	0.576947
$345$	0	0
$346$	−20.0430	−1.07752
$347$	−2.77454	−0.148945	−0.0744724	−	0.997223i	$-0.523727\pi$
$347$	−2.77454	−0.148945	−0.0744724	+	0.997223i	$0.523727\pi$
$348$	0	0
$349$	−22.8887	−1.22520	−0.612602	−	0.790391i	$-0.709877\pi$
$349$	−22.8887	−1.22520	−0.612602	+	0.790391i	$0.709877\pi$
$350$	11.8185	0.631725
$351$	0	0
$352$	13.5717	0.723372
$353$	−5.01067	−0.266691	−0.133346	−	0.991070i	$-0.542572\pi$
$353$	−5.01067	−0.266691	−0.133346	+	0.991070i	$0.542572\pi$
$354$	0	0
$355$	3.79022	0.201164
$356$	1.60095	0.0848503
$357$	0	0
$358$	25.1035	1.32676
$359$	7.48801	0.395202	0.197601	−	0.980283i	$-0.436685\pi$
$359$	7.48801	0.395202	0.197601	+	0.980283i	$0.436685\pi$
$360$	0	0
$361$	−5.90983	−0.311044
$362$	−27.4068	−1.44047
$363$	0	0
$364$	0.149008	0.00781016
$365$	3.78109	0.197912
$366$	0	0
$367$	−24.4417	−1.27585	−0.637924	−	0.770100i	$-0.720206\pi$
$367$	−24.4417	−1.27585	−0.637924	+	0.770100i	$0.720206\pi$
$368$	−17.0285	−0.887672
$369$	0	0
$370$	−0.00567217	−0.000294882 0
$371$	14.8491	0.770927
$372$	0	0
$373$	−11.6191	−0.601615	−0.300808	−	0.953685i	$-0.597256\pi$
$373$	−11.6191	−0.601615	−0.300808	+	0.953685i	$0.597256\pi$
$374$	−2.36424	−0.122252
$375$	0	0
$376$	−10.1974	−0.525890
$377$	0	0
$378$	0	0
$379$	12.1055	0.621816	0.310908	−	0.950440i	$-0.399367\pi$
$379$	12.1055	0.621816	0.310908	+	0.950440i	$0.399367\pi$
$380$	−1.05286	−0.0540108
$381$	0	0
$382$	−7.19981	−0.368374
$383$	−15.8595	−0.810381	−0.405191	−	0.914232i	$-0.632795\pi$
$383$	−15.8595	−0.810381	−0.405191	+	0.914232i	$0.632795\pi$
$384$	0	0
$385$	3.11923	0.158971
$386$	20.3489	1.03573
$387$	0	0
$388$	−4.09395	−0.207839
$389$	27.1255	1.37532	0.687658	−	0.726035i	$-0.258639\pi$
$389$	27.1255	1.37532	0.687658	+	0.726035i	$0.258639\pi$
$390$	0	0
$391$	−4.85298	−0.245426
$392$	−7.23193	−0.365267
$393$	0	0
$394$	4.70662	0.237116
$395$	4.49545	0.226191
$396$	0	0
$397$	−14.9952	−0.752590	−0.376295	−	0.926500i	$-0.622802\pi$
$397$	−14.9952	−0.752590	−0.376295	+	0.926500i	$0.622802\pi$
$398$	7.98236	0.400119
$399$	0	0
$400$	9.95729	0.497865
$401$	−13.5045	−0.674384	−0.337192	−	0.941436i	$-0.609477\pi$
$401$	−13.5045	−0.674384	−0.337192	+	0.941436i	$0.609477\pi$
$402$	0	0
$403$	−0.162210	−0.00808027
$404$	−11.4351	−0.568917
$405$	0	0
$406$	0	0
$407$	0.0436544	0.00216387
$408$	0	0
$409$	−27.8752	−1.37834	−0.689171	−	0.724599i	$-0.742025\pi$
$409$	−27.8752	−1.37834	−0.689171	+	0.724599i	$0.742025\pi$
$410$	4.41866	0.218222
$411$	0	0
$412$	8.02391	0.395310
$413$	−13.9120	−0.684563
$414$	0	0
$415$	−5.24360	−0.257398
$416$	0.369343	0.0181085
$417$	0	0
$418$	−14.5719	−0.712736
$419$	−9.01435	−0.440380	−0.220190	−	0.975457i	$-0.570668\pi$
$419$	−9.01435	−0.440380	−0.220190	+	0.975457i	$0.570668\pi$
$420$	0	0
$421$	−1.95260	−0.0951639	−0.0475820	−	0.998867i	$-0.515152\pi$
$421$	−1.95260	−0.0951639	−0.0475820	+	0.998867i	$0.515152\pi$
$422$	1.78917	0.0870955
$423$	0	0
$424$	21.1928	1.02921
$425$	2.83775	0.137651
$426$	0	0
$427$	−20.8695	−1.00995
$428$	7.50611	0.362821
$429$	0	0
$430$	1.60494	0.0773969
$431$	−28.0908	−1.35308	−0.676542	−	0.736404i	$-0.736522\pi$
$431$	−28.0908	−1.35308	−0.676542	+	0.736404i	$0.736522\pi$
$432$	0	0
$433$	4.16154	0.199991	0.0999954	−	0.994988i	$-0.468117\pi$
$433$	4.16154	0.199991	0.0999954	+	0.994988i	$0.468117\pi$
$434$	−4.10178	−0.196892
$435$	0	0
$436$	−2.76677	−0.132504
$437$	−29.9112	−1.43085
$438$	0	0
$439$	−33.5455	−1.60104	−0.800520	−	0.599305i	$-0.795444\pi$
$439$	−33.5455	−1.60104	−0.800520	+	0.599305i	$0.795444\pi$
$440$	4.45180	0.212231
$441$	0	0
$442$	−0.0643411	−0.00306039
$443$	7.70722	0.366181	0.183091	−	0.983096i	$-0.441390\pi$
$443$	7.70722	0.366181	0.183091	+	0.983096i	$0.441390\pi$
$444$	0	0
$445$	0.912037	0.0432347
$446$	−26.2252	−1.24180
$447$	0	0
$448$	18.2229	0.860953
$449$	−26.2196	−1.23738	−0.618690	−	0.785635i	$-0.712336\pi$
$449$	−26.2196	−1.23738	−0.618690	+	0.785635i	$0.712336\pi$
$450$	0	0
$451$	−34.0070	−1.60133
$452$	8.25943	0.388491
$453$	0	0
$454$	−16.4013	−0.769750
$455$	0.0848876	0.00397959
$456$	0	0
$457$	18.0732	0.845428	0.422714	−	0.906263i	$-0.361077\pi$
$457$	18.0732	0.845428	0.422714	+	0.906263i	$0.361077\pi$
$458$	7.68480	0.359087
$459$	0	0
$460$	2.40581	0.112171
$461$	−7.43554	−0.346308	−0.173154	−	0.984895i	$-0.555396\pi$
$461$	−7.43554	−0.346308	−0.173154	+	0.984895i	$0.555396\pi$
$462$	0	0
$463$	4.33818	0.201612	0.100806	−	0.994906i	$-0.467858\pi$
$463$	4.33818	0.201612	0.100806	+	0.994906i	$0.467858\pi$
$464$	0	0
$465$	0	0
$466$	14.0971	0.653035
$467$	−1.68084	−0.0777799	−0.0388899	−	0.999244i	$-0.512382\pi$
$467$	−1.68084	−0.0777799	−0.0388899	+	0.999244i	$0.512382\pi$
$468$	0	0
$469$	17.3007	0.798871
$470$	−1.52944	−0.0705476
$471$	0	0
$472$	−19.8553	−0.913914
$473$	−12.3520	−0.567944
$474$	0	0
$475$	17.4904	0.802513
$476$	0.904724	0.0414680
$477$	0	0
$478$	0.302331	0.0138283
$479$	−5.54859	−0.253521	−0.126761	−	0.991933i	$-0.540458\pi$
$479$	−5.54859	−0.253521	−0.126761	+	0.991933i	$0.540458\pi$
$480$	0	0
$481$	0.00118802	5.41692e−5 0
$482$	−26.8783	−1.22427
$483$	0	0
$484$	−1.15846	−0.0526573
$485$	−2.33226	−0.105902
$486$	0	0
$487$	22.1699	1.00461	0.502307	−	0.864689i	$-0.332485\pi$
$487$	22.1699	1.00461	0.502307	+	0.864689i	$0.332485\pi$
$488$	−29.7852	−1.34831
$489$	0	0
$490$	−1.08467	−0.0490003
$491$	−15.8015	−0.713113	−0.356557	−	0.934274i	$-0.616049\pi$
$491$	−15.8015	−0.713113	−0.356557	+	0.934274i	$0.616049\pi$
$492$	0	0
$493$	0	0
$494$	−0.396565	−0.0178423
$495$	0	0
$496$	−3.45582	−0.155171
$497$	−20.0740	−0.900442
$498$	0	0
$499$	−0.152409	−0.00682277	−0.00341139	−	0.999994i	$-0.501086\pi$
$499$	−0.152409	−0.00682277	−0.00341139	+	0.999994i	$0.501086\pi$
$500$	−2.86180	−0.127984
$501$	0	0
$502$	30.3226	1.35336
$503$	−4.52672	−0.201837	−0.100918	−	0.994895i	$-0.532178\pi$
$503$	−4.52672	−0.201837	−0.100918	+	0.994895i	$0.532178\pi$
$504$	0	0
$505$	−6.51439	−0.289886
$506$	33.2970	1.48023
$507$	0	0
$508$	−6.93192	−0.307554
$509$	−42.2346	−1.87202	−0.936008	−	0.351979i	$-0.885509\pi$
$509$	−42.2346	−1.87202	−0.936008	+	0.351979i	$0.885509\pi$
$510$	0	0
$511$	−20.0257	−0.885883
$512$	20.5472	0.908068
$513$	0	0
$514$	−17.5376	−0.773549
$515$	4.57109	0.201426
$516$	0	0
$517$	11.7709	0.517683
$518$	0.0300413	0.00131994
$519$	0	0
$520$	0.121152	0.00531289
$521$	−37.1917	−1.62940	−0.814698	−	0.579885i	$-0.803097\pi$
$521$	−37.1917	−1.62940	−0.814698	+	0.579885i	$0.803097\pi$
$522$	0	0
$523$	7.71884	0.337521	0.168761	−	0.985657i	$-0.446023\pi$
$523$	7.71884	0.337521	0.168761	+	0.985657i	$0.446023\pi$
$524$	−12.3995	−0.541673
$525$	0	0
$526$	−6.48568	−0.282789
$527$	−0.984881	−0.0429021
$528$	0	0
$529$	45.3475	1.97163
$530$	3.17856	0.138068
$531$	0	0
$532$	5.57624	0.241761
$533$	−0.925477	−0.0400869
$534$	0	0
$535$	4.27611	0.184872
$536$	24.6917	1.06652
$537$	0	0
$538$	−11.5901	−0.499683
$539$	8.34785	0.359567
$540$	0	0
$541$	−38.1817	−1.64156	−0.820780	−	0.571245i	$-0.806461\pi$
$541$	−38.1817	−1.64156	−0.820780	+	0.571245i	$0.806461\pi$
$542$	−16.5858	−0.712422
$543$	0	0
$544$	2.24251	0.0961470
$545$	−1.57618	−0.0675163
$546$	0	0
$547$	−28.5182	−1.21935	−0.609675	−	0.792651i	$-0.708700\pi$
$547$	−28.5182	−1.21935	−0.609675	+	0.792651i	$0.708700\pi$
$548$	14.7990	0.632180
$549$	0	0
$550$	−19.4702	−0.830213
$551$	0	0
$552$	0	0
$553$	−23.8091	−1.01247
$554$	8.40146	0.356944
$555$	0	0
$556$	−11.6632	−0.494631
$557$	−20.1026	−0.851773	−0.425887	−	0.904777i	$-0.640038\pi$
$557$	−20.1026	−0.851773	−0.425887	+	0.904777i	$0.640038\pi$
$558$	0	0
$559$	−0.336150	−0.0142176
$560$	1.80850	0.0764229
$561$	0	0
$562$	−33.9105	−1.43043
$563$	−23.8844	−1.00661	−0.503303	−	0.864110i	$-0.667882\pi$
$563$	−23.8844	−1.00661	−0.503303	+	0.864110i	$0.667882\pi$
$564$	0	0
$565$	4.70526	0.197952
$566$	15.7969	0.663994
$567$	0	0
$568$	−28.6498	−1.20212
$569$	23.9195	1.00276	0.501378	−	0.865228i	$-0.332827\pi$
$569$	23.9195	1.00276	0.501378	+	0.865228i	$0.332827\pi$
$570$	0	0
$571$	0.245415	0.0102703	0.00513515	−	0.999987i	$-0.498365\pi$
$571$	0.245415	0.0102703	0.00513515	+	0.999987i	$0.498365\pi$
$572$	−0.245482	−0.0102641
$573$	0	0
$574$	−23.4024	−0.976796
$575$	−39.9657	−1.66669
$576$	0	0
$577$	−19.1645	−0.797830	−0.398915	−	0.916988i	$-0.630613\pi$
$577$	−19.1645	−0.797830	−0.398915	+	0.916988i	$0.630613\pi$
$578$	18.8823	0.785401
$579$	0	0
$580$	0	0
$581$	27.7715	1.15215
$582$	0	0
$583$	−24.4630	−1.01315
$584$	−28.5808	−1.18268
$585$	0	0
$586$	19.5341	0.806944
$587$	−18.2018	−0.751269	−0.375634	−	0.926768i	$-0.622575\pi$
$587$	−18.2018	−0.751269	−0.375634	+	0.926768i	$0.622575\pi$
$588$	0	0
$589$	−6.07029	−0.250122
$590$	−2.97796	−0.122601
$591$	0	0
$592$	0.0253103	0.00104025
$593$	−39.2972	−1.61374	−0.806870	−	0.590729i	$-0.798840\pi$
$593$	−39.2972	−1.61374	−0.806870	+	0.590729i	$0.798840\pi$
$594$	0	0
$595$	0.515407	0.0211296
$596$	−4.08567	−0.167355
$597$	0	0
$598$	0.906155	0.0370554
$599$	18.7676	0.766821	0.383411	−	0.923578i	$-0.374749\pi$
$599$	18.7676	0.766821	0.383411	+	0.923578i	$0.374749\pi$
$600$	0	0
$601$	10.6145	0.432976	0.216488	−	0.976285i	$-0.430540\pi$
$601$	10.6145	0.432976	0.216488	+	0.976285i	$0.430540\pi$
$602$	−8.50016	−0.346441
$603$	0	0
$604$	0.529682	0.0215524
$605$	−0.659956	−0.0268310
$606$	0	0
$607$	−31.5882	−1.28213	−0.641063	−	0.767488i	$-0.721506\pi$
$607$	−31.5882	−1.28213	−0.641063	+	0.767488i	$0.721506\pi$
$608$	13.8217	0.560543
$609$	0	0
$610$	−4.46727	−0.180875
$611$	0.320337	0.0129594
$612$	0	0
$613$	10.9585	0.442609	0.221304	−	0.975205i	$-0.428969\pi$
$613$	10.9585	0.442609	0.221304	+	0.975205i	$0.428969\pi$
$614$	28.0304	1.13121
$615$	0	0
$616$	−23.5779	−0.949980
$617$	−33.1167	−1.33323	−0.666615	−	0.745402i	$-0.732257\pi$
$617$	−33.1167	−1.33323	−0.666615	+	0.745402i	$0.732257\pi$
$618$	0	0
$619$	−1.81592	−0.0729880	−0.0364940	−	0.999334i	$-0.511619\pi$
$619$	−1.81592	−0.0729880	−0.0364940	+	0.999334i	$0.511619\pi$
$620$	0.488243	0.0196083
$621$	0	0
$622$	−19.7406	−0.791528
$623$	−4.83038	−0.193525
$624$	0	0
$625$	22.5408	0.901631
$626$	−10.5333	−0.420996
$627$	0	0
$628$	14.4028	0.574735
$629$	0.00721324	0.000287611 0
$630$	0	0
$631$	23.3350	0.928952	0.464476	−	0.885586i	$-0.346243\pi$
$631$	23.3350	0.928952	0.464476	+	0.885586i	$0.346243\pi$
$632$	−33.9806	−1.35167
$633$	0	0
$634$	−4.96401	−0.197146
$635$	−3.94900	−0.156711
$636$	0	0
$637$	0.227181	0.00900123
$638$	0	0
$639$	0	0
$640$	0.789866	0.0312222
$641$	−5.77018	−0.227908	−0.113954	−	0.993486i	$-0.536352\pi$
$641$	−5.77018	−0.227908	−0.113954	+	0.993486i	$0.536352\pi$
$642$	0	0
$643$	26.0363	1.02677	0.513385	−	0.858158i	$-0.328391\pi$
$643$	26.0363	1.02677	0.513385	+	0.858158i	$0.328391\pi$
$644$	−12.7418	−0.502097
$645$	0	0
$646$	−2.40779	−0.0947334
$647$	−30.4369	−1.19660	−0.598299	−	0.801273i	$-0.704157\pi$
$647$	−30.4369	−1.19660	−0.598299	+	0.801273i	$0.704157\pi$
$648$	0	0
$649$	22.9191	0.899652
$650$	−0.529868	−0.0207831
$651$	0	0
$652$	16.5423	0.647847
$653$	−43.2292	−1.69169	−0.845845	−	0.533429i	$-0.820903\pi$
$653$	−43.2292	−1.69169	−0.845845	+	0.533429i	$0.820903\pi$
$654$	0	0
$655$	−7.06378	−0.276005
$656$	−19.7169	−0.769816
$657$	0	0
$658$	8.10029	0.315782
$659$	5.52426	0.215195	0.107597	−	0.994195i	$-0.465684\pi$
$659$	5.52426	0.215195	0.107597	+	0.994195i	$0.465684\pi$
$660$	0	0
$661$	2.45323	0.0954197	0.0477098	−	0.998861i	$-0.484808\pi$
$661$	2.45323	0.0954197	0.0477098	+	0.998861i	$0.484808\pi$
$662$	34.0610	1.32382
$663$	0	0
$664$	39.6357	1.53816
$665$	3.17669	0.123187
$666$	0	0
$667$	0	0
$668$	−11.2870	−0.436705
$669$	0	0
$670$	3.70334	0.143073
$671$	34.3812	1.32727
$672$	0	0
$673$	−17.3823	−0.670039	−0.335020	−	0.942211i	$-0.608743\pi$
$673$	−17.3823	−0.670039	−0.335020	+	0.942211i	$0.608743\pi$
$674$	14.6943	0.566003
$675$	0	0
$676$	9.28462	0.357101
$677$	49.5563	1.90460	0.952302	−	0.305156i	$-0.0987087\pi$
$677$	49.5563	1.90460	0.952302	+	0.305156i	$0.0987087\pi$
$678$	0	0
$679$	12.3522	0.474035
$680$	0.735593	0.0282087
$681$	0	0
$682$	6.75742	0.258755
$683$	21.4856	0.822124	0.411062	−	0.911607i	$-0.365158\pi$
$683$	21.4856	0.822124	0.411062	+	0.911607i	$0.365158\pi$
$684$	0	0
$685$	8.43072	0.322121
$686$	22.8580	0.872722
$687$	0	0
$688$	−7.16154	−0.273031
$689$	−0.665742	−0.0253627
$690$	0	0
$691$	3.31570	0.126135	0.0630676	−	0.998009i	$-0.479912\pi$
$691$	3.31570	0.126135	0.0630676	+	0.998009i	$0.479912\pi$
$692$	−12.6356	−0.480333
$693$	0	0
$694$	3.14550	0.119402
$695$	−6.64435	−0.252035
$696$	0	0
$697$	−5.61916	−0.212841
$698$	25.9490	0.982185
$699$	0	0
$700$	7.45067	0.281609
$701$	34.0148	1.28472	0.642361	−	0.766402i	$-0.277955\pi$
$701$	34.0148	1.28472	0.642361	+	0.766402i	$0.277955\pi$
$702$	0	0
$703$	0.0444586	0.00167679
$704$	−30.0211	−1.13146
$705$	0	0
$706$	5.68062	0.213793
$707$	34.5019	1.29758
$708$	0	0
$709$	27.6139	1.03706	0.518531	−	0.855059i	$-0.326479\pi$
$709$	27.6139	1.03706	0.518531	+	0.855059i	$0.326479\pi$
$710$	−4.29699	−0.161263
$711$	0	0
$712$	−6.89397	−0.258363
$713$	13.8707	0.519461
$714$	0	0
$715$	−0.139847	−0.00522998
$716$	15.8259	0.591441
$717$	0	0
$718$	−8.48918	−0.316813
$719$	−42.2582	−1.57596	−0.787982	−	0.615698i	$-0.788874\pi$
$719$	−42.2582	−1.57596	−0.787982	+	0.615698i	$0.788874\pi$
$720$	0	0
$721$	−24.2097	−0.901615
$722$	6.70000	0.249348
$723$	0	0
$724$	−17.2779	−0.642129
$725$	0	0
$726$	0	0
$727$	22.4119	0.831212	0.415606	−	0.909545i	$-0.363569\pi$
$727$	22.4119	0.831212	0.415606	+	0.909545i	$0.363569\pi$
$728$	−0.641655	−0.0237813
$729$	0	0
$730$	−4.28664	−0.158656
$731$	−2.04098	−0.0754884
$732$	0	0
$733$	9.43062	0.348328	0.174164	−	0.984717i	$-0.444278\pi$
$733$	9.43062	0.348328	0.174164	+	0.984717i	$0.444278\pi$
$734$	27.7097	1.02278
$735$	0	0
$736$	−31.5827	−1.16415
$737$	−28.5018	−1.04988
$738$	0	0
$739$	1.26732	0.0466193	0.0233096	−	0.999728i	$-0.492580\pi$
$739$	1.26732	0.0466193	0.0233096	+	0.999728i	$0.492580\pi$
$740$	−0.00357588	−0.000131452 0
$741$	0	0
$742$	−16.8345	−0.618014
$743$	38.5725	1.41509	0.707543	−	0.706670i	$-0.249803\pi$
$743$	38.5725	1.41509	0.707543	+	0.706670i	$0.249803\pi$
$744$	0	0
$745$	−2.32754	−0.0852744
$746$	13.1726	0.482285
$747$	0	0
$748$	−1.49048	−0.0544972
$749$	−22.6474	−0.827517
$750$	0	0
$751$	−25.6989	−0.937765	−0.468882	−	0.883261i	$-0.655343\pi$
$751$	−25.6989	−0.937765	−0.468882	+	0.883261i	$0.655343\pi$
$752$	6.82464	0.248869
$753$	0	0
$754$	0	0
$755$	0.301751	0.0109818
$756$	0	0
$757$	26.2239	0.953126	0.476563	−	0.879140i	$-0.341882\pi$
$757$	26.2239	0.953126	0.476563	+	0.879140i	$0.341882\pi$
$758$	−13.7240	−0.498479
$759$	0	0
$760$	4.53381	0.164459
$761$	17.2203	0.624235	0.312117	−	0.950044i	$-0.398962\pi$
$761$	17.2203	0.624235	0.312117	+	0.950044i	$0.398962\pi$
$762$	0	0
$763$	8.34787	0.302213
$764$	−4.53894	−0.164213
$765$	0	0
$766$	17.9799	0.649642
$767$	0.623726	0.0225214
$768$	0	0
$769$	−17.1271	−0.617620	−0.308810	−	0.951124i	$-0.599931\pi$
$769$	−17.1271	−0.617620	−0.308810	+	0.951124i	$0.599931\pi$
$770$	−3.53628	−0.127439
$771$	0	0
$772$	12.8284	0.461706
$773$	−10.5156	−0.378220	−0.189110	−	0.981956i	$-0.560560\pi$
$773$	−10.5156	−0.378220	−0.189110	+	0.981956i	$0.560560\pi$
$774$	0	0
$775$	−8.11079	−0.291348
$776$	17.6292	0.632853
$777$	0	0
$778$	−30.7523	−1.10252
$779$	−34.6335	−1.24087
$780$	0	0
$781$	33.0706	1.18336
$782$	5.50184	0.196745
$783$	0	0
$784$	4.84000	0.172857
$785$	8.20506	0.292851
$786$	0	0
$787$	−18.2069	−0.649006	−0.324503	−	0.945885i	$-0.605197\pi$
$787$	−18.2069	−0.649006	−0.324503	+	0.945885i	$0.605197\pi$
$788$	2.96717	0.105701
$789$	0	0
$790$	−5.09651	−0.181326
$791$	−24.9203	−0.886064
$792$	0	0
$793$	0.935659	0.0332262
$794$	17.0002	0.603313
$795$	0	0
$796$	5.03228	0.178364
$797$	2.30221	0.0815485	0.0407742	−	0.999168i	$-0.487018\pi$
$797$	2.30221	0.0815485	0.0407742	+	0.999168i	$0.487018\pi$
$798$	0	0
$799$	1.94497	0.0688080
$800$	18.4678	0.652934
$801$	0	0
$802$	15.3101	0.540620
$803$	32.9910	1.16423
$804$	0	0
$805$	−7.25879	−0.255839
$806$	0.183898	0.00647754
$807$	0	0
$808$	49.2414	1.73231
$809$	33.1485	1.16544	0.582719	−	0.812674i	$-0.301989\pi$
$809$	33.1485	1.16544	0.582719	+	0.812674i	$0.301989\pi$
$810$	0	0
$811$	30.0826	1.05634	0.528171	−	0.849138i	$-0.322878\pi$
$811$	30.0826	1.05634	0.528171	+	0.849138i	$0.322878\pi$
$812$	0	0
$813$	0	0
$814$	−0.0494911	−0.00173466
$815$	9.42389	0.330104
$816$	0	0
$817$	−12.5795	−0.440102
$818$	31.6023	1.10495
$819$	0	0
$820$	2.78563	0.0972785
$821$	−47.6872	−1.66430	−0.832148	−	0.554554i	$-0.812889\pi$
$821$	−47.6872	−1.66430	−0.832148	+	0.554554i	$0.812889\pi$
$822$	0	0
$823$	22.3860	0.780327	0.390163	−	0.920746i	$-0.372419\pi$
$823$	22.3860	0.780327	0.390163	+	0.920746i	$0.372419\pi$
$824$	−34.5523	−1.20369
$825$	0	0
$826$	15.7720	0.548780
$827$	−4.65264	−0.161788	−0.0808940	−	0.996723i	$-0.525778\pi$
$827$	−4.65264	−0.161788	−0.0808940	+	0.996723i	$0.525778\pi$
$828$	0	0
$829$	42.4282	1.47359	0.736796	−	0.676115i	$-0.236338\pi$
$829$	42.4282	1.47359	0.736796	+	0.676115i	$0.236338\pi$
$830$	5.94469	0.206343
$831$	0	0
$832$	−0.817004	−0.0283245
$833$	1.37936	0.0477919
$834$	0	0
$835$	−6.42999	−0.222519
$836$	−9.18650	−0.317722
$837$	0	0
$838$	10.2196	0.353030
$839$	−30.4870	−1.05253	−0.526264	−	0.850321i	$-0.676408\pi$
$839$	−30.4870	−1.05253	−0.526264	+	0.850321i	$0.676408\pi$
$840$	0	0
$841$	0	0
$842$	2.21367	0.0762881
$843$	0	0
$844$	1.12794	0.0388252
$845$	5.28929	0.181957
$846$	0	0
$847$	3.49530	0.120100
$848$	−14.1834	−0.487059
$849$	0	0
$850$	−3.21716	−0.110348
$851$	−0.101588	−0.00348241
$852$	0	0
$853$	−10.8699	−0.372178	−0.186089	−	0.982533i	$-0.559581\pi$
$853$	−10.8699	−0.372178	−0.186089	+	0.982533i	$0.559581\pi$
$854$	23.6598	0.809623
$855$	0	0
$856$	−32.3226	−1.10476
$857$	−10.1511	−0.346754	−0.173377	−	0.984856i	$-0.555468\pi$
$857$	−10.1511	−0.346754	−0.173377	+	0.984856i	$0.555468\pi$
$858$	0	0
$859$	−14.8219	−0.505716	−0.252858	−	0.967503i	$-0.581370\pi$
$859$	−14.8219	−0.505716	−0.252858	+	0.967503i	$0.581370\pi$
$860$	1.01179	0.0345018
$861$	0	0
$862$	31.8466	1.08470
$863$	−47.3684	−1.61244	−0.806219	−	0.591617i	$-0.798490\pi$
$863$	−47.3684	−1.61244	−0.806219	+	0.591617i	$0.798490\pi$
$864$	0	0
$865$	−7.19829	−0.244749
$866$	−4.71796	−0.160323
$867$	0	0
$868$	−2.58586	−0.0877699
$869$	39.2239	1.33058
$870$	0	0
$871$	−0.775655	−0.0262821
$872$	11.9142	0.403465
$873$	0	0
$874$	33.9105	1.14704
$875$	8.63461	0.291903
$876$	0	0
$877$	15.0287	0.507483	0.253742	−	0.967272i	$-0.418339\pi$
$877$	15.0287	0.507483	0.253742	+	0.967272i	$0.418339\pi$
$878$	38.0307	1.28347
$879$	0	0
$880$	−2.97938	−0.100435
$881$	−28.4507	−0.958530	−0.479265	−	0.877670i	$-0.659097\pi$
$881$	−28.4507	−0.958530	−0.479265	+	0.877670i	$0.659097\pi$
$882$	0	0
$883$	−53.5603	−1.80245	−0.901224	−	0.433353i	$-0.857330\pi$
$883$	−53.5603	−1.80245	−0.901224	+	0.433353i	$0.857330\pi$
$884$	−0.0405622	−0.00136425
$885$	0	0
$886$	−8.73771	−0.293549
$887$	−35.1959	−1.18176	−0.590882	−	0.806758i	$-0.701220\pi$
$887$	−35.1959	−1.18176	−0.590882	+	0.806758i	$0.701220\pi$
$888$	0	0
$889$	20.9149	0.701464
$890$	−1.03398	−0.0346591
$891$	0	0
$892$	−16.5330	−0.553566
$893$	11.9877	0.401155
$894$	0	0
$895$	9.01575	0.301363
$896$	−4.18333	−0.139755
$897$	0	0
$898$	29.7253	0.991945
$899$	0	0
$900$	0	0
$901$	−4.04214	−0.134663
$902$	38.5539	1.28370
$903$	0	0
$904$	−35.5665	−1.18292
$905$	−9.84295	−0.327191
$906$	0	0
$907$	13.0105	0.432006	0.216003	−	0.976393i	$-0.430698\pi$
$907$	13.0105	0.432006	0.216003	+	0.976393i	$0.430698\pi$
$908$	−10.3398	−0.343137
$909$	0	0
$910$	−0.0962374	−0.00319024
$911$	−25.7084	−0.851756	−0.425878	−	0.904781i	$-0.640035\pi$
$911$	−25.7084	−0.851756	−0.425878	+	0.904781i	$0.640035\pi$
$912$	0	0
$913$	−45.7517	−1.51416
$914$	−20.4896	−0.677737
$915$	0	0
$916$	4.84469	0.160073
$917$	37.4116	1.23544
$918$	0	0
$919$	−13.5150	−0.445819	−0.222910	−	0.974839i	$-0.571555\pi$
$919$	−13.5150	−0.445819	−0.222910	+	0.974839i	$0.571555\pi$
$920$	−10.3598	−0.341553
$921$	0	0
$922$	8.42970	0.277618
$923$	0.899993	0.0296236
$924$	0	0
$925$	0.0594031	0.00195316
$926$	−4.91821	−0.161622
$927$	0	0
$928$	0	0
$929$	8.35952	0.274267	0.137134	−	0.990553i	$-0.456211\pi$
$929$	8.35952	0.274267	0.137134	+	0.990553i	$0.456211\pi$
$930$	0	0
$931$	8.50164	0.278630
$932$	8.88716	0.291109
$933$	0	0
$934$	1.90557	0.0623522
$935$	−0.849099	−0.0277685
$936$	0	0
$937$	−59.9449	−1.95831	−0.979157	−	0.203107i	$-0.934896\pi$
$937$	−59.9449	−1.95831	−0.979157	+	0.203107i	$0.934896\pi$
$938$	−19.6139	−0.640415
$939$	0	0
$940$	−0.964194	−0.0314486
$941$	54.4563	1.77523	0.887613	−	0.460591i	$-0.152362\pi$
$941$	54.4563	1.77523	0.887613	+	0.460591i	$0.152362\pi$
$942$	0	0
$943$	79.1380	2.57709
$944$	13.2882	0.432495
$945$	0	0
$946$	14.0035	0.455292
$947$	19.1728	0.623032	0.311516	−	0.950241i	$-0.399163\pi$
$947$	19.1728	0.623032	0.311516	+	0.950241i	$0.399163\pi$
$948$	0	0
$949$	0.897826	0.0291447
$950$	−19.8289	−0.643335
$951$	0	0
$952$	−3.89589	−0.126267
$953$	−48.5810	−1.57369	−0.786846	−	0.617149i	$-0.788288\pi$
$953$	−48.5810	−1.57369	−0.786846	+	0.617149i	$0.788288\pi$
$954$	0	0
$955$	−2.58576	−0.0836732
$956$	0.190597	0.00616435
$957$	0	0
$958$	6.29046	0.203235
$959$	−44.6513	−1.44187
$960$	0	0
$961$	−28.1850	−0.909195
$962$	−0.00134687	−4.34247e−5 0
$963$	0	0
$964$	−16.9447	−0.545753
$965$	7.30815	0.235258
$966$	0	0
$967$	−33.9742	−1.09254	−0.546268	−	0.837610i	$-0.683952\pi$
$967$	−33.9742	−1.09254	−0.546268	+	0.837610i	$0.683952\pi$
$968$	4.98853	0.160337
$969$	0	0
$970$	2.64409	0.0848966
$971$	−19.3548	−0.621126	−0.310563	−	0.950553i	$-0.600518\pi$
$971$	−19.3548	−0.621126	−0.310563	+	0.950553i	$0.600518\pi$
$972$	0	0
$973$	35.1902	1.12815
$974$	−25.1341	−0.805349
$975$	0	0
$976$	19.9338	0.638067
$977$	14.4746	0.463085	0.231542	−	0.972825i	$-0.425623\pi$
$977$	14.4746	0.463085	0.231542	+	0.972825i	$0.425623\pi$
$978$	0	0
$979$	7.95775	0.254331
$980$	−0.683801	−0.0218432
$981$	0	0
$982$	17.9143	0.571667
$983$	16.3005	0.519906	0.259953	−	0.965621i	$-0.416293\pi$
$983$	16.3005	0.519906	0.259953	+	0.965621i	$0.416293\pi$
$984$	0	0
$985$	1.69035	0.0538590
$986$	0	0
$987$	0	0
$988$	−0.250004	−0.00795369
$989$	28.7444	0.914018
$990$	0	0
$991$	29.7360	0.944594	0.472297	−	0.881440i	$-0.343425\pi$
$991$	29.7360	0.944594	0.472297	+	0.881440i	$0.343425\pi$
$992$	−6.40951	−0.203502
$993$	0	0
$994$	22.7580	0.721839
$995$	2.86681	0.0908838
$996$	0	0
$997$	45.7094	1.44763	0.723815	−	0.689994i	$-0.242387\pi$
$997$	45.7094	1.44763	0.723815	+	0.689994i	$0.242387\pi$
$998$	0.172787	0.00546947
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7569.2.a.bg.1.3	yes	8
3.2	odd	2	inner	7569.2.a.bg.1.6	yes	8
29.28	even	2		7569.2.a.bf.1.6	yes	8
87.86	odd	2		7569.2.a.bf.1.3	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7569.2.a.bf.1.3	✓	8	87.86	odd	2
7569.2.a.bf.1.6	yes	8	29.28	even	2
7569.2.a.bg.1.3	yes	8	1.1	even	1	trivial
7569.2.a.bg.1.6	yes	8	3.2	odd	2	inner

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)

\(f(q)\)	\(=\)	\(q-1.13370 q^{2} -0.714715 q^{4} -0.407162 q^{5} +2.15643 q^{7} +3.07768 q^{8} +0.461601 q^{10} -3.55259 q^{11} -0.0966811 q^{13} -2.44476 q^{14} -2.05975 q^{16} -0.587012 q^{17} -3.61803 q^{19} +0.291004 q^{20} +4.02758 q^{22} +8.26725 q^{23} -4.83422 q^{25} +0.109608 q^{26} -1.54124 q^{28} +1.67779 q^{31} -3.82022 q^{32} +0.665498 q^{34} -0.878017 q^{35} -0.0122881 q^{37} +4.10178 q^{38} -1.25311 q^{40} +9.57247 q^{41} +3.47689 q^{43} +2.53909 q^{44} -9.37262 q^{46} -3.31333 q^{47} -2.34980 q^{49} +5.48057 q^{50} +0.0690994 q^{52} +6.88596 q^{53} +1.44648 q^{55} +6.63682 q^{56} -6.45137 q^{59} -9.67779 q^{61} -1.90211 q^{62} +8.45050 q^{64} +0.0393648 q^{65} +8.02282 q^{67} +0.419546 q^{68} +0.995411 q^{70} -9.30889 q^{71} -9.28647 q^{73} +0.0139310 q^{74} +2.58586 q^{76} -7.66092 q^{77} -11.0410 q^{79} +0.838652 q^{80} -10.8523 q^{82} +12.8784 q^{83} +0.239009 q^{85} -3.94177 q^{86} -10.9337 q^{88} -2.23999 q^{89} -0.208486 q^{91} -5.90873 q^{92} +3.75634 q^{94} +1.47312 q^{95} +5.72809 q^{97} +2.66397 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 6 q^{4} + 12 q^{10} + 2 q^{13} - 2 q^{16} - 20 q^{19} - 14 q^{22} + 2 q^{25} - 20 q^{28} - 10 q^{31} - 36 q^{34} - 18 q^{37} + 10 q^{40} - 28 q^{43} - 26 q^{46} + 4 q^{49} + 44 q^{52} - 14 q^{55}+ \cdots + 6 q^{97}+O(q^{100}) \) 8 * q + 6 * q^4 + 12 * q^10 + 2 * q^13 - 2 * q^16 - 20 * q^19 - 14 * q^22 + 2 * q^25 - 20 * q^28 - 10 * q^31 - 36 * q^34 - 18 * q^37 + 10 * q^40 - 28 * q^43 - 26 * q^46 + 4 * q^49 + 44 * q^52 - 14 * q^55 - 54 * q^61 - 44 * q^64 + 38 * q^67 - 70 * q^70 + 16 * q^73 - 20 * q^76 - 14 * q^79 + 2 * q^82 - 56 * q^85 - 60 * q^88 - 30 * q^91 + 52 * q^94 + 6 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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