LMFDB - Embedded newform 9075.2.a.bk.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9075,2,Mod(1,9075)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9075, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("9075.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9075 = 3 \cdot 5^{2} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9075.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:	$72.4642398343$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	9075.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.23607 q^{2} -1.00000 q^{3} +3.00000 q^{4} -2.23607 q^{6} +2.23607 q^{7} +2.23607 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+2.23607 q^{2} -1.00000 q^{3} +3.00000 q^{4} -2.23607 q^{6} +2.23607 q^{7} +2.23607 q^{8} +1.00000 q^{9} -3.00000 q^{12} +4.47214 q^{13} +5.00000 q^{14} -1.00000 q^{16} +2.23607 q^{17} +2.23607 q^{18} -2.23607 q^{19} -2.23607 q^{21} -1.00000 q^{23} -2.23607 q^{24} +10.0000 q^{26} -1.00000 q^{27} +6.70820 q^{28} +4.47214 q^{29} +10.0000 q^{31} -6.70820 q^{32} +5.00000 q^{34} +3.00000 q^{36} -7.00000 q^{37} -5.00000 q^{38} -4.47214 q^{39} +6.70820 q^{41} -5.00000 q^{42} -2.23607 q^{46} -3.00000 q^{47} +1.00000 q^{48} -2.00000 q^{49} -2.23607 q^{51} +13.4164 q^{52} +14.0000 q^{53} -2.23607 q^{54} +5.00000 q^{56} +2.23607 q^{57} +10.0000 q^{58} -5.00000 q^{59} +4.47214 q^{61} +22.3607 q^{62} +2.23607 q^{63} -13.0000 q^{64} +2.00000 q^{67} +6.70820 q^{68} +1.00000 q^{69} -13.0000 q^{71} +2.23607 q^{72} +13.4164 q^{73} -15.6525 q^{74} -6.70820 q^{76} -10.0000 q^{78} -15.6525 q^{79} +1.00000 q^{81} +15.0000 q^{82} +4.47214 q^{83} -6.70820 q^{84} -4.47214 q^{87} +6.00000 q^{89} +10.0000 q^{91} -3.00000 q^{92} -10.0000 q^{93} -6.70820 q^{94} +6.70820 q^{96} -17.0000 q^{97} -4.47214 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} + 6 q^{4} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 + 6 * q^4 + 2 * q^9	$ 2 q - 2 q^{3} + 6 q^{4} + 2 q^{9} - 6 q^{12} + 10 q^{14} - 2 q^{16} - 2 q^{23} + 20 q^{26} - 2 q^{27} + 20 q^{31} + 10 q^{34} + 6 q^{36} - 14 q^{37} - 10 q^{38} - 10 q^{42} - 6 q^{47} + 2 q^{48} - 4 q^{49} + 28 q^{53} + 10 q^{56} + 20 q^{58} - 10 q^{59} - 26 q^{64} + 4 q^{67} + 2 q^{69} - 26 q^{71} - 20 q^{78} + 2 q^{81} + 30 q^{82} + 12 q^{89} + 20 q^{91} - 6 q^{92} - 20 q^{93} - 34 q^{97}+O(q^{100}) $ 2 * q - 2 * q^3 + 6 * q^4 + 2 * q^9 - 6 * q^12 + 10 * q^14 - 2 * q^16 - 2 * q^23 + 20 * q^26 - 2 * q^27 + 20 * q^31 + 10 * q^34 + 6 * q^36 - 14 * q^37 - 10 * q^38 - 10 * q^42 - 6 * q^47 + 2 * q^48 - 4 * q^49 + 28 * q^53 + 10 * q^56 + 20 * q^58 - 10 * q^59 - 26 * q^64 + 4 * q^67 + 2 * q^69 - 26 * q^71 - 20 * q^78 + 2 * q^81 + 30 * q^82 + 12 * q^89 + 20 * q^91 - 6 * q^92 - 20 * q^93 - 34 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	2.23607	1.58114	0.790569	−	0.612372i	$-0.209785\pi$
$2$	2.23607	1.58114	0.790569	+	0.612372i	$0.209785\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	3.00000	1.50000
$5$	0	0
$5$	0	0
$6$	−2.23607	−0.912871
$7$	2.23607	0.845154	0.422577	−	0.906327i	$-0.361126\pi$
$7$	2.23607	0.845154	0.422577	+	0.906327i	$0.361126\pi$
$8$	2.23607	0.790569
$9$	1.00000	0.333333
$10$	0	0
$11$	0	0
$11$	0	0
$12$	−3.00000	−0.866025
$13$	4.47214	1.24035	0.620174	−	0.784465i	$-0.287062\pi$
$13$	4.47214	1.24035	0.620174	+	0.784465i	$0.287062\pi$
$14$	5.00000	1.33631
$15$	0	0
$16$	−1.00000	−0.250000
$17$	2.23607	0.542326	0.271163	−	0.962533i	$-0.412592\pi$
$17$	2.23607	0.542326	0.271163	+	0.962533i	$0.412592\pi$
$18$	2.23607	0.527046
$19$	−2.23607	−0.512989	−0.256495	−	0.966546i	$-0.582568\pi$
$19$	−2.23607	−0.512989	−0.256495	+	0.966546i	$0.582568\pi$
$20$	0	0
$21$	−2.23607	−0.487950
$22$	0	0
$23$	−1.00000	−0.208514	−0.104257	−	0.994550i	$-0.533247\pi$
$23$	−1.00000	−0.208514	−0.104257	+	0.994550i	$0.533247\pi$
$24$	−2.23607	−0.456435
$25$	0	0
$26$	10.0000	1.96116
$27$	−1.00000	−0.192450
$28$	6.70820	1.26773
$29$	4.47214	0.830455	0.415227	−	0.909718i	$-0.363702\pi$
$29$	4.47214	0.830455	0.415227	+	0.909718i	$0.363702\pi$
$30$	0	0
$31$	10.0000	1.79605	0.898027	−	0.439941i	$-0.145001\pi$
$31$	10.0000	1.79605	0.898027	+	0.439941i	$0.145001\pi$
$32$	−6.70820	−1.18585
$33$	0	0
$34$	5.00000	0.857493
$35$	0	0
$36$	3.00000	0.500000
$37$	−7.00000	−1.15079	−0.575396	−	0.817875i	$-0.695152\pi$
$37$	−7.00000	−1.15079	−0.575396	+	0.817875i	$0.695152\pi$
$38$	−5.00000	−0.811107
$39$	−4.47214	−0.716115
$40$	0	0
$41$	6.70820	1.04765	0.523823	−	0.851827i	$-0.324505\pi$
$41$	6.70820	1.04765	0.523823	+	0.851827i	$0.324505\pi$
$42$	−5.00000	−0.771517
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	0	0
$45$	0	0
$46$	−2.23607	−0.329690
$47$	−3.00000	−0.437595	−0.218797	−	0.975770i	$-0.570213\pi$
$47$	−3.00000	−0.437595	−0.218797	+	0.975770i	$0.570213\pi$
$48$	1.00000	0.144338
$49$	−2.00000	−0.285714
$50$	0	0
$51$	−2.23607	−0.313112
$52$	13.4164	1.86052
$53$	14.0000	1.92305	0.961524	−	0.274721i	$-0.0885855\pi$
$53$	14.0000	1.92305	0.961524	+	0.274721i	$0.0885855\pi$
$54$	−2.23607	−0.304290
$55$	0	0
$56$	5.00000	0.668153
$57$	2.23607	0.296174
$58$	10.0000	1.31306
$59$	−5.00000	−0.650945	−0.325472	−	0.945552i	$-0.605523\pi$
$59$	−5.00000	−0.650945	−0.325472	+	0.945552i	$0.605523\pi$
$60$	0	0
$61$	4.47214	0.572598	0.286299	−	0.958140i	$-0.407575\pi$
$61$	4.47214	0.572598	0.286299	+	0.958140i	$0.407575\pi$
$62$	22.3607	2.83981
$63$	2.23607	0.281718
$64$	−13.0000	−1.62500
$65$	0	0
$66$	0	0
$67$	2.00000	0.244339	0.122169	−	0.992509i	$-0.461015\pi$
$67$	2.00000	0.244339	0.122169	+	0.992509i	$0.461015\pi$
$68$	6.70820	0.813489
$69$	1.00000	0.120386
$70$	0	0
$71$	−13.0000	−1.54282	−0.771408	−	0.636341i	$-0.780447\pi$
$71$	−13.0000	−1.54282	−0.771408	+	0.636341i	$0.780447\pi$
$72$	2.23607	0.263523
$73$	13.4164	1.57027	0.785136	−	0.619324i	$-0.212593\pi$
$73$	13.4164	1.57027	0.785136	+	0.619324i	$0.212593\pi$
$74$	−15.6525	−1.81956
$75$	0	0
$76$	−6.70820	−0.769484
$77$	0	0
$78$	−10.0000	−1.13228
$79$	−15.6525	−1.76104	−0.880521	−	0.474008i	$-0.842807\pi$
$79$	−15.6525	−1.76104	−0.880521	+	0.474008i	$0.842807\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	15.0000	1.65647
$83$	4.47214	0.490881	0.245440	−	0.969412i	$-0.421067\pi$
$83$	4.47214	0.490881	0.245440	+	0.969412i	$0.421067\pi$
$84$	−6.70820	−0.731925
$85$	0	0
$86$	0	0
$87$	−4.47214	−0.479463
$88$	0	0
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	0	0
$91$	10.0000	1.04828
$92$	−3.00000	−0.312772
$93$	−10.0000	−1.03695
$94$	−6.70820	−0.691898
$95$	0	0
$96$	6.70820	0.684653
$97$	−17.0000	−1.72609	−0.863044	−	0.505128i	$-0.831445\pi$
$97$	−17.0000	−1.72609	−0.863044	+	0.505128i	$0.831445\pi$
$98$	−4.47214	−0.451754
$99$	0	0
$100$	0	0
$101$	11.1803	1.11249	0.556243	−	0.831020i	$-0.312242\pi$
$101$	11.1803	1.11249	0.556243	+	0.831020i	$0.312242\pi$
$102$	−5.00000	−0.495074
$103$	4.00000	0.394132	0.197066	−	0.980390i	$-0.436859\pi$
$103$	4.00000	0.394132	0.197066	+	0.980390i	$0.436859\pi$
$104$	10.0000	0.980581
$105$	0	0
$106$	31.3050	3.04061
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	−3.00000	−0.288675
$109$	17.8885	1.71341	0.856706	−	0.515805i	$-0.172507\pi$
$109$	17.8885	1.71341	0.856706	+	0.515805i	$0.172507\pi$
$110$	0	0
$111$	7.00000	0.664411
$112$	−2.23607	−0.211289
$113$	14.0000	1.31701	0.658505	−	0.752577i	$-0.271189\pi$
$113$	14.0000	1.31701	0.658505	+	0.752577i	$0.271189\pi$
$114$	5.00000	0.468293
$115$	0	0
$116$	13.4164	1.24568
$117$	4.47214	0.413449
$118$	−11.1803	−1.02923
$119$	5.00000	0.458349
$120$	0	0
$121$	0	0
$122$	10.0000	0.905357
$123$	−6.70820	−0.604858
$124$	30.0000	2.69408
$125$	0	0
$126$	5.00000	0.445435
$127$	15.6525	1.38893	0.694466	−	0.719525i	$-0.255641\pi$
$127$	15.6525	1.38893	0.694466	+	0.719525i	$0.255641\pi$
$128$	−15.6525	−1.38350
$129$	0	0
$130$	0	0
$131$	−13.4164	−1.17220	−0.586098	−	0.810240i	$-0.699337\pi$
$131$	−13.4164	−1.17220	−0.586098	+	0.810240i	$0.699337\pi$
$132$	0	0
$133$	−5.00000	−0.433555
$134$	4.47214	0.386334
$135$	0	0
$136$	5.00000	0.428746
$137$	8.00000	0.683486	0.341743	−	0.939793i	$-0.388983\pi$
$137$	8.00000	0.683486	0.341743	+	0.939793i	$0.388983\pi$
$138$	2.23607	0.190347
$139$	0	0		−	1.00000i	$-0.5\pi$
$139$	0	0			1.00000i	$0.5\pi$
$140$	0	0
$141$	3.00000	0.252646
$142$	−29.0689	−2.43941
$143$	0	0
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	30.0000	2.48282
$147$	2.00000	0.164957
$148$	−21.0000	−1.72619
$149$	−20.1246	−1.64867	−0.824336	−	0.566101i	$-0.808451\pi$
$149$	−20.1246	−1.64867	−0.824336	+	0.566101i	$0.808451\pi$
$150$	0	0
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	−5.00000	−0.405554
$153$	2.23607	0.180775
$154$	0	0
$155$	0	0
$156$	−13.4164	−1.07417
$157$	−2.00000	−0.159617	−0.0798087	−	0.996810i	$-0.525431\pi$
$157$	−2.00000	−0.159617	−0.0798087	+	0.996810i	$0.525431\pi$
$158$	−35.0000	−2.78445
$159$	−14.0000	−1.11027
$160$	0	0
$161$	−2.23607	−0.176227
$162$	2.23607	0.175682
$163$	−14.0000	−1.09656	−0.548282	−	0.836293i	$-0.684718\pi$
$163$	−14.0000	−1.09656	−0.548282	+	0.836293i	$0.684718\pi$
$164$	20.1246	1.57147
$165$	0	0
$166$	10.0000	0.776151
$167$	4.47214	0.346064	0.173032	−	0.984916i	$-0.444644\pi$
$167$	4.47214	0.346064	0.173032	+	0.984916i	$0.444644\pi$
$168$	−5.00000	−0.385758
$169$	7.00000	0.538462
$170$	0	0
$171$	−2.23607	−0.170996
$172$	0	0
$173$	20.1246	1.53005	0.765023	−	0.644003i	$-0.222728\pi$
$173$	20.1246	1.53005	0.765023	+	0.644003i	$0.222728\pi$
$174$	−10.0000	−0.758098
$175$	0	0
$176$	0	0
$177$	5.00000	0.375823
$178$	13.4164	1.00560
$179$	11.0000	0.822179	0.411089	−	0.911595i	$-0.365148\pi$
$179$	11.0000	0.822179	0.411089	+	0.911595i	$0.365148\pi$
$180$	0	0
$181$	−5.00000	−0.371647	−0.185824	−	0.982583i	$-0.559495\pi$
$181$	−5.00000	−0.371647	−0.185824	+	0.982583i	$0.559495\pi$
$182$	22.3607	1.65748
$183$	−4.47214	−0.330590
$184$	−2.23607	−0.164845
$185$	0	0
$186$	−22.3607	−1.63956
$187$	0	0
$188$	−9.00000	−0.656392
$189$	−2.23607	−0.162650
$190$	0	0
$191$	15.0000	1.08536	0.542681	−	0.839939i	$-0.317409\pi$
$191$	15.0000	1.08536	0.542681	+	0.839939i	$0.317409\pi$
$192$	13.0000	0.938194
$193$	−8.94427	−0.643823	−0.321911	−	0.946770i	$-0.604325\pi$
$193$	−8.94427	−0.643823	−0.321911	+	0.946770i	$0.604325\pi$
$194$	−38.0132	−2.72919
$195$	0	0
$196$	−6.00000	−0.428571
$197$	−15.6525	−1.11519	−0.557596	−	0.830112i	$-0.688276\pi$
$197$	−15.6525	−1.11519	−0.557596	+	0.830112i	$0.688276\pi$
$198$	0	0
$199$	10.0000	0.708881	0.354441	−	0.935079i	$-0.384671\pi$
$199$	10.0000	0.708881	0.354441	+	0.935079i	$0.384671\pi$
$200$	0	0
$201$	−2.00000	−0.141069
$202$	25.0000	1.75899
$203$	10.0000	0.701862
$204$	−6.70820	−0.469668
$205$	0	0
$206$	8.94427	0.623177
$207$	−1.00000	−0.0695048
$208$	−4.47214	−0.310087
$209$	0	0
$210$	0	0
$211$	26.8328	1.84725	0.923624	−	0.383301i	$-0.125213\pi$
$211$	26.8328	1.84725	0.923624	+	0.383301i	$0.125213\pi$
$212$	42.0000	2.88457
$213$	13.0000	0.890745
$214$	0	0
$215$	0	0
$216$	−2.23607	−0.152145
$217$	22.3607	1.51794
$218$	40.0000	2.70914
$219$	−13.4164	−0.906597
$220$	0	0
$221$	10.0000	0.672673
$222$	15.6525	1.05053
$223$	−14.0000	−0.937509	−0.468755	−	0.883328i	$-0.655297\pi$
$223$	−14.0000	−0.937509	−0.468755	+	0.883328i	$0.655297\pi$
$224$	−15.0000	−1.00223
$225$	0	0
$226$	31.3050	2.08237
$227$	13.4164	0.890478	0.445239	−	0.895412i	$-0.353119\pi$
$227$	13.4164	0.890478	0.445239	+	0.895412i	$0.353119\pi$
$228$	6.70820	0.444262
$229$	−5.00000	−0.330409	−0.165205	−	0.986259i	$-0.552828\pi$
$229$	−5.00000	−0.330409	−0.165205	+	0.986259i	$0.552828\pi$
$230$	0	0
$231$	0	0
$232$	10.0000	0.656532
$233$	−20.1246	−1.31841	−0.659204	−	0.751965i	$-0.729106\pi$
$233$	−20.1246	−1.31841	−0.659204	+	0.751965i	$0.729106\pi$
$234$	10.0000	0.653720
$235$	0	0
$236$	−15.0000	−0.976417
$237$	15.6525	1.01674
$238$	11.1803	0.724714
$239$	8.94427	0.578557	0.289278	−	0.957245i	$-0.406585\pi$
$239$	8.94427	0.578557	0.289278	+	0.957245i	$0.406585\pi$
$240$	0	0
$241$	22.3607	1.44038	0.720189	−	0.693778i	$-0.244055\pi$
$241$	22.3607	1.44038	0.720189	+	0.693778i	$0.244055\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	13.4164	0.858898
$245$	0	0
$246$	−15.0000	−0.956365
$247$	−10.0000	−0.636285
$248$	22.3607	1.41990
$249$	−4.47214	−0.283410
$250$	0	0
$251$	−12.0000	−0.757433	−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000	−0.757433	−0.378717	+	0.925513i	$0.623635\pi$
$252$	6.70820	0.422577
$253$	0	0
$254$	35.0000	2.19610
$255$	0	0
$256$	−9.00000	−0.562500
$257$	−18.0000	−1.12281	−0.561405	−	0.827541i	$-0.689739\pi$
$257$	−18.0000	−1.12281	−0.561405	+	0.827541i	$0.689739\pi$
$258$	0	0
$259$	−15.6525	−0.972598
$260$	0	0
$261$	4.47214	0.276818
$262$	−30.0000	−1.85341
$263$	−17.8885	−1.10305	−0.551527	−	0.834157i	$-0.685955\pi$
$263$	−17.8885	−1.10305	−0.551527	+	0.834157i	$0.685955\pi$
$264$	0	0
$265$	0	0
$266$	−11.1803	−0.685511
$267$	−6.00000	−0.367194
$268$	6.00000	0.366508
$269$	30.0000	1.82913	0.914566	−	0.404436i	$-0.132532\pi$
$269$	30.0000	1.82913	0.914566	+	0.404436i	$0.132532\pi$
$270$	0	0
$271$	20.1246	1.22248	0.611242	−	0.791444i	$-0.290670\pi$
$271$	20.1246	1.22248	0.611242	+	0.791444i	$0.290670\pi$
$272$	−2.23607	−0.135582
$273$	−10.0000	−0.605228
$274$	17.8885	1.08069
$275$	0	0
$276$	3.00000	0.180579
$277$	−13.4164	−0.806114	−0.403057	−	0.915175i	$-0.632052\pi$
$277$	−13.4164	−0.806114	−0.403057	+	0.915175i	$0.632052\pi$
$278$	0	0
$279$	10.0000	0.598684
$280$	0	0
$281$	24.5967	1.46732	0.733659	−	0.679517i	$-0.237811\pi$
$281$	24.5967	1.46732	0.733659	+	0.679517i	$0.237811\pi$
$282$	6.70820	0.399468
$283$	−6.70820	−0.398761	−0.199381	−	0.979922i	$-0.563893\pi$
$283$	−6.70820	−0.398761	−0.199381	+	0.979922i	$0.563893\pi$
$284$	−39.0000	−2.31422
$285$	0	0
$286$	0	0
$287$	15.0000	0.885422
$288$	−6.70820	−0.395285
$289$	−12.0000	−0.705882
$290$	0	0
$291$	17.0000	0.996558
$292$	40.2492	2.35541
$293$	15.6525	0.914427	0.457214	−	0.889357i	$-0.348847\pi$
$293$	15.6525	0.914427	0.457214	+	0.889357i	$0.348847\pi$
$294$	4.47214	0.260820
$295$	0	0
$296$	−15.6525	−0.909782
$297$	0	0
$298$	−45.0000	−2.60678
$299$	−4.47214	−0.258630
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−11.1803	−0.642294
$304$	2.23607	0.128247
$305$	0	0
$306$	5.00000	0.285831
$307$	−17.8885	−1.02095	−0.510477	−	0.859892i	$-0.670531\pi$
$307$	−17.8885	−1.02095	−0.510477	+	0.859892i	$0.670531\pi$
$308$	0	0
$309$	−4.00000	−0.227552
$310$	0	0
$311$	−32.0000	−1.81455	−0.907277	−	0.420534i	$-0.861843\pi$
$311$	−32.0000	−1.81455	−0.907277	+	0.420534i	$0.861843\pi$
$312$	−10.0000	−0.566139
$313$	−9.00000	−0.508710	−0.254355	−	0.967111i	$-0.581863\pi$
$313$	−9.00000	−0.508710	−0.254355	+	0.967111i	$0.581863\pi$
$314$	−4.47214	−0.252377
$315$	0	0
$316$	−46.9574	−2.64156
$317$	22.0000	1.23564	0.617822	−	0.786318i	$-0.288015\pi$
$317$	22.0000	1.23564	0.617822	+	0.786318i	$0.288015\pi$
$318$	−31.3050	−1.75549
$319$	0	0
$320$	0	0
$321$	0	0
$322$	−5.00000	−0.278639
$323$	−5.00000	−0.278207
$324$	3.00000	0.166667
$325$	0	0
$326$	−31.3050	−1.73382
$327$	−17.8885	−0.989239
$328$	15.0000	0.828236
$329$	−6.70820	−0.369835
$330$	0	0
$331$	20.0000	1.09930	0.549650	−	0.835395i	$-0.314761\pi$
$331$	20.0000	1.09930	0.549650	+	0.835395i	$0.314761\pi$
$332$	13.4164	0.736321
$333$	−7.00000	−0.383598
$334$	10.0000	0.547176
$335$	0	0
$336$	2.23607	0.121988
$337$	−13.4164	−0.730838	−0.365419	−	0.930843i	$-0.619074\pi$
$337$	−13.4164	−0.730838	−0.365419	+	0.930843i	$0.619074\pi$
$338$	15.6525	0.851382
$339$	−14.0000	−0.760376
$340$	0	0
$341$	0	0
$342$	−5.00000	−0.270369
$343$	−20.1246	−1.08663
$344$	0	0
$345$	0	0
$346$	45.0000	2.41921
$347$	13.4164	0.720231	0.360115	−	0.932908i	$-0.382737\pi$
$347$	13.4164	0.720231	0.360115	+	0.932908i	$0.382737\pi$
$348$	−13.4164	−0.719195
$349$	−13.4164	−0.718164	−0.359082	−	0.933306i	$-0.616910\pi$
$349$	−13.4164	−0.718164	−0.359082	+	0.933306i	$0.616910\pi$
$350$	0	0
$351$	−4.47214	−0.238705
$352$	0	0
$353$	34.0000	1.80964	0.904819	−	0.425797i	$-0.140006\pi$
$353$	34.0000	1.80964	0.904819	+	0.425797i	$0.140006\pi$
$354$	11.1803	0.594228
$355$	0	0
$356$	18.0000	0.953998
$357$	−5.00000	−0.264628
$358$	24.5967	1.29998
$359$	4.47214	0.236030	0.118015	−	0.993012i	$-0.462347\pi$
$359$	4.47214	0.236030	0.118015	+	0.993012i	$0.462347\pi$
$360$	0	0
$361$	−14.0000	−0.736842
$362$	−11.1803	−0.587626
$363$	0	0
$364$	30.0000	1.57243
$365$	0	0
$366$	−10.0000	−0.522708
$367$	−18.0000	−0.939592	−0.469796	−	0.882775i	$-0.655673\pi$
$367$	−18.0000	−0.939592	−0.469796	+	0.882775i	$0.655673\pi$
$368$	1.00000	0.0521286
$369$	6.70820	0.349215
$370$	0	0
$371$	31.3050	1.62527
$372$	−30.0000	−1.55543
$373$	−8.94427	−0.463117	−0.231558	−	0.972821i	$-0.574382\pi$
$373$	−8.94427	−0.463117	−0.231558	+	0.972821i	$0.574382\pi$
$374$	0	0
$375$	0	0
$376$	−6.70820	−0.345949
$377$	20.0000	1.03005
$378$	−5.00000	−0.257172
$379$	10.0000	0.513665	0.256833	−	0.966456i	$-0.417321\pi$
$379$	10.0000	0.513665	0.256833	+	0.966456i	$0.417321\pi$
$380$	0	0
$381$	−15.6525	−0.801901
$382$	33.5410	1.71611
$383$	16.0000	0.817562	0.408781	−	0.912633i	$-0.365954\pi$
$383$	16.0000	0.817562	0.408781	+	0.912633i	$0.365954\pi$
$384$	15.6525	0.798762
$385$	0	0
$386$	−20.0000	−1.01797
$387$	0	0
$388$	−51.0000	−2.58913
$389$	0	0		−	1.00000i	$-0.5\pi$
$389$	0	0			1.00000i	$0.5\pi$
$390$	0	0
$391$	−2.23607	−0.113083
$392$	−4.47214	−0.225877
$393$	13.4164	0.676768
$394$	−35.0000	−1.76327
$395$	0	0
$396$	0	0
$397$	−22.0000	−1.10415	−0.552074	−	0.833795i	$-0.686163\pi$
$397$	−22.0000	−1.10415	−0.552074	+	0.833795i	$0.686163\pi$
$398$	22.3607	1.12084
$399$	5.00000	0.250313
$400$	0	0
$401$	10.0000	0.499376	0.249688	−	0.968326i	$-0.419672\pi$
$401$	10.0000	0.499376	0.249688	+	0.968326i	$0.419672\pi$
$402$	−4.47214	−0.223050
$403$	44.7214	2.22773
$404$	33.5410	1.66873
$405$	0	0
$406$	22.3607	1.10974
$407$	0	0
$408$	−5.00000	−0.247537
$409$	13.4164	0.663399	0.331699	−	0.943385i	$-0.392378\pi$
$409$	13.4164	0.663399	0.331699	+	0.943385i	$0.392378\pi$
$410$	0	0
$411$	−8.00000	−0.394611
$412$	12.0000	0.591198
$413$	−11.1803	−0.550149
$414$	−2.23607	−0.109897
$415$	0	0
$416$	−30.0000	−1.47087
$417$	0	0
$418$	0	0
$419$	9.00000	0.439679	0.219839	−	0.975536i	$-0.429447\pi$
$419$	9.00000	0.439679	0.219839	+	0.975536i	$0.429447\pi$
$420$	0	0
$421$	−15.0000	−0.731055	−0.365528	−	0.930800i	$-0.619111\pi$
$421$	−15.0000	−0.731055	−0.365528	+	0.930800i	$0.619111\pi$
$422$	60.0000	2.92075
$423$	−3.00000	−0.145865
$424$	31.3050	1.52030
$425$	0	0
$426$	29.0689	1.40839
$427$	10.0000	0.483934
$428$	0	0
$429$	0	0
$430$	0	0
$431$	22.3607	1.07708	0.538538	−	0.842601i	$-0.318977\pi$
$431$	22.3607	1.07708	0.538538	+	0.842601i	$0.318977\pi$
$432$	1.00000	0.0481125
$433$	−14.0000	−0.672797	−0.336399	−	0.941720i	$-0.609209\pi$
$433$	−14.0000	−0.672797	−0.336399	+	0.941720i	$0.609209\pi$
$434$	50.0000	2.40008
$435$	0	0
$436$	53.6656	2.57012
$437$	2.23607	0.106966
$438$	−30.0000	−1.43346
$439$	20.1246	0.960495	0.480248	−	0.877133i	$-0.340547\pi$
$439$	20.1246	0.960495	0.480248	+	0.877133i	$0.340547\pi$
$440$	0	0
$441$	−2.00000	−0.0952381
$442$	22.3607	1.06359
$443$	−21.0000	−0.997740	−0.498870	−	0.866677i	$-0.666252\pi$
$443$	−21.0000	−0.997740	−0.498870	+	0.866677i	$0.666252\pi$
$444$	21.0000	0.996616
$445$	0	0
$446$	−31.3050	−1.48233
$447$	20.1246	0.951861
$448$	−29.0689	−1.37338
$449$	4.00000	0.188772	0.0943858	−	0.995536i	$-0.469911\pi$
$449$	4.00000	0.188772	0.0943858	+	0.995536i	$0.469911\pi$
$450$	0	0
$451$	0	0
$452$	42.0000	1.97551
$453$	0	0
$454$	30.0000	1.40797
$455$	0	0
$456$	5.00000	0.234146
$457$	−22.3607	−1.04599	−0.522994	−	0.852336i	$-0.675185\pi$
$457$	−22.3607	−1.04599	−0.522994	+	0.852336i	$0.675185\pi$
$458$	−11.1803	−0.522423
$459$	−2.23607	−0.104371
$460$	0	0
$461$	−22.3607	−1.04144	−0.520720	−	0.853727i	$-0.674337\pi$
$461$	−22.3607	−1.04144	−0.520720	+	0.853727i	$0.674337\pi$
$462$	0	0
$463$	−14.0000	−0.650635	−0.325318	−	0.945605i	$-0.605471\pi$
$463$	−14.0000	−0.650635	−0.325318	+	0.945605i	$0.605471\pi$
$464$	−4.47214	−0.207614
$465$	0	0
$466$	−45.0000	−2.08458
$467$	−32.0000	−1.48078	−0.740392	−	0.672176i	$-0.765360\pi$
$467$	−32.0000	−1.48078	−0.740392	+	0.672176i	$0.765360\pi$
$468$	13.4164	0.620174
$469$	4.47214	0.206504
$470$	0	0
$471$	2.00000	0.0921551
$472$	−11.1803	−0.514617
$473$	0	0
$474$	35.0000	1.60760
$475$	0	0
$476$	15.0000	0.687524
$477$	14.0000	0.641016
$478$	20.0000	0.914779
$479$	22.3607	1.02169	0.510843	−	0.859674i	$-0.329333\pi$
$479$	22.3607	1.02169	0.510843	+	0.859674i	$0.329333\pi$
$480$	0	0
$481$	−31.3050	−1.42738
$482$	50.0000	2.27744
$483$	2.23607	0.101745
$484$	0	0
$485$	0	0
$486$	−2.23607	−0.101430
$487$	32.0000	1.45006	0.725029	−	0.688718i	$-0.241826\pi$
$487$	32.0000	1.45006	0.725029	+	0.688718i	$0.241826\pi$
$488$	10.0000	0.452679
$489$	14.0000	0.633102
$490$	0	0
$491$	−4.47214	−0.201825	−0.100912	−	0.994895i	$-0.532176\pi$
$491$	−4.47214	−0.201825	−0.100912	+	0.994895i	$0.532176\pi$
$492$	−20.1246	−0.907288
$493$	10.0000	0.450377
$494$	−22.3607	−1.00605
$495$	0	0
$496$	−10.0000	−0.449013
$497$	−29.0689	−1.30392
$498$	−10.0000	−0.448111
$499$	20.0000	0.895323	0.447661	−	0.894203i	$-0.352257\pi$
$499$	20.0000	0.895323	0.447661	+	0.894203i	$0.352257\pi$
$500$	0	0
$501$	−4.47214	−0.199800
$502$	−26.8328	−1.19761
$503$	−35.7771	−1.59522	−0.797611	−	0.603173i	$-0.793903\pi$
$503$	−35.7771	−1.59522	−0.797611	+	0.603173i	$0.793903\pi$
$504$	5.00000	0.222718
$505$	0	0
$506$	0	0
$507$	−7.00000	−0.310881
$508$	46.9574	2.08340
$509$	10.0000	0.443242	0.221621	−	0.975133i	$-0.428865\pi$
$509$	10.0000	0.443242	0.221621	+	0.975133i	$0.428865\pi$
$510$	0	0
$511$	30.0000	1.32712
$512$	11.1803	0.494106
$513$	2.23607	0.0987248
$514$	−40.2492	−1.77532
$515$	0	0
$516$	0	0
$517$	0	0
$518$	−35.0000	−1.53781
$519$	−20.1246	−0.883372
$520$	0	0
$521$	20.0000	0.876216	0.438108	−	0.898922i	$-0.355649\pi$
$521$	20.0000	0.876216	0.438108	+	0.898922i	$0.355649\pi$
$522$	10.0000	0.437688
$523$	6.70820	0.293329	0.146665	−	0.989186i	$-0.453146\pi$
$523$	6.70820	0.293329	0.146665	+	0.989186i	$0.453146\pi$
$524$	−40.2492	−1.75830
$525$	0	0
$526$	−40.0000	−1.74408
$527$	22.3607	0.974047
$528$	0	0
$529$	−22.0000	−0.956522
$530$	0	0
$531$	−5.00000	−0.216982
$532$	−15.0000	−0.650332
$533$	30.0000	1.29944
$534$	−13.4164	−0.580585
$535$	0	0
$536$	4.47214	0.193167
$537$	−11.0000	−0.474685
$538$	67.0820	2.89211
$539$	0	0
$540$	0	0
$541$	17.8885	0.769089	0.384544	−	0.923107i	$-0.374359\pi$
$541$	17.8885	0.769089	0.384544	+	0.923107i	$0.374359\pi$
$542$	45.0000	1.93292
$543$	5.00000	0.214571
$544$	−15.0000	−0.643120
$545$	0	0
$546$	−22.3607	−0.956949
$547$	−20.1246	−0.860466	−0.430233	−	0.902718i	$-0.641569\pi$
$547$	−20.1246	−0.860466	−0.430233	+	0.902718i	$0.641569\pi$
$548$	24.0000	1.02523
$549$	4.47214	0.190866
$550$	0	0
$551$	−10.0000	−0.426014
$552$	2.23607	0.0951734
$553$	−35.0000	−1.48835
$554$	−30.0000	−1.27458
$555$	0	0
$556$	0	0
$557$	22.3607	0.947452	0.473726	−	0.880672i	$-0.342909\pi$
$557$	22.3607	0.947452	0.473726	+	0.880672i	$0.342909\pi$
$558$	22.3607	0.946603
$559$	0	0
$560$	0	0
$561$	0	0
$562$	55.0000	2.32003
$563$	−35.7771	−1.50782	−0.753912	−	0.656975i	$-0.771836\pi$
$563$	−35.7771	−1.50782	−0.753912	+	0.656975i	$0.771836\pi$
$564$	9.00000	0.378968
$565$	0	0
$566$	−15.0000	−0.630497
$567$	2.23607	0.0939060
$568$	−29.0689	−1.21970
$569$	11.1803	0.468704	0.234352	−	0.972152i	$-0.424703\pi$
$569$	11.1803	0.468704	0.234352	+	0.972152i	$0.424703\pi$
$570$	0	0
$571$	35.7771	1.49722	0.748612	−	0.663008i	$-0.230720\pi$
$571$	35.7771	1.49722	0.748612	+	0.663008i	$0.230720\pi$
$572$	0	0
$573$	−15.0000	−0.626634
$574$	33.5410	1.39998
$575$	0	0
$576$	−13.0000	−0.541667
$577$	−3.00000	−0.124892	−0.0624458	−	0.998048i	$-0.519890\pi$
$577$	−3.00000	−0.124892	−0.0624458	+	0.998048i	$0.519890\pi$
$578$	−26.8328	−1.11610
$579$	8.94427	0.371711
$580$	0	0
$581$	10.0000	0.414870
$582$	38.0132	1.57570
$583$	0	0
$584$	30.0000	1.24141
$585$	0	0
$586$	35.0000	1.44584
$587$	23.0000	0.949312	0.474656	−	0.880172i	$-0.342573\pi$
$587$	23.0000	0.949312	0.474656	+	0.880172i	$0.342573\pi$
$588$	6.00000	0.247436
$589$	−22.3607	−0.921356
$590$	0	0
$591$	15.6525	0.643857
$592$	7.00000	0.287698
$593$	4.47214	0.183649	0.0918243	−	0.995775i	$-0.470730\pi$
$593$	4.47214	0.183649	0.0918243	+	0.995775i	$0.470730\pi$
$594$	0	0
$595$	0	0
$596$	−60.3738	−2.47301
$597$	−10.0000	−0.409273
$598$	−10.0000	−0.408930
$599$	9.00000	0.367730	0.183865	−	0.982952i	$-0.441139\pi$
$599$	9.00000	0.367730	0.183865	+	0.982952i	$0.441139\pi$
$600$	0	0
$601$	−13.4164	−0.547267	−0.273633	−	0.961834i	$-0.588225\pi$
$601$	−13.4164	−0.547267	−0.273633	+	0.961834i	$0.588225\pi$
$602$	0	0
$603$	2.00000	0.0814463
$604$	0	0
$605$	0	0
$606$	−25.0000	−1.01556
$607$	−26.8328	−1.08911	−0.544555	−	0.838725i	$-0.683302\pi$
$607$	−26.8328	−1.08911	−0.544555	+	0.838725i	$0.683302\pi$
$608$	15.0000	0.608330
$609$	−10.0000	−0.405220
$610$	0	0
$611$	−13.4164	−0.542770
$612$	6.70820	0.271163
$613$	−8.94427	−0.361256	−0.180628	−	0.983552i	$-0.557813\pi$
$613$	−8.94427	−0.361256	−0.180628	+	0.983552i	$0.557813\pi$
$614$	−40.0000	−1.61427
$615$	0	0
$616$	0	0
$617$	22.0000	0.885687	0.442843	−	0.896599i	$-0.353970\pi$
$617$	22.0000	0.885687	0.442843	+	0.896599i	$0.353970\pi$
$618$	−8.94427	−0.359791
$619$	−16.0000	−0.643094	−0.321547	−	0.946894i	$-0.604203\pi$
$619$	−16.0000	−0.643094	−0.321547	+	0.946894i	$0.604203\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	−71.5542	−2.86906
$623$	13.4164	0.537517
$624$	4.47214	0.179029
$625$	0	0
$626$	−20.1246	−0.804341
$627$	0	0
$628$	−6.00000	−0.239426
$629$	−15.6525	−0.624105
$630$	0	0
$631$	−28.0000	−1.11466	−0.557331	−	0.830290i	$-0.688175\pi$
$631$	−28.0000	−1.11466	−0.557331	+	0.830290i	$0.688175\pi$
$632$	−35.0000	−1.39223
$633$	−26.8328	−1.06651
$634$	49.1935	1.95372
$635$	0	0
$636$	−42.0000	−1.66541
$637$	−8.94427	−0.354385
$638$	0	0
$639$	−13.0000	−0.514272
$640$	0	0
$641$	0	0		−	1.00000i	$-0.5\pi$
$641$	0	0			1.00000i	$0.5\pi$
$642$	0	0
$643$	−16.0000	−0.630978	−0.315489	−	0.948929i	$-0.602169\pi$
$643$	−16.0000	−0.630978	−0.315489	+	0.948929i	$0.602169\pi$
$644$	−6.70820	−0.264340
$645$	0	0
$646$	−11.1803	−0.439885
$647$	−43.0000	−1.69050	−0.845252	−	0.534368i	$-0.820550\pi$
$647$	−43.0000	−1.69050	−0.845252	+	0.534368i	$0.820550\pi$
$648$	2.23607	0.0878410
$649$	0	0
$650$	0	0
$651$	−22.3607	−0.876384
$652$	−42.0000	−1.64485
$653$	−26.0000	−1.01746	−0.508729	−	0.860927i	$-0.669885\pi$
$653$	−26.0000	−1.01746	−0.508729	+	0.860927i	$0.669885\pi$
$654$	−40.0000	−1.56412
$655$	0	0
$656$	−6.70820	−0.261911
$657$	13.4164	0.523424
$658$	−15.0000	−0.584761
$659$	−31.3050	−1.21947	−0.609734	−	0.792606i	$-0.708724\pi$
$659$	−31.3050	−1.21947	−0.609734	+	0.792606i	$0.708724\pi$
$660$	0	0
$661$	3.00000	0.116686	0.0583432	−	0.998297i	$-0.481418\pi$
$661$	3.00000	0.116686	0.0583432	+	0.998297i	$0.481418\pi$
$662$	44.7214	1.73814
$663$	−10.0000	−0.388368
$664$	10.0000	0.388075
$665$	0	0
$666$	−15.6525	−0.606521
$667$	−4.47214	−0.173162
$668$	13.4164	0.519096
$669$	14.0000	0.541271
$670$	0	0
$671$	0	0
$672$	15.0000	0.578638
$673$	−8.94427	−0.344776	−0.172388	−	0.985029i	$-0.555148\pi$
$673$	−8.94427	−0.344776	−0.172388	+	0.985029i	$0.555148\pi$
$674$	−30.0000	−1.15556
$675$	0	0
$676$	21.0000	0.807692
$677$	−49.1935	−1.89066	−0.945330	−	0.326116i	$-0.894260\pi$
$677$	−49.1935	−1.89066	−0.945330	+	0.326116i	$0.894260\pi$
$678$	−31.3050	−1.20226
$679$	−38.0132	−1.45881
$680$	0	0
$681$	−13.4164	−0.514118
$682$	0	0
$683$	21.0000	0.803543	0.401771	−	0.915740i	$-0.368395\pi$
$683$	21.0000	0.803543	0.401771	+	0.915740i	$0.368395\pi$
$684$	−6.70820	−0.256495
$685$	0	0
$686$	−45.0000	−1.71811
$687$	5.00000	0.190762
$688$	0	0
$689$	62.6099	2.38525
$690$	0	0
$691$	−8.00000	−0.304334	−0.152167	−	0.988355i	$-0.548625\pi$
$691$	−8.00000	−0.304334	−0.152167	+	0.988355i	$0.548625\pi$
$692$	60.3738	2.29507
$693$	0	0
$694$	30.0000	1.13878
$695$	0	0
$696$	−10.0000	−0.379049
$697$	15.0000	0.568166
$698$	−30.0000	−1.13552
$699$	20.1246	0.761183
$700$	0	0
$701$	42.4853	1.60465	0.802324	−	0.596889i	$-0.203597\pi$
$701$	42.4853	1.60465	0.802324	+	0.596889i	$0.203597\pi$
$702$	−10.0000	−0.377426
$703$	15.6525	0.590344
$704$	0	0
$705$	0	0
$706$	76.0263	2.86129
$707$	25.0000	0.940222
$708$	15.0000	0.563735
$709$	19.0000	0.713560	0.356780	−	0.934188i	$-0.383875\pi$
$709$	19.0000	0.713560	0.356780	+	0.934188i	$0.383875\pi$
$710$	0	0
$711$	−15.6525	−0.587014
$712$	13.4164	0.502801
$713$	−10.0000	−0.374503
$714$	−11.1803	−0.418414
$715$	0	0
$716$	33.0000	1.23327
$717$	−8.94427	−0.334030
$718$	10.0000	0.373197
$719$	−4.00000	−0.149175	−0.0745874	−	0.997214i	$-0.523764\pi$
$719$	−4.00000	−0.149175	−0.0745874	+	0.997214i	$0.523764\pi$
$720$	0	0
$721$	8.94427	0.333102
$722$	−31.3050	−1.16505
$723$	−22.3607	−0.831603
$724$	−15.0000	−0.557471
$725$	0	0
$726$	0	0
$727$	8.00000	0.296704	0.148352	−	0.988935i	$-0.452603\pi$
$727$	8.00000	0.296704	0.148352	+	0.988935i	$0.452603\pi$
$728$	22.3607	0.828742
$729$	1.00000	0.0370370
$730$	0	0
$731$	0	0
$732$	−13.4164	−0.495885
$733$	40.2492	1.48664	0.743319	−	0.668937i	$-0.233250\pi$
$733$	40.2492	1.48664	0.743319	+	0.668937i	$0.233250\pi$
$734$	−40.2492	−1.48563
$735$	0	0
$736$	6.70820	0.247268
$737$	0	0
$738$	15.0000	0.552158
$739$	−24.5967	−0.904806	−0.452403	−	0.891814i	$-0.649433\pi$
$739$	−24.5967	−0.904806	−0.452403	+	0.891814i	$0.649433\pi$
$740$	0	0
$741$	10.0000	0.367359
$742$	70.0000	2.56978
$743$	−40.2492	−1.47660	−0.738300	−	0.674472i	$-0.764371\pi$
$743$	−40.2492	−1.47660	−0.738300	+	0.674472i	$0.764371\pi$
$744$	−22.3607	−0.819782
$745$	0	0
$746$	−20.0000	−0.732252
$747$	4.47214	0.163627
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−48.0000	−1.75154	−0.875772	−	0.482724i	$-0.839647\pi$
$751$	−48.0000	−1.75154	−0.875772	+	0.482724i	$0.839647\pi$
$752$	3.00000	0.109399
$753$	12.0000	0.437304
$754$	44.7214	1.62866
$755$	0	0
$756$	−6.70820	−0.243975
$757$	−22.0000	−0.799604	−0.399802	−	0.916602i	$-0.630921\pi$
$757$	−22.0000	−0.799604	−0.399802	+	0.916602i	$0.630921\pi$
$758$	22.3607	0.812176
$759$	0	0
$760$	0	0
$761$	−40.2492	−1.45903	−0.729517	−	0.683963i	$-0.760255\pi$
$761$	−40.2492	−1.45903	−0.729517	+	0.683963i	$0.760255\pi$
$762$	−35.0000	−1.26792
$763$	40.0000	1.44810
$764$	45.0000	1.62804
$765$	0	0
$766$	35.7771	1.29268
$767$	−22.3607	−0.807397
$768$	9.00000	0.324760
$769$	−35.7771	−1.29015	−0.645077	−	0.764117i	$-0.723175\pi$
$769$	−35.7771	−1.29015	−0.645077	+	0.764117i	$0.723175\pi$
$770$	0	0
$771$	18.0000	0.648254
$772$	−26.8328	−0.965734
$773$	−24.0000	−0.863220	−0.431610	−	0.902060i	$-0.642054\pi$
$773$	−24.0000	−0.863220	−0.431610	+	0.902060i	$0.642054\pi$
$774$	0	0
$775$	0	0
$776$	−38.0132	−1.36459
$777$	15.6525	0.561529
$778$	0	0
$779$	−15.0000	−0.537431
$780$	0	0
$781$	0	0
$782$	−5.00000	−0.178800
$783$	−4.47214	−0.159821
$784$	2.00000	0.0714286
$785$	0	0
$786$	30.0000	1.07006
$787$	−46.9574	−1.67385	−0.836926	−	0.547316i	$-0.815649\pi$
$787$	−46.9574	−1.67385	−0.836926	+	0.547316i	$0.815649\pi$
$788$	−46.9574	−1.67279
$789$	17.8885	0.636849
$790$	0	0
$791$	31.3050	1.11308
$792$	0	0
$793$	20.0000	0.710221
$794$	−49.1935	−1.74581
$795$	0	0
$796$	30.0000	1.06332
$797$	8.00000	0.283375	0.141687	−	0.989911i	$-0.454747\pi$
$797$	8.00000	0.283375	0.141687	+	0.989911i	$0.454747\pi$
$798$	11.1803	0.395780
$799$	−6.70820	−0.237319
$800$	0	0
$801$	6.00000	0.212000
$802$	22.3607	0.789583
$803$	0	0
$804$	−6.00000	−0.211604
$805$	0	0
$806$	100.000	3.52235
$807$	−30.0000	−1.05605
$808$	25.0000	0.879497
$809$	−24.5967	−0.864776	−0.432388	−	0.901688i	$-0.642329\pi$
$809$	−24.5967	−0.864776	−0.432388	+	0.901688i	$0.642329\pi$
$810$	0	0
$811$	−46.9574	−1.64890	−0.824449	−	0.565936i	$-0.808515\pi$
$811$	−46.9574	−1.64890	−0.824449	+	0.565936i	$0.808515\pi$
$812$	30.0000	1.05279
$813$	−20.1246	−0.705801
$814$	0	0
$815$	0	0
$816$	2.23607	0.0782780
$817$	0	0
$818$	30.0000	1.04893
$819$	10.0000	0.349428
$820$	0	0
$821$	−31.3050	−1.09255	−0.546275	−	0.837606i	$-0.683955\pi$
$821$	−31.3050	−1.09255	−0.546275	+	0.837606i	$0.683955\pi$
$822$	−17.8885	−0.623935
$823$	14.0000	0.488009	0.244005	−	0.969774i	$-0.421539\pi$
$823$	14.0000	0.488009	0.244005	+	0.969774i	$0.421539\pi$
$824$	8.94427	0.311588
$825$	0	0
$826$	−25.0000	−0.869861
$827$	17.8885	0.622046	0.311023	−	0.950402i	$-0.399328\pi$
$827$	17.8885	0.622046	0.311023	+	0.950402i	$0.399328\pi$
$828$	−3.00000	−0.104257
$829$	34.0000	1.18087	0.590434	−	0.807086i	$-0.298956\pi$
$829$	34.0000	1.18087	0.590434	+	0.807086i	$0.298956\pi$
$830$	0	0
$831$	13.4164	0.465410
$832$	−58.1378	−2.01556
$833$	−4.47214	−0.154950
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−10.0000	−0.345651
$838$	20.1246	0.695193
$839$	−40.0000	−1.38095	−0.690477	−	0.723355i	$-0.742599\pi$
$839$	−40.0000	−1.38095	−0.690477	+	0.723355i	$0.742599\pi$
$840$	0	0
$841$	−9.00000	−0.310345
$842$	−33.5410	−1.15590
$843$	−24.5967	−0.847157
$844$	80.4984	2.77087
$845$	0	0
$846$	−6.70820	−0.230633
$847$	0	0
$848$	−14.0000	−0.480762
$849$	6.70820	0.230225
$850$	0	0
$851$	7.00000	0.239957
$852$	39.0000	1.33612
$853$	−22.3607	−0.765615	−0.382808	−	0.923828i	$-0.625043\pi$
$853$	−22.3607	−0.765615	−0.382808	+	0.923828i	$0.625043\pi$
$854$	22.3607	0.765167
$855$	0	0
$856$	0	0
$857$	11.1803	0.381913	0.190957	−	0.981598i	$-0.438841\pi$
$857$	11.1803	0.381913	0.190957	+	0.981598i	$0.438841\pi$
$858$	0	0
$859$	14.0000	0.477674	0.238837	−	0.971060i	$-0.423234\pi$
$859$	14.0000	0.477674	0.238837	+	0.971060i	$0.423234\pi$
$860$	0	0
$861$	−15.0000	−0.511199
$862$	50.0000	1.70301
$863$	−24.0000	−0.816970	−0.408485	−	0.912765i	$-0.633943\pi$
$863$	−24.0000	−0.816970	−0.408485	+	0.912765i	$0.633943\pi$
$864$	6.70820	0.228218
$865$	0	0
$866$	−31.3050	−1.06379
$867$	12.0000	0.407541
$868$	67.0820	2.27691
$869$	0	0
$870$	0	0
$871$	8.94427	0.303065
$872$	40.0000	1.35457
$873$	−17.0000	−0.575363
$874$	5.00000	0.169128
$875$	0	0
$876$	−40.2492	−1.35990
$877$	−31.3050	−1.05709	−0.528547	−	0.848904i	$-0.677263\pi$
$877$	−31.3050	−1.05709	−0.528547	+	0.848904i	$0.677263\pi$
$878$	45.0000	1.51868
$879$	−15.6525	−0.527945
$880$	0	0
$881$	12.0000	0.404290	0.202145	−	0.979356i	$-0.435209\pi$
$881$	12.0000	0.404290	0.202145	+	0.979356i	$0.435209\pi$
$882$	−4.47214	−0.150585
$883$	36.0000	1.21150	0.605748	−	0.795656i	$-0.292874\pi$
$883$	36.0000	1.21150	0.605748	+	0.795656i	$0.292874\pi$
$884$	30.0000	1.00901
$885$	0	0
$886$	−46.9574	−1.57757
$887$	4.47214	0.150160	0.0750798	−	0.997178i	$-0.476079\pi$
$887$	4.47214	0.150160	0.0750798	+	0.997178i	$0.476079\pi$
$888$	15.6525	0.525263
$889$	35.0000	1.17386
$890$	0	0
$891$	0	0
$892$	−42.0000	−1.40626
$893$	6.70820	0.224481
$894$	45.0000	1.50503
$895$	0	0
$896$	−35.0000	−1.16927
$897$	4.47214	0.149320
$898$	8.94427	0.298474
$899$	44.7214	1.49154
$900$	0	0
$901$	31.3050	1.04292
$902$	0	0
$903$	0	0
$904$	31.3050	1.04119
$905$	0	0
$906$	0	0
$907$	−8.00000	−0.265636	−0.132818	−	0.991140i	$-0.542403\pi$
$907$	−8.00000	−0.265636	−0.132818	+	0.991140i	$0.542403\pi$
$908$	40.2492	1.33572
$909$	11.1803	0.370828
$910$	0	0
$911$	55.0000	1.82223	0.911116	−	0.412151i	$-0.135222\pi$
$911$	55.0000	1.82223	0.911116	+	0.412151i	$0.135222\pi$
$912$	−2.23607	−0.0740436
$913$	0	0
$914$	−50.0000	−1.65385
$915$	0	0
$916$	−15.0000	−0.495614
$917$	−30.0000	−0.990687
$918$	−5.00000	−0.165025
$919$	−24.5967	−0.811372	−0.405686	−	0.914013i	$-0.632967\pi$
$919$	−24.5967	−0.811372	−0.405686	+	0.914013i	$0.632967\pi$
$920$	0	0
$921$	17.8885	0.589448
$922$	−50.0000	−1.64666
$923$	−58.1378	−1.91363
$924$	0	0
$925$	0	0
$926$	−31.3050	−1.02874
$927$	4.00000	0.131377
$928$	−30.0000	−0.984798
$929$	−20.0000	−0.656179	−0.328089	−	0.944647i	$-0.606405\pi$
$929$	−20.0000	−0.656179	−0.328089	+	0.944647i	$0.606405\pi$
$930$	0	0
$931$	4.47214	0.146568
$932$	−60.3738	−1.97761
$933$	32.0000	1.04763
$934$	−71.5542	−2.34132
$935$	0	0
$936$	10.0000	0.326860
$937$	−26.8328	−0.876590	−0.438295	−	0.898831i	$-0.644417\pi$
$937$	−26.8328	−0.876590	−0.438295	+	0.898831i	$0.644417\pi$
$938$	10.0000	0.326512
$939$	9.00000	0.293704
$940$	0	0
$941$	−24.5967	−0.801831	−0.400916	−	0.916115i	$-0.631308\pi$
$941$	−24.5967	−0.801831	−0.400916	+	0.916115i	$0.631308\pi$
$942$	4.47214	0.145710
$943$	−6.70820	−0.218449
$944$	5.00000	0.162736
$945$	0	0
$946$	0	0
$947$	−17.0000	−0.552426	−0.276213	−	0.961096i	$-0.589079\pi$
$947$	−17.0000	−0.552426	−0.276213	+	0.961096i	$0.589079\pi$
$948$	46.9574	1.52511
$949$	60.0000	1.94768
$950$	0	0
$951$	−22.0000	−0.713399
$952$	11.1803	0.362357
$953$	−42.4853	−1.37623	−0.688117	−	0.725600i	$-0.741562\pi$
$953$	−42.4853	−1.37623	−0.688117	+	0.725600i	$0.741562\pi$
$954$	31.3050	1.01354
$955$	0	0
$956$	26.8328	0.867835
$957$	0	0
$958$	50.0000	1.61543
$959$	17.8885	0.577651
$960$	0	0
$961$	69.0000	2.22581
$962$	−70.0000	−2.25689
$963$	0	0
$964$	67.0820	2.16057
$965$	0	0
$966$	5.00000	0.160872
$967$	8.94427	0.287628	0.143814	−	0.989605i	$-0.454063\pi$
$967$	8.94427	0.287628	0.143814	+	0.989605i	$0.454063\pi$
$968$	0	0
$969$	5.00000	0.160623
$970$	0	0
$971$	5.00000	0.160458	0.0802288	−	0.996776i	$-0.474435\pi$
$971$	5.00000	0.160458	0.0802288	+	0.996776i	$0.474435\pi$
$972$	−3.00000	−0.0962250
$973$	0	0
$974$	71.5542	2.29274
$975$	0	0
$976$	−4.47214	−0.143150
$977$	12.0000	0.383914	0.191957	−	0.981403i	$-0.438517\pi$
$977$	12.0000	0.383914	0.191957	+	0.981403i	$0.438517\pi$
$978$	31.3050	1.00102
$979$	0	0
$980$	0	0
$981$	17.8885	0.571137
$982$	−10.0000	−0.319113
$983$	49.0000	1.56286	0.781429	−	0.623995i	$-0.214491\pi$
$983$	49.0000	1.56286	0.781429	+	0.623995i	$0.214491\pi$
$984$	−15.0000	−0.478183
$985$	0	0
$986$	22.3607	0.712109
$987$	6.70820	0.213524
$988$	−30.0000	−0.954427
$989$	0	0
$990$	0	0
$991$	10.0000	0.317660	0.158830	−	0.987306i	$-0.449228\pi$
$991$	10.0000	0.317660	0.158830	+	0.987306i	$0.449228\pi$
$992$	−67.0820	−2.12986
$993$	−20.0000	−0.634681
$994$	−65.0000	−2.06167
$995$	0	0
$996$	−13.4164	−0.425115
$997$	−62.6099	−1.98288	−0.991438	−	0.130580i	$-0.958316\pi$
$997$	−62.6099	−1.98288	−0.991438	+	0.130580i	$0.958316\pi$
$998$	44.7214	1.41563
$999$	7.00000	0.221470

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9075.2.a.bk.1.2	yes	2
5.4	even	2		9075.2.a.br.1.1	yes	2
11.10	odd	2	inner	9075.2.a.bk.1.1	✓	2
55.54	odd	2		9075.2.a.br.1.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9075.2.a.bk.1.1	✓	2	11.10	odd	2	inner
9075.2.a.bk.1.2	yes	2	1.1	even	1	trivial
9075.2.a.br.1.1	yes	2	5.4	even	2
9075.2.a.br.1.2	yes	2	55.54	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 9075 = 3 \cdot 5^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9075.a (trivial)

\(f(q)\)	\(=\)	\(q+2.23607 q^{2} -1.00000 q^{3} +3.00000 q^{4} -2.23607 q^{6} +2.23607 q^{7} +2.23607 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+2.23607 q^{2} -1.00000 q^{3} +3.00000 q^{4} -2.23607 q^{6} +2.23607 q^{7} +2.23607 q^{8} +1.00000 q^{9} -3.00000 q^{12} +4.47214 q^{13} +5.00000 q^{14} -1.00000 q^{16} +2.23607 q^{17} +2.23607 q^{18} -2.23607 q^{19} -2.23607 q^{21} -1.00000 q^{23} -2.23607 q^{24} +10.0000 q^{26} -1.00000 q^{27} +6.70820 q^{28} +4.47214 q^{29} +10.0000 q^{31} -6.70820 q^{32} +5.00000 q^{34} +3.00000 q^{36} -7.00000 q^{37} -5.00000 q^{38} -4.47214 q^{39} +6.70820 q^{41} -5.00000 q^{42} -2.23607 q^{46} -3.00000 q^{47} +1.00000 q^{48} -2.00000 q^{49} -2.23607 q^{51} +13.4164 q^{52} +14.0000 q^{53} -2.23607 q^{54} +5.00000 q^{56} +2.23607 q^{57} +10.0000 q^{58} -5.00000 q^{59} +4.47214 q^{61} +22.3607 q^{62} +2.23607 q^{63} -13.0000 q^{64} +2.00000 q^{67} +6.70820 q^{68} +1.00000 q^{69} -13.0000 q^{71} +2.23607 q^{72} +13.4164 q^{73} -15.6525 q^{74} -6.70820 q^{76} -10.0000 q^{78} -15.6525 q^{79} +1.00000 q^{81} +15.0000 q^{82} +4.47214 q^{83} -6.70820 q^{84} -4.47214 q^{87} +6.00000 q^{89} +10.0000 q^{91} -3.00000 q^{92} -10.0000 q^{93} -6.70820 q^{94} +6.70820 q^{96} -17.0000 q^{97} -4.47214 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 6 q^{4} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 + 6 * q^4 + 2 * q^9	\( 2 q - 2 q^{3} + 6 q^{4} + 2 q^{9} - 6 q^{12} + 10 q^{14} - 2 q^{16} - 2 q^{23} + 20 q^{26} - 2 q^{27} + 20 q^{31} + 10 q^{34} + 6 q^{36} - 14 q^{37} - 10 q^{38} - 10 q^{42} - 6 q^{47} + 2 q^{48} - 4 q^{49} + 28 q^{53} + 10 q^{56} + 20 q^{58} - 10 q^{59} - 26 q^{64} + 4 q^{67} + 2 q^{69} - 26 q^{71} - 20 q^{78} + 2 q^{81} + 30 q^{82} + 12 q^{89} + 20 q^{91} - 6 q^{92} - 20 q^{93} - 34 q^{97}+O(q^{100}) \) 2 * q - 2 * q^3 + 6 * q^4 + 2 * q^9 - 6 * q^12 + 10 * q^14 - 2 * q^16 - 2 * q^23 + 20 * q^26 - 2 * q^27 + 20 * q^31 + 10 * q^34 + 6 * q^36 - 14 * q^37 - 10 * q^38 - 10 * q^42 - 6 * q^47 + 2 * q^48 - 4 * q^49 + 28 * q^53 + 10 * q^56 + 20 * q^58 - 10 * q^59 - 26 * q^64 + 4 * q^67 + 2 * q^69 - 26 * q^71 - 20 * q^78 + 2 * q^81 + 30 * q^82 + 12 * q^89 + 20 * q^91 - 6 * q^92 - 20 * q^93 - 34 * q^97

Introduction

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