LMFDB - Embedded newform 9522.2.a.bt.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9522,2,Mod(1,9522)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9522, 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("9522.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9522 = 2 \cdot 3^{2} \cdot 23^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9522.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:	$76.0335528047$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	$\Q(\zeta_{22})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 4x^{3} + 3x^{2} + 3x - 1 $ x^5 - x^4 - 4x^3 + 3x^2 + 3*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 138)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-1.68251$ of defining polynomial
Character	$\chi$	$=$	9522.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} +2.59435 q^{5} -1.23648 q^{7} -1.00000 q^{8} +O(q^{10})$	$q-1.00000 q^{2} +1.00000 q^{4} +2.59435 q^{5} -1.23648 q^{7} -1.00000 q^{8} -2.59435 q^{10} +0.622970 q^{11} -1.32463 q^{13} +1.23648 q^{14} +1.00000 q^{16} +4.70760 q^{17} +3.79020 q^{19} +2.59435 q^{20} -0.622970 q^{22} +1.73066 q^{25} +1.32463 q^{26} -1.23648 q^{28} +1.11204 q^{29} +9.45679 q^{31} -1.00000 q^{32} -4.70760 q^{34} -3.20786 q^{35} -1.62721 q^{37} -3.79020 q^{38} -2.59435 q^{40} +11.8497 q^{41} -1.67879 q^{43} +0.622970 q^{44} +6.94494 q^{47} -5.47112 q^{49} -1.73066 q^{50} -1.32463 q^{52} +11.2537 q^{53} +1.61620 q^{55} +1.23648 q^{56} -1.11204 q^{58} +9.74629 q^{59} -14.3736 q^{61} -9.45679 q^{62} +1.00000 q^{64} -3.43657 q^{65} -6.12125 q^{67} +4.70760 q^{68} +3.20786 q^{70} -6.49255 q^{71} -13.0900 q^{73} +1.62721 q^{74} +3.79020 q^{76} -0.770289 q^{77} -12.0823 q^{79} +2.59435 q^{80} -11.8497 q^{82} -7.39068 q^{83} +12.2132 q^{85} +1.67879 q^{86} -0.622970 q^{88} -3.38500 q^{89} +1.63788 q^{91} -6.94494 q^{94} +9.83310 q^{95} +11.4458 q^{97} +5.47112 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 5 q^{2} + 5 q^{4} + 7 q^{5} - 7 q^{7} - 5 q^{8}+O(q^{10}) $ 5 * q - 5 * q^2 + 5 * q^4 + 7 * q^5 - 7 * q^7 - 5 * q^8	$ 5 q - 5 q^{2} + 5 q^{4} + 7 q^{5} - 7 q^{7} - 5 q^{8} - 7 q^{10} + 13 q^{11} - 4 q^{13} + 7 q^{14} + 5 q^{16} + 9 q^{17} - 11 q^{19} + 7 q^{20} - 13 q^{22} - 2 q^{25} + 4 q^{26} - 7 q^{28} + 7 q^{29} - 8 q^{31} - 5 q^{32} - 9 q^{34} - q^{35} - 12 q^{37} + 11 q^{38} - 7 q^{40} + 10 q^{41} - 4 q^{43} + 13 q^{44} + 24 q^{47} - 12 q^{49} + 2 q^{50} - 4 q^{52} + 9 q^{53} + 16 q^{55} + 7 q^{56} - 7 q^{58} + 14 q^{59} - 5 q^{61} + 8 q^{62} + 5 q^{64} + 12 q^{65} - 13 q^{67} + 9 q^{68} + q^{70} + 19 q^{71} + 4 q^{73} + 12 q^{74} - 11 q^{76} - 5 q^{77} - 4 q^{79} + 7 q^{80} - 10 q^{82} + 24 q^{83} + 17 q^{85} + 4 q^{86} - 13 q^{88} + 4 q^{89} + 21 q^{91} - 24 q^{94} + 11 q^{95} + 9 q^{97} + 12 q^{98}+O(q^{100}) $ 5 * q - 5 * q^2 + 5 * q^4 + 7 * q^5 - 7 * q^7 - 5 * q^8 - 7 * q^10 + 13 * q^11 - 4 * q^13 + 7 * q^14 + 5 * q^16 + 9 * q^17 - 11 * q^19 + 7 * q^20 - 13 * q^22 - 2 * q^25 + 4 * q^26 - 7 * q^28 + 7 * q^29 - 8 * q^31 - 5 * q^32 - 9 * q^34 - q^35 - 12 * q^37 + 11 * q^38 - 7 * q^40 + 10 * q^41 - 4 * q^43 + 13 * q^44 + 24 * q^47 - 12 * q^49 + 2 * q^50 - 4 * q^52 + 9 * q^53 + 16 * q^55 + 7 * q^56 - 7 * q^58 + 14 * q^59 - 5 * q^61 + 8 * q^62 + 5 * q^64 + 12 * q^65 - 13 * q^67 + 9 * q^68 + q^70 + 19 * q^71 + 4 * q^73 + 12 * q^74 - 11 * q^76 - 5 * q^77 - 4 * q^79 + 7 * q^80 - 10 * q^82 + 24 * q^83 + 17 * q^85 + 4 * q^86 - 13 * q^88 + 4 * q^89 + 21 * q^91 - 24 * q^94 + 11 * q^95 + 9 * q^97 + 12 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	2.59435	1.16023	0.580115	−	0.814535i	$-0.303008\pi$
$5$	2.59435	1.16023	0.580115	+	0.814535i	$0.303008\pi$
$6$	0	0
$7$	−1.23648	−0.467345	−0.233673	−	0.972315i	$-0.575074\pi$
$7$	−1.23648	−0.467345	−0.233673	+	0.972315i	$0.575074\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	−2.59435	−0.820406
$11$	0.622970	0.187832	0.0939162	−	0.995580i	$-0.470061\pi$
$11$	0.622970	0.187832	0.0939162	+	0.995580i	$0.470061\pi$
$12$	0	0
$13$	−1.32463	−0.367388	−0.183694	−	0.982984i	$-0.558805\pi$
$13$	−1.32463	−0.367388	−0.183694	+	0.982984i	$0.558805\pi$
$14$	1.23648	0.330463
$15$	0	0
$16$	1.00000	0.250000
$17$	4.70760	1.14176	0.570880	−	0.821033i	$-0.306602\pi$
$17$	4.70760	1.14176	0.570880	+	0.821033i	$0.306602\pi$
$18$	0	0
$19$	3.79020	0.869530	0.434765	−	0.900544i	$-0.356831\pi$
$19$	3.79020	0.869530	0.434765	+	0.900544i	$0.356831\pi$
$20$	2.59435	0.580115
$21$	0	0
$22$	−0.622970	−0.132818
$23$	0	0
$23$	0	0
$24$	0	0
$25$	1.73066	0.346132
$26$	1.32463	0.259782
$27$	0	0
$28$	−1.23648	−0.233673
$29$	1.11204	0.206501	0.103250	−	0.994655i	$-0.467076\pi$
$29$	1.11204	0.206501	0.103250	+	0.994655i	$0.467076\pi$
$30$	0	0
$31$	9.45679	1.69849	0.849244	−	0.528000i	$-0.177058\pi$
$31$	9.45679	1.69849	0.849244	+	0.528000i	$0.177058\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−4.70760	−0.807347
$35$	−3.20786	−0.542227
$36$	0	0
$37$	−1.62721	−0.267512	−0.133756	−	0.991014i	$-0.542704\pi$
$37$	−1.62721	−0.267512	−0.133756	+	0.991014i	$0.542704\pi$
$38$	−3.79020	−0.614851
$39$	0	0
$40$	−2.59435	−0.410203
$41$	11.8497	1.85062	0.925309	−	0.379215i	$-0.123806\pi$
$41$	11.8497	1.85062	0.925309	+	0.379215i	$0.123806\pi$
$42$	0	0
$43$	−1.67879	−0.256012	−0.128006	−	0.991773i	$-0.540858\pi$
$43$	−1.67879	−0.256012	−0.128006	+	0.991773i	$0.540858\pi$
$44$	0.622970	0.0939162
$45$	0	0
$46$	0	0
$47$	6.94494	1.01302	0.506512	−	0.862233i	$-0.330935\pi$
$47$	6.94494	1.01302	0.506512	+	0.862233i	$0.330935\pi$
$48$	0	0
$49$	−5.47112	−0.781589
$50$	−1.73066	−0.244752
$51$	0	0
$52$	−1.32463	−0.183694
$53$	11.2537	1.54581	0.772907	−	0.634520i	$-0.218802\pi$
$53$	11.2537	1.54581	0.772907	+	0.634520i	$0.218802\pi$
$54$	0	0
$55$	1.61620	0.217929
$56$	1.23648	0.165231
$57$	0	0
$58$	−1.11204	−0.146018
$59$	9.74629	1.26886	0.634429	−	0.772981i	$-0.281235\pi$
$59$	9.74629	1.26886	0.634429	+	0.772981i	$0.281235\pi$
$60$	0	0
$61$	−14.3736	−1.84035	−0.920173	−	0.391512i	$-0.871952\pi$
$61$	−14.3736	−1.84035	−0.920173	+	0.391512i	$0.871952\pi$
$62$	−9.45679	−1.20101
$63$	0	0
$64$	1.00000	0.125000
$65$	−3.43657	−0.426254
$66$	0	0
$67$	−6.12125	−0.747830	−0.373915	−	0.927463i	$-0.621985\pi$
$67$	−6.12125	−0.747830	−0.373915	+	0.927463i	$0.621985\pi$
$68$	4.70760	0.570880
$69$	0	0
$70$	3.20786	0.383413
$71$	−6.49255	−0.770523	−0.385262	−	0.922807i	$-0.625889\pi$
$71$	−6.49255	−0.770523	−0.385262	+	0.922807i	$0.625889\pi$
$72$	0	0
$73$	−13.0900	−1.53207	−0.766035	−	0.642799i	$-0.777773\pi$
$73$	−13.0900	−1.53207	−0.766035	+	0.642799i	$0.777773\pi$
$74$	1.62721	0.189160
$75$	0	0
$76$	3.79020	0.434765
$77$	−0.770289	−0.0877826
$78$	0	0
$79$	−12.0823	−1.35937	−0.679685	−	0.733504i	$-0.737883\pi$
$79$	−12.0823	−1.35937	−0.679685	+	0.733504i	$0.737883\pi$
$80$	2.59435	0.290057
$81$	0	0
$82$	−11.8497	−1.30858
$83$	−7.39068	−0.811233	−0.405617	−	0.914043i	$-0.632943\pi$
$83$	−7.39068	−0.811233	−0.405617	+	0.914043i	$0.632943\pi$
$84$	0	0
$85$	12.2132	1.32470
$86$	1.67879	0.181028
$87$	0	0
$88$	−0.622970	−0.0664088
$89$	−3.38500	−0.358809	−0.179404	−	0.983775i	$-0.557417\pi$
$89$	−3.38500	−0.358809	−0.179404	+	0.983775i	$0.557417\pi$
$90$	0	0
$91$	1.63788	0.171697
$92$	0	0
$93$	0	0
$94$	−6.94494	−0.716316
$95$	9.83310	1.00885
$96$	0	0
$97$	11.4458	1.16214	0.581072	−	0.813852i	$-0.302634\pi$
$97$	11.4458	1.16214	0.581072	+	0.813852i	$0.302634\pi$
$98$	5.47112	0.552667
$99$	0	0
$100$	1.73066	0.173066
$101$	1.78374	0.177489	0.0887443	−	0.996054i	$-0.471715\pi$
$101$	1.78374	0.177489	0.0887443	+	0.996054i	$0.471715\pi$
$102$	0	0
$103$	19.8505	1.95593	0.977963	−	0.208776i	$-0.0669480\pi$
$103$	19.8505	1.95593	0.977963	+	0.208776i	$0.0669480\pi$
$104$	1.32463	0.129891
$105$	0	0
$106$	−11.2537	−1.09306
$107$	4.35922	0.421422	0.210711	−	0.977548i	$-0.432422\pi$
$107$	4.35922	0.421422	0.210711	+	0.977548i	$0.432422\pi$
$108$	0	0
$109$	6.46538	0.619271	0.309636	−	0.950855i	$-0.399793\pi$
$109$	6.46538	0.619271	0.309636	+	0.950855i	$0.399793\pi$
$110$	−1.61620	−0.154099
$111$	0	0
$112$	−1.23648	−0.116836
$113$	1.36243	0.128166	0.0640832	−	0.997945i	$-0.479588\pi$
$113$	1.36243	0.128166	0.0640832	+	0.997945i	$0.479588\pi$
$114$	0	0
$115$	0	0
$116$	1.11204	0.103250
$117$	0	0
$118$	−9.74629	−0.897219
$119$	−5.82085	−0.533596
$120$	0	0
$121$	−10.6119	−0.964719
$122$	14.3736	1.30132
$123$	0	0
$124$	9.45679	0.849244
$125$	−8.48182	−0.758637
$126$	0	0
$127$	3.45519	0.306598	0.153299	−	0.988180i	$-0.451010\pi$
$127$	3.45519	0.306598	0.153299	+	0.988180i	$0.451010\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	3.43657	0.301407
$131$	−18.9474	−1.65544	−0.827721	−	0.561140i	$-0.810363\pi$
$131$	−18.9474	−1.65544	−0.827721	+	0.561140i	$0.810363\pi$
$132$	0	0
$133$	−4.68650	−0.406371
$134$	6.12125	0.528796
$135$	0	0
$136$	−4.70760	−0.403673
$137$	11.3960	0.973629	0.486815	−	0.873505i	$-0.338159\pi$
$137$	11.3960	0.973629	0.486815	+	0.873505i	$0.338159\pi$
$138$	0	0
$139$	5.30267	0.449766	0.224883	−	0.974386i	$-0.427800\pi$
$139$	5.30267	0.449766	0.224883	+	0.974386i	$0.427800\pi$
$140$	−3.20786	−0.271114
$141$	0	0
$142$	6.49255	0.544842
$143$	−0.825207	−0.0690073
$144$	0	0
$145$	2.88502	0.239588
$146$	13.0900	1.08334
$147$	0	0
$148$	−1.62721	−0.133756
$149$	−13.8186	−1.13206	−0.566030	−	0.824385i	$-0.691521\pi$
$149$	−13.8186	−1.13206	−0.566030	+	0.824385i	$0.691521\pi$
$150$	0	0
$151$	−5.43503	−0.442296	−0.221148	−	0.975240i	$-0.570980\pi$
$151$	−5.43503	−0.442296	−0.221148	+	0.975240i	$0.570980\pi$
$152$	−3.79020	−0.307425
$153$	0	0
$154$	0.770289	0.0620716
$155$	24.5342	1.97064
$156$	0	0
$157$	5.28204	0.421553	0.210776	−	0.977534i	$-0.432401\pi$
$157$	5.28204	0.421553	0.210776	+	0.977534i	$0.432401\pi$
$158$	12.0823	0.961219
$159$	0	0
$160$	−2.59435	−0.205101
$161$	0	0
$162$	0	0
$163$	9.09064	0.712034	0.356017	−	0.934480i	$-0.384135\pi$
$163$	9.09064	0.712034	0.356017	+	0.934480i	$0.384135\pi$
$164$	11.8497	0.925309
$165$	0	0
$166$	7.39068	0.573628
$167$	−2.87790	−0.222699	−0.111349	−	0.993781i	$-0.535517\pi$
$167$	−2.87790	−0.222699	−0.111349	+	0.993781i	$0.535517\pi$
$168$	0	0
$169$	−11.2453	−0.865026
$170$	−12.2132	−0.936707
$171$	0	0
$172$	−1.67879	−0.128006
$173$	−13.6627	−1.03876	−0.519378	−	0.854545i	$-0.673836\pi$
$173$	−13.6627	−1.03876	−0.519378	+	0.854545i	$0.673836\pi$
$174$	0	0
$175$	−2.13992	−0.161763
$176$	0.622970	0.0469581
$177$	0	0
$178$	3.38500	0.253716
$179$	20.3995	1.52473	0.762367	−	0.647145i	$-0.224037\pi$
$179$	20.3995	1.52473	0.762367	+	0.647145i	$0.224037\pi$
$180$	0	0
$181$	9.53656	0.708847	0.354423	−	0.935085i	$-0.384677\pi$
$181$	9.53656	0.708847	0.354423	+	0.935085i	$0.384677\pi$
$182$	−1.63788	−0.121408
$183$	0	0
$184$	0	0
$185$	−4.22157	−0.310376
$186$	0	0
$187$	2.93269	0.214460
$188$	6.94494	0.506512
$189$	0	0
$190$	−9.83310	−0.713368
$191$	−1.59304	−0.115268	−0.0576340	−	0.998338i	$-0.518356\pi$
$191$	−1.59304	−0.115268	−0.0576340	+	0.998338i	$0.518356\pi$
$192$	0	0
$193$	17.5781	1.26530	0.632651	−	0.774437i	$-0.281967\pi$
$193$	17.5781	1.26530	0.632651	+	0.774437i	$0.281967\pi$
$194$	−11.4458	−0.821759
$195$	0	0
$196$	−5.47112	−0.390794
$197$	−1.22161	−0.0870358	−0.0435179	−	0.999053i	$-0.513857\pi$
$197$	−1.22161	−0.0870358	−0.0435179	+	0.999053i	$0.513857\pi$
$198$	0	0
$199$	−0.100228	−0.00710495	−0.00355248	−	0.999994i	$-0.501131\pi$
$199$	−0.100228	−0.00710495	−0.00355248	+	0.999994i	$0.501131\pi$
$200$	−1.73066	−0.122376
$201$	0	0
$202$	−1.78374	−0.125503
$203$	−1.37501	−0.0965070
$204$	0	0
$205$	30.7424	2.14714
$206$	−19.8505	−1.38305
$207$	0	0
$208$	−1.32463	−0.0918469
$209$	2.36118	0.163326
$210$	0	0
$211$	−16.6971	−1.14948	−0.574738	−	0.818337i	$-0.694896\pi$
$211$	−16.6971	−1.14948	−0.574738	+	0.818337i	$0.694896\pi$
$212$	11.2537	0.772907
$213$	0	0
$214$	−4.35922	−0.297990
$215$	−4.35536	−0.297033
$216$	0	0
$217$	−11.6931	−0.793780
$218$	−6.46538	−0.437891
$219$	0	0
$220$	1.61620	0.108964
$221$	−6.23585	−0.419469
$222$	0	0
$223$	18.2081	1.21931	0.609654	−	0.792668i	$-0.291308\pi$
$223$	18.2081	1.21931	0.609654	+	0.792668i	$0.291308\pi$
$224$	1.23648	0.0826157
$225$	0	0
$226$	−1.36243	−0.0906274
$227$	−15.9853	−1.06098	−0.530490	−	0.847691i	$-0.677992\pi$
$227$	−15.9853	−1.06098	−0.530490	+	0.847691i	$0.677992\pi$
$228$	0	0
$229$	−17.7317	−1.17174	−0.585872	−	0.810404i	$-0.699248\pi$
$229$	−17.7317	−1.17174	−0.585872	+	0.810404i	$0.699248\pi$
$230$	0	0
$231$	0	0
$232$	−1.11204	−0.0730090
$233$	9.34447	0.612176	0.306088	−	0.952003i	$-0.400980\pi$
$233$	9.34447	0.612176	0.306088	+	0.952003i	$0.400980\pi$
$234$	0	0
$235$	18.0176	1.17534
$236$	9.74629	0.634429
$237$	0	0
$238$	5.82085	0.377309
$239$	5.00904	0.324008	0.162004	−	0.986790i	$-0.448204\pi$
$239$	5.00904	0.324008	0.162004	+	0.986790i	$0.448204\pi$
$240$	0	0
$241$	14.5761	0.938932	0.469466	−	0.882951i	$-0.344446\pi$
$241$	14.5761	0.938932	0.469466	+	0.882951i	$0.344446\pi$
$242$	10.6119	0.682159
$243$	0	0
$244$	−14.3736	−0.920173
$245$	−14.1940	−0.906822
$246$	0	0
$247$	−5.02062	−0.319455
$248$	−9.45679	−0.600506
$249$	0	0
$250$	8.48182	0.536437
$251$	18.7355	1.18258	0.591288	−	0.806460i	$-0.298620\pi$
$251$	18.7355	1.18258	0.591288	+	0.806460i	$0.298620\pi$
$252$	0	0
$253$	0	0
$254$	−3.45519	−0.216798
$255$	0	0
$256$	1.00000	0.0625000
$257$	1.74983	0.109151	0.0545756	−	0.998510i	$-0.482619\pi$
$257$	1.74983	0.109151	0.0545756	+	0.998510i	$0.482619\pi$
$258$	0	0
$259$	2.01202	0.125021
$260$	−3.43657	−0.213127
$261$	0	0
$262$	18.9474	1.17057
$263$	16.6550	1.02699	0.513494	−	0.858093i	$-0.328351\pi$
$263$	16.6550	1.02699	0.513494	+	0.858093i	$0.328351\pi$
$264$	0	0
$265$	29.1960	1.79350
$266$	4.68650	0.287348
$267$	0	0
$268$	−6.12125	−0.373915
$269$	26.7029	1.62810	0.814051	−	0.580793i	$-0.197257\pi$
$269$	26.7029	1.62810	0.814051	+	0.580793i	$0.197257\pi$
$270$	0	0
$271$	26.9079	1.63454	0.817269	−	0.576256i	$-0.195487\pi$
$271$	26.9079	1.63454	0.817269	+	0.576256i	$0.195487\pi$
$272$	4.70760	0.285440
$273$	0	0
$274$	−11.3960	−0.688460
$275$	1.07815	0.0650147
$276$	0	0
$277$	−2.13714	−0.128408	−0.0642042	−	0.997937i	$-0.520451\pi$
$277$	−2.13714	−0.128408	−0.0642042	+	0.997937i	$0.520451\pi$
$278$	−5.30267	−0.318033
$279$	0	0
$280$	3.20786	0.191706
$281$	27.7857	1.65756	0.828778	−	0.559578i	$-0.189037\pi$
$281$	27.7857	1.65756	0.828778	+	0.559578i	$0.189037\pi$
$282$	0	0
$283$	23.4814	1.39582	0.697911	−	0.716185i	$-0.254113\pi$
$283$	23.4814	1.39582	0.697911	+	0.716185i	$0.254113\pi$
$284$	−6.49255	−0.385262
$285$	0	0
$286$	0.825207	0.0487955
$287$	−14.6519	−0.864877
$288$	0	0
$289$	5.16149	0.303617
$290$	−2.88502	−0.169414
$291$	0	0
$292$	−13.0900	−0.766035
$293$	18.9804	1.10884	0.554422	−	0.832236i	$-0.312939\pi$
$293$	18.9804	1.10884	0.554422	+	0.832236i	$0.312939\pi$
$294$	0	0
$295$	25.2853	1.47217
$296$	1.62721	0.0945799
$297$	0	0
$298$	13.8186	0.800488
$299$	0	0
$300$	0	0
$301$	2.07578	0.119646
$302$	5.43503	0.312751
$303$	0	0
$304$	3.79020	0.217383
$305$	−37.2901	−2.13522
$306$	0	0
$307$	14.9115	0.851043	0.425522	−	0.904948i	$-0.360091\pi$
$307$	14.9115	0.851043	0.425522	+	0.904948i	$0.360091\pi$
$308$	−0.770289	−0.0438913
$309$	0	0
$310$	−24.5342	−1.39345
$311$	5.28838	0.299876	0.149938	−	0.988695i	$-0.452093\pi$
$311$	5.28838	0.299876	0.149938	+	0.988695i	$0.452093\pi$
$312$	0	0
$313$	−8.87040	−0.501385	−0.250692	−	0.968067i	$-0.580658\pi$
$313$	−8.87040	−0.501385	−0.250692	+	0.968067i	$0.580658\pi$
$314$	−5.28204	−0.298083
$315$	0	0
$316$	−12.0823	−0.679685
$317$	−3.03971	−0.170727	−0.0853636	−	0.996350i	$-0.527205\pi$
$317$	−3.03971	−0.170727	−0.0853636	+	0.996350i	$0.527205\pi$
$318$	0	0
$319$	0.692767	0.0387875
$320$	2.59435	0.145029
$321$	0	0
$322$	0	0
$323$	17.8427	0.992795
$324$	0	0
$325$	−2.29249	−0.127164
$326$	−9.09064	−0.503484
$327$	0	0
$328$	−11.8497	−0.654292
$329$	−8.58727	−0.473432
$330$	0	0
$331$	1.64126	0.0902119	0.0451059	−	0.998982i	$-0.485637\pi$
$331$	1.64126	0.0902119	0.0451059	+	0.998982i	$0.485637\pi$
$332$	−7.39068	−0.405617
$333$	0	0
$334$	2.87790	0.157472
$335$	−15.8807	−0.867654
$336$	0	0
$337$	17.9706	0.978921	0.489461	−	0.872025i	$-0.337194\pi$
$337$	17.9706	0.978921	0.489461	+	0.872025i	$0.337194\pi$
$338$	11.2453	0.611666
$339$	0	0
$340$	12.2132	0.662352
$341$	5.89129	0.319031
$342$	0	0
$343$	15.4203	0.832617
$344$	1.67879	0.0905140
$345$	0	0
$346$	13.6627	0.734511
$347$	−0.223203	−0.0119822	−0.00599109	−	0.999982i	$-0.501907\pi$
$347$	−0.223203	−0.0119822	−0.00599109	+	0.999982i	$0.501907\pi$
$348$	0	0
$349$	−20.2398	−1.08341	−0.541705	−	0.840568i	$-0.682221\pi$
$349$	−20.2398	−1.08341	−0.541705	+	0.840568i	$0.682221\pi$
$350$	2.13992	0.114384
$351$	0	0
$352$	−0.622970	−0.0332044
$353$	−19.0138	−1.01200	−0.506001	−	0.862533i	$-0.668877\pi$
$353$	−19.0138	−1.01200	−0.506001	+	0.862533i	$0.668877\pi$
$354$	0	0
$355$	−16.8439	−0.893984
$356$	−3.38500	−0.179404
$357$	0	0
$358$	−20.3995	−1.07815
$359$	−5.13597	−0.271066	−0.135533	−	0.990773i	$-0.543275\pi$
$359$	−5.13597	−0.271066	−0.135533	+	0.990773i	$0.543275\pi$
$360$	0	0
$361$	−4.63442	−0.243917
$362$	−9.53656	−0.501231
$363$	0	0
$364$	1.63788	0.0858484
$365$	−33.9601	−1.77755
$366$	0	0
$367$	24.7772	1.29336	0.646679	−	0.762762i	$-0.276157\pi$
$367$	24.7772	1.29336	0.646679	+	0.762762i	$0.276157\pi$
$368$	0	0
$369$	0	0
$370$	4.22157	0.219469
$371$	−13.9150	−0.722428
$372$	0	0
$373$	37.1140	1.92169	0.960844	−	0.277090i	$-0.0893701\pi$
$373$	37.1140	1.92169	0.960844	+	0.277090i	$0.0893701\pi$
$374$	−2.93269	−0.151646
$375$	0	0
$376$	−6.94494	−0.358158
$377$	−1.47305	−0.0758657
$378$	0	0
$379$	−31.1816	−1.60169	−0.800846	−	0.598870i	$-0.795617\pi$
$379$	−31.1816	−1.60169	−0.800846	+	0.598870i	$0.795617\pi$
$380$	9.83310	0.504427
$381$	0	0
$382$	1.59304	0.0815068
$383$	11.5562	0.590494	0.295247	−	0.955421i	$-0.404598\pi$
$383$	11.5562	0.590494	0.295247	+	0.955421i	$0.404598\pi$
$384$	0	0
$385$	−1.99840	−0.101848
$386$	−17.5781	−0.894704
$387$	0	0
$388$	11.4458	0.581072
$389$	−23.8039	−1.20691	−0.603453	−	0.797399i	$-0.706209\pi$
$389$	−23.8039	−1.20691	−0.603453	+	0.797399i	$0.706209\pi$
$390$	0	0
$391$	0	0
$392$	5.47112	0.276333
$393$	0	0
$394$	1.22161	0.0615436
$395$	−31.3458	−1.57718
$396$	0	0
$397$	−22.1253	−1.11044	−0.555220	−	0.831704i	$-0.687366\pi$
$397$	−22.1253	−1.11044	−0.555220	+	0.831704i	$0.687366\pi$
$398$	0.100228	0.00502396
$399$	0	0
$400$	1.73066	0.0865329
$401$	10.6876	0.533714	0.266857	−	0.963736i	$-0.414015\pi$
$401$	10.6876	0.533714	0.266857	+	0.963736i	$0.414015\pi$
$402$	0	0
$403$	−12.5268	−0.624004
$404$	1.78374	0.0887443
$405$	0	0
$406$	1.37501	0.0682408
$407$	−1.01371	−0.0502475
$408$	0	0
$409$	−15.8801	−0.785219	−0.392610	−	0.919705i	$-0.628428\pi$
$409$	−15.8801	−0.785219	−0.392610	+	0.919705i	$0.628428\pi$
$410$	−30.7424	−1.51826
$411$	0	0
$412$	19.8505	0.977963
$413$	−12.0511	−0.592995
$414$	0	0
$415$	−19.1740	−0.941216
$416$	1.32463	0.0649456
$417$	0	0
$418$	−2.36118	−0.115489
$419$	4.63014	0.226197	0.113099	−	0.993584i	$-0.463922\pi$
$419$	4.63014	0.226197	0.113099	+	0.993584i	$0.463922\pi$
$420$	0	0
$421$	−10.5088	−0.512170	−0.256085	−	0.966654i	$-0.582433\pi$
$421$	−10.5088	−0.512170	−0.256085	+	0.966654i	$0.582433\pi$
$422$	16.6971	0.812803
$423$	0	0
$424$	−11.2537	−0.546528
$425$	8.14724	0.395199
$426$	0	0
$427$	17.7726	0.860077
$428$	4.35922	0.210711
$429$	0	0
$430$	4.35536	0.210034
$431$	−13.6284	−0.656458	−0.328229	−	0.944598i	$-0.606452\pi$
$431$	−13.6284	−0.656458	−0.328229	+	0.944598i	$0.606452\pi$
$432$	0	0
$433$	−8.61044	−0.413791	−0.206896	−	0.978363i	$-0.566336\pi$
$433$	−8.61044	−0.413791	−0.206896	+	0.978363i	$0.566336\pi$
$434$	11.6931	0.561287
$435$	0	0
$436$	6.46538	0.309636
$437$	0	0
$438$	0	0
$439$	−37.0323	−1.76745	−0.883727	−	0.468003i	$-0.844974\pi$
$439$	−37.0323	−1.76745	−0.883727	+	0.468003i	$0.844974\pi$
$440$	−1.61620	−0.0770494
$441$	0	0
$442$	6.23585	0.296609
$443$	−14.4007	−0.684196	−0.342098	−	0.939664i	$-0.611137\pi$
$443$	−14.4007	−0.684196	−0.342098	+	0.939664i	$0.611137\pi$
$444$	0	0
$445$	−8.78187	−0.416301
$446$	−18.2081	−0.862181
$447$	0	0
$448$	−1.23648	−0.0584181
$449$	−14.7629	−0.696706	−0.348353	−	0.937363i	$-0.613259\pi$
$449$	−14.7629	−0.696706	−0.348353	+	0.937363i	$0.613259\pi$
$450$	0	0
$451$	7.38202	0.347606
$452$	1.36243	0.0640832
$453$	0	0
$454$	15.9853	0.750226
$455$	4.24924	0.199208
$456$	0	0
$457$	6.62444	0.309878	0.154939	−	0.987924i	$-0.450482\pi$
$457$	6.62444	0.309878	0.154939	+	0.987924i	$0.450482\pi$
$458$	17.7317	0.828548
$459$	0	0
$460$	0	0
$461$	−7.35627	−0.342616	−0.171308	−	0.985218i	$-0.554799\pi$
$461$	−7.35627	−0.342616	−0.171308	+	0.985218i	$0.554799\pi$
$462$	0	0
$463$	34.6358	1.60966	0.804832	−	0.593503i	$-0.202256\pi$
$463$	34.6358	1.60966	0.804832	+	0.593503i	$0.202256\pi$
$464$	1.11204	0.0516251
$465$	0	0
$466$	−9.34447	−0.432874
$467$	14.1030	0.652608	0.326304	−	0.945265i	$-0.394197\pi$
$467$	14.1030	0.652608	0.326304	+	0.945265i	$0.394197\pi$
$468$	0	0
$469$	7.56880	0.349495
$470$	−18.0176	−0.831091
$471$	0	0
$472$	−9.74629	−0.448609
$473$	−1.04583	−0.0480874
$474$	0	0
$475$	6.55953	0.300972
$476$	−5.82085	−0.266798
$477$	0	0
$478$	−5.00904	−0.229108
$479$	17.0476	0.778924	0.389462	−	0.921043i	$-0.372661\pi$
$479$	17.0476	0.778924	0.389462	+	0.921043i	$0.372661\pi$
$480$	0	0
$481$	2.15546	0.0982807
$482$	−14.5761	−0.663925
$483$	0	0
$484$	−10.6119	−0.482359
$485$	29.6944	1.34835
$486$	0	0
$487$	−12.3279	−0.558629	−0.279314	−	0.960200i	$-0.590107\pi$
$487$	−12.3279	−0.558629	−0.279314	+	0.960200i	$0.590107\pi$
$488$	14.3736	0.650660
$489$	0	0
$490$	14.1940	0.641220
$491$	6.49393	0.293067	0.146534	−	0.989206i	$-0.453188\pi$
$491$	6.49393	0.293067	0.146534	+	0.989206i	$0.453188\pi$
$492$	0	0
$493$	5.23504	0.235774
$494$	5.02062	0.225889
$495$	0	0
$496$	9.45679	0.424622
$497$	8.02790	0.360100
$498$	0	0
$499$	−13.6005	−0.608841	−0.304420	−	0.952538i	$-0.598463\pi$
$499$	−13.6005	−0.608841	−0.304420	+	0.952538i	$0.598463\pi$
$500$	−8.48182	−0.379319
$501$	0	0
$502$	−18.7355	−0.836208
$503$	−3.01244	−0.134318	−0.0671590	−	0.997742i	$-0.521393\pi$
$503$	−3.01244	−0.134318	−0.0671590	+	0.997742i	$0.521393\pi$
$504$	0	0
$505$	4.62764	0.205927
$506$	0	0
$507$	0	0
$508$	3.45519	0.153299
$509$	−43.0295	−1.90725	−0.953625	−	0.300996i	$-0.902681\pi$
$509$	−43.0295	−1.90725	−0.953625	+	0.300996i	$0.902681\pi$
$510$	0	0
$511$	16.1855	0.716005
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−1.74983	−0.0771815
$515$	51.4991	2.26932
$516$	0	0
$517$	4.32649	0.190279
$518$	−2.01202	−0.0884029
$519$	0	0
$520$	3.43657	0.150703
$521$	15.1561	0.664003	0.332001	−	0.943279i	$-0.392276\pi$
$521$	15.1561	0.664003	0.332001	+	0.943279i	$0.392276\pi$
$522$	0	0
$523$	15.6900	0.686075	0.343037	−	0.939322i	$-0.388544\pi$
$523$	15.6900	0.686075	0.343037	+	0.939322i	$0.388544\pi$
$524$	−18.9474	−0.827721
$525$	0	0
$526$	−16.6550	−0.726191
$527$	44.5188	1.93927
$528$	0	0
$529$	0	0
$530$	−29.1960	−1.26819
$531$	0	0
$532$	−4.68650	−0.203185
$533$	−15.6966	−0.679894
$534$	0	0
$535$	11.3093	0.488946
$536$	6.12125	0.264398
$537$	0	0
$538$	−26.7029	−1.15124
$539$	−3.40834	−0.146808
$540$	0	0
$541$	−21.7189	−0.933769	−0.466884	−	0.884318i	$-0.654624\pi$
$541$	−21.7189	−0.933769	−0.466884	+	0.884318i	$0.654624\pi$
$542$	−26.9079	−1.15579
$543$	0	0
$544$	−4.70760	−0.201837
$545$	16.7735	0.718496
$546$	0	0
$547$	12.2110	0.522105	0.261053	−	0.965325i	$-0.415930\pi$
$547$	12.2110	0.522105	0.261053	+	0.965325i	$0.415930\pi$
$548$	11.3960	0.486815
$549$	0	0
$550$	−1.07815	−0.0459724
$551$	4.21485	0.179559
$552$	0	0
$553$	14.9396	0.635295
$554$	2.13714	0.0907984
$555$	0	0
$556$	5.30267	0.224883
$557$	−16.2652	−0.689180	−0.344590	−	0.938753i	$-0.611982\pi$
$557$	−16.2652	−0.689180	−0.344590	+	0.938753i	$0.611982\pi$
$558$	0	0
$559$	2.22378	0.0940558
$560$	−3.20786	−0.135557
$561$	0	0
$562$	−27.7857	−1.17207
$563$	−31.6637	−1.33447	−0.667233	−	0.744849i	$-0.732522\pi$
$563$	−31.6637	−1.33447	−0.667233	+	0.744849i	$0.732522\pi$
$564$	0	0
$565$	3.53462	0.148702
$566$	−23.4814	−0.986995
$567$	0	0
$568$	6.49255	0.272421
$569$	−36.8598	−1.54524	−0.772621	−	0.634868i	$-0.781054\pi$
$569$	−36.8598	−1.54524	−0.772621	+	0.634868i	$0.781054\pi$
$570$	0	0
$571$	30.8533	1.29117	0.645586	−	0.763688i	$-0.276613\pi$
$571$	30.8533	1.29117	0.645586	+	0.763688i	$0.276613\pi$
$572$	−0.825207	−0.0345036
$573$	0	0
$574$	14.6519	0.611560
$575$	0	0
$576$	0	0
$577$	−45.6832	−1.90182	−0.950908	−	0.309474i	$-0.899847\pi$
$577$	−45.6832	−1.90182	−0.950908	+	0.309474i	$0.899847\pi$
$578$	−5.16149	−0.214690
$579$	0	0
$580$	2.88502	0.119794
$581$	9.13843	0.379126
$582$	0	0
$583$	7.01071	0.290354
$584$	13.0900	0.541668
$585$	0	0
$586$	−18.9804	−0.784071
$587$	30.4171	1.25545	0.627723	−	0.778437i	$-0.283987\pi$
$587$	30.4171	1.25545	0.627723	+	0.778437i	$0.283987\pi$
$588$	0	0
$589$	35.8431	1.47689
$590$	−25.2853	−1.04098
$591$	0	0
$592$	−1.62721	−0.0668781
$593$	27.8211	1.14248	0.571238	−	0.820784i	$-0.306463\pi$
$593$	27.8211	1.14248	0.571238	+	0.820784i	$0.306463\pi$
$594$	0	0
$595$	−15.1013	−0.619094
$596$	−13.8186	−0.566030
$597$	0	0
$598$	0	0
$599$	11.1914	0.457270	0.228635	−	0.973512i	$-0.426574\pi$
$599$	11.1914	0.457270	0.228635	+	0.973512i	$0.426574\pi$
$600$	0	0
$601$	−41.0283	−1.67358	−0.836789	−	0.547525i	$-0.815570\pi$
$601$	−41.0283	−1.67358	−0.836789	+	0.547525i	$0.815570\pi$
$602$	−2.07578	−0.0846026
$603$	0	0
$604$	−5.43503	−0.221148
$605$	−27.5310	−1.11930
$606$	0	0
$607$	36.6692	1.48836	0.744179	−	0.667981i	$-0.232841\pi$
$607$	36.6692	1.48836	0.744179	+	0.667981i	$0.232841\pi$
$608$	−3.79020	−0.153713
$609$	0	0
$610$	37.2901	1.50983
$611$	−9.19951	−0.372172
$612$	0	0
$613$	−0.233905	−0.00944733	−0.00472367	−	0.999989i	$-0.501504\pi$
$613$	−0.233905	−0.00944733	−0.00472367	+	0.999989i	$0.501504\pi$
$614$	−14.9115	−0.601778
$615$	0	0
$616$	0.770289	0.0310358
$617$	43.6491	1.75725	0.878623	−	0.477517i	$-0.158463\pi$
$617$	43.6491	1.75725	0.878623	+	0.477517i	$0.158463\pi$
$618$	0	0
$619$	−1.18895	−0.0477881	−0.0238941	−	0.999714i	$-0.507606\pi$
$619$	−1.18895	−0.0477881	−0.0238941	+	0.999714i	$0.507606\pi$
$620$	24.5342	0.985318
$621$	0	0
$622$	−5.28838	−0.212045
$623$	4.18548	0.167688
$624$	0	0
$625$	−30.6581	−1.22632
$626$	8.87040	0.354533
$627$	0	0
$628$	5.28204	0.210776
$629$	−7.66027	−0.305435
$630$	0	0
$631$	16.6637	0.663371	0.331686	−	0.943390i	$-0.392383\pi$
$631$	16.6637	0.663371	0.331686	+	0.943390i	$0.392383\pi$
$632$	12.0823	0.480610
$633$	0	0
$634$	3.03971	0.120722
$635$	8.96397	0.355724
$636$	0	0
$637$	7.24724	0.287146
$638$	−0.692767	−0.0274269
$639$	0	0
$640$	−2.59435	−0.102551
$641$	29.8418	1.17868	0.589340	−	0.807885i	$-0.299388\pi$
$641$	29.8418	1.17868	0.589340	+	0.807885i	$0.299388\pi$
$642$	0	0
$643$	−36.9937	−1.45889	−0.729445	−	0.684040i	$-0.760221\pi$
$643$	−36.9937	−1.45889	−0.729445	+	0.684040i	$0.760221\pi$
$644$	0	0
$645$	0	0
$646$	−17.8427	−0.702012
$647$	−30.8934	−1.21454	−0.607272	−	0.794494i	$-0.707736\pi$
$647$	−30.8934	−1.21454	−0.607272	+	0.794494i	$0.707736\pi$
$648$	0	0
$649$	6.07164	0.238333
$650$	2.29249	0.0899188
$651$	0	0
$652$	9.09064	0.356017
$653$	25.9281	1.01465	0.507323	−	0.861756i	$-0.330635\pi$
$653$	25.9281	1.01465	0.507323	+	0.861756i	$0.330635\pi$
$654$	0	0
$655$	−49.1562	−1.92069
$656$	11.8497	0.462654
$657$	0	0
$658$	8.58727	0.334767
$659$	20.3688	0.793455	0.396727	−	0.917937i	$-0.370146\pi$
$659$	20.3688	0.793455	0.396727	+	0.917937i	$0.370146\pi$
$660$	0	0
$661$	42.0770	1.63661	0.818303	−	0.574787i	$-0.194915\pi$
$661$	42.0770	1.63661	0.818303	+	0.574787i	$0.194915\pi$
$662$	−1.64126	−0.0637894
$663$	0	0
$664$	7.39068	0.286814
$665$	−12.1584	−0.471483
$666$	0	0
$667$	0	0
$668$	−2.87790	−0.111349
$669$	0	0
$670$	15.8807	0.613524
$671$	−8.95429	−0.345677
$672$	0	0
$673$	20.3437	0.784192	0.392096	−	0.919924i	$-0.371750\pi$
$673$	20.3437	0.784192	0.392096	+	0.919924i	$0.371750\pi$
$674$	−17.9706	−0.692202
$675$	0	0
$676$	−11.2453	−0.432513
$677$	−11.0440	−0.424456	−0.212228	−	0.977220i	$-0.568072\pi$
$677$	−11.0440	−0.424456	−0.212228	+	0.977220i	$0.568072\pi$
$678$	0	0
$679$	−14.1525	−0.543122
$680$	−12.2132	−0.468353
$681$	0	0
$682$	−5.89129	−0.225589
$683$	10.7474	0.411239	0.205619	−	0.978632i	$-0.434079\pi$
$683$	10.7474	0.411239	0.205619	+	0.978632i	$0.434079\pi$
$684$	0	0
$685$	29.5653	1.12963
$686$	−15.4203	−0.588749
$687$	0	0
$688$	−1.67879	−0.0640031
$689$	−14.9070	−0.567913
$690$	0	0
$691$	−21.3444	−0.811980	−0.405990	−	0.913878i	$-0.633073\pi$
$691$	−21.3444	−0.811980	−0.405990	+	0.913878i	$0.633073\pi$
$692$	−13.6627	−0.519378
$693$	0	0
$694$	0.223203	0.00847269
$695$	13.7570	0.521832
$696$	0	0
$697$	55.7838	2.11296
$698$	20.2398	0.766087
$699$	0	0
$700$	−2.13992	−0.0808814
$701$	32.9410	1.24416	0.622082	−	0.782952i	$-0.286287\pi$
$701$	32.9410	1.24416	0.622082	+	0.782952i	$0.286287\pi$
$702$	0	0
$703$	−6.16746	−0.232610
$704$	0.622970	0.0234791
$705$	0	0
$706$	19.0138	0.715593
$707$	−2.20556	−0.0829484
$708$	0	0
$709$	−41.4933	−1.55832	−0.779158	−	0.626828i	$-0.784353\pi$
$709$	−41.4933	−1.55832	−0.779158	+	0.626828i	$0.784353\pi$
$710$	16.8439	0.632142
$711$	0	0
$712$	3.38500	0.126858
$713$	0	0
$714$	0	0
$715$	−2.14088	−0.0800643
$716$	20.3995	0.762367
$717$	0	0
$718$	5.13597	0.191673
$719$	28.0271	1.04523	0.522617	−	0.852568i	$-0.324956\pi$
$719$	28.0271	1.04523	0.522617	+	0.852568i	$0.324956\pi$
$720$	0	0
$721$	−24.5447	−0.914093
$722$	4.63442	0.172475
$723$	0	0
$724$	9.53656	0.354423
$725$	1.92456	0.0714764
$726$	0	0
$727$	9.92976	0.368274	0.184137	−	0.982901i	$-0.441051\pi$
$727$	9.92976	0.368274	0.184137	+	0.982901i	$0.441051\pi$
$728$	−1.63788	−0.0607040
$729$	0	0
$730$	33.9601	1.25692
$731$	−7.90305	−0.292305
$732$	0	0
$733$	17.8060	0.657678	0.328839	−	0.944386i	$-0.393343\pi$
$733$	17.8060	0.657678	0.328839	+	0.944386i	$0.393343\pi$
$734$	−24.7772	−0.914543
$735$	0	0
$736$	0	0
$737$	−3.81336	−0.140467
$738$	0	0
$739$	21.7628	0.800557	0.400279	−	0.916393i	$-0.368913\pi$
$739$	21.7628	0.800557	0.400279	+	0.916393i	$0.368913\pi$
$740$	−4.22157	−0.155188
$741$	0	0
$742$	13.9150	0.510834
$743$	16.5987	0.608947	0.304474	−	0.952521i	$-0.401519\pi$
$743$	16.5987	0.608947	0.304474	+	0.952521i	$0.401519\pi$
$744$	0	0
$745$	−35.8502	−1.31345
$746$	−37.1140	−1.35884
$747$	0	0
$748$	2.93269	0.107230
$749$	−5.39008	−0.196949
$750$	0	0
$751$	−4.09051	−0.149265	−0.0746324	−	0.997211i	$-0.523778\pi$
$751$	−4.09051	−0.149265	−0.0746324	+	0.997211i	$0.523778\pi$
$752$	6.94494	0.253256
$753$	0	0
$754$	1.47305	0.0536452
$755$	−14.1004	−0.513165
$756$	0	0
$757$	9.97094	0.362400	0.181200	−	0.983446i	$-0.442002\pi$
$757$	9.97094	0.362400	0.181200	+	0.983446i	$0.442002\pi$
$758$	31.1816	1.13257
$759$	0	0
$760$	−9.83310	−0.356684
$761$	−46.0080	−1.66779	−0.833894	−	0.551925i	$-0.813893\pi$
$761$	−46.0080	−1.66779	−0.833894	+	0.551925i	$0.813893\pi$
$762$	0	0
$763$	−7.99431	−0.289413
$764$	−1.59304	−0.0576340
$765$	0	0
$766$	−11.5562	−0.417542
$767$	−12.9103	−0.466163
$768$	0	0
$769$	27.4543	0.990027	0.495013	−	0.868885i	$-0.335163\pi$
$769$	27.4543	0.990027	0.495013	+	0.868885i	$0.335163\pi$
$770$	1.99840	0.0720173
$771$	0	0
$772$	17.5781	0.632651
$773$	−9.07298	−0.326333	−0.163166	−	0.986599i	$-0.552171\pi$
$773$	−9.07298	−0.326333	−0.163166	+	0.986599i	$0.552171\pi$
$774$	0	0
$775$	16.3665	0.587901
$776$	−11.4458	−0.410880
$777$	0	0
$778$	23.8039	0.853411
$779$	44.9128	1.60917
$780$	0	0
$781$	−4.04466	−0.144729
$782$	0	0
$783$	0	0
$784$	−5.47112	−0.195397
$785$	13.7035	0.489098
$786$	0	0
$787$	−52.0178	−1.85424	−0.927118	−	0.374770i	$-0.877722\pi$
$787$	−52.0178	−1.85424	−0.927118	+	0.374770i	$0.877722\pi$
$788$	−1.22161	−0.0435179
$789$	0	0
$790$	31.3458	1.11523
$791$	−1.68461	−0.0598980
$792$	0	0
$793$	19.0397	0.676120
$794$	22.1253	0.785199
$795$	0	0
$796$	−0.100228	−0.00355248
$797$	−26.5148	−0.939202	−0.469601	−	0.882879i	$-0.655602\pi$
$797$	−26.5148	−0.939202	−0.469601	+	0.882879i	$0.655602\pi$
$798$	0	0
$799$	32.6940	1.15663
$800$	−1.73066	−0.0611880
$801$	0	0
$802$	−10.6876	−0.377393
$803$	−8.15468	−0.287772
$804$	0	0
$805$	0	0
$806$	12.5268	0.441237
$807$	0	0
$808$	−1.78374	−0.0627517
$809$	21.0758	0.740987	0.370493	−	0.928835i	$-0.379189\pi$
$809$	21.0758	0.740987	0.370493	+	0.928835i	$0.379189\pi$
$810$	0	0
$811$	−35.3708	−1.24204	−0.621018	−	0.783796i	$-0.713281\pi$
$811$	−35.3708	−1.24204	−0.621018	+	0.783796i	$0.713281\pi$
$812$	−1.37501	−0.0482535
$813$	0	0
$814$	1.01371	0.0355303
$815$	23.5843	0.826122
$816$	0	0
$817$	−6.36292	−0.222611
$818$	15.8801	0.555234
$819$	0	0
$820$	30.7424	1.07357
$821$	49.2971	1.72048	0.860241	−	0.509888i	$-0.170313\pi$
$821$	49.2971	1.72048	0.860241	+	0.509888i	$0.170313\pi$
$822$	0	0
$823$	−12.3852	−0.431720	−0.215860	−	0.976424i	$-0.569255\pi$
$823$	−12.3852	−0.431720	−0.215860	+	0.976424i	$0.569255\pi$
$824$	−19.8505	−0.691525
$825$	0	0
$826$	12.0511	0.419311
$827$	−30.0768	−1.04587	−0.522936	−	0.852372i	$-0.675163\pi$
$827$	−30.0768	−1.04587	−0.522936	+	0.852372i	$0.675163\pi$
$828$	0	0
$829$	48.4553	1.68292	0.841462	−	0.540317i	$-0.181696\pi$
$829$	48.4553	1.68292	0.841462	+	0.540317i	$0.181696\pi$
$830$	19.1740	0.665540
$831$	0	0
$832$	−1.32463	−0.0459234
$833$	−25.7558	−0.892387
$834$	0	0
$835$	−7.46629	−0.258381
$836$	2.36118	0.0816630
$837$	0	0
$838$	−4.63014	−0.159946
$839$	45.3388	1.56527	0.782635	−	0.622481i	$-0.213875\pi$
$839$	45.3388	1.56527	0.782635	+	0.622481i	$0.213875\pi$
$840$	0	0
$841$	−27.7634	−0.957358
$842$	10.5088	0.362159
$843$	0	0
$844$	−16.6971	−0.574738
$845$	−29.1744	−1.00363
$846$	0	0
$847$	13.1214	0.450857
$848$	11.2537	0.386453
$849$	0	0
$850$	−8.14724	−0.279448
$851$	0	0
$852$	0	0
$853$	−14.5044	−0.496621	−0.248310	−	0.968681i	$-0.579875\pi$
$853$	−14.5044	−0.496621	−0.248310	+	0.968681i	$0.579875\pi$
$854$	−17.7726	−0.608166
$855$	0	0
$856$	−4.35922	−0.148995
$857$	−49.9110	−1.70493	−0.852464	−	0.522785i	$-0.824893\pi$
$857$	−49.9110	−1.70493	−0.852464	+	0.522785i	$0.824893\pi$
$858$	0	0
$859$	55.0208	1.87729	0.938643	−	0.344890i	$-0.112084\pi$
$859$	55.0208	1.87729	0.938643	+	0.344890i	$0.112084\pi$
$860$	−4.35536	−0.148517
$861$	0	0
$862$	13.6284	0.464186
$863$	−39.8912	−1.35791	−0.678955	−	0.734180i	$-0.737567\pi$
$863$	−39.8912	−1.35791	−0.678955	+	0.734180i	$0.737567\pi$
$864$	0	0
$865$	−35.4458	−1.20519
$866$	8.61044	0.292595
$867$	0	0
$868$	−11.6931	−0.396890
$869$	−7.52693	−0.255334
$870$	0	0
$871$	8.10843	0.274744
$872$	−6.46538	−0.218945
$873$	0	0
$874$	0	0
$875$	10.4876	0.354545
$876$	0	0
$877$	3.81653	0.128875	0.0644375	−	0.997922i	$-0.479475\pi$
$877$	3.81653	0.128875	0.0644375	+	0.997922i	$0.479475\pi$
$878$	37.0323	1.24978
$879$	0	0
$880$	1.61620	0.0544822
$881$	−20.7016	−0.697455	−0.348727	−	0.937224i	$-0.613386\pi$
$881$	−20.7016	−0.697455	−0.348727	+	0.937224i	$0.613386\pi$
$882$	0	0
$883$	18.1201	0.609789	0.304894	−	0.952386i	$-0.401379\pi$
$883$	18.1201	0.609789	0.304894	+	0.952386i	$0.401379\pi$
$884$	−6.23585	−0.209734
$885$	0	0
$886$	14.4007	0.483800
$887$	28.1843	0.946336	0.473168	−	0.880972i	$-0.343110\pi$
$887$	28.1843	0.946336	0.473168	+	0.880972i	$0.343110\pi$
$888$	0	0
$889$	−4.27227	−0.143287
$890$	8.78187	0.294369
$891$	0	0
$892$	18.2081	0.609654
$893$	26.3227	0.880855
$894$	0	0
$895$	52.9236	1.76904
$896$	1.23648	0.0413079
$897$	0	0
$898$	14.7629	0.492646
$899$	10.5163	0.350739
$900$	0	0
$901$	52.9779	1.76495
$902$	−7.38202	−0.245794
$903$	0	0
$904$	−1.36243	−0.0453137
$905$	24.7412	0.822425
$906$	0	0
$907$	24.8123	0.823880	0.411940	−	0.911211i	$-0.364851\pi$
$907$	24.8123	0.823880	0.411940	+	0.911211i	$0.364851\pi$
$908$	−15.9853	−0.530490
$909$	0	0
$910$	−4.24924	−0.140861
$911$	37.6655	1.24791	0.623957	−	0.781459i	$-0.285524\pi$
$911$	37.6655	1.24791	0.623957	+	0.781459i	$0.285524\pi$
$912$	0	0
$913$	−4.60417	−0.152376
$914$	−6.62444	−0.219117
$915$	0	0
$916$	−17.7317	−0.585872
$917$	23.4281	0.773663
$918$	0	0
$919$	51.5748	1.70129	0.850647	−	0.525737i	$-0.176210\pi$
$919$	51.5748	1.70129	0.850647	+	0.525737i	$0.176210\pi$
$920$	0	0
$921$	0	0
$922$	7.35627	0.242266
$923$	8.60025	0.283081
$924$	0	0
$925$	−2.81615	−0.0925945
$926$	−34.6358	−1.13820
$927$	0	0
$928$	−1.11204	−0.0365045
$929$	13.5989	0.446165	0.223082	−	0.974800i	$-0.428388\pi$
$929$	13.5989	0.446165	0.223082	+	0.974800i	$0.428388\pi$
$930$	0	0
$931$	−20.7366	−0.679615
$932$	9.34447	0.306088
$933$	0	0
$934$	−14.1030	−0.461463
$935$	7.60843	0.248822
$936$	0	0
$937$	−10.4764	−0.342248	−0.171124	−	0.985250i	$-0.554740\pi$
$937$	−10.4764	−0.342248	−0.171124	+	0.985250i	$0.554740\pi$
$938$	−7.56880	−0.247130
$939$	0	0
$940$	18.0176	0.587670
$941$	−14.3525	−0.467877	−0.233938	−	0.972251i	$-0.575161\pi$
$941$	−14.3525	−0.467877	−0.233938	+	0.972251i	$0.575161\pi$
$942$	0	0
$943$	0	0
$944$	9.74629	0.317215
$945$	0	0
$946$	1.04583	0.0340029
$947$	2.21149	0.0718639	0.0359319	−	0.999354i	$-0.488560\pi$
$947$	2.21149	0.0718639	0.0359319	+	0.999354i	$0.488560\pi$
$948$	0	0
$949$	17.3395	0.562863
$950$	−6.55953	−0.212819
$951$	0	0
$952$	5.82085	0.188655
$953$	−22.9994	−0.745023	−0.372511	−	0.928028i	$-0.621503\pi$
$953$	−22.9994	−0.745023	−0.372511	+	0.928028i	$0.621503\pi$
$954$	0	0
$955$	−4.13290	−0.133737
$956$	5.00904	0.162004
$957$	0	0
$958$	−17.0476	−0.550782
$959$	−14.0910	−0.455021
$960$	0	0
$961$	58.4308	1.88486
$962$	−2.15546	−0.0694950
$963$	0	0
$964$	14.5761	0.469466
$965$	45.6039	1.46804
$966$	0	0
$967$	−8.31090	−0.267261	−0.133630	−	0.991031i	$-0.542663\pi$
$967$	−8.31090	−0.267261	−0.133630	+	0.991031i	$0.542663\pi$
$968$	10.6119	0.341080
$969$	0	0
$970$	−29.6944	−0.953429
$971$	−14.9920	−0.481115	−0.240557	−	0.970635i	$-0.577330\pi$
$971$	−14.9920	−0.481115	−0.240557	+	0.970635i	$0.577330\pi$
$972$	0	0
$973$	−6.55664	−0.210196
$974$	12.3279	0.395010
$975$	0	0
$976$	−14.3736	−0.460086
$977$	52.3758	1.67565	0.837824	−	0.545940i	$-0.183827\pi$
$977$	52.3758	1.67565	0.837824	+	0.545940i	$0.183827\pi$
$978$	0	0
$979$	−2.10875	−0.0673959
$980$	−14.1940	−0.453411
$981$	0	0
$982$	−6.49393	−0.207230
$983$	30.1510	0.961666	0.480833	−	0.876812i	$-0.340334\pi$
$983$	30.1510	0.961666	0.480833	+	0.876812i	$0.340334\pi$
$984$	0	0
$985$	−3.16927	−0.100981
$986$	−5.23504	−0.166718
$987$	0	0
$988$	−5.02062	−0.159727
$989$	0	0
$990$	0	0
$991$	2.70097	0.0857992	0.0428996	−	0.999079i	$-0.486340\pi$
$991$	2.70097	0.0857992	0.0428996	+	0.999079i	$0.486340\pi$
$992$	−9.45679	−0.300253
$993$	0	0
$994$	−8.02790	−0.254629
$995$	−0.260026	−0.00824337
$996$	0	0
$997$	9.07176	0.287305	0.143653	−	0.989628i	$-0.454115\pi$
$997$	9.07176	0.287305	0.143653	+	0.989628i	$0.454115\pi$
$998$	13.6005	0.430515
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9522.2.a.bt.1.4		5
3.2	odd	2		3174.2.a.bc.1.2		5
23.13	even	11		414.2.i.d.307.1		10
23.16	even	11		414.2.i.d.325.1		10
23.22	odd	2		9522.2.a.bq.1.2		5
69.59	odd	22		138.2.e.a.31.1	✓	10
69.62	odd	22		138.2.e.a.49.1	yes	10
69.68	even	2		3174.2.a.bd.1.4		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
138.2.e.a.31.1	✓	10	69.59	odd	22
138.2.e.a.49.1	yes	10	69.62	odd	22
414.2.i.d.307.1		10	23.13	even	11
414.2.i.d.325.1		10	23.16	even	11
3174.2.a.bc.1.2		5	3.2	odd	2
3174.2.a.bd.1.4		5	69.68	even	2
9522.2.a.bq.1.2		5	23.22	odd	2
9522.2.a.bt.1.4		5	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 9522 = 2 \cdot 3^{2} \cdot 23^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9522.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} +2.59435 q^{5} -1.23648 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} +2.59435 q^{5} -1.23648 q^{7} -1.00000 q^{8} -2.59435 q^{10} +0.622970 q^{11} -1.32463 q^{13} +1.23648 q^{14} +1.00000 q^{16} +4.70760 q^{17} +3.79020 q^{19} +2.59435 q^{20} -0.622970 q^{22} +1.73066 q^{25} +1.32463 q^{26} -1.23648 q^{28} +1.11204 q^{29} +9.45679 q^{31} -1.00000 q^{32} -4.70760 q^{34} -3.20786 q^{35} -1.62721 q^{37} -3.79020 q^{38} -2.59435 q^{40} +11.8497 q^{41} -1.67879 q^{43} +0.622970 q^{44} +6.94494 q^{47} -5.47112 q^{49} -1.73066 q^{50} -1.32463 q^{52} +11.2537 q^{53} +1.61620 q^{55} +1.23648 q^{56} -1.11204 q^{58} +9.74629 q^{59} -14.3736 q^{61} -9.45679 q^{62} +1.00000 q^{64} -3.43657 q^{65} -6.12125 q^{67} +4.70760 q^{68} +3.20786 q^{70} -6.49255 q^{71} -13.0900 q^{73} +1.62721 q^{74} +3.79020 q^{76} -0.770289 q^{77} -12.0823 q^{79} +2.59435 q^{80} -11.8497 q^{82} -7.39068 q^{83} +12.2132 q^{85} +1.67879 q^{86} -0.622970 q^{88} -3.38500 q^{89} +1.63788 q^{91} -6.94494 q^{94} +9.83310 q^{95} +11.4458 q^{97} +5.47112 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 5 q^{2} + 5 q^{4} + 7 q^{5} - 7 q^{7} - 5 q^{8}+O(q^{10}) \) 5 * q - 5 * q^2 + 5 * q^4 + 7 * q^5 - 7 * q^7 - 5 * q^8	\( 5 q - 5 q^{2} + 5 q^{4} + 7 q^{5} - 7 q^{7} - 5 q^{8} - 7 q^{10} + 13 q^{11} - 4 q^{13} + 7 q^{14} + 5 q^{16} + 9 q^{17} - 11 q^{19} + 7 q^{20} - 13 q^{22} - 2 q^{25} + 4 q^{26} - 7 q^{28} + 7 q^{29} - 8 q^{31} - 5 q^{32} - 9 q^{34} - q^{35} - 12 q^{37} + 11 q^{38} - 7 q^{40} + 10 q^{41} - 4 q^{43} + 13 q^{44} + 24 q^{47} - 12 q^{49} + 2 q^{50} - 4 q^{52} + 9 q^{53} + 16 q^{55} + 7 q^{56} - 7 q^{58} + 14 q^{59} - 5 q^{61} + 8 q^{62} + 5 q^{64} + 12 q^{65} - 13 q^{67} + 9 q^{68} + q^{70} + 19 q^{71} + 4 q^{73} + 12 q^{74} - 11 q^{76} - 5 q^{77} - 4 q^{79} + 7 q^{80} - 10 q^{82} + 24 q^{83} + 17 q^{85} + 4 q^{86} - 13 q^{88} + 4 q^{89} + 21 q^{91} - 24 q^{94} + 11 q^{95} + 9 q^{97} + 12 q^{98}+O(q^{100}) \) 5 * q - 5 * q^2 + 5 * q^4 + 7 * q^5 - 7 * q^7 - 5 * q^8 - 7 * q^10 + 13 * q^11 - 4 * q^13 + 7 * q^14 + 5 * q^16 + 9 * q^17 - 11 * q^19 + 7 * q^20 - 13 * q^22 - 2 * q^25 + 4 * q^26 - 7 * q^28 + 7 * q^29 - 8 * q^31 - 5 * q^32 - 9 * q^34 - q^35 - 12 * q^37 + 11 * q^38 - 7 * q^40 + 10 * q^41 - 4 * q^43 + 13 * q^44 + 24 * q^47 - 12 * q^49 + 2 * q^50 - 4 * q^52 + 9 * q^53 + 16 * q^55 + 7 * q^56 - 7 * q^58 + 14 * q^59 - 5 * q^61 + 8 * q^62 + 5 * q^64 + 12 * q^65 - 13 * q^67 + 9 * q^68 + q^70 + 19 * q^71 + 4 * q^73 + 12 * q^74 - 11 * q^76 - 5 * q^77 - 4 * q^79 + 7 * q^80 - 10 * q^82 + 24 * q^83 + 17 * q^85 + 4 * q^86 - 13 * q^88 + 4 * q^89 + 21 * q^91 - 24 * q^94 + 11 * q^95 + 9 * q^97 + 12 * q^98

Introduction

L-functions

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