LMFDB - Embedded newform 9075.2.a.cl.1.3

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:	$1$
Dimension:	$4$
Coefficient field:	4.4.725.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 3x^{2} + x + 1 $ x^4 - x^3 - 3*x^2 + x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 165)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.35567$ 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)

$f(q)$	$=$	$q-1.16215 q^{2} -1.00000 q^{3} -0.649414 q^{4} +1.16215 q^{6} +4.28684 q^{7} +3.07901 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.16215 q^{2} -1.00000 q^{3} -0.649414 q^{4} +1.16215 q^{6} +4.28684 q^{7} +3.07901 q^{8} +1.00000 q^{9} +0.649414 q^{12} -5.16724 q^{13} -4.98194 q^{14} -2.27943 q^{16} -5.00000 q^{17} -1.16215 q^{18} +5.59998 q^{19} -4.28684 q^{21} +0.219819 q^{23} -3.07901 q^{24} +6.00509 q^{26} -1.00000 q^{27} -2.78393 q^{28} +6.41843 q^{29} -2.83095 q^{31} -3.50898 q^{32} +5.81074 q^{34} -0.649414 q^{36} -3.92802 q^{37} -6.50800 q^{38} +5.16724 q^{39} +5.86100 q^{41} +4.98194 q^{42} +8.90173 q^{43} -0.255462 q^{46} +0.237878 q^{47} +2.27943 q^{48} +11.3770 q^{49} +5.00000 q^{51} +3.35567 q^{52} -2.53671 q^{53} +1.16215 q^{54} +13.1992 q^{56} -5.59998 q^{57} -7.45917 q^{58} -7.87035 q^{59} -4.85807 q^{61} +3.28999 q^{62} +4.28684 q^{63} +8.63682 q^{64} -12.1280 q^{67} +3.24707 q^{68} -0.219819 q^{69} -9.98194 q^{71} +3.07901 q^{72} -14.5625 q^{73} +4.56494 q^{74} -3.63670 q^{76} -6.00509 q^{78} +8.00194 q^{79} +1.00000 q^{81} -6.81134 q^{82} -12.5530 q^{83} +2.78393 q^{84} -10.3451 q^{86} -6.41843 q^{87} -2.56545 q^{89} -22.1511 q^{91} -0.142753 q^{92} +2.83095 q^{93} -0.276449 q^{94} +3.50898 q^{96} -2.01199 q^{97} -13.2218 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 5 q^{2} - 4 q^{3} + 9 q^{4} + 5 q^{6} + 2 q^{7} - 15 q^{8} + 4 q^{9}+O(q^{10}) $ 4 * q - 5 * q^2 - 4 * q^3 + 9 * q^4 + 5 * q^6 + 2 * q^7 - 15 * q^8 + 4 * q^9	$ 4 q - 5 q^{2} - 4 q^{3} + 9 q^{4} + 5 q^{6} + 2 q^{7} - 15 q^{8} + 4 q^{9} - 9 q^{12} - 3 q^{13} - 5 q^{14} + 15 q^{16} - 20 q^{17} - 5 q^{18} + 3 q^{19} - 2 q^{21} + 5 q^{23} + 15 q^{24} + 6 q^{26} - 4 q^{27} - 3 q^{28} + 5 q^{29} - q^{31} - 30 q^{32} + 25 q^{34} + 9 q^{36} + 7 q^{37} - q^{38} + 3 q^{39} + 20 q^{41} + 5 q^{42} + 2 q^{43} + 7 q^{46} + 20 q^{47} - 15 q^{48} + 8 q^{49} + 20 q^{51} + 7 q^{52} - 6 q^{53} + 5 q^{54} + 10 q^{56} - 3 q^{57} + 21 q^{58} - 5 q^{59} - 7 q^{61} + 12 q^{62} + 2 q^{63} + 49 q^{64} + 13 q^{67} - 45 q^{68} - 5 q^{69} - 25 q^{71} - 15 q^{72} - 23 q^{73} - 7 q^{74} - 7 q^{76} - 6 q^{78} + 4 q^{81} - 11 q^{82} - 33 q^{83} + 3 q^{84} - 12 q^{86} - 5 q^{87} + 16 q^{89} - 24 q^{91} + q^{93} - 17 q^{94} + 30 q^{96} - 25 q^{98}+O(q^{100}) $ 4 * q - 5 * q^2 - 4 * q^3 + 9 * q^4 + 5 * q^6 + 2 * q^7 - 15 * q^8 + 4 * q^9 - 9 * q^12 - 3 * q^13 - 5 * q^14 + 15 * q^16 - 20 * q^17 - 5 * q^18 + 3 * q^19 - 2 * q^21 + 5 * q^23 + 15 * q^24 + 6 * q^26 - 4 * q^27 - 3 * q^28 + 5 * q^29 - q^31 - 30 * q^32 + 25 * q^34 + 9 * q^36 + 7 * q^37 - q^38 + 3 * q^39 + 20 * q^41 + 5 * q^42 + 2 * q^43 + 7 * q^46 + 20 * q^47 - 15 * q^48 + 8 * q^49 + 20 * q^51 + 7 * q^52 - 6 * q^53 + 5 * q^54 + 10 * q^56 - 3 * q^57 + 21 * q^58 - 5 * q^59 - 7 * q^61 + 12 * q^62 + 2 * q^63 + 49 * q^64 + 13 * q^67 - 45 * q^68 - 5 * q^69 - 25 * q^71 - 15 * q^72 - 23 * q^73 - 7 * q^74 - 7 * q^76 - 6 * q^78 + 4 * q^81 - 11 * q^82 - 33 * q^83 + 3 * q^84 - 12 * q^86 - 5 * q^87 + 16 * q^89 - 24 * q^91 + q^93 - 17 * q^94 + 30 * q^96 - 25 * 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 $

$2$	−1.16215	−0.821762	−0.410881	−	0.911689i	$-0.634779\pi$
$2$	−1.16215	−0.821762	−0.410881	+	0.911689i	$0.634779\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−0.649414	−0.324707
$5$	0	0
$5$	0	0
$6$	1.16215	0.474445
$7$	4.28684	1.62027	0.810137	−	0.586241i	$-0.199393\pi$
$7$	4.28684	1.62027	0.810137	+	0.586241i	$0.199393\pi$
$8$	3.07901	1.08859
$9$	1.00000	0.333333
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0.649414	0.187470
$13$	−5.16724	−1.43313	−0.716567	−	0.697519i	$-0.754287\pi$
$13$	−5.16724	−1.43313	−0.716567	+	0.697519i	$0.754287\pi$
$14$	−4.98194	−1.33148
$15$	0	0
$16$	−2.27943	−0.569858
$17$	−5.00000	−1.21268	−0.606339	−	0.795206i	$-0.707363\pi$
$17$	−5.00000	−1.21268	−0.606339	+	0.795206i	$0.707363\pi$
$18$	−1.16215	−0.273921
$19$	5.59998	1.28472	0.642361	−	0.766402i	$-0.277955\pi$
$19$	5.59998	1.28472	0.642361	+	0.766402i	$0.277955\pi$
$20$	0	0
$21$	−4.28684	−0.935466
$22$	0	0
$23$	0.219819	0.0458354	0.0229177	−	0.999737i	$-0.492704\pi$
$23$	0.219819	0.0458354	0.0229177	+	0.999737i	$0.492704\pi$
$24$	−3.07901	−0.628500
$25$	0	0
$26$	6.00509	1.17769
$27$	−1.00000	−0.192450
$28$	−2.78393	−0.526114
$29$	6.41843	1.19187	0.595937	−	0.803031i	$-0.296781\pi$
$29$	6.41843	1.19187	0.595937	+	0.803031i	$0.296781\pi$
$30$	0	0
$31$	−2.83095	−0.508455	−0.254227	−	0.967145i	$-0.581821\pi$
$31$	−2.83095	−0.508455	−0.254227	+	0.967145i	$0.581821\pi$
$32$	−3.50898	−0.620306
$33$	0	0
$34$	5.81074	0.996533
$35$	0	0
$36$	−0.649414	−0.108236
$37$	−3.92802	−0.645763	−0.322881	−	0.946439i	$-0.604652\pi$
$37$	−3.92802	−0.645763	−0.322881	+	0.946439i	$0.604652\pi$
$38$	−6.50800	−1.05574
$39$	5.16724	0.827420
$40$	0	0
$41$	5.86100	0.915334	0.457667	−	0.889124i	$-0.348685\pi$
$41$	5.86100	0.915334	0.457667	+	0.889124i	$0.348685\pi$
$42$	4.98194	0.768730
$43$	8.90173	1.35750	0.678751	−	0.734369i	$-0.262522\pi$
$43$	8.90173	1.35750	0.678751	+	0.734369i	$0.262522\pi$
$44$	0	0
$45$	0	0
$46$	−0.255462	−0.0376658
$47$	0.237878	0.0346980	0.0173490	−	0.999849i	$-0.494477\pi$
$47$	0.237878	0.0346980	0.0173490	+	0.999849i	$0.494477\pi$
$48$	2.27943	0.329008
$49$	11.3770	1.62529
$50$	0	0
$51$	5.00000	0.700140
$52$	3.35567	0.465348
$53$	−2.53671	−0.348443	−0.174222	−	0.984706i	$-0.555741\pi$
$53$	−2.53671	−0.348443	−0.174222	+	0.984706i	$0.555741\pi$
$54$	1.16215	0.158148
$55$	0	0
$56$	13.1992	1.76382
$57$	−5.59998	−0.741735
$58$	−7.45917	−0.979436
$59$	−7.87035	−1.02463	−0.512316	−	0.858797i	$-0.671212\pi$
$59$	−7.87035	−1.02463	−0.512316	+	0.858797i	$0.671212\pi$
$60$	0	0
$61$	−4.85807	−0.622012	−0.311006	−	0.950408i	$-0.600666\pi$
$61$	−4.85807	−0.622012	−0.311006	+	0.950408i	$0.600666\pi$
$62$	3.28999	0.417829
$63$	4.28684	0.540091
$64$	8.63682	1.07960
$65$	0	0
$66$	0	0
$67$	−12.1280	−1.48167	−0.740834	−	0.671688i	$-0.765569\pi$
$67$	−12.1280	−1.48167	−0.740834	+	0.671688i	$0.765569\pi$
$68$	3.24707	0.393765
$69$	−0.219819	−0.0264631
$70$	0	0
$71$	−9.98194	−1.18464	−0.592319	−	0.805703i	$-0.701788\pi$
$71$	−9.98194	−1.18464	−0.592319	+	0.805703i	$0.701788\pi$
$72$	3.07901	0.362865
$73$	−14.5625	−1.70441	−0.852207	−	0.523205i	$-0.824736\pi$
$73$	−14.5625	−1.70441	−0.852207	+	0.523205i	$0.824736\pi$
$74$	4.56494	0.530664
$75$	0	0
$76$	−3.63670	−0.417158
$77$	0	0
$78$	−6.00509	−0.679942
$79$	8.00194	0.900289	0.450144	−	0.892956i	$-0.351372\pi$
$79$	8.00194	0.900289	0.450144	+	0.892956i	$0.351372\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−6.81134	−0.752187
$83$	−12.5530	−1.37787	−0.688933	−	0.724825i	$-0.741921\pi$
$83$	−12.5530	−1.37787	−0.688933	+	0.724825i	$0.741921\pi$
$84$	2.78393	0.303752
$85$	0	0
$86$	−10.3451	−1.11554
$87$	−6.41843	−0.688128
$88$	0	0
$89$	−2.56545	−0.271937	−0.135969	−	0.990713i	$-0.543415\pi$
$89$	−2.56545	−0.271937	−0.135969	+	0.990713i	$0.543415\pi$
$90$	0	0
$91$	−22.1511	−2.32207
$92$	−0.142753	−0.0148831
$93$	2.83095	0.293556
$94$	−0.276449	−0.0285135
$95$	0	0
$96$	3.50898	0.358134
$97$	−2.01199	−0.204286	−0.102143	−	0.994770i	$-0.532570\pi$
$97$	−2.01199	−0.204286	−0.102143	+	0.994770i	$0.532570\pi$
$98$	−13.2218	−1.33560
$99$	0	0
$100$	0	0
$101$	−7.58484	−0.754720	−0.377360	−	0.926067i	$-0.623168\pi$
$101$	−7.58484	−0.754720	−0.377360	+	0.926067i	$0.623168\pi$
$102$	−5.81074	−0.575349
$103$	16.8685	1.66211	0.831053	−	0.556193i	$-0.187738\pi$
$103$	16.8685	1.66211	0.831053	+	0.556193i	$0.187738\pi$
$104$	−15.9100	−1.56010
$105$	0	0
$106$	2.94803	0.286338
$107$	−7.76161	−0.750343	−0.375172	−	0.926955i	$-0.622416\pi$
$107$	−7.76161	−0.750343	−0.375172	+	0.926955i	$0.622416\pi$
$108$	0.649414	0.0624899
$109$	4.45671	0.426876	0.213438	−	0.976957i	$-0.431534\pi$
$109$	4.45671	0.426876	0.213438	+	0.976957i	$0.431534\pi$
$110$	0	0
$111$	3.92802	0.372831
$112$	−9.77157	−0.923327
$113$	7.71247	0.725528	0.362764	−	0.931881i	$-0.381833\pi$
$113$	7.71247	0.725528	0.362764	+	0.931881i	$0.381833\pi$
$114$	6.50800	0.609530
$115$	0	0
$116$	−4.16822	−0.387010
$117$	−5.16724	−0.477711
$118$	9.14651	0.842004
$119$	−21.4342	−1.96487
$120$	0	0
$121$	0	0
$122$	5.64579	0.511146
$123$	−5.86100	−0.528469
$124$	1.83846	0.165099
$125$	0	0
$126$	−4.98194	−0.443827
$127$	−1.53114	−0.135867	−0.0679335	−	0.997690i	$-0.521641\pi$
$127$	−1.53114	−0.135867	−0.0679335	+	0.997690i	$0.521641\pi$
$128$	−3.01930	−0.266871
$129$	−8.90173	−0.783754
$130$	0	0
$131$	2.66108	0.232500	0.116250	−	0.993220i	$-0.462913\pi$
$131$	2.66108	0.232500	0.116250	+	0.993220i	$0.462913\pi$
$132$	0	0
$133$	24.0062	2.08160
$134$	14.0945	1.21758
$135$	0	0
$136$	−15.3950	−1.32011
$137$	−10.2555	−0.876183	−0.438092	−	0.898930i	$-0.644345\pi$
$137$	−10.2555	−0.876183	−0.438092	+	0.898930i	$0.644345\pi$
$138$	0.255462	0.0217464
$139$	−5.29507	−0.449122	−0.224561	−	0.974460i	$-0.572095\pi$
$139$	−5.29507	−0.449122	−0.224561	+	0.974460i	$0.572095\pi$
$140$	0	0
$141$	−0.237878	−0.0200329
$142$	11.6005	0.973491
$143$	0	0
$144$	−2.27943	−0.189953
$145$	0	0
$146$	16.9238	1.40062
$147$	−11.3770	−0.938360
$148$	2.55091	0.209684
$149$	3.78841	0.310359	0.155179	−	0.987886i	$-0.450404\pi$
$149$	3.78841	0.310359	0.155179	+	0.987886i	$0.450404\pi$
$150$	0	0
$151$	−5.92001	−0.481763	−0.240882	−	0.970554i	$-0.577437\pi$
$151$	−5.92001	−0.481763	−0.240882	+	0.970554i	$0.577437\pi$
$152$	17.2424	1.39854
$153$	−5.00000	−0.404226
$154$	0	0
$155$	0	0
$156$	−3.35567	−0.268669
$157$	7.42909	0.592906	0.296453	−	0.955048i	$-0.404196\pi$
$157$	7.42909	0.592906	0.296453	+	0.955048i	$0.404196\pi$
$158$	−9.29944	−0.739823
$159$	2.53671	0.201174
$160$	0	0
$161$	0.942328	0.0742659
$162$	−1.16215	−0.0913069
$163$	−5.32086	−0.416762	−0.208381	−	0.978048i	$-0.566819\pi$
$163$	−5.32086	−0.416762	−0.208381	+	0.978048i	$0.566819\pi$
$164$	−3.80621	−0.297215
$165$	0	0
$166$	14.5884	1.13228
$167$	1.97139	0.152551	0.0762753	−	0.997087i	$-0.475697\pi$
$167$	1.97139	0.152551	0.0762753	+	0.997087i	$0.475697\pi$
$168$	−13.1992	−1.01834
$169$	13.7003	1.05387
$170$	0	0
$171$	5.59998	0.428241
$172$	−5.78091	−0.440790
$173$	1.97504	0.150160	0.0750799	−	0.997178i	$-0.476079\pi$
$173$	1.97504	0.150160	0.0750799	+	0.997178i	$0.476079\pi$
$174$	7.45917	0.565478
$175$	0	0
$176$	0	0
$177$	7.87035	0.591572
$178$	2.98143	0.223468
$179$	−2.02315	−0.151217	−0.0756086	−	0.997138i	$-0.524090\pi$
$179$	−2.02315	−0.151217	−0.0756086	+	0.997138i	$0.524090\pi$
$180$	0	0
$181$	20.1380	1.49685	0.748423	−	0.663221i	$-0.230811\pi$
$181$	20.1380	1.49685	0.748423	+	0.663221i	$0.230811\pi$
$182$	25.7429	1.90819
$183$	4.85807	0.359119
$184$	0.676824	0.0498961
$185$	0	0
$186$	−3.28999	−0.241234
$187$	0	0
$188$	−0.154481	−0.0112667
$189$	−4.28684	−0.311822
$190$	0	0
$191$	12.6652	0.916418	0.458209	−	0.888844i	$-0.348491\pi$
$191$	12.6652	0.916418	0.458209	+	0.888844i	$0.348491\pi$
$192$	−8.63682	−0.623309
$193$	13.9321	1.00286	0.501428	−	0.865199i	$-0.332808\pi$
$193$	13.9321	1.00286	0.501428	+	0.865199i	$0.332808\pi$
$194$	2.33822	0.167875
$195$	0	0
$196$	−7.38839	−0.527742
$197$	−14.6060	−1.04064	−0.520319	−	0.853972i	$-0.674187\pi$
$197$	−14.6060	−1.04064	−0.520319	+	0.853972i	$0.674187\pi$
$198$	0	0
$199$	−11.8748	−0.841784	−0.420892	−	0.907111i	$-0.638283\pi$
$199$	−11.8748	−0.841784	−0.420892	+	0.907111i	$0.638283\pi$
$200$	0	0
$201$	12.1280	0.855441
$202$	8.81471	0.620201
$203$	27.5148	1.93116
$204$	−3.24707	−0.227340
$205$	0	0
$206$	−19.6037	−1.36586
$207$	0.219819	0.0152785
$208$	11.7784	0.816683
$209$	0	0
$210$	0	0
$211$	−22.3518	−1.53876	−0.769382	−	0.638789i	$-0.779436\pi$
$211$	−22.3518	−1.53876	−0.769382	+	0.638789i	$0.779436\pi$
$212$	1.64737	0.113142
$213$	9.98194	0.683951
$214$	9.02014	0.616604
$215$	0	0
$216$	−3.07901	−0.209500
$217$	−12.1359	−0.823836
$218$	−5.17936	−0.350790
$219$	14.5625	0.984044
$220$	0	0
$221$	25.8362	1.73793
$222$	−4.56494	−0.306379
$223$	19.1583	1.28294	0.641468	−	0.767150i	$-0.278326\pi$
$223$	19.1583	1.28294	0.641468	+	0.767150i	$0.278326\pi$
$224$	−15.0424	−1.00507
$225$	0	0
$226$	−8.96302	−0.596211
$227$	−15.3759	−1.02053	−0.510267	−	0.860016i	$-0.670453\pi$
$227$	−15.3759	−1.02053	−0.510267	+	0.860016i	$0.670453\pi$
$228$	3.63670	0.240846
$229$	8.90126	0.588212	0.294106	−	0.955773i	$-0.404978\pi$
$229$	8.90126	0.588212	0.294106	+	0.955773i	$0.404978\pi$
$230$	0	0
$231$	0	0
$232$	19.7624	1.29747
$233$	−22.8362	−1.49605	−0.748024	−	0.663672i	$-0.768997\pi$
$233$	−22.8362	−1.49605	−0.748024	+	0.663672i	$0.768997\pi$
$234$	6.00509	0.392565
$235$	0	0
$236$	5.11112	0.332705
$237$	−8.00194	−0.519782
$238$	24.9097	1.61466
$239$	16.8605	1.09062	0.545308	−	0.838236i	$-0.316413\pi$
$239$	16.8605	1.09062	0.545308	+	0.838236i	$0.316413\pi$
$240$	0	0
$241$	−11.9607	−0.770459	−0.385229	−	0.922821i	$-0.625878\pi$
$241$	−11.9607	−0.770459	−0.385229	+	0.922821i	$0.625878\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	3.15490	0.201972
$245$	0	0
$246$	6.81134	0.434275
$247$	−28.9364	−1.84118
$248$	−8.71654	−0.553501
$249$	12.5530	0.795511
$250$	0	0
$251$	16.0034	1.01013	0.505063	−	0.863082i	$-0.331469\pi$
$251$	16.0034	1.01013	0.505063	+	0.863082i	$0.331469\pi$
$252$	−2.78393	−0.175371
$253$	0	0
$254$	1.77941	0.111650
$255$	0	0
$256$	−13.7648	−0.860298
$257$	−5.47446	−0.341487	−0.170744	−	0.985315i	$-0.554617\pi$
$257$	−5.47446	−0.341487	−0.170744	+	0.985315i	$0.554617\pi$
$258$	10.3451	0.644059
$259$	−16.8388	−1.04631
$260$	0	0
$261$	6.41843	0.397291
$262$	−3.09257	−0.191060
$263$	−11.9841	−0.738971	−0.369486	−	0.929236i	$-0.620466\pi$
$263$	−11.9841	−0.738971	−0.369486	+	0.929236i	$0.620466\pi$
$264$	0	0
$265$	0	0
$266$	−27.8987	−1.71058
$267$	2.56545	0.157003
$268$	7.87607	0.481108
$269$	13.6465	0.832041	0.416020	−	0.909355i	$-0.363424\pi$
$269$	13.6465	0.832041	0.416020	+	0.909355i	$0.363424\pi$
$270$	0	0
$271$	11.8379	0.719098	0.359549	−	0.933126i	$-0.382930\pi$
$271$	11.8379	0.719098	0.359549	+	0.933126i	$0.382930\pi$
$272$	11.3972	0.691055
$273$	22.1511	1.34065
$274$	11.9184	0.720014
$275$	0	0
$276$	0.142753	0.00859274
$277$	−10.2638	−0.616694	−0.308347	−	0.951274i	$-0.599776\pi$
$277$	−10.2638	−0.616694	−0.308347	+	0.951274i	$0.599776\pi$
$278$	6.15366	0.369072
$279$	−2.83095	−0.169485
$280$	0	0
$281$	−2.93889	−0.175319	−0.0876597	−	0.996150i	$-0.527939\pi$
$281$	−2.93889	−0.175319	−0.0876597	+	0.996150i	$0.527939\pi$
$282$	0.276449	0.0164623
$283$	−21.8279	−1.29754	−0.648768	−	0.760986i	$-0.724716\pi$
$283$	−21.8279	−1.29754	−0.648768	+	0.760986i	$0.724716\pi$
$284$	6.48241	0.384660
$285$	0	0
$286$	0	0
$287$	25.1252	1.48309
$288$	−3.50898	−0.206769
$289$	8.00000	0.470588
$290$	0	0
$291$	2.01199	0.117945
$292$	9.45710	0.553435
$293$	24.7665	1.44688	0.723438	−	0.690390i	$-0.242561\pi$
$293$	24.7665	1.44688	0.723438	+	0.690390i	$0.242561\pi$
$294$	13.2218	0.771109
$295$	0	0
$296$	−12.0944	−0.702974
$297$	0	0
$298$	−4.40269	−0.255041
$299$	−1.13586	−0.0656882
$300$	0	0
$301$	38.1603	2.19952
$302$	6.87992	0.395895
$303$	7.58484	0.435738
$304$	−12.7648	−0.732110
$305$	0	0
$306$	5.81074	0.332178
$307$	4.94023	0.281954	0.140977	−	0.990013i	$-0.454976\pi$
$307$	4.94023	0.281954	0.140977	+	0.990013i	$0.454976\pi$
$308$	0	0
$309$	−16.8685	−0.959618
$310$	0	0
$311$	−13.8096	−0.783070	−0.391535	−	0.920163i	$-0.628056\pi$
$311$	−13.8096	−0.783070	−0.391535	+	0.920163i	$0.628056\pi$
$312$	15.9100	0.900724
$313$	4.06934	0.230013	0.115006	−	0.993365i	$-0.463311\pi$
$313$	4.06934	0.230013	0.115006	+	0.993365i	$0.463311\pi$
$314$	−8.63369	−0.487227
$315$	0	0
$316$	−5.19657	−0.292330
$317$	−25.8199	−1.45019	−0.725096	−	0.688648i	$-0.758205\pi$
$317$	−25.8199	−1.45019	−0.725096	+	0.688648i	$0.758205\pi$
$318$	−2.94803	−0.165317
$319$	0	0
$320$	0	0
$321$	7.76161	0.433211
$322$	−1.09512	−0.0610289
$323$	−27.9999	−1.55795
$324$	−0.649414	−0.0360785
$325$	0	0
$326$	6.18362	0.342479
$327$	−4.45671	−0.246457
$328$	18.0461	0.996428
$329$	1.01974	0.0562203
$330$	0	0
$331$	−3.35008	−0.184137	−0.0920684	−	0.995753i	$-0.529348\pi$
$331$	−3.35008	−0.184137	−0.0920684	+	0.995753i	$0.529348\pi$
$332$	8.15206	0.447403
$333$	−3.92802	−0.215254
$334$	−2.29104	−0.125360
$335$	0	0
$336$	9.77157	0.533083
$337$	27.3998	1.49256	0.746282	−	0.665630i	$-0.231837\pi$
$337$	27.3998	1.49256	0.746282	+	0.665630i	$0.231837\pi$
$338$	−15.9218	−0.866031
$339$	−7.71247	−0.418884
$340$	0	0
$341$	0	0
$342$	−6.50800	−0.351912
$343$	18.7636	1.01314
$344$	27.4085	1.47777
$345$	0	0
$346$	−2.29529	−0.123396
$347$	−15.8265	−0.849610	−0.424805	−	0.905285i	$-0.639657\pi$
$347$	−15.8265	−0.849610	−0.424805	+	0.905285i	$0.639657\pi$
$348$	4.16822	0.223440
$349$	−3.65930	−0.195878	−0.0979388	−	0.995192i	$-0.531225\pi$
$349$	−3.65930	−0.195878	−0.0979388	+	0.995192i	$0.531225\pi$
$350$	0	0
$351$	5.16724	0.275807
$352$	0	0
$353$	27.4937	1.46334	0.731671	−	0.681658i	$-0.238741\pi$
$353$	27.4937	1.46334	0.731671	+	0.681658i	$0.238741\pi$
$354$	−9.14651	−0.486131
$355$	0	0
$356$	1.66604	0.0882999
$357$	21.4342	1.13442
$358$	2.35119	0.124265
$359$	0.478740	0.0252670	0.0126335	−	0.999920i	$-0.495979\pi$
$359$	0.478740	0.0252670	0.0126335	+	0.999920i	$0.495979\pi$
$360$	0	0
$361$	12.3597	0.650512
$362$	−23.4033	−1.23005
$363$	0	0
$364$	14.3852	0.753992
$365$	0	0
$366$	−5.64579	−0.295110
$367$	25.4582	1.32891	0.664453	−	0.747330i	$-0.268665\pi$
$367$	25.4582	1.32891	0.664453	+	0.747330i	$0.268665\pi$
$368$	−0.501062	−0.0261197
$369$	5.86100	0.305111
$370$	0	0
$371$	−10.8745	−0.564574
$372$	−1.83846	−0.0953198
$373$	2.75967	0.142890	0.0714451	−	0.997445i	$-0.477239\pi$
$373$	2.75967	0.142890	0.0714451	+	0.997445i	$0.477239\pi$
$374$	0	0
$375$	0	0
$376$	0.732428	0.0377721
$377$	−33.1656	−1.70811
$378$	4.98194	0.256243
$379$	−21.6535	−1.11227	−0.556133	−	0.831093i	$-0.687716\pi$
$379$	−21.6535	−1.11227	−0.556133	+	0.831093i	$0.687716\pi$
$380$	0	0
$381$	1.53114	0.0784428
$382$	−14.7188	−0.753078
$383$	−16.2295	−0.829287	−0.414643	−	0.909984i	$-0.636094\pi$
$383$	−16.2295	−0.829287	−0.414643	+	0.909984i	$0.636094\pi$
$384$	3.01930	0.154078
$385$	0	0
$386$	−16.1912	−0.824109
$387$	8.90173	0.452500
$388$	1.30661	0.0663332
$389$	−16.2916	−0.826019	−0.413009	−	0.910727i	$-0.635522\pi$
$389$	−16.2916	−0.826019	−0.413009	+	0.910727i	$0.635522\pi$
$390$	0	0
$391$	−1.09909	−0.0555836
$392$	35.0299	1.76928
$393$	−2.66108	−0.134234
$394$	16.9744	0.855157
$395$	0	0
$396$	0	0
$397$	19.9673	1.00213	0.501064	−	0.865410i	$-0.332942\pi$
$397$	19.9673	1.00213	0.501064	+	0.865410i	$0.332942\pi$
$398$	13.8003	0.691747
$399$	−24.0062	−1.20181
$400$	0	0
$401$	−27.3269	−1.36464	−0.682320	−	0.731053i	$-0.739029\pi$
$401$	−27.3269	−1.36464	−0.682320	+	0.731053i	$0.739029\pi$
$402$	−14.0945	−0.702969
$403$	14.6282	0.728683
$404$	4.92570	0.245063
$405$	0	0
$406$	−31.9763	−1.58696
$407$	0	0
$408$	15.3950	0.762168
$409$	24.9380	1.23310	0.616552	−	0.787314i	$-0.288529\pi$
$409$	24.9380	1.23310	0.616552	+	0.787314i	$0.288529\pi$
$410$	0	0
$411$	10.2555	0.505865
$412$	−10.9547	−0.539698
$413$	−33.7389	−1.66019
$414$	−0.255462	−0.0125553
$415$	0	0
$416$	18.1317	0.888981
$417$	5.29507	0.259301
$418$	0	0
$419$	16.8256	0.821986	0.410993	−	0.911639i	$-0.365182\pi$
$419$	16.8256	0.821986	0.410993	+	0.911639i	$0.365182\pi$
$420$	0	0
$421$	23.6006	1.15022	0.575112	−	0.818074i	$-0.304958\pi$
$421$	23.6006	1.15022	0.575112	+	0.818074i	$0.304958\pi$
$422$	25.9761	1.26450
$423$	0.237878	0.0115660
$424$	−7.81054	−0.379313
$425$	0	0
$426$	−11.6005	−0.562045
$427$	−20.8258	−1.00783
$428$	5.04050	0.243642
$429$	0	0
$430$	0	0
$431$	−37.6556	−1.81381	−0.906903	−	0.421339i	$-0.861560\pi$
$431$	−37.6556	−1.81381	−0.906903	+	0.421339i	$0.861560\pi$
$432$	2.27943	0.109669
$433$	30.9752	1.48857	0.744286	−	0.667861i	$-0.232790\pi$
$433$	30.9752	1.48857	0.744286	+	0.667861i	$0.232790\pi$
$434$	14.1037	0.676997
$435$	0	0
$436$	−2.89425	−0.138609
$437$	1.23098	0.0588858
$438$	−16.9238	−0.808650
$439$	−28.2131	−1.34654	−0.673268	−	0.739399i	$-0.735110\pi$
$439$	−28.2131	−1.34654	−0.673268	+	0.739399i	$0.735110\pi$
$440$	0	0
$441$	11.3770	0.541762
$442$	−30.0254	−1.42816
$443$	−14.6420	−0.695663	−0.347831	−	0.937557i	$-0.613082\pi$
$443$	−14.6420	−0.695663	−0.347831	+	0.937557i	$0.613082\pi$
$444$	−2.55091	−0.121061
$445$	0	0
$446$	−22.2648	−1.05427
$447$	−3.78841	−0.179186
$448$	37.0247	1.74925
$449$	11.9977	0.566206	0.283103	−	0.959090i	$-0.408636\pi$
$449$	11.9977	0.566206	0.283103	+	0.959090i	$0.408636\pi$
$450$	0	0
$451$	0	0
$452$	−5.00858	−0.235584
$453$	5.92001	0.278146
$454$	17.8691	0.838636
$455$	0	0
$456$	−17.2424	−0.807448
$457$	−10.0437	−0.469824	−0.234912	−	0.972017i	$-0.575480\pi$
$457$	−10.0437	−0.469824	−0.234912	+	0.972017i	$0.575480\pi$
$458$	−10.3446	−0.483370
$459$	5.00000	0.233380
$460$	0	0
$461$	−13.4491	−0.626386	−0.313193	−	0.949689i	$-0.601399\pi$
$461$	−13.4491	−0.626386	−0.313193	+	0.949689i	$0.601399\pi$
$462$	0	0
$463$	−18.7836	−0.872946	−0.436473	−	0.899717i	$-0.643773\pi$
$463$	−18.7836	−0.872946	−0.436473	+	0.899717i	$0.643773\pi$
$464$	−14.6304	−0.679199
$465$	0	0
$466$	26.5390	1.22940
$467$	−26.5510	−1.22863	−0.614316	−	0.789060i	$-0.710568\pi$
$467$	−26.5510	−1.22863	−0.614316	+	0.789060i	$0.710568\pi$
$468$	3.35567	0.155116
$469$	−51.9907	−2.40071
$470$	0	0
$471$	−7.42909	−0.342314
$472$	−24.2329	−1.11541
$473$	0	0
$474$	9.29944	0.427137
$475$	0	0
$476$	13.9197	0.638007
$477$	−2.53671	−0.116148
$478$	−19.5944	−0.896228
$479$	10.7830	0.492689	0.246345	−	0.969182i	$-0.420771\pi$
$479$	10.7830	0.492689	0.246345	+	0.969182i	$0.420771\pi$
$480$	0	0
$481$	20.2970	0.925464
$482$	13.9001	0.633134
$483$	−0.942328	−0.0428774
$484$	0	0
$485$	0	0
$486$	1.16215	0.0527161
$487$	−6.81095	−0.308634	−0.154317	−	0.988021i	$-0.549318\pi$
$487$	−6.81095	−0.308634	−0.154317	+	0.988021i	$0.549318\pi$
$488$	−14.9580	−0.677119
$489$	5.32086	0.240617
$490$	0	0
$491$	26.0954	1.17767	0.588834	−	0.808254i	$-0.299587\pi$
$491$	26.0954	1.17767	0.588834	+	0.808254i	$0.299587\pi$
$492$	3.80621	0.171597
$493$	−32.0922	−1.44536
$494$	33.6283	1.51301
$495$	0	0
$496$	6.45297	0.289747
$497$	−42.7910	−1.91944
$498$	−14.5884	−0.653721
$499$	19.3834	0.867719	0.433860	−	0.900981i	$-0.357151\pi$
$499$	19.3834	0.867719	0.433860	+	0.900981i	$0.357151\pi$
$500$	0	0
$501$	−1.97139	−0.0880751
$502$	−18.5983	−0.830084
$503$	−18.8422	−0.840134	−0.420067	−	0.907493i	$-0.637993\pi$
$503$	−18.8422	−0.840134	−0.420067	+	0.907493i	$0.637993\pi$
$504$	13.1992	0.587940
$505$	0	0
$506$	0	0
$507$	−13.7003	−0.608453
$508$	0.994345	0.0441169
$509$	−5.15120	−0.228323	−0.114162	−	0.993462i	$-0.536418\pi$
$509$	−5.15120	−0.228323	−0.114162	+	0.993462i	$0.536418\pi$
$510$	0	0
$511$	−62.4272	−2.76162
$512$	22.0353	0.973831
$513$	−5.59998	−0.247245
$514$	6.36212	0.280621
$515$	0	0
$516$	5.78091	0.254490
$517$	0	0
$518$	19.5692	0.859820
$519$	−1.97504	−0.0866948
$520$	0	0
$521$	−41.5022	−1.81824	−0.909121	−	0.416532i	$-0.863246\pi$
$521$	−41.5022	−1.81824	−0.909121	+	0.416532i	$0.863246\pi$
$522$	−7.45917	−0.326479
$523$	6.92280	0.302713	0.151356	−	0.988479i	$-0.451636\pi$
$523$	6.92280	0.302713	0.151356	+	0.988479i	$0.451636\pi$
$524$	−1.72815	−0.0754944
$525$	0	0
$526$	13.9273	0.607259
$527$	14.1548	0.616592
$528$	0	0
$529$	−22.9517	−0.997899
$530$	0	0
$531$	−7.87035	−0.341544
$532$	−15.5900	−0.675911
$533$	−30.2852	−1.31180
$534$	−2.98143	−0.129019
$535$	0	0
$536$	−37.3421	−1.61293
$537$	2.02315	0.0873052
$538$	−15.8592	−0.683740
$539$	0	0
$540$	0	0
$541$	−28.4183	−1.22180	−0.610899	−	0.791709i	$-0.709192\pi$
$541$	−28.4183	−1.22180	−0.610899	+	0.791709i	$0.709192\pi$
$542$	−13.7573	−0.590928
$543$	−20.1380	−0.864205
$544$	17.5449	0.752231
$545$	0	0
$546$	−25.7429	−1.10169
$547$	−25.8592	−1.10566	−0.552829	−	0.833295i	$-0.686452\pi$
$547$	−25.8592	−1.10566	−0.552829	+	0.833295i	$0.686452\pi$
$548$	6.66004	0.284503
$549$	−4.85807	−0.207337
$550$	0	0
$551$	35.9431	1.53123
$552$	−0.676824	−0.0288075
$553$	34.3031	1.45871
$554$	11.9281	0.506776
$555$	0	0
$556$	3.43869	0.145833
$557$	8.06409	0.341687	0.170843	−	0.985298i	$-0.445351\pi$
$557$	8.06409	0.341687	0.170843	+	0.985298i	$0.445351\pi$
$558$	3.28999	0.139276
$559$	−45.9973	−1.94548
$560$	0	0
$561$	0	0
$562$	3.41542	0.144071
$563$	−29.6516	−1.24966	−0.624832	−	0.780759i	$-0.714833\pi$
$563$	−29.6516	−1.24966	−0.624832	+	0.780759i	$0.714833\pi$
$564$	0.154481	0.00650483
$565$	0	0
$566$	25.3673	1.06627
$567$	4.28684	0.180030
$568$	−30.7345	−1.28959
$569$	−42.1026	−1.76503	−0.882517	−	0.470280i	$-0.844153\pi$
$569$	−42.1026	−1.76503	−0.882517	+	0.470280i	$0.844153\pi$
$570$	0	0
$571$	−26.6823	−1.11662	−0.558311	−	0.829632i	$-0.688550\pi$
$571$	−26.6823	−1.11662	−0.558311	+	0.829632i	$0.688550\pi$
$572$	0	0
$573$	−12.6652	−0.529094
$574$	−29.1992	−1.21875
$575$	0	0
$576$	8.63682	0.359867
$577$	3.30818	0.137721	0.0688606	−	0.997626i	$-0.478064\pi$
$577$	3.30818	0.137721	0.0688606	+	0.997626i	$0.478064\pi$
$578$	−9.29718	−0.386712
$579$	−13.9321	−0.578999
$580$	0	0
$581$	−53.8125	−2.23252
$582$	−2.33822	−0.0969225
$583$	0	0
$584$	−44.8381	−1.85542
$585$	0	0
$586$	−28.7823	−1.18899
$587$	8.52770	0.351976	0.175988	−	0.984392i	$-0.443688\pi$
$587$	8.52770	0.351976	0.175988	+	0.984392i	$0.443688\pi$
$588$	7.38839	0.304692
$589$	−15.8533	−0.653223
$590$	0	0
$591$	14.6060	0.600813
$592$	8.95367	0.367993
$593$	23.4343	0.962333	0.481167	−	0.876629i	$-0.340213\pi$
$593$	23.4343	0.962333	0.481167	+	0.876629i	$0.340213\pi$
$594$	0	0
$595$	0	0
$596$	−2.46025	−0.100776
$597$	11.8748	0.486004
$598$	1.32003	0.0539801
$599$	18.8344	0.769555	0.384777	−	0.923009i	$-0.374278\pi$
$599$	18.8344	0.769555	0.384777	+	0.923009i	$0.374278\pi$
$600$	0	0
$601$	−37.6585	−1.53612	−0.768062	−	0.640376i	$-0.778779\pi$
$601$	−37.6585	−1.53612	−0.768062	+	0.640376i	$0.778779\pi$
$602$	−44.3479	−1.80749
$603$	−12.1280	−0.493889
$604$	3.84453	0.156432
$605$	0	0
$606$	−8.81471	−0.358073
$607$	29.3680	1.19201	0.596004	−	0.802981i	$-0.296754\pi$
$607$	29.3680	1.19201	0.596004	+	0.802981i	$0.296754\pi$
$608$	−19.6502	−0.796921
$609$	−27.5148	−1.11496
$610$	0	0
$611$	−1.22917	−0.0497269
$612$	3.24707	0.131255
$613$	12.9111	0.521474	0.260737	−	0.965410i	$-0.416035\pi$
$613$	12.9111	0.521474	0.260737	+	0.965410i	$0.416035\pi$
$614$	−5.74127	−0.231699
$615$	0	0
$616$	0	0
$617$	−41.6041	−1.67492	−0.837459	−	0.546500i	$-0.815960\pi$
$617$	−41.6041	−1.67492	−0.837459	+	0.546500i	$0.815960\pi$
$618$	19.6037	0.788578
$619$	4.27117	0.171673	0.0858363	−	0.996309i	$-0.472644\pi$
$619$	4.27117	0.171673	0.0858363	+	0.996309i	$0.472644\pi$
$620$	0	0
$621$	−0.219819	−0.00882102
$622$	16.0488	0.643497
$623$	−10.9977	−0.440613
$624$	−11.7784	−0.471512
$625$	0	0
$626$	−4.72918	−0.189016
$627$	0	0
$628$	−4.82455	−0.192521
$629$	19.6401	0.783103
$630$	0	0
$631$	−27.5510	−1.09679	−0.548393	−	0.836221i	$-0.684760\pi$
$631$	−27.5510	−1.09679	−0.548393	+	0.836221i	$0.684760\pi$
$632$	24.6381	0.980049
$633$	22.3518	0.888406
$634$	30.0066	1.19171
$635$	0	0
$636$	−1.64737	−0.0653225
$637$	−58.7877	−2.32925
$638$	0	0
$639$	−9.98194	−0.394879
$640$	0	0
$641$	9.16806	0.362117	0.181058	−	0.983472i	$-0.442048\pi$
$641$	9.16806	0.362117	0.181058	+	0.983472i	$0.442048\pi$
$642$	−9.02014	−0.355996
$643$	3.40908	0.134441	0.0672206	−	0.997738i	$-0.478587\pi$
$643$	3.40908	0.134441	0.0672206	+	0.997738i	$0.478587\pi$
$644$	−0.611961	−0.0241146
$645$	0	0
$646$	32.5400	1.28027
$647$	−22.3608	−0.879092	−0.439546	−	0.898220i	$-0.644861\pi$
$647$	−22.3608	−0.879092	−0.439546	+	0.898220i	$0.644861\pi$
$648$	3.07901	0.120955
$649$	0	0
$650$	0	0
$651$	12.1359	0.475642
$652$	3.45544	0.135325
$653$	14.7957	0.578999	0.289500	−	0.957178i	$-0.406511\pi$
$653$	14.7957	0.578999	0.289500	+	0.957178i	$0.406511\pi$
$654$	5.17936	0.202529
$655$	0	0
$656$	−13.3598	−0.521611
$657$	−14.5625	−0.568138
$658$	−1.18509	−0.0461997
$659$	−47.4724	−1.84926	−0.924631	−	0.380864i	$-0.875627\pi$
$659$	−47.4724	−1.84926	−0.924631	+	0.380864i	$0.875627\pi$
$660$	0	0
$661$	−21.6525	−0.842184	−0.421092	−	0.907018i	$-0.638353\pi$
$661$	−21.6525	−0.842184	−0.421092	+	0.907018i	$0.638353\pi$
$662$	3.89328	0.151317
$663$	−25.8362	−1.00339
$664$	−38.6507	−1.49994
$665$	0	0
$666$	4.56494	0.176888
$667$	1.41089	0.0546300
$668$	−1.28025	−0.0495342
$669$	−19.1583	−0.740703
$670$	0	0
$671$	0	0
$672$	15.0424	0.580275
$673$	38.3923	1.47991	0.739957	−	0.672654i	$-0.234846\pi$
$673$	38.3923	1.47991	0.739957	+	0.672654i	$0.234846\pi$
$674$	−31.8426	−1.22653
$675$	0	0
$676$	−8.89718	−0.342199
$677$	22.9333	0.881398	0.440699	−	0.897655i	$-0.354731\pi$
$677$	22.9333	0.881398	0.440699	+	0.897655i	$0.354731\pi$
$678$	8.96302	0.344223
$679$	−8.62507	−0.331000
$680$	0	0
$681$	15.3759	0.589206
$682$	0	0
$683$	−25.9084	−0.991359	−0.495680	−	0.868505i	$-0.665081\pi$
$683$	−25.9084	−0.991359	−0.495680	+	0.868505i	$0.665081\pi$
$684$	−3.63670	−0.139053
$685$	0	0
$686$	−21.8060	−0.832557
$687$	−8.90126	−0.339604
$688$	−20.2909	−0.773584
$689$	13.1078	0.499366
$690$	0	0
$691$	14.8155	0.563608	0.281804	−	0.959472i	$-0.409067\pi$
$691$	14.8155	0.563608	0.281804	+	0.959472i	$0.409067\pi$
$692$	−1.28262	−0.0487579
$693$	0	0
$694$	18.3927	0.698177
$695$	0	0
$696$	−19.7624	−0.749092
$697$	−29.3050	−1.11001
$698$	4.25264	0.160965
$699$	22.8362	0.863744
$700$	0	0
$701$	5.50632	0.207971	0.103985	−	0.994579i	$-0.466840\pi$
$701$	5.50632	0.207971	0.103985	+	0.994579i	$0.466840\pi$
$702$	−6.00509	−0.226647
$703$	−21.9968	−0.829626
$704$	0	0
$705$	0	0
$706$	−31.9517	−1.20252
$707$	−32.5150	−1.22285
$708$	−5.11112	−0.192087
$709$	−17.9007	−0.672274	−0.336137	−	0.941813i	$-0.609120\pi$
$709$	−17.9007	−0.672274	−0.336137	+	0.941813i	$0.609120\pi$
$710$	0	0
$711$	8.00194	0.300096
$712$	−7.89905	−0.296029
$713$	−0.622297	−0.0233052
$714$	−24.9097	−0.932222
$715$	0	0
$716$	1.31386	0.0491012
$717$	−16.8605	−0.629668
$718$	−0.556367	−0.0207634
$719$	32.5918	1.21547	0.607734	−	0.794141i	$-0.292079\pi$
$719$	32.5918	1.21547	0.607734	+	0.794141i	$0.292079\pi$
$720$	0	0
$721$	72.3128	2.69307
$722$	−14.3638	−0.534566
$723$	11.9607	0.444825
$724$	−13.0779	−0.486037
$725$	0	0
$726$	0	0
$727$	43.0199	1.59552	0.797759	−	0.602976i	$-0.206018\pi$
$727$	43.0199	1.59552	0.797759	+	0.602976i	$0.206018\pi$
$728$	−68.2035	−2.52779
$729$	1.00000	0.0370370
$730$	0	0
$731$	−44.5087	−1.64621
$732$	−3.15490	−0.116608
$733$	−40.7066	−1.50353	−0.751766	−	0.659429i	$-0.770798\pi$
$733$	−40.7066	−1.50353	−0.751766	+	0.659429i	$0.770798\pi$
$734$	−29.5862	−1.09204
$735$	0	0
$736$	−0.771340	−0.0284320
$737$	0	0
$738$	−6.81134	−0.250729
$739$	38.1240	1.40241	0.701207	−	0.712958i	$-0.252645\pi$
$739$	38.1240	1.40241	0.701207	+	0.712958i	$0.252645\pi$
$740$	0	0
$741$	28.9364	1.06300
$742$	12.6377	0.463945
$743$	18.0381	0.661754	0.330877	−	0.943674i	$-0.392655\pi$
$743$	18.0381	0.661754	0.330877	+	0.943674i	$0.392655\pi$
$744$	8.71654	0.319564
$745$	0	0
$746$	−3.20714	−0.117422
$747$	−12.5530	−0.459289
$748$	0	0
$749$	−33.2728	−1.21576
$750$	0	0
$751$	−17.5349	−0.639859	−0.319929	−	0.947441i	$-0.603659\pi$
$751$	−17.5349	−0.639859	−0.319929	+	0.947441i	$0.603659\pi$
$752$	−0.542227	−0.0197730
$753$	−16.0034	−0.583197
$754$	38.5433	1.40366
$755$	0	0
$756$	2.78393	0.101251
$757$	25.9609	0.943565	0.471782	−	0.881715i	$-0.343611\pi$
$757$	25.9609	0.943565	0.471782	+	0.881715i	$0.343611\pi$
$758$	25.1646	0.914019
$759$	0	0
$760$	0	0
$761$	28.0059	1.01521	0.507607	−	0.861589i	$-0.330530\pi$
$761$	28.0059	1.01521	0.507607	+	0.861589i	$0.330530\pi$
$762$	−1.77941	−0.0644613
$763$	19.1052	0.691655
$764$	−8.22493	−0.297567
$765$	0	0
$766$	18.8610	0.681477
$767$	40.6680	1.46843
$768$	13.7648	0.496693
$769$	22.9537	0.827732	0.413866	−	0.910338i	$-0.364178\pi$
$769$	22.9537	0.827732	0.413866	+	0.910338i	$0.364178\pi$
$770$	0	0
$771$	5.47446	0.197158
$772$	−9.04772	−0.325634
$773$	23.5282	0.846251	0.423126	−	0.906071i	$-0.360933\pi$
$773$	23.5282	0.846251	0.423126	+	0.906071i	$0.360933\pi$
$774$	−10.3451	−0.371848
$775$	0	0
$776$	−6.19492	−0.222385
$777$	16.8388	0.604089
$778$	18.9333	0.678791
$779$	32.8215	1.17595
$780$	0	0
$781$	0	0
$782$	1.27731	0.0456765
$783$	−6.41843	−0.229376
$784$	−25.9331	−0.926184
$785$	0	0
$786$	3.09257	0.110308
$787$	25.2725	0.900869	0.450434	−	0.892810i	$-0.351269\pi$
$787$	25.2725	0.900869	0.450434	+	0.892810i	$0.351269\pi$
$788$	9.48537	0.337902
$789$	11.9841	0.426645
$790$	0	0
$791$	33.0621	1.17555
$792$	0	0
$793$	25.1028	0.891427
$794$	−23.2049	−0.823511
$795$	0	0
$796$	7.71168	0.273333
$797$	−8.65645	−0.306627	−0.153314	−	0.988178i	$-0.548994\pi$
$797$	−8.65645	−0.306627	−0.153314	+	0.988178i	$0.548994\pi$
$798$	27.8987	0.987605
$799$	−1.18939	−0.0420775
$800$	0	0
$801$	−2.56545	−0.0906457
$802$	31.7579	1.12141
$803$	0	0
$804$	−7.87607	−0.277768
$805$	0	0
$806$	−17.0001	−0.598804
$807$	−13.6465	−0.480379
$808$	−23.3538	−0.821584
$809$	−10.2124	−0.359050	−0.179525	−	0.983753i	$-0.557456\pi$
$809$	−10.2124	−0.359050	−0.179525	+	0.983753i	$0.557456\pi$
$810$	0	0
$811$	−47.6224	−1.67225	−0.836124	−	0.548540i	$-0.815184\pi$
$811$	−47.6224	−1.67225	−0.836124	+	0.548540i	$0.815184\pi$
$812$	−17.8685	−0.627061
$813$	−11.8379	−0.415172
$814$	0	0
$815$	0	0
$816$	−11.3972	−0.398981
$817$	49.8495	1.74401
$818$	−28.9816	−1.01332
$819$	−22.1511	−0.774023
$820$	0	0
$821$	−47.5598	−1.65985	−0.829925	−	0.557875i	$-0.811617\pi$
$821$	−47.5598	−1.65985	−0.829925	+	0.557875i	$0.811617\pi$
$822$	−11.9184	−0.415700
$823$	24.7360	0.862242	0.431121	−	0.902294i	$-0.358118\pi$
$823$	24.7360	0.862242	0.431121	+	0.902294i	$0.358118\pi$
$824$	51.9384	1.80936
$825$	0	0
$826$	39.2096	1.36428
$827$	1.56166	0.0543043	0.0271522	−	0.999631i	$-0.491356\pi$
$827$	1.56166	0.0543043	0.0271522	+	0.999631i	$0.491356\pi$
$828$	−0.142753	−0.00496102
$829$	42.1950	1.46549	0.732747	−	0.680501i	$-0.238238\pi$
$829$	42.1950	1.46549	0.732747	+	0.680501i	$0.238238\pi$
$830$	0	0
$831$	10.2638	0.356048
$832$	−44.6285	−1.54721
$833$	−56.8851	−1.97095
$834$	−6.15366	−0.213084
$835$	0	0
$836$	0	0
$837$	2.83095	0.0978521
$838$	−19.5539	−0.675477
$839$	−6.29451	−0.217311	−0.108655	−	0.994079i	$-0.534654\pi$
$839$	−6.29451	−0.217311	−0.108655	+	0.994079i	$0.534654\pi$
$840$	0	0
$841$	12.1963	0.420562
$842$	−27.4274	−0.945211
$843$	2.93889	0.101221
$844$	14.5156	0.499647
$845$	0	0
$846$	−0.276449	−0.00950451
$847$	0	0
$848$	5.78225	0.198563
$849$	21.8279	0.749133
$850$	0	0
$851$	−0.863453	−0.0295988
$852$	−6.48241	−0.222084
$853$	7.27120	0.248961	0.124481	−	0.992222i	$-0.460274\pi$
$853$	7.27120	0.248961	0.124481	+	0.992222i	$0.460274\pi$
$854$	24.2026	0.828197
$855$	0	0
$856$	−23.8981	−0.816819
$857$	−39.8590	−1.36156	−0.680779	−	0.732489i	$-0.738359\pi$
$857$	−39.8590	−1.36156	−0.680779	+	0.732489i	$0.738359\pi$
$858$	0	0
$859$	13.5278	0.461564	0.230782	−	0.973006i	$-0.425872\pi$
$859$	13.5278	0.461564	0.230782	+	0.973006i	$0.425872\pi$
$860$	0	0
$861$	−25.1252	−0.856264
$862$	43.7613	1.49052
$863$	−50.3117	−1.71263	−0.856316	−	0.516453i	$-0.827252\pi$
$863$	−50.3117	−1.71263	−0.856316	+	0.516453i	$0.827252\pi$
$864$	3.50898	0.119378
$865$	0	0
$866$	−35.9977	−1.22325
$867$	−8.00000	−0.271694
$868$	7.88119	0.267505
$869$	0	0
$870$	0	0
$871$	62.6681	2.12343
$872$	13.7223	0.464694
$873$	−2.01199	−0.0680954
$874$	−1.43058	−0.0483901
$875$	0	0
$876$	−9.45710	−0.319526
$877$	1.20736	0.0407696	0.0203848	−	0.999792i	$-0.493511\pi$
$877$	1.20736	0.0407696	0.0203848	+	0.999792i	$0.493511\pi$
$878$	32.7877	1.10653
$879$	−24.7665	−0.835354
$880$	0	0
$881$	1.91816	0.0646245	0.0323123	−	0.999478i	$-0.489713\pi$
$881$	1.91816	0.0646245	0.0323123	+	0.999478i	$0.489713\pi$
$882$	−13.2218	−0.445200
$883$	−58.0890	−1.95485	−0.977425	−	0.211281i	$-0.932237\pi$
$883$	−58.0890	−1.95485	−0.977425	+	0.211281i	$0.932237\pi$
$884$	−16.7784	−0.564318
$885$	0	0
$886$	17.0162	0.571669
$887$	−30.2427	−1.01545	−0.507725	−	0.861519i	$-0.669513\pi$
$887$	−30.2427	−1.01545	−0.507725	+	0.861519i	$0.669513\pi$
$888$	12.0944	0.405862
$889$	−6.56377	−0.220142
$890$	0	0
$891$	0	0
$892$	−12.4417	−0.416578
$893$	1.33211	0.0445773
$894$	4.40269	0.147248
$895$	0	0
$896$	−12.9432	−0.432403
$897$	1.13586	0.0379251
$898$	−13.9431	−0.465286
$899$	−18.1703	−0.606013
$900$	0	0
$901$	12.6835	0.422550
$902$	0	0
$903$	−38.1603	−1.26990
$904$	23.7468	0.789805
$905$	0	0
$906$	−6.87992	−0.228570
$907$	31.9669	1.06144	0.530721	−	0.847547i	$-0.321921\pi$
$907$	31.9669	1.06144	0.530721	+	0.847547i	$0.321921\pi$
$908$	9.98532	0.331374
$909$	−7.58484	−0.251573
$910$	0	0
$911$	−51.8454	−1.71771	−0.858857	−	0.512216i	$-0.828825\pi$
$911$	−51.8454	−1.71771	−0.858857	+	0.512216i	$0.828825\pi$
$912$	12.7648	0.422684
$913$	0	0
$914$	11.6722	0.386083
$915$	0	0
$916$	−5.78060	−0.190996
$917$	11.4076	0.376714
$918$	−5.81074	−0.191783
$919$	−34.2084	−1.12843	−0.564215	−	0.825628i	$-0.690821\pi$
$919$	−34.2084	−1.12843	−0.564215	+	0.825628i	$0.690821\pi$
$920$	0	0
$921$	−4.94023	−0.162786
$922$	15.6298	0.514741
$923$	51.5790	1.69774
$924$	0	0
$925$	0	0
$926$	21.8293	0.717354
$927$	16.8685	0.554036
$928$	−22.5222	−0.739326
$929$	18.3482	0.601984	0.300992	−	0.953627i	$-0.402682\pi$
$929$	18.3482	0.601984	0.300992	+	0.953627i	$0.402682\pi$
$930$	0	0
$931$	63.7110	2.08804
$932$	14.8301	0.485777
$933$	13.8096	0.452106
$934$	30.8561	1.00964
$935$	0	0
$936$	−15.9100	−0.520033
$937$	−3.38415	−0.110555	−0.0552777	−	0.998471i	$-0.517604\pi$
$937$	−3.38415	−0.110555	−0.0552777	+	0.998471i	$0.517604\pi$
$938$	60.4208	1.97281
$939$	−4.06934	−0.132798
$940$	0	0
$941$	−35.1002	−1.14423	−0.572116	−	0.820172i	$-0.693877\pi$
$941$	−35.1002	−1.14423	−0.572116	+	0.820172i	$0.693877\pi$
$942$	8.63369	0.281301
$943$	1.28836	0.0419547
$944$	17.9399	0.583895
$945$	0	0
$946$	0	0
$947$	39.0512	1.26899	0.634497	−	0.772925i	$-0.281207\pi$
$947$	39.0512	1.26899	0.634497	+	0.772925i	$0.281207\pi$
$948$	5.19657	0.168777
$949$	75.2480	2.44265
$950$	0	0
$951$	25.8199	0.837269
$952$	−65.9961	−2.13895
$953$	−24.6783	−0.799409	−0.399705	−	0.916644i	$-0.630887\pi$
$953$	−24.6783	−0.799409	−0.399705	+	0.916644i	$0.630887\pi$
$954$	2.94803	0.0954458
$955$	0	0
$956$	−10.9495	−0.354131
$957$	0	0
$958$	−12.5315	−0.404873
$959$	−43.9635	−1.41966
$960$	0	0
$961$	−22.9857	−0.741474
$962$	−23.5881	−0.760512
$963$	−7.76161	−0.250114
$964$	7.76747	0.250173
$965$	0	0
$966$	1.09512	0.0352350
$967$	−1.20724	−0.0388222	−0.0194111	−	0.999812i	$-0.506179\pi$
$967$	−1.20724	−0.0388222	−0.0194111	+	0.999812i	$0.506179\pi$
$968$	0	0
$969$	27.9999	0.899486
$970$	0	0
$971$	50.3821	1.61684	0.808418	−	0.588608i	$-0.200324\pi$
$971$	50.3821	1.61684	0.808418	+	0.588608i	$0.200324\pi$
$972$	0.649414	0.0208300
$973$	−22.6991	−0.727701
$974$	7.91533	0.253624
$975$	0	0
$976$	11.0737	0.354459
$977$	5.88330	0.188223	0.0941117	−	0.995562i	$-0.469999\pi$
$977$	5.88330	0.188223	0.0941117	+	0.995562i	$0.469999\pi$
$978$	−6.18362	−0.197730
$979$	0	0
$980$	0	0
$981$	4.45671	0.142292
$982$	−30.3267	−0.967763
$983$	−36.4899	−1.16385	−0.581924	−	0.813243i	$-0.697700\pi$
$983$	−36.4899	−1.16385	−0.581924	+	0.813243i	$0.697700\pi$
$984$	−18.0461	−0.575288
$985$	0	0
$986$	37.2958	1.18774
$987$	−1.01974	−0.0324588
$988$	18.7917	0.597843
$989$	1.95677	0.0622216
$990$	0	0
$991$	−36.8404	−1.17027	−0.585137	−	0.810934i	$-0.698959\pi$
$991$	−36.8404	−1.17027	−0.585137	+	0.810934i	$0.698959\pi$
$992$	9.93376	0.315397
$993$	3.35008	0.106311
$994$	49.7294	1.57732
$995$	0	0
$996$	−8.15206	−0.258308
$997$	−34.8694	−1.10432	−0.552162	−	0.833737i	$-0.686197\pi$
$997$	−34.8694	−1.10432	−0.552162	+	0.833737i	$0.686197\pi$
$998$	−22.5263	−0.713059
$999$	3.92802	0.124277

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9075.2.a.cl.1.3		4
5.4	even	2		1815.2.a.x.1.2		4
11.3	even	5		825.2.n.k.526.2		8
11.4	even	5		825.2.n.k.676.2		8
11.10	odd	2		9075.2.a.dj.1.2		4
15.14	odd	2		5445.2.a.be.1.3		4
55.3	odd	20		825.2.bx.h.724.3		16
55.4	even	10		165.2.m.a.16.1	✓	8
55.14	even	10		165.2.m.a.31.1	yes	8
55.37	odd	20		825.2.bx.h.49.3		16
55.47	odd	20		825.2.bx.h.724.2		16
55.48	odd	20		825.2.bx.h.49.2		16
55.54	odd	2		1815.2.a.o.1.3		4
165.14	odd	10		495.2.n.d.361.2		8
165.59	odd	10		495.2.n.d.181.2		8
165.164	even	2		5445.2.a.bv.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.2.m.a.16.1	✓	8	55.4	even	10
165.2.m.a.31.1	yes	8	55.14	even	10
495.2.n.d.181.2		8	165.59	odd	10
495.2.n.d.361.2		8	165.14	odd	10
825.2.n.k.526.2		8	11.3	even	5
825.2.n.k.676.2		8	11.4	even	5
825.2.bx.h.49.2		16	55.48	odd	20
825.2.bx.h.49.3		16	55.37	odd	20
825.2.bx.h.724.2		16	55.47	odd	20
825.2.bx.h.724.3		16	55.3	odd	20
1815.2.a.o.1.3		4	55.54	odd	2
1815.2.a.x.1.2		4	5.4	even	2
5445.2.a.be.1.3		4	15.14	odd	2
5445.2.a.bv.1.2		4	165.164	even	2
9075.2.a.cl.1.3		4	1.1	even	1	trivial
9075.2.a.dj.1.2		4	11.10	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)

\(f(q)\)	\(=\)	\(q-1.16215 q^{2} -1.00000 q^{3} -0.649414 q^{4} +1.16215 q^{6} +4.28684 q^{7} +3.07901 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.16215 q^{2} -1.00000 q^{3} -0.649414 q^{4} +1.16215 q^{6} +4.28684 q^{7} +3.07901 q^{8} +1.00000 q^{9} +0.649414 q^{12} -5.16724 q^{13} -4.98194 q^{14} -2.27943 q^{16} -5.00000 q^{17} -1.16215 q^{18} +5.59998 q^{19} -4.28684 q^{21} +0.219819 q^{23} -3.07901 q^{24} +6.00509 q^{26} -1.00000 q^{27} -2.78393 q^{28} +6.41843 q^{29} -2.83095 q^{31} -3.50898 q^{32} +5.81074 q^{34} -0.649414 q^{36} -3.92802 q^{37} -6.50800 q^{38} +5.16724 q^{39} +5.86100 q^{41} +4.98194 q^{42} +8.90173 q^{43} -0.255462 q^{46} +0.237878 q^{47} +2.27943 q^{48} +11.3770 q^{49} +5.00000 q^{51} +3.35567 q^{52} -2.53671 q^{53} +1.16215 q^{54} +13.1992 q^{56} -5.59998 q^{57} -7.45917 q^{58} -7.87035 q^{59} -4.85807 q^{61} +3.28999 q^{62} +4.28684 q^{63} +8.63682 q^{64} -12.1280 q^{67} +3.24707 q^{68} -0.219819 q^{69} -9.98194 q^{71} +3.07901 q^{72} -14.5625 q^{73} +4.56494 q^{74} -3.63670 q^{76} -6.00509 q^{78} +8.00194 q^{79} +1.00000 q^{81} -6.81134 q^{82} -12.5530 q^{83} +2.78393 q^{84} -10.3451 q^{86} -6.41843 q^{87} -2.56545 q^{89} -22.1511 q^{91} -0.142753 q^{92} +2.83095 q^{93} -0.276449 q^{94} +3.50898 q^{96} -2.01199 q^{97} -13.2218 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 5 q^{2} - 4 q^{3} + 9 q^{4} + 5 q^{6} + 2 q^{7} - 15 q^{8} + 4 q^{9}+O(q^{10}) \) 4 * q - 5 * q^2 - 4 * q^3 + 9 * q^4 + 5 * q^6 + 2 * q^7 - 15 * q^8 + 4 * q^9	\( 4 q - 5 q^{2} - 4 q^{3} + 9 q^{4} + 5 q^{6} + 2 q^{7} - 15 q^{8} + 4 q^{9} - 9 q^{12} - 3 q^{13} - 5 q^{14} + 15 q^{16} - 20 q^{17} - 5 q^{18} + 3 q^{19} - 2 q^{21} + 5 q^{23} + 15 q^{24} + 6 q^{26} - 4 q^{27} - 3 q^{28} + 5 q^{29} - q^{31} - 30 q^{32} + 25 q^{34} + 9 q^{36} + 7 q^{37} - q^{38} + 3 q^{39} + 20 q^{41} + 5 q^{42} + 2 q^{43} + 7 q^{46} + 20 q^{47} - 15 q^{48} + 8 q^{49} + 20 q^{51} + 7 q^{52} - 6 q^{53} + 5 q^{54} + 10 q^{56} - 3 q^{57} + 21 q^{58} - 5 q^{59} - 7 q^{61} + 12 q^{62} + 2 q^{63} + 49 q^{64} + 13 q^{67} - 45 q^{68} - 5 q^{69} - 25 q^{71} - 15 q^{72} - 23 q^{73} - 7 q^{74} - 7 q^{76} - 6 q^{78} + 4 q^{81} - 11 q^{82} - 33 q^{83} + 3 q^{84} - 12 q^{86} - 5 q^{87} + 16 q^{89} - 24 q^{91} + q^{93} - 17 q^{94} + 30 q^{96} - 25 q^{98}+O(q^{100}) \) 4 * q - 5 * q^2 - 4 * q^3 + 9 * q^4 + 5 * q^6 + 2 * q^7 - 15 * q^8 + 4 * q^9 - 9 * q^12 - 3 * q^13 - 5 * q^14 + 15 * q^16 - 20 * q^17 - 5 * q^18 + 3 * q^19 - 2 * q^21 + 5 * q^23 + 15 * q^24 + 6 * q^26 - 4 * q^27 - 3 * q^28 + 5 * q^29 - q^31 - 30 * q^32 + 25 * q^34 + 9 * q^36 + 7 * q^37 - q^38 + 3 * q^39 + 20 * q^41 + 5 * q^42 + 2 * q^43 + 7 * q^46 + 20 * q^47 - 15 * q^48 + 8 * q^49 + 20 * q^51 + 7 * q^52 - 6 * q^53 + 5 * q^54 + 10 * q^56 - 3 * q^57 + 21 * q^58 - 5 * q^59 - 7 * q^61 + 12 * q^62 + 2 * q^63 + 49 * q^64 + 13 * q^67 - 45 * q^68 - 5 * q^69 - 25 * q^71 - 15 * q^72 - 23 * q^73 - 7 * q^74 - 7 * q^76 - 6 * q^78 + 4 * q^81 - 11 * q^82 - 33 * q^83 + 3 * q^84 - 12 * q^86 - 5 * q^87 + 16 * q^89 - 24 * q^91 + q^93 - 17 * q^94 + 30 * q^96 - 25 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more