LMFDB - Embedded newform 9801.2.a.bq.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9801,2,Mod(1,9801)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9801, 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("9801.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9801 = 3^{4} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9801.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:	$78.2613790211$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.5144904.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 9x^{3} + 4x^{2} + 20x + 3 $ x^5 - x^4 - 9x^3 + 4x^2 + 20*x + 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.52027$ of defining polynomial
Character	$\chi$	$=$	9801.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.52027 q^{2} +0.311228 q^{4} -1.08767 q^{5} +2.25617 q^{7} +2.56739 q^{8} +O(q^{10})$	$q-1.52027 q^{2} +0.311228 q^{4} -1.08767 q^{5} +2.25617 q^{7} +2.56739 q^{8} +1.65355 q^{10} +6.66038 q^{13} -3.42999 q^{14} -4.52559 q^{16} -5.92449 q^{17} +2.94494 q^{19} -0.338512 q^{20} +1.36367 q^{23} -3.81698 q^{25} -10.1256 q^{26} +0.702181 q^{28} +4.90971 q^{29} +1.12796 q^{31} +1.74535 q^{32} +9.00683 q^{34} -2.45396 q^{35} +0.220942 q^{37} -4.47711 q^{38} -2.79247 q^{40} -4.79628 q^{41} +7.34915 q^{43} -2.07315 q^{46} +9.42999 q^{47} -1.90971 q^{49} +5.80285 q^{50} +2.07289 q^{52} -2.06189 q^{53} +5.79247 q^{56} -7.46410 q^{58} -8.60794 q^{59} +12.5676 q^{61} -1.71480 q^{62} +6.39779 q^{64} -7.24427 q^{65} -2.90971 q^{67} -1.84386 q^{68} +3.73068 q^{70} +8.76192 q^{71} -1.17644 q^{73} -0.335892 q^{74} +0.916546 q^{76} +13.9650 q^{79} +4.92233 q^{80} +7.29164 q^{82} -15.8901 q^{83} +6.44386 q^{85} -11.1727 q^{86} -0.635716 q^{89} +15.0269 q^{91} +0.424412 q^{92} -14.3361 q^{94} -3.20311 q^{95} +4.77493 q^{97} +2.90329 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + q^{2} + 9 q^{4} + 4 q^{5} + 3 q^{7} + 12 q^{8}+O(q^{10}) $ 5 * q + q^2 + 9 * q^4 + 4 * q^5 + 3 * q^7 + 12 * q^8	$ 5 q + q^{2} + 9 q^{4} + 4 q^{5} + 3 q^{7} + 12 q^{8} - 5 q^{10} + 13 q^{14} + 9 q^{16} + 4 q^{17} - q^{19} - 7 q^{20} - q^{23} + 15 q^{25} + 7 q^{26} + q^{28} + 3 q^{29} + 24 q^{31} + 20 q^{32} + 25 q^{34} - 26 q^{35} - 13 q^{37} - 33 q^{40} - 17 q^{41} - 4 q^{43} + 32 q^{46} + 17 q^{47} + 12 q^{49} + 47 q^{50} + 13 q^{52} - 6 q^{53} + 48 q^{56} + 15 q^{58} - 25 q^{59} + 17 q^{61} + 49 q^{62} + 2 q^{64} - 26 q^{65} + 7 q^{67} + 2 q^{68} - 23 q^{70} - 6 q^{71} - 15 q^{73} + 24 q^{74} - 37 q^{76} + 19 q^{79} - 52 q^{80} - 11 q^{82} - 3 q^{83} - 5 q^{85} - 6 q^{86} - 3 q^{89} + 20 q^{91} + 40 q^{92} - q^{94} - 15 q^{95} + 4 q^{97} - 12 q^{98}+O(q^{100}) $ 5 * q + q^2 + 9 * q^4 + 4 * q^5 + 3 * q^7 + 12 * q^8 - 5 * q^10 + 13 * q^14 + 9 * q^16 + 4 * q^17 - q^19 - 7 * q^20 - q^23 + 15 * q^25 + 7 * q^26 + q^28 + 3 * q^29 + 24 * q^31 + 20 * q^32 + 25 * q^34 - 26 * q^35 - 13 * q^37 - 33 * q^40 - 17 * q^41 - 4 * q^43 + 32 * q^46 + 17 * q^47 + 12 * q^49 + 47 * q^50 + 13 * q^52 - 6 * q^53 + 48 * q^56 + 15 * q^58 - 25 * q^59 + 17 * q^61 + 49 * q^62 + 2 * q^64 - 26 * q^65 + 7 * q^67 + 2 * q^68 - 23 * q^70 - 6 * q^71 - 15 * q^73 + 24 * q^74 - 37 * q^76 + 19 * q^79 - 52 * q^80 - 11 * q^82 - 3 * q^83 - 5 * q^85 - 6 * q^86 - 3 * q^89 + 20 * q^91 + 40 * q^92 - q^94 - 15 * q^95 + 4 * 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 $

$2$	−1.52027	−1.07499	−0.537497	−	0.843265i	$-0.680630\pi$
$2$	−1.52027	−1.07499	−0.537497	+	0.843265i	$0.680630\pi$
$3$	0	0
$3$	0	0
$4$	0.311228	0.155614
$5$	−1.08767	−0.486419	−0.243210	−	0.969974i	$-0.578200\pi$
$5$	−1.08767	−0.486419	−0.243210	+	0.969974i	$0.578200\pi$
$6$	0	0
$7$	2.25617	0.852751	0.426375	−	0.904546i	$-0.359790\pi$
$7$	2.25617	0.852751	0.426375	+	0.904546i	$0.359790\pi$
$8$	2.56739	0.907711
$9$	0	0
$10$	1.65355	0.522898
$11$	0	0
$11$	0	0
$12$	0	0
$13$	6.66038	1.84726	0.923629	−	0.383289i	$-0.125209\pi$
$13$	6.66038	1.84726	0.923629	+	0.383289i	$0.125209\pi$
$14$	−3.42999	−0.916702
$15$	0	0
$16$	−4.52559	−1.13140
$17$	−5.92449	−1.43690	−0.718449	−	0.695579i	$-0.755148\pi$
$17$	−5.92449	−1.43690	−0.718449	+	0.695579i	$0.755148\pi$
$18$	0	0
$19$	2.94494	0.675615	0.337808	−	0.941215i	$-0.390315\pi$
$19$	2.94494	0.675615	0.337808	+	0.941215i	$0.390315\pi$
$20$	−0.338512	−0.0756935
$21$	0	0
$22$	0	0
$23$	1.36367	0.284345	0.142172	−	0.989842i	$-0.454591\pi$
$23$	1.36367	0.284345	0.142172	+	0.989842i	$0.454591\pi$
$24$	0	0
$25$	−3.81698	−0.763397
$26$	−10.1256	−1.98579
$27$	0	0
$28$	0.702181	0.132700
$29$	4.90971	0.911711	0.455856	−	0.890054i	$-0.349333\pi$
$29$	4.90971	0.911711	0.455856	+	0.890054i	$0.349333\pi$
$30$	0	0
$31$	1.12796	0.202587	0.101293	−	0.994857i	$-0.467702\pi$
$31$	1.12796	0.202587	0.101293	+	0.994857i	$0.467702\pi$
$32$	1.74535	0.308536
$33$	0	0
$34$	9.00683	1.54466
$35$	−2.45396	−0.414794
$36$	0	0
$37$	0.220942	0.0363227	0.0181613	−	0.999835i	$-0.494219\pi$
$37$	0.220942	0.0363227	0.0181613	+	0.999835i	$0.494219\pi$
$38$	−4.47711	−0.726283
$39$	0	0
$40$	−2.79247	−0.441528
$41$	−4.79628	−0.749052	−0.374526	−	0.927216i	$-0.622195\pi$
$41$	−4.79628	−0.749052	−0.374526	+	0.927216i	$0.622195\pi$
$42$	0	0
$43$	7.34915	1.12074	0.560368	−	0.828244i	$-0.310660\pi$
$43$	7.34915	1.12074	0.560368	+	0.828244i	$0.310660\pi$
$44$	0	0
$45$	0	0
$46$	−2.07315	−0.305669
$47$	9.42999	1.37550	0.687752	−	0.725945i	$-0.258597\pi$
$47$	9.42999	1.37550	0.687752	+	0.725945i	$0.258597\pi$
$48$	0	0
$49$	−1.90971	−0.272816
$50$	5.80285	0.820647
$51$	0	0
$52$	2.07289	0.287459
$53$	−2.06189	−0.283223	−0.141611	−	0.989922i	$-0.545228\pi$
$53$	−2.06189	−0.283223	−0.141611	+	0.989922i	$0.545228\pi$
$54$	0	0
$55$	0	0
$56$	5.79247	0.774051
$57$	0	0
$58$	−7.46410	−0.980085
$59$	−8.60794	−1.12066	−0.560329	−	0.828270i	$-0.689325\pi$
$59$	−8.60794	−1.12066	−0.560329	+	0.828270i	$0.689325\pi$
$60$	0	0
$61$	12.5676	1.60912	0.804561	−	0.593870i	$-0.202401\pi$
$61$	12.5676	1.60912	0.804561	+	0.593870i	$0.202401\pi$
$62$	−1.71480	−0.217780
$63$	0	0
$64$	6.39779	0.799723
$65$	−7.24427	−0.898541
$66$	0	0
$67$	−2.90971	−0.355478	−0.177739	−	0.984078i	$-0.556878\pi$
$67$	−2.90971	−0.355478	−0.177739	+	0.984078i	$0.556878\pi$
$68$	−1.84386	−0.223601
$69$	0	0
$70$	3.73068	0.445902
$71$	8.76192	1.03985	0.519924	−	0.854212i	$-0.325960\pi$
$71$	8.76192	1.03985	0.519924	+	0.854212i	$0.325960\pi$
$72$	0	0
$73$	−1.17644	−0.137692	−0.0688460	−	0.997627i	$-0.521932\pi$
$73$	−1.17644	−0.137692	−0.0688460	+	0.997627i	$0.521932\pi$
$74$	−0.335892	−0.0390467
$75$	0	0
$76$	0.916546	0.105135
$77$	0	0
$78$	0	0
$79$	13.9650	1.57119	0.785594	−	0.618742i	$-0.212357\pi$
$79$	13.9650	1.57119	0.785594	+	0.618742i	$0.212357\pi$
$80$	4.92233	0.550334
$81$	0	0
$82$	7.29164	0.805227
$83$	−15.8901	−1.74417	−0.872084	−	0.489356i	$-0.837232\pi$
$83$	−15.8901	−1.74417	−0.872084	+	0.489356i	$0.837232\pi$
$84$	0	0
$85$	6.44386	0.698935
$86$	−11.1727	−1.20478
$87$	0	0
$88$	0	0
$89$	−0.635716	−0.0673857	−0.0336929	−	0.999432i	$-0.510727\pi$
$89$	−0.635716	−0.0673857	−0.0336929	+	0.999432i	$0.510727\pi$
$90$	0	0
$91$	15.0269	1.57525
$92$	0.424412	0.0442480
$93$	0	0
$94$	−14.3361	−1.47866
$95$	−3.20311	−0.328632
$96$	0	0
$97$	4.77493	0.484820	0.242410	−	0.970174i	$-0.422062\pi$
$97$	4.77493	0.484820	0.242410	+	0.970174i	$0.422062\pi$
$98$	2.90329	0.293276
$99$	0	0
$100$	−1.18795	−0.118795
$101$	8.92449	0.888020	0.444010	−	0.896022i	$-0.353556\pi$
$101$	8.92449	0.888020	0.444010	+	0.896022i	$0.353556\pi$
$102$	0	0
$103$	2.93811	0.289500	0.144750	−	0.989468i	$-0.453762\pi$
$103$	2.93811	0.289500	0.144750	+	0.989468i	$0.453762\pi$
$104$	17.0998	1.67678
$105$	0	0
$106$	3.13464	0.304463
$107$	11.0619	1.06939	0.534697	−	0.845044i	$-0.320426\pi$
$107$	11.0619	1.06939	0.534697	+	0.845044i	$0.320426\pi$
$108$	0	0
$109$	−15.3142	−1.46683	−0.733416	−	0.679780i	$-0.762075\pi$
$109$	−15.3142	−1.46683	−0.733416	+	0.679780i	$0.762075\pi$
$110$	0	0
$111$	0	0
$112$	−10.2105	−0.964800
$113$	12.8987	1.21341	0.606704	−	0.794927i	$-0.292491\pi$
$113$	12.8987	1.21341	0.606704	+	0.794927i	$0.292491\pi$
$114$	0	0
$115$	−1.48322	−0.138311
$116$	1.52804	0.141875
$117$	0	0
$118$	13.0864	1.20470
$119$	−13.3666	−1.22532
$120$	0	0
$121$	0	0
$122$	−19.1062	−1.72980
$123$	0	0
$124$	0.351051	0.0315253
$125$	9.58993	0.857750
$126$	0	0
$127$	−20.7553	−1.84174	−0.920869	−	0.389871i	$-0.872520\pi$
$127$	−20.7553	−1.84174	−0.920869	+	0.389871i	$0.872520\pi$
$128$	−13.2171	−1.16823
$129$	0	0
$130$	11.0133	0.965927
$131$	10.5500	0.921758	0.460879	−	0.887463i	$-0.347534\pi$
$131$	10.5500	0.921758	0.460879	+	0.887463i	$0.347534\pi$
$132$	0	0
$133$	6.64427	0.576131
$134$	4.42356	0.382137
$135$	0	0
$136$	−15.2105	−1.30429
$137$	11.9624	1.02202	0.511009	−	0.859575i	$-0.329272\pi$
$137$	11.9624	1.02202	0.511009	+	0.859575i	$0.329272\pi$
$138$	0	0
$139$	−12.6631	−1.07407	−0.537034	−	0.843560i	$-0.680455\pi$
$139$	−12.6631	−1.07407	−0.537034	+	0.843560i	$0.680455\pi$
$140$	−0.763739	−0.0645477
$141$	0	0
$142$	−13.3205	−1.11783
$143$	0	0
$144$	0	0
$145$	−5.34013	−0.443474
$146$	1.78851	0.148018
$147$	0	0
$148$	0.0687633	0.00565231
$149$	−13.3131	−1.09065	−0.545325	−	0.838225i	$-0.683594\pi$
$149$	−13.3131	−1.09065	−0.545325	+	0.838225i	$0.683594\pi$
$150$	0	0
$151$	8.26300	0.672433	0.336217	−	0.941785i	$-0.390853\pi$
$151$	8.26300	0.672433	0.336217	+	0.941785i	$0.390853\pi$
$152$	7.56082	0.613263
$153$	0	0
$154$	0	0
$155$	−1.22684	−0.0985421
$156$	0	0
$157$	22.3628	1.78475	0.892374	−	0.451298i	$-0.149039\pi$
$157$	22.3628	1.78475	0.892374	+	0.451298i	$0.149039\pi$
$158$	−21.2306	−1.68902
$159$	0	0
$160$	−1.89835	−0.150078
$161$	3.07666	0.242475
$162$	0	0
$163$	9.15930	0.717412	0.358706	−	0.933451i	$-0.383218\pi$
$163$	9.15930	0.717412	0.358706	+	0.933451i	$0.383218\pi$
$164$	−1.49273	−0.116563
$165$	0	0
$166$	24.1573	1.87497
$167$	−19.9346	−1.54259	−0.771294	−	0.636479i	$-0.780390\pi$
$167$	−19.9346	−1.54259	−0.771294	+	0.636479i	$0.780390\pi$
$168$	0	0
$169$	31.3607	2.41236
$170$	−9.79642	−0.751351
$171$	0	0
$172$	2.28726	0.174402
$173$	23.3672	1.77658	0.888289	−	0.459285i	$-0.151894\pi$
$173$	23.3672	1.77658	0.888289	+	0.459285i	$0.151894\pi$
$174$	0	0
$175$	−8.61175	−0.650987
$176$	0	0
$177$	0	0
$178$	0.966461	0.0724393
$179$	−10.8595	−0.811678	−0.405839	−	0.913945i	$-0.633021\pi$
$179$	−10.8595	−0.811678	−0.405839	+	0.913945i	$0.633021\pi$
$180$	0	0
$181$	12.4309	0.923980	0.461990	−	0.886885i	$-0.347136\pi$
$181$	12.4309	0.923980	0.461990	+	0.886885i	$0.347136\pi$
$182$	−22.8450	−1.69339
$183$	0	0
$184$	3.50108	0.258103
$185$	−0.240311	−0.0176680
$186$	0	0
$187$	0	0
$188$	2.93487	0.214048
$189$	0	0
$190$	4.86960	0.353278
$191$	−2.34494	−0.169674	−0.0848370	−	0.996395i	$-0.527037\pi$
$191$	−2.34494	−0.169674	−0.0848370	+	0.996395i	$0.527037\pi$
$192$	0	0
$193$	−4.17644	−0.300627	−0.150313	−	0.988638i	$-0.548028\pi$
$193$	−4.17644	−0.300627	−0.150313	+	0.988638i	$0.548028\pi$
$194$	−7.25919	−0.521179
$195$	0	0
$196$	−0.594356	−0.0424540
$197$	−15.4170	−1.09841	−0.549207	−	0.835686i	$-0.685070\pi$
$197$	−15.4170	−1.09841	−0.549207	+	0.835686i	$0.685070\pi$
$198$	0	0
$199$	−5.76678	−0.408796	−0.204398	−	0.978888i	$-0.565524\pi$
$199$	−5.76678	−0.408796	−0.204398	+	0.978888i	$0.565524\pi$
$200$	−9.79970	−0.692943
$201$	0	0
$202$	−13.5676	−0.954616
$203$	11.0771	0.777462
$204$	0	0
$205$	5.21675	0.364353
$206$	−4.46672	−0.311211
$207$	0	0
$208$	−30.1422	−2.08998
$209$	0	0
$210$	0	0
$211$	10.6959	0.736333	0.368167	−	0.929760i	$-0.379986\pi$
$211$	10.6959	0.736333	0.368167	+	0.929760i	$0.379986\pi$
$212$	−0.641718	−0.0440734
$213$	0	0
$214$	−16.8171	−1.14959
$215$	−7.99342	−0.545147
$216$	0	0
$217$	2.54486	0.172756
$218$	23.2817	1.57684
$219$	0	0
$220$	0	0
$221$	−39.4593	−2.65432
$222$	0	0
$223$	−21.5772	−1.44491	−0.722457	−	0.691415i	$-0.756987\pi$
$223$	−21.5772	−1.44491	−0.722457	+	0.691415i	$0.756987\pi$
$224$	3.93779	0.263105
$225$	0	0
$226$	−19.6096	−1.30441
$227$	4.17914	0.277379	0.138690	−	0.990336i	$-0.455711\pi$
$227$	4.17914	0.277379	0.138690	+	0.990336i	$0.455711\pi$
$228$	0	0
$229$	−19.5701	−1.29323	−0.646614	−	0.762817i	$-0.723816\pi$
$229$	−19.5701	−1.29323	−0.646614	+	0.762817i	$0.723816\pi$
$230$	2.25489	0.148683
$231$	0	0
$232$	12.6052	0.827570
$233$	−10.4641	−0.685526	−0.342763	−	0.939422i	$-0.611363\pi$
$233$	−10.4641	−0.685526	−0.342763	+	0.939422i	$0.611363\pi$
$234$	0	0
$235$	−10.2567	−0.669072
$236$	−2.67903	−0.174390
$237$	0	0
$238$	20.3209	1.31721
$239$	−13.8412	−0.895316	−0.447658	−	0.894205i	$-0.647742\pi$
$239$	−13.8412	−0.895316	−0.447658	+	0.894205i	$0.647742\pi$
$240$	0	0
$241$	2.13234	0.137356	0.0686781	−	0.997639i	$-0.478122\pi$
$241$	2.13234	0.137356	0.0686781	+	0.997639i	$0.478122\pi$
$242$	0	0
$243$	0	0
$244$	3.91140	0.250402
$245$	2.07713	0.132703
$246$	0	0
$247$	19.6144	1.24804
$248$	2.89591	0.183890
$249$	0	0
$250$	−14.5793	−0.922076
$251$	−15.5736	−0.982996	−0.491498	−	0.870879i	$-0.663550\pi$
$251$	−15.5736	−0.982996	−0.491498	+	0.870879i	$0.663550\pi$
$252$	0	0
$253$	0	0
$254$	31.5538	1.97986
$255$	0	0
$256$	7.29797	0.456123
$257$	−6.94630	−0.433298	−0.216649	−	0.976250i	$-0.569513\pi$
$257$	−6.94630	−0.433298	−0.216649	+	0.976250i	$0.569513\pi$
$258$	0	0
$259$	0.498482	0.0309742
$260$	−2.25462	−0.139825
$261$	0	0
$262$	−16.0389	−0.990885
$263$	−28.7587	−1.77334	−0.886668	−	0.462406i	$-0.846986\pi$
$263$	−28.7587	−1.77334	−0.886668	+	0.462406i	$0.846986\pi$
$264$	0	0
$265$	2.24265	0.137765
$266$	−10.1011	−0.619338
$267$	0	0
$268$	−0.905583	−0.0553173
$269$	7.91233	0.482424	0.241212	−	0.970473i	$-0.422455\pi$
$269$	7.91233	0.482424	0.241212	+	0.970473i	$0.422455\pi$
$270$	0	0
$271$	−14.4421	−0.877297	−0.438649	−	0.898659i	$-0.644543\pi$
$271$	−14.4421	−0.877297	−0.438649	+	0.898659i	$0.644543\pi$
$272$	26.8118	1.62570
$273$	0	0
$274$	−18.1861	−1.09866
$275$	0	0
$276$	0	0
$277$	13.5300	0.812940	0.406470	−	0.913664i	$-0.366760\pi$
$277$	13.5300	0.812940	0.406470	+	0.913664i	$0.366760\pi$
$278$	19.2513	1.15462
$279$	0	0
$280$	−6.30027	−0.376513
$281$	−5.25347	−0.313395	−0.156698	−	0.987647i	$-0.550085\pi$
$281$	−5.25347	−0.313395	−0.156698	+	0.987647i	$0.550085\pi$
$282$	0	0
$283$	17.8919	1.06356	0.531781	−	0.846882i	$-0.321523\pi$
$283$	17.8919	1.06356	0.531781	+	0.846882i	$0.321523\pi$
$284$	2.72695	0.161815
$285$	0	0
$286$	0	0
$287$	−10.8212	−0.638755
$288$	0	0
$289$	18.0995	1.06468
$290$	8.11845	0.476732
$291$	0	0
$292$	−0.366141	−0.0214268
$293$	−9.22950	−0.539193	−0.269596	−	0.962973i	$-0.586890\pi$
$293$	−9.22950	−0.539193	−0.269596	+	0.962973i	$0.586890\pi$
$294$	0	0
$295$	9.36256	0.545109
$296$	0.567245	0.0329705
$297$	0	0
$298$	20.2395	1.17244
$299$	9.08256	0.525258
$300$	0	0
$301$	16.5809	0.955708
$302$	−12.5620	−0.722862
$303$	0	0
$304$	−13.3276	−0.764390
$305$	−13.6694	−0.782708
$306$	0	0
$307$	30.2755	1.72791	0.863957	−	0.503566i	$-0.167979\pi$
$307$	30.2755	1.72791	0.863957	+	0.503566i	$0.167979\pi$
$308$	0	0
$309$	0	0
$310$	1.86513	0.105932
$311$	−7.17472	−0.406841	−0.203420	−	0.979091i	$-0.565206\pi$
$311$	−7.17472	−0.406841	−0.203420	+	0.979091i	$0.565206\pi$
$312$	0	0
$313$	−29.7516	−1.68166	−0.840831	−	0.541297i	$-0.817933\pi$
$313$	−29.7516	−1.68166	−0.840831	+	0.541297i	$0.817933\pi$
$314$	−33.9976	−1.91859
$315$	0	0
$316$	4.34630	0.244499
$317$	29.2162	1.64095	0.820473	−	0.571685i	$-0.193710\pi$
$317$	29.2162	1.64095	0.820473	+	0.571685i	$0.193710\pi$
$318$	0	0
$319$	0	0
$320$	−6.95865	−0.389001
$321$	0	0
$322$	−4.67737	−0.260660
$323$	−17.4472	−0.970791
$324$	0	0
$325$	−25.4226	−1.41019
$326$	−13.9246	−0.771214
$327$	0	0
$328$	−12.3139	−0.679923
$329$	21.2756	1.17296
$330$	0	0
$331$	30.5439	1.67885	0.839423	−	0.543478i	$-0.182893\pi$
$331$	30.5439	1.67885	0.839423	+	0.543478i	$0.182893\pi$
$332$	−4.94545	−0.271417
$333$	0	0
$334$	30.3061	1.65827
$335$	3.16480	0.172911
$336$	0	0
$337$	23.1138	1.25909	0.629544	−	0.776965i	$-0.283242\pi$
$337$	23.1138	1.25909	0.629544	+	0.776965i	$0.283242\pi$
$338$	−47.6767	−2.59327
$339$	0	0
$340$	2.00551	0.108764
$341$	0	0
$342$	0	0
$343$	−20.1018	−1.08539
$344$	18.8682	1.01730
$345$	0	0
$346$	−35.5246	−1.90981
$347$	25.5935	1.37393	0.686966	−	0.726690i	$-0.258942\pi$
$347$	25.5935	1.37393	0.686966	+	0.726690i	$0.258942\pi$
$348$	0	0
$349$	−2.43088	−0.130122	−0.0650611	−	0.997881i	$-0.520724\pi$
$349$	−2.43088	−0.130122	−0.0650611	+	0.997881i	$0.520724\pi$
$350$	13.0922	0.699808
$351$	0	0
$352$	0	0
$353$	−0.825819	−0.0439539	−0.0219770	−	0.999758i	$-0.506996\pi$
$353$	−0.825819	−0.0439539	−0.0219770	+	0.999758i	$0.506996\pi$
$354$	0	0
$355$	−9.53004	−0.505802
$356$	−0.197852	−0.0104861
$357$	0	0
$358$	16.5094	0.872549
$359$	−29.7329	−1.56924	−0.784622	−	0.619975i	$-0.787143\pi$
$359$	−29.7329	−1.56924	−0.784622	+	0.619975i	$0.787143\pi$
$360$	0	0
$361$	−10.3273	−0.543544
$362$	−18.8983	−0.993274
$363$	0	0
$364$	4.67679	0.245131
$365$	1.27957	0.0669760
$366$	0	0
$367$	36.3364	1.89674	0.948372	−	0.317159i	$-0.102729\pi$
$367$	36.3364	1.89674	0.948372	+	0.317159i	$0.102729\pi$
$368$	−6.17141	−0.321707
$369$	0	0
$370$	0.365338	0.0189930
$371$	−4.65197	−0.241518
$372$	0	0
$373$	0.309250	0.0160123	0.00800617	−	0.999968i	$-0.497452\pi$
$373$	0.309250	0.0160123	0.00800617	+	0.999968i	$0.497452\pi$
$374$	0	0
$375$	0	0
$376$	24.2105	1.24856
$377$	32.7006	1.68416
$378$	0	0
$379$	−16.7314	−0.859434	−0.429717	−	0.902964i	$-0.641387\pi$
$379$	−16.7314	−0.859434	−0.429717	+	0.902964i	$0.641387\pi$
$380$	−0.996896	−0.0511397
$381$	0	0
$382$	3.56495	0.182399
$383$	−12.3894	−0.633071	−0.316535	−	0.948581i	$-0.602520\pi$
$383$	−12.3894	−0.633071	−0.316535	+	0.948581i	$0.602520\pi$
$384$	0	0
$385$	0	0
$386$	6.34933	0.323172
$387$	0	0
$388$	1.48609	0.0754447
$389$	−21.8529	−1.10799	−0.553994	−	0.832521i	$-0.686897\pi$
$389$	−21.8529	−1.10799	−0.553994	+	0.832521i	$0.686897\pi$
$390$	0	0
$391$	−8.07904	−0.408575
$392$	−4.90299	−0.247638
$393$	0	0
$394$	23.4380	1.18079
$395$	−15.1893	−0.764256
$396$	0	0
$397$	21.9721	1.10275	0.551374	−	0.834258i	$-0.314104\pi$
$397$	21.9721	1.10275	0.551374	+	0.834258i	$0.314104\pi$
$398$	8.76707	0.439453
$399$	0	0
$400$	17.2741	0.863705
$401$	16.2137	0.809675	0.404837	−	0.914389i	$-0.367328\pi$
$401$	16.2137	0.809675	0.404837	+	0.914389i	$0.367328\pi$
$402$	0	0
$403$	7.51261	0.374230
$404$	2.77755	0.138188
$405$	0	0
$406$	−16.8403	−0.835768
$407$	0	0
$408$	0	0
$409$	9.02007	0.446014	0.223007	−	0.974817i	$-0.428413\pi$
$409$	9.02007	0.446014	0.223007	+	0.974817i	$0.428413\pi$
$410$	−7.93087	−0.391678
$411$	0	0
$412$	0.914420	0.0450502
$413$	−19.4209	−0.955642
$414$	0	0
$415$	17.2832	0.848397
$416$	11.6247	0.569946
$417$	0	0
$418$	0	0
$419$	−22.7915	−1.11344	−0.556718	−	0.830702i	$-0.687940\pi$
$419$	−22.7915	−1.11344	−0.556718	+	0.830702i	$0.687940\pi$
$420$	0	0
$421$	20.5745	1.00274	0.501370	−	0.865233i	$-0.332829\pi$
$421$	20.5745	1.00274	0.501370	+	0.865233i	$0.332829\pi$
$422$	−16.2606	−0.791554
$423$	0	0
$424$	−5.29369	−0.257084
$425$	22.6137	1.09692
$426$	0	0
$427$	28.3547	1.37218
$428$	3.44277	0.166412
$429$	0	0
$430$	12.1522	0.586030
$431$	5.19653	0.250308	0.125154	−	0.992137i	$-0.460057\pi$
$431$	5.19653	0.250308	0.125154	+	0.992137i	$0.460057\pi$
$432$	0	0
$433$	3.76824	0.181090	0.0905451	−	0.995892i	$-0.471139\pi$
$433$	3.76824	0.181090	0.0905451	+	0.995892i	$0.471139\pi$
$434$	−3.86887	−0.185712
$435$	0	0
$436$	−4.76620	−0.228259
$437$	4.01592	0.192108
$438$	0	0
$439$	−11.1730	−0.533257	−0.266628	−	0.963799i	$-0.585910\pi$
$439$	−11.1730	−0.533257	−0.266628	+	0.963799i	$0.585910\pi$
$440$	0	0
$441$	0	0
$442$	59.9889	2.85338
$443$	−16.1963	−0.769509	−0.384755	−	0.923019i	$-0.625714\pi$
$443$	−16.1963	−0.769509	−0.384755	+	0.923019i	$0.625714\pi$
$444$	0	0
$445$	0.691446	0.0327777
$446$	32.8032	1.55328
$447$	0	0
$448$	14.4345	0.681964
$449$	−3.27004	−0.154323	−0.0771614	−	0.997019i	$-0.524586\pi$
$449$	−3.27004	−0.154323	−0.0771614	+	0.997019i	$0.524586\pi$
$450$	0	0
$451$	0	0
$452$	4.01444	0.188823
$453$	0	0
$454$	−6.35343	−0.298181
$455$	−16.3443	−0.766231
$456$	0	0
$457$	−15.2626	−0.713954	−0.356977	−	0.934113i	$-0.616193\pi$
$457$	−15.2626	−0.713954	−0.356977	+	0.934113i	$0.616193\pi$
$458$	29.7519	1.39021
$459$	0	0
$460$	−0.461618	−0.0215231
$461$	7.82454	0.364425	0.182213	−	0.983259i	$-0.441674\pi$
$461$	7.82454	0.364425	0.182213	+	0.983259i	$0.441674\pi$
$462$	0	0
$463$	16.8395	0.782599	0.391299	−	0.920263i	$-0.372026\pi$
$463$	16.8395	0.782599	0.391299	+	0.920263i	$0.372026\pi$
$464$	−22.2194	−1.03151
$465$	0	0
$466$	15.9083	0.736937
$467$	38.0169	1.75921	0.879606	−	0.475704i	$-0.157807\pi$
$467$	38.0169	1.75921	0.879606	+	0.475704i	$0.157807\pi$
$468$	0	0
$469$	−6.56480	−0.303134
$470$	15.5929	0.719249
$471$	0	0
$472$	−22.1000	−1.01723
$473$	0	0
$474$	0	0
$475$	−11.2408	−0.515762
$476$	−4.16006	−0.190676
$477$	0	0
$478$	21.0425	0.962460
$479$	−15.5460	−0.710317	−0.355158	−	0.934806i	$-0.615573\pi$
$479$	−15.5460	−0.710317	−0.355158	+	0.934806i	$0.615573\pi$
$480$	0	0
$481$	1.47156	0.0670973
$482$	−3.24174	−0.147657
$483$	0	0
$484$	0	0
$485$	−5.19353	−0.235826
$486$	0	0
$487$	−21.1203	−0.957053	−0.478526	−	0.878073i	$-0.658829\pi$
$487$	−21.1203	−0.957053	−0.478526	+	0.878073i	$0.658829\pi$
$488$	32.2661	1.46062
$489$	0	0
$490$	−3.15781	−0.142655
$491$	41.3904	1.86792	0.933960	−	0.357377i	$-0.116329\pi$
$491$	41.3904	1.86792	0.933960	+	0.357377i	$0.116329\pi$
$492$	0	0
$493$	−29.0875	−1.31004
$494$	−29.8192	−1.34163
$495$	0	0
$496$	−5.10467	−0.229206
$497$	19.7683	0.886732
$498$	0	0
$499$	7.07932	0.316914	0.158457	−	0.987366i	$-0.449348\pi$
$499$	7.07932	0.316914	0.158457	+	0.987366i	$0.449348\pi$
$500$	2.98465	0.133478
$501$	0	0
$502$	23.6761	1.05672
$503$	44.1749	1.96966	0.984830	−	0.173524i	$-0.0555154\pi$
$503$	44.1749	1.96966	0.984830	+	0.173524i	$0.0555154\pi$
$504$	0	0
$505$	−9.70686	−0.431950
$506$	0	0
$507$	0	0
$508$	−6.45964	−0.286600
$509$	−25.8703	−1.14668	−0.573341	−	0.819317i	$-0.694353\pi$
$509$	−25.8703	−1.14668	−0.573341	+	0.819317i	$0.694353\pi$
$510$	0	0
$511$	−2.65424	−0.117417
$512$	15.3392	0.677905
$513$	0	0
$514$	10.5603	0.465793
$515$	−3.19568	−0.140818
$516$	0	0
$517$	0	0
$518$	−0.757828	−0.0332971
$519$	0	0
$520$	−18.5989	−0.815615
$521$	25.5813	1.12074	0.560369	−	0.828243i	$-0.310659\pi$
$521$	25.5813	1.12074	0.560369	+	0.828243i	$0.310659\pi$
$522$	0	0
$523$	−7.58921	−0.331853	−0.165926	−	0.986138i	$-0.553061\pi$
$523$	−7.58921	−0.331853	−0.165926	+	0.986138i	$0.553061\pi$
$524$	3.28345	0.143438
$525$	0	0
$526$	43.7210	1.90633
$527$	−6.68256	−0.291097
$528$	0	0
$529$	−21.1404	−0.919148
$530$	−3.40944	−0.148097
$531$	0	0
$532$	2.06788	0.0896540
$533$	−31.9450	−1.38369
$534$	0	0
$535$	−12.0316	−0.520173
$536$	−7.47038	−0.322671
$537$	0	0
$538$	−12.0289	−0.518603
$539$	0	0
$540$	0	0
$541$	−31.6585	−1.36111	−0.680553	−	0.732699i	$-0.738260\pi$
$541$	−31.6585	−1.36111	−0.680553	+	0.732699i	$0.738260\pi$
$542$	21.9560	0.943090
$543$	0	0
$544$	−10.3403	−0.443336
$545$	16.6567	0.713495
$546$	0	0
$547$	30.8160	1.31760	0.658798	−	0.752320i	$-0.271065\pi$
$547$	30.8160	1.31760	0.658798	+	0.752320i	$0.271065\pi$
$548$	3.72303	0.159040
$549$	0	0
$550$	0	0
$551$	14.4588	0.615966
$552$	0	0
$553$	31.5074	1.33983
$554$	−20.5693	−0.873906
$555$	0	0
$556$	−3.94110	−0.167140
$557$	4.98818	0.211356	0.105678	−	0.994400i	$-0.466299\pi$
$557$	4.98818	0.211356	0.105678	+	0.994400i	$0.466299\pi$
$558$	0	0
$559$	48.9481	2.07029
$560$	11.1056	0.469297
$561$	0	0
$562$	7.98670	0.336898
$563$	−3.03235	−0.127798	−0.0638992	−	0.997956i	$-0.520354\pi$
$563$	−3.03235	−0.127798	−0.0638992	+	0.997956i	$0.520354\pi$
$564$	0	0
$565$	−14.0295	−0.590225
$566$	−27.2005	−1.14332
$567$	0	0
$568$	22.4953	0.943882
$569$	47.5101	1.99173	0.995863	−	0.0908639i	$-0.0289628\pi$
$569$	47.5101	1.99173	0.995863	+	0.0908639i	$0.0289628\pi$
$570$	0	0
$571$	8.47272	0.354572	0.177286	−	0.984159i	$-0.443268\pi$
$571$	8.47272	0.354572	0.177286	+	0.984159i	$0.443268\pi$
$572$	0	0
$573$	0	0
$574$	16.4512	0.686658
$575$	−5.20510	−0.217068
$576$	0	0
$577$	−35.2635	−1.46804	−0.734020	−	0.679128i	$-0.762358\pi$
$577$	−35.2635	−1.46804	−0.734020	+	0.679128i	$0.762358\pi$
$578$	−27.5162	−1.14452
$579$	0	0
$580$	−1.66200	−0.0690106
$581$	−35.8508	−1.48734
$582$	0	0
$583$	0	0
$584$	−3.02039	−0.124984
$585$	0	0
$586$	14.0313	0.579630
$587$	25.3219	1.04515	0.522574	−	0.852594i	$-0.324972\pi$
$587$	25.3219	1.04515	0.522574	+	0.852594i	$0.324972\pi$
$588$	0	0
$589$	3.32176	0.136871
$590$	−14.2336	−0.585990
$591$	0	0
$592$	−0.999894	−0.0410954
$593$	39.8626	1.63696	0.818480	−	0.574535i	$-0.194817\pi$
$593$	39.8626	1.63696	0.818480	+	0.574535i	$0.194817\pi$
$594$	0	0
$595$	14.5384	0.596017
$596$	−4.14340	−0.169720
$597$	0	0
$598$	−13.8080	−0.564649
$599$	31.9030	1.30352	0.651761	−	0.758425i	$-0.274031\pi$
$599$	31.9030	1.30352	0.651761	+	0.758425i	$0.274031\pi$
$600$	0	0
$601$	18.0780	0.737417	0.368709	−	0.929545i	$-0.379800\pi$
$601$	18.0780	0.737417	0.368709	+	0.929545i	$0.379800\pi$
$602$	−25.2075	−1.02738
$603$	0	0
$604$	2.57167	0.104640
$605$	0	0
$606$	0	0
$607$	−1.54758	−0.0628145	−0.0314072	−	0.999507i	$-0.509999\pi$
$607$	−1.54758	−0.0628145	−0.0314072	+	0.999507i	$0.509999\pi$
$608$	5.13993	0.208452
$609$	0	0
$610$	20.7812	0.841407
$611$	62.8073	2.54091
$612$	0	0
$613$	10.7046	0.432355	0.216177	−	0.976354i	$-0.430641\pi$
$613$	10.7046	0.432355	0.216177	+	0.976354i	$0.430641\pi$
$614$	−46.0270	−1.85750
$615$	0	0
$616$	0	0
$617$	−32.0693	−1.29106	−0.645530	−	0.763735i	$-0.723364\pi$
$617$	−32.0693	−1.29106	−0.645530	+	0.763735i	$0.723364\pi$
$618$	0	0
$619$	14.2834	0.574098	0.287049	−	0.957916i	$-0.407326\pi$
$619$	14.2834	0.574098	0.287049	+	0.957916i	$0.407326\pi$
$620$	−0.381826	−0.0153345
$621$	0	0
$622$	10.9075	0.437352
$623$	−1.43428	−0.0574632
$624$	0	0
$625$	8.65427	0.346171
$626$	45.2306	1.80778
$627$	0	0
$628$	6.95993	0.277731
$629$	−1.30897	−0.0521920
$630$	0	0
$631$	−20.6442	−0.821833	−0.410917	−	0.911673i	$-0.634791\pi$
$631$	−20.6442	−0.821833	−0.410917	+	0.911673i	$0.634791\pi$
$632$	35.8537	1.42618
$633$	0	0
$634$	−44.4166	−1.76401
$635$	22.5749	0.895857
$636$	0	0
$637$	−12.7194	−0.503962
$638$	0	0
$639$	0	0
$640$	14.3758	0.568252
$641$	11.7046	0.462304	0.231152	−	0.972918i	$-0.425751\pi$
$641$	11.7046	0.462304	0.231152	+	0.972918i	$0.425751\pi$
$642$	0	0
$643$	18.4574	0.727888	0.363944	−	0.931421i	$-0.381430\pi$
$643$	18.4574	0.727888	0.363944	+	0.931421i	$0.381430\pi$
$644$	0.957543	0.0377325
$645$	0	0
$646$	26.5246	1.04360
$647$	26.2873	1.03346	0.516730	−	0.856148i	$-0.327149\pi$
$647$	26.2873	1.03346	0.516730	+	0.856148i	$0.327149\pi$
$648$	0	0
$649$	0	0
$650$	38.6492	1.51595
$651$	0	0
$652$	2.85063	0.111639
$653$	27.9054	1.09202	0.546011	−	0.837778i	$-0.316146\pi$
$653$	27.9054	1.09202	0.546011	+	0.837778i	$0.316146\pi$
$654$	0	0
$655$	−11.4749	−0.448361
$656$	21.7060	0.847477
$657$	0	0
$658$	−32.3447	−1.26093
$659$	−45.9712	−1.79078	−0.895392	−	0.445278i	$-0.853105\pi$
$659$	−45.9712	−1.79078	−0.895392	+	0.445278i	$0.853105\pi$
$660$	0	0
$661$	−12.1654	−0.473181	−0.236590	−	0.971609i	$-0.576030\pi$
$661$	−12.1654	−0.473181	−0.236590	+	0.971609i	$0.576030\pi$
$662$	−46.4351	−1.80475
$663$	0	0
$664$	−40.7962	−1.58320
$665$	−7.22675	−0.280241
$666$	0	0
$667$	6.69523	0.259240
$668$	−6.20421	−0.240048
$669$	0	0
$670$	−4.81135	−0.185879
$671$	0	0
$672$	0	0
$673$	−38.4548	−1.48232	−0.741161	−	0.671327i	$-0.765725\pi$
$673$	−38.4548	−1.48232	−0.741161	+	0.671327i	$0.765725\pi$
$674$	−35.1392	−1.35351
$675$	0	0
$676$	9.76030	0.375396
$677$	−2.09086	−0.0803583	−0.0401791	−	0.999192i	$-0.512793\pi$
$677$	−2.09086	−0.0803583	−0.0401791	+	0.999192i	$0.512793\pi$
$678$	0	0
$679$	10.7730	0.413431
$680$	16.5439	0.634431
$681$	0	0
$682$	0	0
$683$	41.6302	1.59294	0.796468	−	0.604681i	$-0.206699\pi$
$683$	41.6302	1.59294	0.796468	+	0.604681i	$0.206699\pi$
$684$	0	0
$685$	−13.0111	−0.497129
$686$	30.5602	1.16679
$687$	0	0
$688$	−33.2593	−1.26800
$689$	−13.7330	−0.523185
$690$	0	0
$691$	20.5196	0.780603	0.390301	−	0.920687i	$-0.372371\pi$
$691$	20.5196	0.780603	0.390301	+	0.920687i	$0.372371\pi$
$692$	7.27253	0.276460
$693$	0	0
$694$	−38.9091	−1.47697
$695$	13.7732	0.522447
$696$	0	0
$697$	28.4155	1.07631
$698$	3.69560	0.139881
$699$	0	0
$700$	−2.68021	−0.101303
$701$	12.6814	0.478972	0.239486	−	0.970900i	$-0.423021\pi$
$701$	12.6814	0.478972	0.239486	+	0.970900i	$0.423021\pi$
$702$	0	0
$703$	0.650661	0.0245401
$704$	0	0
$705$	0	0
$706$	1.25547	0.0472502
$707$	20.1351	0.757259
$708$	0	0
$709$	19.5544	0.734380	0.367190	−	0.930146i	$-0.380320\pi$
$709$	19.5544	0.734380	0.367190	+	0.930146i	$0.380320\pi$
$710$	14.4883	0.543735
$711$	0	0
$712$	−1.63213	−0.0611667
$713$	1.53816	0.0576045
$714$	0	0
$715$	0	0
$716$	−3.37978	−0.126308
$717$	0	0
$718$	45.2021	1.68693
$719$	32.9171	1.22760	0.613800	−	0.789462i	$-0.289640\pi$
$719$	32.9171	1.22760	0.613800	+	0.789462i	$0.289640\pi$
$720$	0	0
$721$	6.62886	0.246872
$722$	15.7004	0.584307
$723$	0	0
$724$	3.86883	0.143784
$725$	−18.7403	−0.695997
$726$	0	0
$727$	5.34049	0.198068	0.0990339	−	0.995084i	$-0.468425\pi$
$727$	5.34049	0.198068	0.0990339	+	0.995084i	$0.468425\pi$
$728$	38.5800	1.42987
$729$	0	0
$730$	−1.94530	−0.0719988
$731$	−43.5400	−1.61038
$732$	0	0
$733$	1.98487	0.0733128	0.0366564	−	0.999328i	$-0.488329\pi$
$733$	1.98487	0.0733128	0.0366564	+	0.999328i	$0.488329\pi$
$734$	−55.2412	−2.03899
$735$	0	0
$736$	2.38007	0.0877307
$737$	0	0
$738$	0	0
$739$	6.52951	0.240192	0.120096	−	0.992762i	$-0.461680\pi$
$739$	6.52951	0.240192	0.120096	+	0.992762i	$0.461680\pi$
$740$	−0.0747915	−0.00274939
$741$	0	0
$742$	7.07227	0.259631
$743$	7.07132	0.259421	0.129711	−	0.991552i	$-0.458595\pi$
$743$	7.07132	0.259421	0.129711	+	0.991552i	$0.458595\pi$
$744$	0	0
$745$	14.4802	0.530513
$746$	−0.470144	−0.0172132
$747$	0	0
$748$	0	0
$749$	24.9575	0.911926
$750$	0	0
$751$	−24.6750	−0.900403	−0.450202	−	0.892927i	$-0.648648\pi$
$751$	−24.6750	−0.900403	−0.450202	+	0.892927i	$0.648648\pi$
$752$	−42.6763	−1.55624
$753$	0	0
$754$	−49.7138	−1.81047
$755$	−8.98738	−0.327084
$756$	0	0
$757$	4.53023	0.164654	0.0823271	−	0.996605i	$-0.473765\pi$
$757$	4.53023	0.164654	0.0823271	+	0.996605i	$0.473765\pi$
$758$	25.4363	0.923888
$759$	0	0
$760$	−8.22364	−0.298303
$761$	33.9151	1.22942	0.614710	−	0.788753i	$-0.289273\pi$
$761$	33.9151	1.22942	0.614710	+	0.788753i	$0.289273\pi$
$762$	0	0
$763$	−34.5513	−1.25084
$764$	−0.729810	−0.0264036
$765$	0	0
$766$	18.8353	0.680548
$767$	−57.3321	−2.07014
$768$	0	0
$769$	−2.78366	−0.100381	−0.0501906	−	0.998740i	$-0.515983\pi$
$769$	−2.78366	−0.100381	−0.0501906	+	0.998740i	$0.515983\pi$
$770$	0	0
$771$	0	0
$772$	−1.29982	−0.0467817
$773$	7.15532	0.257359	0.128680	−	0.991686i	$-0.458926\pi$
$773$	7.15532	0.257359	0.128680	+	0.991686i	$0.458926\pi$
$774$	0	0
$775$	−4.30539	−0.154654
$776$	12.2591	0.440077
$777$	0	0
$778$	33.2224	1.19108
$779$	−14.1247	−0.506071
$780$	0	0
$781$	0	0
$782$	12.2823	0.439216
$783$	0	0
$784$	8.64259	0.308664
$785$	−24.3233	−0.868135
$786$	0	0
$787$	−20.0978	−0.716411	−0.358205	−	0.933643i	$-0.616611\pi$
$787$	−20.0978	−0.716411	−0.358205	+	0.933643i	$0.616611\pi$
$788$	−4.79819	−0.170928
$789$	0	0
$790$	23.0919	0.821571
$791$	29.1016	1.03474
$792$	0	0
$793$	83.7053	2.97246
$794$	−33.4036	−1.18545
$795$	0	0
$796$	−1.79478	−0.0636143
$797$	−21.8979	−0.775663	−0.387832	−	0.921730i	$-0.626776\pi$
$797$	−21.8979	−0.775663	−0.387832	+	0.921730i	$0.626776\pi$
$798$	0	0
$799$	−55.8678	−1.97646
$800$	−6.66195	−0.235536
$801$	0	0
$802$	−24.6493	−0.870396
$803$	0	0
$804$	0	0
$805$	−3.34638	−0.117945
$806$	−11.4212	−0.402295
$807$	0	0
$808$	22.9127	0.806065
$809$	−5.33571	−0.187593	−0.0937967	−	0.995591i	$-0.529900\pi$
$809$	−5.33571	−0.187593	−0.0937967	+	0.995591i	$0.529900\pi$
$810$	0	0
$811$	−7.04918	−0.247530	−0.123765	−	0.992312i	$-0.539497\pi$
$811$	−7.04918	−0.247530	−0.123765	+	0.992312i	$0.539497\pi$
$812$	3.44751	0.120984
$813$	0	0
$814$	0	0
$815$	−9.96226	−0.348963
$816$	0	0
$817$	21.6428	0.757186
$818$	−13.7130	−0.479462
$819$	0	0
$820$	1.62360	0.0566984
$821$	2.35318	0.0821264	0.0410632	−	0.999157i	$-0.486926\pi$
$821$	2.35318	0.0821264	0.0410632	+	0.999157i	$0.486926\pi$
$822$	0	0
$823$	−48.7811	−1.70040	−0.850201	−	0.526457i	$-0.823520\pi$
$823$	−48.7811	−1.70040	−0.850201	+	0.526457i	$0.823520\pi$
$824$	7.54328	0.262783
$825$	0	0
$826$	29.5251	1.02731
$827$	21.2670	0.739527	0.369764	−	0.929126i	$-0.379439\pi$
$827$	21.2670	0.739527	0.369764	+	0.929126i	$0.379439\pi$
$828$	0	0
$829$	24.7732	0.860408	0.430204	−	0.902732i	$-0.358442\pi$
$829$	24.7732	0.860408	0.430204	+	0.902732i	$0.358442\pi$
$830$	−26.2751	−0.912022
$831$	0	0
$832$	42.6117	1.47729
$833$	11.3141	0.392010
$834$	0	0
$835$	21.6822	0.750344
$836$	0	0
$837$	0	0
$838$	34.6492	1.19694
$839$	3.50888	0.121140	0.0605701	−	0.998164i	$-0.480708\pi$
$839$	3.50888	0.121140	0.0605701	+	0.998164i	$0.480708\pi$
$840$	0	0
$841$	−4.89470	−0.168783
$842$	−31.2789	−1.07794
$843$	0	0
$844$	3.32885	0.114584
$845$	−34.1099	−1.17342
$846$	0	0
$847$	0	0
$848$	9.33129	0.320438
$849$	0	0
$850$	−34.3789	−1.17919
$851$	0.301292	0.0103282
$852$	0	0
$853$	−2.29563	−0.0786008	−0.0393004	−	0.999227i	$-0.512513\pi$
$853$	−2.29563	−0.0786008	−0.0393004	+	0.999227i	$0.512513\pi$
$854$	−43.1069	−1.47509
$855$	0	0
$856$	28.4002	0.970700
$857$	−15.8536	−0.541548	−0.270774	−	0.962643i	$-0.587280\pi$
$857$	−15.8536	−0.541548	−0.270774	+	0.962643i	$0.587280\pi$
$858$	0	0
$859$	4.97552	0.169763	0.0848813	−	0.996391i	$-0.472949\pi$
$859$	4.97552	0.169763	0.0848813	+	0.996391i	$0.472949\pi$
$860$	−2.48777	−0.0848324
$861$	0	0
$862$	−7.90014	−0.269080
$863$	−40.3432	−1.37330	−0.686650	−	0.726988i	$-0.740919\pi$
$863$	−40.3432	−1.37330	−0.686650	+	0.726988i	$0.740919\pi$
$864$	0	0
$865$	−25.4158	−0.864161
$866$	−5.72876	−0.194671
$867$	0	0
$868$	0.792029	0.0268832
$869$	0	0
$870$	0	0
$871$	−19.3798	−0.656660
$872$	−39.3175	−1.33146
$873$	0	0
$874$	−6.10530	−0.206515
$875$	21.6365	0.731447
$876$	0	0
$877$	44.3405	1.49727	0.748637	−	0.662980i	$-0.230709\pi$
$877$	44.3405	1.49727	0.748637	+	0.662980i	$0.230709\pi$
$878$	16.9860	0.573248
$879$	0	0
$880$	0	0
$881$	48.6174	1.63796	0.818981	−	0.573820i	$-0.194539\pi$
$881$	48.6174	1.63796	0.818981	+	0.573820i	$0.194539\pi$
$882$	0	0
$883$	−11.8277	−0.398033	−0.199017	−	0.979996i	$-0.563775\pi$
$883$	−11.8277	−0.398033	−0.199017	+	0.979996i	$0.563775\pi$
$884$	−12.2808	−0.413049
$885$	0	0
$886$	24.6228	0.827218
$887$	−17.6043	−0.591094	−0.295547	−	0.955328i	$-0.595502\pi$
$887$	−17.6043	−0.591094	−0.295547	+	0.955328i	$0.595502\pi$
$888$	0	0
$889$	−46.8275	−1.57054
$890$	−1.05119	−0.0352359
$891$	0	0
$892$	−6.71541	−0.224849
$893$	27.7707	0.929312
$894$	0	0
$895$	11.8115	0.394815
$896$	−29.8199	−0.996213
$897$	0	0
$898$	4.97135	0.165896
$899$	5.53794	0.184701
$900$	0	0
$901$	12.2157	0.406963
$902$	0	0
$903$	0	0
$904$	33.1161	1.10142
$905$	−13.5206	−0.449442
$906$	0	0
$907$	31.4287	1.04357	0.521786	−	0.853076i	$-0.325266\pi$
$907$	31.4287	1.04357	0.521786	+	0.853076i	$0.325266\pi$
$908$	1.30066	0.0431641
$909$	0	0
$910$	24.8477	0.823695
$911$	25.4380	0.842798	0.421399	−	0.906875i	$-0.361539\pi$
$911$	25.4380	0.842798	0.421399	+	0.906875i	$0.361539\pi$
$912$	0	0
$913$	0	0
$914$	23.2033	0.767497
$915$	0	0
$916$	−6.09075	−0.201244
$917$	23.8026	0.786030
$918$	0	0
$919$	33.3699	1.10077	0.550386	−	0.834910i	$-0.314481\pi$
$919$	33.3699	1.10077	0.550386	+	0.834910i	$0.314481\pi$
$920$	−3.80800	−0.125546
$921$	0	0
$922$	−11.8954	−0.391755
$923$	58.3577	1.92087
$924$	0	0
$925$	−0.843332	−0.0277286
$926$	−25.6007	−0.841290
$927$	0	0
$928$	8.56915	0.281296
$929$	14.4401	0.473765	0.236883	−	0.971538i	$-0.423874\pi$
$929$	14.4401	0.473765	0.236883	+	0.971538i	$0.423874\pi$
$930$	0	0
$931$	−5.62399	−0.184319
$932$	−3.25672	−0.106677
$933$	0	0
$934$	−57.7960	−1.89114
$935$	0	0
$936$	0	0
$937$	−29.6541	−0.968757	−0.484379	−	0.874859i	$-0.660954\pi$
$937$	−29.6541	−0.968757	−0.484379	+	0.874859i	$0.660954\pi$
$938$	9.98028	0.325868
$939$	0	0
$940$	−3.19216	−0.104117
$941$	43.1420	1.40639	0.703194	−	0.710998i	$-0.251756\pi$
$941$	43.1420	1.40639	0.703194	+	0.710998i	$0.251756\pi$
$942$	0	0
$943$	−6.54054	−0.212989
$944$	38.9560	1.26791
$945$	0	0
$946$	0	0
$947$	−37.0608	−1.20432	−0.602158	−	0.798377i	$-0.705692\pi$
$947$	−37.0608	−1.20432	−0.602158	+	0.798377i	$0.705692\pi$
$948$	0	0
$949$	−7.83554	−0.254352
$950$	17.0890	0.554442
$951$	0	0
$952$	−34.3174	−1.11223
$953$	25.1122	0.813463	0.406731	−	0.913548i	$-0.366669\pi$
$953$	25.1122	0.813463	0.406731	+	0.913548i	$0.366669\pi$
$954$	0	0
$955$	2.55051	0.0825326
$956$	−4.30778	−0.139323
$957$	0	0
$958$	23.6342	0.763587
$959$	26.9892	0.871526
$960$	0	0
$961$	−29.7277	−0.958959
$962$	−2.23717	−0.0721292
$963$	0	0
$964$	0.663644	0.0213745
$965$	4.54257	0.146231
$966$	0	0
$967$	−7.41108	−0.238324	−0.119162	−	0.992875i	$-0.538021\pi$
$967$	−7.41108	−0.238324	−0.119162	+	0.992875i	$0.538021\pi$
$968$	0	0
$969$	0	0
$970$	7.89557	0.253512
$971$	−2.88466	−0.0925733	−0.0462866	−	0.998928i	$-0.514739\pi$
$971$	−2.88466	−0.0925733	−0.0462866	+	0.998928i	$0.514739\pi$
$972$	0	0
$973$	−28.5700	−0.915913
$974$	32.1086	1.02883
$975$	0	0
$976$	−56.8761	−1.82056
$977$	17.9632	0.574693	0.287346	−	0.957827i	$-0.407227\pi$
$977$	17.9632	0.574693	0.287346	+	0.957827i	$0.407227\pi$
$978$	0	0
$979$	0	0
$980$	0.646461	0.0206504
$981$	0	0
$982$	−62.9246	−2.00800
$983$	31.5601	1.00661	0.503305	−	0.864109i	$-0.332117\pi$
$983$	31.5601	1.00661	0.503305	+	0.864109i	$0.332117\pi$
$984$	0	0
$985$	16.7685	0.534290
$986$	44.2210	1.40828
$987$	0	0
$988$	6.10455	0.194211
$989$	10.0218	0.318675
$990$	0	0
$991$	28.4307	0.903132	0.451566	−	0.892238i	$-0.350866\pi$
$991$	28.4307	0.903132	0.451566	+	0.892238i	$0.350866\pi$
$992$	1.96867	0.0625054
$993$	0	0
$994$	−30.0533	−0.953232
$995$	6.27233	0.198846
$996$	0	0
$997$	−41.4489	−1.31270	−0.656350	−	0.754457i	$-0.727901\pi$
$997$	−41.4489	−1.31270	−0.656350	+	0.754457i	$0.727901\pi$
$998$	−10.7625	−0.340681
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9801.2.a.bq.1.2	yes	5
3.2	odd	2		9801.2.a.bn.1.4	✓	5
11.10	odd	2		9801.2.a.bo.1.4	yes	5
33.32	even	2		9801.2.a.bp.1.2	yes	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9801.2.a.bn.1.4	✓	5	3.2	odd	2
9801.2.a.bo.1.4	yes	5	11.10	odd	2
9801.2.a.bp.1.2	yes	5	33.32	even	2
9801.2.a.bq.1.2	yes	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:	\( N \)	\(=\)	\( 9801 = 3^{4} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9801.a (trivial)

\(f(q)\)	\(=\)	\(q-1.52027 q^{2} +0.311228 q^{4} -1.08767 q^{5} +2.25617 q^{7} +2.56739 q^{8} +O(q^{10})\)	\(q-1.52027 q^{2} +0.311228 q^{4} -1.08767 q^{5} +2.25617 q^{7} +2.56739 q^{8} +1.65355 q^{10} +6.66038 q^{13} -3.42999 q^{14} -4.52559 q^{16} -5.92449 q^{17} +2.94494 q^{19} -0.338512 q^{20} +1.36367 q^{23} -3.81698 q^{25} -10.1256 q^{26} +0.702181 q^{28} +4.90971 q^{29} +1.12796 q^{31} +1.74535 q^{32} +9.00683 q^{34} -2.45396 q^{35} +0.220942 q^{37} -4.47711 q^{38} -2.79247 q^{40} -4.79628 q^{41} +7.34915 q^{43} -2.07315 q^{46} +9.42999 q^{47} -1.90971 q^{49} +5.80285 q^{50} +2.07289 q^{52} -2.06189 q^{53} +5.79247 q^{56} -7.46410 q^{58} -8.60794 q^{59} +12.5676 q^{61} -1.71480 q^{62} +6.39779 q^{64} -7.24427 q^{65} -2.90971 q^{67} -1.84386 q^{68} +3.73068 q^{70} +8.76192 q^{71} -1.17644 q^{73} -0.335892 q^{74} +0.916546 q^{76} +13.9650 q^{79} +4.92233 q^{80} +7.29164 q^{82} -15.8901 q^{83} +6.44386 q^{85} -11.1727 q^{86} -0.635716 q^{89} +15.0269 q^{91} +0.424412 q^{92} -14.3361 q^{94} -3.20311 q^{95} +4.77493 q^{97} +2.90329 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + q^{2} + 9 q^{4} + 4 q^{5} + 3 q^{7} + 12 q^{8}+O(q^{10}) \) 5 * q + q^2 + 9 * q^4 + 4 * q^5 + 3 * q^7 + 12 * q^8	\( 5 q + q^{2} + 9 q^{4} + 4 q^{5} + 3 q^{7} + 12 q^{8} - 5 q^{10} + 13 q^{14} + 9 q^{16} + 4 q^{17} - q^{19} - 7 q^{20} - q^{23} + 15 q^{25} + 7 q^{26} + q^{28} + 3 q^{29} + 24 q^{31} + 20 q^{32} + 25 q^{34} - 26 q^{35} - 13 q^{37} - 33 q^{40} - 17 q^{41} - 4 q^{43} + 32 q^{46} + 17 q^{47} + 12 q^{49} + 47 q^{50} + 13 q^{52} - 6 q^{53} + 48 q^{56} + 15 q^{58} - 25 q^{59} + 17 q^{61} + 49 q^{62} + 2 q^{64} - 26 q^{65} + 7 q^{67} + 2 q^{68} - 23 q^{70} - 6 q^{71} - 15 q^{73} + 24 q^{74} - 37 q^{76} + 19 q^{79} - 52 q^{80} - 11 q^{82} - 3 q^{83} - 5 q^{85} - 6 q^{86} - 3 q^{89} + 20 q^{91} + 40 q^{92} - q^{94} - 15 q^{95} + 4 q^{97} - 12 q^{98}+O(q^{100}) \) 5 * q + q^2 + 9 * q^4 + 4 * q^5 + 3 * q^7 + 12 * q^8 - 5 * q^10 + 13 * q^14 + 9 * q^16 + 4 * q^17 - q^19 - 7 * q^20 - q^23 + 15 * q^25 + 7 * q^26 + q^28 + 3 * q^29 + 24 * q^31 + 20 * q^32 + 25 * q^34 - 26 * q^35 - 13 * q^37 - 33 * q^40 - 17 * q^41 - 4 * q^43 + 32 * q^46 + 17 * q^47 + 12 * q^49 + 47 * q^50 + 13 * q^52 - 6 * q^53 + 48 * q^56 + 15 * q^58 - 25 * q^59 + 17 * q^61 + 49 * q^62 + 2 * q^64 - 26 * q^65 + 7 * q^67 + 2 * q^68 - 23 * q^70 - 6 * q^71 - 15 * q^73 + 24 * q^74 - 37 * q^76 + 19 * q^79 - 52 * q^80 - 11 * q^82 - 3 * q^83 - 5 * q^85 - 6 * q^86 - 3 * q^89 + 20 * q^91 + 40 * q^92 - q^94 - 15 * q^95 + 4 * q^97 - 12 * q^98

Introduction

L-functions

Modular forms

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Database

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