LMFDB - Embedded newform 8001.2.a.x.1.19

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8001,2,Mod(1,8001)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8001, 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("8001.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8001 = 3^{2} \cdot 7 \cdot 127 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8001.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:	$63.8883066572$
Analytic rank:	$1$
Dimension:	$22$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.19
Character	$\chi$	$=$	8001.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.94436 q^{2} +1.78053 q^{4} +1.34248 q^{5} +1.00000 q^{7} -0.426737 q^{8} +O(q^{10})$	\(q+1.94436 q^{2} +1.78053 q^{4} +1.34248 q^{5} +1.00000 q^{7} -0.426737 q^{8} +2.61025 q^{10} -0.671920 q^{11} -2.12397 q^{13} +1.94436 q^{14} -4.39078 q^{16} +1.91283 q^{17} -4.48056 q^{19} +2.39031 q^{20} -1.30645 q^{22} -4.40373 q^{23} -3.19776 q^{25} -4.12976 q^{26} +1.78053 q^{28} -2.36148 q^{29} +1.43789 q^{31} -7.68377 q^{32} +3.71923 q^{34} +1.34248 q^{35} +5.60291 q^{37} -8.71182 q^{38} -0.572884 q^{40} -4.45855 q^{41} -8.18323 q^{43} -1.19637 q^{44} -8.56242 q^{46} -13.1888 q^{47} +1.00000 q^{49} -6.21758 q^{50} -3.78179 q^{52} +11.7277 q^{53} -0.902037 q^{55} -0.426737 q^{56} -4.59157 q^{58} -7.06826 q^{59} +3.58432 q^{61} +2.79577 q^{62} -6.15844 q^{64} -2.85138 q^{65} +1.70366 q^{67} +3.40584 q^{68} +2.61025 q^{70} -5.36282 q^{71} -0.516847 q^{73} +10.8941 q^{74} -7.97776 q^{76} -0.671920 q^{77} -6.93892 q^{79} -5.89452 q^{80} -8.66901 q^{82} -8.45270 q^{83} +2.56793 q^{85} -15.9111 q^{86} +0.286733 q^{88} +13.3759 q^{89} -2.12397 q^{91} -7.84095 q^{92} -25.6437 q^{94} -6.01505 q^{95} -12.9250 q^{97} +1.94436 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 22 q + 10 q^{4} + 22 q^{7}+O(q^{10}) $ 22 * q + 10 * q^4 + 22 * q^7	$ 22 q + 10 q^{4} + 22 q^{7} - 4 q^{10} - 10 q^{13} - 6 q^{16} - 18 q^{19} - 34 q^{22} - 26 q^{25} + 10 q^{28} - 42 q^{31} - 16 q^{34} - 36 q^{37} - 46 q^{40} - 54 q^{43} - 12 q^{46} + 22 q^{49} - 22 q^{52} - 28 q^{55} - 26 q^{58} - 22 q^{61} - 76 q^{64} - 48 q^{67} - 4 q^{70} - 48 q^{73} - 20 q^{76} - 94 q^{79} - 8 q^{82} - 40 q^{85} - 26 q^{88} - 10 q^{91} - 26 q^{94} - 56 q^{97}+O(q^{100}) $ 22 * q + 10 * q^4 + 22 * q^7 - 4 * q^10 - 10 * q^13 - 6 * q^16 - 18 * q^19 - 34 * q^22 - 26 * q^25 + 10 * q^28 - 42 * q^31 - 16 * q^34 - 36 * q^37 - 46 * q^40 - 54 * q^43 - 12 * q^46 + 22 * q^49 - 22 * q^52 - 28 * q^55 - 26 * q^58 - 22 * q^61 - 76 * q^64 - 48 * q^67 - 4 * q^70 - 48 * q^73 - 20 * q^76 - 94 * q^79 - 8 * q^82 - 40 * q^85 - 26 * q^88 - 10 * q^91 - 26 * q^94 - 56 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.94436	1.37487	0.687434	−	0.726247i	$-0.258737\pi$
$2$	1.94436	1.37487	0.687434	+	0.726247i	$0.258737\pi$
$3$	0	0
$3$	0	0
$4$	1.78053	0.890263
$5$	1.34248	0.600374	0.300187	−	0.953880i	$-0.402951\pi$
$5$	1.34248	0.600374	0.300187	+	0.953880i	$0.402951\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−0.426737	−0.150874
$9$	0	0
$10$	2.61025	0.825435
$11$	−0.671920	−0.202591	−0.101296	−	0.994856i	$-0.532299\pi$
$11$	−0.671920	−0.202591	−0.101296	+	0.994856i	$0.532299\pi$
$12$	0	0
$13$	−2.12397	−0.589084	−0.294542	−	0.955639i	$-0.595167\pi$
$13$	−2.12397	−0.589084	−0.294542	+	0.955639i	$0.595167\pi$
$14$	1.94436	0.519651
$15$	0	0
$16$	−4.39078	−1.09769
$17$	1.91283	0.463930	0.231965	−	0.972724i	$-0.425485\pi$
$17$	1.91283	0.463930	0.231965	+	0.972724i	$0.425485\pi$
$18$	0	0
$19$	−4.48056	−1.02791	−0.513956	−	0.857817i	$-0.671821\pi$
$19$	−4.48056	−1.02791	−0.513956	+	0.857817i	$0.671821\pi$
$20$	2.39031	0.534490
$21$	0	0
$22$	−1.30645	−0.278537
$23$	−4.40373	−0.918241	−0.459121	−	0.888374i	$-0.651835\pi$
$23$	−4.40373	−0.918241	−0.459121	+	0.888374i	$0.651835\pi$
$24$	0	0
$25$	−3.19776	−0.639551
$26$	−4.12976	−0.809913
$27$	0	0
$28$	1.78053	0.336488
$29$	−2.36148	−0.438516	−0.219258	−	0.975667i	$-0.570364\pi$
$29$	−2.36148	−0.438516	−0.219258	+	0.975667i	$0.570364\pi$
$30$	0	0
$31$	1.43789	0.258253	0.129126	−	0.991628i	$-0.458783\pi$
$31$	1.43789	0.258253	0.129126	+	0.991628i	$0.458783\pi$
$32$	−7.68377	−1.35831
$33$	0	0
$34$	3.71923	0.637842
$35$	1.34248	0.226920
$36$	0	0
$37$	5.60291	0.921113	0.460556	−	0.887630i	$-0.347650\pi$
$37$	5.60291	0.921113	0.460556	+	0.887630i	$0.347650\pi$
$38$	−8.71182	−1.41324
$39$	0	0
$40$	−0.572884	−0.0905809
$41$	−4.45855	−0.696308	−0.348154	−	0.937437i	$-0.613191\pi$
$41$	−4.45855	−0.696308	−0.348154	+	0.937437i	$0.613191\pi$
$42$	0	0
$43$	−8.18323	−1.24793	−0.623965	−	0.781452i	$-0.714479\pi$
$43$	−8.18323	−1.24793	−0.623965	+	0.781452i	$0.714479\pi$
$44$	−1.19637	−0.180360
$45$	0	0
$46$	−8.56242	−1.26246
$47$	−13.1888	−1.92378	−0.961889	−	0.273440i	$-0.911839\pi$
$47$	−13.1888	−1.92378	−0.961889	+	0.273440i	$0.911839\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−6.21758	−0.879299
$51$	0	0
$52$	−3.78179	−0.524439
$53$	11.7277	1.61093	0.805464	−	0.592644i	$-0.201916\pi$
$53$	11.7277	1.61093	0.805464	+	0.592644i	$0.201916\pi$
$54$	0	0
$55$	−0.902037	−0.121631
$56$	−0.426737	−0.0570251
$57$	0	0
$58$	−4.59157	−0.602902
$59$	−7.06826	−0.920209	−0.460104	−	0.887865i	$-0.652188\pi$
$59$	−7.06826	−0.920209	−0.460104	+	0.887865i	$0.652188\pi$
$60$	0	0
$61$	3.58432	0.458925	0.229462	−	0.973318i	$-0.426303\pi$
$61$	3.58432	0.458925	0.229462	+	0.973318i	$0.426303\pi$
$62$	2.79577	0.355063
$63$	0	0
$64$	−6.15844	−0.769805
$65$	−2.85138	−0.353671
$66$	0	0
$67$	1.70366	0.208135	0.104068	−	0.994570i	$-0.466814\pi$
$67$	1.70366	0.208135	0.104068	+	0.994570i	$0.466814\pi$
$68$	3.40584	0.413019
$69$	0	0
$70$	2.61025	0.311985
$71$	−5.36282	−0.636450	−0.318225	−	0.948015i	$-0.603087\pi$
$71$	−5.36282	−0.636450	−0.318225	+	0.948015i	$0.603087\pi$
$72$	0	0
$73$	−0.516847	−0.0604924	−0.0302462	−	0.999542i	$-0.509629\pi$
$73$	−0.516847	−0.0604924	−0.0302462	+	0.999542i	$0.509629\pi$
$74$	10.8941	1.26641
$75$	0	0
$76$	−7.97776	−0.915112
$77$	−0.671920	−0.0765724
$78$	0	0
$79$	−6.93892	−0.780689	−0.390345	−	0.920669i	$-0.627644\pi$
$79$	−6.93892	−0.780689	−0.390345	+	0.920669i	$0.627644\pi$
$80$	−5.89452	−0.659027
$81$	0	0
$82$	−8.66901	−0.957332
$83$	−8.45270	−0.927805	−0.463902	−	0.885886i	$-0.653551\pi$
$83$	−8.45270	−0.927805	−0.463902	+	0.885886i	$0.653551\pi$
$84$	0	0
$85$	2.56793	0.278531
$86$	−15.9111	−1.71574
$87$	0	0
$88$	0.286733	0.0305658
$89$	13.3759	1.41784	0.708922	−	0.705287i	$-0.249182\pi$
$89$	13.3759	1.41784	0.708922	+	0.705287i	$0.249182\pi$
$90$	0	0
$91$	−2.12397	−0.222653
$92$	−7.84095	−0.817476
$93$	0	0
$94$	−25.6437	−2.64494
$95$	−6.01505	−0.617131
$96$	0	0
$97$	−12.9250	−1.31233	−0.656166	−	0.754617i	$-0.727823\pi$
$97$	−12.9250	−1.31233	−0.656166	+	0.754617i	$0.727823\pi$
$98$	1.94436	0.196410
$99$	0	0
$100$	−5.69369	−0.569369
$101$	3.29760	0.328123	0.164062	−	0.986450i	$-0.447540\pi$
$101$	3.29760	0.328123	0.164062	+	0.986450i	$0.447540\pi$
$102$	0	0
$103$	−7.91791	−0.780175	−0.390088	−	0.920778i	$-0.627555\pi$
$103$	−7.91791	−0.780175	−0.390088	+	0.920778i	$0.627555\pi$
$104$	0.906377	0.0888776
$105$	0	0
$106$	22.8029	2.21481
$107$	1.84047	0.177925	0.0889626	−	0.996035i	$-0.471645\pi$
$107$	1.84047	0.177925	0.0889626	+	0.996035i	$0.471645\pi$
$108$	0	0
$109$	15.4450	1.47936	0.739680	−	0.672959i	$-0.234977\pi$
$109$	15.4450	1.47936	0.739680	+	0.672959i	$0.234977\pi$
$110$	−1.75388	−0.167226
$111$	0	0
$112$	−4.39078	−0.414890
$113$	10.5350	0.991053	0.495527	−	0.868593i	$-0.334975\pi$
$113$	10.5350	0.991053	0.495527	+	0.868593i	$0.334975\pi$
$114$	0	0
$115$	−5.91190	−0.551288
$116$	−4.20468	−0.390395
$117$	0	0
$118$	−13.7432	−1.26517
$119$	1.91283	0.175349
$120$	0	0
$121$	−10.5485	−0.958957
$122$	6.96919	0.630961
$123$	0	0
$124$	2.56020	0.229913
$125$	−11.0053	−0.984344
$126$	0	0
$127$	1.00000	0.0887357
$127$	1.00000	0.0887357
$128$	3.39334	0.299932
$129$	0	0
$130$	−5.54411	−0.486250
$131$	21.6997	1.89591	0.947957	−	0.318398i	$-0.103145\pi$
$131$	21.6997	1.89591	0.947957	+	0.318398i	$0.103145\pi$
$132$	0	0
$133$	−4.48056	−0.388514
$134$	3.31253	0.286159
$135$	0	0
$136$	−0.816275	−0.0699950
$137$	−8.54803	−0.730307	−0.365154	−	0.930947i	$-0.618984\pi$
$137$	−8.54803	−0.730307	−0.365154	+	0.930947i	$0.618984\pi$
$138$	0	0
$139$	−11.4438	−0.970651	−0.485326	−	0.874334i	$-0.661299\pi$
$139$	−11.4438	−0.970651	−0.485326	+	0.874334i	$0.661299\pi$
$140$	2.39031	0.202018
$141$	0	0
$142$	−10.4272	−0.875035
$143$	1.42714	0.119343
$144$	0	0
$145$	−3.17024	−0.263274
$146$	−1.00494	−0.0831690
$147$	0	0
$148$	9.97613	0.820033
$149$	−6.76371	−0.554104	−0.277052	−	0.960855i	$-0.589358\pi$
$149$	−6.76371	−0.554104	−0.277052	+	0.960855i	$0.589358\pi$
$150$	0	0
$151$	12.0326	0.979196	0.489598	−	0.871948i	$-0.337144\pi$
$151$	12.0326	0.979196	0.489598	+	0.871948i	$0.337144\pi$
$152$	1.91202	0.155085
$153$	0	0
$154$	−1.30645	−0.105277
$155$	1.93033	0.155048
$156$	0	0
$157$	19.4980	1.55611	0.778055	−	0.628196i	$-0.216207\pi$
$157$	19.4980	1.55611	0.778055	+	0.628196i	$0.216207\pi$
$158$	−13.4917	−1.07334
$159$	0	0
$160$	−10.3153	−0.815495
$161$	−4.40373	−0.347063
$162$	0	0
$163$	−6.06631	−0.475150	−0.237575	−	0.971369i	$-0.576353\pi$
$163$	−6.06631	−0.475150	−0.237575	+	0.971369i	$0.576353\pi$
$164$	−7.93855	−0.619897
$165$	0	0
$166$	−16.4351	−1.27561
$167$	−6.63895	−0.513738	−0.256869	−	0.966446i	$-0.582691\pi$
$167$	−6.63895	−0.513738	−0.256869	+	0.966446i	$0.582691\pi$
$168$	0	0
$169$	−8.48874	−0.652980
$170$	4.99297	0.382944
$171$	0	0
$172$	−14.5704	−1.11099
$173$	−0.951266	−0.0723234	−0.0361617	−	0.999346i	$-0.511513\pi$
$173$	−0.951266	−0.0723234	−0.0361617	+	0.999346i	$0.511513\pi$
$174$	0	0
$175$	−3.19776	−0.241728
$176$	2.95025	0.222384
$177$	0	0
$178$	26.0076	1.94935
$179$	−13.7723	−1.02939	−0.514696	−	0.857373i	$-0.672095\pi$
$179$	−13.7723	−1.02939	−0.514696	+	0.857373i	$0.672095\pi$
$180$	0	0
$181$	10.3370	0.768345	0.384173	−	0.923261i	$-0.374487\pi$
$181$	10.3370	0.768345	0.384173	+	0.923261i	$0.374487\pi$
$182$	−4.12976	−0.306118
$183$	0	0
$184$	1.87923	0.138539
$185$	7.52178	0.553012
$186$	0	0
$187$	−1.28527	−0.0939882
$188$	−23.4829	−1.71267
$189$	0	0
$190$	−11.6954	−0.848474
$191$	−19.1142	−1.38306	−0.691528	−	0.722350i	$-0.743062\pi$
$191$	−19.1142	−1.38306	−0.691528	+	0.722350i	$0.743062\pi$
$192$	0	0
$193$	−0.102612	−0.00738619	−0.00369309	−	0.999993i	$-0.501176\pi$
$193$	−0.102612	−0.00738619	−0.00369309	+	0.999993i	$0.501176\pi$
$194$	−25.1308	−1.80428
$195$	0	0
$196$	1.78053	0.127180
$197$	−3.80038	−0.270766	−0.135383	−	0.990793i	$-0.543226\pi$
$197$	−3.80038	−0.270766	−0.135383	+	0.990793i	$0.543226\pi$
$198$	0	0
$199$	15.2697	1.08244	0.541219	−	0.840881i	$-0.317963\pi$
$199$	15.2697	1.08244	0.541219	+	0.840881i	$0.317963\pi$
$200$	1.36460	0.0964918
$201$	0	0
$202$	6.41171	0.451126
$203$	−2.36148	−0.165744
$204$	0	0
$205$	−5.98549	−0.418045
$206$	−15.3953	−1.07264
$207$	0	0
$208$	9.32589	0.646634
$209$	3.01058	0.208246
$210$	0	0
$211$	−19.5104	−1.34315	−0.671577	−	0.740934i	$-0.734383\pi$
$211$	−19.5104	−1.34315	−0.671577	+	0.740934i	$0.734383\pi$
$212$	20.8815	1.43415
$213$	0	0
$214$	3.57853	0.244624
$215$	−10.9858	−0.749225
$216$	0	0
$217$	1.43789	0.0976103
$218$	30.0305	2.03393
$219$	0	0
$220$	−1.60610	−0.108283
$221$	−4.06280	−0.273293
$222$	0	0
$223$	25.8049	1.72802	0.864012	−	0.503472i	$-0.167944\pi$
$223$	25.8049	1.72802	0.864012	+	0.503472i	$0.167944\pi$
$224$	−7.68377	−0.513394
$225$	0	0
$226$	20.4839	1.36257
$227$	−19.9150	−1.32181	−0.660904	−	0.750471i	$-0.729827\pi$
$227$	−19.9150	−1.32181	−0.660904	+	0.750471i	$0.729827\pi$
$228$	0	0
$229$	−17.2646	−1.14088	−0.570440	−	0.821339i	$-0.693227\pi$
$229$	−17.2646	−1.14088	−0.570440	+	0.821339i	$0.693227\pi$
$230$	−11.4949	−0.757948
$231$	0	0
$232$	1.00773	0.0661608
$233$	29.8061	1.95267	0.976333	−	0.216275i	$-0.0693906\pi$
$233$	29.8061	1.95267	0.976333	+	0.216275i	$0.0693906\pi$
$234$	0	0
$235$	−17.7056	−1.15499
$236$	−12.5852	−0.819228
$237$	0	0
$238$	3.71923	0.241082
$239$	8.83517	0.571500	0.285750	−	0.958304i	$-0.407757\pi$
$239$	8.83517	0.571500	0.285750	+	0.958304i	$0.407757\pi$
$240$	0	0
$241$	0.377401	0.0243106	0.0121553	−	0.999926i	$-0.496131\pi$
$241$	0.377401	0.0243106	0.0121553	+	0.999926i	$0.496131\pi$
$242$	−20.5101	−1.31844
$243$	0	0
$244$	6.38197	0.408564
$245$	1.34248	0.0857677
$246$	0	0
$247$	9.51659	0.605526
$248$	−0.613600	−0.0389637
$249$	0	0
$250$	−21.3982	−1.35334
$251$	−11.8270	−0.746512	−0.373256	−	0.927728i	$-0.621759\pi$
$251$	−11.8270	−0.746512	−0.373256	+	0.927728i	$0.621759\pi$
$252$	0	0
$253$	2.95895	0.186028
$254$	1.94436	0.122000
$255$	0	0
$256$	18.9147	1.18217
$257$	23.0339	1.43682	0.718409	−	0.695621i	$-0.244871\pi$
$257$	23.0339	1.43682	0.718409	+	0.695621i	$0.244871\pi$
$258$	0	0
$259$	5.60291	0.348148
$260$	−5.07696	−0.314860
$261$	0	0
$262$	42.1920	2.60663
$263$	24.6731	1.52141	0.760704	−	0.649098i	$-0.224854\pi$
$263$	24.6731	1.52141	0.760704	+	0.649098i	$0.224854\pi$
$264$	0	0
$265$	15.7442	0.967159
$266$	−8.71182	−0.534156
$267$	0	0
$268$	3.03341	0.185295
$269$	−14.9956	−0.914298	−0.457149	−	0.889390i	$-0.651129\pi$
$269$	−14.9956	−0.914298	−0.457149	+	0.889390i	$0.651129\pi$
$270$	0	0
$271$	9.69915	0.589182	0.294591	−	0.955623i	$-0.404817\pi$
$271$	9.69915	0.589182	0.294591	+	0.955623i	$0.404817\pi$
$272$	−8.39882	−0.509253
$273$	0	0
$274$	−16.6204	−1.00408
$275$	2.14864	0.129568
$276$	0	0
$277$	−12.0320	−0.722935	−0.361468	−	0.932385i	$-0.617724\pi$
$277$	−12.0320	−0.722935	−0.361468	+	0.932385i	$0.617724\pi$
$278$	−22.2509	−1.33452
$279$	0	0
$280$	−0.572884	−0.0342364
$281$	−8.05023	−0.480237	−0.240118	−	0.970744i	$-0.577186\pi$
$281$	−8.05023	−0.480237	−0.240118	+	0.970744i	$0.577186\pi$
$282$	0	0
$283$	22.4524	1.33466	0.667329	−	0.744763i	$-0.267438\pi$
$283$	22.4524	1.33466	0.667329	+	0.744763i	$0.267438\pi$
$284$	−9.54864	−0.566608
$285$	0	0
$286$	2.77487	0.164081
$287$	−4.45855	−0.263180
$288$	0	0
$289$	−13.3411	−0.784769
$290$	−6.16407	−0.361967
$291$	0	0
$292$	−0.920259	−0.0538541
$293$	11.9622	0.698842	0.349421	−	0.936966i	$-0.386378\pi$
$293$	11.9622	0.698842	0.349421	+	0.936966i	$0.386378\pi$
$294$	0	0
$295$	−9.48897	−0.552469
$296$	−2.39097	−0.138972
$297$	0	0
$298$	−13.1511	−0.761821
$299$	9.35340	0.540921
$300$	0	0
$301$	−8.18323	−0.471674
$302$	23.3956	1.34627
$303$	0	0
$304$	19.6732	1.12833
$305$	4.81186	0.275526
$306$	0	0
$307$	−21.9768	−1.25428	−0.627140	−	0.778906i	$-0.715775\pi$
$307$	−21.9768	−1.25428	−0.627140	+	0.778906i	$0.715775\pi$
$308$	−1.19637	−0.0681695
$309$	0	0
$310$	3.75326	0.213171
$311$	28.3108	1.60536	0.802678	−	0.596413i	$-0.203408\pi$
$311$	28.3108	1.60536	0.802678	+	0.596413i	$0.203408\pi$
$312$	0	0
$313$	−3.01192	−0.170244	−0.0851219	−	0.996371i	$-0.527128\pi$
$313$	−3.01192	−0.170244	−0.0851219	+	0.996371i	$0.527128\pi$
$314$	37.9111	2.13945
$315$	0	0
$316$	−12.3549	−0.695019
$317$	3.65038	0.205026	0.102513	−	0.994732i	$-0.467312\pi$
$317$	3.65038	0.205026	0.102513	+	0.994732i	$0.467312\pi$
$318$	0	0
$319$	1.58673	0.0888397
$320$	−8.26756	−0.462171
$321$	0	0
$322$	−8.56242	−0.477165
$323$	−8.57056	−0.476879
$324$	0	0
$325$	6.79195	0.376749
$326$	−11.7951	−0.653269
$327$	0	0
$328$	1.90263	0.105055
$329$	−13.1888	−0.727120
$330$	0	0
$331$	−26.6291	−1.46367	−0.731834	−	0.681483i	$-0.761335\pi$
$331$	−26.6291	−1.46367	−0.731834	+	0.681483i	$0.761335\pi$
$332$	−15.0503	−0.825990
$333$	0	0
$334$	−12.9085	−0.706322
$335$	2.28713	0.124959
$336$	0	0
$337$	−33.2796	−1.81286	−0.906429	−	0.422359i	$-0.861202\pi$
$337$	−33.2796	−1.81286	−0.906429	+	0.422359i	$0.861202\pi$
$338$	−16.5051	−0.897762
$339$	0	0
$340$	4.57227	0.247966
$341$	−0.966146	−0.0523198
$342$	0	0
$343$	1.00000	0.0539949
$344$	3.49208	0.188281
$345$	0	0
$346$	−1.84960	−0.0994351
$347$	−30.1843	−1.62038	−0.810189	−	0.586168i	$-0.800636\pi$
$347$	−30.1843	−1.62038	−0.810189	+	0.586168i	$0.800636\pi$
$348$	0	0
$349$	18.0746	0.967511	0.483756	−	0.875203i	$-0.339272\pi$
$349$	18.0746	0.967511	0.483756	+	0.875203i	$0.339272\pi$
$350$	−6.21758	−0.332344
$351$	0	0
$352$	5.16288	0.275182
$353$	−16.5042	−0.878431	−0.439216	−	0.898382i	$-0.644744\pi$
$353$	−16.5042	−0.878431	−0.439216	+	0.898382i	$0.644744\pi$
$354$	0	0
$355$	−7.19946	−0.382108
$356$	23.8162	1.26225
$357$	0	0
$358$	−26.7783	−1.41528
$359$	−10.1195	−0.534088	−0.267044	−	0.963684i	$-0.586047\pi$
$359$	−10.1195	−0.534088	−0.267044	+	0.963684i	$0.586047\pi$
$360$	0	0
$361$	1.07546	0.0566030
$362$	20.0989	1.05637
$363$	0	0
$364$	−3.78179	−0.198219
$365$	−0.693855	−0.0363180
$366$	0	0
$367$	−26.6629	−1.39179	−0.695896	−	0.718142i	$-0.744993\pi$
$367$	−26.6629	−1.39179	−0.695896	+	0.718142i	$0.744993\pi$
$368$	19.3358	1.00795
$369$	0	0
$370$	14.6250	0.760319
$371$	11.7277	0.608874
$372$	0	0
$373$	−24.4276	−1.26481	−0.632406	−	0.774637i	$-0.717932\pi$
$373$	−24.4276	−1.26481	−0.632406	+	0.774637i	$0.717932\pi$
$374$	−2.49902	−0.129221
$375$	0	0
$376$	5.62813	0.290249
$377$	5.01572	0.258323
$378$	0	0
$379$	27.7301	1.42440	0.712200	−	0.701976i	$-0.247699\pi$
$379$	27.7301	1.42440	0.712200	+	0.701976i	$0.247699\pi$
$380$	−10.7100	−0.549409
$381$	0	0
$382$	−37.1648	−1.90152
$383$	−8.00622	−0.409099	−0.204549	−	0.978856i	$-0.565573\pi$
$383$	−8.00622	−0.409099	−0.204549	+	0.978856i	$0.565573\pi$
$384$	0	0
$385$	−0.902037	−0.0459720
$386$	−0.199515	−0.0101550
$387$	0	0
$388$	−23.0132	−1.16832
$389$	−21.3761	−1.08381	−0.541906	−	0.840439i	$-0.682297\pi$
$389$	−21.3761	−1.08381	−0.541906	+	0.840439i	$0.682297\pi$
$390$	0	0
$391$	−8.42359	−0.425999
$392$	−0.426737	−0.0215535
$393$	0	0
$394$	−7.38930	−0.372268
$395$	−9.31533	−0.468705
$396$	0	0
$397$	−8.45298	−0.424243	−0.212122	−	0.977243i	$-0.568037\pi$
$397$	−8.45298	−0.424243	−0.212122	+	0.977243i	$0.568037\pi$
$398$	29.6897	1.48821
$399$	0	0
$400$	14.0406	0.702032
$401$	23.5609	1.17657	0.588287	−	0.808652i	$-0.299803\pi$
$401$	23.5609	1.17657	0.588287	+	0.808652i	$0.299803\pi$
$402$	0	0
$403$	−3.05404	−0.152132
$404$	5.87146	0.292116
$405$	0	0
$406$	−4.59157	−0.227876
$407$	−3.76471	−0.186610
$408$	0	0
$409$	−8.81664	−0.435955	−0.217977	−	0.975954i	$-0.569946\pi$
$409$	−8.81664	−0.435955	−0.217977	+	0.975954i	$0.569946\pi$
$410$	−11.6379	−0.574757
$411$	0	0
$412$	−14.0980	−0.694561
$413$	−7.06826	−0.347806
$414$	0	0
$415$	−11.3476	−0.557030
$416$	16.3201	0.800160
$417$	0	0
$418$	5.85364	0.286311
$419$	33.1820	1.62104	0.810522	−	0.585708i	$-0.199183\pi$
$419$	33.1820	1.62104	0.810522	+	0.585708i	$0.199183\pi$
$420$	0	0
$421$	−31.0973	−1.51559	−0.757795	−	0.652493i	$-0.773723\pi$
$421$	−31.0973	−1.51559	−0.757795	+	0.652493i	$0.773723\pi$
$422$	−37.9353	−1.84666
$423$	0	0
$424$	−5.00466	−0.243048
$425$	−6.11677	−0.296707
$426$	0	0
$427$	3.58432	0.173457
$428$	3.27701	0.158400
$429$	0	0
$430$	−21.3603	−1.03009
$431$	28.9047	1.39229	0.696145	−	0.717901i	$-0.254897\pi$
$431$	28.9047	1.39229	0.696145	+	0.717901i	$0.254897\pi$
$432$	0	0
$433$	30.4141	1.46161	0.730803	−	0.682588i	$-0.239146\pi$
$433$	30.4141	1.46161	0.730803	+	0.682588i	$0.239146\pi$
$434$	2.79577	0.134201
$435$	0	0
$436$	27.5002	1.31702
$437$	19.7312	0.943871
$438$	0	0
$439$	13.4997	0.644303	0.322152	−	0.946688i	$-0.395594\pi$
$439$	13.4997	0.644303	0.322152	+	0.946688i	$0.395594\pi$
$440$	0.384932	0.0183509
$441$	0	0
$442$	−7.89953	−0.375743
$443$	18.1731	0.863432	0.431716	−	0.902010i	$-0.357908\pi$
$443$	18.1731	0.863432	0.431716	+	0.902010i	$0.357908\pi$
$444$	0	0
$445$	17.9569	0.851237
$446$	50.1739	2.37580
$447$	0	0
$448$	−6.15844	−0.290959
$449$	4.42578	0.208865	0.104433	−	0.994532i	$-0.466697\pi$
$449$	4.42578	0.208865	0.104433	+	0.994532i	$0.466697\pi$
$450$	0	0
$451$	2.99579	0.141066
$452$	18.7579	0.882298
$453$	0	0
$454$	−38.7219	−1.81731
$455$	−2.85138	−0.133675
$456$	0	0
$457$	20.9525	0.980116	0.490058	−	0.871690i	$-0.336976\pi$
$457$	20.9525	0.980116	0.490058	+	0.871690i	$0.336976\pi$
$458$	−33.5686	−1.56856
$459$	0	0
$460$	−10.5263	−0.490791
$461$	−30.2835	−1.41044	−0.705222	−	0.708986i	$-0.749153\pi$
$461$	−30.2835	−1.41044	−0.705222	+	0.708986i	$0.749153\pi$
$462$	0	0
$463$	24.2977	1.12921	0.564606	−	0.825361i	$-0.309028\pi$
$463$	24.2977	1.12921	0.564606	+	0.825361i	$0.309028\pi$
$464$	10.3688	0.481357
$465$	0	0
$466$	57.9538	2.68466
$467$	1.83190	0.0847703	0.0423852	−	0.999101i	$-0.486504\pi$
$467$	1.83190	0.0847703	0.0423852	+	0.999101i	$0.486504\pi$
$468$	0	0
$469$	1.70366	0.0786678
$470$	−34.4260	−1.58795
$471$	0	0
$472$	3.01629	0.138836
$473$	5.49847	0.252820
$474$	0	0
$475$	14.3278	0.657402
$476$	3.40584	0.156107
$477$	0	0
$478$	17.1787	0.785737
$479$	30.4527	1.39142	0.695711	−	0.718322i	$-0.255090\pi$
$479$	30.4527	1.39142	0.695711	+	0.718322i	$0.255090\pi$
$480$	0	0
$481$	−11.9004	−0.542613
$482$	0.733803	0.0334238
$483$	0	0
$484$	−18.7819	−0.853723
$485$	−17.3515	−0.787890
$486$	0	0
$487$	−25.6116	−1.16057	−0.580286	−	0.814413i	$-0.697059\pi$
$487$	−25.6116	−1.16057	−0.580286	+	0.814413i	$0.697059\pi$
$488$	−1.52956	−0.0692399
$489$	0	0
$490$	2.61025	0.117919
$491$	25.2765	1.14071	0.570356	−	0.821398i	$-0.306805\pi$
$491$	25.2765	1.14071	0.570356	+	0.821398i	$0.306805\pi$
$492$	0	0
$493$	−4.51712	−0.203441
$494$	18.5037	0.832519
$495$	0	0
$496$	−6.31346	−0.283483
$497$	−5.36282	−0.240555
$498$	0	0
$499$	−15.6164	−0.699088	−0.349544	−	0.936920i	$-0.613663\pi$
$499$	−15.6164	−0.699088	−0.349544	+	0.936920i	$0.613663\pi$
$500$	−19.5952	−0.876325
$501$	0	0
$502$	−22.9959	−1.02636
$503$	−6.09933	−0.271956	−0.135978	−	0.990712i	$-0.543418\pi$
$503$	−6.09933	−0.271956	−0.135978	+	0.990712i	$0.543418\pi$
$504$	0	0
$505$	4.42695	0.196997
$506$	5.75326	0.255764
$507$	0	0
$508$	1.78053	0.0789980
$509$	−0.424702	−0.0188246	−0.00941230	−	0.999956i	$-0.502996\pi$
$509$	−0.424702	−0.0188246	−0.00941230	+	0.999956i	$0.502996\pi$
$510$	0	0
$511$	−0.516847	−0.0228640
$512$	29.9903	1.32540
$513$	0	0
$514$	44.7862	1.97543
$515$	−10.6296	−0.468397
$516$	0	0
$517$	8.86179	0.389741
$518$	10.8941	0.478658
$519$	0	0
$520$	1.21679	0.0533598
$521$	−32.4537	−1.42182	−0.710912	−	0.703281i	$-0.751718\pi$
$521$	−32.4537	−1.42182	−0.710912	+	0.703281i	$0.751718\pi$
$522$	0	0
$523$	−43.7148	−1.91151	−0.955757	−	0.294156i	$-0.904961\pi$
$523$	−43.7148	−1.91151	−0.955757	+	0.294156i	$0.904961\pi$
$524$	38.6369	1.68786
$525$	0	0
$526$	47.9733	2.09174
$527$	2.75044	0.119811
$528$	0	0
$529$	−3.60717	−0.156833
$530$	30.6124	1.32972
$531$	0	0
$532$	−7.97776	−0.345880
$533$	9.46983	0.410184
$534$	0	0
$535$	2.47079	0.106822
$536$	−0.727015	−0.0314023
$537$	0	0
$538$	−29.1568	−1.25704
$539$	−0.671920	−0.0289416
$540$	0	0
$541$	−28.6352	−1.23113	−0.615563	−	0.788088i	$-0.711071\pi$
$541$	−28.6352	−1.23113	−0.615563	+	0.788088i	$0.711071\pi$
$542$	18.8586	0.810047
$543$	0	0
$544$	−14.6978	−0.630161
$545$	20.7345	0.888169
$546$	0	0
$547$	31.1793	1.33313	0.666565	−	0.745447i	$-0.267764\pi$
$547$	31.1793	1.33313	0.666565	+	0.745447i	$0.267764\pi$
$548$	−15.2200	−0.650165
$549$	0	0
$550$	4.17772	0.178138
$551$	10.5808	0.450756
$552$	0	0
$553$	−6.93892	−0.295073
$554$	−23.3946	−0.993941
$555$	0	0
$556$	−20.3760	−0.864134
$557$	−21.4442	−0.908618	−0.454309	−	0.890844i	$-0.650114\pi$
$557$	−21.4442	−0.908618	−0.454309	+	0.890844i	$0.650114\pi$
$558$	0	0
$559$	17.3810	0.735136
$560$	−5.89452	−0.249089
$561$	0	0
$562$	−15.6525	−0.660262
$563$	3.29577	0.138900	0.0694500	−	0.997585i	$-0.477876\pi$
$563$	3.29577	0.138900	0.0694500	+	0.997585i	$0.477876\pi$
$564$	0	0
$565$	14.1430	0.595002
$566$	43.6555	1.83498
$567$	0	0
$568$	2.28851	0.0960239
$569$	−8.46859	−0.355022	−0.177511	−	0.984119i	$-0.556805\pi$
$569$	−8.46859	−0.355022	−0.177511	+	0.984119i	$0.556805\pi$
$570$	0	0
$571$	−27.9586	−1.17003	−0.585016	−	0.811021i	$-0.698912\pi$
$571$	−27.9586	−1.17003	−0.585016	+	0.811021i	$0.698912\pi$
$572$	2.54106	0.106247
$573$	0	0
$574$	−8.66901	−0.361837
$575$	14.0821	0.587262
$576$	0	0
$577$	−16.0599	−0.668583	−0.334292	−	0.942470i	$-0.608497\pi$
$577$	−16.0599	−0.668583	−0.334292	+	0.942470i	$0.608497\pi$
$578$	−25.9398	−1.07895
$579$	0	0
$580$	−5.64469	−0.234383
$581$	−8.45270	−0.350677
$582$	0	0
$583$	−7.88010	−0.326360
$584$	0.220558	0.00912674
$585$	0	0
$586$	23.2589	0.960816
$587$	−41.1062	−1.69664	−0.848318	−	0.529487i	$-0.822384\pi$
$587$	−41.1062	−1.69664	−0.848318	+	0.529487i	$0.822384\pi$
$588$	0	0
$589$	−6.44256	−0.265461
$590$	−18.4500	−0.759573
$591$	0	0
$592$	−24.6012	−1.01110
$593$	35.2860	1.44902	0.724511	−	0.689263i	$-0.242066\pi$
$593$	35.2860	1.44902	0.724511	+	0.689263i	$0.242066\pi$
$594$	0	0
$595$	2.56793	0.105275
$596$	−12.0430	−0.493299
$597$	0	0
$598$	18.1863	0.743695
$599$	−19.8649	−0.811656	−0.405828	−	0.913949i	$-0.633017\pi$
$599$	−19.8649	−0.811656	−0.405828	+	0.913949i	$0.633017\pi$
$600$	0	0
$601$	−4.83491	−0.197220	−0.0986101	−	0.995126i	$-0.531440\pi$
$601$	−4.83491	−0.197220	−0.0986101	+	0.995126i	$0.531440\pi$
$602$	−15.9111	−0.648489
$603$	0	0
$604$	21.4243	0.871742
$605$	−14.1611	−0.575732
$606$	0	0
$607$	−9.91972	−0.402629	−0.201315	−	0.979527i	$-0.564521\pi$
$607$	−9.91972	−0.402629	−0.201315	+	0.979527i	$0.564521\pi$
$608$	34.4276	1.39622
$609$	0	0
$610$	9.35598	0.378812
$611$	28.0126	1.13327
$612$	0	0
$613$	−4.74214	−0.191533	−0.0957666	−	0.995404i	$-0.530530\pi$
$613$	−4.74214	−0.191533	−0.0957666	+	0.995404i	$0.530530\pi$
$614$	−42.7307	−1.72447
$615$	0	0
$616$	0.286733	0.0115528
$617$	21.7621	0.876109	0.438055	−	0.898948i	$-0.355668\pi$
$617$	21.7621	0.876109	0.438055	+	0.898948i	$0.355668\pi$
$618$	0	0
$619$	9.52363	0.382787	0.191394	−	0.981513i	$-0.438699\pi$
$619$	9.52363	0.382787	0.191394	+	0.981513i	$0.438699\pi$
$620$	3.43701	0.138034
$621$	0	0
$622$	55.0462	2.20715
$623$	13.3759	0.535895
$624$	0	0
$625$	1.21443	0.0485772
$626$	−5.85625	−0.234063
$627$	0	0
$628$	34.7167	1.38535
$629$	10.7174	0.427332
$630$	0	0
$631$	−22.7939	−0.907412	−0.453706	−	0.891151i	$-0.649898\pi$
$631$	−22.7939	−0.907412	−0.453706	+	0.891151i	$0.649898\pi$
$632$	2.96109	0.117786
$633$	0	0
$634$	7.09765	0.281884
$635$	1.34248	0.0532746
$636$	0	0
$637$	−2.12397	−0.0841548
$638$	3.08517	0.122143
$639$	0	0
$640$	4.55548	0.180071
$641$	26.6372	1.05211	0.526053	−	0.850452i	$-0.323671\pi$
$641$	26.6372	1.05211	0.526053	+	0.850452i	$0.323671\pi$
$642$	0	0
$643$	15.6748	0.618155	0.309077	−	0.951037i	$-0.399980\pi$
$643$	15.6748	0.618155	0.309077	+	0.951037i	$0.399980\pi$
$644$	−7.84095	−0.308977
$645$	0	0
$646$	−16.6642	−0.655646
$647$	27.0353	1.06287	0.531433	−	0.847100i	$-0.321654\pi$
$647$	27.0353	1.06287	0.531433	+	0.847100i	$0.321654\pi$
$648$	0	0
$649$	4.74930	0.186426
$650$	13.2060	0.517981
$651$	0	0
$652$	−10.8012	−0.423009
$653$	13.7013	0.536174	0.268087	−	0.963395i	$-0.413609\pi$
$653$	13.7013	0.536174	0.268087	+	0.963395i	$0.413609\pi$
$654$	0	0
$655$	29.1314	1.13826
$656$	19.5765	0.764334
$657$	0	0
$658$	−25.6437	−0.999694
$659$	−5.11654	−0.199312	−0.0996560	−	0.995022i	$-0.531774\pi$
$659$	−5.11654	−0.199312	−0.0996560	+	0.995022i	$0.531774\pi$
$660$	0	0
$661$	−19.8743	−0.773022	−0.386511	−	0.922285i	$-0.626320\pi$
$661$	−19.8743	−0.773022	−0.386511	+	0.922285i	$0.626320\pi$
$662$	−51.7765	−2.01235
$663$	0	0
$664$	3.60708	0.139982
$665$	−6.01505	−0.233254
$666$	0	0
$667$	10.3993	0.402664
$668$	−11.8208	−0.457361
$669$	0	0
$670$	4.44699	0.171802
$671$	−2.40837	−0.0929742
$672$	0	0
$673$	−12.6067	−0.485951	−0.242975	−	0.970032i	$-0.578123\pi$
$673$	−12.6067	−0.485951	−0.242975	+	0.970032i	$0.578123\pi$
$674$	−64.7075	−2.49244
$675$	0	0
$676$	−15.1144	−0.581324
$677$	44.6723	1.71690	0.858449	−	0.512899i	$-0.171429\pi$
$677$	44.6723	1.71690	0.858449	+	0.512899i	$0.171429\pi$
$678$	0	0
$679$	−12.9250	−0.496015
$680$	−1.09583	−0.0420232
$681$	0	0
$682$	−1.87853	−0.0719328
$683$	−38.3924	−1.46905	−0.734523	−	0.678584i	$-0.762594\pi$
$683$	−38.3924	−1.46905	−0.734523	+	0.678584i	$0.762594\pi$
$684$	0	0
$685$	−11.4755	−0.438457
$686$	1.94436	0.0742359
$687$	0	0
$688$	35.9308	1.36985
$689$	−24.9094	−0.948972
$690$	0	0
$691$	36.4516	1.38668	0.693342	−	0.720608i	$-0.256137\pi$
$691$	36.4516	1.38668	0.693342	+	0.720608i	$0.256137\pi$
$692$	−1.69375	−0.0643868
$693$	0	0
$694$	−58.6891	−2.22781
$695$	−15.3630	−0.582753
$696$	0	0
$697$	−8.52844	−0.323038
$698$	35.1435	1.33020
$699$	0	0
$700$	−5.69369	−0.215201
$701$	37.4865	1.41585	0.707923	−	0.706289i	$-0.249632\pi$
$701$	37.4865	1.41585	0.707923	+	0.706289i	$0.249632\pi$
$702$	0	0
$703$	−25.1042	−0.946823
$704$	4.13798	0.155956
$705$	0	0
$706$	−32.0901	−1.20773
$707$	3.29760	0.124019
$708$	0	0
$709$	45.0970	1.69365	0.846827	−	0.531869i	$-0.178510\pi$
$709$	45.0970	1.69365	0.846827	+	0.531869i	$0.178510\pi$
$710$	−13.9983	−0.525348
$711$	0	0
$712$	−5.70800	−0.213916
$713$	−6.33208	−0.237138
$714$	0	0
$715$	1.91590	0.0716506
$716$	−24.5220	−0.916429
$717$	0	0
$718$	−19.6760	−0.734300
$719$	39.5236	1.47398	0.736991	−	0.675903i	$-0.236246\pi$
$719$	39.5236	1.47398	0.736991	+	0.675903i	$0.236246\pi$
$720$	0	0
$721$	−7.91791	−0.294878
$722$	2.09107	0.0778217
$723$	0	0
$724$	18.4053	0.684029
$725$	7.55145	0.280454
$726$	0	0
$727$	10.1237	0.375466	0.187733	−	0.982220i	$-0.439886\pi$
$727$	10.1237	0.375466	0.187733	+	0.982220i	$0.439886\pi$
$728$	0.906377	0.0335926
$729$	0	0
$730$	−1.34910	−0.0499325
$731$	−15.6531	−0.578952
$732$	0	0
$733$	19.2111	0.709579	0.354789	−	0.934946i	$-0.384553\pi$
$733$	19.2111	0.709579	0.354789	+	0.934946i	$0.384553\pi$
$734$	−51.8422	−1.91353
$735$	0	0
$736$	33.8373	1.24726
$737$	−1.14472	−0.0421665
$738$	0	0
$739$	−44.1795	−1.62517	−0.812585	−	0.582843i	$-0.801940\pi$
$739$	−44.1795	−1.62517	−0.812585	+	0.582843i	$0.801940\pi$
$740$	13.3927	0.492326
$741$	0	0
$742$	22.8029	0.837121
$743$	10.6927	0.392278	0.196139	−	0.980576i	$-0.437160\pi$
$743$	10.6927	0.392278	0.196139	+	0.980576i	$0.437160\pi$
$744$	0	0
$745$	−9.08012	−0.332670
$746$	−47.4959	−1.73895
$747$	0	0
$748$	−2.28845	−0.0836742
$749$	1.84047	0.0672494
$750$	0	0
$751$	−11.0632	−0.403703	−0.201852	−	0.979416i	$-0.564696\pi$
$751$	−11.0632	−0.403703	−0.201852	+	0.979416i	$0.564696\pi$
$752$	57.9089	2.11172
$753$	0	0
$754$	9.75236	0.355160
$755$	16.1534	0.587884
$756$	0	0
$757$	−53.6535	−1.95007	−0.975034	−	0.222055i	$-0.928724\pi$
$757$	−53.6535	−1.95007	−0.975034	+	0.222055i	$0.928724\pi$
$758$	53.9173	1.95836
$759$	0	0
$760$	2.56684	0.0931092
$761$	40.9670	1.48505	0.742526	−	0.669818i	$-0.233628\pi$
$761$	40.9670	1.48505	0.742526	+	0.669818i	$0.233628\pi$
$762$	0	0
$763$	15.4450	0.559146
$764$	−34.0333	−1.23128
$765$	0	0
$766$	−15.5670	−0.562457
$767$	15.0128	0.542080
$768$	0	0
$769$	−3.83517	−0.138300	−0.0691499	−	0.997606i	$-0.522029\pi$
$769$	−3.83517	−0.138300	−0.0691499	+	0.997606i	$0.522029\pi$
$770$	−1.75388	−0.0632055
$771$	0	0
$772$	−0.182704	−0.00657565
$773$	−21.2549	−0.764487	−0.382243	−	0.924062i	$-0.624848\pi$
$773$	−21.2549	−0.764487	−0.382243	+	0.924062i	$0.624848\pi$
$774$	0	0
$775$	−4.59802	−0.165166
$776$	5.51556	0.197997
$777$	0	0
$778$	−41.5628	−1.49010
$779$	19.9768	0.715743
$780$	0	0
$781$	3.60339	0.128939
$782$	−16.3785	−0.585693
$783$	0	0
$784$	−4.39078	−0.156814
$785$	26.1756	0.934247
$786$	0	0
$787$	13.4079	0.477938	0.238969	−	0.971027i	$-0.423191\pi$
$787$	13.4079	0.477938	0.238969	+	0.971027i	$0.423191\pi$
$788$	−6.76668	−0.241053
$789$	0	0
$790$	−18.1123	−0.644408
$791$	10.5350	0.374583
$792$	0	0
$793$	−7.61299	−0.270345
$794$	−16.4356	−0.583278
$795$	0	0
$796$	27.1881	0.963655
$797$	10.0542	0.356137	0.178069	−	0.984018i	$-0.443015\pi$
$797$	10.0542	0.356137	0.178069	+	0.984018i	$0.443015\pi$
$798$	0	0
$799$	−25.2279	−0.892498
$800$	24.5708	0.868710
$801$	0	0
$802$	45.8108	1.61763
$803$	0.347280	0.0122552
$804$	0	0
$805$	−5.91190	−0.208367
$806$	−5.93814	−0.209162
$807$	0	0
$808$	−1.40721	−0.0495053
$809$	−25.4204	−0.893733	−0.446867	−	0.894601i	$-0.647460\pi$
$809$	−25.4204	−0.893733	−0.446867	+	0.894601i	$0.647460\pi$
$810$	0	0
$811$	32.2864	1.13373	0.566865	−	0.823811i	$-0.308156\pi$
$811$	32.2864	1.13373	0.566865	+	0.823811i	$0.308156\pi$
$812$	−4.20468	−0.147555
$813$	0	0
$814$	−7.31994	−0.256564
$815$	−8.14388	−0.285268
$816$	0	0
$817$	36.6655	1.28276
$818$	−17.1427	−0.599380
$819$	0	0
$820$	−10.6573	−0.372170
$821$	2.38698	0.0833062	0.0416531	−	0.999132i	$-0.486738\pi$
$821$	2.38698	0.0833062	0.0416531	+	0.999132i	$0.486738\pi$
$822$	0	0
$823$	−7.37746	−0.257162	−0.128581	−	0.991699i	$-0.541042\pi$
$823$	−7.37746	−0.257162	−0.128581	+	0.991699i	$0.541042\pi$
$824$	3.37886	0.117708
$825$	0	0
$826$	−13.7432	−0.478188
$827$	10.1241	0.352051	0.176025	−	0.984386i	$-0.443676\pi$
$827$	10.1241	0.352051	0.176025	+	0.984386i	$0.443676\pi$
$828$	0	0
$829$	27.9788	0.971743	0.485871	−	0.874030i	$-0.338502\pi$
$829$	27.9788	0.971743	0.485871	+	0.874030i	$0.338502\pi$
$830$	−22.0637	−0.765842
$831$	0	0
$832$	13.0804	0.453480
$833$	1.91283	0.0662757
$834$	0	0
$835$	−8.91264	−0.308435
$836$	5.36041	0.185394
$837$	0	0
$838$	64.5176	2.22872
$839$	−20.3383	−0.702154	−0.351077	−	0.936347i	$-0.614185\pi$
$839$	−20.3383	−0.702154	−0.351077	+	0.936347i	$0.614185\pi$
$840$	0	0
$841$	−23.4234	−0.807703
$842$	−60.4643	−2.08374
$843$	0	0
$844$	−34.7389	−1.19576
$845$	−11.3959	−0.392032
$846$	0	0
$847$	−10.5485	−0.362452
$848$	−51.4939	−1.76831
$849$	0	0
$850$	−11.8932	−0.407933
$851$	−24.6737	−0.845804
$852$	0	0
$853$	19.2870	0.660376	0.330188	−	0.943915i	$-0.392888\pi$
$853$	19.2870	0.660376	0.330188	+	0.943915i	$0.392888\pi$
$854$	6.96919	0.238481
$855$	0	0
$856$	−0.785397	−0.0268443
$857$	22.4053	0.765350	0.382675	−	0.923883i	$-0.375003\pi$
$857$	22.4053	0.765350	0.382675	+	0.923883i	$0.375003\pi$
$858$	0	0
$859$	−28.3277	−0.966529	−0.483265	−	0.875474i	$-0.660549\pi$
$859$	−28.3277	−0.966529	−0.483265	+	0.875474i	$0.660549\pi$
$860$	−19.5605	−0.667007
$861$	0	0
$862$	56.2010	1.91421
$863$	2.87085	0.0977248	0.0488624	−	0.998806i	$-0.484440\pi$
$863$	2.87085	0.0977248	0.0488624	+	0.998806i	$0.484440\pi$
$864$	0	0
$865$	−1.27705	−0.0434211
$866$	59.1358	2.00952
$867$	0	0
$868$	2.56020	0.0868988
$869$	4.66240	0.158161
$870$	0	0
$871$	−3.61853	−0.122609
$872$	−6.59094	−0.223197
$873$	0	0
$874$	38.3645	1.29770
$875$	−11.0053	−0.372047
$876$	0	0
$877$	−13.5296	−0.456861	−0.228430	−	0.973560i	$-0.573359\pi$
$877$	−13.5296	−0.456861	−0.228430	+	0.973560i	$0.573359\pi$
$878$	26.2481	0.885832
$879$	0	0
$880$	3.96064	0.133513
$881$	31.4414	1.05929	0.529644	−	0.848220i	$-0.322325\pi$
$881$	31.4414	1.05929	0.529644	+	0.848220i	$0.322325\pi$
$882$	0	0
$883$	48.1820	1.62145	0.810727	−	0.585424i	$-0.199072\pi$
$883$	48.1820	1.62145	0.810727	+	0.585424i	$0.199072\pi$
$884$	−7.23392	−0.243303
$885$	0	0
$886$	35.3351	1.18711
$887$	−21.8868	−0.734887	−0.367443	−	0.930046i	$-0.619767\pi$
$887$	−21.8868	−0.734887	−0.367443	+	0.930046i	$0.619767\pi$
$888$	0	0
$889$	1.00000	0.0335389
$890$	34.9145	1.17034
$891$	0	0
$892$	45.9463	1.53840
$893$	59.0931	1.97747
$894$	0	0
$895$	−18.4890	−0.618019
$896$	3.39334	0.113363
$897$	0	0
$898$	8.60529	0.287162
$899$	−3.39555	−0.113248
$900$	0	0
$901$	22.4332	0.747357
$902$	5.82488	0.193947
$903$	0	0
$904$	−4.49569	−0.149524
$905$	13.8772	0.461294
$906$	0	0
$907$	17.2138	0.571575	0.285788	−	0.958293i	$-0.407745\pi$
$907$	17.2138	0.571575	0.285788	+	0.958293i	$0.407745\pi$
$908$	−35.4592	−1.17676
$909$	0	0
$910$	−5.54411	−0.183785
$911$	48.6408	1.61154	0.805770	−	0.592228i	$-0.201752\pi$
$911$	48.6408	1.61154	0.805770	+	0.592228i	$0.201752\pi$
$912$	0	0
$913$	5.67954	0.187965
$914$	40.7391	1.34753
$915$	0	0
$916$	−30.7401	−1.01568
$917$	21.6997	0.716588
$918$	0	0
$919$	16.3448	0.539165	0.269583	−	0.962977i	$-0.413114\pi$
$919$	16.3448	0.539165	0.269583	+	0.962977i	$0.413114\pi$
$920$	2.52283	0.0831751
$921$	0	0
$922$	−58.8820	−1.93918
$923$	11.3905	0.374922
$924$	0	0
$925$	−17.9167	−0.589099
$926$	47.2435	1.55252
$927$	0	0
$928$	18.1451	0.595642
$929$	28.5730	0.937451	0.468725	−	0.883344i	$-0.344713\pi$
$929$	28.5730	0.937451	0.468725	+	0.883344i	$0.344713\pi$
$930$	0	0
$931$	−4.48056	−0.146845
$932$	53.0706	1.73839
$933$	0	0
$934$	3.56187	0.116548
$935$	−1.72544	−0.0564280
$936$	0	0
$937$	−10.5805	−0.345650	−0.172825	−	0.984953i	$-0.555290\pi$
$937$	−10.5805	−0.345650	−0.172825	+	0.984953i	$0.555290\pi$
$938$	3.31253	0.108158
$939$	0	0
$940$	−31.5253	−1.02824
$941$	16.7748	0.546842	0.273421	−	0.961894i	$-0.411845\pi$
$941$	16.7748	0.546842	0.273421	+	0.961894i	$0.411845\pi$
$942$	0	0
$943$	19.6342	0.639378
$944$	31.0352	1.01011
$945$	0	0
$946$	10.6910	0.347594
$947$	−32.0250	−1.04067	−0.520336	−	0.853961i	$-0.674193\pi$
$947$	−32.0250	−1.04067	−0.520336	+	0.853961i	$0.674193\pi$
$948$	0	0
$949$	1.09777	0.0356351
$950$	27.8583	0.903842
$951$	0	0
$952$	−0.816275	−0.0264556
$953$	30.0313	0.972809	0.486405	−	0.873734i	$-0.338308\pi$
$953$	30.0313	0.972809	0.486405	+	0.873734i	$0.338308\pi$
$954$	0	0
$955$	−25.6604	−0.830350
$956$	15.7312	0.508785
$957$	0	0
$958$	59.2110	1.91302
$959$	−8.54803	−0.276030
$960$	0	0
$961$	−28.9325	−0.933306
$962$	−23.1387	−0.746021
$963$	0	0
$964$	0.671973	0.0216428
$965$	−0.137754	−0.00443447
$966$	0	0
$967$	−12.7947	−0.411451	−0.205725	−	0.978610i	$-0.565955\pi$
$967$	−12.7947	−0.411451	−0.205725	+	0.978610i	$0.565955\pi$
$968$	4.50144	0.144682
$969$	0	0
$970$	−33.7375	−1.08324
$971$	−30.9532	−0.993334	−0.496667	−	0.867941i	$-0.665443\pi$
$971$	−30.9532	−0.993334	−0.496667	+	0.867941i	$0.665443\pi$
$972$	0	0
$973$	−11.4438	−0.366872
$974$	−49.7981	−1.59563
$975$	0	0
$976$	−15.7379	−0.503759
$977$	−18.5496	−0.593456	−0.296728	−	0.954962i	$-0.595895\pi$
$977$	−18.5496	−0.593456	−0.296728	+	0.954962i	$0.595895\pi$
$978$	0	0
$979$	−8.98754	−0.287243
$980$	2.39031	0.0763558
$981$	0	0
$982$	49.1465	1.56833
$983$	−9.51156	−0.303372	−0.151686	−	0.988429i	$-0.548470\pi$
$983$	−9.51156	−0.303372	−0.151686	+	0.988429i	$0.548470\pi$
$984$	0	0
$985$	−5.10192	−0.162561
$986$	−8.78289	−0.279704
$987$	0	0
$988$	16.9445	0.539078
$989$	36.0367	1.14590
$990$	0	0
$991$	−46.4713	−1.47621	−0.738105	−	0.674686i	$-0.764279\pi$
$991$	−46.4713	−1.47621	−0.738105	+	0.674686i	$0.764279\pi$
$992$	−11.0484	−0.350788
$993$	0	0
$994$	−10.4272	−0.330732
$995$	20.4992	0.649868
$996$	0	0
$997$	−0.682471	−0.0216141	−0.0108070	−	0.999942i	$-0.503440\pi$
$997$	−0.682471	−0.0216141	−0.0108070	+	0.999942i	$0.503440\pi$
$998$	−30.3639	−0.961153
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.x.1.19	yes	22
3.2	odd	2	inner	8001.2.a.x.1.4	✓	22

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8001.2.a.x.1.4	✓	22	3.2	odd	2	inner
8001.2.a.x.1.19	yes	22	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:	\( N \)	\(=\)	\( 8001 = 3^{2} \cdot 7 \cdot 127 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8001.a (trivial)

\(f(q)\)	\(=\)	\(q+1.94436 q^{2} +1.78053 q^{4} +1.34248 q^{5} +1.00000 q^{7} -0.426737 q^{8} +O(q^{10})\)	\(q+1.94436 q^{2} +1.78053 q^{4} +1.34248 q^{5} +1.00000 q^{7} -0.426737 q^{8} +2.61025 q^{10} -0.671920 q^{11} -2.12397 q^{13} +1.94436 q^{14} -4.39078 q^{16} +1.91283 q^{17} -4.48056 q^{19} +2.39031 q^{20} -1.30645 q^{22} -4.40373 q^{23} -3.19776 q^{25} -4.12976 q^{26} +1.78053 q^{28} -2.36148 q^{29} +1.43789 q^{31} -7.68377 q^{32} +3.71923 q^{34} +1.34248 q^{35} +5.60291 q^{37} -8.71182 q^{38} -0.572884 q^{40} -4.45855 q^{41} -8.18323 q^{43} -1.19637 q^{44} -8.56242 q^{46} -13.1888 q^{47} +1.00000 q^{49} -6.21758 q^{50} -3.78179 q^{52} +11.7277 q^{53} -0.902037 q^{55} -0.426737 q^{56} -4.59157 q^{58} -7.06826 q^{59} +3.58432 q^{61} +2.79577 q^{62} -6.15844 q^{64} -2.85138 q^{65} +1.70366 q^{67} +3.40584 q^{68} +2.61025 q^{70} -5.36282 q^{71} -0.516847 q^{73} +10.8941 q^{74} -7.97776 q^{76} -0.671920 q^{77} -6.93892 q^{79} -5.89452 q^{80} -8.66901 q^{82} -8.45270 q^{83} +2.56793 q^{85} -15.9111 q^{86} +0.286733 q^{88} +13.3759 q^{89} -2.12397 q^{91} -7.84095 q^{92} -25.6437 q^{94} -6.01505 q^{95} -12.9250 q^{97} +1.94436 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q + 10 q^{4} + 22 q^{7}+O(q^{10}) \) 22 * q + 10 * q^4 + 22 * q^7	\( 22 q + 10 q^{4} + 22 q^{7} - 4 q^{10} - 10 q^{13} - 6 q^{16} - 18 q^{19} - 34 q^{22} - 26 q^{25} + 10 q^{28} - 42 q^{31} - 16 q^{34} - 36 q^{37} - 46 q^{40} - 54 q^{43} - 12 q^{46} + 22 q^{49} - 22 q^{52} - 28 q^{55} - 26 q^{58} - 22 q^{61} - 76 q^{64} - 48 q^{67} - 4 q^{70} - 48 q^{73} - 20 q^{76} - 94 q^{79} - 8 q^{82} - 40 q^{85} - 26 q^{88} - 10 q^{91} - 26 q^{94} - 56 q^{97}+O(q^{100}) \) 22 * q + 10 * q^4 + 22 * q^7 - 4 * q^10 - 10 * q^13 - 6 * q^16 - 18 * q^19 - 34 * q^22 - 26 * q^25 + 10 * q^28 - 42 * q^31 - 16 * q^34 - 36 * q^37 - 46 * q^40 - 54 * q^43 - 12 * q^46 + 22 * q^49 - 22 * q^52 - 28 * q^55 - 26 * q^58 - 22 * q^61 - 76 * q^64 - 48 * q^67 - 4 * q^70 - 48 * q^73 - 20 * q^76 - 94 * q^79 - 8 * q^82 - 40 * q^85 - 26 * q^88 - 10 * q^91 - 26 * q^94 - 56 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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