LMFDB - Embedded newform 8001.2.a.z.1.15

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:	$32$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.15
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)

$f(q)$	$=$	$q-0.242500 q^{2} -1.94119 q^{4} +2.79671 q^{5} -1.00000 q^{7} +0.955741 q^{8} +O(q^{10})$	\(q-0.242500 q^{2} -1.94119 q^{4} +2.79671 q^{5} -1.00000 q^{7} +0.955741 q^{8} -0.678204 q^{10} +4.59836 q^{11} -4.92098 q^{13} +0.242500 q^{14} +3.65062 q^{16} -2.21339 q^{17} +0.299740 q^{19} -5.42896 q^{20} -1.11510 q^{22} -2.16929 q^{23} +2.82161 q^{25} +1.19334 q^{26} +1.94119 q^{28} -5.57762 q^{29} +6.00125 q^{31} -2.79676 q^{32} +0.536748 q^{34} -2.79671 q^{35} +8.81952 q^{37} -0.0726869 q^{38} +2.67293 q^{40} -10.5020 q^{41} -6.69985 q^{43} -8.92631 q^{44} +0.526055 q^{46} +7.58321 q^{47} +1.00000 q^{49} -0.684241 q^{50} +9.55257 q^{52} -1.04200 q^{53} +12.8603 q^{55} -0.955741 q^{56} +1.35257 q^{58} -5.65706 q^{59} +0.161967 q^{61} -1.45530 q^{62} -6.62302 q^{64} -13.7626 q^{65} +2.60257 q^{67} +4.29662 q^{68} +0.678204 q^{70} -2.49985 q^{71} -5.94931 q^{73} -2.13874 q^{74} -0.581853 q^{76} -4.59836 q^{77} -13.1275 q^{79} +10.2097 q^{80} +2.54675 q^{82} -9.40919 q^{83} -6.19022 q^{85} +1.62472 q^{86} +4.39484 q^{88} +9.33298 q^{89} +4.92098 q^{91} +4.21102 q^{92} -1.83893 q^{94} +0.838286 q^{95} -11.8534 q^{97} -0.242500 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q + 30 q^{4} - 32 q^{7}+O(q^{10}) $ 32 * q + 30 * q^4 - 32 * q^7	$ 32 q + 30 q^{4} - 32 q^{7} - 16 q^{10} - 14 q^{13} + 18 q^{16} - 30 q^{19} - 10 q^{22} + 36 q^{25} - 30 q^{28} - 58 q^{31} - 34 q^{34} + 8 q^{37} - 34 q^{40} + 6 q^{43} - 36 q^{46} + 32 q^{49} - 56 q^{52} - 88 q^{55} - 22 q^{58} - 46 q^{61} + 20 q^{64} - 8 q^{67} + 16 q^{70} - 60 q^{73} - 128 q^{76} - 74 q^{79} - 52 q^{82} - 16 q^{85} - 64 q^{88} + 14 q^{91} - 58 q^{94} - 44 q^{97}+O(q^{100}) $ 32 * q + 30 * q^4 - 32 * q^7 - 16 * q^10 - 14 * q^13 + 18 * q^16 - 30 * q^19 - 10 * q^22 + 36 * q^25 - 30 * q^28 - 58 * q^31 - 34 * q^34 + 8 * q^37 - 34 * q^40 + 6 * q^43 - 36 * q^46 + 32 * q^49 - 56 * q^52 - 88 * q^55 - 22 * q^58 - 46 * q^61 + 20 * q^64 - 8 * q^67 + 16 * q^70 - 60 * q^73 - 128 * q^76 - 74 * q^79 - 52 * q^82 - 16 * q^85 - 64 * q^88 + 14 * q^91 - 58 * q^94 - 44 * 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 $

$2$	−0.242500	−0.171474	−0.0857368	−	0.996318i	$-0.527324\pi$
$2$	−0.242500	−0.171474	−0.0857368	+	0.996318i	$0.527324\pi$
$3$	0	0
$3$	0	0
$4$	−1.94119	−0.970597
$5$	2.79671	1.25073	0.625364	−	0.780333i	$-0.284950\pi$
$5$	2.79671	1.25073	0.625364	+	0.780333i	$0.284950\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0.955741	0.337905
$9$	0	0
$10$	−0.678204	−0.214467
$11$	4.59836	1.38646	0.693229	−	0.720718i	$-0.256188\pi$
$11$	4.59836	1.38646	0.693229	+	0.720718i	$0.256188\pi$
$12$	0	0
$13$	−4.92098	−1.36483	−0.682417	−	0.730963i	$-0.739071\pi$
$13$	−4.92098	−1.36483	−0.682417	+	0.730963i	$0.739071\pi$
$14$	0.242500	0.0648109
$15$	0	0
$16$	3.65062	0.912655
$17$	−2.21339	−0.536826	−0.268413	−	0.963304i	$-0.586499\pi$
$17$	−2.21339	−0.536826	−0.268413	+	0.963304i	$0.586499\pi$
$18$	0	0
$19$	0.299740	0.0687650	0.0343825	−	0.999409i	$-0.489054\pi$
$19$	0.299740	0.0687650	0.0343825	+	0.999409i	$0.489054\pi$
$20$	−5.42896	−1.21395
$21$	0	0
$22$	−1.11510	−0.237741
$23$	−2.16929	−0.452329	−0.226165	−	0.974089i	$-0.572619\pi$
$23$	−2.16929	−0.452329	−0.226165	+	0.974089i	$0.572619\pi$
$24$	0	0
$25$	2.82161	0.564321
$26$	1.19334	0.234033
$27$	0	0
$28$	1.94119	0.366851
$29$	−5.57762	−1.03574	−0.517869	−	0.855460i	$-0.673274\pi$
$29$	−5.57762	−1.03574	−0.517869	+	0.855460i	$0.673274\pi$
$30$	0	0
$31$	6.00125	1.07786	0.538928	−	0.842352i	$-0.318829\pi$
$31$	6.00125	1.07786	0.538928	+	0.842352i	$0.318829\pi$
$32$	−2.79676	−0.494402
$33$	0	0
$34$	0.536748	0.0920515
$35$	−2.79671	−0.472731
$36$	0	0
$37$	8.81952	1.44992	0.724960	−	0.688791i	$-0.241858\pi$
$37$	8.81952	1.44992	0.724960	+	0.688791i	$0.241858\pi$
$38$	−0.0726869	−0.0117914
$39$	0	0
$40$	2.67293	0.422628
$41$	−10.5020	−1.64014	−0.820071	−	0.572262i	$-0.806066\pi$
$41$	−10.5020	−1.64014	−0.820071	+	0.572262i	$0.806066\pi$
$42$	0	0
$43$	−6.69985	−1.02172	−0.510859	−	0.859665i	$-0.670673\pi$
$43$	−6.69985	−1.02172	−0.510859	+	0.859665i	$0.670673\pi$
$44$	−8.92631	−1.34569
$45$	0	0
$46$	0.526055	0.0775625
$47$	7.58321	1.10613	0.553063	−	0.833140i	$-0.313459\pi$
$47$	7.58321	1.10613	0.553063	+	0.833140i	$0.313459\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−0.684241	−0.0967662
$51$	0	0
$52$	9.55257	1.32470
$53$	−1.04200	−0.143130	−0.0715649	−	0.997436i	$-0.522799\pi$
$53$	−1.04200	−0.143130	−0.0715649	+	0.997436i	$0.522799\pi$
$54$	0	0
$55$	12.8603	1.73408
$56$	−0.955741	−0.127716
$57$	0	0
$58$	1.35257	0.177602
$59$	−5.65706	−0.736487	−0.368243	−	0.929729i	$-0.620041\pi$
$59$	−5.65706	−0.736487	−0.368243	+	0.929729i	$0.620041\pi$
$60$	0	0
$61$	0.161967	0.0207378	0.0103689	−	0.999946i	$-0.496699\pi$
$61$	0.161967	0.0207378	0.0103689	+	0.999946i	$0.496699\pi$
$62$	−1.45530	−0.184824
$63$	0	0
$64$	−6.62302	−0.827878
$65$	−13.7626	−1.70704
$66$	0	0
$67$	2.60257	0.317954	0.158977	−	0.987282i	$-0.449180\pi$
$67$	2.60257	0.317954	0.158977	+	0.987282i	$0.449180\pi$
$68$	4.29662	0.521041
$69$	0	0
$70$	0.678204	0.0810609
$71$	−2.49985	−0.296678	−0.148339	−	0.988937i	$-0.547393\pi$
$71$	−2.49985	−0.296678	−0.148339	+	0.988937i	$0.547393\pi$
$72$	0	0
$73$	−5.94931	−0.696314	−0.348157	−	0.937436i	$-0.613192\pi$
$73$	−5.94931	−0.696314	−0.348157	+	0.937436i	$0.613192\pi$
$74$	−2.13874	−0.248623
$75$	0	0
$76$	−0.581853	−0.0667431
$77$	−4.59836	−0.524032
$78$	0	0
$79$	−13.1275	−1.47696	−0.738482	−	0.674273i	$-0.764457\pi$
$79$	−13.1275	−1.47696	−0.738482	+	0.674273i	$0.764457\pi$
$80$	10.2097	1.14148
$81$	0	0
$82$	2.54675	0.281241
$83$	−9.40919	−1.03279	−0.516396	−	0.856350i	$-0.672727\pi$
$83$	−9.40919	−1.03279	−0.516396	+	0.856350i	$0.672727\pi$
$84$	0	0
$85$	−6.19022	−0.671423
$86$	1.62472	0.175198
$87$	0	0
$88$	4.39484	0.468492
$89$	9.33298	0.989294	0.494647	−	0.869094i	$-0.335297\pi$
$89$	9.33298	0.989294	0.494647	+	0.869094i	$0.335297\pi$
$90$	0	0
$91$	4.92098	0.515859
$92$	4.21102	0.439029
$93$	0	0
$94$	−1.83893	−0.189671
$95$	0.838286	0.0860063
$96$	0	0
$97$	−11.8534	−1.20353	−0.601767	−	0.798672i	$-0.705536\pi$
$97$	−11.8534	−1.20353	−0.601767	+	0.798672i	$0.705536\pi$
$98$	−0.242500	−0.0244962
$99$	0	0
$100$	−5.47729	−0.547729
$101$	−5.46441	−0.543729	−0.271864	−	0.962336i	$-0.587640\pi$
$101$	−5.46441	−0.543729	−0.271864	+	0.962336i	$0.587640\pi$
$102$	0	0
$103$	0.754962	0.0743886	0.0371943	−	0.999308i	$-0.488158\pi$
$103$	0.754962	0.0743886	0.0371943	+	0.999308i	$0.488158\pi$
$104$	−4.70318	−0.461185
$105$	0	0
$106$	0.252685	0.0245430
$107$	10.6424	1.02884	0.514419	−	0.857539i	$-0.328008\pi$
$107$	10.6424	1.02884	0.514419	+	0.857539i	$0.328008\pi$
$108$	0	0
$109$	6.61161	0.633277	0.316639	−	0.948546i	$-0.397446\pi$
$109$	6.61161	0.633277	0.316639	+	0.948546i	$0.397446\pi$
$110$	−3.11863	−0.297349
$111$	0	0
$112$	−3.65062	−0.344951
$113$	−3.73009	−0.350897	−0.175449	−	0.984489i	$-0.556138\pi$
$113$	−3.73009	−0.350897	−0.175449	+	0.984489i	$0.556138\pi$
$114$	0	0
$115$	−6.06690	−0.565741
$116$	10.8272	1.00528
$117$	0	0
$118$	1.37184	0.126288
$119$	2.21339	0.202901
$120$	0	0
$121$	10.1449	0.922265
$122$	−0.0392771	−0.00355598
$123$	0	0
$124$	−11.6496	−1.04616
$125$	−6.09234	−0.544916
$126$	0	0
$127$	−1.00000	−0.0887357
$127$	−1.00000	−0.0887357
$128$	7.19960	0.636361
$129$	0	0
$130$	3.33743	0.292712
$131$	4.24308	0.370720	0.185360	−	0.982671i	$-0.440655\pi$
$131$	4.24308	0.370720	0.185360	+	0.982671i	$0.440655\pi$
$132$	0	0
$133$	−0.299740	−0.0259907
$134$	−0.631124	−0.0545208
$135$	0	0
$136$	−2.11543	−0.181396
$137$	−6.12330	−0.523148	−0.261574	−	0.965183i	$-0.584242\pi$
$137$	−6.12330	−0.523148	−0.261574	+	0.965183i	$0.584242\pi$
$138$	0	0
$139$	−5.79278	−0.491337	−0.245668	−	0.969354i	$-0.579007\pi$
$139$	−5.79278	−0.491337	−0.245668	+	0.969354i	$0.579007\pi$
$140$	5.42896	0.458831
$141$	0	0
$142$	0.606215	0.0508725
$143$	−22.6284	−1.89228
$144$	0	0
$145$	−15.5990	−1.29543
$146$	1.44271	0.119399
$147$	0	0
$148$	−17.1204	−1.40729
$149$	−8.83924	−0.724139	−0.362070	−	0.932151i	$-0.617930\pi$
$149$	−8.83924	−0.724139	−0.362070	+	0.932151i	$0.617930\pi$
$150$	0	0
$151$	−8.09628	−0.658866	−0.329433	−	0.944179i	$-0.606858\pi$
$151$	−8.09628	−0.658866	−0.329433	+	0.944179i	$0.606858\pi$
$152$	0.286473	0.0232361
$153$	0	0
$154$	1.11510	0.0898576
$155$	16.7838	1.34811
$156$	0	0
$157$	−0.0956181	−0.00763116	−0.00381558	−	0.999993i	$-0.501215\pi$
$157$	−0.0956181	−0.00763116	−0.00381558	+	0.999993i	$0.501215\pi$
$158$	3.18344	0.253260
$159$	0	0
$160$	−7.82173	−0.618362
$161$	2.16929	0.170964
$162$	0	0
$163$	15.4482	1.21000	0.604998	−	0.796227i	$-0.293174\pi$
$163$	15.4482	1.21000	0.604998	+	0.796227i	$0.293174\pi$
$164$	20.3865	1.59192
$165$	0	0
$166$	2.28173	0.177097
$167$	15.5614	1.20418	0.602089	−	0.798429i	$-0.294335\pi$
$167$	15.5614	1.20418	0.602089	+	0.798429i	$0.294335\pi$
$168$	0	0
$169$	11.2160	0.862772
$170$	1.50113	0.115131
$171$	0	0
$172$	13.0057	0.991676
$173$	3.17589	0.241459	0.120729	−	0.992685i	$-0.461477\pi$
$173$	3.17589	0.241459	0.120729	+	0.992685i	$0.461477\pi$
$174$	0	0
$175$	−2.82161	−0.213293
$176$	16.7869	1.26536
$177$	0	0
$178$	−2.26325	−0.169638
$179$	−0.645010	−0.0482103	−0.0241051	−	0.999709i	$-0.507674\pi$
$179$	−0.645010	−0.0482103	−0.0241051	+	0.999709i	$0.507674\pi$
$180$	0	0
$181$	1.26616	0.0941132	0.0470566	−	0.998892i	$-0.485016\pi$
$181$	1.26616	0.0941132	0.0470566	+	0.998892i	$0.485016\pi$
$182$	−1.19334	−0.0884562
$183$	0	0
$184$	−2.07328	−0.152845
$185$	24.6657	1.81346
$186$	0	0
$187$	−10.1780	−0.744286
$188$	−14.7205	−1.07360
$189$	0	0
$190$	−0.203285	−0.0147478
$191$	18.1421	1.31272	0.656359	−	0.754449i	$-0.272096\pi$
$191$	18.1421	1.31272	0.656359	+	0.754449i	$0.272096\pi$
$192$	0	0
$193$	20.8712	1.50234	0.751170	−	0.660109i	$-0.229490\pi$
$193$	20.8712	1.50234	0.751170	+	0.660109i	$0.229490\pi$
$194$	2.87446	0.206374
$195$	0	0
$196$	−1.94119	−0.138657
$197$	−14.1011	−1.00466	−0.502331	−	0.864676i	$-0.667524\pi$
$197$	−14.1011	−1.00466	−0.502331	+	0.864676i	$0.667524\pi$
$198$	0	0
$199$	−21.2186	−1.50415	−0.752073	−	0.659080i	$-0.770946\pi$
$199$	−21.2186	−1.50415	−0.752073	+	0.659080i	$0.770946\pi$
$200$	2.69672	0.190687
$201$	0	0
$202$	1.32512	0.0932352
$203$	5.57762	0.391472
$204$	0	0
$205$	−29.3712	−2.05137
$206$	−0.183079	−0.0127557
$207$	0	0
$208$	−17.9646	−1.24562
$209$	1.37831	0.0953397
$210$	0	0
$211$	−1.78981	−0.123215	−0.0616077	−	0.998100i	$-0.519623\pi$
$211$	−1.78981	−0.123215	−0.0616077	+	0.998100i	$0.519623\pi$
$212$	2.02272	0.138921
$213$	0	0
$214$	−2.58078	−0.176419
$215$	−18.7376	−1.27789
$216$	0	0
$217$	−6.00125	−0.407391
$218$	−1.60332	−0.108590
$219$	0	0
$220$	−24.9643	−1.68309
$221$	10.8920	0.732678
$222$	0	0
$223$	−2.33108	−0.156101	−0.0780504	−	0.996949i	$-0.524870\pi$
$223$	−2.33108	−0.156101	−0.0780504	+	0.996949i	$0.524870\pi$
$224$	2.79676	0.186866
$225$	0	0
$226$	0.904548	0.0601696
$227$	−17.3413	−1.15098	−0.575490	−	0.817809i	$-0.695189\pi$
$227$	−17.3413	−1.15098	−0.575490	+	0.817809i	$0.695189\pi$
$228$	0	0
$229$	−11.1731	−0.738342	−0.369171	−	0.929361i	$-0.620358\pi$
$229$	−11.1731	−0.738342	−0.369171	+	0.929361i	$0.620358\pi$
$230$	1.47122	0.0970097
$231$	0	0
$232$	−5.33076	−0.349981
$233$	−12.6112	−0.826189	−0.413094	−	0.910688i	$-0.635552\pi$
$233$	−12.6112	−0.826189	−0.413094	+	0.910688i	$0.635552\pi$
$234$	0	0
$235$	21.2081	1.38346
$236$	10.9815	0.714832
$237$	0	0
$238$	−0.536748	−0.0347922
$239$	6.93025	0.448280	0.224140	−	0.974557i	$-0.428043\pi$
$239$	6.93025	0.448280	0.224140	+	0.974557i	$0.428043\pi$
$240$	0	0
$241$	−9.07372	−0.584489	−0.292245	−	0.956344i	$-0.594402\pi$
$241$	−9.07372	−0.584489	−0.292245	+	0.956344i	$0.594402\pi$
$242$	−2.46015	−0.158144
$243$	0	0
$244$	−0.314410	−0.0201280
$245$	2.79671	0.178675
$246$	0	0
$247$	−1.47501	−0.0938528
$248$	5.73564	0.364213
$249$	0	0
$250$	1.47739	0.0934387
$251$	−17.1122	−1.08011	−0.540057	−	0.841628i	$-0.681597\pi$
$251$	−17.1122	−1.08011	−0.540057	+	0.841628i	$0.681597\pi$
$252$	0	0
$253$	−9.97520	−0.627135
$254$	0.242500	0.0152158
$255$	0	0
$256$	11.5001	0.718759
$257$	−9.41351	−0.587199	−0.293599	−	0.955929i	$-0.594853\pi$
$257$	−9.41351	−0.587199	−0.293599	+	0.955929i	$0.594853\pi$
$258$	0	0
$259$	−8.81952	−0.548018
$260$	26.7158	1.65684
$261$	0	0
$262$	−1.02895	−0.0635686
$263$	6.64471	0.409730	0.204865	−	0.978790i	$-0.434324\pi$
$263$	6.64471	0.409730	0.204865	+	0.978790i	$0.434324\pi$
$264$	0	0
$265$	−2.91418	−0.179016
$266$	0.0726869	0.00445672
$267$	0	0
$268$	−5.05209	−0.308605
$269$	16.0652	0.979514	0.489757	−	0.871859i	$-0.337085\pi$
$269$	16.0652	0.979514	0.489757	+	0.871859i	$0.337085\pi$
$270$	0	0
$271$	29.0605	1.76530	0.882651	−	0.470030i	$-0.155757\pi$
$271$	29.0605	1.76530	0.882651	+	0.470030i	$0.155757\pi$
$272$	−8.08024	−0.489937
$273$	0	0
$274$	1.48490	0.0897062
$275$	12.9748	0.782408
$276$	0	0
$277$	12.0489	0.723950	0.361975	−	0.932188i	$-0.382103\pi$
$277$	12.0489	0.723950	0.361975	+	0.932188i	$0.382103\pi$
$278$	1.40475	0.0842513
$279$	0	0
$280$	−2.67293	−0.159738
$281$	−0.729114	−0.0434953	−0.0217476	−	0.999763i	$-0.506923\pi$
$281$	−0.729114	−0.0434953	−0.0217476	+	0.999763i	$0.506923\pi$
$282$	0	0
$283$	5.70007	0.338834	0.169417	−	0.985544i	$-0.445811\pi$
$283$	5.70007	0.338834	0.169417	+	0.985544i	$0.445811\pi$
$284$	4.85270	0.287955
$285$	0	0
$286$	5.48740	0.324477
$287$	10.5020	0.619915
$288$	0	0
$289$	−12.1009	−0.711818
$290$	3.78276	0.222131
$291$	0	0
$292$	11.5488	0.675840
$293$	−16.3296	−0.953983	−0.476991	−	0.878908i	$-0.658273\pi$
$293$	−16.3296	−0.953983	−0.476991	+	0.878908i	$0.658273\pi$
$294$	0	0
$295$	−15.8212	−0.921145
$296$	8.42918	0.489936
$297$	0	0
$298$	2.14352	0.124171
$299$	10.6751	0.617354
$300$	0	0
$301$	6.69985	0.386173
$302$	1.96335	0.112978
$303$	0	0
$304$	1.09424	0.0627587
$305$	0.452976	0.0259373
$306$	0	0
$307$	3.16022	0.180363	0.0901816	−	0.995925i	$-0.471255\pi$
$307$	3.16022	0.180363	0.0901816	+	0.995925i	$0.471255\pi$
$308$	8.92631	0.508624
$309$	0	0
$310$	−4.07007	−0.231165
$311$	−27.0887	−1.53606	−0.768029	−	0.640415i	$-0.778763\pi$
$311$	−27.0887	−1.53606	−0.768029	+	0.640415i	$0.778763\pi$
$312$	0	0
$313$	−18.0755	−1.02169	−0.510844	−	0.859673i	$-0.670667\pi$
$313$	−18.0755	−1.02169	−0.510844	+	0.859673i	$0.670667\pi$
$314$	0.0231874	0.00130854
$315$	0	0
$316$	25.4831	1.43354
$317$	−14.8515	−0.834142	−0.417071	−	0.908874i	$-0.636943\pi$
$317$	−14.8515	−0.834142	−0.417071	+	0.908874i	$0.636943\pi$
$318$	0	0
$319$	−25.6479	−1.43601
$320$	−18.5227	−1.03545
$321$	0	0
$322$	−0.526055	−0.0293159
$323$	−0.663440	−0.0369148
$324$	0	0
$325$	−13.8851	−0.770205
$326$	−3.74620	−0.207483
$327$	0	0
$328$	−10.0372	−0.554213
$329$	−7.58321	−0.418076
$330$	0	0
$331$	−29.7585	−1.63568	−0.817839	−	0.575447i	$-0.804828\pi$
$331$	−29.7585	−1.63568	−0.817839	+	0.575447i	$0.804828\pi$
$332$	18.2651	1.00243
$333$	0	0
$334$	−3.77365	−0.206485
$335$	7.27864	0.397675
$336$	0	0
$337$	17.3368	0.944397	0.472198	−	0.881492i	$-0.343461\pi$
$337$	17.3368	0.944397	0.472198	+	0.881492i	$0.343461\pi$
$338$	−2.71989	−0.147943
$339$	0	0
$340$	12.0164	0.651681
$341$	27.5959	1.49440
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	−6.40332	−0.345244
$345$	0	0
$346$	−0.770156	−0.0414038
$347$	13.4672	0.722959	0.361479	−	0.932380i	$-0.382272\pi$
$347$	13.4672	0.722959	0.361479	+	0.932380i	$0.382272\pi$
$348$	0	0
$349$	−16.6140	−0.889328	−0.444664	−	0.895697i	$-0.646677\pi$
$349$	−16.6140	−0.889328	−0.444664	+	0.895697i	$0.646677\pi$
$350$	0.684241	0.0365742
$351$	0	0
$352$	−12.8605	−0.685467
$353$	0.705030	0.0375249	0.0187625	−	0.999824i	$-0.494027\pi$
$353$	0.705030	0.0375249	0.0187625	+	0.999824i	$0.494027\pi$
$354$	0	0
$355$	−6.99137	−0.371064
$356$	−18.1171	−0.960205
$357$	0	0
$358$	0.156415	0.00826679
$359$	−6.08609	−0.321211	−0.160606	−	0.987019i	$-0.551345\pi$
$359$	−6.08609	−0.321211	−0.160606	+	0.987019i	$0.551345\pi$
$360$	0	0
$361$	−18.9102	−0.995271
$362$	−0.307045	−0.0161379
$363$	0	0
$364$	−9.55257	−0.500691
$365$	−16.6385	−0.870900
$366$	0	0
$367$	0.822218	0.0429194	0.0214597	−	0.999770i	$-0.493169\pi$
$367$	0.822218	0.0429194	0.0214597	+	0.999770i	$0.493169\pi$
$368$	−7.91927	−0.412821
$369$	0	0
$370$	−5.98143	−0.310960
$371$	1.04200	0.0540980
$372$	0	0
$373$	−3.32058	−0.171933	−0.0859665	−	0.996298i	$-0.527398\pi$
$373$	−3.32058	−0.171933	−0.0859665	+	0.996298i	$0.527398\pi$
$374$	2.46816	0.127625
$375$	0	0
$376$	7.24759	0.373766
$377$	27.4473	1.41361
$378$	0	0
$379$	8.11526	0.416853	0.208426	−	0.978038i	$-0.433166\pi$
$379$	8.11526	0.416853	0.208426	+	0.978038i	$0.433166\pi$
$380$	−1.62727	−0.0834774
$381$	0	0
$382$	−4.39947	−0.225096
$383$	6.77315	0.346092	0.173046	−	0.984914i	$-0.444639\pi$
$383$	6.77315	0.346092	0.173046	+	0.984914i	$0.444639\pi$
$384$	0	0
$385$	−12.8603	−0.655421
$386$	−5.06127	−0.257612
$387$	0	0
$388$	23.0098	1.16815
$389$	−8.89791	−0.451142	−0.225571	−	0.974227i	$-0.572425\pi$
$389$	−8.89791	−0.451142	−0.225571	+	0.974227i	$0.572425\pi$
$390$	0	0
$391$	4.80149	0.242822
$392$	0.955741	0.0482722
$393$	0	0
$394$	3.41952	0.172273
$395$	−36.7140	−1.84728
$396$	0	0
$397$	6.55375	0.328923	0.164462	−	0.986383i	$-0.447411\pi$
$397$	6.55375	0.328923	0.164462	+	0.986383i	$0.447411\pi$
$398$	5.14551	0.257921
$399$	0	0
$400$	10.3006	0.515031
$401$	−2.19709	−0.109717	−0.0548586	−	0.998494i	$-0.517471\pi$
$401$	−2.19709	−0.109717	−0.0548586	+	0.998494i	$0.517471\pi$
$402$	0	0
$403$	−29.5320	−1.47109
$404$	10.6075	0.527742
$405$	0	0
$406$	−1.35257	−0.0671271
$407$	40.5553	2.01025
$408$	0	0
$409$	−38.3816	−1.89785	−0.948923	−	0.315507i	$-0.897825\pi$
$409$	−38.3816	−1.89785	−0.948923	+	0.315507i	$0.897825\pi$
$410$	7.12252	0.351756
$411$	0	0
$412$	−1.46553	−0.0722014
$413$	5.65706	0.278366
$414$	0	0
$415$	−26.3148	−1.29174
$416$	13.7628	0.674776
$417$	0	0
$418$	−0.334241	−0.0163482
$419$	−22.1140	−1.08034	−0.540169	−	0.841556i	$-0.681640\pi$
$419$	−22.1140	−1.08034	−0.540169	+	0.841556i	$0.681640\pi$
$420$	0	0
$421$	22.8139	1.11188	0.555940	−	0.831223i	$-0.312359\pi$
$421$	22.8139	1.11188	0.555940	+	0.831223i	$0.312359\pi$
$422$	0.434029	0.0211282
$423$	0	0
$424$	−0.995882	−0.0483643
$425$	−6.24531	−0.302942
$426$	0	0
$427$	−0.161967	−0.00783815
$428$	−20.6589	−0.998587
$429$	0	0
$430$	4.54387	0.219125
$431$	−31.7123	−1.52753	−0.763764	−	0.645495i	$-0.776651\pi$
$431$	−31.7123	−1.52753	−0.763764	+	0.645495i	$0.776651\pi$
$432$	0	0
$433$	30.3802	1.45998	0.729988	−	0.683459i	$-0.239525\pi$
$433$	30.3802	1.45998	0.729988	+	0.683459i	$0.239525\pi$
$434$	1.45530	0.0698569
$435$	0	0
$436$	−12.8344	−0.614657
$437$	−0.650224	−0.0311044
$438$	0	0
$439$	−27.8747	−1.33038	−0.665192	−	0.746672i	$-0.731650\pi$
$439$	−27.8747	−1.33038	−0.665192	+	0.746672i	$0.731650\pi$
$440$	12.2911	0.585956
$441$	0	0
$442$	−2.64132	−0.125635
$443$	−33.6373	−1.59816	−0.799078	−	0.601227i	$-0.794679\pi$
$443$	−33.6373	−1.59816	−0.799078	+	0.601227i	$0.794679\pi$
$444$	0	0
$445$	26.1017	1.23734
$446$	0.565289	0.0267672
$447$	0	0
$448$	6.62302	0.312908
$449$	2.86855	0.135375	0.0676876	−	0.997707i	$-0.478438\pi$
$449$	2.86855	0.135375	0.0676876	+	0.997707i	$0.478438\pi$
$450$	0	0
$451$	−48.2921	−2.27399
$452$	7.24082	0.340580
$453$	0	0
$454$	4.20526	0.197363
$455$	13.7626	0.645199
$456$	0	0
$457$	−27.3334	−1.27860	−0.639301	−	0.768957i	$-0.720776\pi$
$457$	−27.3334	−1.27860	−0.639301	+	0.768957i	$0.720776\pi$
$458$	2.70949	0.126606
$459$	0	0
$460$	11.7770	0.549106
$461$	−3.54928	−0.165306	−0.0826531	−	0.996578i	$-0.526339\pi$
$461$	−3.54928	−0.165306	−0.0826531	+	0.996578i	$0.526339\pi$
$462$	0	0
$463$	−26.7780	−1.24448	−0.622239	−	0.782828i	$-0.713777\pi$
$463$	−26.7780	−1.24448	−0.622239	+	0.782828i	$0.713777\pi$
$464$	−20.3618	−0.945271
$465$	0	0
$466$	3.05823	0.141670
$467$	13.4294	0.621438	0.310719	−	0.950502i	$-0.399430\pi$
$467$	13.4294	0.621438	0.310719	+	0.950502i	$0.399430\pi$
$468$	0	0
$469$	−2.60257	−0.120175
$470$	−5.14297	−0.237227
$471$	0	0
$472$	−5.40668	−0.248863
$473$	−30.8083	−1.41657
$474$	0	0
$475$	0.845747	0.0388055
$476$	−4.29662	−0.196935
$477$	0	0
$478$	−1.68059	−0.0768683
$479$	24.0582	1.09925	0.549624	−	0.835412i	$-0.314771\pi$
$479$	24.0582	1.09925	0.549624	+	0.835412i	$0.314771\pi$
$480$	0	0
$481$	−43.4007	−1.97890
$482$	2.20038	0.100225
$483$	0	0
$484$	−19.6932	−0.895147
$485$	−33.1507	−1.50529
$486$	0	0
$487$	1.40323	0.0635864	0.0317932	−	0.999494i	$-0.489878\pi$
$487$	1.40323	0.0635864	0.0317932	+	0.999494i	$0.489878\pi$
$488$	0.154799	0.00700741
$489$	0	0
$490$	−0.678204	−0.0306381
$491$	−18.9765	−0.856398	−0.428199	−	0.903684i	$-0.640852\pi$
$491$	−18.9765	−0.856398	−0.428199	+	0.903684i	$0.640852\pi$
$492$	0	0
$493$	12.3454	0.556011
$494$	0.357691	0.0160933
$495$	0	0
$496$	21.9083	0.983711
$497$	2.49985	0.112134
$498$	0	0
$499$	−3.91564	−0.175288	−0.0876441	−	0.996152i	$-0.527934\pi$
$499$	−3.91564	−0.175288	−0.0876441	+	0.996152i	$0.527934\pi$
$500$	11.8264	0.528893
$501$	0	0
$502$	4.14972	0.185211
$503$	4.39051	0.195763	0.0978815	−	0.995198i	$-0.468793\pi$
$503$	4.39051	0.195763	0.0978815	+	0.995198i	$0.468793\pi$
$504$	0	0
$505$	−15.2824	−0.680057
$506$	2.41899	0.107537
$507$	0	0
$508$	1.94119	0.0861265
$509$	3.77299	0.167235	0.0836175	−	0.996498i	$-0.473353\pi$
$509$	3.77299	0.167235	0.0836175	+	0.996498i	$0.473353\pi$
$510$	0	0
$511$	5.94931	0.263182
$512$	−17.1880	−0.759609
$513$	0	0
$514$	2.28278	0.100689
$515$	2.11141	0.0930400
$516$	0	0
$517$	34.8703	1.53360
$518$	2.13874	0.0939707
$519$	0	0
$520$	−13.1534	−0.576817
$521$	−40.5257	−1.77546	−0.887731	−	0.460362i	$-0.847719\pi$
$521$	−40.5257	−1.77546	−0.887731	+	0.460362i	$0.847719\pi$
$522$	0	0
$523$	1.15256	0.0503979	0.0251990	−	0.999682i	$-0.491978\pi$
$523$	1.15256	0.0503979	0.0251990	+	0.999682i	$0.491978\pi$
$524$	−8.23664	−0.359819
$525$	0	0
$526$	−1.61134	−0.0702580
$527$	−13.2831	−0.578621
$528$	0	0
$529$	−18.2942	−0.795398
$530$	0.706689	0.0306966
$531$	0	0
$532$	0.581853	0.0252265
$533$	51.6803	2.23852
$534$	0	0
$535$	29.7637	1.28680
$536$	2.48738	0.107438
$537$	0	0
$538$	−3.89582	−0.167961
$539$	4.59836	0.198065
$540$	0	0
$541$	23.7874	1.02270	0.511349	−	0.859373i	$-0.329146\pi$
$541$	23.7874	1.02270	0.511349	+	0.859373i	$0.329146\pi$
$542$	−7.04719	−0.302703
$543$	0	0
$544$	6.19031	0.265408
$545$	18.4908	0.792058
$546$	0	0
$547$	−11.2951	−0.482945	−0.241473	−	0.970408i	$-0.577630\pi$
$547$	−11.2951	−0.482945	−0.241473	+	0.970408i	$0.577630\pi$
$548$	11.8865	0.507766
$549$	0	0
$550$	−3.14638	−0.134162
$551$	−1.67183	−0.0712225
$552$	0	0
$553$	13.1275	0.558240
$554$	−2.92187	−0.124138
$555$	0	0
$556$	11.2449	0.476890
$557$	13.6757	0.579459	0.289730	−	0.957109i	$-0.406435\pi$
$557$	13.6757	0.579459	0.289730	+	0.957109i	$0.406435\pi$
$558$	0	0
$559$	32.9698	1.39448
$560$	−10.2097	−0.431440
$561$	0	0
$562$	0.176810	0.00745829
$563$	23.5207	0.991278	0.495639	−	0.868529i	$-0.334934\pi$
$563$	23.5207	0.991278	0.495639	+	0.868529i	$0.334934\pi$
$564$	0	0
$565$	−10.4320	−0.438877
$566$	−1.38227	−0.0581011
$567$	0	0
$568$	−2.38921	−0.100249
$569$	−4.30680	−0.180550	−0.0902752	−	0.995917i	$-0.528775\pi$
$569$	−4.30680	−0.180550	−0.0902752	+	0.995917i	$0.528775\pi$
$570$	0	0
$571$	−17.3440	−0.725822	−0.362911	−	0.931824i	$-0.618217\pi$
$571$	−17.3440	−0.725822	−0.362911	+	0.931824i	$0.618217\pi$
$572$	43.9262	1.83665
$573$	0	0
$574$	−2.54675	−0.106299
$575$	−6.12090	−0.255259
$576$	0	0
$577$	−2.97581	−0.123885	−0.0619423	−	0.998080i	$-0.519729\pi$
$577$	−2.97581	−0.123885	−0.0619423	+	0.998080i	$0.519729\pi$
$578$	2.93447	0.122058
$579$	0	0
$580$	30.2807	1.25734
$581$	9.40919	0.390359
$582$	0	0
$583$	−4.79149	−0.198443
$584$	−5.68600	−0.235288
$585$	0	0
$586$	3.95992	0.163583
$587$	35.0446	1.44645	0.723224	−	0.690614i	$-0.242660\pi$
$587$	35.0446	1.44645	0.723224	+	0.690614i	$0.242660\pi$
$588$	0	0
$589$	1.79881	0.0741187
$590$	3.83664	0.157952
$591$	0	0
$592$	32.1967	1.32328
$593$	13.3444	0.547988	0.273994	−	0.961731i	$-0.411655\pi$
$593$	13.3444	0.547988	0.273994	+	0.961731i	$0.411655\pi$
$594$	0	0
$595$	6.19022	0.253774
$596$	17.1587	0.702847
$597$	0	0
$598$	−2.58870	−0.105860
$599$	25.1804	1.02884	0.514421	−	0.857538i	$-0.328007\pi$
$599$	25.1804	1.02884	0.514421	+	0.857538i	$0.328007\pi$
$600$	0	0
$601$	−31.7897	−1.29673	−0.648364	−	0.761331i	$-0.724546\pi$
$601$	−31.7897	−1.29673	−0.648364	+	0.761331i	$0.724546\pi$
$602$	−1.62472	−0.0662185
$603$	0	0
$604$	15.7164	0.639493
$605$	28.3724	1.15350
$606$	0	0
$607$	−27.3805	−1.11134	−0.555669	−	0.831403i	$-0.687538\pi$
$607$	−27.3805	−1.11134	−0.555669	+	0.831403i	$0.687538\pi$
$608$	−0.838299	−0.0339975
$609$	0	0
$610$	−0.109847	−0.00444757
$611$	−37.3168	−1.50968
$612$	0	0
$613$	−47.9879	−1.93821	−0.969107	−	0.246642i	$-0.920673\pi$
$613$	−47.9879	−1.93821	−0.969107	+	0.246642i	$0.920673\pi$
$614$	−0.766355	−0.0309276
$615$	0	0
$616$	−4.39484	−0.177073
$617$	23.4031	0.942174	0.471087	−	0.882087i	$-0.343862\pi$
$617$	23.4031	0.942174	0.471087	+	0.882087i	$0.343862\pi$
$618$	0	0
$619$	−3.26112	−0.131076	−0.0655378	−	0.997850i	$-0.520876\pi$
$619$	−3.26112	−0.131076	−0.0655378	+	0.997850i	$0.520876\pi$
$620$	−32.5806	−1.30847
$621$	0	0
$622$	6.56902	0.263394
$623$	−9.33298	−0.373918
$624$	0	0
$625$	−31.1466	−1.24586
$626$	4.38332	0.175193
$627$	0	0
$628$	0.185613	0.00740678
$629$	−19.5210	−0.778355
$630$	0	0
$631$	−19.8046	−0.788407	−0.394203	−	0.919023i	$-0.628979\pi$
$631$	−19.8046	−0.788407	−0.394203	+	0.919023i	$0.628979\pi$
$632$	−12.5465	−0.499074
$633$	0	0
$634$	3.60149	0.143033
$635$	−2.79671	−0.110984
$636$	0	0
$637$	−4.92098	−0.194976
$638$	6.21962	0.246237
$639$	0	0
$640$	20.1352	0.795915
$641$	45.0976	1.78125	0.890624	−	0.454741i	$-0.150268\pi$
$641$	45.0976	1.78125	0.890624	+	0.454741i	$0.150268\pi$
$642$	0	0
$643$	6.19892	0.244461	0.122231	−	0.992502i	$-0.460995\pi$
$643$	6.19892	0.244461	0.122231	+	0.992502i	$0.460995\pi$
$644$	−4.21102	−0.165937
$645$	0	0
$646$	0.160885	0.00632992
$647$	12.8151	0.503815	0.251907	−	0.967751i	$-0.418942\pi$
$647$	12.8151	0.503815	0.251907	+	0.967751i	$0.418942\pi$
$648$	0	0
$649$	−26.0132	−1.02111
$650$	3.36713	0.132070
$651$	0	0
$652$	−29.9880	−1.17442
$653$	6.35355	0.248633	0.124317	−	0.992243i	$-0.460326\pi$
$653$	6.35355	0.248633	0.124317	+	0.992243i	$0.460326\pi$
$654$	0	0
$655$	11.8667	0.463669
$656$	−38.3389	−1.49688
$657$	0	0
$658$	1.83893	0.0716890
$659$	−29.5580	−1.15141	−0.575707	−	0.817656i	$-0.695273\pi$
$659$	−29.5580	−1.15141	−0.575707	+	0.817656i	$0.695273\pi$
$660$	0	0
$661$	−30.8995	−1.20185	−0.600926	−	0.799305i	$-0.705201\pi$
$661$	−30.8995	−1.20185	−0.600926	+	0.799305i	$0.705201\pi$
$662$	7.21646	0.280476
$663$	0	0
$664$	−8.99275	−0.348986
$665$	−0.838286	−0.0325073
$666$	0	0
$667$	12.0995	0.468494
$668$	−30.2077	−1.16877
$669$	0	0
$670$	−1.76507	−0.0681907
$671$	0.744784	0.0287521
$672$	0	0
$673$	37.1773	1.43308	0.716540	−	0.697546i	$-0.245725\pi$
$673$	37.1773	1.43308	0.716540	+	0.697546i	$0.245725\pi$
$674$	−4.20418	−0.161939
$675$	0	0
$676$	−21.7725	−0.837404
$677$	0.0588742	0.00226272	0.00113136	−	0.999999i	$-0.499640\pi$
$677$	0.0588742	0.00226272	0.00113136	+	0.999999i	$0.499640\pi$
$678$	0	0
$679$	11.8534	0.454893
$680$	−5.91624	−0.226878
$681$	0	0
$682$	−6.69202	−0.256251
$683$	7.13511	0.273018	0.136509	−	0.990639i	$-0.456412\pi$
$683$	7.13511	0.273018	0.136509	+	0.990639i	$0.456412\pi$
$684$	0	0
$685$	−17.1251	−0.654317
$686$	0.242500	0.00925871
$687$	0	0
$688$	−24.4586	−0.932476
$689$	5.12766	0.195348
$690$	0	0
$691$	−23.8916	−0.908878	−0.454439	−	0.890778i	$-0.650160\pi$
$691$	−23.8916	−0.908878	−0.454439	+	0.890778i	$0.650160\pi$
$692$	−6.16503	−0.234359
$693$	0	0
$694$	−3.26581	−0.123968
$695$	−16.2007	−0.614529
$696$	0	0
$697$	23.2451	0.880470
$698$	4.02891	0.152496
$699$	0	0
$700$	5.47729	0.207022
$701$	42.6818	1.61207	0.806034	−	0.591869i	$-0.201610\pi$
$701$	42.6818	1.61207	0.806034	+	0.591869i	$0.201610\pi$
$702$	0	0
$703$	2.64356	0.0997037
$704$	−30.4551	−1.14782
$705$	0	0
$706$	−0.170970	−0.00643454
$707$	5.46441	0.205510
$708$	0	0
$709$	−29.4112	−1.10456	−0.552280	−	0.833658i	$-0.686242\pi$
$709$	−29.4112	−1.10456	−0.552280	+	0.833658i	$0.686242\pi$
$710$	1.69541	0.0636276
$711$	0	0
$712$	8.91991	0.334288
$713$	−13.0185	−0.487546
$714$	0	0
$715$	−63.2852	−2.36673
$716$	1.25209	0.0467927
$717$	0	0
$718$	1.47588	0.0550793
$719$	48.1805	1.79683	0.898416	−	0.439146i	$-0.144719\pi$
$719$	48.1805	1.79683	0.898416	+	0.439146i	$0.144719\pi$
$720$	0	0
$721$	−0.754962	−0.0281163
$722$	4.58572	0.170663
$723$	0	0
$724$	−2.45787	−0.0913459
$725$	−15.7378	−0.584489
$726$	0	0
$727$	41.1964	1.52789	0.763944	−	0.645282i	$-0.223260\pi$
$727$	41.1964	1.52789	0.763944	+	0.645282i	$0.223260\pi$
$728$	4.70318	0.174311
$729$	0	0
$730$	4.03485	0.149336
$731$	14.8294	0.548484
$732$	0	0
$733$	−51.8699	−1.91586	−0.957929	−	0.287004i	$-0.907341\pi$
$733$	−51.8699	−1.91586	−0.957929	+	0.287004i	$0.907341\pi$
$734$	−0.199388	−0.00735955
$735$	0	0
$736$	6.06699	0.223632
$737$	11.9675	0.440830
$738$	0	0
$739$	25.1518	0.925222	0.462611	−	0.886561i	$-0.346913\pi$
$739$	25.1518	0.925222	0.462611	+	0.886561i	$0.346913\pi$
$740$	−47.8808	−1.76013
$741$	0	0
$742$	−0.252685	−0.00927637
$743$	−39.8449	−1.46177	−0.730884	−	0.682502i	$-0.760892\pi$
$743$	−39.8449	−1.46177	−0.730884	+	0.682502i	$0.760892\pi$
$744$	0	0
$745$	−24.7208	−0.905701
$746$	0.805242	0.0294820
$747$	0	0
$748$	19.7574	0.722402
$749$	−10.6424	−0.388864
$750$	0	0
$751$	13.6188	0.496958	0.248479	−	0.968637i	$-0.420069\pi$
$751$	13.6188	0.496958	0.248479	+	0.968637i	$0.420069\pi$
$752$	27.6834	1.00951
$753$	0	0
$754$	−6.65599	−0.242397
$755$	−22.6430	−0.824062
$756$	0	0
$757$	37.3375	1.35705	0.678527	−	0.734576i	$-0.262619\pi$
$757$	37.3375	1.35705	0.678527	+	0.734576i	$0.262619\pi$
$758$	−1.96795	−0.0714793
$759$	0	0
$760$	0.801184	0.0290620
$761$	−7.96910	−0.288880	−0.144440	−	0.989514i	$-0.546138\pi$
$761$	−7.96910	−0.288880	−0.144440	+	0.989514i	$0.546138\pi$
$762$	0	0
$763$	−6.61161	−0.239356
$764$	−35.2174	−1.27412
$765$	0	0
$766$	−1.64249	−0.0593456
$767$	27.8383	1.00518
$768$	0	0
$769$	32.8938	1.18618	0.593091	−	0.805136i	$-0.297908\pi$
$769$	32.8938	1.18618	0.593091	+	0.805136i	$0.297908\pi$
$770$	3.11863	0.112387
$771$	0	0
$772$	−40.5150	−1.45817
$773$	−3.09221	−0.111219	−0.0556095	−	0.998453i	$-0.517710\pi$
$773$	−3.09221	−0.111219	−0.0556095	+	0.998453i	$0.517710\pi$
$774$	0	0
$775$	16.9332	0.608257
$776$	−11.3288	−0.406681
$777$	0	0
$778$	2.15775	0.0773590
$779$	−3.14787	−0.112784
$780$	0	0
$781$	−11.4952	−0.411332
$782$	−1.16436	−0.0416376
$783$	0	0
$784$	3.65062	0.130379
$785$	−0.267416	−0.00954450
$786$	0	0
$787$	−18.1780	−0.647975	−0.323987	−	0.946061i	$-0.605024\pi$
$787$	−18.1780	−0.647975	−0.323987	+	0.946061i	$0.605024\pi$
$788$	27.3730	0.975121
$789$	0	0
$790$	8.90316	0.316760
$791$	3.73009	0.132627
$792$	0	0
$793$	−0.797038	−0.0283036
$794$	−1.58929	−0.0564017
$795$	0	0
$796$	41.1894	1.45992
$797$	−12.4294	−0.440273	−0.220136	−	0.975469i	$-0.570650\pi$
$797$	−12.4294	−0.440273	−0.220136	+	0.975469i	$0.570650\pi$
$798$	0	0
$799$	−16.7846	−0.593797
$800$	−7.89135	−0.279001
$801$	0	0
$802$	0.532794	0.0188136
$803$	−27.3571	−0.965410
$804$	0	0
$805$	6.06690	0.213830
$806$	7.16153	0.252254
$807$	0	0
$808$	−5.22256	−0.183729
$809$	−38.2936	−1.34633	−0.673165	−	0.739492i	$-0.735066\pi$
$809$	−38.2936	−1.34633	−0.673165	+	0.739492i	$0.735066\pi$
$810$	0	0
$811$	−43.5810	−1.53034	−0.765168	−	0.643830i	$-0.777344\pi$
$811$	−43.5810	−1.53034	−0.765168	+	0.643830i	$0.777344\pi$
$812$	−10.8272	−0.379962
$813$	0	0
$814$	−9.83468	−0.344705
$815$	43.2042	1.51338
$816$	0	0
$817$	−2.00821	−0.0702584
$818$	9.30754	0.325431
$819$	0	0
$820$	57.0151	1.99105
$821$	34.7141	1.21153	0.605765	−	0.795644i	$-0.292867\pi$
$821$	34.7141	1.21153	0.605765	+	0.795644i	$0.292867\pi$
$822$	0	0
$823$	−30.6184	−1.06729	−0.533645	−	0.845708i	$-0.679178\pi$
$823$	−30.6184	−1.06729	−0.533645	+	0.845708i	$0.679178\pi$
$824$	0.721548	0.0251363
$825$	0	0
$826$	−1.37184	−0.0477324
$827$	−48.9512	−1.70220	−0.851099	−	0.525005i	$-0.824064\pi$
$827$	−48.9512	−1.70220	−0.851099	+	0.525005i	$0.824064\pi$
$828$	0	0
$829$	−3.52359	−0.122379	−0.0611897	−	0.998126i	$-0.519489\pi$
$829$	−3.52359	−0.122379	−0.0611897	+	0.998126i	$0.519489\pi$
$830$	6.38135	0.221500
$831$	0	0
$832$	32.5918	1.12992
$833$	−2.21339	−0.0766894
$834$	0	0
$835$	43.5208	1.50610
$836$	−2.67557	−0.0925364
$837$	0	0
$838$	5.36265	0.185250
$839$	−11.5384	−0.398349	−0.199175	−	0.979964i	$-0.563826\pi$
$839$	−11.5384	−0.398349	−0.199175	+	0.979964i	$0.563826\pi$
$840$	0	0
$841$	2.10982	0.0727525
$842$	−5.53237	−0.190658
$843$	0	0
$844$	3.47436	0.119592
$845$	31.3680	1.07909
$846$	0	0
$847$	−10.1449	−0.348583
$848$	−3.80395	−0.130628
$849$	0	0
$850$	1.51449	0.0519466
$851$	−19.1321	−0.655841
$852$	0	0
$853$	5.47685	0.187524	0.0937618	−	0.995595i	$-0.470111\pi$
$853$	5.47685	0.187524	0.0937618	+	0.995595i	$0.470111\pi$
$854$	0.0392771	0.00134404
$855$	0	0
$856$	10.1714	0.347650
$857$	−9.38512	−0.320590	−0.160295	−	0.987069i	$-0.551244\pi$
$857$	−9.38512	−0.320590	−0.160295	+	0.987069i	$0.551244\pi$
$858$	0	0
$859$	−8.15150	−0.278126	−0.139063	−	0.990284i	$-0.544409\pi$
$859$	−8.15150	−0.278126	−0.139063	+	0.990284i	$0.544409\pi$
$860$	36.3732	1.24032
$861$	0	0
$862$	7.69025	0.261931
$863$	−17.5317	−0.596784	−0.298392	−	0.954443i	$-0.596450\pi$
$863$	−17.5317	−0.596784	−0.298392	+	0.954443i	$0.596450\pi$
$864$	0	0
$865$	8.88207	0.301999
$866$	−7.36720	−0.250348
$867$	0	0
$868$	11.6496	0.395413
$869$	−60.3652	−2.04775
$870$	0	0
$871$	−12.8072	−0.433955
$872$	6.31898	0.213988
$873$	0	0
$874$	0.157679	0.00533359
$875$	6.09234	0.205959
$876$	0	0
$877$	25.2611	0.853008	0.426504	−	0.904486i	$-0.359745\pi$
$877$	25.2611	0.853008	0.426504	+	0.904486i	$0.359745\pi$
$878$	6.75961	0.228126
$879$	0	0
$880$	46.9480	1.58262
$881$	−8.27215	−0.278696	−0.139348	−	0.990243i	$-0.544501\pi$
$881$	−8.27215	−0.278696	−0.139348	+	0.990243i	$0.544501\pi$
$882$	0	0
$883$	53.8500	1.81220	0.906099	−	0.423066i	$-0.139046\pi$
$883$	53.8500	1.81220	0.906099	+	0.423066i	$0.139046\pi$
$884$	−21.1436	−0.711135
$885$	0	0
$886$	8.15706	0.274042
$887$	−26.0125	−0.873414	−0.436707	−	0.899604i	$-0.643855\pi$
$887$	−26.0125	−0.873414	−0.436707	+	0.899604i	$0.643855\pi$
$888$	0	0
$889$	1.00000	0.0335389
$890$	−6.32966	−0.212171
$891$	0	0
$892$	4.52508	0.151511
$893$	2.27299	0.0760627
$894$	0	0
$895$	−1.80391	−0.0602980
$896$	−7.19960	−0.240522
$897$	0	0
$898$	−0.695624	−0.0232133
$899$	−33.4727	−1.11638
$900$	0	0
$901$	2.30635	0.0768357
$902$	11.7109	0.389929
$903$	0	0
$904$	−3.56500	−0.118570
$905$	3.54110	0.117710
$906$	0	0
$907$	43.7207	1.45172	0.725861	−	0.687842i	$-0.241442\pi$
$907$	43.7207	1.45172	0.725861	+	0.687842i	$0.241442\pi$
$908$	33.6628	1.11714
$909$	0	0
$910$	−3.33743	−0.110635
$911$	−56.2800	−1.86464	−0.932320	−	0.361634i	$-0.882219\pi$
$911$	−56.2800	−1.86464	−0.932320	+	0.361634i	$0.882219\pi$
$912$	0	0
$913$	−43.2669	−1.43192
$914$	6.62836	0.219247
$915$	0	0
$916$	21.6892	0.716633
$917$	−4.24308	−0.140119
$918$	0	0
$919$	−49.1675	−1.62189	−0.810943	−	0.585125i	$-0.801045\pi$
$919$	−49.1675	−1.62189	−0.810943	+	0.585125i	$0.801045\pi$
$920$	−5.79838	−0.191167
$921$	0	0
$922$	0.860701	0.0283457
$923$	12.3017	0.404916
$924$	0	0
$925$	24.8852	0.818221
$926$	6.49367	0.213395
$927$	0	0
$928$	15.5992	0.512070
$929$	18.0024	0.590640	0.295320	−	0.955398i	$-0.404574\pi$
$929$	18.0024	0.590640	0.295320	+	0.955398i	$0.404574\pi$
$930$	0	0
$931$	0.299740	0.00982357
$932$	24.4808	0.801896
$933$	0	0
$934$	−3.25663	−0.106560
$935$	−28.4648	−0.930900
$936$	0	0
$937$	6.11724	0.199842	0.0999208	−	0.994995i	$-0.468141\pi$
$937$	6.11724	0.199842	0.0999208	+	0.994995i	$0.468141\pi$
$938$	0.631124	0.0206069
$939$	0	0
$940$	−41.1690	−1.34278
$941$	−39.7665	−1.29635	−0.648175	−	0.761492i	$-0.724467\pi$
$941$	−39.7665	−1.29635	−0.648175	+	0.761492i	$0.724467\pi$
$942$	0	0
$943$	22.7820	0.741884
$944$	−20.6518	−0.672158
$945$	0	0
$946$	7.47103	0.242904
$947$	−17.0699	−0.554698	−0.277349	−	0.960769i	$-0.589456\pi$
$947$	−17.0699	−0.554698	−0.277349	+	0.960769i	$0.589456\pi$
$948$	0	0
$949$	29.2764	0.950353
$950$	−0.205094	−0.00665413
$951$	0	0
$952$	2.11543	0.0685614
$953$	−14.4999	−0.469698	−0.234849	−	0.972032i	$-0.575460\pi$
$953$	−14.4999	−0.469698	−0.234849	+	0.972032i	$0.575460\pi$
$954$	0	0
$955$	50.7383	1.64185
$956$	−13.4529	−0.435099
$957$	0	0
$958$	−5.83412	−0.188492
$959$	6.12330	0.197732
$960$	0	0
$961$	5.01499	0.161774
$962$	10.5247	0.339329
$963$	0	0
$964$	17.6138	0.567304
$965$	58.3707	1.87902
$966$	0	0
$967$	−19.8725	−0.639056	−0.319528	−	0.947577i	$-0.603524\pi$
$967$	−19.8725	−0.639056	−0.319528	+	0.947577i	$0.603524\pi$
$968$	9.69591	0.311638
$969$	0	0
$970$	8.03905	0.258118
$971$	16.7784	0.538446	0.269223	−	0.963078i	$-0.413233\pi$
$971$	16.7784	0.538446	0.269223	+	0.963078i	$0.413233\pi$
$972$	0	0
$973$	5.79278	0.185708
$974$	−0.340284	−0.0109034
$975$	0	0
$976$	0.591281	0.0189264
$977$	58.6614	1.87675	0.938373	−	0.345625i	$-0.112333\pi$
$977$	58.6614	1.87675	0.938373	+	0.345625i	$0.112333\pi$
$978$	0	0
$979$	42.9164	1.37161
$980$	−5.42896	−0.173422
$981$	0	0
$982$	4.60181	0.146850
$983$	4.99877	0.159436	0.0797181	−	0.996817i	$-0.474598\pi$
$983$	4.99877	0.159436	0.0797181	+	0.996817i	$0.474598\pi$
$984$	0	0
$985$	−39.4367	−1.25656
$986$	−2.99377	−0.0953412
$987$	0	0
$988$	2.86328	0.0910932
$989$	14.5340	0.462153
$990$	0	0
$991$	−1.62682	−0.0516775	−0.0258388	−	0.999666i	$-0.508226\pi$
$991$	−1.62682	−0.0516775	−0.0258388	+	0.999666i	$0.508226\pi$
$992$	−16.7840	−0.532894
$993$	0	0
$994$	−0.606215	−0.0192280
$995$	−59.3423	−1.88128
$996$	0	0
$997$	34.8421	1.10346	0.551730	−	0.834023i	$-0.313968\pi$
$997$	34.8421	1.10346	0.551730	+	0.834023i	$0.313968\pi$
$998$	0.949544	0.0300573
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.z.1.15	✓	32
3.2	odd	2	inner	8001.2.a.z.1.18	yes	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8001.2.a.z.1.15	✓	32	1.1	even	1	trivial
8001.2.a.z.1.18	yes	32	3.2	odd	2	inner

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)

\(f(q)\)	\(=\)	\(q-0.242500 q^{2} -1.94119 q^{4} +2.79671 q^{5} -1.00000 q^{7} +0.955741 q^{8} +O(q^{10})\)	\(q-0.242500 q^{2} -1.94119 q^{4} +2.79671 q^{5} -1.00000 q^{7} +0.955741 q^{8} -0.678204 q^{10} +4.59836 q^{11} -4.92098 q^{13} +0.242500 q^{14} +3.65062 q^{16} -2.21339 q^{17} +0.299740 q^{19} -5.42896 q^{20} -1.11510 q^{22} -2.16929 q^{23} +2.82161 q^{25} +1.19334 q^{26} +1.94119 q^{28} -5.57762 q^{29} +6.00125 q^{31} -2.79676 q^{32} +0.536748 q^{34} -2.79671 q^{35} +8.81952 q^{37} -0.0726869 q^{38} +2.67293 q^{40} -10.5020 q^{41} -6.69985 q^{43} -8.92631 q^{44} +0.526055 q^{46} +7.58321 q^{47} +1.00000 q^{49} -0.684241 q^{50} +9.55257 q^{52} -1.04200 q^{53} +12.8603 q^{55} -0.955741 q^{56} +1.35257 q^{58} -5.65706 q^{59} +0.161967 q^{61} -1.45530 q^{62} -6.62302 q^{64} -13.7626 q^{65} +2.60257 q^{67} +4.29662 q^{68} +0.678204 q^{70} -2.49985 q^{71} -5.94931 q^{73} -2.13874 q^{74} -0.581853 q^{76} -4.59836 q^{77} -13.1275 q^{79} +10.2097 q^{80} +2.54675 q^{82} -9.40919 q^{83} -6.19022 q^{85} +1.62472 q^{86} +4.39484 q^{88} +9.33298 q^{89} +4.92098 q^{91} +4.21102 q^{92} -1.83893 q^{94} +0.838286 q^{95} -11.8534 q^{97} -0.242500 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 30 q^{4} - 32 q^{7}+O(q^{10}) \) 32 * q + 30 * q^4 - 32 * q^7	\( 32 q + 30 q^{4} - 32 q^{7} - 16 q^{10} - 14 q^{13} + 18 q^{16} - 30 q^{19} - 10 q^{22} + 36 q^{25} - 30 q^{28} - 58 q^{31} - 34 q^{34} + 8 q^{37} - 34 q^{40} + 6 q^{43} - 36 q^{46} + 32 q^{49} - 56 q^{52} - 88 q^{55} - 22 q^{58} - 46 q^{61} + 20 q^{64} - 8 q^{67} + 16 q^{70} - 60 q^{73} - 128 q^{76} - 74 q^{79} - 52 q^{82} - 16 q^{85} - 64 q^{88} + 14 q^{91} - 58 q^{94} - 44 q^{97}+O(q^{100}) \) 32 * q + 30 * q^4 - 32 * q^7 - 16 * q^10 - 14 * q^13 + 18 * q^16 - 30 * q^19 - 10 * q^22 + 36 * q^25 - 30 * q^28 - 58 * q^31 - 34 * q^34 + 8 * q^37 - 34 * q^40 + 6 * q^43 - 36 * q^46 + 32 * q^49 - 56 * q^52 - 88 * q^55 - 22 * q^58 - 46 * q^61 + 20 * q^64 - 8 * q^67 + 16 * q^70 - 60 * q^73 - 128 * q^76 - 74 * q^79 - 52 * q^82 - 16 * q^85 - 64 * q^88 + 14 * q^91 - 58 * q^94 - 44 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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