LMFDB - Embedded newform 8001.2.a.z.1.18

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			1.18
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	$-0.472676\pi$
$2$	0.242500	0.171474	0.0857368	+	0.996318i	$0.472676\pi$
$3$	0	0
$3$	0	0
$4$	−1.94119	−0.970597
$5$	−2.79671	−1.25073	−0.625364	−	0.780333i	$-0.715050\pi$
$5$	−2.79671	−1.25073	−0.625364	+	0.780333i	$0.715050\pi$
$6$	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	$-0.743812\pi$
$11$	−4.59836	−1.38646	−0.693229	+	0.720718i	$0.743812\pi$
$12$	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	$-0.413501\pi$
$17$	2.21339	0.536826	0.268413	+	0.963304i	$0.413501\pi$
$18$	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	$-0.427381\pi$
$23$	2.16929	0.452329	0.226165	+	0.974089i	$0.427381\pi$
$24$	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	$-0.326726\pi$
$29$	5.57762	1.03574	0.517869	+	0.855460i	$0.326726\pi$
$30$	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	$-0.193934\pi$
$41$	10.5020	1.64014	0.820071	+	0.572262i	$0.193934\pi$
$42$	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	$-0.686541\pi$
$47$	−7.58321	−1.10613	−0.553063	+	0.833140i	$0.686541\pi$
$48$	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	$-0.477201\pi$
$53$	1.04200	0.143130	0.0715649	+	0.997436i	$0.477201\pi$
$54$	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	$-0.379959\pi$
$59$	5.65706	0.736487	0.368243	+	0.929729i	$0.379959\pi$
$60$	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	$-0.452607\pi$
$71$	2.49985	0.296678	0.148339	+	0.988937i	$0.452607\pi$
$72$	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	$-0.327273\pi$
$83$	9.40919	1.03279	0.516396	+	0.856350i	$0.327273\pi$
$84$	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	$-0.664703\pi$
$89$	−9.33298	−0.989294	−0.494647	+	0.869094i	$0.664703\pi$
$90$	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	$-0.412360\pi$
$101$	5.46441	0.543729	0.271864	+	0.962336i	$0.412360\pi$
$102$	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	$-0.671992\pi$
$107$	−10.6424	−1.02884	−0.514419	+	0.857539i	$0.671992\pi$
$108$	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	$-0.443862\pi$
$113$	3.73009	0.350897	0.175449	+	0.984489i	$0.443862\pi$
$114$	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	$-0.559345\pi$
$131$	−4.24308	−0.370720	−0.185360	+	0.982671i	$0.559345\pi$
$132$	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	$-0.415758\pi$
$137$	6.12330	0.523148	0.261574	+	0.965183i	$0.415758\pi$
$138$	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	$-0.382070\pi$
$149$	8.83924	0.724139	0.362070	+	0.932151i	$0.382070\pi$
$150$	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	$-0.705665\pi$
$167$	−15.5614	−1.20418	−0.602089	+	0.798429i	$0.705665\pi$
$168$	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	$-0.538523\pi$
$173$	−3.17589	−0.241459	−0.120729	+	0.992685i	$0.538523\pi$
$174$	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	$-0.492326\pi$
$179$	0.645010	0.0482103	0.0241051	+	0.999709i	$0.492326\pi$
$180$	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	$-0.727904\pi$
$191$	−18.1421	−1.31272	−0.656359	+	0.754449i	$0.727904\pi$
$192$	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	$-0.332476\pi$
$197$	14.1011	1.00466	0.502331	+	0.864676i	$0.332476\pi$
$198$	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	$-0.304811\pi$
$227$	17.3413	1.15098	0.575490	+	0.817809i	$0.304811\pi$
$228$	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	$-0.364448\pi$
$233$	12.6112	0.826189	0.413094	+	0.910688i	$0.364448\pi$
$234$	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	$-0.571957\pi$
$239$	−6.93025	−0.448280	−0.224140	+	0.974557i	$0.571957\pi$
$240$	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	$-0.318403\pi$
$251$	17.1122	1.08011	0.540057	+	0.841628i	$0.318403\pi$
$252$	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	$-0.405147\pi$
$257$	9.41351	0.587199	0.293599	+	0.955929i	$0.405147\pi$
$258$	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	$-0.565676\pi$
$263$	−6.64471	−0.409730	−0.204865	+	0.978790i	$0.565676\pi$
$264$	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	$-0.662915\pi$
$269$	−16.0652	−0.979514	−0.489757	+	0.871859i	$0.662915\pi$
$270$	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	$-0.493077\pi$
$281$	0.729114	0.0434953	0.0217476	+	0.999763i	$0.493077\pi$
$282$	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	$-0.341727\pi$
$293$	16.3296	0.953983	0.476991	+	0.878908i	$0.341727\pi$
$294$	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	$-0.221237\pi$
$311$	27.0887	1.53606	0.768029	+	0.640415i	$0.221237\pi$
$312$	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	$-0.363057\pi$
$317$	14.8515	0.834142	0.417071	+	0.908874i	$0.363057\pi$
$318$	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	$-0.617728\pi$
$347$	−13.4672	−0.722959	−0.361479	+	0.932380i	$0.617728\pi$
$348$	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	$-0.505973\pi$
$353$	−0.705030	−0.0375249	−0.0187625	+	0.999824i	$0.505973\pi$
$354$	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	$-0.448655\pi$
$359$	6.08609	0.321211	0.160606	+	0.987019i	$0.448655\pi$
$360$	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	$-0.555361\pi$
$383$	−6.77315	−0.346092	−0.173046	+	0.984914i	$0.555361\pi$
$384$	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	$-0.427575\pi$
$389$	8.89791	0.451142	0.225571	+	0.974227i	$0.427575\pi$
$390$	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	$-0.482529\pi$
$401$	2.19709	0.109717	0.0548586	+	0.998494i	$0.482529\pi$
$402$	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	$-0.318360\pi$
$419$	22.1140	1.08034	0.540169	+	0.841556i	$0.318360\pi$
$420$	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	$-0.223349\pi$
$431$	31.7123	1.52753	0.763764	+	0.645495i	$0.223349\pi$
$432$	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	$-0.205321\pi$
$443$	33.6373	1.59816	0.799078	+	0.601227i	$0.205321\pi$
$444$	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	$-0.521562\pi$
$449$	−2.86855	−0.135375	−0.0676876	+	0.997707i	$0.521562\pi$
$450$	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	$-0.473661\pi$
$461$	3.54928	0.165306	0.0826531	+	0.996578i	$0.473661\pi$
$462$	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	$-0.600570\pi$
$467$	−13.4294	−0.621438	−0.310719	+	0.950502i	$0.600570\pi$
$468$	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	$-0.685229\pi$
$479$	−24.0582	−1.09925	−0.549624	+	0.835412i	$0.685229\pi$
$480$	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	$-0.359148\pi$
$491$	18.9765	0.856398	0.428199	+	0.903684i	$0.359148\pi$
$492$	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	$-0.531207\pi$
$503$	−4.39051	−0.195763	−0.0978815	+	0.995198i	$0.531207\pi$
$504$	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	$-0.526647\pi$
$509$	−3.77299	−0.167235	−0.0836175	+	0.996498i	$0.526647\pi$
$510$	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	$-0.152281\pi$
$521$	40.5257	1.77546	0.887731	+	0.460362i	$0.152281\pi$
$522$	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	$-0.593565\pi$
$557$	−13.6757	−0.579459	−0.289730	+	0.957109i	$0.593565\pi$
$558$	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	$-0.665066\pi$
$563$	−23.5207	−0.991278	−0.495639	+	0.868529i	$0.665066\pi$
$564$	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	$-0.471225\pi$
$569$	4.30680	0.180550	0.0902752	+	0.995917i	$0.471225\pi$
$570$	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	$-0.757340\pi$
$587$	−35.0446	−1.44645	−0.723224	+	0.690614i	$0.757340\pi$
$588$	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	$-0.588345\pi$
$593$	−13.3444	−0.547988	−0.273994	+	0.961731i	$0.588345\pi$
$594$	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	$-0.671993\pi$
$599$	−25.1804	−1.02884	−0.514421	+	0.857538i	$0.671993\pi$
$600$	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	$-0.656138\pi$
$617$	−23.4031	−0.942174	−0.471087	+	0.882087i	$0.656138\pi$
$618$	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	$-0.849732\pi$
$641$	−45.0976	−1.78125	−0.890624	+	0.454741i	$0.849732\pi$
$642$	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	$-0.581058\pi$
$647$	−12.8151	−0.503815	−0.251907	+	0.967751i	$0.581058\pi$
$648$	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	$-0.539674\pi$
$653$	−6.35355	−0.248633	−0.124317	+	0.992243i	$0.539674\pi$
$654$	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	$-0.304727\pi$
$659$	29.5580	1.15141	0.575707	+	0.817656i	$0.304727\pi$
$660$	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	$-0.500360\pi$
$677$	−0.0588742	−0.00226272	−0.00113136	+	0.999999i	$0.500360\pi$
$678$	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	$-0.543588\pi$
$683$	−7.13511	−0.273018	−0.136509	+	0.990639i	$0.543588\pi$
$684$	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	$-0.798390\pi$
$701$	−42.6818	−1.61207	−0.806034	+	0.591869i	$0.798390\pi$
$702$	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	$-0.855281\pi$
$719$	−48.1805	−1.79683	−0.898416	+	0.439146i	$0.855281\pi$
$720$	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	$-0.239108\pi$
$743$	39.8449	1.46177	0.730884	+	0.682502i	$0.239108\pi$
$744$	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	$-0.453862\pi$
$761$	7.96910	0.288880	0.144440	+	0.989514i	$0.453862\pi$
$762$	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	$-0.482290\pi$
$773$	3.09221	0.111219	0.0556095	+	0.998453i	$0.482290\pi$
$774$	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	$-0.429350\pi$
$797$	12.4294	0.440273	0.220136	+	0.975469i	$0.429350\pi$
$798$	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	$-0.264934\pi$
$809$	38.2936	1.34633	0.673165	+	0.739492i	$0.264934\pi$
$810$	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	$-0.707133\pi$
$821$	−34.7141	−1.21153	−0.605765	+	0.795644i	$0.707133\pi$
$822$	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	$-0.175936\pi$
$827$	48.9512	1.70220	0.851099	+	0.525005i	$0.175936\pi$
$828$	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	$-0.436174\pi$
$839$	11.5384	0.398349	0.199175	+	0.979964i	$0.436174\pi$
$840$	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	$-0.448756\pi$
$857$	9.38512	0.320590	0.160295	+	0.987069i	$0.448756\pi$
$858$	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	$-0.403550\pi$
$863$	17.5317	0.596784	0.298392	+	0.954443i	$0.403550\pi$
$864$	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	$-0.455499\pi$
$881$	8.27215	0.278696	0.139348	+	0.990243i	$0.455499\pi$
$882$	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	$-0.356145\pi$
$887$	26.0125	0.873414	0.436707	+	0.899604i	$0.356145\pi$
$888$	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	$-0.117781\pi$
$911$	56.2800	1.86464	0.932320	+	0.361634i	$0.117781\pi$
$912$	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	$-0.595426\pi$
$929$	−18.0024	−0.590640	−0.295320	+	0.955398i	$0.595426\pi$
$930$	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	$-0.275533\pi$
$941$	39.7665	1.29635	0.648175	+	0.761492i	$0.275533\pi$
$942$	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	$-0.410544\pi$
$947$	17.0699	0.554698	0.277349	+	0.960769i	$0.410544\pi$
$948$	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	$-0.424540\pi$
$953$	14.4999	0.469698	0.234849	+	0.972032i	$0.424540\pi$
$954$	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	$-0.586767\pi$
$971$	−16.7784	−0.538446	−0.269223	+	0.963078i	$0.586767\pi$
$972$	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	$-0.887667\pi$
$977$	−58.6614	−1.87675	−0.938373	+	0.345625i	$0.887667\pi$
$978$	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	$-0.525402\pi$
$983$	−4.99877	−0.159436	−0.0797181	+	0.996817i	$0.525402\pi$
$984$	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.1.18	yes	32
3.2	odd	2	inner	8001.2.a.z.1.15	✓	32

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

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