LMFDB - Embedded newform 8001.2.a.r.1.11

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:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 5 x^{15} - 13 x^{14} + 98 x^{13} + 9 x^{12} - 712 x^{11} + 565 x^{10} + 2282 x^{9} - 3082 x^{8} + \cdots + 4 $ x^16 - 5x^15 - 13x^14 + 98x^13 + 9x^12 - 712x^11 + 565x^10 + 2282x^9 - 3082x^8 - 2747x^7 + 5821x^6 - 158x^5 - 3341x^4 + 1002x^3 + 416x^2 - 148*x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 2667)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
Root			$-0.415886$ of defining polynomial
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)

$f(q)$	$=$	$q+0.415886 q^{2} -1.82704 q^{4} +0.233776 q^{5} -1.00000 q^{7} -1.59161 q^{8} +O(q^{10})$	\(q+0.415886 q^{2} -1.82704 q^{4} +0.233776 q^{5} -1.00000 q^{7} -1.59161 q^{8} +0.0972242 q^{10} -2.55512 q^{11} +1.50814 q^{13} -0.415886 q^{14} +2.99215 q^{16} -0.262289 q^{17} +0.541081 q^{19} -0.427118 q^{20} -1.06264 q^{22} +3.11874 q^{23} -4.94535 q^{25} +0.627215 q^{26} +1.82704 q^{28} +2.23156 q^{29} +8.80983 q^{31} +4.42762 q^{32} -0.109083 q^{34} -0.233776 q^{35} -9.13854 q^{37} +0.225028 q^{38} -0.372081 q^{40} +7.98771 q^{41} -2.63460 q^{43} +4.66830 q^{44} +1.29704 q^{46} -5.48952 q^{47} +1.00000 q^{49} -2.05670 q^{50} -2.75543 q^{52} -6.41058 q^{53} -0.597325 q^{55} +1.59161 q^{56} +0.928077 q^{58} -0.296590 q^{59} -6.94324 q^{61} +3.66389 q^{62} -4.14291 q^{64} +0.352567 q^{65} +11.7484 q^{67} +0.479213 q^{68} -0.0972242 q^{70} -2.31407 q^{71} +9.14504 q^{73} -3.80059 q^{74} -0.988575 q^{76} +2.55512 q^{77} +3.82656 q^{79} +0.699492 q^{80} +3.32198 q^{82} +2.11859 q^{83} -0.0613169 q^{85} -1.09569 q^{86} +4.06676 q^{88} +15.5991 q^{89} -1.50814 q^{91} -5.69806 q^{92} -2.28302 q^{94} +0.126492 q^{95} +16.3042 q^{97} +0.415886 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 5 q^{2} + 19 q^{4} + q^{5} - 16 q^{7} - 6 q^{8}+O(q^{10}) $ 16 * q - 5 * q^2 + 19 * q^4 + q^5 - 16 * q^7 - 6 * q^8	$ 16 q - 5 q^{2} + 19 q^{4} + q^{5} - 16 q^{7} - 6 q^{8} - 12 q^{10} - 11 q^{11} + 18 q^{13} + 5 q^{14} + 25 q^{16} + 5 q^{17} - 11 q^{19} + q^{20} + q^{22} - 13 q^{23} + 33 q^{25} - 8 q^{26} - 19 q^{28} - 24 q^{29} - 42 q^{31} - 42 q^{32} + 9 q^{34} - q^{35} + 40 q^{37} - 38 q^{38} - 61 q^{40} - 9 q^{41} + 7 q^{43} - 3 q^{44} + 24 q^{46} - 31 q^{47} + 16 q^{49} - 6 q^{50} + 52 q^{52} - 66 q^{53} - 36 q^{55} + 6 q^{56} + 19 q^{58} + 7 q^{59} + 6 q^{61} - 52 q^{62} + 10 q^{64} - 51 q^{65} + 16 q^{67} - 14 q^{68} + 12 q^{70} - 46 q^{71} + 39 q^{73} - 72 q^{74} + 24 q^{76} + 11 q^{77} + 4 q^{79} + 2 q^{80} - 18 q^{82} - 15 q^{83} - 4 q^{85} - 14 q^{86} + 58 q^{88} + q^{89} - 18 q^{91} - 26 q^{92} + 5 q^{94} - 44 q^{95} + 41 q^{97} - 5 q^{98}+O(q^{100}) $ 16 * q - 5 * q^2 + 19 * q^4 + q^5 - 16 * q^7 - 6 * q^8 - 12 * q^10 - 11 * q^11 + 18 * q^13 + 5 * q^14 + 25 * q^16 + 5 * q^17 - 11 * q^19 + q^20 + q^22 - 13 * q^23 + 33 * q^25 - 8 * q^26 - 19 * q^28 - 24 * q^29 - 42 * q^31 - 42 * q^32 + 9 * q^34 - q^35 + 40 * q^37 - 38 * q^38 - 61 * q^40 - 9 * q^41 + 7 * q^43 - 3 * q^44 + 24 * q^46 - 31 * q^47 + 16 * q^49 - 6 * q^50 + 52 * q^52 - 66 * q^53 - 36 * q^55 + 6 * q^56 + 19 * q^58 + 7 * q^59 + 6 * q^61 - 52 * q^62 + 10 * q^64 - 51 * q^65 + 16 * q^67 - 14 * q^68 + 12 * q^70 - 46 * q^71 + 39 * q^73 - 72 * q^74 + 24 * q^76 + 11 * q^77 + 4 * q^79 + 2 * q^80 - 18 * q^82 - 15 * q^83 - 4 * q^85 - 14 * q^86 + 58 * q^88 + q^89 - 18 * q^91 - 26 * q^92 + 5 * q^94 - 44 * q^95 + 41 * q^97 - 5 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0.415886	0.294076	0.147038	−	0.989131i	$-0.453026\pi$
$2$	0.415886	0.294076	0.147038	+	0.989131i	$0.453026\pi$
$3$	0	0
$3$	0	0
$4$	−1.82704	−0.913519
$5$	0.233776	0.104548	0.0522739	−	0.998633i	$-0.483353\pi$
$5$	0.233776	0.104548	0.0522739	+	0.998633i	$0.483353\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−1.59161	−0.562720
$9$	0	0
$10$	0.0972242	0.0307450
$11$	−2.55512	−0.770397	−0.385198	−	0.922834i	$-0.625867\pi$
$11$	−2.55512	−0.770397	−0.385198	+	0.922834i	$0.625867\pi$
$12$	0	0
$13$	1.50814	0.418283	0.209142	−	0.977885i	$-0.432933\pi$
$13$	1.50814	0.418283	0.209142	+	0.977885i	$0.432933\pi$
$14$	−0.415886	−0.111150
$15$	0	0
$16$	2.99215	0.748037
$17$	−0.262289	−0.0636145	−0.0318073	−	0.999494i	$-0.510126\pi$
$17$	−0.262289	−0.0636145	−0.0318073	+	0.999494i	$0.510126\pi$
$18$	0	0
$19$	0.541081	0.124132	0.0620662	−	0.998072i	$-0.480231\pi$
$19$	0.541081	0.124132	0.0620662	+	0.998072i	$0.480231\pi$
$20$	−0.427118	−0.0955064
$21$	0	0
$22$	−1.06264	−0.226555
$23$	3.11874	0.650303	0.325151	−	0.945662i	$-0.394585\pi$
$23$	3.11874	0.650303	0.325151	+	0.945662i	$0.394585\pi$
$24$	0	0
$25$	−4.94535	−0.989070
$26$	0.627215	0.123007
$27$	0	0
$28$	1.82704	0.345278
$29$	2.23156	0.414391	0.207196	−	0.978300i	$-0.433566\pi$
$29$	2.23156	0.414391	0.207196	+	0.978300i	$0.433566\pi$
$30$	0	0
$31$	8.80983	1.58229	0.791146	−	0.611627i	$-0.209485\pi$
$31$	8.80983	1.58229	0.791146	+	0.611627i	$0.209485\pi$
$32$	4.42762	0.782700
$33$	0	0
$34$	−0.109083	−0.0187075
$35$	−0.233776	−0.0395153
$36$	0	0
$37$	−9.13854	−1.50237	−0.751183	−	0.660094i	$-0.770517\pi$
$37$	−9.13854	−1.50237	−0.751183	+	0.660094i	$0.770517\pi$
$38$	0.225028	0.0365044
$39$	0	0
$40$	−0.372081	−0.0588311
$41$	7.98771	1.24747	0.623735	−	0.781636i	$-0.285614\pi$
$41$	7.98771	1.24747	0.623735	+	0.781636i	$0.285614\pi$
$42$	0	0
$43$	−2.63460	−0.401773	−0.200886	−	0.979615i	$-0.564382\pi$
$43$	−2.63460	−0.401773	−0.200886	+	0.979615i	$0.564382\pi$
$44$	4.66830	0.703772
$45$	0	0
$46$	1.29704	0.191238
$47$	−5.48952	−0.800729	−0.400365	−	0.916356i	$-0.631117\pi$
$47$	−5.48952	−0.800729	−0.400365	+	0.916356i	$0.631117\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−2.05670	−0.290862
$51$	0	0
$52$	−2.75543	−0.382110
$53$	−6.41058	−0.880561	−0.440280	−	0.897860i	$-0.645121\pi$
$53$	−6.41058	−0.880561	−0.440280	+	0.897860i	$0.645121\pi$
$54$	0	0
$55$	−0.597325	−0.0805433
$56$	1.59161	0.212688
$57$	0	0
$58$	0.928077	0.121862
$59$	−0.296590	−0.0386127	−0.0193064	−	0.999814i	$-0.506146\pi$
$59$	−0.296590	−0.0386127	−0.0193064	+	0.999814i	$0.506146\pi$
$60$	0	0
$61$	−6.94324	−0.888990	−0.444495	−	0.895781i	$-0.646617\pi$
$61$	−6.94324	−0.888990	−0.444495	+	0.895781i	$0.646617\pi$
$62$	3.66389	0.465314
$63$	0	0
$64$	−4.14291	−0.517864
$65$	0.352567	0.0437305
$66$	0	0
$67$	11.7484	1.43530	0.717649	−	0.696405i	$-0.245218\pi$
$67$	11.7484	1.43530	0.717649	+	0.696405i	$0.245218\pi$
$68$	0.479213	0.0581131
$69$	0	0
$70$	−0.0972242	−0.0116205
$71$	−2.31407	−0.274630	−0.137315	−	0.990527i	$-0.543847\pi$
$71$	−2.31407	−0.274630	−0.137315	+	0.990527i	$0.543847\pi$
$72$	0	0
$73$	9.14504	1.07035	0.535173	−	0.844743i	$-0.320246\pi$
$73$	9.14504	1.07035	0.535173	+	0.844743i	$0.320246\pi$
$74$	−3.80059	−0.441810
$75$	0	0
$76$	−0.988575	−0.113397
$77$	2.55512	0.291183
$78$	0	0
$79$	3.82656	0.430522	0.215261	−	0.976557i	$-0.430940\pi$
$79$	3.82656	0.430522	0.215261	+	0.976557i	$0.430940\pi$
$80$	0.699492	0.0782056
$81$	0	0
$82$	3.32198	0.366851
$83$	2.11859	0.232546	0.116273	−	0.993217i	$-0.462905\pi$
$83$	2.11859	0.232546	0.116273	+	0.993217i	$0.462905\pi$
$84$	0	0
$85$	−0.0613169	−0.00665075
$86$	−1.09569	−0.118152
$87$	0	0
$88$	4.06676	0.433518
$89$	15.5991	1.65350	0.826750	−	0.562569i	$-0.190187\pi$
$89$	15.5991	1.65350	0.826750	+	0.562569i	$0.190187\pi$
$90$	0	0
$91$	−1.50814	−0.158096
$92$	−5.69806	−0.594064
$93$	0	0
$94$	−2.28302	−0.235475
$95$	0.126492	0.0129778
$96$	0	0
$97$	16.3042	1.65544	0.827722	−	0.561138i	$-0.189636\pi$
$97$	16.3042	1.65544	0.827722	+	0.561138i	$0.189636\pi$
$98$	0.415886	0.0420109
$99$	0	0
$100$	9.03534	0.903534
$101$	−5.23614	−0.521016	−0.260508	−	0.965472i	$-0.583890\pi$
$101$	−5.23614	−0.521016	−0.260508	+	0.965472i	$0.583890\pi$
$102$	0	0
$103$	−6.04580	−0.595710	−0.297855	−	0.954611i	$-0.596271\pi$
$103$	−6.04580	−0.595710	−0.297855	+	0.954611i	$0.596271\pi$
$104$	−2.40038	−0.235376
$105$	0	0
$106$	−2.66607	−0.258952
$107$	−0.930174	−0.0899233	−0.0449617	−	0.998989i	$-0.514317\pi$
$107$	−0.930174	−0.0899233	−0.0449617	+	0.998989i	$0.514317\pi$
$108$	0	0
$109$	14.5050	1.38933	0.694665	−	0.719333i	$-0.255552\pi$
$109$	14.5050	1.38933	0.694665	+	0.719333i	$0.255552\pi$
$110$	−0.248419	−0.0236858
$111$	0	0
$112$	−2.99215	−0.282731
$113$	−20.2664	−1.90650	−0.953250	−	0.302184i	$-0.902284\pi$
$113$	−20.2664	−1.90650	−0.953250	+	0.302184i	$0.902284\pi$
$114$	0	0
$115$	0.729087	0.0679877
$116$	−4.07715	−0.378554
$117$	0	0
$118$	−0.123348	−0.0113551
$119$	0.262289	0.0240440
$120$	0	0
$121$	−4.47137	−0.406489
$122$	−2.88760	−0.261431
$123$	0	0
$124$	−16.0959	−1.44545
$125$	−2.32498	−0.207953
$126$	0	0
$127$	1.00000	0.0887357
$127$	1.00000	0.0887357
$128$	−10.5782	−0.934991
$129$	0	0
$130$	0.146628	0.0128601
$131$	−13.3874	−1.16966	−0.584832	−	0.811154i	$-0.698840\pi$
$131$	−13.3874	−1.16966	−0.584832	+	0.811154i	$0.698840\pi$
$132$	0	0
$133$	−0.541081	−0.0469176
$134$	4.88600	0.422087
$135$	0	0
$136$	0.417463	0.0357972
$137$	−19.7793	−1.68986	−0.844930	−	0.534878i	$-0.820358\pi$
$137$	−19.7793	−1.68986	−0.844930	+	0.534878i	$0.820358\pi$
$138$	0	0
$139$	−19.1936	−1.62798	−0.813992	−	0.580876i	$-0.802710\pi$
$139$	−19.1936	−1.62798	−0.813992	+	0.580876i	$0.802710\pi$
$140$	0.427118	0.0360980
$141$	0	0
$142$	−0.962391	−0.0807621
$143$	−3.85348	−0.322244
$144$	0	0
$145$	0.521686	0.0433237
$146$	3.80329	0.314763
$147$	0	0
$148$	16.6965	1.37244
$149$	−19.2699	−1.57865	−0.789324	−	0.613977i	$-0.789569\pi$
$149$	−19.2699	−1.57865	−0.789324	+	0.613977i	$0.789569\pi$
$150$	0	0
$151$	−9.79088	−0.796770	−0.398385	−	0.917218i	$-0.630429\pi$
$151$	−9.79088	−0.796770	−0.398385	+	0.917218i	$0.630429\pi$
$152$	−0.861191	−0.0698518
$153$	0	0
$154$	1.06264	0.0856298
$155$	2.05953	0.165425
$156$	0	0
$157$	7.92860	0.632772	0.316386	−	0.948631i	$-0.397531\pi$
$157$	7.92860	0.632772	0.316386	+	0.948631i	$0.397531\pi$
$158$	1.59142	0.126606
$159$	0	0
$160$	1.03507	0.0818295
$161$	−3.11874	−0.245791
$162$	0	0
$163$	−4.12383	−0.323003	−0.161502	−	0.986872i	$-0.551634\pi$
$163$	−4.12383	−0.323003	−0.161502	+	0.986872i	$0.551634\pi$
$164$	−14.5939	−1.13959
$165$	0	0
$166$	0.881093	0.0683861
$167$	13.3468	1.03281	0.516405	−	0.856345i	$-0.327270\pi$
$167$	13.3468	1.03281	0.516405	+	0.856345i	$0.327270\pi$
$168$	0	0
$169$	−10.7255	−0.825039
$170$	−0.0255009	−0.00195583
$171$	0	0
$172$	4.81352	0.367027
$173$	−11.2737	−0.857121	−0.428561	−	0.903513i	$-0.640979\pi$
$173$	−11.2737	−0.857121	−0.428561	+	0.903513i	$0.640979\pi$
$174$	0	0
$175$	4.94535	0.373833
$176$	−7.64529	−0.576285
$177$	0	0
$178$	6.48745	0.486255
$179$	19.0428	1.42333	0.711663	−	0.702521i	$-0.247942\pi$
$179$	19.0428	1.42333	0.711663	+	0.702521i	$0.247942\pi$
$180$	0	0
$181$	11.6032	0.862456	0.431228	−	0.902243i	$-0.358080\pi$
$181$	11.6032	0.862456	0.431228	+	0.902243i	$0.358080\pi$
$182$	−0.627215	−0.0464923
$183$	0	0
$184$	−4.96383	−0.365938
$185$	−2.13637	−0.157069
$186$	0	0
$187$	0.670180	0.0490084
$188$	10.0296	0.731482
$189$	0	0
$190$	0.0526061	0.00381645
$191$	−18.7637	−1.35769	−0.678846	−	0.734281i	$-0.737519\pi$
$191$	−18.7637	−1.35769	−0.678846	+	0.734281i	$0.737519\pi$
$192$	0	0
$193$	−0.321177	−0.0231188	−0.0115594	−	0.999933i	$-0.503680\pi$
$193$	−0.321177	−0.0231188	−0.0115594	+	0.999933i	$0.503680\pi$
$194$	6.78071	0.486826
$195$	0	0
$196$	−1.82704	−0.130503
$197$	−15.8224	−1.12730	−0.563648	−	0.826015i	$-0.690603\pi$
$197$	−15.8224	−1.12730	−0.563648	+	0.826015i	$0.690603\pi$
$198$	0	0
$199$	−23.0558	−1.63438	−0.817192	−	0.576365i	$-0.804471\pi$
$199$	−23.0558	−1.63438	−0.817192	+	0.576365i	$0.804471\pi$
$200$	7.87108	0.556569
$201$	0	0
$202$	−2.17764	−0.153218
$203$	−2.23156	−0.156625
$204$	0	0
$205$	1.86733	0.130420
$206$	−2.51436	−0.175184
$207$	0	0
$208$	4.51258	0.312891
$209$	−1.38252	−0.0956312
$210$	0	0
$211$	17.7515	1.22206	0.611031	−	0.791607i	$-0.290755\pi$
$211$	17.7515	1.22206	0.611031	+	0.791607i	$0.290755\pi$
$212$	11.7124	0.804409
$213$	0	0
$214$	−0.386847	−0.0264443
$215$	−0.615906	−0.0420044
$216$	0	0
$217$	−8.80983	−0.598050
$218$	6.03245	0.408569
$219$	0	0
$220$	1.09134	0.0735778
$221$	−0.395569	−0.0266089
$222$	0	0
$223$	−8.62336	−0.577463	−0.288731	−	0.957410i	$-0.593233\pi$
$223$	−8.62336	−0.577463	−0.288731	+	0.957410i	$0.593233\pi$
$224$	−4.42762	−0.295833
$225$	0	0
$226$	−8.42850	−0.560656
$227$	5.68235	0.377151	0.188575	−	0.982059i	$-0.439613\pi$
$227$	5.68235	0.377151	0.188575	+	0.982059i	$0.439613\pi$
$228$	0	0
$229$	−9.61432	−0.635332	−0.317666	−	0.948203i	$-0.602899\pi$
$229$	−9.61432	−0.635332	−0.317666	+	0.948203i	$0.602899\pi$
$230$	0.303217	0.0199935
$231$	0	0
$232$	−3.55179	−0.233186
$233$	−11.3140	−0.741208	−0.370604	−	0.928791i	$-0.620849\pi$
$233$	−11.3140	−0.741208	−0.370604	+	0.928791i	$0.620849\pi$
$234$	0	0
$235$	−1.28332	−0.0837144
$236$	0.541882	0.0352735
$237$	0	0
$238$	0.109083	0.00707077
$239$	−11.3788	−0.736033	−0.368016	−	0.929819i	$-0.619963\pi$
$239$	−11.3788	−0.736033	−0.368016	+	0.929819i	$0.619963\pi$
$240$	0	0
$241$	29.3396	1.88993	0.944965	−	0.327171i	$-0.106095\pi$
$241$	29.3396	1.88993	0.944965	+	0.327171i	$0.106095\pi$
$242$	−1.85958	−0.119539
$243$	0	0
$244$	12.6856	0.812110
$245$	0.233776	0.0149354
$246$	0	0
$247$	0.816026	0.0519225
$248$	−14.0218	−0.890388
$249$	0	0
$250$	−0.966928	−0.0611539
$251$	−20.2496	−1.27814	−0.639072	−	0.769147i	$-0.720682\pi$
$251$	−20.2496	−1.27814	−0.639072	+	0.769147i	$0.720682\pi$
$252$	0	0
$253$	−7.96875	−0.500991
$254$	0.415886	0.0260950
$255$	0	0
$256$	3.88649	0.242905
$257$	−17.2732	−1.07747	−0.538737	−	0.842474i	$-0.681098\pi$
$257$	−17.2732	−1.07747	−0.538737	+	0.842474i	$0.681098\pi$
$258$	0	0
$259$	9.13854	0.567841
$260$	−0.644153	−0.0399487
$261$	0	0
$262$	−5.56764	−0.343970
$263$	−0.909253	−0.0560670	−0.0280335	−	0.999607i	$-0.508925\pi$
$263$	−0.909253	−0.0560670	−0.0280335	+	0.999607i	$0.508925\pi$
$264$	0	0
$265$	−1.49864	−0.0920606
$266$	−0.225028	−0.0137974
$267$	0	0
$268$	−21.4648	−1.31117
$269$	−18.4398	−1.12429	−0.562146	−	0.827038i	$-0.690024\pi$
$269$	−18.4398	−1.12429	−0.562146	+	0.827038i	$0.690024\pi$
$270$	0	0
$271$	−22.6557	−1.37624	−0.688118	−	0.725599i	$-0.741563\pi$
$271$	−22.6557	−1.37624	−0.688118	+	0.725599i	$0.741563\pi$
$272$	−0.784808	−0.0475860
$273$	0	0
$274$	−8.22594	−0.496947
$275$	12.6359	0.761976
$276$	0	0
$277$	26.7313	1.60613	0.803063	−	0.595894i	$-0.203202\pi$
$277$	26.7313	1.60613	0.803063	+	0.595894i	$0.203202\pi$
$278$	−7.98237	−0.478751
$279$	0	0
$280$	0.372081	0.0222361
$281$	−29.5348	−1.76190	−0.880948	−	0.473213i	$-0.843094\pi$
$281$	−29.5348	−1.76190	−0.880948	+	0.473213i	$0.843094\pi$
$282$	0	0
$283$	31.5045	1.87275	0.936375	−	0.351002i	$-0.114159\pi$
$283$	31.5045	1.87275	0.936375	+	0.351002i	$0.114159\pi$
$284$	4.22790	0.250880
$285$	0	0
$286$	−1.60261	−0.0947642
$287$	−7.98771	−0.471500
$288$	0	0
$289$	−16.9312	−0.995953
$290$	0.216962	0.0127404
$291$	0	0
$292$	−16.7083	−0.977781
$293$	13.5309	0.790486	0.395243	−	0.918577i	$-0.370660\pi$
$293$	13.5309	0.790486	0.395243	+	0.918577i	$0.370660\pi$
$294$	0	0
$295$	−0.0693356	−0.00403688
$296$	14.5450	0.845412
$297$	0	0
$298$	−8.01407	−0.464242
$299$	4.70350	0.272011
$300$	0	0
$301$	2.63460	0.151856
$302$	−4.07189	−0.234311
$303$	0	0
$304$	1.61899	0.0928556
$305$	−1.62316	−0.0929419
$306$	0	0
$307$	−18.7542	−1.07036	−0.535179	−	0.844739i	$-0.679756\pi$
$307$	−18.7542	−1.07036	−0.535179	+	0.844739i	$0.679756\pi$
$308$	−4.66830	−0.266001
$309$	0	0
$310$	0.856528	0.0486475
$311$	9.23956	0.523927	0.261964	−	0.965078i	$-0.415630\pi$
$311$	9.23956	0.523927	0.261964	+	0.965078i	$0.415630\pi$
$312$	0	0
$313$	19.3735	1.09505	0.547527	−	0.836788i	$-0.315569\pi$
$313$	19.3735	1.09505	0.547527	+	0.836788i	$0.315569\pi$
$314$	3.29740	0.186083
$315$	0	0
$316$	−6.99128	−0.393290
$317$	34.6176	1.94432	0.972159	−	0.234323i	$-0.0752874\pi$
$317$	34.6176	1.94432	0.972159	+	0.234323i	$0.0752874\pi$
$318$	0	0
$319$	−5.70191	−0.319246
$320$	−0.968512	−0.0541415
$321$	0	0
$322$	−1.29704	−0.0722813
$323$	−0.141920	−0.00789662
$324$	0	0
$325$	−7.45828	−0.413711
$326$	−1.71504	−0.0949875
$327$	0	0
$328$	−12.7133	−0.701977
$329$	5.48952	0.302647
$330$	0	0
$331$	−21.2380	−1.16734	−0.583672	−	0.811990i	$-0.698385\pi$
$331$	−21.2380	−1.16734	−0.583672	+	0.811990i	$0.698385\pi$
$332$	−3.87075	−0.212435
$333$	0	0
$334$	5.55077	0.303724
$335$	2.74650	0.150057
$336$	0	0
$337$	−19.4542	−1.05974	−0.529868	−	0.848080i	$-0.677759\pi$
$337$	−19.4542	−1.05974	−0.529868	+	0.848080i	$0.677759\pi$
$338$	−4.46059	−0.242624
$339$	0	0
$340$	0.112028	0.00607559
$341$	−22.5102	−1.21899
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	4.19326	0.226086
$345$	0	0
$346$	−4.68857	−0.252059
$347$	−1.01218	−0.0543368	−0.0271684	−	0.999631i	$-0.508649\pi$
$347$	−1.01218	−0.0543368	−0.0271684	+	0.999631i	$0.508649\pi$
$348$	0	0
$349$	−10.2536	−0.548862	−0.274431	−	0.961607i	$-0.588489\pi$
$349$	−10.2536	−0.548862	−0.274431	+	0.961607i	$0.588489\pi$
$350$	2.05670	0.109935
$351$	0	0
$352$	−11.3131	−0.602989
$353$	−0.637692	−0.0339409	−0.0169705	−	0.999856i	$-0.505402\pi$
$353$	−0.637692	−0.0339409	−0.0169705	+	0.999856i	$0.505402\pi$
$354$	0	0
$355$	−0.540974	−0.0287119
$356$	−28.5001	−1.51050
$357$	0	0
$358$	7.91964	0.418566
$359$	−5.89606	−0.311182	−0.155591	−	0.987822i	$-0.549728\pi$
$359$	−5.89606	−0.311182	−0.155591	+	0.987822i	$0.549728\pi$
$360$	0	0
$361$	−18.7072	−0.984591
$362$	4.82560	0.253628
$363$	0	0
$364$	2.75543	0.144424
$365$	2.13789	0.111902
$366$	0	0
$367$	20.2487	1.05698	0.528488	−	0.848941i	$-0.322759\pi$
$367$	20.2487	1.05698	0.528488	+	0.848941i	$0.322759\pi$
$368$	9.33174	0.486450
$369$	0	0
$370$	−0.888487	−0.0461902
$371$	6.41058	0.332821
$372$	0	0
$373$	1.45351	0.0752599	0.0376300	−	0.999292i	$-0.488019\pi$
$373$	1.45351	0.0752599	0.0376300	+	0.999292i	$0.488019\pi$
$374$	0.278719	0.0144122
$375$	0	0
$376$	8.73719	0.450586
$377$	3.36551	0.173333
$378$	0	0
$379$	6.90345	0.354606	0.177303	−	0.984156i	$-0.443263\pi$
$379$	6.90345	0.354606	0.177303	+	0.984156i	$0.443263\pi$
$380$	−0.231105	−0.0118554
$381$	0	0
$382$	−7.80355	−0.399264
$383$	−34.0654	−1.74066	−0.870332	−	0.492466i	$-0.836095\pi$
$383$	−34.0654	−1.74066	−0.870332	+	0.492466i	$0.836095\pi$
$384$	0	0
$385$	0.597325	0.0304425
$386$	−0.133573	−0.00679869
$387$	0	0
$388$	−29.7885	−1.51228
$389$	−30.8976	−1.56657	−0.783284	−	0.621664i	$-0.786457\pi$
$389$	−30.8976	−1.56657	−0.783284	+	0.621664i	$0.786457\pi$
$390$	0	0
$391$	−0.818013	−0.0413687
$392$	−1.59161	−0.0803886
$393$	0	0
$394$	−6.58030	−0.331511
$395$	0.894558	0.0450101
$396$	0	0
$397$	−3.47389	−0.174350	−0.0871749	−	0.996193i	$-0.527784\pi$
$397$	−3.47389	−0.174350	−0.0871749	+	0.996193i	$0.527784\pi$
$398$	−9.58860	−0.480633
$399$	0	0
$400$	−14.7972	−0.739861
$401$	−17.7552	−0.886650	−0.443325	−	0.896361i	$-0.646201\pi$
$401$	−17.7552	−0.886650	−0.443325	+	0.896361i	$0.646201\pi$
$402$	0	0
$403$	13.2865	0.661846
$404$	9.56663	0.475958
$405$	0	0
$406$	−0.928077	−0.0460597
$407$	23.3500	1.15742
$408$	0	0
$409$	34.1586	1.68904	0.844518	−	0.535527i	$-0.179887\pi$
$409$	34.1586	1.68904	0.844518	+	0.535527i	$0.179887\pi$
$410$	0.776598	0.0383535
$411$	0	0
$412$	11.0459	0.544193
$413$	0.296590	0.0145942
$414$	0	0
$415$	0.495276	0.0243121
$416$	6.67747	0.327390
$417$	0	0
$418$	−0.574973	−0.0281228
$419$	24.8237	1.21272	0.606358	−	0.795192i	$-0.292630\pi$
$419$	24.8237	1.21272	0.606358	+	0.795192i	$0.292630\pi$
$420$	0	0
$421$	28.5585	1.39185	0.695927	−	0.718112i	$-0.254994\pi$
$421$	28.5585	1.39185	0.695927	+	0.718112i	$0.254994\pi$
$422$	7.38259	0.359379
$423$	0	0
$424$	10.2032	0.495509
$425$	1.29711	0.0629192
$426$	0	0
$427$	6.94324	0.336007
$428$	1.69946	0.0821467
$429$	0	0
$430$	−0.256147	−0.0123525
$431$	−5.59470	−0.269487	−0.134744	−	0.990880i	$-0.543021\pi$
$431$	−5.59470	−0.269487	−0.134744	+	0.990880i	$0.543021\pi$
$432$	0	0
$433$	−18.4625	−0.887252	−0.443626	−	0.896212i	$-0.646308\pi$
$433$	−18.4625	−0.887252	−0.443626	+	0.896212i	$0.646308\pi$
$434$	−3.66389	−0.175872
$435$	0	0
$436$	−26.5013	−1.26918
$437$	1.68749	0.0807236
$438$	0	0
$439$	−32.1653	−1.53516	−0.767582	−	0.640950i	$-0.778540\pi$
$439$	−32.1653	−1.53516	−0.767582	+	0.640950i	$0.778540\pi$
$440$	0.950710	0.0453233
$441$	0	0
$442$	−0.164512	−0.00782503
$443$	−37.7878	−1.79535	−0.897675	−	0.440658i	$-0.854745\pi$
$443$	−37.7878	−1.79535	−0.897675	+	0.440658i	$0.854745\pi$
$444$	0	0
$445$	3.64669	0.172870
$446$	−3.58634	−0.169818
$447$	0	0
$448$	4.14291	0.195734
$449$	−22.6568	−1.06924	−0.534620	−	0.845092i	$-0.679545\pi$
$449$	−22.6568	−1.06924	−0.534620	+	0.845092i	$0.679545\pi$
$450$	0	0
$451$	−20.4095	−0.961047
$452$	37.0274	1.74162
$453$	0	0
$454$	2.36321	0.110911
$455$	−0.352567	−0.0165286
$456$	0	0
$457$	33.4798	1.56612	0.783059	−	0.621947i	$-0.213658\pi$
$457$	33.4798	1.56612	0.783059	+	0.621947i	$0.213658\pi$
$458$	−3.99846	−0.186836
$459$	0	0
$460$	−1.33207	−0.0621081
$461$	18.3167	0.853092	0.426546	−	0.904466i	$-0.359730\pi$
$461$	18.3167	0.853092	0.426546	+	0.904466i	$0.359730\pi$
$462$	0	0
$463$	−12.6537	−0.588065	−0.294033	−	0.955795i	$-0.594997\pi$
$463$	−12.6537	−0.588065	−0.294033	+	0.955795i	$0.594997\pi$
$464$	6.67717	0.309980
$465$	0	0
$466$	−4.70536	−0.217971
$467$	32.6838	1.51243	0.756213	−	0.654325i	$-0.227047\pi$
$467$	32.6838	1.51243	0.756213	+	0.654325i	$0.227047\pi$
$468$	0	0
$469$	−11.7484	−0.542491
$470$	−0.533714	−0.0246184
$471$	0	0
$472$	0.472057	0.0217282
$473$	6.73171	0.309524
$474$	0	0
$475$	−2.67583	−0.122776
$476$	−0.479213	−0.0219647
$477$	0	0
$478$	−4.73228	−0.216450
$479$	−19.5405	−0.892829	−0.446415	−	0.894826i	$-0.647299\pi$
$479$	−19.5405	−0.892829	−0.446415	+	0.894826i	$0.647299\pi$
$480$	0	0
$481$	−13.7822	−0.628415
$482$	12.2019	0.555783
$483$	0	0
$484$	8.16937	0.371335
$485$	3.81154	0.173073
$486$	0	0
$487$	12.0567	0.546340	0.273170	−	0.961966i	$-0.411928\pi$
$487$	12.0567	0.546340	0.273170	+	0.961966i	$0.411928\pi$
$488$	11.0509	0.500253
$489$	0	0
$490$	0.0972242	0.00439214
$491$	−16.5951	−0.748928	−0.374464	−	0.927241i	$-0.622173\pi$
$491$	−16.5951	−0.748928	−0.374464	+	0.927241i	$0.622173\pi$
$492$	0	0
$493$	−0.585316	−0.0263613
$494$	0.339374	0.0152692
$495$	0	0
$496$	26.3603	1.18361
$497$	2.31407	0.103800
$498$	0	0
$499$	−11.0152	−0.493106	−0.246553	−	0.969129i	$-0.579298\pi$
$499$	−11.0152	−0.493106	−0.246553	+	0.969129i	$0.579298\pi$
$500$	4.24783	0.189969
$501$	0	0
$502$	−8.42154	−0.375872
$503$	−33.2843	−1.48407	−0.742037	−	0.670359i	$-0.766140\pi$
$503$	−33.2843	−1.48407	−0.742037	+	0.670359i	$0.766140\pi$
$504$	0	0
$505$	−1.22408	−0.0544710
$506$	−3.31409	−0.147329
$507$	0	0
$508$	−1.82704	−0.0810617
$509$	20.6251	0.914191	0.457096	−	0.889418i	$-0.348890\pi$
$509$	20.6251	0.914191	0.457096	+	0.889418i	$0.348890\pi$
$510$	0	0
$511$	−9.14504	−0.404553
$512$	22.7728	1.00642
$513$	0	0
$514$	−7.18369	−0.316859
$515$	−1.41336	−0.0622802
$516$	0	0
$517$	14.0264	0.616879
$518$	3.80059	0.166988
$519$	0	0
$520$	−0.561150	−0.0246081
$521$	14.3461	0.628515	0.314258	−	0.949338i	$-0.398244\pi$
$521$	14.3461	0.628515	0.314258	+	0.949338i	$0.398244\pi$
$522$	0	0
$523$	−27.0993	−1.18497	−0.592484	−	0.805582i	$-0.701853\pi$
$523$	−27.0993	−1.18497	−0.592484	+	0.805582i	$0.701853\pi$
$524$	24.4593	1.06851
$525$	0	0
$526$	−0.378146	−0.0164879
$527$	−2.31072	−0.100657
$528$	0	0
$529$	−13.2734	−0.577106
$530$	−0.623263	−0.0270728
$531$	0	0
$532$	0.988575	0.0428602
$533$	12.0466	0.521796
$534$	0	0
$535$	−0.217452	−0.00940128
$536$	−18.6989	−0.807671
$537$	0	0
$538$	−7.66884	−0.330627
$539$	−2.55512	−0.110057
$540$	0	0
$541$	−7.68896	−0.330574	−0.165287	−	0.986245i	$-0.552855\pi$
$541$	−7.68896	−0.330574	−0.165287	+	0.986245i	$0.552855\pi$
$542$	−9.42219	−0.404718
$543$	0	0
$544$	−1.16132	−0.0497911
$545$	3.39093	0.145251
$546$	0	0
$547$	1.35436	0.0579084	0.0289542	−	0.999581i	$-0.490782\pi$
$547$	1.35436	0.0579084	0.0289542	+	0.999581i	$0.490782\pi$
$548$	36.1375	1.54372
$549$	0	0
$550$	5.25512	0.224079
$551$	1.20746	0.0514394
$552$	0	0
$553$	−3.82656	−0.162722
$554$	11.1172	0.472323
$555$	0	0
$556$	35.0675	1.48719
$557$	−2.07592	−0.0879594	−0.0439797	−	0.999032i	$-0.514004\pi$
$557$	−2.07592	−0.0879594	−0.0439797	+	0.999032i	$0.514004\pi$
$558$	0	0
$559$	−3.97335	−0.168055
$560$	−0.699492	−0.0295589
$561$	0	0
$562$	−12.2831	−0.518131
$563$	−32.5774	−1.37297	−0.686486	−	0.727143i	$-0.740848\pi$
$563$	−32.5774	−1.37297	−0.686486	+	0.727143i	$0.740848\pi$
$564$	0	0
$565$	−4.73779	−0.199320
$566$	13.1023	0.550731
$567$	0	0
$568$	3.68311	0.154540
$569$	−28.1702	−1.18095	−0.590477	−	0.807054i	$-0.701060\pi$
$569$	−28.1702	−1.18095	−0.590477	+	0.807054i	$0.701060\pi$
$570$	0	0
$571$	20.9774	0.877876	0.438938	−	0.898517i	$-0.355355\pi$
$571$	20.9774	0.877876	0.438938	+	0.898517i	$0.355355\pi$
$572$	7.04045	0.294376
$573$	0	0
$574$	−3.32198	−0.138657
$575$	−15.4233	−0.643195
$576$	0	0
$577$	−22.7704	−0.947943	−0.473972	−	0.880540i	$-0.657180\pi$
$577$	−22.7704	−0.947943	−0.473972	+	0.880540i	$0.657180\pi$
$578$	−7.04145	−0.292886
$579$	0	0
$580$	−0.953140	−0.0395770
$581$	−2.11859	−0.0878940
$582$	0	0
$583$	16.3798	0.678381
$584$	−14.5554	−0.602305
$585$	0	0
$586$	5.62733	0.232463
$587$	27.4915	1.13469	0.567347	−	0.823479i	$-0.307970\pi$
$587$	27.4915	1.13469	0.567347	+	0.823479i	$0.307970\pi$
$588$	0	0
$589$	4.76683	0.196414
$590$	−0.0288357	−0.00118715
$591$	0	0
$592$	−27.3439	−1.12383
$593$	5.82344	0.239140	0.119570	−	0.992826i	$-0.461848\pi$
$593$	5.82344	0.239140	0.119570	+	0.992826i	$0.461848\pi$
$594$	0	0
$595$	0.0613169	0.00251375
$596$	35.2068	1.44213
$597$	0	0
$598$	1.95612	0.0799918
$599$	33.8888	1.38466	0.692328	−	0.721582i	$-0.256585\pi$
$599$	33.8888	1.38466	0.692328	+	0.721582i	$0.256585\pi$
$600$	0	0
$601$	−16.2335	−0.662179	−0.331090	−	0.943599i	$-0.607416\pi$
$601$	−16.2335	−0.662179	−0.331090	+	0.943599i	$0.607416\pi$
$602$	1.09569	0.0446571
$603$	0	0
$604$	17.8883	0.727865
$605$	−1.04530	−0.0424975
$606$	0	0
$607$	−16.0952	−0.653283	−0.326642	−	0.945148i	$-0.605917\pi$
$607$	−16.0952	−0.653283	−0.326642	+	0.945148i	$0.605917\pi$
$608$	2.39570	0.0971584
$609$	0	0
$610$	−0.675050	−0.0273320
$611$	−8.27897	−0.334931
$612$	0	0
$613$	−42.7403	−1.72627	−0.863133	−	0.504976i	$-0.831501\pi$
$613$	−42.7403	−1.72627	−0.863133	+	0.504976i	$0.831501\pi$
$614$	−7.79960	−0.314766
$615$	0	0
$616$	−4.06676	−0.163854
$617$	−15.1918	−0.611599	−0.305799	−	0.952096i	$-0.598924\pi$
$617$	−15.1918	−0.611599	−0.305799	+	0.952096i	$0.598924\pi$
$618$	0	0
$619$	−22.9392	−0.922005	−0.461003	−	0.887399i	$-0.652510\pi$
$619$	−22.9392	−0.922005	−0.461003	+	0.887399i	$0.652510\pi$
$620$	−3.76283	−0.151119
$621$	0	0
$622$	3.84261	0.154074
$623$	−15.5991	−0.624964
$624$	0	0
$625$	24.1832	0.967329
$626$	8.05717	0.322029
$627$	0	0
$628$	−14.4859	−0.578049
$629$	2.39694	0.0955723
$630$	0	0
$631$	7.21212	0.287110	0.143555	−	0.989642i	$-0.454147\pi$
$631$	7.21212	0.287110	0.143555	+	0.989642i	$0.454147\pi$
$632$	−6.09041	−0.242263
$633$	0	0
$634$	14.3970	0.571777
$635$	0.233776	0.00927711
$636$	0	0
$637$	1.50814	0.0597547
$638$	−2.37135	−0.0938825
$639$	0	0
$640$	−2.47293	−0.0977512
$641$	−4.74405	−0.187379	−0.0936893	−	0.995601i	$-0.529866\pi$
$641$	−4.74405	−0.187379	−0.0936893	+	0.995601i	$0.529866\pi$
$642$	0	0
$643$	21.8862	0.863107	0.431554	−	0.902087i	$-0.357966\pi$
$643$	21.8862	0.863107	0.431554	+	0.902087i	$0.357966\pi$
$644$	5.69806	0.224535
$645$	0	0
$646$	−0.0590224	−0.00232221
$647$	−30.4209	−1.19597	−0.597984	−	0.801508i	$-0.704031\pi$
$647$	−30.4209	−1.19597	−0.597984	+	0.801508i	$0.704031\pi$
$648$	0	0
$649$	0.757823	0.0297471
$650$	−3.10180	−0.121662
$651$	0	0
$652$	7.53439	0.295070
$653$	26.1136	1.02190	0.510952	−	0.859609i	$-0.329293\pi$
$653$	26.1136	1.02190	0.510952	+	0.859609i	$0.329293\pi$
$654$	0	0
$655$	−3.12965	−0.122286
$656$	23.9004	0.933154
$657$	0	0
$658$	2.28302	0.0890013
$659$	−34.7355	−1.35310	−0.676552	−	0.736395i	$-0.736527\pi$
$659$	−34.7355	−1.35310	−0.676552	+	0.736395i	$0.736527\pi$
$660$	0	0
$661$	19.5074	0.758751	0.379375	−	0.925243i	$-0.376139\pi$
$661$	19.5074	0.758751	0.379375	+	0.925243i	$0.376139\pi$
$662$	−8.83257	−0.343288
$663$	0	0
$664$	−3.37198	−0.130858
$665$	−0.126492	−0.00490513
$666$	0	0
$667$	6.95967	0.269480
$668$	−24.3852	−0.943491
$669$	0	0
$670$	1.14223	0.0441282
$671$	17.7408	0.684875
$672$	0	0
$673$	5.79306	0.223306	0.111653	−	0.993747i	$-0.464386\pi$
$673$	5.79306	0.223306	0.111653	+	0.993747i	$0.464386\pi$
$674$	−8.09073	−0.311643
$675$	0	0
$676$	19.5959	0.753689
$677$	27.0216	1.03852	0.519261	−	0.854616i	$-0.326207\pi$
$677$	27.0216	1.03852	0.519261	+	0.854616i	$0.326207\pi$
$678$	0	0
$679$	−16.3042	−0.625699
$680$	0.0975928	0.00374251
$681$	0	0
$682$	−9.36166	−0.358477
$683$	20.5033	0.784537	0.392268	−	0.919851i	$-0.371690\pi$
$683$	20.5033	0.784537	0.392268	+	0.919851i	$0.371690\pi$
$684$	0	0
$685$	−4.62392	−0.176671
$686$	−0.415886	−0.0158786
$687$	0	0
$688$	−7.88311	−0.300541
$689$	−9.66805	−0.368324
$690$	0	0
$691$	0.801681	0.0304974	0.0152487	−	0.999884i	$-0.495146\pi$
$691$	0.801681	0.0304974	0.0152487	+	0.999884i	$0.495146\pi$
$692$	20.5974	0.782997
$693$	0	0
$694$	−0.420953	−0.0159792
$695$	−4.48701	−0.170202
$696$	0	0
$697$	−2.09509	−0.0793572
$698$	−4.26433	−0.161407
$699$	0	0
$700$	−9.03534	−0.341504
$701$	10.8290	0.409005	0.204503	−	0.978866i	$-0.434442\pi$
$701$	10.8290	0.409005	0.204503	+	0.978866i	$0.434442\pi$
$702$	0	0
$703$	−4.94469	−0.186492
$704$	10.5856	0.398961
$705$	0	0
$706$	−0.265207	−0.00998121
$707$	5.23614	0.196925
$708$	0	0
$709$	25.9381	0.974125	0.487062	−	0.873367i	$-0.338068\pi$
$709$	25.9381	0.974125	0.487062	+	0.873367i	$0.338068\pi$
$710$	−0.224984	−0.00844349
$711$	0	0
$712$	−24.8277	−0.930458
$713$	27.4756	1.02897
$714$	0	0
$715$	−0.900850	−0.0336899
$716$	−34.7920	−1.30024
$717$	0	0
$718$	−2.45209	−0.0915112
$719$	−34.7457	−1.29580	−0.647898	−	0.761727i	$-0.724352\pi$
$719$	−34.7457	−1.29580	−0.647898	+	0.761727i	$0.724352\pi$
$720$	0	0
$721$	6.04580	0.225157
$722$	−7.78008	−0.289545
$723$	0	0
$724$	−21.1994	−0.787870
$725$	−11.0359	−0.409862
$726$	0	0
$727$	−0.970577	−0.0359967	−0.0179984	−	0.999838i	$-0.505729\pi$
$727$	−0.970577	−0.0359967	−0.0179984	+	0.999838i	$0.505729\pi$
$728$	2.40038	0.0889639
$729$	0	0
$730$	0.889118	0.0329078
$731$	0.691028	0.0255586
$732$	0	0
$733$	31.8418	1.17610	0.588051	−	0.808824i	$-0.299895\pi$
$733$	31.8418	1.17610	0.588051	+	0.808824i	$0.299895\pi$
$734$	8.42117	0.310831
$735$	0	0
$736$	13.8086	0.508992
$737$	−30.0186	−1.10575
$738$	0	0
$739$	−15.4180	−0.567160	−0.283580	−	0.958949i	$-0.591522\pi$
$739$	−15.4180	−0.567160	−0.283580	+	0.958949i	$0.591522\pi$
$740$	3.90323	0.143486
$741$	0	0
$742$	2.66607	0.0978745
$743$	−29.0370	−1.06526	−0.532632	−	0.846347i	$-0.678797\pi$
$743$	−29.0370	−1.06526	−0.532632	+	0.846347i	$0.678797\pi$
$744$	0	0
$745$	−4.50483	−0.165044
$746$	0.604495	0.0221321
$747$	0	0
$748$	−1.22444	−0.0447701
$749$	0.930174	0.0339878
$750$	0	0
$751$	−28.2756	−1.03179	−0.515896	−	0.856651i	$-0.672541\pi$
$751$	−28.2756	−1.03179	−0.515896	+	0.856651i	$0.672541\pi$
$752$	−16.4255	−0.598975
$753$	0	0
$754$	1.39967	0.0509730
$755$	−2.28887	−0.0833005
$756$	0	0
$757$	45.8305	1.66574	0.832868	−	0.553471i	$-0.186697\pi$
$757$	45.8305	1.66574	0.832868	+	0.553471i	$0.186697\pi$
$758$	2.87105	0.104281
$759$	0	0
$760$	−0.201326	−0.00730285
$761$	18.3092	0.663708	0.331854	−	0.943331i	$-0.392326\pi$
$761$	18.3092	0.663708	0.331854	+	0.943331i	$0.392326\pi$
$762$	0	0
$763$	−14.5050	−0.525118
$764$	34.2819	1.24028
$765$	0	0
$766$	−14.1674	−0.511887
$767$	−0.447300	−0.0161511
$768$	0	0
$769$	33.1824	1.19659	0.598294	−	0.801277i	$-0.295846\pi$
$769$	33.1824	1.19659	0.598294	+	0.801277i	$0.295846\pi$
$770$	0.248419	0.00895240
$771$	0	0
$772$	0.586803	0.0211195
$773$	12.1762	0.437946	0.218973	−	0.975731i	$-0.429729\pi$
$773$	12.1762	0.437946	0.218973	+	0.975731i	$0.429729\pi$
$774$	0	0
$775$	−43.5677	−1.56500
$776$	−25.9500	−0.931552
$777$	0	0
$778$	−12.8499	−0.460690
$779$	4.32200	0.154852
$780$	0	0
$781$	5.91273	0.211574
$782$	−0.340200	−0.0121655
$783$	0	0
$784$	2.99215	0.106862
$785$	1.85352	0.0661548
$786$	0	0
$787$	−15.3085	−0.545690	−0.272845	−	0.962058i	$-0.587965\pi$
$787$	−15.3085	−0.545690	−0.272845	+	0.962058i	$0.587965\pi$
$788$	28.9081	1.02981
$789$	0	0
$790$	0.372034	0.0132364
$791$	20.2664	0.720589
$792$	0	0
$793$	−10.4714	−0.371850
$794$	−1.44474	−0.0512721
$795$	0	0
$796$	42.1239	1.49304
$797$	−55.5963	−1.96932	−0.984662	−	0.174474i	$-0.944178\pi$
$797$	−55.5963	−1.96932	−0.984662	+	0.174474i	$0.944178\pi$
$798$	0	0
$799$	1.43984	0.0509380
$800$	−21.8961	−0.774145
$801$	0	0
$802$	−7.38413	−0.260743
$803$	−23.3666	−0.824591
$804$	0	0
$805$	−0.729087	−0.0256969
$806$	5.52566	0.194633
$807$	0	0
$808$	8.33391	0.293186
$809$	28.7363	1.01032	0.505158	−	0.863027i	$-0.331434\pi$
$809$	28.7363	1.01032	0.505158	+	0.863027i	$0.331434\pi$
$810$	0	0
$811$	−27.0970	−0.951504	−0.475752	−	0.879579i	$-0.657824\pi$
$811$	−27.0970	−0.951504	−0.475752	+	0.879579i	$0.657824\pi$
$812$	4.07715	0.143080
$813$	0	0
$814$	9.71096	0.340369
$815$	−0.964051	−0.0337693
$816$	0	0
$817$	−1.42553	−0.0498730
$818$	14.2061	0.496705
$819$	0	0
$820$	−3.41169	−0.119141
$821$	8.53361	0.297825	0.148913	−	0.988850i	$-0.452423\pi$
$821$	8.53361	0.297825	0.148913	+	0.988850i	$0.452423\pi$
$822$	0	0
$823$	19.1860	0.668782	0.334391	−	0.942434i	$-0.391469\pi$
$823$	19.1860	0.668782	0.334391	+	0.942434i	$0.391469\pi$
$824$	9.62257	0.335218
$825$	0	0
$826$	0.123348	0.00429182
$827$	−19.5848	−0.681032	−0.340516	−	0.940239i	$-0.610602\pi$
$827$	−19.5848	−0.681032	−0.340516	+	0.940239i	$0.610602\pi$
$828$	0	0
$829$	10.2487	0.355954	0.177977	−	0.984035i	$-0.443045\pi$
$829$	10.2487	0.355954	0.177977	+	0.984035i	$0.443045\pi$
$830$	0.205978	0.00714961
$831$	0	0
$832$	−6.24809	−0.216614
$833$	−0.262289	−0.00908779
$834$	0	0
$835$	3.12017	0.107978
$836$	2.52593	0.0873610
$837$	0	0
$838$	10.3238	0.356630
$839$	6.85135	0.236535	0.118267	−	0.992982i	$-0.462266\pi$
$839$	6.85135	0.236535	0.118267	+	0.992982i	$0.462266\pi$
$840$	0	0
$841$	−24.0201	−0.828280
$842$	11.8771	0.409311
$843$	0	0
$844$	−32.4326	−1.11638
$845$	−2.50737	−0.0862560
$846$	0	0
$847$	4.47137	0.153638
$848$	−19.1814	−0.658692
$849$	0	0
$850$	0.539451	0.0185030
$851$	−28.5008	−0.976993
$852$	0	0
$853$	−35.2516	−1.20699	−0.603496	−	0.797366i	$-0.706226\pi$
$853$	−35.2516	−1.20699	−0.603496	+	0.797366i	$0.706226\pi$
$854$	2.88760	0.0988115
$855$	0	0
$856$	1.48048	0.0506016
$857$	47.7812	1.63217	0.816087	−	0.577929i	$-0.196139\pi$
$857$	47.7812	1.63217	0.816087	+	0.577929i	$0.196139\pi$
$858$	0	0
$859$	−8.64937	−0.295113	−0.147556	−	0.989054i	$-0.547141\pi$
$859$	−8.64937	−0.295113	−0.147556	+	0.989054i	$0.547141\pi$
$860$	1.12528	0.0383719
$861$	0	0
$862$	−2.32676	−0.0792497
$863$	−37.8764	−1.28933	−0.644664	−	0.764466i	$-0.723003\pi$
$863$	−37.8764	−1.28933	−0.644664	+	0.764466i	$0.723003\pi$
$864$	0	0
$865$	−2.63551	−0.0896101
$866$	−7.67831	−0.260919
$867$	0	0
$868$	16.0959	0.546331
$869$	−9.77732	−0.331673
$870$	0	0
$871$	17.7183	0.600361
$872$	−23.0864	−0.781804
$873$	0	0
$874$	0.701804	0.0237389
$875$	2.32498	0.0785987
$876$	0	0
$877$	−19.8055	−0.668785	−0.334393	−	0.942434i	$-0.608531\pi$
$877$	−19.8055	−0.668785	−0.334393	+	0.942434i	$0.608531\pi$
$878$	−13.3771	−0.451455
$879$	0	0
$880$	−1.78728	−0.0602493
$881$	24.5225	0.826183	0.413091	−	0.910690i	$-0.364449\pi$
$881$	24.5225	0.826183	0.413091	+	0.910690i	$0.364449\pi$
$882$	0	0
$883$	16.5522	0.557027	0.278513	−	0.960432i	$-0.410158\pi$
$883$	16.5522	0.557027	0.278513	+	0.960432i	$0.410158\pi$
$884$	0.722720	0.0243077
$885$	0	0
$886$	−15.7154	−0.527969
$887$	5.01125	0.168261	0.0841307	−	0.996455i	$-0.473189\pi$
$887$	5.01125	0.168261	0.0841307	+	0.996455i	$0.473189\pi$
$888$	0	0
$889$	−1.00000	−0.0335389
$890$	1.51661	0.0508368
$891$	0	0
$892$	15.7552	0.527523
$893$	−2.97027	−0.0993964
$894$	0	0
$895$	4.45175	0.148806
$896$	10.5782	0.353393
$897$	0	0
$898$	−9.42266	−0.314438
$899$	19.6597	0.655688
$900$	0	0
$901$	1.68143	0.0560164
$902$	−8.48804	−0.282621
$903$	0	0
$904$	32.2562	1.07283
$905$	2.71254	0.0901678
$906$	0	0
$907$	−2.46395	−0.0818140	−0.0409070	−	0.999163i	$-0.513025\pi$
$907$	−2.46395	−0.0818140	−0.0409070	+	0.999163i	$0.513025\pi$
$908$	−10.3819	−0.344535
$909$	0	0
$910$	−0.146628	−0.00486066
$911$	−50.1497	−1.66153	−0.830766	−	0.556622i	$-0.812097\pi$
$911$	−50.1497	−1.66153	−0.830766	+	0.556622i	$0.812097\pi$
$912$	0	0
$913$	−5.41325	−0.179153
$914$	13.9238	0.460558
$915$	0	0
$916$	17.5657	0.580388
$917$	13.3874	0.442091
$918$	0	0
$919$	−35.3629	−1.16651	−0.583257	−	0.812288i	$-0.698222\pi$
$919$	−35.3629	−1.16651	−0.583257	+	0.812288i	$0.698222\pi$
$920$	−1.16042	−0.0382580
$921$	0	0
$922$	7.61765	0.250874
$923$	−3.48995	−0.114873
$924$	0	0
$925$	45.1933	1.48595
$926$	−5.26248	−0.172936
$927$	0	0
$928$	9.88052	0.324344
$929$	47.8519	1.56997	0.784984	−	0.619516i	$-0.212671\pi$
$929$	47.8519	1.56997	0.784984	+	0.619516i	$0.212671\pi$
$930$	0	0
$931$	0.541081	0.0177332
$932$	20.6712	0.677108
$933$	0	0
$934$	13.5927	0.444768
$935$	0.156672	0.00512372
$936$	0	0
$937$	−25.5485	−0.834632	−0.417316	−	0.908761i	$-0.637029\pi$
$937$	−25.5485	−0.834632	−0.417316	+	0.908761i	$0.637029\pi$
$938$	−4.88600	−0.159534
$939$	0	0
$940$	2.34467	0.0764747
$941$	−14.6471	−0.477482	−0.238741	−	0.971083i	$-0.576735\pi$
$941$	−14.6471	−0.477482	−0.238741	+	0.971083i	$0.576735\pi$
$942$	0	0
$943$	24.9116	0.811234
$944$	−0.887441	−0.0288838
$945$	0	0
$946$	2.79963	0.0910237
$947$	37.3948	1.21517	0.607584	−	0.794256i	$-0.292139\pi$
$947$	37.3948	1.21517	0.607584	+	0.794256i	$0.292139\pi$
$948$	0	0
$949$	13.7920	0.447707
$950$	−1.11284	−0.0361054
$951$	0	0
$952$	−0.417463	−0.0135301
$953$	13.2720	0.429921	0.214961	−	0.976623i	$-0.431038\pi$
$953$	13.2720	0.429921	0.214961	+	0.976623i	$0.431038\pi$
$954$	0	0
$955$	−4.38649	−0.141944
$956$	20.7895	0.672380
$957$	0	0
$958$	−8.12663	−0.262560
$959$	19.7793	0.638707
$960$	0	0
$961$	46.6131	1.50365
$962$	−5.73183	−0.184802
$963$	0	0
$964$	−53.6046	−1.72649
$965$	−0.0750834	−0.00241702
$966$	0	0
$967$	11.2385	0.361406	0.180703	−	0.983538i	$-0.442163\pi$
$967$	11.2385	0.361406	0.180703	+	0.983538i	$0.442163\pi$
$968$	7.11670	0.228739
$969$	0	0
$970$	1.58517	0.0508966
$971$	38.5997	1.23872	0.619361	−	0.785106i	$-0.287392\pi$
$971$	38.5997	1.23872	0.619361	+	0.785106i	$0.287392\pi$
$972$	0	0
$973$	19.1936	0.615320
$974$	5.01421	0.160666
$975$	0	0
$976$	−20.7752	−0.664998
$977$	−55.3476	−1.77073	−0.885363	−	0.464900i	$-0.846090\pi$
$977$	−55.3476	−1.77073	−0.885363	+	0.464900i	$0.846090\pi$
$978$	0	0
$979$	−39.8575	−1.27385
$980$	−0.427118	−0.0136438
$981$	0	0
$982$	−6.90169	−0.220242
$983$	22.5714	0.719916	0.359958	−	0.932969i	$-0.382791\pi$
$983$	22.5714	0.719916	0.359958	+	0.932969i	$0.382791\pi$
$984$	0	0
$985$	−3.69889	−0.117856
$986$	−0.243425	−0.00775222
$987$	0	0
$988$	−1.49091	−0.0474322
$989$	−8.21664	−0.261274
$990$	0	0
$991$	−13.6847	−0.434709	−0.217354	−	0.976093i	$-0.569743\pi$
$991$	−13.6847	−0.434709	−0.217354	+	0.976093i	$0.569743\pi$
$992$	39.0066	1.23846
$993$	0	0
$994$	0.962391	0.0305252
$995$	−5.38990	−0.170871
$996$	0	0
$997$	−12.8075	−0.405617	−0.202808	−	0.979218i	$-0.565007\pi$
$997$	−12.8075	−0.405617	−0.202808	+	0.979218i	$0.565007\pi$
$998$	−4.58105	−0.145011
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.r.1.11		16
3.2	odd	2		2667.2.a.o.1.6	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2667.2.a.o.1.6	✓	16	3.2	odd	2
8001.2.a.r.1.11		16	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)

\(f(q)\)	\(=\)	\(q+0.415886 q^{2} -1.82704 q^{4} +0.233776 q^{5} -1.00000 q^{7} -1.59161 q^{8} +O(q^{10})\)	\(q+0.415886 q^{2} -1.82704 q^{4} +0.233776 q^{5} -1.00000 q^{7} -1.59161 q^{8} +0.0972242 q^{10} -2.55512 q^{11} +1.50814 q^{13} -0.415886 q^{14} +2.99215 q^{16} -0.262289 q^{17} +0.541081 q^{19} -0.427118 q^{20} -1.06264 q^{22} +3.11874 q^{23} -4.94535 q^{25} +0.627215 q^{26} +1.82704 q^{28} +2.23156 q^{29} +8.80983 q^{31} +4.42762 q^{32} -0.109083 q^{34} -0.233776 q^{35} -9.13854 q^{37} +0.225028 q^{38} -0.372081 q^{40} +7.98771 q^{41} -2.63460 q^{43} +4.66830 q^{44} +1.29704 q^{46} -5.48952 q^{47} +1.00000 q^{49} -2.05670 q^{50} -2.75543 q^{52} -6.41058 q^{53} -0.597325 q^{55} +1.59161 q^{56} +0.928077 q^{58} -0.296590 q^{59} -6.94324 q^{61} +3.66389 q^{62} -4.14291 q^{64} +0.352567 q^{65} +11.7484 q^{67} +0.479213 q^{68} -0.0972242 q^{70} -2.31407 q^{71} +9.14504 q^{73} -3.80059 q^{74} -0.988575 q^{76} +2.55512 q^{77} +3.82656 q^{79} +0.699492 q^{80} +3.32198 q^{82} +2.11859 q^{83} -0.0613169 q^{85} -1.09569 q^{86} +4.06676 q^{88} +15.5991 q^{89} -1.50814 q^{91} -5.69806 q^{92} -2.28302 q^{94} +0.126492 q^{95} +16.3042 q^{97} +0.415886 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 5 q^{2} + 19 q^{4} + q^{5} - 16 q^{7} - 6 q^{8}+O(q^{10}) \) 16 * q - 5 * q^2 + 19 * q^4 + q^5 - 16 * q^7 - 6 * q^8	\( 16 q - 5 q^{2} + 19 q^{4} + q^{5} - 16 q^{7} - 6 q^{8} - 12 q^{10} - 11 q^{11} + 18 q^{13} + 5 q^{14} + 25 q^{16} + 5 q^{17} - 11 q^{19} + q^{20} + q^{22} - 13 q^{23} + 33 q^{25} - 8 q^{26} - 19 q^{28} - 24 q^{29} - 42 q^{31} - 42 q^{32} + 9 q^{34} - q^{35} + 40 q^{37} - 38 q^{38} - 61 q^{40} - 9 q^{41} + 7 q^{43} - 3 q^{44} + 24 q^{46} - 31 q^{47} + 16 q^{49} - 6 q^{50} + 52 q^{52} - 66 q^{53} - 36 q^{55} + 6 q^{56} + 19 q^{58} + 7 q^{59} + 6 q^{61} - 52 q^{62} + 10 q^{64} - 51 q^{65} + 16 q^{67} - 14 q^{68} + 12 q^{70} - 46 q^{71} + 39 q^{73} - 72 q^{74} + 24 q^{76} + 11 q^{77} + 4 q^{79} + 2 q^{80} - 18 q^{82} - 15 q^{83} - 4 q^{85} - 14 q^{86} + 58 q^{88} + q^{89} - 18 q^{91} - 26 q^{92} + 5 q^{94} - 44 q^{95} + 41 q^{97} - 5 q^{98}+O(q^{100}) \) 16 * q - 5 * q^2 + 19 * q^4 + q^5 - 16 * q^7 - 6 * q^8 - 12 * q^10 - 11 * q^11 + 18 * q^13 + 5 * q^14 + 25 * q^16 + 5 * q^17 - 11 * q^19 + q^20 + q^22 - 13 * q^23 + 33 * q^25 - 8 * q^26 - 19 * q^28 - 24 * q^29 - 42 * q^31 - 42 * q^32 + 9 * q^34 - q^35 + 40 * q^37 - 38 * q^38 - 61 * q^40 - 9 * q^41 + 7 * q^43 - 3 * q^44 + 24 * q^46 - 31 * q^47 + 16 * q^49 - 6 * q^50 + 52 * q^52 - 66 * q^53 - 36 * q^55 + 6 * q^56 + 19 * q^58 + 7 * q^59 + 6 * q^61 - 52 * q^62 + 10 * q^64 - 51 * q^65 + 16 * q^67 - 14 * q^68 + 12 * q^70 - 46 * q^71 + 39 * q^73 - 72 * q^74 + 24 * q^76 + 11 * q^77 + 4 * q^79 + 2 * q^80 - 18 * q^82 - 15 * q^83 - 4 * q^85 - 14 * q^86 + 58 * q^88 + q^89 - 18 * q^91 - 26 * q^92 + 5 * q^94 - 44 * q^95 + 41 * q^97 - 5 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more