LMFDB - Embedded newform 8001.2.a.s.1.7

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:	$0$
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} - 4 x^{15} - 18 x^{14} + 83 x^{13} + 112 x^{12} - 668 x^{11} - 235 x^{10} + 2648 x^{9} + \cdots - 20 $ x^16 - 4x^15 - 18x^14 + 83x^13 + 112x^12 - 668x^11 - 235x^10 + 2648x^9 - 298x^8 - 5422x^7 + 2075x^6 + 5385x^5 - 3163x^4 - 1882x^3 + 1614x^2 - 200*x - 20
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 2667)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$0.825012$ 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)

$f(q)$	$=$	$q-0.825012 q^{2} -1.31936 q^{4} +1.66882 q^{5} -1.00000 q^{7} +2.73851 q^{8} +O(q^{10})$	\(q-0.825012 q^{2} -1.31936 q^{4} +1.66882 q^{5} -1.00000 q^{7} +2.73851 q^{8} -1.37680 q^{10} -4.60134 q^{11} -0.261360 q^{13} +0.825012 q^{14} +0.379411 q^{16} -1.30909 q^{17} -5.83541 q^{19} -2.20177 q^{20} +3.79616 q^{22} +1.54469 q^{23} -2.21504 q^{25} +0.215625 q^{26} +1.31936 q^{28} -4.76454 q^{29} -0.627450 q^{31} -5.79003 q^{32} +1.08001 q^{34} -1.66882 q^{35} +6.11849 q^{37} +4.81428 q^{38} +4.57008 q^{40} -11.5755 q^{41} +8.48685 q^{43} +6.07080 q^{44} -1.27439 q^{46} +4.72063 q^{47} +1.00000 q^{49} +1.82743 q^{50} +0.344827 q^{52} -13.5754 q^{53} -7.67881 q^{55} -2.73851 q^{56} +3.93080 q^{58} +2.92411 q^{59} +3.22354 q^{61} +0.517653 q^{62} +4.01802 q^{64} -0.436164 q^{65} -5.93511 q^{67} +1.72715 q^{68} +1.37680 q^{70} -0.659163 q^{71} -0.0147131 q^{73} -5.04783 q^{74} +7.69898 q^{76} +4.60134 q^{77} +3.62188 q^{79} +0.633169 q^{80} +9.54991 q^{82} -6.07333 q^{83} -2.18463 q^{85} -7.00175 q^{86} -12.6008 q^{88} +0.737302 q^{89} +0.261360 q^{91} -2.03800 q^{92} -3.89457 q^{94} -9.73825 q^{95} +6.68248 q^{97} -0.825012 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 4 q^{2} + 20 q^{4} - 5 q^{5} - 16 q^{7} - 15 q^{8}+O(q^{10}) $ 16 * q - 4 * q^2 + 20 * q^4 - 5 * q^5 - 16 * q^7 - 15 * q^8	$ 16 q - 4 q^{2} + 20 q^{4} - 5 q^{5} - 16 q^{7} - 15 q^{8} - 4 q^{10} - q^{11} + 20 q^{13} + 4 q^{14} + 32 q^{16} - 3 q^{17} + 13 q^{19} - 17 q^{20} + 13 q^{22} - 5 q^{23} + 17 q^{25} + 2 q^{26} - 20 q^{28} - 22 q^{29} + 26 q^{31} - 54 q^{32} - 6 q^{34} + 5 q^{35} + 30 q^{37} - 5 q^{38} + 13 q^{40} - q^{41} + 31 q^{43} - 22 q^{44} - 2 q^{46} + q^{47} + 16 q^{49} - 5 q^{50} + 31 q^{52} - 24 q^{53} + 8 q^{55} + 15 q^{56} + 13 q^{58} + 17 q^{59} + 32 q^{61} + 5 q^{62} + 61 q^{64} + 3 q^{65} + 16 q^{67} + 10 q^{68} + 4 q^{70} + 10 q^{71} + 23 q^{73} - q^{74} + 18 q^{76} + q^{77} + 48 q^{79} - 38 q^{80} + 12 q^{82} - 9 q^{83} + 22 q^{85} + 4 q^{86} + 27 q^{88} - 17 q^{89} - 20 q^{91} - 16 q^{92} + 13 q^{94} - 22 q^{95} + 17 q^{97} - 4 q^{98}+O(q^{100}) $ 16 * q - 4 * q^2 + 20 * q^4 - 5 * q^5 - 16 * q^7 - 15 * q^8 - 4 * q^10 - q^11 + 20 * q^13 + 4 * q^14 + 32 * q^16 - 3 * q^17 + 13 * q^19 - 17 * q^20 + 13 * q^22 - 5 * q^23 + 17 * q^25 + 2 * q^26 - 20 * q^28 - 22 * q^29 + 26 * q^31 - 54 * q^32 - 6 * q^34 + 5 * q^35 + 30 * q^37 - 5 * q^38 + 13 * q^40 - q^41 + 31 * q^43 - 22 * q^44 - 2 * q^46 + q^47 + 16 * q^49 - 5 * q^50 + 31 * q^52 - 24 * q^53 + 8 * q^55 + 15 * q^56 + 13 * q^58 + 17 * q^59 + 32 * q^61 + 5 * q^62 + 61 * q^64 + 3 * q^65 + 16 * q^67 + 10 * q^68 + 4 * q^70 + 10 * q^71 + 23 * q^73 - q^74 + 18 * q^76 + q^77 + 48 * q^79 - 38 * q^80 + 12 * q^82 - 9 * q^83 + 22 * q^85 + 4 * q^86 + 27 * q^88 - 17 * q^89 - 20 * q^91 - 16 * q^92 + 13 * q^94 - 22 * q^95 + 17 * q^97 - 4 * 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 $

$2$	−0.825012	−0.583371	−0.291686	−	0.956514i	$-0.594216\pi$
$2$	−0.825012	−0.583371	−0.291686	+	0.956514i	$0.594216\pi$
$3$	0	0
$3$	0	0
$4$	−1.31936	−0.659678
$5$	1.66882	0.746319	0.373160	−	0.927767i	$-0.378274\pi$
$5$	1.66882	0.746319	0.373160	+	0.927767i	$0.378274\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	2.73851	0.968209
$9$	0	0
$10$	−1.37680	−0.435381
$11$	−4.60134	−1.38736	−0.693678	−	0.720285i	$-0.744011\pi$
$11$	−4.60134	−1.38736	−0.693678	+	0.720285i	$0.744011\pi$
$12$	0	0
$13$	−0.261360	−0.0724884	−0.0362442	−	0.999343i	$-0.511539\pi$
$13$	−0.261360	−0.0724884	−0.0362442	+	0.999343i	$0.511539\pi$
$14$	0.825012	0.220494
$15$	0	0
$16$	0.379411	0.0948527
$17$	−1.30909	−0.317500	−0.158750	−	0.987319i	$-0.550746\pi$
$17$	−1.30909	−0.317500	−0.158750	+	0.987319i	$0.550746\pi$
$18$	0	0
$19$	−5.83541	−1.33873	−0.669367	−	0.742932i	$-0.733435\pi$
$19$	−5.83541	−1.33873	−0.669367	+	0.742932i	$0.733435\pi$
$20$	−2.20177	−0.492330
$21$	0	0
$22$	3.79616	0.809344
$23$	1.54469	0.322091	0.161045	−	0.986947i	$-0.448513\pi$
$23$	1.54469	0.322091	0.161045	+	0.986947i	$0.448513\pi$
$24$	0	0
$25$	−2.21504	−0.443007
$26$	0.215625	0.0422876
$27$	0	0
$28$	1.31936	0.249335
$29$	−4.76454	−0.884753	−0.442377	−	0.896829i	$-0.645865\pi$
$29$	−4.76454	−0.884753	−0.442377	+	0.896829i	$0.645865\pi$
$30$	0	0
$31$	−0.627450	−0.112693	−0.0563467	−	0.998411i	$-0.517945\pi$
$31$	−0.627450	−0.112693	−0.0563467	+	0.998411i	$0.517945\pi$
$32$	−5.79003	−1.02354
$33$	0	0
$34$	1.08001	0.185221
$35$	−1.66882	−0.282082
$36$	0	0
$37$	6.11849	1.00587	0.502937	−	0.864323i	$-0.332253\pi$
$37$	6.11849	1.00587	0.502937	+	0.864323i	$0.332253\pi$
$38$	4.81428	0.780979
$39$	0	0
$40$	4.57008	0.722593
$41$	−11.5755	−1.80779	−0.903893	−	0.427758i	$-0.859304\pi$
$41$	−11.5755	−1.80779	−0.903893	+	0.427758i	$0.859304\pi$
$42$	0	0
$43$	8.48685	1.29423	0.647117	−	0.762391i	$-0.275975\pi$
$43$	8.48685	1.29423	0.647117	+	0.762391i	$0.275975\pi$
$44$	6.07080	0.915208
$45$	0	0
$46$	−1.27439	−0.187898
$47$	4.72063	0.688574	0.344287	−	0.938864i	$-0.388121\pi$
$47$	4.72063	0.688574	0.344287	+	0.938864i	$0.388121\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	1.82743	0.258438
$51$	0	0
$52$	0.344827	0.0478190
$53$	−13.5754	−1.86472	−0.932360	−	0.361531i	$-0.882254\pi$
$53$	−13.5754	−1.86472	−0.932360	+	0.361531i	$0.882254\pi$
$54$	0	0
$55$	−7.67881	−1.03541
$56$	−2.73851	−0.365948
$57$	0	0
$58$	3.93080	0.516140
$59$	2.92411	0.380687	0.190344	−	0.981718i	$-0.439040\pi$
$59$	2.92411	0.380687	0.190344	+	0.981718i	$0.439040\pi$
$60$	0	0
$61$	3.22354	0.412732	0.206366	−	0.978475i	$-0.433836\pi$
$61$	3.22354	0.412732	0.206366	+	0.978475i	$0.433836\pi$
$62$	0.517653	0.0657420
$63$	0	0
$64$	4.01802	0.502253
$65$	−0.436164	−0.0540995
$66$	0	0
$67$	−5.93511	−0.725089	−0.362544	−	0.931966i	$-0.618092\pi$
$67$	−5.93511	−0.725089	−0.362544	+	0.931966i	$0.618092\pi$
$68$	1.72715	0.209448
$69$	0	0
$70$	1.37680	0.164559
$71$	−0.659163	−0.0782282	−0.0391141	−	0.999235i	$-0.512454\pi$
$71$	−0.659163	−0.0782282	−0.0391141	+	0.999235i	$0.512454\pi$
$72$	0	0
$73$	−0.0147131	−0.00172204	−0.000861021	−	1.00000i	$-0.500274\pi$
$73$	−0.0147131	−0.00172204	−0.000861021		1.00000i	$0.500274\pi$
$74$	−5.04783	−0.586798
$75$	0	0
$76$	7.69898	0.883133
$77$	4.60134	0.524371
$78$	0	0
$79$	3.62188	0.407493	0.203746	−	0.979024i	$-0.434688\pi$
$79$	3.62188	0.407493	0.203746	+	0.979024i	$0.434688\pi$
$80$	0.633169	0.0707904
$81$	0	0
$82$	9.54991	1.05461
$83$	−6.07333	−0.666635	−0.333318	−	0.942815i	$-0.608168\pi$
$83$	−6.07333	−0.666635	−0.333318	+	0.942815i	$0.608168\pi$
$84$	0	0
$85$	−2.18463	−0.236957
$86$	−7.00175	−0.755019
$87$	0	0
$88$	−12.6008	−1.34325
$89$	0.737302	0.0781539	0.0390769	−	0.999236i	$-0.487558\pi$
$89$	0.737302	0.0781539	0.0390769	+	0.999236i	$0.487558\pi$
$90$	0	0
$91$	0.261360	0.0273980
$92$	−2.03800	−0.212476
$93$	0	0
$94$	−3.89457	−0.401695
$95$	−9.73825	−0.999123
$96$	0	0
$97$	6.68248	0.678504	0.339252	−	0.940696i	$-0.389826\pi$
$97$	6.68248	0.678504	0.339252	+	0.940696i	$0.389826\pi$
$98$	−0.825012	−0.0833388
$99$	0	0
$100$	2.92242	0.292242
$101$	13.6759	1.36080	0.680400	−	0.732841i	$-0.261806\pi$
$101$	13.6759	1.36080	0.680400	+	0.732841i	$0.261806\pi$
$102$	0	0
$103$	18.4629	1.81921	0.909603	−	0.415478i	$-0.136386\pi$
$103$	18.4629	1.81921	0.909603	+	0.415478i	$0.136386\pi$
$104$	−0.715738	−0.0701838
$105$	0	0
$106$	11.1998	1.08782
$107$	12.0876	1.16855	0.584277	−	0.811554i	$-0.301378\pi$
$107$	12.0876	1.16855	0.584277	+	0.811554i	$0.301378\pi$
$108$	0	0
$109$	17.7946	1.70441	0.852205	−	0.523208i	$-0.175265\pi$
$109$	17.7946	1.70441	0.852205	+	0.523208i	$0.175265\pi$
$110$	6.33511	0.604029
$111$	0	0
$112$	−0.379411	−0.0358510
$113$	−7.21643	−0.678864	−0.339432	−	0.940631i	$-0.610235\pi$
$113$	−7.21643	−0.678864	−0.339432	+	0.940631i	$0.610235\pi$
$114$	0	0
$115$	2.57781	0.240382
$116$	6.28612	0.583652
$117$	0	0
$118$	−2.41243	−0.222082
$119$	1.30909	0.120004
$120$	0	0
$121$	10.1723	0.924758
$122$	−2.65946	−0.240776
$123$	0	0
$124$	0.827829	0.0743413
$125$	−12.0406	−1.07694
$126$	0	0
$127$	−1.00000	−0.0887357
$127$	−1.00000	−0.0887357
$128$	8.26515	0.730543
$129$	0	0
$130$	0.359840	0.0315601
$131$	−11.4020	−0.996200	−0.498100	−	0.867120i	$-0.665969\pi$
$131$	−11.4020	−0.996200	−0.498100	+	0.867120i	$0.665969\pi$
$132$	0	0
$133$	5.83541	0.505994
$134$	4.89653	0.422996
$135$	0	0
$136$	−3.58495	−0.307407
$137$	5.14751	0.439782	0.219891	−	0.975524i	$-0.429430\pi$
$137$	5.14751	0.439782	0.219891	+	0.975524i	$0.429430\pi$
$138$	0	0
$139$	−10.1543	−0.861275	−0.430638	−	0.902525i	$-0.641711\pi$
$139$	−10.1543	−0.861275	−0.430638	+	0.902525i	$0.641711\pi$
$140$	2.20177	0.186083
$141$	0	0
$142$	0.543817	0.0456361
$143$	1.20261	0.100567
$144$	0	0
$145$	−7.95117	−0.660308
$146$	0.0121385	0.00100459
$147$	0	0
$148$	−8.07247	−0.663553
$149$	−20.6154	−1.68888	−0.844440	−	0.535651i	$-0.820066\pi$
$149$	−20.6154	−1.68888	−0.844440	+	0.535651i	$0.820066\pi$
$150$	0	0
$151$	14.3082	1.16438	0.582191	−	0.813052i	$-0.302196\pi$
$151$	14.3082	1.16438	0.582191	+	0.813052i	$0.302196\pi$
$152$	−15.9803	−1.29617
$153$	0	0
$154$	−3.79616	−0.305903
$155$	−1.04710	−0.0841052
$156$	0	0
$157$	2.17987	0.173972	0.0869862	−	0.996210i	$-0.472276\pi$
$157$	2.17987	0.173972	0.0869862	+	0.996210i	$0.472276\pi$
$158$	−2.98809	−0.237720
$159$	0	0
$160$	−9.66253	−0.763890
$161$	−1.54469	−0.121739
$162$	0	0
$163$	−9.93117	−0.777869	−0.388935	−	0.921265i	$-0.627157\pi$
$163$	−9.93117	−0.777869	−0.388935	+	0.921265i	$0.627157\pi$
$164$	15.2722	1.19256
$165$	0	0
$166$	5.01057	0.388896
$167$	4.00791	0.310141	0.155071	−	0.987903i	$-0.450439\pi$
$167$	4.00791	0.310141	0.155071	+	0.987903i	$0.450439\pi$
$168$	0	0
$169$	−12.9317	−0.994745
$170$	1.80235	0.138234
$171$	0	0
$172$	−11.1972	−0.853777
$173$	8.01508	0.609375	0.304688	−	0.952452i	$-0.401448\pi$
$173$	8.01508	0.609375	0.304688	+	0.952452i	$0.401448\pi$
$174$	0	0
$175$	2.21504	0.167441
$176$	−1.74580	−0.131595
$177$	0	0
$178$	−0.608283	−0.0455927
$179$	11.8618	0.886595	0.443297	−	0.896375i	$-0.353809\pi$
$179$	11.8618	0.886595	0.443297	+	0.896375i	$0.353809\pi$
$180$	0	0
$181$	−0.265509	−0.0197351	−0.00986756	−	0.999951i	$-0.503141\pi$
$181$	−0.265509	−0.0197351	−0.00986756	+	0.999951i	$0.503141\pi$
$182$	−0.215625	−0.0159832
$183$	0	0
$184$	4.23015	0.311851
$185$	10.2107	0.750703
$186$	0	0
$187$	6.02356	0.440486
$188$	−6.22819	−0.454237
$189$	0	0
$190$	8.03417	0.582860
$191$	−11.2881	−0.816780	−0.408390	−	0.912807i	$-0.633910\pi$
$191$	−11.2881	−0.816780	−0.408390	+	0.912807i	$0.633910\pi$
$192$	0	0
$193$	13.0967	0.942722	0.471361	−	0.881940i	$-0.343763\pi$
$193$	13.0967	0.942722	0.471361	+	0.881940i	$0.343763\pi$
$194$	−5.51313	−0.395820
$195$	0	0
$196$	−1.31936	−0.0942397
$197$	4.84097	0.344905	0.172453	−	0.985018i	$-0.444831\pi$
$197$	4.84097	0.344905	0.172453	+	0.985018i	$0.444831\pi$
$198$	0	0
$199$	−0.364592	−0.0258452	−0.0129226	−	0.999916i	$-0.504114\pi$
$199$	−0.364592	−0.0258452	−0.0129226	+	0.999916i	$0.504114\pi$
$200$	−6.06589	−0.428924
$201$	0	0
$202$	−11.2827	−0.793851
$203$	4.76454	0.334405
$204$	0	0
$205$	−19.3174	−1.34919
$206$	−15.2321	−1.06127
$207$	0	0
$208$	−0.0991630	−0.00687572
$209$	26.8507	1.85730
$210$	0	0
$211$	−5.23697	−0.360528	−0.180264	−	0.983618i	$-0.557695\pi$
$211$	−5.23697	−0.360528	−0.180264	+	0.983618i	$0.557695\pi$
$212$	17.9107	1.23011
$213$	0	0
$214$	−9.97243	−0.681701
$215$	14.1630	0.965911
$216$	0	0
$217$	0.627450	0.0425941
$218$	−14.6807	−0.994304
$219$	0	0
$220$	10.1311	0.683038
$221$	0.342144	0.0230151
$222$	0	0
$223$	26.8209	1.79606	0.898030	−	0.439934i	$-0.144998\pi$
$223$	26.8209	1.79606	0.898030	+	0.439934i	$0.144998\pi$
$224$	5.79003	0.386863
$225$	0	0
$226$	5.95364	0.396030
$227$	19.3813	1.28638	0.643192	−	0.765705i	$-0.277610\pi$
$227$	19.3813	1.28638	0.643192	+	0.765705i	$0.277610\pi$
$228$	0	0
$229$	−12.8206	−0.847208	−0.423604	−	0.905847i	$-0.639235\pi$
$229$	−12.8206	−0.847208	−0.423604	+	0.905847i	$0.639235\pi$
$230$	−2.12673	−0.140232
$231$	0	0
$232$	−13.0477	−0.856626
$233$	13.5367	0.886816	0.443408	−	0.896320i	$-0.353769\pi$
$233$	13.5367	0.886816	0.443408	+	0.896320i	$0.353769\pi$
$234$	0	0
$235$	7.87788	0.513896
$236$	−3.85795	−0.251131
$237$	0	0
$238$	−1.08001	−0.0700068
$239$	−26.9411	−1.74268	−0.871338	−	0.490683i	$-0.836747\pi$
$239$	−26.9411	−1.74268	−0.871338	+	0.490683i	$0.836747\pi$
$240$	0	0
$241$	8.15394	0.525242	0.262621	−	0.964899i	$-0.415413\pi$
$241$	8.15394	0.525242	0.262621	+	0.964899i	$0.415413\pi$
$242$	−8.39230	−0.539477
$243$	0	0
$244$	−4.25299	−0.272270
$245$	1.66882	0.106617
$246$	0	0
$247$	1.52514	0.0970427
$248$	−1.71828	−0.109111
$249$	0	0
$250$	9.93364	0.628259
$251$	15.7468	0.993931	0.496966	−	0.867770i	$-0.334448\pi$
$251$	15.7468	0.993931	0.496966	+	0.867770i	$0.334448\pi$
$252$	0	0
$253$	−7.10766	−0.446854
$254$	0.825012	0.0517658
$255$	0	0
$256$	−14.8549	−0.928431
$257$	14.7201	0.918213	0.459106	−	0.888381i	$-0.348170\pi$
$257$	14.7201	0.918213	0.459106	+	0.888381i	$0.348170\pi$
$258$	0	0
$259$	−6.11849	−0.380185
$260$	0.575455	0.0356882
$261$	0	0
$262$	9.40681	0.581155
$263$	−4.72618	−0.291429	−0.145714	−	0.989327i	$-0.546548\pi$
$263$	−4.72618	−0.291429	−0.145714	+	0.989327i	$0.546548\pi$
$264$	0	0
$265$	−22.6549	−1.39168
$266$	−4.81428	−0.295182
$267$	0	0
$268$	7.83052	0.478325
$269$	−22.5445	−1.37456	−0.687282	−	0.726391i	$-0.741196\pi$
$269$	−22.5445	−1.37456	−0.687282	+	0.726391i	$0.741196\pi$
$270$	0	0
$271$	−27.1598	−1.64984	−0.824921	−	0.565247i	$-0.808781\pi$
$271$	−27.1598	−1.64984	−0.824921	+	0.565247i	$0.808781\pi$
$272$	−0.496682	−0.0301158
$273$	0	0
$274$	−4.24676	−0.256556
$275$	10.1921	0.614609
$276$	0	0
$277$	−21.8823	−1.31478	−0.657389	−	0.753551i	$-0.728339\pi$
$277$	−21.8823	−1.31478	−0.657389	+	0.753551i	$0.728339\pi$
$278$	8.37741	0.502443
$279$	0	0
$280$	−4.57008	−0.273114
$281$	21.4596	1.28017	0.640087	−	0.768303i	$-0.278898\pi$
$281$	21.4596	1.28017	0.640087	+	0.768303i	$0.278898\pi$
$282$	0	0
$283$	13.9478	0.829110	0.414555	−	0.910024i	$-0.363937\pi$
$283$	13.9478	0.829110	0.414555	+	0.910024i	$0.363937\pi$
$284$	0.869670	0.0516054
$285$	0	0
$286$	−0.992166	−0.0586680
$287$	11.5755	0.683279
$288$	0	0
$289$	−15.2863	−0.899194
$290$	6.55981	0.385205
$291$	0	0
$292$	0.0194119	0.00113599
$293$	5.10858	0.298446	0.149223	−	0.988804i	$-0.452323\pi$
$293$	5.10858	0.298446	0.149223	+	0.988804i	$0.452323\pi$
$294$	0	0
$295$	4.87982	0.284114
$296$	16.7555	0.973896
$297$	0	0
$298$	17.0079	0.985244
$299$	−0.403722	−0.0233478
$300$	0	0
$301$	−8.48685	−0.489174
$302$	−11.8044	−0.679267
$303$	0	0
$304$	−2.21402	−0.126983
$305$	5.37951	0.308030
$306$	0	0
$307$	−0.707271	−0.0403661	−0.0201830	−	0.999796i	$-0.506425\pi$
$307$	−0.707271	−0.0403661	−0.0201830	+	0.999796i	$0.506425\pi$
$308$	−6.07080	−0.345916
$309$	0	0
$310$	0.863871	0.0490646
$311$	−29.2375	−1.65791	−0.828953	−	0.559318i	$-0.811063\pi$
$311$	−29.2375	−1.65791	−0.828953	+	0.559318i	$0.811063\pi$
$312$	0	0
$313$	12.6997	0.717828	0.358914	−	0.933371i	$-0.383147\pi$
$313$	12.6997	0.717828	0.358914	+	0.933371i	$0.383147\pi$
$314$	−1.79842	−0.101490
$315$	0	0
$316$	−4.77854	−0.268814
$317$	30.0297	1.68663	0.843317	−	0.537416i	$-0.180599\pi$
$317$	30.0297	1.68663	0.843317	+	0.537416i	$0.180599\pi$
$318$	0	0
$319$	21.9233	1.22747
$320$	6.70536	0.374841
$321$	0	0
$322$	1.27439	0.0710189
$323$	7.63906	0.425049
$324$	0	0
$325$	0.578923	0.0321129
$326$	8.19333	0.453787
$327$	0	0
$328$	−31.6995	−1.75031
$329$	−4.72063	−0.260257
$330$	0	0
$331$	−12.3497	−0.678801	−0.339400	−	0.940642i	$-0.610224\pi$
$331$	−12.3497	−0.678801	−0.339400	+	0.940642i	$0.610224\pi$
$332$	8.01289	0.439764
$333$	0	0
$334$	−3.30657	−0.180928
$335$	−9.90463	−0.541148
$336$	0	0
$337$	17.2242	0.938261	0.469130	−	0.883129i	$-0.344567\pi$
$337$	17.2242	0.938261	0.469130	+	0.883129i	$0.344567\pi$
$338$	10.6688	0.580306
$339$	0	0
$340$	2.88231	0.156315
$341$	2.88711	0.156346
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	23.2413	1.25309
$345$	0	0
$346$	−6.61253	−0.355492
$347$	−8.56790	−0.459949	−0.229975	−	0.973197i	$-0.573864\pi$
$347$	−8.56790	−0.459949	−0.229975	+	0.973197i	$0.573864\pi$
$348$	0	0
$349$	−24.7735	−1.32609	−0.663047	−	0.748578i	$-0.730737\pi$
$349$	−24.7735	−1.32609	−0.663047	+	0.748578i	$0.730737\pi$
$350$	−1.82743	−0.0976803
$351$	0	0
$352$	26.6419	1.42002
$353$	−8.61224	−0.458383	−0.229192	−	0.973381i	$-0.573608\pi$
$353$	−8.61224	−0.458383	−0.229192	+	0.973381i	$0.573608\pi$
$354$	0	0
$355$	−1.10002	−0.0583832
$356$	−0.972764	−0.0515564
$357$	0	0
$358$	−9.78615	−0.517214
$359$	35.5800	1.87784	0.938919	−	0.344137i	$-0.111828\pi$
$359$	35.5800	1.87784	0.938919	+	0.344137i	$0.111828\pi$
$360$	0	0
$361$	15.0520	0.792210
$362$	0.219048	0.0115129
$363$	0	0
$364$	−0.344827	−0.0180739
$365$	−0.0245536	−0.00128519
$366$	0	0
$367$	2.98841	0.155994	0.0779968	−	0.996954i	$-0.475148\pi$
$367$	2.98841	0.155994	0.0779968	+	0.996954i	$0.475148\pi$
$368$	0.586073	0.0305512
$369$	0	0
$370$	−8.42392	−0.437939
$371$	13.5754	0.704798
$372$	0	0
$373$	19.9109	1.03095	0.515474	−	0.856905i	$-0.327616\pi$
$373$	19.9109	1.03095	0.515474	+	0.856905i	$0.327616\pi$
$374$	−4.96950	−0.256967
$375$	0	0
$376$	12.9275	0.666684
$377$	1.24526	0.0641343
$378$	0	0
$379$	11.2608	0.578427	0.289213	−	0.957265i	$-0.406606\pi$
$379$	11.2608	0.578427	0.289213	+	0.957265i	$0.406606\pi$
$380$	12.8482	0.659100
$381$	0	0
$382$	9.31284	0.476486
$383$	−3.80104	−0.194224	−0.0971121	−	0.995273i	$-0.530961\pi$
$383$	−3.80104	−0.194224	−0.0971121	+	0.995273i	$0.530961\pi$
$384$	0	0
$385$	7.67881	0.391349
$386$	−10.8049	−0.549957
$387$	0	0
$388$	−8.81657	−0.447594
$389$	−6.73824	−0.341642	−0.170821	−	0.985302i	$-0.554642\pi$
$389$	−6.73824	−0.341642	−0.170821	+	0.985302i	$0.554642\pi$
$390$	0	0
$391$	−2.02214	−0.102264
$392$	2.73851	0.138316
$393$	0	0
$394$	−3.99386	−0.201208
$395$	6.04426	0.304120
$396$	0	0
$397$	17.2347	0.864983	0.432491	−	0.901638i	$-0.357635\pi$
$397$	17.2347	0.864983	0.432491	+	0.901638i	$0.357635\pi$
$398$	0.300792	0.0150774
$399$	0	0
$400$	−0.840409	−0.0420205
$401$	4.91867	0.245627	0.122813	−	0.992430i	$-0.460808\pi$
$401$	4.91867	0.245627	0.122813	+	0.992430i	$0.460808\pi$
$402$	0	0
$403$	0.163991	0.00816895
$404$	−18.0433	−0.897689
$405$	0	0
$406$	−3.93080	−0.195082
$407$	−28.1533	−1.39551
$408$	0	0
$409$	−31.4953	−1.55734	−0.778671	−	0.627432i	$-0.784106\pi$
$409$	−31.4953	−1.55734	−0.778671	+	0.627432i	$0.784106\pi$
$410$	15.9371	0.787077
$411$	0	0
$412$	−24.3592	−1.20009
$413$	−2.92411	−0.143886
$414$	0	0
$415$	−10.1353	−0.497523
$416$	1.51329	0.0741949
$417$	0	0
$418$	−22.1521	−1.08350
$419$	6.88895	0.336547	0.168274	−	0.985740i	$-0.446181\pi$
$419$	6.88895	0.336547	0.168274	+	0.985740i	$0.446181\pi$
$420$	0	0
$421$	15.0039	0.731248	0.365624	−	0.930763i	$-0.380856\pi$
$421$	15.0039	0.731248	0.365624	+	0.930763i	$0.380856\pi$
$422$	4.32056	0.210322
$423$	0	0
$424$	−37.1762	−1.80544
$425$	2.89968	0.140655
$426$	0	0
$427$	−3.22354	−0.155998
$428$	−15.9479	−0.770869
$429$	0	0
$430$	−11.6847	−0.563485
$431$	−5.48485	−0.264196	−0.132098	−	0.991237i	$-0.542171\pi$
$431$	−5.48485	−0.264196	−0.132098	+	0.991237i	$0.542171\pi$
$432$	0	0
$433$	5.45047	0.261933	0.130967	−	0.991387i	$-0.458192\pi$
$433$	5.45047	0.261933	0.130967	+	0.991387i	$0.458192\pi$
$434$	−0.517653	−0.0248482
$435$	0	0
$436$	−23.4774	−1.12436
$437$	−9.01391	−0.431194
$438$	0	0
$439$	−24.0806	−1.14931	−0.574653	−	0.818397i	$-0.694863\pi$
$439$	−24.0806	−1.14931	−0.574653	+	0.818397i	$0.694863\pi$
$440$	−21.0285	−1.00249
$441$	0	0
$442$	−0.282273	−0.0134263
$443$	39.4432	1.87400	0.937001	−	0.349327i	$-0.113590\pi$
$443$	39.4432	1.87400	0.937001	+	0.349327i	$0.113590\pi$
$444$	0	0
$445$	1.23043	0.0583278
$446$	−22.1276	−1.04777
$447$	0	0
$448$	−4.01802	−0.189834
$449$	−24.2900	−1.14632	−0.573159	−	0.819444i	$-0.694282\pi$
$449$	−24.2900	−1.14632	−0.573159	+	0.819444i	$0.694282\pi$
$450$	0	0
$451$	53.2627	2.50804
$452$	9.52103	0.447832
$453$	0	0
$454$	−15.9898	−0.750439
$455$	0.436164	0.0204477
$456$	0	0
$457$	27.3742	1.28051	0.640255	−	0.768162i	$-0.278828\pi$
$457$	27.3742	1.28051	0.640255	+	0.768162i	$0.278828\pi$
$458$	10.5771	0.494237
$459$	0	0
$460$	−3.40105	−0.158575
$461$	21.4437	0.998732	0.499366	−	0.866391i	$-0.333566\pi$
$461$	21.4437	0.998732	0.499366	+	0.866391i	$0.333566\pi$
$462$	0	0
$463$	25.2351	1.17278	0.586388	−	0.810031i	$-0.300550\pi$
$463$	25.2351	1.17278	0.586388	+	0.810031i	$0.300550\pi$
$464$	−1.80772	−0.0839212
$465$	0	0
$466$	−11.1679	−0.517343
$467$	25.2535	1.16859	0.584296	−	0.811540i	$-0.301371\pi$
$467$	25.2535	1.16859	0.584296	+	0.811540i	$0.301371\pi$
$468$	0	0
$469$	5.93511	0.274058
$470$	−6.49934	−0.299792
$471$	0	0
$472$	8.00771	0.368585
$473$	−39.0509	−1.79556
$474$	0	0
$475$	12.9256	0.593069
$476$	−1.72715	−0.0791639
$477$	0	0
$478$	22.2267	1.01663
$479$	23.7391	1.08467	0.542334	−	0.840163i	$-0.317541\pi$
$479$	23.7391	1.08467	0.542334	+	0.840163i	$0.317541\pi$
$480$	0	0
$481$	−1.59913	−0.0729142
$482$	−6.72710	−0.306411
$483$	0	0
$484$	−13.4209	−0.610042
$485$	11.1519	0.506380
$486$	0	0
$487$	17.2184	0.780238	0.390119	−	0.920764i	$-0.372434\pi$
$487$	17.2184	0.780238	0.390119	+	0.920764i	$0.372434\pi$
$488$	8.82768	0.399611
$489$	0	0
$490$	−1.37680	−0.0621973
$491$	−0.879006	−0.0396690	−0.0198345	−	0.999803i	$-0.506314\pi$
$491$	−0.879006	−0.0396690	−0.0198345	+	0.999803i	$0.506314\pi$
$492$	0	0
$493$	6.23720	0.280909
$494$	−1.25826	−0.0566119
$495$	0	0
$496$	−0.238061	−0.0106893
$497$	0.659163	0.0295675
$498$	0	0
$499$	−7.84992	−0.351411	−0.175705	−	0.984443i	$-0.556221\pi$
$499$	−7.84992	−0.351411	−0.175705	+	0.984443i	$0.556221\pi$
$500$	15.8858	0.710436
$501$	0	0
$502$	−12.9913	−0.579831
$503$	32.2673	1.43873	0.719363	−	0.694635i	$-0.244434\pi$
$503$	32.2673	1.43873	0.719363	+	0.694635i	$0.244434\pi$
$504$	0	0
$505$	22.8226	1.01559
$506$	5.86390	0.260682
$507$	0	0
$508$	1.31936	0.0585369
$509$	−29.0330	−1.28687	−0.643433	−	0.765502i	$-0.722491\pi$
$509$	−29.0330	−1.28687	−0.643433	+	0.765502i	$0.722491\pi$
$510$	0	0
$511$	0.0147131	0.000650871 0
$512$	−4.27484	−0.188923
$513$	0	0
$514$	−12.1442	−0.535659
$515$	30.8113	1.35771
$516$	0	0
$517$	−21.7212	−0.955298
$518$	5.04783	0.221789
$519$	0	0
$520$	−1.19444	−0.0523796
$521$	−4.96697	−0.217607	−0.108804	−	0.994063i	$-0.534702\pi$
$521$	−4.96697	−0.217607	−0.108804	+	0.994063i	$0.534702\pi$
$522$	0	0
$523$	4.89328	0.213968	0.106984	−	0.994261i	$-0.465881\pi$
$523$	4.89328	0.213968	0.106984	+	0.994261i	$0.465881\pi$
$524$	15.0433	0.657171
$525$	0	0
$526$	3.89915	0.170011
$527$	0.821387	0.0357802
$528$	0	0
$529$	−20.6139	−0.896258
$530$	18.6905	0.811864
$531$	0	0
$532$	−7.69898	−0.333793
$533$	3.02537	0.131043
$534$	0	0
$535$	20.1721	0.872115
$536$	−16.2533	−0.702037
$537$	0	0
$538$	18.5995	0.801881
$539$	−4.60134	−0.198194
$540$	0	0
$541$	31.5095	1.35470	0.677350	−	0.735661i	$-0.263128\pi$
$541$	31.5095	1.35470	0.677350	+	0.735661i	$0.263128\pi$
$542$	22.4072	0.962471
$543$	0	0
$544$	7.57966	0.324975
$545$	29.6959	1.27203
$546$	0	0
$547$	9.94998	0.425430	0.212715	−	0.977114i	$-0.431769\pi$
$547$	9.94998	0.425430	0.212715	+	0.977114i	$0.431769\pi$
$548$	−6.79140	−0.290114
$549$	0	0
$550$	−8.40863	−0.358545
$551$	27.8030	1.18445
$552$	0	0
$553$	−3.62188	−0.154018
$554$	18.0531	0.767004
$555$	0	0
$556$	13.3971	0.568164
$557$	−33.3031	−1.41110	−0.705550	−	0.708660i	$-0.749300\pi$
$557$	−33.3031	−1.41110	−0.705550	+	0.708660i	$0.749300\pi$
$558$	0	0
$559$	−2.21813	−0.0938168
$560$	−0.633169	−0.0267563
$561$	0	0
$562$	−17.7044	−0.746816
$563$	28.4895	1.20069	0.600345	−	0.799741i	$-0.295030\pi$
$563$	28.4895	1.20069	0.600345	+	0.799741i	$0.295030\pi$
$564$	0	0
$565$	−12.0429	−0.506650
$566$	−11.5071	−0.483679
$567$	0	0
$568$	−1.80512	−0.0757412
$569$	27.9887	1.17335	0.586675	−	0.809823i	$-0.300437\pi$
$569$	27.9887	1.17335	0.586675	+	0.809823i	$0.300437\pi$
$570$	0	0
$571$	34.3765	1.43861	0.719307	−	0.694692i	$-0.244460\pi$
$571$	34.3765	1.43861	0.719307	+	0.694692i	$0.244460\pi$
$572$	−1.58667	−0.0663419
$573$	0	0
$574$	−9.54991	−0.398606
$575$	−3.42155	−0.142689
$576$	0	0
$577$	17.5721	0.731538	0.365769	−	0.930706i	$-0.380806\pi$
$577$	17.5721	0.731538	0.365769	+	0.930706i	$0.380806\pi$
$578$	12.6114	0.524564
$579$	0	0
$580$	10.4904	0.435591
$581$	6.07333	0.251964
$582$	0	0
$583$	62.4649	2.58703
$584$	−0.0402920	−0.00166730
$585$	0	0
$586$	−4.21464	−0.174105
$587$	−25.7237	−1.06173	−0.530866	−	0.847456i	$-0.678133\pi$
$587$	−25.7237	−1.06173	−0.530866	+	0.847456i	$0.678133\pi$
$588$	0	0
$589$	3.66143	0.150866
$590$	−4.02591	−0.165744
$591$	0	0
$592$	2.32142	0.0954099
$593$	18.8109	0.772470	0.386235	−	0.922400i	$-0.373775\pi$
$593$	18.8109	0.772470	0.386235	+	0.922400i	$0.373775\pi$
$594$	0	0
$595$	2.18463	0.0895612
$596$	27.1990	1.11412
$597$	0	0
$598$	0.333075	0.0136204
$599$	8.88720	0.363121	0.181561	−	0.983380i	$-0.441885\pi$
$599$	8.88720	0.363121	0.181561	+	0.983380i	$0.441885\pi$
$600$	0	0
$601$	−26.9014	−1.09733	−0.548666	−	0.836042i	$-0.684864\pi$
$601$	−26.9014	−1.09733	−0.548666	+	0.836042i	$0.684864\pi$
$602$	7.00175	0.285370
$603$	0	0
$604$	−18.8776	−0.768117
$605$	16.9758	0.690165
$606$	0	0
$607$	−4.40142	−0.178648	−0.0893241	−	0.996003i	$-0.528471\pi$
$607$	−4.40142	−0.178648	−0.0893241	+	0.996003i	$0.528471\pi$
$608$	33.7872	1.37025
$609$	0	0
$610$	−4.43816	−0.179696
$611$	−1.23379	−0.0499136
$612$	0	0
$613$	11.1060	0.448569	0.224284	−	0.974524i	$-0.427996\pi$
$613$	11.1060	0.448569	0.224284	+	0.974524i	$0.427996\pi$
$614$	0.583507	0.0235484
$615$	0	0
$616$	12.6008	0.507701
$617$	−33.5372	−1.35016	−0.675078	−	0.737747i	$-0.735890\pi$
$617$	−33.5372	−1.35016	−0.675078	+	0.737747i	$0.735890\pi$
$618$	0	0
$619$	36.3793	1.46221	0.731105	−	0.682265i	$-0.239005\pi$
$619$	36.3793	1.46221	0.731105	+	0.682265i	$0.239005\pi$
$620$	1.38150	0.0554823
$621$	0	0
$622$	24.1213	0.967175
$623$	−0.737302	−0.0295394
$624$	0	0
$625$	−9.01843	−0.360737
$626$	−10.4774	−0.418760
$627$	0	0
$628$	−2.87602	−0.114766
$629$	−8.00964	−0.319365
$630$	0	0
$631$	7.90960	0.314876	0.157438	−	0.987529i	$-0.449677\pi$
$631$	7.90960	0.314876	0.157438	+	0.987529i	$0.449677\pi$
$632$	9.91853	0.394538
$633$	0	0
$634$	−24.7748	−0.983934
$635$	−1.66882	−0.0662251
$636$	0	0
$637$	−0.261360	−0.0103555
$638$	−18.0870	−0.716070
$639$	0	0
$640$	13.7931	0.545218
$641$	−35.5663	−1.40478	−0.702392	−	0.711790i	$-0.747885\pi$
$641$	−35.5663	−1.40478	−0.702392	+	0.711790i	$0.747885\pi$
$642$	0	0
$643$	23.7154	0.935245	0.467622	−	0.883928i	$-0.345111\pi$
$643$	23.7154	0.935245	0.467622	+	0.883928i	$0.345111\pi$
$644$	2.03800	0.0803084
$645$	0	0
$646$	−6.30231	−0.247961
$647$	−14.2349	−0.559632	−0.279816	−	0.960054i	$-0.590273\pi$
$647$	−14.2349	−0.559632	−0.279816	+	0.960054i	$0.590273\pi$
$648$	0	0
$649$	−13.4548	−0.528149
$650$	−0.477618	−0.0187337
$651$	0	0
$652$	13.1027	0.513143
$653$	−6.48456	−0.253760	−0.126880	−	0.991918i	$-0.540496\pi$
$653$	−6.48456	−0.253760	−0.126880	+	0.991918i	$0.540496\pi$
$654$	0	0
$655$	−19.0279	−0.743484
$656$	−4.39186	−0.171474
$657$	0	0
$658$	3.89457	0.151826
$659$	33.4585	1.30336	0.651679	−	0.758495i	$-0.274065\pi$
$659$	33.4585	1.30336	0.651679	+	0.758495i	$0.274065\pi$
$660$	0	0
$661$	−24.9902	−0.972007	−0.486004	−	0.873957i	$-0.661546\pi$
$661$	−24.9902	−0.972007	−0.486004	+	0.873957i	$0.661546\pi$
$662$	10.1886	0.395993
$663$	0	0
$664$	−16.6319	−0.645442
$665$	9.73825	0.377633
$666$	0	0
$667$	−7.35975	−0.284971
$668$	−5.28786	−0.204593
$669$	0	0
$670$	8.17144	0.315690
$671$	−14.8326	−0.572606
$672$	0	0
$673$	21.5025	0.828859	0.414429	−	0.910081i	$-0.363981\pi$
$673$	21.5025	0.828859	0.414429	+	0.910081i	$0.363981\pi$
$674$	−14.2101	−0.547354
$675$	0	0
$676$	17.0615	0.656212
$677$	−39.9889	−1.53690	−0.768449	−	0.639910i	$-0.778971\pi$
$677$	−39.9889	−1.53690	−0.768449	+	0.639910i	$0.778971\pi$
$678$	0	0
$679$	−6.68248	−0.256450
$680$	−5.98263	−0.229423
$681$	0	0
$682$	−2.38190	−0.0912077
$683$	−18.5347	−0.709209	−0.354605	−	0.935016i	$-0.615385\pi$
$683$	−18.5347	−0.709209	−0.354605	+	0.935016i	$0.615385\pi$
$684$	0	0
$685$	8.59028	0.328218
$686$	0.825012	0.0314991
$687$	0	0
$688$	3.22000	0.122762
$689$	3.54806	0.135171
$690$	0	0
$691$	28.9570	1.10158	0.550788	−	0.834645i	$-0.314327\pi$
$691$	28.9570	1.10158	0.550788	+	0.834645i	$0.314327\pi$
$692$	−10.5747	−0.401991
$693$	0	0
$694$	7.06862	0.268321
$695$	−16.9457	−0.642786
$696$	0	0
$697$	15.1533	0.573973
$698$	20.4384	0.773605
$699$	0	0
$700$	−2.92242	−0.110457
$701$	17.4162	0.657800	0.328900	−	0.944365i	$-0.393322\pi$
$701$	17.4162	0.657800	0.328900	+	0.944365i	$0.393322\pi$
$702$	0	0
$703$	−35.7039	−1.34660
$704$	−18.4883	−0.696804
$705$	0	0
$706$	7.10520	0.267408
$707$	−13.6759	−0.514334
$708$	0	0
$709$	−6.05329	−0.227336	−0.113668	−	0.993519i	$-0.536260\pi$
$709$	−6.05329	−0.227336	−0.113668	+	0.993519i	$0.536260\pi$
$710$	0.907533	0.0340591
$711$	0	0
$712$	2.01911	0.0756693
$713$	−0.969217	−0.0362975
$714$	0	0
$715$	2.00694	0.0750552
$716$	−15.6500	−0.584867
$717$	0	0
$718$	−29.3539	−1.09548
$719$	−1.07573	−0.0401178	−0.0200589	−	0.999799i	$-0.506385\pi$
$719$	−1.07573	−0.0401178	−0.0200589	+	0.999799i	$0.506385\pi$
$720$	0	0
$721$	−18.4629	−0.687595
$722$	−12.4181	−0.462152
$723$	0	0
$724$	0.350301	0.0130188
$725$	10.5536	0.391952
$726$	0	0
$727$	12.6441	0.468944	0.234472	−	0.972123i	$-0.424664\pi$
$727$	12.6441	0.468944	0.234472	+	0.972123i	$0.424664\pi$
$728$	0.715738	0.0265270
$729$	0	0
$730$	0.0202570	0.000749745 0
$731$	−11.1100	−0.410919
$732$	0	0
$733$	−11.6888	−0.431735	−0.215867	−	0.976423i	$-0.569258\pi$
$733$	−11.6888	−0.431735	−0.215867	+	0.976423i	$0.569258\pi$
$734$	−2.46547	−0.0910022
$735$	0	0
$736$	−8.94382	−0.329674
$737$	27.3095	1.00596
$738$	0	0
$739$	35.2377	1.29624	0.648120	−	0.761538i	$-0.275555\pi$
$739$	35.2377	1.29624	0.648120	+	0.761538i	$0.275555\pi$
$740$	−13.4715	−0.495222
$741$	0	0
$742$	−11.1998	−0.411159
$743$	−12.8123	−0.470039	−0.235019	−	0.971991i	$-0.575515\pi$
$743$	−12.8123	−0.470039	−0.235019	+	0.971991i	$0.575515\pi$
$744$	0	0
$745$	−34.4034	−1.26044
$746$	−16.4267	−0.601426
$747$	0	0
$748$	−7.94721	−0.290579
$749$	−12.0876	−0.441672
$750$	0	0
$751$	42.8878	1.56500	0.782500	−	0.622651i	$-0.213944\pi$
$751$	42.8878	1.56500	0.782500	+	0.622651i	$0.213944\pi$
$752$	1.79106	0.0653132
$753$	0	0
$754$	−1.02736	−0.0374141
$755$	23.8778	0.869001
$756$	0	0
$757$	14.0589	0.510981	0.255490	−	0.966812i	$-0.417763\pi$
$757$	14.0589	0.510981	0.255490	+	0.966812i	$0.417763\pi$
$758$	−9.29026	−0.337438
$759$	0	0
$760$	−26.6683	−0.967360
$761$	39.2406	1.42247	0.711236	−	0.702954i	$-0.248136\pi$
$761$	39.2406	1.42247	0.711236	+	0.702954i	$0.248136\pi$
$762$	0	0
$763$	−17.7946	−0.644206
$764$	14.8931	0.538812
$765$	0	0
$766$	3.13590	0.113305
$767$	−0.764248	−0.0275954
$768$	0	0
$769$	21.7458	0.784173	0.392086	−	0.919928i	$-0.371753\pi$
$769$	21.7458	0.784173	0.392086	+	0.919928i	$0.371753\pi$
$770$	−6.33511	−0.228302
$771$	0	0
$772$	−17.2792	−0.621893
$773$	28.9195	1.04016	0.520081	−	0.854117i	$-0.325902\pi$
$773$	28.9195	1.04016	0.520081	+	0.854117i	$0.325902\pi$
$774$	0	0
$775$	1.38982	0.0499240
$776$	18.3000	0.656933
$777$	0	0
$778$	5.55913	0.199304
$779$	67.5477	2.42015
$780$	0	0
$781$	3.03303	0.108530
$782$	1.66829	0.0596578
$783$	0	0
$784$	0.379411	0.0135504
$785$	3.63781	0.129839
$786$	0	0
$787$	31.9684	1.13955	0.569775	−	0.821800i	$-0.307030\pi$
$787$	31.9684	1.13955	0.569775	+	0.821800i	$0.307030\pi$
$788$	−6.38696	−0.227526
$789$	0	0
$790$	−4.98659	−0.177415
$791$	7.21643	0.256587
$792$	0	0
$793$	−0.842506	−0.0299183
$794$	−14.2188	−0.504606
$795$	0	0
$796$	0.481026	0.0170495
$797$	0.776415	0.0275020	0.0137510	−	0.999905i	$-0.495623\pi$
$797$	0.776415	0.0275020	0.0137510	+	0.999905i	$0.495623\pi$
$798$	0	0
$799$	−6.17971	−0.218623
$800$	12.8251	0.453437
$801$	0	0
$802$	−4.05796	−0.143292
$803$	0.0677001	0.00238909
$804$	0	0
$805$	−2.57781	−0.0908560
$806$	−0.135294	−0.00476553
$807$	0	0
$808$	37.4514	1.31754
$809$	2.48477	0.0873600	0.0436800	−	0.999046i	$-0.486092\pi$
$809$	2.48477	0.0873600	0.0436800	+	0.999046i	$0.486092\pi$
$810$	0	0
$811$	5.07685	0.178272	0.0891362	−	0.996019i	$-0.471589\pi$
$811$	5.07685	0.178272	0.0891362	+	0.996019i	$0.471589\pi$
$812$	−6.28612	−0.220600
$813$	0	0
$814$	23.2268	0.814098
$815$	−16.5733	−0.580539
$816$	0	0
$817$	−49.5243	−1.73263
$818$	25.9840	0.908509
$819$	0	0
$820$	25.4865	0.890028
$821$	−35.7778	−1.24865	−0.624326	−	0.781164i	$-0.714626\pi$
$821$	−35.7778	−1.24865	−0.624326	+	0.781164i	$0.714626\pi$
$822$	0	0
$823$	0.319264	0.0111289	0.00556443	−	0.999985i	$-0.498229\pi$
$823$	0.319264	0.0111289	0.00556443	+	0.999985i	$0.498229\pi$
$824$	50.5609	1.76137
$825$	0	0
$826$	2.41243	0.0839391
$827$	−19.9032	−0.692102	−0.346051	−	0.938216i	$-0.612478\pi$
$827$	−19.9032	−0.692102	−0.346051	+	0.938216i	$0.612478\pi$
$828$	0	0
$829$	24.1189	0.837683	0.418842	−	0.908059i	$-0.362436\pi$
$829$	24.1189	0.837683	0.418842	+	0.908059i	$0.362436\pi$
$830$	8.36175	0.290241
$831$	0	0
$832$	−1.05015	−0.0364075
$833$	−1.30909	−0.0453572
$834$	0	0
$835$	6.68848	0.231464
$836$	−35.4256	−1.22522
$837$	0	0
$838$	−5.68346	−0.196332
$839$	50.7155	1.75090	0.875448	−	0.483313i	$-0.160567\pi$
$839$	50.7155	1.75090	0.875448	+	0.483313i	$0.160567\pi$
$840$	0	0
$841$	−6.29915	−0.217212
$842$	−12.3784	−0.426589
$843$	0	0
$844$	6.90943	0.237832
$845$	−21.5807	−0.742398
$846$	0	0
$847$	−10.1723	−0.349526
$848$	−5.15064	−0.176874
$849$	0	0
$850$	−2.39227	−0.0820541
$851$	9.45119	0.323983
$852$	0	0
$853$	−43.6913	−1.49596	−0.747980	−	0.663721i	$-0.768976\pi$
$853$	−43.6913	−1.49596	−0.747980	+	0.663721i	$0.768976\pi$
$854$	2.65946	0.0910048
$855$	0	0
$856$	33.1020	1.13140
$857$	−2.91619	−0.0996152	−0.0498076	−	0.998759i	$-0.515861\pi$
$857$	−2.91619	−0.0996152	−0.0498076	+	0.998759i	$0.515861\pi$
$858$	0	0
$859$	−57.6247	−1.96613	−0.983065	−	0.183256i	$-0.941336\pi$
$859$	−57.6247	−1.96613	−0.983065	+	0.183256i	$0.941336\pi$
$860$	−18.6861	−0.637190
$861$	0	0
$862$	4.52506	0.154124
$863$	−8.90463	−0.303117	−0.151559	−	0.988448i	$-0.548429\pi$
$863$	−8.90463	−0.303117	−0.151559	+	0.988448i	$0.548429\pi$
$864$	0	0
$865$	13.3757	0.454789
$866$	−4.49671	−0.152804
$867$	0	0
$868$	−0.827829	−0.0280984
$869$	−16.6655	−0.565338
$870$	0	0
$871$	1.55120	0.0525605
$872$	48.7305	1.65022
$873$	0	0
$874$	7.43658	0.251546
$875$	12.0406	0.407047
$876$	0	0
$877$	33.8901	1.14439	0.572194	−	0.820118i	$-0.306092\pi$
$877$	33.8901	1.14439	0.572194	+	0.820118i	$0.306092\pi$
$878$	19.8668	0.670472
$879$	0	0
$880$	−2.91343	−0.0982116
$881$	−32.1116	−1.08187	−0.540934	−	0.841065i	$-0.681929\pi$
$881$	−32.1116	−1.08187	−0.540934	+	0.841065i	$0.681929\pi$
$882$	0	0
$883$	−11.1539	−0.375360	−0.187680	−	0.982230i	$-0.560097\pi$
$883$	−11.1539	−0.375360	−0.187680	+	0.982230i	$0.560097\pi$
$884$	−0.451409	−0.0151825
$885$	0	0
$886$	−32.5411	−1.09324
$887$	50.5917	1.69870	0.849351	−	0.527829i	$-0.176994\pi$
$887$	50.5917	1.69870	0.849351	+	0.527829i	$0.176994\pi$
$888$	0	0
$889$	1.00000	0.0335389
$890$	−1.01512	−0.0340267
$891$	0	0
$892$	−35.3863	−1.18482
$893$	−27.5468	−0.921818
$894$	0	0
$895$	19.7953	0.661683
$896$	−8.26515	−0.276119
$897$	0	0
$898$	20.0396	0.668729
$899$	2.98951	0.0997058
$900$	0	0
$901$	17.7713	0.592049
$902$	−43.9424	−1.46312
$903$	0	0
$904$	−19.7622	−0.657282
$905$	−0.443087	−0.0147287
$906$	0	0
$907$	−33.2740	−1.10485	−0.552423	−	0.833564i	$-0.686296\pi$
$907$	−33.2740	−1.10485	−0.552423	+	0.833564i	$0.686296\pi$
$908$	−25.5709	−0.848599
$909$	0	0
$910$	−0.359840	−0.0119286
$911$	19.4515	0.644459	0.322229	−	0.946662i	$-0.395568\pi$
$911$	19.4515	0.644459	0.322229	+	0.946662i	$0.395568\pi$
$912$	0	0
$913$	27.9455	0.924860
$914$	−22.5840	−0.747013
$915$	0	0
$916$	16.9149	0.558884
$917$	11.4020	0.376528
$918$	0	0
$919$	56.0874	1.85015	0.925076	−	0.379783i	$-0.124001\pi$
$919$	56.0874	1.85015	0.925076	+	0.379783i	$0.124001\pi$
$920$	7.05936	0.232740
$921$	0	0
$922$	−17.6913	−0.582632
$923$	0.172279	0.00567063
$924$	0	0
$925$	−13.5527	−0.445610
$926$	−20.8193	−0.684164
$927$	0	0
$928$	27.5869	0.905583
$929$	−33.4550	−1.09762	−0.548811	−	0.835947i	$-0.684919\pi$
$929$	−33.4550	−1.09762	−0.548811	+	0.835947i	$0.684919\pi$
$930$	0	0
$931$	−5.83541	−0.191248
$932$	−17.8597	−0.585013
$933$	0	0
$934$	−20.8344	−0.681723
$935$	10.0522	0.328743
$936$	0	0
$937$	−32.0756	−1.04786	−0.523932	−	0.851760i	$-0.675535\pi$
$937$	−32.0756	−1.04786	−0.523932	+	0.851760i	$0.675535\pi$
$938$	−4.89653	−0.159877
$939$	0	0
$940$	−10.3937	−0.339006
$941$	24.2661	0.791054	0.395527	−	0.918454i	$-0.370562\pi$
$941$	24.2661	0.791054	0.395527	+	0.918454i	$0.370562\pi$
$942$	0	0
$943$	−17.8806	−0.582271
$944$	1.10944	0.0361092
$945$	0	0
$946$	32.2175	1.04748
$947$	−35.8418	−1.16470	−0.582350	−	0.812938i	$-0.697867\pi$
$947$	−35.8418	−1.16470	−0.582350	+	0.812938i	$0.697867\pi$
$948$	0	0
$949$	0.00384543	0.000124828 0
$950$	−10.6638	−0.345980
$951$	0	0
$952$	3.58495	0.116189
$953$	44.1257	1.42937	0.714685	−	0.699446i	$-0.246570\pi$
$953$	44.1257	1.42937	0.714685	+	0.699446i	$0.246570\pi$
$954$	0	0
$955$	−18.8379	−0.609579
$956$	35.5449	1.14961
$957$	0	0
$958$	−19.5851	−0.632764
$959$	−5.14751	−0.166222
$960$	0	0
$961$	−30.6063	−0.987300
$962$	1.31930	0.0425360
$963$	0	0
$964$	−10.7579	−0.346490
$965$	21.8561	0.703572
$966$	0	0
$967$	1.50881	0.0485201	0.0242601	−	0.999706i	$-0.492277\pi$
$967$	1.50881	0.0485201	0.0242601	+	0.999706i	$0.492277\pi$
$968$	27.8570	0.895358
$969$	0	0
$970$	−9.20042	−0.295408
$971$	3.85131	0.123594	0.0617972	−	0.998089i	$-0.480317\pi$
$971$	3.85131	0.123594	0.0617972	+	0.998089i	$0.480317\pi$
$972$	0	0
$973$	10.1543	0.325531
$974$	−14.2053	−0.455169
$975$	0	0
$976$	1.22305	0.0391487
$977$	22.7654	0.728329	0.364164	−	0.931335i	$-0.381355\pi$
$977$	22.7654	0.728329	0.364164	+	0.931335i	$0.381355\pi$
$978$	0	0
$979$	−3.39258	−0.108427
$980$	−2.20177	−0.0703329
$981$	0	0
$982$	0.725190	0.0231417
$983$	42.8788	1.36762	0.683810	−	0.729660i	$-0.260322\pi$
$983$	42.8788	1.36762	0.683810	+	0.729660i	$0.260322\pi$
$984$	0	0
$985$	8.07872	0.257409
$986$	−5.14576	−0.163874
$987$	0	0
$988$	−2.01221	−0.0640169
$989$	13.1096	0.416860
$990$	0	0
$991$	34.8386	1.10668	0.553342	−	0.832954i	$-0.313352\pi$
$991$	34.8386	1.10668	0.553342	+	0.832954i	$0.313352\pi$
$992$	3.63295	0.115346
$993$	0	0
$994$	−0.543817	−0.0172488
$995$	−0.608438	−0.0192888
$996$	0	0
$997$	−14.7823	−0.468160	−0.234080	−	0.972217i	$-0.575208\pi$
$997$	−14.7823	−0.468160	−0.234080	+	0.972217i	$0.575208\pi$
$998$	6.47628	0.205003
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.s.1.7		16
3.2	odd	2		2667.2.a.n.1.10	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2667.2.a.n.1.10	✓	16	3.2	odd	2
8001.2.a.s.1.7		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)

\(f(q)\)	\(=\)	\(q-0.825012 q^{2} -1.31936 q^{4} +1.66882 q^{5} -1.00000 q^{7} +2.73851 q^{8} +O(q^{10})\)	\(q-0.825012 q^{2} -1.31936 q^{4} +1.66882 q^{5} -1.00000 q^{7} +2.73851 q^{8} -1.37680 q^{10} -4.60134 q^{11} -0.261360 q^{13} +0.825012 q^{14} +0.379411 q^{16} -1.30909 q^{17} -5.83541 q^{19} -2.20177 q^{20} +3.79616 q^{22} +1.54469 q^{23} -2.21504 q^{25} +0.215625 q^{26} +1.31936 q^{28} -4.76454 q^{29} -0.627450 q^{31} -5.79003 q^{32} +1.08001 q^{34} -1.66882 q^{35} +6.11849 q^{37} +4.81428 q^{38} +4.57008 q^{40} -11.5755 q^{41} +8.48685 q^{43} +6.07080 q^{44} -1.27439 q^{46} +4.72063 q^{47} +1.00000 q^{49} +1.82743 q^{50} +0.344827 q^{52} -13.5754 q^{53} -7.67881 q^{55} -2.73851 q^{56} +3.93080 q^{58} +2.92411 q^{59} +3.22354 q^{61} +0.517653 q^{62} +4.01802 q^{64} -0.436164 q^{65} -5.93511 q^{67} +1.72715 q^{68} +1.37680 q^{70} -0.659163 q^{71} -0.0147131 q^{73} -5.04783 q^{74} +7.69898 q^{76} +4.60134 q^{77} +3.62188 q^{79} +0.633169 q^{80} +9.54991 q^{82} -6.07333 q^{83} -2.18463 q^{85} -7.00175 q^{86} -12.6008 q^{88} +0.737302 q^{89} +0.261360 q^{91} -2.03800 q^{92} -3.89457 q^{94} -9.73825 q^{95} +6.68248 q^{97} -0.825012 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 4 q^{2} + 20 q^{4} - 5 q^{5} - 16 q^{7} - 15 q^{8}+O(q^{10}) \) 16 * q - 4 * q^2 + 20 * q^4 - 5 * q^5 - 16 * q^7 - 15 * q^8	\( 16 q - 4 q^{2} + 20 q^{4} - 5 q^{5} - 16 q^{7} - 15 q^{8} - 4 q^{10} - q^{11} + 20 q^{13} + 4 q^{14} + 32 q^{16} - 3 q^{17} + 13 q^{19} - 17 q^{20} + 13 q^{22} - 5 q^{23} + 17 q^{25} + 2 q^{26} - 20 q^{28} - 22 q^{29} + 26 q^{31} - 54 q^{32} - 6 q^{34} + 5 q^{35} + 30 q^{37} - 5 q^{38} + 13 q^{40} - q^{41} + 31 q^{43} - 22 q^{44} - 2 q^{46} + q^{47} + 16 q^{49} - 5 q^{50} + 31 q^{52} - 24 q^{53} + 8 q^{55} + 15 q^{56} + 13 q^{58} + 17 q^{59} + 32 q^{61} + 5 q^{62} + 61 q^{64} + 3 q^{65} + 16 q^{67} + 10 q^{68} + 4 q^{70} + 10 q^{71} + 23 q^{73} - q^{74} + 18 q^{76} + q^{77} + 48 q^{79} - 38 q^{80} + 12 q^{82} - 9 q^{83} + 22 q^{85} + 4 q^{86} + 27 q^{88} - 17 q^{89} - 20 q^{91} - 16 q^{92} + 13 q^{94} - 22 q^{95} + 17 q^{97} - 4 q^{98}+O(q^{100}) \) 16 * q - 4 * q^2 + 20 * q^4 - 5 * q^5 - 16 * q^7 - 15 * q^8 - 4 * q^10 - q^11 + 20 * q^13 + 4 * q^14 + 32 * q^16 - 3 * q^17 + 13 * q^19 - 17 * q^20 + 13 * q^22 - 5 * q^23 + 17 * q^25 + 2 * q^26 - 20 * q^28 - 22 * q^29 + 26 * q^31 - 54 * q^32 - 6 * q^34 + 5 * q^35 + 30 * q^37 - 5 * q^38 + 13 * q^40 - q^41 + 31 * q^43 - 22 * q^44 - 2 * q^46 + q^47 + 16 * q^49 - 5 * q^50 + 31 * q^52 - 24 * q^53 + 8 * q^55 + 15 * q^56 + 13 * q^58 + 17 * q^59 + 32 * q^61 + 5 * q^62 + 61 * q^64 + 3 * q^65 + 16 * q^67 + 10 * q^68 + 4 * q^70 + 10 * q^71 + 23 * q^73 - q^74 + 18 * q^76 + q^77 + 48 * q^79 - 38 * q^80 + 12 * q^82 - 9 * q^83 + 22 * q^85 + 4 * q^86 + 27 * q^88 - 17 * q^89 - 20 * q^91 - 16 * q^92 + 13 * q^94 - 22 * q^95 + 17 * q^97 - 4 * q^98

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

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