LMFDB - Embedded newform 8001.2.a.s.1.13

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			1.13
Root			$-1.40936$ 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)

$f(q)$	$=$	$q+1.40936 q^{2} -0.0136985 q^{4} -2.06019 q^{5} -1.00000 q^{7} -2.83803 q^{8} +O(q^{10})$	\(q+1.40936 q^{2} -0.0136985 q^{4} -2.06019 q^{5} -1.00000 q^{7} -2.83803 q^{8} -2.90356 q^{10} -5.38331 q^{11} +1.72882 q^{13} -1.40936 q^{14} -3.97242 q^{16} -4.40780 q^{17} -2.86650 q^{19} +0.0282216 q^{20} -7.58703 q^{22} -5.22653 q^{23} -0.755610 q^{25} +2.43653 q^{26} +0.0136985 q^{28} -7.87724 q^{29} -3.76214 q^{31} +0.0774892 q^{32} -6.21218 q^{34} +2.06019 q^{35} +9.55099 q^{37} -4.03993 q^{38} +5.84689 q^{40} +3.83006 q^{41} +0.492883 q^{43} +0.0737433 q^{44} -7.36607 q^{46} +4.21432 q^{47} +1.00000 q^{49} -1.06493 q^{50} -0.0236822 q^{52} +11.2226 q^{53} +11.0906 q^{55} +2.83803 q^{56} -11.1019 q^{58} -1.97993 q^{59} -7.35952 q^{61} -5.30221 q^{62} +8.05404 q^{64} -3.56169 q^{65} +1.47644 q^{67} +0.0603803 q^{68} +2.90356 q^{70} -1.17019 q^{71} -11.8919 q^{73} +13.4608 q^{74} +0.0392668 q^{76} +5.38331 q^{77} +8.50434 q^{79} +8.18394 q^{80} +5.39794 q^{82} +6.25858 q^{83} +9.08091 q^{85} +0.694651 q^{86} +15.2780 q^{88} -4.17500 q^{89} -1.72882 q^{91} +0.0715957 q^{92} +5.93951 q^{94} +5.90554 q^{95} +2.30828 q^{97} +1.40936 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	1.40936	0.996569	0.498285	−	0.867013i	$-0.333963\pi$
$2$	1.40936	0.996569	0.498285	+	0.867013i	$0.333963\pi$
$3$	0	0
$3$	0	0
$4$	−0.0136985	−0.00684926
$5$	−2.06019	−0.921346	−0.460673	−	0.887570i	$-0.652392\pi$
$5$	−2.06019	−0.921346	−0.460673	+	0.887570i	$0.652392\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−2.83803	−1.00340
$9$	0	0
$10$	−2.90356	−0.918185
$11$	−5.38331	−1.62313	−0.811564	−	0.584263i	$-0.801384\pi$
$11$	−5.38331	−1.62313	−0.811564	+	0.584263i	$0.801384\pi$
$12$	0	0
$13$	1.72882	0.479488	0.239744	−	0.970836i	$-0.422937\pi$
$13$	1.72882	0.479488	0.239744	+	0.970836i	$0.422937\pi$
$14$	−1.40936	−0.376668
$15$	0	0
$16$	−3.97242	−0.993104
$17$	−4.40780	−1.06905	−0.534524	−	0.845153i	$-0.679509\pi$
$17$	−4.40780	−1.06905	−0.534524	+	0.845153i	$0.679509\pi$
$18$	0	0
$19$	−2.86650	−0.657620	−0.328810	−	0.944396i	$-0.606648\pi$
$19$	−2.86650	−0.657620	−0.328810	+	0.944396i	$0.606648\pi$
$20$	0.0282216	0.00631054
$21$	0	0
$22$	−7.58703	−1.61756
$23$	−5.22653	−1.08981	−0.544903	−	0.838499i	$-0.683434\pi$
$23$	−5.22653	−1.08981	−0.544903	+	0.838499i	$0.683434\pi$
$24$	0	0
$25$	−0.755610	−0.151122
$26$	2.43653	0.477843
$27$	0	0
$28$	0.0136985	0.00258878
$29$	−7.87724	−1.46277	−0.731383	−	0.681966i	$-0.761125\pi$
$29$	−7.87724	−1.46277	−0.731383	+	0.681966i	$0.761125\pi$
$30$	0	0
$31$	−3.76214	−0.675700	−0.337850	−	0.941200i	$-0.609700\pi$
$31$	−3.76214	−0.675700	−0.337850	+	0.941200i	$0.609700\pi$
$32$	0.0774892	0.0136983
$33$	0	0
$34$	−6.21218	−1.06538
$35$	2.06019	0.348236
$36$	0	0
$37$	9.55099	1.57017	0.785087	−	0.619386i	$-0.212618\pi$
$37$	9.55099	1.57017	0.785087	+	0.619386i	$0.212618\pi$
$38$	−4.03993	−0.655364
$39$	0	0
$40$	5.84689	0.924474
$41$	3.83006	0.598155	0.299078	−	0.954229i	$-0.403321\pi$
$41$	3.83006	0.598155	0.299078	+	0.954229i	$0.403321\pi$
$42$	0	0
$43$	0.492883	0.0751640	0.0375820	−	0.999294i	$-0.488034\pi$
$43$	0.492883	0.0751640	0.0375820	+	0.999294i	$0.488034\pi$
$44$	0.0737433	0.0111172
$45$	0	0
$46$	−7.36607	−1.08607
$47$	4.21432	0.614722	0.307361	−	0.951593i	$-0.400554\pi$
$47$	4.21432	0.614722	0.307361	+	0.951593i	$0.400554\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−1.06493	−0.150604
$51$	0	0
$52$	−0.0236822	−0.00328413
$53$	11.2226	1.54155	0.770774	−	0.637109i	$-0.219870\pi$
$53$	11.2226	1.54155	0.770774	+	0.637109i	$0.219870\pi$
$54$	0	0
$55$	11.0906	1.49546
$56$	2.83803	0.379248
$57$	0	0
$58$	−11.1019	−1.45775
$59$	−1.97993	−0.257765	−0.128883	−	0.991660i	$-0.541139\pi$
$59$	−1.97993	−0.257765	−0.128883	+	0.991660i	$0.541139\pi$
$60$	0	0
$61$	−7.35952	−0.942290	−0.471145	−	0.882056i	$-0.656159\pi$
$61$	−7.35952	−0.942290	−0.471145	+	0.882056i	$0.656159\pi$
$62$	−5.30221	−0.673382
$63$	0	0
$64$	8.05404	1.00676
$65$	−3.56169	−0.441774
$66$	0	0
$67$	1.47644	0.180376	0.0901880	−	0.995925i	$-0.471253\pi$
$67$	1.47644	0.180376	0.0901880	+	0.995925i	$0.471253\pi$
$68$	0.0603803	0.00732219
$69$	0	0
$70$	2.90356	0.347041
$71$	−1.17019	−0.138876	−0.0694380	−	0.997586i	$-0.522121\pi$
$71$	−1.17019	−0.138876	−0.0694380	+	0.997586i	$0.522121\pi$
$72$	0	0
$73$	−11.8919	−1.39184	−0.695918	−	0.718121i	$-0.745002\pi$
$73$	−11.8919	−1.39184	−0.695918	+	0.718121i	$0.745002\pi$
$74$	13.4608	1.56479
$75$	0	0
$76$	0.0392668	0.00450421
$77$	5.38331	0.613485
$78$	0	0
$79$	8.50434	0.956814	0.478407	−	0.878138i	$-0.341214\pi$
$79$	8.50434	0.956814	0.478407	+	0.878138i	$0.341214\pi$
$80$	8.18394	0.914992
$81$	0	0
$82$	5.39794	0.596103
$83$	6.25858	0.686969	0.343484	−	0.939158i	$-0.388393\pi$
$83$	6.25858	0.686969	0.343484	+	0.939158i	$0.388393\pi$
$84$	0	0
$85$	9.08091	0.984963
$86$	0.694651	0.0749062
$87$	0	0
$88$	15.2780	1.62864
$89$	−4.17500	−0.442549	−0.221274	−	0.975212i	$-0.571022\pi$
$89$	−4.17500	−0.442549	−0.221274	+	0.975212i	$0.571022\pi$
$90$	0	0
$91$	−1.72882	−0.181229
$92$	0.0715957	0.00746437
$93$	0	0
$94$	5.93951	0.612613
$95$	5.90554	0.605895
$96$	0	0
$97$	2.30828	0.234370	0.117185	−	0.993110i	$-0.462613\pi$
$97$	2.30828	0.234370	0.117185	+	0.993110i	$0.462613\pi$
$98$	1.40936	0.142367
$99$	0	0
$100$	0.0103507	0.00103507
$101$	−10.3151	−1.02639	−0.513193	−	0.858273i	$-0.671538\pi$
$101$	−10.3151	−1.02639	−0.513193	+	0.858273i	$0.671538\pi$
$102$	0	0
$103$	−17.5850	−1.73270	−0.866351	−	0.499435i	$-0.833541\pi$
$103$	−17.5850	−1.73270	−0.866351	+	0.499435i	$0.833541\pi$
$104$	−4.90644	−0.481116
$105$	0	0
$106$	15.8168	1.53626
$107$	5.04288	0.487513	0.243757	−	0.969836i	$-0.421620\pi$
$107$	5.04288	0.487513	0.243757	+	0.969836i	$0.421620\pi$
$108$	0	0
$109$	9.06561	0.868328	0.434164	−	0.900834i	$-0.357044\pi$
$109$	9.06561	0.868328	0.434164	+	0.900834i	$0.357044\pi$
$110$	15.6307	1.49033
$111$	0	0
$112$	3.97242	0.375358
$113$	−11.6369	−1.09470	−0.547352	−	0.836903i	$-0.684364\pi$
$113$	−11.6369	−1.09470	−0.547352	+	0.836903i	$0.684364\pi$
$114$	0	0
$115$	10.7676	1.00409
$116$	0.107907	0.0100189
$117$	0	0
$118$	−2.79044	−0.256881
$119$	4.40780	0.404062
$120$	0	0
$121$	17.9800	1.63455
$122$	−10.3722	−0.939057
$123$	0	0
$124$	0.0515357	0.00462804
$125$	11.8577	1.06058
$126$	0	0
$127$	−1.00000	−0.0887357
$127$	−1.00000	−0.0887357
$128$	11.1961	0.989603
$129$	0	0
$130$	−5.01972	−0.440258
$131$	12.1670	1.06303	0.531517	−	0.847048i	$-0.321622\pi$
$131$	12.1670	1.06303	0.531517	+	0.847048i	$0.321622\pi$
$132$	0	0
$133$	2.86650	0.248557
$134$	2.08084	0.179757
$135$	0	0
$136$	12.5095	1.07268
$137$	−12.7729	−1.09127	−0.545633	−	0.838024i	$-0.683711\pi$
$137$	−12.7729	−1.09127	−0.545633	+	0.838024i	$0.683711\pi$
$138$	0	0
$139$	−2.40348	−0.203861	−0.101930	−	0.994792i	$-0.532502\pi$
$139$	−2.40348	−0.203861	−0.101930	+	0.994792i	$0.532502\pi$
$140$	−0.0282216	−0.00238516
$141$	0	0
$142$	−1.64922	−0.138400
$143$	−9.30675	−0.778270
$144$	0	0
$145$	16.2286	1.34771
$146$	−16.7599	−1.38706
$147$	0	0
$148$	−0.130834	−0.0107545
$149$	0.417337	0.0341896	0.0170948	−	0.999854i	$-0.494558\pi$
$149$	0.417337	0.0341896	0.0170948	+	0.999854i	$0.494558\pi$
$150$	0	0
$151$	−9.76270	−0.794477	−0.397239	−	0.917715i	$-0.630032\pi$
$151$	−9.76270	−0.794477	−0.397239	+	0.917715i	$0.630032\pi$
$152$	8.13521	0.659853
$153$	0	0
$154$	7.58703	0.611380
$155$	7.75073	0.622553
$156$	0	0
$157$	−1.35676	−0.108282	−0.0541408	−	0.998533i	$-0.517242\pi$
$157$	−1.35676	−0.108282	−0.0541408	+	0.998533i	$0.517242\pi$
$158$	11.9857	0.953531
$159$	0	0
$160$	−0.159643	−0.0126208
$161$	5.22653	0.411908
$162$	0	0
$163$	9.87658	0.773594	0.386797	−	0.922165i	$-0.373581\pi$
$163$	9.87658	0.773594	0.386797	+	0.922165i	$0.373581\pi$
$164$	−0.0524662	−0.00409692
$165$	0	0
$166$	8.82061	0.684612
$167$	2.01384	0.155835	0.0779177	−	0.996960i	$-0.475173\pi$
$167$	2.01384	0.155835	0.0779177	+	0.996960i	$0.475173\pi$
$168$	0	0
$169$	−10.0112	−0.770092
$170$	12.7983	0.981584
$171$	0	0
$172$	−0.00675177	−0.000514818 0
$173$	−0.396000	−0.0301073	−0.0150537	−	0.999887i	$-0.504792\pi$
$173$	−0.396000	−0.0301073	−0.0150537	+	0.999887i	$0.504792\pi$
$174$	0	0
$175$	0.755610	0.0571187
$176$	21.3847	1.61193
$177$	0	0
$178$	−5.88408	−0.441031
$179$	−6.32647	−0.472862	−0.236431	−	0.971648i	$-0.575978\pi$
$179$	−6.32647	−0.472862	−0.236431	+	0.971648i	$0.575978\pi$
$180$	0	0
$181$	−4.37635	−0.325292	−0.162646	−	0.986685i	$-0.552003\pi$
$181$	−4.37635	−0.325292	−0.162646	+	0.986685i	$0.552003\pi$
$182$	−2.43653	−0.180608
$183$	0	0
$184$	14.8330	1.09351
$185$	−19.6769	−1.44667
$186$	0	0
$187$	23.7285	1.73520
$188$	−0.0577300	−0.00421039
$189$	0	0
$190$	8.32304	0.603817
$191$	−1.56654	−0.113351	−0.0566754	−	0.998393i	$-0.518050\pi$
$191$	−1.56654	−0.113351	−0.0566754	+	0.998393i	$0.518050\pi$
$192$	0	0
$193$	−21.5721	−1.55279	−0.776397	−	0.630244i	$-0.782955\pi$
$193$	−21.5721	−1.55279	−0.776397	+	0.630244i	$0.782955\pi$
$194$	3.25320	0.233566
$195$	0	0
$196$	−0.0136985	−0.000978466 0
$197$	−19.5087	−1.38994	−0.694968	−	0.719041i	$-0.744581\pi$
$197$	−19.5087	−1.38994	−0.694968	+	0.719041i	$0.744581\pi$
$198$	0	0
$199$	−18.1478	−1.28646	−0.643231	−	0.765672i	$-0.722407\pi$
$199$	−18.1478	−1.28646	−0.643231	+	0.765672i	$0.722407\pi$
$200$	2.14444	0.151635
$201$	0	0
$202$	−14.5377	−1.02287
$203$	7.87724	0.552874
$204$	0	0
$205$	−7.89066	−0.551108
$206$	−24.7837	−1.72676
$207$	0	0
$208$	−6.86758	−0.476181
$209$	15.4312	1.06740
$210$	0	0
$211$	26.2749	1.80884	0.904419	−	0.426645i	$-0.140305\pi$
$211$	26.2749	1.80884	0.904419	+	0.426645i	$0.140305\pi$
$212$	−0.153734	−0.0105585
$213$	0	0
$214$	7.10724	0.485841
$215$	−1.01543	−0.0692520
$216$	0	0
$217$	3.76214	0.255391
$218$	12.7767	0.865349
$219$	0	0
$220$	−0.151925	−0.0102428
$221$	−7.62027	−0.512595
$222$	0	0
$223$	13.8791	0.929414	0.464707	−	0.885464i	$-0.346160\pi$
$223$	13.8791	0.929414	0.464707	+	0.885464i	$0.346160\pi$
$224$	−0.0774892	−0.00517746
$225$	0	0
$226$	−16.4005	−1.09095
$227$	2.88838	0.191709	0.0958544	−	0.995395i	$-0.469442\pi$
$227$	2.88838	0.191709	0.0958544	+	0.995395i	$0.469442\pi$
$228$	0	0
$229$	9.63743	0.636859	0.318430	−	0.947947i	$-0.396845\pi$
$229$	9.63743	0.636859	0.318430	+	0.947947i	$0.396845\pi$
$230$	15.1755	1.00064
$231$	0	0
$232$	22.3558	1.46773
$233$	−4.60154	−0.301457	−0.150729	−	0.988575i	$-0.548162\pi$
$233$	−4.60154	−0.301457	−0.150729	+	0.988575i	$0.548162\pi$
$234$	0	0
$235$	−8.68231	−0.566372
$236$	0.0271221	0.00176550
$237$	0	0
$238$	6.21218	0.402676
$239$	−15.6725	−1.01377	−0.506885	−	0.862014i	$-0.669203\pi$
$239$	−15.6725	−1.01377	−0.506885	+	0.862014i	$0.669203\pi$
$240$	0	0
$241$	25.4620	1.64015	0.820076	−	0.572255i	$-0.193931\pi$
$241$	25.4620	1.64015	0.820076	+	0.572255i	$0.193931\pi$
$242$	25.3403	1.62894
$243$	0	0
$244$	0.100814	0.00645399
$245$	−2.06019	−0.131621
$246$	0	0
$247$	−4.95565	−0.315321
$248$	10.6771	0.677994
$249$	0	0
$250$	16.7117	1.05694
$251$	−0.264556	−0.0166986	−0.00834931	−	0.999965i	$-0.502658\pi$
$251$	−0.264556	−0.0166986	−0.00834931	+	0.999965i	$0.502658\pi$
$252$	0	0
$253$	28.1360	1.76890
$254$	−1.40936	−0.0884312
$255$	0	0
$256$	−0.328749	−0.0205468
$257$	9.94917	0.620612	0.310306	−	0.950637i	$-0.399568\pi$
$257$	9.94917	0.620612	0.310306	+	0.950637i	$0.399568\pi$
$258$	0	0
$259$	−9.55099	−0.593470
$260$	0.0487899	0.00302582
$261$	0	0
$262$	17.1477	1.05939
$263$	12.3121	0.759197	0.379599	−	0.925151i	$-0.376062\pi$
$263$	12.3121	0.759197	0.379599	+	0.925151i	$0.376062\pi$
$264$	0	0
$265$	−23.1208	−1.42030
$266$	4.03993	0.247704
$267$	0	0
$268$	−0.0202251	−0.00123544
$269$	30.4350	1.85565	0.927826	−	0.373013i	$-0.121675\pi$
$269$	30.4350	1.85565	0.927826	+	0.373013i	$0.121675\pi$
$270$	0	0
$271$	5.90111	0.358467	0.179233	−	0.983807i	$-0.442638\pi$
$271$	5.90111	0.358467	0.179233	+	0.983807i	$0.442638\pi$
$272$	17.5096	1.06168
$273$	0	0
$274$	−18.0017	−1.08752
$275$	4.06768	0.245290
$276$	0	0
$277$	−0.276794	−0.0166310	−0.00831548	−	0.999965i	$-0.502647\pi$
$277$	−0.276794	−0.0166310	−0.00831548	+	0.999965i	$0.502647\pi$
$278$	−3.38738	−0.203161
$279$	0	0
$280$	−5.84689	−0.349418
$281$	−24.0152	−1.43262	−0.716312	−	0.697780i	$-0.754171\pi$
$281$	−24.0152	−1.43262	−0.716312	+	0.697780i	$0.754171\pi$
$282$	0	0
$283$	22.9723	1.36556	0.682780	−	0.730624i	$-0.260771\pi$
$283$	22.9723	1.36556	0.682780	+	0.730624i	$0.260771\pi$
$284$	0.0160299	0.000951198 0
$285$	0	0
$286$	−13.1166	−0.775600
$287$	−3.83006	−0.226081
$288$	0	0
$289$	2.42868	0.142863
$290$	22.8720	1.34309
$291$	0	0
$292$	0.162901	0.00953305
$293$	−1.25849	−0.0735220	−0.0367610	−	0.999324i	$-0.511704\pi$
$293$	−1.25849	−0.0735220	−0.0367610	+	0.999324i	$0.511704\pi$
$294$	0	0
$295$	4.07904	0.237491
$296$	−27.1060	−1.57550
$297$	0	0
$298$	0.588179	0.0340723
$299$	−9.03571	−0.522549
$300$	0	0
$301$	−0.492883	−0.0284093
$302$	−13.7592	−0.791752
$303$	0	0
$304$	11.3869	0.653085
$305$	15.1620	0.868175
$306$	0	0
$307$	28.1027	1.60391	0.801954	−	0.597386i	$-0.203794\pi$
$307$	28.1027	1.60391	0.801954	+	0.597386i	$0.203794\pi$
$308$	−0.0737433	−0.00420192
$309$	0	0
$310$	10.9236	0.620418
$311$	−8.65626	−0.490851	−0.245426	−	0.969415i	$-0.578928\pi$
$311$	−8.65626	−0.490851	−0.245426	+	0.969415i	$0.578928\pi$
$312$	0	0
$313$	17.4600	0.986895	0.493448	−	0.869776i	$-0.335737\pi$
$313$	17.4600	0.986895	0.493448	+	0.869776i	$0.335737\pi$
$314$	−1.91217	−0.107910
$315$	0	0
$316$	−0.116497	−0.00655346
$317$	0.931902	0.0523408	0.0261704	−	0.999657i	$-0.491669\pi$
$317$	0.931902	0.0523408	0.0261704	+	0.999657i	$0.491669\pi$
$318$	0	0
$319$	42.4056	2.37426
$320$	−16.5929	−0.927570
$321$	0	0
$322$	7.36607	0.410495
$323$	12.6349	0.703027
$324$	0	0
$325$	−1.30631	−0.0724611
$326$	13.9197	0.770940
$327$	0	0
$328$	−10.8698	−0.600186
$329$	−4.21432	−0.232343
$330$	0	0
$331$	−0.101149	−0.00555963	−0.00277982	−	0.999996i	$-0.500885\pi$
$331$	−0.101149	−0.00555963	−0.00277982	+	0.999996i	$0.500885\pi$
$332$	−0.0857333	−0.00470523
$333$	0	0
$334$	2.83822	0.155301
$335$	−3.04175	−0.166189
$336$	0	0
$337$	27.5769	1.50221	0.751106	−	0.660182i	$-0.229521\pi$
$337$	27.5769	1.50221	0.751106	+	0.660182i	$0.229521\pi$
$338$	−14.1094	−0.767450
$339$	0	0
$340$	−0.124395	−0.00674626
$341$	20.2527	1.09675
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	−1.39882	−0.0754192
$345$	0	0
$346$	−0.558107	−0.0300040
$347$	14.5950	0.783503	0.391752	−	0.920071i	$-0.371869\pi$
$347$	14.5950	0.783503	0.391752	+	0.920071i	$0.371869\pi$
$348$	0	0
$349$	−14.7769	−0.790992	−0.395496	−	0.918468i	$-0.629427\pi$
$349$	−14.7769	−0.790992	−0.395496	+	0.918468i	$0.629427\pi$
$350$	1.06493	0.0569228
$351$	0	0
$352$	−0.417148	−0.0222341
$353$	−9.32705	−0.496429	−0.248214	−	0.968705i	$-0.579844\pi$
$353$	−9.32705	−0.496429	−0.248214	+	0.968705i	$0.579844\pi$
$354$	0	0
$355$	2.41082	0.127953
$356$	0.0571913	0.00303113
$357$	0	0
$358$	−8.91628	−0.471240
$359$	19.4912	1.02870	0.514352	−	0.857579i	$-0.328032\pi$
$359$	19.4912	1.02870	0.514352	+	0.857579i	$0.328032\pi$
$360$	0	0
$361$	−10.7832	−0.567536
$362$	−6.16787	−0.324176
$363$	0	0
$364$	0.0236822	0.00124129
$365$	24.4995	1.28236
$366$	0	0
$367$	30.7571	1.60551	0.802754	−	0.596310i	$-0.203367\pi$
$367$	30.7571	1.60551	0.802754	+	0.596310i	$0.203367\pi$
$368$	20.7619	1.08229
$369$	0	0
$370$	−27.7318	−1.44171
$371$	−11.2226	−0.582650
$372$	0	0
$373$	−36.5595	−1.89298	−0.946490	−	0.322733i	$-0.895399\pi$
$373$	−36.5595	−1.89298	−0.946490	+	0.322733i	$0.895399\pi$
$374$	33.4421	1.72925
$375$	0	0
$376$	−11.9604	−0.616809
$377$	−13.6183	−0.701379
$378$	0	0
$379$	−30.7010	−1.57700	−0.788502	−	0.615032i	$-0.789143\pi$
$379$	−30.7010	−1.57700	−0.788502	+	0.615032i	$0.789143\pi$
$380$	−0.0808971	−0.00414993
$381$	0	0
$382$	−2.20782	−0.112962
$383$	30.5496	1.56101	0.780506	−	0.625148i	$-0.214962\pi$
$383$	30.5496	1.56101	0.780506	+	0.625148i	$0.214962\pi$
$384$	0	0
$385$	−11.0906	−0.565232
$386$	−30.4029	−1.54747
$387$	0	0
$388$	−0.0316200	−0.00160526
$389$	−37.2201	−1.88713	−0.943566	−	0.331185i	$-0.892552\pi$
$389$	−37.2201	−1.88713	−0.943566	+	0.331185i	$0.892552\pi$
$390$	0	0
$391$	23.0375	1.16506
$392$	−2.83803	−0.143342
$393$	0	0
$394$	−27.4948	−1.38517
$395$	−17.5206	−0.881556
$396$	0	0
$397$	26.0035	1.30508	0.652538	−	0.757756i	$-0.273704\pi$
$397$	26.0035	1.30508	0.652538	+	0.757756i	$0.273704\pi$
$398$	−25.5768	−1.28205
$399$	0	0
$400$	3.00160	0.150080
$401$	18.0905	0.903398	0.451699	−	0.892170i	$-0.350818\pi$
$401$	18.0905	0.903398	0.451699	+	0.892170i	$0.350818\pi$
$402$	0	0
$403$	−6.50405	−0.323990
$404$	0.141301	0.00702999
$405$	0	0
$406$	11.1019	0.550977
$407$	−51.4159	−2.54859
$408$	0	0
$409$	−8.87935	−0.439056	−0.219528	−	0.975606i	$-0.570452\pi$
$409$	−8.87935	−0.439056	−0.219528	+	0.975606i	$0.570452\pi$
$410$	−11.1208	−0.549217
$411$	0	0
$412$	0.240889	0.0118677
$413$	1.97993	0.0974261
$414$	0	0
$415$	−12.8939	−0.632936
$416$	0.133965	0.00656815
$417$	0	0
$418$	21.7482	1.06374
$419$	20.2505	0.989300	0.494650	−	0.869092i	$-0.335296\pi$
$419$	20.2505	0.989300	0.494650	+	0.869092i	$0.335296\pi$
$420$	0	0
$421$	−26.8954	−1.31080	−0.655402	−	0.755281i	$-0.727501\pi$
$421$	−26.8954	−1.31080	−0.655402	+	0.755281i	$0.727501\pi$
$422$	37.0308	1.80263
$423$	0	0
$424$	−31.8502	−1.54678
$425$	3.33057	0.161557
$426$	0	0
$427$	7.35952	0.356152
$428$	−0.0690799	−0.00333911
$429$	0	0
$430$	−1.43111	−0.0690145
$431$	5.14387	0.247771	0.123886	−	0.992296i	$-0.460464\pi$
$431$	5.14387	0.247771	0.123886	+	0.992296i	$0.460464\pi$
$432$	0	0
$433$	30.0478	1.44400	0.722002	−	0.691891i	$-0.243222\pi$
$433$	30.0478	1.44400	0.722002	+	0.691891i	$0.243222\pi$
$434$	5.30221	0.254514
$435$	0	0
$436$	−0.124185	−0.00594740
$437$	14.9818	0.716678
$438$	0	0
$439$	−18.4644	−0.881256	−0.440628	−	0.897690i	$-0.645244\pi$
$439$	−18.4644	−0.881256	−0.440628	+	0.897690i	$0.645244\pi$
$440$	−31.4756	−1.50054
$441$	0	0
$442$	−10.7397	−0.510837
$443$	22.9376	1.08980	0.544898	−	0.838502i	$-0.316568\pi$
$443$	22.9376	1.08980	0.544898	+	0.838502i	$0.316568\pi$
$444$	0	0
$445$	8.60130	0.407741
$446$	19.5607	0.926226
$447$	0	0
$448$	−8.05404	−0.380518
$449$	−14.4781	−0.683266	−0.341633	−	0.939834i	$-0.610980\pi$
$449$	−14.4781	−0.683266	−0.341633	+	0.939834i	$0.610980\pi$
$450$	0	0
$451$	−20.6184	−0.970883
$452$	0.159408	0.00749791
$453$	0	0
$454$	4.07078	0.191051
$455$	3.56169	0.166975
$456$	0	0
$457$	−1.89432	−0.0886126	−0.0443063	−	0.999018i	$-0.514108\pi$
$457$	−1.89432	−0.0886126	−0.0443063	+	0.999018i	$0.514108\pi$
$458$	13.5826	0.634674
$459$	0	0
$460$	−0.147501	−0.00687726
$461$	−15.7984	−0.735804	−0.367902	−	0.929865i	$-0.619924\pi$
$461$	−15.7984	−0.735804	−0.367902	+	0.929865i	$0.619924\pi$
$462$	0	0
$463$	27.3251	1.26991	0.634953	−	0.772550i	$-0.281019\pi$
$463$	27.3251	1.26991	0.634953	+	0.772550i	$0.281019\pi$
$464$	31.2917	1.45268
$465$	0	0
$466$	−6.48524	−0.300423
$467$	14.7989	0.684813	0.342406	−	0.939552i	$-0.388758\pi$
$467$	14.7989	0.684813	0.342406	+	0.939552i	$0.388758\pi$
$468$	0	0
$469$	−1.47644	−0.0681758
$470$	−12.2365	−0.564429
$471$	0	0
$472$	5.61911	0.258640
$473$	−2.65334	−0.122001
$474$	0	0
$475$	2.16595	0.0993808
$476$	−0.0603803	−0.00276753
$477$	0	0
$478$	−22.0882	−1.01029
$479$	0.594997	0.0271861	0.0135930	−	0.999908i	$-0.495673\pi$
$479$	0.594997	0.0271861	0.0135930	+	0.999908i	$0.495673\pi$
$480$	0	0
$481$	16.5119	0.752879
$482$	35.8852	1.63452
$483$	0	0
$484$	−0.246299	−0.0111954
$485$	−4.75549	−0.215936
$486$	0	0
$487$	−21.3144	−0.965849	−0.482925	−	0.875662i	$-0.660426\pi$
$487$	−21.3144	−0.965849	−0.482925	+	0.875662i	$0.660426\pi$
$488$	20.8865	0.945489
$489$	0	0
$490$	−2.90356	−0.131169
$491$	24.6949	1.11446	0.557232	−	0.830357i	$-0.311863\pi$
$491$	24.6949	1.11446	0.557232	+	0.830357i	$0.311863\pi$
$492$	0	0
$493$	34.7213	1.56377
$494$	−6.98431	−0.314239
$495$	0	0
$496$	14.9448	0.671040
$497$	1.17019	0.0524902
$498$	0	0
$499$	−9.26885	−0.414931	−0.207465	−	0.978242i	$-0.566521\pi$
$499$	−9.26885	−0.414931	−0.207465	+	0.978242i	$0.566521\pi$
$500$	−0.162432	−0.00726420
$501$	0	0
$502$	−0.372855	−0.0166413
$503$	−39.6311	−1.76706	−0.883532	−	0.468371i	$-0.844841\pi$
$503$	−39.6311	−1.76706	−0.883532	+	0.468371i	$0.844841\pi$
$504$	0	0
$505$	21.2510	0.945657
$506$	39.6538	1.76283
$507$	0	0
$508$	0.0136985	0.000607773 0
$509$	−42.5693	−1.88685	−0.943426	−	0.331582i	$-0.892418\pi$
$509$	−42.5693	−1.88685	−0.943426	+	0.331582i	$0.892418\pi$
$510$	0	0
$511$	11.8919	0.526065
$512$	−22.8555	−1.01008
$513$	0	0
$514$	14.0220	0.618483
$515$	36.2285	1.59642
$516$	0	0
$517$	−22.6870	−0.997773
$518$	−13.4608	−0.591434
$519$	0	0
$520$	10.1082	0.443274
$521$	6.76444	0.296356	0.148178	−	0.988961i	$-0.452659\pi$
$521$	6.76444	0.296356	0.148178	+	0.988961i	$0.452659\pi$
$522$	0	0
$523$	−8.50567	−0.371927	−0.185963	−	0.982557i	$-0.559541\pi$
$523$	−8.50567	−0.371927	−0.185963	+	0.982557i	$0.559541\pi$
$524$	−0.166670	−0.00728099
$525$	0	0
$526$	17.3522	0.756593
$527$	16.5827	0.722356
$528$	0	0
$529$	4.31659	0.187678
$530$	−32.5856	−1.41543
$531$	0	0
$532$	−0.0392668	−0.00170243
$533$	6.62148	0.286808
$534$	0	0
$535$	−10.3893	−0.449168
$536$	−4.19019	−0.180989
$537$	0	0
$538$	42.8939	1.84929
$539$	−5.38331	−0.231875
$540$	0	0
$541$	−42.4367	−1.82450	−0.912248	−	0.409638i	$-0.865655\pi$
$541$	−42.4367	−1.82450	−0.912248	+	0.409638i	$0.865655\pi$
$542$	8.31680	0.357237
$543$	0	0
$544$	−0.341556	−0.0146441
$545$	−18.6769	−0.800030
$546$	0	0
$547$	−30.0725	−1.28581	−0.642904	−	0.765947i	$-0.722271\pi$
$547$	−30.0725	−1.28581	−0.642904	+	0.765947i	$0.722271\pi$
$548$	0.174970	0.00747436
$549$	0	0
$550$	5.73283	0.244449
$551$	22.5801	0.961945
$552$	0	0
$553$	−8.50434	−0.361642
$554$	−0.390103	−0.0165739
$555$	0	0
$556$	0.0329241	0.00139629
$557$	−42.3083	−1.79266	−0.896329	−	0.443389i	$-0.853776\pi$
$557$	−42.3083	−1.79266	−0.896329	+	0.443389i	$0.853776\pi$
$558$	0	0
$559$	0.852105	0.0360402
$560$	−8.18394	−0.345834
$561$	0	0
$562$	−33.8461	−1.42771
$563$	−9.01205	−0.379813	−0.189906	−	0.981802i	$-0.560818\pi$
$563$	−9.01205	−0.379813	−0.189906	+	0.981802i	$0.560818\pi$
$564$	0	0
$565$	23.9742	1.00860
$566$	32.3763	1.36088
$567$	0	0
$568$	3.32104	0.139348
$569$	−9.95547	−0.417355	−0.208677	−	0.977985i	$-0.566916\pi$
$569$	−9.95547	−0.417355	−0.208677	+	0.977985i	$0.566916\pi$
$570$	0	0
$571$	−32.5570	−1.36247	−0.681235	−	0.732065i	$-0.738557\pi$
$571$	−32.5570	−1.36247	−0.681235	+	0.732065i	$0.738557\pi$
$572$	0.127489	0.00533057
$573$	0	0
$574$	−5.39794	−0.225306
$575$	3.94922	0.164694
$576$	0	0
$577$	−31.1269	−1.29583	−0.647914	−	0.761714i	$-0.724358\pi$
$577$	−31.1269	−1.29583	−0.647914	+	0.761714i	$0.724358\pi$
$578$	3.42288	0.142373
$579$	0	0
$580$	−0.222308	−0.00923084
$581$	−6.25858	−0.259650
$582$	0	0
$583$	−60.4149	−2.50213
$584$	33.7495	1.39656
$585$	0	0
$586$	−1.77367	−0.0732698
$587$	−38.8330	−1.60281	−0.801405	−	0.598122i	$-0.795914\pi$
$587$	−38.8330	−1.60281	−0.801405	+	0.598122i	$0.795914\pi$
$588$	0	0
$589$	10.7842	0.444354
$590$	5.74885	0.236676
$591$	0	0
$592$	−37.9405	−1.55935
$593$	43.7929	1.79836	0.899180	−	0.437579i	$-0.144164\pi$
$593$	43.7929	1.79836	0.899180	+	0.437579i	$0.144164\pi$
$594$	0	0
$595$	−9.08091	−0.372281
$596$	−0.00571690	−0.000234173 0
$597$	0	0
$598$	−12.7346	−0.520756
$599$	29.5799	1.20860	0.604301	−	0.796756i	$-0.293452\pi$
$599$	29.5799	1.20860	0.604301	+	0.796756i	$0.293452\pi$
$600$	0	0
$601$	−9.25462	−0.377504	−0.188752	−	0.982025i	$-0.560444\pi$
$601$	−9.25462	−0.377504	−0.188752	+	0.982025i	$0.560444\pi$
$602$	−0.694651	−0.0283119
$603$	0	0
$604$	0.133735	0.00544158
$605$	−37.0422	−1.50598
$606$	0	0
$607$	−9.55386	−0.387779	−0.193890	−	0.981023i	$-0.562110\pi$
$607$	−9.55386	−0.387779	−0.193890	+	0.981023i	$0.562110\pi$
$608$	−0.222123	−0.00900826
$609$	0	0
$610$	21.3688	0.865196
$611$	7.28579	0.294752
$612$	0	0
$613$	−25.5408	−1.03159	−0.515793	−	0.856714i	$-0.672502\pi$
$613$	−25.5408	−1.03159	−0.515793	+	0.856714i	$0.672502\pi$
$614$	39.6069	1.59841
$615$	0	0
$616$	−15.2780	−0.615568
$617$	22.3625	0.900281	0.450141	−	0.892958i	$-0.351374\pi$
$617$	22.3625	0.900281	0.450141	+	0.892958i	$0.351374\pi$
$618$	0	0
$619$	11.2085	0.450506	0.225253	−	0.974300i	$-0.427679\pi$
$619$	11.2085	0.450506	0.225253	+	0.974300i	$0.427679\pi$
$620$	−0.106173	−0.00426403
$621$	0	0
$622$	−12.1998	−0.489168
$623$	4.17500	0.167268
$624$	0	0
$625$	−20.6510	−0.826040
$626$	24.6074	0.983509
$627$	0	0
$628$	0.0185857	0.000741649 0
$629$	−42.0988	−1.67859
$630$	0	0
$631$	15.5742	0.620001	0.310000	−	0.950736i	$-0.399671\pi$
$631$	15.5742	0.620001	0.310000	+	0.950736i	$0.399671\pi$
$632$	−24.1356	−0.960062
$633$	0	0
$634$	1.31339	0.0521612
$635$	2.06019	0.0817562
$636$	0	0
$637$	1.72882	0.0684982
$638$	59.7649	2.36611
$639$	0	0
$640$	−23.0661	−0.911767
$641$	−16.5542	−0.653852	−0.326926	−	0.945050i	$-0.606013\pi$
$641$	−16.5542	−0.653852	−0.326926	+	0.945050i	$0.606013\pi$
$642$	0	0
$643$	−32.5175	−1.28236	−0.641182	−	0.767389i	$-0.721556\pi$
$643$	−32.5175	−1.28236	−0.641182	+	0.767389i	$0.721556\pi$
$644$	−0.0715957	−0.00282127
$645$	0	0
$646$	17.8072	0.700615
$647$	−14.6904	−0.577538	−0.288769	−	0.957399i	$-0.593246\pi$
$647$	−14.6904	−0.577538	−0.288769	+	0.957399i	$0.593246\pi$
$648$	0	0
$649$	10.6586	0.418386
$650$	−1.84107	−0.0722125
$651$	0	0
$652$	−0.135295	−0.00529854
$653$	−3.54948	−0.138902	−0.0694509	−	0.997585i	$-0.522125\pi$
$653$	−3.54948	−0.138902	−0.0694509	+	0.997585i	$0.522125\pi$
$654$	0	0
$655$	−25.0663	−0.979421
$656$	−15.2146	−0.594030
$657$	0	0
$658$	−5.93951	−0.231546
$659$	19.9283	0.776298	0.388149	−	0.921597i	$-0.373115\pi$
$659$	19.9283	0.776298	0.388149	+	0.921597i	$0.373115\pi$
$660$	0	0
$661$	−33.1130	−1.28795	−0.643973	−	0.765048i	$-0.722715\pi$
$661$	−33.1130	−1.28795	−0.643973	+	0.765048i	$0.722715\pi$
$662$	−0.142555	−0.00554056
$663$	0	0
$664$	−17.7621	−0.689301
$665$	−5.90554	−0.229007
$666$	0	0
$667$	41.1706	1.59413
$668$	−0.0275866	−0.00106736
$669$	0	0
$670$	−4.28693	−0.165619
$671$	39.6185	1.52946
$672$	0	0
$673$	25.8845	0.997774	0.498887	−	0.866667i	$-0.333742\pi$
$673$	25.8845	0.997774	0.498887	+	0.866667i	$0.333742\pi$
$674$	38.8659	1.49706
$675$	0	0
$676$	0.137138	0.00527456
$677$	21.1876	0.814304	0.407152	−	0.913360i	$-0.366522\pi$
$677$	21.1876	0.814304	0.407152	+	0.913360i	$0.366522\pi$
$678$	0	0
$679$	−2.30828	−0.0885835
$680$	−25.7719	−0.988307
$681$	0	0
$682$	28.5435	1.09299
$683$	50.5021	1.93241	0.966204	−	0.257778i	$-0.0829904\pi$
$683$	50.5021	1.93241	0.966204	+	0.257778i	$0.0829904\pi$
$684$	0	0
$685$	26.3147	1.00543
$686$	−1.40936	−0.0538097
$687$	0	0
$688$	−1.95794	−0.0746457
$689$	19.4019	0.739153
$690$	0	0
$691$	−44.6565	−1.69881	−0.849407	−	0.527738i	$-0.823040\pi$
$691$	−44.6565	−1.69881	−0.849407	+	0.527738i	$0.823040\pi$
$692$	0.00542461	0.000206213 0
$693$	0	0
$694$	20.5697	0.780816
$695$	4.95163	0.187826
$696$	0	0
$697$	−16.8821	−0.639457
$698$	−20.8261	−0.788278
$699$	0	0
$700$	−0.0103507	−0.000391221 0
$701$	−30.8064	−1.16354	−0.581770	−	0.813353i	$-0.697640\pi$
$701$	−30.8064	−1.16354	−0.581770	+	0.813353i	$0.697640\pi$
$702$	0	0
$703$	−27.3779	−1.03258
$704$	−43.3574	−1.63409
$705$	0	0
$706$	−13.1452	−0.494726
$707$	10.3151	0.387938
$708$	0	0
$709$	12.0253	0.451619	0.225809	−	0.974172i	$-0.427497\pi$
$709$	12.0253	0.451619	0.225809	+	0.974172i	$0.427497\pi$
$710$	3.39771	0.127514
$711$	0	0
$712$	11.8488	0.444051
$713$	19.6629	0.736382
$714$	0	0
$715$	19.1737	0.717056
$716$	0.0866632	0.00323876
$717$	0	0
$718$	27.4701	1.02518
$719$	−16.2337	−0.605413	−0.302707	−	0.953084i	$-0.597890\pi$
$719$	−16.2337	−0.605413	−0.302707	+	0.953084i	$0.597890\pi$
$720$	0	0
$721$	17.5850	0.654900
$722$	−15.1974	−0.565589
$723$	0	0
$724$	0.0599496	0.00222801
$725$	5.95212	0.221056
$726$	0	0
$727$	10.1895	0.377909	0.188955	−	0.981986i	$-0.439490\pi$
$727$	10.1895	0.377909	0.188955	+	0.981986i	$0.439490\pi$
$728$	4.90644	0.181845
$729$	0	0
$730$	34.5287	1.27796
$731$	−2.17253	−0.0803539
$732$	0	0
$733$	−6.33953	−0.234156	−0.117078	−	0.993123i	$-0.537353\pi$
$733$	−6.33953	−0.234156	−0.117078	+	0.993123i	$0.537353\pi$
$734$	43.3479	1.60000
$735$	0	0
$736$	−0.404999	−0.0149285
$737$	−7.94814	−0.292774
$738$	0	0
$739$	−15.0939	−0.555237	−0.277618	−	0.960691i	$-0.589545\pi$
$739$	−15.0939	−0.555237	−0.277618	+	0.960691i	$0.589545\pi$
$740$	0.269544	0.00990864
$741$	0	0
$742$	−15.8168	−0.580652
$743$	18.7671	0.688499	0.344249	−	0.938878i	$-0.388133\pi$
$743$	18.7671	0.688499	0.344249	+	0.938878i	$0.388133\pi$
$744$	0	0
$745$	−0.859794	−0.0315004
$746$	−51.5256	−1.88649
$747$	0	0
$748$	−0.325046	−0.0118848
$749$	−5.04288	−0.184263
$750$	0	0
$751$	7.16617	0.261497	0.130749	−	0.991416i	$-0.458262\pi$
$751$	7.16617	0.261497	0.130749	+	0.991416i	$0.458262\pi$
$752$	−16.7410	−0.610483
$753$	0	0
$754$	−19.1931	−0.698972
$755$	20.1130	0.731988
$756$	0	0
$757$	−1.13515	−0.0412578	−0.0206289	−	0.999787i	$-0.506567\pi$
$757$	−1.13515	−0.0412578	−0.0206289	+	0.999787i	$0.506567\pi$
$758$	−43.2688	−1.57159
$759$	0	0
$760$	−16.7601	−0.607952
$761$	33.8563	1.22729	0.613646	−	0.789582i	$-0.289702\pi$
$761$	33.8563	1.22729	0.613646	+	0.789582i	$0.289702\pi$
$762$	0	0
$763$	−9.06561	−0.328197
$764$	0.0214593	0.000776370 0
$765$	0	0
$766$	43.0555	1.55566
$767$	−3.42294	−0.123595
$768$	0	0
$769$	−51.8292	−1.86901	−0.934505	−	0.355950i	$-0.884157\pi$
$769$	−51.8292	−1.86901	−0.934505	+	0.355950i	$0.884157\pi$
$770$	−15.6307	−0.563293
$771$	0	0
$772$	0.295506	0.0106355
$773$	30.4395	1.09483	0.547417	−	0.836860i	$-0.315611\pi$
$773$	30.4395	1.09483	0.547417	+	0.836860i	$0.315611\pi$
$774$	0	0
$775$	2.84271	0.102113
$776$	−6.55096	−0.235166
$777$	0	0
$778$	−52.4565	−1.88066
$779$	−10.9789	−0.393359
$780$	0	0
$781$	6.29950	0.225414
$782$	32.4681	1.16106
$783$	0	0
$784$	−3.97242	−0.141872
$785$	2.79519	0.0997648
$786$	0	0
$787$	−25.2749	−0.900952	−0.450476	−	0.892788i	$-0.648746\pi$
$787$	−25.2749	−0.900952	−0.450476	+	0.892788i	$0.648746\pi$
$788$	0.267240	0.00952003
$789$	0	0
$790$	−24.6928	−0.878532
$791$	11.6369	0.413759
$792$	0	0
$793$	−12.7233	−0.451816
$794$	36.6483	1.30060
$795$	0	0
$796$	0.248598	0.00881131
$797$	−19.6712	−0.696791	−0.348396	−	0.937348i	$-0.613273\pi$
$797$	−19.6712	−0.696791	−0.348396	+	0.937348i	$0.613273\pi$
$798$	0	0
$799$	−18.5759	−0.657167
$800$	−0.0585516	−0.00207011
$801$	0	0
$802$	25.4961	0.900299
$803$	64.0175	2.25913
$804$	0	0
$805$	−10.7676	−0.379510
$806$	−9.16656	−0.322878
$807$	0	0
$808$	29.2745	1.02987
$809$	−2.47525	−0.0870251	−0.0435125	−	0.999053i	$-0.513855\pi$
$809$	−2.47525	−0.0870251	−0.0435125	+	0.999053i	$0.513855\pi$
$810$	0	0
$811$	8.24799	0.289626	0.144813	−	0.989459i	$-0.453742\pi$
$811$	8.24799	0.289626	0.144813	+	0.989459i	$0.453742\pi$
$812$	−0.107907	−0.00378678
$813$	0	0
$814$	−72.4637	−2.53985
$815$	−20.3477	−0.712747
$816$	0	0
$817$	−1.41285	−0.0494294
$818$	−12.5142	−0.437549
$819$	0	0
$820$	0.108090	0.00377468
$821$	−22.5630	−0.787453	−0.393726	−	0.919228i	$-0.628814\pi$
$821$	−22.5630	−0.787453	−0.393726	+	0.919228i	$0.628814\pi$
$822$	0	0
$823$	2.17124	0.0756848	0.0378424	−	0.999284i	$-0.487952\pi$
$823$	2.17124	0.0756848	0.0378424	+	0.999284i	$0.487952\pi$
$824$	49.9068	1.73859
$825$	0	0
$826$	2.79044	0.0970919
$827$	4.28156	0.148884	0.0744422	−	0.997225i	$-0.476282\pi$
$827$	4.28156	0.148884	0.0744422	+	0.997225i	$0.476282\pi$
$828$	0	0
$829$	−11.8023	−0.409910	−0.204955	−	0.978771i	$-0.565705\pi$
$829$	−11.8023	−0.409910	−0.204955	+	0.978771i	$0.565705\pi$
$830$	−18.1721	−0.630765
$831$	0	0
$832$	13.9240	0.482727
$833$	−4.40780	−0.152721
$834$	0	0
$835$	−4.14889	−0.143578
$836$	−0.211385	−0.00731091
$837$	0	0
$838$	28.5402	0.985906
$839$	23.5250	0.812175	0.406087	−	0.913834i	$-0.366893\pi$
$839$	23.5250	0.812175	0.406087	+	0.913834i	$0.366893\pi$
$840$	0	0
$841$	33.0509	1.13969
$842$	−37.9054	−1.30631
$843$	0	0
$844$	−0.359927	−0.0123892
$845$	20.6250	0.709521
$846$	0	0
$847$	−17.9800	−0.617800
$848$	−44.5810	−1.53092
$849$	0	0
$850$	4.69399	0.161002
$851$	−49.9185	−1.71119
$852$	0	0
$853$	37.8419	1.29568	0.647840	−	0.761776i	$-0.275672\pi$
$853$	37.8419	1.29568	0.647840	+	0.761776i	$0.275672\pi$
$854$	10.3722	0.354930
$855$	0	0
$856$	−14.3118	−0.489169
$857$	8.28248	0.282924	0.141462	−	0.989944i	$-0.454820\pi$
$857$	8.28248	0.282924	0.141462	+	0.989944i	$0.454820\pi$
$858$	0	0
$859$	−11.2310	−0.383198	−0.191599	−	0.981473i	$-0.561367\pi$
$859$	−11.2310	−0.383198	−0.191599	+	0.981473i	$0.561367\pi$
$860$	0.0139099	0.000474325 0
$861$	0	0
$862$	7.24958	0.246922
$863$	−4.09250	−0.139310	−0.0696551	−	0.997571i	$-0.522190\pi$
$863$	−4.09250	−0.139310	−0.0696551	+	0.997571i	$0.522190\pi$
$864$	0	0
$865$	0.815835	0.0277392
$866$	42.3482	1.43905
$867$	0	0
$868$	−0.0515357	−0.00174924
$869$	−45.7815	−1.55303
$870$	0	0
$871$	2.55250	0.0864881
$872$	−25.7285	−0.871276
$873$	0	0
$874$	21.1148	0.714220
$875$	−11.8577	−0.400862
$876$	0	0
$877$	10.1271	0.341968	0.170984	−	0.985274i	$-0.445305\pi$
$877$	10.1271	0.341968	0.170984	+	0.985274i	$0.445305\pi$
$878$	−26.0230	−0.878233
$879$	0	0
$880$	−44.0567	−1.48515
$881$	27.8564	0.938504	0.469252	−	0.883064i	$-0.344524\pi$
$881$	27.8564	0.938504	0.469252	+	0.883064i	$0.344524\pi$
$882$	0	0
$883$	−8.65692	−0.291329	−0.145664	−	0.989334i	$-0.546532\pi$
$883$	−8.65692	−0.291329	−0.145664	+	0.989334i	$0.546532\pi$
$884$	0.104386	0.00351090
$885$	0	0
$886$	32.3273	1.08606
$887$	42.1151	1.41409	0.707044	−	0.707170i	$-0.250028\pi$
$887$	42.1151	1.41409	0.707044	+	0.707170i	$0.250028\pi$
$888$	0	0
$889$	1.00000	0.0335389
$890$	12.1223	0.406342
$891$	0	0
$892$	−0.190123	−0.00636580
$893$	−12.0803	−0.404253
$894$	0	0
$895$	13.0337	0.435670
$896$	−11.1961	−0.374035
$897$	0	0
$898$	−20.4049	−0.680922
$899$	29.6353	0.988392
$900$	0	0
$901$	−49.4671	−1.64799
$902$	−29.0588	−0.967552
$903$	0	0
$904$	33.0258	1.09842
$905$	9.01613	0.299706
$906$	0	0
$907$	−35.7631	−1.18749	−0.593746	−	0.804652i	$-0.702352\pi$
$907$	−35.7631	−1.18749	−0.593746	+	0.804652i	$0.702352\pi$
$908$	−0.0395666	−0.00131306
$909$	0	0
$910$	5.01972	0.166402
$911$	−27.7615	−0.919781	−0.459890	−	0.887976i	$-0.652111\pi$
$911$	−27.7615	−0.919781	−0.459890	+	0.887976i	$0.652111\pi$
$912$	0	0
$913$	−33.6919	−1.11504
$914$	−2.66978	−0.0883086
$915$	0	0
$916$	−0.132018	−0.00436201
$917$	−12.1670	−0.401789
$918$	0	0
$919$	−23.2528	−0.767040	−0.383520	−	0.923533i	$-0.625288\pi$
$919$	−23.2528	−0.767040	−0.383520	+	0.923533i	$0.625288\pi$
$920$	−30.5589	−1.00750
$921$	0	0
$922$	−22.2656	−0.733280
$923$	−2.02305	−0.0665894
$924$	0	0
$925$	−7.21682	−0.237288
$926$	38.5110	1.26555
$927$	0	0
$928$	−0.610401	−0.0200374
$929$	28.8896	0.947836	0.473918	−	0.880569i	$-0.342839\pi$
$929$	28.8896	0.947836	0.473918	+	0.880569i	$0.342839\pi$
$930$	0	0
$931$	−2.86650	−0.0939457
$932$	0.0630343	0.00206476
$933$	0	0
$934$	20.8570	0.682463
$935$	−48.8853	−1.59872
$936$	0	0
$937$	48.1915	1.57435	0.787174	−	0.616731i	$-0.211543\pi$
$937$	48.1915	1.57435	0.787174	+	0.616731i	$0.211543\pi$
$938$	−2.08084	−0.0679419
$939$	0	0
$940$	0.118935	0.00387923
$941$	3.87248	0.126239	0.0631195	−	0.998006i	$-0.479895\pi$
$941$	3.87248	0.126239	0.0631195	+	0.998006i	$0.479895\pi$
$942$	0	0
$943$	−20.0179	−0.651873
$944$	7.86511	0.255988
$945$	0	0
$946$	−3.73952	−0.121582
$947$	−12.1136	−0.393638	−0.196819	−	0.980440i	$-0.563061\pi$
$947$	−12.1136	−0.393638	−0.196819	+	0.980440i	$0.563061\pi$
$948$	0	0
$949$	−20.5588	−0.667368
$950$	3.05261	0.0990399
$951$	0	0
$952$	−12.5095	−0.405434
$953$	6.51285	0.210972	0.105486	−	0.994421i	$-0.466360\pi$
$953$	6.51285	0.210972	0.105486	+	0.994421i	$0.466360\pi$
$954$	0	0
$955$	3.22737	0.104435
$956$	0.214690	0.00694357
$957$	0	0
$958$	0.838566	0.0270928
$959$	12.7729	0.412460
$960$	0	0
$961$	−16.8463	−0.543430
$962$	23.2713	0.750296
$963$	0	0
$964$	−0.348792	−0.0112338
$965$	44.4427	1.43066
$966$	0	0
$967$	20.1251	0.647179	0.323589	−	0.946198i	$-0.395110\pi$
$967$	20.1251	0.647179	0.323589	+	0.946198i	$0.395110\pi$
$968$	−51.0278	−1.64010
$969$	0	0
$970$	−6.70221	−0.215195
$971$	−24.9523	−0.800757	−0.400378	−	0.916350i	$-0.631121\pi$
$971$	−24.9523	−0.800757	−0.400378	+	0.916350i	$0.631121\pi$
$972$	0	0
$973$	2.40348	0.0770521
$974$	−30.0398	−0.962536
$975$	0	0
$976$	29.2351	0.935791
$977$	4.81212	0.153953	0.0769767	−	0.997033i	$-0.475473\pi$
$977$	4.81212	0.153953	0.0769767	+	0.997033i	$0.475473\pi$
$978$	0	0
$979$	22.4753	0.718314
$980$	0.0282216	0.000901505 0
$981$	0	0
$982$	34.8040	1.11064
$983$	−55.5892	−1.77302	−0.886511	−	0.462708i	$-0.846878\pi$
$983$	−55.5892	−1.77302	−0.886511	+	0.462708i	$0.846878\pi$
$984$	0	0
$985$	40.1916	1.28061
$986$	48.9349	1.55840
$987$	0	0
$988$	0.0678851	0.00215971
$989$	−2.57607	−0.0819142
$990$	0	0
$991$	40.9535	1.30093	0.650466	−	0.759536i	$-0.274574\pi$
$991$	40.9535	1.30093	0.650466	+	0.759536i	$0.274574\pi$
$992$	−0.291525	−0.00925593
$993$	0	0
$994$	1.64922	0.0523102
$995$	37.3879	1.18528
$996$	0	0
$997$	54.6480	1.73072	0.865360	−	0.501151i	$-0.167090\pi$
$997$	54.6480	1.73072	0.865360	+	0.501151i	$0.167090\pi$
$998$	−13.0632	−0.413507
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.s.1.13		16
3.2	odd	2		2667.2.a.n.1.4	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2667.2.a.n.1.4	✓	16	3.2	odd	2
8001.2.a.s.1.13		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)

\(f(q)\)	\(=\)	\(q+1.40936 q^{2} -0.0136985 q^{4} -2.06019 q^{5} -1.00000 q^{7} -2.83803 q^{8} +O(q^{10})\)	\(q+1.40936 q^{2} -0.0136985 q^{4} -2.06019 q^{5} -1.00000 q^{7} -2.83803 q^{8} -2.90356 q^{10} -5.38331 q^{11} +1.72882 q^{13} -1.40936 q^{14} -3.97242 q^{16} -4.40780 q^{17} -2.86650 q^{19} +0.0282216 q^{20} -7.58703 q^{22} -5.22653 q^{23} -0.755610 q^{25} +2.43653 q^{26} +0.0136985 q^{28} -7.87724 q^{29} -3.76214 q^{31} +0.0774892 q^{32} -6.21218 q^{34} +2.06019 q^{35} +9.55099 q^{37} -4.03993 q^{38} +5.84689 q^{40} +3.83006 q^{41} +0.492883 q^{43} +0.0737433 q^{44} -7.36607 q^{46} +4.21432 q^{47} +1.00000 q^{49} -1.06493 q^{50} -0.0236822 q^{52} +11.2226 q^{53} +11.0906 q^{55} +2.83803 q^{56} -11.1019 q^{58} -1.97993 q^{59} -7.35952 q^{61} -5.30221 q^{62} +8.05404 q^{64} -3.56169 q^{65} +1.47644 q^{67} +0.0603803 q^{68} +2.90356 q^{70} -1.17019 q^{71} -11.8919 q^{73} +13.4608 q^{74} +0.0392668 q^{76} +5.38331 q^{77} +8.50434 q^{79} +8.18394 q^{80} +5.39794 q^{82} +6.25858 q^{83} +9.08091 q^{85} +0.694651 q^{86} +15.2780 q^{88} -4.17500 q^{89} -1.72882 q^{91} +0.0715957 q^{92} +5.93951 q^{94} +5.90554 q^{95} +2.30828 q^{97} +1.40936 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 4 q^{2} + 20 q^{4} - 5 q^{5} - 16 q^{7} - 15 q^{8}+O(q^{10}) \) 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

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more