LMFDB - Embedded newform 8001.2.a.u.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:	$0$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - 6 x^{17} - 11 x^{16} + 123 x^{15} - 35 x^{14} - 982 x^{13} + 988 x^{12} + 3872 x^{11} + \cdots + 16 $ x^18 - 6x^17 - 11x^16 + 123x^15 - 35x^14 - 982x^13 + 988x^12 + 3872x^11 - 5421x^10 - 7882x^9 + 13376x^8 + 7948x^7 - 15795x^6 - 3858x^5 + 8199x^4 + 1453x^3 - 1610x^2 - 380*x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2\cdot 5 $
Twist minimal:	no (minimal twist has level 2667)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
Root			$0.981962$ 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)

$f(q)$	$=$	$q+0.981962 q^{2} -1.03575 q^{4} -2.59441 q^{5} +1.00000 q^{7} -2.98099 q^{8} +O(q^{10})$	\(q+0.981962 q^{2} -1.03575 q^{4} -2.59441 q^{5} +1.00000 q^{7} -2.98099 q^{8} -2.54761 q^{10} +0.832401 q^{11} -5.10508 q^{13} +0.981962 q^{14} -0.855717 q^{16} -7.10660 q^{17} -7.54643 q^{19} +2.68717 q^{20} +0.817386 q^{22} -8.33771 q^{23} +1.73097 q^{25} -5.01299 q^{26} -1.03575 q^{28} +8.62369 q^{29} -6.02299 q^{31} +5.12170 q^{32} -6.97841 q^{34} -2.59441 q^{35} -5.52425 q^{37} -7.41031 q^{38} +7.73392 q^{40} +8.68580 q^{41} +4.88483 q^{43} -0.862161 q^{44} -8.18731 q^{46} -2.27297 q^{47} +1.00000 q^{49} +1.69975 q^{50} +5.28759 q^{52} -3.81937 q^{53} -2.15959 q^{55} -2.98099 q^{56} +8.46813 q^{58} +3.88402 q^{59} -7.64885 q^{61} -5.91434 q^{62} +6.74075 q^{64} +13.2447 q^{65} +9.03188 q^{67} +7.36067 q^{68} -2.54761 q^{70} +9.47044 q^{71} -6.75232 q^{73} -5.42460 q^{74} +7.81623 q^{76} +0.832401 q^{77} +5.46202 q^{79} +2.22008 q^{80} +8.52913 q^{82} -9.63296 q^{83} +18.4374 q^{85} +4.79672 q^{86} -2.48138 q^{88} -9.34318 q^{89} -5.10508 q^{91} +8.63580 q^{92} -2.23196 q^{94} +19.5786 q^{95} -0.647975 q^{97} +0.981962 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q + 6 q^{2} + 22 q^{4} + 10 q^{5} + 18 q^{7} + 21 q^{8}+O(q^{10}) $ 18 * q + 6 * q^2 + 22 * q^4 + 10 * q^5 + 18 * q^7 + 21 * q^8	$ 18 q + 6 q^{2} + 22 q^{4} + 10 q^{5} + 18 q^{7} + 21 q^{8} - 4 q^{10} + 9 q^{11} - 25 q^{13} + 6 q^{14} + 34 q^{16} + 17 q^{17} - 5 q^{19} + 21 q^{20} + 5 q^{22} + 14 q^{23} + 28 q^{25} + 8 q^{26} + 22 q^{28} + 17 q^{29} + 5 q^{31} + 53 q^{32} - 19 q^{34} + 10 q^{35} - 15 q^{37} + 22 q^{38} - q^{40} + 17 q^{41} + q^{43} + 33 q^{44} + 10 q^{46} + 31 q^{47} + 18 q^{49} + 35 q^{50} - 70 q^{52} + 35 q^{53} + 4 q^{55} + 21 q^{56} + 3 q^{58} + 46 q^{59} - 5 q^{61} + 10 q^{62} + 63 q^{64} + 12 q^{65} + 6 q^{67} + 56 q^{68} - 4 q^{70} + 22 q^{71} - 16 q^{73} - 18 q^{74} + 32 q^{76} + 9 q^{77} + 46 q^{79} + 30 q^{80} - 12 q^{82} + 46 q^{83} + 4 q^{85} - 18 q^{86} + 30 q^{88} + 42 q^{89} - 25 q^{91} + 48 q^{92} + 3 q^{94} + 2 q^{95} - 35 q^{97} + 6 q^{98}+O(q^{100}) $ 18 * q + 6 * q^2 + 22 * q^4 + 10 * q^5 + 18 * q^7 + 21 * q^8 - 4 * q^10 + 9 * q^11 - 25 * q^13 + 6 * q^14 + 34 * q^16 + 17 * q^17 - 5 * q^19 + 21 * q^20 + 5 * q^22 + 14 * q^23 + 28 * q^25 + 8 * q^26 + 22 * q^28 + 17 * q^29 + 5 * q^31 + 53 * q^32 - 19 * q^34 + 10 * q^35 - 15 * q^37 + 22 * q^38 - q^40 + 17 * q^41 + q^43 + 33 * q^44 + 10 * q^46 + 31 * q^47 + 18 * q^49 + 35 * q^50 - 70 * q^52 + 35 * q^53 + 4 * q^55 + 21 * q^56 + 3 * q^58 + 46 * q^59 - 5 * q^61 + 10 * q^62 + 63 * q^64 + 12 * q^65 + 6 * q^67 + 56 * q^68 - 4 * q^70 + 22 * q^71 - 16 * q^73 - 18 * q^74 + 32 * q^76 + 9 * q^77 + 46 * q^79 + 30 * q^80 - 12 * q^82 + 46 * q^83 + 4 * q^85 - 18 * q^86 + 30 * q^88 + 42 * q^89 - 25 * q^91 + 48 * q^92 + 3 * q^94 + 2 * q^95 - 35 * q^97 + 6 * 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 $

$2$	0.981962	0.694352	0.347176	−	0.937800i	$-0.387141\pi$
$2$	0.981962	0.694352	0.347176	+	0.937800i	$0.387141\pi$
$3$	0	0
$3$	0	0
$4$	−1.03575	−0.517876
$5$	−2.59441	−1.16026	−0.580128	−	0.814525i	$-0.696998\pi$
$5$	−2.59441	−1.16026	−0.580128	+	0.814525i	$0.696998\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−2.98099	−1.05394
$9$	0	0
$10$	−2.54761	−0.805626
$11$	0.832401	0.250978	0.125489	−	0.992095i	$-0.459950\pi$
$11$	0.832401	0.250978	0.125489	+	0.992095i	$0.459950\pi$
$12$	0	0
$13$	−5.10508	−1.41589	−0.707947	−	0.706265i	$-0.750379\pi$
$13$	−5.10508	−1.41589	−0.707947	+	0.706265i	$0.750379\pi$
$14$	0.981962	0.262440
$15$	0	0
$16$	−0.855717	−0.213929
$17$	−7.10660	−1.72360	−0.861802	−	0.507245i	$-0.830664\pi$
$17$	−7.10660	−1.72360	−0.861802	+	0.507245i	$0.830664\pi$
$18$	0	0
$19$	−7.54643	−1.73127	−0.865635	−	0.500675i	$-0.833085\pi$
$19$	−7.54643	−1.73127	−0.865635	+	0.500675i	$0.833085\pi$
$20$	2.68717	0.600868
$21$	0	0
$22$	0.817386	0.174267
$23$	−8.33771	−1.73853	−0.869267	−	0.494344i	$-0.835408\pi$
$23$	−8.33771	−1.73853	−0.869267	+	0.494344i	$0.835408\pi$
$24$	0	0
$25$	1.73097	0.346195
$26$	−5.01299	−0.983129
$27$	0	0
$28$	−1.03575	−0.195739
$29$	8.62369	1.60138	0.800689	−	0.599080i	$-0.204467\pi$
$29$	8.62369	1.60138	0.800689	+	0.599080i	$0.204467\pi$
$30$	0	0
$31$	−6.02299	−1.08176	−0.540880	−	0.841100i	$-0.681909\pi$
$31$	−6.02299	−1.08176	−0.540880	+	0.841100i	$0.681909\pi$
$32$	5.12170	0.905397
$33$	0	0
$34$	−6.97841	−1.19679
$35$	−2.59441	−0.438536
$36$	0	0
$37$	−5.52425	−0.908181	−0.454090	−	0.890956i	$-0.650036\pi$
$37$	−5.52425	−0.908181	−0.454090	+	0.890956i	$0.650036\pi$
$38$	−7.41031	−1.20211
$39$	0	0
$40$	7.73392	1.22284
$41$	8.68580	1.35649	0.678247	−	0.734834i	$-0.262740\pi$
$41$	8.68580	1.35649	0.678247	+	0.734834i	$0.262740\pi$
$42$	0	0
$43$	4.88483	0.744930	0.372465	−	0.928046i	$-0.378513\pi$
$43$	4.88483	0.744930	0.372465	+	0.928046i	$0.378513\pi$
$44$	−0.862161	−0.129976
$45$	0	0
$46$	−8.18731	−1.20715
$47$	−2.27297	−0.331546	−0.165773	−	0.986164i	$-0.553012\pi$
$47$	−2.27297	−0.331546	−0.165773	+	0.986164i	$0.553012\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	1.69975	0.240381
$51$	0	0
$52$	5.28759	0.733257
$53$	−3.81937	−0.524631	−0.262316	−	0.964982i	$-0.584486\pi$
$53$	−3.81937	−0.524631	−0.262316	+	0.964982i	$0.584486\pi$
$54$	0	0
$55$	−2.15959	−0.291199
$56$	−2.98099	−0.398352
$57$	0	0
$58$	8.46813	1.11192
$59$	3.88402	0.505656	0.252828	−	0.967511i	$-0.418639\pi$
$59$	3.88402	0.505656	0.252828	+	0.967511i	$0.418639\pi$
$60$	0	0
$61$	−7.64885	−0.979335	−0.489668	−	0.871909i	$-0.662882\pi$
$61$	−7.64885	−0.979335	−0.489668	+	0.871909i	$0.662882\pi$
$62$	−5.91434	−0.751122
$63$	0	0
$64$	6.74075	0.842594
$65$	13.2447	1.64280
$66$	0	0
$67$	9.03188	1.10342	0.551710	−	0.834036i	$-0.313975\pi$
$67$	9.03188	1.10342	0.551710	+	0.834036i	$0.313975\pi$
$68$	7.36067	0.892612
$69$	0	0
$70$	−2.54761	−0.304498
$71$	9.47044	1.12393	0.561967	−	0.827160i	$-0.310045\pi$
$71$	9.47044	1.12393	0.561967	+	0.827160i	$0.310045\pi$
$72$	0	0
$73$	−6.75232	−0.790300	−0.395150	−	0.918617i	$-0.629307\pi$
$73$	−6.75232	−0.790300	−0.395150	+	0.918617i	$0.629307\pi$
$74$	−5.42460	−0.630597
$75$	0	0
$76$	7.81623	0.896583
$77$	0.832401	0.0948609
$78$	0	0
$79$	5.46202	0.614525	0.307262	−	0.951625i	$-0.400587\pi$
$79$	5.46202	0.614525	0.307262	+	0.951625i	$0.400587\pi$
$80$	2.22008	0.248213
$81$	0	0
$82$	8.52913	0.941885
$83$	−9.63296	−1.05735	−0.528677	−	0.848823i	$-0.677312\pi$
$83$	−9.63296	−1.05735	−0.528677	+	0.848823i	$0.677312\pi$
$84$	0	0
$85$	18.4374	1.99982
$86$	4.79672	0.517244
$87$	0	0
$88$	−2.48138	−0.264516
$89$	−9.34318	−0.990375	−0.495188	−	0.868786i	$-0.664901\pi$
$89$	−9.34318	−0.990375	−0.495188	+	0.868786i	$0.664901\pi$
$90$	0	0
$91$	−5.10508	−0.535158
$92$	8.63580	0.900344
$93$	0	0
$94$	−2.23196	−0.230210
$95$	19.5786	2.00872
$96$	0	0
$97$	−0.647975	−0.0657918	−0.0328959	−	0.999459i	$-0.510473\pi$
$97$	−0.647975	−0.0657918	−0.0328959	+	0.999459i	$0.510473\pi$
$98$	0.981962	0.0991931
$99$	0	0
$100$	−1.79286	−0.179286
$101$	−8.32814	−0.828681	−0.414340	−	0.910122i	$-0.635988\pi$
$101$	−8.32814	−0.828681	−0.414340	+	0.910122i	$0.635988\pi$
$102$	0	0
$103$	−18.0685	−1.78034	−0.890169	−	0.455631i	$-0.849414\pi$
$103$	−18.0685	−1.78034	−0.890169	+	0.455631i	$0.849414\pi$
$104$	15.2182	1.49227
$105$	0	0
$106$	−3.75048	−0.364279
$107$	−11.8426	−1.14487	−0.572434	−	0.819951i	$-0.694001\pi$
$107$	−11.8426	−1.14487	−0.572434	+	0.819951i	$0.694001\pi$
$108$	0	0
$109$	2.81629	0.269752	0.134876	−	0.990863i	$-0.456936\pi$
$109$	2.81629	0.269752	0.134876	+	0.990863i	$0.456936\pi$
$110$	−2.12064	−0.202195
$111$	0	0
$112$	−0.855717	−0.0808577
$113$	−19.7700	−1.85980	−0.929901	−	0.367811i	$-0.880107\pi$
$113$	−19.7700	−1.85980	−0.929901	+	0.367811i	$0.880107\pi$
$114$	0	0
$115$	21.6315	2.01714
$116$	−8.93199	−0.829315
$117$	0	0
$118$	3.81395	0.351103
$119$	−7.10660	−0.651461
$120$	0	0
$121$	−10.3071	−0.937010
$122$	−7.51088	−0.680003
$123$	0	0
$124$	6.23832	0.560217
$125$	8.48120	0.758582
$126$	0	0
$127$	1.00000	0.0887357
$127$	1.00000	0.0887357
$128$	−3.62425	−0.320341
$129$	0	0
$130$	13.0058	1.14068
$131$	22.1245	1.93302	0.966512	−	0.256622i	$-0.0826095\pi$
$131$	22.1245	1.93302	0.966512	+	0.256622i	$0.0826095\pi$
$132$	0	0
$133$	−7.54643	−0.654359
$134$	8.86896	0.766162
$135$	0	0
$136$	21.1847	1.81657
$137$	9.68182	0.827174	0.413587	−	0.910465i	$-0.364276\pi$
$137$	9.68182	0.827174	0.413587	+	0.910465i	$0.364276\pi$
$138$	0	0
$139$	4.55040	0.385960	0.192980	−	0.981203i	$-0.438185\pi$
$139$	4.55040	0.385960	0.192980	+	0.981203i	$0.438185\pi$
$140$	2.68717	0.227107
$141$	0	0
$142$	9.29961	0.780406
$143$	−4.24948	−0.355359
$144$	0	0
$145$	−22.3734	−1.85801
$146$	−6.63052	−0.548746
$147$	0	0
$148$	5.72175	0.470325
$149$	23.9402	1.96126	0.980628	−	0.195880i	$-0.0627562\pi$
$149$	23.9402	1.96126	0.980628	+	0.195880i	$0.0627562\pi$
$150$	0	0
$151$	−11.2109	−0.912329	−0.456164	−	0.889896i	$-0.650777\pi$
$151$	−11.2109	−0.912329	−0.456164	+	0.889896i	$0.650777\pi$
$152$	22.4959	1.82465
$153$	0	0
$154$	0.817386	0.0658669
$155$	15.6261	1.25512
$156$	0	0
$157$	−0.154475	−0.0123284	−0.00616422	−	0.999981i	$-0.501962\pi$
$157$	−0.154475	−0.0123284	−0.00616422	+	0.999981i	$0.501962\pi$
$158$	5.36349	0.426696
$159$	0	0
$160$	−13.2878	−1.05049
$161$	−8.33771	−0.657104
$162$	0	0
$163$	−11.4755	−0.898831	−0.449415	−	0.893323i	$-0.648368\pi$
$163$	−11.4755	−0.898831	−0.449415	+	0.893323i	$0.648368\pi$
$164$	−8.99633	−0.702496
$165$	0	0
$166$	−9.45920	−0.734176
$167$	1.57208	0.121651	0.0608255	−	0.998148i	$-0.480627\pi$
$167$	1.57208	0.121651	0.0608255	+	0.998148i	$0.480627\pi$
$168$	0	0
$169$	13.0619	1.00476
$170$	18.1049	1.38858
$171$	0	0
$172$	−5.05947	−0.385781
$173$	24.4378	1.85797	0.928985	−	0.370117i	$-0.120682\pi$
$173$	24.4378	1.85797	0.928985	+	0.370117i	$0.120682\pi$
$174$	0	0
$175$	1.73097	0.130849
$176$	−0.712300	−0.0536916
$177$	0	0
$178$	−9.17464	−0.687669
$179$	8.38681	0.626860	0.313430	−	0.949611i	$-0.398522\pi$
$179$	8.38681	0.626860	0.313430	+	0.949611i	$0.398522\pi$
$180$	0	0
$181$	−11.9980	−0.891806	−0.445903	−	0.895081i	$-0.647117\pi$
$181$	−11.9980	−0.891806	−0.445903	+	0.895081i	$0.647117\pi$
$182$	−5.01299	−0.371588
$183$	0	0
$184$	24.8546	1.83231
$185$	14.3322	1.05372
$186$	0	0
$187$	−5.91554	−0.432587
$188$	2.35423	0.171700
$189$	0	0
$190$	19.2254	1.39476
$191$	3.62612	0.262377	0.131188	−	0.991357i	$-0.458121\pi$
$191$	3.62612	0.262377	0.131188	+	0.991357i	$0.458121\pi$
$192$	0	0
$193$	−20.6742	−1.48816	−0.744081	−	0.668089i	$-0.767112\pi$
$193$	−20.6742	−1.48816	−0.744081	+	0.668089i	$0.767112\pi$
$194$	−0.636286	−0.0456827
$195$	0	0
$196$	−1.03575	−0.0739822
$197$	22.7186	1.61863	0.809315	−	0.587374i	$-0.199838\pi$
$197$	22.7186	1.61863	0.809315	+	0.587374i	$0.199838\pi$
$198$	0	0
$199$	13.5397	0.959803	0.479901	−	0.877322i	$-0.340672\pi$
$199$	13.5397	0.959803	0.479901	+	0.877322i	$0.340672\pi$
$200$	−5.16002	−0.364868
$201$	0	0
$202$	−8.17791	−0.575396
$203$	8.62369	0.605264
$204$	0	0
$205$	−22.5346	−1.57388
$206$	−17.7425	−1.23618
$207$	0	0
$208$	4.36850	0.302901
$209$	−6.28166	−0.434512
$210$	0	0
$211$	−0.844287	−0.0581231	−0.0290615	−	0.999578i	$-0.509252\pi$
$211$	−0.844287	−0.0581231	−0.0290615	+	0.999578i	$0.509252\pi$
$212$	3.95592	0.271694
$213$	0	0
$214$	−11.6290	−0.794941
$215$	−12.6733	−0.864310
$216$	0	0
$217$	−6.02299	−0.408867
$218$	2.76549	0.187303
$219$	0	0
$220$	2.23680	0.150805
$221$	36.2798	2.44044
$222$	0	0
$223$	1.45728	0.0975867	0.0487934	−	0.998809i	$-0.484462\pi$
$223$	1.45728	0.0975867	0.0487934	+	0.998809i	$0.484462\pi$
$224$	5.12170	0.342208
$225$	0	0
$226$	−19.4133	−1.29136
$227$	−19.2589	−1.27825	−0.639127	−	0.769101i	$-0.720704\pi$
$227$	−19.2589	−1.27825	−0.639127	+	0.769101i	$0.720704\pi$
$228$	0	0
$229$	−9.93419	−0.656470	−0.328235	−	0.944596i	$-0.606454\pi$
$229$	−9.93419	−0.656470	−0.328235	+	0.944596i	$0.606454\pi$
$230$	21.2413	1.40061
$231$	0	0
$232$	−25.7071	−1.68776
$233$	16.6806	1.09278	0.546390	−	0.837531i	$-0.316001\pi$
$233$	16.6806	1.09278	0.546390	+	0.837531i	$0.316001\pi$
$234$	0	0
$235$	5.89701	0.384678
$236$	−4.02287	−0.261867
$237$	0	0
$238$	−6.97841	−0.452343
$239$	−7.56639	−0.489429	−0.244715	−	0.969595i	$-0.578694\pi$
$239$	−7.56639	−0.489429	−0.244715	+	0.969595i	$0.578694\pi$
$240$	0	0
$241$	3.45538	0.222580	0.111290	−	0.993788i	$-0.464502\pi$
$241$	3.45538	0.222580	0.111290	+	0.993788i	$0.464502\pi$
$242$	−10.1212	−0.650614
$243$	0	0
$244$	7.92231	0.507174
$245$	−2.59441	−0.165751
$246$	0	0
$247$	38.5251	2.45130
$248$	17.9545	1.14011
$249$	0	0
$250$	8.32822	0.526723
$251$	3.24187	0.204625	0.102313	−	0.994752i	$-0.467376\pi$
$251$	3.24187	0.204625	0.102313	+	0.994752i	$0.467376\pi$
$252$	0	0
$253$	−6.94032	−0.436334
$254$	0.981962	0.0616138
$255$	0	0
$256$	−17.0404	−1.06502
$257$	−2.15313	−0.134308	−0.0671542	−	0.997743i	$-0.521392\pi$
$257$	−2.15313	−0.134308	−0.0671542	+	0.997743i	$0.521392\pi$
$258$	0	0
$259$	−5.52425	−0.343260
$260$	−13.7182	−0.850766
$261$	0	0
$262$	21.7254	1.34220
$263$	−12.2970	−0.758264	−0.379132	−	0.925343i	$-0.623777\pi$
$263$	−12.2970	−0.758264	−0.379132	+	0.925343i	$0.623777\pi$
$264$	0	0
$265$	9.90903	0.608707
$266$	−7.41031	−0.454355
$267$	0	0
$268$	−9.35478	−0.571434
$269$	−28.2317	−1.72132	−0.860659	−	0.509183i	$-0.829948\pi$
$269$	−28.2317	−1.72132	−0.860659	+	0.509183i	$0.829948\pi$
$270$	0	0
$271$	8.79041	0.533980	0.266990	−	0.963699i	$-0.413971\pi$
$271$	8.79041	0.533980	0.266990	+	0.963699i	$0.413971\pi$
$272$	6.08124	0.368729
$273$	0	0
$274$	9.50718	0.574350
$275$	1.44086	0.0868874
$276$	0	0
$277$	−7.19056	−0.432039	−0.216019	−	0.976389i	$-0.569307\pi$
$277$	−7.19056	−0.432039	−0.216019	+	0.976389i	$0.569307\pi$
$278$	4.46832	0.267992
$279$	0	0
$280$	7.73392	0.462190
$281$	0.601253	0.0358678	0.0179339	−	0.999839i	$-0.494291\pi$
$281$	0.601253	0.0358678	0.0179339	+	0.999839i	$0.494291\pi$
$282$	0	0
$283$	32.0154	1.90312	0.951560	−	0.307462i	$-0.0994797\pi$
$283$	32.0154	1.90312	0.951560	+	0.307462i	$0.0994797\pi$
$284$	−9.80902	−0.582058
$285$	0	0
$286$	−4.17282	−0.246744
$287$	8.68580	0.512707
$288$	0	0
$289$	33.5038	1.97081
$290$	−21.9698	−1.29011
$291$	0	0
$292$	6.99373	0.409277
$293$	23.4909	1.37235	0.686175	−	0.727436i	$-0.259288\pi$
$293$	23.4909	1.37235	0.686175	+	0.727436i	$0.259288\pi$
$294$	0	0
$295$	−10.0767	−0.586690
$296$	16.4677	0.957168
$297$	0	0
$298$	23.5083	1.36180
$299$	42.5647	2.46158
$300$	0	0
$301$	4.88483	0.281557
$302$	−11.0087	−0.633477
$303$	0	0
$304$	6.45761	0.370369
$305$	19.8443	1.13628
$306$	0	0
$307$	−17.5946	−1.00418	−0.502088	−	0.864816i	$-0.667435\pi$
$307$	−17.5946	−1.00418	−0.502088	+	0.864816i	$0.667435\pi$
$308$	−0.862161	−0.0491262
$309$	0	0
$310$	15.3442	0.871494
$311$	19.6113	1.11205	0.556026	−	0.831165i	$-0.312325\pi$
$311$	19.6113	1.11205	0.556026	+	0.831165i	$0.312325\pi$
$312$	0	0
$313$	−16.1312	−0.911787	−0.455893	−	0.890034i	$-0.650680\pi$
$313$	−16.1312	−0.911787	−0.455893	+	0.890034i	$0.650680\pi$
$314$	−0.151688	−0.00856028
$315$	0	0
$316$	−5.65729	−0.318247
$317$	−6.34583	−0.356417	−0.178209	−	0.983993i	$-0.557030\pi$
$317$	−6.34583	−0.356417	−0.178209	+	0.983993i	$0.557030\pi$
$318$	0	0
$319$	7.17837	0.401912
$320$	−17.4883	−0.977625
$321$	0	0
$322$	−8.18731	−0.456261
$323$	53.6295	2.98402
$324$	0	0
$325$	−8.83676	−0.490175
$326$	−11.2685	−0.624105
$327$	0	0
$328$	−25.8923	−1.42966
$329$	−2.27297	−0.125313
$330$	0	0
$331$	−1.36803	−0.0751938	−0.0375969	−	0.999293i	$-0.511970\pi$
$331$	−1.36803	−0.0751938	−0.0375969	+	0.999293i	$0.511970\pi$
$332$	9.97735	0.547578
$333$	0	0
$334$	1.54372	0.0844686
$335$	−23.4324	−1.28025
$336$	0	0
$337$	−10.0731	−0.548719	−0.274359	−	0.961627i	$-0.588466\pi$
$337$	−10.0731	−0.548719	−0.274359	+	0.961627i	$0.588466\pi$
$338$	12.8262	0.697655
$339$	0	0
$340$	−19.0966	−1.03566
$341$	−5.01354	−0.271499
$342$	0	0
$343$	1.00000	0.0539949
$344$	−14.5616	−0.785111
$345$	0	0
$346$	23.9970	1.29009
$347$	−11.8342	−0.635295	−0.317647	−	0.948209i	$-0.602893\pi$
$347$	−11.8342	−0.635295	−0.317647	+	0.948209i	$0.602893\pi$
$348$	0	0
$349$	13.7100	0.733880	0.366940	−	0.930245i	$-0.380405\pi$
$349$	13.7100	0.733880	0.366940	+	0.930245i	$0.380405\pi$
$350$	1.69975	0.0908554
$351$	0	0
$352$	4.26331	0.227235
$353$	24.2687	1.29169	0.645846	−	0.763468i	$-0.276505\pi$
$353$	24.2687	1.29169	0.645846	+	0.763468i	$0.276505\pi$
$354$	0	0
$355$	−24.5702	−1.30405
$356$	9.67721	0.512891
$357$	0	0
$358$	8.23553	0.435261
$359$	−4.37238	−0.230765	−0.115383	−	0.993321i	$-0.536809\pi$
$359$	−4.37238	−0.230765	−0.115383	+	0.993321i	$0.536809\pi$
$360$	0	0
$361$	37.9486	1.99730
$362$	−11.7816	−0.619227
$363$	0	0
$364$	5.28759	0.277145
$365$	17.5183	0.916950
$366$	0	0
$367$	22.4911	1.17403	0.587013	−	0.809578i	$-0.300304\pi$
$367$	22.4911	1.17403	0.587013	+	0.809578i	$0.300304\pi$
$368$	7.13472	0.371923
$369$	0	0
$370$	14.0736	0.731654
$371$	−3.81937	−0.198292
$372$	0	0
$373$	12.0199	0.622367	0.311183	−	0.950350i	$-0.399275\pi$
$373$	12.0199	0.622367	0.311183	+	0.950350i	$0.399275\pi$
$374$	−5.80884	−0.300368
$375$	0	0
$376$	6.77569	0.349430
$377$	−44.0246	−2.26738
$378$	0	0
$379$	−3.53055	−0.181352	−0.0906761	−	0.995880i	$-0.528903\pi$
$379$	−3.53055	−0.181352	−0.0906761	+	0.995880i	$0.528903\pi$
$380$	−20.2785	−1.04027
$381$	0	0
$382$	3.56071	0.182182
$383$	−27.7234	−1.41660	−0.708299	−	0.705913i	$-0.750537\pi$
$383$	−27.7234	−1.41660	−0.708299	+	0.705913i	$0.750537\pi$
$384$	0	0
$385$	−2.15959	−0.110063
$386$	−20.3013	−1.03331
$387$	0	0
$388$	0.671140	0.0340720
$389$	3.79519	0.192424	0.0962118	−	0.995361i	$-0.469327\pi$
$389$	3.79519	0.192424	0.0962118	+	0.995361i	$0.469327\pi$
$390$	0	0
$391$	59.2528	2.99654
$392$	−2.98099	−0.150563
$393$	0	0
$394$	22.3088	1.12390
$395$	−14.1707	−0.713006
$396$	0	0
$397$	−21.7337	−1.09078	−0.545392	−	0.838181i	$-0.683619\pi$
$397$	−21.7337	−1.09078	−0.545392	+	0.838181i	$0.683619\pi$
$398$	13.2955	0.666441
$399$	0	0
$400$	−1.48122	−0.0740612
$401$	1.98537	0.0991445	0.0495723	−	0.998771i	$-0.484214\pi$
$401$	1.98537	0.0991445	0.0495723	+	0.998771i	$0.484214\pi$
$402$	0	0
$403$	30.7478	1.53166
$404$	8.62588	0.429154
$405$	0	0
$406$	8.46813	0.420266
$407$	−4.59839	−0.227934
$408$	0	0
$409$	−36.7841	−1.81886	−0.909429	−	0.415859i	$-0.863481\pi$
$409$	−36.7841	−1.81886	−0.909429	+	0.415859i	$0.863481\pi$
$410$	−22.1281	−1.09283
$411$	0	0
$412$	18.7144	0.921993
$413$	3.88402	0.191120
$414$	0	0
$415$	24.9919	1.22680
$416$	−26.1467	−1.28195
$417$	0	0
$418$	−6.16835	−0.301704
$419$	1.04076	0.0508444	0.0254222	−	0.999677i	$-0.491907\pi$
$419$	1.04076	0.0508444	0.0254222	+	0.999677i	$0.491907\pi$
$420$	0	0
$421$	−7.51888	−0.366448	−0.183224	−	0.983071i	$-0.558653\pi$
$421$	−7.51888	−0.366448	−0.183224	+	0.983071i	$0.558653\pi$
$422$	−0.829057	−0.0403579
$423$	0	0
$424$	11.3855	0.552930
$425$	−12.3013	−0.596702
$426$	0	0
$427$	−7.64885	−0.370154
$428$	12.2660	0.592899
$429$	0	0
$430$	−12.4447	−0.600135
$431$	27.1064	1.30567	0.652834	−	0.757501i	$-0.273580\pi$
$431$	27.1064	1.30567	0.652834	+	0.757501i	$0.273580\pi$
$432$	0	0
$433$	26.6125	1.27891	0.639457	−	0.768827i	$-0.279159\pi$
$433$	26.6125	1.27891	0.639457	+	0.768827i	$0.279159\pi$
$434$	−5.91434	−0.283898
$435$	0	0
$436$	−2.91698	−0.139698
$437$	62.9200	3.00987
$438$	0	0
$439$	−3.69046	−0.176136	−0.0880681	−	0.996114i	$-0.528069\pi$
$439$	−3.69046	−0.176136	−0.0880681	+	0.996114i	$0.528069\pi$
$440$	6.43773	0.306907
$441$	0	0
$442$	35.6253	1.69452
$443$	9.76106	0.463762	0.231881	−	0.972744i	$-0.425512\pi$
$443$	9.76106	0.463762	0.231881	+	0.972744i	$0.425512\pi$
$444$	0	0
$445$	24.2401	1.14909
$446$	1.43099	0.0677595
$447$	0	0
$448$	6.74075	0.318470
$449$	−41.3985	−1.95372	−0.976858	−	0.213887i	$-0.931388\pi$
$449$	−41.3985	−1.95372	−0.976858	+	0.213887i	$0.931388\pi$
$450$	0	0
$451$	7.23008	0.340451
$452$	20.4768	0.963146
$453$	0	0
$454$	−18.9115	−0.887559
$455$	13.2447	0.620920
$456$	0	0
$457$	26.3294	1.23164	0.615820	−	0.787887i	$-0.288825\pi$
$457$	26.3294	1.23164	0.615820	+	0.787887i	$0.288825\pi$
$458$	−9.75500	−0.455821
$459$	0	0
$460$	−22.4048	−1.04463
$461$	27.2734	1.27025	0.635125	−	0.772409i	$-0.280948\pi$
$461$	27.2734	1.27025	0.635125	+	0.772409i	$0.280948\pi$
$462$	0	0
$463$	−37.5923	−1.74706	−0.873531	−	0.486769i	$-0.838175\pi$
$463$	−37.5923	−1.74706	−0.873531	+	0.486769i	$0.838175\pi$
$464$	−7.37944	−0.342582
$465$	0	0
$466$	16.3797	0.758774
$467$	−25.9161	−1.19925	−0.599626	−	0.800280i	$-0.704684\pi$
$467$	−25.9161	−1.19925	−0.599626	+	0.800280i	$0.704684\pi$
$468$	0	0
$469$	9.03188	0.417054
$470$	5.79064	0.267102
$471$	0	0
$472$	−11.5782	−0.532931
$473$	4.06614	0.186961
$474$	0	0
$475$	−13.0627	−0.599357
$476$	7.36067	0.337376
$477$	0	0
$478$	−7.42991	−0.339836
$479$	13.5335	0.618361	0.309180	−	0.951003i	$-0.399945\pi$
$479$	13.5335	0.618361	0.309180	+	0.951003i	$0.399945\pi$
$480$	0	0
$481$	28.2017	1.28589
$482$	3.39305	0.154549
$483$	0	0
$484$	10.6756	0.485255
$485$	1.68111	0.0763354
$486$	0	0
$487$	17.0640	0.773243	0.386622	−	0.922238i	$-0.373642\pi$
$487$	17.0640	0.773243	0.386622	+	0.922238i	$0.373642\pi$
$488$	22.8012	1.03216
$489$	0	0
$490$	−2.54761	−0.115089
$491$	−40.2689	−1.81731	−0.908655	−	0.417547i	$-0.862890\pi$
$491$	−40.2689	−1.81731	−0.908655	+	0.417547i	$0.862890\pi$
$492$	0	0
$493$	−61.2851	−2.76014
$494$	37.8302	1.70206
$495$	0	0
$496$	5.15397	0.231420
$497$	9.47044	0.424807
$498$	0	0
$499$	−11.2792	−0.504927	−0.252463	−	0.967606i	$-0.581241\pi$
$499$	−11.2792	−0.504927	−0.252463	+	0.967606i	$0.581241\pi$
$500$	−8.78441	−0.392851
$501$	0	0
$502$	3.18339	0.142082
$503$	0.345174	0.0153905	0.00769527	−	0.999970i	$-0.497550\pi$
$503$	0.345174	0.0153905	0.00769527	+	0.999970i	$0.497550\pi$
$504$	0	0
$505$	21.6066	0.961482
$506$	−6.81513	−0.302970
$507$	0	0
$508$	−1.03575	−0.0459540
$509$	36.5704	1.62095	0.810476	−	0.585771i	$-0.199208\pi$
$509$	36.5704	1.62095	0.810476	+	0.585771i	$0.199208\pi$
$510$	0	0
$511$	−6.75232	−0.298705
$512$	−9.48450	−0.419160
$513$	0	0
$514$	−2.11429	−0.0932573
$515$	46.8770	2.06565
$516$	0	0
$517$	−1.89202	−0.0832109
$518$	−5.42460	−0.238343
$519$	0	0
$520$	−39.4823	−1.73141
$521$	−5.93817	−0.260156	−0.130078	−	0.991504i	$-0.541523\pi$
$521$	−5.93817	−0.260156	−0.130078	+	0.991504i	$0.541523\pi$
$522$	0	0
$523$	−40.8155	−1.78474	−0.892368	−	0.451309i	$-0.850957\pi$
$523$	−40.8155	−1.78474	−0.892368	+	0.451309i	$0.850957\pi$
$524$	−22.9154	−1.00107
$525$	0	0
$526$	−12.0752	−0.526502
$527$	42.8030	1.86453
$528$	0	0
$529$	46.5175	2.02250
$530$	9.73029	0.422657
$531$	0	0
$532$	7.81623	0.338876
$533$	−44.3417	−1.92065
$534$	0	0
$535$	30.7246	1.32834
$536$	−26.9240	−1.16294
$537$	0	0
$538$	−27.7224	−1.19520
$539$	0.832401	0.0358541
$540$	0	0
$541$	−29.7773	−1.28023	−0.640114	−	0.768280i	$-0.721113\pi$
$541$	−29.7773	−1.28023	−0.640114	+	0.768280i	$0.721113\pi$
$542$	8.63185	0.370770
$543$	0	0
$544$	−36.3979	−1.56055
$545$	−7.30662	−0.312981
$546$	0	0
$547$	−32.4777	−1.38865	−0.694323	−	0.719663i	$-0.744296\pi$
$547$	−32.4777	−1.38865	−0.694323	+	0.719663i	$0.744296\pi$
$548$	−10.0280	−0.428373
$549$	0	0
$550$	1.41487	0.0603304
$551$	−65.0781	−2.77242
$552$	0	0
$553$	5.46202	0.232269
$554$	−7.06085	−0.299987
$555$	0	0
$556$	−4.71308	−0.199879
$557$	−7.30357	−0.309462	−0.154731	−	0.987957i	$-0.549451\pi$
$557$	−7.30357	−0.309462	−0.154731	+	0.987957i	$0.549451\pi$
$558$	0	0
$559$	−24.9375	−1.05474
$560$	2.22008	0.0938156
$561$	0	0
$562$	0.590408	0.0249049
$563$	−20.9917	−0.884695	−0.442348	−	0.896844i	$-0.645854\pi$
$563$	−20.9917	−0.884695	−0.442348	+	0.896844i	$0.645854\pi$
$564$	0	0
$565$	51.2914	2.15785
$566$	31.4379	1.32144
$567$	0	0
$568$	−28.2313	−1.18456
$569$	9.59641	0.402302	0.201151	−	0.979560i	$-0.435532\pi$
$569$	9.59641	0.402302	0.201151	+	0.979560i	$0.435532\pi$
$570$	0	0
$571$	−29.0968	−1.21766	−0.608831	−	0.793300i	$-0.708361\pi$
$571$	−29.0968	−1.21766	−0.608831	+	0.793300i	$0.708361\pi$
$572$	4.40140	0.184032
$573$	0	0
$574$	8.52913	0.355999
$575$	−14.4324	−0.601871
$576$	0	0
$577$	35.6413	1.48377	0.741883	−	0.670530i	$-0.233933\pi$
$577$	35.6413	1.48377	0.741883	+	0.670530i	$0.233933\pi$
$578$	32.8994	1.36843
$579$	0	0
$580$	23.1733	0.962218
$581$	−9.63296	−0.399642
$582$	0	0
$583$	−3.17925	−0.131671
$584$	20.1286	0.832928
$585$	0	0
$586$	23.0671	0.952894
$587$	21.8864	0.903349	0.451674	−	0.892183i	$-0.350827\pi$
$587$	21.8864	0.903349	0.451674	+	0.892183i	$0.350827\pi$
$588$	0	0
$589$	45.4521	1.87282
$590$	−9.89497	−0.407369
$591$	0	0
$592$	4.72719	0.194286
$593$	6.80864	0.279597	0.139799	−	0.990180i	$-0.455354\pi$
$593$	6.80864	0.279597	0.139799	+	0.990180i	$0.455354\pi$
$594$	0	0
$595$	18.4374	0.755862
$596$	−24.7961	−1.01569
$597$	0	0
$598$	41.7969	1.70920
$599$	27.9609	1.14245	0.571225	−	0.820794i	$-0.306468\pi$
$599$	27.9609	1.14245	0.571225	+	0.820794i	$0.306468\pi$
$600$	0	0
$601$	9.23744	0.376803	0.188401	−	0.982092i	$-0.439669\pi$
$601$	9.23744	0.376803	0.188401	+	0.982092i	$0.439669\pi$
$602$	4.79672	0.195500
$603$	0	0
$604$	11.6117	0.472473
$605$	26.7409	1.08717
$606$	0	0
$607$	−6.28025	−0.254907	−0.127454	−	0.991845i	$-0.540680\pi$
$607$	−6.28025	−0.254907	−0.127454	+	0.991845i	$0.540680\pi$
$608$	−38.6506	−1.56749
$609$	0	0
$610$	19.4863	0.788978
$611$	11.6037	0.469434
$612$	0	0
$613$	−15.5564	−0.628316	−0.314158	−	0.949371i	$-0.601722\pi$
$613$	−15.5564	−0.628316	−0.314158	+	0.949371i	$0.601722\pi$
$614$	−17.2772	−0.697252
$615$	0	0
$616$	−2.48138	−0.0999777
$617$	−19.0545	−0.767106	−0.383553	−	0.923519i	$-0.625300\pi$
$617$	−19.0545	−0.767106	−0.383553	+	0.923519i	$0.625300\pi$
$618$	0	0
$619$	24.7065	0.993037	0.496519	−	0.868026i	$-0.334611\pi$
$619$	24.7065	0.993037	0.496519	+	0.868026i	$0.334611\pi$
$620$	−16.1848	−0.649996
$621$	0	0
$622$	19.2575	0.772155
$623$	−9.34318	−0.374327
$624$	0	0
$625$	−30.6586	−1.22634
$626$	−15.8402	−0.633101
$627$	0	0
$628$	0.159998	0.00638460
$629$	39.2586	1.56534
$630$	0	0
$631$	−12.9423	−0.515224	−0.257612	−	0.966248i	$-0.582936\pi$
$631$	−12.9423	−0.515224	−0.257612	+	0.966248i	$0.582936\pi$
$632$	−16.2822	−0.647672
$633$	0	0
$634$	−6.23136	−0.247479
$635$	−2.59441	−0.102956
$636$	0	0
$637$	−5.10508	−0.202271
$638$	7.04888	0.279068
$639$	0	0
$640$	9.40279	0.371678
$641$	−27.1939	−1.07409	−0.537047	−	0.843553i	$-0.680460\pi$
$641$	−27.1939	−1.07409	−0.537047	+	0.843553i	$0.680460\pi$
$642$	0	0
$643$	4.58228	0.180707	0.0903537	−	0.995910i	$-0.471200\pi$
$643$	4.58228	0.180707	0.0903537	+	0.995910i	$0.471200\pi$
$644$	8.63580	0.340298
$645$	0	0
$646$	52.6621	2.07196
$647$	12.4292	0.488642	0.244321	−	0.969694i	$-0.421435\pi$
$647$	12.4292	0.488642	0.244321	+	0.969694i	$0.421435\pi$
$648$	0	0
$649$	3.23306	0.126909
$650$	−8.67736	−0.340354
$651$	0	0
$652$	11.8858	0.465482
$653$	−7.35908	−0.287983	−0.143992	−	0.989579i	$-0.545994\pi$
$653$	−7.35908	−0.287983	−0.143992	+	0.989579i	$0.545994\pi$
$654$	0	0
$655$	−57.4000	−2.24280
$656$	−7.43259	−0.290194
$657$	0	0
$658$	−2.23196	−0.0870111
$659$	−22.2982	−0.868613	−0.434307	−	0.900765i	$-0.643007\pi$
$659$	−22.2982	−0.868613	−0.434307	+	0.900765i	$0.643007\pi$
$660$	0	0
$661$	−31.4767	−1.22430	−0.612151	−	0.790741i	$-0.709695\pi$
$661$	−31.4767	−1.22430	−0.612151	+	0.790741i	$0.709695\pi$
$662$	−1.34335	−0.0522109
$663$	0	0
$664$	28.7158	1.11439
$665$	19.5786	0.759224
$666$	0	0
$667$	−71.9018	−2.78405
$668$	−1.62828	−0.0630001
$669$	0	0
$670$	−23.0097	−0.888944
$671$	−6.36692	−0.245792
$672$	0	0
$673$	26.0157	1.00283	0.501416	−	0.865206i	$-0.332813\pi$
$673$	26.0157	1.00283	0.501416	+	0.865206i	$0.332813\pi$
$674$	−9.89144	−0.381004
$675$	0	0
$676$	−13.5288	−0.520340
$677$	−50.1304	−1.92667	−0.963334	−	0.268306i	$-0.913536\pi$
$677$	−50.1304	−1.92667	−0.963334	+	0.268306i	$0.913536\pi$
$678$	0	0
$679$	−0.647975	−0.0248670
$680$	−54.9619	−2.10769
$681$	0	0
$682$	−4.92311	−0.188515
$683$	−44.8648	−1.71670	−0.858352	−	0.513060i	$-0.828512\pi$
$683$	−44.8648	−1.71670	−0.858352	+	0.513060i	$0.828512\pi$
$684$	0	0
$685$	−25.1186	−0.959733
$686$	0.981962	0.0374915
$687$	0	0
$688$	−4.18003	−0.159362
$689$	19.4982	0.742823
$690$	0	0
$691$	29.4106	1.11883	0.559416	−	0.828887i	$-0.311026\pi$
$691$	29.4106	1.11883	0.559416	+	0.828887i	$0.311026\pi$
$692$	−25.3115	−0.962198
$693$	0	0
$694$	−11.6208	−0.441118
$695$	−11.8056	−0.447812
$696$	0	0
$697$	−61.7265	−2.33806
$698$	13.4627	0.509571
$699$	0	0
$700$	−1.79286	−0.0677637
$701$	3.93713	0.148703	0.0743517	−	0.997232i	$-0.476311\pi$
$701$	3.93713	0.148703	0.0743517	+	0.997232i	$0.476311\pi$
$702$	0	0
$703$	41.6884	1.57231
$704$	5.61101	0.211473
$705$	0	0
$706$	23.8309	0.896888
$707$	−8.32814	−0.313212
$708$	0	0
$709$	28.9928	1.08885	0.544423	−	0.838811i	$-0.316749\pi$
$709$	28.9928	1.08885	0.544423	+	0.838811i	$0.316749\pi$
$710$	−24.1270	−0.905470
$711$	0	0
$712$	27.8519	1.04380
$713$	50.2179	1.88068
$714$	0	0
$715$	11.0249	0.412308
$716$	−8.68665	−0.324635
$717$	0	0
$718$	−4.29351	−0.160232
$719$	33.2336	1.23940	0.619701	−	0.784838i	$-0.287254\pi$
$719$	33.2336	1.23940	0.619701	+	0.784838i	$0.287254\pi$
$720$	0	0
$721$	−18.0685	−0.672904
$722$	37.2641	1.38683
$723$	0	0
$724$	12.4270	0.461845
$725$	14.9274	0.554389
$726$	0	0
$727$	−19.0117	−0.705105	−0.352552	−	0.935792i	$-0.614686\pi$
$727$	−19.0117	−0.705105	−0.352552	+	0.935792i	$0.614686\pi$
$728$	15.2182	0.564024
$729$	0	0
$730$	17.2023	0.636686
$731$	−34.7146	−1.28396
$732$	0	0
$733$	−30.5354	−1.12785	−0.563925	−	0.825826i	$-0.690709\pi$
$733$	−30.5354	−1.12785	−0.563925	+	0.825826i	$0.690709\pi$
$734$	22.0854	0.815187
$735$	0	0
$736$	−42.7033	−1.57406
$737$	7.51815	0.276935
$738$	0	0
$739$	−2.93682	−0.108033	−0.0540163	−	0.998540i	$-0.517202\pi$
$739$	−2.93682	−0.108033	−0.0540163	+	0.998540i	$0.517202\pi$
$740$	−14.8446	−0.545697
$741$	0	0
$742$	−3.75048	−0.137684
$743$	−37.7443	−1.38470	−0.692352	−	0.721560i	$-0.743426\pi$
$743$	−37.7443	−1.38470	−0.692352	+	0.721560i	$0.743426\pi$
$744$	0	0
$745$	−62.1107	−2.27556
$746$	11.8031	0.432141
$747$	0	0
$748$	6.12703	0.224026
$749$	−11.8426	−0.432719
$750$	0	0
$751$	2.36132	0.0861657	0.0430828	−	0.999072i	$-0.486282\pi$
$751$	2.36132	0.0861657	0.0430828	+	0.999072i	$0.486282\pi$
$752$	1.94502	0.0709274
$753$	0	0
$754$	−43.2305	−1.57436
$755$	29.0857	1.05854
$756$	0	0
$757$	42.6822	1.55131	0.775655	−	0.631157i	$-0.217420\pi$
$757$	42.6822	1.55131	0.775655	+	0.631157i	$0.217420\pi$
$758$	−3.46687	−0.125922
$759$	0	0
$760$	−58.3635	−2.11707
$761$	43.9063	1.59160	0.795802	−	0.605558i	$-0.207050\pi$
$761$	43.9063	1.59160	0.795802	+	0.605558i	$0.207050\pi$
$762$	0	0
$763$	2.81629	0.101957
$764$	−3.75576	−0.135878
$765$	0	0
$766$	−27.2233	−0.983617
$767$	−19.8282	−0.715955
$768$	0	0
$769$	34.9812	1.26145	0.630727	−	0.776005i	$-0.282757\pi$
$769$	34.9812	1.26145	0.630727	+	0.776005i	$0.282757\pi$
$770$	−2.12064	−0.0764224
$771$	0	0
$772$	21.4133	0.770683
$773$	3.10373	0.111633	0.0558166	−	0.998441i	$-0.482224\pi$
$773$	3.10373	0.111633	0.0558166	+	0.998441i	$0.482224\pi$
$774$	0	0
$775$	−10.4256	−0.374500
$776$	1.93161	0.0693406
$777$	0	0
$778$	3.72673	0.133610
$779$	−65.5468	−2.34846
$780$	0	0
$781$	7.88320	0.282083
$782$	58.1840	2.08065
$783$	0	0
$784$	−0.855717	−0.0305613
$785$	0.400772	0.0143042
$786$	0	0
$787$	−36.8082	−1.31207	−0.656036	−	0.754730i	$-0.727768\pi$
$787$	−36.8082	−1.31207	−0.656036	+	0.754730i	$0.727768\pi$
$788$	−23.5308	−0.838250
$789$	0	0
$790$	−13.9151	−0.495077
$791$	−19.7700	−0.702939
$792$	0	0
$793$	39.0480	1.38664
$794$	−21.3417	−0.757388
$795$	0	0
$796$	−14.0237	−0.497059
$797$	−38.1550	−1.35152	−0.675759	−	0.737122i	$-0.736184\pi$
$797$	−38.1550	−1.35152	−0.675759	+	0.737122i	$0.736184\pi$
$798$	0	0
$799$	16.1531	0.571454
$800$	8.86553	0.313444
$801$	0	0
$802$	1.94956	0.0688412
$803$	−5.62064	−0.198348
$804$	0	0
$805$	21.6315	0.762409
$806$	30.1932	1.06351
$807$	0	0
$808$	24.8261	0.873379
$809$	−22.3583	−0.786076	−0.393038	−	0.919522i	$-0.628576\pi$
$809$	−22.3583	−0.786076	−0.393038	+	0.919522i	$0.628576\pi$
$810$	0	0
$811$	−47.2102	−1.65777	−0.828887	−	0.559416i	$-0.811025\pi$
$811$	−47.2102	−1.65777	−0.828887	+	0.559416i	$0.811025\pi$
$812$	−8.93199	−0.313452
$813$	0	0
$814$	−4.51544	−0.158266
$815$	29.7722	1.04287
$816$	0	0
$817$	−36.8631	−1.28968
$818$	−36.1206	−1.26293
$819$	0	0
$820$	23.3402	0.815075
$821$	−2.65321	−0.0925975	−0.0462988	−	0.998928i	$-0.514743\pi$
$821$	−2.65321	−0.0925975	−0.0462988	+	0.998928i	$0.514743\pi$
$822$	0	0
$823$	46.5374	1.62219	0.811095	−	0.584914i	$-0.198872\pi$
$823$	46.5374	1.62219	0.811095	+	0.584914i	$0.198872\pi$
$824$	53.8619	1.87637
$825$	0	0
$826$	3.81395	0.132704
$827$	−23.3068	−0.810458	−0.405229	−	0.914215i	$-0.632808\pi$
$827$	−23.3068	−0.810458	−0.405229	+	0.914215i	$0.632808\pi$
$828$	0	0
$829$	−20.0978	−0.698025	−0.349012	−	0.937118i	$-0.613483\pi$
$829$	−20.0978	−0.698025	−0.349012	+	0.937118i	$0.613483\pi$
$830$	24.5411	0.851832
$831$	0	0
$832$	−34.4121	−1.19302
$833$	−7.10660	−0.246229
$834$	0	0
$835$	−4.07862	−0.141146
$836$	6.50624	0.225023
$837$	0	0
$838$	1.02199	0.0353039
$839$	49.4669	1.70779	0.853894	−	0.520447i	$-0.174235\pi$
$839$	49.4669	1.70779	0.853894	+	0.520447i	$0.174235\pi$
$840$	0	0
$841$	45.3680	1.56441
$842$	−7.38325	−0.254444
$843$	0	0
$844$	0.874471	0.0301005
$845$	−33.8878	−1.16578
$846$	0	0
$847$	−10.3071	−0.354156
$848$	3.26830	0.112234
$849$	0	0
$850$	−12.0794	−0.414321
$851$	46.0596	1.57890
$852$	0	0
$853$	7.99651	0.273795	0.136898	−	0.990585i	$-0.456287\pi$
$853$	7.99651	0.273795	0.136898	+	0.990585i	$0.456287\pi$
$854$	−7.51088	−0.257017
$855$	0	0
$856$	35.3027	1.20662
$857$	21.7094	0.741579	0.370790	−	0.928717i	$-0.379087\pi$
$857$	21.7094	0.741579	0.370790	+	0.928717i	$0.379087\pi$
$858$	0	0
$859$	8.50670	0.290245	0.145122	−	0.989414i	$-0.453642\pi$
$859$	8.50670	0.290245	0.145122	+	0.989414i	$0.453642\pi$
$860$	13.1264	0.447605
$861$	0	0
$862$	26.6174	0.906593
$863$	−31.9233	−1.08668	−0.543341	−	0.839512i	$-0.682841\pi$
$863$	−31.9233	−1.08668	−0.543341	+	0.839512i	$0.682841\pi$
$864$	0	0
$865$	−63.4017	−2.15572
$866$	26.1324	0.888016
$867$	0	0
$868$	6.23832	0.211742
$869$	4.54659	0.154232
$870$	0	0
$871$	−46.1085	−1.56233
$872$	−8.39534	−0.284302
$873$	0	0
$874$	61.7850	2.08991
$875$	8.48120	0.286717
$876$	0	0
$877$	−42.5730	−1.43759	−0.718793	−	0.695224i	$-0.755305\pi$
$877$	−42.5730	−1.43759	−0.718793	+	0.695224i	$0.755305\pi$
$878$	−3.62389	−0.122300
$879$	0	0
$880$	1.84800	0.0622961
$881$	35.2691	1.18825	0.594124	−	0.804374i	$-0.297499\pi$
$881$	35.2691	1.18825	0.594124	+	0.804374i	$0.297499\pi$
$882$	0	0
$883$	−21.3815	−0.719546	−0.359773	−	0.933040i	$-0.617146\pi$
$883$	−21.3815	−0.719546	−0.359773	+	0.933040i	$0.617146\pi$
$884$	−37.5768	−1.26384
$885$	0	0
$886$	9.58498	0.322014
$887$	−37.3935	−1.25555	−0.627776	−	0.778394i	$-0.716035\pi$
$887$	−37.3935	−1.25555	−0.627776	+	0.778394i	$0.716035\pi$
$888$	0	0
$889$	1.00000	0.0335389
$890$	23.8028	0.797872
$891$	0	0
$892$	−1.50938	−0.0505378
$893$	17.1528	0.573996
$894$	0	0
$895$	−21.7588	−0.727318
$896$	−3.62425	−0.121078
$897$	0	0
$898$	−40.6518	−1.35657
$899$	−51.9404	−1.73231
$900$	0	0
$901$	27.1428	0.904257
$902$	7.09966	0.236393
$903$	0	0
$904$	58.9341	1.96012
$905$	31.1278	1.03472
$906$	0	0
$907$	9.53781	0.316698	0.158349	−	0.987383i	$-0.449383\pi$
$907$	9.53781	0.316698	0.158349	+	0.987383i	$0.449383\pi$
$908$	19.9474	0.661977
$909$	0	0
$910$	13.0058	0.431137
$911$	47.6465	1.57860	0.789300	−	0.614008i	$-0.210444\pi$
$911$	47.6465	1.57860	0.789300	+	0.614008i	$0.210444\pi$
$912$	0	0
$913$	−8.01849	−0.265373
$914$	25.8545	0.855191
$915$	0	0
$916$	10.2894	0.339970
$917$	22.1245	0.730614
$918$	0	0
$919$	32.1422	1.06027	0.530136	−	0.847913i	$-0.322141\pi$
$919$	32.1422	1.06027	0.530136	+	0.847913i	$0.322141\pi$
$920$	−64.4832	−2.12595
$921$	0	0
$922$	26.7815	0.882001
$923$	−48.3473	−1.59137
$924$	0	0
$925$	−9.56233	−0.314407
$926$	−36.9142	−1.21307
$927$	0	0
$928$	44.1680	1.44988
$929$	−21.6067	−0.708892	−0.354446	−	0.935076i	$-0.615331\pi$
$929$	−21.6067	−0.708892	−0.354446	+	0.935076i	$0.615331\pi$
$930$	0	0
$931$	−7.54643	−0.247324
$932$	−17.2769	−0.565925
$933$	0	0
$934$	−25.4486	−0.832703
$935$	15.3474	0.501912
$936$	0	0
$937$	−2.03535	−0.0664920	−0.0332460	−	0.999447i	$-0.510584\pi$
$937$	−2.03535	−0.0664920	−0.0332460	+	0.999447i	$0.510584\pi$
$938$	8.86896	0.289582
$939$	0	0
$940$	−6.10783	−0.199216
$941$	−30.7135	−1.00123	−0.500616	−	0.865670i	$-0.666893\pi$
$941$	−30.7135	−1.00123	−0.500616	+	0.865670i	$0.666893\pi$
$942$	0	0
$943$	−72.4197	−2.35831
$944$	−3.32362	−0.108175
$945$	0	0
$946$	3.99280	0.129817
$947$	29.7046	0.965271	0.482636	−	0.875821i	$-0.339680\pi$
$947$	29.7046	0.965271	0.482636	+	0.875821i	$0.339680\pi$
$948$	0	0
$949$	34.4712	1.11898
$950$	−12.8270	−0.416164
$951$	0	0
$952$	21.1847	0.686600
$953$	3.51868	0.113981	0.0569907	−	0.998375i	$-0.481849\pi$
$953$	3.51868	0.113981	0.0569907	+	0.998375i	$0.481849\pi$
$954$	0	0
$955$	−9.40764	−0.304424
$956$	7.83690	0.253463
$957$	0	0
$958$	13.2894	0.429360
$959$	9.68182	0.312642
$960$	0	0
$961$	5.27637	0.170205
$962$	27.6930	0.892859
$963$	0	0
$964$	−3.57891	−0.115269
$965$	53.6374	1.72665
$966$	0	0
$967$	−14.7718	−0.475028	−0.237514	−	0.971384i	$-0.576333\pi$
$967$	−14.7718	−0.475028	−0.237514	+	0.971384i	$0.576333\pi$
$968$	30.7254	0.987552
$969$	0	0
$970$	1.65079	0.0530036
$971$	−6.76299	−0.217035	−0.108517	−	0.994095i	$-0.534610\pi$
$971$	−6.76299	−0.217035	−0.108517	+	0.994095i	$0.534610\pi$
$972$	0	0
$973$	4.55040	0.145879
$974$	16.7562	0.536903
$975$	0	0
$976$	6.54525	0.209508
$977$	−7.10433	−0.227288	−0.113644	−	0.993522i	$-0.536252\pi$
$977$	−7.10433	−0.227288	−0.113644	+	0.993522i	$0.536252\pi$
$978$	0	0
$979$	−7.77728	−0.248563
$980$	2.68717	0.0858383
$981$	0	0
$982$	−39.5425	−1.26185
$983$	−15.7760	−0.503175	−0.251588	−	0.967835i	$-0.580953\pi$
$983$	−15.7760	−0.503175	−0.251588	+	0.967835i	$0.580953\pi$
$984$	0	0
$985$	−58.9413	−1.87803
$986$	−60.1796	−1.91651
$987$	0	0
$988$	−39.9025	−1.26947
$989$	−40.7283	−1.29509
$990$	0	0
$991$	50.7213	1.61121	0.805607	−	0.592450i	$-0.201839\pi$
$991$	50.7213	1.61121	0.805607	+	0.592450i	$0.201839\pi$
$992$	−30.8479	−0.979423
$993$	0	0
$994$	9.29961	0.294966
$995$	−35.1275	−1.11362
$996$	0	0
$997$	11.3933	0.360829	0.180415	−	0.983591i	$-0.442256\pi$
$997$	11.3933	0.360829	0.180415	+	0.983591i	$0.442256\pi$
$998$	−11.0758	−0.350597
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.u.1.11		18
3.2	odd	2		2667.2.a.p.1.8	✓	18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2667.2.a.p.1.8	✓	18	3.2	odd	2
8001.2.a.u.1.11		18	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)

\(f(q)\)	\(=\)	\(q+0.981962 q^{2} -1.03575 q^{4} -2.59441 q^{5} +1.00000 q^{7} -2.98099 q^{8} +O(q^{10})\)	\(q+0.981962 q^{2} -1.03575 q^{4} -2.59441 q^{5} +1.00000 q^{7} -2.98099 q^{8} -2.54761 q^{10} +0.832401 q^{11} -5.10508 q^{13} +0.981962 q^{14} -0.855717 q^{16} -7.10660 q^{17} -7.54643 q^{19} +2.68717 q^{20} +0.817386 q^{22} -8.33771 q^{23} +1.73097 q^{25} -5.01299 q^{26} -1.03575 q^{28} +8.62369 q^{29} -6.02299 q^{31} +5.12170 q^{32} -6.97841 q^{34} -2.59441 q^{35} -5.52425 q^{37} -7.41031 q^{38} +7.73392 q^{40} +8.68580 q^{41} +4.88483 q^{43} -0.862161 q^{44} -8.18731 q^{46} -2.27297 q^{47} +1.00000 q^{49} +1.69975 q^{50} +5.28759 q^{52} -3.81937 q^{53} -2.15959 q^{55} -2.98099 q^{56} +8.46813 q^{58} +3.88402 q^{59} -7.64885 q^{61} -5.91434 q^{62} +6.74075 q^{64} +13.2447 q^{65} +9.03188 q^{67} +7.36067 q^{68} -2.54761 q^{70} +9.47044 q^{71} -6.75232 q^{73} -5.42460 q^{74} +7.81623 q^{76} +0.832401 q^{77} +5.46202 q^{79} +2.22008 q^{80} +8.52913 q^{82} -9.63296 q^{83} +18.4374 q^{85} +4.79672 q^{86} -2.48138 q^{88} -9.34318 q^{89} -5.10508 q^{91} +8.63580 q^{92} -2.23196 q^{94} +19.5786 q^{95} -0.647975 q^{97} +0.981962 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 6 q^{2} + 22 q^{4} + 10 q^{5} + 18 q^{7} + 21 q^{8}+O(q^{10}) \) 18 * q + 6 * q^2 + 22 * q^4 + 10 * q^5 + 18 * q^7 + 21 * q^8	\( 18 q + 6 q^{2} + 22 q^{4} + 10 q^{5} + 18 q^{7} + 21 q^{8} - 4 q^{10} + 9 q^{11} - 25 q^{13} + 6 q^{14} + 34 q^{16} + 17 q^{17} - 5 q^{19} + 21 q^{20} + 5 q^{22} + 14 q^{23} + 28 q^{25} + 8 q^{26} + 22 q^{28} + 17 q^{29} + 5 q^{31} + 53 q^{32} - 19 q^{34} + 10 q^{35} - 15 q^{37} + 22 q^{38} - q^{40} + 17 q^{41} + q^{43} + 33 q^{44} + 10 q^{46} + 31 q^{47} + 18 q^{49} + 35 q^{50} - 70 q^{52} + 35 q^{53} + 4 q^{55} + 21 q^{56} + 3 q^{58} + 46 q^{59} - 5 q^{61} + 10 q^{62} + 63 q^{64} + 12 q^{65} + 6 q^{67} + 56 q^{68} - 4 q^{70} + 22 q^{71} - 16 q^{73} - 18 q^{74} + 32 q^{76} + 9 q^{77} + 46 q^{79} + 30 q^{80} - 12 q^{82} + 46 q^{83} + 4 q^{85} - 18 q^{86} + 30 q^{88} + 42 q^{89} - 25 q^{91} + 48 q^{92} + 3 q^{94} + 2 q^{95} - 35 q^{97} + 6 q^{98}+O(q^{100}) \) 18 * q + 6 * q^2 + 22 * q^4 + 10 * q^5 + 18 * q^7 + 21 * q^8 - 4 * q^10 + 9 * q^11 - 25 * q^13 + 6 * q^14 + 34 * q^16 + 17 * q^17 - 5 * q^19 + 21 * q^20 + 5 * q^22 + 14 * q^23 + 28 * q^25 + 8 * q^26 + 22 * q^28 + 17 * q^29 + 5 * q^31 + 53 * q^32 - 19 * q^34 + 10 * q^35 - 15 * q^37 + 22 * q^38 - q^40 + 17 * q^41 + q^43 + 33 * q^44 + 10 * q^46 + 31 * q^47 + 18 * q^49 + 35 * q^50 - 70 * q^52 + 35 * q^53 + 4 * q^55 + 21 * q^56 + 3 * q^58 + 46 * q^59 - 5 * q^61 + 10 * q^62 + 63 * q^64 + 12 * q^65 + 6 * q^67 + 56 * q^68 - 4 * q^70 + 22 * q^71 - 16 * q^73 - 18 * q^74 + 32 * q^76 + 9 * q^77 + 46 * q^79 + 30 * q^80 - 12 * q^82 + 46 * q^83 + 4 * q^85 - 18 * q^86 + 30 * q^88 + 42 * q^89 - 25 * q^91 + 48 * q^92 + 3 * q^94 + 2 * q^95 - 35 * q^97 + 6 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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