LMFDB - Embedded newform 8008.2.a.j.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8008,2,Mod(1,8008)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8008, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("8008.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8008 = 2^{3} \cdot 7 \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8008.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:	$63.9442019386$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.668973.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 9x^{3} - x^{2} + 7x - 2 $ x^5 - x^4 - 9x^3 - x^2 + 7x - 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 3 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$0.553533$ of defining polynomial
Character	$\chi$	$=$	8008.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.18752 q^{3} -2.05962 q^{5} +1.00000 q^{7} +1.78523 q^{9} +O(q^{10})$	$q+2.18752 q^{3} -2.05962 q^{5} +1.00000 q^{7} +1.78523 q^{9} -1.00000 q^{11} -1.00000 q^{13} -4.50545 q^{15} +1.94038 q^{17} +2.63335 q^{19} +2.18752 q^{21} -2.37755 q^{23} -0.757976 q^{25} -2.65733 q^{27} +8.17116 q^{29} +0.915644 q^{31} -2.18752 q^{33} -2.05962 q^{35} +0.853512 q^{37} -2.18752 q^{39} +6.33422 q^{41} -8.24965 q^{43} -3.67689 q^{45} -4.20381 q^{47} +1.00000 q^{49} +4.24462 q^{51} +6.84485 q^{53} +2.05962 q^{55} +5.76049 q^{57} +12.3639 q^{59} +3.68207 q^{61} +1.78523 q^{63} +2.05962 q^{65} +13.4099 q^{67} -5.20093 q^{69} -2.58135 q^{71} -9.32883 q^{73} -1.65809 q^{75} -1.00000 q^{77} +11.9316 q^{79} -11.1686 q^{81} +3.62756 q^{83} -3.99645 q^{85} +17.8745 q^{87} -7.74169 q^{89} -1.00000 q^{91} +2.00299 q^{93} -5.42369 q^{95} +4.78272 q^{97} -1.78523 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + q^{3} - q^{5} + 5 q^{7} + 6 q^{9}+O(q^{10}) $ 5 * q + q^3 - q^5 + 5 * q^7 + 6 * q^9	$ 5 q + q^{3} - q^{5} + 5 q^{7} + 6 q^{9} - 5 q^{11} - 5 q^{13} - 5 q^{15} + 19 q^{17} - 5 q^{19} + q^{21} + 5 q^{23} + 12 q^{25} - 11 q^{27} - 17 q^{29} + 4 q^{31} - q^{33} - q^{35} + 10 q^{37} - q^{39} + 10 q^{41} - 25 q^{43} + q^{45} + 3 q^{47} + 5 q^{49} - q^{51} + 22 q^{53} + q^{55} + 16 q^{57} + 21 q^{59} + 26 q^{61} + 6 q^{63} + q^{65} + 28 q^{67} + 16 q^{69} + 28 q^{71} - 4 q^{73} - 5 q^{77} - 11 q^{79} + 5 q^{81} + 33 q^{85} + 31 q^{87} - 37 q^{89} - 5 q^{91} - 49 q^{93} + 29 q^{95} + 18 q^{97} - 6 q^{99}+O(q^{100}) $ 5 * q + q^3 - q^5 + 5 * q^7 + 6 * q^9 - 5 * q^11 - 5 * q^13 - 5 * q^15 + 19 * q^17 - 5 * q^19 + q^21 + 5 * q^23 + 12 * q^25 - 11 * q^27 - 17 * q^29 + 4 * q^31 - q^33 - q^35 + 10 * q^37 - q^39 + 10 * q^41 - 25 * q^43 + q^45 + 3 * q^47 + 5 * q^49 - q^51 + 22 * q^53 + q^55 + 16 * q^57 + 21 * q^59 + 26 * q^61 + 6 * q^63 + q^65 + 28 * q^67 + 16 * q^69 + 28 * q^71 - 4 * q^73 - 5 * q^77 - 11 * q^79 + 5 * q^81 + 33 * q^85 + 31 * q^87 - 37 * q^89 - 5 * q^91 - 49 * q^93 + 29 * q^95 + 18 * q^97 - 6 * q^99

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	0
$2$	0	0
$3$	2.18752	1.26296	0.631482	−	0.775391i	$-0.282447\pi$
$3$	2.18752	1.26296	0.631482	+	0.775391i	$0.282447\pi$
$4$	0	0
$5$	−2.05962	−0.921089	−0.460544	−	0.887637i	$-0.652346\pi$
$5$	−2.05962	−0.921089	−0.460544	+	0.887637i	$0.652346\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	1.78523	0.595077
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	−4.50545	−1.16330
$16$	0	0
$17$	1.94038	0.470612	0.235306	−	0.971921i	$-0.424391\pi$
$17$	1.94038	0.470612	0.235306	+	0.971921i	$0.424391\pi$
$18$	0	0
$19$	2.63335	0.604131	0.302066	−	0.953287i	$-0.402324\pi$
$19$	2.63335	0.604131	0.302066	+	0.953287i	$0.402324\pi$
$20$	0	0
$21$	2.18752	0.477355
$22$	0	0
$23$	−2.37755	−0.495753	−0.247877	−	0.968792i	$-0.579733\pi$
$23$	−2.37755	−0.495753	−0.247877	+	0.968792i	$0.579733\pi$
$24$	0	0
$25$	−0.757976	−0.151595
$26$	0	0
$27$	−2.65733	−0.511403
$28$	0	0
$29$	8.17116	1.51735	0.758673	−	0.651472i	$-0.225848\pi$
$29$	8.17116	1.51735	0.758673	+	0.651472i	$0.225848\pi$
$30$	0	0
$31$	0.915644	0.164454	0.0822272	−	0.996614i	$-0.473797\pi$
$31$	0.915644	0.164454	0.0822272	+	0.996614i	$0.473797\pi$
$32$	0	0
$33$	−2.18752	−0.380798
$34$	0	0
$35$	−2.05962	−0.348139
$36$	0	0
$37$	0.853512	0.140316	0.0701582	−	0.997536i	$-0.477650\pi$
$37$	0.853512	0.140316	0.0701582	+	0.997536i	$0.477650\pi$
$38$	0	0
$39$	−2.18752	−0.350283
$40$	0	0
$41$	6.33422	0.989239	0.494620	−	0.869110i	$-0.335307\pi$
$41$	6.33422	0.989239	0.494620	+	0.869110i	$0.335307\pi$
$42$	0	0
$43$	−8.24965	−1.25806	−0.629030	−	0.777381i	$-0.716548\pi$
$43$	−8.24965	−1.25806	−0.629030	+	0.777381i	$0.716548\pi$
$44$	0	0
$45$	−3.67689	−0.548118
$46$	0	0
$47$	−4.20381	−0.613188	−0.306594	−	0.951840i	$-0.599189\pi$
$47$	−4.20381	−0.613188	−0.306594	+	0.951840i	$0.599189\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	4.24462	0.594366
$52$	0	0
$53$	6.84485	0.940212	0.470106	−	0.882610i	$-0.344216\pi$
$53$	6.84485	0.940212	0.470106	+	0.882610i	$0.344216\pi$
$54$	0	0
$55$	2.05962	0.277719
$56$	0	0
$57$	5.76049	0.762996
$58$	0	0
$59$	12.3639	1.60964	0.804818	−	0.593521i	$-0.202263\pi$
$59$	12.3639	1.60964	0.804818	+	0.593521i	$0.202263\pi$
$60$	0	0
$61$	3.68207	0.471441	0.235720	−	0.971821i	$-0.424255\pi$
$61$	3.68207	0.471441	0.235720	+	0.971821i	$0.424255\pi$
$62$	0	0
$63$	1.78523	0.224918
$64$	0	0
$65$	2.05962	0.255464
$66$	0	0
$67$	13.4099	1.63828	0.819141	−	0.573593i	$-0.194451\pi$
$67$	13.4099	1.63828	0.819141	+	0.573593i	$0.194451\pi$
$68$	0	0
$69$	−5.20093	−0.626118
$70$	0	0
$71$	−2.58135	−0.306350	−0.153175	−	0.988199i	$-0.548950\pi$
$71$	−2.58135	−0.306350	−0.153175	+	0.988199i	$0.548950\pi$
$72$	0	0
$73$	−9.32883	−1.09186	−0.545928	−	0.837832i	$-0.683823\pi$
$73$	−9.32883	−1.09186	−0.545928	+	0.837832i	$0.683823\pi$
$74$	0	0
$75$	−1.65809	−0.191459
$76$	0	0
$77$	−1.00000	−0.113961
$78$	0	0
$79$	11.9316	1.34242	0.671208	−	0.741269i	$-0.265776\pi$
$79$	11.9316	1.34242	0.671208	+	0.741269i	$0.265776\pi$
$80$	0	0
$81$	−11.1686	−1.24096
$82$	0	0
$83$	3.62756	0.398177	0.199088	−	0.979982i	$-0.436202\pi$
$83$	3.62756	0.398177	0.199088	+	0.979982i	$0.436202\pi$
$84$	0	0
$85$	−3.99645	−0.433475
$86$	0	0
$87$	17.8745	1.91635
$88$	0	0
$89$	−7.74169	−0.820617	−0.410309	−	0.911947i	$-0.634579\pi$
$89$	−7.74169	−0.820617	−0.410309	+	0.911947i	$0.634579\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	2.00299	0.207700
$94$	0	0
$95$	−5.42369	−0.556459
$96$	0	0
$97$	4.78272	0.485611	0.242806	−	0.970075i	$-0.421932\pi$
$97$	4.78272	0.485611	0.242806	+	0.970075i	$0.421932\pi$
$98$	0	0
$99$	−1.78523	−0.179422
$100$	0	0
$101$	9.95966	0.991023	0.495512	−	0.868601i	$-0.334981\pi$
$101$	9.95966	0.991023	0.495512	+	0.868601i	$0.334981\pi$
$102$	0	0
$103$	6.10036	0.601087	0.300543	−	0.953768i	$-0.402832\pi$
$103$	6.10036	0.601087	0.300543	+	0.953768i	$0.402832\pi$
$104$	0	0
$105$	−4.50545	−0.439687
$106$	0	0
$107$	−12.7093	−1.22865	−0.614325	−	0.789053i	$-0.710572\pi$
$107$	−12.7093	−1.22865	−0.614325	+	0.789053i	$0.710572\pi$
$108$	0	0
$109$	8.17731	0.783244	0.391622	−	0.920126i	$-0.371914\pi$
$109$	8.17731	0.783244	0.391622	+	0.920126i	$0.371914\pi$
$110$	0	0
$111$	1.86707	0.177215
$112$	0	0
$113$	4.02747	0.378872	0.189436	−	0.981893i	$-0.439334\pi$
$113$	4.02747	0.378872	0.189436	+	0.981893i	$0.439334\pi$
$114$	0	0
$115$	4.89684	0.456633
$116$	0	0
$117$	−1.78523	−0.165045
$118$	0	0
$119$	1.94038	0.177875
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	13.8562	1.24937
$124$	0	0
$125$	11.8592	1.06072
$126$	0	0
$127$	18.6388	1.65393	0.826965	−	0.562253i	$-0.190065\pi$
$127$	18.6388	1.65393	0.826965	+	0.562253i	$0.190065\pi$
$128$	0	0
$129$	−18.0462	−1.58888
$130$	0	0
$131$	5.34512	0.467005	0.233502	−	0.972356i	$-0.424981\pi$
$131$	5.34512	0.467005	0.233502	+	0.972356i	$0.424981\pi$
$132$	0	0
$133$	2.63335	0.228340
$134$	0	0
$135$	5.47308	0.471048
$136$	0	0
$137$	−21.5592	−1.84193	−0.920964	−	0.389648i	$-0.872597\pi$
$137$	−21.5592	−1.84193	−0.920964	+	0.389648i	$0.872597\pi$
$138$	0	0
$139$	−21.3696	−1.81254	−0.906272	−	0.422696i	$-0.861084\pi$
$139$	−21.3696	−1.81254	−0.906272	+	0.422696i	$0.861084\pi$
$140$	0	0
$141$	−9.19590	−0.774434
$142$	0	0
$143$	1.00000	0.0836242
$144$	0	0
$145$	−16.8295	−1.39761
$146$	0	0
$147$	2.18752	0.180423
$148$	0	0
$149$	17.1326	1.40356	0.701780	−	0.712393i	$-0.252389\pi$
$149$	17.1326	1.40356	0.701780	+	0.712393i	$0.252389\pi$
$150$	0	0
$151$	18.4106	1.49823	0.749117	−	0.662438i	$-0.230478\pi$
$151$	18.4106	1.49823	0.749117	+	0.662438i	$0.230478\pi$
$152$	0	0
$153$	3.46403	0.280050
$154$	0	0
$155$	−1.88588	−0.151477
$156$	0	0
$157$	−1.70141	−0.135788	−0.0678938	−	0.997693i	$-0.521628\pi$
$157$	−1.70141	−0.135788	−0.0678938	+	0.997693i	$0.521628\pi$
$158$	0	0
$159$	14.9732	1.18745
$160$	0	0
$161$	−2.37755	−0.187377
$162$	0	0
$163$	0.673620	0.0527620	0.0263810	−	0.999652i	$-0.491602\pi$
$163$	0.673620	0.0527620	0.0263810	+	0.999652i	$0.491602\pi$
$164$	0	0
$165$	4.50545	0.350749
$166$	0	0
$167$	9.53270	0.737663	0.368831	−	0.929496i	$-0.379758\pi$
$167$	9.53270	0.737663	0.368831	+	0.929496i	$0.379758\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	4.70113	0.359504
$172$	0	0
$173$	3.73651	0.284081	0.142041	−	0.989861i	$-0.454634\pi$
$173$	3.73651	0.284081	0.142041	+	0.989861i	$0.454634\pi$
$174$	0	0
$175$	−0.757976	−0.0572976
$176$	0	0
$177$	27.0461	2.03291
$178$	0	0
$179$	−13.2525	−0.990535	−0.495268	−	0.868740i	$-0.664930\pi$
$179$	−13.2525	−0.990535	−0.495268	+	0.868740i	$0.664930\pi$
$180$	0	0
$181$	−0.353565	−0.0262803	−0.0131401	−	0.999914i	$-0.504183\pi$
$181$	−0.353565	−0.0262803	−0.0131401	+	0.999914i	$0.504183\pi$
$182$	0	0
$183$	8.05459	0.595412
$184$	0	0
$185$	−1.75791	−0.129244
$186$	0	0
$187$	−1.94038	−0.141895
$188$	0	0
$189$	−2.65733	−0.193292
$190$	0	0
$191$	−8.22049	−0.594814	−0.297407	−	0.954751i	$-0.596122\pi$
$191$	−8.22049	−0.594814	−0.297407	+	0.954751i	$0.596122\pi$
$192$	0	0
$193$	26.7045	1.92224	0.961118	−	0.276139i	$-0.0890551\pi$
$193$	26.7045	1.92224	0.961118	+	0.276139i	$0.0890551\pi$
$194$	0	0
$195$	4.50545	0.322642
$196$	0	0
$197$	5.17655	0.368814	0.184407	−	0.982850i	$-0.440964\pi$
$197$	5.17655	0.368814	0.184407	+	0.982850i	$0.440964\pi$
$198$	0	0
$199$	−3.62238	−0.256784	−0.128392	−	0.991724i	$-0.540982\pi$
$199$	−3.62238	−0.256784	−0.128392	+	0.991724i	$0.540982\pi$
$200$	0	0
$201$	29.3344	2.06909
$202$	0	0
$203$	8.17116	0.573503
$204$	0	0
$205$	−13.0461	−0.911177
$206$	0	0
$207$	−4.24447	−0.295011
$208$	0	0
$209$	−2.63335	−0.182152
$210$	0	0
$211$	−0.926323	−0.0637707	−0.0318854	−	0.999492i	$-0.510151\pi$
$211$	−0.926323	−0.0637707	−0.0318854	+	0.999492i	$0.510151\pi$
$212$	0	0
$213$	−5.64676	−0.386909
$214$	0	0
$215$	16.9911	1.15878
$216$	0	0
$217$	0.915644	0.0621579
$218$	0	0
$219$	−20.4070	−1.37897
$220$	0	0
$221$	−1.94038	−0.130524
$222$	0	0
$223$	23.1088	1.54748	0.773740	−	0.633503i	$-0.218384\pi$
$223$	23.1088	1.54748	0.773740	+	0.633503i	$0.218384\pi$
$224$	0	0
$225$	−1.35316	−0.0902108
$226$	0	0
$227$	26.8203	1.78013	0.890063	−	0.455838i	$-0.150660\pi$
$227$	26.8203	1.78013	0.890063	+	0.455838i	$0.150660\pi$
$228$	0	0
$229$	−29.0541	−1.91995	−0.959974	−	0.280089i	$-0.909636\pi$
$229$	−29.0541	−1.91995	−0.959974	+	0.280089i	$0.909636\pi$
$230$	0	0
$231$	−2.18752	−0.143928
$232$	0	0
$233$	26.0397	1.70592	0.852959	−	0.521978i	$-0.174806\pi$
$233$	26.0397	1.70592	0.852959	+	0.521978i	$0.174806\pi$
$234$	0	0
$235$	8.65823	0.564801
$236$	0	0
$237$	26.1007	1.69542
$238$	0	0
$239$	−11.9480	−0.772852	−0.386426	−	0.922320i	$-0.626291\pi$
$239$	−11.9480	−0.772852	−0.386426	+	0.922320i	$0.626291\pi$
$240$	0	0
$241$	12.8182	0.825693	0.412846	−	0.910801i	$-0.364535\pi$
$241$	12.8182	0.825693	0.412846	+	0.910801i	$0.364535\pi$
$242$	0	0
$243$	−16.4596	−1.05588
$244$	0	0
$245$	−2.05962	−0.131584
$246$	0	0
$247$	−2.63335	−0.167556
$248$	0	0
$249$	7.93535	0.502883
$250$	0	0
$251$	28.3473	1.78927	0.894634	−	0.446799i	$-0.147436\pi$
$251$	28.3473	1.78927	0.894634	+	0.446799i	$0.147436\pi$
$252$	0	0
$253$	2.37755	0.149475
$254$	0	0
$255$	−8.74229	−0.547464
$256$	0	0
$257$	−14.1222	−0.880916	−0.440458	−	0.897773i	$-0.645184\pi$
$257$	−14.1222	−0.880916	−0.440458	+	0.897773i	$0.645184\pi$
$258$	0	0
$259$	0.853512	0.0530346
$260$	0	0
$261$	14.5874	0.902937
$262$	0	0
$263$	−17.2667	−1.06471	−0.532355	−	0.846521i	$-0.678693\pi$
$263$	−17.2667	−1.06471	−0.532355	+	0.846521i	$0.678693\pi$
$264$	0	0
$265$	−14.0978	−0.866019
$266$	0	0
$267$	−16.9351	−1.03641
$268$	0	0
$269$	4.71695	0.287597	0.143799	−	0.989607i	$-0.454068\pi$
$269$	4.71695	0.287597	0.143799	+	0.989607i	$0.454068\pi$
$270$	0	0
$271$	−28.2552	−1.71638	−0.858189	−	0.513333i	$-0.828411\pi$
$271$	−28.2552	−1.71638	−0.858189	+	0.513333i	$0.828411\pi$
$272$	0	0
$273$	−2.18752	−0.132395
$274$	0	0
$275$	0.757976	0.0457077
$276$	0	0
$277$	5.00578	0.300768	0.150384	−	0.988628i	$-0.451949\pi$
$277$	5.00578	0.300768	0.150384	+	0.988628i	$0.451949\pi$
$278$	0	0
$279$	1.63463	0.0978630
$280$	0	0
$281$	−17.2182	−1.02715	−0.513575	−	0.858044i	$-0.671679\pi$
$281$	−17.2182	−1.02715	−0.513575	+	0.858044i	$0.671679\pi$
$282$	0	0
$283$	−9.84309	−0.585111	−0.292555	−	0.956249i	$-0.594506\pi$
$283$	−9.84309	−0.585111	−0.292555	+	0.956249i	$0.594506\pi$
$284$	0	0
$285$	−11.8644	−0.702787
$286$	0	0
$287$	6.33422	0.373897
$288$	0	0
$289$	−13.2349	−0.778524
$290$	0	0
$291$	10.4623	0.613309
$292$	0	0
$293$	−18.3613	−1.07268	−0.536338	−	0.844003i	$-0.680193\pi$
$293$	−18.3613	−1.07268	−0.536338	+	0.844003i	$0.680193\pi$
$294$	0	0
$295$	−25.4648	−1.48262
$296$	0	0
$297$	2.65733	0.154194
$298$	0	0
$299$	2.37755	0.137497
$300$	0	0
$301$	−8.24965	−0.475502
$302$	0	0
$303$	21.7869	1.25163
$304$	0	0
$305$	−7.58365	−0.434239
$306$	0	0
$307$	13.4378	0.766934	0.383467	−	0.923555i	$-0.374730\pi$
$307$	13.4378	0.766934	0.383467	+	0.923555i	$0.374730\pi$
$308$	0	0
$309$	13.3446	0.759150
$310$	0	0
$311$	28.0625	1.59128	0.795640	−	0.605770i	$-0.207135\pi$
$311$	28.0625	1.59128	0.795640	+	0.605770i	$0.207135\pi$
$312$	0	0
$313$	−14.4622	−0.817452	−0.408726	−	0.912657i	$-0.634027\pi$
$313$	−14.4622	−0.817452	−0.408726	+	0.912657i	$0.634027\pi$
$314$	0	0
$315$	−3.67689	−0.207169
$316$	0	0
$317$	18.2981	1.02772	0.513861	−	0.857873i	$-0.328215\pi$
$317$	18.2981	1.02772	0.513861	+	0.857873i	$0.328215\pi$
$318$	0	0
$319$	−8.17116	−0.457497
$320$	0	0
$321$	−27.8017	−1.55174
$322$	0	0
$323$	5.10970	0.284311
$324$	0	0
$325$	0.757976	0.0420450
$326$	0	0
$327$	17.8880	0.989209
$328$	0	0
$329$	−4.20381	−0.231763
$330$	0	0
$331$	9.35127	0.513992	0.256996	−	0.966412i	$-0.417267\pi$
$331$	9.35127	0.513992	0.256996	+	0.966412i	$0.417267\pi$
$332$	0	0
$333$	1.52371	0.0834991
$334$	0	0
$335$	−27.6193	−1.50900
$336$	0	0
$337$	11.6926	0.636937	0.318469	−	0.947933i	$-0.396831\pi$
$337$	11.6926	0.636937	0.318469	+	0.947933i	$0.396831\pi$
$338$	0	0
$339$	8.81016	0.478502
$340$	0	0
$341$	−0.915644	−0.0495849
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	10.7119	0.576710
$346$	0	0
$347$	5.72583	0.307379	0.153689	−	0.988119i	$-0.450885\pi$
$347$	5.72583	0.307379	0.153689	+	0.988119i	$0.450885\pi$
$348$	0	0
$349$	−36.9425	−1.97749	−0.988743	−	0.149626i	$-0.952193\pi$
$349$	−36.9425	−1.97749	−0.988743	+	0.149626i	$0.952193\pi$
$350$	0	0
$351$	2.65733	0.141838
$352$	0	0
$353$	1.74420	0.0928345	0.0464172	−	0.998922i	$-0.485220\pi$
$353$	1.74420	0.0928345	0.0464172	+	0.998922i	$0.485220\pi$
$354$	0	0
$355$	5.31660	0.282176
$356$	0	0
$357$	4.24462	0.224649
$358$	0	0
$359$	36.3846	1.92031	0.960153	−	0.279475i	$-0.0901604\pi$
$359$	36.3846	1.92031	0.960153	+	0.279475i	$0.0901604\pi$
$360$	0	0
$361$	−12.0655	−0.635025
$362$	0	0
$363$	2.18752	0.114815
$364$	0	0
$365$	19.2138	1.00570
$366$	0	0
$367$	20.0647	1.04737	0.523683	−	0.851913i	$-0.324558\pi$
$367$	20.0647	1.04737	0.523683	+	0.851913i	$0.324558\pi$
$368$	0	0
$369$	11.3080	0.588673
$370$	0	0
$371$	6.84485	0.355367
$372$	0	0
$373$	−25.5948	−1.32525	−0.662625	−	0.748951i	$-0.730558\pi$
$373$	−25.5948	−1.32525	−0.662625	+	0.748951i	$0.730558\pi$
$374$	0	0
$375$	25.9423	1.33965
$376$	0	0
$377$	−8.17116	−0.420836
$378$	0	0
$379$	11.2631	0.578544	0.289272	−	0.957247i	$-0.406587\pi$
$379$	11.2631	0.578544	0.289272	+	0.957247i	$0.406587\pi$
$380$	0	0
$381$	40.7728	2.08885
$382$	0	0
$383$	11.8962	0.607866	0.303933	−	0.952693i	$-0.401700\pi$
$383$	11.8962	0.607866	0.303933	+	0.952693i	$0.401700\pi$
$384$	0	0
$385$	2.05962	0.104968
$386$	0	0
$387$	−14.7275	−0.748642
$388$	0	0
$389$	14.2863	0.724342	0.362171	−	0.932112i	$-0.382036\pi$
$389$	14.2863	0.724342	0.362171	+	0.932112i	$0.382036\pi$
$390$	0	0
$391$	−4.61335	−0.233307
$392$	0	0
$393$	11.6925	0.589810
$394$	0	0
$395$	−24.5746	−1.23648
$396$	0	0
$397$	14.6623	0.735879	0.367940	−	0.929850i	$-0.380063\pi$
$397$	14.6623	0.735879	0.367940	+	0.929850i	$0.380063\pi$
$398$	0	0
$399$	5.76049	0.288385
$400$	0	0
$401$	12.1559	0.607037	0.303519	−	0.952826i	$-0.401839\pi$
$401$	12.1559	0.607037	0.303519	+	0.952826i	$0.401839\pi$
$402$	0	0
$403$	−0.915644	−0.0456115
$404$	0	0
$405$	23.0031	1.14303
$406$	0	0
$407$	−0.853512	−0.0423070
$408$	0	0
$409$	12.3792	0.612111	0.306055	−	0.952014i	$-0.400991\pi$
$409$	12.3792	0.612111	0.306055	+	0.952014i	$0.400991\pi$
$410$	0	0
$411$	−47.1611	−2.32629
$412$	0	0
$413$	12.3639	0.608385
$414$	0	0
$415$	−7.47139	−0.366756
$416$	0	0
$417$	−46.7463	−2.28918
$418$	0	0
$419$	−21.6126	−1.05584	−0.527921	−	0.849293i	$-0.677028\pi$
$419$	−21.6126	−1.05584	−0.527921	+	0.849293i	$0.677028\pi$
$420$	0	0
$421$	−28.3147	−1.37997	−0.689987	−	0.723822i	$-0.742384\pi$
$421$	−28.3147	−1.37997	−0.689987	+	0.723822i	$0.742384\pi$
$422$	0	0
$423$	−7.50476	−0.364894
$424$	0	0
$425$	−1.47076	−0.0713425
$426$	0	0
$427$	3.68207	0.178188
$428$	0	0
$429$	2.18752	0.105614
$430$	0	0
$431$	22.0242	1.06087	0.530435	−	0.847726i	$-0.322029\pi$
$431$	22.0242	1.06087	0.530435	+	0.847726i	$0.322029\pi$
$432$	0	0
$433$	32.5563	1.56455	0.782277	−	0.622931i	$-0.214058\pi$
$433$	32.5563	1.56455	0.782277	+	0.622931i	$0.214058\pi$
$434$	0	0
$435$	−36.8147	−1.76513
$436$	0	0
$437$	−6.26091	−0.299500
$438$	0	0
$439$	−37.2907	−1.77979	−0.889894	−	0.456168i	$-0.849222\pi$
$439$	−37.2907	−1.77979	−0.889894	+	0.456168i	$0.849222\pi$
$440$	0	0
$441$	1.78523	0.0850109
$442$	0	0
$443$	12.6741	0.602164	0.301082	−	0.953598i	$-0.402652\pi$
$443$	12.6741	0.602164	0.301082	+	0.953598i	$0.402652\pi$
$444$	0	0
$445$	15.9449	0.755861
$446$	0	0
$447$	37.4779	1.77265
$448$	0	0
$449$	31.2717	1.47580	0.737902	−	0.674908i	$-0.235817\pi$
$449$	31.2717	1.47580	0.737902	+	0.674908i	$0.235817\pi$
$450$	0	0
$451$	−6.33422	−0.298267
$452$	0	0
$453$	40.2735	1.89221
$454$	0	0
$455$	2.05962	0.0965564
$456$	0	0
$457$	18.8173	0.880237	0.440118	−	0.897940i	$-0.354937\pi$
$457$	18.8173	0.880237	0.440118	+	0.897940i	$0.354937\pi$
$458$	0	0
$459$	−5.15624	−0.240673
$460$	0	0
$461$	28.0432	1.30610	0.653050	−	0.757314i	$-0.273489\pi$
$461$	28.0432	1.30610	0.653050	+	0.757314i	$0.273489\pi$
$462$	0	0
$463$	−7.45939	−0.346667	−0.173334	−	0.984863i	$-0.555454\pi$
$463$	−7.45939	−0.346667	−0.173334	+	0.984863i	$0.555454\pi$
$464$	0	0
$465$	−4.12538	−0.191310
$466$	0	0
$467$	12.4758	0.577312	0.288656	−	0.957433i	$-0.406792\pi$
$467$	12.4758	0.577312	0.288656	+	0.957433i	$0.406792\pi$
$468$	0	0
$469$	13.4099	0.619212
$470$	0	0
$471$	−3.72187	−0.171495
$472$	0	0
$473$	8.24965	0.379319
$474$	0	0
$475$	−1.99601	−0.0915834
$476$	0	0
$477$	12.2196	0.559498
$478$	0	0
$479$	11.9292	0.545059	0.272530	−	0.962147i	$-0.412140\pi$
$479$	11.9292	0.545059	0.272530	+	0.962147i	$0.412140\pi$
$480$	0	0
$481$	−0.853512	−0.0389168
$482$	0	0
$483$	−5.20093	−0.236650
$484$	0	0
$485$	−9.85056	−0.447291
$486$	0	0
$487$	5.15523	0.233606	0.116803	−	0.993155i	$-0.462735\pi$
$487$	5.15523	0.233606	0.116803	+	0.993155i	$0.462735\pi$
$488$	0	0
$489$	1.47356	0.0666365
$490$	0	0
$491$	−39.8312	−1.79756	−0.898778	−	0.438404i	$-0.855544\pi$
$491$	−39.8312	−1.79756	−0.898778	+	0.438404i	$0.855544\pi$
$492$	0	0
$493$	15.8552	0.714081
$494$	0	0
$495$	3.67689	0.165264
$496$	0	0
$497$	−2.58135	−0.115790
$498$	0	0
$499$	−41.6137	−1.86289	−0.931443	−	0.363887i	$-0.881450\pi$
$499$	−41.6137	−1.86289	−0.931443	+	0.363887i	$0.881450\pi$
$500$	0	0
$501$	20.8529	0.931641
$502$	0	0
$503$	28.6242	1.27629	0.638146	−	0.769916i	$-0.279702\pi$
$503$	28.6242	1.27629	0.638146	+	0.769916i	$0.279702\pi$
$504$	0	0
$505$	−20.5131	−0.912820
$506$	0	0
$507$	2.18752	0.0971510
$508$	0	0
$509$	−12.5655	−0.556954	−0.278477	−	0.960443i	$-0.589830\pi$
$509$	−12.5655	−0.556954	−0.278477	+	0.960443i	$0.589830\pi$
$510$	0	0
$511$	−9.32883	−0.412683
$512$	0	0
$513$	−6.99767	−0.308955
$514$	0	0
$515$	−12.5644	−0.553654
$516$	0	0
$517$	4.20381	0.184883
$518$	0	0
$519$	8.17367	0.358784
$520$	0	0
$521$	−4.96660	−0.217591	−0.108795	−	0.994064i	$-0.534699\pi$
$521$	−4.96660	−0.217591	−0.108795	+	0.994064i	$0.534699\pi$
$522$	0	0
$523$	−43.5177	−1.90290	−0.951449	−	0.307806i	$-0.900405\pi$
$523$	−43.5177	−1.90290	−0.951449	+	0.307806i	$0.900405\pi$
$524$	0	0
$525$	−1.65809	−0.0723648
$526$	0	0
$527$	1.77670	0.0773942
$528$	0	0
$529$	−17.3473	−0.754229
$530$	0	0
$531$	22.0723	0.957857
$532$	0	0
$533$	−6.33422	−0.274366
$534$	0	0
$535$	26.1762	1.13170
$536$	0	0
$537$	−28.9900	−1.25101
$538$	0	0
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	−10.2877	−0.442304	−0.221152	−	0.975239i	$-0.570982\pi$
$541$	−10.2877	−0.442304	−0.221152	+	0.975239i	$0.570982\pi$
$542$	0	0
$543$	−0.773429	−0.0331910
$544$	0	0
$545$	−16.8421	−0.721437
$546$	0	0
$547$	−22.0693	−0.943613	−0.471807	−	0.881702i	$-0.656398\pi$
$547$	−22.0693	−0.943613	−0.471807	+	0.881702i	$0.656398\pi$
$548$	0	0
$549$	6.57334	0.280543
$550$	0	0
$551$	21.5175	0.916676
$552$	0	0
$553$	11.9316	0.507385
$554$	0	0
$555$	−3.84545	−0.163230
$556$	0	0
$557$	−19.7691	−0.837643	−0.418822	−	0.908069i	$-0.637557\pi$
$557$	−19.7691	−0.837643	−0.418822	+	0.908069i	$0.637557\pi$
$558$	0	0
$559$	8.24965	0.348923
$560$	0	0
$561$	−4.24462	−0.179208
$562$	0	0
$563$	−9.17954	−0.386871	−0.193436	−	0.981113i	$-0.561963\pi$
$563$	−9.17954	−0.386871	−0.193436	+	0.981113i	$0.561963\pi$
$564$	0	0
$565$	−8.29505	−0.348975
$566$	0	0
$567$	−11.1686	−0.469039
$568$	0	0
$569$	40.4185	1.69443	0.847217	−	0.531247i	$-0.178276\pi$
$569$	40.4185	1.69443	0.847217	+	0.531247i	$0.178276\pi$
$570$	0	0
$571$	−33.9688	−1.42155	−0.710776	−	0.703419i	$-0.751656\pi$
$571$	−33.9688	−1.42155	−0.710776	+	0.703419i	$0.751656\pi$
$572$	0	0
$573$	−17.9825	−0.751228
$574$	0	0
$575$	1.80213	0.0751538
$576$	0	0
$577$	43.4229	1.80772	0.903860	−	0.427829i	$-0.140722\pi$
$577$	43.4229	1.80772	0.903860	+	0.427829i	$0.140722\pi$
$578$	0	0
$579$	58.4166	2.42771
$580$	0	0
$581$	3.62756	0.150497
$582$	0	0
$583$	−6.84485	−0.283485
$584$	0	0
$585$	3.67689	0.152021
$586$	0	0
$587$	−41.6860	−1.72056	−0.860282	−	0.509819i	$-0.829712\pi$
$587$	−41.6860	−1.72056	−0.860282	+	0.509819i	$0.829712\pi$
$588$	0	0
$589$	2.41121	0.0993521
$590$	0	0
$591$	11.3238	0.465799
$592$	0	0
$593$	16.8018	0.689969	0.344984	−	0.938608i	$-0.387884\pi$
$593$	16.8018	0.689969	0.344984	+	0.938608i	$0.387884\pi$
$594$	0	0
$595$	−3.99645	−0.163838
$596$	0	0
$597$	−7.92402	−0.324309
$598$	0	0
$599$	−37.6019	−1.53637	−0.768186	−	0.640227i	$-0.778840\pi$
$599$	−37.6019	−1.53637	−0.768186	+	0.640227i	$0.778840\pi$
$600$	0	0
$601$	−40.0375	−1.63316	−0.816582	−	0.577229i	$-0.804134\pi$
$601$	−40.0375	−1.63316	−0.816582	+	0.577229i	$0.804134\pi$
$602$	0	0
$603$	23.9398	0.974903
$604$	0	0
$605$	−2.05962	−0.0837354
$606$	0	0
$607$	38.6907	1.57041	0.785204	−	0.619238i	$-0.212558\pi$
$607$	38.6907	1.57041	0.785204	+	0.619238i	$0.212558\pi$
$608$	0	0
$609$	17.8745	0.724313
$610$	0	0
$611$	4.20381	0.170068
$612$	0	0
$613$	−43.6420	−1.76269	−0.881343	−	0.472477i	$-0.843360\pi$
$613$	−43.6420	−1.76269	−0.881343	+	0.472477i	$0.843360\pi$
$614$	0	0
$615$	−28.5385	−1.15078
$616$	0	0
$617$	14.4978	0.583660	0.291830	−	0.956470i	$-0.405736\pi$
$617$	14.4978	0.583660	0.291830	+	0.956470i	$0.405736\pi$
$618$	0	0
$619$	27.1430	1.09097	0.545485	−	0.838121i	$-0.316346\pi$
$619$	27.1430	1.09097	0.545485	+	0.838121i	$0.316346\pi$
$620$	0	0
$621$	6.31793	0.253530
$622$	0	0
$623$	−7.74169	−0.310164
$624$	0	0
$625$	−20.6356	−0.825424
$626$	0	0
$627$	−5.76049	−0.230052
$628$	0	0
$629$	1.65614	0.0660346
$630$	0	0
$631$	−8.83123	−0.351566	−0.175783	−	0.984429i	$-0.556246\pi$
$631$	−8.83123	−0.351566	−0.175783	+	0.984429i	$0.556246\pi$
$632$	0	0
$633$	−2.02635	−0.0805401
$634$	0	0
$635$	−38.3889	−1.52342
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	−4.60831	−0.182302
$640$	0	0
$641$	6.93557	0.273939	0.136969	−	0.990575i	$-0.456264\pi$
$641$	6.93557	0.273939	0.136969	+	0.990575i	$0.456264\pi$
$642$	0	0
$643$	−3.44674	−0.135926	−0.0679630	−	0.997688i	$-0.521650\pi$
$643$	−3.44674	−0.135926	−0.0679630	+	0.997688i	$0.521650\pi$
$644$	0	0
$645$	37.1684	1.46350
$646$	0	0
$647$	−1.99442	−0.0784087	−0.0392043	−	0.999231i	$-0.512482\pi$
$647$	−1.99442	−0.0784087	−0.0392043	+	0.999231i	$0.512482\pi$
$648$	0	0
$649$	−12.3639	−0.485324
$650$	0	0
$651$	2.00299	0.0785032
$652$	0	0
$653$	41.2655	1.61484	0.807422	−	0.589974i	$-0.200862\pi$
$653$	41.2655	1.61484	0.807422	+	0.589974i	$0.200862\pi$
$654$	0	0
$655$	−11.0089	−0.430153
$656$	0	0
$657$	−16.6541	−0.649738
$658$	0	0
$659$	−15.4748	−0.602812	−0.301406	−	0.953496i	$-0.597456\pi$
$659$	−15.4748	−0.602812	−0.301406	+	0.953496i	$0.597456\pi$
$660$	0	0
$661$	8.07135	0.313939	0.156970	−	0.987603i	$-0.449828\pi$
$661$	8.07135	0.313939	0.156970	+	0.987603i	$0.449828\pi$
$662$	0	0
$663$	−4.24462	−0.164847
$664$	0	0
$665$	−5.42369	−0.210322
$666$	0	0
$667$	−19.4273	−0.752229
$668$	0	0
$669$	50.5509	1.95441
$670$	0	0
$671$	−3.68207	−0.142145
$672$	0	0
$673$	−26.7774	−1.03219	−0.516097	−	0.856530i	$-0.672616\pi$
$673$	−26.7774	−1.03219	−0.516097	+	0.856530i	$0.672616\pi$
$674$	0	0
$675$	2.01419	0.0775263
$676$	0	0
$677$	46.3091	1.77980	0.889902	−	0.456151i	$-0.150772\pi$
$677$	46.3091	1.77980	0.889902	+	0.456151i	$0.150772\pi$
$678$	0	0
$679$	4.78272	0.183544
$680$	0	0
$681$	58.6698	2.24823
$682$	0	0
$683$	8.40909	0.321765	0.160882	−	0.986974i	$-0.448566\pi$
$683$	8.40909	0.321765	0.160882	+	0.986974i	$0.448566\pi$
$684$	0	0
$685$	44.4037	1.69658
$686$	0	0
$687$	−63.5563	−2.42482
$688$	0	0
$689$	−6.84485	−0.260768
$690$	0	0
$691$	37.1060	1.41158	0.705789	−	0.708422i	$-0.250593\pi$
$691$	37.1060	1.41158	0.705789	+	0.708422i	$0.250593\pi$
$692$	0	0
$693$	−1.78523	−0.0678153
$694$	0	0
$695$	44.0131	1.66951
$696$	0	0
$697$	12.2908	0.465548
$698$	0	0
$699$	56.9623	2.15451
$700$	0	0
$701$	−49.9600	−1.88696	−0.943481	−	0.331425i	$-0.892470\pi$
$701$	−49.9600	−1.88696	−0.943481	+	0.331425i	$0.892470\pi$
$702$	0	0
$703$	2.24759	0.0847696
$704$	0	0
$705$	18.9400	0.713323
$706$	0	0
$707$	9.95966	0.374572
$708$	0	0
$709$	42.5349	1.59743	0.798717	−	0.601707i	$-0.205513\pi$
$709$	42.5349	1.59743	0.798717	+	0.601707i	$0.205513\pi$
$710$	0	0
$711$	21.3007	0.798840
$712$	0	0
$713$	−2.17699	−0.0815288
$714$	0	0
$715$	−2.05962	−0.0770253
$716$	0	0
$717$	−26.1365	−0.976084
$718$	0	0
$719$	−29.5845	−1.10332	−0.551658	−	0.834070i	$-0.686005\pi$
$719$	−29.5845	−1.10332	−0.551658	+	0.834070i	$0.686005\pi$
$720$	0	0
$721$	6.10036	0.227189
$722$	0	0
$723$	28.0400	1.04282
$724$	0	0
$725$	−6.19354	−0.230022
$726$	0	0
$727$	−43.6553	−1.61908	−0.809542	−	0.587061i	$-0.800285\pi$
$727$	−43.6553	−1.61908	−0.809542	+	0.587061i	$0.800285\pi$
$728$	0	0
$729$	−2.49973	−0.0925826
$730$	0	0
$731$	−16.0075	−0.592058
$732$	0	0
$733$	19.5254	0.721187	0.360594	−	0.932723i	$-0.382574\pi$
$733$	19.5254	0.721187	0.360594	+	0.932723i	$0.382574\pi$
$734$	0	0
$735$	−4.50545	−0.166186
$736$	0	0
$737$	−13.4099	−0.493960
$738$	0	0
$739$	−45.1902	−1.66235	−0.831175	−	0.556012i	$-0.812331\pi$
$739$	−45.1902	−1.66235	−0.831175	+	0.556012i	$0.812331\pi$
$740$	0	0
$741$	−5.76049	−0.211617
$742$	0	0
$743$	−2.25278	−0.0826467	−0.0413233	−	0.999146i	$-0.513157\pi$
$743$	−2.25278	−0.0826467	−0.0413233	+	0.999146i	$0.513157\pi$
$744$	0	0
$745$	−35.2867	−1.29280
$746$	0	0
$747$	6.47603	0.236946
$748$	0	0
$749$	−12.7093	−0.464386
$750$	0	0
$751$	−16.2216	−0.591935	−0.295968	−	0.955198i	$-0.595642\pi$
$751$	−16.2216	−0.591935	−0.295968	+	0.955198i	$0.595642\pi$
$752$	0	0
$753$	62.0103	2.25978
$754$	0	0
$755$	−37.9188	−1.38001
$756$	0	0
$757$	−12.1966	−0.443292	−0.221646	−	0.975127i	$-0.571143\pi$
$757$	−12.1966	−0.443292	−0.221646	+	0.975127i	$0.571143\pi$
$758$	0	0
$759$	5.20093	0.188782
$760$	0	0
$761$	−34.7661	−1.26027	−0.630136	−	0.776485i	$-0.717001\pi$
$761$	−34.7661	−1.26027	−0.630136	+	0.776485i	$0.717001\pi$
$762$	0	0
$763$	8.17731	0.296038
$764$	0	0
$765$	−7.13457	−0.257951
$766$	0	0
$767$	−12.3639	−0.446433
$768$	0	0
$769$	12.1295	0.437402	0.218701	−	0.975792i	$-0.429818\pi$
$769$	12.1295	0.437402	0.218701	+	0.975792i	$0.429818\pi$
$770$	0	0
$771$	−30.8924	−1.11256
$772$	0	0
$773$	21.5202	0.774026	0.387013	−	0.922074i	$-0.373507\pi$
$773$	21.5202	0.774026	0.387013	+	0.922074i	$0.373507\pi$
$774$	0	0
$775$	−0.694036	−0.0249305
$776$	0	0
$777$	1.86707	0.0669808
$778$	0	0
$779$	16.6802	0.597630
$780$	0	0
$781$	2.58135	0.0923681
$782$	0	0
$783$	−21.7135	−0.775976
$784$	0	0
$785$	3.50426	0.125072
$786$	0	0
$787$	−39.4297	−1.40552	−0.702759	−	0.711428i	$-0.748049\pi$
$787$	−39.4297	−1.40552	−0.702759	+	0.711428i	$0.748049\pi$
$788$	0	0
$789$	−37.7712	−1.34469
$790$	0	0
$791$	4.02747	0.143200
$792$	0	0
$793$	−3.68207	−0.130754
$794$	0	0
$795$	−30.8391	−1.09375
$796$	0	0
$797$	4.27518	0.151435	0.0757174	−	0.997129i	$-0.475875\pi$
$797$	4.27518	0.151435	0.0757174	+	0.997129i	$0.475875\pi$
$798$	0	0
$799$	−8.15699	−0.288574
$800$	0	0
$801$	−13.8207	−0.488330
$802$	0	0
$803$	9.32883	0.329207
$804$	0	0
$805$	4.89684	0.172591
$806$	0	0
$807$	10.3184	0.363225
$808$	0	0
$809$	20.2794	0.712984	0.356492	−	0.934298i	$-0.383973\pi$
$809$	20.2794	0.712984	0.356492	+	0.934298i	$0.383973\pi$
$810$	0	0
$811$	−35.4650	−1.24535	−0.622673	−	0.782482i	$-0.713953\pi$
$811$	−35.4650	−1.24535	−0.622673	+	0.782482i	$0.713953\pi$
$812$	0	0
$813$	−61.8086	−2.16772
$814$	0	0
$815$	−1.38740	−0.0485985
$816$	0	0
$817$	−21.7242	−0.760033
$818$	0	0
$819$	−1.78523	−0.0623810
$820$	0	0
$821$	−19.6107	−0.684417	−0.342208	−	0.939624i	$-0.611175\pi$
$821$	−19.6107	−0.684417	−0.342208	+	0.939624i	$0.611175\pi$
$822$	0	0
$823$	18.5669	0.647202	0.323601	−	0.946194i	$-0.395106\pi$
$823$	18.5669	0.647202	0.323601	+	0.946194i	$0.395106\pi$
$824$	0	0
$825$	1.65809	0.0577271
$826$	0	0
$827$	−50.4514	−1.75437	−0.877184	−	0.480155i	$-0.840580\pi$
$827$	−50.4514	−1.75437	−0.877184	+	0.480155i	$0.840580\pi$
$828$	0	0
$829$	4.12771	0.143361	0.0716806	−	0.997428i	$-0.477164\pi$
$829$	4.12771	0.143361	0.0716806	+	0.997428i	$0.477164\pi$
$830$	0	0
$831$	10.9502	0.379860
$832$	0	0
$833$	1.94038	0.0672303
$834$	0	0
$835$	−19.6337	−0.679453
$836$	0	0
$837$	−2.43317	−0.0841026
$838$	0	0
$839$	−25.7910	−0.890403	−0.445201	−	0.895430i	$-0.646868\pi$
$839$	−25.7910	−0.890403	−0.445201	+	0.895430i	$0.646868\pi$
$840$	0	0
$841$	37.7678	1.30234
$842$	0	0
$843$	−37.6651	−1.29725
$844$	0	0
$845$	−2.05962	−0.0708530
$846$	0	0
$847$	1.00000	0.0343604
$848$	0	0
$849$	−21.5319	−0.738974
$850$	0	0
$851$	−2.02927	−0.0695623
$852$	0	0
$853$	28.4982	0.975761	0.487881	−	0.872910i	$-0.337770\pi$
$853$	28.4982	0.975761	0.487881	+	0.872910i	$0.337770\pi$
$854$	0	0
$855$	−9.68253	−0.331135
$856$	0	0
$857$	−1.38098	−0.0471734	−0.0235867	−	0.999722i	$-0.507509\pi$
$857$	−1.38098	−0.0471734	−0.0235867	+	0.999722i	$0.507509\pi$
$858$	0	0
$859$	−29.0451	−0.991006	−0.495503	−	0.868606i	$-0.665016\pi$
$859$	−29.0451	−0.991006	−0.495503	+	0.868606i	$0.665016\pi$
$860$	0	0
$861$	13.8562	0.472219
$862$	0	0
$863$	12.5657	0.427742	0.213871	−	0.976862i	$-0.431393\pi$
$863$	12.5657	0.427742	0.213871	+	0.976862i	$0.431393\pi$
$864$	0	0
$865$	−7.69578	−0.261664
$866$	0	0
$867$	−28.9516	−0.983248
$868$	0	0
$869$	−11.9316	−0.404753
$870$	0	0
$871$	−13.4099	−0.454378
$872$	0	0
$873$	8.53825	0.288976
$874$	0	0
$875$	11.8592	0.400915
$876$	0	0
$877$	33.8689	1.14367	0.571837	−	0.820368i	$-0.306231\pi$
$877$	33.8689	1.14367	0.571837	+	0.820368i	$0.306231\pi$
$878$	0	0
$879$	−40.1656	−1.35475
$880$	0	0
$881$	4.45741	0.150174	0.0750870	−	0.997177i	$-0.476077\pi$
$881$	4.45741	0.150174	0.0750870	+	0.997177i	$0.476077\pi$
$882$	0	0
$883$	−22.2039	−0.747220	−0.373610	−	0.927586i	$-0.621880\pi$
$883$	−22.2039	−0.747220	−0.373610	+	0.927586i	$0.621880\pi$
$884$	0	0
$885$	−55.7047	−1.87249
$886$	0	0
$887$	42.9156	1.44096	0.720482	−	0.693474i	$-0.243921\pi$
$887$	42.9156	1.44096	0.720482	+	0.693474i	$0.243921\pi$
$888$	0	0
$889$	18.6388	0.625127
$890$	0	0
$891$	11.1686	0.374164
$892$	0	0
$893$	−11.0701	−0.370446
$894$	0	0
$895$	27.2950	0.912371
$896$	0	0
$897$	5.20093	0.173654
$898$	0	0
$899$	7.48187	0.249534
$900$	0	0
$901$	13.2816	0.442475
$902$	0	0
$903$	−18.0462	−0.600541
$904$	0	0
$905$	0.728208	0.0242065
$906$	0	0
$907$	−8.18957	−0.271930	−0.135965	−	0.990714i	$-0.543414\pi$
$907$	−8.18957	−0.271930	−0.135965	+	0.990714i	$0.543414\pi$
$908$	0	0
$909$	17.7803	0.589735
$910$	0	0
$911$	32.1220	1.06425	0.532125	−	0.846666i	$-0.321394\pi$
$911$	32.1220	1.06425	0.532125	+	0.846666i	$0.321394\pi$
$912$	0	0
$913$	−3.62756	−0.120055
$914$	0	0
$915$	−16.5894	−0.548428
$916$	0	0
$917$	5.34512	0.176511
$918$	0	0
$919$	−39.4195	−1.30033	−0.650165	−	0.759793i	$-0.725300\pi$
$919$	−39.4195	−1.30033	−0.650165	+	0.759793i	$0.725300\pi$
$920$	0	0
$921$	29.3954	0.968610
$922$	0	0
$923$	2.58135	0.0849663
$924$	0	0
$925$	−0.646942	−0.0212713
$926$	0	0
$927$	10.8905	0.357693
$928$	0	0
$929$	−21.8852	−0.718029	−0.359014	−	0.933332i	$-0.616887\pi$
$929$	−21.8852	−0.718029	−0.359014	+	0.933332i	$0.616887\pi$
$930$	0	0
$931$	2.63335	0.0863045
$932$	0	0
$933$	61.3873	2.00973
$934$	0	0
$935$	3.99645	0.130698
$936$	0	0
$937$	18.6548	0.609426	0.304713	−	0.952444i	$-0.401439\pi$
$937$	18.6548	0.609426	0.304713	+	0.952444i	$0.401439\pi$
$938$	0	0
$939$	−31.6363	−1.03241
$940$	0	0
$941$	−2.91191	−0.0949255	−0.0474627	−	0.998873i	$-0.515114\pi$
$941$	−2.91191	−0.0949255	−0.0474627	+	0.998873i	$0.515114\pi$
$942$	0	0
$943$	−15.0599	−0.490418
$944$	0	0
$945$	5.47308	0.178039
$946$	0	0
$947$	44.8350	1.45694	0.728470	−	0.685077i	$-0.240232\pi$
$947$	44.8350	1.45694	0.728470	+	0.685077i	$0.240232\pi$
$948$	0	0
$949$	9.32883	0.302827
$950$	0	0
$951$	40.0274	1.29798
$952$	0	0
$953$	−48.0339	−1.55597	−0.777986	−	0.628282i	$-0.783758\pi$
$953$	−48.0339	−1.55597	−0.777986	+	0.628282i	$0.783758\pi$
$954$	0	0
$955$	16.9311	0.547876
$956$	0	0
$957$	−17.8745	−0.577802
$958$	0	0
$959$	−21.5592	−0.696183
$960$	0	0
$961$	−30.1616	−0.972955
$962$	0	0
$963$	−22.6889	−0.731141
$964$	0	0
$965$	−55.0011	−1.77055
$966$	0	0
$967$	−19.8944	−0.639760	−0.319880	−	0.947458i	$-0.603643\pi$
$967$	−19.8944	−0.639760	−0.319880	+	0.947458i	$0.603643\pi$
$968$	0	0
$969$	11.1776	0.359075
$970$	0	0
$971$	−31.9804	−1.02630	−0.513150	−	0.858299i	$-0.671522\pi$
$971$	−31.9804	−1.02630	−0.513150	+	0.858299i	$0.671522\pi$
$972$	0	0
$973$	−21.3696	−0.685077
$974$	0	0
$975$	1.65809	0.0531012
$976$	0	0
$977$	19.4945	0.623684	0.311842	−	0.950134i	$-0.399054\pi$
$977$	19.4945	0.623684	0.311842	+	0.950134i	$0.399054\pi$
$978$	0	0
$979$	7.74169	0.247425
$980$	0	0
$981$	14.5984	0.466090
$982$	0	0
$983$	4.21514	0.134442	0.0672210	−	0.997738i	$-0.478587\pi$
$983$	4.21514	0.134442	0.0672210	+	0.997738i	$0.478587\pi$
$984$	0	0
$985$	−10.6617	−0.339711
$986$	0	0
$987$	−9.19590	−0.292709
$988$	0	0
$989$	19.6139	0.623687
$990$	0	0
$991$	35.9947	1.14341	0.571704	−	0.820460i	$-0.306282\pi$
$991$	35.9947	1.14341	0.571704	+	0.820460i	$0.306282\pi$
$992$	0	0
$993$	20.4561	0.649153
$994$	0	0
$995$	7.46072	0.236521
$996$	0	0
$997$	−56.2334	−1.78093	−0.890464	−	0.455053i	$-0.849620\pi$
$997$	−56.2334	−1.78093	−0.890464	+	0.455053i	$0.849620\pi$
$998$	0	0
$999$	−2.26806	−0.0717583

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8008.2.a.j.1.4	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8008.2.a.j.1.4	✓	5	1.1	even	1	trivial

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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:	\( N \)	\(=\)	\( 8008 = 2^{3} \cdot 7 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8008.a (trivial)

\(f(q)\)	\(=\)	\(q+2.18752 q^{3} -2.05962 q^{5} +1.00000 q^{7} +1.78523 q^{9} +O(q^{10})\)	\(q+2.18752 q^{3} -2.05962 q^{5} +1.00000 q^{7} +1.78523 q^{9} -1.00000 q^{11} -1.00000 q^{13} -4.50545 q^{15} +1.94038 q^{17} +2.63335 q^{19} +2.18752 q^{21} -2.37755 q^{23} -0.757976 q^{25} -2.65733 q^{27} +8.17116 q^{29} +0.915644 q^{31} -2.18752 q^{33} -2.05962 q^{35} +0.853512 q^{37} -2.18752 q^{39} +6.33422 q^{41} -8.24965 q^{43} -3.67689 q^{45} -4.20381 q^{47} +1.00000 q^{49} +4.24462 q^{51} +6.84485 q^{53} +2.05962 q^{55} +5.76049 q^{57} +12.3639 q^{59} +3.68207 q^{61} +1.78523 q^{63} +2.05962 q^{65} +13.4099 q^{67} -5.20093 q^{69} -2.58135 q^{71} -9.32883 q^{73} -1.65809 q^{75} -1.00000 q^{77} +11.9316 q^{79} -11.1686 q^{81} +3.62756 q^{83} -3.99645 q^{85} +17.8745 q^{87} -7.74169 q^{89} -1.00000 q^{91} +2.00299 q^{93} -5.42369 q^{95} +4.78272 q^{97} -1.78523 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + q^{3} - q^{5} + 5 q^{7} + 6 q^{9}+O(q^{10}) \) 5 * q + q^3 - q^5 + 5 * q^7 + 6 * q^9	\( 5 q + q^{3} - q^{5} + 5 q^{7} + 6 q^{9} - 5 q^{11} - 5 q^{13} - 5 q^{15} + 19 q^{17} - 5 q^{19} + q^{21} + 5 q^{23} + 12 q^{25} - 11 q^{27} - 17 q^{29} + 4 q^{31} - q^{33} - q^{35} + 10 q^{37} - q^{39} + 10 q^{41} - 25 q^{43} + q^{45} + 3 q^{47} + 5 q^{49} - q^{51} + 22 q^{53} + q^{55} + 16 q^{57} + 21 q^{59} + 26 q^{61} + 6 q^{63} + q^{65} + 28 q^{67} + 16 q^{69} + 28 q^{71} - 4 q^{73} - 5 q^{77} - 11 q^{79} + 5 q^{81} + 33 q^{85} + 31 q^{87} - 37 q^{89} - 5 q^{91} - 49 q^{93} + 29 q^{95} + 18 q^{97} - 6 q^{99}+O(q^{100}) \) 5 * q + q^3 - q^5 + 5 * q^7 + 6 * q^9 - 5 * q^11 - 5 * q^13 - 5 * q^15 + 19 * q^17 - 5 * q^19 + q^21 + 5 * q^23 + 12 * q^25 - 11 * q^27 - 17 * q^29 + 4 * q^31 - q^33 - q^35 + 10 * q^37 - q^39 + 10 * q^41 - 25 * q^43 + q^45 + 3 * q^47 + 5 * q^49 - q^51 + 22 * q^53 + q^55 + 16 * q^57 + 21 * q^59 + 26 * q^61 + 6 * q^63 + q^65 + 28 * q^67 + 16 * q^69 + 28 * q^71 - 4 * q^73 - 5 * q^77 - 11 * q^79 + 5 * q^81 + 33 * q^85 + 31 * q^87 - 37 * q^89 - 5 * q^91 - 49 * q^93 + 29 * q^95 + 18 * q^97 - 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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