LMFDB - Embedded newform 8004.2.a.e.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8004,2,Mod(1,8004)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8004, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("8004.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8004 = 2^{2} \cdot 3 \cdot 23 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8004.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:	$63.9122617778$
Analytic rank:	$1$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 3x^{8} - 13x^{7} + 32x^{6} + 40x^{5} - 79x^{4} - 39x^{3} + 58x^{2} + 9x - 11 $ x^9 - 3x^8 - 13x^7 + 32x^6 + 40x^5 - 79x^4 - 39x^3 + 58x^2 + 9x - 11
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$3.93556$ of defining polynomial
Character	$\chi$	$=$	8004.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -2.36120 q^{5} +2.82157 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -2.36120 q^{5} +2.82157 q^{7} +1.00000 q^{9} +2.46337 q^{11} -2.24447 q^{13} +2.36120 q^{15} -1.17707 q^{17} +5.00771 q^{19} -2.82157 q^{21} -1.00000 q^{23} +0.575276 q^{25} -1.00000 q^{27} +1.00000 q^{29} -6.16676 q^{31} -2.46337 q^{33} -6.66229 q^{35} +4.43163 q^{37} +2.24447 q^{39} -3.21444 q^{41} -12.1113 q^{43} -2.36120 q^{45} -3.40978 q^{47} +0.961232 q^{49} +1.17707 q^{51} -3.20671 q^{53} -5.81651 q^{55} -5.00771 q^{57} -2.13703 q^{59} +5.45934 q^{61} +2.82157 q^{63} +5.29964 q^{65} +11.5503 q^{67} +1.00000 q^{69} +6.80571 q^{71} +0.535914 q^{73} -0.575276 q^{75} +6.95055 q^{77} -9.08449 q^{79} +1.00000 q^{81} -13.4663 q^{83} +2.77929 q^{85} -1.00000 q^{87} +10.9138 q^{89} -6.33291 q^{91} +6.16676 q^{93} -11.8242 q^{95} +18.3762 q^{97} +2.46337 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - 9 q^{3} - 3 q^{5} + 7 q^{7} + 9 q^{9}+O(q^{10}) $ 9 * q - 9 * q^3 - 3 * q^5 + 7 * q^7 + 9 * q^9	$ 9 q - 9 q^{3} - 3 q^{5} + 7 q^{7} + 9 q^{9} - 2 q^{11} - 7 q^{13} + 3 q^{15} - q^{19} - 7 q^{21} - 9 q^{23} + 2 q^{25} - 9 q^{27} + 9 q^{29} + 8 q^{31} + 2 q^{33} - 5 q^{35} - 8 q^{37} + 7 q^{39} - 19 q^{41} - 3 q^{43} - 3 q^{45} - 3 q^{47} - 18 q^{49} - 17 q^{53} + 9 q^{55} + q^{57} - 10 q^{59} + q^{61} + 7 q^{63} - 16 q^{65} + 12 q^{67} + 9 q^{69} - 7 q^{71} + 13 q^{73} - 2 q^{75} - 15 q^{77} - 10 q^{79} + 9 q^{81} + 9 q^{83} - 6 q^{85} - 9 q^{87} - 5 q^{89} - 18 q^{91} - 8 q^{93} + 31 q^{95} - 7 q^{97} - 2 q^{99}+O(q^{100}) $ 9 * q - 9 * q^3 - 3 * q^5 + 7 * q^7 + 9 * q^9 - 2 * q^11 - 7 * q^13 + 3 * q^15 - q^19 - 7 * q^21 - 9 * q^23 + 2 * q^25 - 9 * q^27 + 9 * q^29 + 8 * q^31 + 2 * q^33 - 5 * q^35 - 8 * q^37 + 7 * q^39 - 19 * q^41 - 3 * q^43 - 3 * q^45 - 3 * q^47 - 18 * q^49 - 17 * q^53 + 9 * q^55 + q^57 - 10 * q^59 + q^61 + 7 * q^63 - 16 * q^65 + 12 * q^67 + 9 * q^69 - 7 * q^71 + 13 * q^73 - 2 * q^75 - 15 * q^77 - 10 * q^79 + 9 * q^81 + 9 * q^83 - 6 * q^85 - 9 * q^87 - 5 * q^89 - 18 * q^91 - 8 * q^93 + 31 * q^95 - 7 * q^97 - 2 * 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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	−2.36120	−1.05596	−0.527981	−	0.849256i	$-0.677051\pi$
$5$	−2.36120	−1.05596	−0.527981	+	0.849256i	$0.677051\pi$
$6$	0	0
$7$	2.82157	1.06645	0.533226	−	0.845973i	$-0.320980\pi$
$7$	2.82157	1.06645	0.533226	+	0.845973i	$0.320980\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	2.46337	0.742733	0.371367	−	0.928486i	$-0.378889\pi$
$11$	2.46337	0.742733	0.371367	+	0.928486i	$0.378889\pi$
$12$	0	0
$13$	−2.24447	−0.622503	−0.311252	−	0.950328i	$-0.600748\pi$
$13$	−2.24447	−0.622503	−0.311252	+	0.950328i	$0.600748\pi$
$14$	0	0
$15$	2.36120	0.609660
$16$	0	0
$17$	−1.17707	−0.285480	−0.142740	−	0.989760i	$-0.545591\pi$
$17$	−1.17707	−0.285480	−0.142740	+	0.989760i	$0.545591\pi$
$18$	0	0
$19$	5.00771	1.14885	0.574424	−	0.818558i	$-0.305226\pi$
$19$	5.00771	1.14885	0.574424	+	0.818558i	$0.305226\pi$
$20$	0	0
$21$	−2.82157	−0.615716
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	0.575276	0.115055
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	−6.16676	−1.10758	−0.553792	−	0.832655i	$-0.686820\pi$
$31$	−6.16676	−1.10758	−0.553792	+	0.832655i	$0.686820\pi$
$32$	0	0
$33$	−2.46337	−0.428817
$34$	0	0
$35$	−6.66229	−1.12613
$36$	0	0
$37$	4.43163	0.728555	0.364278	−	0.931290i	$-0.381316\pi$
$37$	4.43163	0.728555	0.364278	+	0.931290i	$0.381316\pi$
$38$	0	0
$39$	2.24447	0.359402
$40$	0	0
$41$	−3.21444	−0.502011	−0.251006	−	0.967986i	$-0.580761\pi$
$41$	−3.21444	−0.502011	−0.251006	+	0.967986i	$0.580761\pi$
$42$	0	0
$43$	−12.1113	−1.84696	−0.923478	−	0.383652i	$-0.874666\pi$
$43$	−12.1113	−1.84696	−0.923478	+	0.383652i	$0.874666\pi$
$44$	0	0
$45$	−2.36120	−0.351987
$46$	0	0
$47$	−3.40978	−0.497367	−0.248683	−	0.968585i	$-0.579998\pi$
$47$	−3.40978	−0.497367	−0.248683	+	0.968585i	$0.579998\pi$
$48$	0	0
$49$	0.961232	0.137319
$50$	0	0
$51$	1.17707	0.164822
$52$	0	0
$53$	−3.20671	−0.440476	−0.220238	−	0.975446i	$-0.570683\pi$
$53$	−3.20671	−0.440476	−0.220238	+	0.975446i	$0.570683\pi$
$54$	0	0
$55$	−5.81651	−0.784298
$56$	0	0
$57$	−5.00771	−0.663288
$58$	0	0
$59$	−2.13703	−0.278218	−0.139109	−	0.990277i	$-0.544424\pi$
$59$	−2.13703	−0.278218	−0.139109	+	0.990277i	$0.544424\pi$
$60$	0	0
$61$	5.45934	0.698997	0.349498	−	0.936937i	$-0.386352\pi$
$61$	5.45934	0.698997	0.349498	+	0.936937i	$0.386352\pi$
$62$	0	0
$63$	2.82157	0.355484
$64$	0	0
$65$	5.29964	0.657339
$66$	0	0
$67$	11.5503	1.41109	0.705546	−	0.708664i	$-0.250702\pi$
$67$	11.5503	1.41109	0.705546	+	0.708664i	$0.250702\pi$
$68$	0	0
$69$	1.00000	0.120386
$70$	0	0
$71$	6.80571	0.807690	0.403845	−	0.914827i	$-0.367674\pi$
$71$	6.80571	0.807690	0.403845	+	0.914827i	$0.367674\pi$
$72$	0	0
$73$	0.535914	0.0627240	0.0313620	−	0.999508i	$-0.490016\pi$
$73$	0.535914	0.0627240	0.0313620	+	0.999508i	$0.490016\pi$
$74$	0	0
$75$	−0.575276	−0.0664271
$76$	0	0
$77$	6.95055	0.792089
$78$	0	0
$79$	−9.08449	−1.02209	−0.511043	−	0.859555i	$-0.670741\pi$
$79$	−9.08449	−1.02209	−0.511043	+	0.859555i	$0.670741\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−13.4663	−1.47811	−0.739057	−	0.673643i	$-0.764729\pi$
$83$	−13.4663	−1.47811	−0.739057	+	0.673643i	$0.764729\pi$
$84$	0	0
$85$	2.77929	0.301456
$86$	0	0
$87$	−1.00000	−0.107211
$88$	0	0
$89$	10.9138	1.15686	0.578431	−	0.815731i	$-0.303665\pi$
$89$	10.9138	1.15686	0.578431	+	0.815731i	$0.303665\pi$
$90$	0	0
$91$	−6.33291	−0.663869
$92$	0	0
$93$	6.16676	0.639464
$94$	0	0
$95$	−11.8242	−1.21314
$96$	0	0
$97$	18.3762	1.86582	0.932908	−	0.360115i	$-0.117262\pi$
$97$	18.3762	1.86582	0.932908	+	0.360115i	$0.117262\pi$
$98$	0	0
$99$	2.46337	0.247578
$100$	0	0
$101$	−8.04365	−0.800374	−0.400187	−	0.916434i	$-0.631055\pi$
$101$	−8.04365	−0.800374	−0.400187	+	0.916434i	$0.631055\pi$
$102$	0	0
$103$	12.1332	1.19552	0.597758	−	0.801677i	$-0.296058\pi$
$103$	12.1332	1.19552	0.597758	+	0.801677i	$0.296058\pi$
$104$	0	0
$105$	6.66229	0.650173
$106$	0	0
$107$	−5.53795	−0.535374	−0.267687	−	0.963506i	$-0.586259\pi$
$107$	−5.53795	−0.535374	−0.267687	+	0.963506i	$0.586259\pi$
$108$	0	0
$109$	3.56615	0.341575	0.170788	−	0.985308i	$-0.445369\pi$
$109$	3.56615	0.341575	0.170788	+	0.985308i	$0.445369\pi$
$110$	0	0
$111$	−4.43163	−0.420632
$112$	0	0
$113$	0.165574	0.0155759	0.00778794	−	0.999970i	$-0.497521\pi$
$113$	0.165574	0.0155759	0.00778794	+	0.999970i	$0.497521\pi$
$114$	0	0
$115$	2.36120	0.220183
$116$	0	0
$117$	−2.24447	−0.207501
$118$	0	0
$119$	−3.32117	−0.304451
$120$	0	0
$121$	−4.93182	−0.448348
$122$	0	0
$123$	3.21444	0.289836
$124$	0	0
$125$	10.4477	0.934468
$126$	0	0
$127$	13.0244	1.15573	0.577863	−	0.816134i	$-0.303887\pi$
$127$	13.0244	1.15573	0.577863	+	0.816134i	$0.303887\pi$
$128$	0	0
$129$	12.1113	1.06634
$130$	0	0
$131$	−20.4285	−1.78485	−0.892423	−	0.451199i	$-0.850996\pi$
$131$	−20.4285	−1.78485	−0.892423	+	0.451199i	$0.850996\pi$
$132$	0	0
$133$	14.1296	1.22519
$134$	0	0
$135$	2.36120	0.203220
$136$	0	0
$137$	12.2888	1.04990	0.524952	−	0.851132i	$-0.324083\pi$
$137$	12.2888	1.04990	0.524952	+	0.851132i	$0.324083\pi$
$138$	0	0
$139$	12.1965	1.03450	0.517248	−	0.855836i	$-0.326957\pi$
$139$	12.1965	1.03450	0.517248	+	0.855836i	$0.326957\pi$
$140$	0	0
$141$	3.40978	0.287155
$142$	0	0
$143$	−5.52895	−0.462354
$144$	0	0
$145$	−2.36120	−0.196087
$146$	0	0
$147$	−0.961232	−0.0792811
$148$	0	0
$149$	−21.7213	−1.77947	−0.889737	−	0.456473i	$-0.849113\pi$
$149$	−21.7213	−1.77947	−0.889737	+	0.456473i	$0.849113\pi$
$150$	0	0
$151$	−1.74182	−0.141747	−0.0708737	−	0.997485i	$-0.522579\pi$
$151$	−1.74182	−0.141747	−0.0708737	+	0.997485i	$0.522579\pi$
$152$	0	0
$153$	−1.17707	−0.0951601
$154$	0	0
$155$	14.5610	1.16957
$156$	0	0
$157$	−18.6182	−1.48589	−0.742947	−	0.669350i	$-0.766573\pi$
$157$	−18.6182	−1.48589	−0.742947	+	0.669350i	$0.766573\pi$
$158$	0	0
$159$	3.20671	0.254309
$160$	0	0
$161$	−2.82157	−0.222371
$162$	0	0
$163$	−12.4124	−0.972214	−0.486107	−	0.873899i	$-0.661583\pi$
$163$	−12.4124	−0.972214	−0.486107	+	0.873899i	$0.661583\pi$
$164$	0	0
$165$	5.81651	0.452815
$166$	0	0
$167$	2.02150	0.156428	0.0782140	−	0.996937i	$-0.475078\pi$
$167$	2.02150	0.156428	0.0782140	+	0.996937i	$0.475078\pi$
$168$	0	0
$169$	−7.96237	−0.612490
$170$	0	0
$171$	5.00771	0.382949
$172$	0	0
$173$	−1.55676	−0.118358	−0.0591792	−	0.998247i	$-0.518848\pi$
$173$	−1.55676	−0.118358	−0.0591792	+	0.998247i	$0.518848\pi$
$174$	0	0
$175$	1.62318	0.122701
$176$	0	0
$177$	2.13703	0.160629
$178$	0	0
$179$	−22.6826	−1.69538	−0.847689	−	0.530494i	$-0.822006\pi$
$179$	−22.6826	−1.69538	−0.847689	+	0.530494i	$0.822006\pi$
$180$	0	0
$181$	−2.39045	−0.177680	−0.0888402	−	0.996046i	$-0.528316\pi$
$181$	−2.39045	−0.177680	−0.0888402	+	0.996046i	$0.528316\pi$
$182$	0	0
$183$	−5.45934	−0.403566
$184$	0	0
$185$	−10.4640	−0.769327
$186$	0	0
$187$	−2.89954	−0.212036
$188$	0	0
$189$	−2.82157	−0.205239
$190$	0	0
$191$	4.58330	0.331636	0.165818	−	0.986156i	$-0.446974\pi$
$191$	4.58330	0.331636	0.165818	+	0.986156i	$0.446974\pi$
$192$	0	0
$193$	−17.3826	−1.25123	−0.625613	−	0.780133i	$-0.715151\pi$
$193$	−17.3826	−1.25123	−0.625613	+	0.780133i	$0.715151\pi$
$194$	0	0
$195$	−5.29964	−0.379515
$196$	0	0
$197$	3.86402	0.275300	0.137650	−	0.990481i	$-0.456045\pi$
$197$	3.86402	0.275300	0.137650	+	0.990481i	$0.456045\pi$
$198$	0	0
$199$	−1.66919	−0.118326	−0.0591628	−	0.998248i	$-0.518843\pi$
$199$	−1.66919	−0.118326	−0.0591628	+	0.998248i	$0.518843\pi$
$200$	0	0
$201$	−11.5503	−0.814695
$202$	0	0
$203$	2.82157	0.198035
$204$	0	0
$205$	7.58995	0.530105
$206$	0	0
$207$	−1.00000	−0.0695048
$208$	0	0
$209$	12.3358	0.853288
$210$	0	0
$211$	2.03379	0.140012	0.0700058	−	0.997547i	$-0.477698\pi$
$211$	2.03379	0.140012	0.0700058	+	0.997547i	$0.477698\pi$
$212$	0	0
$213$	−6.80571	−0.466320
$214$	0	0
$215$	28.5972	1.95031
$216$	0	0
$217$	−17.3999	−1.18118
$218$	0	0
$219$	−0.535914	−0.0362137
$220$	0	0
$221$	2.64188	0.177712
$222$	0	0
$223$	−7.15464	−0.479110	−0.239555	−	0.970883i	$-0.577002\pi$
$223$	−7.15464	−0.479110	−0.239555	+	0.970883i	$0.577002\pi$
$224$	0	0
$225$	0.575276	0.0383517
$226$	0	0
$227$	9.46474	0.628197	0.314099	−	0.949390i	$-0.398298\pi$
$227$	9.46474	0.628197	0.314099	+	0.949390i	$0.398298\pi$
$228$	0	0
$229$	19.9346	1.31731	0.658657	−	0.752443i	$-0.271125\pi$
$229$	19.9346	1.31731	0.658657	+	0.752443i	$0.271125\pi$
$230$	0	0
$231$	−6.95055	−0.457313
$232$	0	0
$233$	−20.0069	−1.31069	−0.655347	−	0.755328i	$-0.727478\pi$
$233$	−20.0069	−1.31069	−0.655347	+	0.755328i	$0.727478\pi$
$234$	0	0
$235$	8.05117	0.525200
$236$	0	0
$237$	9.08449	0.590101
$238$	0	0
$239$	−24.1719	−1.56355	−0.781777	−	0.623559i	$-0.785686\pi$
$239$	−24.1719	−1.56355	−0.781777	+	0.623559i	$0.785686\pi$
$240$	0	0
$241$	24.6368	1.58700	0.793499	−	0.608571i	$-0.208257\pi$
$241$	24.6368	1.58700	0.793499	+	0.608571i	$0.208257\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−2.26966	−0.145003
$246$	0	0
$247$	−11.2396	−0.715162
$248$	0	0
$249$	13.4663	0.853390
$250$	0	0
$251$	−21.1137	−1.33269	−0.666343	−	0.745645i	$-0.732141\pi$
$251$	−21.1137	−1.33269	−0.666343	+	0.745645i	$0.732141\pi$
$252$	0	0
$253$	−2.46337	−0.154871
$254$	0	0
$255$	−2.77929	−0.174046
$256$	0	0
$257$	15.9018	0.991929	0.495965	−	0.868343i	$-0.334815\pi$
$257$	15.9018	0.991929	0.495965	+	0.868343i	$0.334815\pi$
$258$	0	0
$259$	12.5041	0.776969
$260$	0	0
$261$	1.00000	0.0618984
$262$	0	0
$263$	−26.1463	−1.61225	−0.806124	−	0.591747i	$-0.798438\pi$
$263$	−26.1463	−1.61225	−0.806124	+	0.591747i	$0.798438\pi$
$264$	0	0
$265$	7.57169	0.465125
$266$	0	0
$267$	−10.9138	−0.667915
$268$	0	0
$269$	−2.39482	−0.146015	−0.0730073	−	0.997331i	$-0.523260\pi$
$269$	−2.39482	−0.146015	−0.0730073	+	0.997331i	$0.523260\pi$
$270$	0	0
$271$	4.23346	0.257164	0.128582	−	0.991699i	$-0.458957\pi$
$271$	4.23346	0.257164	0.128582	+	0.991699i	$0.458957\pi$
$272$	0	0
$273$	6.33291	0.383285
$274$	0	0
$275$	1.41712	0.0854553
$276$	0	0
$277$	14.8325	0.891199	0.445599	−	0.895233i	$-0.352991\pi$
$277$	14.8325	0.891199	0.445599	+	0.895233i	$0.352991\pi$
$278$	0	0
$279$	−6.16676	−0.369195
$280$	0	0
$281$	−27.3067	−1.62898	−0.814491	−	0.580176i	$-0.802984\pi$
$281$	−27.3067	−1.62898	−0.814491	+	0.580176i	$0.802984\pi$
$282$	0	0
$283$	13.7600	0.817950	0.408975	−	0.912546i	$-0.365886\pi$
$283$	13.7600	0.817950	0.408975	+	0.912546i	$0.365886\pi$
$284$	0	0
$285$	11.8242	0.700407
$286$	0	0
$287$	−9.06976	−0.535371
$288$	0	0
$289$	−15.6145	−0.918501
$290$	0	0
$291$	−18.3762	−1.07723
$292$	0	0
$293$	14.0909	0.823201	0.411600	−	0.911364i	$-0.364970\pi$
$293$	14.0909	0.823201	0.411600	+	0.911364i	$0.364970\pi$
$294$	0	0
$295$	5.04596	0.293787
$296$	0	0
$297$	−2.46337	−0.142939
$298$	0	0
$299$	2.24447	0.129801
$300$	0	0
$301$	−34.1728	−1.96969
$302$	0	0
$303$	8.04365	0.462096
$304$	0	0
$305$	−12.8906	−0.738114
$306$	0	0
$307$	16.7805	0.957716	0.478858	−	0.877893i	$-0.341051\pi$
$307$	16.7805	0.957716	0.478858	+	0.877893i	$0.341051\pi$
$308$	0	0
$309$	−12.1332	−0.690231
$310$	0	0
$311$	32.9428	1.86802	0.934008	−	0.357251i	$-0.116286\pi$
$311$	32.9428	1.86802	0.934008	+	0.357251i	$0.116286\pi$
$312$	0	0
$313$	0.604859	0.0341886	0.0170943	−	0.999854i	$-0.494558\pi$
$313$	0.604859	0.0341886	0.0170943	+	0.999854i	$0.494558\pi$
$314$	0	0
$315$	−6.66229	−0.375377
$316$	0	0
$317$	−31.1888	−1.75174	−0.875869	−	0.482549i	$-0.839711\pi$
$317$	−31.1888	−1.75174	−0.875869	+	0.482549i	$0.839711\pi$
$318$	0	0
$319$	2.46337	0.137922
$320$	0	0
$321$	5.53795	0.309098
$322$	0	0
$323$	−5.89440	−0.327973
$324$	0	0
$325$	−1.29119	−0.0716222
$326$	0	0
$327$	−3.56615	−0.197209
$328$	0	0
$329$	−9.62091	−0.530418
$330$	0	0
$331$	−21.4478	−1.17888	−0.589440	−	0.807812i	$-0.700652\pi$
$331$	−21.4478	−1.17888	−0.589440	+	0.807812i	$0.700652\pi$
$332$	0	0
$333$	4.43163	0.242852
$334$	0	0
$335$	−27.2726	−1.49006
$336$	0	0
$337$	25.7310	1.40166	0.700829	−	0.713330i	$-0.252814\pi$
$337$	25.7310	1.40166	0.700829	+	0.713330i	$0.252814\pi$
$338$	0	0
$339$	−0.165574	−0.00899274
$340$	0	0
$341$	−15.1910	−0.822639
$342$	0	0
$343$	−17.0388	−0.920008
$344$	0	0
$345$	−2.36120	−0.127123
$346$	0	0
$347$	7.72527	0.414714	0.207357	−	0.978265i	$-0.433514\pi$
$347$	7.72527	0.414714	0.207357	+	0.978265i	$0.433514\pi$
$348$	0	0
$349$	10.0404	0.537449	0.268725	−	0.963217i	$-0.413398\pi$
$349$	10.0404	0.537449	0.268725	+	0.963217i	$0.413398\pi$
$350$	0	0
$351$	2.24447	0.119801
$352$	0	0
$353$	−17.0110	−0.905405	−0.452702	−	0.891662i	$-0.649540\pi$
$353$	−17.0110	−0.905405	−0.452702	+	0.891662i	$0.649540\pi$
$354$	0	0
$355$	−16.0697	−0.852889
$356$	0	0
$357$	3.32117	0.175775
$358$	0	0
$359$	6.26101	0.330443	0.165222	−	0.986256i	$-0.447166\pi$
$359$	6.26101	0.330443	0.165222	+	0.986256i	$0.447166\pi$
$360$	0	0
$361$	6.07720	0.319852
$362$	0	0
$363$	4.93182	0.258854
$364$	0	0
$365$	−1.26540	−0.0662341
$366$	0	0
$367$	15.2884	0.798049	0.399025	−	0.916940i	$-0.369349\pi$
$367$	15.2884	0.798049	0.399025	+	0.916940i	$0.369349\pi$
$368$	0	0
$369$	−3.21444	−0.167337
$370$	0	0
$371$	−9.04794	−0.469746
$372$	0	0
$373$	1.18321	0.0612643	0.0306321	−	0.999531i	$-0.490248\pi$
$373$	1.18321	0.0612643	0.0306321	+	0.999531i	$0.490248\pi$
$374$	0	0
$375$	−10.4477	−0.539515
$376$	0	0
$377$	−2.24447	−0.115596
$378$	0	0
$379$	−13.4193	−0.689301	−0.344650	−	0.938731i	$-0.612003\pi$
$379$	−13.4193	−0.689301	−0.344650	+	0.938731i	$0.612003\pi$
$380$	0	0
$381$	−13.0244	−0.667258
$382$	0	0
$383$	3.52064	0.179896	0.0899481	−	0.995946i	$-0.471330\pi$
$383$	3.52064	0.179896	0.0899481	+	0.995946i	$0.471330\pi$
$384$	0	0
$385$	−16.4117	−0.836416
$386$	0	0
$387$	−12.1113	−0.615652
$388$	0	0
$389$	−23.1423	−1.17336	−0.586680	−	0.809819i	$-0.699565\pi$
$389$	−23.1423	−1.17336	−0.586680	+	0.809819i	$0.699565\pi$
$390$	0	0
$391$	1.17707	0.0595267
$392$	0	0
$393$	20.4285	1.03048
$394$	0	0
$395$	21.4503	1.07928
$396$	0	0
$397$	−35.7587	−1.79468	−0.897339	−	0.441342i	$-0.854502\pi$
$397$	−35.7587	−1.79468	−0.897339	+	0.441342i	$0.854502\pi$
$398$	0	0
$399$	−14.1296	−0.707364
$400$	0	0
$401$	−27.4606	−1.37132	−0.685659	−	0.727923i	$-0.740486\pi$
$401$	−27.4606	−1.37132	−0.685659	+	0.727923i	$0.740486\pi$
$402$	0	0
$403$	13.8411	0.689474
$404$	0	0
$405$	−2.36120	−0.117329
$406$	0	0
$407$	10.9167	0.541122
$408$	0	0
$409$	−34.4448	−1.70319	−0.851593	−	0.524204i	$-0.824363\pi$
$409$	−34.4448	−1.70319	−0.851593	+	0.524204i	$0.824363\pi$
$410$	0	0
$411$	−12.2888	−0.606162
$412$	0	0
$413$	−6.02977	−0.296706
$414$	0	0
$415$	31.7966	1.56083
$416$	0	0
$417$	−12.1965	−0.597266
$418$	0	0
$419$	−15.2570	−0.745354	−0.372677	−	0.927961i	$-0.621560\pi$
$419$	−15.2570	−0.745354	−0.372677	+	0.927961i	$0.621560\pi$
$420$	0	0
$421$	−10.2774	−0.500888	−0.250444	−	0.968131i	$-0.580577\pi$
$421$	−10.2774	−0.500888	−0.250444	+	0.968131i	$0.580577\pi$
$422$	0	0
$423$	−3.40978	−0.165789
$424$	0	0
$425$	−0.677137	−0.0328460
$426$	0	0
$427$	15.4039	0.745446
$428$	0	0
$429$	5.52895	0.266940
$430$	0	0
$431$	13.9360	0.671271	0.335636	−	0.941992i	$-0.391049\pi$
$431$	13.9360	0.671271	0.335636	+	0.941992i	$0.391049\pi$
$432$	0	0
$433$	−8.93234	−0.429261	−0.214630	−	0.976695i	$-0.568855\pi$
$433$	−8.93234	−0.429261	−0.214630	+	0.976695i	$0.568855\pi$
$434$	0	0
$435$	2.36120	0.113211
$436$	0	0
$437$	−5.00771	−0.239551
$438$	0	0
$439$	15.8164	0.754877	0.377439	−	0.926035i	$-0.376805\pi$
$439$	15.8164	0.754877	0.377439	+	0.926035i	$0.376805\pi$
$440$	0	0
$441$	0.961232	0.0457729
$442$	0	0
$443$	0.957836	0.0455081	0.0227541	−	0.999741i	$-0.492757\pi$
$443$	0.957836	0.0455081	0.0227541	+	0.999741i	$0.492757\pi$
$444$	0	0
$445$	−25.7697	−1.22160
$446$	0	0
$447$	21.7213	1.02738
$448$	0	0
$449$	−0.612674	−0.0289139	−0.0144569	−	0.999895i	$-0.504602\pi$
$449$	−0.612674	−0.0289139	−0.0144569	+	0.999895i	$0.504602\pi$
$450$	0	0
$451$	−7.91835	−0.372861
$452$	0	0
$453$	1.74182	0.0818380
$454$	0	0
$455$	14.9533	0.701021
$456$	0	0
$457$	13.7397	0.642715	0.321358	−	0.946958i	$-0.395861\pi$
$457$	13.7397	0.642715	0.321358	+	0.946958i	$0.395861\pi$
$458$	0	0
$459$	1.17707	0.0549407
$460$	0	0
$461$	−27.6910	−1.28970	−0.644848	−	0.764311i	$-0.723079\pi$
$461$	−27.6910	−1.28970	−0.644848	+	0.764311i	$0.723079\pi$
$462$	0	0
$463$	16.0140	0.744232	0.372116	−	0.928186i	$-0.378632\pi$
$463$	16.0140	0.744232	0.372116	+	0.928186i	$0.378632\pi$
$464$	0	0
$465$	−14.5610	−0.675249
$466$	0	0
$467$	21.1599	0.979162	0.489581	−	0.871958i	$-0.337150\pi$
$467$	21.1599	0.979162	0.489581	+	0.871958i	$0.337150\pi$
$468$	0	0
$469$	32.5899	1.50486
$470$	0	0
$471$	18.6182	0.857882
$472$	0	0
$473$	−29.8346	−1.37180
$474$	0	0
$475$	2.88082	0.132181
$476$	0	0
$477$	−3.20671	−0.146825
$478$	0	0
$479$	−11.6140	−0.530657	−0.265329	−	0.964158i	$-0.585480\pi$
$479$	−11.6140	−0.530657	−0.265329	+	0.964158i	$0.585480\pi$
$480$	0	0
$481$	−9.94665	−0.453528
$482$	0	0
$483$	2.82157	0.128386
$484$	0	0
$485$	−43.3898	−1.97023
$486$	0	0
$487$	−26.5870	−1.20477	−0.602385	−	0.798206i	$-0.705783\pi$
$487$	−26.5870	−1.20477	−0.602385	+	0.798206i	$0.705783\pi$
$488$	0	0
$489$	12.4124	0.561308
$490$	0	0
$491$	−32.7224	−1.47674	−0.738371	−	0.674395i	$-0.764405\pi$
$491$	−32.7224	−1.47674	−0.738371	+	0.674395i	$0.764405\pi$
$492$	0	0
$493$	−1.17707	−0.0530123
$494$	0	0
$495$	−5.81651	−0.261433
$496$	0	0
$497$	19.2028	0.861362
$498$	0	0
$499$	−36.0244	−1.61267	−0.806337	−	0.591456i	$-0.798553\pi$
$499$	−36.0244	−1.61267	−0.806337	+	0.591456i	$0.798553\pi$
$500$	0	0
$501$	−2.02150	−0.0903138
$502$	0	0
$503$	32.6779	1.45704	0.728518	−	0.685026i	$-0.240209\pi$
$503$	32.6779	1.45704	0.728518	+	0.685026i	$0.240209\pi$
$504$	0	0
$505$	18.9927	0.845164
$506$	0	0
$507$	7.96237	0.353621
$508$	0	0
$509$	−12.9207	−0.572700	−0.286350	−	0.958125i	$-0.592442\pi$
$509$	−12.9207	−0.572700	−0.286350	+	0.958125i	$0.592442\pi$
$510$	0	0
$511$	1.51212	0.0668921
$512$	0	0
$513$	−5.00771	−0.221096
$514$	0	0
$515$	−28.6488	−1.26242
$516$	0	0
$517$	−8.39953	−0.369411
$518$	0	0
$519$	1.55676	0.0683343
$520$	0	0
$521$	0.00620199	0.000271714 0	0.000135857	−	1.00000i	$-0.499957\pi$
$521$	0.00620199	0.000271714 0	0.000135857		1.00000i	$0.499957\pi$
$522$	0	0
$523$	16.6212	0.726794	0.363397	−	0.931634i	$-0.381617\pi$
$523$	16.6212	0.726794	0.363397	+	0.931634i	$0.381617\pi$
$524$	0	0
$525$	−1.62318	−0.0708413
$526$	0	0
$527$	7.25868	0.316193
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−2.13703	−0.0927392
$532$	0	0
$533$	7.21471	0.312504
$534$	0	0
$535$	13.0762	0.565334
$536$	0	0
$537$	22.6826	0.978827
$538$	0	0
$539$	2.36787	0.101991
$540$	0	0
$541$	−34.2809	−1.47385	−0.736926	−	0.675973i	$-0.763724\pi$
$541$	−34.2809	−1.47385	−0.736926	+	0.675973i	$0.763724\pi$
$542$	0	0
$543$	2.39045	0.102584
$544$	0	0
$545$	−8.42040	−0.360690
$546$	0	0
$547$	−12.1453	−0.519294	−0.259647	−	0.965704i	$-0.583606\pi$
$547$	−12.1453	−0.519294	−0.259647	+	0.965704i	$0.583606\pi$
$548$	0	0
$549$	5.45934	0.232999
$550$	0	0
$551$	5.00771	0.213336
$552$	0	0
$553$	−25.6325	−1.09000
$554$	0	0
$555$	10.4640	0.444171
$556$	0	0
$557$	34.6002	1.46606	0.733029	−	0.680198i	$-0.238106\pi$
$557$	34.6002	1.46606	0.733029	+	0.680198i	$0.238106\pi$
$558$	0	0
$559$	27.1834	1.14974
$560$	0	0
$561$	2.89954	0.122419
$562$	0	0
$563$	−38.9129	−1.63998	−0.819992	−	0.572376i	$-0.806022\pi$
$563$	−38.9129	−1.63998	−0.819992	+	0.572376i	$0.806022\pi$
$564$	0	0
$565$	−0.390953	−0.0164475
$566$	0	0
$567$	2.82157	0.118495
$568$	0	0
$569$	−45.2467	−1.89684	−0.948421	−	0.317014i	$-0.897320\pi$
$569$	−45.2467	−1.89684	−0.948421	+	0.317014i	$0.897320\pi$
$570$	0	0
$571$	−21.9622	−0.919088	−0.459544	−	0.888155i	$-0.651987\pi$
$571$	−21.9622	−0.919088	−0.459544	+	0.888155i	$0.651987\pi$
$572$	0	0
$573$	−4.58330	−0.191470
$574$	0	0
$575$	−0.575276	−0.0239907
$576$	0	0
$577$	−27.5096	−1.14524	−0.572620	−	0.819821i	$-0.694073\pi$
$577$	−27.5096	−1.14524	−0.572620	+	0.819821i	$0.694073\pi$
$578$	0	0
$579$	17.3826	0.722396
$580$	0	0
$581$	−37.9959	−1.57634
$582$	0	0
$583$	−7.89930	−0.327156
$584$	0	0
$585$	5.29964	0.219113
$586$	0	0
$587$	−4.60137	−0.189919	−0.0949595	−	0.995481i	$-0.530272\pi$
$587$	−4.60137	−0.189919	−0.0949595	+	0.995481i	$0.530272\pi$
$588$	0	0
$589$	−30.8814	−1.27245
$590$	0	0
$591$	−3.86402	−0.158944
$592$	0	0
$593$	−24.9518	−1.02465	−0.512323	−	0.858793i	$-0.671215\pi$
$593$	−24.9518	−1.02465	−0.512323	+	0.858793i	$0.671215\pi$
$594$	0	0
$595$	7.84194	0.321488
$596$	0	0
$597$	1.66919	0.0683154
$598$	0	0
$599$	−24.8140	−1.01387	−0.506937	−	0.861983i	$-0.669222\pi$
$599$	−24.8140	−1.01387	−0.506937	+	0.861983i	$0.669222\pi$
$600$	0	0
$601$	39.7618	1.62192	0.810959	−	0.585103i	$-0.198946\pi$
$601$	39.7618	1.62192	0.810959	+	0.585103i	$0.198946\pi$
$602$	0	0
$603$	11.5503	0.470364
$604$	0	0
$605$	11.6450	0.473438
$606$	0	0
$607$	−18.6818	−0.758272	−0.379136	−	0.925341i	$-0.623779\pi$
$607$	−18.6818	−0.758272	−0.379136	+	0.925341i	$0.623779\pi$
$608$	0	0
$609$	−2.82157	−0.114336
$610$	0	0
$611$	7.65313	0.309612
$612$	0	0
$613$	−10.4059	−0.420292	−0.210146	−	0.977670i	$-0.567394\pi$
$613$	−10.4059	−0.420292	−0.210146	+	0.977670i	$0.567394\pi$
$614$	0	0
$615$	−7.58995	−0.306056
$616$	0	0
$617$	−26.8185	−1.07967	−0.539837	−	0.841770i	$-0.681514\pi$
$617$	−26.8185	−1.07967	−0.539837	+	0.841770i	$0.681514\pi$
$618$	0	0
$619$	1.86410	0.0749245	0.0374622	−	0.999298i	$-0.488073\pi$
$619$	1.86410	0.0749245	0.0374622	+	0.999298i	$0.488073\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	0	0
$623$	30.7940	1.23374
$624$	0	0
$625$	−27.5454	−1.10182
$626$	0	0
$627$	−12.3358	−0.492646
$628$	0	0
$629$	−5.21632	−0.207988
$630$	0	0
$631$	−8.76571	−0.348958	−0.174479	−	0.984661i	$-0.555824\pi$
$631$	−8.76571	−0.348958	−0.174479	+	0.984661i	$0.555824\pi$
$632$	0	0
$633$	−2.03379	−0.0808357
$634$	0	0
$635$	−30.7532	−1.22040
$636$	0	0
$637$	−2.15745	−0.0854814
$638$	0	0
$639$	6.80571	0.269230
$640$	0	0
$641$	3.96717	0.156694	0.0783469	−	0.996926i	$-0.475036\pi$
$641$	3.96717	0.156694	0.0783469	+	0.996926i	$0.475036\pi$
$642$	0	0
$643$	−18.1768	−0.716823	−0.358412	−	0.933564i	$-0.616682\pi$
$643$	−18.1768	−0.716823	−0.358412	+	0.933564i	$0.616682\pi$
$644$	0	0
$645$	−28.5972	−1.12601
$646$	0	0
$647$	34.0048	1.33687	0.668433	−	0.743772i	$-0.266965\pi$
$647$	34.0048	1.33687	0.668433	+	0.743772i	$0.266965\pi$
$648$	0	0
$649$	−5.26429	−0.206641
$650$	0	0
$651$	17.3999	0.681957
$652$	0	0
$653$	−24.4414	−0.956466	−0.478233	−	0.878233i	$-0.658723\pi$
$653$	−24.4414	−0.956466	−0.478233	+	0.878233i	$0.658723\pi$
$654$	0	0
$655$	48.2358	1.88473
$656$	0	0
$657$	0.535914	0.0209080
$658$	0	0
$659$	13.8977	0.541379	0.270690	−	0.962667i	$-0.412748\pi$
$659$	13.8977	0.541379	0.270690	+	0.962667i	$0.412748\pi$
$660$	0	0
$661$	−4.70133	−0.182860	−0.0914302	−	0.995811i	$-0.529144\pi$
$661$	−4.70133	−0.182860	−0.0914302	+	0.995811i	$0.529144\pi$
$662$	0	0
$663$	−2.64188	−0.102602
$664$	0	0
$665$	−33.3628	−1.29375
$666$	0	0
$667$	−1.00000	−0.0387202
$668$	0	0
$669$	7.15464	0.276614
$670$	0	0
$671$	13.4484	0.519168
$672$	0	0
$673$	5.03790	0.194197	0.0970983	−	0.995275i	$-0.469044\pi$
$673$	5.03790	0.194197	0.0970983	+	0.995275i	$0.469044\pi$
$674$	0	0
$675$	−0.575276	−0.0221424
$676$	0	0
$677$	−2.76911	−0.106426	−0.0532128	−	0.998583i	$-0.516946\pi$
$677$	−2.76911	−0.106426	−0.0532128	+	0.998583i	$0.516946\pi$
$678$	0	0
$679$	51.8495	1.98980
$680$	0	0
$681$	−9.46474	−0.362690
$682$	0	0
$683$	−8.25556	−0.315890	−0.157945	−	0.987448i	$-0.550487\pi$
$683$	−8.25556	−0.315890	−0.157945	+	0.987448i	$0.550487\pi$
$684$	0	0
$685$	−29.0163	−1.10866
$686$	0	0
$687$	−19.9346	−0.760552
$688$	0	0
$689$	7.19736	0.274197
$690$	0	0
$691$	−31.7949	−1.20954	−0.604768	−	0.796401i	$-0.706734\pi$
$691$	−31.7949	−1.20954	−0.604768	+	0.796401i	$0.706734\pi$
$692$	0	0
$693$	6.95055	0.264030
$694$	0	0
$695$	−28.7985	−1.09239
$696$	0	0
$697$	3.78361	0.143314
$698$	0	0
$699$	20.0069	0.756730
$700$	0	0
$701$	42.8623	1.61889	0.809444	−	0.587197i	$-0.199769\pi$
$701$	42.8623	1.61889	0.809444	+	0.587197i	$0.199769\pi$
$702$	0	0
$703$	22.1923	0.837000
$704$	0	0
$705$	−8.05117	−0.303225
$706$	0	0
$707$	−22.6957	−0.853560
$708$	0	0
$709$	27.2260	1.02249	0.511247	−	0.859434i	$-0.329184\pi$
$709$	27.2260	1.02249	0.511247	+	0.859434i	$0.329184\pi$
$710$	0	0
$711$	−9.08449	−0.340695
$712$	0	0
$713$	6.16676	0.230947
$714$	0	0
$715$	13.0550	0.488228
$716$	0	0
$717$	24.1719	0.902718
$718$	0	0
$719$	−8.21601	−0.306405	−0.153203	−	0.988195i	$-0.548959\pi$
$719$	−8.21601	−0.306405	−0.153203	+	0.988195i	$0.548959\pi$
$720$	0	0
$721$	34.2345	1.27496
$722$	0	0
$723$	−24.6368	−0.916254
$724$	0	0
$725$	0.575276	0.0213652
$726$	0	0
$727$	36.2756	1.34539	0.672694	−	0.739921i	$-0.265137\pi$
$727$	36.2756	1.34539	0.672694	+	0.739921i	$0.265137\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	14.2558	0.527269
$732$	0	0
$733$	16.1792	0.597592	0.298796	−	0.954317i	$-0.403415\pi$
$733$	16.1792	0.597592	0.298796	+	0.954317i	$0.403415\pi$
$734$	0	0
$735$	2.26966	0.0837178
$736$	0	0
$737$	28.4526	1.04807
$738$	0	0
$739$	23.8855	0.878642	0.439321	−	0.898330i	$-0.355219\pi$
$739$	23.8855	0.878642	0.439321	+	0.898330i	$0.355219\pi$
$740$	0	0
$741$	11.2396	0.412899
$742$	0	0
$743$	−1.62100	−0.0594686	−0.0297343	−	0.999558i	$-0.509466\pi$
$743$	−1.62100	−0.0594686	−0.0297343	+	0.999558i	$0.509466\pi$
$744$	0	0
$745$	51.2883	1.87906
$746$	0	0
$747$	−13.4663	−0.492705
$748$	0	0
$749$	−15.6257	−0.570950
$750$	0	0
$751$	15.7260	0.573849	0.286925	−	0.957953i	$-0.407367\pi$
$751$	15.7260	0.573849	0.286925	+	0.957953i	$0.407367\pi$
$752$	0	0
$753$	21.1137	0.769427
$754$	0	0
$755$	4.11279	0.149680
$756$	0	0
$757$	0.718152	0.0261017	0.0130508	−	0.999915i	$-0.495846\pi$
$757$	0.718152	0.0261017	0.0130508	+	0.999915i	$0.495846\pi$
$758$	0	0
$759$	2.46337	0.0894146
$760$	0	0
$761$	−18.6447	−0.675869	−0.337935	−	0.941170i	$-0.609728\pi$
$761$	−18.6447	−0.675869	−0.337935	+	0.941170i	$0.609728\pi$
$762$	0	0
$763$	10.0621	0.364273
$764$	0	0
$765$	2.77929	0.100485
$766$	0	0
$767$	4.79649	0.173191
$768$	0	0
$769$	−51.6366	−1.86206	−0.931031	−	0.364940i	$-0.881089\pi$
$769$	−51.6366	−1.86206	−0.931031	+	0.364940i	$0.881089\pi$
$770$	0	0
$771$	−15.9018	−0.572691
$772$	0	0
$773$	50.5927	1.81969	0.909846	−	0.414945i	$-0.136199\pi$
$773$	50.5927	1.81969	0.909846	+	0.414945i	$0.136199\pi$
$774$	0	0
$775$	−3.54759	−0.127433
$776$	0	0
$777$	−12.5041	−0.448583
$778$	0	0
$779$	−16.0970	−0.576735
$780$	0	0
$781$	16.7650	0.599898
$782$	0	0
$783$	−1.00000	−0.0357371
$784$	0	0
$785$	43.9614	1.56905
$786$	0	0
$787$	−40.1277	−1.43040	−0.715199	−	0.698921i	$-0.753664\pi$
$787$	−40.1277	−1.43040	−0.715199	+	0.698921i	$0.753664\pi$
$788$	0	0
$789$	26.1463	0.930831
$790$	0	0
$791$	0.467177	0.0166109
$792$	0	0
$793$	−12.2533	−0.435128
$794$	0	0
$795$	−7.57169	−0.268540
$796$	0	0
$797$	−22.2487	−0.788088	−0.394044	−	0.919092i	$-0.628924\pi$
$797$	−22.2487	−0.788088	−0.394044	+	0.919092i	$0.628924\pi$
$798$	0	0
$799$	4.01353	0.141988
$800$	0	0
$801$	10.9138	0.385621
$802$	0	0
$803$	1.32015	0.0465872
$804$	0	0
$805$	6.66229	0.234815
$806$	0	0
$807$	2.39482	0.0843015
$808$	0	0
$809$	52.6354	1.85056	0.925281	−	0.379282i	$-0.123829\pi$
$809$	52.6354	1.85056	0.925281	+	0.379282i	$0.123829\pi$
$810$	0	0
$811$	21.3831	0.750863	0.375431	−	0.926850i	$-0.377495\pi$
$811$	21.3831	0.750863	0.375431	+	0.926850i	$0.377495\pi$
$812$	0	0
$813$	−4.23346	−0.148474
$814$	0	0
$815$	29.3082	1.02662
$816$	0	0
$817$	−60.6499	−2.12187
$818$	0	0
$819$	−6.33291	−0.221290
$820$	0	0
$821$	38.0037	1.32634	0.663169	−	0.748469i	$-0.269211\pi$
$821$	38.0037	1.32634	0.663169	+	0.748469i	$0.269211\pi$
$822$	0	0
$823$	−22.2845	−0.776789	−0.388394	−	0.921493i	$-0.626970\pi$
$823$	−22.2845	−0.776789	−0.388394	+	0.921493i	$0.626970\pi$
$824$	0	0
$825$	−1.41712	−0.0493376
$826$	0	0
$827$	−17.9745	−0.625034	−0.312517	−	0.949912i	$-0.601172\pi$
$827$	−17.9745	−0.625034	−0.312517	+	0.949912i	$0.601172\pi$
$828$	0	0
$829$	−39.6008	−1.37539	−0.687697	−	0.725998i	$-0.741378\pi$
$829$	−39.6008	−1.37539	−0.687697	+	0.725998i	$0.741378\pi$
$830$	0	0
$831$	−14.8325	−0.514534
$832$	0	0
$833$	−1.13143	−0.0392018
$834$	0	0
$835$	−4.77316	−0.165182
$836$	0	0
$837$	6.16676	0.213155
$838$	0	0
$839$	−36.8250	−1.27134	−0.635669	−	0.771961i	$-0.719276\pi$
$839$	−36.8250	−1.27134	−0.635669	+	0.771961i	$0.719276\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	27.3067	0.940494
$844$	0	0
$845$	18.8008	0.646766
$846$	0	0
$847$	−13.9155	−0.478141
$848$	0	0
$849$	−13.7600	−0.472243
$850$	0	0
$851$	−4.43163	−0.151914
$852$	0	0
$853$	−53.9826	−1.84833	−0.924164	−	0.381995i	$-0.875237\pi$
$853$	−53.9826	−1.84833	−0.924164	+	0.381995i	$0.875237\pi$
$854$	0	0
$855$	−11.8242	−0.404380
$856$	0	0
$857$	27.0718	0.924755	0.462378	−	0.886683i	$-0.346996\pi$
$857$	27.0718	0.924755	0.462378	+	0.886683i	$0.346996\pi$
$858$	0	0
$859$	−29.5478	−1.00816	−0.504079	−	0.863658i	$-0.668168\pi$
$859$	−29.5478	−1.00816	−0.504079	+	0.863658i	$0.668168\pi$
$860$	0	0
$861$	9.06976	0.309097
$862$	0	0
$863$	−33.1310	−1.12779	−0.563897	−	0.825845i	$-0.690698\pi$
$863$	−33.1310	−1.12779	−0.563897	+	0.825845i	$0.690698\pi$
$864$	0	0
$865$	3.67583	0.124982
$866$	0	0
$867$	15.6145	0.530297
$868$	0	0
$869$	−22.3784	−0.759137
$870$	0	0
$871$	−25.9242	−0.878409
$872$	0	0
$873$	18.3762	0.621939
$874$	0	0
$875$	29.4788	0.996565
$876$	0	0
$877$	22.4928	0.759527	0.379764	−	0.925084i	$-0.376005\pi$
$877$	22.4928	0.759527	0.379764	+	0.925084i	$0.376005\pi$
$878$	0	0
$879$	−14.0909	−0.475275
$880$	0	0
$881$	−20.2861	−0.683456	−0.341728	−	0.939799i	$-0.611012\pi$
$881$	−20.2861	−0.683456	−0.341728	+	0.939799i	$0.611012\pi$
$882$	0	0
$883$	28.1579	0.947588	0.473794	−	0.880636i	$-0.342884\pi$
$883$	28.1579	0.947588	0.473794	+	0.880636i	$0.342884\pi$
$884$	0	0
$885$	−5.04596	−0.169618
$886$	0	0
$887$	23.7823	0.798532	0.399266	−	0.916835i	$-0.369265\pi$
$887$	23.7823	0.798532	0.399266	+	0.916835i	$0.369265\pi$
$888$	0	0
$889$	36.7491	1.23252
$890$	0	0
$891$	2.46337	0.0825259
$892$	0	0
$893$	−17.0752	−0.571399
$894$	0	0
$895$	53.5582	1.79025
$896$	0	0
$897$	−2.24447	−0.0749406
$898$	0	0
$899$	−6.16676	−0.205673
$900$	0	0
$901$	3.77451	0.125747
$902$	0	0
$903$	34.1728	1.13720
$904$	0	0
$905$	5.64433	0.187624
$906$	0	0
$907$	31.7392	1.05388	0.526942	−	0.849901i	$-0.323339\pi$
$907$	31.7392	1.05388	0.526942	+	0.849901i	$0.323339\pi$
$908$	0	0
$909$	−8.04365	−0.266791
$910$	0	0
$911$	36.1257	1.19690	0.598449	−	0.801161i	$-0.295784\pi$
$911$	36.1257	1.19690	0.598449	+	0.801161i	$0.295784\pi$
$912$	0	0
$913$	−33.1724	−1.09784
$914$	0	0
$915$	12.8906	0.426150
$916$	0	0
$917$	−57.6404	−1.90345
$918$	0	0
$919$	3.92619	0.129513	0.0647565	−	0.997901i	$-0.479373\pi$
$919$	3.92619	0.129513	0.0647565	+	0.997901i	$0.479373\pi$
$920$	0	0
$921$	−16.7805	−0.552937
$922$	0	0
$923$	−15.2752	−0.502789
$924$	0	0
$925$	2.54941	0.0838241
$926$	0	0
$927$	12.1332	0.398505
$928$	0	0
$929$	29.8059	0.977898	0.488949	−	0.872312i	$-0.337380\pi$
$929$	29.8059	0.977898	0.488949	+	0.872312i	$0.337380\pi$
$930$	0	0
$931$	4.81357	0.157758
$932$	0	0
$933$	−32.9428	−1.07850
$934$	0	0
$935$	6.84641	0.223901
$936$	0	0
$937$	24.9343	0.814569	0.407285	−	0.913301i	$-0.366476\pi$
$937$	24.9343	0.814569	0.407285	+	0.913301i	$0.366476\pi$
$938$	0	0
$939$	−0.604859	−0.0197388
$940$	0	0
$941$	4.77338	0.155608	0.0778038	−	0.996969i	$-0.475209\pi$
$941$	4.77338	0.155608	0.0778038	+	0.996969i	$0.475209\pi$
$942$	0	0
$943$	3.21444	0.104677
$944$	0	0
$945$	6.66229	0.216724
$946$	0	0
$947$	49.0605	1.59425	0.797126	−	0.603813i	$-0.206353\pi$
$947$	49.0605	1.59425	0.797126	+	0.603813i	$0.206353\pi$
$948$	0	0
$949$	−1.20284	−0.0390459
$950$	0	0
$951$	31.1888	1.01137
$952$	0	0
$953$	−42.5128	−1.37712	−0.688562	−	0.725177i	$-0.741758\pi$
$953$	−42.5128	−1.37712	−0.688562	+	0.725177i	$0.741758\pi$
$954$	0	0
$955$	−10.8221	−0.350195
$956$	0	0
$957$	−2.46337	−0.0796293
$958$	0	0
$959$	34.6737	1.11967
$960$	0	0
$961$	7.02899	0.226742
$962$	0	0
$963$	−5.53795	−0.178458
$964$	0	0
$965$	41.0438	1.32125
$966$	0	0
$967$	−16.6675	−0.535989	−0.267995	−	0.963420i	$-0.586361\pi$
$967$	−16.6675	−0.535989	−0.267995	+	0.963420i	$0.586361\pi$
$968$	0	0
$969$	5.89440	0.189356
$970$	0	0
$971$	37.6485	1.20820	0.604098	−	0.796910i	$-0.293533\pi$
$971$	37.6485	1.20820	0.604098	+	0.796910i	$0.293533\pi$
$972$	0	0
$973$	34.4133	1.10324
$974$	0	0
$975$	1.29119	0.0413511
$976$	0	0
$977$	48.0575	1.53750	0.768748	−	0.639552i	$-0.220880\pi$
$977$	48.0575	1.53750	0.768748	+	0.639552i	$0.220880\pi$
$978$	0	0
$979$	26.8847	0.859240
$980$	0	0
$981$	3.56615	0.113858
$982$	0	0
$983$	−4.10065	−0.130790	−0.0653952	−	0.997859i	$-0.520831\pi$
$983$	−4.10065	−0.130790	−0.0653952	+	0.997859i	$0.520831\pi$
$984$	0	0
$985$	−9.12372	−0.290706
$986$	0	0
$987$	9.62091	0.306237
$988$	0	0
$989$	12.1113	0.385117
$990$	0	0
$991$	20.8892	0.663567	0.331784	−	0.943356i	$-0.392350\pi$
$991$	20.8892	0.663567	0.331784	+	0.943356i	$0.392350\pi$
$992$	0	0
$993$	21.4478	0.680627
$994$	0	0
$995$	3.94129	0.124947
$996$	0	0
$997$	−61.6324	−1.95192	−0.975958	−	0.217958i	$-0.930060\pi$
$997$	−61.6324	−1.95192	−0.975958	+	0.217958i	$0.930060\pi$
$998$	0	0
$999$	−4.43163	−0.140211

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8004.2.a.e.1.3	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8004.2.a.e.1.3	✓	9	1.1	even	1	trivial

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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:	\( N \)	\(=\)	\( 8004 = 2^{2} \cdot 3 \cdot 23 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8004.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -2.36120 q^{5} +2.82157 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -2.36120 q^{5} +2.82157 q^{7} +1.00000 q^{9} +2.46337 q^{11} -2.24447 q^{13} +2.36120 q^{15} -1.17707 q^{17} +5.00771 q^{19} -2.82157 q^{21} -1.00000 q^{23} +0.575276 q^{25} -1.00000 q^{27} +1.00000 q^{29} -6.16676 q^{31} -2.46337 q^{33} -6.66229 q^{35} +4.43163 q^{37} +2.24447 q^{39} -3.21444 q^{41} -12.1113 q^{43} -2.36120 q^{45} -3.40978 q^{47} +0.961232 q^{49} +1.17707 q^{51} -3.20671 q^{53} -5.81651 q^{55} -5.00771 q^{57} -2.13703 q^{59} +5.45934 q^{61} +2.82157 q^{63} +5.29964 q^{65} +11.5503 q^{67} +1.00000 q^{69} +6.80571 q^{71} +0.535914 q^{73} -0.575276 q^{75} +6.95055 q^{77} -9.08449 q^{79} +1.00000 q^{81} -13.4663 q^{83} +2.77929 q^{85} -1.00000 q^{87} +10.9138 q^{89} -6.33291 q^{91} +6.16676 q^{93} -11.8242 q^{95} +18.3762 q^{97} +2.46337 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - 9 q^{3} - 3 q^{5} + 7 q^{7} + 9 q^{9}+O(q^{10}) \) 9 * q - 9 * q^3 - 3 * q^5 + 7 * q^7 + 9 * q^9	\( 9 q - 9 q^{3} - 3 q^{5} + 7 q^{7} + 9 q^{9} - 2 q^{11} - 7 q^{13} + 3 q^{15} - q^{19} - 7 q^{21} - 9 q^{23} + 2 q^{25} - 9 q^{27} + 9 q^{29} + 8 q^{31} + 2 q^{33} - 5 q^{35} - 8 q^{37} + 7 q^{39} - 19 q^{41} - 3 q^{43} - 3 q^{45} - 3 q^{47} - 18 q^{49} - 17 q^{53} + 9 q^{55} + q^{57} - 10 q^{59} + q^{61} + 7 q^{63} - 16 q^{65} + 12 q^{67} + 9 q^{69} - 7 q^{71} + 13 q^{73} - 2 q^{75} - 15 q^{77} - 10 q^{79} + 9 q^{81} + 9 q^{83} - 6 q^{85} - 9 q^{87} - 5 q^{89} - 18 q^{91} - 8 q^{93} + 31 q^{95} - 7 q^{97} - 2 q^{99}+O(q^{100}) \) 9 * q - 9 * q^3 - 3 * q^5 + 7 * q^7 + 9 * q^9 - 2 * q^11 - 7 * q^13 + 3 * q^15 - q^19 - 7 * q^21 - 9 * q^23 + 2 * q^25 - 9 * q^27 + 9 * q^29 + 8 * q^31 + 2 * q^33 - 5 * q^35 - 8 * q^37 + 7 * q^39 - 19 * q^41 - 3 * q^43 - 3 * q^45 - 3 * q^47 - 18 * q^49 - 17 * q^53 + 9 * q^55 + q^57 - 10 * q^59 + q^61 + 7 * q^63 - 16 * q^65 + 12 * q^67 + 9 * q^69 - 7 * q^71 + 13 * q^73 - 2 * q^75 - 15 * q^77 - 10 * q^79 + 9 * q^81 + 9 * q^83 - 6 * q^85 - 9 * q^87 - 5 * q^89 - 18 * q^91 - 8 * q^93 + 31 * q^95 - 7 * q^97 - 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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