LMFDB - Embedded newform 8004.2.a.d.1.6

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:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{7} - 8x^{6} + 19x^{5} + 19x^{4} - 35x^{3} - 10x^{2} + 18x - 4 $ x^8 - 3x^7 - 8x^6 + 19x^5 + 19x^4 - 35x^3 - 10x^2 + 18*x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$1.24887$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} +0.138622 q^{5} -3.33662 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +0.138622 q^{5} -3.33662 q^{7} +1.00000 q^{9} -2.06616 q^{11} -0.702105 q^{13} +0.138622 q^{15} +4.32538 q^{17} +2.44146 q^{19} -3.33662 q^{21} +1.00000 q^{23} -4.98078 q^{25} +1.00000 q^{27} +1.00000 q^{29} +7.41338 q^{31} -2.06616 q^{33} -0.462529 q^{35} -2.90399 q^{37} -0.702105 q^{39} -4.05761 q^{41} +0.235390 q^{43} +0.138622 q^{45} -2.05531 q^{47} +4.13301 q^{49} +4.32538 q^{51} -3.22303 q^{53} -0.286416 q^{55} +2.44146 q^{57} -10.7016 q^{59} -2.78827 q^{61} -3.33662 q^{63} -0.0973274 q^{65} -11.6076 q^{67} +1.00000 q^{69} +7.47417 q^{71} +9.16987 q^{73} -4.98078 q^{75} +6.89398 q^{77} +11.0456 q^{79} +1.00000 q^{81} -12.3394 q^{83} +0.599594 q^{85} +1.00000 q^{87} +3.05613 q^{89} +2.34265 q^{91} +7.41338 q^{93} +0.338441 q^{95} +2.74522 q^{97} -2.06616 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{3} - 5 q^{5} - 4 q^{7} + 8 q^{9}+O(q^{10}) $ 8 * q + 8 * q^3 - 5 * q^5 - 4 * q^7 + 8 * q^9	$ 8 q + 8 q^{3} - 5 q^{5} - 4 q^{7} + 8 q^{9} - 5 q^{11} - 4 q^{13} - 5 q^{15} - 3 q^{17} - 5 q^{19} - 4 q^{21} + 8 q^{23} - 5 q^{25} + 8 q^{27} + 8 q^{29} - 2 q^{31} - 5 q^{33} - 15 q^{35} - 10 q^{37} - 4 q^{39} - 11 q^{41} - 7 q^{43} - 5 q^{45} - 14 q^{47} - 18 q^{49} - 3 q^{51} - 15 q^{53} - 17 q^{55} - 5 q^{57} + 4 q^{59} + q^{61} - 4 q^{63} - 5 q^{67} + 8 q^{69} - q^{71} - 21 q^{73} - 5 q^{75} - 8 q^{79} + 8 q^{81} + 3 q^{83} + 8 q^{87} - 20 q^{89} - 7 q^{91} - 2 q^{93} - 3 q^{95} - 7 q^{97} - 5 q^{99}+O(q^{100}) $ 8 * q + 8 * q^3 - 5 * q^5 - 4 * q^7 + 8 * q^9 - 5 * q^11 - 4 * q^13 - 5 * q^15 - 3 * q^17 - 5 * q^19 - 4 * q^21 + 8 * q^23 - 5 * q^25 + 8 * q^27 + 8 * q^29 - 2 * q^31 - 5 * q^33 - 15 * q^35 - 10 * q^37 - 4 * q^39 - 11 * q^41 - 7 * q^43 - 5 * q^45 - 14 * q^47 - 18 * q^49 - 3 * q^51 - 15 * q^53 - 17 * q^55 - 5 * q^57 + 4 * q^59 + q^61 - 4 * q^63 - 5 * q^67 + 8 * q^69 - q^71 - 21 * q^73 - 5 * q^75 - 8 * q^79 + 8 * q^81 + 3 * q^83 + 8 * q^87 - 20 * q^89 - 7 * q^91 - 2 * q^93 - 3 * q^95 - 7 * q^97 - 5 * 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
$5$	0.138622	0.0619938	0.0309969	−	0.999519i	$-0.490132\pi$
$5$	0.138622	0.0619938	0.0309969	+	0.999519i	$0.490132\pi$
$6$	0	0
$7$	−3.33662	−1.26112	−0.630561	−	0.776139i	$-0.717175\pi$
$7$	−3.33662	−1.26112	−0.630561	+	0.776139i	$0.717175\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−2.06616	−0.622971	−0.311485	−	0.950251i	$-0.600826\pi$
$11$	−2.06616	−0.622971	−0.311485	+	0.950251i	$0.600826\pi$
$12$	0	0
$13$	−0.702105	−0.194729	−0.0973644	−	0.995249i	$-0.531041\pi$
$13$	−0.702105	−0.194729	−0.0973644	+	0.995249i	$0.531041\pi$
$14$	0	0
$15$	0.138622	0.0357921
$16$	0	0
$17$	4.32538	1.04906	0.524529	−	0.851392i	$-0.324241\pi$
$17$	4.32538	1.04906	0.524529	+	0.851392i	$0.324241\pi$
$18$	0	0
$19$	2.44146	0.560110	0.280055	−	0.959984i	$-0.409647\pi$
$19$	2.44146	0.560110	0.280055	+	0.959984i	$0.409647\pi$
$20$	0	0
$21$	−3.33662	−0.728109
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	−4.98078	−0.996157
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	7.41338	1.33148	0.665741	−	0.746183i	$-0.268116\pi$
$31$	7.41338	1.33148	0.665741	+	0.746183i	$0.268116\pi$
$32$	0	0
$33$	−2.06616	−0.359672
$34$	0	0
$35$	−0.462529	−0.0781817
$36$	0	0
$37$	−2.90399	−0.477413	−0.238707	−	0.971092i	$-0.576723\pi$
$37$	−2.90399	−0.477413	−0.238707	+	0.971092i	$0.576723\pi$
$38$	0	0
$39$	−0.702105	−0.112427
$40$	0	0
$41$	−4.05761	−0.633692	−0.316846	−	0.948477i	$-0.602624\pi$
$41$	−4.05761	−0.633692	−0.316846	+	0.948477i	$0.602624\pi$
$42$	0	0
$43$	0.235390	0.0358966	0.0179483	−	0.999839i	$-0.494287\pi$
$43$	0.235390	0.0358966	0.0179483	+	0.999839i	$0.494287\pi$
$44$	0	0
$45$	0.138622	0.0206646
$46$	0	0
$47$	−2.05531	−0.299798	−0.149899	−	0.988701i	$-0.547895\pi$
$47$	−2.05531	−0.299798	−0.149899	+	0.988701i	$0.547895\pi$
$48$	0	0
$49$	4.13301	0.590430
$50$	0	0
$51$	4.32538	0.605674
$52$	0	0
$53$	−3.22303	−0.442718	−0.221359	−	0.975192i	$-0.571049\pi$
$53$	−3.22303	−0.442718	−0.221359	+	0.975192i	$0.571049\pi$
$54$	0	0
$55$	−0.286416	−0.0386203
$56$	0	0
$57$	2.44146	0.323379
$58$	0	0
$59$	−10.7016	−1.39323	−0.696614	−	0.717446i	$-0.745311\pi$
$59$	−10.7016	−1.39323	−0.696614	+	0.717446i	$0.745311\pi$
$60$	0	0
$61$	−2.78827	−0.357001	−0.178501	−	0.983940i	$-0.557125\pi$
$61$	−2.78827	−0.357001	−0.178501	+	0.983940i	$0.557125\pi$
$62$	0	0
$63$	−3.33662	−0.420374
$64$	0	0
$65$	−0.0973274	−0.0120720
$66$	0	0
$67$	−11.6076	−1.41809	−0.709046	−	0.705163i	$-0.750874\pi$
$67$	−11.6076	−1.41809	−0.709046	+	0.705163i	$0.750874\pi$
$68$	0	0
$69$	1.00000	0.120386
$70$	0	0
$71$	7.47417	0.887021	0.443510	−	0.896269i	$-0.353733\pi$
$71$	7.47417	0.887021	0.443510	+	0.896269i	$0.353733\pi$
$72$	0	0
$73$	9.16987	1.07325	0.536626	−	0.843820i	$-0.319699\pi$
$73$	9.16987	1.07325	0.536626	+	0.843820i	$0.319699\pi$
$74$	0	0
$75$	−4.98078	−0.575131
$76$	0	0
$77$	6.89398	0.785642
$78$	0	0
$79$	11.0456	1.24273	0.621364	−	0.783522i	$-0.286579\pi$
$79$	11.0456	1.24273	0.621364	+	0.783522i	$0.286579\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−12.3394	−1.35442	−0.677210	−	0.735790i	$-0.736811\pi$
$83$	−12.3394	−1.35442	−0.677210	+	0.735790i	$0.736811\pi$
$84$	0	0
$85$	0.599594	0.0650351
$86$	0	0
$87$	1.00000	0.107211
$88$	0	0
$89$	3.05613	0.323950	0.161975	−	0.986795i	$-0.448214\pi$
$89$	3.05613	0.323950	0.161975	+	0.986795i	$0.448214\pi$
$90$	0	0
$91$	2.34265	0.245577
$92$	0	0
$93$	7.41338	0.768731
$94$	0	0
$95$	0.338441	0.0347233
$96$	0	0
$97$	2.74522	0.278735	0.139368	−	0.990241i	$-0.455493\pi$
$97$	2.74522	0.278735	0.139368	+	0.990241i	$0.455493\pi$
$98$	0	0
$99$	−2.06616	−0.207657
$100$	0	0
$101$	−13.4238	−1.33572	−0.667861	−	0.744286i	$-0.732790\pi$
$101$	−13.4238	−1.33572	−0.667861	+	0.744286i	$0.732790\pi$
$102$	0	0
$103$	5.32865	0.525047	0.262524	−	0.964926i	$-0.415445\pi$
$103$	5.32865	0.525047	0.262524	+	0.964926i	$0.415445\pi$
$104$	0	0
$105$	−0.462529	−0.0451383
$106$	0	0
$107$	13.7453	1.32880	0.664402	−	0.747375i	$-0.268686\pi$
$107$	13.7453	1.32880	0.664402	+	0.747375i	$0.268686\pi$
$108$	0	0
$109$	−6.22293	−0.596049	−0.298024	−	0.954558i	$-0.596328\pi$
$109$	−6.22293	−0.596049	−0.298024	+	0.954558i	$0.596328\pi$
$110$	0	0
$111$	−2.90399	−0.275635
$112$	0	0
$113$	0.191841	0.0180468	0.00902342	−	0.999959i	$-0.497128\pi$
$113$	0.191841	0.0180468	0.00902342	+	0.999959i	$0.497128\pi$
$114$	0	0
$115$	0.138622	0.0129266
$116$	0	0
$117$	−0.702105	−0.0649096
$118$	0	0
$119$	−14.4321	−1.32299
$120$	0	0
$121$	−6.73099	−0.611908
$122$	0	0
$123$	−4.05761	−0.365862
$124$	0	0
$125$	−1.38356	−0.123749
$126$	0	0
$127$	−19.8935	−1.76526	−0.882632	−	0.470065i	$-0.844231\pi$
$127$	−19.8935	−1.76526	−0.882632	+	0.470065i	$0.844231\pi$
$128$	0	0
$129$	0.235390	0.0207249
$130$	0	0
$131$	−9.36944	−0.818611	−0.409306	−	0.912397i	$-0.634229\pi$
$131$	−9.36944	−0.818611	−0.409306	+	0.912397i	$0.634229\pi$
$132$	0	0
$133$	−8.14622	−0.706367
$134$	0	0
$135$	0.138622	0.0119307
$136$	0	0
$137$	−4.64779	−0.397088	−0.198544	−	0.980092i	$-0.563621\pi$
$137$	−4.64779	−0.397088	−0.198544	+	0.980092i	$0.563621\pi$
$138$	0	0
$139$	1.53574	0.130260	0.0651300	−	0.997877i	$-0.479254\pi$
$139$	1.53574	0.130260	0.0651300	+	0.997877i	$0.479254\pi$
$140$	0	0
$141$	−2.05531	−0.173089
$142$	0	0
$143$	1.45066	0.121310
$144$	0	0
$145$	0.138622	0.0115120
$146$	0	0
$147$	4.13301	0.340885
$148$	0	0
$149$	−15.6363	−1.28097	−0.640486	−	0.767970i	$-0.721267\pi$
$149$	−15.6363	−1.28097	−0.640486	+	0.767970i	$0.721267\pi$
$150$	0	0
$151$	−0.131361	−0.0106900	−0.00534500	−	0.999986i	$-0.501701\pi$
$151$	−0.131361	−0.0106900	−0.00534500	+	0.999986i	$0.501701\pi$
$152$	0	0
$153$	4.32538	0.349686
$154$	0	0
$155$	1.02766	0.0825436
$156$	0	0
$157$	−22.4628	−1.79272	−0.896362	−	0.443324i	$-0.853799\pi$
$157$	−22.4628	−1.79272	−0.896362	+	0.443324i	$0.853799\pi$
$158$	0	0
$159$	−3.22303	−0.255603
$160$	0	0
$161$	−3.33662	−0.262962
$162$	0	0
$163$	16.0695	1.25866	0.629329	−	0.777139i	$-0.283330\pi$
$163$	16.0695	1.25866	0.629329	+	0.777139i	$0.283330\pi$
$164$	0	0
$165$	−0.286416	−0.0222974
$166$	0	0
$167$	14.0563	1.08771	0.543856	−	0.839178i	$-0.316964\pi$
$167$	14.0563	1.08771	0.543856	+	0.839178i	$0.316964\pi$
$168$	0	0
$169$	−12.5070	−0.962081
$170$	0	0
$171$	2.44146	0.186703
$172$	0	0
$173$	−19.4446	−1.47835	−0.739174	−	0.673514i	$-0.764784\pi$
$173$	−19.4446	−1.47835	−0.739174	+	0.673514i	$0.764784\pi$
$174$	0	0
$175$	16.6190	1.25628
$176$	0	0
$177$	−10.7016	−0.804380
$178$	0	0
$179$	−6.84250	−0.511432	−0.255716	−	0.966752i	$-0.582311\pi$
$179$	−6.84250	−0.511432	−0.255716	+	0.966752i	$0.582311\pi$
$180$	0	0
$181$	17.2255	1.28036	0.640182	−	0.768223i	$-0.278859\pi$
$181$	17.2255	1.28036	0.640182	+	0.768223i	$0.278859\pi$
$182$	0	0
$183$	−2.78827	−0.206115
$184$	0	0
$185$	−0.402558	−0.0295966
$186$	0	0
$187$	−8.93692	−0.653532
$188$	0	0
$189$	−3.33662	−0.242703
$190$	0	0
$191$	6.76311	0.489361	0.244681	−	0.969604i	$-0.421317\pi$
$191$	6.76311	0.489361	0.244681	+	0.969604i	$0.421317\pi$
$192$	0	0
$193$	−19.1555	−1.37884	−0.689420	−	0.724362i	$-0.742135\pi$
$193$	−19.1555	−1.37884	−0.689420	+	0.724362i	$0.742135\pi$
$194$	0	0
$195$	−0.0973274	−0.00696976
$196$	0	0
$197$	22.7833	1.62324	0.811622	−	0.584183i	$-0.198585\pi$
$197$	22.7833	1.62324	0.811622	+	0.584183i	$0.198585\pi$
$198$	0	0
$199$	17.0668	1.20983	0.604916	−	0.796289i	$-0.293207\pi$
$199$	17.0668	1.20983	0.604916	+	0.796289i	$0.293207\pi$
$200$	0	0
$201$	−11.6076	−0.818735
$202$	0	0
$203$	−3.33662	−0.234185
$204$	0	0
$205$	−0.562475	−0.0392849
$206$	0	0
$207$	1.00000	0.0695048
$208$	0	0
$209$	−5.04445	−0.348932
$210$	0	0
$211$	−21.7265	−1.49572	−0.747858	−	0.663859i	$-0.768918\pi$
$211$	−21.7265	−1.49572	−0.747858	+	0.663859i	$0.768918\pi$
$212$	0	0
$213$	7.47417	0.512122
$214$	0	0
$215$	0.0326303	0.00222537
$216$	0	0
$217$	−24.7356	−1.67916
$218$	0	0
$219$	9.16987	0.619643
$220$	0	0
$221$	−3.03687	−0.204282
$222$	0	0
$223$	1.66917	0.111776	0.0558880	−	0.998437i	$-0.482201\pi$
$223$	1.66917	0.111776	0.0558880	+	0.998437i	$0.482201\pi$
$224$	0	0
$225$	−4.98078	−0.332052
$226$	0	0
$227$	7.86528	0.522037	0.261019	−	0.965334i	$-0.415942\pi$
$227$	7.86528	0.522037	0.261019	+	0.965334i	$0.415942\pi$
$228$	0	0
$229$	−25.6280	−1.69355	−0.846774	−	0.531954i	$-0.821458\pi$
$229$	−25.6280	−1.69355	−0.846774	+	0.531954i	$0.821458\pi$
$230$	0	0
$231$	6.89398	0.453591
$232$	0	0
$233$	−11.8273	−0.774833	−0.387416	−	0.921905i	$-0.626632\pi$
$233$	−11.8273	−0.774833	−0.387416	+	0.921905i	$0.626632\pi$
$234$	0	0
$235$	−0.284912	−0.0185856
$236$	0	0
$237$	11.0456	0.717489
$238$	0	0
$239$	−15.0934	−0.976314	−0.488157	−	0.872756i	$-0.662331\pi$
$239$	−15.0934	−0.976314	−0.488157	+	0.872756i	$0.662331\pi$
$240$	0	0
$241$	−4.05076	−0.260933	−0.130466	−	0.991453i	$-0.541647\pi$
$241$	−4.05076	−0.260933	−0.130466	+	0.991453i	$0.541647\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	0.572928	0.0366030
$246$	0	0
$247$	−1.71416	−0.109069
$248$	0	0
$249$	−12.3394	−0.781975
$250$	0	0
$251$	−12.3880	−0.781923	−0.390961	−	0.920407i	$-0.627857\pi$
$251$	−12.3880	−0.781923	−0.390961	+	0.920407i	$0.627857\pi$
$252$	0	0
$253$	−2.06616	−0.129898
$254$	0	0
$255$	0.599594	0.0375480
$256$	0	0
$257$	−3.36413	−0.209848	−0.104924	−	0.994480i	$-0.533460\pi$
$257$	−3.36413	−0.209848	−0.104924	+	0.994480i	$0.533460\pi$
$258$	0	0
$259$	9.68950	0.602076
$260$	0	0
$261$	1.00000	0.0618984
$262$	0	0
$263$	−2.66208	−0.164151	−0.0820754	−	0.996626i	$-0.526155\pi$
$263$	−2.66208	−0.164151	−0.0820754	+	0.996626i	$0.526155\pi$
$264$	0	0
$265$	−0.446784	−0.0274457
$266$	0	0
$267$	3.05613	0.187032
$268$	0	0
$269$	−27.7521	−1.69208	−0.846038	−	0.533122i	$-0.821019\pi$
$269$	−27.7521	−1.69208	−0.846038	+	0.533122i	$0.821019\pi$
$270$	0	0
$271$	−14.0820	−0.855420	−0.427710	−	0.903916i	$-0.640679\pi$
$271$	−14.0820	−0.855420	−0.427710	+	0.903916i	$0.640679\pi$
$272$	0	0
$273$	2.34265	0.141784
$274$	0	0
$275$	10.2911	0.620576
$276$	0	0
$277$	−21.3082	−1.28029	−0.640143	−	0.768256i	$-0.721125\pi$
$277$	−21.3082	−1.28029	−0.640143	+	0.768256i	$0.721125\pi$
$278$	0	0
$279$	7.41338	0.443827
$280$	0	0
$281$	6.82977	0.407430	0.203715	−	0.979030i	$-0.434698\pi$
$281$	6.82977	0.407430	0.203715	+	0.979030i	$0.434698\pi$
$282$	0	0
$283$	−15.5966	−0.927123	−0.463562	−	0.886065i	$-0.653429\pi$
$283$	−15.5966	−0.927123	−0.463562	+	0.886065i	$0.653429\pi$
$284$	0	0
$285$	0.338441	0.0200475
$286$	0	0
$287$	13.5387	0.799163
$288$	0	0
$289$	1.70890	0.100524
$290$	0	0
$291$	2.74522	0.160928
$292$	0	0
$293$	−4.78475	−0.279528	−0.139764	−	0.990185i	$-0.544634\pi$
$293$	−4.78475	−0.279528	−0.139764	+	0.990185i	$0.544634\pi$
$294$	0	0
$295$	−1.48348	−0.0863714
$296$	0	0
$297$	−2.06616	−0.119891
$298$	0	0
$299$	−0.702105	−0.0406038
$300$	0	0
$301$	−0.785406	−0.0452700
$302$	0	0
$303$	−13.4238	−0.771179
$304$	0	0
$305$	−0.386516	−0.0221318
$306$	0	0
$307$	1.56614	0.0893842	0.0446921	−	0.999001i	$-0.485769\pi$
$307$	1.56614	0.0893842	0.0446921	+	0.999001i	$0.485769\pi$
$308$	0	0
$309$	5.32865	0.303136
$310$	0	0
$311$	−23.2734	−1.31972	−0.659858	−	0.751390i	$-0.729384\pi$
$311$	−23.2734	−1.31972	−0.659858	+	0.751390i	$0.729384\pi$
$312$	0	0
$313$	−9.54782	−0.539675	−0.269837	−	0.962906i	$-0.586970\pi$
$313$	−9.54782	−0.539675	−0.269837	+	0.962906i	$0.586970\pi$
$314$	0	0
$315$	−0.462529	−0.0260606
$316$	0	0
$317$	4.20816	0.236354	0.118177	−	0.992993i	$-0.462295\pi$
$317$	4.20816	0.236354	0.118177	+	0.992993i	$0.462295\pi$
$318$	0	0
$319$	−2.06616	−0.115683
$320$	0	0
$321$	13.7453	0.767186
$322$	0	0
$323$	10.5602	0.587588
$324$	0	0
$325$	3.49703	0.193980
$326$	0	0
$327$	−6.22293	−0.344129
$328$	0	0
$329$	6.85779	0.378082
$330$	0	0
$331$	−5.28355	−0.290410	−0.145205	−	0.989402i	$-0.546384\pi$
$331$	−5.28355	−0.290410	−0.145205	+	0.989402i	$0.546384\pi$
$332$	0	0
$333$	−2.90399	−0.159138
$334$	0	0
$335$	−1.60907	−0.0879128
$336$	0	0
$337$	−11.0431	−0.601554	−0.300777	−	0.953695i	$-0.597246\pi$
$337$	−11.0431	−0.601554	−0.300777	+	0.953695i	$0.597246\pi$
$338$	0	0
$339$	0.191841	0.0104193
$340$	0	0
$341$	−15.3172	−0.829474
$342$	0	0
$343$	9.56604	0.516518
$344$	0	0
$345$	0.138622	0.00746317
$346$	0	0
$347$	35.2662	1.89319	0.946595	−	0.322425i	$-0.104498\pi$
$347$	35.2662	1.89319	0.946595	+	0.322425i	$0.104498\pi$
$348$	0	0
$349$	28.9854	1.55155	0.775776	−	0.631008i	$-0.217359\pi$
$349$	28.9854	1.55155	0.775776	+	0.631008i	$0.217359\pi$
$350$	0	0
$351$	−0.702105	−0.0374756
$352$	0	0
$353$	−9.29498	−0.494722	−0.247361	−	0.968923i	$-0.579563\pi$
$353$	−9.29498	−0.494722	−0.247361	+	0.968923i	$0.579563\pi$
$354$	0	0
$355$	1.03609	0.0549898
$356$	0	0
$357$	−14.4321	−0.763829
$358$	0	0
$359$	−21.9567	−1.15883	−0.579416	−	0.815032i	$-0.696719\pi$
$359$	−21.9567	−1.15883	−0.579416	+	0.815032i	$0.696719\pi$
$360$	0	0
$361$	−13.0393	−0.686277
$362$	0	0
$363$	−6.73099	−0.353285
$364$	0	0
$365$	1.27115	0.0665350
$366$	0	0
$367$	21.2236	1.10786	0.553932	−	0.832562i	$-0.313127\pi$
$367$	21.2236	1.10786	0.553932	+	0.832562i	$0.313127\pi$
$368$	0	0
$369$	−4.05761	−0.211231
$370$	0	0
$371$	10.7540	0.558322
$372$	0	0
$373$	−4.56449	−0.236340	−0.118170	−	0.992993i	$-0.537703\pi$
$373$	−4.56449	−0.236340	−0.118170	+	0.992993i	$0.537703\pi$
$374$	0	0
$375$	−1.38356	−0.0714467
$376$	0	0
$377$	−0.702105	−0.0361602
$378$	0	0
$379$	−24.0049	−1.23305	−0.616524	−	0.787336i	$-0.711460\pi$
$379$	−24.0049	−1.23305	−0.616524	+	0.787336i	$0.711460\pi$
$380$	0	0
$381$	−19.8935	−1.01918
$382$	0	0
$383$	23.2380	1.18741	0.593704	−	0.804684i	$-0.297665\pi$
$383$	23.2380	1.18741	0.593704	+	0.804684i	$0.297665\pi$
$384$	0	0
$385$	0.955660	0.0487049
$386$	0	0
$387$	0.235390	0.0119655
$388$	0	0
$389$	27.5066	1.39464	0.697321	−	0.716759i	$-0.254375\pi$
$389$	27.5066	1.39464	0.697321	+	0.716759i	$0.254375\pi$
$390$	0	0
$391$	4.32538	0.218744
$392$	0	0
$393$	−9.36944	−0.472626
$394$	0	0
$395$	1.53117	0.0770414
$396$	0	0
$397$	−11.7160	−0.588008	−0.294004	−	0.955804i	$-0.594988\pi$
$397$	−11.7160	−0.588008	−0.294004	+	0.955804i	$0.594988\pi$
$398$	0	0
$399$	−8.14622	−0.407821
$400$	0	0
$401$	23.4290	1.16999	0.584994	−	0.811037i	$-0.301097\pi$
$401$	23.4290	1.16999	0.584994	+	0.811037i	$0.301097\pi$
$402$	0	0
$403$	−5.20497	−0.259278
$404$	0	0
$405$	0.138622	0.00688820
$406$	0	0
$407$	6.00011	0.297414
$408$	0	0
$409$	1.07003	0.0529096	0.0264548	−	0.999650i	$-0.491578\pi$
$409$	1.07003	0.0529096	0.0264548	+	0.999650i	$0.491578\pi$
$410$	0	0
$411$	−4.64779	−0.229259
$412$	0	0
$413$	35.7071	1.75703
$414$	0	0
$415$	−1.71051	−0.0839656
$416$	0	0
$417$	1.53574	0.0752057
$418$	0	0
$419$	−10.9312	−0.534024	−0.267012	−	0.963693i	$-0.586036\pi$
$419$	−10.9312	−0.534024	−0.267012	+	0.963693i	$0.586036\pi$
$420$	0	0
$421$	4.63235	0.225767	0.112883	−	0.993608i	$-0.463991\pi$
$421$	4.63235	0.225767	0.112883	+	0.993608i	$0.463991\pi$
$422$	0	0
$423$	−2.05531	−0.0999327
$424$	0	0
$425$	−21.5438	−1.04503
$426$	0	0
$427$	9.30338	0.450222
$428$	0	0
$429$	1.45066	0.0700385
$430$	0	0
$431$	−0.537917	−0.0259106	−0.0129553	−	0.999916i	$-0.504124\pi$
$431$	−0.537917	−0.0259106	−0.0129553	+	0.999916i	$0.504124\pi$
$432$	0	0
$433$	8.35692	0.401608	0.200804	−	0.979631i	$-0.435645\pi$
$433$	8.35692	0.401608	0.200804	+	0.979631i	$0.435645\pi$
$434$	0	0
$435$	0.138622	0.00664643
$436$	0	0
$437$	2.44146	0.116791
$438$	0	0
$439$	−18.5296	−0.884372	−0.442186	−	0.896923i	$-0.645797\pi$
$439$	−18.5296	−0.884372	−0.442186	+	0.896923i	$0.645797\pi$
$440$	0	0
$441$	4.13301	0.196810
$442$	0	0
$443$	−22.9601	−1.09087	−0.545435	−	0.838153i	$-0.683635\pi$
$443$	−22.9601	−1.09087	−0.545435	+	0.838153i	$0.683635\pi$
$444$	0	0
$445$	0.423648	0.0200829
$446$	0	0
$447$	−15.6363	−0.739570
$448$	0	0
$449$	−12.8847	−0.608069	−0.304034	−	0.952661i	$-0.598334\pi$
$449$	−12.8847	−0.608069	−0.304034	+	0.952661i	$0.598334\pi$
$450$	0	0
$451$	8.38366	0.394771
$452$	0	0
$453$	−0.131361	−0.00617187
$454$	0	0
$455$	0.324744	0.0152242
$456$	0	0
$457$	−6.25690	−0.292685	−0.146343	−	0.989234i	$-0.546750\pi$
$457$	−6.25690	−0.292685	−0.146343	+	0.989234i	$0.546750\pi$
$458$	0	0
$459$	4.32538	0.201891
$460$	0	0
$461$	33.5984	1.56483	0.782416	−	0.622756i	$-0.213987\pi$
$461$	33.5984	1.56483	0.782416	+	0.622756i	$0.213987\pi$
$462$	0	0
$463$	−8.53294	−0.396559	−0.198280	−	0.980145i	$-0.563535\pi$
$463$	−8.53294	−0.396559	−0.198280	+	0.980145i	$0.563535\pi$
$464$	0	0
$465$	1.02766	0.0476565
$466$	0	0
$467$	16.5443	0.765579	0.382790	−	0.923836i	$-0.374963\pi$
$467$	16.5443	0.765579	0.382790	+	0.923836i	$0.374963\pi$
$468$	0	0
$469$	38.7300	1.78839
$470$	0	0
$471$	−22.4628	−1.03503
$472$	0	0
$473$	−0.486353	−0.0223625
$474$	0	0
$475$	−12.1604	−0.557957
$476$	0	0
$477$	−3.22303	−0.147573
$478$	0	0
$479$	32.6154	1.49024	0.745119	−	0.666932i	$-0.232393\pi$
$479$	32.6154	1.49024	0.745119	+	0.666932i	$0.232393\pi$
$480$	0	0
$481$	2.03891	0.0929661
$482$	0	0
$483$	−3.33662	−0.151821
$484$	0	0
$485$	0.380549	0.0172799
$486$	0	0
$487$	26.8002	1.21443	0.607217	−	0.794536i	$-0.292286\pi$
$487$	26.8002	1.21443	0.607217	+	0.794536i	$0.292286\pi$
$488$	0	0
$489$	16.0695	0.726686
$490$	0	0
$491$	−1.18043	−0.0532719	−0.0266359	−	0.999645i	$-0.508479\pi$
$491$	−1.18043	−0.0532719	−0.0266359	+	0.999645i	$0.508479\pi$
$492$	0	0
$493$	4.32538	0.194805
$494$	0	0
$495$	−0.286416	−0.0128734
$496$	0	0
$497$	−24.9384	−1.11864
$498$	0	0
$499$	−36.0023	−1.61168	−0.805841	−	0.592132i	$-0.798286\pi$
$499$	−36.0023	−1.61168	−0.805841	+	0.592132i	$0.798286\pi$
$500$	0	0
$501$	14.0563	0.627991
$502$	0	0
$503$	21.0968	0.940661	0.470330	−	0.882490i	$-0.344135\pi$
$503$	21.0968	0.940661	0.470330	+	0.882490i	$0.344135\pi$
$504$	0	0
$505$	−1.86084	−0.0828064
$506$	0	0
$507$	−12.5070	−0.555458
$508$	0	0
$509$	−20.3383	−0.901478	−0.450739	−	0.892656i	$-0.648839\pi$
$509$	−20.3383	−0.901478	−0.450739	+	0.892656i	$0.648839\pi$
$510$	0	0
$511$	−30.5964	−1.35350
$512$	0	0
$513$	2.44146	0.107793
$514$	0	0
$515$	0.738669	0.0325496
$516$	0	0
$517$	4.24660	0.186765
$518$	0	0
$519$	−19.4446	−0.853525
$520$	0	0
$521$	−16.3917	−0.718135	−0.359067	−	0.933312i	$-0.616905\pi$
$521$	−16.3917	−0.718135	−0.359067	+	0.933312i	$0.616905\pi$
$522$	0	0
$523$	32.2893	1.41191	0.705956	−	0.708256i	$-0.250518\pi$
$523$	32.2893	1.41191	0.705956	+	0.708256i	$0.250518\pi$
$524$	0	0
$525$	16.6190	0.725311
$526$	0	0
$527$	32.0657	1.39680
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−10.7016	−0.464409
$532$	0	0
$533$	2.84887	0.123398
$534$	0	0
$535$	1.90540	0.0823776
$536$	0	0
$537$	−6.84250	−0.295276
$538$	0	0
$539$	−8.53946	−0.367821
$540$	0	0
$541$	38.2985	1.64658	0.823290	−	0.567622i	$-0.192136\pi$
$541$	38.2985	1.64658	0.823290	+	0.567622i	$0.192136\pi$
$542$	0	0
$543$	17.2255	0.739219
$544$	0	0
$545$	−0.862637	−0.0369513
$546$	0	0
$547$	−4.29502	−0.183642	−0.0918208	−	0.995776i	$-0.529269\pi$
$547$	−4.29502	−0.183642	−0.0918208	+	0.995776i	$0.529269\pi$
$548$	0	0
$549$	−2.78827	−0.119000
$550$	0	0
$551$	2.44146	0.104010
$552$	0	0
$553$	−36.8550	−1.56723
$554$	0	0
$555$	−0.402558	−0.0170876
$556$	0	0
$557$	−28.5287	−1.20880	−0.604400	−	0.796681i	$-0.706587\pi$
$557$	−28.5287	−1.20880	−0.604400	+	0.796681i	$0.706587\pi$
$558$	0	0
$559$	−0.165268	−0.00699011
$560$	0	0
$561$	−8.93692	−0.377317
$562$	0	0
$563$	−30.7568	−1.29625	−0.648123	−	0.761536i	$-0.724446\pi$
$563$	−30.7568	−1.29625	−0.648123	+	0.761536i	$0.724446\pi$
$564$	0	0
$565$	0.0265934	0.00111879
$566$	0	0
$567$	−3.33662	−0.140125
$568$	0	0
$569$	18.5814	0.778973	0.389486	−	0.921032i	$-0.372653\pi$
$569$	18.5814	0.778973	0.389486	+	0.921032i	$0.372653\pi$
$570$	0	0
$571$	−14.3754	−0.601593	−0.300797	−	0.953688i	$-0.597253\pi$
$571$	−14.3754	−0.601593	−0.300797	+	0.953688i	$0.597253\pi$
$572$	0	0
$573$	6.76311	0.282533
$574$	0	0
$575$	−4.98078	−0.207713
$576$	0	0
$577$	17.8952	0.744986	0.372493	−	0.928035i	$-0.378503\pi$
$577$	17.8952	0.744986	0.372493	+	0.928035i	$0.378503\pi$
$578$	0	0
$579$	−19.1555	−0.796074
$580$	0	0
$581$	41.1717	1.70809
$582$	0	0
$583$	6.65930	0.275800
$584$	0	0
$585$	−0.0973274	−0.00402399
$586$	0	0
$587$	−9.83427	−0.405904	−0.202952	−	0.979189i	$-0.565053\pi$
$587$	−9.83427	−0.405904	−0.202952	+	0.979189i	$0.565053\pi$
$588$	0	0
$589$	18.0995	0.745776
$590$	0	0
$591$	22.7833	0.937180
$592$	0	0
$593$	18.1723	0.746247	0.373124	−	0.927782i	$-0.378287\pi$
$593$	18.1723	0.746247	0.373124	+	0.927782i	$0.378287\pi$
$594$	0	0
$595$	−2.00062	−0.0820172
$596$	0	0
$597$	17.0668	0.698497
$598$	0	0
$599$	14.4396	0.589985	0.294992	−	0.955500i	$-0.404683\pi$
$599$	14.4396	0.589985	0.294992	+	0.955500i	$0.404683\pi$
$600$	0	0
$601$	−22.5937	−0.921617	−0.460808	−	0.887500i	$-0.652440\pi$
$601$	−22.5937	−0.921617	−0.460808	+	0.887500i	$0.652440\pi$
$602$	0	0
$603$	−11.6076	−0.472697
$604$	0	0
$605$	−0.933064	−0.0379345
$606$	0	0
$607$	34.3011	1.39224	0.696120	−	0.717925i	$-0.254908\pi$
$607$	34.3011	1.39224	0.696120	+	0.717925i	$0.254908\pi$
$608$	0	0
$609$	−3.33662	−0.135207
$610$	0	0
$611$	1.44305	0.0583794
$612$	0	0
$613$	41.0333	1.65732	0.828659	−	0.559753i	$-0.189104\pi$
$613$	41.0333	1.65732	0.828659	+	0.559753i	$0.189104\pi$
$614$	0	0
$615$	−0.562475	−0.0226812
$616$	0	0
$617$	−23.0204	−0.926767	−0.463384	−	0.886158i	$-0.653365\pi$
$617$	−23.0204	−0.926767	−0.463384	+	0.886158i	$0.653365\pi$
$618$	0	0
$619$	−29.5961	−1.18957	−0.594785	−	0.803885i	$-0.702763\pi$
$619$	−29.5961	−1.18957	−0.594785	+	0.803885i	$0.702763\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	0	0
$623$	−10.1972	−0.408540
$624$	0	0
$625$	24.7121	0.988485
$626$	0	0
$627$	−5.04445	−0.201456
$628$	0	0
$629$	−12.5609	−0.500834
$630$	0	0
$631$	9.57230	0.381067	0.190534	−	0.981681i	$-0.438978\pi$
$631$	9.57230	0.381067	0.190534	+	0.981681i	$0.438978\pi$
$632$	0	0
$633$	−21.7265	−0.863552
$634$	0	0
$635$	−2.75768	−0.109435
$636$	0	0
$637$	−2.90181	−0.114974
$638$	0	0
$639$	7.47417	0.295674
$640$	0	0
$641$	26.5793	1.04982	0.524909	−	0.851158i	$-0.324099\pi$
$641$	26.5793	1.04982	0.524909	+	0.851158i	$0.324099\pi$
$642$	0	0
$643$	−12.8133	−0.505305	−0.252653	−	0.967557i	$-0.581303\pi$
$643$	−12.8133	−0.505305	−0.252653	+	0.967557i	$0.581303\pi$
$644$	0	0
$645$	0.0326303	0.00128482
$646$	0	0
$647$	15.3443	0.603246	0.301623	−	0.953427i	$-0.402472\pi$
$647$	15.3443	0.603246	0.301623	+	0.953427i	$0.402472\pi$
$648$	0	0
$649$	22.1112	0.867940
$650$	0	0
$651$	−24.7356	−0.969464
$652$	0	0
$653$	30.6963	1.20124	0.600619	−	0.799535i	$-0.294921\pi$
$653$	30.6963	1.20124	0.600619	+	0.799535i	$0.294921\pi$
$654$	0	0
$655$	−1.29881	−0.0507488
$656$	0	0
$657$	9.16987	0.357751
$658$	0	0
$659$	36.3908	1.41758	0.708792	−	0.705418i	$-0.249241\pi$
$659$	36.3908	1.41758	0.708792	+	0.705418i	$0.249241\pi$
$660$	0	0
$661$	−14.8679	−0.578294	−0.289147	−	0.957285i	$-0.593372\pi$
$661$	−14.8679	−0.578294	−0.289147	+	0.957285i	$0.593372\pi$
$662$	0	0
$663$	−3.03687	−0.117942
$664$	0	0
$665$	−1.12925	−0.0437903
$666$	0	0
$667$	1.00000	0.0387202
$668$	0	0
$669$	1.66917	0.0645338
$670$	0	0
$671$	5.76101	0.222401
$672$	0	0
$673$	−12.0592	−0.464848	−0.232424	−	0.972615i	$-0.574666\pi$
$673$	−12.0592	−0.464848	−0.232424	+	0.972615i	$0.574666\pi$
$674$	0	0
$675$	−4.98078	−0.191710
$676$	0	0
$677$	−25.0028	−0.960937	−0.480469	−	0.877012i	$-0.659533\pi$
$677$	−25.0028	−0.960937	−0.480469	+	0.877012i	$0.659533\pi$
$678$	0	0
$679$	−9.15976	−0.351519
$680$	0	0
$681$	7.86528	0.301398
$682$	0	0
$683$	11.6252	0.444826	0.222413	−	0.974953i	$-0.428607\pi$
$683$	11.6252	0.444826	0.222413	+	0.974953i	$0.428607\pi$
$684$	0	0
$685$	−0.644288	−0.0246170
$686$	0	0
$687$	−25.6280	−0.977770
$688$	0	0
$689$	2.26291	0.0862099
$690$	0	0
$691$	6.77933	0.257898	0.128949	−	0.991651i	$-0.458840\pi$
$691$	6.77933	0.257898	0.128949	+	0.991651i	$0.458840\pi$
$692$	0	0
$693$	6.89398	0.261881
$694$	0	0
$695$	0.212888	0.00807531
$696$	0	0
$697$	−17.5507	−0.664780
$698$	0	0
$699$	−11.8273	−0.447350
$700$	0	0
$701$	16.0754	0.607160	0.303580	−	0.952806i	$-0.401818\pi$
$701$	16.0754	0.607160	0.303580	+	0.952806i	$0.401818\pi$
$702$	0	0
$703$	−7.08998	−0.267404
$704$	0	0
$705$	−0.284912	−0.0107304
$706$	0	0
$707$	44.7902	1.68451
$708$	0	0
$709$	−15.6323	−0.587083	−0.293542	−	0.955946i	$-0.594834\pi$
$709$	−15.6323	−0.587083	−0.293542	+	0.955946i	$0.594834\pi$
$710$	0	0
$711$	11.0456	0.414243
$712$	0	0
$713$	7.41338	0.277633
$714$	0	0
$715$	0.201094	0.00752048
$716$	0	0
$717$	−15.0934	−0.563675
$718$	0	0
$719$	49.2658	1.83730	0.918652	−	0.395067i	$-0.129279\pi$
$719$	49.2658	1.83730	0.918652	+	0.395067i	$0.129279\pi$
$720$	0	0
$721$	−17.7796	−0.662149
$722$	0	0
$723$	−4.05076	−0.150649
$724$	0	0
$725$	−4.98078	−0.184982
$726$	0	0
$727$	−29.4731	−1.09310	−0.546549	−	0.837427i	$-0.684059\pi$
$727$	−29.4731	−1.09310	−0.546549	+	0.837427i	$0.684059\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	1.01815	0.0376577
$732$	0	0
$733$	−48.2391	−1.78175	−0.890875	−	0.454248i	$-0.849908\pi$
$733$	−48.2391	−1.78175	−0.890875	+	0.454248i	$0.849908\pi$
$734$	0	0
$735$	0.572928	0.0211328
$736$	0	0
$737$	23.9831	0.883429
$738$	0	0
$739$	−12.6485	−0.465281	−0.232640	−	0.972563i	$-0.574737\pi$
$739$	−12.6485	−0.465281	−0.232640	+	0.972563i	$0.574737\pi$
$740$	0	0
$741$	−1.71416	−0.0629713
$742$	0	0
$743$	−24.9291	−0.914561	−0.457280	−	0.889323i	$-0.651176\pi$
$743$	−24.9291	−0.914561	−0.457280	+	0.889323i	$0.651176\pi$
$744$	0	0
$745$	−2.16753	−0.0794123
$746$	0	0
$747$	−12.3394	−0.451473
$748$	0	0
$749$	−45.8627	−1.67579
$750$	0	0
$751$	31.5561	1.15150	0.575749	−	0.817626i	$-0.304711\pi$
$751$	31.5561	1.15150	0.575749	+	0.817626i	$0.304711\pi$
$752$	0	0
$753$	−12.3880	−0.451443
$754$	0	0
$755$	−0.0182095	−0.000662713 0
$756$	0	0
$757$	−50.8340	−1.84759	−0.923797	−	0.382882i	$-0.874932\pi$
$757$	−50.8340	−1.84759	−0.923797	+	0.382882i	$0.874932\pi$
$758$	0	0
$759$	−2.06616	−0.0749968
$760$	0	0
$761$	−42.0280	−1.52351	−0.761757	−	0.647862i	$-0.775663\pi$
$761$	−42.0280	−1.52351	−0.761757	+	0.647862i	$0.775663\pi$
$762$	0	0
$763$	20.7635	0.751691
$764$	0	0
$765$	0.599594	0.0216784
$766$	0	0
$767$	7.51363	0.271302
$768$	0	0
$769$	−23.7074	−0.854910	−0.427455	−	0.904037i	$-0.640590\pi$
$769$	−23.7074	−0.854910	−0.427455	+	0.904037i	$0.640590\pi$
$770$	0	0
$771$	−3.36413	−0.121156
$772$	0	0
$773$	45.9892	1.65412	0.827058	−	0.562117i	$-0.190013\pi$
$773$	45.9892	1.65412	0.827058	+	0.562117i	$0.190013\pi$
$774$	0	0
$775$	−36.9244	−1.32636
$776$	0	0
$777$	9.68950	0.347609
$778$	0	0
$779$	−9.90649	−0.354937
$780$	0	0
$781$	−15.4428	−0.552588
$782$	0	0
$783$	1.00000	0.0357371
$784$	0	0
$785$	−3.11384	−0.111138
$786$	0	0
$787$	−26.6405	−0.949633	−0.474817	−	0.880085i	$-0.657486\pi$
$787$	−26.6405	−0.949633	−0.474817	+	0.880085i	$0.657486\pi$
$788$	0	0
$789$	−2.66208	−0.0947725
$790$	0	0
$791$	−0.640098	−0.0227593
$792$	0	0
$793$	1.95766	0.0695184
$794$	0	0
$795$	−0.446784	−0.0158458
$796$	0	0
$797$	−13.0712	−0.463005	−0.231503	−	0.972834i	$-0.574364\pi$
$797$	−13.0712	−0.463005	−0.231503	+	0.972834i	$0.574364\pi$
$798$	0	0
$799$	−8.89001	−0.314506
$800$	0	0
$801$	3.05613	0.107983
$802$	0	0
$803$	−18.9464	−0.668605
$804$	0	0
$805$	−0.462529	−0.0163020
$806$	0	0
$807$	−27.7521	−0.976921
$808$	0	0
$809$	−46.1007	−1.62081	−0.810407	−	0.585867i	$-0.800754\pi$
$809$	−46.1007	−1.62081	−0.810407	+	0.585867i	$0.800754\pi$
$810$	0	0
$811$	51.4502	1.80666	0.903331	−	0.428944i	$-0.141114\pi$
$811$	51.4502	1.80666	0.903331	+	0.428944i	$0.141114\pi$
$812$	0	0
$813$	−14.0820	−0.493877
$814$	0	0
$815$	2.22759	0.0780289
$816$	0	0
$817$	0.574695	0.0201060
$818$	0	0
$819$	2.34265	0.0818590
$820$	0	0
$821$	18.8958	0.659468	0.329734	−	0.944074i	$-0.393041\pi$
$821$	18.8958	0.659468	0.329734	+	0.944074i	$0.393041\pi$
$822$	0	0
$823$	−44.2997	−1.54419	−0.772095	−	0.635508i	$-0.780791\pi$
$823$	−44.2997	−1.54419	−0.772095	+	0.635508i	$0.780791\pi$
$824$	0	0
$825$	10.2911	0.358290
$826$	0	0
$827$	55.2164	1.92006	0.960031	−	0.279894i	$-0.0902994\pi$
$827$	55.2164	1.92006	0.960031	+	0.279894i	$0.0902994\pi$
$828$	0	0
$829$	20.0445	0.696173	0.348086	−	0.937462i	$-0.386832\pi$
$829$	20.0445	0.696173	0.348086	+	0.937462i	$0.386832\pi$
$830$	0	0
$831$	−21.3082	−0.739173
$832$	0	0
$833$	17.8768	0.619396
$834$	0	0
$835$	1.94852	0.0674314
$836$	0	0
$837$	7.41338	0.256244
$838$	0	0
$839$	36.2113	1.25015	0.625077	−	0.780563i	$-0.285068\pi$
$839$	36.2113	1.25015	0.625077	+	0.780563i	$0.285068\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	6.82977	0.235230
$844$	0	0
$845$	−1.73376	−0.0596430
$846$	0	0
$847$	22.4587	0.771691
$848$	0	0
$849$	−15.5966	−0.535275
$850$	0	0
$851$	−2.90399	−0.0995475
$852$	0	0
$853$	10.3414	0.354083	0.177041	−	0.984203i	$-0.443347\pi$
$853$	10.3414	0.354083	0.177041	+	0.984203i	$0.443347\pi$
$854$	0	0
$855$	0.338441	0.0115744
$856$	0	0
$857$	−39.0306	−1.33326	−0.666630	−	0.745389i	$-0.732264\pi$
$857$	−39.0306	−1.33326	−0.666630	+	0.745389i	$0.732264\pi$
$858$	0	0
$859$	−30.8566	−1.05281	−0.526407	−	0.850232i	$-0.676461\pi$
$859$	−30.8566	−1.05281	−0.526407	+	0.850232i	$0.676461\pi$
$860$	0	0
$861$	13.5387	0.461397
$862$	0	0
$863$	26.4991	0.902041	0.451021	−	0.892514i	$-0.351060\pi$
$863$	26.4991	0.902041	0.451021	+	0.892514i	$0.351060\pi$
$864$	0	0
$865$	−2.69546	−0.0916484
$866$	0	0
$867$	1.70890	0.0580373
$868$	0	0
$869$	−22.8220	−0.774183
$870$	0	0
$871$	8.14973	0.276143
$872$	0	0
$873$	2.74522	0.0929118
$874$	0	0
$875$	4.61641	0.156063
$876$	0	0
$877$	48.2089	1.62790	0.813950	−	0.580935i	$-0.197313\pi$
$877$	48.2089	1.62790	0.813950	+	0.580935i	$0.197313\pi$
$878$	0	0
$879$	−4.78475	−0.161385
$880$	0	0
$881$	1.44657	0.0487363	0.0243681	−	0.999703i	$-0.492243\pi$
$881$	1.44657	0.0487363	0.0243681	+	0.999703i	$0.492243\pi$
$882$	0	0
$883$	−53.2500	−1.79200	−0.896002	−	0.444049i	$-0.853541\pi$
$883$	−53.2500	−1.79200	−0.896002	+	0.444049i	$0.853541\pi$
$884$	0	0
$885$	−1.48348	−0.0498666
$886$	0	0
$887$	−16.8685	−0.566390	−0.283195	−	0.959062i	$-0.591394\pi$
$887$	−16.8685	−0.566390	−0.283195	+	0.959062i	$0.591394\pi$
$888$	0	0
$889$	66.3770	2.22621
$890$	0	0
$891$	−2.06616	−0.0692189
$892$	0	0
$893$	−5.01797	−0.167920
$894$	0	0
$895$	−0.948523	−0.0317056
$896$	0	0
$897$	−0.702105	−0.0234426
$898$	0	0
$899$	7.41338	0.247250
$900$	0	0
$901$	−13.9408	−0.464437
$902$	0	0
$903$	−0.785406	−0.0261367
$904$	0	0
$905$	2.38784	0.0793746
$906$	0	0
$907$	−20.3282	−0.674988	−0.337494	−	0.941328i	$-0.609579\pi$
$907$	−20.3282	−0.674988	−0.337494	+	0.941328i	$0.609579\pi$
$908$	0	0
$909$	−13.4238	−0.445240
$910$	0	0
$911$	18.6443	0.617713	0.308856	−	0.951109i	$-0.400054\pi$
$911$	18.6443	0.617713	0.308856	+	0.951109i	$0.400054\pi$
$912$	0	0
$913$	25.4951	0.843764
$914$	0	0
$915$	−0.386516	−0.0127778
$916$	0	0
$917$	31.2622	1.03237
$918$	0	0
$919$	−37.5774	−1.23956	−0.619782	−	0.784774i	$-0.712779\pi$
$919$	−37.5774	−1.23956	−0.619782	+	0.784774i	$0.712779\pi$
$920$	0	0
$921$	1.56614	0.0516060
$922$	0	0
$923$	−5.24765	−0.172729
$924$	0	0
$925$	14.4641	0.475578
$926$	0	0
$927$	5.32865	0.175016
$928$	0	0
$929$	24.3674	0.799468	0.399734	−	0.916631i	$-0.369103\pi$
$929$	24.3674	0.799468	0.399734	+	0.916631i	$0.369103\pi$
$930$	0	0
$931$	10.0906	0.330706
$932$	0	0
$933$	−23.2734	−0.761939
$934$	0	0
$935$	−1.23886	−0.0405149
$936$	0	0
$937$	28.7809	0.940231	0.470116	−	0.882605i	$-0.344212\pi$
$937$	28.7809	0.940231	0.470116	+	0.882605i	$0.344212\pi$
$938$	0	0
$939$	−9.54782	−0.311581
$940$	0	0
$941$	25.9181	0.844906	0.422453	−	0.906385i	$-0.361169\pi$
$941$	25.9181	0.844906	0.422453	+	0.906385i	$0.361169\pi$
$942$	0	0
$943$	−4.05761	−0.132134
$944$	0	0
$945$	−0.462529	−0.0150461
$946$	0	0
$947$	52.4650	1.70488	0.852442	−	0.522822i	$-0.175121\pi$
$947$	52.4650	1.70488	0.852442	+	0.522822i	$0.175121\pi$
$948$	0	0
$949$	−6.43821	−0.208993
$950$	0	0
$951$	4.20816	0.136459
$952$	0	0
$953$	37.1505	1.20342	0.601711	−	0.798714i	$-0.294486\pi$
$953$	37.1505	1.20342	0.601711	+	0.798714i	$0.294486\pi$
$954$	0	0
$955$	0.937517	0.0303373
$956$	0	0
$957$	−2.06616	−0.0667894
$958$	0	0
$959$	15.5079	0.500776
$960$	0	0
$961$	23.9581	0.772843
$962$	0	0
$963$	13.7453	0.442935
$964$	0	0
$965$	−2.65537	−0.0854795
$966$	0	0
$967$	25.9982	0.836046	0.418023	−	0.908436i	$-0.362723\pi$
$967$	25.9982	0.836046	0.418023	+	0.908436i	$0.362723\pi$
$968$	0	0
$969$	10.5602	0.339244
$970$	0	0
$971$	22.7888	0.731327	0.365664	−	0.930747i	$-0.380842\pi$
$971$	22.7888	0.731327	0.365664	+	0.930747i	$0.380842\pi$
$972$	0	0
$973$	−5.12419	−0.164274
$974$	0	0
$975$	3.49703	0.111995
$976$	0	0
$977$	36.4695	1.16676	0.583382	−	0.812198i	$-0.301729\pi$
$977$	36.4695	1.16676	0.583382	+	0.812198i	$0.301729\pi$
$978$	0	0
$979$	−6.31446	−0.201811
$980$	0	0
$981$	−6.22293	−0.198683
$982$	0	0
$983$	−3.18723	−0.101657	−0.0508284	−	0.998707i	$-0.516186\pi$
$983$	−3.18723	−0.101657	−0.0508284	+	0.998707i	$0.516186\pi$
$984$	0	0
$985$	3.15827	0.100631
$986$	0	0
$987$	6.85779	0.218286
$988$	0	0
$989$	0.235390	0.00748496
$990$	0	0
$991$	53.1446	1.68819	0.844097	−	0.536191i	$-0.180137\pi$
$991$	53.1446	1.68819	0.844097	+	0.536191i	$0.180137\pi$
$992$	0	0
$993$	−5.28355	−0.167668
$994$	0	0
$995$	2.36584	0.0750020
$996$	0	0
$997$	−51.0173	−1.61573	−0.807867	−	0.589364i	$-0.799378\pi$
$997$	−51.0173	−1.61573	−0.807867	+	0.589364i	$0.799378\pi$
$998$	0	0
$999$	−2.90399	−0.0918782

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8004.2.a.d.1.6	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8004.2.a.d.1.6	✓	8	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:	\( N \)	\(=\)	\( 8004 = 2^{2} \cdot 3 \cdot 23 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8004.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +0.138622 q^{5} -3.33662 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +0.138622 q^{5} -3.33662 q^{7} +1.00000 q^{9} -2.06616 q^{11} -0.702105 q^{13} +0.138622 q^{15} +4.32538 q^{17} +2.44146 q^{19} -3.33662 q^{21} +1.00000 q^{23} -4.98078 q^{25} +1.00000 q^{27} +1.00000 q^{29} +7.41338 q^{31} -2.06616 q^{33} -0.462529 q^{35} -2.90399 q^{37} -0.702105 q^{39} -4.05761 q^{41} +0.235390 q^{43} +0.138622 q^{45} -2.05531 q^{47} +4.13301 q^{49} +4.32538 q^{51} -3.22303 q^{53} -0.286416 q^{55} +2.44146 q^{57} -10.7016 q^{59} -2.78827 q^{61} -3.33662 q^{63} -0.0973274 q^{65} -11.6076 q^{67} +1.00000 q^{69} +7.47417 q^{71} +9.16987 q^{73} -4.98078 q^{75} +6.89398 q^{77} +11.0456 q^{79} +1.00000 q^{81} -12.3394 q^{83} +0.599594 q^{85} +1.00000 q^{87} +3.05613 q^{89} +2.34265 q^{91} +7.41338 q^{93} +0.338441 q^{95} +2.74522 q^{97} -2.06616 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{3} - 5 q^{5} - 4 q^{7} + 8 q^{9}+O(q^{10}) \) 8 * q + 8 * q^3 - 5 * q^5 - 4 * q^7 + 8 * q^9	\( 8 q + 8 q^{3} - 5 q^{5} - 4 q^{7} + 8 q^{9} - 5 q^{11} - 4 q^{13} - 5 q^{15} - 3 q^{17} - 5 q^{19} - 4 q^{21} + 8 q^{23} - 5 q^{25} + 8 q^{27} + 8 q^{29} - 2 q^{31} - 5 q^{33} - 15 q^{35} - 10 q^{37} - 4 q^{39} - 11 q^{41} - 7 q^{43} - 5 q^{45} - 14 q^{47} - 18 q^{49} - 3 q^{51} - 15 q^{53} - 17 q^{55} - 5 q^{57} + 4 q^{59} + q^{61} - 4 q^{63} - 5 q^{67} + 8 q^{69} - q^{71} - 21 q^{73} - 5 q^{75} - 8 q^{79} + 8 q^{81} + 3 q^{83} + 8 q^{87} - 20 q^{89} - 7 q^{91} - 2 q^{93} - 3 q^{95} - 7 q^{97} - 5 q^{99}+O(q^{100}) \) 8 * q + 8 * q^3 - 5 * q^5 - 4 * q^7 + 8 * q^9 - 5 * q^11 - 4 * q^13 - 5 * q^15 - 3 * q^17 - 5 * q^19 - 4 * q^21 + 8 * q^23 - 5 * q^25 + 8 * q^27 + 8 * q^29 - 2 * q^31 - 5 * q^33 - 15 * q^35 - 10 * q^37 - 4 * q^39 - 11 * q^41 - 7 * q^43 - 5 * q^45 - 14 * q^47 - 18 * q^49 - 3 * q^51 - 15 * q^53 - 17 * q^55 - 5 * q^57 + 4 * q^59 + q^61 - 4 * q^63 - 5 * q^67 + 8 * q^69 - q^71 - 21 * q^73 - 5 * q^75 - 8 * q^79 + 8 * q^81 + 3 * q^83 + 8 * q^87 - 20 * q^89 - 7 * q^91 - 2 * q^93 - 3 * q^95 - 7 * q^97 - 5 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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