LMFDB - Embedded newform 8004.2.a.i.1.13

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:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 5 x^{15} - 43 x^{14} + 234 x^{13} + 634 x^{12} - 4048 x^{11} - 3483 x^{10} + 32512 x^{9} + \cdots - 208 $ x^16 - 5x^15 - 43x^14 + 234x^13 + 634x^12 - 4048x^11 - 3483x^10 + 32512x^9 - 137x^8 - 121665x^7 + 60168x^6 + 172218x^5 - 138024x^4 - 17844x^3 + 14352x^2 + 72*x - 208
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.13
Root			$3.01816$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} +3.01816 q^{5} +3.11628 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +3.01816 q^{5} +3.11628 q^{7} +1.00000 q^{9} +4.01390 q^{11} +2.08194 q^{13} -3.01816 q^{15} +1.51150 q^{17} -3.39944 q^{19} -3.11628 q^{21} +1.00000 q^{23} +4.10928 q^{25} -1.00000 q^{27} +1.00000 q^{29} -0.311954 q^{31} -4.01390 q^{33} +9.40544 q^{35} -0.640134 q^{37} -2.08194 q^{39} +4.74044 q^{41} +2.47895 q^{43} +3.01816 q^{45} -8.97233 q^{47} +2.71123 q^{49} -1.51150 q^{51} +5.80006 q^{53} +12.1146 q^{55} +3.39944 q^{57} -11.3298 q^{59} +10.1299 q^{61} +3.11628 q^{63} +6.28362 q^{65} +9.23512 q^{67} -1.00000 q^{69} +10.8687 q^{71} +13.2814 q^{73} -4.10928 q^{75} +12.5084 q^{77} +15.2895 q^{79} +1.00000 q^{81} -7.22584 q^{83} +4.56194 q^{85} -1.00000 q^{87} -8.32140 q^{89} +6.48791 q^{91} +0.311954 q^{93} -10.2601 q^{95} -19.6100 q^{97} +4.01390 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 16 q^{3} + 5 q^{5} - 4 q^{7} + 16 q^{9}+O(q^{10}) $ 16 * q - 16 * q^3 + 5 * q^5 - 4 * q^7 + 16 * q^9	$ 16 q - 16 q^{3} + 5 q^{5} - 4 q^{7} + 16 q^{9} + 5 q^{11} + 8 q^{13} - 5 q^{15} + 7 q^{17} + q^{19} + 4 q^{21} + 16 q^{23} + 31 q^{25} - 16 q^{27} + 16 q^{29} - 2 q^{31} - 5 q^{33} + 5 q^{35} + 14 q^{37} - 8 q^{39} - q^{41} - 13 q^{43} + 5 q^{45} - 4 q^{47} + 30 q^{49} - 7 q^{51} + 19 q^{53} - 37 q^{55} - q^{57} + 12 q^{59} + 21 q^{61} - 4 q^{63} + 26 q^{65} - 11 q^{67} - 16 q^{69} + 7 q^{71} - 13 q^{73} - 31 q^{75} + 4 q^{77} - 18 q^{79} + 16 q^{81} + 25 q^{83} + 48 q^{85} - 16 q^{87} + 12 q^{89} - 11 q^{91} + 2 q^{93} - q^{95} + 5 q^{97} + 5 q^{99}+O(q^{100}) $ 16 * q - 16 * q^3 + 5 * q^5 - 4 * q^7 + 16 * q^9 + 5 * q^11 + 8 * q^13 - 5 * q^15 + 7 * q^17 + q^19 + 4 * q^21 + 16 * q^23 + 31 * q^25 - 16 * q^27 + 16 * q^29 - 2 * q^31 - 5 * q^33 + 5 * q^35 + 14 * q^37 - 8 * q^39 - q^41 - 13 * q^43 + 5 * q^45 - 4 * q^47 + 30 * q^49 - 7 * q^51 + 19 * q^53 - 37 * q^55 - q^57 + 12 * q^59 + 21 * q^61 - 4 * q^63 + 26 * q^65 - 11 * q^67 - 16 * q^69 + 7 * q^71 - 13 * q^73 - 31 * q^75 + 4 * q^77 - 18 * q^79 + 16 * q^81 + 25 * q^83 + 48 * q^85 - 16 * q^87 + 12 * q^89 - 11 * q^91 + 2 * q^93 - q^95 + 5 * 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
$5$	3.01816	1.34976	0.674881	−	0.737927i	$-0.264195\pi$
$5$	3.01816	1.34976	0.674881	+	0.737927i	$0.264195\pi$
$6$	0	0
$7$	3.11628	1.17784	0.588922	−	0.808190i	$-0.299552\pi$
$7$	3.11628	1.17784	0.588922	+	0.808190i	$0.299552\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	4.01390	1.21023	0.605117	−	0.796136i	$-0.293126\pi$
$11$	4.01390	1.21023	0.605117	+	0.796136i	$0.293126\pi$
$12$	0	0
$13$	2.08194	0.577426	0.288713	−	0.957416i	$-0.406773\pi$
$13$	2.08194	0.577426	0.288713	+	0.957416i	$0.406773\pi$
$14$	0	0
$15$	−3.01816	−0.779285
$16$	0	0
$17$	1.51150	0.366592	0.183296	−	0.983058i	$-0.441323\pi$
$17$	1.51150	0.366592	0.183296	+	0.983058i	$0.441323\pi$
$18$	0	0
$19$	−3.39944	−0.779885	−0.389943	−	0.920839i	$-0.627505\pi$
$19$	−3.39944	−0.779885	−0.389943	+	0.920839i	$0.627505\pi$
$20$	0	0
$21$	−3.11628	−0.680029
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	4.10928	0.821857
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	−0.311954	−0.0560287	−0.0280143	−	0.999608i	$-0.508918\pi$
$31$	−0.311954	−0.0560287	−0.0280143	+	0.999608i	$0.508918\pi$
$32$	0	0
$33$	−4.01390	−0.698729
$34$	0	0
$35$	9.40544	1.58981
$36$	0	0
$37$	−0.640134	−0.105237	−0.0526187	−	0.998615i	$-0.516757\pi$
$37$	−0.640134	−0.105237	−0.0526187	+	0.998615i	$0.516757\pi$
$38$	0	0
$39$	−2.08194	−0.333377
$40$	0	0
$41$	4.74044	0.740333	0.370166	−	0.928965i	$-0.379301\pi$
$41$	4.74044	0.740333	0.370166	+	0.928965i	$0.379301\pi$
$42$	0	0
$43$	2.47895	0.378036	0.189018	−	0.981974i	$-0.439470\pi$
$43$	2.47895	0.378036	0.189018	+	0.981974i	$0.439470\pi$
$44$	0	0
$45$	3.01816	0.449921
$46$	0	0
$47$	−8.97233	−1.30875	−0.654374	−	0.756171i	$-0.727068\pi$
$47$	−8.97233	−1.30875	−0.654374	+	0.756171i	$0.727068\pi$
$48$	0	0
$49$	2.71123	0.387318
$50$	0	0
$51$	−1.51150	−0.211652
$52$	0	0
$53$	5.80006	0.796700	0.398350	−	0.917233i	$-0.369583\pi$
$53$	5.80006	0.796700	0.398350	+	0.917233i	$0.369583\pi$
$54$	0	0
$55$	12.1146	1.63353
$56$	0	0
$57$	3.39944	0.450267
$58$	0	0
$59$	−11.3298	−1.47501	−0.737505	−	0.675341i	$-0.763996\pi$
$59$	−11.3298	−1.47501	−0.737505	+	0.675341i	$0.763996\pi$
$60$	0	0
$61$	10.1299	1.29701	0.648503	−	0.761212i	$-0.275396\pi$
$61$	10.1299	1.29701	0.648503	+	0.761212i	$0.275396\pi$
$62$	0	0
$63$	3.11628	0.392615
$64$	0	0
$65$	6.28362	0.779388
$66$	0	0
$67$	9.23512	1.12825	0.564125	−	0.825689i	$-0.309214\pi$
$67$	9.23512	1.12825	0.564125	+	0.825689i	$0.309214\pi$
$68$	0	0
$69$	−1.00000	−0.120386
$70$	0	0
$71$	10.8687	1.28988	0.644938	−	0.764235i	$-0.276883\pi$
$71$	10.8687	1.28988	0.644938	+	0.764235i	$0.276883\pi$
$72$	0	0
$73$	13.2814	1.55447	0.777234	−	0.629212i	$-0.216622\pi$
$73$	13.2814	1.55447	0.777234	+	0.629212i	$0.216622\pi$
$74$	0	0
$75$	−4.10928	−0.474499
$76$	0	0
$77$	12.5084	1.42547
$78$	0	0
$79$	15.2895	1.72021	0.860103	−	0.510120i	$-0.170399\pi$
$79$	15.2895	1.72021	0.860103	+	0.510120i	$0.170399\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−7.22584	−0.793139	−0.396570	−	0.918005i	$-0.629799\pi$
$83$	−7.22584	−0.793139	−0.396570	+	0.918005i	$0.629799\pi$
$84$	0	0
$85$	4.56194	0.494812
$86$	0	0
$87$	−1.00000	−0.107211
$88$	0	0
$89$	−8.32140	−0.882067	−0.441033	−	0.897491i	$-0.645388\pi$
$89$	−8.32140	−0.882067	−0.441033	+	0.897491i	$0.645388\pi$
$90$	0	0
$91$	6.48791	0.680118
$92$	0	0
$93$	0.311954	0.0323482
$94$	0	0
$95$	−10.2601	−1.05266
$96$	0	0
$97$	−19.6100	−1.99109	−0.995544	−	0.0942940i	$-0.969941\pi$
$97$	−19.6100	−1.99109	−0.995544	+	0.0942940i	$0.969941\pi$
$98$	0	0
$99$	4.01390	0.403412
$100$	0	0
$101$	−19.8128	−1.97145	−0.985725	−	0.168366i	$-0.946151\pi$
$101$	−19.8128	−1.97145	−0.985725	+	0.168366i	$0.946151\pi$
$102$	0	0
$103$	−11.1184	−1.09553	−0.547765	−	0.836632i	$-0.684521\pi$
$103$	−11.1184	−1.09553	−0.547765	+	0.836632i	$0.684521\pi$
$104$	0	0
$105$	−9.40544	−0.917877
$106$	0	0
$107$	12.1490	1.17448	0.587242	−	0.809412i	$-0.300214\pi$
$107$	12.1490	1.17448	0.587242	+	0.809412i	$0.300214\pi$
$108$	0	0
$109$	12.5557	1.20262	0.601308	−	0.799017i	$-0.294646\pi$
$109$	12.5557	1.20262	0.601308	+	0.799017i	$0.294646\pi$
$110$	0	0
$111$	0.640134	0.0607588
$112$	0	0
$113$	4.44375	0.418033	0.209017	−	0.977912i	$-0.432974\pi$
$113$	4.44375	0.418033	0.209017	+	0.977912i	$0.432974\pi$
$114$	0	0
$115$	3.01816	0.281445
$116$	0	0
$117$	2.08194	0.192475
$118$	0	0
$119$	4.71026	0.431789
$120$	0	0
$121$	5.11135	0.464668
$122$	0	0
$123$	−4.74044	−0.427431
$124$	0	0
$125$	−2.68832	−0.240451
$126$	0	0
$127$	−21.3241	−1.89221	−0.946106	−	0.323858i	$-0.895020\pi$
$127$	−21.3241	−1.89221	−0.946106	+	0.323858i	$0.895020\pi$
$128$	0	0
$129$	−2.47895	−0.218259
$130$	0	0
$131$	−9.00172	−0.786484	−0.393242	−	0.919435i	$-0.628647\pi$
$131$	−9.00172	−0.786484	−0.393242	+	0.919435i	$0.628647\pi$
$132$	0	0
$133$	−10.5936	−0.918584
$134$	0	0
$135$	−3.01816	−0.259762
$136$	0	0
$137$	21.8217	1.86435	0.932177	−	0.362004i	$-0.117907\pi$
$137$	21.8217	1.86435	0.932177	+	0.362004i	$0.117907\pi$
$138$	0	0
$139$	10.5613	0.895796	0.447898	−	0.894085i	$-0.352173\pi$
$139$	10.5613	0.895796	0.447898	+	0.894085i	$0.352173\pi$
$140$	0	0
$141$	8.97233	0.755606
$142$	0	0
$143$	8.35668	0.698821
$144$	0	0
$145$	3.01816	0.250644
$146$	0	0
$147$	−2.71123	−0.223618
$148$	0	0
$149$	5.60764	0.459396	0.229698	−	0.973262i	$-0.426226\pi$
$149$	5.60764	0.459396	0.229698	+	0.973262i	$0.426226\pi$
$150$	0	0
$151$	−2.14664	−0.174691	−0.0873456	−	0.996178i	$-0.527838\pi$
$151$	−2.14664	−0.174691	−0.0873456	+	0.996178i	$0.527838\pi$
$152$	0	0
$153$	1.51150	0.122197
$154$	0	0
$155$	−0.941528	−0.0756253
$156$	0	0
$157$	−9.18809	−0.733290	−0.366645	−	0.930361i	$-0.619494\pi$
$157$	−9.18809	−0.733290	−0.366645	+	0.930361i	$0.619494\pi$
$158$	0	0
$159$	−5.80006	−0.459975
$160$	0	0
$161$	3.11628	0.245598
$162$	0	0
$163$	−22.6642	−1.77520	−0.887598	−	0.460619i	$-0.847627\pi$
$163$	−22.6642	−1.77520	−0.887598	+	0.460619i	$0.847627\pi$
$164$	0	0
$165$	−12.1146	−0.943118
$166$	0	0
$167$	15.2030	1.17644	0.588220	−	0.808701i	$-0.299829\pi$
$167$	15.2030	1.17644	0.588220	+	0.808701i	$0.299829\pi$
$168$	0	0
$169$	−8.66553	−0.666579
$170$	0	0
$171$	−3.39944	−0.259962
$172$	0	0
$173$	−0.0577373	−0.00438969	−0.00219484	−	0.999998i	$-0.500699\pi$
$173$	−0.0577373	−0.00438969	−0.00219484	+	0.999998i	$0.500699\pi$
$174$	0	0
$175$	12.8057	0.968020
$176$	0	0
$177$	11.3298	0.851598
$178$	0	0
$179$	19.6721	1.47036	0.735182	−	0.677869i	$-0.237096\pi$
$179$	19.6721	1.47036	0.735182	+	0.677869i	$0.237096\pi$
$180$	0	0
$181$	−4.55761	−0.338765	−0.169382	−	0.985550i	$-0.554177\pi$
$181$	−4.55761	−0.338765	−0.169382	+	0.985550i	$0.554177\pi$
$182$	0	0
$183$	−10.1299	−0.748826
$184$	0	0
$185$	−1.93203	−0.142045
$186$	0	0
$187$	6.06700	0.443663
$188$	0	0
$189$	−3.11628	−0.226676
$190$	0	0
$191$	4.68909	0.339291	0.169645	−	0.985505i	$-0.445738\pi$
$191$	4.68909	0.339291	0.169645	+	0.985505i	$0.445738\pi$
$192$	0	0
$193$	12.3245	0.887134	0.443567	−	0.896241i	$-0.353713\pi$
$193$	12.3245	0.887134	0.443567	+	0.896241i	$0.353713\pi$
$194$	0	0
$195$	−6.28362	−0.449980
$196$	0	0
$197$	8.15404	0.580951	0.290476	−	0.956882i	$-0.406186\pi$
$197$	8.15404	0.580951	0.290476	+	0.956882i	$0.406186\pi$
$198$	0	0
$199$	−1.42058	−0.100702	−0.0503510	−	0.998732i	$-0.516034\pi$
$199$	−1.42058	−0.100702	−0.0503510	+	0.998732i	$0.516034\pi$
$200$	0	0
$201$	−9.23512	−0.651395
$202$	0	0
$203$	3.11628	0.218720
$204$	0	0
$205$	14.3074	0.999273
$206$	0	0
$207$	1.00000	0.0695048
$208$	0	0
$209$	−13.6450	−0.943844
$210$	0	0
$211$	3.98242	0.274161	0.137081	−	0.990560i	$-0.456228\pi$
$211$	3.98242	0.274161	0.137081	+	0.990560i	$0.456228\pi$
$212$	0	0
$213$	−10.8687	−0.744710
$214$	0	0
$215$	7.48186	0.510258
$216$	0	0
$217$	−0.972138	−0.0659930
$218$	0	0
$219$	−13.2814	−0.897473
$220$	0	0
$221$	3.14685	0.211680
$222$	0	0
$223$	−7.29555	−0.488546	−0.244273	−	0.969706i	$-0.578549\pi$
$223$	−7.29555	−0.488546	−0.244273	+	0.969706i	$0.578549\pi$
$224$	0	0
$225$	4.10928	0.273952
$226$	0	0
$227$	−19.4143	−1.28857	−0.644285	−	0.764786i	$-0.722845\pi$
$227$	−19.4143	−1.28857	−0.644285	+	0.764786i	$0.722845\pi$
$228$	0	0
$229$	−16.7203	−1.10491	−0.552453	−	0.833544i	$-0.686308\pi$
$229$	−16.7203	−1.10491	−0.552453	+	0.833544i	$0.686308\pi$
$230$	0	0
$231$	−12.5084	−0.822995
$232$	0	0
$233$	−0.394749	−0.0258608	−0.0129304	−	0.999916i	$-0.504116\pi$
$233$	−0.394749	−0.0258608	−0.0129304	+	0.999916i	$0.504116\pi$
$234$	0	0
$235$	−27.0799	−1.76650
$236$	0	0
$237$	−15.2895	−0.993162
$238$	0	0
$239$	4.88353	0.315889	0.157945	−	0.987448i	$-0.449513\pi$
$239$	4.88353	0.315889	0.157945	+	0.987448i	$0.449513\pi$
$240$	0	0
$241$	1.59740	0.102898	0.0514488	−	0.998676i	$-0.483616\pi$
$241$	1.59740	0.102898	0.0514488	+	0.998676i	$0.483616\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	8.18291	0.522787
$246$	0	0
$247$	−7.07743	−0.450326
$248$	0	0
$249$	7.22584	0.457919
$250$	0	0
$251$	−30.2217	−1.90758	−0.953789	−	0.300476i	$-0.902854\pi$
$251$	−30.2217	−1.90758	−0.953789	+	0.300476i	$0.902854\pi$
$252$	0	0
$253$	4.01390	0.252351
$254$	0	0
$255$	−4.56194	−0.285680
$256$	0	0
$257$	−19.9318	−1.24331	−0.621656	−	0.783290i	$-0.713540\pi$
$257$	−19.9318	−1.24331	−0.621656	+	0.783290i	$0.713540\pi$
$258$	0	0
$259$	−1.99484	−0.123953
$260$	0	0
$261$	1.00000	0.0618984
$262$	0	0
$263$	−21.1395	−1.30352	−0.651759	−	0.758426i	$-0.725969\pi$
$263$	−21.1395	−1.30352	−0.651759	+	0.758426i	$0.725969\pi$
$264$	0	0
$265$	17.5055	1.07536
$266$	0	0
$267$	8.32140	0.509261
$268$	0	0
$269$	−22.7862	−1.38930	−0.694650	−	0.719348i	$-0.744441\pi$
$269$	−22.7862	−1.38930	−0.694650	+	0.719348i	$0.744441\pi$
$270$	0	0
$271$	16.0246	0.973428	0.486714	−	0.873561i	$-0.338195\pi$
$271$	16.0246	0.973428	0.486714	+	0.873561i	$0.338195\pi$
$272$	0	0
$273$	−6.48791	−0.392666
$274$	0	0
$275$	16.4942	0.994640
$276$	0	0
$277$	1.48741	0.0893699	0.0446850	−	0.999001i	$-0.485772\pi$
$277$	1.48741	0.0893699	0.0446850	+	0.999001i	$0.485772\pi$
$278$	0	0
$279$	−0.311954	−0.0186762
$280$	0	0
$281$	9.54063	0.569146	0.284573	−	0.958654i	$-0.408148\pi$
$281$	9.54063	0.569146	0.284573	+	0.958654i	$0.408148\pi$
$282$	0	0
$283$	−18.6641	−1.10946	−0.554732	−	0.832029i	$-0.687179\pi$
$283$	−18.6641	−1.10946	−0.554732	+	0.832029i	$0.687179\pi$
$284$	0	0
$285$	10.2601	0.607753
$286$	0	0
$287$	14.7726	0.871997
$288$	0	0
$289$	−14.7154	−0.865610
$290$	0	0
$291$	19.6100	1.14956
$292$	0	0
$293$	24.4969	1.43112	0.715562	−	0.698550i	$-0.246171\pi$
$293$	24.4969	1.43112	0.715562	+	0.698550i	$0.246171\pi$
$294$	0	0
$295$	−34.1951	−1.99091
$296$	0	0
$297$	−4.01390	−0.232910
$298$	0	0
$299$	2.08194	0.120402
$300$	0	0
$301$	7.72510	0.445267
$302$	0	0
$303$	19.8128	1.13822
$304$	0	0
$305$	30.5738	1.75065
$306$	0	0
$307$	28.1002	1.60376	0.801881	−	0.597483i	$-0.203833\pi$
$307$	28.1002	1.60376	0.801881	+	0.597483i	$0.203833\pi$
$308$	0	0
$309$	11.1184	0.632504
$310$	0	0
$311$	−22.7191	−1.28828	−0.644140	−	0.764907i	$-0.722785\pi$
$311$	−22.7191	−1.28828	−0.644140	+	0.764907i	$0.722785\pi$
$312$	0	0
$313$	21.1963	1.19809	0.599044	−	0.800716i	$-0.295547\pi$
$313$	21.1963	1.19809	0.599044	+	0.800716i	$0.295547\pi$
$314$	0	0
$315$	9.40544	0.529937
$316$	0	0
$317$	28.0936	1.57790	0.788948	−	0.614460i	$-0.210626\pi$
$317$	28.0936	1.57790	0.788948	+	0.614460i	$0.210626\pi$
$318$	0	0
$319$	4.01390	0.224735
$320$	0	0
$321$	−12.1490	−0.678088
$322$	0	0
$323$	−5.13825	−0.285900
$324$	0	0
$325$	8.55528	0.474562
$326$	0	0
$327$	−12.5557	−0.694331
$328$	0	0
$329$	−27.9603	−1.54150
$330$	0	0
$331$	−13.6408	−0.749765	−0.374882	−	0.927072i	$-0.622317\pi$
$331$	−13.6408	−0.749765	−0.374882	+	0.927072i	$0.622317\pi$
$332$	0	0
$333$	−0.640134	−0.0350791
$334$	0	0
$335$	27.8731	1.52287
$336$	0	0
$337$	31.4994	1.71588	0.857942	−	0.513747i	$-0.171743\pi$
$337$	31.4994	1.71588	0.857942	+	0.513747i	$0.171743\pi$
$338$	0	0
$339$	−4.44375	−0.241352
$340$	0	0
$341$	−1.25215	−0.0678078
$342$	0	0
$343$	−13.3650	−0.721644
$344$	0	0
$345$	−3.01816	−0.162492
$346$	0	0
$347$	9.35775	0.502351	0.251175	−	0.967942i	$-0.419183\pi$
$347$	9.35775	0.502351	0.251175	+	0.967942i	$0.419183\pi$
$348$	0	0
$349$	21.8298	1.16852	0.584262	−	0.811565i	$-0.301384\pi$
$349$	21.8298	1.16852	0.584262	+	0.811565i	$0.301384\pi$
$350$	0	0
$351$	−2.08194	−0.111126
$352$	0	0
$353$	−14.5936	−0.776741	−0.388371	−	0.921503i	$-0.626962\pi$
$353$	−14.5936	−0.776741	−0.388371	+	0.921503i	$0.626962\pi$
$354$	0	0
$355$	32.8034	1.74102
$356$	0	0
$357$	−4.71026	−0.249293
$358$	0	0
$359$	−27.7744	−1.46588	−0.732939	−	0.680294i	$-0.761852\pi$
$359$	−27.7744	−1.46588	−0.732939	+	0.680294i	$0.761852\pi$
$360$	0	0
$361$	−7.44380	−0.391779
$362$	0	0
$363$	−5.11135	−0.268276
$364$	0	0
$365$	40.0853	2.09816
$366$	0	0
$367$	−18.1887	−0.949444	−0.474722	−	0.880136i	$-0.657451\pi$
$367$	−18.1887	−0.949444	−0.474722	+	0.880136i	$0.657451\pi$
$368$	0	0
$369$	4.74044	0.246778
$370$	0	0
$371$	18.0746	0.938389
$372$	0	0
$373$	−25.6001	−1.32552	−0.662760	−	0.748832i	$-0.730615\pi$
$373$	−25.6001	−1.32552	−0.662760	+	0.748832i	$0.730615\pi$
$374$	0	0
$375$	2.68832	0.138824
$376$	0	0
$377$	2.08194	0.107225
$378$	0	0
$379$	−20.1736	−1.03625	−0.518125	−	0.855305i	$-0.673370\pi$
$379$	−20.1736	−1.03625	−0.518125	+	0.855305i	$0.673370\pi$
$380$	0	0
$381$	21.3241	1.09247
$382$	0	0
$383$	−10.4984	−0.536443	−0.268222	−	0.963357i	$-0.586436\pi$
$383$	−10.4984	−0.536443	−0.268222	+	0.963357i	$0.586436\pi$
$384$	0	0
$385$	37.7525	1.92404
$386$	0	0
$387$	2.47895	0.126012
$388$	0	0
$389$	3.11093	0.157730	0.0788652	−	0.996885i	$-0.474870\pi$
$389$	3.11093	0.157730	0.0788652	+	0.996885i	$0.474870\pi$
$390$	0	0
$391$	1.51150	0.0764398
$392$	0	0
$393$	9.00172	0.454077
$394$	0	0
$395$	46.1462	2.32187
$396$	0	0
$397$	28.9655	1.45374	0.726869	−	0.686776i	$-0.240975\pi$
$397$	28.9655	1.45374	0.726869	+	0.686776i	$0.240975\pi$
$398$	0	0
$399$	10.5936	0.530344
$400$	0	0
$401$	−2.06390	−0.103066	−0.0515330	−	0.998671i	$-0.516411\pi$
$401$	−2.06390	−0.103066	−0.0515330	+	0.998671i	$0.516411\pi$
$402$	0	0
$403$	−0.649470	−0.0323524
$404$	0	0
$405$	3.01816	0.149974
$406$	0	0
$407$	−2.56943	−0.127362
$408$	0	0
$409$	−32.6300	−1.61345	−0.806724	−	0.590928i	$-0.798762\pi$
$409$	−32.6300	−1.61345	−0.806724	+	0.590928i	$0.798762\pi$
$410$	0	0
$411$	−21.8217	−1.07638
$412$	0	0
$413$	−35.3068	−1.73733
$414$	0	0
$415$	−21.8087	−1.07055
$416$	0	0
$417$	−10.5613	−0.517188
$418$	0	0
$419$	−12.9519	−0.632740	−0.316370	−	0.948636i	$-0.602464\pi$
$419$	−12.9519	−0.632740	−0.316370	+	0.948636i	$0.602464\pi$
$420$	0	0
$421$	9.36719	0.456529	0.228265	−	0.973599i	$-0.426695\pi$
$421$	9.36719	0.456529	0.228265	+	0.973599i	$0.426695\pi$
$422$	0	0
$423$	−8.97233	−0.436250
$424$	0	0
$425$	6.21118	0.301286
$426$	0	0
$427$	31.5678	1.52767
$428$	0	0
$429$	−8.35668	−0.403465
$430$	0	0
$431$	15.6930	0.755907	0.377954	−	0.925825i	$-0.376628\pi$
$431$	15.6930	0.755907	0.377954	+	0.925825i	$0.376628\pi$
$432$	0	0
$433$	15.8098	0.759772	0.379886	−	0.925033i	$-0.375963\pi$
$433$	15.8098	0.759772	0.379886	+	0.925033i	$0.375963\pi$
$434$	0	0
$435$	−3.01816	−0.144710
$436$	0	0
$437$	−3.39944	−0.162617
$438$	0	0
$439$	−14.6868	−0.700964	−0.350482	−	0.936569i	$-0.613982\pi$
$439$	−14.6868	−0.700964	−0.350482	+	0.936569i	$0.613982\pi$
$440$	0	0
$441$	2.71123	0.129106
$442$	0	0
$443$	−4.79849	−0.227983	−0.113991	−	0.993482i	$-0.536364\pi$
$443$	−4.79849	−0.227983	−0.113991	+	0.993482i	$0.536364\pi$
$444$	0	0
$445$	−25.1153	−1.19058
$446$	0	0
$447$	−5.60764	−0.265232
$448$	0	0
$449$	2.59321	0.122381	0.0611905	−	0.998126i	$-0.480510\pi$
$449$	2.59321	0.122381	0.0611905	+	0.998126i	$0.480510\pi$
$450$	0	0
$451$	19.0276	0.895976
$452$	0	0
$453$	2.14664	0.100858
$454$	0	0
$455$	19.5816	0.917997
$456$	0	0
$457$	29.1359	1.36292	0.681460	−	0.731855i	$-0.261345\pi$
$457$	29.1359	1.36292	0.681460	+	0.731855i	$0.261345\pi$
$458$	0	0
$459$	−1.51150	−0.0705507
$460$	0	0
$461$	15.8875	0.739954	0.369977	−	0.929041i	$-0.379366\pi$
$461$	15.8875	0.739954	0.369977	+	0.929041i	$0.379366\pi$
$462$	0	0
$463$	28.1147	1.30660	0.653301	−	0.757099i	$-0.273384\pi$
$463$	28.1147	1.30660	0.653301	+	0.757099i	$0.273384\pi$
$464$	0	0
$465$	0.941528	0.0436623
$466$	0	0
$467$	9.49080	0.439182	0.219591	−	0.975592i	$-0.429528\pi$
$467$	9.49080	0.439182	0.219591	+	0.975592i	$0.429528\pi$
$468$	0	0
$469$	28.7793	1.32890
$470$	0	0
$471$	9.18809	0.423365
$472$	0	0
$473$	9.95023	0.457512
$474$	0	0
$475$	−13.9693	−0.640954
$476$	0	0
$477$	5.80006	0.265567
$478$	0	0
$479$	18.4200	0.841633	0.420816	−	0.907146i	$-0.361744\pi$
$479$	18.4200	0.841633	0.420816	+	0.907146i	$0.361744\pi$
$480$	0	0
$481$	−1.33272	−0.0607668
$482$	0	0
$483$	−3.11628	−0.141796
$484$	0	0
$485$	−59.1859	−2.68750
$486$	0	0
$487$	−40.7305	−1.84568	−0.922838	−	0.385189i	$-0.874136\pi$
$487$	−40.7305	−1.84568	−0.922838	+	0.385189i	$0.874136\pi$
$488$	0	0
$489$	22.6642	1.02491
$490$	0	0
$491$	−12.0643	−0.544455	−0.272227	−	0.962233i	$-0.587760\pi$
$491$	−12.0643	−0.544455	−0.272227	+	0.962233i	$0.587760\pi$
$492$	0	0
$493$	1.51150	0.0680745
$494$	0	0
$495$	12.1146	0.544510
$496$	0	0
$497$	33.8699	1.51927
$498$	0	0
$499$	37.1644	1.66371	0.831855	−	0.554994i	$-0.187279\pi$
$499$	37.1644	1.66371	0.831855	+	0.554994i	$0.187279\pi$
$500$	0	0
$501$	−15.2030	−0.679218
$502$	0	0
$503$	37.1622	1.65698	0.828491	−	0.560003i	$-0.189200\pi$
$503$	37.1622	1.65698	0.828491	+	0.560003i	$0.189200\pi$
$504$	0	0
$505$	−59.7982	−2.66099
$506$	0	0
$507$	8.66553	0.384850
$508$	0	0
$509$	0.747383	0.0331272	0.0165636	−	0.999863i	$-0.494727\pi$
$509$	0.747383	0.0331272	0.0165636	+	0.999863i	$0.494727\pi$
$510$	0	0
$511$	41.3886	1.83092
$512$	0	0
$513$	3.39944	0.150089
$514$	0	0
$515$	−33.5571	−1.47870
$516$	0	0
$517$	−36.0140	−1.58389
$518$	0	0
$519$	0.0577373	0.00253439
$520$	0	0
$521$	33.4372	1.46491	0.732455	−	0.680815i	$-0.238374\pi$
$521$	33.4372	1.46491	0.732455	+	0.680815i	$0.238374\pi$
$522$	0	0
$523$	5.12177	0.223960	0.111980	−	0.993710i	$-0.464281\pi$
$523$	5.12177	0.223960	0.111980	+	0.993710i	$0.464281\pi$
$524$	0	0
$525$	−12.8057	−0.558886
$526$	0	0
$527$	−0.471518	−0.0205397
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−11.3298	−0.491670
$532$	0	0
$533$	9.86931	0.427487
$534$	0	0
$535$	36.6675	1.58527
$536$	0	0
$537$	−19.6721	−0.848916
$538$	0	0
$539$	10.8826	0.468746
$540$	0	0
$541$	−23.8365	−1.02481	−0.512405	−	0.858744i	$-0.671245\pi$
$541$	−23.8365	−1.02481	−0.512405	+	0.858744i	$0.671245\pi$
$542$	0	0
$543$	4.55761	0.195586
$544$	0	0
$545$	37.8951	1.62325
$546$	0	0
$547$	−3.60333	−0.154067	−0.0770336	−	0.997029i	$-0.524545\pi$
$547$	−3.60333	−0.154067	−0.0770336	+	0.997029i	$0.524545\pi$
$548$	0	0
$549$	10.1299	0.432335
$550$	0	0
$551$	−3.39944	−0.144821
$552$	0	0
$553$	47.6465	2.02614
$554$	0	0
$555$	1.93203	0.0820100
$556$	0	0
$557$	36.9165	1.56420	0.782102	−	0.623151i	$-0.214148\pi$
$557$	36.9165	1.56420	0.782102	+	0.623151i	$0.214148\pi$
$558$	0	0
$559$	5.16102	0.218288
$560$	0	0
$561$	−6.06700	−0.256149
$562$	0	0
$563$	32.7021	1.37823	0.689115	−	0.724652i	$-0.258000\pi$
$563$	32.7021	1.37823	0.689115	+	0.724652i	$0.258000\pi$
$564$	0	0
$565$	13.4120	0.564245
$566$	0	0
$567$	3.11628	0.130872
$568$	0	0
$569$	38.3521	1.60780	0.803901	−	0.594763i	$-0.202754\pi$
$569$	38.3521	1.60780	0.803901	+	0.594763i	$0.202754\pi$
$570$	0	0
$571$	−9.73555	−0.407420	−0.203710	−	0.979031i	$-0.565300\pi$
$571$	−9.73555	−0.407420	−0.203710	+	0.979031i	$0.565300\pi$
$572$	0	0
$573$	−4.68909	−0.195890
$574$	0	0
$575$	4.10928	0.171369
$576$	0	0
$577$	−43.0538	−1.79235	−0.896176	−	0.443698i	$-0.853666\pi$
$577$	−43.0538	−1.79235	−0.896176	+	0.443698i	$0.853666\pi$
$578$	0	0
$579$	−12.3245	−0.512187
$580$	0	0
$581$	−22.5178	−0.934195
$582$	0	0
$583$	23.2808	0.964194
$584$	0	0
$585$	6.28362	0.259796
$586$	0	0
$587$	−25.5360	−1.05398	−0.526992	−	0.849870i	$-0.676680\pi$
$587$	−25.5360	−1.05398	−0.526992	+	0.849870i	$0.676680\pi$
$588$	0	0
$589$	1.06047	0.0436959
$590$	0	0
$591$	−8.15404	−0.335412
$592$	0	0
$593$	−17.7807	−0.730165	−0.365083	−	0.930975i	$-0.618959\pi$
$593$	−17.7807	−0.730165	−0.365083	+	0.930975i	$0.618959\pi$
$594$	0	0
$595$	14.2163	0.582812
$596$	0	0
$597$	1.42058	0.0581404
$598$	0	0
$599$	26.4509	1.08075	0.540377	−	0.841423i	$-0.318281\pi$
$599$	26.4509	1.08075	0.540377	+	0.841423i	$0.318281\pi$
$600$	0	0
$601$	31.8168	1.29783	0.648916	−	0.760860i	$-0.275223\pi$
$601$	31.8168	1.29783	0.648916	+	0.760860i	$0.275223\pi$
$602$	0	0
$603$	9.23512	0.376083
$604$	0	0
$605$	15.4269	0.627192
$606$	0	0
$607$	−36.9352	−1.49916	−0.749578	−	0.661917i	$-0.769743\pi$
$607$	−36.9352	−1.49916	−0.749578	+	0.661917i	$0.769743\pi$
$608$	0	0
$609$	−3.11628	−0.126278
$610$	0	0
$611$	−18.6798	−0.755706
$612$	0	0
$613$	14.9193	0.602587	0.301293	−	0.953531i	$-0.402582\pi$
$613$	14.9193	0.602587	0.301293	+	0.953531i	$0.402582\pi$
$614$	0	0
$615$	−14.3074	−0.576930
$616$	0	0
$617$	7.63942	0.307552	0.153776	−	0.988106i	$-0.450857\pi$
$617$	7.63942	0.307552	0.153776	+	0.988106i	$0.450857\pi$
$618$	0	0
$619$	5.64294	0.226809	0.113404	−	0.993549i	$-0.463824\pi$
$619$	5.64294	0.226809	0.113404	+	0.993549i	$0.463824\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	0	0
$623$	−25.9318	−1.03894
$624$	0	0
$625$	−28.6602	−1.14641
$626$	0	0
$627$	13.6450	0.544929
$628$	0	0
$629$	−0.967561	−0.0385792
$630$	0	0
$631$	−28.4240	−1.13154	−0.565772	−	0.824562i	$-0.691422\pi$
$631$	−28.4240	−1.13154	−0.565772	+	0.824562i	$0.691422\pi$
$632$	0	0
$633$	−3.98242	−0.158287
$634$	0	0
$635$	−64.3597	−2.55403
$636$	0	0
$637$	5.64461	0.223647
$638$	0	0
$639$	10.8687	0.429959
$640$	0	0
$641$	46.3130	1.82925	0.914626	−	0.404301i	$-0.132485\pi$
$641$	46.3130	1.82925	0.914626	+	0.404301i	$0.132485\pi$
$642$	0	0
$643$	−27.0406	−1.06638	−0.533188	−	0.845997i	$-0.679006\pi$
$643$	−27.0406	−1.06638	−0.533188	+	0.845997i	$0.679006\pi$
$644$	0	0
$645$	−7.48186	−0.294598
$646$	0	0
$647$	27.8531	1.09502	0.547510	−	0.836799i	$-0.315576\pi$
$647$	27.8531	1.09502	0.547510	+	0.836799i	$0.315576\pi$
$648$	0	0
$649$	−45.4765	−1.78511
$650$	0	0
$651$	0.972138	0.0381011
$652$	0	0
$653$	−16.8048	−0.657622	−0.328811	−	0.944396i	$-0.606648\pi$
$653$	−16.8048	−0.657622	−0.328811	+	0.944396i	$0.606648\pi$
$654$	0	0
$655$	−27.1686	−1.06157
$656$	0	0
$657$	13.2814	0.518156
$658$	0	0
$659$	7.66296	0.298506	0.149253	−	0.988799i	$-0.452313\pi$
$659$	7.66296	0.298506	0.149253	+	0.988799i	$0.452313\pi$
$660$	0	0
$661$	21.2426	0.826240	0.413120	−	0.910677i	$-0.364439\pi$
$661$	21.2426	0.826240	0.413120	+	0.910677i	$0.364439\pi$
$662$	0	0
$663$	−3.14685	−0.122213
$664$	0	0
$665$	−31.9732	−1.23987
$666$	0	0
$667$	1.00000	0.0387202
$668$	0	0
$669$	7.29555	0.282062
$670$	0	0
$671$	40.6605	1.56968
$672$	0	0
$673$	−26.9404	−1.03848	−0.519238	−	0.854630i	$-0.673784\pi$
$673$	−26.9404	−1.03848	−0.519238	+	0.854630i	$0.673784\pi$
$674$	0	0
$675$	−4.10928	−0.158166
$676$	0	0
$677$	15.3473	0.589845	0.294923	−	0.955521i	$-0.404706\pi$
$677$	15.3473	0.589845	0.294923	+	0.955521i	$0.404706\pi$
$678$	0	0
$679$	−61.1102	−2.34519
$680$	0	0
$681$	19.4143	0.743956
$682$	0	0
$683$	−22.9220	−0.877088	−0.438544	−	0.898710i	$-0.644506\pi$
$683$	−22.9220	−0.877088	−0.438544	+	0.898710i	$0.644506\pi$
$684$	0	0
$685$	65.8614	2.51643
$686$	0	0
$687$	16.7203	0.637918
$688$	0	0
$689$	12.0754	0.460035
$690$	0	0
$691$	8.94873	0.340426	0.170213	−	0.985407i	$-0.445554\pi$
$691$	8.94873	0.340426	0.170213	+	0.985407i	$0.445554\pi$
$692$	0	0
$693$	12.5084	0.475156
$694$	0	0
$695$	31.8756	1.20911
$696$	0	0
$697$	7.16517	0.271400
$698$	0	0
$699$	0.394749	0.0149308
$700$	0	0
$701$	−0.809113	−0.0305598	−0.0152799	−	0.999883i	$-0.504864\pi$
$701$	−0.809113	−0.0305598	−0.0152799	+	0.999883i	$0.504864\pi$
$702$	0	0
$703$	2.17610	0.0820731
$704$	0	0
$705$	27.0799	1.01989
$706$	0	0
$707$	−61.7424	−2.32206
$708$	0	0
$709$	26.8125	1.00696	0.503482	−	0.864006i	$-0.332052\pi$
$709$	26.8125	1.00696	0.503482	+	0.864006i	$0.332052\pi$
$710$	0	0
$711$	15.2895	0.573402
$712$	0	0
$713$	−0.311954	−0.0116828
$714$	0	0
$715$	25.2218	0.943242
$716$	0	0
$717$	−4.88353	−0.182379
$718$	0	0
$719$	−1.28642	−0.0479752	−0.0239876	−	0.999712i	$-0.507636\pi$
$719$	−1.28642	−0.0479752	−0.0239876	+	0.999712i	$0.507636\pi$
$720$	0	0
$721$	−34.6481	−1.29036
$722$	0	0
$723$	−1.59740	−0.0594079
$724$	0	0
$725$	4.10928	0.152615
$726$	0	0
$727$	−34.4980	−1.27946	−0.639730	−	0.768600i	$-0.720954\pi$
$727$	−34.4980	−1.27946	−0.639730	+	0.768600i	$0.720954\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	3.74692	0.138585
$732$	0	0
$733$	−17.9446	−0.662800	−0.331400	−	0.943490i	$-0.607521\pi$
$733$	−17.9446	−0.662800	−0.331400	+	0.943490i	$0.607521\pi$
$734$	0	0
$735$	−8.18291	−0.301831
$736$	0	0
$737$	37.0688	1.36545
$738$	0	0
$739$	−47.9158	−1.76261	−0.881306	−	0.472546i	$-0.843335\pi$
$739$	−47.9158	−1.76261	−0.881306	+	0.472546i	$0.843335\pi$
$740$	0	0
$741$	7.07743	0.259996
$742$	0	0
$743$	28.1651	1.03328	0.516638	−	0.856204i	$-0.327183\pi$
$743$	28.1651	1.03328	0.516638	+	0.856204i	$0.327183\pi$
$744$	0	0
$745$	16.9247	0.620075
$746$	0	0
$747$	−7.22584	−0.264380
$748$	0	0
$749$	37.8596	1.38336
$750$	0	0
$751$	−10.7976	−0.394009	−0.197005	−	0.980403i	$-0.563121\pi$
$751$	−10.7976	−0.394009	−0.197005	+	0.980403i	$0.563121\pi$
$752$	0	0
$753$	30.2217	1.10134
$754$	0	0
$755$	−6.47890	−0.235791
$756$	0	0
$757$	21.1049	0.767072	0.383536	−	0.923526i	$-0.374706\pi$
$757$	21.1049	0.767072	0.383536	+	0.923526i	$0.374706\pi$
$758$	0	0
$759$	−4.01390	−0.145695
$760$	0	0
$761$	49.3580	1.78923	0.894613	−	0.446841i	$-0.147451\pi$
$761$	49.3580	1.78923	0.894613	+	0.446841i	$0.147451\pi$
$762$	0	0
$763$	39.1271	1.41650
$764$	0	0
$765$	4.56194	0.164937
$766$	0	0
$767$	−23.5879	−0.851709
$768$	0	0
$769$	−38.1872	−1.37707	−0.688533	−	0.725205i	$-0.741745\pi$
$769$	−38.1872	−1.37707	−0.688533	+	0.725205i	$0.741745\pi$
$770$	0	0
$771$	19.9318	0.717826
$772$	0	0
$773$	25.7530	0.926272	0.463136	−	0.886287i	$-0.346724\pi$
$773$	25.7530	0.926272	0.463136	+	0.886287i	$0.346724\pi$
$774$	0	0
$775$	−1.28191	−0.0460475
$776$	0	0
$777$	1.99484	0.0715645
$778$	0	0
$779$	−16.1149	−0.577374
$780$	0	0
$781$	43.6258	1.56105
$782$	0	0
$783$	−1.00000	−0.0357371
$784$	0	0
$785$	−27.7311	−0.989766
$786$	0	0
$787$	0.966498	0.0344519	0.0172260	−	0.999852i	$-0.494517\pi$
$787$	0.966498	0.0344519	0.0172260	+	0.999852i	$0.494517\pi$
$788$	0	0
$789$	21.1395	0.752587
$790$	0	0
$791$	13.8480	0.492378
$792$	0	0
$793$	21.0899	0.748925
$794$	0	0
$795$	−17.5055	−0.620857
$796$	0	0
$797$	26.2327	0.929210	0.464605	−	0.885518i	$-0.346196\pi$
$797$	26.2327	0.929210	0.464605	+	0.885518i	$0.346196\pi$
$798$	0	0
$799$	−13.5617	−0.479777
$800$	0	0
$801$	−8.32140	−0.294022
$802$	0	0
$803$	53.3101	1.88127
$804$	0	0
$805$	9.40544	0.331498
$806$	0	0
$807$	22.7862	0.802113
$808$	0	0
$809$	−29.2417	−1.02808	−0.514041	−	0.857765i	$-0.671852\pi$
$809$	−29.2417	−1.02808	−0.514041	+	0.857765i	$0.671852\pi$
$810$	0	0
$811$	6.61433	0.232261	0.116130	−	0.993234i	$-0.462951\pi$
$811$	6.61433	0.232261	0.116130	+	0.993234i	$0.462951\pi$
$812$	0	0
$813$	−16.0246	−0.562009
$814$	0	0
$815$	−68.4041	−2.39609
$816$	0	0
$817$	−8.42703	−0.294825
$818$	0	0
$819$	6.48791	0.226706
$820$	0	0
$821$	25.7326	0.898075	0.449038	−	0.893513i	$-0.351767\pi$
$821$	25.7326	0.898075	0.449038	+	0.893513i	$0.351767\pi$
$822$	0	0
$823$	36.3098	1.26568	0.632841	−	0.774282i	$-0.281889\pi$
$823$	36.3098	1.26568	0.632841	+	0.774282i	$0.281889\pi$
$824$	0	0
$825$	−16.4942	−0.574256
$826$	0	0
$827$	−48.8534	−1.69880	−0.849400	−	0.527749i	$-0.823036\pi$
$827$	−48.8534	−1.69880	−0.849400	+	0.527749i	$0.823036\pi$
$828$	0	0
$829$	−52.1550	−1.81142	−0.905709	−	0.423900i	$-0.860661\pi$
$829$	−52.1550	−1.81142	−0.905709	+	0.423900i	$0.860661\pi$
$830$	0	0
$831$	−1.48741	−0.0515978
$832$	0	0
$833$	4.09801	0.141988
$834$	0	0
$835$	45.8850	1.58791
$836$	0	0
$837$	0.311954	0.0107827
$838$	0	0
$839$	−27.9594	−0.965264	−0.482632	−	0.875823i	$-0.660319\pi$
$839$	−27.9594	−0.965264	−0.482632	+	0.875823i	$0.660319\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	−9.54063	−0.328597
$844$	0	0
$845$	−26.1539	−0.899723
$846$	0	0
$847$	15.9284	0.547307
$848$	0	0
$849$	18.6641	0.640550
$850$	0	0
$851$	−0.640134	−0.0219435
$852$	0	0
$853$	−51.1221	−1.75039	−0.875193	−	0.483774i	$-0.839266\pi$
$853$	−51.1221	−1.75039	−0.875193	+	0.483774i	$0.839266\pi$
$854$	0	0
$855$	−10.2601	−0.350886
$856$	0	0
$857$	−32.9031	−1.12395	−0.561975	−	0.827154i	$-0.689958\pi$
$857$	−32.9031	−1.12395	−0.561975	+	0.827154i	$0.689958\pi$
$858$	0	0
$859$	−21.1992	−0.723308	−0.361654	−	0.932312i	$-0.617788\pi$
$859$	−21.1992	−0.723308	−0.361654	+	0.932312i	$0.617788\pi$
$860$	0	0
$861$	−14.7726	−0.503448
$862$	0	0
$863$	−48.3542	−1.64599	−0.822997	−	0.568045i	$-0.807700\pi$
$863$	−48.3542	−1.64599	−0.822997	+	0.568045i	$0.807700\pi$
$864$	0	0
$865$	−0.174260	−0.00592503
$866$	0	0
$867$	14.7154	0.499760
$868$	0	0
$869$	61.3706	2.08185
$870$	0	0
$871$	19.2270	0.651481
$872$	0	0
$873$	−19.6100	−0.663696
$874$	0	0
$875$	−8.37757	−0.283214
$876$	0	0
$877$	14.0704	0.475123	0.237562	−	0.971372i	$-0.423652\pi$
$877$	14.0704	0.475123	0.237562	+	0.971372i	$0.423652\pi$
$878$	0	0
$879$	−24.4969	−0.826259
$880$	0	0
$881$	32.7327	1.10279	0.551396	−	0.834244i	$-0.314096\pi$
$881$	32.7327	1.10279	0.551396	+	0.834244i	$0.314096\pi$
$882$	0	0
$883$	−40.1331	−1.35059	−0.675294	−	0.737549i	$-0.735983\pi$
$883$	−40.1331	−1.35059	−0.675294	+	0.737549i	$0.735983\pi$
$884$	0	0
$885$	34.1951	1.14945
$886$	0	0
$887$	−1.74123	−0.0584648	−0.0292324	−	0.999573i	$-0.509306\pi$
$887$	−1.74123	−0.0584648	−0.0292324	+	0.999573i	$0.509306\pi$
$888$	0	0
$889$	−66.4521	−2.22873
$890$	0	0
$891$	4.01390	0.134471
$892$	0	0
$893$	30.5009	1.02067
$894$	0	0
$895$	59.3736	1.98464
$896$	0	0
$897$	−2.08194	−0.0695139
$898$	0	0
$899$	−0.311954	−0.0104043
$900$	0	0
$901$	8.76679	0.292064
$902$	0	0
$903$	−7.72510	−0.257075
$904$	0	0
$905$	−13.7556	−0.457252
$906$	0	0
$907$	−7.92339	−0.263092	−0.131546	−	0.991310i	$-0.541994\pi$
$907$	−7.92339	−0.263092	−0.131546	+	0.991310i	$0.541994\pi$
$908$	0	0
$909$	−19.8128	−0.657150
$910$	0	0
$911$	−11.9492	−0.395894	−0.197947	−	0.980213i	$-0.563427\pi$
$911$	−11.9492	−0.395894	−0.197947	+	0.980213i	$0.563427\pi$
$912$	0	0
$913$	−29.0038	−0.959885
$914$	0	0
$915$	−30.5738	−1.01074
$916$	0	0
$917$	−28.0519	−0.926356
$918$	0	0
$919$	35.3329	1.16553	0.582763	−	0.812642i	$-0.301972\pi$
$919$	35.3329	1.16553	0.582763	+	0.812642i	$0.301972\pi$
$920$	0	0
$921$	−28.1002	−0.925933
$922$	0	0
$923$	22.6279	0.744808
$924$	0	0
$925$	−2.63049	−0.0864901
$926$	0	0
$927$	−11.1184	−0.365176
$928$	0	0
$929$	27.3999	0.898961	0.449480	−	0.893290i	$-0.351609\pi$
$929$	27.3999	0.898961	0.449480	+	0.893290i	$0.351609\pi$
$930$	0	0
$931$	−9.21665	−0.302064
$932$	0	0
$933$	22.7191	0.743789
$934$	0	0
$935$	18.3112	0.598839
$936$	0	0
$937$	22.7656	0.743718	0.371859	−	0.928289i	$-0.378720\pi$
$937$	22.7656	0.743718	0.371859	+	0.928289i	$0.378720\pi$
$938$	0	0
$939$	−21.1963	−0.691717
$940$	0	0
$941$	41.6999	1.35938	0.679688	−	0.733501i	$-0.262115\pi$
$941$	41.6999	1.35938	0.679688	+	0.733501i	$0.262115\pi$
$942$	0	0
$943$	4.74044	0.154370
$944$	0	0
$945$	−9.40544	−0.305959
$946$	0	0
$947$	−44.4725	−1.44516	−0.722581	−	0.691286i	$-0.757044\pi$
$947$	−44.4725	−1.44516	−0.722581	+	0.691286i	$0.757044\pi$
$948$	0	0
$949$	27.6510	0.897590
$950$	0	0
$951$	−28.0936	−0.910999
$952$	0	0
$953$	16.8109	0.544560	0.272280	−	0.962218i	$-0.412222\pi$
$953$	16.8109	0.544560	0.272280	+	0.962218i	$0.412222\pi$
$954$	0	0
$955$	14.1524	0.457962
$956$	0	0
$957$	−4.01390	−0.129751
$958$	0	0
$959$	68.0026	2.19592
$960$	0	0
$961$	−30.9027	−0.996861
$962$	0	0
$963$	12.1490	0.391495
$964$	0	0
$965$	37.1972	1.19742
$966$	0	0
$967$	−19.1942	−0.617244	−0.308622	−	0.951185i	$-0.599868\pi$
$967$	−19.1942	−0.617244	−0.308622	+	0.951185i	$0.599868\pi$
$968$	0	0
$969$	5.13825	0.165064
$970$	0	0
$971$	−14.4563	−0.463925	−0.231962	−	0.972725i	$-0.574515\pi$
$971$	−14.4563	−0.463925	−0.231962	+	0.972725i	$0.574515\pi$
$972$	0	0
$973$	32.9120	1.05511
$974$	0	0
$975$	−8.55528	−0.273988
$976$	0	0
$977$	−25.9966	−0.831705	−0.415852	−	0.909432i	$-0.636517\pi$
$977$	−25.9966	−0.831705	−0.415852	+	0.909432i	$0.636517\pi$
$978$	0	0
$979$	−33.4012	−1.06751
$980$	0	0
$981$	12.5557	0.400872
$982$	0	0
$983$	47.6020	1.51827	0.759135	−	0.650934i	$-0.225622\pi$
$983$	47.6020	1.51827	0.759135	+	0.650934i	$0.225622\pi$
$984$	0	0
$985$	24.6102	0.784146
$986$	0	0
$987$	27.9603	0.889987
$988$	0	0
$989$	2.47895	0.0788259
$990$	0	0
$991$	19.6633	0.624626	0.312313	−	0.949979i	$-0.398896\pi$
$991$	19.6633	0.624626	0.312313	+	0.949979i	$0.398896\pi$
$992$	0	0
$993$	13.6408	0.432877
$994$	0	0
$995$	−4.28753	−0.135924
$996$	0	0
$997$	16.2571	0.514868	0.257434	−	0.966296i	$-0.417123\pi$
$997$	16.2571	0.514868	0.257434	+	0.966296i	$0.417123\pi$
$998$	0	0
$999$	0.640134	0.0202529

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8004.2.a.i.1.13	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8004.2.a.i.1.13	✓	16	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +3.01816 q^{5} +3.11628 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +3.01816 q^{5} +3.11628 q^{7} +1.00000 q^{9} +4.01390 q^{11} +2.08194 q^{13} -3.01816 q^{15} +1.51150 q^{17} -3.39944 q^{19} -3.11628 q^{21} +1.00000 q^{23} +4.10928 q^{25} -1.00000 q^{27} +1.00000 q^{29} -0.311954 q^{31} -4.01390 q^{33} +9.40544 q^{35} -0.640134 q^{37} -2.08194 q^{39} +4.74044 q^{41} +2.47895 q^{43} +3.01816 q^{45} -8.97233 q^{47} +2.71123 q^{49} -1.51150 q^{51} +5.80006 q^{53} +12.1146 q^{55} +3.39944 q^{57} -11.3298 q^{59} +10.1299 q^{61} +3.11628 q^{63} +6.28362 q^{65} +9.23512 q^{67} -1.00000 q^{69} +10.8687 q^{71} +13.2814 q^{73} -4.10928 q^{75} +12.5084 q^{77} +15.2895 q^{79} +1.00000 q^{81} -7.22584 q^{83} +4.56194 q^{85} -1.00000 q^{87} -8.32140 q^{89} +6.48791 q^{91} +0.311954 q^{93} -10.2601 q^{95} -19.6100 q^{97} +4.01390 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 16 q^{3} + 5 q^{5} - 4 q^{7} + 16 q^{9}+O(q^{10}) \) 16 * q - 16 * q^3 + 5 * q^5 - 4 * q^7 + 16 * q^9	\( 16 q - 16 q^{3} + 5 q^{5} - 4 q^{7} + 16 q^{9} + 5 q^{11} + 8 q^{13} - 5 q^{15} + 7 q^{17} + q^{19} + 4 q^{21} + 16 q^{23} + 31 q^{25} - 16 q^{27} + 16 q^{29} - 2 q^{31} - 5 q^{33} + 5 q^{35} + 14 q^{37} - 8 q^{39} - q^{41} - 13 q^{43} + 5 q^{45} - 4 q^{47} + 30 q^{49} - 7 q^{51} + 19 q^{53} - 37 q^{55} - q^{57} + 12 q^{59} + 21 q^{61} - 4 q^{63} + 26 q^{65} - 11 q^{67} - 16 q^{69} + 7 q^{71} - 13 q^{73} - 31 q^{75} + 4 q^{77} - 18 q^{79} + 16 q^{81} + 25 q^{83} + 48 q^{85} - 16 q^{87} + 12 q^{89} - 11 q^{91} + 2 q^{93} - q^{95} + 5 q^{97} + 5 q^{99}+O(q^{100}) \) 16 * q - 16 * q^3 + 5 * q^5 - 4 * q^7 + 16 * q^9 + 5 * q^11 + 8 * q^13 - 5 * q^15 + 7 * q^17 + q^19 + 4 * q^21 + 16 * q^23 + 31 * q^25 - 16 * q^27 + 16 * q^29 - 2 * q^31 - 5 * q^33 + 5 * q^35 + 14 * q^37 - 8 * q^39 - q^41 - 13 * q^43 + 5 * q^45 - 4 * q^47 + 30 * q^49 - 7 * q^51 + 19 * q^53 - 37 * q^55 - q^57 + 12 * q^59 + 21 * q^61 - 4 * q^63 + 26 * q^65 - 11 * q^67 - 16 * q^69 + 7 * q^71 - 13 * q^73 - 31 * q^75 + 4 * q^77 - 18 * q^79 + 16 * q^81 + 25 * q^83 + 48 * q^85 - 16 * q^87 + 12 * q^89 - 11 * q^91 + 2 * q^93 - q^95 + 5 * q^97 + 5 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

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