LMFDB - Embedded newform 4008.2.a.m.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4008,2,Mod(1,4008)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4008, 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("4008.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4008 = 2^{3} \cdot 3 \cdot 167 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4008.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:	$32.0040411301$
Analytic rank:	$0$
Dimension:	$13$
Coefficient field:	$\mathbb{Q}[x]/(x^{13} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{13} - 2 x^{12} - 49 x^{11} + 99 x^{10} + 901 x^{9} - 1879 x^{8} - 7582 x^{7} + 16968 x^{6} + \cdots + 12144 $ x^13 - 2x^12 - 49x^11 + 99x^10 + 901x^9 - 1879x^8 - 7582x^7 + 16968x^6 + 26911x^5 - 72240x^4 - 14532x^3 + 112850x^2 - 72184x + 12144
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			1.7
Root			$0.649069$ of defining polynomial
Character	$\chi$	$=$	4008.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +0.649069 q^{5} +4.19646 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +0.649069 q^{5} +4.19646 q^{7} +1.00000 q^{9} +1.95286 q^{11} +3.07299 q^{13} +0.649069 q^{15} +6.85258 q^{17} +8.04597 q^{19} +4.19646 q^{21} -6.50672 q^{23} -4.57871 q^{25} +1.00000 q^{27} -2.52529 q^{29} +4.14955 q^{31} +1.95286 q^{33} +2.72380 q^{35} -4.94625 q^{37} +3.07299 q^{39} +2.05737 q^{41} +4.35750 q^{43} +0.649069 q^{45} -11.2109 q^{47} +10.6103 q^{49} +6.85258 q^{51} -1.69789 q^{53} +1.26754 q^{55} +8.04597 q^{57} -7.31745 q^{59} -12.5360 q^{61} +4.19646 q^{63} +1.99458 q^{65} -6.97495 q^{67} -6.50672 q^{69} +2.84782 q^{71} -7.78221 q^{73} -4.57871 q^{75} +8.19511 q^{77} -3.29528 q^{79} +1.00000 q^{81} +1.99647 q^{83} +4.44780 q^{85} -2.52529 q^{87} -15.9262 q^{89} +12.8957 q^{91} +4.14955 q^{93} +5.22239 q^{95} +13.7538 q^{97} +1.95286 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 13 q + 13 q^{3} + 2 q^{5} + q^{7} + 13 q^{9}+O(q^{10}) $ 13 * q + 13 * q^3 + 2 * q^5 + q^7 + 13 * q^9	$ 13 q + 13 q^{3} + 2 q^{5} + q^{7} + 13 q^{9} + 11 q^{11} + 12 q^{13} + 2 q^{15} + 15 q^{17} + 14 q^{19} + q^{21} + 9 q^{23} + 37 q^{25} + 13 q^{27} - 3 q^{29} - 17 q^{31} + 11 q^{33} + 15 q^{35} + 16 q^{37} + 12 q^{39} + 12 q^{41} + 20 q^{43} + 2 q^{45} - 6 q^{47} + 26 q^{49} + 15 q^{51} - 12 q^{53} + 7 q^{55} + 14 q^{57} + 14 q^{59} + 24 q^{61} + q^{63} + 8 q^{65} + 3 q^{67} + 9 q^{69} + 17 q^{71} + 34 q^{73} + 37 q^{75} + 30 q^{77} + 10 q^{79} + 13 q^{81} + 44 q^{83} + 25 q^{85} - 3 q^{87} + 25 q^{89} + 29 q^{91} - 17 q^{93} - 15 q^{95} + 38 q^{97} + 11 q^{99}+O(q^{100}) $ 13 * q + 13 * q^3 + 2 * q^5 + q^7 + 13 * q^9 + 11 * q^11 + 12 * q^13 + 2 * q^15 + 15 * q^17 + 14 * q^19 + q^21 + 9 * q^23 + 37 * q^25 + 13 * q^27 - 3 * q^29 - 17 * q^31 + 11 * q^33 + 15 * q^35 + 16 * q^37 + 12 * q^39 + 12 * q^41 + 20 * q^43 + 2 * q^45 - 6 * q^47 + 26 * q^49 + 15 * q^51 - 12 * q^53 + 7 * q^55 + 14 * q^57 + 14 * q^59 + 24 * q^61 + q^63 + 8 * q^65 + 3 * q^67 + 9 * q^69 + 17 * q^71 + 34 * q^73 + 37 * q^75 + 30 * q^77 + 10 * q^79 + 13 * q^81 + 44 * q^83 + 25 * q^85 - 3 * q^87 + 25 * q^89 + 29 * q^91 - 17 * q^93 - 15 * q^95 + 38 * q^97 + 11 * 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
$5$	0.649069	0.290273	0.145136	−	0.989412i	$-0.453638\pi$
$5$	0.649069	0.290273	0.145136	+	0.989412i	$0.453638\pi$
$6$	0	0
$7$	4.19646	1.58611	0.793057	−	0.609147i	$-0.208488\pi$
$7$	4.19646	1.58611	0.793057	+	0.609147i	$0.208488\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	1.95286	0.588810	0.294405	−	0.955681i	$-0.404879\pi$
$11$	1.95286	0.588810	0.294405	+	0.955681i	$0.404879\pi$
$12$	0	0
$13$	3.07299	0.852293	0.426146	−	0.904654i	$-0.359871\pi$
$13$	3.07299	0.852293	0.426146	+	0.904654i	$0.359871\pi$
$14$	0	0
$15$	0.649069	0.167589
$16$	0	0
$17$	6.85258	1.66200	0.830998	−	0.556276i	$-0.187770\pi$
$17$	6.85258	1.66200	0.830998	+	0.556276i	$0.187770\pi$
$18$	0	0
$19$	8.04597	1.84587	0.922936	−	0.384954i	$-0.125783\pi$
$19$	8.04597	1.84587	0.922936	+	0.384954i	$0.125783\pi$
$20$	0	0
$21$	4.19646	0.915744
$22$	0	0
$23$	−6.50672	−1.35675	−0.678373	−	0.734718i	$-0.737314\pi$
$23$	−6.50672	−1.35675	−0.678373	+	0.734718i	$0.737314\pi$
$24$	0	0
$25$	−4.57871	−0.915742
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−2.52529	−0.468935	−0.234468	−	0.972124i	$-0.575335\pi$
$29$	−2.52529	−0.468935	−0.234468	+	0.972124i	$0.575335\pi$
$30$	0	0
$31$	4.14955	0.745281	0.372640	−	0.927976i	$-0.378453\pi$
$31$	4.14955	0.745281	0.372640	+	0.927976i	$0.378453\pi$
$32$	0	0
$33$	1.95286	0.339949
$34$	0	0
$35$	2.72380	0.460406
$36$	0	0
$37$	−4.94625	−0.813159	−0.406579	−	0.913615i	$-0.633279\pi$
$37$	−4.94625	−0.813159	−0.406579	+	0.913615i	$0.633279\pi$
$38$	0	0
$39$	3.07299	0.492072
$40$	0	0
$41$	2.05737	0.321308	0.160654	−	0.987011i	$-0.448640\pi$
$41$	2.05737	0.321308	0.160654	+	0.987011i	$0.448640\pi$
$42$	0	0
$43$	4.35750	0.664512	0.332256	−	0.943189i	$-0.392190\pi$
$43$	4.35750	0.664512	0.332256	+	0.943189i	$0.392190\pi$
$44$	0	0
$45$	0.649069	0.0967576
$46$	0	0
$47$	−11.2109	−1.63528	−0.817641	−	0.575728i	$-0.804719\pi$
$47$	−11.2109	−1.63528	−0.817641	+	0.575728i	$0.804719\pi$
$48$	0	0
$49$	10.6103	1.51576
$50$	0	0
$51$	6.85258	0.959554
$52$	0	0
$53$	−1.69789	−0.233224	−0.116612	−	0.993178i	$-0.537203\pi$
$53$	−1.69789	−0.233224	−0.116612	+	0.993178i	$0.537203\pi$
$54$	0	0
$55$	1.26754	0.170915
$56$	0	0
$57$	8.04597	1.06571
$58$	0	0
$59$	−7.31745	−0.952651	−0.476326	−	0.879269i	$-0.658032\pi$
$59$	−7.31745	−0.952651	−0.476326	+	0.879269i	$0.658032\pi$
$60$	0	0
$61$	−12.5360	−1.60507	−0.802534	−	0.596607i	$-0.796515\pi$
$61$	−12.5360	−1.60507	−0.802534	+	0.596607i	$0.796515\pi$
$62$	0	0
$63$	4.19646	0.528705
$64$	0	0
$65$	1.99458	0.247397
$66$	0	0
$67$	−6.97495	−0.852126	−0.426063	−	0.904694i	$-0.640100\pi$
$67$	−6.97495	−0.852126	−0.426063	+	0.904694i	$0.640100\pi$
$68$	0	0
$69$	−6.50672	−0.783317
$70$	0	0
$71$	2.84782	0.337974	0.168987	−	0.985618i	$-0.445950\pi$
$71$	2.84782	0.337974	0.168987	+	0.985618i	$0.445950\pi$
$72$	0	0
$73$	−7.78221	−0.910839	−0.455420	−	0.890277i	$-0.650511\pi$
$73$	−7.78221	−0.910839	−0.455420	+	0.890277i	$0.650511\pi$
$74$	0	0
$75$	−4.57871	−0.528704
$76$	0	0
$77$	8.19511	0.933920
$78$	0	0
$79$	−3.29528	−0.370747	−0.185374	−	0.982668i	$-0.559350\pi$
$79$	−3.29528	−0.370747	−0.185374	+	0.982668i	$0.559350\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	1.99647	0.219141	0.109571	−	0.993979i	$-0.465052\pi$
$83$	1.99647	0.219141	0.109571	+	0.993979i	$0.465052\pi$
$84$	0	0
$85$	4.44780	0.482432
$86$	0	0
$87$	−2.52529	−0.270740
$88$	0	0
$89$	−15.9262	−1.68817	−0.844086	−	0.536207i	$-0.819857\pi$
$89$	−15.9262	−1.68817	−0.844086	+	0.536207i	$0.819857\pi$
$90$	0	0
$91$	12.8957	1.35183
$92$	0	0
$93$	4.14955	0.430288
$94$	0	0
$95$	5.22239	0.535806
$96$	0	0
$97$	13.7538	1.39649	0.698243	−	0.715861i	$-0.253966\pi$
$97$	13.7538	1.39649	0.698243	+	0.715861i	$0.253966\pi$
$98$	0	0
$99$	1.95286	0.196270
$100$	0	0
$101$	−2.39371	−0.238183	−0.119092	−	0.992883i	$-0.537998\pi$
$101$	−2.39371	−0.238183	−0.119092	+	0.992883i	$0.537998\pi$
$102$	0	0
$103$	−0.185299	−0.0182581	−0.00912904	−	0.999958i	$-0.502906\pi$
$103$	−0.185299	−0.0182581	−0.00912904	+	0.999958i	$0.502906\pi$
$104$	0	0
$105$	2.72380	0.265815
$106$	0	0
$107$	−10.1993	−0.986002	−0.493001	−	0.870029i	$-0.664100\pi$
$107$	−10.1993	−0.986002	−0.493001	+	0.870029i	$0.664100\pi$
$108$	0	0
$109$	−11.3264	−1.08487	−0.542437	−	0.840097i	$-0.682498\pi$
$109$	−11.3264	−1.08487	−0.542437	+	0.840097i	$0.682498\pi$
$110$	0	0
$111$	−4.94625	−0.469477
$112$	0	0
$113$	1.85199	0.174221	0.0871104	−	0.996199i	$-0.472237\pi$
$113$	1.85199	0.174221	0.0871104	+	0.996199i	$0.472237\pi$
$114$	0	0
$115$	−4.22331	−0.393826
$116$	0	0
$117$	3.07299	0.284098
$118$	0	0
$119$	28.7566	2.63612
$120$	0	0
$121$	−7.18633	−0.653303
$122$	0	0
$123$	2.05737	0.185507
$124$	0	0
$125$	−6.21725	−0.556087
$126$	0	0
$127$	10.4107	0.923802	0.461901	−	0.886931i	$-0.347167\pi$
$127$	10.4107	0.923802	0.461901	+	0.886931i	$0.347167\pi$
$128$	0	0
$129$	4.35750	0.383656
$130$	0	0
$131$	−15.5093	−1.35506	−0.677528	−	0.735497i	$-0.736949\pi$
$131$	−15.5093	−1.35506	−0.677528	+	0.735497i	$0.736949\pi$
$132$	0	0
$133$	33.7646	2.92776
$134$	0	0
$135$	0.649069	0.0558630
$136$	0	0
$137$	16.4624	1.40648	0.703239	−	0.710954i	$-0.251736\pi$
$137$	16.4624	1.40648	0.703239	+	0.710954i	$0.251736\pi$
$138$	0	0
$139$	19.8190	1.68102	0.840512	−	0.541793i	$-0.182254\pi$
$139$	19.8190	1.68102	0.840512	+	0.541793i	$0.182254\pi$
$140$	0	0
$141$	−11.2109	−0.944131
$142$	0	0
$143$	6.00111	0.501838
$144$	0	0
$145$	−1.63909	−0.136119
$146$	0	0
$147$	10.6103	0.875124
$148$	0	0
$149$	9.90281	0.811269	0.405635	−	0.914035i	$-0.367051\pi$
$149$	9.90281	0.811269	0.405635	+	0.914035i	$0.367051\pi$
$150$	0	0
$151$	3.84338	0.312770	0.156385	−	0.987696i	$-0.450016\pi$
$151$	3.84338	0.312770	0.156385	+	0.987696i	$0.450016\pi$
$152$	0	0
$153$	6.85258	0.553999
$154$	0	0
$155$	2.69334	0.216335
$156$	0	0
$157$	6.26403	0.499924	0.249962	−	0.968256i	$-0.419582\pi$
$157$	6.26403	0.499924	0.249962	+	0.968256i	$0.419582\pi$
$158$	0	0
$159$	−1.69789	−0.134652
$160$	0	0
$161$	−27.3052	−2.15195
$162$	0	0
$163$	−15.4625	−1.21111	−0.605557	−	0.795802i	$-0.707050\pi$
$163$	−15.4625	−1.21111	−0.605557	+	0.795802i	$0.707050\pi$
$164$	0	0
$165$	1.26754	0.0986780
$166$	0	0
$167$	−1.00000	−0.0773823
$167$	−1.00000	−0.0773823
$168$	0	0
$169$	−3.55676	−0.273597
$170$	0	0
$171$	8.04597	0.615290
$172$	0	0
$173$	−4.68343	−0.356075	−0.178037	−	0.984024i	$-0.556975\pi$
$173$	−4.68343	−0.356075	−0.178037	+	0.984024i	$0.556975\pi$
$174$	0	0
$175$	−19.2144	−1.45247
$176$	0	0
$177$	−7.31745	−0.550014
$178$	0	0
$179$	−23.5932	−1.76344	−0.881720	−	0.471773i	$-0.843614\pi$
$179$	−23.5932	−1.76344	−0.881720	+	0.471773i	$0.843614\pi$
$180$	0	0
$181$	6.72533	0.499890	0.249945	−	0.968260i	$-0.419588\pi$
$181$	6.72533	0.499890	0.249945	+	0.968260i	$0.419588\pi$
$182$	0	0
$183$	−12.5360	−0.926686
$184$	0	0
$185$	−3.21046	−0.236038
$186$	0	0
$187$	13.3821	0.978599
$188$	0	0
$189$	4.19646	0.305248
$190$	0	0
$191$	10.7649	0.778922	0.389461	−	0.921043i	$-0.372661\pi$
$191$	10.7649	0.778922	0.389461	+	0.921043i	$0.372661\pi$
$192$	0	0
$193$	17.0021	1.22384	0.611918	−	0.790921i	$-0.290398\pi$
$193$	17.0021	1.22384	0.611918	+	0.790921i	$0.290398\pi$
$194$	0	0
$195$	1.99458	0.142835
$196$	0	0
$197$	10.7297	0.764461	0.382230	−	0.924067i	$-0.375156\pi$
$197$	10.7297	0.764461	0.382230	+	0.924067i	$0.375156\pi$
$198$	0	0
$199$	−12.9427	−0.917486	−0.458743	−	0.888569i	$-0.651700\pi$
$199$	−12.9427	−0.917486	−0.458743	+	0.888569i	$0.651700\pi$
$200$	0	0
$201$	−6.97495	−0.491975
$202$	0	0
$203$	−10.5973	−0.743785
$204$	0	0
$205$	1.33538	0.0932669
$206$	0	0
$207$	−6.50672	−0.452248
$208$	0	0
$209$	15.7127	1.08687
$210$	0	0
$211$	−0.727181	−0.0500612	−0.0250306	−	0.999687i	$-0.507968\pi$
$211$	−0.727181	−0.0500612	−0.0250306	+	0.999687i	$0.507968\pi$
$212$	0	0
$213$	2.84782	0.195129
$214$	0	0
$215$	2.82832	0.192890
$216$	0	0
$217$	17.4134	1.18210
$218$	0	0
$219$	−7.78221	−0.525873
$220$	0	0
$221$	21.0579	1.41651
$222$	0	0
$223$	−22.1726	−1.48478	−0.742392	−	0.669966i	$-0.766309\pi$
$223$	−22.1726	−1.48478	−0.742392	+	0.669966i	$0.766309\pi$
$224$	0	0
$225$	−4.57871	−0.305247
$226$	0	0
$227$	6.28074	0.416867	0.208434	−	0.978037i	$-0.433163\pi$
$227$	6.28074	0.416867	0.208434	+	0.978037i	$0.433163\pi$
$228$	0	0
$229$	21.3981	1.41402	0.707012	−	0.707201i	$-0.250043\pi$
$229$	21.3981	1.41402	0.707012	+	0.707201i	$0.250043\pi$
$230$	0	0
$231$	8.19511	0.539199
$232$	0	0
$233$	−22.8556	−1.49732	−0.748659	−	0.662956i	$-0.769302\pi$
$233$	−22.8556	−1.49732	−0.748659	+	0.662956i	$0.769302\pi$
$234$	0	0
$235$	−7.27667	−0.474678
$236$	0	0
$237$	−3.29528	−0.214051
$238$	0	0
$239$	−21.7956	−1.40984	−0.704920	−	0.709287i	$-0.749017\pi$
$239$	−21.7956	−1.40984	−0.704920	+	0.709287i	$0.749017\pi$
$240$	0	0
$241$	−3.28924	−0.211878	−0.105939	−	0.994373i	$-0.533785\pi$
$241$	−3.28924	−0.211878	−0.105939	+	0.994373i	$0.533785\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	6.88683	0.439983
$246$	0	0
$247$	24.7251	1.57322
$248$	0	0
$249$	1.99647	0.126521
$250$	0	0
$251$	12.5801	0.794050	0.397025	−	0.917808i	$-0.370043\pi$
$251$	12.5801	0.794050	0.397025	+	0.917808i	$0.370043\pi$
$252$	0	0
$253$	−12.7067	−0.798865
$254$	0	0
$255$	4.44780	0.278532
$256$	0	0
$257$	4.03469	0.251677	0.125839	−	0.992051i	$-0.459838\pi$
$257$	4.03469	0.251677	0.125839	+	0.992051i	$0.459838\pi$
$258$	0	0
$259$	−20.7568	−1.28976
$260$	0	0
$261$	−2.52529	−0.156312
$262$	0	0
$263$	−12.0597	−0.743631	−0.371815	−	0.928307i	$-0.621265\pi$
$263$	−12.0597	−0.743631	−0.371815	+	0.928307i	$0.621265\pi$
$264$	0	0
$265$	−1.10205	−0.0676984
$266$	0	0
$267$	−15.9262	−0.974667
$268$	0	0
$269$	−0.441761	−0.0269346	−0.0134673	−	0.999909i	$-0.504287\pi$
$269$	−0.441761	−0.0269346	−0.0134673	+	0.999909i	$0.504287\pi$
$270$	0	0
$271$	−21.7472	−1.32105	−0.660523	−	0.750806i	$-0.729666\pi$
$271$	−21.7472	−1.32105	−0.660523	+	0.750806i	$0.729666\pi$
$272$	0	0
$273$	12.8957	0.780482
$274$	0	0
$275$	−8.94158	−0.539198
$276$	0	0
$277$	−12.7062	−0.763442	−0.381721	−	0.924278i	$-0.624669\pi$
$277$	−12.7062	−0.763442	−0.381721	+	0.924278i	$0.624669\pi$
$278$	0	0
$279$	4.14955	0.248427
$280$	0	0
$281$	24.6981	1.47336	0.736682	−	0.676240i	$-0.236392\pi$
$281$	24.6981	1.47336	0.736682	+	0.676240i	$0.236392\pi$
$282$	0	0
$283$	5.77605	0.343350	0.171675	−	0.985154i	$-0.445082\pi$
$283$	5.77605	0.343350	0.171675	+	0.985154i	$0.445082\pi$
$284$	0	0
$285$	5.22239	0.309348
$286$	0	0
$287$	8.63370	0.509631
$288$	0	0
$289$	29.9579	1.76223
$290$	0	0
$291$	13.7538	0.806261
$292$	0	0
$293$	14.9087	0.870973	0.435486	−	0.900195i	$-0.356576\pi$
$293$	14.9087	0.870973	0.435486	+	0.900195i	$0.356576\pi$
$294$	0	0
$295$	−4.74954	−0.276529
$296$	0	0
$297$	1.95286	0.113316
$298$	0	0
$299$	−19.9951	−1.15634
$300$	0	0
$301$	18.2861	1.05399
$302$	0	0
$303$	−2.39371	−0.137515
$304$	0	0
$305$	−8.13672	−0.465907
$306$	0	0
$307$	2.60592	0.148728	0.0743638	−	0.997231i	$-0.476307\pi$
$307$	2.60592	0.148728	0.0743638	+	0.997231i	$0.476307\pi$
$308$	0	0
$309$	−0.185299	−0.0105413
$310$	0	0
$311$	−5.88680	−0.333810	−0.166905	−	0.985973i	$-0.553377\pi$
$311$	−5.88680	−0.333810	−0.166905	+	0.985973i	$0.553377\pi$
$312$	0	0
$313$	29.5589	1.67077	0.835384	−	0.549667i	$-0.185245\pi$
$313$	29.5589	1.67077	0.835384	+	0.549667i	$0.185245\pi$
$314$	0	0
$315$	2.72380	0.153469
$316$	0	0
$317$	27.3897	1.53836	0.769180	−	0.639032i	$-0.220665\pi$
$317$	27.3897	1.53836	0.769180	+	0.639032i	$0.220665\pi$
$318$	0	0
$319$	−4.93155	−0.276114
$320$	0	0
$321$	−10.1993	−0.569268
$322$	0	0
$323$	55.1357	3.06783
$324$	0	0
$325$	−14.0703	−0.780480
$326$	0	0
$327$	−11.3264	−0.626352
$328$	0	0
$329$	−47.0463	−2.59375
$330$	0	0
$331$	25.0680	1.37786	0.688931	−	0.724827i	$-0.258080\pi$
$331$	25.0680	1.37786	0.688931	+	0.724827i	$0.258080\pi$
$332$	0	0
$333$	−4.94625	−0.271053
$334$	0	0
$335$	−4.52723	−0.247349
$336$	0	0
$337$	−20.0351	−1.09138	−0.545691	−	0.837987i	$-0.683733\pi$
$337$	−20.0351	−1.09138	−0.545691	+	0.837987i	$0.683733\pi$
$338$	0	0
$339$	1.85199	0.100586
$340$	0	0
$341$	8.10349	0.438829
$342$	0	0
$343$	15.1505	0.818052
$344$	0	0
$345$	−4.22331	−0.227376
$346$	0	0
$347$	27.6121	1.48229	0.741147	−	0.671343i	$-0.234282\pi$
$347$	27.6121	1.48229	0.741147	+	0.671343i	$0.234282\pi$
$348$	0	0
$349$	23.2868	1.24652	0.623258	−	0.782016i	$-0.285809\pi$
$349$	23.2868	1.24652	0.623258	+	0.782016i	$0.285809\pi$
$350$	0	0
$351$	3.07299	0.164024
$352$	0	0
$353$	−27.0763	−1.44112	−0.720562	−	0.693390i	$-0.756116\pi$
$353$	−27.0763	−1.44112	−0.720562	+	0.693390i	$0.756116\pi$
$354$	0	0
$355$	1.84843	0.0981045
$356$	0	0
$357$	28.7566	1.52196
$358$	0	0
$359$	−10.5945	−0.559156	−0.279578	−	0.960123i	$-0.590195\pi$
$359$	−10.5945	−0.559156	−0.279578	+	0.960123i	$0.590195\pi$
$360$	0	0
$361$	45.7376	2.40724
$362$	0	0
$363$	−7.18633	−0.377185
$364$	0	0
$365$	−5.05120	−0.264392
$366$	0	0
$367$	8.01577	0.418420	0.209210	−	0.977871i	$-0.432911\pi$
$367$	8.01577	0.418420	0.209210	+	0.977871i	$0.432911\pi$
$368$	0	0
$369$	2.05737	0.107103
$370$	0	0
$371$	−7.12515	−0.369919
$372$	0	0
$373$	−0.337791	−0.0174902	−0.00874508	−	0.999962i	$-0.502784\pi$
$373$	−0.337791	−0.0174902	−0.00874508	+	0.999962i	$0.502784\pi$
$374$	0	0
$375$	−6.21725	−0.321057
$376$	0	0
$377$	−7.76019	−0.399670
$378$	0	0
$379$	31.4233	1.61411	0.807054	−	0.590477i	$-0.201060\pi$
$379$	31.4233	1.61411	0.807054	+	0.590477i	$0.201060\pi$
$380$	0	0
$381$	10.4107	0.533357
$382$	0	0
$383$	−23.6461	−1.20826	−0.604129	−	0.796887i	$-0.706479\pi$
$383$	−23.6461	−1.20826	−0.604129	+	0.796887i	$0.706479\pi$
$384$	0	0
$385$	5.31920	0.271091
$386$	0	0
$387$	4.35750	0.221504
$388$	0	0
$389$	9.30638	0.471852	0.235926	−	0.971771i	$-0.424188\pi$
$389$	9.30638	0.471852	0.235926	+	0.971771i	$0.424188\pi$
$390$	0	0
$391$	−44.5879	−2.25490
$392$	0	0
$393$	−15.5093	−0.782342
$394$	0	0
$395$	−2.13886	−0.107618
$396$	0	0
$397$	5.29743	0.265870	0.132935	−	0.991125i	$-0.457560\pi$
$397$	5.29743	0.265870	0.132935	+	0.991125i	$0.457560\pi$
$398$	0	0
$399$	33.7646	1.69034
$400$	0	0
$401$	24.1903	1.20801	0.604004	−	0.796982i	$-0.293571\pi$
$401$	24.1903	1.20801	0.604004	+	0.796982i	$0.293571\pi$
$402$	0	0
$403$	12.7515	0.635198
$404$	0	0
$405$	0.649069	0.0322525
$406$	0	0
$407$	−9.65934	−0.478796
$408$	0	0
$409$	−26.6321	−1.31687	−0.658436	−	0.752637i	$-0.728782\pi$
$409$	−26.6321	−1.31687	−0.658436	+	0.752637i	$0.728782\pi$
$410$	0	0
$411$	16.4624	0.812030
$412$	0	0
$413$	−30.7074	−1.51101
$414$	0	0
$415$	1.29585	0.0636107
$416$	0	0
$417$	19.8190	0.970540
$418$	0	0
$419$	−16.8970	−0.825473	−0.412737	−	0.910850i	$-0.635427\pi$
$419$	−16.8970	−0.825473	−0.412737	+	0.910850i	$0.635427\pi$
$420$	0	0
$421$	13.6534	0.665428	0.332714	−	0.943028i	$-0.392036\pi$
$421$	13.6534	0.665428	0.332714	+	0.943028i	$0.392036\pi$
$422$	0	0
$423$	−11.2109	−0.545094
$424$	0	0
$425$	−31.3760	−1.52196
$426$	0	0
$427$	−52.6068	−2.54582
$428$	0	0
$429$	6.00111	0.289737
$430$	0	0
$431$	3.15760	0.152096	0.0760481	−	0.997104i	$-0.475770\pi$
$431$	3.15760	0.152096	0.0760481	+	0.997104i	$0.475770\pi$
$432$	0	0
$433$	−3.19183	−0.153390	−0.0766948	−	0.997055i	$-0.524437\pi$
$433$	−3.19183	−0.153390	−0.0766948	+	0.997055i	$0.524437\pi$
$434$	0	0
$435$	−1.63909	−0.0785884
$436$	0	0
$437$	−52.3529	−2.50438
$438$	0	0
$439$	24.3383	1.16160	0.580801	−	0.814046i	$-0.302739\pi$
$439$	24.3383	1.16160	0.580801	+	0.814046i	$0.302739\pi$
$440$	0	0
$441$	10.6103	0.505253
$442$	0	0
$443$	−2.32690	−0.110554	−0.0552772	−	0.998471i	$-0.517604\pi$
$443$	−2.32690	−0.110554	−0.0552772	+	0.998471i	$0.517604\pi$
$444$	0	0
$445$	−10.3372	−0.490030
$446$	0	0
$447$	9.90281	0.468387
$448$	0	0
$449$	18.9408	0.893869	0.446935	−	0.894567i	$-0.352516\pi$
$449$	18.9408	0.893869	0.446935	+	0.894567i	$0.352516\pi$
$450$	0	0
$451$	4.01777	0.189189
$452$	0	0
$453$	3.84338	0.180578
$454$	0	0
$455$	8.37019	0.392400
$456$	0	0
$457$	−7.00613	−0.327733	−0.163867	−	0.986483i	$-0.552397\pi$
$457$	−7.00613	−0.327733	−0.163867	+	0.986483i	$0.552397\pi$
$458$	0	0
$459$	6.85258	0.319851
$460$	0	0
$461$	29.9382	1.39436	0.697182	−	0.716895i	$-0.254437\pi$
$461$	29.9382	1.39436	0.697182	+	0.716895i	$0.254437\pi$
$462$	0	0
$463$	−28.6425	−1.33113	−0.665564	−	0.746341i	$-0.731809\pi$
$463$	−28.6425	−1.33113	−0.665564	+	0.746341i	$0.731809\pi$
$464$	0	0
$465$	2.69334	0.124901
$466$	0	0
$467$	−10.2858	−0.475970	−0.237985	−	0.971269i	$-0.576487\pi$
$467$	−10.2858	−0.475970	−0.237985	+	0.971269i	$0.576487\pi$
$468$	0	0
$469$	−29.2701	−1.35157
$470$	0	0
$471$	6.26403	0.288631
$472$	0	0
$473$	8.50959	0.391271
$474$	0	0
$475$	−36.8401	−1.69034
$476$	0	0
$477$	−1.69789	−0.0777412
$478$	0	0
$479$	−35.9258	−1.64149	−0.820745	−	0.571295i	$-0.806441\pi$
$479$	−35.9258	−1.64149	−0.820745	+	0.571295i	$0.806441\pi$
$480$	0	0
$481$	−15.1998	−0.693050
$482$	0	0
$483$	−27.3052	−1.24243
$484$	0	0
$485$	8.92716	0.405362
$486$	0	0
$487$	−24.5681	−1.11329	−0.556644	−	0.830751i	$-0.687911\pi$
$487$	−24.5681	−1.11329	−0.556644	+	0.830751i	$0.687911\pi$
$488$	0	0
$489$	−15.4625	−0.699237
$490$	0	0
$491$	−14.0673	−0.634848	−0.317424	−	0.948284i	$-0.602818\pi$
$491$	−14.0673	−0.634848	−0.317424	+	0.948284i	$0.602818\pi$
$492$	0	0
$493$	−17.3048	−0.779368
$494$	0	0
$495$	1.26754	0.0569718
$496$	0	0
$497$	11.9508	0.536065
$498$	0	0
$499$	28.3654	1.26981	0.634905	−	0.772590i	$-0.281039\pi$
$499$	28.3654	1.26981	0.634905	+	0.772590i	$0.281039\pi$
$500$	0	0
$501$	−1.00000	−0.0446767
$502$	0	0
$503$	−3.29871	−0.147082	−0.0735412	−	0.997292i	$-0.523430\pi$
$503$	−3.29871	−0.147082	−0.0735412	+	0.997292i	$0.523430\pi$
$504$	0	0
$505$	−1.55369	−0.0691381
$506$	0	0
$507$	−3.55676	−0.157961
$508$	0	0
$509$	−13.2600	−0.587738	−0.293869	−	0.955846i	$-0.594943\pi$
$509$	−13.2600	−0.587738	−0.293869	+	0.955846i	$0.594943\pi$
$510$	0	0
$511$	−32.6578	−1.44470
$512$	0	0
$513$	8.04597	0.355238
$514$	0	0
$515$	−0.120272	−0.00529982
$516$	0	0
$517$	−21.8934	−0.962870
$518$	0	0
$519$	−4.68343	−0.205580
$520$	0	0
$521$	23.0586	1.01022	0.505108	−	0.863056i	$-0.331453\pi$
$521$	23.0586	1.01022	0.505108	+	0.863056i	$0.331453\pi$
$522$	0	0
$523$	−5.08722	−0.222449	−0.111224	−	0.993795i	$-0.535477\pi$
$523$	−5.08722	−0.222449	−0.111224	+	0.993795i	$0.535477\pi$
$524$	0	0
$525$	−19.2144	−0.838585
$526$	0	0
$527$	28.4351	1.23865
$528$	0	0
$529$	19.3374	0.840758
$530$	0	0
$531$	−7.31745	−0.317550
$532$	0	0
$533$	6.32228	0.273848
$534$	0	0
$535$	−6.62004	−0.286209
$536$	0	0
$537$	−23.5932	−1.01812
$538$	0	0
$539$	20.7205	0.892493
$540$	0	0
$541$	6.83762	0.293972	0.146986	−	0.989139i	$-0.453043\pi$
$541$	6.83762	0.293972	0.146986	+	0.989139i	$0.453043\pi$
$542$	0	0
$543$	6.72533	0.288611
$544$	0	0
$545$	−7.35163	−0.314909
$546$	0	0
$547$	−2.89764	−0.123894	−0.0619470	−	0.998079i	$-0.519731\pi$
$547$	−2.89764	−0.123894	−0.0619470	+	0.998079i	$0.519731\pi$
$548$	0	0
$549$	−12.5360	−0.535022
$550$	0	0
$551$	−20.3184	−0.865594
$552$	0	0
$553$	−13.8285	−0.588048
$554$	0	0
$555$	−3.21046	−0.136276
$556$	0	0
$557$	1.11407	0.0472045	0.0236023	−	0.999721i	$-0.492486\pi$
$557$	1.11407	0.0472045	0.0236023	+	0.999721i	$0.492486\pi$
$558$	0	0
$559$	13.3905	0.566359
$560$	0	0
$561$	13.3821	0.564995
$562$	0	0
$563$	−26.2894	−1.10797	−0.553984	−	0.832527i	$-0.686893\pi$
$563$	−26.2894	−1.10797	−0.553984	+	0.832527i	$0.686893\pi$
$564$	0	0
$565$	1.20207	0.0505715
$566$	0	0
$567$	4.19646	0.176235
$568$	0	0
$569$	29.5405	1.23840	0.619200	−	0.785233i	$-0.287457\pi$
$569$	29.5405	1.23840	0.619200	+	0.785233i	$0.287457\pi$
$570$	0	0
$571$	−17.4949	−0.732139	−0.366069	−	0.930588i	$-0.619297\pi$
$571$	−17.4949	−0.732139	−0.366069	+	0.930588i	$0.619297\pi$
$572$	0	0
$573$	10.7649	0.449711
$574$	0	0
$575$	29.7924	1.24243
$576$	0	0
$577$	−24.5885	−1.02363	−0.511816	−	0.859095i	$-0.671027\pi$
$577$	−24.5885	−1.02363	−0.511816	+	0.859095i	$0.671027\pi$
$578$	0	0
$579$	17.0021	0.706582
$580$	0	0
$581$	8.37812	0.347583
$582$	0	0
$583$	−3.31575	−0.137324
$584$	0	0
$585$	1.99458	0.0824658
$586$	0	0
$587$	7.38624	0.304863	0.152431	−	0.988314i	$-0.451290\pi$
$587$	7.38624	0.304863	0.152431	+	0.988314i	$0.451290\pi$
$588$	0	0
$589$	33.3871	1.37569
$590$	0	0
$591$	10.7297	0.441361
$592$	0	0
$593$	−3.49026	−0.143328	−0.0716640	−	0.997429i	$-0.522831\pi$
$593$	−3.49026	−0.143328	−0.0716640	+	0.997429i	$0.522831\pi$
$594$	0	0
$595$	18.6650	0.765192
$596$	0	0
$597$	−12.9427	−0.529711
$598$	0	0
$599$	19.5847	0.800211	0.400105	−	0.916469i	$-0.368974\pi$
$599$	19.5847	0.800211	0.400105	+	0.916469i	$0.368974\pi$
$600$	0	0
$601$	9.07252	0.370076	0.185038	−	0.982731i	$-0.440759\pi$
$601$	9.07252	0.370076	0.185038	+	0.982731i	$0.440759\pi$
$602$	0	0
$603$	−6.97495	−0.284042
$604$	0	0
$605$	−4.66443	−0.189636
$606$	0	0
$607$	42.6413	1.73076	0.865379	−	0.501119i	$-0.167078\pi$
$607$	42.6413	1.73076	0.865379	+	0.501119i	$0.167078\pi$
$608$	0	0
$609$	−10.5973	−0.429424
$610$	0	0
$611$	−34.4510	−1.39374
$612$	0	0
$613$	30.2567	1.22206	0.611028	−	0.791609i	$-0.290756\pi$
$613$	30.2567	1.22206	0.611028	+	0.791609i	$0.290756\pi$
$614$	0	0
$615$	1.33538	0.0538477
$616$	0	0
$617$	22.7488	0.915831	0.457915	−	0.888996i	$-0.348596\pi$
$617$	22.7488	0.915831	0.457915	+	0.888996i	$0.348596\pi$
$618$	0	0
$619$	−3.82335	−0.153674	−0.0768368	−	0.997044i	$-0.524482\pi$
$619$	−3.82335	−0.153674	−0.0768368	+	0.997044i	$0.524482\pi$
$620$	0	0
$621$	−6.50672	−0.261106
$622$	0	0
$623$	−66.8337	−2.67764
$624$	0	0
$625$	18.8581	0.754325
$626$	0	0
$627$	15.7127	0.627503
$628$	0	0
$629$	−33.8946	−1.35147
$630$	0	0
$631$	−10.5670	−0.420666	−0.210333	−	0.977630i	$-0.567455\pi$
$631$	−10.5670	−0.420666	−0.210333	+	0.977630i	$0.567455\pi$
$632$	0	0
$633$	−0.727181	−0.0289028
$634$	0	0
$635$	6.75728	0.268155
$636$	0	0
$637$	32.6053	1.29187
$638$	0	0
$639$	2.84782	0.112658
$640$	0	0
$641$	−24.3166	−0.960447	−0.480223	−	0.877146i	$-0.659444\pi$
$641$	−24.3166	−0.960447	−0.480223	+	0.877146i	$0.659444\pi$
$642$	0	0
$643$	12.6719	0.499733	0.249866	−	0.968280i	$-0.419613\pi$
$643$	12.6719	0.499733	0.249866	+	0.968280i	$0.419613\pi$
$644$	0	0
$645$	2.82832	0.111365
$646$	0	0
$647$	11.2943	0.444024	0.222012	−	0.975044i	$-0.428738\pi$
$647$	11.2943	0.444024	0.222012	+	0.975044i	$0.428738\pi$
$648$	0	0
$649$	−14.2900	−0.560930
$650$	0	0
$651$	17.4134	0.682486
$652$	0	0
$653$	−40.4706	−1.58374	−0.791869	−	0.610691i	$-0.790892\pi$
$653$	−40.4706	−1.58374	−0.791869	+	0.610691i	$0.790892\pi$
$654$	0	0
$655$	−10.0666	−0.393336
$656$	0	0
$657$	−7.78221	−0.303613
$658$	0	0
$659$	−13.5779	−0.528921	−0.264460	−	0.964397i	$-0.585194\pi$
$659$	−13.5779	−0.528921	−0.264460	+	0.964397i	$0.585194\pi$
$660$	0	0
$661$	−38.4128	−1.49408	−0.747041	−	0.664777i	$-0.768526\pi$
$661$	−38.4128	−1.49408	−0.747041	+	0.664777i	$0.768526\pi$
$662$	0	0
$663$	21.0579	0.817821
$664$	0	0
$665$	21.9156	0.849849
$666$	0	0
$667$	16.4314	0.636226
$668$	0	0
$669$	−22.1726	−0.857240
$670$	0	0
$671$	−24.4810	−0.945079
$672$	0	0
$673$	33.7593	1.30132	0.650662	−	0.759367i	$-0.274491\pi$
$673$	33.7593	1.30132	0.650662	+	0.759367i	$0.274491\pi$
$674$	0	0
$675$	−4.57871	−0.176235
$676$	0	0
$677$	−0.268807	−0.0103311	−0.00516555	−	0.999987i	$-0.501644\pi$
$677$	−0.268807	−0.0103311	−0.00516555	+	0.999987i	$0.501644\pi$
$678$	0	0
$679$	57.7173	2.21499
$680$	0	0
$681$	6.28074	0.240678
$682$	0	0
$683$	−42.6332	−1.63131	−0.815656	−	0.578537i	$-0.803624\pi$
$683$	−42.6332	−1.63131	−0.815656	+	0.578537i	$0.803624\pi$
$684$	0	0
$685$	10.6852	0.408262
$686$	0	0
$687$	21.3981	0.816387
$688$	0	0
$689$	−5.21760	−0.198775
$690$	0	0
$691$	14.9041	0.566979	0.283490	−	0.958975i	$-0.408508\pi$
$691$	14.9041	0.566979	0.283490	+	0.958975i	$0.408508\pi$
$692$	0	0
$693$	8.19511	0.311307
$694$	0	0
$695$	12.8639	0.487955
$696$	0	0
$697$	14.0983	0.534012
$698$	0	0
$699$	−22.8556	−0.864476
$700$	0	0
$701$	−46.9108	−1.77180	−0.885898	−	0.463880i	$-0.846457\pi$
$701$	−46.9108	−1.77180	−0.885898	+	0.463880i	$0.846457\pi$
$702$	0	0
$703$	−39.7974	−1.50099
$704$	0	0
$705$	−7.27667	−0.274055
$706$	0	0
$707$	−10.0451	−0.377786
$708$	0	0
$709$	38.2929	1.43812	0.719059	−	0.694949i	$-0.244573\pi$
$709$	38.2929	1.43812	0.719059	+	0.694949i	$0.244573\pi$
$710$	0	0
$711$	−3.29528	−0.123582
$712$	0	0
$713$	−27.0000	−1.01116
$714$	0	0
$715$	3.89514	0.145670
$716$	0	0
$717$	−21.7956	−0.813971
$718$	0	0
$719$	15.4361	0.575670	0.287835	−	0.957680i	$-0.407064\pi$
$719$	15.4361	0.575670	0.287835	+	0.957680i	$0.407064\pi$
$720$	0	0
$721$	−0.777601	−0.0289594
$722$	0	0
$723$	−3.28924	−0.122328
$724$	0	0
$725$	11.5626	0.429424
$726$	0	0
$727$	28.7752	1.06721	0.533606	−	0.845733i	$-0.320837\pi$
$727$	28.7752	1.06721	0.533606	+	0.845733i	$0.320837\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	29.8601	1.10442
$732$	0	0
$733$	16.2197	0.599088	0.299544	−	0.954082i	$-0.403165\pi$
$733$	16.2197	0.599088	0.299544	+	0.954082i	$0.403165\pi$
$734$	0	0
$735$	6.88683	0.254024
$736$	0	0
$737$	−13.6211	−0.501740
$738$	0	0
$739$	40.9725	1.50720	0.753599	−	0.657334i	$-0.228316\pi$
$739$	40.9725	1.50720	0.753599	+	0.657334i	$0.228316\pi$
$740$	0	0
$741$	24.7251	0.908301
$742$	0	0
$743$	−9.94574	−0.364874	−0.182437	−	0.983218i	$-0.558399\pi$
$743$	−9.94574	−0.364874	−0.182437	+	0.983218i	$0.558399\pi$
$744$	0	0
$745$	6.42761	0.235489
$746$	0	0
$747$	1.99647	0.0730471
$748$	0	0
$749$	−42.8009	−1.56391
$750$	0	0
$751$	−36.5092	−1.33224	−0.666121	−	0.745844i	$-0.732046\pi$
$751$	−36.5092	−1.33224	−0.666121	+	0.745844i	$0.732046\pi$
$752$	0	0
$753$	12.5801	0.458445
$754$	0	0
$755$	2.49462	0.0907885
$756$	0	0
$757$	45.9732	1.67092	0.835462	−	0.549549i	$-0.185200\pi$
$757$	45.9732	1.67092	0.835462	+	0.549549i	$0.185200\pi$
$758$	0	0
$759$	−12.7067	−0.461225
$760$	0	0
$761$	−9.48336	−0.343771	−0.171886	−	0.985117i	$-0.554986\pi$
$761$	−9.48336	−0.343771	−0.171886	+	0.985117i	$0.554986\pi$
$762$	0	0
$763$	−47.5309	−1.72073
$764$	0	0
$765$	4.44780	0.160811
$766$	0	0
$767$	−22.4864	−0.811938
$768$	0	0
$769$	11.3149	0.408027	0.204014	−	0.978968i	$-0.434601\pi$
$769$	11.3149	0.408027	0.204014	+	0.978968i	$0.434601\pi$
$770$	0	0
$771$	4.03469	0.145306
$772$	0	0
$773$	22.0938	0.794659	0.397329	−	0.917676i	$-0.369937\pi$
$773$	22.0938	0.794659	0.397329	+	0.917676i	$0.369937\pi$
$774$	0	0
$775$	−18.9996	−0.682485
$776$	0	0
$777$	−20.7568	−0.744645
$778$	0	0
$779$	16.5536	0.593093
$780$	0	0
$781$	5.56139	0.199002
$782$	0	0
$783$	−2.52529	−0.0902466
$784$	0	0
$785$	4.06579	0.145114
$786$	0	0
$787$	−27.3069	−0.973385	−0.486692	−	0.873573i	$-0.661797\pi$
$787$	−27.3069	−0.973385	−0.486692	+	0.873573i	$0.661797\pi$
$788$	0	0
$789$	−12.0597	−0.429335
$790$	0	0
$791$	7.77182	0.276334
$792$	0	0
$793$	−38.5229	−1.36799
$794$	0	0
$795$	−1.10205	−0.0390857
$796$	0	0
$797$	−28.3415	−1.00391	−0.501953	−	0.864895i	$-0.667385\pi$
$797$	−28.3415	−1.00391	−0.501953	+	0.864895i	$0.667385\pi$
$798$	0	0
$799$	−76.8239	−2.71783
$800$	0	0
$801$	−15.9262	−0.562724
$802$	0	0
$803$	−15.1976	−0.536311
$804$	0	0
$805$	−17.7230	−0.624653
$806$	0	0
$807$	−0.441761	−0.0155507
$808$	0	0
$809$	33.8936	1.19163	0.595817	−	0.803120i	$-0.296828\pi$
$809$	33.8936	1.19163	0.595817	+	0.803120i	$0.296828\pi$
$810$	0	0
$811$	22.7048	0.797275	0.398637	−	0.917109i	$-0.369483\pi$
$811$	22.7048	0.797275	0.398637	+	0.917109i	$0.369483\pi$
$812$	0	0
$813$	−21.7472	−0.762707
$814$	0	0
$815$	−10.0362	−0.351553
$816$	0	0
$817$	35.0603	1.22660
$818$	0	0
$819$	12.8957	0.450611
$820$	0	0
$821$	0.663314	0.0231498	0.0115749	−	0.999933i	$-0.496316\pi$
$821$	0.663314	0.0231498	0.0115749	+	0.999933i	$0.496316\pi$
$822$	0	0
$823$	−41.4039	−1.44325	−0.721624	−	0.692285i	$-0.756604\pi$
$823$	−41.4039	−1.44325	−0.721624	+	0.692285i	$0.756604\pi$
$824$	0	0
$825$	−8.94158	−0.311306
$826$	0	0
$827$	17.6369	0.613296	0.306648	−	0.951823i	$-0.400793\pi$
$827$	17.6369	0.613296	0.306648	+	0.951823i	$0.400793\pi$
$828$	0	0
$829$	11.8064	0.410053	0.205027	−	0.978756i	$-0.434272\pi$
$829$	11.8064	0.410053	0.205027	+	0.978756i	$0.434272\pi$
$830$	0	0
$831$	−12.7062	−0.440774
$832$	0	0
$833$	72.7080	2.51918
$834$	0	0
$835$	−0.649069	−0.0224620
$836$	0	0
$837$	4.14955	0.143429
$838$	0	0
$839$	−42.9529	−1.48290	−0.741449	−	0.671010i	$-0.765861\pi$
$839$	−42.9529	−1.48290	−0.741449	+	0.671010i	$0.765861\pi$
$840$	0	0
$841$	−22.6229	−0.780100
$842$	0	0
$843$	24.6981	0.850647
$844$	0	0
$845$	−2.30858	−0.0794177
$846$	0	0
$847$	−30.1572	−1.03621
$848$	0	0
$849$	5.77605	0.198233
$850$	0	0
$851$	32.1839	1.10325
$852$	0	0
$853$	31.7110	1.08576	0.542882	−	0.839809i	$-0.317333\pi$
$853$	31.7110	1.08576	0.542882	+	0.839809i	$0.317333\pi$
$854$	0	0
$855$	5.22239	0.178602
$856$	0	0
$857$	−43.9632	−1.50175	−0.750877	−	0.660442i	$-0.770369\pi$
$857$	−43.9632	−1.50175	−0.750877	+	0.660442i	$0.770369\pi$
$858$	0	0
$859$	−28.7001	−0.979234	−0.489617	−	0.871938i	$-0.662863\pi$
$859$	−28.7001	−0.979234	−0.489617	+	0.871938i	$0.662863\pi$
$860$	0	0
$861$	8.63370	0.294236
$862$	0	0
$863$	−42.6438	−1.45161	−0.725806	−	0.687899i	$-0.758533\pi$
$863$	−42.6438	−1.45161	−0.725806	+	0.687899i	$0.758533\pi$
$864$	0	0
$865$	−3.03987	−0.103359
$866$	0	0
$867$	29.9579	1.01742
$868$	0	0
$869$	−6.43521	−0.218300
$870$	0	0
$871$	−21.4339	−0.726261
$872$	0	0
$873$	13.7538	0.465495
$874$	0	0
$875$	−26.0905	−0.882018
$876$	0	0
$877$	−15.1091	−0.510199	−0.255099	−	0.966915i	$-0.582108\pi$
$877$	−15.1091	−0.510199	−0.255099	+	0.966915i	$0.582108\pi$
$878$	0	0
$879$	14.9087	0.502856
$880$	0	0
$881$	−14.5387	−0.489820	−0.244910	−	0.969546i	$-0.578758\pi$
$881$	−14.5387	−0.489820	−0.244910	+	0.969546i	$0.578758\pi$
$882$	0	0
$883$	6.92350	0.232994	0.116497	−	0.993191i	$-0.462833\pi$
$883$	6.92350	0.232994	0.116497	+	0.993191i	$0.462833\pi$
$884$	0	0
$885$	−4.74954	−0.159654
$886$	0	0
$887$	−45.7047	−1.53461	−0.767306	−	0.641281i	$-0.778403\pi$
$887$	−45.7047	−1.53461	−0.767306	+	0.641281i	$0.778403\pi$
$888$	0	0
$889$	43.6882	1.46526
$890$	0	0
$891$	1.95286	0.0654233
$892$	0	0
$893$	−90.2028	−3.01852
$894$	0	0
$895$	−15.3136	−0.511878
$896$	0	0
$897$	−19.9951	−0.667616
$898$	0	0
$899$	−10.4788	−0.349488
$900$	0	0
$901$	−11.6350	−0.387616
$902$	0	0
$903$	18.2861	0.608523
$904$	0	0
$905$	4.36520	0.145104
$906$	0	0
$907$	28.7584	0.954907	0.477453	−	0.878657i	$-0.341560\pi$
$907$	28.7584	0.954907	0.477453	+	0.878657i	$0.341560\pi$
$908$	0	0
$909$	−2.39371	−0.0793945
$910$	0	0
$911$	3.33280	0.110421	0.0552103	−	0.998475i	$-0.482417\pi$
$911$	3.33280	0.110421	0.0552103	+	0.998475i	$0.482417\pi$
$912$	0	0
$913$	3.89883	0.129032
$914$	0	0
$915$	−8.13672	−0.268992
$916$	0	0
$917$	−65.0843	−2.14927
$918$	0	0
$919$	45.6696	1.50650	0.753250	−	0.657735i	$-0.228485\pi$
$919$	45.6696	1.50650	0.753250	+	0.657735i	$0.228485\pi$
$920$	0	0
$921$	2.60592	0.0858679
$922$	0	0
$923$	8.75130	0.288052
$924$	0	0
$925$	22.6474	0.744643
$926$	0	0
$927$	−0.185299	−0.00608602
$928$	0	0
$929$	41.0210	1.34586	0.672928	−	0.739708i	$-0.265037\pi$
$929$	41.0210	1.34586	0.672928	+	0.739708i	$0.265037\pi$
$930$	0	0
$931$	85.3702	2.79789
$932$	0	0
$933$	−5.88680	−0.192725
$934$	0	0
$935$	8.68594	0.284061
$936$	0	0
$937$	0.449676	0.0146903	0.00734514	−	0.999973i	$-0.497662\pi$
$937$	0.449676	0.0146903	0.00734514	+	0.999973i	$0.497662\pi$
$938$	0	0
$939$	29.5589	0.964618
$940$	0	0
$941$	−24.2530	−0.790626	−0.395313	−	0.918546i	$-0.629364\pi$
$941$	−24.2530	−0.790626	−0.395313	+	0.918546i	$0.629364\pi$
$942$	0	0
$943$	−13.3868	−0.435933
$944$	0	0
$945$	2.72380	0.0886051
$946$	0	0
$947$	42.4970	1.38097	0.690484	−	0.723348i	$-0.257398\pi$
$947$	42.4970	1.38097	0.690484	+	0.723348i	$0.257398\pi$
$948$	0	0
$949$	−23.9146	−0.776302
$950$	0	0
$951$	27.3897	0.888173
$952$	0	0
$953$	17.5025	0.566961	0.283480	−	0.958978i	$-0.408511\pi$
$953$	17.5025	0.566961	0.283480	+	0.958978i	$0.408511\pi$
$954$	0	0
$955$	6.98718	0.226100
$956$	0	0
$957$	−4.93155	−0.159414
$958$	0	0
$959$	69.0839	2.23083
$960$	0	0
$961$	−13.7813	−0.444557
$962$	0	0
$963$	−10.1993	−0.328667
$964$	0	0
$965$	11.0355	0.355246
$966$	0	0
$967$	−16.8865	−0.543032	−0.271516	−	0.962434i	$-0.587525\pi$
$967$	−16.8865	−0.543032	−0.271516	+	0.962434i	$0.587525\pi$
$968$	0	0
$969$	55.1357	1.77121
$970$	0	0
$971$	−54.6352	−1.75333	−0.876664	−	0.481103i	$-0.840236\pi$
$971$	−54.6352	−1.75333	−0.876664	+	0.481103i	$0.840236\pi$
$972$	0	0
$973$	83.1696	2.66630
$974$	0	0
$975$	−14.0703	−0.450610
$976$	0	0
$977$	−18.6725	−0.597386	−0.298693	−	0.954349i	$-0.596551\pi$
$977$	−18.6725	−0.597386	−0.298693	+	0.954349i	$0.596551\pi$
$978$	0	0
$979$	−31.1016	−0.994013
$980$	0	0
$981$	−11.3264	−0.361625
$982$	0	0
$983$	46.7507	1.49112	0.745558	−	0.666441i	$-0.232183\pi$
$983$	46.7507	1.49112	0.745558	+	0.666441i	$0.232183\pi$
$984$	0	0
$985$	6.96433	0.221902
$986$	0	0
$987$	−47.0463	−1.49750
$988$	0	0
$989$	−28.3530	−0.901574
$990$	0	0
$991$	−46.7105	−1.48381	−0.741904	−	0.670506i	$-0.766077\pi$
$991$	−46.7105	−1.48381	−0.741904	+	0.670506i	$0.766077\pi$
$992$	0	0
$993$	25.0680	0.795509
$994$	0	0
$995$	−8.40074	−0.266321
$996$	0	0
$997$	−59.1332	−1.87277	−0.936383	−	0.350979i	$-0.885849\pi$
$997$	−59.1332	−1.87277	−0.936383	+	0.350979i	$0.885849\pi$
$998$	0	0
$999$	−4.94625	−0.156492

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4008.2.a.m.1.7	✓	13
4.3	odd	2		8016.2.a.bg.1.7		13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4008.2.a.m.1.7	✓	13	1.1	even	1	trivial
8016.2.a.bg.1.7		13	4.3	odd	2

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:	\( N \)	\(=\)	\( 4008 = 2^{3} \cdot 3 \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4008.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +0.649069 q^{5} +4.19646 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +0.649069 q^{5} +4.19646 q^{7} +1.00000 q^{9} +1.95286 q^{11} +3.07299 q^{13} +0.649069 q^{15} +6.85258 q^{17} +8.04597 q^{19} +4.19646 q^{21} -6.50672 q^{23} -4.57871 q^{25} +1.00000 q^{27} -2.52529 q^{29} +4.14955 q^{31} +1.95286 q^{33} +2.72380 q^{35} -4.94625 q^{37} +3.07299 q^{39} +2.05737 q^{41} +4.35750 q^{43} +0.649069 q^{45} -11.2109 q^{47} +10.6103 q^{49} +6.85258 q^{51} -1.69789 q^{53} +1.26754 q^{55} +8.04597 q^{57} -7.31745 q^{59} -12.5360 q^{61} +4.19646 q^{63} +1.99458 q^{65} -6.97495 q^{67} -6.50672 q^{69} +2.84782 q^{71} -7.78221 q^{73} -4.57871 q^{75} +8.19511 q^{77} -3.29528 q^{79} +1.00000 q^{81} +1.99647 q^{83} +4.44780 q^{85} -2.52529 q^{87} -15.9262 q^{89} +12.8957 q^{91} +4.14955 q^{93} +5.22239 q^{95} +13.7538 q^{97} +1.95286 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q + 13 q^{3} + 2 q^{5} + q^{7} + 13 q^{9}+O(q^{10}) \) 13 * q + 13 * q^3 + 2 * q^5 + q^7 + 13 * q^9	\( 13 q + 13 q^{3} + 2 q^{5} + q^{7} + 13 q^{9} + 11 q^{11} + 12 q^{13} + 2 q^{15} + 15 q^{17} + 14 q^{19} + q^{21} + 9 q^{23} + 37 q^{25} + 13 q^{27} - 3 q^{29} - 17 q^{31} + 11 q^{33} + 15 q^{35} + 16 q^{37} + 12 q^{39} + 12 q^{41} + 20 q^{43} + 2 q^{45} - 6 q^{47} + 26 q^{49} + 15 q^{51} - 12 q^{53} + 7 q^{55} + 14 q^{57} + 14 q^{59} + 24 q^{61} + q^{63} + 8 q^{65} + 3 q^{67} + 9 q^{69} + 17 q^{71} + 34 q^{73} + 37 q^{75} + 30 q^{77} + 10 q^{79} + 13 q^{81} + 44 q^{83} + 25 q^{85} - 3 q^{87} + 25 q^{89} + 29 q^{91} - 17 q^{93} - 15 q^{95} + 38 q^{97} + 11 q^{99}+O(q^{100}) \) 13 * q + 13 * q^3 + 2 * q^5 + q^7 + 13 * q^9 + 11 * q^11 + 12 * q^13 + 2 * q^15 + 15 * q^17 + 14 * q^19 + q^21 + 9 * q^23 + 37 * q^25 + 13 * q^27 - 3 * q^29 - 17 * q^31 + 11 * q^33 + 15 * q^35 + 16 * q^37 + 12 * q^39 + 12 * q^41 + 20 * q^43 + 2 * q^45 - 6 * q^47 + 26 * q^49 + 15 * q^51 - 12 * q^53 + 7 * q^55 + 14 * q^57 + 14 * q^59 + 24 * q^61 + q^63 + 8 * q^65 + 3 * q^67 + 9 * q^69 + 17 * q^71 + 34 * q^73 + 37 * q^75 + 30 * q^77 + 10 * q^79 + 13 * q^81 + 44 * q^83 + 25 * q^85 - 3 * q^87 + 25 * q^89 + 29 * q^91 - 17 * q^93 - 15 * q^95 + 38 * q^97 + 11 * q^99

Introduction

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