LMFDB - Embedded newform 8008.2.a.y.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("8008.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$63.9442019386$
Analytic rank:	$1$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - 3 x^{13} - 27 x^{12} + 78 x^{11} + 273 x^{10} - 750 x^{9} - 1306 x^{8} + 3378 x^{7} + \cdots - 128 $ x^14 - 3x^13 - 27x^12 + 78x^11 + 273x^10 - 750x^9 - 1306x^8 + 3378x^7 + 2996x^6 - 7275x^5 - 2804x^4 + 6417x^3 + 538x^2 - 1032*x - 128
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$1.36889$ of defining polynomial
Character	$\chi$	$=$	8008.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.36889 q^{3} -3.92349 q^{5} +1.00000 q^{7} -1.12615 q^{9} +O(q^{10})$	$q-1.36889 q^{3} -3.92349 q^{5} +1.00000 q^{7} -1.12615 q^{9} -1.00000 q^{11} -1.00000 q^{13} +5.37081 q^{15} -2.95970 q^{17} -3.52044 q^{19} -1.36889 q^{21} +4.25065 q^{23} +10.3938 q^{25} +5.64823 q^{27} -9.21747 q^{29} +1.84881 q^{31} +1.36889 q^{33} -3.92349 q^{35} +0.471628 q^{37} +1.36889 q^{39} +6.70492 q^{41} +12.6251 q^{43} +4.41844 q^{45} +5.58340 q^{47} +1.00000 q^{49} +4.05149 q^{51} +12.0869 q^{53} +3.92349 q^{55} +4.81908 q^{57} -7.88278 q^{59} -11.8317 q^{61} -1.12615 q^{63} +3.92349 q^{65} -0.782757 q^{67} -5.81866 q^{69} -0.831881 q^{71} -7.35993 q^{73} -14.2279 q^{75} -1.00000 q^{77} +3.31354 q^{79} -4.35334 q^{81} +12.9354 q^{83} +11.6123 q^{85} +12.6177 q^{87} -9.73498 q^{89} -1.00000 q^{91} -2.53081 q^{93} +13.8124 q^{95} -0.699621 q^{97} +1.12615 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q - 3 q^{3} - 6 q^{5} + 14 q^{7} + 21 q^{9}+O(q^{10}) $ 14 * q - 3 * q^3 - 6 * q^5 + 14 * q^7 + 21 * q^9	$ 14 q - 3 q^{3} - 6 q^{5} + 14 q^{7} + 21 q^{9} - 14 q^{11} - 14 q^{13} - 6 q^{15} - 6 q^{17} - 13 q^{19} - 3 q^{21} - 9 q^{23} + 22 q^{25} - 18 q^{27} + 2 q^{29} - 2 q^{31} + 3 q^{33} - 6 q^{35} - q^{37} + 3 q^{39} - 16 q^{41} - 15 q^{43} - 44 q^{45} - 8 q^{47} + 14 q^{49} - 14 q^{51} - 6 q^{53} + 6 q^{55} - 10 q^{57} - 36 q^{59} - 19 q^{61} + 21 q^{63} + 6 q^{65} - 34 q^{67} - q^{69} - 10 q^{71} + 9 q^{73} - 44 q^{75} - 14 q^{77} - q^{79} + 42 q^{81} - 56 q^{83} + 21 q^{85} - 5 q^{87} - 14 q^{89} - 14 q^{91} - 20 q^{93} + q^{95} - 14 q^{97} - 21 q^{99}+O(q^{100}) $ 14 * q - 3 * q^3 - 6 * q^5 + 14 * q^7 + 21 * q^9 - 14 * q^11 - 14 * q^13 - 6 * q^15 - 6 * q^17 - 13 * q^19 - 3 * q^21 - 9 * q^23 + 22 * q^25 - 18 * q^27 + 2 * q^29 - 2 * q^31 + 3 * q^33 - 6 * q^35 - q^37 + 3 * q^39 - 16 * q^41 - 15 * q^43 - 44 * q^45 - 8 * q^47 + 14 * q^49 - 14 * q^51 - 6 * q^53 + 6 * q^55 - 10 * q^57 - 36 * q^59 - 19 * q^61 + 21 * q^63 + 6 * q^65 - 34 * q^67 - q^69 - 10 * q^71 + 9 * q^73 - 44 * q^75 - 14 * q^77 - q^79 + 42 * q^81 - 56 * q^83 + 21 * q^85 - 5 * q^87 - 14 * q^89 - 14 * q^91 - 20 * q^93 + q^95 - 14 * q^97 - 21 * 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
$3$	−1.36889	−0.790327	−0.395163	−	0.918611i	$-0.629312\pi$
$3$	−1.36889	−0.790327	−0.395163	+	0.918611i	$0.629312\pi$
$4$	0	0
$5$	−3.92349	−1.75464	−0.877318	−	0.479909i	$-0.840670\pi$
$5$	−3.92349	−1.75464	−0.877318	+	0.479909i	$0.840670\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	−1.12615	−0.375383
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	5.37081	1.38674
$16$	0	0
$17$	−2.95970	−0.717832	−0.358916	−	0.933370i	$-0.616854\pi$
$17$	−2.95970	−0.717832	−0.358916	+	0.933370i	$0.616854\pi$
$18$	0	0
$19$	−3.52044	−0.807644	−0.403822	−	0.914838i	$-0.632319\pi$
$19$	−3.52044	−0.807644	−0.403822	+	0.914838i	$0.632319\pi$
$20$	0	0
$21$	−1.36889	−0.298715
$22$	0	0
$23$	4.25065	0.886322	0.443161	−	0.896442i	$-0.353857\pi$
$23$	4.25065	0.886322	0.443161	+	0.896442i	$0.353857\pi$
$24$	0	0
$25$	10.3938	2.07875
$26$	0	0
$27$	5.64823	1.08700
$28$	0	0
$29$	−9.21747	−1.71164	−0.855821	−	0.517273i	$-0.826947\pi$
$29$	−9.21747	−1.71164	−0.855821	+	0.517273i	$0.826947\pi$
$30$	0	0
$31$	1.84881	0.332056	0.166028	−	0.986121i	$-0.446906\pi$
$31$	1.84881	0.332056	0.166028	+	0.986121i	$0.446906\pi$
$32$	0	0
$33$	1.36889	0.238293
$34$	0	0
$35$	−3.92349	−0.663190
$36$	0	0
$37$	0.471628	0.0775352	0.0387676	−	0.999248i	$-0.487657\pi$
$37$	0.471628	0.0775352	0.0387676	+	0.999248i	$0.487657\pi$
$38$	0	0
$39$	1.36889	0.219197
$40$	0	0
$41$	6.70492	1.04713	0.523567	−	0.851985i	$-0.324601\pi$
$41$	6.70492	1.04713	0.523567	+	0.851985i	$0.324601\pi$
$42$	0	0
$43$	12.6251	1.92531	0.962657	−	0.270725i	$-0.0872635\pi$
$43$	12.6251	1.92531	0.962657	+	0.270725i	$0.0872635\pi$
$44$	0	0
$45$	4.41844	0.658662
$46$	0	0
$47$	5.58340	0.814422	0.407211	−	0.913334i	$-0.366501\pi$
$47$	5.58340	0.814422	0.407211	+	0.913334i	$0.366501\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	4.05149	0.567322
$52$	0	0
$53$	12.0869	1.66027	0.830134	−	0.557565i	$-0.188264\pi$
$53$	12.0869	1.66027	0.830134	+	0.557565i	$0.188264\pi$
$54$	0	0
$55$	3.92349	0.529043
$56$	0	0
$57$	4.81908	0.638303
$58$	0	0
$59$	−7.88278	−1.02625	−0.513125	−	0.858314i	$-0.671512\pi$
$59$	−7.88278	−1.02625	−0.513125	+	0.858314i	$0.671512\pi$
$60$	0	0
$61$	−11.8317	−1.51490	−0.757450	−	0.652893i	$-0.773555\pi$
$61$	−11.8317	−1.51490	−0.757450	+	0.652893i	$0.773555\pi$
$62$	0	0
$63$	−1.12615	−0.141882
$64$	0	0
$65$	3.92349	0.486649
$66$	0	0
$67$	−0.782757	−0.0956290	−0.0478145	−	0.998856i	$-0.515226\pi$
$67$	−0.782757	−0.0956290	−0.0478145	+	0.998856i	$0.515226\pi$
$68$	0	0
$69$	−5.81866	−0.700484
$70$	0	0
$71$	−0.831881	−0.0987261	−0.0493630	−	0.998781i	$-0.515719\pi$
$71$	−0.831881	−0.0987261	−0.0493630	+	0.998781i	$0.515719\pi$
$72$	0	0
$73$	−7.35993	−0.861415	−0.430707	−	0.902492i	$-0.641736\pi$
$73$	−7.35993	−0.861415	−0.430707	+	0.902492i	$0.641736\pi$
$74$	0	0
$75$	−14.2279	−1.64289
$76$	0	0
$77$	−1.00000	−0.113961
$78$	0	0
$79$	3.31354	0.372803	0.186401	−	0.982474i	$-0.440318\pi$
$79$	3.31354	0.372803	0.186401	+	0.982474i	$0.440318\pi$
$80$	0	0
$81$	−4.35334	−0.483704
$82$	0	0
$83$	12.9354	1.41984	0.709920	−	0.704283i	$-0.248731\pi$
$83$	12.9354	1.41984	0.709920	+	0.704283i	$0.248731\pi$
$84$	0	0
$85$	11.6123	1.25954
$86$	0	0
$87$	12.6177	1.35276
$88$	0	0
$89$	−9.73498	−1.03191	−0.515953	−	0.856617i	$-0.672562\pi$
$89$	−9.73498	−1.03191	−0.515953	+	0.856617i	$0.672562\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	−2.53081	−0.262433
$94$	0	0
$95$	13.8124	1.41712
$96$	0	0
$97$	−0.699621	−0.0710357	−0.0355179	−	0.999369i	$-0.511308\pi$
$97$	−0.699621	−0.0710357	−0.0355179	+	0.999369i	$0.511308\pi$
$98$	0	0
$99$	1.12615	0.113182
$100$	0	0
$101$	−9.86753	−0.981856	−0.490928	−	0.871200i	$-0.663342\pi$
$101$	−9.86753	−0.981856	−0.490928	+	0.871200i	$0.663342\pi$
$102$	0	0
$103$	5.51293	0.543205	0.271603	−	0.962409i	$-0.412446\pi$
$103$	5.51293	0.543205	0.271603	+	0.962409i	$0.412446\pi$
$104$	0	0
$105$	5.37081	0.524137
$106$	0	0
$107$	1.72204	0.166476	0.0832382	−	0.996530i	$-0.473474\pi$
$107$	1.72204	0.166476	0.0832382	+	0.996530i	$0.473474\pi$
$108$	0	0
$109$	7.44416	0.713021	0.356511	−	0.934291i	$-0.383966\pi$
$109$	7.44416	0.713021	0.356511	+	0.934291i	$0.383966\pi$
$110$	0	0
$111$	−0.645605	−0.0612782
$112$	0	0
$113$	6.99925	0.658434	0.329217	−	0.944254i	$-0.393215\pi$
$113$	6.99925	0.658434	0.329217	+	0.944254i	$0.393215\pi$
$114$	0	0
$115$	−16.6774	−1.55517
$116$	0	0
$117$	1.12615	0.104113
$118$	0	0
$119$	−2.95970	−0.271315
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−9.17828	−0.827578
$124$	0	0
$125$	−21.1623	−1.89282
$126$	0	0
$127$	−1.81379	−0.160947	−0.0804737	−	0.996757i	$-0.525643\pi$
$127$	−1.81379	−0.160947	−0.0804737	+	0.996757i	$0.525643\pi$
$128$	0	0
$129$	−17.2824	−1.52163
$130$	0	0
$131$	−4.48032	−0.391447	−0.195723	−	0.980659i	$-0.562706\pi$
$131$	−4.48032	−0.391447	−0.195723	+	0.980659i	$0.562706\pi$
$132$	0	0
$133$	−3.52044	−0.305261
$134$	0	0
$135$	−22.1608	−1.90729
$136$	0	0
$137$	−7.35564	−0.628435	−0.314217	−	0.949351i	$-0.601742\pi$
$137$	−7.35564	−0.628435	−0.314217	+	0.949351i	$0.601742\pi$
$138$	0	0
$139$	4.79409	0.406629	0.203315	−	0.979113i	$-0.434829\pi$
$139$	4.79409	0.406629	0.203315	+	0.979113i	$0.434829\pi$
$140$	0	0
$141$	−7.64304	−0.643660
$142$	0	0
$143$	1.00000	0.0836242
$144$	0	0
$145$	36.1646	3.00331
$146$	0	0
$147$	−1.36889	−0.112904
$148$	0	0
$149$	4.53609	0.371611	0.185805	−	0.982587i	$-0.440511\pi$
$149$	4.53609	0.371611	0.185805	+	0.982587i	$0.440511\pi$
$150$	0	0
$151$	11.5746	0.941924	0.470962	−	0.882153i	$-0.343907\pi$
$151$	11.5746	0.941924	0.470962	+	0.882153i	$0.343907\pi$
$152$	0	0
$153$	3.33307	0.269462
$154$	0	0
$155$	−7.25379	−0.582638
$156$	0	0
$157$	23.1828	1.85019	0.925096	−	0.379734i	$-0.123984\pi$
$157$	23.1828	1.85019	0.925096	+	0.379734i	$0.123984\pi$
$158$	0	0
$159$	−16.5456	−1.31215
$160$	0	0
$161$	4.25065	0.334998
$162$	0	0
$163$	−9.86881	−0.772985	−0.386492	−	0.922293i	$-0.626313\pi$
$163$	−9.86881	−0.772985	−0.386492	+	0.922293i	$0.626313\pi$
$164$	0	0
$165$	−5.37081	−0.418117
$166$	0	0
$167$	15.7798	1.22108	0.610541	−	0.791985i	$-0.290952\pi$
$167$	15.7798	1.22108	0.610541	+	0.791985i	$0.290952\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	3.96454	0.303176
$172$	0	0
$173$	15.1450	1.15146	0.575728	−	0.817641i	$-0.304719\pi$
$173$	15.1450	1.15146	0.575728	+	0.817641i	$0.304719\pi$
$174$	0	0
$175$	10.3938	0.785694
$176$	0	0
$177$	10.7906	0.811073
$178$	0	0
$179$	−19.7436	−1.47571	−0.737853	−	0.674961i	$-0.764160\pi$
$179$	−19.7436	−1.47571	−0.737853	+	0.674961i	$0.764160\pi$
$180$	0	0
$181$	5.07812	0.377454	0.188727	−	0.982030i	$-0.439564\pi$
$181$	5.07812	0.377454	0.188727	+	0.982030i	$0.439564\pi$
$182$	0	0
$183$	16.1963	1.19727
$184$	0	0
$185$	−1.85043	−0.136046
$186$	0	0
$187$	2.95970	0.216435
$188$	0	0
$189$	5.64823	0.410848
$190$	0	0
$191$	14.5000	1.04918	0.524590	−	0.851355i	$-0.324219\pi$
$191$	14.5000	1.04918	0.524590	+	0.851355i	$0.324219\pi$
$192$	0	0
$193$	21.6987	1.56191	0.780954	−	0.624589i	$-0.214733\pi$
$193$	21.6987	1.56191	0.780954	+	0.624589i	$0.214733\pi$
$194$	0	0
$195$	−5.37081	−0.384612
$196$	0	0
$197$	−24.4856	−1.74453	−0.872265	−	0.489033i	$-0.837350\pi$
$197$	−24.4856	−1.74453	−0.872265	+	0.489033i	$0.837350\pi$
$198$	0	0
$199$	6.62802	0.469848	0.234924	−	0.972014i	$-0.424516\pi$
$199$	6.62802	0.469848	0.234924	+	0.972014i	$0.424516\pi$
$200$	0	0
$201$	1.07151	0.0755782
$202$	0	0
$203$	−9.21747	−0.646940
$204$	0	0
$205$	−26.3067	−1.83734
$206$	0	0
$207$	−4.78687	−0.332711
$208$	0	0
$209$	3.52044	0.243514
$210$	0	0
$211$	−1.48137	−0.101981	−0.0509907	−	0.998699i	$-0.516238\pi$
$211$	−1.48137	−0.101981	−0.0509907	+	0.998699i	$0.516238\pi$
$212$	0	0
$213$	1.13875	0.0780259
$214$	0	0
$215$	−49.5345	−3.37823
$216$	0	0
$217$	1.84881	0.125506
$218$	0	0
$219$	10.0749	0.680799
$220$	0	0
$221$	2.95970	0.199091
$222$	0	0
$223$	9.88882	0.662204	0.331102	−	0.943595i	$-0.392580\pi$
$223$	9.88882	0.662204	0.331102	+	0.943595i	$0.392580\pi$
$224$	0	0
$225$	−11.7049	−0.780329
$226$	0	0
$227$	−3.28655	−0.218136	−0.109068	−	0.994034i	$-0.534787\pi$
$227$	−3.28655	−0.218136	−0.109068	+	0.994034i	$0.534787\pi$
$228$	0	0
$229$	−24.3860	−1.61147	−0.805737	−	0.592274i	$-0.798230\pi$
$229$	−24.3860	−1.61147	−0.805737	+	0.592274i	$0.798230\pi$
$230$	0	0
$231$	1.36889	0.0900661
$232$	0	0
$233$	−19.5317	−1.27956	−0.639782	−	0.768556i	$-0.720975\pi$
$233$	−19.5317	−1.27956	−0.639782	+	0.768556i	$0.720975\pi$
$234$	0	0
$235$	−21.9064	−1.42902
$236$	0	0
$237$	−4.53586	−0.294636
$238$	0	0
$239$	5.47876	0.354392	0.177196	−	0.984176i	$-0.443297\pi$
$239$	5.47876	0.354392	0.177196	+	0.984176i	$0.443297\pi$
$240$	0	0
$241$	14.7097	0.947536	0.473768	−	0.880650i	$-0.342893\pi$
$241$	14.7097	0.947536	0.473768	+	0.880650i	$0.342893\pi$
$242$	0	0
$243$	−10.9855	−0.704718
$244$	0	0
$245$	−3.92349	−0.250662
$246$	0	0
$247$	3.52044	0.224000
$248$	0	0
$249$	−17.7070	−1.12214
$250$	0	0
$251$	−31.0379	−1.95910	−0.979548	−	0.201209i	$-0.935513\pi$
$251$	−31.0379	−1.95910	−0.979548	+	0.201209i	$0.935513\pi$
$252$	0	0
$253$	−4.25065	−0.267236
$254$	0	0
$255$	−15.8960	−0.995445
$256$	0	0
$257$	−2.89779	−0.180759	−0.0903797	−	0.995907i	$-0.528808\pi$
$257$	−2.89779	−0.180759	−0.0903797	+	0.995907i	$0.528808\pi$
$258$	0	0
$259$	0.471628	0.0293056
$260$	0	0
$261$	10.3803	0.642522
$262$	0	0
$263$	−13.9612	−0.860887	−0.430444	−	0.902617i	$-0.641643\pi$
$263$	−13.9612	−0.860887	−0.430444	+	0.902617i	$0.641643\pi$
$264$	0	0
$265$	−47.4229	−2.91317
$266$	0	0
$267$	13.3261	0.815543
$268$	0	0
$269$	−6.34620	−0.386935	−0.193467	−	0.981107i	$-0.561973\pi$
$269$	−6.34620	−0.386935	−0.193467	+	0.981107i	$0.561973\pi$
$270$	0	0
$271$	−23.6989	−1.43961	−0.719803	−	0.694178i	$-0.755768\pi$
$271$	−23.6989	−1.43961	−0.719803	+	0.694178i	$0.755768\pi$
$272$	0	0
$273$	1.36889	0.0828488
$274$	0	0
$275$	−10.3938	−0.626767
$276$	0	0
$277$	26.7582	1.60775	0.803873	−	0.594802i	$-0.202769\pi$
$277$	26.7582	1.60775	0.803873	+	0.594802i	$0.202769\pi$
$278$	0	0
$279$	−2.08204	−0.124648
$280$	0	0
$281$	12.4095	0.740290	0.370145	−	0.928974i	$-0.379308\pi$
$281$	12.4095	0.740290	0.370145	+	0.928974i	$0.379308\pi$
$282$	0	0
$283$	7.10729	0.422484	0.211242	−	0.977434i	$-0.432249\pi$
$283$	7.10729	0.422484	0.211242	+	0.977434i	$0.432249\pi$
$284$	0	0
$285$	−18.9076	−1.11999
$286$	0	0
$287$	6.70492	0.395779
$288$	0	0
$289$	−8.24018	−0.484717
$290$	0	0
$291$	0.957701	0.0561414
$292$	0	0
$293$	−22.9499	−1.34075	−0.670374	−	0.742023i	$-0.733866\pi$
$293$	−22.9499	−1.34075	−0.670374	+	0.742023i	$0.733866\pi$
$294$	0	0
$295$	30.9280	1.80070
$296$	0	0
$297$	−5.64823	−0.327744
$298$	0	0
$299$	−4.25065	−0.245822
$300$	0	0
$301$	12.6251	0.727700
$302$	0	0
$303$	13.5075	0.775987
$304$	0	0
$305$	46.4217	2.65810
$306$	0	0
$307$	−17.6540	−1.00757	−0.503785	−	0.863829i	$-0.668059\pi$
$307$	−17.6540	−1.00757	−0.503785	+	0.863829i	$0.668059\pi$
$308$	0	0
$309$	−7.54658	−0.429310
$310$	0	0
$311$	−26.3068	−1.49172	−0.745861	−	0.666102i	$-0.767962\pi$
$311$	−26.3068	−1.49172	−0.745861	+	0.666102i	$0.767962\pi$
$312$	0	0
$313$	33.5958	1.89895	0.949475	−	0.313844i	$-0.101617\pi$
$313$	33.5958	1.89895	0.949475	+	0.313844i	$0.101617\pi$
$314$	0	0
$315$	4.41844	0.248951
$316$	0	0
$317$	−13.9822	−0.785317	−0.392659	−	0.919684i	$-0.628445\pi$
$317$	−13.9822	−0.785317	−0.392659	+	0.919684i	$0.628445\pi$
$318$	0	0
$319$	9.21747	0.516079
$320$	0	0
$321$	−2.35728	−0.131571
$322$	0	0
$323$	10.4194	0.579753
$324$	0	0
$325$	−10.3938	−0.576542
$326$	0	0
$327$	−10.1902	−0.563520
$328$	0	0
$329$	5.58340	0.307823
$330$	0	0
$331$	−6.93059	−0.380940	−0.190470	−	0.981693i	$-0.561001\pi$
$331$	−6.93059	−0.380940	−0.190470	+	0.981693i	$0.561001\pi$
$332$	0	0
$333$	−0.531124	−0.0291054
$334$	0	0
$335$	3.07114	0.167794
$336$	0	0
$337$	26.6642	1.45249	0.726245	−	0.687436i	$-0.241264\pi$
$337$	26.6642	1.45249	0.726245	+	0.687436i	$0.241264\pi$
$338$	0	0
$339$	−9.58118	−0.520378
$340$	0	0
$341$	−1.84881	−0.100119
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	22.8294	1.22910
$346$	0	0
$347$	−9.51291	−0.510680	−0.255340	−	0.966851i	$-0.582187\pi$
$347$	−9.51291	−0.510680	−0.255340	+	0.966851i	$0.582187\pi$
$348$	0	0
$349$	23.7941	1.27367	0.636835	−	0.771000i	$-0.280243\pi$
$349$	23.7941	1.27367	0.636835	+	0.771000i	$0.280243\pi$
$350$	0	0
$351$	−5.64823	−0.301480
$352$	0	0
$353$	−11.2681	−0.599740	−0.299870	−	0.953980i	$-0.596943\pi$
$353$	−11.2681	−0.599740	−0.299870	+	0.953980i	$0.596943\pi$
$354$	0	0
$355$	3.26387	0.173228
$356$	0	0
$357$	4.05149	0.214428
$358$	0	0
$359$	−34.9300	−1.84353	−0.921767	−	0.387745i	$-0.873254\pi$
$359$	−34.9300	−1.84353	−0.921767	+	0.387745i	$0.873254\pi$
$360$	0	0
$361$	−6.60650	−0.347711
$362$	0	0
$363$	−1.36889	−0.0718479
$364$	0	0
$365$	28.8766	1.51147
$366$	0	0
$367$	−27.2940	−1.42473	−0.712367	−	0.701807i	$-0.752377\pi$
$367$	−27.2940	−1.42473	−0.712367	+	0.701807i	$0.752377\pi$
$368$	0	0
$369$	−7.55075	−0.393076
$370$	0	0
$371$	12.0869	0.627522
$372$	0	0
$373$	−35.9483	−1.86133	−0.930665	−	0.365872i	$-0.880771\pi$
$373$	−35.9483	−1.86133	−0.930665	+	0.365872i	$0.880771\pi$
$374$	0	0
$375$	28.9688	1.49594
$376$	0	0
$377$	9.21747	0.474724
$378$	0	0
$379$	8.57316	0.440374	0.220187	−	0.975458i	$-0.429333\pi$
$379$	8.57316	0.440374	0.220187	+	0.975458i	$0.429333\pi$
$380$	0	0
$381$	2.48287	0.127201
$382$	0	0
$383$	−11.7435	−0.600065	−0.300033	−	0.953929i	$-0.596998\pi$
$383$	−11.7435	−0.600065	−0.300033	+	0.953929i	$0.596998\pi$
$384$	0	0
$385$	3.92349	0.199959
$386$	0	0
$387$	−14.2178	−0.722731
$388$	0	0
$389$	−18.6387	−0.945021	−0.472510	−	0.881325i	$-0.656652\pi$
$389$	−18.6387	−0.945021	−0.472510	+	0.881325i	$0.656652\pi$
$390$	0	0
$391$	−12.5806	−0.636231
$392$	0	0
$393$	6.13304	0.309371
$394$	0	0
$395$	−13.0006	−0.654133
$396$	0	0
$397$	−14.8418	−0.744890	−0.372445	−	0.928054i	$-0.621480\pi$
$397$	−14.8418	−0.744890	−0.372445	+	0.928054i	$0.621480\pi$
$398$	0	0
$399$	4.81908	0.241256
$400$	0	0
$401$	−6.72889	−0.336024	−0.168012	−	0.985785i	$-0.553735\pi$
$401$	−6.72889	−0.336024	−0.168012	+	0.985785i	$0.553735\pi$
$402$	0	0
$403$	−1.84881	−0.0920959
$404$	0	0
$405$	17.0803	0.848725
$406$	0	0
$407$	−0.471628	−0.0233777
$408$	0	0
$409$	−8.27349	−0.409098	−0.204549	−	0.978856i	$-0.565573\pi$
$409$	−8.27349	−0.409098	−0.204549	+	0.978856i	$0.565573\pi$
$410$	0	0
$411$	10.0690	0.496669
$412$	0	0
$413$	−7.88278	−0.387886
$414$	0	0
$415$	−50.7517	−2.49130
$416$	0	0
$417$	−6.56256	−0.321370
$418$	0	0
$419$	1.11189	0.0543191	0.0271596	−	0.999631i	$-0.491354\pi$
$419$	1.11189	0.0543191	0.0271596	+	0.999631i	$0.491354\pi$
$420$	0	0
$421$	15.0839	0.735146	0.367573	−	0.929995i	$-0.380189\pi$
$421$	15.0839	0.735146	0.367573	+	0.929995i	$0.380189\pi$
$422$	0	0
$423$	−6.28774	−0.305721
$424$	0	0
$425$	−30.7624	−1.49219
$426$	0	0
$427$	−11.8317	−0.572578
$428$	0	0
$429$	−1.36889	−0.0660905
$430$	0	0
$431$	38.2980	1.84475	0.922374	−	0.386298i	$-0.126246\pi$
$431$	38.2980	1.84475	0.922374	+	0.386298i	$0.126246\pi$
$432$	0	0
$433$	28.5031	1.36977	0.684885	−	0.728651i	$-0.259853\pi$
$433$	28.5031	1.36977	0.684885	+	0.728651i	$0.259853\pi$
$434$	0	0
$435$	−49.5053	−2.37360
$436$	0	0
$437$	−14.9642	−0.715833
$438$	0	0
$439$	−32.1423	−1.53407	−0.767034	−	0.641607i	$-0.778268\pi$
$439$	−32.1423	−1.53407	−0.767034	+	0.641607i	$0.778268\pi$
$440$	0	0
$441$	−1.12615	−0.0536262
$442$	0	0
$443$	29.7332	1.41267	0.706333	−	0.707880i	$-0.250348\pi$
$443$	29.7332	1.41267	0.706333	+	0.707880i	$0.250348\pi$
$444$	0	0
$445$	38.1951	1.81062
$446$	0	0
$447$	−6.20939	−0.293694
$448$	0	0
$449$	8.20721	0.387322	0.193661	−	0.981068i	$-0.437964\pi$
$449$	8.20721	0.387322	0.193661	+	0.981068i	$0.437964\pi$
$450$	0	0
$451$	−6.70492	−0.315723
$452$	0	0
$453$	−15.8443	−0.744428
$454$	0	0
$455$	3.92349	0.183936
$456$	0	0
$457$	−21.9390	−1.02626	−0.513132	−	0.858310i	$-0.671515\pi$
$457$	−21.9390	−1.02626	−0.513132	+	0.858310i	$0.671515\pi$
$458$	0	0
$459$	−16.7171	−0.780286
$460$	0	0
$461$	−33.0049	−1.53719	−0.768597	−	0.639734i	$-0.779045\pi$
$461$	−33.0049	−1.53719	−0.768597	+	0.639734i	$0.779045\pi$
$462$	0	0
$463$	−28.0210	−1.30225	−0.651123	−	0.758973i	$-0.725702\pi$
$463$	−28.0210	−1.30225	−0.651123	+	0.758973i	$0.725702\pi$
$464$	0	0
$465$	9.92961	0.460475
$466$	0	0
$467$	1.92140	0.0889116	0.0444558	−	0.999011i	$-0.485845\pi$
$467$	1.92140	0.0889116	0.0444558	+	0.999011i	$0.485845\pi$
$468$	0	0
$469$	−0.782757	−0.0361444
$470$	0	0
$471$	−31.7347	−1.46226
$472$	0	0
$473$	−12.6251	−0.580504
$474$	0	0
$475$	−36.5906	−1.67889
$476$	0	0
$477$	−13.6117	−0.623237
$478$	0	0
$479$	14.8084	0.676613	0.338307	−	0.941036i	$-0.390146\pi$
$479$	14.8084	0.676613	0.338307	+	0.941036i	$0.390146\pi$
$480$	0	0
$481$	−0.471628	−0.0215044
$482$	0	0
$483$	−5.81866	−0.264758
$484$	0	0
$485$	2.74495	0.124642
$486$	0	0
$487$	10.2831	0.465972	0.232986	−	0.972480i	$-0.425150\pi$
$487$	10.2831	0.465972	0.232986	+	0.972480i	$0.425150\pi$
$488$	0	0
$489$	13.5093	0.610911
$490$	0	0
$491$	20.0218	0.903573	0.451787	−	0.892126i	$-0.350787\pi$
$491$	20.0218	0.903573	0.451787	+	0.892126i	$0.350787\pi$
$492$	0	0
$493$	27.2809	1.22867
$494$	0	0
$495$	−4.41844	−0.198594
$496$	0	0
$497$	−0.831881	−0.0373149
$498$	0	0
$499$	−32.3216	−1.44691	−0.723456	−	0.690370i	$-0.757448\pi$
$499$	−32.3216	−1.44691	−0.723456	+	0.690370i	$0.757448\pi$
$500$	0	0
$501$	−21.6008	−0.965053
$502$	0	0
$503$	−12.8365	−0.572350	−0.286175	−	0.958177i	$-0.592384\pi$
$503$	−12.8365	−0.572350	−0.286175	+	0.958177i	$0.592384\pi$
$504$	0	0
$505$	38.7151	1.72280
$506$	0	0
$507$	−1.36889	−0.0607944
$508$	0	0
$509$	−35.4870	−1.57294	−0.786468	−	0.617632i	$-0.788092\pi$
$509$	−35.4870	−1.57294	−0.786468	+	0.617632i	$0.788092\pi$
$510$	0	0
$511$	−7.35993	−0.325584
$512$	0	0
$513$	−19.8843	−0.877911
$514$	0	0
$515$	−21.6299	−0.953128
$516$	0	0
$517$	−5.58340	−0.245558
$518$	0	0
$519$	−20.7318	−0.910026
$520$	0	0
$521$	−10.5604	−0.462658	−0.231329	−	0.972876i	$-0.574307\pi$
$521$	−10.5604	−0.462658	−0.231329	+	0.972876i	$0.574307\pi$
$522$	0	0
$523$	19.6286	0.858300	0.429150	−	0.903233i	$-0.358813\pi$
$523$	19.6286	0.858300	0.429150	+	0.903233i	$0.358813\pi$
$524$	0	0
$525$	−14.2279	−0.620955
$526$	0	0
$527$	−5.47193	−0.238361
$528$	0	0
$529$	−4.93197	−0.214433
$530$	0	0
$531$	8.87719	0.385237
$532$	0	0
$533$	−6.70492	−0.290422
$534$	0	0
$535$	−6.75642	−0.292106
$536$	0	0
$537$	27.0268	1.16629
$538$	0	0
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	36.9553	1.58883	0.794416	−	0.607375i	$-0.207777\pi$
$541$	36.9553	1.58883	0.794416	+	0.607375i	$0.207777\pi$
$542$	0	0
$543$	−6.95137	−0.298312
$544$	0	0
$545$	−29.2071	−1.25109
$546$	0	0
$547$	−14.6151	−0.624895	−0.312448	−	0.949935i	$-0.601149\pi$
$547$	−14.6151	−0.624895	−0.312448	+	0.949935i	$0.601149\pi$
$548$	0	0
$549$	13.3243	0.568668
$550$	0	0
$551$	32.4495	1.38240
$552$	0	0
$553$	3.31354	0.140906
$554$	0	0
$555$	2.53303	0.107521
$556$	0	0
$557$	−11.1401	−0.472021	−0.236011	−	0.971750i	$-0.575840\pi$
$557$	−11.1401	−0.472021	−0.236011	+	0.971750i	$0.575840\pi$
$558$	0	0
$559$	−12.6251	−0.533986
$560$	0	0
$561$	−4.05149	−0.171054
$562$	0	0
$563$	2.07994	0.0876591	0.0438296	−	0.999039i	$-0.486044\pi$
$563$	2.07994	0.0876591	0.0438296	+	0.999039i	$0.486044\pi$
$564$	0	0
$565$	−27.4615	−1.15531
$566$	0	0
$567$	−4.35334	−0.182823
$568$	0	0
$569$	47.3776	1.98617	0.993087	−	0.117383i	$-0.0374506\pi$
$569$	47.3776	1.98617	0.993087	+	0.117383i	$0.0374506\pi$
$570$	0	0
$571$	−26.4238	−1.10580	−0.552900	−	0.833248i	$-0.686479\pi$
$571$	−26.4238	−1.10580	−0.552900	+	0.833248i	$0.686479\pi$
$572$	0	0
$573$	−19.8488	−0.829195
$574$	0	0
$575$	44.1802	1.84244
$576$	0	0
$577$	−16.0595	−0.668564	−0.334282	−	0.942473i	$-0.608494\pi$
$577$	−16.0595	−0.668564	−0.334282	+	0.942473i	$0.608494\pi$
$578$	0	0
$579$	−29.7031	−1.23442
$580$	0	0
$581$	12.9354	0.536649
$582$	0	0
$583$	−12.0869	−0.500589
$584$	0	0
$585$	−4.41844	−0.182680
$586$	0	0
$587$	−9.47574	−0.391106	−0.195553	−	0.980693i	$-0.562650\pi$
$587$	−9.47574	−0.391106	−0.195553	+	0.980693i	$0.562650\pi$
$588$	0	0
$589$	−6.50863	−0.268183
$590$	0	0
$591$	33.5181	1.37875
$592$	0	0
$593$	−33.6277	−1.38092	−0.690462	−	0.723368i	$-0.742593\pi$
$593$	−33.6277	−1.38092	−0.690462	+	0.723368i	$0.742593\pi$
$594$	0	0
$595$	11.6123	0.476060
$596$	0	0
$597$	−9.07301	−0.371334
$598$	0	0
$599$	−5.38046	−0.219840	−0.109920	−	0.993940i	$-0.535059\pi$
$599$	−5.38046	−0.219840	−0.109920	+	0.993940i	$0.535059\pi$
$600$	0	0
$601$	−26.4364	−1.07836	−0.539182	−	0.842189i	$-0.681266\pi$
$601$	−26.4364	−1.07836	−0.539182	+	0.842189i	$0.681266\pi$
$602$	0	0
$603$	0.881502	0.0358976
$604$	0	0
$605$	−3.92349	−0.159512
$606$	0	0
$607$	8.89838	0.361174	0.180587	−	0.983559i	$-0.442200\pi$
$607$	8.89838	0.361174	0.180587	+	0.983559i	$0.442200\pi$
$608$	0	0
$609$	12.6177	0.511294
$610$	0	0
$611$	−5.58340	−0.225880
$612$	0	0
$613$	35.9224	1.45089	0.725446	−	0.688279i	$-0.241633\pi$
$613$	35.9224	1.45089	0.725446	+	0.688279i	$0.241633\pi$
$614$	0	0
$615$	36.0109	1.45210
$616$	0	0
$617$	−25.4334	−1.02391	−0.511956	−	0.859012i	$-0.671079\pi$
$617$	−25.4334	−1.02391	−0.511956	+	0.859012i	$0.671079\pi$
$618$	0	0
$619$	9.83143	0.395158	0.197579	−	0.980287i	$-0.436692\pi$
$619$	9.83143	0.395158	0.197579	+	0.980287i	$0.436692\pi$
$620$	0	0
$621$	24.0087	0.963434
$622$	0	0
$623$	−9.73498	−0.390024
$624$	0	0
$625$	31.0614	1.24245
$626$	0	0
$627$	−4.81908	−0.192456
$628$	0	0
$629$	−1.39588	−0.0556573
$630$	0	0
$631$	−1.26016	−0.0501661	−0.0250831	−	0.999685i	$-0.507985\pi$
$631$	−1.26016	−0.0501661	−0.0250831	+	0.999685i	$0.507985\pi$
$632$	0	0
$633$	2.02782	0.0805987
$634$	0	0
$635$	7.11637	0.282404
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	0.936822	0.0370601
$640$	0	0
$641$	19.1849	0.757760	0.378880	−	0.925446i	$-0.376309\pi$
$641$	19.1849	0.757760	0.378880	+	0.925446i	$0.376309\pi$
$642$	0	0
$643$	−15.3088	−0.603720	−0.301860	−	0.953352i	$-0.597607\pi$
$643$	−15.3088	−0.603720	−0.301860	+	0.953352i	$0.597607\pi$
$644$	0	0
$645$	67.8071	2.66990
$646$	0	0
$647$	−17.2197	−0.676976	−0.338488	−	0.940971i	$-0.609915\pi$
$647$	−17.2197	−0.676976	−0.338488	+	0.940971i	$0.609915\pi$
$648$	0	0
$649$	7.88278	0.309426
$650$	0	0
$651$	−2.53081	−0.0991904
$652$	0	0
$653$	30.9933	1.21286	0.606431	−	0.795136i	$-0.292601\pi$
$653$	30.9933	1.21286	0.606431	+	0.795136i	$0.292601\pi$
$654$	0	0
$655$	17.5785	0.686847
$656$	0	0
$657$	8.28839	0.323361
$658$	0	0
$659$	−30.2889	−1.17989	−0.589944	−	0.807444i	$-0.700850\pi$
$659$	−30.2889	−1.17989	−0.589944	+	0.807444i	$0.700850\pi$
$660$	0	0
$661$	18.7447	0.729086	0.364543	−	0.931187i	$-0.381225\pi$
$661$	18.7447	0.729086	0.364543	+	0.931187i	$0.381225\pi$
$662$	0	0
$663$	−4.05149	−0.157347
$664$	0	0
$665$	13.8124	0.535622
$666$	0	0
$667$	−39.1802	−1.51707
$668$	0	0
$669$	−13.5367	−0.523358
$670$	0	0
$671$	11.8317	0.456759
$672$	0	0
$673$	−3.31762	−0.127885	−0.0639423	−	0.997954i	$-0.520367\pi$
$673$	−3.31762	−0.127885	−0.0639423	+	0.997954i	$0.520367\pi$
$674$	0	0
$675$	58.7063	2.25961
$676$	0	0
$677$	−5.10434	−0.196176	−0.0980878	−	0.995178i	$-0.531273\pi$
$677$	−5.10434	−0.196176	−0.0980878	+	0.995178i	$0.531273\pi$
$678$	0	0
$679$	−0.699621	−0.0268490
$680$	0	0
$681$	4.49892	0.172399
$682$	0	0
$683$	49.7613	1.90406	0.952032	−	0.305999i	$-0.0989904\pi$
$683$	49.7613	1.90406	0.952032	+	0.305999i	$0.0989904\pi$
$684$	0	0
$685$	28.8598	1.10267
$686$	0	0
$687$	33.3817	1.27359
$688$	0	0
$689$	−12.0869	−0.460475
$690$	0	0
$691$	−41.3988	−1.57488	−0.787441	−	0.616390i	$-0.788595\pi$
$691$	−41.3988	−1.57488	−0.787441	+	0.616390i	$0.788595\pi$
$692$	0	0
$693$	1.12615	0.0427789
$694$	0	0
$695$	−18.8095	−0.713487
$696$	0	0
$697$	−19.8446	−0.751666
$698$	0	0
$699$	26.7367	1.01127
$700$	0	0
$701$	33.3662	1.26022	0.630111	−	0.776505i	$-0.283009\pi$
$701$	33.3662	1.26022	0.630111	+	0.776505i	$0.283009\pi$
$702$	0	0
$703$	−1.66034	−0.0626209
$704$	0	0
$705$	29.9874	1.12939
$706$	0	0
$707$	−9.86753	−0.371107
$708$	0	0
$709$	39.3320	1.47714	0.738572	−	0.674174i	$-0.235500\pi$
$709$	39.3320	1.47714	0.738572	+	0.674174i	$0.235500\pi$
$710$	0	0
$711$	−3.73155	−0.139944
$712$	0	0
$713$	7.85865	0.294309
$714$	0	0
$715$	−3.92349	−0.146730
$716$	0	0
$717$	−7.49980	−0.280085
$718$	0	0
$719$	−31.6552	−1.18054	−0.590270	−	0.807206i	$-0.700979\pi$
$719$	−31.6552	−1.18054	−0.590270	+	0.807206i	$0.700979\pi$
$720$	0	0
$721$	5.51293	0.205312
$722$	0	0
$723$	−20.1359	−0.748864
$724$	0	0
$725$	−95.8041	−3.55808
$726$	0	0
$727$	25.3027	0.938424	0.469212	−	0.883086i	$-0.344538\pi$
$727$	25.3027	0.938424	0.469212	+	0.883086i	$0.344538\pi$
$728$	0	0
$729$	28.0979	1.04066
$730$	0	0
$731$	−37.3666	−1.38205
$732$	0	0
$733$	40.4044	1.49237	0.746185	−	0.665739i	$-0.231884\pi$
$733$	40.4044	1.49237	0.746185	+	0.665739i	$0.231884\pi$
$734$	0	0
$735$	5.37081	0.198105
$736$	0	0
$737$	0.782757	0.0288332
$738$	0	0
$739$	22.9622	0.844678	0.422339	−	0.906438i	$-0.361209\pi$
$739$	22.9622	0.844678	0.422339	+	0.906438i	$0.361209\pi$
$740$	0	0
$741$	−4.81908	−0.177033
$742$	0	0
$743$	−45.4301	−1.66667	−0.833335	−	0.552769i	$-0.813571\pi$
$743$	−45.4301	−1.66667	−0.833335	+	0.552769i	$0.813571\pi$
$744$	0	0
$745$	−17.7973	−0.652042
$746$	0	0
$747$	−14.5671	−0.532984
$748$	0	0
$749$	1.72204	0.0629222
$750$	0	0
$751$	2.04380	0.0745792	0.0372896	−	0.999305i	$-0.488128\pi$
$751$	2.04380	0.0745792	0.0372896	+	0.999305i	$0.488128\pi$
$752$	0	0
$753$	42.4874	1.54833
$754$	0	0
$755$	−45.4126	−1.65273
$756$	0	0
$757$	−18.7041	−0.679813	−0.339906	−	0.940459i	$-0.610395\pi$
$757$	−18.7041	−0.679813	−0.339906	+	0.940459i	$0.610395\pi$
$758$	0	0
$759$	5.81866	0.211204
$760$	0	0
$761$	−26.1854	−0.949221	−0.474611	−	0.880196i	$-0.657411\pi$
$761$	−26.1854	−0.949221	−0.474611	+	0.880196i	$0.657411\pi$
$762$	0	0
$763$	7.44416	0.269497
$764$	0	0
$765$	−13.0772	−0.472809
$766$	0	0
$767$	7.88278	0.284631
$768$	0	0
$769$	5.11583	0.184481	0.0922407	−	0.995737i	$-0.470597\pi$
$769$	5.11583	0.184481	0.0922407	+	0.995737i	$0.470597\pi$
$770$	0	0
$771$	3.96675	0.142859
$772$	0	0
$773$	14.8675	0.534747	0.267373	−	0.963593i	$-0.413844\pi$
$773$	14.8675	0.534747	0.267373	+	0.963593i	$0.413844\pi$
$774$	0	0
$775$	19.2161	0.690263
$776$	0	0
$777$	−0.645605	−0.0231610
$778$	0	0
$779$	−23.6043	−0.845711
$780$	0	0
$781$	0.831881	0.0297670
$782$	0	0
$783$	−52.0624	−1.86056
$784$	0	0
$785$	−90.9576	−3.24641
$786$	0	0
$787$	17.1119	0.609972	0.304986	−	0.952357i	$-0.401348\pi$
$787$	17.1119	0.609972	0.304986	+	0.952357i	$0.401348\pi$
$788$	0	0
$789$	19.1114	0.680382
$790$	0	0
$791$	6.99925	0.248865
$792$	0	0
$793$	11.8317	0.420158
$794$	0	0
$795$	64.9166	2.30235
$796$	0	0
$797$	27.3720	0.969567	0.484783	−	0.874634i	$-0.338898\pi$
$797$	27.3720	0.969567	0.484783	+	0.874634i	$0.338898\pi$
$798$	0	0
$799$	−16.5252	−0.584619
$800$	0	0
$801$	10.9630	0.387360
$802$	0	0
$803$	7.35993	0.259726
$804$	0	0
$805$	−16.6774	−0.587800
$806$	0	0
$807$	8.68722	0.305805
$808$	0	0
$809$	15.6997	0.551971	0.275985	−	0.961162i	$-0.410996\pi$
$809$	15.6997	0.551971	0.275985	+	0.961162i	$0.410996\pi$
$810$	0	0
$811$	−7.63057	−0.267946	−0.133973	−	0.990985i	$-0.542773\pi$
$811$	−7.63057	−0.267946	−0.133973	+	0.990985i	$0.542773\pi$
$812$	0	0
$813$	32.4411	1.13776
$814$	0	0
$815$	38.7202	1.35631
$816$	0	0
$817$	−44.4460	−1.55497
$818$	0	0
$819$	1.12615	0.0393509
$820$	0	0
$821$	47.6814	1.66409	0.832045	−	0.554708i	$-0.187170\pi$
$821$	47.6814	1.66409	0.832045	+	0.554708i	$0.187170\pi$
$822$	0	0
$823$	−39.0778	−1.36217	−0.681084	−	0.732205i	$-0.738491\pi$
$823$	−39.0778	−1.36217	−0.681084	+	0.732205i	$0.738491\pi$
$824$	0	0
$825$	14.2279	0.495351
$826$	0	0
$827$	20.5380	0.714176	0.357088	−	0.934071i	$-0.383770\pi$
$827$	20.5380	0.714176	0.357088	+	0.934071i	$0.383770\pi$
$828$	0	0
$829$	40.0917	1.39244	0.696221	−	0.717827i	$-0.254863\pi$
$829$	40.0917	1.39244	0.696221	+	0.717827i	$0.254863\pi$
$830$	0	0
$831$	−36.6289	−1.27064
$832$	0	0
$833$	−2.95970	−0.102547
$834$	0	0
$835$	−61.9120	−2.14255
$836$	0	0
$837$	10.4425	0.360946
$838$	0	0
$839$	44.7619	1.54535	0.772676	−	0.634800i	$-0.218918\pi$
$839$	44.7619	1.54535	0.772676	+	0.634800i	$0.218918\pi$
$840$	0	0
$841$	55.9617	1.92972
$842$	0	0
$843$	−16.9872	−0.585071
$844$	0	0
$845$	−3.92349	−0.134972
$846$	0	0
$847$	1.00000	0.0343604
$848$	0	0
$849$	−9.72907	−0.333901
$850$	0	0
$851$	2.00473	0.0687212
$852$	0	0
$853$	20.6013	0.705376	0.352688	−	0.935741i	$-0.385268\pi$
$853$	20.6013	0.705376	0.352688	+	0.935741i	$0.385268\pi$
$854$	0	0
$855$	−15.5548	−0.531964
$856$	0	0
$857$	−8.76523	−0.299414	−0.149707	−	0.988730i	$-0.547833\pi$
$857$	−8.76523	−0.299414	−0.149707	+	0.988730i	$0.547833\pi$
$858$	0	0
$859$	34.7334	1.18509	0.592544	−	0.805538i	$-0.298124\pi$
$859$	34.7334	1.18509	0.592544	+	0.805538i	$0.298124\pi$
$860$	0	0
$861$	−9.17828	−0.312795
$862$	0	0
$863$	43.5787	1.48344	0.741719	−	0.670711i	$-0.234011\pi$
$863$	43.5787	1.48344	0.741719	+	0.670711i	$0.234011\pi$
$864$	0	0
$865$	−59.4213	−2.02039
$866$	0	0
$867$	11.2799	0.383085
$868$	0	0
$869$	−3.31354	−0.112404
$870$	0	0
$871$	0.782757	0.0265227
$872$	0	0
$873$	0.787878	0.0266656
$874$	0	0
$875$	−21.1623	−0.715417
$876$	0	0
$877$	32.4574	1.09601	0.548003	−	0.836476i	$-0.315388\pi$
$877$	32.4574	1.09601	0.548003	+	0.836476i	$0.315388\pi$
$878$	0	0
$879$	31.4158	1.05963
$880$	0	0
$881$	11.1530	0.375755	0.187877	−	0.982193i	$-0.439839\pi$
$881$	11.1530	0.375755	0.187877	+	0.982193i	$0.439839\pi$
$882$	0	0
$883$	−34.6085	−1.16467	−0.582335	−	0.812949i	$-0.697861\pi$
$883$	−34.6085	−1.16467	−0.582335	+	0.812949i	$0.697861\pi$
$884$	0	0
$885$	−42.3369	−1.42314
$886$	0	0
$887$	30.0443	1.00879	0.504394	−	0.863474i	$-0.331716\pi$
$887$	30.0443	1.00879	0.504394	+	0.863474i	$0.331716\pi$
$888$	0	0
$889$	−1.81379	−0.0608324
$890$	0	0
$891$	4.35334	0.145842
$892$	0	0
$893$	−19.6560	−0.657764
$894$	0	0
$895$	77.4638	2.58933
$896$	0	0
$897$	5.81866	0.194279
$898$	0	0
$899$	−17.0414	−0.568361
$900$	0	0
$901$	−35.7737	−1.19179
$902$	0	0
$903$	−17.2824	−0.575121
$904$	0	0
$905$	−19.9239	−0.662294
$906$	0	0
$907$	25.8933	0.859773	0.429886	−	0.902883i	$-0.358554\pi$
$907$	25.8933	0.859773	0.429886	+	0.902883i	$0.358554\pi$
$908$	0	0
$909$	11.1123	0.368573
$910$	0	0
$911$	52.5754	1.74190	0.870950	−	0.491372i	$-0.163504\pi$
$911$	52.5754	1.74190	0.870950	+	0.491372i	$0.163504\pi$
$912$	0	0
$913$	−12.9354	−0.428098
$914$	0	0
$915$	−63.5460	−2.10077
$916$	0	0
$917$	−4.48032	−0.147953
$918$	0	0
$919$	−16.9725	−0.559870	−0.279935	−	0.960019i	$-0.590313\pi$
$919$	−16.9725	−0.559870	−0.279935	+	0.960019i	$0.590313\pi$
$920$	0	0
$921$	24.1664	0.796309
$922$	0	0
$923$	0.831881	0.0273817
$924$	0	0
$925$	4.90199	0.161176
$926$	0	0
$927$	−6.20839	−0.203910
$928$	0	0
$929$	26.7855	0.878803	0.439402	−	0.898291i	$-0.355191\pi$
$929$	26.7855	0.878803	0.439402	+	0.898291i	$0.355191\pi$
$930$	0	0
$931$	−3.52044	−0.115378
$932$	0	0
$933$	36.0110	1.17895
$934$	0	0
$935$	−11.6123	−0.379764
$936$	0	0
$937$	−6.40506	−0.209244	−0.104622	−	0.994512i	$-0.533363\pi$
$937$	−6.40506	−0.209244	−0.104622	+	0.994512i	$0.533363\pi$
$938$	0	0
$939$	−45.9889	−1.50079
$940$	0	0
$941$	−35.5346	−1.15839	−0.579197	−	0.815187i	$-0.696634\pi$
$941$	−35.5346	−1.15839	−0.579197	+	0.815187i	$0.696634\pi$
$942$	0	0
$943$	28.5003	0.928097
$944$	0	0
$945$	−22.1608	−0.720890
$946$	0	0
$947$	−31.3711	−1.01943	−0.509713	−	0.860345i	$-0.670248\pi$
$947$	−31.3711	−1.01943	−0.509713	+	0.860345i	$0.670248\pi$
$948$	0	0
$949$	7.35993	0.238914
$950$	0	0
$951$	19.1400	0.620657
$952$	0	0
$953$	−21.3368	−0.691167	−0.345584	−	0.938388i	$-0.612319\pi$
$953$	−21.3368	−0.691167	−0.345584	+	0.938388i	$0.612319\pi$
$954$	0	0
$955$	−56.8904	−1.84093
$956$	0	0
$957$	−12.6177	−0.407871
$958$	0	0
$959$	−7.35564	−0.237526
$960$	0	0
$961$	−27.5819	−0.889739
$962$	0	0
$963$	−1.93928	−0.0624925
$964$	0	0
$965$	−85.1346	−2.74058
$966$	0	0
$967$	23.4016	0.752544	0.376272	−	0.926509i	$-0.377206\pi$
$967$	23.4016	0.752544	0.376272	+	0.926509i	$0.377206\pi$
$968$	0	0
$969$	−14.2630	−0.458195
$970$	0	0
$971$	−18.2293	−0.585007	−0.292503	−	0.956265i	$-0.594488\pi$
$971$	−18.2293	−0.585007	−0.292503	+	0.956265i	$0.594488\pi$
$972$	0	0
$973$	4.79409	0.153691
$974$	0	0
$975$	14.2279	0.455656
$976$	0	0
$977$	−42.0299	−1.34465	−0.672327	−	0.740254i	$-0.734705\pi$
$977$	−42.0299	−1.34465	−0.672327	+	0.740254i	$0.734705\pi$
$978$	0	0
$979$	9.73498	0.311131
$980$	0	0
$981$	−8.38324	−0.267656
$982$	0	0
$983$	−38.0019	−1.21207	−0.606037	−	0.795437i	$-0.707242\pi$
$983$	−38.0019	−1.21207	−0.606037	+	0.795437i	$0.707242\pi$
$984$	0	0
$985$	96.0691	3.06102
$986$	0	0
$987$	−7.64304	−0.243281
$988$	0	0
$989$	53.6650	1.70645
$990$	0	0
$991$	12.1620	0.386338	0.193169	−	0.981165i	$-0.438123\pi$
$991$	12.1620	0.386338	0.193169	+	0.981165i	$0.438123\pi$
$992$	0	0
$993$	9.48720	0.301067
$994$	0	0
$995$	−26.0050	−0.824413
$996$	0	0
$997$	−12.7700	−0.404430	−0.202215	−	0.979341i	$-0.564814\pi$
$997$	−12.7700	−0.404430	−0.202215	+	0.979341i	$0.564814\pi$
$998$	0	0
$999$	2.66387	0.0842810

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8008.2.a.y.1.6	✓	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8008.2.a.y.1.6	✓	14	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:	\( N \)	\(=\)	\( 8008 = 2^{3} \cdot 7 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8008.a (trivial)

\(f(q)\)	\(=\)	\(q-1.36889 q^{3} -3.92349 q^{5} +1.00000 q^{7} -1.12615 q^{9} +O(q^{10})\)	\(q-1.36889 q^{3} -3.92349 q^{5} +1.00000 q^{7} -1.12615 q^{9} -1.00000 q^{11} -1.00000 q^{13} +5.37081 q^{15} -2.95970 q^{17} -3.52044 q^{19} -1.36889 q^{21} +4.25065 q^{23} +10.3938 q^{25} +5.64823 q^{27} -9.21747 q^{29} +1.84881 q^{31} +1.36889 q^{33} -3.92349 q^{35} +0.471628 q^{37} +1.36889 q^{39} +6.70492 q^{41} +12.6251 q^{43} +4.41844 q^{45} +5.58340 q^{47} +1.00000 q^{49} +4.05149 q^{51} +12.0869 q^{53} +3.92349 q^{55} +4.81908 q^{57} -7.88278 q^{59} -11.8317 q^{61} -1.12615 q^{63} +3.92349 q^{65} -0.782757 q^{67} -5.81866 q^{69} -0.831881 q^{71} -7.35993 q^{73} -14.2279 q^{75} -1.00000 q^{77} +3.31354 q^{79} -4.35334 q^{81} +12.9354 q^{83} +11.6123 q^{85} +12.6177 q^{87} -9.73498 q^{89} -1.00000 q^{91} -2.53081 q^{93} +13.8124 q^{95} -0.699621 q^{97} +1.12615 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 3 q^{3} - 6 q^{5} + 14 q^{7} + 21 q^{9}+O(q^{10}) \) 14 * q - 3 * q^3 - 6 * q^5 + 14 * q^7 + 21 * q^9	\( 14 q - 3 q^{3} - 6 q^{5} + 14 q^{7} + 21 q^{9} - 14 q^{11} - 14 q^{13} - 6 q^{15} - 6 q^{17} - 13 q^{19} - 3 q^{21} - 9 q^{23} + 22 q^{25} - 18 q^{27} + 2 q^{29} - 2 q^{31} + 3 q^{33} - 6 q^{35} - q^{37} + 3 q^{39} - 16 q^{41} - 15 q^{43} - 44 q^{45} - 8 q^{47} + 14 q^{49} - 14 q^{51} - 6 q^{53} + 6 q^{55} - 10 q^{57} - 36 q^{59} - 19 q^{61} + 21 q^{63} + 6 q^{65} - 34 q^{67} - q^{69} - 10 q^{71} + 9 q^{73} - 44 q^{75} - 14 q^{77} - q^{79} + 42 q^{81} - 56 q^{83} + 21 q^{85} - 5 q^{87} - 14 q^{89} - 14 q^{91} - 20 q^{93} + q^{95} - 14 q^{97} - 21 q^{99}+O(q^{100}) \) 14 * q - 3 * q^3 - 6 * q^5 + 14 * q^7 + 21 * q^9 - 14 * q^11 - 14 * q^13 - 6 * q^15 - 6 * q^17 - 13 * q^19 - 3 * q^21 - 9 * q^23 + 22 * q^25 - 18 * q^27 + 2 * q^29 - 2 * q^31 + 3 * q^33 - 6 * q^35 - q^37 + 3 * q^39 - 16 * q^41 - 15 * q^43 - 44 * q^45 - 8 * q^47 + 14 * q^49 - 14 * q^51 - 6 * q^53 + 6 * q^55 - 10 * q^57 - 36 * q^59 - 19 * q^61 + 21 * q^63 + 6 * q^65 - 34 * q^67 - q^69 - 10 * q^71 + 9 * q^73 - 44 * q^75 - 14 * q^77 - q^79 + 42 * q^81 - 56 * q^83 + 21 * q^85 - 5 * q^87 - 14 * q^89 - 14 * q^91 - 20 * q^93 + q^95 - 14 * q^97 - 21 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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