LMFDB - Embedded newform 84.12.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [84,12,Mod(1,84)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(84, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 12, names="a")

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

chi := DirichletCharacter("84.1");

S:= CuspForms(chi, 12);

N := Newforms(S);

Level:	$ N $	$=$	$ 84 = 2^{2} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 12 $
Character orbit:	$[\chi]$	$=$	84.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:	$64.5408271670$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{1000465}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 250116 $ x^2 - x - 250116
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			1.1
Root			$500.616$ of defining polynomial
Character	$\chi$	$=$	84.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-243.000 q^{3} -9456.16 q^{5} -16807.0 q^{7} +59049.0 q^{9} +O(q^{10})$	$q-243.000 q^{3} -9456.16 q^{5} -16807.0 q^{7} +59049.0 q^{9} +51590.3 q^{11} +1.56893e6 q^{13} +2.29785e6 q^{15} +1.61665e6 q^{17} +6.80210e6 q^{19} +4.08410e6 q^{21} +1.89255e7 q^{23} +4.05909e7 q^{25} -1.43489e7 q^{27} -8.48521e7 q^{29} +5.37726e6 q^{31} -1.25364e7 q^{33} +1.58930e8 q^{35} -3.36649e8 q^{37} -3.81250e8 q^{39} +4.82004e8 q^{41} +1.47740e9 q^{43} -5.58377e8 q^{45} -2.04886e9 q^{47} +2.82475e8 q^{49} -3.92845e8 q^{51} +9.77346e8 q^{53} -4.87846e8 q^{55} -1.65291e9 q^{57} -6.69092e9 q^{59} +7.25803e9 q^{61} -9.92437e8 q^{63} -1.48361e10 q^{65} -1.01656e10 q^{67} -4.59890e9 q^{69} -1.07514e10 q^{71} -6.06351e9 q^{73} -9.86358e9 q^{75} -8.67078e8 q^{77} -1.20483e10 q^{79} +3.48678e9 q^{81} -1.22082e10 q^{83} -1.52873e10 q^{85} +2.06191e10 q^{87} +5.94996e10 q^{89} -2.63690e10 q^{91} -1.30667e9 q^{93} -6.43217e10 q^{95} -7.84565e10 q^{97} +3.04635e9 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 486 q^{3} - 8910 q^{5} - 33614 q^{7} + 118098 q^{9}+O(q^{10}) $ 2 * q - 486 * q^3 - 8910 * q^5 - 33614 * q^7 + 118098 * q^9	$ 2 q - 486 q^{3} - 8910 q^{5} - 33614 q^{7} + 118098 q^{9} + 333234 q^{11} + 221184 q^{13} + 2165130 q^{15} + 8240454 q^{17} + 9715288 q^{19} + 8168202 q^{21} + 17112222 q^{23} - 7938950 q^{25} - 28697814 q^{27} + 65950524 q^{29} + 79458480 q^{31} - 80975862 q^{33} + 149750370 q^{35} + 294067288 q^{37} - 53747712 q^{39} - 417495870 q^{41} - 298918424 q^{43} - 526126590 q^{45} + 482895108 q^{47} + 564950498 q^{49} - 2002430322 q^{51} - 4269511296 q^{53} - 334022720 q^{55} - 2360814984 q^{57} - 5231599164 q^{59} - 1849747188 q^{61} - 1984873086 q^{63} - 15572154420 q^{65} - 7454789692 q^{67} - 4158269946 q^{69} - 31535521182 q^{71} - 9729183112 q^{73} + 1929164850 q^{75} - 5600663838 q^{77} - 50956356444 q^{79} + 6973568802 q^{81} - 24244144344 q^{83} - 11669583620 q^{85} - 16025977332 q^{87} + 11485149570 q^{89} - 3717439488 q^{91} - 19308410640 q^{93} - 62730647640 q^{95} - 110800663168 q^{97} + 19677134466 q^{99}+O(q^{100}) $ 2 * q - 486 * q^3 - 8910 * q^5 - 33614 * q^7 + 118098 * q^9 + 333234 * q^11 + 221184 * q^13 + 2165130 * q^15 + 8240454 * q^17 + 9715288 * q^19 + 8168202 * q^21 + 17112222 * q^23 - 7938950 * q^25 - 28697814 * q^27 + 65950524 * q^29 + 79458480 * q^31 - 80975862 * q^33 + 149750370 * q^35 + 294067288 * q^37 - 53747712 * q^39 - 417495870 * q^41 - 298918424 * q^43 - 526126590 * q^45 + 482895108 * q^47 + 564950498 * q^49 - 2002430322 * q^51 - 4269511296 * q^53 - 334022720 * q^55 - 2360814984 * q^57 - 5231599164 * q^59 - 1849747188 * q^61 - 1984873086 * q^63 - 15572154420 * q^65 - 7454789692 * q^67 - 4158269946 * q^69 - 31535521182 * q^71 - 9729183112 * q^73 + 1929164850 * q^75 - 5600663838 * q^77 - 50956356444 * q^79 + 6973568802 * q^81 - 24244144344 * q^83 - 11669583620 * q^85 - 16025977332 * q^87 + 11485149570 * q^89 - 3717439488 * q^91 - 19308410640 * q^93 - 62730647640 * q^95 - 110800663168 * q^97 + 19677134466 * 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
$3$	−243.000	−0.577350
$3$	−243.000	−0.577350
$4$	0	0
$5$	−9456.16	−1.35326	−0.676628	−	0.736325i	$-0.736559\pi$
$5$	−9456.16	−1.35326	−0.676628	+	0.736325i	$0.736559\pi$
$6$	0	0
$7$	−16807.0	−0.377964
$7$	−16807.0	−0.377964
$8$	0	0
$9$	59049.0	0.333333
$10$	0	0
$11$	51590.3	0.0965846	0.0482923	−	0.998833i	$-0.484622\pi$
$11$	51590.3	0.0965846	0.0482923	+	0.998833i	$0.484622\pi$
$12$	0	0
$13$	1.56893e6	1.17197	0.585984	−	0.810323i	$-0.300708\pi$
$13$	1.56893e6	1.17197	0.585984	+	0.810323i	$0.300708\pi$
$14$	0	0
$15$	2.29785e6	0.781303
$16$	0	0
$17$	1.61665e6	0.276150	0.138075	−	0.990422i	$-0.455908\pi$
$17$	1.61665e6	0.276150	0.138075	+	0.990422i	$0.455908\pi$
$18$	0	0
$19$	6.80210e6	0.630228	0.315114	−	0.949054i	$-0.397957\pi$
$19$	6.80210e6	0.630228	0.315114	+	0.949054i	$0.397957\pi$
$20$	0	0
$21$	4.08410e6	0.218218
$22$	0	0
$23$	1.89255e7	0.613119	0.306559	−	0.951852i	$-0.400822\pi$
$23$	1.89255e7	0.613119	0.306559	+	0.951852i	$0.400822\pi$
$24$	0	0
$25$	4.05909e7	0.831301
$26$	0	0
$27$	−1.43489e7	−0.192450
$28$	0	0
$29$	−8.48521e7	−0.768199	−0.384099	−	0.923292i	$-0.625488\pi$
$29$	−8.48521e7	−0.768199	−0.384099	+	0.923292i	$0.625488\pi$
$30$	0	0
$31$	5.37726e6	0.0337343	0.0168671	−	0.999858i	$-0.494631\pi$
$31$	5.37726e6	0.0337343	0.0168671	+	0.999858i	$0.494631\pi$
$32$	0	0
$33$	−1.25364e7	−0.0557632
$34$	0	0
$35$	1.58930e8	0.511483
$36$	0	0
$37$	−3.36649e8	−0.798119	−0.399059	−	0.916925i	$-0.630663\pi$
$37$	−3.36649e8	−0.798119	−0.399059	+	0.916925i	$0.630663\pi$
$38$	0	0
$39$	−3.81250e8	−0.676635
$40$	0	0
$41$	4.82004e8	0.649739	0.324870	−	0.945759i	$-0.394680\pi$
$41$	4.82004e8	0.649739	0.324870	+	0.945759i	$0.394680\pi$
$42$	0	0
$43$	1.47740e9	1.53257	0.766287	−	0.642499i	$-0.222102\pi$
$43$	1.47740e9	1.53257	0.766287	+	0.642499i	$0.222102\pi$
$44$	0	0
$45$	−5.58377e8	−0.451085
$46$	0	0
$47$	−2.04886e9	−1.30309	−0.651544	−	0.758611i	$-0.725878\pi$
$47$	−2.04886e9	−1.30309	−0.651544	+	0.758611i	$0.725878\pi$
$48$	0	0
$49$	2.82475e8	0.142857
$50$	0	0
$51$	−3.92845e8	−0.159435
$52$	0	0
$53$	9.77346e8	0.321019	0.160510	−	0.987034i	$-0.448686\pi$
$53$	9.77346e8	0.321019	0.160510	+	0.987034i	$0.448686\pi$
$54$	0	0
$55$	−4.87846e8	−0.130704
$56$	0	0
$57$	−1.65291e9	−0.363863
$58$	0	0
$59$	−6.69092e9	−1.21843	−0.609214	−	0.793006i	$-0.708515\pi$
$59$	−6.69092e9	−1.21843	−0.609214	+	0.793006i	$0.708515\pi$
$60$	0	0
$61$	7.25803e9	1.10028	0.550142	−	0.835071i	$-0.314574\pi$
$61$	7.25803e9	1.10028	0.550142	+	0.835071i	$0.314574\pi$
$62$	0	0
$63$	−9.92437e8	−0.125988
$64$	0	0
$65$	−1.48361e10	−1.58597
$66$	0	0
$67$	−1.01656e10	−0.919863	−0.459932	−	0.887954i	$-0.652126\pi$
$67$	−1.01656e10	−0.919863	−0.459932	+	0.887954i	$0.652126\pi$
$68$	0	0
$69$	−4.59890e9	−0.353984
$70$	0	0
$71$	−1.07514e10	−0.707206	−0.353603	−	0.935396i	$-0.615044\pi$
$71$	−1.07514e10	−0.707206	−0.353603	+	0.935396i	$0.615044\pi$
$72$	0	0
$73$	−6.06351e9	−0.342333	−0.171166	−	0.985242i	$-0.554754\pi$
$73$	−6.06351e9	−0.342333	−0.171166	+	0.985242i	$0.554754\pi$
$74$	0	0
$75$	−9.86358e9	−0.479952
$76$	0	0
$77$	−8.67078e8	−0.0365056
$78$	0	0
$79$	−1.20483e10	−0.440533	−0.220266	−	0.975440i	$-0.570693\pi$
$79$	−1.20483e10	−0.440533	−0.220266	+	0.975440i	$0.570693\pi$
$80$	0	0
$81$	3.48678e9	0.111111
$82$	0	0
$83$	−1.22082e10	−0.340190	−0.170095	−	0.985428i	$-0.554407\pi$
$83$	−1.22082e10	−0.340190	−0.170095	+	0.985428i	$0.554407\pi$
$84$	0	0
$85$	−1.52873e10	−0.373702
$86$	0	0
$87$	2.06191e10	0.443520
$88$	0	0
$89$	5.94996e10	1.12945	0.564727	−	0.825278i	$-0.308981\pi$
$89$	5.94996e10	1.12945	0.564727	+	0.825278i	$0.308981\pi$
$90$	0	0
$91$	−2.63690e10	−0.442962
$92$	0	0
$93$	−1.30667e9	−0.0194765
$94$	0	0
$95$	−6.43217e10	−0.852860
$96$	0	0
$97$	−7.84565e10	−0.927651	−0.463825	−	0.885927i	$-0.653523\pi$
$97$	−7.84565e10	−0.927651	−0.463825	+	0.885927i	$0.653523\pi$
$98$	0	0
$99$	3.04635e9	0.0321949
$100$	0	0
$101$	−6.00046e10	−0.568090	−0.284045	−	0.958811i	$-0.591676\pi$
$101$	−6.00046e10	−0.568090	−0.284045	+	0.958811i	$0.591676\pi$
$102$	0	0
$103$	−4.90508e10	−0.416909	−0.208455	−	0.978032i	$-0.566843\pi$
$103$	−4.90508e10	−0.416909	−0.208455	+	0.978032i	$0.566843\pi$
$104$	0	0
$105$	−3.86199e10	−0.295305
$106$	0	0
$107$	−1.91450e11	−1.31960	−0.659802	−	0.751439i	$-0.729360\pi$
$107$	−1.91450e11	−1.31960	−0.659802	+	0.751439i	$0.729360\pi$
$108$	0	0
$109$	−2.88373e11	−1.79519	−0.897593	−	0.440825i	$-0.854686\pi$
$109$	−2.88373e11	−1.79519	−0.897593	+	0.440825i	$0.854686\pi$
$110$	0	0
$111$	8.18057e10	0.460794
$112$	0	0
$113$	−9.45257e10	−0.482635	−0.241317	−	0.970446i	$-0.577579\pi$
$113$	−9.45257e10	−0.482635	−0.241317	+	0.970446i	$0.577579\pi$
$114$	0	0
$115$	−1.78963e11	−0.829707
$116$	0	0
$117$	9.26438e10	0.390656
$118$	0	0
$119$	−2.71710e10	−0.104375
$120$	0	0
$121$	−2.82650e11	−0.990671
$122$	0	0
$123$	−1.17127e11	−0.375127
$124$	0	0
$125$	7.78927e10	0.228293
$126$	0	0
$127$	4.38040e11	1.17650	0.588251	−	0.808678i	$-0.299817\pi$
$127$	4.38040e11	1.17650	0.588251	+	0.808678i	$0.299817\pi$
$128$	0	0
$129$	−3.59008e11	−0.884832
$130$	0	0
$131$	5.74572e11	1.30122	0.650612	−	0.759410i	$-0.274512\pi$
$131$	5.74572e11	1.30122	0.650612	+	0.759410i	$0.274512\pi$
$132$	0	0
$133$	−1.14323e11	−0.238204
$134$	0	0
$135$	1.35686e11	0.260434
$136$	0	0
$137$	−2.16930e11	−0.384022	−0.192011	−	0.981393i	$-0.561501\pi$
$137$	−2.16930e11	−0.384022	−0.192011	+	0.981393i	$0.561501\pi$
$138$	0	0
$139$	1.15649e12	1.89042	0.945211	−	0.326460i	$-0.105856\pi$
$139$	1.15649e12	1.89042	0.945211	+	0.326460i	$0.105856\pi$
$140$	0	0
$141$	4.97873e11	0.752338
$142$	0	0
$143$	8.09416e10	0.113194
$144$	0	0
$145$	8.02375e11	1.03957
$146$	0	0
$147$	−6.86415e10	−0.0824786
$148$	0	0
$149$	−1.38728e12	−1.54753	−0.773764	−	0.633474i	$-0.781628\pi$
$149$	−1.38728e12	−1.54753	−0.773764	+	0.633474i	$0.781628\pi$
$150$	0	0
$151$	9.55515e11	0.990522	0.495261	−	0.868744i	$-0.335072\pi$
$151$	9.55515e11	0.990522	0.495261	+	0.868744i	$0.335072\pi$
$152$	0	0
$153$	9.54613e10	0.0920501
$154$	0	0
$155$	−5.08482e10	−0.0456511
$156$	0	0
$157$	−1.05473e11	−0.0882453	−0.0441226	−	0.999026i	$-0.514049\pi$
$157$	−1.05473e11	−0.0882453	−0.0441226	+	0.999026i	$0.514049\pi$
$158$	0	0
$159$	−2.37495e11	−0.185340
$160$	0	0
$161$	−3.18081e11	−0.231737
$162$	0	0
$163$	1.37741e12	0.937632	0.468816	−	0.883296i	$-0.344681\pi$
$163$	1.37741e12	0.937632	0.468816	+	0.883296i	$0.344681\pi$
$164$	0	0
$165$	1.18547e11	0.0754618
$166$	0	0
$167$	−1.74727e11	−0.104092	−0.0520462	−	0.998645i	$-0.516574\pi$
$167$	−1.74727e11	−0.104092	−0.0520462	+	0.998645i	$0.516574\pi$
$168$	0	0
$169$	6.69384e11	0.373507
$170$	0	0
$171$	4.01657e11	0.210076
$172$	0	0
$173$	1.81217e12	0.889088	0.444544	−	0.895757i	$-0.353366\pi$
$173$	1.81217e12	0.889088	0.444544	+	0.895757i	$0.353366\pi$
$174$	0	0
$175$	−6.82211e11	−0.314202
$176$	0	0
$177$	1.62589e12	0.703459
$178$	0	0
$179$	6.57146e11	0.267282	0.133641	−	0.991030i	$-0.457333\pi$
$179$	6.57146e11	0.267282	0.133641	+	0.991030i	$0.457333\pi$
$180$	0	0
$181$	2.15308e12	0.823813	0.411906	−	0.911226i	$-0.364863\pi$
$181$	2.15308e12	0.823813	0.411906	+	0.911226i	$0.364863\pi$
$182$	0	0
$183$	−1.76370e12	−0.635249
$184$	0	0
$185$	3.18341e12	1.08006
$186$	0	0
$187$	8.34032e10	0.0266719
$188$	0	0
$189$	2.41162e11	0.0727393
$190$	0	0
$191$	−1.79490e12	−0.510924	−0.255462	−	0.966819i	$-0.582228\pi$
$191$	−1.79490e12	−0.510924	−0.255462	+	0.966819i	$0.582228\pi$
$192$	0	0
$193$	−1.09706e12	−0.294893	−0.147447	−	0.989070i	$-0.547105\pi$
$193$	−1.09706e12	−0.294893	−0.147447	+	0.989070i	$0.547105\pi$
$194$	0	0
$195$	3.60516e12	0.915661
$196$	0	0
$197$	−1.74128e12	−0.418123	−0.209061	−	0.977903i	$-0.567041\pi$
$197$	−1.74128e12	−0.418123	−0.209061	+	0.977903i	$0.567041\pi$
$198$	0	0
$199$	4.07764e12	0.926226	0.463113	−	0.886299i	$-0.346732\pi$
$199$	4.07764e12	0.926226	0.463113	+	0.886299i	$0.346732\pi$
$200$	0	0
$201$	2.47025e12	0.531083
$202$	0	0
$203$	1.42611e12	0.290352
$204$	0	0
$205$	−4.55790e12	−0.879263
$206$	0	0
$207$	1.11753e12	0.204373
$208$	0	0
$209$	3.50922e11	0.0608704
$210$	0	0
$211$	−7.35715e11	−0.121103	−0.0605517	−	0.998165i	$-0.519286\pi$
$211$	−7.35715e11	−0.121103	−0.0605517	+	0.998165i	$0.519286\pi$
$212$	0	0
$213$	2.61260e12	0.408305
$214$	0	0
$215$	−1.39705e13	−2.07396
$216$	0	0
$217$	−9.03755e10	−0.0127504
$218$	0	0
$219$	1.47343e12	0.197646
$220$	0	0
$221$	2.53640e12	0.323639
$222$	0	0
$223$	−2.18012e12	−0.264731	−0.132365	−	0.991201i	$-0.542257\pi$
$223$	−2.18012e12	−0.264731	−0.132365	+	0.991201i	$0.542257\pi$
$224$	0	0
$225$	2.39685e12	0.277100
$226$	0	0
$227$	−9.86718e12	−1.08655	−0.543276	−	0.839554i	$-0.682816\pi$
$227$	−9.86718e12	−1.08655	−0.543276	+	0.839554i	$0.682816\pi$
$228$	0	0
$229$	−1.33029e13	−1.39589	−0.697944	−	0.716152i	$-0.745902\pi$
$229$	−1.33029e13	−1.39589	−0.697944	+	0.716152i	$0.745902\pi$
$230$	0	0
$231$	2.10700e11	0.0210765
$232$	0	0
$233$	−5.02729e12	−0.479597	−0.239798	−	0.970823i	$-0.577081\pi$
$233$	−5.02729e12	−0.479597	−0.239798	+	0.970823i	$0.577081\pi$
$234$	0	0
$235$	1.93743e13	1.76341
$236$	0	0
$237$	2.92775e12	0.254342
$238$	0	0
$239$	−1.95343e13	−1.62036	−0.810178	−	0.586185i	$-0.800629\pi$
$239$	−1.95343e13	−1.62036	−0.810178	+	0.586185i	$0.800629\pi$
$240$	0	0
$241$	2.56454e12	0.203196	0.101598	−	0.994826i	$-0.467604\pi$
$241$	2.56454e12	0.203196	0.101598	+	0.994826i	$0.467604\pi$
$242$	0	0
$243$	−8.47289e11	−0.0641500
$244$	0	0
$245$	−2.67113e12	−0.193322
$246$	0	0
$247$	1.06720e13	0.738607
$248$	0	0
$249$	2.96659e12	0.196409
$250$	0	0
$251$	7.92044e12	0.501815	0.250908	−	0.968011i	$-0.419271\pi$
$251$	7.92044e12	0.501815	0.250908	+	0.968011i	$0.419271\pi$
$252$	0	0
$253$	9.76373e11	0.0592179
$254$	0	0
$255$	3.71480e12	0.215757
$256$	0	0
$257$	−3.08819e13	−1.71819	−0.859097	−	0.511814i	$-0.828974\pi$
$257$	−3.08819e13	−1.71819	−0.859097	+	0.511814i	$0.828974\pi$
$258$	0	0
$259$	5.65806e12	0.301661
$260$	0	0
$261$	−5.01043e12	−0.256066
$262$	0	0
$263$	−2.32929e13	−1.14148	−0.570738	−	0.821132i	$-0.693343\pi$
$263$	−2.32929e13	−1.14148	−0.570738	+	0.821132i	$0.693343\pi$
$264$	0	0
$265$	−9.24194e12	−0.434421
$266$	0	0
$267$	−1.44584e13	−0.652091
$268$	0	0
$269$	−3.20353e13	−1.38673	−0.693365	−	0.720587i	$-0.743873\pi$
$269$	−3.20353e13	−1.38673	−0.693365	+	0.720587i	$0.743873\pi$
$270$	0	0
$271$	−1.93190e13	−0.802885	−0.401442	−	0.915884i	$-0.631491\pi$
$271$	−1.93190e13	−0.802885	−0.401442	+	0.915884i	$0.631491\pi$
$272$	0	0
$273$	6.40767e12	0.255744
$274$	0	0
$275$	2.09409e12	0.0802909
$276$	0	0
$277$	3.31630e12	0.122184	0.0610921	−	0.998132i	$-0.480542\pi$
$277$	3.31630e12	0.122184	0.0610921	+	0.998132i	$0.480542\pi$
$278$	0	0
$279$	3.17522e11	0.0112448
$280$	0	0
$281$	−4.87320e13	−1.65932	−0.829659	−	0.558270i	$-0.811465\pi$
$281$	−4.87320e13	−1.65932	−0.829659	+	0.558270i	$0.811465\pi$
$282$	0	0
$283$	4.35586e13	1.42642	0.713212	−	0.700949i	$-0.247240\pi$
$283$	4.35586e13	1.42642	0.713212	+	0.700949i	$0.247240\pi$
$284$	0	0
$285$	1.56302e13	0.492399
$286$	0	0
$287$	−8.10103e12	−0.245578
$288$	0	0
$289$	−3.16584e13	−0.923741
$290$	0	0
$291$	1.90649e13	0.535579
$292$	0	0
$293$	−2.80530e13	−0.758940	−0.379470	−	0.925204i	$-0.623894\pi$
$293$	−2.80530e13	−0.758940	−0.379470	+	0.925204i	$0.623894\pi$
$294$	0	0
$295$	6.32704e13	1.64884
$296$	0	0
$297$	−7.40264e11	−0.0185877
$298$	0	0
$299$	2.96928e13	0.718555
$300$	0	0
$301$	−2.48306e13	−0.579258
$302$	0	0
$303$	1.45811e13	0.327987
$304$	0	0
$305$	−6.86331e13	−1.48896
$306$	0	0
$307$	4.28989e13	0.897812	0.448906	−	0.893579i	$-0.351814\pi$
$307$	4.28989e13	0.897812	0.448906	+	0.893579i	$0.351814\pi$
$308$	0	0
$309$	1.19194e13	0.240703
$310$	0	0
$311$	2.16502e13	0.421968	0.210984	−	0.977490i	$-0.432333\pi$
$311$	2.16502e13	0.421968	0.210984	+	0.977490i	$0.432333\pi$
$312$	0	0
$313$	7.02884e12	0.132248	0.0661241	−	0.997811i	$-0.478937\pi$
$313$	7.02884e12	0.132248	0.0661241	+	0.997811i	$0.478937\pi$
$314$	0	0
$315$	9.38464e12	0.170494
$316$	0	0
$317$	5.59175e13	0.981119	0.490560	−	0.871408i	$-0.336792\pi$
$317$	5.59175e13	0.981119	0.490560	+	0.871408i	$0.336792\pi$
$318$	0	0
$319$	−4.37754e12	−0.0741962
$320$	0	0
$321$	4.65223e13	0.761874
$322$	0	0
$323$	1.09966e13	0.174038
$324$	0	0
$325$	6.36843e13	0.974258
$326$	0	0
$327$	7.00747e13	1.03645
$328$	0	0
$329$	3.44352e13	0.492521
$330$	0	0
$331$	−6.17491e13	−0.854234	−0.427117	−	0.904196i	$-0.640471\pi$
$331$	−6.17491e13	−0.854234	−0.427117	+	0.904196i	$0.640471\pi$
$332$	0	0
$333$	−1.98788e13	−0.266040
$334$	0	0
$335$	9.61279e13	1.24481
$336$	0	0
$337$	1.24397e14	1.55900	0.779500	−	0.626402i	$-0.215473\pi$
$337$	1.24397e14	1.55900	0.779500	+	0.626402i	$0.215473\pi$
$338$	0	0
$339$	2.29697e13	0.278649
$340$	0	0
$341$	2.77414e11	0.00325821
$342$	0	0
$343$	−4.74756e12	−0.0539949
$344$	0	0
$345$	4.34880e13	0.479031
$346$	0	0
$347$	1.40256e13	0.149662	0.0748309	−	0.997196i	$-0.476158\pi$
$347$	1.40256e13	0.149662	0.0748309	+	0.997196i	$0.476158\pi$
$348$	0	0
$349$	−9.00071e13	−0.930544	−0.465272	−	0.885168i	$-0.654044\pi$
$349$	−9.00071e13	−0.930544	−0.465272	+	0.885168i	$0.654044\pi$
$350$	0	0
$351$	−2.25124e13	−0.225545
$352$	0	0
$353$	−1.42861e14	−1.38724	−0.693620	−	0.720342i	$-0.743985\pi$
$353$	−1.42861e14	−1.38724	−0.693620	+	0.720342i	$0.743985\pi$
$354$	0	0
$355$	1.01667e14	0.957030
$356$	0	0
$357$	6.60254e12	0.0602610
$358$	0	0
$359$	−1.77220e14	−1.56853	−0.784266	−	0.620425i	$-0.786960\pi$
$359$	−1.77220e14	−1.56853	−0.784266	+	0.620425i	$0.786960\pi$
$360$	0	0
$361$	−7.02217e13	−0.602812
$362$	0	0
$363$	6.86840e13	0.571964
$364$	0	0
$365$	5.73375e13	0.463264
$366$	0	0
$367$	−3.45093e13	−0.270566	−0.135283	−	0.990807i	$-0.543194\pi$
$367$	−3.45093e13	−0.270566	−0.135283	+	0.990807i	$0.543194\pi$
$368$	0	0
$369$	2.84618e13	0.216580
$370$	0	0
$371$	−1.64262e13	−0.121334
$372$	0	0
$373$	5.40697e13	0.387753	0.193877	−	0.981026i	$-0.437894\pi$
$373$	5.40697e13	0.387753	0.193877	+	0.981026i	$0.437894\pi$
$374$	0	0
$375$	−1.89279e13	−0.131805
$376$	0	0
$377$	−1.33127e14	−0.900304
$378$	0	0
$379$	2.48575e14	1.63283	0.816416	−	0.577464i	$-0.195958\pi$
$379$	2.48575e14	1.63283	0.816416	+	0.577464i	$0.195958\pi$
$380$	0	0
$381$	−1.06444e14	−0.679254
$382$	0	0
$383$	2.25588e14	1.39869	0.699346	−	0.714783i	$-0.253475\pi$
$383$	2.25588e14	1.39869	0.699346	+	0.714783i	$0.253475\pi$
$384$	0	0
$385$	8.19923e12	0.0494014
$386$	0	0
$387$	8.72389e13	0.510858
$388$	0	0
$389$	−1.63893e14	−0.932908	−0.466454	−	0.884546i	$-0.654469\pi$
$389$	−1.63893e14	−0.932908	−0.466454	+	0.884546i	$0.654469\pi$
$390$	0	0
$391$	3.05959e13	0.169313
$392$	0	0
$393$	−1.39621e14	−0.751263
$394$	0	0
$395$	1.13931e14	0.596153
$396$	0	0
$397$	−2.38650e14	−1.21454	−0.607272	−	0.794494i	$-0.707736\pi$
$397$	−2.38650e14	−1.21454	−0.607272	+	0.794494i	$0.707736\pi$
$398$	0	0
$399$	2.77804e13	0.137527
$400$	0	0
$401$	−1.12123e14	−0.540010	−0.270005	−	0.962859i	$-0.587025\pi$
$401$	−1.12123e14	−0.540010	−0.270005	+	0.962859i	$0.587025\pi$
$402$	0	0
$403$	8.43654e12	0.0395355
$404$	0	0
$405$	−3.29716e13	−0.150362
$406$	0	0
$407$	−1.73678e13	−0.0770860
$408$	0	0
$409$	2.58111e14	1.11514	0.557569	−	0.830130i	$-0.311734\pi$
$409$	2.58111e14	1.11514	0.557569	+	0.830130i	$0.311734\pi$
$410$	0	0
$411$	5.27140e13	0.221715
$412$	0	0
$413$	1.12454e14	0.460522
$414$	0	0
$415$	1.15442e14	0.460364
$416$	0	0
$417$	−2.81026e14	−1.09144
$418$	0	0
$419$	2.32473e14	0.879418	0.439709	−	0.898140i	$-0.355082\pi$
$419$	2.32473e14	0.879418	0.439709	+	0.898140i	$0.355082\pi$
$420$	0	0
$421$	4.07811e14	1.50282	0.751410	−	0.659836i	$-0.229374\pi$
$421$	4.07811e14	1.50282	0.751410	+	0.659836i	$0.229374\pi$
$422$	0	0
$423$	−1.20983e14	−0.434363
$424$	0	0
$425$	6.56211e13	0.229564
$426$	0	0
$427$	−1.21986e14	−0.415868
$428$	0	0
$429$	−1.96688e13	−0.0653526
$430$	0	0
$431$	−3.78524e14	−1.22594	−0.612969	−	0.790107i	$-0.710025\pi$
$431$	−3.78524e14	−1.22594	−0.612969	+	0.790107i	$0.710025\pi$
$432$	0	0
$433$	−5.66112e14	−1.78739	−0.893695	−	0.448675i	$-0.851896\pi$
$433$	−5.66112e14	−1.78739	−0.893695	+	0.448675i	$0.851896\pi$
$434$	0	0
$435$	−1.94977e14	−0.600196
$436$	0	0
$437$	1.28733e14	0.386405
$438$	0	0
$439$	−6.59592e14	−1.93073	−0.965363	−	0.260911i	$-0.915977\pi$
$439$	−6.59592e14	−1.93073	−0.965363	+	0.260911i	$0.915977\pi$
$440$	0	0
$441$	1.66799e13	0.0476190
$442$	0	0
$443$	2.94009e14	0.818729	0.409364	−	0.912371i	$-0.365750\pi$
$443$	2.94009e14	0.818729	0.409364	+	0.912371i	$0.365750\pi$
$444$	0	0
$445$	−5.62638e14	−1.52844
$446$	0	0
$447$	3.37108e14	0.893465
$448$	0	0
$449$	−6.46695e14	−1.67242	−0.836208	−	0.548412i	$-0.815233\pi$
$449$	−6.46695e14	−1.67242	−0.836208	+	0.548412i	$0.815233\pi$
$450$	0	0
$451$	2.48667e13	0.0627548
$452$	0	0
$453$	−2.32190e14	−0.571878
$454$	0	0
$455$	2.49350e14	0.599441
$456$	0	0
$457$	−2.95204e14	−0.692761	−0.346381	−	0.938094i	$-0.612589\pi$
$457$	−2.95204e14	−0.692761	−0.346381	+	0.938094i	$0.612589\pi$
$458$	0	0
$459$	−2.31971e13	−0.0531452
$460$	0	0
$461$	−1.37343e14	−0.307222	−0.153611	−	0.988131i	$-0.549090\pi$
$461$	−1.37343e14	−0.307222	−0.153611	+	0.988131i	$0.549090\pi$
$462$	0	0
$463$	−2.10912e14	−0.460687	−0.230344	−	0.973109i	$-0.573985\pi$
$463$	−2.10912e14	−0.460687	−0.230344	+	0.973109i	$0.573985\pi$
$464$	0	0
$465$	1.23561e13	0.0263567
$466$	0	0
$467$	−8.77133e14	−1.82735	−0.913677	−	0.406442i	$-0.866769\pi$
$467$	−8.77133e14	−1.82735	−0.913677	+	0.406442i	$0.866769\pi$
$468$	0	0
$469$	1.70854e14	0.347676
$470$	0	0
$471$	2.56298e13	0.0509484
$472$	0	0
$473$	7.62194e13	0.148023
$474$	0	0
$475$	2.76103e14	0.523910
$476$	0	0
$477$	5.77113e13	0.107006
$478$	0	0
$479$	7.51105e14	1.36099	0.680496	−	0.732752i	$-0.261764\pi$
$479$	7.51105e14	1.36099	0.680496	+	0.732752i	$0.261764\pi$
$480$	0	0
$481$	−5.28179e14	−0.935369
$482$	0	0
$483$	7.72937e13	0.133794
$484$	0	0
$485$	7.41897e14	1.25535
$486$	0	0
$487$	1.94439e14	0.321642	0.160821	−	0.986984i	$-0.448586\pi$
$487$	1.94439e14	0.321642	0.160821	+	0.986984i	$0.448586\pi$
$488$	0	0
$489$	−3.34711e14	−0.541342
$490$	0	0
$491$	6.52006e14	1.03111	0.515553	−	0.856858i	$-0.327586\pi$
$491$	6.52006e14	1.03111	0.515553	+	0.856858i	$0.327586\pi$
$492$	0	0
$493$	−1.37176e14	−0.212138
$494$	0	0
$495$	−2.88068e13	−0.0435679
$496$	0	0
$497$	1.80699e14	0.267299
$498$	0	0
$499$	−1.19271e15	−1.72577	−0.862885	−	0.505400i	$-0.831345\pi$
$499$	−1.19271e15	−1.72577	−0.862885	+	0.505400i	$0.831345\pi$
$500$	0	0
$501$	4.24587e13	0.0600978
$502$	0	0
$503$	8.51407e14	1.17900	0.589500	−	0.807769i	$-0.299325\pi$
$503$	8.51407e14	1.17900	0.589500	+	0.807769i	$0.299325\pi$
$504$	0	0
$505$	5.67413e14	0.768771
$506$	0	0
$507$	−1.62660e14	−0.215644
$508$	0	0
$509$	9.00215e14	1.16788	0.583940	−	0.811797i	$-0.301510\pi$
$509$	9.00215e14	1.16788	0.583940	+	0.811797i	$0.301510\pi$
$510$	0	0
$511$	1.01909e14	0.129390
$512$	0	0
$513$	−9.76026e13	−0.121288
$514$	0	0
$515$	4.63833e14	0.564185
$516$	0	0
$517$	−1.05701e14	−0.125858
$518$	0	0
$519$	−4.40357e14	−0.513315
$520$	0	0
$521$	−4.12875e12	−0.00471207	−0.00235603	−	0.999997i	$-0.500750\pi$
$521$	−4.12875e12	−0.00471207	−0.00235603	+	0.999997i	$0.500750\pi$
$522$	0	0
$523$	−1.16416e15	−1.30092	−0.650462	−	0.759539i	$-0.725425\pi$
$523$	−1.16416e15	−1.30092	−0.650462	+	0.759539i	$0.725425\pi$
$524$	0	0
$525$	1.65777e14	0.181405
$526$	0	0
$527$	8.69311e12	0.00931573
$528$	0	0
$529$	−5.94634e14	−0.624085
$530$	0	0
$531$	−3.95092e14	−0.406142
$532$	0	0
$533$	7.56230e14	0.761473
$534$	0	0
$535$	1.81038e15	1.78576
$536$	0	0
$537$	−1.59686e14	−0.154315
$538$	0	0
$539$	1.45730e13	0.0137978
$540$	0	0
$541$	3.75530e14	0.348386	0.174193	−	0.984712i	$-0.444268\pi$
$541$	3.75530e14	0.348386	0.174193	+	0.984712i	$0.444268\pi$
$542$	0	0
$543$	−5.23199e14	−0.475629
$544$	0	0
$545$	2.72691e15	2.42935
$546$	0	0
$547$	4.27145e14	0.372945	0.186473	−	0.982460i	$-0.440294\pi$
$547$	4.27145e14	0.372945	0.186473	+	0.982460i	$0.440294\pi$
$548$	0	0
$549$	4.28579e14	0.366761
$550$	0	0
$551$	−5.77172e14	−0.484141
$552$	0	0
$553$	2.02496e14	0.166506
$554$	0	0
$555$	−7.73568e14	−0.623572
$556$	0	0
$557$	−1.90728e15	−1.50734	−0.753670	−	0.657252i	$-0.771719\pi$
$557$	−1.90728e15	−1.50734	−0.753670	+	0.657252i	$0.771719\pi$
$558$	0	0
$559$	2.31794e15	1.79613
$560$	0	0
$561$	−2.02670e13	−0.0153990
$562$	0	0
$563$	4.46693e14	0.332823	0.166411	−	0.986056i	$-0.446782\pi$
$563$	4.46693e14	0.332823	0.166411	+	0.986056i	$0.446782\pi$
$564$	0	0
$565$	8.93850e14	0.653128
$566$	0	0
$567$	−5.86024e13	−0.0419961
$568$	0	0
$569$	6.81877e14	0.479279	0.239640	−	0.970862i	$-0.422971\pi$
$569$	6.81877e14	0.479279	0.239640	+	0.970862i	$0.422971\pi$
$570$	0	0
$571$	−1.06447e15	−0.733896	−0.366948	−	0.930241i	$-0.619597\pi$
$571$	−1.06447e15	−0.733896	−0.366948	+	0.930241i	$0.619597\pi$
$572$	0	0
$573$	4.36161e14	0.294982
$574$	0	0
$575$	7.68204e14	0.509687
$576$	0	0
$577$	1.30190e15	0.847441	0.423720	−	0.905793i	$-0.360724\pi$
$577$	1.30190e15	0.847441	0.423720	+	0.905793i	$0.360724\pi$
$578$	0	0
$579$	2.66585e14	0.170257
$580$	0	0
$581$	2.05183e14	0.128580
$582$	0	0
$583$	5.04215e13	0.0310055
$584$	0	0
$585$	−8.76055e14	−0.528657
$586$	0	0
$587$	1.53237e15	0.907518	0.453759	−	0.891125i	$-0.350083\pi$
$587$	1.53237e15	0.907518	0.453759	+	0.891125i	$0.350083\pi$
$588$	0	0
$589$	3.65766e13	0.0212603
$590$	0	0
$591$	4.23130e14	0.241403
$592$	0	0
$593$	−2.06596e14	−0.115697	−0.0578484	−	0.998325i	$-0.518424\pi$
$593$	−2.06596e14	−0.115697	−0.0578484	+	0.998325i	$0.518424\pi$
$594$	0	0
$595$	2.56933e14	0.141246
$596$	0	0
$597$	−9.90866e14	−0.534757
$598$	0	0
$599$	−4.81951e14	−0.255362	−0.127681	−	0.991815i	$-0.540753\pi$
$599$	−4.81951e14	−0.255362	−0.127681	+	0.991815i	$0.540753\pi$
$600$	0	0
$601$	2.62323e14	0.136467	0.0682333	−	0.997669i	$-0.478264\pi$
$601$	2.62323e14	0.136467	0.0682333	+	0.997669i	$0.478264\pi$
$602$	0	0
$603$	−6.00271e14	−0.306621
$604$	0	0
$605$	2.67279e15	1.34063
$606$	0	0
$607$	1.08014e15	0.532036	0.266018	−	0.963968i	$-0.414292\pi$
$607$	1.08014e15	0.532036	0.266018	+	0.963968i	$0.414292\pi$
$608$	0	0
$609$	−3.46545e14	−0.167635
$610$	0	0
$611$	−3.21452e15	−1.52718
$612$	0	0
$613$	−4.17420e15	−1.94778	−0.973892	−	0.227010i	$-0.927105\pi$
$613$	−4.17420e15	−1.94778	−0.973892	+	0.227010i	$0.927105\pi$
$614$	0	0
$615$	1.10757e15	0.507643
$616$	0	0
$617$	2.81771e15	1.26861	0.634306	−	0.773082i	$-0.281286\pi$
$617$	2.81771e15	1.26861	0.634306	+	0.773082i	$0.281286\pi$
$618$	0	0
$619$	−8.18108e14	−0.361836	−0.180918	−	0.983498i	$-0.557907\pi$
$619$	−8.18108e14	−0.361836	−0.180918	+	0.983498i	$0.557907\pi$
$620$	0	0
$621$	−2.71561e14	−0.117995
$622$	0	0
$623$	−1.00001e15	−0.426894
$624$	0	0
$625$	−2.71854e15	−1.14024
$626$	0	0
$627$	−8.52740e13	−0.0351435
$628$	0	0
$629$	−5.44242e14	−0.220401
$630$	0	0
$631$	−2.52392e15	−1.00442	−0.502208	−	0.864747i	$-0.667479\pi$
$631$	−2.52392e15	−1.00442	−0.502208	+	0.864747i	$0.667479\pi$
$632$	0	0
$633$	1.78779e14	0.0699191
$634$	0	0
$635$	−4.14217e15	−1.59211
$636$	0	0
$637$	4.43184e14	0.167424
$638$	0	0
$639$	−6.34862e14	−0.235735
$640$	0	0
$641$	−5.17940e15	−1.89043	−0.945214	−	0.326451i	$-0.894147\pi$
$641$	−5.17940e15	−1.89043	−0.945214	+	0.326451i	$0.894147\pi$
$642$	0	0
$643$	5.22350e15	1.87414	0.937068	−	0.349147i	$-0.113529\pi$
$643$	5.22350e15	1.87414	0.937068	+	0.349147i	$0.113529\pi$
$644$	0	0
$645$	3.39484e15	1.19740
$646$	0	0
$647$	−1.59957e14	−0.0554663	−0.0277332	−	0.999615i	$-0.508829\pi$
$647$	−1.59957e14	−0.0554663	−0.0277332	+	0.999615i	$0.508829\pi$
$648$	0	0
$649$	−3.45186e14	−0.117681
$650$	0	0
$651$	2.19613e13	0.00736142
$652$	0	0
$653$	9.98003e14	0.328934	0.164467	−	0.986383i	$-0.447410\pi$
$653$	9.98003e14	0.328934	0.164467	+	0.986383i	$0.447410\pi$
$654$	0	0
$655$	−5.43325e15	−1.76089
$656$	0	0
$657$	−3.58044e14	−0.114111
$658$	0	0
$659$	1.00059e15	0.313607	0.156804	−	0.987630i	$-0.449881\pi$
$659$	1.00059e15	0.313607	0.156804	+	0.987630i	$0.449881\pi$
$660$	0	0
$661$	−3.24748e15	−1.00101	−0.500506	−	0.865733i	$-0.666853\pi$
$661$	−3.24748e15	−1.00101	−0.500506	+	0.865733i	$0.666853\pi$
$662$	0	0
$663$	−6.16346e14	−0.186853
$664$	0	0
$665$	1.08106e15	0.322351
$666$	0	0
$667$	−1.60587e15	−0.470997
$668$	0	0
$669$	5.29770e14	0.152842
$670$	0	0
$671$	3.74444e14	0.106270
$672$	0	0
$673$	−2.40342e14	−0.0671038	−0.0335519	−	0.999437i	$-0.510682\pi$
$673$	−2.40342e14	−0.0671038	−0.0335519	+	0.999437i	$0.510682\pi$
$674$	0	0
$675$	−5.82435e14	−0.159984
$676$	0	0
$677$	2.97213e15	0.803212	0.401606	−	0.915813i	$-0.368452\pi$
$677$	2.97213e15	0.803212	0.401606	+	0.915813i	$0.368452\pi$
$678$	0	0
$679$	1.31862e15	0.350619
$680$	0	0
$681$	2.39772e15	0.627321
$682$	0	0
$683$	−1.96947e15	−0.507031	−0.253516	−	0.967331i	$-0.581587\pi$
$683$	−1.96947e15	−0.507031	−0.253516	+	0.967331i	$0.581587\pi$
$684$	0	0
$685$	2.05132e15	0.519680
$686$	0	0
$687$	3.23260e15	0.805916
$688$	0	0
$689$	1.53339e15	0.376224
$690$	0	0
$691$	9.86476e14	0.238209	0.119104	−	0.992882i	$-0.461998\pi$
$691$	9.86476e14	0.238209	0.119104	+	0.992882i	$0.461998\pi$
$692$	0	0
$693$	−5.12001e13	−0.0121685
$694$	0	0
$695$	−1.09359e16	−2.55822
$696$	0	0
$697$	7.79229e14	0.179426
$698$	0	0
$699$	1.22163e15	0.276895
$700$	0	0
$701$	−1.20434e15	−0.268721	−0.134361	−	0.990933i	$-0.542898\pi$
$701$	−1.20434e15	−0.268721	−0.134361	+	0.990933i	$0.542898\pi$
$702$	0	0
$703$	−2.28992e15	−0.502997
$704$	0	0
$705$	−4.70796e15	−1.01811
$706$	0	0
$707$	1.00850e15	0.214718
$708$	0	0
$709$	5.78518e15	1.21273	0.606363	−	0.795188i	$-0.292628\pi$
$709$	5.78518e15	1.21273	0.606363	+	0.795188i	$0.292628\pi$
$710$	0	0
$711$	−7.11442e14	−0.146844
$712$	0	0
$713$	1.01767e14	0.0206831
$714$	0	0
$715$	−7.65397e14	−0.153180
$716$	0	0
$717$	4.74684e15	0.935512
$718$	0	0
$719$	−8.40625e15	−1.63152	−0.815762	−	0.578387i	$-0.803682\pi$
$719$	−8.40625e15	−1.63152	−0.815762	+	0.578387i	$0.803682\pi$
$720$	0	0
$721$	8.24397e14	0.157577
$722$	0	0
$723$	−6.23184e14	−0.117316
$724$	0	0
$725$	−3.44422e15	−0.638605
$726$	0	0
$727$	−1.42770e14	−0.0260734	−0.0130367	−	0.999915i	$-0.504150\pi$
$727$	−1.42770e14	−0.0260734	−0.0130367	+	0.999915i	$0.504150\pi$
$728$	0	0
$729$	2.05891e14	0.0370370
$730$	0	0
$731$	2.38843e15	0.423221
$732$	0	0
$733$	−4.07818e15	−0.711860	−0.355930	−	0.934513i	$-0.615836\pi$
$733$	−4.07818e15	−0.711860	−0.355930	+	0.934513i	$0.615836\pi$
$734$	0	0
$735$	6.49085e14	0.111615
$736$	0	0
$737$	−5.24448e14	−0.0888446
$738$	0	0
$739$	−6.88325e15	−1.14881	−0.574406	−	0.818571i	$-0.694767\pi$
$739$	−6.88325e15	−1.14881	−0.574406	+	0.818571i	$0.694767\pi$
$740$	0	0
$741$	−2.59330e15	−0.426435
$742$	0	0
$743$	−2.34148e15	−0.379360	−0.189680	−	0.981846i	$-0.560745\pi$
$743$	−2.34148e15	−0.379360	−0.189680	+	0.981846i	$0.560745\pi$
$744$	0	0
$745$	1.31183e16	2.09420
$746$	0	0
$747$	−7.20880e14	−0.113397
$748$	0	0
$749$	3.21769e15	0.498764
$750$	0	0
$751$	3.32129e15	0.507326	0.253663	−	0.967293i	$-0.418365\pi$
$751$	3.32129e15	0.507326	0.253663	+	0.967293i	$0.418365\pi$
$752$	0	0
$753$	−1.92467e15	−0.289723
$754$	0	0
$755$	−9.03551e15	−1.34043
$756$	0	0
$757$	1.90119e14	0.0277970	0.0138985	−	0.999903i	$-0.495576\pi$
$757$	1.90119e14	0.0277970	0.0138985	+	0.999903i	$0.495576\pi$
$758$	0	0
$759$	−2.37259e14	−0.0341895
$760$	0	0
$761$	−1.19935e16	−1.70345	−0.851726	−	0.523988i	$-0.824444\pi$
$761$	−1.19935e16	−1.70345	−0.851726	+	0.523988i	$0.824444\pi$
$762$	0	0
$763$	4.84669e15	0.678516
$764$	0	0
$765$	−9.02697e14	−0.124567
$766$	0	0
$767$	−1.04976e16	−1.42796
$768$	0	0
$769$	1.44463e16	1.93714	0.968571	−	0.248736i	$-0.0800152\pi$
$769$	1.44463e16	1.93714	0.968571	+	0.248736i	$0.0800152\pi$
$770$	0	0
$771$	7.50430e15	0.991999
$772$	0	0
$773$	1.29373e16	1.68599	0.842995	−	0.537922i	$-0.180790\pi$
$773$	1.29373e16	1.68599	0.842995	+	0.537922i	$0.180790\pi$
$774$	0	0
$775$	2.18268e14	0.0280433
$776$	0	0
$777$	−1.37491e15	−0.174164
$778$	0	0
$779$	3.27863e15	0.409484
$780$	0	0
$781$	−5.54670e14	−0.0683052
$782$	0	0
$783$	1.21754e15	0.147840
$784$	0	0
$785$	9.97366e14	0.119418
$786$	0	0
$787$	4.36868e15	0.515809	0.257905	−	0.966170i	$-0.416968\pi$
$787$	4.36868e15	0.515809	0.257905	+	0.966170i	$0.416968\pi$
$788$	0	0
$789$	5.66017e15	0.659031
$790$	0	0
$791$	1.58869e15	0.182419
$792$	0	0
$793$	1.13873e16	1.28950
$794$	0	0
$795$	2.24579e15	0.250813
$796$	0	0
$797$	−8.17319e14	−0.0900265	−0.0450133	−	0.998986i	$-0.514333\pi$
$797$	−8.17319e14	−0.0900265	−0.0450133	+	0.998986i	$0.514333\pi$
$798$	0	0
$799$	−3.31228e15	−0.359848
$800$	0	0
$801$	3.51339e15	0.376485
$802$	0	0
$803$	−3.12818e14	−0.0330641
$804$	0	0
$805$	3.00783e15	0.313600
$806$	0	0
$807$	7.78459e15	0.800629
$808$	0	0
$809$	9.35074e15	0.948701	0.474350	−	0.880336i	$-0.342683\pi$
$809$	9.35074e15	0.948701	0.474350	+	0.880336i	$0.342683\pi$
$810$	0	0
$811$	6.93654e15	0.694270	0.347135	−	0.937815i	$-0.387155\pi$
$811$	6.93654e15	0.694270	0.347135	+	0.937815i	$0.387155\pi$
$812$	0	0
$813$	4.69451e15	0.463546
$814$	0	0
$815$	−1.30250e16	−1.26886
$816$	0	0
$817$	1.00494e16	0.965871
$818$	0	0
$819$	−1.55706e15	−0.147654
$820$	0	0
$821$	5.83766e15	0.546200	0.273100	−	0.961986i	$-0.411951\pi$
$821$	5.83766e15	0.546200	0.273100	+	0.961986i	$0.411951\pi$
$822$	0	0
$823$	9.51706e14	0.0878625	0.0439313	−	0.999035i	$-0.486012\pi$
$823$	9.51706e14	0.0878625	0.0439313	+	0.999035i	$0.486012\pi$
$824$	0	0
$825$	−5.08865e14	−0.0463560
$826$	0	0
$827$	1.60751e16	1.44501	0.722507	−	0.691363i	$-0.242990\pi$
$827$	1.60751e16	1.44501	0.722507	+	0.691363i	$0.242990\pi$
$828$	0	0
$829$	−5.69836e14	−0.0505475	−0.0252738	−	0.999681i	$-0.508046\pi$
$829$	−5.69836e14	−0.0505475	−0.0252738	+	0.999681i	$0.508046\pi$
$830$	0	0
$831$	−8.05861e14	−0.0705431
$832$	0	0
$833$	4.56662e14	0.0394501
$834$	0	0
$835$	1.65225e15	0.140864
$836$	0	0
$837$	−7.71577e13	−0.00649216
$838$	0	0
$839$	5.65312e14	0.0469459	0.0234729	−	0.999724i	$-0.492528\pi$
$839$	5.65312e14	0.0469459	0.0234729	+	0.999724i	$0.492528\pi$
$840$	0	0
$841$	−5.00063e15	−0.409870
$842$	0	0
$843$	1.18419e16	0.958008
$844$	0	0
$845$	−6.32980e15	−0.505450
$846$	0	0
$847$	4.75050e15	0.374439
$848$	0	0
$849$	−1.05847e16	−0.823546
$850$	0	0
$851$	−6.37125e15	−0.489342
$852$	0	0
$853$	−1.91236e16	−1.44994	−0.724971	−	0.688779i	$-0.758147\pi$
$853$	−1.91236e16	−1.44994	−0.724971	+	0.688779i	$0.758147\pi$
$854$	0	0
$855$	−3.79813e15	−0.284287
$856$	0	0
$857$	−1.14362e15	−0.0845060	−0.0422530	−	0.999107i	$-0.513454\pi$
$857$	−1.14362e15	−0.0845060	−0.0422530	+	0.999107i	$0.513454\pi$
$858$	0	0
$859$	−5.38558e15	−0.392889	−0.196445	−	0.980515i	$-0.562940\pi$
$859$	−5.38558e15	−0.392889	−0.196445	+	0.980515i	$0.562940\pi$
$860$	0	0
$861$	1.96855e15	0.141785
$862$	0	0
$863$	1.13141e16	0.804566	0.402283	−	0.915515i	$-0.368217\pi$
$863$	1.13141e16	0.804566	0.402283	+	0.915515i	$0.368217\pi$
$864$	0	0
$865$	−1.71361e16	−1.20316
$866$	0	0
$867$	7.69298e15	0.533322
$868$	0	0
$869$	−6.21577e14	−0.0425487
$870$	0	0
$871$	−1.59492e16	−1.07805
$872$	0	0
$873$	−4.63278e15	−0.309217
$874$	0	0
$875$	−1.30914e15	−0.0862865
$876$	0	0
$877$	−2.09313e16	−1.36238	−0.681190	−	0.732107i	$-0.738537\pi$
$877$	−2.09313e16	−1.36238	−0.681190	+	0.732107i	$0.738537\pi$
$878$	0	0
$879$	6.81688e15	0.438174
$880$	0	0
$881$	2.23841e16	1.42093	0.710464	−	0.703734i	$-0.248485\pi$
$881$	2.23841e16	1.42093	0.710464	+	0.703734i	$0.248485\pi$
$882$	0	0
$883$	2.83593e16	1.77792	0.888960	−	0.457985i	$-0.151429\pi$
$883$	2.83593e16	1.77792	0.888960	+	0.457985i	$0.151429\pi$
$884$	0	0
$885$	−1.53747e16	−0.951960
$886$	0	0
$887$	2.19331e15	0.134128	0.0670642	−	0.997749i	$-0.478637\pi$
$887$	2.19331e15	0.134128	0.0670642	+	0.997749i	$0.478637\pi$
$888$	0	0
$889$	−7.36213e15	−0.444676
$890$	0	0
$891$	1.79884e14	0.0107316
$892$	0	0
$893$	−1.39365e16	−0.821243
$894$	0	0
$895$	−6.21407e15	−0.361701
$896$	0	0
$897$	−7.21536e15	−0.414858
$898$	0	0
$899$	−4.56272e14	−0.0259146
$900$	0	0
$901$	1.58002e15	0.0886496
$902$	0	0
$903$	6.03385e15	0.334435
$904$	0	0
$905$	−2.03599e16	−1.11483
$906$	0	0
$907$	1.90609e15	0.103111	0.0515553	−	0.998670i	$-0.483582\pi$
$907$	1.90609e15	0.103111	0.0515553	+	0.998670i	$0.483582\pi$
$908$	0	0
$909$	−3.54321e15	−0.189363
$910$	0	0
$911$	−2.98818e15	−0.157781	−0.0788907	−	0.996883i	$-0.525138\pi$
$911$	−2.98818e15	−0.157781	−0.0788907	+	0.996883i	$0.525138\pi$
$912$	0	0
$913$	−6.29823e14	−0.0328571
$914$	0	0
$915$	1.66778e16	0.859654
$916$	0	0
$917$	−9.65683e15	−0.491817
$918$	0	0
$919$	−9.77177e15	−0.491743	−0.245871	−	0.969302i	$-0.579074\pi$
$919$	−9.77177e15	−0.491743	−0.245871	+	0.969302i	$0.579074\pi$
$920$	0	0
$921$	−1.04244e16	−0.518352
$922$	0	0
$923$	−1.68683e16	−0.828822
$924$	0	0
$925$	−1.36649e16	−0.663477
$926$	0	0
$927$	−2.89640e15	−0.138970
$928$	0	0
$929$	−7.81595e15	−0.370591	−0.185296	−	0.982683i	$-0.559324\pi$
$929$	−7.81595e15	−0.370591	−0.185296	+	0.982683i	$0.559324\pi$
$930$	0	0
$931$	1.92142e15	0.0900326
$932$	0	0
$933$	−5.26099e15	−0.243623
$934$	0	0
$935$	−7.88674e14	−0.0360939
$936$	0	0
$937$	1.80085e16	0.814533	0.407267	−	0.913309i	$-0.366482\pi$
$937$	1.80085e16	0.814533	0.407267	+	0.913309i	$0.366482\pi$
$938$	0	0
$939$	−1.70801e15	−0.0763535
$940$	0	0
$941$	1.68461e16	0.744313	0.372157	−	0.928170i	$-0.378618\pi$
$941$	1.68461e16	0.744313	0.372157	+	0.928170i	$0.378618\pi$
$942$	0	0
$943$	9.12217e15	0.398367
$944$	0	0
$945$	−2.28047e15	−0.0984349
$946$	0	0
$947$	1.14851e16	0.490016	0.245008	−	0.969521i	$-0.421209\pi$
$947$	1.14851e16	0.490016	0.245008	+	0.969521i	$0.421209\pi$
$948$	0	0
$949$	−9.51323e15	−0.401202
$950$	0	0
$951$	−1.35880e16	−0.566450
$952$	0	0
$953$	−5.67295e15	−0.233775	−0.116887	−	0.993145i	$-0.537292\pi$
$953$	−5.67295e15	−0.233775	−0.116887	+	0.993145i	$0.537292\pi$
$954$	0	0
$955$	1.69729e16	0.691412
$956$	0	0
$957$	1.06374e15	0.0428372
$958$	0	0
$959$	3.64594e15	0.145147
$960$	0	0
$961$	−2.53796e16	−0.998862
$962$	0	0
$963$	−1.13049e16	−0.439868
$964$	0	0
$965$	1.03740e16	0.399066
$966$	0	0
$967$	1.89032e16	0.718936	0.359468	−	0.933157i	$-0.382958\pi$
$967$	1.89032e16	0.718936	0.359468	+	0.933157i	$0.382958\pi$
$968$	0	0
$969$	−2.67217e15	−0.100481
$970$	0	0
$971$	1.58508e16	0.589314	0.294657	−	0.955603i	$-0.404795\pi$
$971$	1.58508e16	0.589314	0.294657	+	0.955603i	$0.404795\pi$
$972$	0	0
$973$	−1.94371e16	−0.714512
$974$	0	0
$975$	−1.54753e16	−0.562488
$976$	0	0
$977$	−4.46332e15	−0.160413	−0.0802063	−	0.996778i	$-0.525558\pi$
$977$	−4.46332e15	−0.160413	−0.0802063	+	0.996778i	$0.525558\pi$
$978$	0	0
$979$	3.06960e15	0.109088
$980$	0	0
$981$	−1.70282e16	−0.598395
$982$	0	0
$983$	−2.87942e16	−1.00060	−0.500300	−	0.865852i	$-0.666777\pi$
$983$	−2.87942e16	−1.00060	−0.500300	+	0.865852i	$0.666777\pi$
$984$	0	0
$985$	1.64658e16	0.565827
$986$	0	0
$987$	−8.36775e15	−0.284357
$988$	0	0
$989$	2.79605e16	0.939650
$990$	0	0
$991$	−7.84810e15	−0.260831	−0.130415	−	0.991459i	$-0.541631\pi$
$991$	−7.84810e15	−0.260831	−0.130415	+	0.991459i	$0.541631\pi$
$992$	0	0
$993$	1.50050e16	0.493192
$994$	0	0
$995$	−3.85588e16	−1.25342
$996$	0	0
$997$	−3.16967e16	−1.01904	−0.509519	−	0.860459i	$-0.670177\pi$
$997$	−3.16967e16	−1.01904	−0.509519	+	0.860459i	$0.670177\pi$
$998$	0	0
$999$	4.83054e15	0.153598

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	84.12.a.b.1.1	✓	2
3.2	odd	2		252.12.a.d.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
84.12.a.b.1.1	✓	2	1.1	even	1	trivial
252.12.a.d.1.2		2	3.2	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:	\( N \)	\(=\)	\( 84 = 2^{2} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	84.a (trivial)

\(f(q)\)	\(=\)	\(q-243.000 q^{3} -9456.16 q^{5} -16807.0 q^{7} +59049.0 q^{9} +O(q^{10})\)	\(q-243.000 q^{3} -9456.16 q^{5} -16807.0 q^{7} +59049.0 q^{9} +51590.3 q^{11} +1.56893e6 q^{13} +2.29785e6 q^{15} +1.61665e6 q^{17} +6.80210e6 q^{19} +4.08410e6 q^{21} +1.89255e7 q^{23} +4.05909e7 q^{25} -1.43489e7 q^{27} -8.48521e7 q^{29} +5.37726e6 q^{31} -1.25364e7 q^{33} +1.58930e8 q^{35} -3.36649e8 q^{37} -3.81250e8 q^{39} +4.82004e8 q^{41} +1.47740e9 q^{43} -5.58377e8 q^{45} -2.04886e9 q^{47} +2.82475e8 q^{49} -3.92845e8 q^{51} +9.77346e8 q^{53} -4.87846e8 q^{55} -1.65291e9 q^{57} -6.69092e9 q^{59} +7.25803e9 q^{61} -9.92437e8 q^{63} -1.48361e10 q^{65} -1.01656e10 q^{67} -4.59890e9 q^{69} -1.07514e10 q^{71} -6.06351e9 q^{73} -9.86358e9 q^{75} -8.67078e8 q^{77} -1.20483e10 q^{79} +3.48678e9 q^{81} -1.22082e10 q^{83} -1.52873e10 q^{85} +2.06191e10 q^{87} +5.94996e10 q^{89} -2.63690e10 q^{91} -1.30667e9 q^{93} -6.43217e10 q^{95} -7.84565e10 q^{97} +3.04635e9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 486 q^{3} - 8910 q^{5} - 33614 q^{7} + 118098 q^{9}+O(q^{10}) \) 2 * q - 486 * q^3 - 8910 * q^5 - 33614 * q^7 + 118098 * q^9	\( 2 q - 486 q^{3} - 8910 q^{5} - 33614 q^{7} + 118098 q^{9} + 333234 q^{11} + 221184 q^{13} + 2165130 q^{15} + 8240454 q^{17} + 9715288 q^{19} + 8168202 q^{21} + 17112222 q^{23} - 7938950 q^{25} - 28697814 q^{27} + 65950524 q^{29} + 79458480 q^{31} - 80975862 q^{33} + 149750370 q^{35} + 294067288 q^{37} - 53747712 q^{39} - 417495870 q^{41} - 298918424 q^{43} - 526126590 q^{45} + 482895108 q^{47} + 564950498 q^{49} - 2002430322 q^{51} - 4269511296 q^{53} - 334022720 q^{55} - 2360814984 q^{57} - 5231599164 q^{59} - 1849747188 q^{61} - 1984873086 q^{63} - 15572154420 q^{65} - 7454789692 q^{67} - 4158269946 q^{69} - 31535521182 q^{71} - 9729183112 q^{73} + 1929164850 q^{75} - 5600663838 q^{77} - 50956356444 q^{79} + 6973568802 q^{81} - 24244144344 q^{83} - 11669583620 q^{85} - 16025977332 q^{87} + 11485149570 q^{89} - 3717439488 q^{91} - 19308410640 q^{93} - 62730647640 q^{95} - 110800663168 q^{97} + 19677134466 q^{99}+O(q^{100}) \) 2 * q - 486 * q^3 - 8910 * q^5 - 33614 * q^7 + 118098 * q^9 + 333234 * q^11 + 221184 * q^13 + 2165130 * q^15 + 8240454 * q^17 + 9715288 * q^19 + 8168202 * q^21 + 17112222 * q^23 - 7938950 * q^25 - 28697814 * q^27 + 65950524 * q^29 + 79458480 * q^31 - 80975862 * q^33 + 149750370 * q^35 + 294067288 * q^37 - 53747712 * q^39 - 417495870 * q^41 - 298918424 * q^43 - 526126590 * q^45 + 482895108 * q^47 + 564950498 * q^49 - 2002430322 * q^51 - 4269511296 * q^53 - 334022720 * q^55 - 2360814984 * q^57 - 5231599164 * q^59 - 1849747188 * q^61 - 1984873086 * q^63 - 15572154420 * q^65 - 7454789692 * q^67 - 4158269946 * q^69 - 31535521182 * q^71 - 9729183112 * q^73 + 1929164850 * q^75 - 5600663838 * q^77 - 50956356444 * q^79 + 6973568802 * q^81 - 24244144344 * q^83 - 11669583620 * q^85 - 16025977332 * q^87 + 11485149570 * q^89 - 3717439488 * q^91 - 19308410640 * q^93 - 62730647640 * q^95 - 110800663168 * q^97 + 19677134466 * q^99

Introduction

L-functions

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