LMFDB - Embedded newform 84.8.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [84,8,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, 8, names="a")

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

chi := DirichletCharacter("84.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 84 = 2^{2} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 8 $
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:	$26.2403421407$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{3649}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 912 $ x^2 - x - 912
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2}\cdot 3 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$30.7035$ 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)

$f(q)$	$=$	$q-27.0000 q^{3} -230.442 q^{5} +343.000 q^{7} +729.000 q^{9} -5027.09 q^{11} -13772.6 q^{13} +6221.93 q^{15} +32474.7 q^{17} +9651.95 q^{19} -9261.00 q^{21} +32034.5 q^{23} -25021.6 q^{25} -19683.0 q^{27} +103672. q^{29} +241936. q^{31} +135731. q^{33} -79041.5 q^{35} -127147. q^{37} +371860. q^{39} +607138. q^{41} +443862. q^{43} -167992. q^{45} +691778. q^{47} +117649. q^{49} -876817. q^{51} +672824. q^{53} +1.15845e6 q^{55} -260603. q^{57} -2.58646e6 q^{59} +1.53593e6 q^{61} +250047. q^{63} +3.17378e6 q^{65} -4.20823e6 q^{67} -864931. q^{69} +1.51772e6 q^{71} +5.47779e6 q^{73} +675584. q^{75} -1.72429e6 q^{77} -6.75855e6 q^{79} +531441. q^{81} +8.36612e6 q^{83} -7.48353e6 q^{85} -2.79914e6 q^{87} -5.17007e6 q^{89} -4.72400e6 q^{91} -6.53227e6 q^{93} -2.22421e6 q^{95} -1.06286e7 q^{97} -3.66475e6 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 54 q^{3} + 264 q^{5} + 686 q^{7} + 1458 q^{9} - 4980 q^{11} - 10148 q^{13} - 7128 q^{15} + 17832 q^{17} + 6256 q^{19} - 18522 q^{21} + 14052 q^{23} + 141326 q^{25} - 39366 q^{27} + 243588 q^{29} + 470824 q^{31}+ \cdots - 3630420 q^{99}+O(q^{100}) $ 2 * q - 54 * q^3 + 264 * q^5 + 686 * q^7 + 1458 * q^9 - 4980 * q^11 - 10148 * q^13 - 7128 * q^15 + 17832 * q^17 + 6256 * q^19 - 18522 * q^21 + 14052 * q^23 + 141326 * q^25 - 39366 * q^27 + 243588 * q^29 + 470824 * q^31 + 134460 * q^33 + 90552 * q^35 + 311116 * q^37 + 273996 * q^39 + 919248 * q^41 - 112616 * q^43 + 192456 * q^45 - 102456 * q^47 + 235298 * q^49 - 481464 * q^51 + 2720028 * q^53 + 1181736 * q^55 - 168912 * q^57 - 19008 * q^59 - 925148 * q^61 + 500094 * q^63 + 4965936 * q^65 - 2053424 * q^67 - 379404 * q^69 - 869508 * q^71 + 3505228 * q^73 - 3815802 * q^75 - 1708140 * q^77 - 6640856 * q^79 + 1062882 * q^81 + 7856760 * q^83 - 14723496 * q^85 - 6576876 * q^87 - 9330384 * q^89 - 3480764 * q^91 - 12712248 * q^93 - 3903312 * q^95 - 2220212 * q^97 - 3630420 * q^99

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
$3$	−27.0000	−0.577350
$3$	−27.0000	−0.577350
$4$	0	0
$5$	−230.442	−0.824453	−0.412227	−	0.911081i	$-0.635249\pi$
$5$	−230.442	−0.824453	−0.412227	+	0.911081i	$0.635249\pi$
$6$	0	0
$7$	343.000	0.377964
$7$	343.000	0.377964
$8$	0	0
$9$	729.000	0.333333
$10$	0	0
$11$	−5027.09	−1.13879	−0.569393	−	0.822065i	$-0.692822\pi$
$11$	−5027.09	−1.13879	−0.569393	+	0.822065i	$0.692822\pi$
$12$	0	0
$13$	−13772.6	−1.73866	−0.869329	−	0.494234i	$-0.835449\pi$
$13$	−13772.6	−1.73866	−0.869329	+	0.494234i	$0.835449\pi$
$14$	0	0
$15$	6221.93	0.475998
$16$	0	0
$17$	32474.7	1.60315	0.801575	−	0.597894i	$-0.203996\pi$
$17$	32474.7	1.60315	0.801575	+	0.597894i	$0.203996\pi$
$18$	0	0
$19$	9651.95	0.322833	0.161416	−	0.986886i	$-0.448394\pi$
$19$	9651.95	0.322833	0.161416	+	0.986886i	$0.448394\pi$
$20$	0	0
$21$	−9261.00	−0.218218
$22$	0	0
$23$	32034.5	0.548997	0.274499	−	0.961587i	$-0.411488\pi$
$23$	32034.5	0.548997	0.274499	+	0.961587i	$0.411488\pi$
$24$	0	0
$25$	−25021.6	−0.320277
$26$	0	0
$27$	−19683.0	−0.192450
$28$	0	0
$29$	103672.	0.789347	0.394674	−	0.918821i	$-0.370858\pi$
$29$	103672.	0.789347	0.394674	+	0.918821i	$0.370858\pi$
$30$	0	0
$31$	241936.	1.45859	0.729297	−	0.684197i	$-0.239847\pi$
$31$	241936.	1.45859	0.729297	+	0.684197i	$0.239847\pi$
$32$	0	0
$33$	135731.	0.657479
$34$	0	0
$35$	−79041.5	−0.311614
$36$	0	0
$37$	−127147.	−0.412666	−0.206333	−	0.978482i	$-0.566153\pi$
$37$	−127147.	−0.412666	−0.206333	+	0.978482i	$0.566153\pi$
$38$	0	0
$39$	371860.	1.00381
$40$	0	0
$41$	607138.	1.37576	0.687882	−	0.725823i	$-0.258541\pi$
$41$	607138.	1.37576	0.687882	+	0.725823i	$0.258541\pi$
$42$	0	0
$43$	443862.	0.851350	0.425675	−	0.904876i	$-0.360037\pi$
$43$	443862.	0.851350	0.425675	+	0.904876i	$0.360037\pi$
$44$	0	0
$45$	−167992.	−0.274818
$46$	0	0
$47$	691778.	0.971905	0.485953	−	0.873985i	$-0.338473\pi$
$47$	691778.	0.971905	0.485953	+	0.873985i	$0.338473\pi$
$48$	0	0
$49$	117649.	0.142857
$50$	0	0
$51$	−876817.	−0.925579
$52$	0	0
$53$	672824.	0.620778	0.310389	−	0.950610i	$-0.399541\pi$
$53$	672824.	0.620778	0.310389	+	0.950610i	$0.399541\pi$
$54$	0	0
$55$	1.15845e6	0.938877
$56$	0	0
$57$	−260603.	−0.186388
$58$	0	0
$59$	−2.58646e6	−1.63955	−0.819775	−	0.572686i	$-0.805901\pi$
$59$	−2.58646e6	−1.63955	−0.819775	+	0.572686i	$0.805901\pi$
$60$	0	0
$61$	1.53593e6	0.866397	0.433198	−	0.901299i	$-0.357385\pi$
$61$	1.53593e6	0.866397	0.433198	+	0.901299i	$0.357385\pi$
$62$	0	0
$63$	250047.	0.125988
$64$	0	0
$65$	3.17378e6	1.43344
$66$	0	0
$67$	−4.20823e6	−1.70937	−0.854687	−	0.519143i	$-0.826251\pi$
$67$	−4.20823e6	−1.70937	−0.854687	+	0.519143i	$0.826251\pi$
$68$	0	0
$69$	−864931.	−0.316964
$70$	0	0
$71$	1.51772e6	0.503254	0.251627	−	0.967824i	$-0.419034\pi$
$71$	1.51772e6	0.503254	0.251627	+	0.967824i	$0.419034\pi$
$72$	0	0
$73$	5.47779e6	1.64807	0.824034	−	0.566540i	$-0.191718\pi$
$73$	5.47779e6	1.64807	0.824034	+	0.566540i	$0.191718\pi$
$74$	0	0
$75$	675584.	0.184912
$76$	0	0
$77$	−1.72429e6	−0.430421
$78$	0	0
$79$	−6.75855e6	−1.54226	−0.771132	−	0.636675i	$-0.780309\pi$
$79$	−6.75855e6	−1.54226	−0.771132	+	0.636675i	$0.780309\pi$
$80$	0	0
$81$	531441.	0.111111
$82$	0	0
$83$	8.36612e6	1.60602	0.803009	−	0.595966i	$-0.203231\pi$
$83$	8.36612e6	1.60602	0.803009	+	0.595966i	$0.203231\pi$
$84$	0	0
$85$	−7.48353e6	−1.32172
$86$	0	0
$87$	−2.79914e6	−0.455730
$88$	0	0
$89$	−5.17007e6	−0.777377	−0.388688	−	0.921369i	$-0.627072\pi$
$89$	−5.17007e6	−0.777377	−0.388688	+	0.921369i	$0.627072\pi$
$90$	0	0
$91$	−4.72400e6	−0.657151
$92$	0	0
$93$	−6.53227e6	−0.842120
$94$	0	0
$95$	−2.22421e6	−0.266160
$96$	0	0
$97$	−1.06286e7	−1.18242	−0.591212	−	0.806516i	$-0.701350\pi$
$97$	−1.06286e7	−1.18242	−0.591212	+	0.806516i	$0.701350\pi$
$98$	0	0
$99$	−3.66475e6	−0.379596
$100$	0	0
$101$	4.04115e6	0.390283	0.195142	−	0.980775i	$-0.437483\pi$
$101$	4.04115e6	0.390283	0.195142	+	0.980775i	$0.437483\pi$
$102$	0	0
$103$	−1.44735e7	−1.30509	−0.652547	−	0.757748i	$-0.726300\pi$
$103$	−1.44735e7	−1.30509	−0.652547	+	0.757748i	$0.726300\pi$
$104$	0	0
$105$	2.13412e6	0.179910
$106$	0	0
$107$	−6.89622e6	−0.544212	−0.272106	−	0.962267i	$-0.587720\pi$
$107$	−6.89622e6	−0.544212	−0.272106	+	0.962267i	$0.587720\pi$
$108$	0	0
$109$	1.41017e7	1.04299	0.521493	−	0.853255i	$-0.325375\pi$
$109$	1.41017e7	1.04299	0.521493	+	0.853255i	$0.325375\pi$
$110$	0	0
$111$	3.43296e6	0.238253
$112$	0	0
$113$	9.62581e6	0.627571	0.313785	−	0.949494i	$-0.398403\pi$
$113$	9.62581e6	0.627571	0.313785	+	0.949494i	$0.398403\pi$
$114$	0	0
$115$	−7.38208e6	−0.452623
$116$	0	0
$117$	−1.00402e7	−0.579553
$118$	0	0
$119$	1.11388e7	0.605934
$120$	0	0
$121$	5.78448e6	0.296835
$122$	0	0
$123$	−1.63927e7	−0.794297
$124$	0	0
$125$	2.37693e7	1.08851
$126$	0	0
$127$	1.76901e6	0.0766333	0.0383166	−	0.999266i	$-0.487800\pi$
$127$	1.76901e6	0.0766333	0.0383166	+	0.999266i	$0.487800\pi$
$128$	0	0
$129$	−1.19843e7	−0.491527
$130$	0	0
$131$	1.67408e7	0.650618	0.325309	−	0.945608i	$-0.394532\pi$
$131$	1.67408e7	0.650618	0.325309	+	0.945608i	$0.394532\pi$
$132$	0	0
$133$	3.31062e6	0.122019
$134$	0	0
$135$	4.53578e6	0.158666
$136$	0	0
$137$	3.38365e7	1.12425	0.562125	−	0.827052i	$-0.309984\pi$
$137$	3.38365e7	1.12425	0.562125	+	0.827052i	$0.309984\pi$
$138$	0	0
$139$	1.61726e7	0.510773	0.255386	−	0.966839i	$-0.417797\pi$
$139$	1.61726e7	0.510773	0.255386	+	0.966839i	$0.417797\pi$
$140$	0	0
$141$	−1.86780e7	−0.561130
$142$	0	0
$143$	6.92361e7	1.97996
$144$	0	0
$145$	−2.38903e7	−0.650780
$146$	0	0
$147$	−3.17652e6	−0.0824786
$148$	0	0
$149$	2.85945e7	0.708158	0.354079	−	0.935216i	$-0.384794\pi$
$149$	2.85945e7	0.708158	0.354079	+	0.935216i	$0.384794\pi$
$150$	0	0
$151$	−3.16118e7	−0.747189	−0.373594	−	0.927592i	$-0.621875\pi$
$151$	−3.16118e7	−0.747189	−0.373594	+	0.927592i	$0.621875\pi$
$152$	0	0
$153$	2.36741e7	0.534383
$154$	0	0
$155$	−5.57521e7	−1.20254
$156$	0	0
$157$	3.43563e7	0.708528	0.354264	−	0.935145i	$-0.384731\pi$
$157$	3.43563e7	0.708528	0.354264	+	0.935145i	$0.384731\pi$
$158$	0	0
$159$	−1.81663e7	−0.358406
$160$	0	0
$161$	1.09878e7	0.207501
$162$	0	0
$163$	−2.46318e7	−0.445492	−0.222746	−	0.974877i	$-0.571502\pi$
$163$	−2.46318e7	−0.445492	−0.222746	+	0.974877i	$0.571502\pi$
$164$	0	0
$165$	−3.12782e7	−0.542061
$166$	0	0
$167$	3.01569e7	0.501048	0.250524	−	0.968110i	$-0.419397\pi$
$167$	3.01569e7	0.501048	0.250524	+	0.968110i	$0.419397\pi$
$168$	0	0
$169$	1.26936e8	2.02293
$170$	0	0
$171$	7.03627e6	0.107611
$172$	0	0
$173$	7.66151e7	1.12500	0.562501	−	0.826797i	$-0.309839\pi$
$173$	7.66151e7	1.12500	0.562501	+	0.826797i	$0.309839\pi$
$174$	0	0
$175$	−8.58241e6	−0.121053
$176$	0	0
$177$	6.98345e7	0.946594
$178$	0	0
$179$	−3.55691e7	−0.463541	−0.231770	−	0.972771i	$-0.574452\pi$
$179$	−3.55691e7	−0.463541	−0.231770	+	0.972771i	$0.574452\pi$
$180$	0	0
$181$	6.85132e6	0.0858814	0.0429407	−	0.999078i	$-0.486327\pi$
$181$	6.85132e6	0.0858814	0.0429407	+	0.999078i	$0.486327\pi$
$182$	0	0
$183$	−4.14701e7	−0.500214
$184$	0	0
$185$	2.92999e7	0.340224
$186$	0	0
$187$	−1.63253e8	−1.82565
$188$	0	0
$189$	−6.75127e6	−0.0727393
$190$	0	0
$191$	−3.63357e7	−0.377326	−0.188663	−	0.982042i	$-0.560415\pi$
$191$	−3.63357e7	−0.377326	−0.188663	+	0.982042i	$0.560415\pi$
$192$	0	0
$193$	−1.25918e8	−1.26077	−0.630386	−	0.776281i	$-0.717104\pi$
$193$	−1.25918e8	−1.26077	−0.630386	+	0.776281i	$0.717104\pi$
$194$	0	0
$195$	−8.56921e7	−0.827599
$196$	0	0
$197$	1.66346e8	1.55018	0.775088	−	0.631853i	$-0.217705\pi$
$197$	1.66346e8	1.55018	0.775088	+	0.631853i	$0.217705\pi$
$198$	0	0
$199$	−7.63051e7	−0.686385	−0.343193	−	0.939265i	$-0.611508\pi$
$199$	−7.63051e7	−0.686385	−0.343193	+	0.939265i	$0.611508\pi$
$200$	0	0
$201$	1.13622e8	0.986908
$202$	0	0
$203$	3.55595e7	0.298345
$204$	0	0
$205$	−1.39910e8	−1.13425
$206$	0	0
$207$	2.33531e7	0.182999
$208$	0	0
$209$	−4.85212e7	−0.367638
$210$	0	0
$211$	1.99778e8	1.46406	0.732031	−	0.681271i	$-0.238572\pi$
$211$	1.99778e8	1.46406	0.732031	+	0.681271i	$0.238572\pi$
$212$	0	0
$213$	−4.09784e7	−0.290554
$214$	0	0
$215$	−1.02284e8	−0.701898
$216$	0	0
$217$	8.29840e7	0.551297
$218$	0	0
$219$	−1.47900e8	−0.951513
$220$	0	0
$221$	−4.47261e8	−2.78733
$222$	0	0
$223$	3.84285e7	0.232053	0.116026	−	0.993246i	$-0.462984\pi$
$223$	3.84285e7	0.232053	0.116026	+	0.993246i	$0.462984\pi$
$224$	0	0
$225$	−1.82408e7	−0.106759
$226$	0	0
$227$	−2.11611e8	−1.20074	−0.600370	−	0.799723i	$-0.704980\pi$
$227$	−2.11611e8	−1.20074	−0.600370	+	0.799723i	$0.704980\pi$
$228$	0	0
$229$	1.25230e8	0.689103	0.344552	−	0.938767i	$-0.388031\pi$
$229$	1.25230e8	0.689103	0.344552	+	0.938767i	$0.388031\pi$
$230$	0	0
$231$	4.65559e7	0.248504
$232$	0	0
$233$	6.15511e7	0.318779	0.159390	−	0.987216i	$-0.449047\pi$
$233$	6.15511e7	0.318779	0.159390	+	0.987216i	$0.449047\pi$
$234$	0	0
$235$	−1.59414e8	−0.801291
$236$	0	0
$237$	1.82481e8	0.890427
$238$	0	0
$239$	−2.58334e8	−1.22402	−0.612010	−	0.790850i	$-0.709639\pi$
$239$	−2.58334e8	−1.22402	−0.612010	+	0.790850i	$0.709639\pi$
$240$	0	0
$241$	3.85619e8	1.77459	0.887297	−	0.461199i	$-0.152581\pi$
$241$	3.85619e8	1.77459	0.887297	+	0.461199i	$0.152581\pi$
$242$	0	0
$243$	−1.43489e7	−0.0641500
$244$	0	0
$245$	−2.71112e7	−0.117779
$246$	0	0
$247$	−1.32932e8	−0.561296
$248$	0	0
$249$	−2.25885e8	−0.927235
$250$	0	0
$251$	2.98427e8	1.19119	0.595593	−	0.803286i	$-0.296917\pi$
$251$	2.98427e8	1.19119	0.595593	+	0.803286i	$0.296917\pi$
$252$	0	0
$253$	−1.61040e8	−0.625191
$254$	0	0
$255$	2.02055e8	0.763097
$256$	0	0
$257$	−4.05861e8	−1.49146	−0.745730	−	0.666249i	$-0.767899\pi$
$257$	−4.05861e8	−1.49146	−0.745730	+	0.666249i	$0.767899\pi$
$258$	0	0
$259$	−4.36113e7	−0.155973
$260$	0	0
$261$	7.55768e7	0.263116
$262$	0	0
$263$	−2.64406e8	−0.896243	−0.448121	−	0.893973i	$-0.647907\pi$
$263$	−2.64406e8	−0.896243	−0.448121	+	0.893973i	$0.647907\pi$
$264$	0	0
$265$	−1.55047e8	−0.511802
$266$	0	0
$267$	1.39592e8	0.448819
$268$	0	0
$269$	1.98913e8	0.623060	0.311530	−	0.950236i	$-0.399159\pi$
$269$	1.98913e8	0.623060	0.311530	+	0.950236i	$0.399159\pi$
$270$	0	0
$271$	1.40372e8	0.428436	0.214218	−	0.976786i	$-0.431280\pi$
$271$	1.40372e8	0.428436	0.214218	+	0.976786i	$0.431280\pi$
$272$	0	0
$273$	1.27548e8	0.379406
$274$	0	0
$275$	1.25786e8	0.364727
$276$	0	0
$277$	−1.40282e8	−0.396571	−0.198286	−	0.980144i	$-0.563537\pi$
$277$	−1.40282e8	−0.396571	−0.198286	+	0.980144i	$0.563537\pi$
$278$	0	0
$279$	1.76371e8	0.486198
$280$	0	0
$281$	3.41638e8	0.918532	0.459266	−	0.888299i	$-0.348113\pi$
$281$	3.41638e8	0.918532	0.459266	+	0.888299i	$0.348113\pi$
$282$	0	0
$283$	3.73842e8	0.980473	0.490237	−	0.871589i	$-0.336910\pi$
$283$	3.73842e8	0.980473	0.490237	+	0.871589i	$0.336910\pi$
$284$	0	0
$285$	6.00537e7	0.153668
$286$	0	0
$287$	2.08248e8	0.519990
$288$	0	0
$289$	6.44268e8	1.57009
$290$	0	0
$291$	2.86971e8	0.682672
$292$	0	0
$293$	−7.10879e7	−0.165105	−0.0825523	−	0.996587i	$-0.526307\pi$
$293$	−7.10879e7	−0.165105	−0.0825523	+	0.996587i	$0.526307\pi$
$294$	0	0
$295$	5.96029e8	1.35173
$296$	0	0
$297$	9.89483e7	0.219160
$298$	0	0
$299$	−4.41198e8	−0.954519
$300$	0	0
$301$	1.52245e8	0.321780
$302$	0	0
$303$	−1.09111e8	−0.225330
$304$	0	0
$305$	−3.53942e8	−0.714304
$306$	0	0
$307$	6.85606e8	1.35235	0.676177	−	0.736739i	$-0.263636\pi$
$307$	6.85606e8	1.35235	0.676177	+	0.736739i	$0.263636\pi$
$308$	0	0
$309$	3.90783e8	0.753497
$310$	0	0
$311$	2.64884e8	0.499337	0.249669	−	0.968331i	$-0.419678\pi$
$311$	2.64884e8	0.499337	0.249669	+	0.968331i	$0.419678\pi$
$312$	0	0
$313$	−5.69557e8	−1.04986	−0.524931	−	0.851145i	$-0.675909\pi$
$313$	−5.69557e8	−1.04986	−0.524931	+	0.851145i	$0.675909\pi$
$314$	0	0
$315$	−5.76213e7	−0.103871
$316$	0	0
$317$	−9.13159e8	−1.61005	−0.805025	−	0.593241i	$-0.797848\pi$
$317$	−9.13159e8	−1.61005	−0.805025	+	0.593241i	$0.797848\pi$
$318$	0	0
$319$	−5.21168e8	−0.898898
$320$	0	0
$321$	1.86198e8	0.314201
$322$	0	0
$323$	3.13444e8	0.517549
$324$	0	0
$325$	3.44613e8	0.556852
$326$	0	0
$327$	−3.80746e8	−0.602169
$328$	0	0
$329$	2.37280e8	0.367346
$330$	0	0
$331$	−4.39121e8	−0.665559	−0.332779	−	0.943005i	$-0.607986\pi$
$331$	−4.39121e8	−0.665559	−0.332779	+	0.943005i	$0.607986\pi$
$332$	0	0
$333$	−9.26898e7	−0.137555
$334$	0	0
$335$	9.69751e8	1.40930
$336$	0	0
$337$	−6.75259e8	−0.961094	−0.480547	−	0.876969i	$-0.659562\pi$
$337$	−6.75259e8	−0.961094	−0.480547	+	0.876969i	$0.659562\pi$
$338$	0	0
$339$	−2.59897e8	−0.362328
$340$	0	0
$341$	−1.21623e9	−1.66103
$342$	0	0
$343$	4.03536e7	0.0539949
$344$	0	0
$345$	1.99316e8	0.261322
$346$	0	0
$347$	5.40071e7	0.0693901	0.0346950	−	0.999398i	$-0.488954\pi$
$347$	5.40071e7	0.0693901	0.0346950	+	0.999398i	$0.488954\pi$
$348$	0	0
$349$	1.54651e9	1.94744	0.973722	−	0.227740i	$-0.0731336\pi$
$349$	1.54651e9	1.94744	0.973722	+	0.227740i	$0.0731336\pi$
$350$	0	0
$351$	2.71086e8	0.334605
$352$	0	0
$353$	6.29969e8	0.762269	0.381134	−	0.924520i	$-0.375534\pi$
$353$	6.29969e8	0.762269	0.381134	+	0.924520i	$0.375534\pi$
$354$	0	0
$355$	−3.49746e8	−0.414910
$356$	0	0
$357$	−3.00748e8	−0.349836
$358$	0	0
$359$	−7.64717e8	−0.872309	−0.436155	−	0.899872i	$-0.643660\pi$
$359$	−7.64717e8	−0.872309	−0.436155	+	0.899872i	$0.643660\pi$
$360$	0	0
$361$	−8.00712e8	−0.895779
$362$	0	0
$363$	−1.56181e8	−0.171378
$364$	0	0
$365$	−1.26231e9	−1.35876
$366$	0	0
$367$	5.11425e8	0.540071	0.270035	−	0.962850i	$-0.412965\pi$
$367$	5.11425e8	0.540071	0.270035	+	0.962850i	$0.412965\pi$
$368$	0	0
$369$	4.42603e8	0.458588
$370$	0	0
$371$	2.30779e8	0.234632
$372$	0	0
$373$	−1.44200e8	−0.143875	−0.0719373	−	0.997409i	$-0.522918\pi$
$373$	−1.44200e8	−0.143875	−0.0719373	+	0.997409i	$0.522918\pi$
$374$	0	0
$375$	−6.41771e8	−0.628450
$376$	0	0
$377$	−1.42783e9	−1.37241
$378$	0	0
$379$	−2.26277e8	−0.213502	−0.106751	−	0.994286i	$-0.534045\pi$
$379$	−2.26277e8	−0.213502	−0.106751	+	0.994286i	$0.534045\pi$
$380$	0	0
$381$	−4.77633e7	−0.0442442
$382$	0	0
$383$	1.46072e9	1.32853	0.664263	−	0.747499i	$-0.268745\pi$
$383$	1.46072e9	1.32853	0.664263	+	0.747499i	$0.268745\pi$
$384$	0	0
$385$	3.97349e8	0.354862
$386$	0	0
$387$	3.23575e8	0.283783
$388$	0	0
$389$	−1.04642e9	−0.901324	−0.450662	−	0.892695i	$-0.648812\pi$
$389$	−1.04642e9	−0.901324	−0.450662	+	0.892695i	$0.648812\pi$
$390$	0	0
$391$	1.04031e9	0.880125
$392$	0	0
$393$	−4.52001e8	−0.375635
$394$	0	0
$395$	1.55745e9	1.27152
$396$	0	0
$397$	5.50414e7	0.0441492	0.0220746	−	0.999756i	$-0.492973\pi$
$397$	5.50414e7	0.0441492	0.0220746	+	0.999756i	$0.492973\pi$
$398$	0	0
$399$	−8.93867e7	−0.0704479
$400$	0	0
$401$	6.34403e7	0.0491315	0.0245658	−	0.999698i	$-0.492180\pi$
$401$	6.34403e7	0.0491315	0.0245658	+	0.999698i	$0.492180\pi$
$402$	0	0
$403$	−3.33209e9	−2.53600
$404$	0	0
$405$	−1.22466e8	−0.0916059
$406$	0	0
$407$	6.39177e8	0.469938
$408$	0	0
$409$	−2.13211e9	−1.54091	−0.770455	−	0.637494i	$-0.779971\pi$
$409$	−2.13211e9	−1.54091	−0.770455	+	0.637494i	$0.779971\pi$
$410$	0	0
$411$	−9.13584e8	−0.649086
$412$	0	0
$413$	−8.87157e8	−0.619691
$414$	0	0
$415$	−1.92790e9	−1.32409
$416$	0	0
$417$	−4.36660e8	−0.294895
$418$	0	0
$419$	7.69058e8	0.510752	0.255376	−	0.966842i	$-0.417801\pi$
$419$	7.69058e8	0.510752	0.255376	+	0.966842i	$0.417801\pi$
$420$	0	0
$421$	4.56924e8	0.298440	0.149220	−	0.988804i	$-0.452324\pi$
$421$	4.56924e8	0.298440	0.149220	+	0.988804i	$0.452324\pi$
$422$	0	0
$423$	5.04306e8	0.323968
$424$	0	0
$425$	−8.12570e8	−0.513451
$426$	0	0
$427$	5.26824e8	0.327467
$428$	0	0
$429$	−1.86938e9	−1.14313
$430$	0	0
$431$	2.10451e9	1.26614	0.633069	−	0.774095i	$-0.281795\pi$
$431$	2.10451e9	1.26614	0.633069	+	0.774095i	$0.281795\pi$
$432$	0	0
$433$	−7.87114e8	−0.465940	−0.232970	−	0.972484i	$-0.574844\pi$
$433$	−7.87114e8	−0.465940	−0.232970	+	0.972484i	$0.574844\pi$
$434$	0	0
$435$	6.45039e8	0.375728
$436$	0	0
$437$	3.09195e8	0.177234
$438$	0	0
$439$	1.35373e9	0.763672	0.381836	−	0.924230i	$-0.375292\pi$
$439$	1.35373e9	0.763672	0.381836	+	0.924230i	$0.375292\pi$
$440$	0	0
$441$	8.57661e7	0.0476190
$442$	0	0
$443$	2.31247e9	1.26375	0.631876	−	0.775069i	$-0.282285\pi$
$443$	2.31247e9	1.26375	0.631876	+	0.775069i	$0.282285\pi$
$444$	0	0
$445$	1.19140e9	0.640911
$446$	0	0
$447$	−7.72051e8	−0.408855
$448$	0	0
$449$	3.25028e9	1.69457	0.847283	−	0.531142i	$-0.178237\pi$
$449$	3.25028e9	1.69457	0.847283	+	0.531142i	$0.178237\pi$
$450$	0	0
$451$	−3.05214e9	−1.56670
$452$	0	0
$453$	8.53519e8	0.431390
$454$	0	0
$455$	1.08861e9	0.541790
$456$	0	0
$457$	6.91903e8	0.339108	0.169554	−	0.985521i	$-0.445767\pi$
$457$	6.91903e8	0.339108	0.169554	+	0.985521i	$0.445767\pi$
$458$	0	0
$459$	−6.39200e8	−0.308526
$460$	0	0
$461$	3.55884e9	1.69183	0.845913	−	0.533322i	$-0.179056\pi$
$461$	3.55884e9	1.69183	0.845913	+	0.533322i	$0.179056\pi$
$462$	0	0
$463$	3.71651e8	0.174021	0.0870105	−	0.996207i	$-0.472269\pi$
$463$	3.71651e8	0.174021	0.0870105	+	0.996207i	$0.472269\pi$
$464$	0	0
$465$	1.50531e9	0.694288
$466$	0	0
$467$	−3.67510e9	−1.66978	−0.834891	−	0.550416i	$-0.814469\pi$
$467$	−3.67510e9	−1.66978	−0.834891	+	0.550416i	$0.814469\pi$
$468$	0	0
$469$	−1.44342e9	−0.646083
$470$	0	0
$471$	−9.27619e8	−0.409069
$472$	0	0
$473$	−2.23133e9	−0.969506
$474$	0	0
$475$	−2.41507e8	−0.103396
$476$	0	0
$477$	4.90489e8	0.206926
$478$	0	0
$479$	1.31391e9	0.546248	0.273124	−	0.961979i	$-0.411943\pi$
$479$	1.31391e9	0.546248	0.273124	+	0.961979i	$0.411943\pi$
$480$	0	0
$481$	1.75114e9	0.717485
$482$	0	0
$483$	−2.96671e8	−0.119801
$484$	0	0
$485$	2.44926e9	0.974853
$486$	0	0
$487$	4.98027e9	1.95390	0.976948	−	0.213478i	$-0.0684791\pi$
$487$	4.98027e9	1.95390	0.976948	+	0.213478i	$0.0684791\pi$
$488$	0	0
$489$	6.65059e8	0.257205
$490$	0	0
$491$	2.27145e9	0.866000	0.433000	−	0.901394i	$-0.357455\pi$
$491$	2.27145e9	0.866000	0.433000	+	0.901394i	$0.357455\pi$
$492$	0	0
$493$	3.36672e9	1.26544
$494$	0	0
$495$	8.44511e8	0.312959
$496$	0	0
$497$	5.20578e8	0.190212
$498$	0	0
$499$	−1.08175e9	−0.389741	−0.194870	−	0.980829i	$-0.562429\pi$
$499$	−1.08175e9	−0.389741	−0.194870	+	0.980829i	$0.562429\pi$
$500$	0	0
$501$	−8.14236e8	−0.289280
$502$	0	0
$503$	−2.19858e9	−0.770290	−0.385145	−	0.922856i	$-0.625849\pi$
$503$	−2.19858e9	−0.770290	−0.385145	+	0.922856i	$0.625849\pi$
$504$	0	0
$505$	−9.31249e8	−0.321770
$506$	0	0
$507$	−3.42727e9	−1.16794
$508$	0	0
$509$	1.07364e9	0.360866	0.180433	−	0.983587i	$-0.442250\pi$
$509$	1.07364e9	0.360866	0.180433	+	0.983587i	$0.442250\pi$
$510$	0	0
$511$	1.87888e9	0.622911
$512$	0	0
$513$	−1.89979e8	−0.0621292
$514$	0	0
$515$	3.33529e9	1.07599
$516$	0	0
$517$	−3.47763e9	−1.10679
$518$	0	0
$519$	−2.06861e9	−0.649520
$520$	0	0
$521$	1.85536e7	0.00574773	0.00287386	−	0.999996i	$-0.499085\pi$
$521$	1.85536e7	0.00574773	0.00287386	+	0.999996i	$0.499085\pi$
$522$	0	0
$523$	2.63388e9	0.805081	0.402541	−	0.915402i	$-0.368127\pi$
$523$	2.63388e9	0.805081	0.402541	+	0.915402i	$0.368127\pi$
$524$	0	0
$525$	2.31725e8	0.0698901
$526$	0	0
$527$	7.85680e9	2.33834
$528$	0	0
$529$	−2.37862e9	−0.698602
$530$	0	0
$531$	−1.88553e9	−0.546516
$532$	0	0
$533$	−8.36187e9	−2.39198
$534$	0	0
$535$	1.58918e9	0.448677
$536$	0	0
$537$	9.60367e8	0.267625
$538$	0	0
$539$	−5.91432e8	−0.162684
$540$	0	0
$541$	5.11805e9	1.38968	0.694839	−	0.719165i	$-0.255475\pi$
$541$	5.11805e9	1.38968	0.694839	+	0.719165i	$0.255475\pi$
$542$	0	0
$543$	−1.84986e8	−0.0495836
$544$	0	0
$545$	−3.24962e9	−0.859894
$546$	0	0
$547$	−2.60774e9	−0.681253	−0.340627	−	0.940199i	$-0.610639\pi$
$547$	−2.60774e9	−0.681253	−0.340627	+	0.940199i	$0.610639\pi$
$548$	0	0
$549$	1.11969e9	0.288799
$550$	0	0
$551$	1.00064e9	0.254827
$552$	0	0
$553$	−2.31818e9	−0.582921
$554$	0	0
$555$	−7.91096e8	−0.196428
$556$	0	0
$557$	5.37079e9	1.31688	0.658438	−	0.752635i	$-0.271218\pi$
$557$	5.37079e9	1.31688	0.658438	+	0.752635i	$0.271218\pi$
$558$	0	0
$559$	−6.11313e9	−1.48021
$560$	0	0
$561$	4.40784e9	1.05404
$562$	0	0
$563$	−5.79755e9	−1.36919	−0.684597	−	0.728921i	$-0.740022\pi$
$563$	−5.79755e9	−1.36919	−0.684597	+	0.728921i	$0.740022\pi$
$564$	0	0
$565$	−2.21819e9	−0.517403
$566$	0	0
$567$	1.82284e8	0.0419961
$568$	0	0
$569$	−3.81291e9	−0.867687	−0.433844	−	0.900988i	$-0.642843\pi$
$569$	−3.81291e9	−0.867687	−0.433844	+	0.900988i	$0.642843\pi$
$570$	0	0
$571$	−5.08818e9	−1.14376	−0.571881	−	0.820336i	$-0.693786\pi$
$571$	−5.08818e9	−1.14376	−0.571881	+	0.820336i	$0.693786\pi$
$572$	0	0
$573$	9.81064e8	0.217849
$574$	0	0
$575$	−8.01554e8	−0.175831
$576$	0	0
$577$	−2.24096e9	−0.485645	−0.242823	−	0.970071i	$-0.578073\pi$
$577$	−2.24096e9	−0.485645	−0.242823	+	0.970071i	$0.578073\pi$
$578$	0	0
$579$	3.39978e9	0.727908
$580$	0	0
$581$	2.86958e9	0.607018
$582$	0	0
$583$	−3.38235e9	−0.706934
$584$	0	0
$585$	2.31369e9	0.477814
$586$	0	0
$587$	−7.24473e9	−1.47839	−0.739194	−	0.673492i	$-0.764793\pi$
$587$	−7.24473e9	−1.47839	−0.739194	+	0.673492i	$0.764793\pi$
$588$	0	0
$589$	2.33515e9	0.470882
$590$	0	0
$591$	−4.49135e9	−0.894995
$592$	0	0
$593$	−7.11833e9	−1.40180	−0.700901	−	0.713259i	$-0.747218\pi$
$593$	−7.11833e9	−1.40180	−0.700901	+	0.713259i	$0.747218\pi$
$594$	0	0
$595$	−2.56685e9	−0.499564
$596$	0	0
$597$	2.06024e9	0.396285
$598$	0	0
$599$	9.02747e9	1.71622	0.858109	−	0.513468i	$-0.171639\pi$
$599$	9.02747e9	1.71622	0.858109	+	0.513468i	$0.171639\pi$
$600$	0	0
$601$	9.56566e9	1.79744	0.898719	−	0.438524i	$-0.144499\pi$
$601$	9.56566e9	1.79744	0.898719	+	0.438524i	$0.144499\pi$
$602$	0	0
$603$	−3.06780e9	−0.569791
$604$	0	0
$605$	−1.33299e9	−0.244727
$606$	0	0
$607$	−3.53359e9	−0.641292	−0.320646	−	0.947199i	$-0.603900\pi$
$607$	−3.53359e9	−0.641292	−0.320646	+	0.947199i	$0.603900\pi$
$608$	0	0
$609$	−9.60106e8	−0.172250
$610$	0	0
$611$	−9.52758e9	−1.68981
$612$	0	0
$613$	8.12987e9	1.42552	0.712758	−	0.701410i	$-0.247446\pi$
$613$	8.12987e9	1.42552	0.712758	+	0.701410i	$0.247446\pi$
$614$	0	0
$615$	3.77757e9	0.654861
$616$	0	0
$617$	−9.27957e9	−1.59049	−0.795243	−	0.606291i	$-0.792657\pi$
$617$	−9.27957e9	−1.59049	−0.795243	+	0.606291i	$0.792657\pi$
$618$	0	0
$619$	−1.19776e9	−0.202979	−0.101490	−	0.994837i	$-0.532361\pi$
$619$	−1.19776e9	−0.202979	−0.101490	+	0.994837i	$0.532361\pi$
$620$	0	0
$621$	−6.30535e8	−0.105655
$622$	0	0
$623$	−1.77334e9	−0.293821
$624$	0	0
$625$	−3.52262e9	−0.577146
$626$	0	0
$627$	1.31007e9	0.212256
$628$	0	0
$629$	−4.12905e9	−0.661565
$630$	0	0
$631$	7.29303e9	1.15559	0.577797	−	0.816181i	$-0.303913\pi$
$631$	7.29303e9	1.15559	0.577797	+	0.816181i	$0.303913\pi$
$632$	0	0
$633$	−5.39401e9	−0.845277
$634$	0	0
$635$	−4.07654e8	−0.0631805
$636$	0	0
$637$	−1.62033e9	−0.248380
$638$	0	0
$639$	1.10642e9	0.167751
$640$	0	0
$641$	7.58895e9	1.13809	0.569047	−	0.822305i	$-0.307312\pi$
$641$	7.58895e9	1.13809	0.569047	+	0.822305i	$0.307312\pi$
$642$	0	0
$643$	−6.78950e9	−1.00716	−0.503581	−	0.863948i	$-0.667985\pi$
$643$	−6.78950e9	−1.00716	−0.503581	+	0.863948i	$0.667985\pi$
$644$	0	0
$645$	2.76167e9	0.405241
$646$	0	0
$647$	4.07625e9	0.591693	0.295846	−	0.955236i	$-0.404398\pi$
$647$	4.07625e9	0.591693	0.295846	+	0.955236i	$0.404398\pi$
$648$	0	0
$649$	1.30024e10	1.86710
$650$	0	0
$651$	−2.24057e9	−0.318291
$652$	0	0
$653$	8.55289e9	1.20203	0.601017	−	0.799236i	$-0.294762\pi$
$653$	8.55289e9	1.20203	0.601017	+	0.799236i	$0.294762\pi$
$654$	0	0
$655$	−3.85778e9	−0.536405
$656$	0	0
$657$	3.99331e9	0.549356
$658$	0	0
$659$	−7.08894e9	−0.964901	−0.482450	−	0.875923i	$-0.660253\pi$
$659$	−7.08894e9	−0.964901	−0.482450	+	0.875923i	$0.660253\pi$
$660$	0	0
$661$	−4.59534e9	−0.618889	−0.309444	−	0.950918i	$-0.600143\pi$
$661$	−4.59534e9	−0.618889	−0.309444	+	0.950918i	$0.600143\pi$
$662$	0	0
$663$	1.20761e10	1.60927
$664$	0	0
$665$	−7.62905e8	−0.100599
$666$	0	0
$667$	3.32108e9	0.433350
$668$	0	0
$669$	−1.03757e9	−0.133976
$670$	0	0
$671$	−7.72126e9	−0.986641
$672$	0	0
$673$	1.26787e10	1.60333	0.801663	−	0.597776i	$-0.203949\pi$
$673$	1.26787e10	1.60333	0.801663	+	0.597776i	$0.203949\pi$
$674$	0	0
$675$	4.92500e8	0.0616373
$676$	0	0
$677$	−6.00290e9	−0.743534	−0.371767	−	0.928326i	$-0.621248\pi$
$677$	−6.00290e9	−0.743534	−0.371767	+	0.928326i	$0.621248\pi$
$678$	0	0
$679$	−3.64559e9	−0.446914
$680$	0	0
$681$	5.71351e9	0.693247
$682$	0	0
$683$	−1.44747e9	−0.173835	−0.0869175	−	0.996216i	$-0.527702\pi$
$683$	−1.44747e9	−0.173835	−0.0869175	+	0.996216i	$0.527702\pi$
$684$	0	0
$685$	−7.79733e9	−0.926892
$686$	0	0
$687$	−3.38121e9	−0.397854
$688$	0	0
$689$	−9.26654e9	−1.07932
$690$	0	0
$691$	8.76319e9	1.01039	0.505195	−	0.863005i	$-0.331421\pi$
$691$	8.76319e9	1.01039	0.505195	+	0.863005i	$0.331421\pi$
$692$	0	0
$693$	−1.25701e9	−0.143474
$694$	0	0
$695$	−3.72684e9	−0.421108
$696$	0	0
$697$	1.97166e10	2.20555
$698$	0	0
$699$	−1.66188e9	−0.184047
$700$	0	0
$701$	−7.63748e9	−0.837408	−0.418704	−	0.908123i	$-0.637516\pi$
$701$	−7.63748e9	−0.837408	−0.418704	+	0.908123i	$0.637516\pi$
$702$	0	0
$703$	−1.22721e9	−0.133222
$704$	0	0
$705$	4.30419e9	0.462625
$706$	0	0
$707$	1.38611e9	0.147513
$708$	0	0
$709$	4.09420e9	0.431427	0.215714	−	0.976457i	$-0.430792\pi$
$709$	4.09420e9	0.431427	0.215714	+	0.976457i	$0.430792\pi$
$710$	0	0
$711$	−4.92698e9	−0.514088
$712$	0	0
$713$	7.75029e9	0.800764
$714$	0	0
$715$	−1.59549e10	−1.63239
$716$	0	0
$717$	6.97501e9	0.706688
$718$	0	0
$719$	4.10615e9	0.411987	0.205994	−	0.978553i	$-0.433957\pi$
$719$	4.10615e9	0.411987	0.205994	+	0.978553i	$0.433957\pi$
$720$	0	0
$721$	−4.96439e9	−0.493279
$722$	0	0
$723$	−1.04117e10	−1.02456
$724$	0	0
$725$	−2.59404e9	−0.252810
$726$	0	0
$727$	1.57311e10	1.51840	0.759202	−	0.650855i	$-0.225589\pi$
$727$	1.57311e10	1.51840	0.759202	+	0.650855i	$0.225589\pi$
$728$	0	0
$729$	3.87420e8	0.0370370
$730$	0	0
$731$	1.44143e10	1.36484
$732$	0	0
$733$	1.37988e10	1.29413	0.647066	−	0.762434i	$-0.275996\pi$
$733$	1.37988e10	1.29413	0.647066	+	0.762434i	$0.275996\pi$
$734$	0	0
$735$	7.32003e8	0.0679998
$736$	0	0
$737$	2.11551e10	1.94661
$738$	0	0
$739$	1.89691e10	1.72899	0.864494	−	0.502642i	$-0.167639\pi$
$739$	1.89691e10	1.72899	0.864494	+	0.502642i	$0.167639\pi$
$740$	0	0
$741$	3.58918e9	0.324064
$742$	0	0
$743$	−5.44466e9	−0.486979	−0.243489	−	0.969904i	$-0.578292\pi$
$743$	−5.44466e9	−0.486979	−0.243489	+	0.969904i	$0.578292\pi$
$744$	0	0
$745$	−6.58936e9	−0.583843
$746$	0	0
$747$	6.09890e9	0.535340
$748$	0	0
$749$	−2.36541e9	−0.205693
$750$	0	0
$751$	−9.54161e9	−0.822019	−0.411010	−	0.911631i	$-0.634824\pi$
$751$	−9.54161e9	−0.822019	−0.411010	+	0.911631i	$0.634824\pi$
$752$	0	0
$753$	−8.05752e9	−0.687732
$754$	0	0
$755$	7.28468e9	0.616022
$756$	0	0
$757$	−2.53204e9	−0.212146	−0.106073	−	0.994358i	$-0.533828\pi$
$757$	−2.53204e9	−0.212146	−0.106073	+	0.994358i	$0.533828\pi$
$758$	0	0
$759$	4.34809e9	0.360954
$760$	0	0
$761$	−1.63663e10	−1.34618	−0.673092	−	0.739559i	$-0.735034\pi$
$761$	−1.63663e10	−1.34618	−0.673092	+	0.739559i	$0.735034\pi$
$762$	0	0
$763$	4.83688e9	0.394212
$764$	0	0
$765$	−5.45549e9	−0.440574
$766$	0	0
$767$	3.56223e10	2.85062
$768$	0	0
$769$	−4.39164e9	−0.348245	−0.174123	−	0.984724i	$-0.555709\pi$
$769$	−4.39164e9	−0.348245	−0.174123	+	0.984724i	$0.555709\pi$
$770$	0	0
$771$	1.09582e10	0.861094
$772$	0	0
$773$	−9.74170e9	−0.758589	−0.379294	−	0.925276i	$-0.623833\pi$
$773$	−9.74170e9	−0.758589	−0.379294	+	0.925276i	$0.623833\pi$
$774$	0	0
$775$	−6.05363e9	−0.467154
$776$	0	0
$777$	1.17750e9	0.0900511
$778$	0	0
$779$	5.86006e9	0.444141
$780$	0	0
$781$	−7.62972e9	−0.573099
$782$	0	0
$783$	−2.04057e9	−0.151910
$784$	0	0
$785$	−7.91712e9	−0.584149
$786$	0	0
$787$	−1.30910e10	−0.957331	−0.478665	−	0.877997i	$-0.658879\pi$
$787$	−1.30910e10	−0.957331	−0.478665	+	0.877997i	$0.658879\pi$
$788$	0	0
$789$	7.13895e9	0.517446
$790$	0	0
$791$	3.30165e9	0.237199
$792$	0	0
$793$	−2.11537e10	−1.50637
$794$	0	0
$795$	4.18626e9	0.295489
$796$	0	0
$797$	−2.40324e10	−1.68148	−0.840741	−	0.541438i	$-0.817880\pi$
$797$	−2.40324e10	−1.68148	−0.840741	+	0.541438i	$0.817880\pi$
$798$	0	0
$799$	2.24653e10	1.55811
$800$	0	0
$801$	−3.76898e9	−0.259126
$802$	0	0
$803$	−2.75374e10	−1.87680
$804$	0	0
$805$	−2.53205e9	−0.171075
$806$	0	0
$807$	−5.37065e9	−0.359724
$808$	0	0
$809$	5.56860e9	0.369765	0.184883	−	0.982761i	$-0.440809\pi$
$809$	5.56860e9	0.369765	0.184883	+	0.982761i	$0.440809\pi$
$810$	0	0
$811$	−4.26897e9	−0.281028	−0.140514	−	0.990079i	$-0.544876\pi$
$811$	−4.26897e9	−0.281028	−0.140514	+	0.990079i	$0.544876\pi$
$812$	0	0
$813$	−3.79003e9	−0.247358
$814$	0	0
$815$	5.67620e9	0.367287
$816$	0	0
$817$	4.28413e9	0.274843
$818$	0	0
$819$	−3.44380e9	−0.219050
$820$	0	0
$821$	2.37393e10	1.49715	0.748577	−	0.663048i	$-0.230737\pi$
$821$	2.37393e10	1.49715	0.748577	+	0.663048i	$0.230737\pi$
$822$	0	0
$823$	−2.03110e10	−1.27008	−0.635040	−	0.772479i	$-0.719016\pi$
$823$	−2.03110e10	−1.27008	−0.635040	+	0.772479i	$0.719016\pi$
$824$	0	0
$825$	−3.39622e9	−0.210575
$826$	0	0
$827$	−7.03758e9	−0.432667	−0.216334	−	0.976319i	$-0.569410\pi$
$827$	−7.03758e9	−0.432667	−0.216334	+	0.976319i	$0.569410\pi$
$828$	0	0
$829$	−2.08404e10	−1.27047	−0.635235	−	0.772319i	$-0.719097\pi$
$829$	−2.08404e10	−1.27047	−0.635235	+	0.772319i	$0.719097\pi$
$830$	0	0
$831$	3.78760e9	0.228961
$832$	0	0
$833$	3.82062e9	0.229021
$834$	0	0
$835$	−6.94941e9	−0.413091
$836$	0	0
$837$	−4.76203e9	−0.280707
$838$	0	0
$839$	−8.16531e9	−0.477316	−0.238658	−	0.971104i	$-0.576707\pi$
$839$	−8.16531e9	−0.477316	−0.238658	+	0.971104i	$0.576707\pi$
$840$	0	0
$841$	−6.50201e9	−0.376931
$842$	0	0
$843$	−9.22423e9	−0.530315
$844$	0	0
$845$	−2.92514e10	−1.66781
$846$	0	0
$847$	1.98408e9	0.112193
$848$	0	0
$849$	−1.00937e10	−0.566076
$850$	0	0
$851$	−4.07307e9	−0.226552
$852$	0	0
$853$	−2.02675e10	−1.11809	−0.559047	−	0.829136i	$-0.688833\pi$
$853$	−2.02675e10	−1.11809	−0.559047	+	0.829136i	$0.688833\pi$
$854$	0	0
$855$	−1.62145e9	−0.0887202
$856$	0	0
$857$	5.57784e9	0.302714	0.151357	−	0.988479i	$-0.451636\pi$
$857$	5.57784e9	0.302714	0.151357	+	0.988479i	$0.451636\pi$
$858$	0	0
$859$	1.21117e9	0.0651974	0.0325987	−	0.999469i	$-0.489622\pi$
$859$	1.21117e9	0.0651974	0.0325987	+	0.999469i	$0.489622\pi$
$860$	0	0
$861$	−5.62270e9	−0.300216
$862$	0	0
$863$	2.55888e10	1.35523	0.677615	−	0.735417i	$-0.263014\pi$
$863$	2.55888e10	1.35523	0.677615	+	0.735417i	$0.263014\pi$
$864$	0	0
$865$	−1.76553e10	−0.927511
$866$	0	0
$867$	−1.73952e10	−0.906491
$868$	0	0
$869$	3.39759e10	1.75631
$870$	0	0
$871$	5.79582e10	2.97202
$872$	0	0
$873$	−7.74821e9	−0.394141
$874$	0	0
$875$	8.15286e9	0.411417
$876$	0	0
$877$	1.25211e10	0.626821	0.313411	−	0.949618i	$-0.398528\pi$
$877$	1.25211e10	0.626821	0.313411	+	0.949618i	$0.398528\pi$
$878$	0	0
$879$	1.91937e9	0.0953232
$880$	0	0
$881$	1.18409e10	0.583403	0.291701	−	0.956509i	$-0.405779\pi$
$881$	1.18409e10	0.583403	0.291701	+	0.956509i	$0.405779\pi$
$882$	0	0
$883$	−3.56852e10	−1.74432	−0.872160	−	0.489221i	$-0.837281\pi$
$883$	−3.56852e10	−1.74432	−0.872160	+	0.489221i	$0.837281\pi$
$884$	0	0
$885$	−1.60928e10	−0.780423
$886$	0	0
$887$	4.14449e9	0.199406	0.0997031	−	0.995017i	$-0.468211\pi$
$887$	4.14449e9	0.199406	0.0997031	+	0.995017i	$0.468211\pi$
$888$	0	0
$889$	6.06770e8	0.0289646
$890$	0	0
$891$	−2.67160e9	−0.126532
$892$	0	0
$893$	6.67700e9	0.313763
$894$	0	0
$895$	8.19661e9	0.382168
$896$	0	0
$897$	1.19123e10	0.551092
$898$	0	0
$899$	2.50820e10	1.15134
$900$	0	0
$901$	2.18498e10	0.995200
$902$	0	0
$903$	−4.11060e9	−0.185780
$904$	0	0
$905$	−1.57883e9	−0.0708052
$906$	0	0
$907$	−2.66045e10	−1.18394	−0.591971	−	0.805959i	$-0.701650\pi$
$907$	−2.66045e10	−1.18394	−0.591971	+	0.805959i	$0.701650\pi$
$908$	0	0
$909$	2.94600e9	0.130094
$910$	0	0
$911$	−3.42503e10	−1.50090	−0.750448	−	0.660929i	$-0.770162\pi$
$911$	−3.42503e10	−1.50090	−0.750448	+	0.660929i	$0.770162\pi$
$912$	0	0
$913$	−4.20572e10	−1.82891
$914$	0	0
$915$	9.55644e9	0.412404
$916$	0	0
$917$	5.74209e9	0.245911
$918$	0	0
$919$	2.28724e10	0.972093	0.486046	−	0.873933i	$-0.338439\pi$
$919$	2.28724e10	0.972093	0.486046	+	0.873933i	$0.338439\pi$
$920$	0	0
$921$	−1.85114e10	−0.780782
$922$	0	0
$923$	−2.09029e10	−0.874987
$924$	0	0
$925$	3.18141e9	0.132167
$926$	0	0
$927$	−1.05511e10	−0.435031
$928$	0	0
$929$	−2.31413e10	−0.946964	−0.473482	−	0.880803i	$-0.657003\pi$
$929$	−2.31413e10	−0.946964	−0.473482	+	0.880803i	$0.657003\pi$
$930$	0	0
$931$	1.13554e9	0.0461190
$932$	0	0
$933$	−7.15186e9	−0.288293
$934$	0	0
$935$	3.76204e10	1.50516
$936$	0	0
$937$	4.19161e10	1.66453	0.832266	−	0.554377i	$-0.187043\pi$
$937$	4.19161e10	1.66453	0.832266	+	0.554377i	$0.187043\pi$
$938$	0	0
$939$	1.53780e10	0.606138
$940$	0	0
$941$	4.48967e10	1.75651	0.878255	−	0.478192i	$-0.158708\pi$
$941$	4.48967e10	1.75651	0.878255	+	0.478192i	$0.158708\pi$
$942$	0	0
$943$	1.94493e10	0.755290
$944$	0	0
$945$	1.55577e9	0.0599702
$946$	0	0
$947$	−4.47628e10	−1.71274	−0.856372	−	0.516360i	$-0.827287\pi$
$947$	−4.47628e10	−1.71274	−0.856372	+	0.516360i	$0.827287\pi$
$948$	0	0
$949$	−7.54434e10	−2.86543
$950$	0	0
$951$	2.46553e10	0.929562
$952$	0	0
$953$	2.02761e10	0.758857	0.379428	−	0.925221i	$-0.376121\pi$
$953$	2.02761e10	0.758857	0.379428	+	0.925221i	$0.376121\pi$
$954$	0	0
$955$	8.37326e9	0.311088
$956$	0	0
$957$	1.40715e10	0.518979
$958$	0	0
$959$	1.16059e10	0.424927
$960$	0	0
$961$	3.10204e10	1.12750
$962$	0	0
$963$	−5.02735e9	−0.181404
$964$	0	0
$965$	2.90167e10	1.03945
$966$	0	0
$967$	2.09473e10	0.744965	0.372483	−	0.928039i	$-0.378507\pi$
$967$	2.09473e10	0.744965	0.372483	+	0.928039i	$0.378507\pi$
$968$	0	0
$969$	−8.46300e9	−0.298807
$970$	0	0
$971$	1.77224e10	0.621234	0.310617	−	0.950535i	$-0.399464\pi$
$971$	1.77224e10	0.621234	0.310617	+	0.950535i	$0.399464\pi$
$972$	0	0
$973$	5.54720e9	0.193054
$974$	0	0
$975$	−9.30454e9	−0.321498
$976$	0	0
$977$	1.81082e10	0.621220	0.310610	−	0.950538i	$-0.399467\pi$
$977$	1.81082e10	0.621220	0.310610	+	0.950538i	$0.399467\pi$
$978$	0	0
$979$	2.59904e10	0.885267
$980$	0	0
$981$	1.02801e10	0.347662
$982$	0	0
$983$	6.25321e9	0.209974	0.104987	−	0.994474i	$-0.466520\pi$
$983$	6.25321e9	0.209974	0.104987	+	0.994474i	$0.466520\pi$
$984$	0	0
$985$	−3.83331e10	−1.27805
$986$	0	0
$987$	−6.40655e9	−0.212087
$988$	0	0
$989$	1.42189e10	0.467389
$990$	0	0
$991$	−1.31966e10	−0.430729	−0.215364	−	0.976534i	$-0.569094\pi$
$991$	−1.31966e10	−0.430729	−0.215364	+	0.976534i	$0.569094\pi$
$992$	0	0
$993$	1.18563e10	0.384260
$994$	0	0
$995$	1.75839e10	0.565893
$996$	0	0
$997$	−1.14508e10	−0.365934	−0.182967	−	0.983119i	$-0.558570\pi$
$997$	−1.14508e10	−0.365934	−0.182967	+	0.983119i	$0.558570\pi$
$998$	0	0
$999$	2.50263e9	0.0794176

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	84.8.a.c.1.1	✓	2
3.2	odd	2		252.8.a.c.1.2		2
4.3	odd	2		336.8.a.q.1.1		2
7.2	even	3		588.8.i.k.361.2		4
7.3	odd	6		588.8.i.j.373.1		4
7.4	even	3		588.8.i.k.373.2		4
7.5	odd	6		588.8.i.j.361.1		4
7.6	odd	2		588.8.a.f.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
84.8.a.c.1.1	✓	2	1.1	even	1	trivial
252.8.a.c.1.2		2	3.2	odd	2
336.8.a.q.1.1		2	4.3	odd	2
588.8.a.f.1.2		2	7.6	odd	2
588.8.i.j.361.1		4	7.5	odd	6
588.8.i.j.373.1		4	7.3	odd	6
588.8.i.k.361.2		4	7.2	even	3
588.8.i.k.373.2		4	7.4	even	3

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:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	84.a (trivial)

\(f(q)\)	\(=\)	\(q-27.0000 q^{3} -230.442 q^{5} +343.000 q^{7} +729.000 q^{9} -5027.09 q^{11} -13772.6 q^{13} +6221.93 q^{15} +32474.7 q^{17} +9651.95 q^{19} -9261.00 q^{21} +32034.5 q^{23} -25021.6 q^{25} -19683.0 q^{27} +103672. q^{29} +241936. q^{31} +135731. q^{33} -79041.5 q^{35} -127147. q^{37} +371860. q^{39} +607138. q^{41} +443862. q^{43} -167992. q^{45} +691778. q^{47} +117649. q^{49} -876817. q^{51} +672824. q^{53} +1.15845e6 q^{55} -260603. q^{57} -2.58646e6 q^{59} +1.53593e6 q^{61} +250047. q^{63} +3.17378e6 q^{65} -4.20823e6 q^{67} -864931. q^{69} +1.51772e6 q^{71} +5.47779e6 q^{73} +675584. q^{75} -1.72429e6 q^{77} -6.75855e6 q^{79} +531441. q^{81} +8.36612e6 q^{83} -7.48353e6 q^{85} -2.79914e6 q^{87} -5.17007e6 q^{89} -4.72400e6 q^{91} -6.53227e6 q^{93} -2.22421e6 q^{95} -1.06286e7 q^{97} -3.66475e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 54 q^{3} + 264 q^{5} + 686 q^{7} + 1458 q^{9} - 4980 q^{11} - 10148 q^{13} - 7128 q^{15} + 17832 q^{17} + 6256 q^{19} - 18522 q^{21} + 14052 q^{23} + 141326 q^{25} - 39366 q^{27} + 243588 q^{29} + 470824 q^{31}+ \cdots - 3630420 q^{99}+O(q^{100}) \) 2 * q - 54 * q^3 + 264 * q^5 + 686 * q^7 + 1458 * q^9 - 4980 * q^11 - 10148 * q^13 - 7128 * q^15 + 17832 * q^17 + 6256 * q^19 - 18522 * q^21 + 14052 * q^23 + 141326 * q^25 - 39366 * q^27 + 243588 * q^29 + 470824 * q^31 + 134460 * q^33 + 90552 * q^35 + 311116 * q^37 + 273996 * q^39 + 919248 * q^41 - 112616 * q^43 + 192456 * q^45 - 102456 * q^47 + 235298 * q^49 - 481464 * q^51 + 2720028 * q^53 + 1181736 * q^55 - 168912 * q^57 - 19008 * q^59 - 925148 * q^61 + 500094 * q^63 + 4965936 * q^65 - 2053424 * q^67 - 379404 * q^69 - 869508 * q^71 + 3505228 * q^73 - 3815802 * q^75 - 1708140 * q^77 - 6640856 * q^79 + 1062882 * q^81 + 7856760 * q^83 - 14723496 * q^85 - 6576876 * q^87 - 9330384 * q^89 - 3480764 * q^91 - 12712248 * q^93 - 3903312 * q^95 - 2220212 * q^97 - 3630420 * q^99

Introduction

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