LMFDB - Embedded newform 448.8.a.t.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [448,8,Mod(1,448)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(448, 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("448.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 448 = 2^{6} \cdot 7 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	448.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:	$139.948491417$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{865}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 216 $ x^2 - x - 216
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 7)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$15.2054$ of defining polynomial
Character	$\chi$	$=$	448.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+17.5891 q^{3} -17.9456 q^{5} +343.000 q^{7} -1877.62 q^{9} +O(q^{10})$	$q+17.5891 q^{3} -17.9456 q^{5} +343.000 q^{7} -1877.62 q^{9} +3127.83 q^{11} -14237.2 q^{13} -315.647 q^{15} -5508.56 q^{17} +29022.3 q^{19} +6033.07 q^{21} +3993.86 q^{23} -77803.0 q^{25} -71493.1 q^{27} -148257. q^{29} +232799. q^{31} +55015.8 q^{33} -6155.34 q^{35} +215998. q^{37} -250420. q^{39} +716086. q^{41} -157476. q^{43} +33695.0 q^{45} +1.08718e6 q^{47} +117649. q^{49} -96890.8 q^{51} -1.50566e6 q^{53} -56130.8 q^{55} +510476. q^{57} -708450. q^{59} -414093. q^{61} -644025. q^{63} +255495. q^{65} -913243. q^{67} +70248.5 q^{69} +1.82558e6 q^{71} -671764. q^{73} -1.36849e6 q^{75} +1.07285e6 q^{77} +7.27880e6 q^{79} +2.84886e6 q^{81} +7.77939e6 q^{83} +98854.4 q^{85} -2.60771e6 q^{87} -454651. q^{89} -4.88336e6 q^{91} +4.09473e6 q^{93} -520822. q^{95} +6.77351e6 q^{97} -5.87289e6 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 94 q^{3} - 330 q^{5} + 686 q^{7} + 1774 q^{9}+O(q^{10}) $ 2 * q + 94 * q^3 - 330 * q^5 + 686 * q^7 + 1774 * q^9	$ 2 q + 94 q^{3} - 330 q^{5} + 686 q^{7} + 1774 q^{9} + 2844 q^{11} - 2534 q^{13} - 24160 q^{15} - 1488 q^{17} + 32810 q^{19} + 32242 q^{21} + 6576 q^{23} - 58550 q^{25} + 40420 q^{27} - 20640 q^{29} + 391836 q^{31} + 33328 q^{33} - 113190 q^{35} - 367392 q^{37} + 643832 q^{39} + 734664 q^{41} - 480476 q^{43} - 1105810 q^{45} + 1089108 q^{47} + 235298 q^{49} + 210324 q^{51} - 2858844 q^{53} + 32440 q^{55} + 799900 q^{57} + 160170 q^{59} + 864646 q^{61} + 608482 q^{63} - 3396540 q^{65} - 328648 q^{67} + 267552 q^{69} + 7500216 q^{71} + 4301244 q^{73} + 102650 q^{75} + 975492 q^{77} + 6408440 q^{79} + 3414142 q^{81} + 11659074 q^{83} - 1155780 q^{85} + 7143620 q^{87} + 9772260 q^{89} - 869162 q^{91} + 16246872 q^{93} - 1702800 q^{95} + 10762752 q^{97} - 6909332 q^{99}+O(q^{100}) $ 2 * q + 94 * q^3 - 330 * q^5 + 686 * q^7 + 1774 * q^9 + 2844 * q^11 - 2534 * q^13 - 24160 * q^15 - 1488 * q^17 + 32810 * q^19 + 32242 * q^21 + 6576 * q^23 - 58550 * q^25 + 40420 * q^27 - 20640 * q^29 + 391836 * q^31 + 33328 * q^33 - 113190 * q^35 - 367392 * q^37 + 643832 * q^39 + 734664 * q^41 - 480476 * q^43 - 1105810 * q^45 + 1089108 * q^47 + 235298 * q^49 + 210324 * q^51 - 2858844 * q^53 + 32440 * q^55 + 799900 * q^57 + 160170 * q^59 + 864646 * q^61 + 608482 * q^63 - 3396540 * q^65 - 328648 * q^67 + 267552 * q^69 + 7500216 * q^71 + 4301244 * q^73 + 102650 * q^75 + 975492 * q^77 + 6408440 * q^79 + 3414142 * q^81 + 11659074 * q^83 - 1155780 * q^85 + 7143620 * q^87 + 9772260 * q^89 - 869162 * q^91 + 16246872 * q^93 - 1702800 * q^95 + 10762752 * q^97 - 6909332 * 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
$3$	17.5891	0.376114	0.188057	−	0.982158i	$-0.439781\pi$
$3$	17.5891	0.376114	0.188057	+	0.982158i	$0.439781\pi$
$4$	0	0
$5$	−17.9456	−0.0642041	−0.0321020	−	0.999485i	$-0.510220\pi$
$5$	−17.9456	−0.0642041	−0.0321020	+	0.999485i	$0.510220\pi$
$6$	0	0
$7$	343.000	0.377964
$7$	343.000	0.377964
$8$	0	0
$9$	−1877.62	−0.858538
$10$	0	0
$11$	3127.83	0.708547	0.354274	−	0.935142i	$-0.384728\pi$
$11$	3127.83	0.708547	0.354274	+	0.935142i	$0.384728\pi$
$12$	0	0
$13$	−14237.2	−1.79731	−0.898655	−	0.438657i	$-0.855454\pi$
$13$	−14237.2	−1.79731	−0.898655	+	0.438657i	$0.855454\pi$
$14$	0	0
$15$	−315.647	−0.0241481
$16$	0	0
$17$	−5508.56	−0.271936	−0.135968	−	0.990713i	$-0.543414\pi$
$17$	−5508.56	−0.271936	−0.135968	+	0.990713i	$0.543414\pi$
$18$	0	0
$19$	29022.3	0.970719	0.485360	−	0.874315i	$-0.338689\pi$
$19$	29022.3	0.970719	0.485360	+	0.874315i	$0.338689\pi$
$20$	0	0
$21$	6033.07	0.142158
$22$	0	0
$23$	3993.86	0.0684456	0.0342228	−	0.999414i	$-0.489104\pi$
$23$	3993.86	0.0684456	0.0342228	+	0.999414i	$0.489104\pi$
$24$	0	0
$25$	−77803.0	−0.995878
$26$	0	0
$27$	−71493.1	−0.699022
$28$	0	0
$29$	−148257.	−1.12881	−0.564407	−	0.825497i	$-0.690895\pi$
$29$	−148257.	−1.12881	−0.564407	+	0.825497i	$0.690895\pi$
$30$	0	0
$31$	232799.	1.40351	0.701755	−	0.712418i	$-0.252400\pi$
$31$	232799.	1.40351	0.701755	+	0.712418i	$0.252400\pi$
$32$	0	0
$33$	55015.8	0.266495
$34$	0	0
$35$	−6155.34	−0.0242669
$36$	0	0
$37$	215998.	0.701041	0.350521	−	0.936555i	$-0.386005\pi$
$37$	215998.	0.701041	0.350521	+	0.936555i	$0.386005\pi$
$38$	0	0
$39$	−250420.	−0.675993
$40$	0	0
$41$	716086.	1.62264	0.811319	−	0.584603i	$-0.198750\pi$
$41$	716086.	1.62264	0.811319	+	0.584603i	$0.198750\pi$
$42$	0	0
$43$	−157476.	−0.302047	−0.151023	−	0.988530i	$-0.548257\pi$
$43$	−157476.	−0.302047	−0.151023	+	0.988530i	$0.548257\pi$
$44$	0	0
$45$	33695.0	0.0551217
$46$	0	0
$47$	1.08718e6	1.52743	0.763714	−	0.645555i	$-0.223374\pi$
$47$	1.08718e6	1.52743	0.763714	+	0.645555i	$0.223374\pi$
$48$	0	0
$49$	117649.	0.142857
$50$	0	0
$51$	−96890.8	−0.102279
$52$	0	0
$53$	−1.50566e6	−1.38918	−0.694592	−	0.719404i	$-0.744415\pi$
$53$	−1.50566e6	−1.38918	−0.694592	+	0.719404i	$0.744415\pi$
$54$	0	0
$55$	−56130.8	−0.0454916
$56$	0	0
$57$	510476.	0.365101
$58$	0	0
$59$	−708450.	−0.449084	−0.224542	−	0.974464i	$-0.572089\pi$
$59$	−708450.	−0.449084	−0.224542	+	0.974464i	$0.572089\pi$
$60$	0	0
$61$	−414093.	−0.233584	−0.116792	−	0.993156i	$-0.537261\pi$
$61$	−414093.	−0.233584	−0.116792	+	0.993156i	$0.537261\pi$
$62$	0	0
$63$	−644025.	−0.324497
$64$	0	0
$65$	255495.	0.115395
$66$	0	0
$67$	−913243.	−0.370958	−0.185479	−	0.982648i	$-0.559384\pi$
$67$	−913243.	−0.370958	−0.185479	+	0.982648i	$0.559384\pi$
$68$	0	0
$69$	70248.5	0.0257434
$70$	0	0
$71$	1.82558e6	0.605335	0.302668	−	0.953096i	$-0.402123\pi$
$71$	1.82558e6	0.605335	0.302668	+	0.953096i	$0.402123\pi$
$72$	0	0
$73$	−671764.	−0.202109	−0.101055	−	0.994881i	$-0.532222\pi$
$73$	−671764.	−0.202109	−0.101055	+	0.994881i	$0.532222\pi$
$74$	0	0
$75$	−1.36849e6	−0.374564
$76$	0	0
$77$	1.07285e6	0.267806
$78$	0	0
$79$	7.27880e6	1.66098	0.830491	−	0.557031i	$-0.188060\pi$
$79$	7.27880e6	1.66098	0.830491	+	0.557031i	$0.188060\pi$
$80$	0	0
$81$	2.84886e6	0.595626
$82$	0	0
$83$	7.77939e6	1.49339	0.746693	−	0.665169i	$-0.231640\pi$
$83$	7.77939e6	1.49339	0.746693	+	0.665169i	$0.231640\pi$
$84$	0	0
$85$	98854.4	0.0174594
$86$	0	0
$87$	−2.60771e6	−0.424563
$88$	0	0
$89$	−454651.	−0.0683617	−0.0341809	−	0.999416i	$-0.510882\pi$
$89$	−454651.	−0.0683617	−0.0341809	+	0.999416i	$0.510882\pi$
$90$	0	0
$91$	−4.88336e6	−0.679319
$92$	0	0
$93$	4.09473e6	0.527880
$94$	0	0
$95$	−520822.	−0.0623242
$96$	0	0
$97$	6.77351e6	0.753551	0.376776	−	0.926305i	$-0.377033\pi$
$97$	6.77351e6	0.753551	0.376776	+	0.926305i	$0.377033\pi$
$98$	0	0
$99$	−5.87289e6	−0.608315
$100$	0	0
$101$	−1.28396e7	−1.24002	−0.620009	−	0.784595i	$-0.712871\pi$
$101$	−1.28396e7	−1.24002	−0.620009	+	0.784595i	$0.712871\pi$
$102$	0	0
$103$	−1.48395e7	−1.33811	−0.669053	−	0.743215i	$-0.733300\pi$
$103$	−1.48395e7	−1.33811	−0.669053	+	0.743215i	$0.733300\pi$
$104$	0	0
$105$	−108267.	−0.00912711
$106$	0	0
$107$	1.64129e7	1.29521	0.647606	−	0.761975i	$-0.275770\pi$
$107$	1.64129e7	1.29521	0.647606	+	0.761975i	$0.275770\pi$
$108$	0	0
$109$	1.10131e7	0.814547	0.407273	−	0.913306i	$-0.366480\pi$
$109$	1.10131e7	0.814547	0.407273	+	0.913306i	$0.366480\pi$
$110$	0	0
$111$	3.79921e6	0.263671
$112$	0	0
$113$	−9.72308e6	−0.633912	−0.316956	−	0.948440i	$-0.602661\pi$
$113$	−9.72308e6	−0.633912	−0.316956	+	0.948440i	$0.602661\pi$
$114$	0	0
$115$	−71672.2	−0.00439449
$116$	0	0
$117$	2.67321e7	1.54306
$118$	0	0
$119$	−1.88944e6	−0.102782
$120$	0	0
$121$	−9.70384e6	−0.497961
$122$	0	0
$123$	1.25953e7	0.610297
$124$	0	0
$125$	2.79822e6	0.128144
$126$	0	0
$127$	1.05122e7	0.455388	0.227694	−	0.973733i	$-0.426881\pi$
$127$	1.05122e7	0.455388	0.227694	+	0.973733i	$0.426881\pi$
$128$	0	0
$129$	−2.76986e6	−0.113604
$130$	0	0
$131$	−2.07068e7	−0.804755	−0.402377	−	0.915474i	$-0.631816\pi$
$131$	−2.07068e7	−0.804755	−0.402377	+	0.915474i	$0.631816\pi$
$132$	0	0
$133$	9.95464e6	0.366897
$134$	0	0
$135$	1.28299e6	0.0448801
$136$	0	0
$137$	2.34728e7	0.779907	0.389954	−	0.920834i	$-0.372491\pi$
$137$	2.34728e7	0.779907	0.389954	+	0.920834i	$0.372491\pi$
$138$	0	0
$139$	3.81861e7	1.20602	0.603009	−	0.797734i	$-0.293968\pi$
$139$	3.81861e7	1.20602	0.603009	+	0.797734i	$0.293968\pi$
$140$	0	0
$141$	1.91226e7	0.574487
$142$	0	0
$143$	−4.45316e7	−1.27348
$144$	0	0
$145$	2.66056e6	0.0724745
$146$	0	0
$147$	2.06934e6	0.0537306
$148$	0	0
$149$	1.47686e7	0.365753	0.182877	−	0.983136i	$-0.441459\pi$
$149$	1.47686e7	0.365753	0.182877	+	0.983136i	$0.441459\pi$
$150$	0	0
$151$	7.29854e6	0.172511	0.0862555	−	0.996273i	$-0.472510\pi$
$151$	7.29854e6	0.172511	0.0862555	+	0.996273i	$0.472510\pi$
$152$	0	0
$153$	1.03430e7	0.233468
$154$	0	0
$155$	−4.17772e6	−0.0901111
$156$	0	0
$157$	5.53072e7	1.14060	0.570300	−	0.821437i	$-0.306827\pi$
$157$	5.53072e7	1.14060	0.570300	+	0.821437i	$0.306827\pi$
$158$	0	0
$159$	−2.64831e7	−0.522492
$160$	0	0
$161$	1.36989e6	0.0258700
$162$	0	0
$163$	−1.00185e7	−0.181195	−0.0905975	−	0.995888i	$-0.528878\pi$
$163$	−1.00185e7	−0.181195	−0.0905975	+	0.995888i	$0.528878\pi$
$164$	0	0
$165$	−987291.	−0.0171100
$166$	0	0
$167$	3.98934e7	0.662816	0.331408	−	0.943488i	$-0.392476\pi$
$167$	3.98934e7	0.662816	0.331408	+	0.943488i	$0.392476\pi$
$168$	0	0
$169$	1.39949e8	2.23032
$170$	0	0
$171$	−5.44929e7	−0.833400
$172$	0	0
$173$	−6.60320e7	−0.969602	−0.484801	−	0.874625i	$-0.661108\pi$
$173$	−6.60320e7	−0.969602	−0.484801	+	0.874625i	$0.661108\pi$
$174$	0	0
$175$	−2.66864e7	−0.376406
$176$	0	0
$177$	−1.24610e7	−0.168907
$178$	0	0
$179$	−5.30302e7	−0.691095	−0.345547	−	0.938401i	$-0.612307\pi$
$179$	−5.30302e7	−0.691095	−0.345547	+	0.938401i	$0.612307\pi$
$180$	0	0
$181$	5.37611e7	0.673896	0.336948	−	0.941523i	$-0.390605\pi$
$181$	5.37611e7	0.673896	0.336948	+	0.941523i	$0.390605\pi$
$182$	0	0
$183$	−7.28353e6	−0.0878543
$184$	0	0
$185$	−3.87621e6	−0.0450097
$186$	0	0
$187$	−1.72299e7	−0.192680
$188$	0	0
$189$	−2.45221e7	−0.264206
$190$	0	0
$191$	−9.64979e6	−0.100208	−0.0501038	−	0.998744i	$-0.515955\pi$
$191$	−9.64979e6	−0.100208	−0.0501038	+	0.998744i	$0.515955\pi$
$192$	0	0
$193$	4.31180e7	0.431726	0.215863	−	0.976424i	$-0.430744\pi$
$193$	4.31180e7	0.431726	0.215863	+	0.976424i	$0.430744\pi$
$194$	0	0
$195$	4.49393e6	0.0434015
$196$	0	0
$197$	3.56570e7	0.332287	0.166143	−	0.986102i	$-0.446869\pi$
$197$	3.56570e7	0.332287	0.166143	+	0.986102i	$0.446869\pi$
$198$	0	0
$199$	9.28980e7	0.835643	0.417821	−	0.908529i	$-0.362794\pi$
$199$	9.28980e7	0.835643	0.417821	+	0.908529i	$0.362794\pi$
$200$	0	0
$201$	−1.60631e7	−0.139522
$202$	0	0
$203$	−5.08522e7	−0.426652
$204$	0	0
$205$	−1.28506e7	−0.104180
$206$	0	0
$207$	−7.49897e6	−0.0587632
$208$	0	0
$209$	9.07768e7	0.687801
$210$	0	0
$211$	−9.32366e7	−0.683279	−0.341639	−	0.939831i	$-0.610982\pi$
$211$	−9.32366e7	−0.683279	−0.341639	+	0.939831i	$0.610982\pi$
$212$	0	0
$213$	3.21103e7	0.227675
$214$	0	0
$215$	2.82600e6	0.0193926
$216$	0	0
$217$	7.98501e7	0.530477
$218$	0	0
$219$	−1.18157e7	−0.0760162
$220$	0	0
$221$	7.84265e7	0.488754
$222$	0	0
$223$	2.50199e8	1.51084	0.755421	−	0.655240i	$-0.227433\pi$
$223$	2.50199e8	1.51084	0.755421	+	0.655240i	$0.227433\pi$
$224$	0	0
$225$	1.46085e8	0.854999
$226$	0	0
$227$	2.65443e8	1.50619	0.753096	−	0.657910i	$-0.228559\pi$
$227$	2.65443e8	1.50619	0.753096	+	0.657910i	$0.228559\pi$
$228$	0	0
$229$	2.17020e8	1.19419	0.597097	−	0.802169i	$-0.296321\pi$
$229$	2.17020e8	1.19419	0.597097	+	0.802169i	$0.296321\pi$
$230$	0	0
$231$	1.88704e7	0.100726
$232$	0	0
$233$	2.93897e8	1.52212	0.761062	−	0.648679i	$-0.224678\pi$
$233$	2.93897e8	1.52212	0.761062	+	0.648679i	$0.224678\pi$
$234$	0	0
$235$	−1.95102e7	−0.0980671
$236$	0	0
$237$	1.28028e8	0.624719
$238$	0	0
$239$	−5.98819e7	−0.283729	−0.141864	−	0.989886i	$-0.545310\pi$
$239$	−5.98819e7	−0.283729	−0.141864	+	0.989886i	$0.545310\pi$
$240$	0	0
$241$	4.30347e7	0.198043	0.0990214	−	0.995085i	$-0.468429\pi$
$241$	4.30347e7	0.198043	0.0990214	+	0.995085i	$0.468429\pi$
$242$	0	0
$243$	2.06464e8	0.923046
$244$	0	0
$245$	−2.11128e6	−0.00917201
$246$	0	0
$247$	−4.13196e8	−1.74468
$248$	0	0
$249$	1.36833e8	0.561684
$250$	0	0
$251$	−2.57413e8	−1.02748	−0.513739	−	0.857947i	$-0.671740\pi$
$251$	−2.57413e8	−1.02748	−0.513739	+	0.857947i	$0.671740\pi$
$252$	0	0
$253$	1.24921e7	0.0484970
$254$	0	0
$255$	1.73876e6	0.00656673
$256$	0	0
$257$	2.58130e8	0.948575	0.474288	−	0.880370i	$-0.342706\pi$
$257$	2.58130e8	0.948575	0.474288	+	0.880370i	$0.342706\pi$
$258$	0	0
$259$	7.40873e7	0.264969
$260$	0	0
$261$	2.78371e8	0.969130
$262$	0	0
$263$	5.51866e8	1.87063	0.935316	−	0.353813i	$-0.115115\pi$
$263$	5.51866e8	1.87063	0.935316	+	0.353813i	$0.115115\pi$
$264$	0	0
$265$	2.70199e7	0.0891913
$266$	0	0
$267$	−7.99691e6	−0.0257118
$268$	0	0
$269$	3.96314e8	1.24139	0.620693	−	0.784054i	$-0.286851\pi$
$269$	3.96314e8	1.24139	0.620693	+	0.784054i	$0.286851\pi$
$270$	0	0
$271$	−3.64103e6	−0.0111130	−0.00555651	−	0.999985i	$-0.501769\pi$
$271$	−3.64103e6	−0.0111130	−0.00555651	+	0.999985i	$0.501769\pi$
$272$	0	0
$273$	−8.58940e7	−0.255502
$274$	0	0
$275$	−2.43355e8	−0.705627
$276$	0	0
$277$	−1.03984e7	−0.0293960	−0.0146980	−	0.999892i	$-0.504679\pi$
$277$	−1.03984e7	−0.0293960	−0.0146980	+	0.999892i	$0.504679\pi$
$278$	0	0
$279$	−4.37109e8	−1.20497
$280$	0	0
$281$	−3.67278e8	−0.987467	−0.493733	−	0.869613i	$-0.664368\pi$
$281$	−3.67278e8	−0.987467	−0.493733	+	0.869613i	$0.664368\pi$
$282$	0	0
$283$	6.11224e8	1.60305	0.801527	−	0.597959i	$-0.204021\pi$
$283$	6.11224e8	1.60305	0.801527	+	0.597959i	$0.204021\pi$
$284$	0	0
$285$	−9.16079e6	−0.0234410
$286$	0	0
$287$	2.45618e8	0.613300
$288$	0	0
$289$	−3.79994e8	−0.926051
$290$	0	0
$291$	1.19140e8	0.283421
$292$	0	0
$293$	−4.96531e7	−0.115321	−0.0576607	−	0.998336i	$-0.518364\pi$
$293$	−4.96531e7	−0.115321	−0.0576607	+	0.998336i	$0.518364\pi$
$294$	0	0
$295$	1.27136e7	0.0288330
$296$	0	0
$297$	−2.23618e8	−0.495291
$298$	0	0
$299$	−5.68614e7	−0.123018
$300$	0	0
$301$	−5.40142e7	−0.114163
$302$	0	0
$303$	−2.25838e8	−0.466388
$304$	0	0
$305$	7.43114e6	0.0149971
$306$	0	0
$307$	−9.04736e7	−0.178459	−0.0892293	−	0.996011i	$-0.528440\pi$
$307$	−9.04736e7	−0.178459	−0.0892293	+	0.996011i	$0.528440\pi$
$308$	0	0
$309$	−2.61014e8	−0.503280
$310$	0	0
$311$	−6.45859e8	−1.21752	−0.608760	−	0.793354i	$-0.708333\pi$
$311$	−6.45859e8	−1.21752	−0.608760	+	0.793354i	$0.708333\pi$
$312$	0	0
$313$	1.63102e8	0.300645	0.150322	−	0.988637i	$-0.451969\pi$
$313$	1.63102e8	0.300645	0.150322	+	0.988637i	$0.451969\pi$
$314$	0	0
$315$	1.15574e7	0.0208340
$316$	0	0
$317$	3.76232e8	0.663358	0.331679	−	0.943392i	$-0.392385\pi$
$317$	3.76232e8	0.663358	0.331679	+	0.943392i	$0.392385\pi$
$318$	0	0
$319$	−4.63723e8	−0.799818
$320$	0	0
$321$	2.88688e8	0.487148
$322$	0	0
$323$	−1.59871e8	−0.263974
$324$	0	0
$325$	1.10770e9	1.78990
$326$	0	0
$327$	1.93710e8	0.306363
$328$	0	0
$329$	3.72904e8	0.577314
$330$	0	0
$331$	4.53834e8	0.687859	0.343930	−	0.938995i	$-0.388242\pi$
$331$	4.53834e8	0.687859	0.343930	+	0.938995i	$0.388242\pi$
$332$	0	0
$333$	−4.05563e8	−0.601871
$334$	0	0
$335$	1.63887e7	0.0238170
$336$	0	0
$337$	8.61064e8	1.22555	0.612774	−	0.790258i	$-0.290053\pi$
$337$	8.61064e8	1.22555	0.612774	+	0.790258i	$0.290053\pi$
$338$	0	0
$339$	−1.71020e8	−0.238423
$340$	0	0
$341$	7.28157e8	0.994454
$342$	0	0
$343$	4.03536e7	0.0539949
$344$	0	0
$345$	−1.26065e6	−0.00165283
$346$	0	0
$347$	4.35344e8	0.559345	0.279672	−	0.960095i	$-0.409774\pi$
$347$	4.35344e8	0.559345	0.279672	+	0.960095i	$0.409774\pi$
$348$	0	0
$349$	−5.46030e8	−0.687587	−0.343793	−	0.939045i	$-0.611712\pi$
$349$	−5.46030e8	−0.687587	−0.343793	+	0.939045i	$0.611712\pi$
$350$	0	0
$351$	1.01786e9	1.25636
$352$	0	0
$353$	8.95544e7	0.108362	0.0541808	−	0.998531i	$-0.482745\pi$
$353$	8.95544e7	0.108362	0.0541808	+	0.998531i	$0.482745\pi$
$354$	0	0
$355$	−3.27611e7	−0.0388650
$356$	0	0
$357$	−3.32335e7	−0.0386579
$358$	0	0
$359$	9.16847e8	1.04584	0.522921	−	0.852381i	$-0.324842\pi$
$359$	9.16847e8	1.04584	0.522921	+	0.852381i	$0.324842\pi$
$360$	0	0
$361$	−5.15797e7	−0.0577037
$362$	0	0
$363$	−1.70682e8	−0.187290
$364$	0	0
$365$	1.20552e7	0.0129763
$366$	0	0
$367$	−5.33494e8	−0.563376	−0.281688	−	0.959506i	$-0.590894\pi$
$367$	−5.33494e8	−0.563376	−0.281688	+	0.959506i	$0.590894\pi$
$368$	0	0
$369$	−1.34454e9	−1.39310
$370$	0	0
$371$	−5.16440e8	−0.525062
$372$	0	0
$373$	3.63396e8	0.362576	0.181288	−	0.983430i	$-0.441973\pi$
$373$	3.63396e8	0.362576	0.181288	+	0.983430i	$0.441973\pi$
$374$	0	0
$375$	4.92182e7	0.0481966
$376$	0	0
$377$	2.11076e9	2.02883
$378$	0	0
$379$	−3.89928e8	−0.367915	−0.183958	−	0.982934i	$-0.558891\pi$
$379$	−3.89928e8	−0.367915	−0.183958	+	0.982934i	$0.558891\pi$
$380$	0	0
$381$	1.84901e8	0.171278
$382$	0	0
$383$	3.56788e7	0.0324500	0.0162250	−	0.999868i	$-0.494835\pi$
$383$	3.56788e7	0.0324500	0.0162250	+	0.999868i	$0.494835\pi$
$384$	0	0
$385$	−1.92529e7	−0.0171942
$386$	0	0
$387$	2.95680e8	0.259319
$388$	0	0
$389$	−2.20545e9	−1.89965	−0.949824	−	0.312784i	$-0.898738\pi$
$389$	−2.20545e9	−1.89965	−0.949824	+	0.312784i	$0.898738\pi$
$390$	0	0
$391$	−2.20004e7	−0.0186128
$392$	0	0
$393$	−3.64215e8	−0.302680
$394$	0	0
$395$	−1.30622e8	−0.106642
$396$	0	0
$397$	−1.25366e9	−1.00557	−0.502786	−	0.864411i	$-0.667692\pi$
$397$	−1.25366e9	−1.00557	−0.502786	+	0.864411i	$0.667692\pi$
$398$	0	0
$399$	1.75093e8	0.137995
$400$	0	0
$401$	−8.68826e8	−0.672864	−0.336432	−	0.941708i	$-0.609220\pi$
$401$	−8.68826e8	−0.672864	−0.336432	+	0.941708i	$0.609220\pi$
$402$	0	0
$403$	−3.31441e9	−2.52254
$404$	0	0
$405$	−5.11245e7	−0.0382416
$406$	0	0
$407$	6.75605e8	0.496721
$408$	0	0
$409$	−1.58422e9	−1.14494	−0.572472	−	0.819925i	$-0.694015\pi$
$409$	−1.58422e9	−1.14494	−0.572472	+	0.819925i	$0.694015\pi$
$410$	0	0
$411$	4.12866e8	0.293334
$412$	0	0
$413$	−2.42998e8	−0.169738
$414$	0	0
$415$	−1.39606e8	−0.0958815
$416$	0	0
$417$	6.71660e8	0.453601
$418$	0	0
$419$	−3.45532e8	−0.229477	−0.114738	−	0.993396i	$-0.536603\pi$
$419$	−3.45532e8	−0.229477	−0.114738	+	0.993396i	$0.536603\pi$
$420$	0	0
$421$	−1.68396e9	−1.09987	−0.549937	−	0.835206i	$-0.685348\pi$
$421$	−1.68396e9	−1.09987	−0.549937	+	0.835206i	$0.685348\pi$
$422$	0	0
$423$	−2.04132e9	−1.31136
$424$	0	0
$425$	4.28582e8	0.270815
$426$	0	0
$427$	−1.42034e8	−0.0882865
$428$	0	0
$429$	−7.83271e8	−0.478973
$430$	0	0
$431$	1.37234e9	0.825644	0.412822	−	0.910812i	$-0.364543\pi$
$431$	1.37234e9	0.825644	0.412822	+	0.910812i	$0.364543\pi$
$432$	0	0
$433$	−2.87706e9	−1.70311	−0.851553	−	0.524269i	$-0.824339\pi$
$433$	−2.87706e9	−1.70311	−0.851553	+	0.524269i	$0.824339\pi$
$434$	0	0
$435$	4.67969e7	0.0272587
$436$	0	0
$437$	1.15911e8	0.0664415
$438$	0	0
$439$	−2.19748e9	−1.23965	−0.619825	−	0.784740i	$-0.712797\pi$
$439$	−2.19748e9	−1.23965	−0.619825	+	0.784740i	$0.712797\pi$
$440$	0	0
$441$	−2.20900e8	−0.122648
$442$	0	0
$443$	2.02677e9	1.10762	0.553811	−	0.832642i	$-0.313173\pi$
$443$	2.02677e9	1.10762	0.553811	+	0.832642i	$0.313173\pi$
$444$	0	0
$445$	8.15898e6	0.00438910
$446$	0	0
$447$	2.59767e8	0.137565
$448$	0	0
$449$	2.36608e9	1.23358	0.616791	−	0.787127i	$-0.288432\pi$
$449$	2.36608e9	1.23358	0.616791	+	0.787127i	$0.288432\pi$
$450$	0	0
$451$	2.23980e9	1.14972
$452$	0	0
$453$	1.28375e8	0.0648838
$454$	0	0
$455$	8.76348e7	0.0436151
$456$	0	0
$457$	−8.96708e8	−0.439486	−0.219743	−	0.975558i	$-0.570522\pi$
$457$	−8.96708e8	−0.439486	−0.219743	+	0.975558i	$0.570522\pi$
$458$	0	0
$459$	3.93824e8	0.190090
$460$	0	0
$461$	−1.91173e9	−0.908809	−0.454405	−	0.890795i	$-0.650148\pi$
$461$	−1.91173e9	−0.908809	−0.454405	+	0.890795i	$0.650148\pi$
$462$	0	0
$463$	1.71131e9	0.801299	0.400649	−	0.916231i	$-0.368785\pi$
$463$	1.71131e9	0.801299	0.400649	+	0.916231i	$0.368785\pi$
$464$	0	0
$465$	−7.34824e7	−0.0338921
$466$	0	0
$467$	−8.91644e8	−0.405119	−0.202559	−	0.979270i	$-0.564926\pi$
$467$	−8.91644e8	−0.405119	−0.202559	+	0.979270i	$0.564926\pi$
$468$	0	0
$469$	−3.13242e8	−0.140209
$470$	0	0
$471$	9.72805e8	0.428996
$472$	0	0
$473$	−4.92558e8	−0.214014
$474$	0	0
$475$	−2.25802e9	−0.966718
$476$	0	0
$477$	2.82705e9	1.19267
$478$	0	0
$479$	−4.43678e8	−0.184456	−0.0922282	−	0.995738i	$-0.529399\pi$
$479$	−4.43678e8	−0.184456	−0.0922282	+	0.995738i	$0.529399\pi$
$480$	0	0
$481$	−3.07520e9	−1.25999
$482$	0	0
$483$	2.40952e7	0.00973007
$484$	0	0
$485$	−1.21555e8	−0.0483811
$486$	0	0
$487$	−1.35113e9	−0.530087	−0.265044	−	0.964236i	$-0.585386\pi$
$487$	−1.35113e9	−0.530087	−0.265044	+	0.964236i	$0.585386\pi$
$488$	0	0
$489$	−1.76217e8	−0.0681500
$490$	0	0
$491$	−2.75570e9	−1.05062	−0.525312	−	0.850910i	$-0.676051\pi$
$491$	−2.75570e9	−1.05062	−0.525312	+	0.850910i	$0.676051\pi$
$492$	0	0
$493$	8.16683e8	0.306965
$494$	0	0
$495$	1.05392e8	0.0390563
$496$	0	0
$497$	6.26173e8	0.228795
$498$	0	0
$499$	4.60991e9	1.66089	0.830445	−	0.557101i	$-0.188086\pi$
$499$	4.60991e9	1.66089	0.830445	+	0.557101i	$0.188086\pi$
$500$	0	0
$501$	7.01689e8	0.249294
$502$	0	0
$503$	1.38728e9	0.486045	0.243022	−	0.970021i	$-0.421861\pi$
$503$	1.38728e9	0.486045	0.243022	+	0.970021i	$0.421861\pi$
$504$	0	0
$505$	2.30415e8	0.0796142
$506$	0	0
$507$	2.46159e9	0.838855
$508$	0	0
$509$	−1.73584e9	−0.583441	−0.291720	−	0.956504i	$-0.594228\pi$
$509$	−1.73584e9	−0.583441	−0.291720	+	0.956504i	$0.594228\pi$
$510$	0	0
$511$	−2.30415e8	−0.0763902
$512$	0	0
$513$	−2.07489e9	−0.678555
$514$	0	0
$515$	2.66304e8	0.0859118
$516$	0	0
$517$	3.40053e9	1.08226
$518$	0	0
$519$	−1.16145e9	−0.364681
$520$	0	0
$521$	−5.79637e9	−1.79566	−0.897831	−	0.440341i	$-0.854858\pi$
$521$	−5.79637e9	−1.79566	−0.897831	+	0.440341i	$0.854858\pi$
$522$	0	0
$523$	3.71908e8	0.113679	0.0568394	−	0.998383i	$-0.481898\pi$
$523$	3.71908e8	0.113679	0.0568394	+	0.998383i	$0.481898\pi$
$524$	0	0
$525$	−4.69390e8	−0.141572
$526$	0	0
$527$	−1.28239e9	−0.381665
$528$	0	0
$529$	−3.38887e9	−0.995315
$530$	0	0
$531$	1.33020e9	0.385555
$532$	0	0
$533$	−1.01951e10	−2.91638
$534$	0	0
$535$	−2.94538e8	−0.0831580
$536$	0	0
$537$	−9.32754e8	−0.259930
$538$	0	0
$539$	3.67986e8	0.101221
$540$	0	0
$541$	−2.31256e9	−0.627919	−0.313959	−	0.949436i	$-0.601656\pi$
$541$	−2.31256e9	−0.627919	−0.313959	+	0.949436i	$0.601656\pi$
$542$	0	0
$543$	9.45610e8	0.253462
$544$	0	0
$545$	−1.97636e8	−0.0522972
$546$	0	0
$547$	2.08057e9	0.543533	0.271767	−	0.962363i	$-0.412392\pi$
$547$	2.08057e9	0.543533	0.271767	+	0.962363i	$0.412392\pi$
$548$	0	0
$549$	7.77510e8	0.200541
$550$	0	0
$551$	−4.30276e9	−1.09576
$552$	0	0
$553$	2.49663e9	0.627793
$554$	0	0
$555$	−6.81791e7	−0.0169288
$556$	0	0
$557$	−3.68671e9	−0.903952	−0.451976	−	0.892030i	$-0.649281\pi$
$557$	−3.68671e9	−0.903952	−0.451976	+	0.892030i	$0.649281\pi$
$558$	0	0
$559$	2.24201e9	0.542871
$560$	0	0
$561$	−3.03058e8	−0.0724696
$562$	0	0
$563$	−4.69057e9	−1.10776	−0.553880	−	0.832596i	$-0.686854\pi$
$563$	−4.69057e9	−1.10776	−0.553880	+	0.832596i	$0.686854\pi$
$564$	0	0
$565$	1.74486e8	0.0406998
$566$	0	0
$567$	9.77159e8	0.225125
$568$	0	0
$569$	−4.94800e9	−1.12599	−0.562997	−	0.826459i	$-0.690352\pi$
$569$	−4.94800e9	−1.12599	−0.562997	+	0.826459i	$0.690352\pi$
$570$	0	0
$571$	1.50127e9	0.337468	0.168734	−	0.985662i	$-0.446032\pi$
$571$	1.50127e9	0.337468	0.168734	+	0.985662i	$0.446032\pi$
$572$	0	0
$573$	−1.69731e8	−0.0376895
$574$	0	0
$575$	−3.10734e8	−0.0681635
$576$	0	0
$577$	−2.96898e8	−0.0643416	−0.0321708	−	0.999482i	$-0.510242\pi$
$577$	−2.96898e8	−0.0643416	−0.0321708	+	0.999482i	$0.510242\pi$
$578$	0	0
$579$	7.58408e8	0.162378
$580$	0	0
$581$	2.66833e9	0.564447
$582$	0	0
$583$	−4.70943e9	−0.984303
$584$	0	0
$585$	−4.79723e8	−0.0990707
$586$	0	0
$587$	4.57881e9	0.934370	0.467185	−	0.884160i	$-0.345268\pi$
$587$	4.57881e9	0.934370	0.467185	+	0.884160i	$0.345268\pi$
$588$	0	0
$589$	6.75636e9	1.36241
$590$	0	0
$591$	6.27175e8	0.124978
$592$	0	0
$593$	1.36674e9	0.269149	0.134575	−	0.990903i	$-0.457033\pi$
$593$	1.36674e9	0.269149	0.134575	+	0.990903i	$0.457033\pi$
$594$	0	0
$595$	3.39071e7	0.00659904
$596$	0	0
$597$	1.63399e9	0.314297
$598$	0	0
$599$	9.14960e9	1.73943	0.869717	−	0.493550i	$-0.164301\pi$
$599$	9.14960e9	1.73943	0.869717	+	0.493550i	$0.164301\pi$
$600$	0	0
$601$	5.10215e9	0.958722	0.479361	−	0.877618i	$-0.340868\pi$
$601$	5.10215e9	0.958722	0.479361	+	0.877618i	$0.340868\pi$
$602$	0	0
$603$	1.71473e9	0.318481
$604$	0	0
$605$	1.74141e8	0.0319711
$606$	0	0
$607$	5.80682e9	1.05385	0.526924	−	0.849912i	$-0.323345\pi$
$607$	5.80682e9	1.05385	0.526924	+	0.849912i	$0.323345\pi$
$608$	0	0
$609$	−8.94445e8	−0.160470
$610$	0	0
$611$	−1.54785e10	−2.74526
$612$	0	0
$613$	−8.74919e9	−1.53411	−0.767054	−	0.641583i	$-0.778278\pi$
$613$	−8.74919e9	−1.53411	−0.767054	+	0.641583i	$0.778278\pi$
$614$	0	0
$615$	−2.26031e8	−0.0391836
$616$	0	0
$617$	−1.87154e9	−0.320776	−0.160388	−	0.987054i	$-0.551275\pi$
$617$	−1.87154e9	−0.320776	−0.160388	+	0.987054i	$0.551275\pi$
$618$	0	0
$619$	5.73563e9	0.971995	0.485998	−	0.873960i	$-0.338456\pi$
$619$	5.73563e9	0.971995	0.485998	+	0.873960i	$0.338456\pi$
$620$	0	0
$621$	−2.85534e8	−0.0478450
$622$	0	0
$623$	−1.55945e8	−0.0258383
$624$	0	0
$625$	6.02814e9	0.987650
$626$	0	0
$627$	1.59668e9	0.258692
$628$	0	0
$629$	−1.18984e9	−0.190638
$630$	0	0
$631$	−5.97623e9	−0.946944	−0.473472	−	0.880809i	$-0.657000\pi$
$631$	−5.97623e9	−0.946944	−0.473472	+	0.880809i	$0.657000\pi$
$632$	0	0
$633$	−1.63995e9	−0.256991
$634$	0	0
$635$	−1.88648e8	−0.0292378
$636$	0	0
$637$	−1.67499e9	−0.256758
$638$	0	0
$639$	−3.42775e9	−0.519704
$640$	0	0
$641$	2.41475e9	0.362133	0.181067	−	0.983471i	$-0.442045\pi$
$641$	2.41475e9	0.362133	0.181067	+	0.983471i	$0.442045\pi$
$642$	0	0
$643$	−8.40746e9	−1.24717	−0.623586	−	0.781755i	$-0.714325\pi$
$643$	−8.40746e9	−1.24717	−0.623586	+	0.781755i	$0.714325\pi$
$644$	0	0
$645$	4.97068e7	0.00729384
$646$	0	0
$647$	3.79787e9	0.551284	0.275642	−	0.961260i	$-0.411110\pi$
$647$	3.79787e9	0.551284	0.275642	+	0.961260i	$0.411110\pi$
$648$	0	0
$649$	−2.21591e9	−0.318197
$650$	0	0
$651$	1.40449e9	0.199520
$652$	0	0
$653$	−4.32604e9	−0.607987	−0.303993	−	0.952674i	$-0.598320\pi$
$653$	−4.32604e9	−0.607987	−0.303993	+	0.952674i	$0.598320\pi$
$654$	0	0
$655$	3.71596e8	0.0516686
$656$	0	0
$657$	1.26132e9	0.173519
$658$	0	0
$659$	−1.36877e10	−1.86309	−0.931543	−	0.363630i	$-0.881537\pi$
$659$	−1.36877e10	−1.86309	−0.931543	+	0.363630i	$0.881537\pi$
$660$	0	0
$661$	−2.63108e9	−0.354347	−0.177173	−	0.984180i	$-0.556695\pi$
$661$	−2.63108e9	−0.354347	−0.177173	+	0.984180i	$0.556695\pi$
$662$	0	0
$663$	1.37945e9	0.183827
$664$	0	0
$665$	−1.78642e8	−0.0235563
$666$	0	0
$667$	−5.92118e8	−0.0772623
$668$	0	0
$669$	4.40078e9	0.568249
$670$	0	0
$671$	−1.29521e9	−0.165505
$672$	0	0
$673$	−2.35441e8	−0.0297735	−0.0148867	−	0.999889i	$-0.504739\pi$
$673$	−2.35441e8	−0.0297735	−0.0148867	+	0.999889i	$0.504739\pi$
$674$	0	0
$675$	5.56238e9	0.696141
$676$	0	0
$677$	−1.99784e9	−0.247457	−0.123729	−	0.992316i	$-0.539485\pi$
$677$	−1.99784e9	−0.247457	−0.123729	+	0.992316i	$0.539485\pi$
$678$	0	0
$679$	2.32331e9	0.284816
$680$	0	0
$681$	4.66890e9	0.566500
$682$	0	0
$683$	4.81437e9	0.578186	0.289093	−	0.957301i	$-0.406646\pi$
$683$	4.81437e9	0.578186	0.289093	+	0.957301i	$0.406646\pi$
$684$	0	0
$685$	−4.21233e8	−0.0500732
$686$	0	0
$687$	3.81718e9	0.449153
$688$	0	0
$689$	2.14363e10	2.49679
$690$	0	0
$691$	−1.98244e9	−0.228574	−0.114287	−	0.993448i	$-0.536458\pi$
$691$	−1.98244e9	−0.228574	−0.114287	+	0.993448i	$0.536458\pi$
$692$	0	0
$693$	−2.01440e9	−0.229921
$694$	0	0
$695$	−6.85273e8	−0.0774313
$696$	0	0
$697$	−3.94461e9	−0.441254
$698$	0	0
$699$	5.16940e9	0.572492
$700$	0	0
$701$	5.01091e9	0.549419	0.274710	−	0.961527i	$-0.411418\pi$
$701$	5.01091e9	0.549419	0.274710	+	0.961527i	$0.411418\pi$
$702$	0	0
$703$	6.26875e9	0.680514
$704$	0	0
$705$	−3.43167e8	−0.0368844
$706$	0	0
$707$	−4.40400e9	−0.468683
$708$	0	0
$709$	1.00821e10	1.06240	0.531199	−	0.847247i	$-0.321741\pi$
$709$	1.00821e10	1.06240	0.531199	+	0.847247i	$0.321741\pi$
$710$	0	0
$711$	−1.36668e10	−1.42602
$712$	0	0
$713$	9.29768e8	0.0960641
$714$	0	0
$715$	7.99145e8	0.0817625
$716$	0	0
$717$	−1.05327e9	−0.106714
$718$	0	0
$719$	7.66365e8	0.0768926	0.0384463	−	0.999261i	$-0.487759\pi$
$719$	7.66365e8	0.0768926	0.0384463	+	0.999261i	$0.487759\pi$
$720$	0	0
$721$	−5.08996e9	−0.505756
$722$	0	0
$723$	7.56942e8	0.0744867
$724$	0	0
$725$	1.15348e10	1.12416
$726$	0	0
$727$	−1.32313e10	−1.27712	−0.638561	−	0.769572i	$-0.720470\pi$
$727$	−1.32313e10	−1.27712	−0.638561	+	0.769572i	$0.720470\pi$
$728$	0	0
$729$	−2.59893e9	−0.248455
$730$	0	0
$731$	8.67465e8	0.0821375
$732$	0	0
$733$	−8.07873e8	−0.0757669	−0.0378834	−	0.999282i	$-0.512062\pi$
$733$	−8.07873e8	−0.0757669	−0.0378834	+	0.999282i	$0.512062\pi$
$734$	0	0
$735$	−3.71356e7	−0.00344972
$736$	0	0
$737$	−2.85647e9	−0.262841
$738$	0	0
$739$	7.19738e9	0.656023	0.328011	−	0.944674i	$-0.393622\pi$
$739$	7.19738e9	0.656023	0.328011	+	0.944674i	$0.393622\pi$
$740$	0	0
$741$	−7.26775e9	−0.656200
$742$	0	0
$743$	−1.16008e10	−1.03760	−0.518798	−	0.854897i	$-0.673620\pi$
$743$	−1.16008e10	−1.03760	−0.518798	+	0.854897i	$0.673620\pi$
$744$	0	0
$745$	−2.65032e8	−0.0234828
$746$	0	0
$747$	−1.46068e10	−1.28213
$748$	0	0
$749$	5.62961e9	0.489544
$750$	0	0
$751$	2.03771e10	1.75551	0.877754	−	0.479112i	$-0.159041\pi$
$751$	2.03771e10	1.75551	0.877754	+	0.479112i	$0.159041\pi$
$752$	0	0
$753$	−4.52767e9	−0.386449
$754$	0	0
$755$	−1.30977e8	−0.0110759
$756$	0	0
$757$	1.07153e10	0.897781	0.448891	−	0.893587i	$-0.351819\pi$
$757$	1.07153e10	0.897781	0.448891	+	0.893587i	$0.351819\pi$
$758$	0	0
$759$	2.19725e8	0.0182404
$760$	0	0
$761$	1.64781e10	1.35538	0.677688	−	0.735350i	$-0.262982\pi$
$761$	1.64781e10	1.35538	0.677688	+	0.735350i	$0.262982\pi$
$762$	0	0
$763$	3.77749e9	0.307870
$764$	0	0
$765$	−1.85611e8	−0.0149896
$766$	0	0
$767$	1.00863e10	0.807142
$768$	0	0
$769$	−1.74919e9	−0.138706	−0.0693529	−	0.997592i	$-0.522093\pi$
$769$	−1.74919e9	−0.138706	−0.0693529	+	0.997592i	$0.522093\pi$
$770$	0	0
$771$	4.54027e9	0.356773
$772$	0	0
$773$	1.43910e10	1.12063	0.560316	−	0.828279i	$-0.310680\pi$
$773$	1.43910e10	1.12063	0.560316	+	0.828279i	$0.310680\pi$
$774$	0	0
$775$	−1.81125e10	−1.39772
$776$	0	0
$777$	1.30313e9	0.0996584
$778$	0	0
$779$	2.07824e10	1.57513
$780$	0	0
$781$	5.71010e9	0.428909
$782$	0	0
$783$	1.05994e10	0.789066
$784$	0	0
$785$	−9.92521e8	−0.0732311
$786$	0	0
$787$	−1.77872e10	−1.30076	−0.650378	−	0.759611i	$-0.725389\pi$
$787$	−1.77872e10	−1.30076	−0.650378	+	0.759611i	$0.725389\pi$
$788$	0	0
$789$	9.70683e9	0.703571
$790$	0	0
$791$	−3.33501e9	−0.239596
$792$	0	0
$793$	5.89552e9	0.419823
$794$	0	0
$795$	4.75256e8	0.0335461
$796$	0	0
$797$	1.92091e10	1.34401	0.672006	−	0.740545i	$-0.265433\pi$
$797$	1.92091e10	1.34401	0.672006	+	0.740545i	$0.265433\pi$
$798$	0	0
$799$	−5.98883e9	−0.415363
$800$	0	0
$801$	8.53663e8	0.0586912
$802$	0	0
$803$	−2.10116e9	−0.143204
$804$	0	0
$805$	−2.45836e7	−0.00166096
$806$	0	0
$807$	6.97082e9	0.466903
$808$	0	0
$809$	4.19018e9	0.278236	0.139118	−	0.990276i	$-0.455573\pi$
$809$	4.19018e9	0.278236	0.139118	+	0.990276i	$0.455573\pi$
$810$	0	0
$811$	1.74310e10	1.14749	0.573746	−	0.819033i	$-0.305490\pi$
$811$	1.74310e10	1.14749	0.573746	+	0.819033i	$0.305490\pi$
$812$	0	0
$813$	−6.40425e7	−0.00417976
$814$	0	0
$815$	1.79788e8	0.0116335
$816$	0	0
$817$	−4.57030e9	−0.293203
$818$	0	0
$819$	9.16911e9	0.583221
$820$	0	0
$821$	−6.91999e9	−0.436420	−0.218210	−	0.975902i	$-0.570022\pi$
$821$	−6.91999e9	−0.436420	−0.218210	+	0.975902i	$0.570022\pi$
$822$	0	0
$823$	−1.07474e10	−0.672051	−0.336026	−	0.941853i	$-0.609083\pi$
$823$	−1.07474e10	−0.672051	−0.336026	+	0.941853i	$0.609083\pi$
$824$	0	0
$825$	−4.28039e9	−0.265396
$826$	0	0
$827$	2.45188e10	1.50741	0.753703	−	0.657215i	$-0.228266\pi$
$827$	2.45188e10	1.50741	0.753703	+	0.657215i	$0.228266\pi$
$828$	0	0
$829$	−7.81378e9	−0.476343	−0.238172	−	0.971223i	$-0.576548\pi$
$829$	−7.81378e9	−0.476343	−0.238172	+	0.971223i	$0.576548\pi$
$830$	0	0
$831$	−1.82899e8	−0.0110563
$832$	0	0
$833$	−6.48077e8	−0.0388480
$834$	0	0
$835$	−7.15910e8	−0.0425555
$836$	0	0
$837$	−1.66435e10	−0.981085
$838$	0	0
$839$	−7.99809e9	−0.467540	−0.233770	−	0.972292i	$-0.575106\pi$
$839$	−7.99809e9	−0.467540	−0.233770	+	0.972292i	$0.575106\pi$
$840$	0	0
$841$	4.73027e9	0.274221
$842$	0	0
$843$	−6.46009e9	−0.371400
$844$	0	0
$845$	−2.51147e9	−0.143196
$846$	0	0
$847$	−3.32842e9	−0.188211
$848$	0	0
$849$	1.07509e10	0.602931
$850$	0	0
$851$	8.62666e8	0.0479832
$852$	0	0
$853$	6.11440e9	0.337312	0.168656	−	0.985675i	$-0.446057\pi$
$853$	6.11440e9	0.337312	0.168656	+	0.985675i	$0.446057\pi$
$854$	0	0
$855$	9.77907e8	0.0535077
$856$	0	0
$857$	2.26453e10	1.22898	0.614490	−	0.788924i	$-0.289362\pi$
$857$	2.26453e10	1.22898	0.614490	+	0.788924i	$0.289362\pi$
$858$	0	0
$859$	−1.72723e10	−0.929768	−0.464884	−	0.885372i	$-0.653904\pi$
$859$	−1.72723e10	−0.929768	−0.464884	+	0.885372i	$0.653904\pi$
$860$	0	0
$861$	4.32020e9	0.230671
$862$	0	0
$863$	2.03599e10	1.07830	0.539149	−	0.842210i	$-0.318746\pi$
$863$	2.03599e10	1.07830	0.539149	+	0.842210i	$0.318746\pi$
$864$	0	0
$865$	1.18498e9	0.0622524
$866$	0	0
$867$	−6.68377e9	−0.348301
$868$	0	0
$869$	2.27669e10	1.17689
$870$	0	0
$871$	1.30020e10	0.666726
$872$	0	0
$873$	−1.27181e10	−0.646952
$874$	0	0
$875$	9.59789e8	0.0484337
$876$	0	0
$877$	1.80874e9	0.0905479	0.0452739	−	0.998975i	$-0.485584\pi$
$877$	1.80874e9	0.0905479	0.0452739	+	0.998975i	$0.485584\pi$
$878$	0	0
$879$	−8.73355e8	−0.0433740
$880$	0	0
$881$	4.77980e9	0.235502	0.117751	−	0.993043i	$-0.462432\pi$
$881$	4.77980e9	0.235502	0.117751	+	0.993043i	$0.462432\pi$
$882$	0	0
$883$	−3.86470e9	−0.188910	−0.0944548	−	0.995529i	$-0.530111\pi$
$883$	−3.86470e9	−0.188910	−0.0944548	+	0.995529i	$0.530111\pi$
$884$	0	0
$885$	2.23620e8	0.0108445
$886$	0	0
$887$	−1.45953e10	−0.702229	−0.351115	−	0.936332i	$-0.614197\pi$
$887$	−1.45953e10	−0.702229	−0.351115	+	0.936332i	$0.614197\pi$
$888$	0	0
$889$	3.60570e9	0.172121
$890$	0	0
$891$	8.91075e9	0.422029
$892$	0	0
$893$	3.15526e10	1.48270
$894$	0	0
$895$	9.51658e8	0.0443711
$896$	0	0
$897$	−1.00014e9	−0.0462688
$898$	0	0
$899$	−3.45141e10	−1.58430
$900$	0	0
$901$	8.29400e9	0.377770
$902$	0	0
$903$	−9.50062e8	−0.0429383
$904$	0	0
$905$	−9.64774e8	−0.0432669
$906$	0	0
$907$	1.38176e10	0.614903	0.307452	−	0.951564i	$-0.400524\pi$
$907$	1.38176e10	0.614903	0.307452	+	0.951564i	$0.400524\pi$
$908$	0	0
$909$	2.41080e10	1.06460
$910$	0	0
$911$	−1.31673e10	−0.577008	−0.288504	−	0.957479i	$-0.593158\pi$
$911$	−1.31673e10	−0.577008	−0.288504	+	0.957479i	$0.593158\pi$
$912$	0	0
$913$	2.43326e10	1.05814
$914$	0	0
$915$	1.30707e8	0.00564060
$916$	0	0
$917$	−7.10244e9	−0.304169
$918$	0	0
$919$	−5.23384e9	−0.222442	−0.111221	−	0.993796i	$-0.535476\pi$
$919$	−5.23384e9	−0.222442	−0.111221	+	0.993796i	$0.535476\pi$
$920$	0	0
$921$	−1.59135e9	−0.0671208
$922$	0	0
$923$	−2.59911e10	−1.08797
$924$	0	0
$925$	−1.68053e10	−0.698151
$926$	0	0
$927$	2.78631e10	1.14881
$928$	0	0
$929$	3.80007e10	1.55502	0.777512	−	0.628868i	$-0.216481\pi$
$929$	3.80007e10	1.55502	0.777512	+	0.628868i	$0.216481\pi$
$930$	0	0
$931$	3.41444e9	0.138674
$932$	0	0
$933$	−1.13601e10	−0.457927
$934$	0	0
$935$	3.09200e8	0.0123708
$936$	0	0
$937$	−1.66726e10	−0.662088	−0.331044	−	0.943615i	$-0.607401\pi$
$937$	−1.66726e10	−0.662088	−0.331044	+	0.943615i	$0.607401\pi$
$938$	0	0
$939$	2.86882e9	0.113077
$940$	0	0
$941$	−2.78637e10	−1.09012	−0.545060	−	0.838397i	$-0.683493\pi$
$941$	−2.78637e10	−1.09012	−0.545060	+	0.838397i	$0.683493\pi$
$942$	0	0
$943$	2.85995e9	0.111062
$944$	0	0
$945$	4.40064e8	0.0169631
$946$	0	0
$947$	8.68768e9	0.332414	0.166207	−	0.986091i	$-0.446848\pi$
$947$	8.68768e9	0.332414	0.166207	+	0.986091i	$0.446848\pi$
$948$	0	0
$949$	9.56404e9	0.363253
$950$	0	0
$951$	6.61759e9	0.249498
$952$	0	0
$953$	2.11469e10	0.791448	0.395724	−	0.918370i	$-0.370494\pi$
$953$	2.11469e10	0.791448	0.395724	+	0.918370i	$0.370494\pi$
$954$	0	0
$955$	1.73171e8	0.00643374
$956$	0	0
$957$	−8.15648e9	−0.300823
$958$	0	0
$959$	8.05117e9	0.294777
$960$	0	0
$961$	2.66829e10	0.969842
$962$	0	0
$963$	−3.08172e10	−1.11199
$964$	0	0
$965$	−7.73778e8	−0.0277186
$966$	0	0
$967$	2.28600e10	0.812986	0.406493	−	0.913654i	$-0.366752\pi$
$967$	2.28600e10	0.812986	0.406493	+	0.913654i	$0.366752\pi$
$968$	0	0
$969$	−2.81199e9	−0.0992843
$970$	0	0
$971$	3.20055e10	1.12191	0.560953	−	0.827847i	$-0.310435\pi$
$971$	3.20055e10	1.12191	0.560953	+	0.827847i	$0.310435\pi$
$972$	0	0
$973$	1.30978e10	0.455832
$974$	0	0
$975$	1.94834e10	0.673207
$976$	0	0
$977$	−3.58376e10	−1.22944	−0.614721	−	0.788744i	$-0.710731\pi$
$977$	−3.58376e10	−1.22944	−0.614721	+	0.788744i	$0.710731\pi$
$978$	0	0
$979$	−1.42207e9	−0.0484375
$980$	0	0
$981$	−2.06784e10	−0.699320
$982$	0	0
$983$	−1.21893e10	−0.409300	−0.204650	−	0.978835i	$-0.565606\pi$
$983$	−1.21893e10	−0.409300	−0.204650	+	0.978835i	$0.565606\pi$
$984$	0	0
$985$	−6.39886e8	−0.0213342
$986$	0	0
$987$	6.55906e9	0.217136
$988$	0	0
$989$	−6.28936e8	−0.0206738
$990$	0	0
$991$	1.62521e10	0.530459	0.265229	−	0.964185i	$-0.414552\pi$
$991$	1.62521e10	0.530459	0.265229	+	0.964185i	$0.414552\pi$
$992$	0	0
$993$	7.98255e9	0.258714
$994$	0	0
$995$	−1.66711e9	−0.0536517
$996$	0	0
$997$	−4.79814e10	−1.53334	−0.766672	−	0.642038i	$-0.778089\pi$
$997$	−4.79814e10	−1.53334	−0.766672	+	0.642038i	$0.778089\pi$
$998$	0	0
$999$	−1.54424e10	−0.490043

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	448.8.a.t.1.1		2
4.3	odd	2		448.8.a.k.1.2		2
8.3	odd	2		7.8.a.b.1.2	✓	2
8.5	even	2		112.8.a.f.1.2		2
24.11	even	2		63.8.a.e.1.1		2
40.3	even	4		175.8.b.b.99.2		4
40.19	odd	2		175.8.a.c.1.1		2
40.27	even	4		175.8.b.b.99.3		4
56.3	even	6		49.8.c.f.30.1		4
56.11	odd	6		49.8.c.e.30.1		4
56.19	even	6		49.8.c.f.18.1		4
56.27	even	2		49.8.a.c.1.2		2
56.51	odd	6		49.8.c.e.18.1		4
168.83	odd	2		441.8.a.l.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7.8.a.b.1.2	✓	2	8.3	odd	2
49.8.a.c.1.2		2	56.27	even	2
49.8.c.e.18.1		4	56.51	odd	6
49.8.c.e.30.1		4	56.11	odd	6
49.8.c.f.18.1		4	56.19	even	6
49.8.c.f.30.1		4	56.3	even	6
63.8.a.e.1.1		2	24.11	even	2
112.8.a.f.1.2		2	8.5	even	2
175.8.a.c.1.1		2	40.19	odd	2
175.8.b.b.99.2		4	40.3	even	4
175.8.b.b.99.3		4	40.27	even	4
441.8.a.l.1.1		2	168.83	odd	2
448.8.a.k.1.2		2	4.3	odd	2
448.8.a.t.1.1		2	1.1	even	1	trivial

Overview	Random
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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:	\( N \)	\(=\)	\( 448 = 2^{6} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	448.a (trivial)

\(f(q)\)	\(=\)	\(q+17.5891 q^{3} -17.9456 q^{5} +343.000 q^{7} -1877.62 q^{9} +O(q^{10})\)	\(q+17.5891 q^{3} -17.9456 q^{5} +343.000 q^{7} -1877.62 q^{9} +3127.83 q^{11} -14237.2 q^{13} -315.647 q^{15} -5508.56 q^{17} +29022.3 q^{19} +6033.07 q^{21} +3993.86 q^{23} -77803.0 q^{25} -71493.1 q^{27} -148257. q^{29} +232799. q^{31} +55015.8 q^{33} -6155.34 q^{35} +215998. q^{37} -250420. q^{39} +716086. q^{41} -157476. q^{43} +33695.0 q^{45} +1.08718e6 q^{47} +117649. q^{49} -96890.8 q^{51} -1.50566e6 q^{53} -56130.8 q^{55} +510476. q^{57} -708450. q^{59} -414093. q^{61} -644025. q^{63} +255495. q^{65} -913243. q^{67} +70248.5 q^{69} +1.82558e6 q^{71} -671764. q^{73} -1.36849e6 q^{75} +1.07285e6 q^{77} +7.27880e6 q^{79} +2.84886e6 q^{81} +7.77939e6 q^{83} +98854.4 q^{85} -2.60771e6 q^{87} -454651. q^{89} -4.88336e6 q^{91} +4.09473e6 q^{93} -520822. q^{95} +6.77351e6 q^{97} -5.87289e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 94 q^{3} - 330 q^{5} + 686 q^{7} + 1774 q^{9}+O(q^{10}) \) 2 * q + 94 * q^3 - 330 * q^5 + 686 * q^7 + 1774 * q^9	\( 2 q + 94 q^{3} - 330 q^{5} + 686 q^{7} + 1774 q^{9} + 2844 q^{11} - 2534 q^{13} - 24160 q^{15} - 1488 q^{17} + 32810 q^{19} + 32242 q^{21} + 6576 q^{23} - 58550 q^{25} + 40420 q^{27} - 20640 q^{29} + 391836 q^{31} + 33328 q^{33} - 113190 q^{35} - 367392 q^{37} + 643832 q^{39} + 734664 q^{41} - 480476 q^{43} - 1105810 q^{45} + 1089108 q^{47} + 235298 q^{49} + 210324 q^{51} - 2858844 q^{53} + 32440 q^{55} + 799900 q^{57} + 160170 q^{59} + 864646 q^{61} + 608482 q^{63} - 3396540 q^{65} - 328648 q^{67} + 267552 q^{69} + 7500216 q^{71} + 4301244 q^{73} + 102650 q^{75} + 975492 q^{77} + 6408440 q^{79} + 3414142 q^{81} + 11659074 q^{83} - 1155780 q^{85} + 7143620 q^{87} + 9772260 q^{89} - 869162 q^{91} + 16246872 q^{93} - 1702800 q^{95} + 10762752 q^{97} - 6909332 q^{99}+O(q^{100}) \) 2 * q + 94 * q^3 - 330 * q^5 + 686 * q^7 + 1774 * q^9 + 2844 * q^11 - 2534 * q^13 - 24160 * q^15 - 1488 * q^17 + 32810 * q^19 + 32242 * q^21 + 6576 * q^23 - 58550 * q^25 + 40420 * q^27 - 20640 * q^29 + 391836 * q^31 + 33328 * q^33 - 113190 * q^35 - 367392 * q^37 + 643832 * q^39 + 734664 * q^41 - 480476 * q^43 - 1105810 * q^45 + 1089108 * q^47 + 235298 * q^49 + 210324 * q^51 - 2858844 * q^53 + 32440 * q^55 + 799900 * q^57 + 160170 * q^59 + 864646 * q^61 + 608482 * q^63 - 3396540 * q^65 - 328648 * q^67 + 267552 * q^69 + 7500216 * q^71 + 4301244 * q^73 + 102650 * q^75 + 975492 * q^77 + 6408440 * q^79 + 3414142 * q^81 + 11659074 * q^83 - 1155780 * q^85 + 7143620 * q^87 + 9772260 * q^89 - 869162 * q^91 + 16246872 * q^93 - 1702800 * q^95 + 10762752 * q^97 - 6909332 * q^99

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