LMFDB - Embedded newform 272.10.a.g.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [272,10,Mod(1,272)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(272, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 10, names="a")

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

chi := DirichletCharacter("272.1");

S:= CuspForms(chi, 10);

N := Newforms(S);

Level:	$ N $	$=$	$ 272 = 2^{4} \cdot 17 $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	272.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:	$140.089747437$
Analytic rank:	$1$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - x^{6} - 2986x^{5} + 8252x^{4} + 2252056x^{3} - 10388768x^{2} - 243559296x - 675998208 $ x^7 - x^6 - 2986x^5 + 8252x^4 + 2252056x^3 - 10388768x^2 - 243559296*x - 675998208
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{19}\cdot 3^{4} $
Twist minimal:	no (minimal twist has level 17)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-43.1213$ of defining polynomial
Character	$\chi$	$=$	272.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+171.025 q^{3} +1536.21 q^{5} -3027.69 q^{7} +9566.70 q^{9} +O(q^{10})$	$q+171.025 q^{3} +1536.21 q^{5} -3027.69 q^{7} +9566.70 q^{9} -51964.2 q^{11} -166337. q^{13} +262732. q^{15} +83521.0 q^{17} +1.03482e6 q^{19} -517811. q^{21} +647595. q^{23} +406829. q^{25} -1.73014e6 q^{27} +101038. q^{29} -1.03448e6 q^{31} -8.88720e6 q^{33} -4.65118e6 q^{35} -5.58601e6 q^{37} -2.84479e7 q^{39} -4.18736e6 q^{41} +9.60190e6 q^{43} +1.46965e7 q^{45} -3.98318e7 q^{47} -3.11867e7 q^{49} +1.42842e7 q^{51} +6.22183e7 q^{53} -7.98282e7 q^{55} +1.76980e8 q^{57} -9.60858e7 q^{59} -1.86877e8 q^{61} -2.89650e7 q^{63} -2.55530e8 q^{65} -3.73689e7 q^{67} +1.10755e8 q^{69} -2.04593e8 q^{71} -1.95705e8 q^{73} +6.95782e7 q^{75} +1.57331e8 q^{77} -2.72638e8 q^{79} -4.84200e8 q^{81} +1.96272e8 q^{83} +1.28306e8 q^{85} +1.72801e7 q^{87} -3.93217e8 q^{89} +5.03618e8 q^{91} -1.76922e8 q^{93} +1.58970e9 q^{95} +8.75485e8 q^{97} -4.97126e8 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - 88 q^{3} + 1362 q^{5} - 9388 q^{7} + 81419 q^{9}+O(q^{10}) $ 7 * q - 88 * q^3 + 1362 * q^5 - 9388 * q^7 + 81419 * q^9	$ 7 q - 88 q^{3} + 1362 q^{5} - 9388 q^{7} + 81419 q^{9} - 135536 q^{11} + 166122 q^{13} - 159048 q^{15} + 584647 q^{17} - 777172 q^{19} - 3412104 q^{21} - 1357764 q^{23} + 1065785 q^{25} + 4519064 q^{27} + 967002 q^{29} - 3546740 q^{31} + 11928016 q^{33} + 530736 q^{35} + 18296498 q^{37} - 86306872 q^{39} + 10285686 q^{41} - 21913204 q^{43} + 108916410 q^{45} - 56639800 q^{47} + 27010351 q^{49} - 7349848 q^{51} + 121813562 q^{53} - 40793128 q^{55} + 153612960 q^{57} - 29222388 q^{59} - 49915846 q^{61} + 2185356 q^{63} - 122633668 q^{65} - 301863420 q^{67} + 379683432 q^{69} - 652473940 q^{71} + 306656342 q^{73} - 919071912 q^{75} - 102442536 q^{77} - 959147884 q^{79} - 374486977 q^{81} + 1512945268 q^{83} + 113755602 q^{85} + 1612550856 q^{87} - 1971327114 q^{89} + 1061062864 q^{91} - 798598936 q^{93} + 3249631512 q^{95} + 2006526254 q^{97} + 2579159272 q^{99}+O(q^{100}) $ 7 * q - 88 * q^3 + 1362 * q^5 - 9388 * q^7 + 81419 * q^9 - 135536 * q^11 + 166122 * q^13 - 159048 * q^15 + 584647 * q^17 - 777172 * q^19 - 3412104 * q^21 - 1357764 * q^23 + 1065785 * q^25 + 4519064 * q^27 + 967002 * q^29 - 3546740 * q^31 + 11928016 * q^33 + 530736 * q^35 + 18296498 * q^37 - 86306872 * q^39 + 10285686 * q^41 - 21913204 * q^43 + 108916410 * q^45 - 56639800 * q^47 + 27010351 * q^49 - 7349848 * q^51 + 121813562 * q^53 - 40793128 * q^55 + 153612960 * q^57 - 29222388 * q^59 - 49915846 * q^61 + 2185356 * q^63 - 122633668 * q^65 - 301863420 * q^67 + 379683432 * q^69 - 652473940 * q^71 + 306656342 * q^73 - 919071912 * q^75 - 102442536 * q^77 - 959147884 * q^79 - 374486977 * q^81 + 1512945268 * q^83 + 113755602 * q^85 + 1612550856 * q^87 - 1971327114 * q^89 + 1061062864 * q^91 - 798598936 * q^93 + 3249631512 * q^95 + 2006526254 * q^97 + 2579159272 * 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
$3$	171.025	1.21903	0.609516	−	0.792774i	$-0.291364\pi$
$3$	171.025	1.21903	0.609516	+	0.792774i	$0.291364\pi$
$4$	0	0
$5$	1536.21	1.09923	0.549613	−	0.835420i	$-0.314775\pi$
$5$	1536.21	1.09923	0.549613	+	0.835420i	$0.314775\pi$
$6$	0	0
$7$	−3027.69	−0.476617	−0.238309	−	0.971189i	$-0.576593\pi$
$7$	−3027.69	−0.476617	−0.238309	+	0.971189i	$0.576593\pi$
$8$	0	0
$9$	9566.70	0.486039
$10$	0	0
$11$	−51964.2	−1.07013	−0.535066	−	0.844810i	$-0.679713\pi$
$11$	−51964.2	−1.07013	−0.535066	+	0.844810i	$0.679713\pi$
$12$	0	0
$13$	−166337.	−1.61527	−0.807635	−	0.589683i	$-0.799253\pi$
$13$	−166337.	−1.61527	−0.807635	+	0.589683i	$0.799253\pi$
$14$	0	0
$15$	262732.	1.33999
$16$	0	0
$17$	83521.0	0.242536
$17$	83521.0	0.242536
$18$	0	0
$19$	1.03482e6	1.82168	0.910840	−	0.412759i	$-0.135435\pi$
$19$	1.03482e6	1.82168	0.910840	+	0.412759i	$0.135435\pi$
$20$	0	0
$21$	−517811.	−0.581012
$22$	0	0
$23$	647595.	0.482535	0.241267	−	0.970459i	$-0.422437\pi$
$23$	647595.	0.482535	0.241267	+	0.970459i	$0.422437\pi$
$24$	0	0
$25$	406829.	0.208297
$26$	0	0
$27$	−1.73014e6	−0.626535
$28$	0	0
$29$	101038.	0.0265274	0.0132637	−	0.999912i	$-0.495778\pi$
$29$	101038.	0.0265274	0.0132637	+	0.999912i	$0.495778\pi$
$30$	0	0
$31$	−1.03448e6	−0.201184	−0.100592	−	0.994928i	$-0.532074\pi$
$31$	−1.03448e6	−0.201184	−0.100592	+	0.994928i	$0.532074\pi$
$32$	0	0
$33$	−8.88720e6	−1.30452
$34$	0	0
$35$	−4.65118e6	−0.523910
$36$	0	0
$37$	−5.58601e6	−0.489998	−0.244999	−	0.969523i	$-0.578788\pi$
$37$	−5.58601e6	−0.489998	−0.244999	+	0.969523i	$0.578788\pi$
$38$	0	0
$39$	−2.84479e7	−1.96907
$40$	0	0
$41$	−4.18736e6	−0.231426	−0.115713	−	0.993283i	$-0.536915\pi$
$41$	−4.18736e6	−0.231426	−0.115713	+	0.993283i	$0.536915\pi$
$42$	0	0
$43$	9.60190e6	0.428301	0.214151	−	0.976801i	$-0.431302\pi$
$43$	9.60190e6	0.428301	0.214151	+	0.976801i	$0.431302\pi$
$44$	0	0
$45$	1.46965e7	0.534266
$46$	0	0
$47$	−3.98318e7	−1.19067	−0.595333	−	0.803479i	$-0.702980\pi$
$47$	−3.98318e7	−1.19067	−0.595333	+	0.803479i	$0.702980\pi$
$48$	0	0
$49$	−3.11867e7	−0.772836
$50$	0	0
$51$	1.42842e7	0.295659
$52$	0	0
$53$	6.22183e7	1.08312	0.541560	−	0.840662i	$-0.317834\pi$
$53$	6.22183e7	1.08312	0.541560	+	0.840662i	$0.317834\pi$
$54$	0	0
$55$	−7.98282e7	−1.17632
$56$	0	0
$57$	1.76980e8	2.22069
$58$	0	0
$59$	−9.60858e7	−1.03235	−0.516173	−	0.856484i	$-0.672644\pi$
$59$	−9.60858e7	−1.03235	−0.516173	+	0.856484i	$0.672644\pi$
$60$	0	0
$61$	−1.86877e8	−1.72811	−0.864055	−	0.503398i	$-0.832083\pi$
$61$	−1.86877e8	−1.72811	−0.864055	+	0.503398i	$0.832083\pi$
$62$	0	0
$63$	−2.89650e7	−0.231655
$64$	0	0
$65$	−2.55530e8	−1.77555
$66$	0	0
$67$	−3.73689e7	−0.226555	−0.113278	−	0.993563i	$-0.536135\pi$
$67$	−3.73689e7	−0.226555	−0.113278	+	0.993563i	$0.536135\pi$
$68$	0	0
$69$	1.10755e8	0.588225
$70$	0	0
$71$	−2.04593e8	−0.955495	−0.477748	−	0.878497i	$-0.658547\pi$
$71$	−2.04593e8	−0.955495	−0.477748	+	0.878497i	$0.658547\pi$
$72$	0	0
$73$	−1.95705e8	−0.806581	−0.403291	−	0.915072i	$-0.632134\pi$
$73$	−1.95705e8	−0.806581	−0.403291	+	0.915072i	$0.632134\pi$
$74$	0	0
$75$	6.95782e7	0.253920
$76$	0	0
$77$	1.57331e8	0.510043
$78$	0	0
$79$	−2.72638e8	−0.787524	−0.393762	−	0.919212i	$-0.628827\pi$
$79$	−2.72638e8	−0.787524	−0.393762	+	0.919212i	$0.628827\pi$
$80$	0	0
$81$	−4.84200e8	−1.24981
$82$	0	0
$83$	1.96272e8	0.453950	0.226975	−	0.973901i	$-0.427116\pi$
$83$	1.96272e8	0.453950	0.226975	+	0.973901i	$0.427116\pi$
$84$	0	0
$85$	1.28306e8	0.266601
$86$	0	0
$87$	1.72801e7	0.0323378
$88$	0	0
$89$	−3.93217e8	−0.664319	−0.332160	−	0.943223i	$-0.607777\pi$
$89$	−3.93217e8	−0.664319	−0.332160	+	0.943223i	$0.607777\pi$
$90$	0	0
$91$	5.03618e8	0.769865
$92$	0	0
$93$	−1.76922e8	−0.245249
$94$	0	0
$95$	1.58970e9	2.00244
$96$	0	0
$97$	8.75485e8	1.00410	0.502049	−	0.864839i	$-0.332580\pi$
$97$	8.75485e8	1.00410	0.502049	+	0.864839i	$0.332580\pi$
$98$	0	0
$99$	−4.97126e8	−0.520126
$100$	0	0
$101$	2.76987e8	0.264858	0.132429	−	0.991192i	$-0.457722\pi$
$101$	2.76987e8	0.264858	0.132429	+	0.991192i	$0.457722\pi$
$102$	0	0
$103$	5.47928e7	0.0479685	0.0239842	−	0.999712i	$-0.492365\pi$
$103$	5.47928e7	0.0479685	0.0239842	+	0.999712i	$0.492365\pi$
$104$	0	0
$105$	−7.95469e8	−0.638663
$106$	0	0
$107$	−1.33346e9	−0.983449	−0.491724	−	0.870751i	$-0.663633\pi$
$107$	−1.33346e9	−0.983449	−0.491724	+	0.870751i	$0.663633\pi$
$108$	0	0
$109$	−7.04251e8	−0.477868	−0.238934	−	0.971036i	$-0.576798\pi$
$109$	−7.04251e8	−0.477868	−0.238934	+	0.971036i	$0.576798\pi$
$110$	0	0
$111$	−9.55350e8	−0.597323
$112$	0	0
$113$	1.40406e9	0.810091	0.405045	−	0.914297i	$-0.367256\pi$
$113$	1.40406e9	0.810091	0.405045	+	0.914297i	$0.367256\pi$
$114$	0	0
$115$	9.94845e8	0.530414
$116$	0	0
$117$	−1.59130e9	−0.785084
$118$	0	0
$119$	−2.52875e8	−0.115597
$120$	0	0
$121$	3.42331e8	0.145182
$122$	0	0
$123$	−7.16145e8	−0.282116
$124$	0	0
$125$	−2.37544e9	−0.870261
$126$	0	0
$127$	−4.64719e8	−0.158516	−0.0792581	−	0.996854i	$-0.525255\pi$
$127$	−4.64719e8	−0.158516	−0.0792581	+	0.996854i	$0.525255\pi$
$128$	0	0
$129$	1.64217e9	0.522113
$130$	0	0
$131$	−4.70705e9	−1.39646	−0.698229	−	0.715874i	$-0.746028\pi$
$131$	−4.70705e9	−1.39646	−0.698229	+	0.715874i	$0.746028\pi$
$132$	0	0
$133$	−3.13310e9	−0.868244
$134$	0	0
$135$	−2.65787e9	−0.688703
$136$	0	0
$137$	4.09197e9	0.992407	0.496203	−	0.868206i	$-0.334727\pi$
$137$	4.09197e9	0.992407	0.496203	+	0.868206i	$0.334727\pi$
$138$	0	0
$139$	4.61710e8	0.104907	0.0524533	−	0.998623i	$-0.483296\pi$
$139$	4.61710e8	0.104907	0.0524533	+	0.998623i	$0.483296\pi$
$140$	0	0
$141$	−6.81226e9	−1.45146
$142$	0	0
$143$	8.64359e9	1.72855
$144$	0	0
$145$	1.55216e8	0.0291596
$146$	0	0
$147$	−5.33372e9	−0.942112
$148$	0	0
$149$	1.87018e9	0.310847	0.155423	−	0.987848i	$-0.450326\pi$
$149$	1.87018e9	0.310847	0.155423	+	0.987848i	$0.450326\pi$
$150$	0	0
$151$	6.04895e9	0.946856	0.473428	−	0.880833i	$-0.343016\pi$
$151$	6.04895e9	0.946856	0.473428	+	0.880833i	$0.343016\pi$
$152$	0	0
$153$	7.99021e8	0.117882
$154$	0	0
$155$	−1.58918e9	−0.221146
$156$	0	0
$157$	5.87552e9	0.771788	0.385894	−	0.922543i	$-0.373893\pi$
$157$	5.87552e9	0.771788	0.385894	+	0.922543i	$0.373893\pi$
$158$	0	0
$159$	1.06409e10	1.32036
$160$	0	0
$161$	−1.96072e9	−0.229984
$162$	0	0
$163$	−1.19332e10	−1.32408	−0.662039	−	0.749470i	$-0.730309\pi$
$163$	−1.19332e10	−1.32408	−0.662039	+	0.749470i	$0.730309\pi$
$164$	0	0
$165$	−1.36526e10	−1.43397
$166$	0	0
$167$	−2.11741e9	−0.210659	−0.105330	−	0.994437i	$-0.533590\pi$
$167$	−2.11741e9	−0.210659	−0.105330	+	0.994437i	$0.533590\pi$
$168$	0	0
$169$	1.70637e10	1.60910
$170$	0	0
$171$	9.89978e9	0.885407
$172$	0	0
$173$	4.37524e9	0.371359	0.185680	−	0.982610i	$-0.440551\pi$
$173$	4.37524e9	0.371359	0.185680	+	0.982610i	$0.440551\pi$
$174$	0	0
$175$	−1.23175e9	−0.0992777
$176$	0	0
$177$	−1.64331e10	−1.25846
$178$	0	0
$179$	−2.12791e9	−0.154922	−0.0774612	−	0.996995i	$-0.524681\pi$
$179$	−2.12791e9	−0.154922	−0.0774612	+	0.996995i	$0.524681\pi$
$180$	0	0
$181$	1.80496e10	1.25001	0.625007	−	0.780619i	$-0.285096\pi$
$181$	1.80496e10	1.25001	0.625007	+	0.780619i	$0.285096\pi$
$182$	0	0
$183$	−3.19607e10	−2.10662
$184$	0	0
$185$	−8.58131e9	−0.538618
$186$	0	0
$187$	−4.34010e9	−0.259545
$188$	0	0
$189$	5.23833e9	0.298617
$190$	0	0
$191$	6.31774e8	0.0343488	0.0171744	−	0.999853i	$-0.494533\pi$
$191$	6.31774e8	0.0343488	0.0171744	+	0.999853i	$0.494533\pi$
$192$	0	0
$193$	1.51993e10	0.788524	0.394262	−	0.918998i	$-0.371000\pi$
$193$	1.51993e10	0.788524	0.394262	+	0.918998i	$0.371000\pi$
$194$	0	0
$195$	−4.37021e10	−2.16445
$196$	0	0
$197$	−2.63264e10	−1.24535	−0.622677	−	0.782479i	$-0.713955\pi$
$197$	−2.63264e10	−1.24535	−0.622677	+	0.782479i	$0.713955\pi$
$198$	0	0
$199$	−3.40221e10	−1.53788	−0.768940	−	0.639321i	$-0.779216\pi$
$199$	−3.40221e10	−1.53788	−0.768940	+	0.639321i	$0.779216\pi$
$200$	0	0
$201$	−6.39104e9	−0.276178
$202$	0	0
$203$	−3.05912e8	−0.0126434
$204$	0	0
$205$	−6.43268e9	−0.254390
$206$	0	0
$207$	6.19535e9	0.234531
$208$	0	0
$209$	−5.37734e10	−1.94944
$210$	0	0
$211$	2.29918e10	0.798550	0.399275	−	0.916831i	$-0.369262\pi$
$211$	2.29918e10	0.798550	0.399275	+	0.916831i	$0.369262\pi$
$212$	0	0
$213$	−3.49906e10	−1.16478
$214$	0	0
$215$	1.47506e10	0.470800
$216$	0	0
$217$	3.13207e9	0.0958876
$218$	0	0
$219$	−3.34705e10	−0.983248
$220$	0	0
$221$	−1.38927e10	−0.391760
$222$	0	0
$223$	5.81614e10	1.57494	0.787468	−	0.616355i	$-0.211391\pi$
$223$	5.81614e10	1.57494	0.787468	+	0.616355i	$0.211391\pi$
$224$	0	0
$225$	3.89201e9	0.101240
$226$	0	0
$227$	6.36497e9	0.159104	0.0795518	−	0.996831i	$-0.474651\pi$
$227$	6.36497e9	0.159104	0.0795518	+	0.996831i	$0.474651\pi$
$228$	0	0
$229$	−7.25334e10	−1.74292	−0.871461	−	0.490464i	$-0.836827\pi$
$229$	−7.25334e10	−1.74292	−0.871461	+	0.490464i	$0.836827\pi$
$230$	0	0
$231$	2.69077e10	0.621759
$232$	0	0
$233$	−8.13370e10	−1.80795	−0.903975	−	0.427585i	$-0.859365\pi$
$233$	−8.13370e10	−1.80795	−0.903975	+	0.427585i	$0.859365\pi$
$234$	0	0
$235$	−6.11902e10	−1.30881
$236$	0	0
$237$	−4.66280e10	−0.960018
$238$	0	0
$239$	−7.48651e10	−1.48419	−0.742094	−	0.670296i	$-0.766167\pi$
$239$	−7.48651e10	−1.48419	−0.742094	+	0.670296i	$0.766167\pi$
$240$	0	0
$241$	−1.06255e10	−0.202896	−0.101448	−	0.994841i	$-0.532348\pi$
$241$	−1.06255e10	−0.202896	−0.101448	+	0.994841i	$0.532348\pi$
$242$	0	0
$243$	−4.87561e10	−0.897017
$244$	0	0
$245$	−4.79095e10	−0.849521
$246$	0	0
$247$	−1.72129e11	−2.94250
$248$	0	0
$249$	3.35676e10	0.553379
$250$	0	0
$251$	9.55285e10	1.51915	0.759576	−	0.650418i	$-0.225406\pi$
$251$	9.55285e10	1.51915	0.759576	+	0.650418i	$0.225406\pi$
$252$	0	0
$253$	−3.36518e10	−0.516376
$254$	0	0
$255$	2.19436e10	0.324996
$256$	0	0
$257$	6.50855e10	0.930647	0.465324	−	0.885141i	$-0.345938\pi$
$257$	6.50855e10	0.930647	0.465324	+	0.885141i	$0.345938\pi$
$258$	0	0
$259$	1.69127e10	0.233541
$260$	0	0
$261$	9.66604e8	0.0128934
$262$	0	0
$263$	6.33984e10	0.817104	0.408552	−	0.912735i	$-0.366034\pi$
$263$	6.33984e10	0.817104	0.408552	+	0.912735i	$0.366034\pi$
$264$	0	0
$265$	9.55806e10	1.19059
$266$	0	0
$267$	−6.72501e10	−0.809826
$268$	0	0
$269$	−7.50317e10	−0.873695	−0.436847	−	0.899536i	$-0.643905\pi$
$269$	−7.50317e10	−0.873695	−0.436847	+	0.899536i	$0.643905\pi$
$270$	0	0
$271$	9.55753e10	1.07643	0.538213	−	0.842809i	$-0.319100\pi$
$271$	9.55753e10	1.07643	0.538213	+	0.842809i	$0.319100\pi$
$272$	0	0
$273$	8.61314e10	0.938490
$274$	0	0
$275$	−2.11406e10	−0.222905
$276$	0	0
$277$	−1.40532e10	−0.143422	−0.0717110	−	0.997425i	$-0.522846\pi$
$277$	−1.40532e10	−0.143422	−0.0717110	+	0.997425i	$0.522846\pi$
$278$	0	0
$279$	−9.89653e9	−0.0977831
$280$	0	0
$281$	1.00661e11	0.963124	0.481562	−	0.876412i	$-0.340070\pi$
$281$	1.00661e11	0.963124	0.481562	+	0.876412i	$0.340070\pi$
$282$	0	0
$283$	−7.78714e10	−0.721670	−0.360835	−	0.932630i	$-0.617508\pi$
$283$	−7.78714e10	−0.721670	−0.360835	+	0.932630i	$0.617508\pi$
$284$	0	0
$285$	2.71879e11	2.44103
$286$	0	0
$287$	1.26780e10	0.110302
$288$	0	0
$289$	6.97576e9	0.0588235
$290$	0	0
$291$	1.49730e11	1.22403
$292$	0	0
$293$	−8.86150e10	−0.702429	−0.351215	−	0.936295i	$-0.614231\pi$
$293$	−8.86150e10	−0.702429	−0.351215	+	0.936295i	$0.614231\pi$
$294$	0	0
$295$	−1.47608e11	−1.13478
$296$	0	0
$297$	8.99056e10	0.670475
$298$	0	0
$299$	−1.07719e11	−0.779423
$300$	0	0
$301$	−2.90716e10	−0.204136
$302$	0	0
$303$	4.73718e10	0.322871
$304$	0	0
$305$	−2.87083e11	−1.89958
$306$	0	0
$307$	1.69929e11	1.09180	0.545902	−	0.837849i	$-0.316187\pi$
$307$	1.69929e11	1.09180	0.545902	+	0.837849i	$0.316187\pi$
$308$	0	0
$309$	9.37096e9	0.0584751
$310$	0	0
$311$	−1.18662e11	−0.719265	−0.359633	−	0.933094i	$-0.617098\pi$
$311$	−1.18662e11	−0.719265	−0.359633	+	0.933094i	$0.617098\pi$
$312$	0	0
$313$	2.52103e11	1.48467	0.742333	−	0.670032i	$-0.233719\pi$
$313$	2.52103e11	1.48467	0.742333	+	0.670032i	$0.233719\pi$
$314$	0	0
$315$	−4.44964e10	−0.254641
$316$	0	0
$317$	7.54487e10	0.419648	0.209824	−	0.977739i	$-0.432711\pi$
$317$	7.54487e10	0.419648	0.209824	+	0.977739i	$0.432711\pi$
$318$	0	0
$319$	−5.25038e9	−0.0283878
$320$	0	0
$321$	−2.28055e11	−1.19886
$322$	0	0
$323$	8.64289e10	0.441822
$324$	0	0
$325$	−6.76709e10	−0.336455
$326$	0	0
$327$	−1.20445e11	−0.582537
$328$	0	0
$329$	1.20598e11	0.567492
$330$	0	0
$331$	2.39508e11	1.09672	0.548358	−	0.836244i	$-0.315253\pi$
$331$	2.39508e11	1.09672	0.548358	+	0.836244i	$0.315253\pi$
$332$	0	0
$333$	−5.34397e10	−0.238158
$334$	0	0
$335$	−5.74067e10	−0.249035
$336$	0	0
$337$	8.05715e10	0.340288	0.170144	−	0.985419i	$-0.445577\pi$
$337$	8.05715e10	0.340288	0.170144	+	0.985419i	$0.445577\pi$
$338$	0	0
$339$	2.40131e11	0.987526
$340$	0	0
$341$	5.37557e10	0.215293
$342$	0	0
$343$	2.16602e11	0.844964
$344$	0	0
$345$	1.70144e11	0.646592
$346$	0	0
$347$	3.81897e11	1.41405	0.707024	−	0.707190i	$-0.250037\pi$
$347$	3.81897e11	1.41405	0.707024	+	0.707190i	$0.250037\pi$
$348$	0	0
$349$	−2.86306e11	−1.03304	−0.516518	−	0.856276i	$-0.672772\pi$
$349$	−2.86306e11	−1.03304	−0.516518	+	0.856276i	$0.672772\pi$
$350$	0	0
$351$	2.87788e11	1.01202
$352$	0	0
$353$	3.83234e10	0.131364	0.0656822	−	0.997841i	$-0.479078\pi$
$353$	3.83234e10	0.131364	0.0656822	+	0.997841i	$0.479078\pi$
$354$	0	0
$355$	−3.14299e11	−1.05030
$356$	0	0
$357$	−4.32481e10	−0.140916
$358$	0	0
$359$	8.46163e10	0.268862	0.134431	−	0.990923i	$-0.457079\pi$
$359$	8.46163e10	0.268862	0.134431	+	0.990923i	$0.457079\pi$
$360$	0	0
$361$	7.48157e11	2.31852
$362$	0	0
$363$	5.85473e10	0.176981
$364$	0	0
$365$	−3.00644e11	−0.886615
$366$	0	0
$367$	−6.52825e11	−1.87845	−0.939225	−	0.343301i	$-0.888455\pi$
$367$	−6.52825e11	−1.87845	−0.939225	+	0.343301i	$0.888455\pi$
$368$	0	0
$369$	−4.00592e10	−0.112482
$370$	0	0
$371$	−1.88377e11	−0.516234
$372$	0	0
$373$	−9.69405e10	−0.259308	−0.129654	−	0.991559i	$-0.541387\pi$
$373$	−9.69405e10	−0.259308	−0.129654	+	0.991559i	$0.541387\pi$
$374$	0	0
$375$	−4.06261e11	−1.06088
$376$	0	0
$377$	−1.68065e10	−0.0428489
$378$	0	0
$379$	6.80659e11	1.69454	0.847272	−	0.531159i	$-0.178243\pi$
$379$	6.80659e11	1.69454	0.847272	+	0.531159i	$0.178243\pi$
$380$	0	0
$381$	−7.94788e10	−0.193236
$382$	0	0
$383$	−1.56533e11	−0.371717	−0.185858	−	0.982577i	$-0.559507\pi$
$383$	−1.56533e11	−0.371717	−0.185858	+	0.982577i	$0.559507\pi$
$384$	0	0
$385$	2.41695e11	0.560652
$386$	0	0
$387$	9.18586e10	0.208171
$388$	0	0
$389$	−5.55777e11	−1.23063	−0.615315	−	0.788282i	$-0.710971\pi$
$389$	−5.55777e11	−1.23063	−0.615315	+	0.788282i	$0.710971\pi$
$390$	0	0
$391$	5.40878e10	0.117032
$392$	0	0
$393$	−8.05025e11	−1.70233
$394$	0	0
$395$	−4.18830e11	−0.865667
$396$	0	0
$397$	−2.61410e11	−0.528159	−0.264079	−	0.964501i	$-0.585068\pi$
$397$	−2.61410e11	−0.528159	−0.264079	+	0.964501i	$0.585068\pi$
$398$	0	0
$399$	−5.35840e11	−1.05842
$400$	0	0
$401$	8.04746e10	0.155421	0.0777104	−	0.996976i	$-0.475239\pi$
$401$	8.04746e10	0.155421	0.0777104	+	0.996976i	$0.475239\pi$
$402$	0	0
$403$	1.72072e11	0.324966
$404$	0	0
$405$	−7.43835e11	−1.37382
$406$	0	0
$407$	2.90273e11	0.524362
$408$	0	0
$409$	4.96676e11	0.877643	0.438822	−	0.898574i	$-0.355396\pi$
$409$	4.96676e11	0.877643	0.438822	+	0.898574i	$0.355396\pi$
$410$	0	0
$411$	6.99831e11	1.20978
$412$	0	0
$413$	2.90918e11	0.492034
$414$	0	0
$415$	3.01516e11	0.498993
$416$	0	0
$417$	7.89642e10	0.127885
$418$	0	0
$419$	−5.00093e11	−0.792661	−0.396331	−	0.918108i	$-0.629717\pi$
$419$	−5.00093e11	−0.792661	−0.396331	+	0.918108i	$0.629717\pi$
$420$	0	0
$421$	9.47398e11	1.46982	0.734908	−	0.678167i	$-0.237225\pi$
$421$	9.47398e11	1.46982	0.734908	+	0.678167i	$0.237225\pi$
$422$	0	0
$423$	−3.81059e11	−0.578710
$424$	0	0
$425$	3.39788e10	0.0505193
$426$	0	0
$427$	5.65805e11	0.823647
$428$	0	0
$429$	1.47827e12	2.10716
$430$	0	0
$431$	5.09374e11	0.711032	0.355516	−	0.934670i	$-0.384305\pi$
$431$	5.09374e11	0.711032	0.355516	+	0.934670i	$0.384305\pi$
$432$	0	0
$433$	1.25104e12	1.71032	0.855160	−	0.518365i	$-0.173459\pi$
$433$	1.25104e12	1.71032	0.855160	+	0.518365i	$0.173459\pi$
$434$	0	0
$435$	2.65460e10	0.0355465
$436$	0	0
$437$	6.70142e11	0.879024
$438$	0	0
$439$	1.15261e12	1.48112	0.740561	−	0.671989i	$-0.234560\pi$
$439$	1.15261e12	1.48112	0.740561	+	0.671989i	$0.234560\pi$
$440$	0	0
$441$	−2.98354e11	−0.375628
$442$	0	0
$443$	−1.57617e12	−1.94440	−0.972199	−	0.234155i	$-0.924768\pi$
$443$	−1.57617e12	−1.94440	−0.972199	+	0.234155i	$0.924768\pi$
$444$	0	0
$445$	−6.04065e11	−0.730237
$446$	0	0
$447$	3.19849e11	0.378932
$448$	0	0
$449$	−1.39885e12	−1.62428	−0.812141	−	0.583461i	$-0.801698\pi$
$449$	−1.39885e12	−1.62428	−0.812141	+	0.583461i	$0.801698\pi$
$450$	0	0
$451$	2.17593e11	0.247657
$452$	0	0
$453$	1.03452e12	1.15425
$454$	0	0
$455$	7.73665e11	0.846255
$456$	0	0
$457$	3.76472e11	0.403747	0.201874	−	0.979412i	$-0.435297\pi$
$457$	3.76472e11	0.403747	0.201874	+	0.979412i	$0.435297\pi$
$458$	0	0
$459$	−1.44503e11	−0.151957
$460$	0	0
$461$	2.03996e11	0.210362	0.105181	−	0.994453i	$-0.466458\pi$
$461$	2.03996e11	0.210362	0.105181	+	0.994453i	$0.466458\pi$
$462$	0	0
$463$	1.19166e12	1.20514	0.602571	−	0.798065i	$-0.294143\pi$
$463$	1.19166e12	1.20514	0.602571	+	0.798065i	$0.294143\pi$
$464$	0	0
$465$	−2.71790e11	−0.269584
$466$	0	0
$467$	1.20046e12	1.16795	0.583973	−	0.811773i	$-0.301497\pi$
$467$	1.20046e12	1.16795	0.583973	+	0.811773i	$0.301497\pi$
$468$	0	0
$469$	1.13141e11	0.107980
$470$	0	0
$471$	1.00486e12	0.940834
$472$	0	0
$473$	−4.98955e11	−0.458339
$474$	0	0
$475$	4.20994e11	0.379450
$476$	0	0
$477$	5.95224e11	0.526439
$478$	0	0
$479$	3.78765e10	0.0328746	0.0164373	−	0.999865i	$-0.494768\pi$
$479$	3.78765e10	0.0328746	0.0164373	+	0.999865i	$0.494768\pi$
$480$	0	0
$481$	9.29163e11	0.791478
$482$	0	0
$483$	−3.35332e11	−0.280358
$484$	0	0
$485$	1.34493e12	1.10373
$486$	0	0
$487$	−1.88071e12	−1.51510	−0.757550	−	0.652778i	$-0.773604\pi$
$487$	−1.88071e12	−1.51510	−0.757550	+	0.652778i	$0.773604\pi$
$488$	0	0
$489$	−2.04088e12	−1.61409
$490$	0	0
$491$	2.42874e11	0.188588	0.0942942	−	0.995544i	$-0.469941\pi$
$491$	2.42874e11	0.188588	0.0942942	+	0.995544i	$0.469941\pi$
$492$	0	0
$493$	8.43882e9	0.00643384
$494$	0	0
$495$	−7.63692e11	−0.571735
$496$	0	0
$497$	6.19444e11	0.455405
$498$	0	0
$499$	−7.88648e11	−0.569417	−0.284709	−	0.958614i	$-0.591897\pi$
$499$	−7.88648e11	−0.569417	−0.284709	+	0.958614i	$0.591897\pi$
$500$	0	0
$501$	−3.62131e11	−0.256800
$502$	0	0
$503$	−1.92372e12	−1.33994	−0.669972	−	0.742386i	$-0.733694\pi$
$503$	−1.92372e12	−1.33994	−0.669972	+	0.742386i	$0.733694\pi$
$504$	0	0
$505$	4.25512e11	0.291139
$506$	0	0
$507$	2.91832e12	1.96154
$508$	0	0
$509$	2.87805e12	1.90050	0.950250	−	0.311487i	$-0.100827\pi$
$509$	2.87805e12	1.90050	0.950250	+	0.311487i	$0.100827\pi$
$510$	0	0
$511$	5.92532e11	0.384431
$512$	0	0
$513$	−1.79038e12	−1.14135
$514$	0	0
$515$	8.41734e10	0.0527281
$516$	0	0
$517$	2.06983e12	1.27417
$518$	0	0
$519$	7.48277e11	0.452699
$520$	0	0
$521$	2.41942e12	1.43861	0.719303	−	0.694697i	$-0.244461\pi$
$521$	2.41942e12	1.43861	0.719303	+	0.694697i	$0.244461\pi$
$522$	0	0
$523$	−2.25636e12	−1.31872	−0.659358	−	0.751829i	$-0.729172\pi$
$523$	−2.25636e12	−1.31872	−0.659358	+	0.751829i	$0.729172\pi$
$524$	0	0
$525$	−2.10661e11	−0.121023
$526$	0	0
$527$	−8.64005e10	−0.0487942
$528$	0	0
$529$	−1.38177e12	−0.767160
$530$	0	0
$531$	−9.19225e11	−0.501760
$532$	0	0
$533$	6.96515e11	0.373816
$534$	0	0
$535$	−2.04847e12	−1.08103
$536$	0	0
$537$	−3.63926e11	−0.188855
$538$	0	0
$539$	1.62059e12	0.827036
$540$	0	0
$541$	−2.18649e12	−1.09739	−0.548694	−	0.836023i	$-0.684875\pi$
$541$	−2.18649e12	−1.09739	−0.548694	+	0.836023i	$0.684875\pi$
$542$	0	0
$543$	3.08695e12	1.52381
$544$	0	0
$545$	−1.08188e12	−0.525285
$546$	0	0
$547$	7.53862e11	0.360039	0.180019	−	0.983663i	$-0.442384\pi$
$547$	7.53862e11	0.360039	0.180019	+	0.983663i	$0.442384\pi$
$548$	0	0
$549$	−1.78780e12	−0.839929
$550$	0	0
$551$	1.04556e11	0.0483245
$552$	0	0
$553$	8.25462e11	0.375348
$554$	0	0
$555$	−1.46762e12	−0.656592
$556$	0	0
$557$	−2.08949e12	−0.919797	−0.459898	−	0.887972i	$-0.652114\pi$
$557$	−2.08949e12	−0.919797	−0.459898	+	0.887972i	$0.652114\pi$
$558$	0	0
$559$	−1.59716e12	−0.691822
$560$	0	0
$561$	−7.42268e11	−0.316394
$562$	0	0
$563$	6.61278e11	0.277393	0.138697	−	0.990335i	$-0.455709\pi$
$563$	6.61278e11	0.277393	0.138697	+	0.990335i	$0.455709\pi$
$564$	0	0
$565$	2.15694e12	0.890472
$566$	0	0
$567$	1.46601e12	0.595679
$568$	0	0
$569$	−1.47797e12	−0.591100	−0.295550	−	0.955327i	$-0.595503\pi$
$569$	−1.47797e12	−0.591100	−0.295550	+	0.955327i	$0.595503\pi$
$570$	0	0
$571$	2.27003e12	0.893652	0.446826	−	0.894621i	$-0.352554\pi$
$571$	2.27003e12	0.893652	0.446826	+	0.894621i	$0.352554\pi$
$572$	0	0
$573$	1.08049e11	0.0418723
$574$	0	0
$575$	2.63461e11	0.100510
$576$	0	0
$577$	−1.93038e12	−0.725021	−0.362511	−	0.931980i	$-0.618080\pi$
$577$	−1.93038e12	−0.725021	−0.362511	+	0.931980i	$0.618080\pi$
$578$	0	0
$579$	2.59946e12	0.961235
$580$	0	0
$581$	−5.94251e11	−0.216360
$582$	0	0
$583$	−3.23312e12	−1.15908
$584$	0	0
$585$	−2.44458e12	−0.862984
$586$	0	0
$587$	−2.62778e12	−0.913518	−0.456759	−	0.889590i	$-0.650990\pi$
$587$	−2.62778e12	−0.913518	−0.456759	+	0.889590i	$0.650990\pi$
$588$	0	0
$589$	−1.07049e12	−0.366492
$590$	0	0
$591$	−4.50248e12	−1.51813
$592$	0	0
$593$	5.85157e12	1.94324	0.971620	−	0.236548i	$-0.0760162\pi$
$593$	5.85157e12	1.94324	0.971620	+	0.236548i	$0.0760162\pi$
$594$	0	0
$595$	−3.88471e11	−0.127067
$596$	0	0
$597$	−5.81865e12	−1.87472
$598$	0	0
$599$	−4.56968e11	−0.145033	−0.0725163	−	0.997367i	$-0.523103\pi$
$599$	−4.56968e11	−0.145033	−0.0725163	+	0.997367i	$0.523103\pi$
$600$	0	0
$601$	−4.72333e12	−1.47677	−0.738385	−	0.674379i	$-0.764411\pi$
$601$	−4.72333e12	−1.47677	−0.738385	+	0.674379i	$0.764411\pi$
$602$	0	0
$603$	−3.57497e11	−0.110115
$604$	0	0
$605$	5.25894e11	0.159588
$606$	0	0
$607$	−1.89920e12	−0.567836	−0.283918	−	0.958849i	$-0.591634\pi$
$607$	−1.89920e12	−0.567836	−0.283918	+	0.958849i	$0.591634\pi$
$608$	0	0
$609$	−5.23188e10	−0.0154127
$610$	0	0
$611$	6.62553e12	1.92325
$612$	0	0
$613$	−5.38704e12	−1.54091	−0.770456	−	0.637493i	$-0.779971\pi$
$613$	−5.38704e12	−1.54091	−0.770456	+	0.637493i	$0.779971\pi$
$614$	0	0
$615$	−1.10015e12	−0.310109
$616$	0	0
$617$	−5.80011e11	−0.161121	−0.0805607	−	0.996750i	$-0.525671\pi$
$617$	−5.80011e11	−0.161121	−0.0805607	+	0.996750i	$0.525671\pi$
$618$	0	0
$619$	1.58371e12	0.433579	0.216790	−	0.976218i	$-0.430441\pi$
$619$	1.58371e12	0.433579	0.216790	+	0.976218i	$0.430441\pi$
$620$	0	0
$621$	−1.12043e12	−0.302325
$622$	0	0
$623$	1.19054e12	0.316626
$624$	0	0
$625$	−4.44378e12	−1.16491
$626$	0	0
$627$	−9.19662e12	−2.37643
$628$	0	0
$629$	−4.66549e11	−0.118842
$630$	0	0
$631$	−3.36740e12	−0.845595	−0.422797	−	0.906224i	$-0.638952\pi$
$631$	−3.36740e12	−0.845595	−0.422797	+	0.906224i	$0.638952\pi$
$632$	0	0
$633$	3.93218e12	0.973458
$634$	0	0
$635$	−7.13908e11	−0.174245
$636$	0	0
$637$	5.18752e12	1.24834
$638$	0	0
$639$	−1.95728e12	−0.464408
$640$	0	0
$641$	−7.80868e12	−1.82691	−0.913454	−	0.406943i	$-0.866595\pi$
$641$	−7.80868e12	−1.82691	−0.913454	+	0.406943i	$0.866595\pi$
$642$	0	0
$643$	2.11639e12	0.488255	0.244128	−	0.969743i	$-0.421498\pi$
$643$	2.11639e12	0.488255	0.244128	+	0.969743i	$0.421498\pi$
$644$	0	0
$645$	2.52272e12	0.573920
$646$	0	0
$647$	−3.18454e12	−0.714460	−0.357230	−	0.934016i	$-0.616279\pi$
$647$	−3.18454e12	−0.714460	−0.357230	+	0.934016i	$0.616279\pi$
$648$	0	0
$649$	4.99302e12	1.10475
$650$	0	0
$651$	5.35664e11	0.116890
$652$	0	0
$653$	1.82854e12	0.393545	0.196772	−	0.980449i	$-0.436954\pi$
$653$	1.82854e12	0.393545	0.196772	+	0.980449i	$0.436954\pi$
$654$	0	0
$655$	−7.23104e12	−1.53502
$656$	0	0
$657$	−1.87225e12	−0.392030
$658$	0	0
$659$	−7.92252e11	−0.163636	−0.0818181	−	0.996647i	$-0.526073\pi$
$659$	−7.92252e11	−0.163636	−0.0818181	+	0.996647i	$0.526073\pi$
$660$	0	0
$661$	−7.49133e11	−0.152634	−0.0763172	−	0.997084i	$-0.524316\pi$
$661$	−7.49133e11	−0.152634	−0.0763172	+	0.997084i	$0.524316\pi$
$662$	0	0
$663$	−2.37600e12	−0.477568
$664$	0	0
$665$	−4.81311e12	−0.954396
$666$	0	0
$667$	6.54319e10	0.0128004
$668$	0	0
$669$	9.94708e12	1.91990
$670$	0	0
$671$	9.71091e12	1.84931
$672$	0	0
$673$	7.01486e12	1.31811	0.659055	−	0.752095i	$-0.270957\pi$
$673$	7.01486e12	1.31811	0.659055	+	0.752095i	$0.270957\pi$
$674$	0	0
$675$	−7.03873e11	−0.130505
$676$	0	0
$677$	7.71992e12	1.41242	0.706210	−	0.708002i	$-0.250403\pi$
$677$	7.71992e12	1.41242	0.706210	+	0.708002i	$0.250403\pi$
$678$	0	0
$679$	−2.65069e12	−0.478570
$680$	0	0
$681$	1.08857e12	0.193952
$682$	0	0
$683$	4.25216e12	0.747680	0.373840	−	0.927493i	$-0.378041\pi$
$683$	4.25216e12	0.747680	0.373840	+	0.927493i	$0.378041\pi$
$684$	0	0
$685$	6.28614e12	1.09088
$686$	0	0
$687$	−1.24051e13	−2.12468
$688$	0	0
$689$	−1.03492e13	−1.74953
$690$	0	0
$691$	2.49111e12	0.415664	0.207832	−	0.978165i	$-0.433359\pi$
$691$	2.49111e12	0.415664	0.207832	+	0.978165i	$0.433359\pi$
$692$	0	0
$693$	1.50514e12	0.247901
$694$	0	0
$695$	7.09286e11	0.115316
$696$	0	0
$697$	−3.49732e11	−0.0561291
$698$	0	0
$699$	−1.39107e13	−2.20395
$700$	0	0
$701$	−3.08136e11	−0.0481961	−0.0240980	−	0.999710i	$-0.507671\pi$
$701$	−3.08136e11	−0.0481961	−0.0240980	+	0.999710i	$0.507671\pi$
$702$	0	0
$703$	−5.78049e12	−0.892619
$704$	0	0
$705$	−1.04651e13	−1.59548
$706$	0	0
$707$	−8.38630e11	−0.126236
$708$	0	0
$709$	−6.98388e12	−1.03798	−0.518990	−	0.854780i	$-0.673692\pi$
$709$	−6.98388e12	−1.03798	−0.518990	+	0.854780i	$0.673692\pi$
$710$	0	0
$711$	−2.60824e12	−0.382768
$712$	0	0
$713$	−6.69922e11	−0.0970781
$714$	0	0
$715$	1.32784e13	1.90007
$716$	0	0
$717$	−1.28038e13	−1.80927
$718$	0	0
$719$	1.30686e13	1.82368	0.911842	−	0.410540i	$-0.134660\pi$
$719$	1.30686e13	1.82368	0.911842	+	0.410540i	$0.134660\pi$
$720$	0	0
$721$	−1.65895e11	−0.0228626
$722$	0	0
$723$	−1.81724e12	−0.247337
$724$	0	0
$725$	4.11053e10	0.00552557
$726$	0	0
$727$	1.13650e12	0.150891	0.0754456	−	0.997150i	$-0.475962\pi$
$727$	1.13650e12	0.150891	0.0754456	+	0.997150i	$0.475962\pi$
$728$	0	0
$729$	1.19197e12	0.156312
$730$	0	0
$731$	8.01960e11	0.103878
$732$	0	0
$733$	5.78287e12	0.739904	0.369952	−	0.929051i	$-0.379374\pi$
$733$	5.78287e12	0.739904	0.369952	+	0.929051i	$0.379374\pi$
$734$	0	0
$735$	−8.19374e12	−1.03559
$736$	0	0
$737$	1.94185e12	0.242444
$738$	0	0
$739$	1.05562e13	1.30199	0.650996	−	0.759081i	$-0.274351\pi$
$739$	1.05562e13	1.30199	0.650996	+	0.759081i	$0.274351\pi$
$740$	0	0
$741$	−2.94384e13	−3.58701
$742$	0	0
$743$	−1.33049e13	−1.60163	−0.800815	−	0.598912i	$-0.795600\pi$
$743$	−1.33049e13	−1.60163	−0.800815	+	0.598912i	$0.795600\pi$
$744$	0	0
$745$	2.87300e12	0.341690
$746$	0	0
$747$	1.87768e12	0.220637
$748$	0	0
$749$	4.03729e12	0.468729
$750$	0	0
$751$	2.33606e12	0.267982	0.133991	−	0.990983i	$-0.457221\pi$
$751$	2.33606e12	0.267982	0.133991	+	0.990983i	$0.457221\pi$
$752$	0	0
$753$	1.63378e13	1.85190
$754$	0	0
$755$	9.29248e12	1.04081
$756$	0	0
$757$	−1.17644e13	−1.30208	−0.651039	−	0.759045i	$-0.725666\pi$
$757$	−1.17644e13	−1.30208	−0.651039	+	0.759045i	$0.725666\pi$
$758$	0	0
$759$	−5.75531e12	−0.629478
$760$	0	0
$761$	3.21795e12	0.347815	0.173907	−	0.984762i	$-0.444361\pi$
$761$	3.21795e12	0.347815	0.173907	+	0.984762i	$0.444361\pi$
$762$	0	0
$763$	2.13225e12	0.227760
$764$	0	0
$765$	1.22747e12	0.129579
$766$	0	0
$767$	1.59827e13	1.66752
$768$	0	0
$769$	−9.65995e12	−0.996108	−0.498054	−	0.867146i	$-0.665952\pi$
$769$	−9.65995e12	−0.996108	−0.498054	+	0.867146i	$0.665952\pi$
$770$	0	0
$771$	1.11313e13	1.13449
$772$	0	0
$773$	−3.02219e12	−0.304449	−0.152224	−	0.988346i	$-0.548644\pi$
$773$	−3.02219e12	−0.304449	−0.152224	+	0.988346i	$0.548644\pi$
$774$	0	0
$775$	−4.20855e11	−0.0419059
$776$	0	0
$777$	2.89250e12	0.284694
$778$	0	0
$779$	−4.33315e12	−0.421585
$780$	0	0
$781$	1.06315e13	1.02251
$782$	0	0
$783$	−1.74811e11	−0.0166204
$784$	0	0
$785$	9.02606e12	0.848369
$786$	0	0
$787$	1.47037e13	1.36628	0.683141	−	0.730286i	$-0.260613\pi$
$787$	1.47037e13	1.36628	0.683141	+	0.730286i	$0.260613\pi$
$788$	0	0
$789$	1.08427e13	0.996076
$790$	0	0
$791$	−4.25106e12	−0.386103
$792$	0	0
$793$	3.10846e13	2.79136
$794$	0	0
$795$	1.63467e13	1.45137
$796$	0	0
$797$	−1.44405e13	−1.26771	−0.633856	−	0.773451i	$-0.718529\pi$
$797$	−1.44405e13	−1.26771	−0.633856	+	0.773451i	$0.718529\pi$
$798$	0	0
$799$	−3.32680e12	−0.288779
$800$	0	0
$801$	−3.76179e12	−0.322885
$802$	0	0
$803$	1.01696e13	0.863148
$804$	0	0
$805$	−3.01208e12	−0.252805
$806$	0	0
$807$	−1.28323e13	−1.06506
$808$	0	0
$809$	1.10265e13	0.905041	0.452521	−	0.891754i	$-0.350525\pi$
$809$	1.10265e13	0.905041	0.452521	+	0.891754i	$0.350525\pi$
$810$	0	0
$811$	−1.97221e12	−0.160088	−0.0800440	−	0.996791i	$-0.525506\pi$
$811$	−1.97221e12	−0.160088	−0.0800440	+	0.996791i	$0.525506\pi$
$812$	0	0
$813$	1.63458e13	1.31220
$814$	0	0
$815$	−1.83320e13	−1.45546
$816$	0	0
$817$	9.93621e12	0.780228
$818$	0	0
$819$	4.81796e12	0.374184
$820$	0	0
$821$	−1.24725e13	−0.958093	−0.479047	−	0.877789i	$-0.659018\pi$
$821$	−1.24725e13	−0.958093	−0.479047	+	0.877789i	$0.659018\pi$
$822$	0	0
$823$	−5.63904e12	−0.428455	−0.214228	−	0.976784i	$-0.568723\pi$
$823$	−5.63904e12	−0.428455	−0.214228	+	0.976784i	$0.568723\pi$
$824$	0	0
$825$	−3.61557e12	−0.271728
$826$	0	0
$827$	6.43118e12	0.478097	0.239048	−	0.971008i	$-0.423165\pi$
$827$	6.43118e12	0.478097	0.239048	+	0.971008i	$0.423165\pi$
$828$	0	0
$829$	−4.51844e12	−0.332272	−0.166136	−	0.986103i	$-0.553129\pi$
$829$	−4.51844e12	−0.332272	−0.166136	+	0.986103i	$0.553129\pi$
$830$	0	0
$831$	−2.40345e12	−0.174836
$832$	0	0
$833$	−2.60475e12	−0.187440
$834$	0	0
$835$	−3.25279e12	−0.231562
$836$	0	0
$837$	1.78979e12	0.126049
$838$	0	0
$839$	−1.73764e13	−1.21068	−0.605342	−	0.795965i	$-0.706964\pi$
$839$	−1.73764e13	−1.21068	−0.605342	+	0.795965i	$0.706964\pi$
$840$	0	0
$841$	−1.44969e13	−0.999296
$842$	0	0
$843$	1.72156e13	1.17408
$844$	0	0
$845$	2.62134e13	1.76876
$846$	0	0
$847$	−1.03647e12	−0.0691962
$848$	0	0
$849$	−1.33180e13	−0.879739
$850$	0	0
$851$	−3.61747e12	−0.236441
$852$	0	0
$853$	−3.23489e12	−0.209213	−0.104607	−	0.994514i	$-0.533358\pi$
$853$	−3.23489e12	−0.209213	−0.104607	+	0.994514i	$0.533358\pi$
$854$	0	0
$855$	1.52082e13	0.973262
$856$	0	0
$857$	6.56773e12	0.415912	0.207956	−	0.978138i	$-0.433319\pi$
$857$	6.56773e12	0.415912	0.207956	+	0.978138i	$0.433319\pi$
$858$	0	0
$859$	1.06597e13	0.668001	0.334000	−	0.942573i	$-0.391601\pi$
$859$	1.06597e13	0.668001	0.334000	+	0.942573i	$0.391601\pi$
$860$	0	0
$861$	2.16826e12	0.134461
$862$	0	0
$863$	2.96333e12	0.181857	0.0909287	−	0.995857i	$-0.471016\pi$
$863$	2.96333e12	0.181857	0.0909287	+	0.995857i	$0.471016\pi$
$864$	0	0
$865$	6.72130e12	0.408207
$866$	0	0
$867$	1.19303e12	0.0717078
$868$	0	0
$869$	1.41674e13	0.842755
$870$	0	0
$871$	6.21585e12	0.365948
$872$	0	0
$873$	8.37551e12	0.488030
$874$	0	0
$875$	7.19209e12	0.414781
$876$	0	0
$877$	−1.22652e13	−0.700128	−0.350064	−	0.936726i	$-0.613840\pi$
$877$	−1.22652e13	−0.700128	−0.350064	+	0.936726i	$0.613840\pi$
$878$	0	0
$879$	−1.51554e13	−0.856284
$880$	0	0
$881$	2.36185e13	1.32087	0.660435	−	0.750883i	$-0.270372\pi$
$881$	2.36185e13	1.32087	0.660435	+	0.750883i	$0.270372\pi$
$882$	0	0
$883$	−2.46792e13	−1.36618	−0.683089	−	0.730335i	$-0.739364\pi$
$883$	−2.46792e13	−1.36618	−0.683089	+	0.730335i	$0.739364\pi$
$884$	0	0
$885$	−2.52448e13	−1.38333
$886$	0	0
$887$	8.49974e12	0.461051	0.230526	−	0.973066i	$-0.425955\pi$
$887$	8.49974e12	0.461051	0.230526	+	0.973066i	$0.425955\pi$
$888$	0	0
$889$	1.40702e12	0.0755516
$890$	0	0
$891$	2.51611e13	1.33746
$892$	0	0
$893$	−4.12186e13	−2.16901
$894$	0	0
$895$	−3.26892e12	−0.170295
$896$	0	0
$897$	−1.84228e13	−0.950142
$898$	0	0
$899$	−1.04522e11	−0.00533689
$900$	0	0
$901$	5.19653e12	0.262695
$902$	0	0
$903$	−4.97197e12	−0.248848
$904$	0	0
$905$	2.77281e13	1.37405
$906$	0	0
$907$	1.51717e13	0.744390	0.372195	−	0.928155i	$-0.378605\pi$
$907$	1.51717e13	0.744390	0.372195	+	0.928155i	$0.378605\pi$
$908$	0	0
$909$	2.64985e12	0.128731
$910$	0	0
$911$	2.65044e12	0.127493	0.0637464	−	0.997966i	$-0.479695\pi$
$911$	2.65044e12	0.127493	0.0637464	+	0.997966i	$0.479695\pi$
$912$	0	0
$913$	−1.01991e13	−0.485786
$914$	0	0
$915$	−4.90985e13	−2.31565
$916$	0	0
$917$	1.42515e13	0.665576
$918$	0	0
$919$	1.10783e13	0.512336	0.256168	−	0.966632i	$-0.417540\pi$
$919$	1.10783e13	0.512336	0.256168	+	0.966632i	$0.417540\pi$
$920$	0	0
$921$	2.90622e13	1.33094
$922$	0	0
$923$	3.40315e13	1.54338
$924$	0	0
$925$	−2.27255e12	−0.102065
$926$	0	0
$927$	5.24186e11	0.0233145
$928$	0	0
$929$	−1.10570e13	−0.487044	−0.243522	−	0.969895i	$-0.578303\pi$
$929$	−1.10570e13	−0.487044	−0.243522	+	0.969895i	$0.578303\pi$
$930$	0	0
$931$	−3.22725e13	−1.40786
$932$	0	0
$933$	−2.02942e13	−0.876808
$934$	0	0
$935$	−6.66733e12	−0.285299
$936$	0	0
$937$	−3.12152e13	−1.32293	−0.661466	−	0.749975i	$-0.730065\pi$
$937$	−3.12152e13	−1.32293	−0.661466	+	0.749975i	$0.730065\pi$
$938$	0	0
$939$	4.31160e13	1.80985
$940$	0	0
$941$	3.48495e13	1.44891	0.724457	−	0.689320i	$-0.242090\pi$
$941$	3.48495e13	1.44891	0.724457	+	0.689320i	$0.242090\pi$
$942$	0	0
$943$	−2.71171e12	−0.111671
$944$	0	0
$945$	8.04720e12	0.328248
$946$	0	0
$947$	−2.89060e13	−1.16792	−0.583960	−	0.811783i	$-0.698497\pi$
$947$	−2.89060e13	−1.16792	−0.583960	+	0.811783i	$0.698497\pi$
$948$	0	0
$949$	3.25530e13	1.30285
$950$	0	0
$951$	1.29036e13	0.511564
$952$	0	0
$953$	1.17377e13	0.460963	0.230482	−	0.973077i	$-0.425970\pi$
$953$	1.17377e13	0.460963	0.230482	+	0.973077i	$0.425970\pi$
$954$	0	0
$955$	9.70540e11	0.0377571
$956$	0	0
$957$	−8.97948e11	−0.0346057
$958$	0	0
$959$	−1.23892e13	−0.472998
$960$	0	0
$961$	−2.53695e13	−0.959525
$962$	0	0
$963$	−1.27568e13	−0.477994
$964$	0	0
$965$	2.33493e13	0.866765
$966$	0	0
$967$	−4.64159e13	−1.70705	−0.853527	−	0.521048i	$-0.825541\pi$
$967$	−4.64159e13	−1.70705	−0.853527	+	0.521048i	$0.825541\pi$
$968$	0	0
$969$	1.47815e13	0.538596
$970$	0	0
$971$	3.56315e12	0.128632	0.0643158	−	0.997930i	$-0.479514\pi$
$971$	3.56315e12	0.128632	0.0643158	+	0.997930i	$0.479514\pi$
$972$	0	0
$973$	−1.39791e12	−0.0500003
$974$	0	0
$975$	−1.15735e13	−0.410150
$976$	0	0
$977$	−1.39593e13	−0.490160	−0.245080	−	0.969503i	$-0.578814\pi$
$977$	−1.39593e13	−0.490160	−0.245080	+	0.969503i	$0.578814\pi$
$978$	0	0
$979$	2.04332e13	0.710909
$980$	0	0
$981$	−6.73736e12	−0.232263
$982$	0	0
$983$	6.28271e12	0.214613	0.107306	−	0.994226i	$-0.465777\pi$
$983$	6.28271e12	0.214613	0.107306	+	0.994226i	$0.465777\pi$
$984$	0	0
$985$	−4.04429e13	−1.36893
$986$	0	0
$987$	2.06254e13	0.691791
$988$	0	0
$989$	6.21815e12	0.206670
$990$	0	0
$991$	4.49640e13	1.48093	0.740463	−	0.672097i	$-0.234606\pi$
$991$	4.49640e13	1.48093	0.740463	+	0.672097i	$0.234606\pi$
$992$	0	0
$993$	4.09619e13	1.33693
$994$	0	0
$995$	−5.22653e13	−1.69048
$996$	0	0
$997$	−1.21770e13	−0.390311	−0.195156	−	0.980772i	$-0.562521\pi$
$997$	−1.21770e13	−0.390311	−0.195156	+	0.980772i	$0.562521\pi$
$998$	0	0
$999$	9.66460e12	0.307001

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	272.10.a.g.1.6		7
4.3	odd	2		17.10.a.b.1.7	✓	7
12.11	even	2		153.10.a.f.1.1		7
68.67	odd	2		289.10.a.b.1.7		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
17.10.a.b.1.7	✓	7	4.3	odd	2
153.10.a.f.1.1		7	12.11	even	2
272.10.a.g.1.6		7	1.1	even	1	trivial
289.10.a.b.1.7		7	68.67	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:	\( N \)	\(=\)	\( 272 = 2^{4} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	272.a (trivial)

\(f(q)\)	\(=\)	\(q+171.025 q^{3} +1536.21 q^{5} -3027.69 q^{7} +9566.70 q^{9} +O(q^{10})\)	\(q+171.025 q^{3} +1536.21 q^{5} -3027.69 q^{7} +9566.70 q^{9} -51964.2 q^{11} -166337. q^{13} +262732. q^{15} +83521.0 q^{17} +1.03482e6 q^{19} -517811. q^{21} +647595. q^{23} +406829. q^{25} -1.73014e6 q^{27} +101038. q^{29} -1.03448e6 q^{31} -8.88720e6 q^{33} -4.65118e6 q^{35} -5.58601e6 q^{37} -2.84479e7 q^{39} -4.18736e6 q^{41} +9.60190e6 q^{43} +1.46965e7 q^{45} -3.98318e7 q^{47} -3.11867e7 q^{49} +1.42842e7 q^{51} +6.22183e7 q^{53} -7.98282e7 q^{55} +1.76980e8 q^{57} -9.60858e7 q^{59} -1.86877e8 q^{61} -2.89650e7 q^{63} -2.55530e8 q^{65} -3.73689e7 q^{67} +1.10755e8 q^{69} -2.04593e8 q^{71} -1.95705e8 q^{73} +6.95782e7 q^{75} +1.57331e8 q^{77} -2.72638e8 q^{79} -4.84200e8 q^{81} +1.96272e8 q^{83} +1.28306e8 q^{85} +1.72801e7 q^{87} -3.93217e8 q^{89} +5.03618e8 q^{91} -1.76922e8 q^{93} +1.58970e9 q^{95} +8.75485e8 q^{97} -4.97126e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 88 q^{3} + 1362 q^{5} - 9388 q^{7} + 81419 q^{9}+O(q^{10}) \) 7 * q - 88 * q^3 + 1362 * q^5 - 9388 * q^7 + 81419 * q^9	\( 7 q - 88 q^{3} + 1362 q^{5} - 9388 q^{7} + 81419 q^{9} - 135536 q^{11} + 166122 q^{13} - 159048 q^{15} + 584647 q^{17} - 777172 q^{19} - 3412104 q^{21} - 1357764 q^{23} + 1065785 q^{25} + 4519064 q^{27} + 967002 q^{29} - 3546740 q^{31} + 11928016 q^{33} + 530736 q^{35} + 18296498 q^{37} - 86306872 q^{39} + 10285686 q^{41} - 21913204 q^{43} + 108916410 q^{45} - 56639800 q^{47} + 27010351 q^{49} - 7349848 q^{51} + 121813562 q^{53} - 40793128 q^{55} + 153612960 q^{57} - 29222388 q^{59} - 49915846 q^{61} + 2185356 q^{63} - 122633668 q^{65} - 301863420 q^{67} + 379683432 q^{69} - 652473940 q^{71} + 306656342 q^{73} - 919071912 q^{75} - 102442536 q^{77} - 959147884 q^{79} - 374486977 q^{81} + 1512945268 q^{83} + 113755602 q^{85} + 1612550856 q^{87} - 1971327114 q^{89} + 1061062864 q^{91} - 798598936 q^{93} + 3249631512 q^{95} + 2006526254 q^{97} + 2579159272 q^{99}+O(q^{100}) \) 7 * q - 88 * q^3 + 1362 * q^5 - 9388 * q^7 + 81419 * q^9 - 135536 * q^11 + 166122 * q^13 - 159048 * q^15 + 584647 * q^17 - 777172 * q^19 - 3412104 * q^21 - 1357764 * q^23 + 1065785 * q^25 + 4519064 * q^27 + 967002 * q^29 - 3546740 * q^31 + 11928016 * q^33 + 530736 * q^35 + 18296498 * q^37 - 86306872 * q^39 + 10285686 * q^41 - 21913204 * q^43 + 108916410 * q^45 - 56639800 * q^47 + 27010351 * q^49 - 7349848 * q^51 + 121813562 * q^53 - 40793128 * q^55 + 153612960 * q^57 - 29222388 * q^59 - 49915846 * q^61 + 2185356 * q^63 - 122633668 * q^65 - 301863420 * q^67 + 379683432 * q^69 - 652473940 * q^71 + 306656342 * q^73 - 919071912 * q^75 - 102442536 * q^77 - 959147884 * q^79 - 374486977 * q^81 + 1512945268 * q^83 + 113755602 * q^85 + 1612550856 * q^87 - 1971327114 * q^89 + 1061062864 * q^91 - 798598936 * q^93 + 3249631512 * q^95 + 2006526254 * q^97 + 2579159272 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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