LMFDB - Embedded newform 272.10.a.g.1.7

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			1.7
Root			$-4.12962$ 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)

$f(q)$	$=$	$q+254.074 q^{3} +151.544 q^{5} -9407.97 q^{7} +44870.8 q^{9} +O(q^{10})$	$q+254.074 q^{3} +151.544 q^{5} -9407.97 q^{7} +44870.8 q^{9} +56967.2 q^{11} -60874.5 q^{13} +38503.4 q^{15} +83521.0 q^{17} -1.00994e6 q^{19} -2.39032e6 q^{21} -1.35979e6 q^{23} -1.93016e6 q^{25} +6.39958e6 q^{27} +3.12503e6 q^{29} -2.97426e6 q^{31} +1.44739e7 q^{33} -1.42572e6 q^{35} +681625. q^{37} -1.54666e7 q^{39} -4.09135e6 q^{41} -1.00085e7 q^{43} +6.79989e6 q^{45} -2.54570e7 q^{47} +4.81562e7 q^{49} +2.12206e7 q^{51} -3.14563e7 q^{53} +8.63302e6 q^{55} -2.56601e8 q^{57} +9.03573e7 q^{59} +9.87113e7 q^{61} -4.22143e8 q^{63} -9.22514e6 q^{65} -1.32700e8 q^{67} -3.45488e8 q^{69} -4.18554e8 q^{71} +4.80282e7 q^{73} -4.90404e8 q^{75} -5.35945e8 q^{77} -3.49030e7 q^{79} +7.42778e8 q^{81} +2.05650e8 q^{83} +1.26571e7 q^{85} +7.93991e8 q^{87} -2.03866e8 q^{89} +5.72705e8 q^{91} -7.55684e8 q^{93} -1.53050e8 q^{95} +1.24300e9 q^{97} +2.55616e9 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	254.074	1.81099	0.905493	−	0.424360i	$-0.139501\pi$
$3$	254.074	1.81099	0.905493	+	0.424360i	$0.139501\pi$
$4$	0	0
$5$	151.544	0.108436	0.0542179	−	0.998529i	$-0.482733\pi$
$5$	151.544	0.108436	0.0542179	+	0.998529i	$0.482733\pi$
$6$	0	0
$7$	−9407.97	−1.48100	−0.740499	−	0.672057i	$-0.765411\pi$
$7$	−9407.97	−1.48100	−0.740499	+	0.672057i	$0.765411\pi$
$8$	0	0
$9$	44870.8	2.27967
$10$	0	0
$11$	56967.2	1.17316	0.586581	−	0.809891i	$-0.300474\pi$
$11$	56967.2	1.17316	0.586581	+	0.809891i	$0.300474\pi$
$12$	0	0
$13$	−60874.5	−0.591140	−0.295570	−	0.955321i	$-0.595510\pi$
$13$	−60874.5	−0.591140	−0.295570	+	0.955321i	$0.595510\pi$
$14$	0	0
$15$	38503.4	0.196376
$16$	0	0
$17$	83521.0	0.242536
$17$	83521.0	0.242536
$18$	0	0
$19$	−1.00994e6	−1.77789	−0.888947	−	0.458010i	$-0.848562\pi$
$19$	−1.00994e6	−1.77789	−0.888947	+	0.458010i	$0.848562\pi$
$20$	0	0
$21$	−2.39032e6	−2.68207
$22$	0	0
$23$	−1.35979e6	−1.01320	−0.506602	−	0.862180i	$-0.669099\pi$
$23$	−1.35979e6	−1.01320	−0.506602	+	0.862180i	$0.669099\pi$
$24$	0	0
$25$	−1.93016e6	−0.988242
$26$	0	0
$27$	6.39958e6	2.31747
$28$	0	0
$29$	3.12503e6	0.820472	0.410236	−	0.911979i	$-0.365446\pi$
$29$	3.12503e6	0.820472	0.410236	+	0.911979i	$0.365446\pi$
$30$	0	0
$31$	−2.97426e6	−0.578431	−0.289216	−	0.957264i	$-0.593394\pi$
$31$	−2.97426e6	−0.578431	−0.289216	+	0.957264i	$0.593394\pi$
$32$	0	0
$33$	1.44739e7	2.12458
$34$	0	0
$35$	−1.42572e6	−0.160593
$36$	0	0
$37$	681625.	0.0597913	0.0298956	−	0.999553i	$-0.490483\pi$
$37$	681625.	0.0597913	0.0298956	+	0.999553i	$0.490483\pi$
$38$	0	0
$39$	−1.54666e7	−1.07055
$40$	0	0
$41$	−4.09135e6	−0.226120	−0.113060	−	0.993588i	$-0.536065\pi$
$41$	−4.09135e6	−0.226120	−0.113060	+	0.993588i	$0.536065\pi$
$42$	0	0
$43$	−1.00085e7	−0.446436	−0.223218	−	0.974769i	$-0.571656\pi$
$43$	−1.00085e7	−0.446436	−0.223218	+	0.974769i	$0.571656\pi$
$44$	0	0
$45$	6.79989e6	0.247198
$46$	0	0
$47$	−2.54570e7	−0.760970	−0.380485	−	0.924787i	$-0.624243\pi$
$47$	−2.54570e7	−0.760970	−0.380485	+	0.924787i	$0.624243\pi$
$48$	0	0
$49$	4.81562e7	1.19336
$50$	0	0
$51$	2.12206e7	0.439229
$52$	0	0
$53$	−3.14563e7	−0.547603	−0.273801	−	0.961786i	$-0.588281\pi$
$53$	−3.14563e7	−0.547603	−0.273801	+	0.961786i	$0.588281\pi$
$54$	0	0
$55$	8.63302e6	0.127213
$56$	0	0
$57$	−2.56601e8	−3.21974
$58$	0	0
$59$	9.03573e7	0.970798	0.485399	−	0.874293i	$-0.338674\pi$
$59$	9.03573e7	0.970798	0.485399	+	0.874293i	$0.338674\pi$
$60$	0	0
$61$	9.87113e7	0.912815	0.456408	−	0.889771i	$-0.349136\pi$
$61$	9.87113e7	0.912815	0.456408	+	0.889771i	$0.349136\pi$
$62$	0	0
$63$	−4.22143e8	−3.37619
$64$	0	0
$65$	−9.22514e6	−0.0641007
$66$	0	0
$67$	−1.32700e8	−0.804516	−0.402258	−	0.915526i	$-0.631775\pi$
$67$	−1.32700e8	−0.804516	−0.402258	+	0.915526i	$0.631775\pi$
$68$	0	0
$69$	−3.45488e8	−1.83490
$70$	0	0
$71$	−4.18554e8	−1.95474	−0.977369	−	0.211541i	$-0.932152\pi$
$71$	−4.18554e8	−1.95474	−0.977369	+	0.211541i	$0.932152\pi$
$72$	0	0
$73$	4.80282e7	0.197944	0.0989721	−	0.995090i	$-0.468445\pi$
$73$	4.80282e7	0.197944	0.0989721	+	0.995090i	$0.468445\pi$
$74$	0	0
$75$	−4.90404e8	−1.78969
$76$	0	0
$77$	−5.35945e8	−1.73745
$78$	0	0
$79$	−3.49030e7	−0.100819	−0.0504093	−	0.998729i	$-0.516053\pi$
$79$	−3.49030e7	−0.100819	−0.0504093	+	0.998729i	$0.516053\pi$
$80$	0	0
$81$	7.42778e8	1.91724
$82$	0	0
$83$	2.05650e8	0.475638	0.237819	−	0.971309i	$-0.423567\pi$
$83$	2.05650e8	0.475638	0.237819	+	0.971309i	$0.423567\pi$
$84$	0	0
$85$	1.26571e7	0.0262995
$86$	0	0
$87$	7.93991e8	1.48586
$88$	0	0
$89$	−2.03866e8	−0.344422	−0.172211	−	0.985060i	$-0.555091\pi$
$89$	−2.03866e8	−0.344422	−0.172211	+	0.985060i	$0.555091\pi$
$90$	0	0
$91$	5.72705e8	0.875477
$92$	0	0
$93$	−7.55684e8	−1.04753
$94$	0	0
$95$	−1.53050e8	−0.192787
$96$	0	0
$97$	1.24300e9	1.42560	0.712798	−	0.701369i	$-0.247427\pi$
$97$	1.24300e9	1.42560	0.712798	+	0.701369i	$0.247427\pi$
$98$	0	0
$99$	2.55616e9	2.67443
$100$	0	0
$101$	−2.02496e9	−1.93629	−0.968144	−	0.250395i	$-0.919440\pi$
$101$	−2.02496e9	−1.93629	−0.968144	+	0.250395i	$0.919440\pi$
$102$	0	0
$103$	−1.16276e9	−1.01794	−0.508970	−	0.860785i	$-0.669973\pi$
$103$	−1.16276e9	−1.01794	−0.508970	+	0.860785i	$0.669973\pi$
$104$	0	0
$105$	−3.62238e8	−0.290832
$106$	0	0
$107$	−1.31593e9	−0.970520	−0.485260	−	0.874370i	$-0.661275\pi$
$107$	−1.31593e9	−0.970520	−0.485260	+	0.874370i	$0.661275\pi$
$108$	0	0
$109$	−1.79000e8	−0.121460	−0.0607302	−	0.998154i	$-0.519343\pi$
$109$	−1.79000e8	−0.121460	−0.0607302	+	0.998154i	$0.519343\pi$
$110$	0	0
$111$	1.73184e8	0.108281
$112$	0	0
$113$	2.53572e9	1.46301	0.731506	−	0.681835i	$-0.238818\pi$
$113$	2.53572e9	1.46301	0.731506	+	0.681835i	$0.238818\pi$
$114$	0	0
$115$	−2.06068e8	−0.109868
$116$	0	0
$117$	−2.73149e9	−1.34761
$118$	0	0
$119$	−7.85763e8	−0.359195
$120$	0	0
$121$	8.87314e8	0.376308
$122$	0	0
$123$	−1.03951e9	−0.409500
$124$	0	0
$125$	−5.88487e8	−0.215597
$126$	0	0
$127$	−4.05326e9	−1.38257	−0.691287	−	0.722581i	$-0.742956\pi$
$127$	−4.05326e9	−1.38257	−0.691287	+	0.722581i	$0.742956\pi$
$128$	0	0
$129$	−2.54289e9	−0.808490
$130$	0	0
$131$	2.93543e9	0.870865	0.435432	−	0.900221i	$-0.356595\pi$
$131$	2.93543e9	0.870865	0.435432	+	0.900221i	$0.356595\pi$
$132$	0	0
$133$	9.50151e9	2.63306
$134$	0	0
$135$	9.69816e8	0.251297
$136$	0	0
$137$	−4.82610e9	−1.17045	−0.585226	−	0.810870i	$-0.698994\pi$
$137$	−4.82610e9	−1.17045	−0.585226	+	0.810870i	$0.698994\pi$
$138$	0	0
$139$	−2.65630e9	−0.603546	−0.301773	−	0.953380i	$-0.597579\pi$
$139$	−2.65630e9	−0.603546	−0.301773	+	0.953380i	$0.597579\pi$
$140$	0	0
$141$	−6.46798e9	−1.37811
$142$	0	0
$143$	−3.46785e9	−0.693502
$144$	0	0
$145$	4.73579e8	0.0889685
$146$	0	0
$147$	1.22353e10	2.16115
$148$	0	0
$149$	−2.85114e9	−0.473894	−0.236947	−	0.971523i	$-0.576147\pi$
$149$	−2.85114e9	−0.473894	−0.236947	+	0.971523i	$0.576147\pi$
$150$	0	0
$151$	8.37259e9	1.31058	0.655290	−	0.755378i	$-0.272546\pi$
$151$	8.37259e9	1.31058	0.655290	+	0.755378i	$0.272546\pi$
$152$	0	0
$153$	3.74766e9	0.552902
$154$	0	0
$155$	−4.50731e8	−0.0627227
$156$	0	0
$157$	8.37311e9	1.09986	0.549931	−	0.835210i	$-0.314654\pi$
$157$	8.37311e9	1.09986	0.549931	+	0.835210i	$0.314654\pi$
$158$	0	0
$159$	−7.99223e9	−0.991702
$160$	0	0
$161$	1.27929e10	1.50055
$162$	0	0
$163$	2.12079e8	0.0235317	0.0117659	−	0.999931i	$-0.496255\pi$
$163$	2.12079e8	0.0235317	0.0117659	+	0.999931i	$0.496255\pi$
$164$	0	0
$165$	2.19343e9	0.230381
$166$	0	0
$167$	9.27756e9	0.923017	0.461509	−	0.887136i	$-0.347308\pi$
$167$	9.27756e9	0.923017	0.461509	+	0.887136i	$0.347308\pi$
$168$	0	0
$169$	−6.89880e9	−0.650554
$170$	0	0
$171$	−4.53170e10	−4.05302
$172$	0	0
$173$	−1.29744e10	−1.10123	−0.550617	−	0.834758i	$-0.685608\pi$
$173$	−1.29744e10	−1.10123	−0.550617	+	0.834758i	$0.685608\pi$
$174$	0	0
$175$	1.81589e10	1.46358
$176$	0	0
$177$	2.29575e10	1.75810
$178$	0	0
$179$	−2.58172e10	−1.87962	−0.939812	−	0.341692i	$-0.889000\pi$
$179$	−2.58172e10	−1.87962	−0.939812	+	0.341692i	$0.889000\pi$
$180$	0	0
$181$	2.61713e9	0.181248	0.0906238	−	0.995885i	$-0.471114\pi$
$181$	2.61713e9	0.181248	0.0906238	+	0.995885i	$0.471114\pi$
$182$	0	0
$183$	2.50800e10	1.65310
$184$	0	0
$185$	1.03296e8	0.00648352
$186$	0	0
$187$	4.75796e9	0.284533
$188$	0	0
$189$	−6.02070e10	−3.43217
$190$	0	0
$191$	−8.25575e8	−0.0448855	−0.0224428	−	0.999748i	$-0.507144\pi$
$191$	−8.25575e8	−0.0448855	−0.0224428	+	0.999748i	$0.507144\pi$
$192$	0	0
$193$	1.88810e10	0.979530	0.489765	−	0.871855i	$-0.337083\pi$
$193$	1.88810e10	0.979530	0.489765	+	0.871855i	$0.337083\pi$
$194$	0	0
$195$	−2.34387e9	−0.116086
$196$	0	0
$197$	1.73273e10	0.819659	0.409829	−	0.912162i	$-0.365588\pi$
$197$	1.73273e10	0.819659	0.409829	+	0.912162i	$0.365588\pi$
$198$	0	0
$199$	−2.14638e10	−0.970216	−0.485108	−	0.874454i	$-0.661220\pi$
$199$	−2.14638e10	−0.970216	−0.485108	+	0.874454i	$0.661220\pi$
$200$	0	0
$201$	−3.37157e10	−1.45697
$202$	0	0
$203$	−2.94002e10	−1.21512
$204$	0	0
$205$	−6.20018e8	−0.0245195
$206$	0	0
$207$	−6.10149e10	−2.30977
$208$	0	0
$209$	−5.75336e10	−2.08576
$210$	0	0
$211$	−7.73227e9	−0.268557	−0.134278	−	0.990944i	$-0.542872\pi$
$211$	−7.73227e9	−0.268557	−0.134278	+	0.990944i	$0.542872\pi$
$212$	0	0
$213$	−1.06344e11	−3.54001
$214$	0	0
$215$	−1.51672e9	−0.0484096
$216$	0	0
$217$	2.79818e10	0.856656
$218$	0	0
$219$	1.22027e10	0.358475
$220$	0	0
$221$	−5.08430e9	−0.143372
$222$	0	0
$223$	−9.27651e9	−0.251196	−0.125598	−	0.992081i	$-0.540085\pi$
$223$	−9.27651e9	−0.251196	−0.125598	+	0.992081i	$0.540085\pi$
$224$	0	0
$225$	−8.66078e10	−2.25287
$226$	0	0
$227$	1.00921e10	0.252270	0.126135	−	0.992013i	$-0.459743\pi$
$227$	1.00921e10	0.252270	0.126135	+	0.992013i	$0.459743\pi$
$228$	0	0
$229$	−2.70436e10	−0.649837	−0.324918	−	0.945742i	$-0.605337\pi$
$229$	−2.70436e10	−0.649837	−0.324918	+	0.945742i	$0.605337\pi$
$230$	0	0
$231$	−1.36170e11	−3.14650
$232$	0	0
$233$	6.83620e10	1.51954	0.759771	−	0.650191i	$-0.225311\pi$
$233$	6.83620e10	1.51954	0.759771	+	0.650191i	$0.225311\pi$
$234$	0	0
$235$	−3.85785e9	−0.0825164
$236$	0	0
$237$	−8.86795e9	−0.182581
$238$	0	0
$239$	2.57344e10	0.510179	0.255090	−	0.966917i	$-0.417895\pi$
$239$	2.57344e10	0.510179	0.255090	+	0.966917i	$0.417895\pi$
$240$	0	0
$241$	−1.97738e10	−0.377584	−0.188792	−	0.982017i	$-0.560457\pi$
$241$	−1.97738e10	−0.377584	−0.188792	+	0.982017i	$0.560457\pi$
$242$	0	0
$243$	6.27578e10	1.15462
$244$	0	0
$245$	7.29777e9	0.129402
$246$	0	0
$247$	6.14798e10	1.05098
$248$	0	0
$249$	5.22504e10	0.861375
$250$	0	0
$251$	−3.35205e10	−0.533063	−0.266532	−	0.963826i	$-0.585878\pi$
$251$	−3.35205e10	−0.533063	−0.266532	+	0.963826i	$0.585878\pi$
$252$	0	0
$253$	−7.74635e10	−1.18865
$254$	0	0
$255$	3.21584e9	0.0476281
$256$	0	0
$257$	−9.22091e10	−1.31848	−0.659242	−	0.751931i	$-0.729123\pi$
$257$	−9.22091e10	−1.31848	−0.659242	+	0.751931i	$0.729123\pi$
$258$	0	0
$259$	−6.41271e9	−0.0885508
$260$	0	0
$261$	1.40223e11	1.87041
$262$	0	0
$263$	−1.83512e10	−0.236518	−0.118259	−	0.992983i	$-0.537731\pi$
$263$	−1.83512e10	−0.236518	−0.118259	+	0.992983i	$0.537731\pi$
$264$	0	0
$265$	−4.76700e9	−0.0593798
$266$	0	0
$267$	−5.17972e10	−0.623743
$268$	0	0
$269$	−9.60172e10	−1.11806	−0.559029	−	0.829148i	$-0.688826\pi$
$269$	−9.60172e10	−1.11806	−0.559029	+	0.829148i	$0.688826\pi$
$270$	0	0
$271$	1.61081e11	1.81419	0.907097	−	0.420922i	$-0.138294\pi$
$271$	1.61081e11	1.81419	0.907097	+	0.420922i	$0.138294\pi$
$272$	0	0
$273$	1.45510e11	1.58548
$274$	0	0
$275$	−1.09956e11	−1.15937
$276$	0	0
$277$	5.34317e10	0.545305	0.272653	−	0.962113i	$-0.412099\pi$
$277$	5.34317e10	0.545305	0.272653	+	0.962113i	$0.412099\pi$
$278$	0	0
$279$	−1.33458e11	−1.31863
$280$	0	0
$281$	1.48954e11	1.42520	0.712599	−	0.701572i	$-0.247518\pi$
$281$	1.48954e11	1.42520	0.712599	+	0.701572i	$0.247518\pi$
$282$	0	0
$283$	5.51543e10	0.511140	0.255570	−	0.966790i	$-0.417737\pi$
$283$	5.51543e10	0.511140	0.255570	+	0.966790i	$0.417737\pi$
$284$	0	0
$285$	−3.88862e10	−0.349135
$286$	0	0
$287$	3.84913e10	0.334883
$288$	0	0
$289$	6.97576e9	0.0588235
$290$	0	0
$291$	3.15813e11	2.58174
$292$	0	0
$293$	−1.58399e11	−1.25559	−0.627794	−	0.778379i	$-0.716042\pi$
$293$	−1.58399e11	−1.25559	−0.627794	+	0.778379i	$0.716042\pi$
$294$	0	0
$295$	1.36931e10	0.105269
$296$	0	0
$297$	3.64566e11	2.71877
$298$	0	0
$299$	8.27766e10	0.598945
$300$	0	0
$301$	9.41592e10	0.661171
$302$	0	0
$303$	−5.14490e11	−3.50659
$304$	0	0
$305$	1.49591e10	0.0989818
$306$	0	0
$307$	−2.33754e11	−1.50188	−0.750942	−	0.660368i	$-0.770400\pi$
$307$	−2.33754e11	−1.50188	−0.750942	+	0.660368i	$0.770400\pi$
$308$	0	0
$309$	−2.95427e11	−1.84347
$310$	0	0
$311$	2.24488e11	1.36073	0.680363	−	0.732875i	$-0.261822\pi$
$311$	2.24488e11	1.36073	0.680363	+	0.732875i	$0.261822\pi$
$312$	0	0
$313$	−2.83464e10	−0.166935	−0.0834676	−	0.996510i	$-0.526599\pi$
$313$	−2.83464e10	−0.166935	−0.0834676	+	0.996510i	$0.526599\pi$
$314$	0	0
$315$	−6.39731e10	−0.366100
$316$	0	0
$317$	−1.13413e11	−0.630807	−0.315404	−	0.948958i	$-0.602140\pi$
$317$	−1.13413e11	−0.630807	−0.315404	+	0.948958i	$0.602140\pi$
$318$	0	0
$319$	1.78024e11	0.962546
$320$	0	0
$321$	−3.34343e11	−1.75760
$322$	0	0
$323$	−8.43515e10	−0.431203
$324$	0	0
$325$	1.17497e11	0.584189
$326$	0	0
$327$	−4.54794e10	−0.219963
$328$	0	0
$329$	2.39499e11	1.12699
$330$	0	0
$331$	4.08556e10	0.187079	0.0935397	−	0.995616i	$-0.470182\pi$
$331$	4.08556e10	0.187079	0.0935397	+	0.995616i	$0.470182\pi$
$332$	0	0
$333$	3.05851e10	0.136305
$334$	0	0
$335$	−2.01099e10	−0.0872384
$336$	0	0
$337$	1.14143e11	0.482075	0.241037	−	0.970516i	$-0.422512\pi$
$337$	1.14143e11	0.482075	0.241037	+	0.970516i	$0.422512\pi$
$338$	0	0
$339$	6.44261e11	2.64949
$340$	0	0
$341$	−1.69435e11	−0.678593
$342$	0	0
$343$	−7.34065e10	−0.286359
$344$	0	0
$345$	−5.23565e10	−0.198969
$346$	0	0
$347$	1.30233e11	0.482213	0.241106	−	0.970499i	$-0.422490\pi$
$347$	1.30233e11	0.482213	0.241106	+	0.970499i	$0.422490\pi$
$348$	0	0
$349$	1.52443e11	0.550039	0.275020	−	0.961439i	$-0.411316\pi$
$349$	1.52443e11	0.550039	0.275020	+	0.961439i	$0.411316\pi$
$350$	0	0
$351$	−3.89571e11	−1.36995
$352$	0	0
$353$	4.83570e11	1.65757	0.828787	−	0.559565i	$-0.189032\pi$
$353$	4.83570e11	1.65757	0.828787	+	0.559565i	$0.189032\pi$
$354$	0	0
$355$	−6.34292e10	−0.211964
$356$	0	0
$357$	−1.99642e11	−0.650497
$358$	0	0
$359$	3.20450e11	1.01820	0.509102	−	0.860706i	$-0.329977\pi$
$359$	3.20450e11	1.01820	0.509102	+	0.860706i	$0.329977\pi$
$360$	0	0
$361$	6.97298e11	2.16091
$362$	0	0
$363$	2.25444e11	0.681488
$364$	0	0
$365$	7.27836e9	0.0214642
$366$	0	0
$367$	1.77614e11	0.511070	0.255535	−	0.966800i	$-0.417748\pi$
$367$	1.77614e11	0.511070	0.255535	+	0.966800i	$0.417748\pi$
$368$	0	0
$369$	−1.83582e11	−0.515480
$370$	0	0
$371$	2.95939e11	0.810999
$372$	0	0
$373$	4.80280e11	1.28471	0.642355	−	0.766407i	$-0.277958\pi$
$373$	4.80280e11	1.28471	0.642355	+	0.766407i	$0.277958\pi$
$374$	0	0
$375$	−1.49520e11	−0.390443
$376$	0	0
$377$	−1.90235e11	−0.485014
$378$	0	0
$379$	−3.81929e11	−0.950837	−0.475419	−	0.879760i	$-0.657703\pi$
$379$	−3.81929e11	−0.950837	−0.475419	+	0.879760i	$0.657703\pi$
$380$	0	0
$381$	−1.02983e12	−2.50382
$382$	0	0
$383$	−3.63419e11	−0.863004	−0.431502	−	0.902112i	$-0.642016\pi$
$383$	−3.63419e11	−0.863004	−0.431502	+	0.902112i	$0.642016\pi$
$384$	0	0
$385$	−8.12191e10	−0.188402
$386$	0	0
$387$	−4.49088e11	−1.01773
$388$	0	0
$389$	−6.51726e11	−1.44308	−0.721542	−	0.692370i	$-0.756566\pi$
$389$	−6.51726e11	−1.44308	−0.721542	+	0.692370i	$0.756566\pi$
$390$	0	0
$391$	−1.13571e11	−0.245738
$392$	0	0
$393$	7.45818e11	1.57713
$394$	0	0
$395$	−5.28932e9	−0.0109323
$396$	0	0
$397$	2.04054e11	0.412276	0.206138	−	0.978523i	$-0.433910\pi$
$397$	2.04054e11	0.412276	0.206138	+	0.978523i	$0.433910\pi$
$398$	0	0
$399$	2.41409e12	4.76843
$400$	0	0
$401$	−2.48041e11	−0.479042	−0.239521	−	0.970891i	$-0.576990\pi$
$401$	−2.48041e11	−0.479042	−0.239521	+	0.970891i	$0.576990\pi$
$402$	0	0
$403$	1.81057e11	0.341934
$404$	0	0
$405$	1.12563e11	0.207897
$406$	0	0
$407$	3.88303e10	0.0701448
$408$	0	0
$409$	−2.59982e11	−0.459397	−0.229698	−	0.973262i	$-0.573774\pi$
$409$	−2.59982e11	−0.459397	−0.229698	+	0.973262i	$0.573774\pi$
$410$	0	0
$411$	−1.22619e12	−2.11967
$412$	0	0
$413$	−8.50078e11	−1.43775
$414$	0	0
$415$	3.11649e10	0.0515762
$416$	0	0
$417$	−6.74898e11	−1.09301
$418$	0	0
$419$	5.23470e11	0.829714	0.414857	−	0.909887i	$-0.363832\pi$
$419$	5.23470e11	0.829714	0.414857	+	0.909887i	$0.363832\pi$
$420$	0	0
$421$	4.04503e11	0.627555	0.313778	−	0.949497i	$-0.398405\pi$
$421$	4.04503e11	0.627555	0.313778	+	0.949497i	$0.398405\pi$
$422$	0	0
$423$	−1.14228e12	−1.73476
$424$	0	0
$425$	−1.61209e11	−0.239684
$426$	0	0
$427$	−9.28673e11	−1.35188
$428$	0	0
$429$	−8.81092e11	−1.25592
$430$	0	0
$431$	5.97189e11	0.833612	0.416806	−	0.908995i	$-0.363149\pi$
$431$	5.97189e11	0.833612	0.416806	+	0.908995i	$0.363149\pi$
$432$	0	0
$433$	−8.45782e11	−1.15628	−0.578140	−	0.815938i	$-0.696221\pi$
$433$	−8.45782e11	−1.15628	−0.578140	+	0.815938i	$0.696221\pi$
$434$	0	0
$435$	1.20324e11	0.161121
$436$	0	0
$437$	1.37331e12	1.80137
$438$	0	0
$439$	−4.42347e11	−0.568424	−0.284212	−	0.958761i	$-0.591732\pi$
$439$	−4.42347e11	−0.568424	−0.284212	+	0.958761i	$0.591732\pi$
$440$	0	0
$441$	2.16081e12	2.72046
$442$	0	0
$443$	2.05741e11	0.253807	0.126903	−	0.991915i	$-0.459496\pi$
$443$	2.05741e11	0.253807	0.126903	+	0.991915i	$0.459496\pi$
$444$	0	0
$445$	−3.08946e10	−0.0373476
$446$	0	0
$447$	−7.24403e11	−0.858215
$448$	0	0
$449$	1.29191e12	1.50012	0.750059	−	0.661371i	$-0.230025\pi$
$449$	1.29191e12	1.50012	0.750059	+	0.661371i	$0.230025\pi$
$450$	0	0
$451$	−2.33073e11	−0.265275
$452$	0	0
$453$	2.12726e12	2.37344
$454$	0	0
$455$	8.67898e10	0.0949330
$456$	0	0
$457$	−2.65686e11	−0.284935	−0.142467	−	0.989799i	$-0.545504\pi$
$457$	−2.65686e11	−0.284935	−0.142467	+	0.989799i	$0.545504\pi$
$458$	0	0
$459$	5.34499e11	0.562070
$460$	0	0
$461$	4.14119e11	0.427043	0.213521	−	0.976938i	$-0.431507\pi$
$461$	4.14119e11	0.427043	0.213521	+	0.976938i	$0.431507\pi$
$462$	0	0
$463$	−5.26394e11	−0.532349	−0.266175	−	0.963925i	$-0.585760\pi$
$463$	−5.26394e11	−0.532349	−0.266175	+	0.963925i	$0.585760\pi$
$464$	0	0
$465$	−1.14519e11	−0.113590
$466$	0	0
$467$	−5.24754e11	−0.510540	−0.255270	−	0.966870i	$-0.582164\pi$
$467$	−5.24754e11	−0.510540	−0.255270	+	0.966870i	$0.582164\pi$
$468$	0	0
$469$	1.24844e12	1.19149
$470$	0	0
$471$	2.12739e12	1.99184
$472$	0	0
$473$	−5.70154e11	−0.523741
$474$	0	0
$475$	1.94935e12	1.75699
$476$	0	0
$477$	−1.41147e12	−1.24836
$478$	0	0
$479$	6.90731e11	0.599514	0.299757	−	0.954016i	$-0.403094\pi$
$479$	6.90731e11	0.599514	0.299757	+	0.954016i	$0.403094\pi$
$480$	0	0
$481$	−4.14936e10	−0.0353450
$482$	0	0
$483$	3.25034e12	2.71748
$484$	0	0
$485$	1.88368e11	0.154586
$486$	0	0
$487$	−5.21019e11	−0.419733	−0.209867	−	0.977730i	$-0.567303\pi$
$487$	−5.21019e11	−0.419733	−0.209867	+	0.977730i	$0.567303\pi$
$488$	0	0
$489$	5.38839e10	0.0426157
$490$	0	0
$491$	−3.74675e11	−0.290930	−0.145465	−	0.989363i	$-0.546468\pi$
$491$	−3.74675e11	−0.290930	−0.145465	+	0.989363i	$0.546468\pi$
$492$	0	0
$493$	2.61006e11	0.198994
$494$	0	0
$495$	3.87371e11	0.290003
$496$	0	0
$497$	3.93774e12	2.89496
$498$	0	0
$499$	1.96144e12	1.41619	0.708096	−	0.706116i	$-0.249554\pi$
$499$	1.96144e12	1.41619	0.708096	+	0.706116i	$0.249554\pi$
$500$	0	0
$501$	2.35719e12	1.67157
$502$	0	0
$503$	−7.20856e11	−0.502103	−0.251051	−	0.967974i	$-0.580776\pi$
$503$	−7.20856e11	−0.502103	−0.251051	+	0.967974i	$0.580776\pi$
$504$	0	0
$505$	−3.06870e11	−0.209963
$506$	0	0
$507$	−1.75281e12	−1.17814
$508$	0	0
$509$	4.96970e11	0.328171	0.164085	−	0.986446i	$-0.447533\pi$
$509$	4.96970e11	0.328171	0.164085	+	0.986446i	$0.447533\pi$
$510$	0	0
$511$	−4.51847e11	−0.293155
$512$	0	0
$513$	−6.46321e12	−4.12022
$514$	0	0
$515$	−1.76209e11	−0.110381
$516$	0	0
$517$	−1.45022e12	−0.892740
$518$	0	0
$519$	−3.29646e12	−1.99432
$520$	0	0
$521$	−7.85633e11	−0.467143	−0.233572	−	0.972340i	$-0.575041\pi$
$521$	−7.85633e11	−0.467143	−0.233572	+	0.972340i	$0.575041\pi$
$522$	0	0
$523$	1.28047e12	0.748363	0.374182	−	0.927355i	$-0.377924\pi$
$523$	1.28047e12	0.748363	0.374182	+	0.927355i	$0.377924\pi$
$524$	0	0
$525$	4.61371e12	2.65053
$526$	0	0
$527$	−2.48413e11	−0.140290
$528$	0	0
$529$	4.78790e10	0.0265824
$530$	0	0
$531$	4.05440e12	2.21310
$532$	0	0
$533$	2.49059e11	0.133669
$534$	0	0
$535$	−1.99420e11	−0.105239
$536$	0	0
$537$	−6.55950e12	−3.40397
$538$	0	0
$539$	2.74332e12	1.40000
$540$	0	0
$541$	2.82997e12	1.42035	0.710174	−	0.704026i	$-0.248616\pi$
$541$	2.82997e12	1.42035	0.710174	+	0.704026i	$0.248616\pi$
$542$	0	0
$543$	6.64947e11	0.328237
$544$	0	0
$545$	−2.71264e10	−0.0131707
$546$	0	0
$547$	−3.65687e12	−1.74649	−0.873247	−	0.487278i	$-0.837990\pi$
$547$	−3.65687e12	−1.74649	−0.873247	+	0.487278i	$0.837990\pi$
$548$	0	0
$549$	4.42926e12	2.08092
$550$	0	0
$551$	−3.15611e12	−1.45871
$552$	0	0
$553$	3.28366e11	0.149312
$554$	0	0
$555$	2.62449e10	0.0117416
$556$	0	0
$557$	−8.41201e11	−0.370298	−0.185149	−	0.982710i	$-0.559277\pi$
$557$	−8.41201e11	−0.370298	−0.185149	+	0.982710i	$0.559277\pi$
$558$	0	0
$559$	6.09260e11	0.263906
$560$	0	0
$561$	1.20888e12	0.515286
$562$	0	0
$563$	−2.93812e12	−1.23249	−0.616243	−	0.787556i	$-0.711346\pi$
$563$	−2.93812e12	−1.23249	−0.616243	+	0.787556i	$0.711346\pi$
$564$	0	0
$565$	3.84272e11	0.158643
$566$	0	0
$567$	−6.98803e12	−2.83943
$568$	0	0
$569$	−2.06188e12	−0.824627	−0.412313	−	0.911042i	$-0.635279\pi$
$569$	−2.06188e12	−0.824627	−0.412313	+	0.911042i	$0.635279\pi$
$570$	0	0
$571$	2.23269e12	0.878955	0.439477	−	0.898254i	$-0.355164\pi$
$571$	2.23269e12	0.878955	0.439477	+	0.898254i	$0.355164\pi$
$572$	0	0
$573$	−2.09757e11	−0.0812871
$574$	0	0
$575$	2.62461e12	1.00129
$576$	0	0
$577$	−9.94350e11	−0.373464	−0.186732	−	0.982411i	$-0.559790\pi$
$577$	−9.94350e11	−0.373464	−0.186732	+	0.982411i	$0.559790\pi$
$578$	0	0
$579$	4.79719e12	1.77392
$580$	0	0
$581$	−1.93475e12	−0.704420
$582$	0	0
$583$	−1.79198e12	−0.642427
$584$	0	0
$585$	−4.13940e11	−0.146129
$586$	0	0
$587$	4.22018e12	1.46710	0.733549	−	0.679636i	$-0.237862\pi$
$587$	4.22018e12	1.46710	0.733549	+	0.679636i	$0.237862\pi$
$588$	0	0
$589$	3.00384e12	1.02839
$590$	0	0
$591$	4.40243e12	1.48439
$592$	0	0
$593$	−4.49622e12	−1.49314	−0.746571	−	0.665305i	$-0.768301\pi$
$593$	−4.49622e12	−1.49314	−0.746571	+	0.665305i	$0.768301\pi$
$594$	0	0
$595$	−1.19077e11	−0.0389496
$596$	0	0
$597$	−5.45341e12	−1.75705
$598$	0	0
$599$	−1.61824e11	−0.0513598	−0.0256799	−	0.999670i	$-0.508175\pi$
$599$	−1.61824e11	−0.0513598	−0.0256799	+	0.999670i	$0.508175\pi$
$600$	0	0
$601$	−3.79766e12	−1.18735	−0.593677	−	0.804703i	$-0.702324\pi$
$601$	−3.79766e12	−1.18735	−0.593677	+	0.804703i	$0.702324\pi$
$602$	0	0
$603$	−5.95437e12	−1.83403
$604$	0	0
$605$	1.34467e11	0.0408052
$606$	0	0
$607$	−9.59183e11	−0.286782	−0.143391	−	0.989666i	$-0.545801\pi$
$607$	−9.59183e11	−0.286782	−0.143391	+	0.989666i	$0.545801\pi$
$608$	0	0
$609$	−7.46984e12	−2.20056
$610$	0	0
$611$	1.54968e12	0.449839
$612$	0	0
$613$	2.49059e12	0.712409	0.356205	−	0.934408i	$-0.384071\pi$
$613$	2.49059e12	0.712409	0.356205	+	0.934408i	$0.384071\pi$
$614$	0	0
$615$	−1.57531e11	−0.0444045
$616$	0	0
$617$	−1.31221e12	−0.364519	−0.182259	−	0.983250i	$-0.558341\pi$
$617$	−1.31221e12	−0.364519	−0.182259	+	0.983250i	$0.558341\pi$
$618$	0	0
$619$	−6.10985e12	−1.67272	−0.836359	−	0.548182i	$-0.815320\pi$
$619$	−6.10985e12	−1.67272	−0.836359	+	0.548182i	$0.815320\pi$
$620$	0	0
$621$	−8.70209e12	−2.34807
$622$	0	0
$623$	1.91797e12	0.510088
$624$	0	0
$625$	3.68066e12	0.964863
$626$	0	0
$627$	−1.46178e13	−3.77728
$628$	0	0
$629$	5.69300e10	0.0145015
$630$	0	0
$631$	2.63235e12	0.661016	0.330508	−	0.943803i	$-0.392780\pi$
$631$	2.63235e12	0.661016	0.330508	+	0.943803i	$0.392780\pi$
$632$	0	0
$633$	−1.96457e12	−0.486353
$634$	0	0
$635$	−6.14247e11	−0.149920
$636$	0	0
$637$	−2.93148e12	−0.705440
$638$	0	0
$639$	−1.87808e13	−4.45617
$640$	0	0
$641$	4.56614e12	1.06829	0.534144	−	0.845394i	$-0.320634\pi$
$641$	4.56614e12	1.06829	0.534144	+	0.845394i	$0.320634\pi$
$642$	0	0
$643$	6.83172e12	1.57609	0.788044	−	0.615618i	$-0.211094\pi$
$643$	6.83172e12	1.57609	0.788044	+	0.615618i	$0.211094\pi$
$644$	0	0
$645$	−3.85359e11	−0.0876692
$646$	0	0
$647$	3.87609e12	0.869609	0.434805	−	0.900525i	$-0.356817\pi$
$647$	3.87609e12	0.869609	0.434805	+	0.900525i	$0.356817\pi$
$648$	0	0
$649$	5.14740e12	1.13890
$650$	0	0
$651$	7.10945e12	1.55139
$652$	0	0
$653$	5.89438e12	1.26861	0.634306	−	0.773082i	$-0.281286\pi$
$653$	5.89438e12	1.26861	0.634306	+	0.773082i	$0.281286\pi$
$654$	0	0
$655$	4.44846e11	0.0944329
$656$	0	0
$657$	2.15506e12	0.451248
$658$	0	0
$659$	−6.62732e12	−1.36884	−0.684421	−	0.729087i	$-0.739945\pi$
$659$	−6.62732e12	−1.36884	−0.684421	+	0.729087i	$0.739945\pi$
$660$	0	0
$661$	−1.51309e12	−0.308288	−0.154144	−	0.988048i	$-0.549262\pi$
$661$	−1.51309e12	−0.308288	−0.154144	+	0.988048i	$0.549262\pi$
$662$	0	0
$663$	−1.29179e12	−0.259646
$664$	0	0
$665$	1.43989e12	0.285518
$666$	0	0
$667$	−4.24939e12	−0.831306
$668$	0	0
$669$	−2.35692e12	−0.454913
$670$	0	0
$671$	5.62331e12	1.07088
$672$	0	0
$673$	−7.99861e10	−0.0150296	−0.00751479	−	0.999972i	$-0.502392\pi$
$673$	−7.99861e10	−0.0150296	−0.00751479	+	0.999972i	$0.502392\pi$
$674$	0	0
$675$	−1.23522e13	−2.29022
$676$	0	0
$677$	−1.28946e12	−0.235917	−0.117958	−	0.993019i	$-0.537635\pi$
$677$	−1.28946e12	−0.235917	−0.117958	+	0.993019i	$0.537635\pi$
$678$	0	0
$679$	−1.16941e13	−2.11131
$680$	0	0
$681$	2.56415e12	0.456858
$682$	0	0
$683$	1.41260e12	0.248386	0.124193	−	0.992258i	$-0.460366\pi$
$683$	1.41260e12	0.248386	0.124193	+	0.992258i	$0.460366\pi$
$684$	0	0
$685$	−7.31365e11	−0.126919
$686$	0	0
$687$	−6.87108e12	−1.17685
$688$	0	0
$689$	1.91488e12	0.323710
$690$	0	0
$691$	2.52816e12	0.421846	0.210923	−	0.977503i	$-0.432353\pi$
$691$	2.52816e12	0.421846	0.210923	+	0.977503i	$0.432353\pi$
$692$	0	0
$693$	−2.40483e13	−3.96082
$694$	0	0
$695$	−4.02546e11	−0.0654460
$696$	0	0
$697$	−3.41713e11	−0.0548422
$698$	0	0
$699$	1.73690e13	2.75187
$700$	0	0
$701$	6.98644e12	1.09276	0.546380	−	0.837537i	$-0.316005\pi$
$701$	6.98644e12	1.09276	0.546380	+	0.837537i	$0.316005\pi$
$702$	0	0
$703$	−6.88403e11	−0.106303
$704$	0	0
$705$	−9.80181e11	−0.149436
$706$	0	0
$707$	1.90507e13	2.86764
$708$	0	0
$709$	−4.48848e12	−0.667100	−0.333550	−	0.942732i	$-0.608247\pi$
$709$	−4.48848e12	−0.667100	−0.333550	+	0.942732i	$0.608247\pi$
$710$	0	0
$711$	−1.56612e12	−0.229833
$712$	0	0
$713$	4.04438e12	0.586069
$714$	0	0
$715$	−5.25530e11	−0.0752005
$716$	0	0
$717$	6.53844e12	0.923928
$718$	0	0
$719$	7.44881e12	1.03946	0.519729	−	0.854331i	$-0.326033\pi$
$719$	7.44881e12	1.03946	0.519729	+	0.854331i	$0.326033\pi$
$720$	0	0
$721$	1.09392e13	1.50757
$722$	0	0
$723$	−5.02401e12	−0.683799
$724$	0	0
$725$	−6.03181e12	−0.810825
$726$	0	0
$727$	−3.94611e12	−0.523920	−0.261960	−	0.965079i	$-0.584369\pi$
$727$	−3.94611e12	−0.523920	−0.261960	+	0.965079i	$0.584369\pi$
$728$	0	0
$729$	1.32507e12	0.173766
$730$	0	0
$731$	−8.35917e11	−0.108277
$732$	0	0
$733$	8.83103e12	1.12991	0.564955	−	0.825122i	$-0.308894\pi$
$733$	8.83103e12	1.12991	0.564955	+	0.825122i	$0.308894\pi$
$734$	0	0
$735$	1.85418e12	0.234346
$736$	0	0
$737$	−7.55956e12	−0.943828
$738$	0	0
$739$	6.70494e12	0.826980	0.413490	−	0.910509i	$-0.364310\pi$
$739$	6.70494e12	0.826980	0.413490	+	0.910509i	$0.364310\pi$
$740$	0	0
$741$	1.56204e13	1.90332
$742$	0	0
$743$	8.35278e12	1.00550	0.502749	−	0.864432i	$-0.332322\pi$
$743$	8.35278e12	1.00550	0.502749	+	0.864432i	$0.332322\pi$
$744$	0	0
$745$	−4.32073e11	−0.0513870
$746$	0	0
$747$	9.22768e12	1.08430
$748$	0	0
$749$	1.23802e13	1.43734
$750$	0	0
$751$	1.43369e13	1.64465	0.822327	−	0.569015i	$-0.192675\pi$
$751$	1.43369e13	1.64465	0.822327	+	0.569015i	$0.192675\pi$
$752$	0	0
$753$	−8.51671e12	−0.965371
$754$	0	0
$755$	1.26881e12	0.142114
$756$	0	0
$757$	−1.30485e13	−1.44421	−0.722105	−	0.691784i	$-0.756825\pi$
$757$	−1.30485e13	−1.44421	−0.722105	+	0.691784i	$0.756825\pi$
$758$	0	0
$759$	−1.96815e13	−2.15263
$760$	0	0
$761$	−3.25303e10	−0.00351607	−0.00175804	−	0.999998i	$-0.500560\pi$
$761$	−3.25303e10	−0.00351607	−0.00175804	+	0.999998i	$0.500560\pi$
$762$	0	0
$763$	1.68403e12	0.179883
$764$	0	0
$765$	5.67933e11	0.0599544
$766$	0	0
$767$	−5.50045e12	−0.573877
$768$	0	0
$769$	1.79023e13	1.84603	0.923016	−	0.384761i	$-0.125716\pi$
$769$	1.79023e13	1.84603	0.923016	+	0.384761i	$0.125716\pi$
$770$	0	0
$771$	−2.34280e13	−2.38776
$772$	0	0
$773$	2.45846e12	0.247660	0.123830	−	0.992303i	$-0.460482\pi$
$773$	2.45846e12	0.247660	0.123830	+	0.992303i	$0.460482\pi$
$774$	0	0
$775$	5.74080e12	0.571630
$776$	0	0
$777$	−1.62930e12	−0.160364
$778$	0	0
$779$	4.13203e12	0.402017
$780$	0	0
$781$	−2.38438e13	−2.29322
$782$	0	0
$783$	1.99989e13	1.90142
$784$	0	0
$785$	1.26889e12	0.119264
$786$	0	0
$787$	1.50623e13	1.39961	0.699803	−	0.714336i	$-0.253271\pi$
$787$	1.50623e13	1.39961	0.699803	+	0.714336i	$0.253271\pi$
$788$	0	0
$789$	−4.66257e12	−0.428331
$790$	0	0
$791$	−2.38559e13	−2.16672
$792$	0	0
$793$	−6.00900e12	−0.539601
$794$	0	0
$795$	−1.21117e12	−0.107536
$796$	0	0
$797$	−1.58318e12	−0.138985	−0.0694924	−	0.997582i	$-0.522138\pi$
$797$	−1.58318e12	−0.138985	−0.0694924	+	0.997582i	$0.522138\pi$
$798$	0	0
$799$	−2.12620e12	−0.184562
$800$	0	0
$801$	−9.14765e12	−0.785169
$802$	0	0
$803$	2.73603e12	0.232221
$804$	0	0
$805$	1.93868e12	0.162714
$806$	0	0
$807$	−2.43955e13	−2.02479
$808$	0	0
$809$	1.09269e13	0.896871	0.448435	−	0.893815i	$-0.351981\pi$
$809$	1.09269e13	0.896871	0.448435	+	0.893815i	$0.351981\pi$
$810$	0	0
$811$	1.40564e13	1.14098	0.570492	−	0.821303i	$-0.306753\pi$
$811$	1.40564e13	1.14098	0.570492	+	0.821303i	$0.306753\pi$
$812$	0	0
$813$	4.09267e13	3.28548
$814$	0	0
$815$	3.21393e10	0.00255168
$816$	0	0
$817$	1.01080e13	0.793716
$818$	0	0
$819$	2.56977e13	1.99580
$820$	0	0
$821$	−8.17487e12	−0.627967	−0.313983	−	0.949429i	$-0.601664\pi$
$821$	−8.17487e12	−0.627967	−0.313983	+	0.949429i	$0.601664\pi$
$822$	0	0
$823$	−1.18003e13	−0.896587	−0.448294	−	0.893886i	$-0.647968\pi$
$823$	−1.18003e13	−0.896587	−0.448294	+	0.893886i	$0.647968\pi$
$824$	0	0
$825$	−2.79370e13	−2.09960
$826$	0	0
$827$	−1.45470e13	−1.08143	−0.540716	−	0.841205i	$-0.681847\pi$
$827$	−1.45470e13	−1.08143	−0.540716	+	0.841205i	$0.681847\pi$
$828$	0	0
$829$	−1.06246e13	−0.781301	−0.390650	−	0.920539i	$-0.627750\pi$
$829$	−1.06246e13	−0.781301	−0.390650	+	0.920539i	$0.627750\pi$
$830$	0	0
$831$	1.35756e13	0.987541
$832$	0	0
$833$	4.02205e12	0.289431
$834$	0	0
$835$	1.40596e12	0.100088
$836$	0	0
$837$	−1.90340e13	−1.34050
$838$	0	0
$839$	5.78736e12	0.403229	0.201614	−	0.979465i	$-0.435381\pi$
$839$	5.78736e12	0.403229	0.201614	+	0.979465i	$0.435381\pi$
$840$	0	0
$841$	−4.74131e12	−0.326826
$842$	0	0
$843$	3.78455e13	2.58101
$844$	0	0
$845$	−1.04547e12	−0.0705433
$846$	0	0
$847$	−8.34782e12	−0.557311
$848$	0	0
$849$	1.40133e13	0.925669
$850$	0	0
$851$	−9.26868e11	−0.0605808
$852$	0	0
$853$	1.49513e13	0.966958	0.483479	−	0.875356i	$-0.339373\pi$
$853$	1.49513e13	0.966958	0.483479	+	0.875356i	$0.339373\pi$
$854$	0	0
$855$	−6.86750e12	−0.439492
$856$	0	0
$857$	−1.61752e13	−1.02432	−0.512159	−	0.858890i	$-0.671154\pi$
$857$	−1.61752e13	−1.02432	−0.512159	+	0.858890i	$0.671154\pi$
$858$	0	0
$859$	2.12735e13	1.33312	0.666562	−	0.745450i	$-0.267765\pi$
$859$	2.12735e13	1.33312	0.666562	+	0.745450i	$0.267765\pi$
$860$	0	0
$861$	9.77964e12	0.606469
$862$	0	0
$863$	−1.58563e13	−0.973093	−0.486547	−	0.873655i	$-0.661744\pi$
$863$	−1.58563e13	−0.973093	−0.486547	+	0.873655i	$0.661744\pi$
$864$	0	0
$865$	−1.96619e12	−0.119413
$866$	0	0
$867$	1.77236e12	0.106529
$868$	0	0
$869$	−1.98832e12	−0.118276
$870$	0	0
$871$	8.07805e12	0.475582
$872$	0	0
$873$	5.57742e13	3.24990
$874$	0	0
$875$	5.53647e12	0.319298
$876$	0	0
$877$	−1.59980e13	−0.913203	−0.456601	−	0.889671i	$-0.650933\pi$
$877$	−1.59980e13	−0.913203	−0.456601	+	0.889671i	$0.650933\pi$
$878$	0	0
$879$	−4.02451e13	−2.27385
$880$	0	0
$881$	1.84519e13	1.03193	0.515963	−	0.856611i	$-0.327434\pi$
$881$	1.84519e13	1.03193	0.515963	+	0.856611i	$0.327434\pi$
$882$	0	0
$883$	−2.67143e11	−0.0147884	−0.00739420	−	0.999973i	$-0.502354\pi$
$883$	−2.67143e11	−0.0147884	−0.00739420	+	0.999973i	$0.502354\pi$
$884$	0	0
$885$	3.47906e12	0.190641
$886$	0	0
$887$	−2.38695e13	−1.29475	−0.647377	−	0.762170i	$-0.724134\pi$
$887$	−2.38695e13	−1.29475	−0.647377	+	0.762170i	$0.724134\pi$
$888$	0	0
$889$	3.81330e13	2.04759
$890$	0	0
$891$	4.23140e13	2.24923
$892$	0	0
$893$	2.57102e13	1.35292
$894$	0	0
$895$	−3.91244e12	−0.203819
$896$	0	0
$897$	2.10314e13	1.08468
$898$	0	0
$899$	−9.29467e12	−0.474587
$900$	0	0
$901$	−2.62726e12	−0.132813
$902$	0	0
$903$	2.39235e13	1.19737
$904$	0	0
$905$	3.96610e11	0.0196537
$906$	0	0
$907$	2.77628e13	1.36217	0.681085	−	0.732205i	$-0.261508\pi$
$907$	2.77628e13	1.36217	0.681085	+	0.732205i	$0.261508\pi$
$908$	0	0
$909$	−9.08615e13	−4.41410
$910$	0	0
$911$	−2.28307e13	−1.09821	−0.549106	−	0.835753i	$-0.685032\pi$
$911$	−2.28307e13	−1.09821	−0.549106	+	0.835753i	$0.685032\pi$
$912$	0	0
$913$	1.17153e13	0.558001
$914$	0	0
$915$	3.80072e12	0.179255
$916$	0	0
$917$	−2.76164e13	−1.28975
$918$	0	0
$919$	5.88802e11	0.0272301	0.0136150	−	0.999907i	$-0.495666\pi$
$919$	5.88802e11	0.0272301	0.0136150	+	0.999907i	$0.495666\pi$
$920$	0	0
$921$	−5.93909e13	−2.71989
$922$	0	0
$923$	2.54792e13	1.15552
$924$	0	0
$925$	−1.31565e12	−0.0590882
$926$	0	0
$927$	−5.21739e13	−2.32057
$928$	0	0
$929$	3.31307e13	1.45935	0.729676	−	0.683793i	$-0.239671\pi$
$929$	3.31307e13	1.45935	0.729676	+	0.683793i	$0.239671\pi$
$930$	0	0
$931$	−4.86350e13	−2.12166
$932$	0	0
$933$	5.70366e13	2.46426
$934$	0	0
$935$	7.21038e11	0.0308536
$936$	0	0
$937$	−1.19627e13	−0.506993	−0.253496	−	0.967336i	$-0.581581\pi$
$937$	−1.19627e13	−0.506993	−0.253496	+	0.967336i	$0.581581\pi$
$938$	0	0
$939$	−7.20208e12	−0.302317
$940$	0	0
$941$	−2.09786e13	−0.872213	−0.436106	−	0.899895i	$-0.643643\pi$
$941$	−2.09786e13	−0.872213	−0.436106	+	0.899895i	$0.643643\pi$
$942$	0	0
$943$	5.56338e12	0.229106
$944$	0	0
$945$	−9.12399e12	−0.372170
$946$	0	0
$947$	−2.72802e13	−1.10223	−0.551115	−	0.834429i	$-0.685797\pi$
$947$	−2.72802e13	−1.10223	−0.551115	+	0.834429i	$0.685797\pi$
$948$	0	0
$949$	−2.92369e12	−0.117013
$950$	0	0
$951$	−2.88154e13	−1.14238
$952$	0	0
$953$	−1.60893e13	−0.631858	−0.315929	−	0.948783i	$-0.602316\pi$
$953$	−1.60893e13	−0.631858	−0.315929	+	0.948783i	$0.602316\pi$
$954$	0	0
$955$	−1.25111e11	−0.00486720
$956$	0	0
$957$	4.52315e13	1.74316
$958$	0	0
$959$	4.54038e13	1.73344
$960$	0	0
$961$	−1.75934e13	−0.665417
$962$	0	0
$963$	−5.90467e13	−2.21247
$964$	0	0
$965$	2.86130e12	0.106216
$966$	0	0
$967$	5.02741e13	1.84895	0.924475	−	0.381242i	$-0.124503\pi$
$967$	5.02741e13	1.84895	0.924475	+	0.381242i	$0.124503\pi$
$968$	0	0
$969$	−2.14316e13	−0.780902
$970$	0	0
$971$	−3.00761e13	−1.08576	−0.542881	−	0.839810i	$-0.682667\pi$
$971$	−3.00761e13	−1.08576	−0.542881	+	0.839810i	$0.682667\pi$
$972$	0	0
$973$	2.49904e13	0.893851
$974$	0	0
$975$	2.98531e13	1.05796
$976$	0	0
$977$	−4.15719e13	−1.45974	−0.729868	−	0.683588i	$-0.760419\pi$
$977$	−4.15719e13	−1.45974	−0.729868	+	0.683588i	$0.760419\pi$
$978$	0	0
$979$	−1.16137e13	−0.404062
$980$	0	0
$981$	−8.03190e12	−0.276890
$982$	0	0
$983$	3.65566e12	0.124875	0.0624375	−	0.998049i	$-0.480113\pi$
$983$	3.65566e12	0.124875	0.0624375	+	0.998049i	$0.480113\pi$
$984$	0	0
$985$	2.62584e12	0.0888804
$986$	0	0
$987$	6.08505e13	2.04097
$988$	0	0
$989$	1.36094e13	0.452331
$990$	0	0
$991$	5.35178e13	1.76265	0.881326	−	0.472509i	$-0.156652\pi$
$991$	5.35178e13	1.76265	0.881326	+	0.472509i	$0.156652\pi$
$992$	0	0
$993$	1.03804e13	0.338798
$994$	0	0
$995$	−3.25271e12	−0.105206
$996$	0	0
$997$	6.71487e11	0.0215233	0.0107617	−	0.999942i	$-0.496574\pi$
$997$	6.71487e11	0.0215233	0.0107617	+	0.999942i	$0.496574\pi$
$998$	0	0
$999$	4.36212e12	0.138565

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	272.10.a.g.1.7		7
4.3	odd	2		17.10.a.b.1.4	✓	7
12.11	even	2		153.10.a.f.1.4		7
68.67	odd	2		289.10.a.b.1.4		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
17.10.a.b.1.4	✓	7	4.3	odd	2
153.10.a.f.1.4		7	12.11	even	2
272.10.a.g.1.7		7	1.1	even	1	trivial
289.10.a.b.1.4		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)

\(f(q)\)	\(=\)	\(q+254.074 q^{3} +151.544 q^{5} -9407.97 q^{7} +44870.8 q^{9} +O(q^{10})\)	\(q+254.074 q^{3} +151.544 q^{5} -9407.97 q^{7} +44870.8 q^{9} +56967.2 q^{11} -60874.5 q^{13} +38503.4 q^{15} +83521.0 q^{17} -1.00994e6 q^{19} -2.39032e6 q^{21} -1.35979e6 q^{23} -1.93016e6 q^{25} +6.39958e6 q^{27} +3.12503e6 q^{29} -2.97426e6 q^{31} +1.44739e7 q^{33} -1.42572e6 q^{35} +681625. q^{37} -1.54666e7 q^{39} -4.09135e6 q^{41} -1.00085e7 q^{43} +6.79989e6 q^{45} -2.54570e7 q^{47} +4.81562e7 q^{49} +2.12206e7 q^{51} -3.14563e7 q^{53} +8.63302e6 q^{55} -2.56601e8 q^{57} +9.03573e7 q^{59} +9.87113e7 q^{61} -4.22143e8 q^{63} -9.22514e6 q^{65} -1.32700e8 q^{67} -3.45488e8 q^{69} -4.18554e8 q^{71} +4.80282e7 q^{73} -4.90404e8 q^{75} -5.35945e8 q^{77} -3.49030e7 q^{79} +7.42778e8 q^{81} +2.05650e8 q^{83} +1.26571e7 q^{85} +7.93991e8 q^{87} -2.03866e8 q^{89} +5.72705e8 q^{91} -7.55684e8 q^{93} -1.53050e8 q^{95} +1.24300e9 q^{97} +2.55616e9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 88 q^{3} + 1362 q^{5} - 9388 q^{7} + 81419 q^{9}+O(q^{10}) \) 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

Related objects

Downloads

Learn more