LMFDB - Embedded newform 27.10.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [27,10,Mod(1,27)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(27, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("27.1");

S:= CuspForms(chi, 10);

N := Newforms(S);

Level:	$ N $	$=$	$ 27 = 3^{3} $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	27.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:	$13.9059675764$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.177113.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 118x + 136 $ x^3 - x^2 - 118*x + 136
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2\cdot 3^{4} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-10.9320$ of defining polynomial
Character	$\chi$	$=$	27.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-32.7960 q^{2} +563.577 q^{4} -1315.38 q^{5} +5158.54 q^{7} -1691.52 q^{8} +43139.3 q^{10} +15135.0 q^{11} +180503. q^{13} -169179. q^{14} -233076. q^{16} -595825. q^{17} -785830. q^{19} -741319. q^{20} -496368. q^{22} +1.17544e6 q^{23} -222893. q^{25} -5.91979e6 q^{26} +2.90723e6 q^{28} -1.54282e6 q^{29} -2.27049e6 q^{31} +8.51003e6 q^{32} +1.95407e7 q^{34} -6.78546e6 q^{35} -1.12928e7 q^{37} +2.57721e7 q^{38} +2.22499e6 q^{40} +1.67274e7 q^{41} -3.11877e7 q^{43} +8.52976e6 q^{44} -3.85498e7 q^{46} -1.93554e7 q^{47} -1.37431e7 q^{49} +7.30999e6 q^{50} +1.01728e8 q^{52} +3.77611e7 q^{53} -1.99084e7 q^{55} -8.72576e6 q^{56} +5.05984e7 q^{58} -1.37708e8 q^{59} +1.84447e8 q^{61} +7.44629e7 q^{62} -1.59760e8 q^{64} -2.37431e8 q^{65} -1.43814e8 q^{67} -3.35793e8 q^{68} +2.22536e8 q^{70} +2.34392e7 q^{71} -1.57666e8 q^{73} +3.70357e8 q^{74} -4.42875e8 q^{76} +7.80747e7 q^{77} +4.69594e7 q^{79} +3.06585e8 q^{80} -5.48593e8 q^{82} -2.45495e8 q^{83} +7.83738e8 q^{85} +1.02283e9 q^{86} -2.56012e7 q^{88} -1.10618e9 q^{89} +9.31134e8 q^{91} +6.62452e8 q^{92} +6.34778e8 q^{94} +1.03367e9 q^{95} +2.96985e8 q^{97} +4.50717e8 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} + 597 q^{4} - 1983 q^{5} - 3693 q^{7} - 4503 q^{8} - 18981 q^{10} - 16863 q^{11} + 116916 q^{13} - 503463 q^{14} - 239919 q^{16} - 1014048 q^{17} - 15222 q^{19} - 2548407 q^{20} + 305721 q^{22}+ \cdots + 2592182286 q^{98}+O(q^{100}) $ 3 * q + 3 * q^2 + 597 * q^4 - 1983 * q^5 - 3693 * q^7 - 4503 * q^8 - 18981 * q^10 - 16863 * q^11 + 116916 * q^13 - 503463 * q^14 - 239919 * q^16 - 1014048 * q^17 - 15222 * q^19 - 2548407 * q^20 + 305721 * q^22 - 2927118 * q^23 + 2133732 * q^25 - 4765116 * q^26 - 3535725 * q^28 - 5768790 * q^29 - 6575223 * q^31 + 2687697 * q^32 + 17098128 * q^34 + 17340537 * q^35 - 11686026 * q^37 + 50473374 * q^38 - 4130811 * q^40 + 22213518 * q^41 + 45384414 * q^43 + 57206991 * q^44 - 95638590 * q^46 - 12392034 * q^47 + 18933462 * q^49 + 82984044 * q^50 + 182736492 * q^52 - 80579637 * q^53 - 174735333 * q^55 - 21850521 * q^56 - 102758922 * q^58 - 244026660 * q^59 + 369729960 * q^61 - 166297341 * q^62 - 420692127 * q^64 - 492225684 * q^65 - 252614586 * q^67 - 162270144 * q^68 + 935092377 * q^70 - 403193088 * q^71 - 406626717 * q^73 + 641373558 * q^74 - 39499590 * q^76 - 360443199 * q^77 + 265451856 * q^79 + 1168359837 * q^80 - 884408166 * q^82 - 121625871 * q^83 + 316251216 * q^85 + 2368956570 * q^86 + 219257739 * q^88 - 377904006 * q^89 + 245059140 * q^91 + 1178754558 * q^92 + 86583438 * q^94 - 536878770 * q^95 - 438907539 * q^97 + 2592182286 * q^98

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 $

$2$	−32.7960	−1.44939	−0.724696	−	0.689069i	$-0.758020\pi$
$2$	−32.7960	−1.44939	−0.724696	+	0.689069i	$0.758020\pi$
$3$	0	0
$3$	0	0
$4$	563.577	1.10074
$5$	−1315.38	−0.941211	−0.470606	−	0.882344i	$-0.655965\pi$
$5$	−1315.38	−0.941211	−0.470606	+	0.882344i	$0.655965\pi$
$6$	0	0
$7$	5158.54	0.812055	0.406028	−	0.913861i	$-0.366914\pi$
$7$	5158.54	0.812055	0.406028	+	0.913861i	$0.366914\pi$
$8$	−1691.52	−0.146006
$9$	0	0
$10$	43139.3	1.36418
$11$	15135.0	0.311685	0.155843	−	0.987782i	$-0.450191\pi$
$11$	15135.0	0.311685	0.155843	+	0.987782i	$0.450191\pi$
$12$	0	0
$13$	180503.	1.75283	0.876416	−	0.481555i	$-0.159928\pi$
$13$	180503.	1.75283	0.876416	+	0.481555i	$0.159928\pi$
$14$	−169179.	−1.17699
$15$	0	0
$16$	−233076.	−0.889116
$17$	−595825.	−1.73021	−0.865105	−	0.501591i	$-0.832748\pi$
$17$	−595825.	−1.73021	−0.865105	+	0.501591i	$0.832748\pi$
$18$	0	0
$19$	−785830.	−1.38337	−0.691683	−	0.722201i	$-0.743131\pi$
$19$	−785830.	−1.38337	−0.691683	+	0.722201i	$0.743131\pi$
$20$	−741319.	−1.03603
$21$	0	0
$22$	−496368.	−0.451754
$23$	1.17544e6	0.875842	0.437921	−	0.899013i	$-0.355715\pi$
$23$	1.17544e6	0.875842	0.437921	+	0.899013i	$0.355715\pi$
$24$	0	0
$25$	−222893.	−0.114121
$26$	−5.91979e6	−2.54054
$27$	0	0
$28$	2.90723e6	0.893859
$29$	−1.54282e6	−0.405065	−0.202533	−	0.979276i	$-0.564917\pi$
$29$	−1.54282e6	−0.405065	−0.202533	+	0.979276i	$0.564917\pi$
$30$	0	0
$31$	−2.27049e6	−0.441562	−0.220781	−	0.975323i	$-0.570861\pi$
$31$	−2.27049e6	−0.441562	−0.220781	+	0.975323i	$0.570861\pi$
$32$	8.51003e6	1.43468
$33$	0	0
$34$	1.95407e7	2.50775
$35$	−6.78546e6	−0.764316
$36$	0	0
$37$	−1.12928e7	−0.990586	−0.495293	−	0.868726i	$-0.664939\pi$
$37$	−1.12928e7	−0.990586	−0.495293	+	0.868726i	$0.664939\pi$
$38$	2.57721e7	2.00504
$39$	0	0
$40$	2.22499e6	0.137423
$41$	1.67274e7	0.924489	0.462245	−	0.886752i	$-0.347044\pi$
$41$	1.67274e7	0.924489	0.462245	+	0.886752i	$0.347044\pi$
$42$	0	0
$43$	−3.11877e7	−1.39115	−0.695576	−	0.718452i	$-0.744851\pi$
$43$	−3.11877e7	−1.39115	−0.695576	+	0.718452i	$0.744851\pi$
$44$	8.52976e6	0.343083
$45$	0	0
$46$	−3.85498e7	−1.26944
$47$	−1.93554e7	−0.578577	−0.289288	−	0.957242i	$-0.593419\pi$
$47$	−1.93554e7	−0.578577	−0.289288	+	0.957242i	$0.593419\pi$
$48$	0	0
$49$	−1.37431e7	−0.340566
$50$	7.30999e6	0.165406
$51$	0	0
$52$	1.01728e8	1.92941
$53$	3.77611e7	0.657360	0.328680	−	0.944441i	$-0.393396\pi$
$53$	3.77611e7	0.657360	0.328680	+	0.944441i	$0.393396\pi$
$54$	0	0
$55$	−1.99084e7	−0.293362
$56$	−8.72576e6	−0.118565
$57$	0	0
$58$	5.05984e7	0.587098
$59$	−1.37708e8	−1.47954	−0.739770	−	0.672860i	$-0.765066\pi$
$59$	−1.37708e8	−1.47954	−0.739770	+	0.672860i	$0.765066\pi$
$60$	0	0
$61$	1.84447e8	1.70564	0.852818	−	0.522209i	$-0.174892\pi$
$61$	1.84447e8	1.70564	0.852818	+	0.522209i	$0.174892\pi$
$62$	7.44629e7	0.639996
$63$	0	0
$64$	−1.59760e8	−1.19030
$65$	−2.37431e8	−1.64979
$66$	0	0
$67$	−1.43814e8	−0.871898	−0.435949	−	0.899971i	$-0.643587\pi$
$67$	−1.43814e8	−0.871898	−0.435949	+	0.899971i	$0.643587\pi$
$68$	−3.35793e8	−1.90450
$69$	0	0
$70$	2.22536e8	1.10779
$71$	2.34392e7	0.109466	0.0547332	−	0.998501i	$-0.482569\pi$
$71$	2.34392e7	0.109466	0.0547332	+	0.998501i	$0.482569\pi$
$72$	0	0
$73$	−1.57666e8	−0.649808	−0.324904	−	0.945747i	$-0.605332\pi$
$73$	−1.57666e8	−0.649808	−0.324904	+	0.945747i	$0.605332\pi$
$74$	3.70357e8	1.43575
$75$	0	0
$76$	−4.42875e8	−1.52272
$77$	7.80747e7	0.253106
$78$	0	0
$79$	4.69594e7	0.135644	0.0678220	−	0.997697i	$-0.478395\pi$
$79$	4.69594e7	0.135644	0.0678220	+	0.997697i	$0.478395\pi$
$80$	3.06585e8	0.836846
$81$	0	0
$82$	−5.48593e8	−1.33995
$83$	−2.45495e8	−0.567795	−0.283898	−	0.958855i	$-0.591628\pi$
$83$	−2.45495e8	−0.567795	−0.283898	+	0.958855i	$0.591628\pi$
$84$	0	0
$85$	7.83738e8	1.62849
$86$	1.02283e9	2.01632
$87$	0	0
$88$	−2.56012e7	−0.0455080
$89$	−1.10618e9	−1.86883	−0.934415	−	0.356185i	$-0.884077\pi$
$89$	−1.10618e9	−1.86883	−0.934415	+	0.356185i	$0.884077\pi$
$90$	0	0
$91$	9.31134e8	1.42340
$92$	6.62452e8	0.964071
$93$	0	0
$94$	6.34778e8	0.838584
$95$	1.03367e9	1.30204
$96$	0	0
$97$	2.96985e8	0.340614	0.170307	−	0.985391i	$-0.445524\pi$
$97$	2.96985e8	0.340614	0.170307	+	0.985391i	$0.445524\pi$
$98$	4.50717e8	0.493613
$99$	0	0
$100$	−1.25617e8	−0.125617
$101$	−1.00457e8	−0.0960579	−0.0480289	−	0.998846i	$-0.515294\pi$
$101$	−1.00457e8	−0.0960579	−0.0480289	+	0.998846i	$0.515294\pi$
$102$	0	0
$103$	−8.94524e8	−0.783113	−0.391557	−	0.920154i	$-0.628063\pi$
$103$	−8.94524e8	−0.783113	−0.391557	+	0.920154i	$0.628063\pi$
$104$	−3.05325e8	−0.255924
$105$	0	0
$106$	−1.23841e9	−0.952773
$107$	−1.67004e9	−1.23168	−0.615841	−	0.787870i	$-0.711184\pi$
$107$	−1.67004e9	−1.23168	−0.615841	+	0.787870i	$0.711184\pi$
$108$	0	0
$109$	−7.93643e8	−0.538525	−0.269263	−	0.963067i	$-0.586780\pi$
$109$	−7.93643e8	−0.538525	−0.269263	+	0.963067i	$0.586780\pi$
$110$	6.52915e8	0.425196
$111$	0	0
$112$	−1.20233e9	−0.722012
$113$	−5.83682e8	−0.336762	−0.168381	−	0.985722i	$-0.553854\pi$
$113$	−5.83682e8	−0.336762	−0.168381	+	0.985722i	$0.553854\pi$
$114$	0	0
$115$	−1.54616e9	−0.824353
$116$	−8.69499e8	−0.445870
$117$	0	0
$118$	4.51629e9	2.14443
$119$	−3.07359e9	−1.40503
$120$	0	0
$121$	−2.12888e9	−0.902852
$122$	−6.04911e9	−2.47213
$123$	0	0
$124$	−1.27959e9	−0.486043
$125$	2.86230e9	1.04862
$126$	0	0
$127$	1.00303e9	0.342133	0.171067	−	0.985259i	$-0.445279\pi$
$127$	1.00303e9	0.342133	0.171067	+	0.985259i	$0.445279\pi$
$128$	8.82342e8	0.290531
$129$	0	0
$130$	7.78679e9	2.39119
$131$	4.66779e9	1.38481	0.692406	−	0.721508i	$-0.256551\pi$
$131$	4.66779e9	1.38481	0.692406	+	0.721508i	$0.256551\pi$
$132$	0	0
$133$	−4.05373e9	−1.12337
$134$	4.71654e9	1.26372
$135$	0	0
$136$	1.00785e9	0.252621
$137$	−3.40122e9	−0.824884	−0.412442	−	0.910984i	$-0.635324\pi$
$137$	−3.40122e9	−0.824884	−0.412442	+	0.910984i	$0.635324\pi$
$138$	0	0
$139$	4.61428e9	1.04843	0.524213	−	0.851587i	$-0.324360\pi$
$139$	4.61428e9	1.04843	0.524213	+	0.851587i	$0.324360\pi$
$140$	−3.82413e9	−0.841310
$141$	0	0
$142$	−7.68713e8	−0.158660
$143$	2.73193e9	0.546332
$144$	0	0
$145$	2.02940e9	0.381252
$146$	5.17081e9	0.941826
$147$	0	0
$148$	−6.36434e9	−1.09037
$149$	2.27168e9	0.377580	0.188790	−	0.982017i	$-0.439543\pi$
$149$	2.27168e9	0.377580	0.188790	+	0.982017i	$0.439543\pi$
$150$	0	0
$151$	−9.94969e8	−0.155745	−0.0778723	−	0.996963i	$-0.524813\pi$
$151$	−9.94969e8	−0.155745	−0.0778723	+	0.996963i	$0.524813\pi$
$152$	1.32924e9	0.201980
$153$	0	0
$154$	−2.56054e9	−0.366849
$155$	2.98656e9	0.415603
$156$	0	0
$157$	−1.19315e10	−1.56728	−0.783640	−	0.621216i	$-0.786639\pi$
$157$	−1.19315e10	−1.56728	−0.783640	+	0.621216i	$0.786639\pi$
$158$	−1.54008e9	−0.196601
$159$	0	0
$160$	−1.11939e10	−1.35034
$161$	6.06356e9	0.711232
$162$	0	0
$163$	2.02118e9	0.224264	0.112132	−	0.993693i	$-0.464232\pi$
$163$	2.02118e9	0.224264	0.112132	+	0.993693i	$0.464232\pi$
$164$	9.42719e9	1.01762
$165$	0	0
$166$	8.05126e9	0.822958
$167$	1.26942e10	1.26294	0.631468	−	0.775402i	$-0.282453\pi$
$167$	1.26942e10	1.26294	0.631468	+	0.775402i	$0.282453\pi$
$168$	0	0
$169$	2.19770e10	2.07242
$170$	−2.57035e10	−2.36032
$171$	0	0
$172$	−1.75766e10	−1.53129
$173$	−5.72278e9	−0.485736	−0.242868	−	0.970059i	$-0.578088\pi$
$173$	−5.72278e9	−0.485736	−0.242868	+	0.970059i	$0.578088\pi$
$174$	0	0
$175$	−1.14980e9	−0.0926726
$176$	−3.52762e9	−0.277124
$177$	0	0
$178$	3.62782e10	2.70867
$179$	−1.53037e10	−1.11419	−0.557093	−	0.830450i	$-0.688083\pi$
$179$	−1.53037e10	−1.11419	−0.557093	+	0.830450i	$0.688083\pi$
$180$	0	0
$181$	1.52538e10	1.05639	0.528197	−	0.849122i	$-0.322868\pi$
$181$	1.52538e10	1.05639	0.528197	+	0.849122i	$0.322868\pi$
$182$	−3.05375e10	−2.06306
$183$	0	0
$184$	−1.98828e9	−0.127878
$185$	1.48543e10	0.932351
$186$	0	0
$187$	−9.01783e9	−0.539281
$188$	−1.09082e10	−0.636860
$189$	0	0
$190$	−3.39001e10	−1.88717
$191$	1.92923e10	1.04890	0.524448	−	0.851442i	$-0.324272\pi$
$191$	1.92923e10	1.04890	0.524448	+	0.851442i	$0.324272\pi$
$192$	0	0
$193$	8.81933e9	0.457539	0.228769	−	0.973481i	$-0.426530\pi$
$193$	8.81933e9	0.457539	0.228769	+	0.973481i	$0.426530\pi$
$194$	−9.73993e9	−0.493683
$195$	0	0
$196$	−7.74527e9	−0.374873
$197$	−6.14540e9	−0.290705	−0.145352	−	0.989380i	$-0.546432\pi$
$197$	−6.14540e9	−0.290705	−0.145352	+	0.989380i	$0.546432\pi$
$198$	0	0
$199$	1.32275e10	0.597916	0.298958	−	0.954266i	$-0.403361\pi$
$199$	1.32275e10	0.597916	0.298958	+	0.954266i	$0.403361\pi$
$200$	3.77027e8	0.0166624
$201$	0	0
$202$	3.29458e9	0.139225
$203$	−7.95871e9	−0.328935
$204$	0	0
$205$	−2.20030e10	−0.870140
$206$	2.93368e10	1.13504
$207$	0	0
$208$	−4.20711e10	−1.55847
$209$	−1.18936e10	−0.431175
$210$	0	0
$211$	−2.47717e10	−0.860367	−0.430184	−	0.902741i	$-0.641551\pi$
$211$	−2.47717e10	−0.860367	−0.430184	+	0.902741i	$0.641551\pi$
$212$	2.12813e10	0.723580
$213$	0	0
$214$	5.47705e10	1.78519
$215$	4.10237e10	1.30937
$216$	0	0
$217$	−1.17124e10	−0.358573
$218$	2.60283e10	0.780534
$219$	0	0
$220$	−1.12199e10	−0.322914
$221$	−1.07548e11	−3.03277
$222$	0	0
$223$	1.38481e10	0.374990	0.187495	−	0.982266i	$-0.439963\pi$
$223$	1.38481e10	0.374990	0.187495	+	0.982266i	$0.439963\pi$
$224$	4.38993e10	1.16504
$225$	0	0
$226$	1.91424e10	0.488100
$227$	5.54912e10	1.38710	0.693550	−	0.720409i	$-0.256046\pi$
$227$	5.54912e10	1.38710	0.693550	+	0.720409i	$0.256046\pi$
$228$	0	0
$229$	3.20006e9	0.0768950	0.0384475	−	0.999261i	$-0.487759\pi$
$229$	3.20006e9	0.0768950	0.0384475	+	0.999261i	$0.487759\pi$
$230$	5.07077e10	1.19481
$231$	0	0
$232$	2.60971e9	0.0591420
$233$	1.45987e10	0.324498	0.162249	−	0.986750i	$-0.448125\pi$
$233$	1.45987e10	0.324498	0.162249	+	0.986750i	$0.448125\pi$
$234$	0	0
$235$	2.54597e10	0.544563
$236$	−7.76093e10	−1.62858
$237$	0	0
$238$	1.00801e11	2.03643
$239$	4.57249e10	0.906488	0.453244	−	0.891386i	$-0.350267\pi$
$239$	4.57249e10	0.906488	0.453244	+	0.891386i	$0.350267\pi$
$240$	0	0
$241$	1.86250e9	0.0355647	0.0177824	−	0.999842i	$-0.494339\pi$
$241$	1.86250e9	0.0355647	0.0177824	+	0.999842i	$0.494339\pi$
$242$	6.98187e10	1.30859
$243$	0	0
$244$	1.03950e11	1.87745
$245$	1.80774e10	0.320545
$246$	0	0
$247$	−1.41845e11	−2.42481
$248$	3.84057e9	0.0644708
$249$	0	0
$250$	−9.38718e10	−1.51987
$251$	−1.02269e11	−1.62634	−0.813171	−	0.582025i	$-0.802261\pi$
$251$	−1.02269e11	−1.62634	−0.813171	+	0.582025i	$0.802261\pi$
$252$	0	0
$253$	1.77904e10	0.272987
$254$	−3.28952e10	−0.495885
$255$	0	0
$256$	5.28597e10	0.769209
$257$	−1.26388e11	−1.80721	−0.903603	−	0.428370i	$-0.859088\pi$
$257$	−1.26388e11	−1.80721	−0.903603	+	0.428370i	$0.859088\pi$
$258$	0	0
$259$	−5.82541e10	−0.804411
$260$	−1.33811e11	−1.81598
$261$	0	0
$262$	−1.53085e11	−2.00713
$263$	−9.15762e10	−1.18027	−0.590135	−	0.807304i	$-0.700926\pi$
$263$	−9.15762e10	−1.18027	−0.590135	+	0.807304i	$0.700926\pi$
$264$	0	0
$265$	−4.96703e10	−0.618715
$266$	1.32946e11	1.62820
$267$	0	0
$268$	−8.10505e10	−0.959730
$269$	−3.86326e9	−0.0449851	−0.0224925	−	0.999747i	$-0.507160\pi$
$269$	−3.86326e9	−0.0449851	−0.0224925	+	0.999747i	$0.507160\pi$
$270$	0	0
$271$	1.26432e11	1.42395	0.711976	−	0.702203i	$-0.247800\pi$
$271$	1.26432e11	1.42395	0.711976	+	0.702203i	$0.247800\pi$
$272$	1.38873e11	1.53836
$273$	0	0
$274$	1.11547e11	1.19558
$275$	−3.37349e9	−0.0355699
$276$	0	0
$277$	−1.22275e11	−1.24790	−0.623949	−	0.781465i	$-0.714473\pi$
$277$	−1.22275e11	−1.24790	−0.623949	+	0.781465i	$0.714473\pi$
$278$	−1.51330e11	−1.51958
$279$	0	0
$280$	1.14777e10	0.111595
$281$	−1.78993e10	−0.171261	−0.0856305	−	0.996327i	$-0.527290\pi$
$281$	−1.78993e10	−0.171261	−0.0856305	+	0.996327i	$0.527290\pi$
$282$	0	0
$283$	1.99650e11	1.85025	0.925123	−	0.379667i	$-0.123961\pi$
$283$	1.99650e11	1.85025	0.925123	+	0.379667i	$0.123961\pi$
$284$	1.32098e10	0.120494
$285$	0	0
$286$	−8.95962e10	−0.791849
$287$	8.62891e10	0.750737
$288$	0	0
$289$	2.36420e11	1.99362
$290$	−6.65563e10	−0.552583
$291$	0	0
$292$	−8.88569e10	−0.715267
$293$	5.29403e10	0.419645	0.209822	−	0.977740i	$-0.432711\pi$
$293$	5.29403e10	0.419645	0.209822	+	0.977740i	$0.432711\pi$
$294$	0	0
$295$	1.81139e11	1.39256
$296$	1.91019e10	0.144632
$297$	0	0
$298$	−7.45020e10	−0.547261
$299$	2.12171e11	1.53520
$300$	0	0
$301$	−1.60883e11	−1.12969
$302$	3.26310e10	0.225735
$303$	0	0
$304$	1.83158e11	1.22997
$305$	−2.42618e11	−1.60536
$306$	0	0
$307$	−6.10366e10	−0.392164	−0.196082	−	0.980587i	$-0.562822\pi$
$307$	−6.10366e10	−0.392164	−0.196082	+	0.980587i	$0.562822\pi$
$308$	4.40011e10	0.278603
$309$	0	0
$310$	−9.79472e10	−0.602371
$311$	3.75525e10	0.227623	0.113812	−	0.993502i	$-0.463694\pi$
$311$	3.75525e10	0.227623	0.113812	+	0.993502i	$0.463694\pi$
$312$	0	0
$313$	8.95477e10	0.527357	0.263679	−	0.964611i	$-0.415064\pi$
$313$	8.95477e10	0.527357	0.263679	+	0.964611i	$0.415064\pi$
$314$	3.91305e11	2.27160
$315$	0	0
$316$	2.64652e10	0.149308
$317$	−2.19863e11	−1.22288	−0.611442	−	0.791289i	$-0.709410\pi$
$317$	−2.19863e11	−1.22288	−0.611442	+	0.791289i	$0.709410\pi$
$318$	0	0
$319$	−2.33507e10	−0.126253
$320$	2.10145e11	1.12033
$321$	0	0
$322$	−1.98861e11	−1.03085
$323$	4.68217e11	2.39351
$324$	0	0
$325$	−4.02329e10	−0.200035
$326$	−6.62865e10	−0.325047
$327$	0	0
$328$	−2.82947e10	−0.134981
$329$	−9.98454e10	−0.469836
$330$	0	0
$331$	−1.66891e11	−0.764201	−0.382101	−	0.924121i	$-0.624799\pi$
$331$	−1.66891e11	−0.764201	−0.382101	+	0.924121i	$0.624799\pi$
$332$	−1.38356e11	−0.624993
$333$	0	0
$334$	−4.16319e11	−1.83049
$335$	1.89171e11	0.820641
$336$	0	0
$337$	−2.86023e9	−0.0120800	−0.00604000	−	0.999982i	$-0.501923\pi$
$337$	−2.86023e9	−0.0120800	−0.00604000	+	0.999982i	$0.501923\pi$
$338$	−7.20757e11	−3.00375
$339$	0	0
$340$	4.41697e11	1.79254
$341$	−3.43639e10	−0.137628
$342$	0	0
$343$	−2.79060e11	−1.08861
$344$	5.27544e10	0.203117
$345$	0	0
$346$	1.87684e11	0.704021
$347$	1.56638e11	0.579983	0.289992	−	0.957029i	$-0.406347\pi$
$347$	1.56638e11	0.579983	0.289992	+	0.957029i	$0.406347\pi$
$348$	0	0
$349$	−3.18752e10	−0.115011	−0.0575054	−	0.998345i	$-0.518315\pi$
$349$	−3.18752e10	−0.115011	−0.0575054	+	0.998345i	$0.518315\pi$
$350$	3.77089e10	0.134319
$351$	0	0
$352$	1.28800e11	0.447170
$353$	−1.99632e11	−0.684296	−0.342148	−	0.939646i	$-0.611154\pi$
$353$	−1.99632e11	−0.684296	−0.342148	+	0.939646i	$0.611154\pi$
$354$	0	0
$355$	−3.08316e10	−0.103031
$356$	−6.23416e11	−2.05709
$357$	0	0
$358$	5.01900e11	1.61489
$359$	4.79288e10	0.152290	0.0761451	−	0.997097i	$-0.475739\pi$
$359$	4.79288e10	0.152290	0.0761451	+	0.997097i	$0.475739\pi$
$360$	0	0
$361$	2.94840e11	0.913702
$362$	−5.00265e11	−1.53113
$363$	0	0
$364$	5.24766e11	1.56678
$365$	2.07391e11	0.611607
$366$	0	0
$367$	1.89262e11	0.544587	0.272293	−	0.962214i	$-0.412218\pi$
$367$	1.89262e11	0.544587	0.272293	+	0.962214i	$0.412218\pi$
$368$	−2.73968e11	−0.778725
$369$	0	0
$370$	−4.87161e11	−1.35134
$371$	1.94792e11	0.533813
$372$	0	0
$373$	−9.98250e10	−0.267023	−0.133512	−	0.991047i	$-0.542625\pi$
$373$	−9.98250e10	−0.267023	−0.133512	+	0.991047i	$0.542625\pi$
$374$	2.95749e11	0.781629
$375$	0	0
$376$	3.27399e10	0.0844758
$377$	−2.78485e11	−0.710011
$378$	0	0
$379$	6.79275e10	0.169110	0.0845550	−	0.996419i	$-0.473053\pi$
$379$	6.79275e10	0.169110	0.0845550	+	0.996419i	$0.473053\pi$
$380$	5.82551e11	1.43320
$381$	0	0
$382$	−6.32709e11	−1.52026
$383$	5.15514e11	1.22418	0.612091	−	0.790787i	$-0.290329\pi$
$383$	5.15514e11	1.22418	0.612091	+	0.790787i	$0.290329\pi$
$384$	0	0
$385$	−1.02698e11	−0.238226
$386$	−2.89239e11	−0.663153
$387$	0	0
$388$	1.67374e11	0.374926
$389$	−1.97005e11	−0.436219	−0.218110	−	0.975924i	$-0.569989\pi$
$389$	−1.97005e11	−0.436219	−0.218110	+	0.975924i	$0.569989\pi$
$390$	0	0
$391$	−7.00358e11	−1.51539
$392$	2.32466e10	0.0497247
$393$	0	0
$394$	2.01544e11	0.421345
$395$	−6.17696e10	−0.127670
$396$	0	0
$397$	−5.86747e11	−1.18548	−0.592739	−	0.805395i	$-0.701953\pi$
$397$	−5.86747e11	−1.18548	−0.592739	+	0.805395i	$0.701953\pi$
$398$	−4.33810e11	−0.866615
$399$	0	0
$400$	5.19510e10	0.101467
$401$	5.55697e11	1.07322	0.536609	−	0.843831i	$-0.319705\pi$
$401$	5.55697e11	1.07322	0.536609	+	0.843831i	$0.319705\pi$
$402$	0	0
$403$	−4.09831e11	−0.773984
$404$	−5.66151e10	−0.105734
$405$	0	0
$406$	2.61014e11	0.476756
$407$	−1.70916e11	−0.308751
$408$	0	0
$409$	1.89722e11	0.335245	0.167622	−	0.985851i	$-0.446391\pi$
$409$	1.89722e11	0.335245	0.167622	+	0.985851i	$0.446391\pi$
$410$	7.21609e11	1.26117
$411$	0	0
$412$	−5.04133e11	−0.862001
$413$	−7.10375e11	−1.20147
$414$	0	0
$415$	3.22920e11	0.534416
$416$	1.53609e12	2.51476
$417$	0	0
$418$	3.90061e11	0.624941
$419$	7.24161e11	1.14781	0.573907	−	0.818920i	$-0.305427\pi$
$419$	7.24161e11	1.14781	0.573907	+	0.818920i	$0.305427\pi$
$420$	0	0
$421$	1.09498e12	1.69877	0.849387	−	0.527771i	$-0.176972\pi$
$421$	1.09498e12	1.69877	0.849387	+	0.527771i	$0.176972\pi$
$422$	8.12411e11	1.24701
$423$	0	0
$424$	−6.38736e10	−0.0959787
$425$	1.32805e11	0.197453
$426$	0	0
$427$	9.51475e11	1.38507
$428$	−9.41194e11	−1.35576
$429$	0	0
$430$	−1.34541e12	−1.89779
$431$	5.24483e11	0.732123	0.366061	−	0.930591i	$-0.380706\pi$
$431$	5.24483e11	0.732123	0.366061	+	0.930591i	$0.380706\pi$
$432$	0	0
$433$	−5.33682e11	−0.729604	−0.364802	−	0.931085i	$-0.618863\pi$
$433$	−5.33682e11	−0.729604	−0.364802	+	0.931085i	$0.618863\pi$
$434$	3.84120e11	0.519712
$435$	0	0
$436$	−4.47279e11	−0.592774
$437$	−9.23697e11	−1.21161
$438$	0	0
$439$	−5.42500e11	−0.697123	−0.348562	−	0.937286i	$-0.613330\pi$
$439$	−5.42500e11	−0.697123	−0.348562	+	0.937286i	$0.613330\pi$
$440$	3.36753e10	0.0428326
$441$	0	0
$442$	3.52716e12	4.39567
$443$	−1.01686e12	−1.25442	−0.627209	−	0.778851i	$-0.715803\pi$
$443$	−1.01686e12	−1.25442	−0.627209	+	0.778851i	$0.715803\pi$
$444$	0	0
$445$	1.45505e12	1.75897
$446$	−4.54164e11	−0.543507
$447$	0	0
$448$	−8.24127e11	−0.966592
$449$	7.68289e10	0.0892105	0.0446053	−	0.999005i	$-0.485797\pi$
$449$	7.68289e10	0.0892105	0.0446053	+	0.999005i	$0.485797\pi$
$450$	0	0
$451$	2.53170e11	0.288150
$452$	−3.28950e11	−0.370686
$453$	0	0
$454$	−1.81989e12	−2.01045
$455$	−1.22480e12	−1.33972
$456$	0	0
$457$	−5.54294e11	−0.594453	−0.297226	−	0.954807i	$-0.596062\pi$
$457$	−5.54294e11	−0.594453	−0.297226	+	0.954807i	$0.596062\pi$
$458$	−1.04949e11	−0.111451
$459$	0	0
$460$	−8.71378e11	−0.907395
$461$	−1.09346e12	−1.12759	−0.563794	−	0.825915i	$-0.690659\pi$
$461$	−1.09346e12	−1.12759	−0.563794	+	0.825915i	$0.690659\pi$
$462$	0	0
$463$	1.40031e12	1.41615	0.708077	−	0.706135i	$-0.249563\pi$
$463$	1.40031e12	1.41615	0.708077	+	0.706135i	$0.249563\pi$
$464$	3.59596e11	0.360150
$465$	0	0
$466$	−4.78779e11	−0.470325
$467$	1.59376e12	1.55059	0.775296	−	0.631599i	$-0.217601\pi$
$467$	1.59376e12	1.55059	0.775296	+	0.631599i	$0.217601\pi$
$468$	0	0
$469$	−7.41873e11	−0.708030
$470$	−8.34976e11	−0.789285
$471$	0	0
$472$	2.32936e11	0.216022
$473$	−4.72026e11	−0.433602
$474$	0	0
$475$	1.75156e11	0.157871
$476$	−1.73220e12	−1.54656
$477$	0	0
$478$	−1.49959e12	−1.31386
$479$	8.12123e11	0.704875	0.352438	−	0.935835i	$-0.385353\pi$
$479$	8.12123e11	0.704875	0.352438	+	0.935835i	$0.385353\pi$
$480$	0	0
$481$	−2.03838e12	−1.73633
$482$	−6.10825e10	−0.0515472
$483$	0	0
$484$	−1.19979e12	−0.993802
$485$	−3.90650e11	−0.320590
$486$	0	0
$487$	8.95090e11	0.721085	0.360542	−	0.932743i	$-0.382592\pi$
$487$	8.95090e11	0.721085	0.360542	+	0.932743i	$0.382592\pi$
$488$	−3.11994e11	−0.249033
$489$	0	0
$490$	−5.92866e11	−0.464595
$491$	−1.41149e12	−1.09600	−0.548002	−	0.836477i	$-0.684611\pi$
$491$	−1.41149e12	−1.09600	−0.548002	+	0.836477i	$0.684611\pi$
$492$	0	0
$493$	9.19252e11	0.700848
$494$	4.65194e12	3.51450
$495$	0	0
$496$	5.29197e11	0.392600
$497$	1.20912e11	0.0888928
$498$	0	0
$499$	−1.82428e11	−0.131716	−0.0658580	−	0.997829i	$-0.520978\pi$
$499$	−1.82428e11	−0.131716	−0.0658580	+	0.997829i	$0.520978\pi$
$500$	1.61312e12	1.15426
$501$	0	0
$502$	3.35401e12	2.35721
$503$	−2.33627e12	−1.62730	−0.813649	−	0.581357i	$-0.802522\pi$
$503$	−2.33627e12	−1.62730	−0.813649	+	0.581357i	$0.802522\pi$
$504$	0	0
$505$	1.32139e11	0.0904108
$506$	−5.83452e11	−0.395665
$507$	0	0
$508$	5.65282e11	0.376599
$509$	−3.15862e11	−0.208577	−0.104289	−	0.994547i	$-0.533257\pi$
$509$	−3.15862e11	−0.208577	−0.104289	+	0.994547i	$0.533257\pi$
$510$	0	0
$511$	−8.13326e11	−0.527680
$512$	−2.18534e12	−1.40542
$513$	0	0
$514$	4.14503e12	2.61935
$515$	1.17664e12	0.737075
$516$	0	0
$517$	−2.92944e11	−0.180334
$518$	1.91050e12	1.16591
$519$	0	0
$520$	4.01619e11	0.240879
$521$	−7.03557e11	−0.418340	−0.209170	−	0.977879i	$-0.567076\pi$
$521$	−7.03557e11	−0.418340	−0.209170	+	0.977879i	$0.567076\pi$
$522$	0	0
$523$	−3.23536e12	−1.89088	−0.945441	−	0.325793i	$-0.894369\pi$
$523$	−3.23536e12	−1.89088	−0.945441	+	0.325793i	$0.894369\pi$
$524$	2.63066e12	1.52431
$525$	0	0
$526$	3.00333e12	1.71067
$527$	1.35281e12	0.763994
$528$	0	0
$529$	−4.19489e11	−0.232901
$530$	1.62899e12	0.896760
$531$	0	0
$532$	−2.28459e12	−1.23653
$533$	3.01936e12	1.62047
$534$	0	0
$535$	2.19674e12	1.15927
$536$	2.43265e11	0.127303
$537$	0	0
$538$	1.26699e11	0.0652010
$539$	−2.08002e11	−0.106149
$540$	0	0
$541$	−1.71900e12	−0.862759	−0.431379	−	0.902171i	$-0.641973\pi$
$541$	−1.71900e12	−0.862759	−0.431379	+	0.902171i	$0.641973\pi$
$542$	−4.14647e12	−2.06387
$543$	0	0
$544$	−5.07049e12	−2.48230
$545$	1.04394e12	0.506866
$546$	0	0
$547$	1.38622e12	0.662047	0.331024	−	0.943622i	$-0.392606\pi$
$547$	1.38622e12	0.662047	0.331024	+	0.943622i	$0.392606\pi$
$548$	−1.91685e12	−0.907979
$549$	0	0
$550$	1.10637e11	0.0515547
$551$	1.21240e12	0.560354
$552$	0	0
$553$	2.42242e11	0.110150
$554$	4.01013e12	1.80869
$555$	0	0
$556$	2.60050e12	1.15404
$557$	1.75428e12	0.772236	0.386118	−	0.922449i	$-0.373816\pi$
$557$	1.75428e12	0.772236	0.386118	+	0.922449i	$0.373816\pi$
$558$	0	0
$559$	−5.62948e12	−2.43846
$560$	1.58153e12	0.679565
$561$	0	0
$562$	5.87026e11	0.248224
$563$	−3.55391e12	−1.49080	−0.745399	−	0.666618i	$-0.767741\pi$
$563$	−3.55391e12	−1.49080	−0.745399	+	0.666618i	$0.767741\pi$
$564$	0	0
$565$	7.67765e11	0.316964
$566$	−6.54771e12	−2.68173
$567$	0	0
$568$	−3.96479e10	−0.0159828
$569$	−1.14959e12	−0.459768	−0.229884	−	0.973218i	$-0.573835\pi$
$569$	−1.14959e12	−0.459768	−0.229884	+	0.973218i	$0.573835\pi$
$570$	0	0
$571$	−1.67560e12	−0.659641	−0.329821	−	0.944044i	$-0.606988\pi$
$571$	−1.67560e12	−0.659641	−0.329821	+	0.944044i	$0.606988\pi$
$572$	1.53965e12	0.601367
$573$	0	0
$574$	−2.82994e12	−1.08811
$575$	−2.61997e11	−0.0999520
$576$	0	0
$577$	2.71422e12	1.01942	0.509712	−	0.860345i	$-0.329752\pi$
$577$	2.71422e12	1.01942	0.509712	+	0.860345i	$0.329752\pi$
$578$	−7.75362e12	−2.88954
$579$	0	0
$580$	1.14372e12	0.419658
$581$	−1.26640e12	−0.461081
$582$	0	0
$583$	5.71516e11	0.204890
$584$	2.66695e11	0.0948760
$585$	0	0
$586$	−1.73623e12	−0.608230
$587$	5.77644e11	0.200812	0.100406	−	0.994947i	$-0.467986\pi$
$587$	5.77644e11	0.200812	0.100406	+	0.994947i	$0.467986\pi$
$588$	0	0
$589$	1.78422e12	0.610842
$590$	−5.94064e12	−2.01836
$591$	0	0
$592$	2.63207e12	0.880746
$593$	3.92531e12	1.30355	0.651776	−	0.758411i	$-0.274024\pi$
$593$	3.92531e12	1.30355	0.651776	+	0.758411i	$0.274024\pi$
$594$	0	0
$595$	4.04295e12	1.32243
$596$	1.28027e12	0.415616
$597$	0	0
$598$	−6.95837e12	−2.22511
$599$	3.75910e12	1.19306	0.596532	−	0.802590i	$-0.296545\pi$
$599$	3.75910e12	1.19306	0.596532	+	0.802590i	$0.296545\pi$
$600$	0	0
$601$	1.43232e12	0.447821	0.223910	−	0.974610i	$-0.428118\pi$
$601$	1.43232e12	0.447821	0.223910	+	0.974610i	$0.428118\pi$
$602$	5.27631e12	1.63737
$603$	0	0
$604$	−5.60741e11	−0.171434
$605$	2.80029e12	0.849775
$606$	0	0
$607$	3.68221e12	1.10093	0.550465	−	0.834858i	$-0.314450\pi$
$607$	3.68221e12	1.10093	0.550465	+	0.834858i	$0.314450\pi$
$608$	−6.68743e12	−1.98469
$609$	0	0
$610$	7.95689e12	2.32680
$611$	−3.49371e12	−1.01415
$612$	0	0
$613$	3.14562e12	0.899777	0.449888	−	0.893085i	$-0.351464\pi$
$613$	3.14562e12	0.899777	0.449888	+	0.893085i	$0.351464\pi$
$614$	2.00176e12	0.568399
$615$	0	0
$616$	−1.32065e11	−0.0369550
$617$	−1.84112e12	−0.511444	−0.255722	−	0.966750i	$-0.582313\pi$
$617$	−1.84112e12	−0.511444	−0.255722	+	0.966750i	$0.582313\pi$
$618$	0	0
$619$	−2.16872e11	−0.0593739	−0.0296870	−	0.999559i	$-0.509451\pi$
$619$	−2.16872e11	−0.0593739	−0.0296870	+	0.999559i	$0.509451\pi$
$620$	1.68316e12	0.457469
$621$	0	0
$622$	−1.23157e12	−0.329915
$623$	−5.70626e12	−1.51759
$624$	0	0
$625$	−3.32968e12	−0.872855
$626$	−2.93681e12	−0.764347
$627$	0	0
$628$	−6.72432e12	−1.72516
$629$	6.72851e12	1.71392
$630$	0	0
$631$	3.08505e11	0.0774693	0.0387346	−	0.999250i	$-0.487667\pi$
$631$	3.08505e11	0.0774693	0.0387346	+	0.999250i	$0.487667\pi$
$632$	−7.94326e10	−0.0198049
$633$	0	0
$634$	7.21062e12	1.77244
$635$	−1.31936e12	−0.322020
$636$	0	0
$637$	−2.48067e12	−0.596955
$638$	7.65808e11	0.182990
$639$	0	0
$640$	−1.16062e12	−0.273451
$641$	6.75425e12	1.58022	0.790108	−	0.612968i	$-0.210025\pi$
$641$	6.75425e12	1.58022	0.790108	+	0.612968i	$0.210025\pi$
$642$	0	0
$643$	9.14571e11	0.210993	0.105496	−	0.994420i	$-0.466357\pi$
$643$	9.14571e11	0.210993	0.105496	+	0.994420i	$0.466357\pi$
$644$	3.41728e12	0.782879
$645$	0	0
$646$	−1.53556e13	−3.46914
$647$	1.29528e12	0.290600	0.145300	−	0.989388i	$-0.453585\pi$
$647$	1.29528e12	0.290600	0.145300	+	0.989388i	$0.453585\pi$
$648$	0	0
$649$	−2.08422e12	−0.461151
$650$	1.31948e12	0.289929
$651$	0	0
$652$	1.13909e12	0.246856
$653$	5.22047e12	1.12357	0.561785	−	0.827283i	$-0.310115\pi$
$653$	5.22047e12	1.12357	0.561785	+	0.827283i	$0.310115\pi$
$654$	0	0
$655$	−6.13993e12	−1.30340
$656$	−3.89877e12	−0.821978
$657$	0	0
$658$	3.27453e12	0.680977
$659$	−4.86991e12	−1.00586	−0.502928	−	0.864328i	$-0.667744\pi$
$659$	−4.86991e12	−1.00586	−0.502928	+	0.864328i	$0.667744\pi$
$660$	0	0
$661$	1.73969e12	0.354458	0.177229	−	0.984170i	$-0.443287\pi$
$661$	1.73969e12	0.354458	0.177229	+	0.984170i	$0.443287\pi$
$662$	5.47337e12	1.10763
$663$	0	0
$664$	4.15260e11	0.0829017
$665$	5.33221e12	1.05733
$666$	0	0
$667$	−1.81350e12	−0.354773
$668$	7.15417e12	1.39016
$669$	0	0
$670$	−6.20405e12	−1.18943
$671$	2.79160e12	0.531622
$672$	0	0
$673$	5.00841e12	0.941092	0.470546	−	0.882375i	$-0.344057\pi$
$673$	5.00841e12	0.941092	0.470546	+	0.882375i	$0.344057\pi$
$674$	9.38042e10	0.0175087
$675$	0	0
$676$	1.23857e13	2.28119
$677$	−7.44082e12	−1.36136	−0.680678	−	0.732582i	$-0.738315\pi$
$677$	−7.44082e12	−1.36136	−0.680678	+	0.732582i	$0.738315\pi$
$678$	0	0
$679$	1.53201e12	0.276597
$680$	−1.32571e12	−0.237770
$681$	0	0
$682$	1.12700e12	0.199477
$683$	−9.49272e12	−1.66916	−0.834579	−	0.550888i	$-0.814289\pi$
$683$	−9.49272e12	−1.66916	−0.834579	+	0.550888i	$0.814289\pi$
$684$	0	0
$685$	4.47391e12	0.776390
$686$	9.15205e12	1.57783
$687$	0	0
$688$	7.26911e12	1.23690
$689$	6.81601e12	1.15224
$690$	0	0
$691$	1.30318e12	0.217447	0.108724	−	0.994072i	$-0.465324\pi$
$691$	1.30318e12	0.217447	0.108724	+	0.994072i	$0.465324\pi$
$692$	−3.22523e12	−0.534667
$693$	0	0
$694$	−5.13711e12	−0.840623
$695$	−6.06955e12	−0.986790
$696$	0	0
$697$	−9.96662e12	−1.59956
$698$	1.04538e12	0.166696
$699$	0	0
$700$	−6.48001e11	−0.102008
$701$	−1.24667e13	−1.94993	−0.974966	−	0.222354i	$-0.928626\pi$
$701$	−1.24667e13	−1.94993	−0.974966	+	0.222354i	$0.928626\pi$
$702$	0	0
$703$	8.87418e12	1.37034
$704$	−2.41797e12	−0.371000
$705$	0	0
$706$	6.54713e12	0.991813
$707$	−5.18210e11	−0.0780043
$708$	0	0
$709$	2.99079e12	0.444506	0.222253	−	0.974989i	$-0.428659\pi$
$709$	2.99079e12	0.444506	0.222253	+	0.974989i	$0.428659\pi$
$710$	1.01115e12	0.149332
$711$	0	0
$712$	1.87112e12	0.272861
$713$	−2.66883e12	−0.386738
$714$	0	0
$715$	−3.59353e12	−0.514214
$716$	−8.62481e12	−1.22642
$717$	0	0
$718$	−1.57187e12	−0.220728
$719$	−4.07204e12	−0.568240	−0.284120	−	0.958789i	$-0.591701\pi$
$719$	−4.07204e12	−0.568240	−0.284120	+	0.958789i	$0.591701\pi$
$720$	0	0
$721$	−4.61444e12	−0.635931
$722$	−9.66959e12	−1.32431
$723$	0	0
$724$	8.59672e12	1.16281
$725$	3.43884e11	0.0462265
$726$	0	0
$727$	−1.03208e13	−1.37028	−0.685141	−	0.728410i	$-0.740260\pi$
$727$	−1.03208e13	−1.37028	−0.685141	+	0.728410i	$0.740260\pi$
$728$	−1.57503e12	−0.207825
$729$	0	0
$730$	−6.80159e12	−0.886457
$731$	1.85824e13	2.40698
$732$	0	0
$733$	−1.09370e13	−1.39937	−0.699684	−	0.714452i	$-0.746676\pi$
$733$	−1.09370e13	−1.39937	−0.699684	+	0.714452i	$0.746676\pi$
$734$	−6.20705e12	−0.789320
$735$	0	0
$736$	1.00030e13	1.25656
$737$	−2.17664e12	−0.271758
$738$	0	0
$739$	5.74543e12	0.708635	0.354317	−	0.935125i	$-0.384713\pi$
$739$	5.74543e12	0.708635	0.354317	+	0.935125i	$0.384713\pi$
$740$	8.37154e12	1.02627
$741$	0	0
$742$	−6.38841e12	−0.773704
$743$	−2.53423e12	−0.305067	−0.152534	−	0.988298i	$-0.548743\pi$
$743$	−2.53423e12	−0.305067	−0.152534	+	0.988298i	$0.548743\pi$
$744$	0	0
$745$	−2.98813e12	−0.355383
$746$	3.27386e12	0.387022
$747$	0	0
$748$	−5.08224e12	−0.593606
$749$	−8.61495e12	−1.00019
$750$	0	0
$751$	2.73854e12	0.314151	0.157076	−	0.987587i	$-0.449793\pi$
$751$	2.73854e12	0.314151	0.157076	+	0.987587i	$0.449793\pi$
$752$	4.51128e12	0.514422
$753$	0	0
$754$	9.13318e12	1.02908
$755$	1.30876e12	0.146589
$756$	0	0
$757$	−1.27669e13	−1.41304	−0.706519	−	0.707694i	$-0.749736\pi$
$757$	−1.27669e13	−1.41304	−0.706519	+	0.707694i	$0.749736\pi$
$758$	−2.22775e12	−0.245107
$759$	0	0
$760$	−1.74846e12	−0.190106
$761$	3.97240e12	0.429361	0.214680	−	0.976684i	$-0.431129\pi$
$761$	3.97240e12	0.429361	0.214680	+	0.976684i	$0.431129\pi$
$762$	0	0
$763$	−4.09404e12	−0.437312
$764$	1.08727e13	1.15456
$765$	0	0
$766$	−1.69068e13	−1.77432
$767$	−2.48568e13	−2.59338
$768$	0	0
$769$	−5.85641e11	−0.0603897	−0.0301948	−	0.999544i	$-0.509613\pi$
$769$	−5.85641e11	−0.0603897	−0.0301948	+	0.999544i	$0.509613\pi$
$770$	3.36809e12	0.345283
$771$	0	0
$772$	4.97037e12	0.503629
$773$	4.19065e12	0.422157	0.211079	−	0.977469i	$-0.432302\pi$
$773$	4.19065e12	0.422157	0.211079	+	0.977469i	$0.432302\pi$
$774$	0	0
$775$	5.06075e11	0.0503915
$776$	−5.02356e11	−0.0497317
$777$	0	0
$778$	6.46099e12	0.632253
$779$	−1.31449e13	−1.27891
$780$	0	0
$781$	3.54754e11	0.0341191
$782$	2.29689e13	2.19639
$783$	0	0
$784$	3.20318e12	0.302803
$785$	1.56945e13	1.47514
$786$	0	0
$787$	8.90686e12	0.827634	0.413817	−	0.910360i	$-0.364195\pi$
$787$	8.90686e12	0.827634	0.413817	+	0.910360i	$0.364195\pi$
$788$	−3.46340e12	−0.319989
$789$	0	0
$790$	2.02579e12	0.185043
$791$	−3.01095e12	−0.273469
$792$	0	0
$793$	3.32932e13	2.98969
$794$	1.92429e13	1.71822
$795$	0	0
$796$	7.45474e12	0.658148
$797$	−1.53960e12	−0.135159	−0.0675797	−	0.997714i	$-0.521528\pi$
$797$	−1.53960e12	−0.135159	−0.0675797	+	0.997714i	$0.521528\pi$
$798$	0	0
$799$	1.15324e13	1.00106
$800$	−1.89682e12	−0.163728
$801$	0	0
$802$	−1.82246e13	−1.55551
$803$	−2.38628e12	−0.202536
$804$	0	0
$805$	−7.97591e12	−0.669420
$806$	1.34408e13	1.12181
$807$	0	0
$808$	1.69924e11	0.0140250
$809$	2.05894e13	1.68996	0.844979	−	0.534799i	$-0.179613\pi$
$809$	2.05894e13	1.68996	0.844979	+	0.534799i	$0.179613\pi$
$810$	0	0
$811$	−7.35060e12	−0.596663	−0.298331	−	0.954462i	$-0.596430\pi$
$811$	−7.35060e12	−0.596663	−0.298331	+	0.954462i	$0.596430\pi$
$812$	−4.48535e12	−0.362071
$813$	0	0
$814$	5.60537e12	0.447501
$815$	−2.65862e12	−0.211080
$816$	0	0
$817$	2.45082e13	1.92447
$818$	−6.22210e12	−0.485901
$819$	0	0
$820$	−1.24004e13	−0.957794
$821$	7.03985e12	0.540778	0.270389	−	0.962751i	$-0.412848\pi$
$821$	7.03985e12	0.540778	0.270389	+	0.962751i	$0.412848\pi$
$822$	0	0
$823$	3.39772e11	0.0258159	0.0129080	−	0.999917i	$-0.495891\pi$
$823$	3.39772e11	0.0258159	0.0129080	+	0.999917i	$0.495891\pi$
$824$	1.51310e12	0.114339
$825$	0	0
$826$	2.32974e13	1.74140
$827$	1.68131e13	1.24989	0.624947	−	0.780668i	$-0.285121\pi$
$827$	1.68131e13	1.24989	0.624947	+	0.780668i	$0.285121\pi$
$828$	0	0
$829$	6.50421e12	0.478299	0.239149	−	0.970983i	$-0.423131\pi$
$829$	6.50421e12	0.478299	0.239149	+	0.970983i	$0.423131\pi$
$830$	−1.05905e13	−0.774577
$831$	0	0
$832$	−2.88372e13	−2.08640
$833$	8.18846e12	0.589250
$834$	0	0
$835$	−1.66978e13	−1.18869
$836$	−6.70294e12	−0.474610
$837$	0	0
$838$	−2.37496e13	−1.66363
$839$	2.10850e12	0.146908	0.0734538	−	0.997299i	$-0.476598\pi$
$839$	2.10850e12	0.146908	0.0734538	+	0.997299i	$0.476598\pi$
$840$	0	0
$841$	−1.21268e13	−0.835922
$842$	−3.59108e13	−2.46219
$843$	0	0
$844$	−1.39607e13	−0.947037
$845$	−2.89081e13	−1.95059
$846$	0	0
$847$	−1.09819e13	−0.733166
$848$	−8.80123e12	−0.584470
$849$	0	0
$850$	−4.35547e12	−0.286187
$851$	−1.32740e13	−0.867597
$852$	0	0
$853$	1.74244e13	1.12691	0.563453	−	0.826148i	$-0.309473\pi$
$853$	1.74244e13	1.12691	0.563453	+	0.826148i	$0.309473\pi$
$854$	−3.12046e13	−2.00751
$855$	0	0
$856$	2.82489e12	0.179833
$857$	−1.54986e12	−0.0981475	−0.0490737	−	0.998795i	$-0.515627\pi$
$857$	−1.54986e12	−0.0981475	−0.0490737	+	0.998795i	$0.515627\pi$
$858$	0	0
$859$	1.94553e13	1.21918	0.609592	−	0.792715i	$-0.291333\pi$
$859$	1.94553e13	1.21918	0.609592	+	0.792715i	$0.291333\pi$
$860$	2.31200e13	1.44127
$861$	0	0
$862$	−1.72009e13	−1.06113
$863$	1.23365e13	0.757080	0.378540	−	0.925585i	$-0.376426\pi$
$863$	1.23365e13	0.757080	0.378540	+	0.925585i	$0.376426\pi$
$864$	0	0
$865$	7.52765e12	0.457180
$866$	1.75026e13	1.05748
$867$	0	0
$868$	−6.60084e12	−0.394694
$869$	7.10732e11	0.0422782
$870$	0	0
$871$	−2.59590e13	−1.52829
$872$	1.34246e12	0.0786280
$873$	0	0
$874$	3.02936e13	1.75610
$875$	1.47653e13	0.851540
$876$	0	0
$877$	−2.37694e13	−1.35681	−0.678406	−	0.734687i	$-0.737329\pi$
$877$	−2.37694e13	−1.35681	−0.678406	+	0.734687i	$0.737329\pi$
$878$	1.77918e13	1.01040
$879$	0	0
$880$	4.64017e12	0.260833
$881$	−2.37158e13	−1.32631	−0.663156	−	0.748481i	$-0.730783\pi$
$881$	−2.37158e13	−1.32631	−0.663156	+	0.748481i	$0.730783\pi$
$882$	0	0
$883$	1.17562e13	0.650795	0.325397	−	0.945577i	$-0.394502\pi$
$883$	1.17562e13	0.650795	0.325397	+	0.945577i	$0.394502\pi$
$884$	−6.06118e13	−3.33828
$885$	0	0
$886$	3.33488e13	1.81814
$887$	3.11342e13	1.68881	0.844406	−	0.535704i	$-0.179954\pi$
$887$	3.11342e13	1.68881	0.844406	+	0.535704i	$0.179954\pi$
$888$	0	0
$889$	5.17415e12	0.277831
$890$	−4.77197e13	−2.54943
$891$	0	0
$892$	7.80449e12	0.412765
$893$	1.52100e13	0.800383
$894$	0	0
$895$	2.01302e13	1.04868
$896$	4.55160e12	0.235927
$897$	0	0
$898$	−2.51968e12	−0.129301
$899$	3.50296e12	0.178861
$900$	0	0
$901$	−2.24990e13	−1.13737
$902$	−8.30297e12	−0.417642
$903$	0	0
$904$	9.87307e11	0.0491693
$905$	−2.00647e13	−0.994290
$906$	0	0
$907$	−3.04416e13	−1.49360	−0.746802	−	0.665047i	$-0.768412\pi$
$907$	−3.04416e13	−1.49360	−0.746802	+	0.665047i	$0.768412\pi$
$908$	3.12736e13	1.52683
$909$	0	0
$910$	4.01685e13	1.94178
$911$	1.74487e13	0.839327	0.419664	−	0.907680i	$-0.362148\pi$
$911$	1.74487e13	0.839327	0.419664	+	0.907680i	$0.362148\pi$
$912$	0	0
$913$	−3.71558e12	−0.176973
$914$	1.81786e13	0.861595
$915$	0	0
$916$	1.80348e12	0.0846411
$917$	2.40790e13	1.12454
$918$	0	0
$919$	9.40533e12	0.434965	0.217483	−	0.976064i	$-0.430215\pi$
$919$	9.40533e12	0.434965	0.217483	+	0.976064i	$0.430215\pi$
$920$	2.61535e12	0.120361
$921$	0	0
$922$	3.58613e13	1.63432
$923$	4.23086e12	0.191876
$924$	0	0
$925$	2.51707e12	0.113047
$926$	−4.59247e13	−2.05256
$927$	0	0
$928$	−1.31295e13	−0.581140
$929$	−8.43027e12	−0.371339	−0.185670	−	0.982612i	$-0.559445\pi$
$929$	−8.43027e12	−0.371339	−0.185670	+	0.982612i	$0.559445\pi$
$930$	0	0
$931$	1.07997e13	0.471127
$932$	8.22749e12	0.357187
$933$	0	0
$934$	−5.22690e13	−2.24741
$935$	1.18619e13	0.507577
$936$	0	0
$937$	2.45230e10	0.00103931	0.000519656	−	1.00000i	$-0.499835\pi$
$937$	2.45230e10	0.00103931	0.000519656		1.00000i	$0.499835\pi$
$938$	2.43304e13	1.02621
$939$	0	0
$940$	1.43485e13	0.599420
$941$	−3.52870e13	−1.46711	−0.733553	−	0.679632i	$-0.762139\pi$
$941$	−3.52870e13	−1.46711	−0.733553	+	0.679632i	$0.762139\pi$
$942$	0	0
$943$	1.96621e13	0.809707
$944$	3.20966e13	1.31548
$945$	0	0
$946$	1.54806e13	0.628459
$947$	1.73678e13	0.701730	0.350865	−	0.936426i	$-0.385888\pi$
$947$	1.73678e13	0.701730	0.350865	+	0.936426i	$0.385888\pi$
$948$	0	0
$949$	−2.84592e13	−1.13900
$950$	−5.74440e12	−0.228817
$951$	0	0
$952$	5.19903e12	0.205143
$953$	1.07207e13	0.421024	0.210512	−	0.977591i	$-0.432487\pi$
$953$	1.07207e13	0.421024	0.210512	+	0.977591i	$0.432487\pi$
$954$	0	0
$955$	−2.53767e13	−0.987234
$956$	2.57695e13	0.997805
$957$	0	0
$958$	−2.66344e13	−1.02164
$959$	−1.75454e13	−0.669851
$960$	0	0
$961$	−2.12845e13	−0.805023
$962$	6.68507e13	2.51662
$963$	0	0
$964$	1.04966e12	0.0391474
$965$	−1.16008e13	−0.430641
$966$	0	0
$967$	−4.40236e13	−1.61907	−0.809537	−	0.587068i	$-0.800282\pi$
$967$	−4.40236e13	−1.61907	−0.809537	+	0.587068i	$0.800282\pi$
$968$	3.60103e12	0.131822
$969$	0	0
$970$	1.28117e13	0.464660
$971$	3.18653e13	1.15035	0.575177	−	0.818029i	$-0.304933\pi$
$971$	3.18653e13	1.15035	0.575177	+	0.818029i	$0.304933\pi$
$972$	0	0
$973$	2.38030e13	0.851380
$974$	−2.93554e13	−1.04513
$975$	0	0
$976$	−4.29901e13	−1.51651
$977$	3.67219e13	1.28944	0.644718	−	0.764420i	$-0.276975\pi$
$977$	3.67219e13	1.28944	0.644718	+	0.764420i	$0.276975\pi$
$978$	0	0
$979$	−1.67420e13	−0.582487
$980$	1.01880e13	0.352835
$981$	0	0
$982$	4.62913e13	1.58854
$983$	1.38475e13	0.473020	0.236510	−	0.971629i	$-0.423996\pi$
$983$	1.38475e13	0.473020	0.236510	+	0.971629i	$0.423996\pi$
$984$	0	0
$985$	8.08355e12	0.273614
$986$	−3.01478e13	−1.01580
$987$	0	0
$988$	−7.99405e13	−2.66907
$989$	−3.66593e13	−1.21843
$990$	0	0
$991$	4.28865e12	0.141250	0.0706252	−	0.997503i	$-0.477501\pi$
$991$	4.28865e12	0.141250	0.0706252	+	0.997503i	$0.477501\pi$
$992$	−1.93219e13	−0.633501
$993$	0	0
$994$	−3.96544e12	−0.128841
$995$	−1.73993e13	−0.562766
$996$	0	0
$997$	5.09035e13	1.63162	0.815811	−	0.578319i	$-0.196291\pi$
$997$	5.09035e13	1.63162	0.815811	+	0.578319i	$0.196291\pi$
$998$	5.98290e12	0.190908
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	27.10.a.c.1.1	yes	3
3.2	odd	2		27.10.a.b.1.3	✓	3
9.2	odd	6		81.10.c.h.28.1		6
9.4	even	3		81.10.c.g.55.3		6
9.5	odd	6		81.10.c.h.55.1		6
9.7	even	3		81.10.c.g.28.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.10.a.b.1.3	✓	3	3.2	odd	2
27.10.a.c.1.1	yes	3	1.1	even	1	trivial
81.10.c.g.28.3		6	9.7	even	3
81.10.c.g.55.3		6	9.4	even	3
81.10.c.h.28.1		6	9.2	odd	6
81.10.c.h.55.1		6	9.5	odd	6

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:	\( N \)	\(=\)	\( 27 = 3^{3} \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	27.a (trivial)

\(f(q)\)	\(=\)	\(q-32.7960 q^{2} +563.577 q^{4} -1315.38 q^{5} +5158.54 q^{7} -1691.52 q^{8} +43139.3 q^{10} +15135.0 q^{11} +180503. q^{13} -169179. q^{14} -233076. q^{16} -595825. q^{17} -785830. q^{19} -741319. q^{20} -496368. q^{22} +1.17544e6 q^{23} -222893. q^{25} -5.91979e6 q^{26} +2.90723e6 q^{28} -1.54282e6 q^{29} -2.27049e6 q^{31} +8.51003e6 q^{32} +1.95407e7 q^{34} -6.78546e6 q^{35} -1.12928e7 q^{37} +2.57721e7 q^{38} +2.22499e6 q^{40} +1.67274e7 q^{41} -3.11877e7 q^{43} +8.52976e6 q^{44} -3.85498e7 q^{46} -1.93554e7 q^{47} -1.37431e7 q^{49} +7.30999e6 q^{50} +1.01728e8 q^{52} +3.77611e7 q^{53} -1.99084e7 q^{55} -8.72576e6 q^{56} +5.05984e7 q^{58} -1.37708e8 q^{59} +1.84447e8 q^{61} +7.44629e7 q^{62} -1.59760e8 q^{64} -2.37431e8 q^{65} -1.43814e8 q^{67} -3.35793e8 q^{68} +2.22536e8 q^{70} +2.34392e7 q^{71} -1.57666e8 q^{73} +3.70357e8 q^{74} -4.42875e8 q^{76} +7.80747e7 q^{77} +4.69594e7 q^{79} +3.06585e8 q^{80} -5.48593e8 q^{82} -2.45495e8 q^{83} +7.83738e8 q^{85} +1.02283e9 q^{86} -2.56012e7 q^{88} -1.10618e9 q^{89} +9.31134e8 q^{91} +6.62452e8 q^{92} +6.34778e8 q^{94} +1.03367e9 q^{95} +2.96985e8 q^{97} +4.50717e8 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} + 597 q^{4} - 1983 q^{5} - 3693 q^{7} - 4503 q^{8} - 18981 q^{10} - 16863 q^{11} + 116916 q^{13} - 503463 q^{14} - 239919 q^{16} - 1014048 q^{17} - 15222 q^{19} - 2548407 q^{20} + 305721 q^{22}+ \cdots + 2592182286 q^{98}+O(q^{100}) \) 3 * q + 3 * q^2 + 597 * q^4 - 1983 * q^5 - 3693 * q^7 - 4503 * q^8 - 18981 * q^10 - 16863 * q^11 + 116916 * q^13 - 503463 * q^14 - 239919 * q^16 - 1014048 * q^17 - 15222 * q^19 - 2548407 * q^20 + 305721 * q^22 - 2927118 * q^23 + 2133732 * q^25 - 4765116 * q^26 - 3535725 * q^28 - 5768790 * q^29 - 6575223 * q^31 + 2687697 * q^32 + 17098128 * q^34 + 17340537 * q^35 - 11686026 * q^37 + 50473374 * q^38 - 4130811 * q^40 + 22213518 * q^41 + 45384414 * q^43 + 57206991 * q^44 - 95638590 * q^46 - 12392034 * q^47 + 18933462 * q^49 + 82984044 * q^50 + 182736492 * q^52 - 80579637 * q^53 - 174735333 * q^55 - 21850521 * q^56 - 102758922 * q^58 - 244026660 * q^59 + 369729960 * q^61 - 166297341 * q^62 - 420692127 * q^64 - 492225684 * q^65 - 252614586 * q^67 - 162270144 * q^68 + 935092377 * q^70 - 403193088 * q^71 - 406626717 * q^73 + 641373558 * q^74 - 39499590 * q^76 - 360443199 * q^77 + 265451856 * q^79 + 1168359837 * q^80 - 884408166 * q^82 - 121625871 * q^83 + 316251216 * q^85 + 2368956570 * q^86 + 219257739 * q^88 - 377904006 * q^89 + 245059140 * q^91 + 1178754558 * q^92 + 86583438 * q^94 - 536878770 * q^95 - 438907539 * q^97 + 2592182286 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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