LMFDB - Embedded newform 72.22.a.f.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [72,22,Mod(1,72)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(72, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 22, names="a")

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

chi := DirichletCharacter("72.1");

S:= CuspForms(chi, 22);

N := Newforms(S);

Level:	$ N $	$=$	$ 72 = 2^{3} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 22 $
Character orbit:	$[\chi]$	$=$	72.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:	$201.223687887$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	$\mathbb{Q}[x]/(x^{3} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4963x + 96223 $ x^3 - x^2 - 4963*x + 96223
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{21}\cdot 3^{5}\cdot 5\cdot 7 $
Twist minimal:	no (minimal twist has level 8)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-78.2002$ of defining polynomial
Character	$\chi$	$=$	72.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.39292e7 q^{5} -5.28731e8 q^{7} +O(q^{10})$	$q+3.39292e7 q^{5} -5.28731e8 q^{7} +1.21955e11 q^{11} +4.33605e11 q^{13} +1.31741e13 q^{17} +2.18808e13 q^{19} -1.40779e14 q^{23} +6.74355e14 q^{25} -1.17115e15 q^{29} +9.57156e14 q^{31} -1.79394e16 q^{35} -3.53720e16 q^{37} +1.71456e17 q^{41} +1.35679e17 q^{43} +5.75514e17 q^{47} -2.78989e17 q^{49} +9.62123e17 q^{53} +4.13782e18 q^{55} -4.87671e18 q^{59} +4.59195e18 q^{61} +1.47119e19 q^{65} -1.78611e19 q^{67} -2.93629e19 q^{71} -8.32585e18 q^{73} -6.44812e19 q^{77} -7.05718e19 q^{79} -2.11929e20 q^{83} +4.46988e20 q^{85} -2.62972e20 q^{89} -2.29260e20 q^{91} +7.42397e20 q^{95} +3.84293e20 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 24111774 q^{5} + 295988280 q^{7}+O(q^{10}) $ 3 * q + 24111774 * q^5 + 295988280 * q^7	$ 3 q + 24111774 q^{5} + 295988280 q^{7} + 40335108684 q^{11} + 133734425946 q^{13} - 7797732274422 q^{17} + 35788199781996 q^{19} - 193770761479080 q^{23} + 11\!\cdots\!01 q^{25}+ \cdots - 32\!\cdots\!42 q^{97}+O(q^{100}) $ 3 * q + 24111774 * q^5 + 295988280 * q^7 + 40335108684 * q^11 + 133734425946 * q^13 - 7797732274422 * q^17 + 35788199781996 * q^19 - 193770761479080 * q^23 + 1127564438439501 * q^25 - 5607343422466122 * q^29 + 11246757871503072 * q^31 - 5274251425350096 * q^35 - 24272499791100606 * q^37 + 298159108991869602 * q^41 - 33333932139754860 * q^43 + 120874283547603888 * q^47 - 850403331975639477 * q^49 + 1138443393004854222 * q^53 + 6957552484263571704 * q^55 - 9225624498709937412 * q^59 - 6554902294063924182 * q^61 + 20714581819561144452 * q^65 - 15793054074531629124 * q^67 + 41139582493467997704 * q^71 - 19422167949903851970 * q^73 - 68380237365358617888 * q^77 - 131735321299806049488 * q^79 - 64013993832679681068 * q^83 + 390258202763001297252 * q^85 - 429891446897537246766 * q^89 - 297181701588021496176 * q^91 + 1036372406649019824696 * q^95 - 324059336514148638042 * q^97

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	3.39292e7	1.55378	0.776889	−	0.629638i	$-0.216797\pi$
$5$	3.39292e7	1.55378	0.776889	+	0.629638i	$0.216797\pi$
$6$	0	0
$7$	−5.28731e8	−0.707466	−0.353733	−	0.935346i	$-0.615088\pi$
$7$	−5.28731e8	−0.707466	−0.353733	+	0.935346i	$0.615088\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	1.21955e11	1.41767	0.708835	−	0.705375i	$-0.249221\pi$
$11$	1.21955e11	1.41767	0.708835	+	0.705375i	$0.249221\pi$
$12$	0	0
$13$	4.33605e11	0.872346	0.436173	−	0.899863i	$-0.356334\pi$
$13$	4.33605e11	0.872346	0.436173	+	0.899863i	$0.356334\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1.31741e13	1.58492	0.792461	−	0.609922i	$-0.208799\pi$
$17$	1.31741e13	1.58492	0.792461	+	0.609922i	$0.208799\pi$
$18$	0	0
$19$	2.18808e13	0.818747	0.409373	−	0.912367i	$-0.365747\pi$
$19$	2.18808e13	0.818747	0.409373	+	0.912367i	$0.365747\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−1.40779e14	−0.708589	−0.354294	−	0.935134i	$-0.615279\pi$
$23$	−1.40779e14	−0.708589	−0.354294	+	0.935134i	$0.615279\pi$
$24$	0	0
$25$	6.74355e14	1.41423
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−1.17115e15	−0.516934	−0.258467	−	0.966020i	$-0.583217\pi$
$29$	−1.17115e15	−0.516934	−0.258467	+	0.966020i	$0.583217\pi$
$30$	0	0
$31$	9.57156e14	0.209742	0.104871	−	0.994486i	$-0.466557\pi$
$31$	9.57156e14	0.209742	0.104871	+	0.994486i	$0.466557\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−1.79394e16	−1.09925
$36$	0	0
$37$	−3.53720e16	−1.20932	−0.604661	−	0.796483i	$-0.706691\pi$
$37$	−3.53720e16	−1.20932	−0.604661	+	0.796483i	$0.706691\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	1.71456e17	1.99490	0.997451	−	0.0713504i	$-0.0227309\pi$
$41$	1.71456e17	1.99490	0.997451	+	0.0713504i	$0.0227309\pi$
$42$	0	0
$43$	1.35679e17	0.957398	0.478699	−	0.877979i	$-0.341108\pi$
$43$	1.35679e17	0.957398	0.478699	+	0.877979i	$0.341108\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	5.75514e17	1.59598	0.797991	−	0.602669i	$-0.205896\pi$
$47$	5.75514e17	1.59598	0.797991	+	0.602669i	$0.205896\pi$
$48$	0	0
$49$	−2.78989e17	−0.499492
$50$	0	0
$51$	0	0
$52$	0	0
$53$	9.62123e17	0.755673	0.377837	−	0.925872i	$-0.376668\pi$
$53$	9.62123e17	0.755673	0.377837	+	0.925872i	$0.376668\pi$
$54$	0	0
$55$	4.13782e18	2.20274
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−4.87671e18	−1.24217	−0.621085	−	0.783743i	$-0.713308\pi$
$59$	−4.87671e18	−1.24217	−0.621085	+	0.783743i	$0.713308\pi$
$60$	0	0
$61$	4.59195e18	0.824202	0.412101	−	0.911138i	$-0.364795\pi$
$61$	4.59195e18	0.824202	0.412101	+	0.911138i	$0.364795\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	1.47119e19	1.35543
$66$	0	0
$67$	−1.78611e19	−1.19708	−0.598539	−	0.801094i	$-0.704252\pi$
$67$	−1.78611e19	−1.19708	−0.598539	+	0.801094i	$0.704252\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−2.93629e19	−1.07050	−0.535250	−	0.844694i	$-0.679783\pi$
$71$	−2.93629e19	−1.07050	−0.535250	+	0.844694i	$0.679783\pi$
$72$	0	0
$73$	−8.32585e18	−0.226745	−0.113373	−	0.993553i	$-0.536165\pi$
$73$	−8.32585e18	−0.226745	−0.113373	+	0.993553i	$0.536165\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−6.44812e19	−1.00295
$78$	0	0
$79$	−7.05718e19	−0.838584	−0.419292	−	0.907851i	$-0.637722\pi$
$79$	−7.05718e19	−0.838584	−0.419292	+	0.907851i	$0.637722\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−2.11929e20	−1.49924	−0.749620	−	0.661869i	$-0.769764\pi$
$83$	−2.11929e20	−1.49924	−0.749620	+	0.661869i	$0.769764\pi$
$84$	0	0
$85$	4.46988e20	2.46262
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−2.62972e20	−0.893952	−0.446976	−	0.894546i	$-0.647499\pi$
$89$	−2.62972e20	−0.893952	−0.446976	+	0.894546i	$0.647499\pi$
$90$	0	0
$91$	−2.29260e20	−0.617155
$92$	0	0
$93$	0	0
$94$	0	0
$95$	7.42397e20	1.27215
$96$	0	0
$97$	3.84293e20	0.529126	0.264563	−	0.964368i	$-0.414772\pi$
$97$	3.84293e20	0.529126	0.264563	+	0.964368i	$0.414772\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	1.92132e21	1.73071	0.865356	−	0.501158i	$-0.167092\pi$
$101$	1.92132e21	1.73071	0.865356	+	0.501158i	$0.167092\pi$
$102$	0	0
$103$	6.06205e20	0.444456	0.222228	−	0.974995i	$-0.428667\pi$
$103$	6.06205e20	0.444456	0.222228	+	0.974995i	$0.428667\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−7.79821e20	−0.383235	−0.191618	−	0.981470i	$-0.561373\pi$
$107$	−7.79821e20	−0.383235	−0.191618	+	0.981470i	$0.561373\pi$
$108$	0	0
$109$	−2.58673e20	−0.104658	−0.0523291	−	0.998630i	$-0.516664\pi$
$109$	−2.58673e20	−0.104658	−0.0523291	+	0.998630i	$0.516664\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−1.56041e21	−0.432428	−0.216214	−	0.976346i	$-0.569371\pi$
$113$	−1.56041e21	−0.432428	−0.216214	+	0.976346i	$0.569371\pi$
$114$	0	0
$115$	−4.77651e21	−1.10099
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−6.96557e21	−1.12128
$120$	0	0
$121$	7.47267e21	1.00979
$122$	0	0
$123$	0	0
$124$	0	0
$125$	6.70164e21	0.643615
$126$	0	0
$127$	−7.53317e21	−0.612405	−0.306202	−	0.951966i	$-0.599058\pi$
$127$	−7.53317e21	−0.612405	−0.306202	+	0.951966i	$0.599058\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	1.41613e22	0.831291	0.415645	−	0.909527i	$-0.363556\pi$
$131$	1.41613e22	0.831291	0.415645	+	0.909527i	$0.363556\pi$
$132$	0	0
$133$	−1.15690e22	−0.579236
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−2.76400e22	−1.01385	−0.506924	−	0.861991i	$-0.669218\pi$
$137$	−2.76400e22	−1.01385	−0.506924	+	0.861991i	$0.669218\pi$
$138$	0	0
$139$	1.98007e21	0.0623770	0.0311885	−	0.999514i	$-0.490071\pi$
$139$	1.98007e21	0.0623770	0.0311885	+	0.999514i	$0.490071\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	5.28801e22	1.23670
$144$	0	0
$145$	−3.97363e22	−0.803200
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−2.76024e22	−0.419268	−0.209634	−	0.977780i	$-0.567227\pi$
$149$	−2.76024e22	−0.419268	−0.209634	+	0.977780i	$0.567227\pi$
$150$	0	0
$151$	4.72139e22	0.623465	0.311732	−	0.950170i	$-0.399091\pi$
$151$	4.72139e22	0.623465	0.311732	+	0.950170i	$0.399091\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	3.24756e22	0.325892
$156$	0	0
$157$	1.89837e23	1.66508	0.832540	−	0.553965i	$-0.186886\pi$
$157$	1.89837e23	1.66508	0.832540	+	0.553965i	$0.186886\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	7.44340e22	0.501302
$162$	0	0
$163$	1.59931e23	0.946158	0.473079	−	0.881020i	$-0.343143\pi$
$163$	1.59931e23	0.946158	0.473079	+	0.881020i	$0.343143\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−3.26974e22	−0.149965	−0.0749825	−	0.997185i	$-0.523890\pi$
$167$	−3.26974e22	−0.149965	−0.0749825	+	0.997185i	$0.523890\pi$
$168$	0	0
$169$	−5.90514e22	−0.239012
$170$	0	0
$171$	0	0
$172$	0	0
$173$	5.17436e22	0.163822	0.0819111	−	0.996640i	$-0.473898\pi$
$173$	5.17436e22	0.163822	0.0819111	+	0.996640i	$0.473898\pi$
$174$	0	0
$175$	−3.56553e23	−1.00052
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−3.75079e22	−0.0830167	−0.0415084	−	0.999138i	$-0.513216\pi$
$179$	−3.75079e22	−0.0830167	−0.0415084	+	0.999138i	$0.513216\pi$
$180$	0	0
$181$	−3.65662e22	−0.0720203	−0.0360102	−	0.999351i	$-0.511465\pi$
$181$	−3.65662e22	−0.0720203	−0.0360102	+	0.999351i	$0.511465\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−1.20015e24	−1.87902
$186$	0	0
$187$	1.60664e24	2.24690
$188$	0	0
$189$	0	0
$190$	0	0
$191$	4.61401e23	0.516688	0.258344	−	0.966053i	$-0.416823\pi$
$191$	4.61401e23	0.516688	0.258344	+	0.966053i	$0.416823\pi$
$192$	0	0
$193$	2.52365e23	0.253324	0.126662	−	0.991946i	$-0.459574\pi$
$193$	2.52365e23	0.253324	0.126662	+	0.991946i	$0.459574\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	7.91314e23	0.640403	0.320202	−	0.947349i	$-0.396249\pi$
$197$	7.91314e23	0.640403	0.320202	+	0.947349i	$0.396249\pi$
$198$	0	0
$199$	−2.46245e24	−1.79230	−0.896150	−	0.443752i	$-0.853647\pi$
$199$	−2.46245e24	−1.79230	−0.896150	+	0.443752i	$0.853647\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	6.19226e23	0.365713
$204$	0	0
$205$	5.81736e24	3.09964
$206$	0	0
$207$	0	0
$208$	0	0
$209$	2.66846e24	1.16071
$210$	0	0
$211$	−1.79516e24	−0.706542	−0.353271	−	0.935521i	$-0.614931\pi$
$211$	−1.79516e24	−0.706542	−0.353271	+	0.935521i	$0.614931\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	4.60347e24	1.48758
$216$	0	0
$217$	−5.06079e23	−0.148385
$218$	0	0
$219$	0	0
$220$	0	0
$221$	5.71236e24	1.38260
$222$	0	0
$223$	3.76539e24	0.829103	0.414551	−	0.910026i	$-0.363939\pi$
$223$	3.76539e24	0.829103	0.414551	+	0.910026i	$0.363939\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−8.37475e24	−1.53003	−0.765016	−	0.644011i	$-0.777269\pi$
$227$	−8.37475e24	−1.53003	−0.765016	+	0.644011i	$0.777269\pi$
$228$	0	0
$229$	3.14314e24	0.523710	0.261855	−	0.965107i	$-0.415666\pi$
$229$	3.14314e24	0.523710	0.261855	+	0.965107i	$0.415666\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	4.21087e24	0.584971	0.292486	−	0.956270i	$-0.405518\pi$
$233$	4.21087e24	0.584971	0.292486	+	0.956270i	$0.405518\pi$
$234$	0	0
$235$	1.95267e25	2.47980
$236$	0	0
$237$	0	0
$238$	0	0
$239$	9.27921e24	0.987036	0.493518	−	0.869736i	$-0.335711\pi$
$239$	9.27921e24	0.987036	0.493518	+	0.869736i	$0.335711\pi$
$240$	0	0
$241$	1.45105e24	0.141418	0.0707090	−	0.997497i	$-0.477474\pi$
$241$	1.45105e24	0.141418	0.0707090	+	0.997497i	$0.477474\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−9.46588e24	−0.776099
$246$	0	0
$247$	9.48760e24	0.714231
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−2.04939e25	−1.30332	−0.651658	−	0.758513i	$-0.725926\pi$
$251$	−2.04939e25	−1.30332	−0.651658	+	0.758513i	$0.725926\pi$
$252$	0	0
$253$	−1.71686e25	−1.00454
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−1.77825e25	−0.882462	−0.441231	−	0.897394i	$-0.645458\pi$
$257$	−1.77825e25	−0.882462	−0.441231	+	0.897394i	$0.645458\pi$
$258$	0	0
$259$	1.87023e25	0.855554
$260$	0	0
$261$	0	0
$262$	0	0
$263$	3.76203e25	1.46517	0.732583	−	0.680678i	$-0.238315\pi$
$263$	3.76203e25	1.46517	0.732583	+	0.680678i	$0.238315\pi$
$264$	0	0
$265$	3.26441e25	1.17415
$266$	0	0
$267$	0	0
$268$	0	0
$269$	5.82547e25	1.79033	0.895163	−	0.445739i	$-0.147059\pi$
$269$	5.82547e25	1.79033	0.895163	+	0.445739i	$0.147059\pi$
$270$	0	0
$271$	4.00531e25	1.13883	0.569414	−	0.822051i	$-0.307170\pi$
$271$	4.00531e25	1.13883	0.569414	+	0.822051i	$0.307170\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	8.22407e25	2.00490
$276$	0	0
$277$	5.12820e25	1.15858	0.579292	−	0.815120i	$-0.303329\pi$
$277$	5.12820e25	1.15858	0.579292	+	0.815120i	$0.303329\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	1.22313e25	0.237714	0.118857	−	0.992911i	$-0.462077\pi$
$281$	1.22313e25	0.237714	0.118857	+	0.992911i	$0.462077\pi$
$282$	0	0
$283$	8.45355e25	1.52504	0.762520	−	0.646964i	$-0.223962\pi$
$283$	8.45355e25	1.52504	0.762520	+	0.646964i	$0.223962\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−9.06541e25	−1.41133
$288$	0	0
$289$	1.04466e26	1.51198
$290$	0	0
$291$	0	0
$292$	0	0
$293$	2.19428e25	0.274905	0.137453	−	0.990508i	$-0.456109\pi$
$293$	2.19428e25	0.274905	0.137453	+	0.990508i	$0.456109\pi$
$294$	0	0
$295$	−1.65463e26	−1.93006
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−6.10422e25	−0.618135
$300$	0	0
$301$	−7.17375e25	−0.677327
$302$	0	0
$303$	0	0
$304$	0	0
$305$	1.55801e26	1.28063
$306$	0	0
$307$	3.79642e25	0.291354	0.145677	−	0.989332i	$-0.453464\pi$
$307$	3.79642e25	0.291354	0.145677	+	0.989332i	$0.453464\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−7.02783e25	−0.470801	−0.235401	−	0.971898i	$-0.575640\pi$
$311$	−7.02783e25	−0.470801	−0.235401	+	0.971898i	$0.575640\pi$
$312$	0	0
$313$	−9.09673e25	−0.569731	−0.284865	−	0.958568i	$-0.591949\pi$
$313$	−9.09673e25	−0.569731	−0.284865	+	0.958568i	$0.591949\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	2.99033e25	0.163907	0.0819533	−	0.996636i	$-0.473884\pi$
$317$	2.99033e25	0.163907	0.0819533	+	0.996636i	$0.473884\pi$
$318$	0	0
$319$	−1.42828e26	−0.732841
$320$	0	0
$321$	0	0
$322$	0	0
$323$	2.88260e26	1.29765
$324$	0	0
$325$	2.92404e26	1.23369
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−3.04292e26	−1.12910
$330$	0	0
$331$	−6.61150e25	−0.230200	−0.115100	−	0.993354i	$-0.536719\pi$
$331$	−6.61150e25	−0.230200	−0.115100	+	0.993354i	$0.536719\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−6.06013e26	−1.85999
$336$	0	0
$337$	−1.11607e26	−0.321793	−0.160896	−	0.986971i	$-0.551438\pi$
$337$	−1.11607e26	−0.321793	−0.160896	+	0.986971i	$0.551438\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	1.16730e26	0.297345
$342$	0	0
$343$	4.42831e26	1.06084
$344$	0	0
$345$	0	0
$346$	0	0
$347$	5.94900e26	1.26178	0.630891	−	0.775872i	$-0.282690\pi$
$347$	5.94900e26	1.26178	0.630891	+	0.775872i	$0.282690\pi$
$348$	0	0
$349$	4.45113e26	0.888799	0.444400	−	0.895829i	$-0.353417\pi$
$349$	4.45113e26	0.888799	0.444400	+	0.895829i	$0.353417\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−2.37403e26	−0.420584	−0.210292	−	0.977639i	$-0.567441\pi$
$353$	−2.37403e26	−0.420584	−0.210292	+	0.977639i	$0.567441\pi$
$354$	0	0
$355$	−9.96262e26	−1.66332
$356$	0	0
$357$	0	0
$358$	0	0
$359$	3.45652e26	0.513036	0.256518	−	0.966539i	$-0.417425\pi$
$359$	3.45652e26	0.513036	0.256518	+	0.966539i	$0.417425\pi$
$360$	0	0
$361$	−2.35442e26	−0.329654
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−2.82490e26	−0.352312
$366$	0	0
$367$	−8.76689e26	−1.03241	−0.516205	−	0.856465i	$-0.672656\pi$
$367$	−8.76689e26	−1.03241	−0.516205	+	0.856465i	$0.672656\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−5.08705e26	−0.534613
$372$	0	0
$373$	−1.54618e27	−1.53573	−0.767867	−	0.640609i	$-0.778682\pi$
$373$	−1.54618e27	−1.53573	−0.767867	+	0.640609i	$0.778682\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−5.07818e26	−0.450945
$378$	0	0
$379$	1.48367e27	1.24631	0.623156	−	0.782097i	$-0.285850\pi$
$379$	1.48367e27	1.24631	0.623156	+	0.782097i	$0.285850\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−1.86137e27	−1.40038	−0.700190	−	0.713956i	$-0.746901\pi$
$383$	−1.86137e27	−1.40038	−0.700190	+	0.713956i	$0.746901\pi$
$384$	0	0
$385$	−2.18780e27	−1.55837
$386$	0	0
$387$	0	0
$388$	0	0
$389$	4.43791e26	0.283601	0.141801	−	0.989895i	$-0.454711\pi$
$389$	4.43791e26	0.283601	0.141801	+	0.989895i	$0.454711\pi$
$390$	0	0
$391$	−1.85463e27	−1.12306
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−2.39445e27	−1.30297
$396$	0	0
$397$	−3.85978e26	−0.199188	−0.0995939	−	0.995028i	$-0.531754\pi$
$397$	−3.85978e26	−0.199188	−0.0995939	+	0.995028i	$0.531754\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	8.65938e26	0.402227	0.201113	−	0.979568i	$-0.435544\pi$
$401$	8.65938e26	0.402227	0.201113	+	0.979568i	$0.435544\pi$
$402$	0	0
$403$	4.15028e26	0.182967
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−4.31378e27	−1.71442
$408$	0	0
$409$	6.19682e26	0.233924	0.116962	−	0.993136i	$-0.462684\pi$
$409$	6.19682e26	0.233924	0.116962	+	0.993136i	$0.462684\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	2.57847e27	0.878793
$414$	0	0
$415$	−7.19060e27	−2.32948
$416$	0	0
$417$	0	0
$418$	0	0
$419$	2.87299e27	0.841562	0.420781	−	0.907162i	$-0.361756\pi$
$419$	2.87299e27	0.841562	0.420781	+	0.907162i	$0.361756\pi$
$420$	0	0
$421$	2.25957e27	0.629598	0.314799	−	0.949158i	$-0.398063\pi$
$421$	2.25957e27	0.629598	0.314799	+	0.949158i	$0.398063\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	8.88404e27	2.24144
$426$	0	0
$427$	−2.42791e27	−0.583095
$428$	0	0
$429$	0	0
$430$	0	0
$431$	3.01840e27	0.657303	0.328652	−	0.944451i	$-0.393406\pi$
$431$	3.01840e27	0.657303	0.328652	+	0.944451i	$0.393406\pi$
$432$	0	0
$433$	6.44418e27	1.33673	0.668367	−	0.743832i	$-0.266994\pi$
$433$	6.44418e27	1.33673	0.668367	+	0.743832i	$0.266994\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−3.08034e27	−0.580155
$438$	0	0
$439$	−8.50049e27	−1.52604	−0.763021	−	0.646374i	$-0.776285\pi$
$439$	−8.50049e27	−1.52604	−0.763021	+	0.646374i	$0.776285\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−6.00246e27	−0.979694	−0.489847	−	0.871809i	$-0.662947\pi$
$443$	−6.00246e27	−0.979694	−0.489847	+	0.871809i	$0.662947\pi$
$444$	0	0
$445$	−8.92243e27	−1.38900
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−8.58205e27	−1.21620	−0.608100	−	0.793861i	$-0.708068\pi$
$449$	−8.58205e27	−1.21620	−0.608100	+	0.793861i	$0.708068\pi$
$450$	0	0
$451$	2.09098e28	2.82811
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−7.77863e27	−0.958922
$456$	0	0
$457$	1.09251e28	1.28619	0.643093	−	0.765788i	$-0.277651\pi$
$457$	1.09251e28	1.28619	0.643093	+	0.765788i	$0.277651\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	7.29500e27	0.783728	0.391864	−	0.920023i	$-0.371830\pi$
$461$	7.29500e27	0.783728	0.391864	+	0.920023i	$0.371830\pi$
$462$	0	0
$463$	−1.13930e28	−1.16960	−0.584800	−	0.811178i	$-0.698827\pi$
$463$	−1.13930e28	−1.16960	−0.584800	+	0.811178i	$0.698827\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−1.51114e28	−1.41735	−0.708677	−	0.705533i	$-0.750708\pi$
$467$	−1.51114e28	−1.41735	−0.708677	+	0.705533i	$0.750708\pi$
$468$	0	0
$469$	9.44371e27	0.846892
$470$	0	0
$471$	0	0
$472$	0	0
$473$	1.65466e28	1.35727
$474$	0	0
$475$	1.47554e28	1.15789
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−3.89042e27	−0.279559	−0.139779	−	0.990183i	$-0.544639\pi$
$479$	−3.89042e27	−0.279559	−0.139779	+	0.990183i	$0.544639\pi$
$480$	0	0
$481$	−1.53375e28	−1.05495
$482$	0	0
$483$	0	0
$484$	0	0
$485$	1.30388e28	0.822145
$486$	0	0
$487$	1.89704e28	1.14557	0.572787	−	0.819704i	$-0.305862\pi$
$487$	1.89704e28	1.14557	0.572787	+	0.819704i	$0.305862\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	1.33589e28	0.740313	0.370157	−	0.928969i	$-0.379304\pi$
$491$	1.33589e28	0.740313	0.370157	+	0.928969i	$0.379304\pi$
$492$	0	0
$493$	−1.54289e28	−0.819300
$494$	0	0
$495$	0	0
$496$	0	0
$497$	1.55251e28	0.757343
$498$	0	0
$499$	−2.86135e28	−1.33818	−0.669090	−	0.743181i	$-0.733316\pi$
$499$	−2.86135e28	−1.33818	−0.669090	+	0.743181i	$0.733316\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−9.44655e27	−0.406265	−0.203133	−	0.979151i	$-0.565112\pi$
$503$	−9.44655e27	−0.406265	−0.203133	+	0.979151i	$0.565112\pi$
$504$	0	0
$505$	6.51888e28	2.68914
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−2.06060e28	−0.782452	−0.391226	−	0.920295i	$-0.627949\pi$
$509$	−2.06060e28	−0.782452	−0.391226	+	0.920295i	$0.627949\pi$
$510$	0	0
$511$	4.40214e27	0.160415
$512$	0	0
$513$	0	0
$514$	0	0
$515$	2.05681e28	0.690586
$516$	0	0
$517$	7.01865e28	2.26258
$518$	0	0
$519$	0	0
$520$	0	0
$521$	4.30969e28	1.28130	0.640649	−	0.767834i	$-0.278665\pi$
$521$	4.30969e28	1.28130	0.640649	+	0.767834i	$0.278665\pi$
$522$	0	0
$523$	1.08864e28	0.310897	0.155448	−	0.987844i	$-0.450318\pi$
$523$	1.08864e28	0.310897	0.155448	+	0.987844i	$0.450318\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1.26097e28	0.332425
$528$	0	0
$529$	−1.96530e28	−0.497902
$530$	0	0
$531$	0	0
$532$	0	0
$533$	7.43441e28	1.74025
$534$	0	0
$535$	−2.64587e28	−0.595462
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−3.40240e28	−0.708114
$540$	0	0
$541$	3.99516e28	0.799765	0.399883	−	0.916566i	$-0.369051\pi$
$541$	3.99516e28	0.799765	0.399883	+	0.916566i	$0.369051\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−8.77658e27	−0.162616
$546$	0	0
$547$	−5.58498e28	−0.995760	−0.497880	−	0.867246i	$-0.665888\pi$
$547$	−5.58498e28	−0.995760	−0.497880	+	0.867246i	$0.665888\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−2.56257e28	−0.423238
$552$	0	0
$553$	3.73135e28	0.593270
$554$	0	0
$555$	0	0
$556$	0	0
$557$	8.87710e28	1.30855	0.654277	−	0.756255i	$-0.272973\pi$
$557$	8.87710e28	1.30855	0.654277	+	0.756255i	$0.272973\pi$
$558$	0	0
$559$	5.88309e28	0.835183
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−7.63115e28	−1.00520	−0.502599	−	0.864520i	$-0.667623\pi$
$563$	−7.63115e28	−1.00520	−0.502599	+	0.864520i	$0.667623\pi$
$564$	0	0
$565$	−5.29434e28	−0.671898
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−1.48073e29	−1.74500	−0.872502	−	0.488610i	$-0.837504\pi$
$569$	−1.48073e29	−1.74500	−0.872502	+	0.488610i	$0.837504\pi$
$570$	0	0
$571$	−7.51419e28	−0.853499	−0.426749	−	0.904370i	$-0.640341\pi$
$571$	−7.51419e28	−0.853499	−0.426749	+	0.904370i	$0.640341\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−9.49348e28	−1.00210
$576$	0	0
$577$	1.56608e28	0.159392	0.0796961	−	0.996819i	$-0.474605\pi$
$577$	1.56608e28	0.159392	0.0796961	+	0.996819i	$0.474605\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	1.12054e29	1.06066
$582$	0	0
$583$	1.17335e29	1.07129
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−1.19163e29	−1.01260	−0.506302	−	0.862356i	$-0.668988\pi$
$587$	−1.19163e29	−1.01260	−0.506302	+	0.862356i	$0.668988\pi$
$588$	0	0
$589$	2.09433e28	0.171725
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−3.34770e28	−0.255666	−0.127833	−	0.991796i	$-0.540802\pi$
$593$	−3.34770e28	−0.255666	−0.127833	+	0.991796i	$0.540802\pi$
$594$	0	0
$595$	−2.36337e29	−1.74222
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−1.39785e29	−0.960461	−0.480231	−	0.877142i	$-0.659447\pi$
$599$	−1.39785e29	−0.960461	−0.480231	+	0.877142i	$0.659447\pi$
$600$	0	0
$601$	−1.02954e27	−0.00683061	−0.00341531	−	0.999994i	$-0.501087\pi$
$601$	−1.02954e27	−0.00683061	−0.00341531	+	0.999994i	$0.501087\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	2.53542e29	1.56898
$606$	0	0
$607$	1.61045e29	0.962647	0.481323	−	0.876543i	$-0.340156\pi$
$607$	1.61045e29	0.962647	0.481323	+	0.876543i	$0.340156\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	2.49546e29	1.39225
$612$	0	0
$613$	−1.08394e29	−0.584345	−0.292173	−	0.956366i	$-0.594378\pi$
$613$	−1.08394e29	−0.584345	−0.292173	+	0.956366i	$0.594378\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	1.69220e29	0.852033	0.426017	−	0.904715i	$-0.359916\pi$
$617$	1.69220e29	0.852033	0.426017	+	0.904715i	$0.359916\pi$
$618$	0	0
$619$	4.72000e28	0.229715	0.114858	−	0.993382i	$-0.463359\pi$
$619$	4.72000e28	0.229715	0.114858	+	0.993382i	$0.463359\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	1.39041e29	0.632441
$624$	0	0
$625$	−9.41762e28	−0.414191
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−4.65995e29	−1.91668
$630$	0	0
$631$	−4.15217e29	−1.65184	−0.825918	−	0.563790i	$-0.809343\pi$
$631$	−4.15217e29	−1.65184	−0.825918	+	0.563790i	$0.809343\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−2.55595e29	−0.951541
$636$	0	0
$637$	−1.20971e29	−0.435730
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−9.51731e28	−0.321000	−0.160500	−	0.987036i	$-0.551311\pi$
$641$	−9.51731e28	−0.321000	−0.160500	+	0.987036i	$0.551311\pi$
$642$	0	0
$643$	−3.25707e29	−1.06319	−0.531596	−	0.846998i	$-0.678408\pi$
$643$	−3.25707e29	−1.06319	−0.531596	+	0.846998i	$0.678408\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	3.06325e28	0.0936888	0.0468444	−	0.998902i	$-0.485084\pi$
$647$	3.06325e28	0.0936888	0.0468444	+	0.998902i	$0.485084\pi$
$648$	0	0
$649$	−5.94737e29	−1.76099
$650$	0	0
$651$	0	0
$652$	0	0
$653$	8.66491e28	0.240534	0.120267	−	0.992742i	$-0.461625\pi$
$653$	8.66491e28	0.240534	0.120267	+	0.992742i	$0.461625\pi$
$654$	0	0
$655$	4.80480e29	1.29164
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−7.32944e29	−1.84831	−0.924154	−	0.382021i	$-0.875228\pi$
$659$	−7.32944e29	−1.84831	−0.924154	+	0.382021i	$0.875228\pi$
$660$	0	0
$661$	−5.06204e29	−1.23655	−0.618273	−	0.785964i	$-0.712167\pi$
$661$	−5.06204e29	−1.23655	−0.618273	+	0.785964i	$0.712167\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−3.92529e29	−0.900004
$666$	0	0
$667$	1.64873e29	0.366293
$668$	0	0
$669$	0	0
$670$	0	0
$671$	5.60009e29	1.16845
$672$	0	0
$673$	4.74688e29	0.959953	0.479977	−	0.877281i	$-0.340645\pi$
$673$	4.74688e29	0.959953	0.479977	+	0.877281i	$0.340645\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−6.45182e29	−1.22603	−0.613015	−	0.790071i	$-0.710044\pi$
$677$	−6.45182e29	−1.22603	−0.613015	+	0.790071i	$0.710044\pi$
$678$	0	0
$679$	−2.03188e29	−0.374339
$680$	0	0
$681$	0	0
$682$	0	0
$683$	5.14621e29	0.891394	0.445697	−	0.895184i	$-0.352956\pi$
$683$	5.14621e29	0.891394	0.445697	+	0.895184i	$0.352956\pi$
$684$	0	0
$685$	−9.37805e29	−1.57529
$686$	0	0
$687$	0	0
$688$	0	0
$689$	4.17181e29	0.659209
$690$	0	0
$691$	−1.06801e30	−1.63702	−0.818512	−	0.574489i	$-0.805201\pi$
$691$	−1.06801e30	−1.63702	−0.818512	+	0.574489i	$0.805201\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	6.71821e28	0.0969199
$696$	0	0
$697$	2.25878e30	3.16177
$698$	0	0
$699$	0	0
$700$	0	0
$701$	1.04529e30	1.37784	0.688919	−	0.724839i	$-0.258086\pi$
$701$	1.04529e30	1.37784	0.688919	+	0.724839i	$0.258086\pi$
$702$	0	0
$703$	−7.73967e29	−0.990128
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−1.01586e30	−1.22442
$708$	0	0
$709$	8.68030e28	0.101566	0.0507831	−	0.998710i	$-0.483828\pi$
$709$	8.68030e28	0.101566	0.0507831	+	0.998710i	$0.483828\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−1.34747e29	−0.148621
$714$	0	0
$715$	1.79418e30	1.92155
$716$	0	0
$717$	0	0
$718$	0	0
$719$	1.16156e30	1.17325	0.586624	−	0.809859i	$-0.300457\pi$
$719$	1.16156e30	1.17325	0.586624	+	0.809859i	$0.300457\pi$
$720$	0	0
$721$	−3.20520e29	−0.314438
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−7.89774e29	−0.731061
$726$	0	0
$727$	2.77822e29	0.249836	0.124918	−	0.992167i	$-0.460133\pi$
$727$	2.77822e29	0.249836	0.124918	+	0.992167i	$0.460133\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	1.78745e30	1.51740
$732$	0	0
$733$	2.16010e30	1.78189	0.890947	−	0.454107i	$-0.150042\pi$
$733$	2.16010e30	1.78189	0.890947	+	0.454107i	$0.150042\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−2.17824e30	−1.69706
$738$	0	0
$739$	1.74302e30	1.31988	0.659942	−	0.751317i	$-0.270581\pi$
$739$	1.74302e30	1.31988	0.659942	+	0.751317i	$0.270581\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	5.89639e29	0.421895	0.210947	−	0.977497i	$-0.432345\pi$
$743$	5.89639e29	0.421895	0.210947	+	0.977497i	$0.432345\pi$
$744$	0	0
$745$	−9.36530e29	−0.651449
$746$	0	0
$747$	0	0
$748$	0	0
$749$	4.12316e29	0.271126
$750$	0	0
$751$	−8.31754e28	−0.0531833	−0.0265917	−	0.999646i	$-0.508465\pi$
$751$	−8.31754e28	−0.0531833	−0.0265917	+	0.999646i	$0.508465\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	1.60193e30	0.968726
$756$	0	0
$757$	−2.67582e29	−0.157380	−0.0786902	−	0.996899i	$-0.525074\pi$
$757$	−2.67582e29	−0.157380	−0.0786902	+	0.996899i	$0.525074\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−2.95107e30	−1.64225	−0.821127	−	0.570745i	$-0.806654\pi$
$761$	−2.95107e30	−1.64225	−0.821127	+	0.570745i	$0.806654\pi$
$762$	0	0
$763$	1.36769e29	0.0740421
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−2.11457e30	−1.08360
$768$	0	0
$769$	−2.62362e30	−1.30820	−0.654100	−	0.756408i	$-0.726953\pi$
$769$	−2.62362e30	−1.30820	−0.654100	+	0.756408i	$0.726953\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−9.36482e29	−0.442196	−0.221098	−	0.975252i	$-0.570964\pi$
$773$	−9.36482e29	−0.442196	−0.221098	+	0.975252i	$0.570964\pi$
$774$	0	0
$775$	6.45464e29	0.296622
$776$	0	0
$777$	0	0
$778$	0	0
$779$	3.75158e30	1.63332
$780$	0	0
$781$	−3.58094e30	−1.51762
$782$	0	0
$783$	0	0
$784$	0	0
$785$	6.44103e30	2.58716
$786$	0	0
$787$	2.42850e30	0.949736	0.474868	−	0.880057i	$-0.342496\pi$
$787$	2.42850e30	0.949736	0.474868	+	0.880057i	$0.342496\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	8.25036e29	0.305929
$792$	0	0
$793$	1.99109e30	0.718990
$794$	0	0
$795$	0	0
$796$	0	0
$797$	1.99411e29	0.0683025	0.0341513	−	0.999417i	$-0.489127\pi$
$797$	1.99411e29	0.0683025	0.0341513	+	0.999417i	$0.489127\pi$
$798$	0	0
$799$	7.58189e30	2.52951
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−1.01537e30	−0.321450
$804$	0	0
$805$	2.52549e30	0.778913
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−3.77765e30	−1.10602	−0.553010	−	0.833175i	$-0.686521\pi$
$809$	−3.77765e30	−1.10602	−0.553010	+	0.833175i	$0.686521\pi$
$810$	0	0
$811$	2.72778e30	0.778199	0.389099	−	0.921196i	$-0.372786\pi$
$811$	2.72778e30	0.778199	0.389099	+	0.921196i	$0.372786\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	5.42635e30	1.47012
$816$	0	0
$817$	2.96875e30	0.783867
$818$	0	0
$819$	0	0
$820$	0	0
$821$	1.90653e30	0.478234	0.239117	−	0.970991i	$-0.423142\pi$
$821$	1.90653e30	0.478234	0.239117	+	0.970991i	$0.423142\pi$
$822$	0	0
$823$	−3.39146e30	−0.829255	−0.414628	−	0.909991i	$-0.636088\pi$
$823$	−3.39146e30	−0.829255	−0.414628	+	0.909991i	$0.636088\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	3.99616e30	0.928613	0.464307	−	0.885675i	$-0.346304\pi$
$827$	3.99616e30	0.928613	0.464307	+	0.885675i	$0.346304\pi$
$828$	0	0
$829$	−2.80835e30	−0.636251	−0.318126	−	0.948049i	$-0.603053\pi$
$829$	−2.80835e30	−0.636251	−0.318126	+	0.948049i	$0.603053\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−3.67544e30	−0.791656
$834$	0	0
$835$	−1.10940e30	−0.233012
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−5.63065e30	−1.12475	−0.562377	−	0.826881i	$-0.690113\pi$
$839$	−5.63065e30	−1.12475	−0.562377	+	0.826881i	$0.690113\pi$
$840$	0	0
$841$	−3.76124e30	−0.732779
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−2.00357e30	−0.371372
$846$	0	0
$847$	−3.95103e30	−0.714389
$848$	0	0
$849$	0	0
$850$	0	0
$851$	4.97962e30	0.856911
$852$	0	0
$853$	4.25429e30	0.714269	0.357135	−	0.934053i	$-0.383754\pi$
$853$	4.25429e30	0.714269	0.357135	+	0.934053i	$0.383754\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	5.42388e29	0.0866984	0.0433492	−	0.999060i	$-0.486197\pi$
$857$	5.42388e29	0.0866984	0.0433492	+	0.999060i	$0.486197\pi$
$858$	0	0
$859$	−3.77718e30	−0.589169	−0.294585	−	0.955625i	$-0.595181\pi$
$859$	−3.77718e30	−0.589169	−0.294585	+	0.955625i	$0.595181\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−9.65328e30	−1.43404	−0.717021	−	0.697052i	$-0.754495\pi$
$863$	−9.65328e30	−1.43404	−0.717021	+	0.697052i	$0.754495\pi$
$864$	0	0
$865$	1.75562e30	0.254543
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−8.60655e30	−1.18883
$870$	0	0
$871$	−7.74465e30	−1.04427
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−3.54337e30	−0.455336
$876$	0	0
$877$	−8.26757e30	−1.03725	−0.518623	−	0.855003i	$-0.673555\pi$
$877$	−8.26757e30	−1.03725	−0.518623	+	0.855003i	$0.673555\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−3.71573e30	−0.444423	−0.222212	−	0.974998i	$-0.571328\pi$
$881$	−3.71573e30	−0.444423	−0.222212	+	0.974998i	$0.571328\pi$
$882$	0	0
$883$	3.53555e29	0.0412923	0.0206462	−	0.999787i	$-0.493428\pi$
$883$	3.53555e29	0.0412923	0.0206462	+	0.999787i	$0.493428\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−7.78376e28	−0.00866944	−0.00433472	−	0.999991i	$-0.501380\pi$
$887$	−7.78376e28	−0.00866944	−0.00433472	+	0.999991i	$0.501380\pi$
$888$	0	0
$889$	3.98302e30	0.433256
$890$	0	0
$891$	0	0
$892$	0	0
$893$	1.25927e31	1.30671
$894$	0	0
$895$	−1.27261e30	−0.128990
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−1.12098e30	−0.108423
$900$	0	0
$901$	1.26751e31	1.19768
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−1.24066e30	−0.111904
$906$	0	0
$907$	7.49286e30	0.660345	0.330173	−	0.943921i	$-0.392893\pi$
$907$	7.49286e30	0.660345	0.330173	+	0.943921i	$0.392893\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−1.85880e31	−1.56419	−0.782096	−	0.623159i	$-0.785849\pi$
$911$	−1.85880e31	−1.56419	−0.782096	+	0.623159i	$0.785849\pi$
$912$	0	0
$913$	−2.58457e31	−2.12543
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−7.48750e30	−0.588110
$918$	0	0
$919$	−1.21998e31	−0.936573	−0.468286	−	0.883577i	$-0.655128\pi$
$919$	−1.21998e31	−0.936573	−0.468286	+	0.883577i	$0.655128\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−1.27319e31	−0.933847
$924$	0	0
$925$	−2.38533e31	−1.71025
$926$	0	0
$927$	0	0
$928$	0	0
$929$	1.53725e31	1.05337	0.526684	−	0.850061i	$-0.323435\pi$
$929$	1.53725e31	1.05337	0.526684	+	0.850061i	$0.323435\pi$
$930$	0	0
$931$	−6.10449e30	−0.408957
$932$	0	0
$933$	0	0
$934$	0	0
$935$	5.45122e31	3.49118
$936$	0	0
$937$	1.67237e31	1.04729	0.523645	−	0.851937i	$-0.324572\pi$
$937$	1.67237e31	1.04729	0.523645	+	0.851937i	$0.324572\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−2.77704e30	−0.166299	−0.0831497	−	0.996537i	$-0.526498\pi$
$941$	−2.77704e30	−0.166299	−0.0831497	+	0.996537i	$0.526498\pi$
$942$	0	0
$943$	−2.41373e31	−1.41357
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−1.06419e30	−0.0596132	−0.0298066	−	0.999556i	$-0.509489\pi$
$947$	−1.06419e30	−0.0596132	−0.0298066	+	0.999556i	$0.509489\pi$
$948$	0	0
$949$	−3.61013e30	−0.197800
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−2.76399e31	−1.44897	−0.724487	−	0.689289i	$-0.757923\pi$
$953$	−2.76399e31	−1.44897	−0.724487	+	0.689289i	$0.757923\pi$
$954$	0	0
$955$	1.56550e31	0.802818
$956$	0	0
$957$	0	0
$958$	0	0
$959$	1.46142e31	0.717263
$960$	0	0
$961$	−1.99094e31	−0.956008
$962$	0	0
$963$	0	0
$964$	0	0
$965$	8.56254e30	0.393610
$966$	0	0
$967$	1.28234e31	0.576800	0.288400	−	0.957510i	$-0.406877\pi$
$967$	1.28234e31	0.576800	0.288400	+	0.957510i	$0.406877\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−7.42766e30	−0.319926	−0.159963	−	0.987123i	$-0.551138\pi$
$971$	−7.42766e30	−0.319926	−0.159963	+	0.987123i	$0.551138\pi$
$972$	0	0
$973$	−1.04692e30	−0.0441296
$974$	0	0
$975$	0	0
$976$	0	0
$977$	2.93176e31	1.18368	0.591842	−	0.806054i	$-0.298401\pi$
$977$	2.93176e31	1.18368	0.591842	+	0.806054i	$0.298401\pi$
$978$	0	0
$979$	−3.20706e31	−1.26733
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−4.75461e31	−1.80013	−0.900064	−	0.435758i	$-0.856480\pi$
$983$	−4.75461e31	−1.80013	−0.900064	+	0.435758i	$0.856480\pi$
$984$	0	0
$985$	2.68487e31	0.995044
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−1.91006e31	−0.678401
$990$	0	0
$991$	4.88614e31	1.69900	0.849498	−	0.527592i	$-0.176905\pi$
$991$	4.88614e31	1.69900	0.849498	+	0.527592i	$0.176905\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−8.35491e31	−2.78484
$996$	0	0
$997$	5.22919e31	1.70661	0.853306	−	0.521410i	$-0.174594\pi$
$997$	5.22919e31	1.70661	0.853306	+	0.521410i	$0.174594\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	72.22.a.f.1.3		3
3.2	odd	2		8.22.a.b.1.1	✓	3
12.11	even	2		16.22.a.f.1.3		3
24.5	odd	2		64.22.a.l.1.3		3
24.11	even	2		64.22.a.m.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8.22.a.b.1.1	✓	3	3.2	odd	2
16.22.a.f.1.3		3	12.11	even	2
64.22.a.l.1.3		3	24.5	odd	2
64.22.a.m.1.1		3	24.11	even	2
72.22.a.f.1.3		3	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 72 = 2^{3} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 22 \)
Character orbit:	\([\chi]\)	\(=\)	72.a (trivial)

\(f(q)\)	\(=\)	\(q+3.39292e7 q^{5} -5.28731e8 q^{7} +O(q^{10})\)	\(q+3.39292e7 q^{5} -5.28731e8 q^{7} +1.21955e11 q^{11} +4.33605e11 q^{13} +1.31741e13 q^{17} +2.18808e13 q^{19} -1.40779e14 q^{23} +6.74355e14 q^{25} -1.17115e15 q^{29} +9.57156e14 q^{31} -1.79394e16 q^{35} -3.53720e16 q^{37} +1.71456e17 q^{41} +1.35679e17 q^{43} +5.75514e17 q^{47} -2.78989e17 q^{49} +9.62123e17 q^{53} +4.13782e18 q^{55} -4.87671e18 q^{59} +4.59195e18 q^{61} +1.47119e19 q^{65} -1.78611e19 q^{67} -2.93629e19 q^{71} -8.32585e18 q^{73} -6.44812e19 q^{77} -7.05718e19 q^{79} -2.11929e20 q^{83} +4.46988e20 q^{85} -2.62972e20 q^{89} -2.29260e20 q^{91} +7.42397e20 q^{95} +3.84293e20 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 24111774 q^{5} + 295988280 q^{7}+O(q^{10}) \) 3 * q + 24111774 * q^5 + 295988280 * q^7	\( 3 q + 24111774 q^{5} + 295988280 q^{7} + 40335108684 q^{11} + 133734425946 q^{13} - 7797732274422 q^{17} + 35788199781996 q^{19} - 193770761479080 q^{23} + 11\!\cdots\!01 q^{25}+ \cdots - 32\!\cdots\!42 q^{97}+O(q^{100}) \) 3 * q + 24111774 * q^5 + 295988280 * q^7 + 40335108684 * q^11 + 133734425946 * q^13 - 7797732274422 * q^17 + 35788199781996 * q^19 - 193770761479080 * q^23 + 1127564438439501 * q^25 - 5607343422466122 * q^29 + 11246757871503072 * q^31 - 5274251425350096 * q^35 - 24272499791100606 * q^37 + 298159108991869602 * q^41 - 33333932139754860 * q^43 + 120874283547603888 * q^47 - 850403331975639477 * q^49 + 1138443393004854222 * q^53 + 6957552484263571704 * q^55 - 9225624498709937412 * q^59 - 6554902294063924182 * q^61 + 20714581819561144452 * q^65 - 15793054074531629124 * q^67 + 41139582493467997704 * q^71 - 19422167949903851970 * q^73 - 68380237365358617888 * q^77 - 131735321299806049488 * q^79 - 64013993832679681068 * q^83 + 390258202763001297252 * q^85 - 429891446897537246766 * q^89 - 297181701588021496176 * q^91 + 1036372406649019824696 * q^95 - 324059336514148638042 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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