LMFDB - Embedded newform 144.9.e.c.17.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [144,9,Mod(17,144)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("144.17"); S:= CuspForms(chi, 9); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(144, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 9, names="a")

Level:	$ N $	$=$	$ 144 = 2^{4} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	144.e (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,0,0,-616,0,0,0,0,0,36928] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$58.6625198488$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-2}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 2 $ x^2 + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 3^{3} $
Twist minimal:	no (minimal twist has level 36)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			17.1
Root			$-1.41421i$ of defining polynomial
Character	$\chi$	$=$	144.17
Dual form			144.9.e.c.17.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-38.1838i q^{5} -308.000 q^{7} +9316.84i q^{11} +18464.0 q^{13} -118637. i q^{17} -149552. q^{19} +467217. i q^{23} +389167. q^{25} -585854. i q^{29} -466532. q^{31} +11760.6i q^{35} -964522. q^{37} +3.57350e6i q^{41} +2.06716e6 q^{43} -3.93094e6i q^{47} -5.66994e6 q^{49} +9.41402e6i q^{53} +355752. q^{55} +7.74947e6i q^{59} -3.76639e6 q^{61} -705025. i q^{65} -2.62235e7 q^{67} +4.45401e7i q^{71} +709136. q^{73} -2.86959e6i q^{77} -3.84657e7 q^{79} +6.04066e7i q^{83} -4.53001e6 q^{85} +6.34896e7i q^{89} -5.68691e6 q^{91} +5.71046e6i q^{95} -1.11271e8 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 616 q^{7} + 36928 q^{13} - 299104 q^{19} + 778334 q^{25} - 933064 q^{31} - 1929044 q^{37} + 4134320 q^{43} - 11339874 q^{49} + 711504 q^{55} - 7532780 q^{61} - 52447024 q^{67} + 1418272 q^{73} - 76931320 q^{79}+ \cdots - 222541376 q^{97}+O(q^{100}) $ 2 * q - 616 * q^7 + 36928 * q^13 - 299104 * q^19 + 778334 * q^25 - 933064 * q^31 - 1929044 * q^37 + 4134320 * q^43 - 11339874 * q^49 + 711504 * q^55 - 7532780 * q^61 - 52447024 * q^67 + 1418272 * q^73 - 76931320 * q^79 - 9060012 * q^85 - 11373824 * q^91 - 222541376 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/144\mathbb{Z}\right)^\times$.

$n$	$37$	$65$	$127$
$\chi(n)$	$1$	$-1$	$1$

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
$5$	− 38.1838i	− 0.0610940i	−0.999533	−	0.0305470i	$-0.990275\pi$
$5$	− 38.1838i	− 0.0610940i	0.999533	−	0.0305470i	$-0.00972493\pi$
$6$	0	0
$7$	−308.000	−0.128280	−0.0641399	−	0.997941i	$-0.520430\pi$
$7$	−308.000	−0.128280	−0.0641399	+	0.997941i	$0.520430\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	9316.84i	0.636353i	0.948032	+	0.318176i	$0.103070\pi$
$11$	9316.84i	0.636353i	−0.948032	+	0.318176i	$0.896930\pi$
$12$	0	0
$13$	18464.0	0.646476	0.323238	−	0.946318i	$-0.395229\pi$
$13$	18464.0	0.646476	0.323238	+	0.946318i	$0.395229\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 118637.i	− 1.42044i	−0.703977	−	0.710222i	$-0.748594\pi$
$17$	− 118637.i	− 1.42044i	0.703977	−	0.710222i	$-0.251406\pi$
$18$	0	0
$19$	−149552.	−1.14757	−0.573783	−	0.819007i	$-0.694525\pi$
$19$	−149552.	−1.14757	−0.573783	+	0.819007i	$0.694525\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	467217.i	1.66958i	0.550569	+	0.834789i	$0.314411\pi$
$23$	467217.i	1.66958i	−0.550569	+	0.834789i	$0.685589\pi$
$24$	0	0
$25$	389167.	0.996268
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 585854.i	− 0.828318i	−0.910205	−	0.414159i	$-0.864076\pi$
$29$	− 585854.i	− 0.828318i	0.910205	−	0.414159i	$-0.135924\pi$
$30$	0	0
$31$	−466532.	−0.505167	−0.252583	−	0.967575i	$-0.581280\pi$
$31$	−466532.	−0.505167	−0.252583	+	0.967575i	$0.581280\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	11760.6i	0.00783713i
$36$	0	0
$37$	−964522.	−0.514642	−0.257321	−	0.966326i	$-0.582840\pi$
$37$	−964522.	−0.514642	−0.257321	+	0.966326i	$0.582840\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	3.57350e6i	1.26462i	0.774717	+	0.632308i	$0.217892\pi$
$41$	3.57350e6i	1.26462i	−0.774717	+	0.632308i	$0.782108\pi$
$42$	0	0
$43$	2.06716e6	0.604645	0.302322	−	0.953206i	$-0.402238\pi$
$43$	2.06716e6	0.604645	0.302322	+	0.953206i	$0.402238\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 3.93094e6i	− 0.805574i	−0.915294	−	0.402787i	$-0.868042\pi$
$47$	− 3.93094e6i	− 0.805574i	0.915294	−	0.402787i	$-0.131958\pi$
$48$	0	0
$49$	−5.66994e6	−0.983544
$50$	0	0
$51$	0	0
$52$	0	0
$53$	9.41402e6i	1.19309i	0.802581	+	0.596543i	$0.203459\pi$
$53$	9.41402e6i	1.19309i	−0.802581	+	0.596543i	$0.796541\pi$
$54$	0	0
$55$	355752.	0.0388773
$56$	0	0
$57$	0	0
$58$	0	0
$59$	7.74947e6i	0.639535i	0.947496	+	0.319767i	$0.103605\pi$
$59$	7.74947e6i	0.639535i	−0.947496	+	0.319767i	$0.896395\pi$
$60$	0	0
$61$	−3.76639e6	−0.272023	−0.136012	−	0.990707i	$-0.543428\pi$
$61$	−3.76639e6	−0.272023	−0.136012	+	0.990707i	$0.543428\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 705025.i	− 0.0394958i
$66$	0	0
$67$	−2.62235e7	−1.30134	−0.650671	−	0.759360i	$-0.725512\pi$
$67$	−2.62235e7	−1.30134	−0.650671	+	0.759360i	$0.725512\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	4.45401e7i	1.75274i	0.481636	+	0.876371i	$0.340043\pi$
$71$	4.45401e7i	1.75274i	−0.481636	+	0.876371i	$0.659957\pi$
$72$	0	0
$73$	709136.	0.0249711	0.0124856	−	0.999922i	$-0.496026\pi$
$73$	709136.	0.0249711	0.0124856	+	0.999922i	$0.496026\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 2.86959e6i	− 0.0816312i
$78$	0	0
$79$	−3.84657e7	−0.987563	−0.493782	−	0.869586i	$-0.664386\pi$
$79$	−3.84657e7	−0.987563	−0.493782	+	0.869586i	$0.664386\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	6.04066e7i	1.27283i	0.771345	+	0.636417i	$0.219584\pi$
$83$	6.04066e7i	1.27283i	−0.771345	+	0.636417i	$0.780416\pi$
$84$	0	0
$85$	−4.53001e6	−0.0867807
$86$	0	0
$87$	0	0
$88$	0	0
$89$	6.34896e7i	1.01191i	0.862559	+	0.505956i	$0.168860\pi$
$89$	6.34896e7i	1.01191i	−0.862559	+	0.505956i	$0.831140\pi$
$90$	0	0
$91$	−5.68691e6	−0.0829299
$92$	0	0
$93$	0	0
$94$	0	0
$95$	5.71046e6i	0.0701095i
$96$	0	0
$97$	−1.11271e8	−1.25688	−0.628440	−	0.777858i	$-0.716306\pi$
$97$	−1.11271e8	−1.25688	−0.628440	+	0.777858i	$0.716306\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	1.70351e8i	1.63704i	0.574479	+	0.818519i	$0.305205\pi$
$101$	1.70351e8i	1.63704i	−0.574479	+	0.818519i	$0.694795\pi$
$102$	0	0
$103$	−3.88148e7	−0.344864	−0.172432	−	0.985021i	$-0.555163\pi$
$103$	−3.88148e7	−0.344864	−0.172432	+	0.985021i	$0.555163\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 1.46528e7i	− 0.111785i	−0.998437	−	0.0558927i	$-0.982200\pi$
$107$	− 1.46528e7i	− 0.111785i	0.998437	−	0.0558927i	$-0.0178005\pi$
$108$	0	0
$109$	−1.97164e8	−1.39676	−0.698380	−	0.715727i	$-0.746095\pi$
$109$	−1.97164e8	−1.39676	−0.698380	+	0.715727i	$0.746095\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	1.74536e8i	1.07046i	0.844706	+	0.535231i	$0.179775\pi$
$113$	1.74536e8i	1.07046i	−0.844706	+	0.535231i	$0.820225\pi$
$114$	0	0
$115$	1.78401e7	0.102001
$116$	0	0
$117$	0	0
$118$	0	0
$119$	3.65402e7i	0.182214i
$120$	0	0
$121$	1.27555e8	0.595055
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 2.97754e7i	− 0.121960i
$126$	0	0
$127$	3.19781e7	0.122924	0.0614621	−	0.998109i	$-0.480424\pi$
$127$	3.19781e7	0.122924	0.0614621	+	0.998109i	$0.480424\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	2.86660e8i	0.973380i	0.873575	+	0.486690i	$0.161796\pi$
$131$	2.86660e8i	0.973380i	−0.873575	+	0.486690i	$0.838204\pi$
$132$	0	0
$133$	4.60620e7	0.147210
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 3.22152e8i	− 0.914489i	−0.889341	−	0.457245i	$-0.848836\pi$
$137$	− 3.22152e8i	− 0.914489i	0.889341	−	0.457245i	$-0.151164\pi$
$138$	0	0
$139$	−6.15554e8	−1.64895	−0.824474	−	0.565899i	$-0.808529\pi$
$139$	−6.15554e8	−1.64895	−0.824474	+	0.565899i	$0.808529\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	1.72026e8i	0.411387i
$144$	0	0
$145$	−2.23701e7	−0.0506053
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 3.57382e8i	− 0.725082i	−0.931968	−	0.362541i	$-0.881909\pi$
$149$	− 3.57382e8i	− 0.725082i	0.931968	−	0.362541i	$-0.118091\pi$
$150$	0	0
$151$	2.47603e8	0.476264	0.238132	−	0.971233i	$-0.423465\pi$
$151$	2.47603e8	0.476264	0.238132	+	0.971233i	$0.423465\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	1.78139e7i	0.0308627i
$156$	0	0
$157$	9.46913e8	1.55852	0.779259	−	0.626703i	$-0.215596\pi$
$157$	9.46913e8	1.55852	0.779259	+	0.626703i	$0.215596\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 1.43903e8i	− 0.214173i
$162$	0	0
$163$	8.83120e8	1.25103	0.625517	−	0.780210i	$-0.284888\pi$
$163$	8.83120e8	1.25103	0.625517	+	0.780210i	$0.284888\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 9.79955e8i	− 1.25991i	−0.776631	−	0.629956i	$-0.783073\pi$
$167$	− 9.79955e8i	− 1.25991i	0.776631	−	0.629956i	$-0.216927\pi$
$168$	0	0
$169$	−4.74811e8	−0.582069
$170$	0	0
$171$	0	0
$172$	0	0
$173$	1.87766e8i	0.209620i	0.994492	+	0.104810i	$0.0334234\pi$
$173$	1.87766e8i	0.209620i	−0.994492	+	0.104810i	$0.966577\pi$
$174$	0	0
$175$	−1.19863e8	−0.127801
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 1.19137e9i	− 1.16047i	−0.814447	−	0.580237i	$-0.802960\pi$
$179$	− 1.19137e9i	− 1.16047i	0.814447	−	0.580237i	$-0.197040\pi$
$180$	0	0
$181$	−1.05118e9	−0.979405	−0.489702	−	0.871890i	$-0.662895\pi$
$181$	−1.05118e9	−0.979405	−0.489702	+	0.871890i	$0.662895\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	3.68291e7i	0.0314416i
$186$	0	0
$187$	1.10532e9	0.903904
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 1.87106e9i	− 1.40590i	−0.711240	−	0.702949i	$-0.751866\pi$
$191$	− 1.87106e9i	− 1.40590i	0.711240	−	0.702949i	$-0.248134\pi$
$192$	0	0
$193$	2.04392e9	1.47311	0.736553	−	0.676380i	$-0.236452\pi$
$193$	2.04392e9	1.47311	0.736553	+	0.676380i	$0.236452\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 9.02247e8i	− 0.599047i	−0.954089	−	0.299523i	$-0.903172\pi$
$197$	− 9.02247e8i	− 0.599047i	0.954089	−	0.299523i	$-0.0968276\pi$
$198$	0	0
$199$	2.57553e9	1.64231	0.821154	−	0.570707i	$-0.193331\pi$
$199$	2.57553e9	1.64231	0.821154	+	0.570707i	$0.193331\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	1.80443e8i	0.106257i
$204$	0	0
$205$	1.36450e8	0.0772605
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 1.39335e9i	− 0.730257i
$210$	0	0
$211$	−3.40870e9	−1.71973	−0.859863	−	0.510525i	$-0.829451\pi$
$211$	−3.40870e9	−1.71973	−0.859863	+	0.510525i	$0.829451\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	− 7.89320e7i	− 0.0369402i
$216$	0	0
$217$	1.43692e8	0.0648027
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 2.19051e9i	− 0.918283i
$222$	0	0
$223$	1.02091e9	0.412826	0.206413	−	0.978465i	$-0.433821\pi$
$223$	1.02091e9	0.412826	0.206413	+	0.978465i	$0.433821\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	3.58798e9i	1.35128i	0.737230	+	0.675642i	$0.236134\pi$
$227$	3.58798e9i	1.35128i	−0.737230	+	0.675642i	$0.763866\pi$
$228$	0	0
$229$	4.35815e9	1.58475	0.792375	−	0.610035i	$-0.208845\pi$
$229$	4.35815e9	1.58475	0.792375	+	0.610035i	$0.208845\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 1.29656e9i	− 0.439916i	−0.975509	−	0.219958i	$-0.929408\pi$
$233$	− 1.29656e9i	− 0.439916i	0.975509	−	0.219958i	$-0.0705920\pi$
$234$	0	0
$235$	−1.50098e8	−0.0492157
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 1.47964e9i	− 0.453485i	−0.973955	−	0.226743i	$-0.927192\pi$
$239$	− 1.47964e9i	− 0.453485i	0.973955	−	0.226743i	$-0.0728077\pi$
$240$	0	0
$241$	−1.31031e9	−0.388424	−0.194212	−	0.980960i	$-0.562215\pi$
$241$	−1.31031e9	−0.388424	−0.194212	+	0.980960i	$0.562215\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	2.16500e8i	0.0600887i
$246$	0	0
$247$	−2.76133e9	−0.741874
$248$	0	0
$249$	0	0
$250$	0	0
$251$	2.38442e9i	0.600742i	0.953822	+	0.300371i	$0.0971105\pi$
$251$	2.38442e9i	0.600742i	−0.953822	+	0.300371i	$0.902890\pi$
$252$	0	0
$253$	−4.35298e9	−1.06244
$254$	0	0
$255$	0	0
$256$	0	0
$257$	1.89123e9i	0.433524i	0.976225	+	0.216762i	$0.0695495\pi$
$257$	1.89123e9i	0.433524i	−0.976225	+	0.216762i	$0.930450\pi$
$258$	0	0
$259$	2.97073e8	0.0660182
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 5.33367e9i	− 1.11482i	−0.830239	−	0.557408i	$-0.811796\pi$
$263$	− 5.33367e9i	− 1.11482i	0.830239	−	0.557408i	$-0.188204\pi$
$264$	0	0
$265$	3.59463e8	0.0728904
$266$	0	0
$267$	0	0
$268$	0	0
$269$	7.81908e9i	1.49330i	0.665218	+	0.746650i	$0.268339\pi$
$269$	7.81908e9i	1.49330i	−0.665218	+	0.746650i	$0.731661\pi$
$270$	0	0
$271$	5.87364e9	1.08901	0.544503	−	0.838759i	$-0.316718\pi$
$271$	5.87364e9	1.08901	0.544503	+	0.838759i	$0.316718\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	3.62581e9i	0.633977i
$276$	0	0
$277$	−4.66702e9	−0.792721	−0.396360	−	0.918095i	$-0.629727\pi$
$277$	−4.66702e9	−0.792721	−0.396360	+	0.918095i	$0.629727\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	6.59175e9i	1.05724i	0.848857	+	0.528622i	$0.177291\pi$
$281$	6.59175e9i	1.05724i	−0.848857	+	0.528622i	$0.822709\pi$
$282$	0	0
$283$	−4.70364e9	−0.733311	−0.366655	−	0.930357i	$-0.619497\pi$
$283$	−4.70364e9	−0.733311	−0.366655	+	0.930357i	$0.619497\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 1.10064e9i	− 0.162225i
$288$	0	0
$289$	−7.09897e9	−1.01766
$290$	0	0
$291$	0	0
$292$	0	0
$293$	5.71651e9i	0.775641i	0.921735	+	0.387821i	$0.126772\pi$
$293$	5.71651e9i	0.775641i	−0.921735	+	0.387821i	$0.873228\pi$
$294$	0	0
$295$	2.95904e8	0.0390717
$296$	0	0
$297$	0	0
$298$	0	0
$299$	8.62669e9i	1.07934i
$300$	0	0
$301$	−6.36685e8	−0.0775638
$302$	0	0
$303$	0	0
$304$	0	0
$305$	1.43815e8i	0.0166190i
$306$	0	0
$307$	−1.35171e10	−1.52170	−0.760851	−	0.648926i	$-0.775218\pi$
$307$	−1.35171e10	−1.52170	−0.760851	+	0.648926i	$0.775218\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	3.73908e8i	0.0399690i	0.999800	+	0.0199845i	$0.00636169\pi$
$311$	3.73908e8i	0.0399690i	−0.999800	+	0.0199845i	$0.993638\pi$
$312$	0	0
$313$	−4.77843e8	−0.0497861	−0.0248931	−	0.999690i	$-0.507925\pi$
$313$	−4.77843e8	−0.0497861	−0.0248931	+	0.999690i	$0.507925\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 7.78932e9i	− 0.771370i	−0.922631	−	0.385685i	$-0.873965\pi$
$317$	− 7.78932e9i	− 0.771370i	0.922631	−	0.385685i	$-0.126035\pi$
$318$	0	0
$319$	5.45830e9	0.527102
$320$	0	0
$321$	0	0
$322$	0	0
$323$	1.77424e10i	1.63005i
$324$	0	0
$325$	7.18558e9	0.644063
$326$	0	0
$327$	0	0
$328$	0	0
$329$	1.21073e9i	0.103339i
$330$	0	0
$331$	−1.62253e10	−1.35170	−0.675852	−	0.737037i	$-0.736224\pi$
$331$	−1.62253e10	−1.35170	−0.675852	+	0.737037i	$0.736224\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	1.00131e9i	0.0795043i
$336$	0	0
$337$	4.39525e9	0.340772	0.170386	−	0.985377i	$-0.445498\pi$
$337$	4.39525e9	0.340772	0.170386	+	0.985377i	$0.445498\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 4.34660e9i	− 0.321464i
$342$	0	0
$343$	3.52190e9	0.254449
$344$	0	0
$345$	0	0
$346$	0	0
$347$	2.31605e10i	1.59746i	0.601690	+	0.798730i	$0.294494\pi$
$347$	2.31605e10i	1.59746i	−0.601690	+	0.798730i	$0.705506\pi$
$348$	0	0
$349$	−4.51839e9	−0.304566	−0.152283	−	0.988337i	$-0.548663\pi$
$349$	−4.51839e9	−0.304566	−0.152283	+	0.988337i	$0.548663\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	5.62597e9i	0.362325i	0.983453	+	0.181163i	$0.0579860\pi$
$353$	5.62597e9i	0.362325i	−0.983453	+	0.181163i	$0.942014\pi$
$354$	0	0
$355$	1.70071e9	0.107082
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 3.11191e10i	− 1.87348i	−0.350027	−	0.936740i	$-0.613827\pi$
$359$	− 3.11191e10i	− 1.87348i	0.350027	−	0.936740i	$-0.386173\pi$
$360$	0	0
$361$	5.38224e9	0.316909
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 2.70775e7i	− 0.00152559i
$366$	0	0
$367$	1.56434e9	0.0862318	0.0431159	−	0.999070i	$-0.486272\pi$
$367$	1.56434e9	0.0862318	0.0431159	+	0.999070i	$0.486272\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 2.89952e9i	− 0.153049i
$372$	0	0
$373$	−2.26218e10	−1.16867	−0.584334	−	0.811513i	$-0.698644\pi$
$373$	−2.26218e10	−1.16867	−0.584334	+	0.811513i	$0.698644\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 1.08172e10i	− 0.535488i
$378$	0	0
$379$	3.93940e8	0.0190929	0.00954647	−	0.999954i	$-0.496961\pi$
$379$	3.93940e8	0.0190929	0.00954647	+	0.999954i	$0.496961\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 4.12365e10i	− 1.91640i	−0.286092	−	0.958202i	$-0.592356\pi$
$383$	− 4.12365e10i	− 1.91640i	0.286092	−	0.958202i	$-0.407644\pi$
$384$	0	0
$385$	−1.09572e8	−0.00498718
$386$	0	0
$387$	0	0
$388$	0	0
$389$	2.07290e10i	0.905274i	0.891695	+	0.452637i	$0.149517\pi$
$389$	2.07290e10i	0.905274i	−0.891695	+	0.452637i	$0.850483\pi$
$390$	0	0
$391$	5.54292e10	2.37154
$392$	0	0
$393$	0	0
$394$	0	0
$395$	1.46876e9i	0.0603342i
$396$	0	0
$397$	3.50017e10	1.40905	0.704526	−	0.709678i	$-0.251160\pi$
$397$	3.50017e10	1.40905	0.704526	+	0.709678i	$0.251160\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	2.48313e9i	0.0960333i	0.998847	+	0.0480166i	$0.0152900\pi$
$401$	2.48313e9i	0.0960333i	−0.998847	+	0.0480166i	$0.984710\pi$
$402$	0	0
$403$	−8.61405e9	−0.326578
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 8.98630e9i	− 0.327494i
$408$	0	0
$409$	−2.76424e10	−0.987832	−0.493916	−	0.869510i	$-0.664435\pi$
$409$	−2.76424e10	−0.987832	−0.493916	+	0.869510i	$0.664435\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 2.38684e9i	− 0.0820394i
$414$	0	0
$415$	2.30655e9	0.0777626
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 6.32695e9i	− 0.205276i	−0.994719	−	0.102638i	$-0.967272\pi$
$419$	− 6.32695e9i	− 0.205276i	0.994719	−	0.102638i	$-0.0327283\pi$
$420$	0	0
$421$	1.98153e10	0.630772	0.315386	−	0.948963i	$-0.397866\pi$
$421$	1.98153e10	0.630772	0.315386	+	0.948963i	$0.397866\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 4.61696e10i	− 1.41514i
$426$	0	0
$427$	1.16005e9	0.0348951
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 2.61939e10i	− 0.759086i	−0.925174	−	0.379543i	$-0.876081\pi$
$431$	− 2.61939e10i	− 0.759086i	0.925174	−	0.379543i	$-0.123919\pi$
$432$	0	0
$433$	4.04686e10	1.15124	0.575621	−	0.817717i	$-0.304761\pi$
$433$	4.04686e10	1.15124	0.575621	+	0.817717i	$0.304761\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 6.98732e10i	− 1.91595i
$438$	0	0
$439$	1.22359e9	0.0329441	0.0164721	−	0.999864i	$-0.494757\pi$
$439$	1.22359e9	0.0329441	0.0164721	+	0.999864i	$0.494757\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	1.42135e10i	0.369051i	0.982828	+	0.184526i	$0.0590749\pi$
$443$	1.42135e10i	0.369051i	−0.982828	+	0.184526i	$0.940925\pi$
$444$	0	0
$445$	2.42427e9	0.0618218
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 5.31389e9i	− 0.130746i	−0.997861	−	0.0653728i	$-0.979176\pi$
$449$	− 5.31389e9i	− 0.130746i	0.997861	−	0.0653728i	$-0.0208236\pi$
$450$	0	0
$451$	−3.32938e10	−0.804742
$452$	0	0
$453$	0	0
$454$	0	0
$455$	2.17148e8i	0.00506652i
$456$	0	0
$457$	4.66165e10	1.06875	0.534374	−	0.845248i	$-0.320547\pi$
$457$	4.66165e10	1.06875	0.534374	+	0.845248i	$0.320547\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 2.02949e10i	− 0.449349i	−0.974434	−	0.224674i	$-0.927868\pi$
$461$	− 2.02949e10i	− 0.449349i	0.974434	−	0.224674i	$-0.0721318\pi$
$462$	0	0
$463$	5.91381e10	1.28689	0.643447	−	0.765490i	$-0.277503\pi$
$463$	5.91381e10	1.28689	0.643447	+	0.765490i	$0.277503\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	8.72934e10i	1.83533i	0.397356	+	0.917664i	$0.369928\pi$
$467$	8.72934e10i	1.83533i	−0.397356	+	0.917664i	$0.630072\pi$
$468$	0	0
$469$	8.07684e9	0.166936
$470$	0	0
$471$	0	0
$472$	0	0
$473$	1.92594e10i	0.384767i
$474$	0	0
$475$	−5.82007e10	−1.14328
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 7.77943e10i	− 1.47777i	−0.673833	−	0.738883i	$-0.735353\pi$
$479$	− 7.77943e10i	− 1.47777i	0.673833	−	0.738883i	$-0.264647\pi$
$480$	0	0
$481$	−1.78089e10	−0.332704
$482$	0	0
$483$	0	0
$484$	0	0
$485$	4.24873e9i	0.0767879i
$486$	0	0
$487$	−2.25316e10	−0.400568	−0.200284	−	0.979738i	$-0.564186\pi$
$487$	−2.25316e10	−0.400568	−0.200284	+	0.979738i	$0.564186\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	1.53255e9i	0.0263688i	0.999913	+	0.0131844i	$0.00419684\pi$
$491$	1.53255e9i	0.0263688i	−0.999913	+	0.0131844i	$0.995803\pi$
$492$	0	0
$493$	−6.95039e10	−1.17658
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 1.37184e10i	− 0.224842i
$498$	0	0
$499$	9.06869e10	1.46266	0.731329	−	0.682025i	$-0.238901\pi$
$499$	9.06869e10	1.46266	0.731329	+	0.682025i	$0.238901\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 5.91507e10i	− 0.924034i	−0.886871	−	0.462017i	$-0.847126\pi$
$503$	− 5.91507e10i	− 0.924034i	0.886871	−	0.462017i	$-0.152874\pi$
$504$	0	0
$505$	6.50464e9	0.100013
$506$	0	0
$507$	0	0
$508$	0	0
$509$	9.78868e10i	1.45832i	0.684343	+	0.729160i	$0.260089\pi$
$509$	9.78868e10i	1.45832i	−0.684343	+	0.729160i	$0.739911\pi$
$510$	0	0
$511$	−2.18414e8	−0.00320329
$512$	0	0
$513$	0	0
$514$	0	0
$515$	1.48209e9i	0.0210691i
$516$	0	0
$517$	3.66240e10	0.512629
$518$	0	0
$519$	0	0
$520$	0	0
$521$	7.22778e10i	0.980967i	0.871451	+	0.490483i	$0.163180\pi$
$521$	7.22778e10i	0.980967i	−0.871451	+	0.490483i	$0.836820\pi$
$522$	0	0
$523$	1.12539e10	0.150417	0.0752085	−	0.997168i	$-0.476038\pi$
$523$	1.12539e10	0.150417	0.0752085	+	0.997168i	$0.476038\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	5.53479e10i	0.717561i
$528$	0	0
$529$	−1.39980e11	−1.78749
$530$	0	0
$531$	0	0
$532$	0	0
$533$	6.59812e10i	0.817544i
$534$	0	0
$535$	−5.59499e8	−0.00682942
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 5.28259e10i	− 0.625881i
$540$	0	0
$541$	−4.32921e10	−0.505382	−0.252691	−	0.967547i	$-0.581316\pi$
$541$	−4.32921e10	−0.505382	−0.252691	+	0.967547i	$0.581316\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	7.52847e9i	0.0853337i
$546$	0	0
$547$	−1.40671e11	−1.57128	−0.785641	−	0.618682i	$-0.787667\pi$
$547$	−1.40671e11	−1.57128	−0.785641	+	0.618682i	$0.787667\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	8.76156e10i	0.950550i
$552$	0	0
$553$	1.18474e10	0.126684
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 5.91458e10i	− 0.614473i	−0.951633	−	0.307237i	$-0.900596\pi$
$557$	− 5.91458e10i	− 0.614473i	0.951633	−	0.307237i	$-0.0994043\pi$
$558$	0	0
$559$	3.81680e10	0.390888
$560$	0	0
$561$	0	0
$562$	0	0
$563$	2.18458e10i	0.217438i	0.994073	+	0.108719i	$0.0346748\pi$
$563$	2.18458e10i	0.217438i	−0.994073	+	0.108719i	$0.965325\pi$
$564$	0	0
$565$	6.66444e9	0.0653988
$566$	0	0
$567$	0	0
$568$	0	0
$569$	8.81079e10i	0.840555i	0.907396	+	0.420277i	$0.138067\pi$
$569$	8.81079e10i	0.840555i	−0.907396	+	0.420277i	$0.861933\pi$
$570$	0	0
$571$	−4.66558e10	−0.438895	−0.219448	−	0.975624i	$-0.570426\pi$
$571$	−4.66558e10	−0.438895	−0.219448	+	0.975624i	$0.570426\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	1.81825e11i	1.66335i
$576$	0	0
$577$	5.51677e10	0.497716	0.248858	−	0.968540i	$-0.419945\pi$
$577$	5.51677e10	0.497716	0.248858	+	0.968540i	$0.419945\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 1.86052e10i	− 0.163279i
$582$	0	0
$583$	−8.77089e10	−0.759223
$584$	0	0
$585$	0	0
$586$	0	0
$587$	2.11137e11i	1.77833i	0.457590	+	0.889163i	$0.348713\pi$
$587$	2.11137e11i	1.77833i	−0.457590	+	0.889163i	$0.651287\pi$
$588$	0	0
$589$	6.97708e10	0.579712
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 1.54170e10i	− 0.124675i	−0.998055	−	0.0623376i	$-0.980144\pi$
$593$	− 1.54170e10i	− 0.124675i	0.998055	−	0.0623376i	$-0.0198555\pi$
$594$	0	0
$595$	1.39524e9	0.0111322
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 3.08975e10i	− 0.240003i	−0.992774	−	0.120001i	$-0.961710\pi$
$599$	− 3.08975e10i	− 0.240003i	0.992774	−	0.120001i	$-0.0382899\pi$
$600$	0	0
$601$	−8.71875e10	−0.668277	−0.334138	−	0.942524i	$-0.608445\pi$
$601$	−8.71875e10	−0.668277	−0.334138	+	0.942524i	$0.608445\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 4.87055e9i	− 0.0363543i
$606$	0	0
$607$	−2.50783e11	−1.84732	−0.923662	−	0.383208i	$-0.874819\pi$
$607$	−2.50783e11	−1.84732	−0.923662	+	0.383208i	$0.874819\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 7.25809e10i	− 0.520784i
$612$	0	0
$613$	−1.15871e11	−0.820599	−0.410300	−	0.911951i	$-0.634576\pi$
$613$	−1.15871e11	−0.820599	−0.410300	+	0.911951i	$0.634576\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 1.94008e11i	− 1.33869i	−0.742954	−	0.669343i	$-0.766576\pi$
$617$	− 1.94008e11i	− 1.33869i	0.742954	−	0.669343i	$-0.233424\pi$
$618$	0	0
$619$	8.55898e10	0.582988	0.291494	−	0.956573i	$-0.405848\pi$
$619$	8.55898e10	0.582988	0.291494	+	0.956573i	$0.405848\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 1.95548e10i	− 0.129808i
$624$	0	0
$625$	1.50881e11	0.988816
$626$	0	0
$627$	0	0
$628$	0	0
$629$	1.14428e11i	0.731021i
$630$	0	0
$631$	1.01422e11	0.639758	0.319879	−	0.947458i	$-0.396358\pi$
$631$	1.01422e11	0.639758	0.319879	+	0.947458i	$0.396358\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 1.22104e9i	− 0.00750993i
$636$	0	0
$637$	−1.04690e11	−0.635838
$638$	0	0
$639$	0	0
$640$	0	0
$641$	1.64922e11i	0.976893i	0.872594	+	0.488446i	$0.162436\pi$
$641$	1.64922e11i	0.976893i	−0.872594	+	0.488446i	$0.837564\pi$
$642$	0	0
$643$	7.82285e10	0.457637	0.228818	−	0.973469i	$-0.426514\pi$
$643$	7.82285e10	0.457637	0.228818	+	0.973469i	$0.426514\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	8.07501e10i	0.460814i	0.973094	+	0.230407i	$0.0740058\pi$
$647$	8.07501e10i	0.460814i	−0.973094	+	0.230407i	$0.925994\pi$
$648$	0	0
$649$	−7.22006e10	−0.406970
$650$	0	0
$651$	0	0
$652$	0	0
$653$	2.31263e11i	1.27190i	0.771731	+	0.635949i	$0.219391\pi$
$653$	2.31263e11i	1.27190i	−0.771731	+	0.635949i	$0.780609\pi$
$654$	0	0
$655$	1.09458e10	0.0594677
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 3.23651e11i	− 1.71607i	−0.513589	−	0.858036i	$-0.671684\pi$
$659$	− 3.23651e11i	− 1.71607i	0.513589	−	0.858036i	$-0.328316\pi$
$660$	0	0
$661$	1.62923e11	0.853448	0.426724	−	0.904382i	$-0.359668\pi$
$661$	1.62923e11	0.853448	0.426724	+	0.904382i	$0.359668\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 1.75882e9i	− 0.00899363i
$666$	0	0
$667$	2.73720e11	1.38294
$668$	0	0
$669$	0	0
$670$	0	0
$671$	− 3.50908e10i	− 0.173103i
$672$	0	0
$673$	−2.27514e11	−1.10904	−0.554521	−	0.832170i	$-0.687099\pi$
$673$	−2.27514e11	−1.10904	−0.554521	+	0.832170i	$0.687099\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 3.40015e11i	− 1.61861i	−0.587387	−	0.809306i	$-0.699843\pi$
$677$	− 3.40015e11i	− 1.61861i	0.587387	−	0.809306i	$-0.300157\pi$
$678$	0	0
$679$	3.42714e10	0.161232
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 3.79283e11i	− 1.74293i	−0.490455	−	0.871467i	$-0.663169\pi$
$683$	− 3.79283e11i	− 1.74293i	0.490455	−	0.871467i	$-0.336831\pi$
$684$	0	0
$685$	−1.23010e10	−0.0558698
$686$	0	0
$687$	0	0
$688$	0	0
$689$	1.73820e11i	0.771301i
$690$	0	0
$691$	−3.21427e10	−0.140984	−0.0704921	−	0.997512i	$-0.522457\pi$
$691$	−3.21427e10	−0.140984	−0.0704921	+	0.997512i	$0.522457\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	2.35042e10i	0.100741i
$696$	0	0
$697$	4.23950e11	1.79632
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 1.42670e11i	− 0.590828i	−0.955369	−	0.295414i	$-0.904542\pi$
$701$	− 1.42670e11i	− 0.590828i	0.955369	−	0.295414i	$-0.0954576\pi$
$702$	0	0
$703$	1.44246e11	0.590586
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 5.24681e10i	− 0.209999i
$708$	0	0
$709$	−2.64697e11	−1.04752	−0.523762	−	0.851865i	$-0.675472\pi$
$709$	−2.64697e11	−1.04752	−0.523762	+	0.851865i	$0.675472\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 2.17971e11i	− 0.843415i
$714$	0	0
$715$	6.56860e9	0.0251333
$716$	0	0
$717$	0	0
$718$	0	0
$719$	2.37840e11i	0.889959i	0.895541	+	0.444979i	$0.146789\pi$
$719$	2.37840e11i	0.889959i	−0.895541	+	0.444979i	$0.853211\pi$
$720$	0	0
$721$	1.19549e10	0.0442391
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 2.27995e11i	− 0.825226i
$726$	0	0
$727$	−4.73481e11	−1.69498	−0.847490	−	0.530811i	$-0.821887\pi$
$727$	−4.73481e11	−1.69498	−0.847490	+	0.530811i	$0.821887\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 2.45242e11i	− 0.858864i
$732$	0	0
$733$	5.42473e11	1.87915	0.939577	−	0.342338i	$-0.111219\pi$
$733$	5.42473e11	1.87915	0.939577	+	0.342338i	$0.111219\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 2.44320e11i	− 0.828113i
$738$	0	0
$739$	4.30205e11	1.44244	0.721220	−	0.692706i	$-0.243582\pi$
$739$	4.30205e11	1.44244	0.721220	+	0.692706i	$0.243582\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	2.89115e11i	0.948671i	0.880344	+	0.474336i	$0.157312\pi$
$743$	2.89115e11i	0.948671i	−0.880344	+	0.474336i	$0.842688\pi$
$744$	0	0
$745$	−1.36462e10	−0.0442982
$746$	0	0
$747$	0	0
$748$	0	0
$749$	4.51306e9i	0.0143398i
$750$	0	0
$751$	−1.20953e10	−0.0380240	−0.0190120	−	0.999819i	$-0.506052\pi$
$751$	−1.20953e10	−0.0380240	−0.0190120	+	0.999819i	$0.506052\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 9.45441e9i	− 0.0290969i
$756$	0	0
$757$	4.84047e11	1.47402	0.737012	−	0.675880i	$-0.236236\pi$
$757$	4.84047e11	1.47402	0.737012	+	0.675880i	$0.236236\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 5.43604e11i	− 1.62085i	−0.585839	−	0.810427i	$-0.699235\pi$
$761$	− 5.43604e11i	− 1.62085i	0.585839	−	0.810427i	$-0.300765\pi$
$762$	0	0
$763$	6.07265e10	0.179176
$764$	0	0
$765$	0	0
$766$	0	0
$767$	1.43086e11i	0.413444i
$768$	0	0
$769$	1.11279e11	0.318206	0.159103	−	0.987262i	$-0.449140\pi$
$769$	1.11279e11	0.318206	0.159103	+	0.987262i	$0.449140\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 8.62777e9i	− 0.0241647i	−0.999927	−	0.0120823i	$-0.996154\pi$
$773$	− 8.62777e9i	− 0.0241647i	0.999927	−	0.0120823i	$-0.00384602\pi$
$774$	0	0
$775$	−1.81559e11	−0.503281
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 5.34425e11i	− 1.45123i
$780$	0	0
$781$	−4.14973e11	−1.11536
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 3.61567e10i	− 0.0952161i
$786$	0	0
$787$	3.44168e11	0.897163	0.448582	−	0.893742i	$-0.351929\pi$
$787$	3.44168e11	0.897163	0.448582	+	0.893742i	$0.351929\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 5.37571e10i	− 0.137319i
$792$	0	0
$793$	−6.95426e10	−0.175856
$794$	0	0
$795$	0	0
$796$	0	0
$797$	8.55307e10i	0.211977i	0.994367	+	0.105989i	$0.0338007\pi$
$797$	8.55307e10i	0.211977i	−0.994367	+	0.105989i	$0.966199\pi$
$798$	0	0
$799$	−4.66355e11	−1.14427
$800$	0	0
$801$	0	0
$802$	0	0
$803$	6.60691e9i	0.0158904i
$804$	0	0
$805$	−5.49475e9	−0.0130847
$806$	0	0
$807$	0	0
$808$	0	0
$809$	4.54082e11i	1.06008i	0.847972	+	0.530041i	$0.177824\pi$
$809$	4.54082e11i	1.06008i	−0.847972	+	0.530041i	$0.822176\pi$
$810$	0	0
$811$	1.65545e11	0.382677	0.191339	−	0.981524i	$-0.438717\pi$
$811$	1.65545e11	0.382677	0.191339	+	0.981524i	$0.438717\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	− 3.37209e10i	− 0.0764308i
$816$	0	0
$817$	−3.09148e11	−0.693870
$818$	0	0
$819$	0	0
$820$	0	0
$821$	3.33968e11i	0.735076i	0.930008	+	0.367538i	$0.119799\pi$
$821$	3.33968e11i	0.735076i	−0.930008	+	0.367538i	$0.880201\pi$
$822$	0	0
$823$	5.69842e11	1.24210	0.621048	−	0.783772i	$-0.286707\pi$
$823$	5.69842e11	1.24210	0.621048	+	0.783772i	$0.286707\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 1.20093e11i	− 0.256742i	−0.991726	−	0.128371i	$-0.959025\pi$
$827$	− 1.20093e11i	− 0.256742i	0.991726	−	0.128371i	$-0.0409748\pi$
$828$	0	0
$829$	−3.28795e11	−0.696158	−0.348079	−	0.937465i	$-0.613166\pi$
$829$	−3.28795e11	−0.696158	−0.348079	+	0.937465i	$0.613166\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	6.72664e11i	1.39707i
$834$	0	0
$835$	−3.74184e10	−0.0769731
$836$	0	0
$837$	0	0
$838$	0	0
$839$	2.31570e11i	0.467341i	0.972316	+	0.233670i	$0.0750737\pi$
$839$	2.31570e11i	0.467341i	−0.972316	+	0.233670i	$0.924926\pi$
$840$	0	0
$841$	1.57022e11	0.313889
$842$	0	0
$843$	0	0
$844$	0	0
$845$	1.81301e10i	0.0355609i
$846$	0	0
$847$	−3.92871e10	−0.0763336
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 4.50641e11i	− 0.859235i
$852$	0	0
$853$	−2.07386e11	−0.391727	−0.195864	−	0.980631i	$-0.562751\pi$
$853$	−2.07386e11	−0.391727	−0.195864	+	0.980631i	$0.562751\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	2.44983e11i	0.454164i	0.973876	+	0.227082i	$0.0729185\pi$
$857$	2.44983e11i	0.454164i	−0.973876	+	0.227082i	$0.927082\pi$
$858$	0	0
$859$	1.57885e11	0.289981	0.144991	−	0.989433i	$-0.453685\pi$
$859$	1.57885e11	0.289981	0.144991	+	0.989433i	$0.453685\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	1.15251e11i	0.207780i	0.994589	+	0.103890i	$0.0331290\pi$
$863$	1.15251e11i	0.207780i	−0.994589	+	0.103890i	$0.966871\pi$
$864$	0	0
$865$	7.16960e9	0.0128065
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 3.58378e11i	− 0.628438i
$870$	0	0
$871$	−4.84191e11	−0.841287
$872$	0	0
$873$	0	0
$874$	0	0
$875$	9.17082e9i	0.0156450i
$876$	0	0
$877$	−7.38618e11	−1.24859	−0.624297	−	0.781187i	$-0.714615\pi$
$877$	−7.38618e11	−1.24859	−0.624297	+	0.781187i	$0.714615\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 2.88079e10i	− 0.0478198i	−0.999714	−	0.0239099i	$-0.992389\pi$
$881$	− 2.88079e10i	− 0.0478198i	0.999714	−	0.0239099i	$-0.00761148\pi$
$882$	0	0
$883$	−5.52423e11	−0.908718	−0.454359	−	0.890819i	$-0.650132\pi$
$883$	−5.52423e11	−0.908718	−0.454359	+	0.890819i	$0.650132\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	1.11814e12i	1.80635i	0.429275	+	0.903174i	$0.358769\pi$
$887$	1.11814e12i	1.80635i	−0.429275	+	0.903174i	$0.641231\pi$
$888$	0	0
$889$	−9.84924e9	−0.0157687
$890$	0	0
$891$	0	0
$892$	0	0
$893$	5.87880e11i	0.924449i
$894$	0	0
$895$	−4.54911e10	−0.0708981
$896$	0	0
$897$	0	0
$898$	0	0
$899$	2.73319e11i	0.418439i
$900$	0	0
$901$	1.11685e12	1.69471
$902$	0	0
$903$	0	0
$904$	0	0
$905$	4.01380e10i	0.0598358i
$906$	0	0
$907$	−5.83483e11	−0.862183	−0.431091	−	0.902308i	$-0.641871\pi$
$907$	−5.83483e11	−0.862183	−0.431091	+	0.902308i	$0.641871\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	9.16715e11i	1.33095i	0.746422	+	0.665473i	$0.231770\pi$
$911$	9.16715e11i	1.33095i	−0.746422	+	0.665473i	$0.768230\pi$
$912$	0	0
$913$	−5.62798e11	−0.809971
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 8.82913e10i	− 0.124865i
$918$	0	0
$919$	−6.20521e11	−0.869951	−0.434975	−	0.900442i	$-0.643243\pi$
$919$	−6.20521e11	−0.869951	−0.434975	+	0.900442i	$0.643243\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	8.22389e11i	1.13311i
$924$	0	0
$925$	−3.75360e11	−0.512721
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 7.45299e11i	− 1.00062i	−0.865847	−	0.500309i	$-0.833220\pi$
$929$	− 7.45299e11i	− 1.00062i	0.865847	−	0.500309i	$-0.166780\pi$
$930$	0	0
$931$	8.47950e11	1.12868
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 4.22053e10i	− 0.0552231i
$936$	0	0
$937$	5.37546e11	0.697360	0.348680	−	0.937242i	$-0.386630\pi$
$937$	5.37546e11	0.697360	0.348680	+	0.937242i	$0.386630\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	5.84470e11i	0.745424i	0.927947	+	0.372712i	$0.121572\pi$
$941$	5.84470e11i	0.745424i	−0.927947	+	0.372712i	$0.878428\pi$
$942$	0	0
$943$	−1.66960e12	−2.11138
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 4.46879e11i	− 0.555636i	−0.960634	−	0.277818i	$-0.910389\pi$
$947$	− 4.46879e11i	− 0.555636i	0.960634	−	0.277818i	$-0.0896112\pi$
$948$	0	0
$949$	1.30935e10	0.0161432
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 7.28928e11i	− 0.883717i	−0.897085	−	0.441858i	$-0.854319\pi$
$953$	− 7.28928e11i	− 0.883717i	0.897085	−	0.441858i	$-0.145681\pi$
$954$	0	0
$955$	−7.14441e10	−0.0858920
$956$	0	0
$957$	0	0
$958$	0	0
$959$	9.92228e10i	0.117311i
$960$	0	0
$961$	−6.35239e11	−0.744807
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 7.80444e10i	− 0.0899980i
$966$	0	0
$967$	1.11861e12	1.27930	0.639650	−	0.768666i	$-0.279079\pi$
$967$	1.11861e12	1.27930	0.639650	+	0.768666i	$0.279079\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	8.66426e11i	0.974663i	0.873217	+	0.487331i	$0.162030\pi$
$971$	8.66426e11i	0.974663i	−0.873217	+	0.487331i	$0.837970\pi$
$972$	0	0
$973$	1.89591e11	0.211527
$974$	0	0
$975$	0	0
$976$	0	0
$977$	1.78648e12i	1.96074i	0.197176	+	0.980368i	$0.436823\pi$
$977$	1.78648e12i	1.96074i	−0.197176	+	0.980368i	$0.563177\pi$
$978$	0	0
$979$	−5.91523e11	−0.643933
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 5.77446e11i	− 0.618439i	−0.950991	−	0.309220i	$-0.899932\pi$
$983$	− 5.77446e11i	− 0.618439i	0.950991	−	0.309220i	$-0.100068\pi$
$984$	0	0
$985$	−3.44512e10	−0.0365982
$986$	0	0
$987$	0	0
$988$	0	0
$989$	9.65811e11i	1.00950i
$990$	0	0
$991$	1.02797e12	1.06582	0.532912	−	0.846171i	$-0.321098\pi$
$991$	1.02797e12	1.06582	0.532912	+	0.846171i	$0.321098\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 9.83435e10i	− 0.100335i
$996$	0	0
$997$	−6.19055e11	−0.626540	−0.313270	−	0.949664i	$-0.601424\pi$
$997$	−6.19055e11	−0.626540	−0.313270	+	0.949664i	$0.601424\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	144.9.e.c.17.1		2
3.2	odd	2	inner	144.9.e.c.17.2		2
4.3	odd	2		36.9.c.a.17.1	✓	2
12.11	even	2		36.9.c.a.17.2	yes	2
36.7	odd	6		324.9.g.d.53.2		4
36.11	even	6		324.9.g.d.53.1		4
36.23	even	6		324.9.g.d.269.2		4
36.31	odd	6		324.9.g.d.269.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
36.9.c.a.17.1	✓	2	4.3	odd	2
36.9.c.a.17.2	yes	2	12.11	even	2
144.9.e.c.17.1		2	1.1	even	1	trivial
144.9.e.c.17.2		2	3.2	odd	2	inner
324.9.g.d.53.1		4	36.11	even	6
324.9.g.d.53.2		4	36.7	odd	6
324.9.g.d.269.1		4	36.31	odd	6
324.9.g.d.269.2		4	36.23	even	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 144 = 2^{4} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	144.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-38.1838i q^{5} -308.000 q^{7} +9316.84i q^{11} +18464.0 q^{13} -118637. i q^{17} -149552. q^{19} +467217. i q^{23} +389167. q^{25} -585854. i q^{29} -466532. q^{31} +11760.6i q^{35} -964522. q^{37} +3.57350e6i q^{41} +2.06716e6 q^{43} -3.93094e6i q^{47} -5.66994e6 q^{49} +9.41402e6i q^{53} +355752. q^{55} +7.74947e6i q^{59} -3.76639e6 q^{61} -705025. i q^{65} -2.62235e7 q^{67} +4.45401e7i q^{71} +709136. q^{73} -2.86959e6i q^{77} -3.84657e7 q^{79} +6.04066e7i q^{83} -4.53001e6 q^{85} +6.34896e7i q^{89} -5.68691e6 q^{91} +5.71046e6i q^{95} -1.11271e8 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 616 q^{7} + 36928 q^{13} - 299104 q^{19} + 778334 q^{25} - 933064 q^{31} - 1929044 q^{37} + 4134320 q^{43} - 11339874 q^{49} + 711504 q^{55} - 7532780 q^{61} - 52447024 q^{67} + 1418272 q^{73} - 76931320 q^{79}+ \cdots - 222541376 q^{97}+O(q^{100}) \) 2 * q - 616 * q^7 + 36928 * q^13 - 299104 * q^19 + 778334 * q^25 - 933064 * q^31 - 1929044 * q^37 + 4134320 * q^43 - 11339874 * q^49 + 711504 * q^55 - 7532780 * q^61 - 52447024 * q^67 + 1418272 * q^73 - 76931320 * q^79 - 9060012 * q^85 - 11373824 * q^91 - 222541376 * q^97

Introduction

L-functions

Modular forms

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Representations

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Database

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