LMFDB - Embedded newform 1620.4.a.h.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1620,4,Mod(1,1620)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1620, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1620.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1620 = 2^{2} \cdot 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1620.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:	$95.5830942093$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.438516.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 20x^{2} + 48 $ x^4 - x^3 - 20*x^2 + 48
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 180)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.58654$ of defining polynomial
Character	$\chi$	$=$	1620.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+5.00000 q^{5} +7.61161 q^{7} -12.5749 q^{11} -20.3581 q^{13} +22.6483 q^{17} -18.8592 q^{19} +6.29244 q^{23} +25.0000 q^{25} -206.960 q^{29} +33.7288 q^{31} +38.0580 q^{35} +13.4727 q^{37} +290.559 q^{41} -338.477 q^{43} -263.145 q^{47} -285.063 q^{49} -240.338 q^{53} -62.8746 q^{55} +373.694 q^{59} -454.553 q^{61} -101.791 q^{65} +442.622 q^{67} -371.467 q^{71} +901.209 q^{73} -95.7154 q^{77} -1034.81 q^{79} -34.6531 q^{83} +113.241 q^{85} -1070.24 q^{89} -154.958 q^{91} -94.2962 q^{95} -1382.50 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 20 q^{5} - 13 q^{7} - 57 q^{11} + 14 q^{13} - 3 q^{17} - 31 q^{19} + 69 q^{23} + 100 q^{25} + 69 q^{29} - 58 q^{31} - 65 q^{35} - 388 q^{37} - 396 q^{41} + 371 q^{43} - 129 q^{47} + 111 q^{49} - 1356 q^{53}+ \cdots - 1495 q^{97}+O(q^{100}) $ 4 * q + 20 * q^5 - 13 * q^7 - 57 * q^11 + 14 * q^13 - 3 * q^17 - 31 * q^19 + 69 * q^23 + 100 * q^25 + 69 * q^29 - 58 * q^31 - 65 * q^35 - 388 * q^37 - 396 * q^41 + 371 * q^43 - 129 * q^47 + 111 * q^49 - 1356 * q^53 - 285 * q^55 - 15 * q^59 - 1441 * q^61 + 70 * q^65 + 368 * q^67 - 168 * q^71 - 955 * q^73 - 342 * q^77 - 1408 * q^79 + 789 * q^83 - 15 * q^85 - 1617 * q^89 - 1406 * q^91 - 155 * q^95 - 1495 * q^97

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
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	7.61161	0.410988	0.205494	−	0.978658i	$-0.434120\pi$
$7$	7.61161	0.410988	0.205494	+	0.978658i	$0.434120\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−12.5749	−0.344680	−0.172340	−	0.985038i	$-0.555133\pi$
$11$	−12.5749	−0.344680	−0.172340	+	0.985038i	$0.555133\pi$
$12$	0	0
$13$	−20.3581	−0.434333	−0.217166	−	0.976135i	$-0.569681\pi$
$13$	−20.3581	−0.434333	−0.217166	+	0.976135i	$0.569681\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	22.6483	0.323119	0.161559	−	0.986863i	$-0.448348\pi$
$17$	22.6483	0.323119	0.161559	+	0.986863i	$0.448348\pi$
$18$	0	0
$19$	−18.8592	−0.227716	−0.113858	−	0.993497i	$-0.536321\pi$
$19$	−18.8592	−0.227716	−0.113858	+	0.993497i	$0.536321\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	6.29244	0.0570463	0.0285232	−	0.999593i	$-0.490920\pi$
$23$	6.29244	0.0570463	0.0285232	+	0.999593i	$0.490920\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−206.960	−1.32522	−0.662612	−	0.748963i	$-0.730552\pi$
$29$	−206.960	−1.32522	−0.662612	+	0.748963i	$0.730552\pi$
$30$	0	0
$31$	33.7288	0.195415	0.0977077	−	0.995215i	$-0.468849\pi$
$31$	33.7288	0.195415	0.0977077	+	0.995215i	$0.468849\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	38.0580	0.183800
$36$	0	0
$37$	13.4727	0.0598623	0.0299312	−	0.999552i	$-0.490471\pi$
$37$	13.4727	0.0598623	0.0299312	+	0.999552i	$0.490471\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	290.559	1.10677	0.553387	−	0.832924i	$-0.313335\pi$
$41$	290.559	1.10677	0.553387	+	0.832924i	$0.313335\pi$
$42$	0	0
$43$	−338.477	−1.20040	−0.600200	−	0.799850i	$-0.704912\pi$
$43$	−338.477	−1.20040	−0.600200	+	0.799850i	$0.704912\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−263.145	−0.816674	−0.408337	−	0.912831i	$-0.633891\pi$
$47$	−263.145	−0.816674	−0.408337	+	0.912831i	$0.633891\pi$
$48$	0	0
$49$	−285.063	−0.831089
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−240.338	−0.622887	−0.311444	−	0.950265i	$-0.600812\pi$
$53$	−240.338	−0.622887	−0.311444	+	0.950265i	$0.600812\pi$
$54$	0	0
$55$	−62.8746	−0.154146
$56$	0	0
$57$	0	0
$58$	0	0
$59$	373.694	0.824590	0.412295	−	0.911050i	$-0.364727\pi$
$59$	373.694	0.824590	0.412295	+	0.911050i	$0.364727\pi$
$60$	0	0
$61$	−454.553	−0.954091	−0.477045	−	0.878879i	$-0.658292\pi$
$61$	−454.553	−0.954091	−0.477045	+	0.878879i	$0.658292\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−101.791	−0.194240
$66$	0	0
$67$	442.622	0.807088	0.403544	−	0.914960i	$-0.367778\pi$
$67$	442.622	0.807088	0.403544	+	0.914960i	$0.367778\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−371.467	−0.620917	−0.310458	−	0.950587i	$-0.600483\pi$
$71$	−371.467	−0.620917	−0.310458	+	0.950587i	$0.600483\pi$
$72$	0	0
$73$	901.209	1.44491	0.722456	−	0.691417i	$-0.243013\pi$
$73$	901.209	1.44491	0.722456	+	0.691417i	$0.243013\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−95.7154	−0.141660
$78$	0	0
$79$	−1034.81	−1.47373	−0.736866	−	0.676039i	$-0.763695\pi$
$79$	−1034.81	−1.47373	−0.736866	+	0.676039i	$0.763695\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−34.6531	−0.0458273	−0.0229137	−	0.999737i	$-0.507294\pi$
$83$	−34.6531	−0.0458273	−0.0229137	+	0.999737i	$0.507294\pi$
$84$	0	0
$85$	113.241	0.144503
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−1070.24	−1.27467	−0.637333	−	0.770588i	$-0.719963\pi$
$89$	−1070.24	−1.27467	−0.637333	+	0.770588i	$0.719963\pi$
$90$	0	0
$91$	−154.958	−0.178506
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−94.2962	−0.101838
$96$	0	0
$97$	−1382.50	−1.44713	−0.723567	−	0.690254i	$-0.757499\pi$
$97$	−1382.50	−1.44713	−0.723567	+	0.690254i	$0.757499\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−138.480	−0.136429	−0.0682144	−	0.997671i	$-0.521730\pi$
$101$	−138.480	−0.136429	−0.0682144	+	0.997671i	$0.521730\pi$
$102$	0	0
$103$	−51.1086	−0.0488920	−0.0244460	−	0.999701i	$-0.507782\pi$
$103$	−51.1086	−0.0488920	−0.0244460	+	0.999701i	$0.507782\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1519.23	1.37261	0.686304	−	0.727315i	$-0.259232\pi$
$107$	1519.23	1.37261	0.686304	+	0.727315i	$0.259232\pi$
$108$	0	0
$109$	1692.75	1.48748	0.743742	−	0.668467i	$-0.233049\pi$
$109$	1692.75	1.48748	0.743742	+	0.668467i	$0.233049\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	1308.17	1.08905	0.544523	−	0.838746i	$-0.316711\pi$
$113$	1308.17	1.08905	0.544523	+	0.838746i	$0.316711\pi$
$114$	0	0
$115$	31.4622	0.0255119
$116$	0	0
$117$	0	0
$118$	0	0
$119$	172.390	0.132798
$120$	0	0
$121$	−1172.87	−0.881196
$122$	0	0
$123$	0	0
$124$	0	0
$125$	125.000	0.0894427
$126$	0	0
$127$	−1977.69	−1.38182	−0.690912	−	0.722939i	$-0.742791\pi$
$127$	−1977.69	−1.38182	−0.690912	+	0.722939i	$0.742791\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	1947.63	1.29897	0.649486	−	0.760374i	$-0.274984\pi$
$131$	1947.63	1.29897	0.649486	+	0.760374i	$0.274984\pi$
$132$	0	0
$133$	−143.549	−0.0935886
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−1260.91	−0.786325	−0.393163	−	0.919469i	$-0.628619\pi$
$137$	−1260.91	−0.786325	−0.393163	+	0.919469i	$0.628619\pi$
$138$	0	0
$139$	−661.071	−0.403391	−0.201695	−	0.979448i	$-0.564645\pi$
$139$	−661.071	−0.403391	−0.201695	+	0.979448i	$0.564645\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	256.002	0.149706
$144$	0	0
$145$	−1034.80	−0.592658
$146$	0	0
$147$	0	0
$148$	0	0
$149$	86.9994	0.0478340	0.0239170	−	0.999714i	$-0.492386\pi$
$149$	86.9994	0.0478340	0.0239170	+	0.999714i	$0.492386\pi$
$150$	0	0
$151$	1348.25	0.726618	0.363309	−	0.931669i	$-0.381647\pi$
$151$	1348.25	0.726618	0.363309	+	0.931669i	$0.381647\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	168.644	0.0873924
$156$	0	0
$157$	−516.367	−0.262488	−0.131244	−	0.991350i	$-0.541897\pi$
$157$	−516.367	−0.262488	−0.131244	+	0.991350i	$0.541897\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	47.8956	0.0234454
$162$	0	0
$163$	−2464.89	−1.18445	−0.592224	−	0.805774i	$-0.701750\pi$
$163$	−2464.89	−1.18445	−0.592224	+	0.805774i	$0.701750\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−2981.02	−1.38131	−0.690653	−	0.723187i	$-0.742677\pi$
$167$	−2981.02	−1.38131	−0.690653	+	0.723187i	$0.742677\pi$
$168$	0	0
$169$	−1782.55	−0.811355
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−1026.45	−0.451096	−0.225548	−	0.974232i	$-0.572417\pi$
$173$	−1026.45	−0.451096	−0.225548	+	0.974232i	$0.572417\pi$
$174$	0	0
$175$	190.290	0.0821977
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−492.431	−0.205620	−0.102810	−	0.994701i	$-0.532783\pi$
$179$	−492.431	−0.205620	−0.102810	+	0.994701i	$0.532783\pi$
$180$	0	0
$181$	−3523.74	−1.44706	−0.723529	−	0.690294i	$-0.757481\pi$
$181$	−3523.74	−1.44706	−0.723529	+	0.690294i	$0.757481\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	67.3637	0.0267712
$186$	0	0
$187$	−284.801	−0.111373
$188$	0	0
$189$	0	0
$190$	0	0
$191$	2981.86	1.12963	0.564816	−	0.825217i	$-0.308947\pi$
$191$	2981.86	1.12963	0.564816	+	0.825217i	$0.308947\pi$
$192$	0	0
$193$	−3252.42	−1.21303	−0.606514	−	0.795073i	$-0.707432\pi$
$193$	−3252.42	−1.21303	−0.606514	+	0.795073i	$0.707432\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	4300.46	1.55531	0.777653	−	0.628694i	$-0.216410\pi$
$197$	4300.46	1.55531	0.777653	+	0.628694i	$0.216410\pi$
$198$	0	0
$199$	−1549.59	−0.551997	−0.275998	−	0.961158i	$-0.589008\pi$
$199$	−1549.59	−0.551997	−0.275998	+	0.961158i	$0.589008\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−1575.30	−0.544652
$204$	0	0
$205$	1452.80	0.494964
$206$	0	0
$207$	0	0
$208$	0	0
$209$	237.153	0.0784892
$210$	0	0
$211$	−2564.99	−0.836877	−0.418439	−	0.908245i	$-0.637422\pi$
$211$	−2564.99	−0.836877	−0.418439	+	0.908245i	$0.637422\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−1692.38	−0.536835
$216$	0	0
$217$	256.731	0.0803134
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−461.077	−0.140341
$222$	0	0
$223$	−2499.47	−0.750570	−0.375285	−	0.926909i	$-0.622455\pi$
$223$	−2499.47	−0.750570	−0.375285	+	0.926909i	$0.622455\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	2917.54	0.853057	0.426528	−	0.904474i	$-0.359736\pi$
$227$	2917.54	0.853057	0.426528	+	0.904474i	$0.359736\pi$
$228$	0	0
$229$	−3904.29	−1.12665	−0.563324	−	0.826236i	$-0.690478\pi$
$229$	−3904.29	−1.12665	−0.563324	+	0.826236i	$0.690478\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−529.427	−0.148858	−0.0744290	−	0.997226i	$-0.523713\pi$
$233$	−529.427	−0.148858	−0.0744290	+	0.997226i	$0.523713\pi$
$234$	0	0
$235$	−1315.73	−0.365228
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−3854.43	−1.04319	−0.521595	−	0.853193i	$-0.674663\pi$
$239$	−3854.43	−1.04319	−0.521595	+	0.853193i	$0.674663\pi$
$240$	0	0
$241$	1713.23	0.457921	0.228960	−	0.973436i	$-0.426467\pi$
$241$	1713.23	0.457921	0.228960	+	0.973436i	$0.426467\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−1425.32	−0.371674
$246$	0	0
$247$	383.939	0.0989046
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−7789.93	−1.95895	−0.979474	−	0.201573i	$-0.935395\pi$
$251$	−7789.93	−1.95895	−0.979474	+	0.201573i	$0.935395\pi$
$252$	0	0
$253$	−79.1270	−0.0196627
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−4350.88	−1.05603	−0.528016	−	0.849234i	$-0.677064\pi$
$257$	−4350.88	−1.05603	−0.528016	+	0.849234i	$0.677064\pi$
$258$	0	0
$259$	102.549	0.0246027
$260$	0	0
$261$	0	0
$262$	0	0
$263$	6189.73	1.45124	0.725618	−	0.688098i	$-0.241554\pi$
$263$	6189.73	1.45124	0.725618	+	0.688098i	$0.241554\pi$
$264$	0	0
$265$	−1201.69	−0.278564
$266$	0	0
$267$	0	0
$268$	0	0
$269$	5497.65	1.24609	0.623044	−	0.782187i	$-0.285896\pi$
$269$	5497.65	1.24609	0.623044	+	0.782187i	$0.285896\pi$
$270$	0	0
$271$	−6473.63	−1.45109	−0.725545	−	0.688175i	$-0.758412\pi$
$271$	−6473.63	−1.45109	−0.725545	+	0.688175i	$0.758412\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−314.373	−0.0689360
$276$	0	0
$277$	−6639.69	−1.44022	−0.720109	−	0.693861i	$-0.755908\pi$
$277$	−6639.69	−1.44022	−0.720109	+	0.693861i	$0.755908\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	2423.44	0.514486	0.257243	−	0.966347i	$-0.417186\pi$
$281$	2423.44	0.514486	0.257243	+	0.966347i	$0.417186\pi$
$282$	0	0
$283$	2182.30	0.458390	0.229195	−	0.973380i	$-0.426391\pi$
$283$	2182.30	0.458390	0.229195	+	0.973380i	$0.426391\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	2211.62	0.454871
$288$	0	0
$289$	−4400.05	−0.895594
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−5783.64	−1.15319	−0.576594	−	0.817031i	$-0.695618\pi$
$293$	−5783.64	−1.15319	−0.576594	+	0.817031i	$0.695618\pi$
$294$	0	0
$295$	1868.47	0.368768
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−128.102	−0.0247771
$300$	0	0
$301$	−2576.35	−0.493351
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−2272.76	−0.426682
$306$	0	0
$307$	3781.94	0.703084	0.351542	−	0.936172i	$-0.385657\pi$
$307$	3781.94	0.703084	0.351542	+	0.936172i	$0.385657\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	5358.79	0.977072	0.488536	−	0.872544i	$-0.337531\pi$
$311$	5358.79	0.977072	0.488536	+	0.872544i	$0.337531\pi$
$312$	0	0
$313$	−1146.63	−0.207066	−0.103533	−	0.994626i	$-0.533015\pi$
$313$	−1146.63	−0.207066	−0.103533	+	0.994626i	$0.533015\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−1328.96	−0.235464	−0.117732	−	0.993045i	$-0.537562\pi$
$317$	−1328.96	−0.235464	−0.117732	+	0.993045i	$0.537562\pi$
$318$	0	0
$319$	2602.51	0.456779
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−427.129	−0.0735793
$324$	0	0
$325$	−508.953	−0.0868666
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−2002.96	−0.335643
$330$	0	0
$331$	10544.9	1.75106	0.875528	−	0.483167i	$-0.160514\pi$
$331$	10544.9	1.75106	0.875528	+	0.483167i	$0.160514\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	2213.11	0.360941
$336$	0	0
$337$	−3318.42	−0.536398	−0.268199	−	0.963364i	$-0.586428\pi$
$337$	−3318.42	−0.536398	−0.268199	+	0.963364i	$0.586428\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−424.138	−0.0673558
$342$	0	0
$343$	−4780.57	−0.752556
$344$	0	0
$345$	0	0
$346$	0	0
$347$	5230.96	0.809258	0.404629	−	0.914481i	$-0.367401\pi$
$347$	5230.96	0.809258	0.404629	+	0.914481i	$0.367401\pi$
$348$	0	0
$349$	−2859.66	−0.438608	−0.219304	−	0.975657i	$-0.570379\pi$
$349$	−2859.66	−0.438608	−0.219304	+	0.975657i	$0.570379\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−10993.4	−1.65757	−0.828784	−	0.559569i	$-0.810967\pi$
$353$	−10993.4	−1.65757	−0.828784	+	0.559569i	$0.810967\pi$
$354$	0	0
$355$	−1857.34	−0.277682
$356$	0	0
$357$	0	0
$358$	0	0
$359$	3419.15	0.502662	0.251331	−	0.967901i	$-0.419132\pi$
$359$	3419.15	0.502662	0.251331	+	0.967901i	$0.419132\pi$
$360$	0	0
$361$	−6503.33	−0.948145
$362$	0	0
$363$	0	0
$364$	0	0
$365$	4506.05	0.646184
$366$	0	0
$367$	−7785.47	−1.10735	−0.553676	−	0.832732i	$-0.686775\pi$
$367$	−7785.47	−1.10735	−0.553676	+	0.832732i	$0.686775\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−1829.36	−0.255999
$372$	0	0
$373$	−9123.79	−1.26652	−0.633260	−	0.773939i	$-0.718284\pi$
$373$	−9123.79	−1.26652	−0.633260	+	0.773939i	$0.718284\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	4213.32	0.575589
$378$	0	0
$379$	2117.60	0.287003	0.143501	−	0.989650i	$-0.454164\pi$
$379$	2117.60	0.287003	0.143501	+	0.989650i	$0.454164\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−3239.27	−0.432165	−0.216082	−	0.976375i	$-0.569328\pi$
$383$	−3239.27	−0.432165	−0.216082	+	0.976375i	$0.569328\pi$
$384$	0	0
$385$	−478.577	−0.0633521
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−6103.04	−0.795466	−0.397733	−	0.917501i	$-0.630203\pi$
$389$	−6103.04	−0.795466	−0.397733	+	0.917501i	$0.630203\pi$
$390$	0	0
$391$	142.513	0.0184327
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−5174.03	−0.659073
$396$	0	0
$397$	9935.69	1.25607	0.628033	−	0.778187i	$-0.283860\pi$
$397$	9935.69	1.25607	0.628033	+	0.778187i	$0.283860\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	11715.9	1.45902	0.729508	−	0.683973i	$-0.239749\pi$
$401$	11715.9	1.45902	0.729508	+	0.683973i	$0.239749\pi$
$402$	0	0
$403$	−686.656	−0.0848753
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−169.419	−0.0206334
$408$	0	0
$409$	7690.95	0.929812	0.464906	−	0.885360i	$-0.346088\pi$
$409$	7690.95	0.929812	0.464906	+	0.885360i	$0.346088\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	2844.41	0.338897
$414$	0	0
$415$	−173.265	−0.0204946
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−4633.54	−0.540246	−0.270123	−	0.962826i	$-0.587064\pi$
$419$	−4633.54	−0.540246	−0.270123	+	0.962826i	$0.587064\pi$
$420$	0	0
$421$	3867.10	0.447675	0.223837	−	0.974626i	$-0.428142\pi$
$421$	3867.10	0.447675	0.223837	+	0.974626i	$0.428142\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	566.207	0.0646237
$426$	0	0
$427$	−3459.88	−0.392120
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−5735.22	−0.640965	−0.320482	−	0.947254i	$-0.603845\pi$
$431$	−5735.22	−0.640965	−0.320482	+	0.947254i	$0.603845\pi$
$432$	0	0
$433$	2696.51	0.299274	0.149637	−	0.988741i	$-0.452189\pi$
$433$	2696.51	0.299274	0.149637	+	0.988741i	$0.452189\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−118.671	−0.0129904
$438$	0	0
$439$	−9039.61	−0.982773	−0.491386	−	0.870942i	$-0.663510\pi$
$439$	−9039.61	−0.982773	−0.491386	+	0.870942i	$0.663510\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−8959.37	−0.960886	−0.480443	−	0.877026i	$-0.659524\pi$
$443$	−8959.37	−0.960886	−0.480443	+	0.877026i	$0.659524\pi$
$444$	0	0
$445$	−5351.21	−0.570048
$446$	0	0
$447$	0	0
$448$	0	0
$449$	10716.1	1.12634	0.563169	−	0.826342i	$-0.309582\pi$
$449$	10716.1	1.12634	0.563169	+	0.826342i	$0.309582\pi$
$450$	0	0
$451$	−3653.76	−0.381483
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−774.790	−0.0798302
$456$	0	0
$457$	−15370.6	−1.57331	−0.786657	−	0.617390i	$-0.788190\pi$
$457$	−15370.6	−1.57331	−0.786657	+	0.617390i	$0.788190\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	7155.35	0.722902	0.361451	−	0.932391i	$-0.382281\pi$
$461$	7155.35	0.722902	0.361451	+	0.932391i	$0.382281\pi$
$462$	0	0
$463$	18448.9	1.85182	0.925911	−	0.377743i	$-0.123300\pi$
$463$	18448.9	1.85182	0.925911	+	0.377743i	$0.123300\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−3276.70	−0.324684	−0.162342	−	0.986735i	$-0.551905\pi$
$467$	−3276.70	−0.324684	−0.162342	+	0.986735i	$0.551905\pi$
$468$	0	0
$469$	3369.07	0.331704
$470$	0	0
$471$	0	0
$472$	0	0
$473$	4256.32	0.413754
$474$	0	0
$475$	−471.481	−0.0455432
$476$	0	0
$477$	0	0
$478$	0	0
$479$	7742.87	0.738582	0.369291	−	0.929314i	$-0.379601\pi$
$479$	7742.87	0.738582	0.369291	+	0.929314i	$0.379601\pi$
$480$	0	0
$481$	−274.280	−0.0260002
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−6912.52	−0.647178
$486$	0	0
$487$	18292.5	1.70208	0.851041	−	0.525099i	$-0.175972\pi$
$487$	18292.5	1.70208	0.851041	+	0.525099i	$0.175972\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	19322.1	1.77596	0.887979	−	0.459883i	$-0.152109\pi$
$491$	19322.1	1.77596	0.887979	+	0.459883i	$0.152109\pi$
$492$	0	0
$493$	−4687.29	−0.428205
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−2827.47	−0.255189
$498$	0	0
$499$	2480.64	0.222543	0.111271	−	0.993790i	$-0.464508\pi$
$499$	2480.64	0.222543	0.111271	+	0.993790i	$0.464508\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−5598.09	−0.496236	−0.248118	−	0.968730i	$-0.579812\pi$
$503$	−5598.09	−0.496236	−0.248118	+	0.968730i	$0.579812\pi$
$504$	0	0
$505$	−692.402	−0.0610128
$506$	0	0
$507$	0	0
$508$	0	0
$509$	8550.84	0.744616	0.372308	−	0.928109i	$-0.378567\pi$
$509$	8550.84	0.744616	0.372308	+	0.928109i	$0.378567\pi$
$510$	0	0
$511$	6859.65	0.593842
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−255.543	−0.0218652
$516$	0	0
$517$	3309.03	0.281491
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−9207.53	−0.774260	−0.387130	−	0.922025i	$-0.626533\pi$
$521$	−9207.53	−0.774260	−0.387130	+	0.922025i	$0.626533\pi$
$522$	0	0
$523$	4862.02	0.406503	0.203252	−	0.979127i	$-0.434849\pi$
$523$	4862.02	0.406503	0.203252	+	0.979127i	$0.434849\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	763.901	0.0631424
$528$	0	0
$529$	−12127.4	−0.996746
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−5915.24	−0.480708
$534$	0	0
$535$	7596.13	0.613849
$536$	0	0
$537$	0	0
$538$	0	0
$539$	3584.65	0.286460
$540$	0	0
$541$	5654.94	0.449399	0.224700	−	0.974428i	$-0.427860\pi$
$541$	5654.94	0.449399	0.224700	+	0.974428i	$0.427860\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	8463.74	0.665223
$546$	0	0
$547$	−3574.52	−0.279407	−0.139703	−	0.990193i	$-0.544615\pi$
$547$	−3574.52	−0.279407	−0.139703	+	0.990193i	$0.544615\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	3903.11	0.301775
$552$	0	0
$553$	−7876.55	−0.605687
$554$	0	0
$555$	0	0
$556$	0	0
$557$	10361.8	0.788230	0.394115	−	0.919061i	$-0.371051\pi$
$557$	10361.8	0.788230	0.394115	+	0.919061i	$0.371051\pi$
$558$	0	0
$559$	6890.75	0.521373
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−12751.9	−0.954580	−0.477290	−	0.878746i	$-0.658381\pi$
$563$	−12751.9	−0.954580	−0.477290	+	0.878746i	$0.658381\pi$
$564$	0	0
$565$	6540.84	0.487036
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−5557.24	−0.409441	−0.204720	−	0.978821i	$-0.565628\pi$
$569$	−5557.24	−0.409441	−0.204720	+	0.978821i	$0.565628\pi$
$570$	0	0
$571$	1543.35	0.113112	0.0565562	−	0.998399i	$-0.481988\pi$
$571$	1543.35	0.113112	0.0565562	+	0.998399i	$0.481988\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	157.311	0.0114093
$576$	0	0
$577$	4507.17	0.325192	0.162596	−	0.986693i	$-0.448013\pi$
$577$	4507.17	0.325192	0.162596	+	0.986693i	$0.448013\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−263.766	−0.0188345
$582$	0	0
$583$	3022.24	0.214697
$584$	0	0
$585$	0	0
$586$	0	0
$587$	1968.90	0.138442	0.0692208	−	0.997601i	$-0.477949\pi$
$587$	1968.90	0.138442	0.0692208	+	0.997601i	$0.477949\pi$
$588$	0	0
$589$	−636.100	−0.0444992
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−18751.9	−1.29857	−0.649283	−	0.760547i	$-0.724931\pi$
$593$	−18751.9	−1.29857	−0.649283	+	0.760547i	$0.724931\pi$
$594$	0	0
$595$	861.950	0.0593891
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−2736.59	−0.186668	−0.0933340	−	0.995635i	$-0.529752\pi$
$599$	−2736.59	−0.186668	−0.0933340	+	0.995635i	$0.529752\pi$
$600$	0	0
$601$	−6214.47	−0.421786	−0.210893	−	0.977509i	$-0.567637\pi$
$601$	−6214.47	−0.421786	−0.210893	+	0.977509i	$0.567637\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−5864.36	−0.394083
$606$	0	0
$607$	15237.7	1.01891	0.509457	−	0.860496i	$-0.329846\pi$
$607$	15237.7	1.01891	0.509457	+	0.860496i	$0.329846\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	5357.14	0.354708
$612$	0	0
$613$	9390.77	0.618743	0.309371	−	0.950941i	$-0.399881\pi$
$613$	9390.77	0.618743	0.309371	+	0.950941i	$0.399881\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−9597.64	−0.626234	−0.313117	−	0.949715i	$-0.601373\pi$
$617$	−9597.64	−0.626234	−0.313117	+	0.949715i	$0.601373\pi$
$618$	0	0
$619$	20136.4	1.30751	0.653756	−	0.756705i	$-0.273192\pi$
$619$	20136.4	1.30751	0.653756	+	0.756705i	$0.273192\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−8146.26	−0.523873
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	305.135	0.0193426
$630$	0	0
$631$	−6466.60	−0.407974	−0.203987	−	0.978974i	$-0.565390\pi$
$631$	−6466.60	−0.407974	−0.203987	+	0.978974i	$0.565390\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−9888.45	−0.617971
$636$	0	0
$637$	5803.36	0.360969
$638$	0	0
$639$	0	0
$640$	0	0
$641$	22555.5	1.38984	0.694921	−	0.719086i	$-0.255440\pi$
$641$	22555.5	1.38984	0.694921	+	0.719086i	$0.255440\pi$
$642$	0	0
$643$	18114.3	1.11098	0.555488	−	0.831525i	$-0.312532\pi$
$643$	18114.3	1.11098	0.555488	+	0.831525i	$0.312532\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	23680.5	1.43891	0.719456	−	0.694538i	$-0.244391\pi$
$647$	23680.5	1.43891	0.719456	+	0.694538i	$0.244391\pi$
$648$	0	0
$649$	−4699.17	−0.284220
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−10761.9	−0.644939	−0.322469	−	0.946580i	$-0.604513\pi$
$653$	−10761.9	−0.644939	−0.322469	+	0.946580i	$0.604513\pi$
$654$	0	0
$655$	9738.15	0.580918
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−3650.17	−0.215767	−0.107883	−	0.994164i	$-0.534407\pi$
$659$	−3650.17	−0.215767	−0.107883	+	0.994164i	$0.534407\pi$
$660$	0	0
$661$	−3031.97	−0.178411	−0.0892056	−	0.996013i	$-0.528433\pi$
$661$	−3031.97	−0.178411	−0.0892056	+	0.996013i	$0.528433\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−717.746	−0.0418541
$666$	0	0
$667$	−1302.28	−0.0755992
$668$	0	0
$669$	0	0
$670$	0	0
$671$	5715.97	0.328856
$672$	0	0
$673$	7852.53	0.449766	0.224883	−	0.974386i	$-0.427800\pi$
$673$	7852.53	0.449766	0.224883	+	0.974386i	$0.427800\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	1629.74	0.0925202	0.0462601	−	0.998929i	$-0.485270\pi$
$677$	1629.74	0.0925202	0.0462601	+	0.998929i	$0.485270\pi$
$678$	0	0
$679$	−10523.1	−0.594755
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−14384.1	−0.805847	−0.402923	−	0.915234i	$-0.632006\pi$
$683$	−14384.1	−0.805847	−0.402923	+	0.915234i	$0.632006\pi$
$684$	0	0
$685$	−6304.54	−0.351655
$686$	0	0
$687$	0	0
$688$	0	0
$689$	4892.84	0.270540
$690$	0	0
$691$	33385.6	1.83799	0.918993	−	0.394273i	$-0.129004\pi$
$691$	33385.6	1.83799	0.918993	+	0.394273i	$0.129004\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−3305.36	−0.180402
$696$	0	0
$697$	6580.67	0.357619
$698$	0	0
$699$	0	0
$700$	0	0
$701$	13436.1	0.723929	0.361965	−	0.932192i	$-0.382106\pi$
$701$	13436.1	0.723929	0.361965	+	0.932192i	$0.382106\pi$
$702$	0	0
$703$	−254.086	−0.0136316
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−1054.06	−0.0560707
$708$	0	0
$709$	25316.2	1.34100	0.670501	−	0.741908i	$-0.266079\pi$
$709$	25316.2	1.34100	0.670501	+	0.741908i	$0.266079\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	212.237	0.0111477
$714$	0	0
$715$	1280.01	0.0669505
$716$	0	0
$717$	0	0
$718$	0	0
$719$	1384.99	0.0718378	0.0359189	−	0.999355i	$-0.488564\pi$
$719$	1384.99	0.0718378	0.0359189	+	0.999355i	$0.488564\pi$
$720$	0	0
$721$	−389.019	−0.0200940
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−5174.00	−0.265045
$726$	0	0
$727$	34360.0	1.75288	0.876439	−	0.481513i	$-0.159912\pi$
$727$	34360.0	1.75288	0.876439	+	0.481513i	$0.159912\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−7665.92	−0.387872
$732$	0	0
$733$	−12812.9	−0.645642	−0.322821	−	0.946460i	$-0.604631\pi$
$733$	−12812.9	−0.645642	−0.322821	+	0.946460i	$0.604631\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−5565.94	−0.278187
$738$	0	0
$739$	32839.5	1.63467	0.817335	−	0.576163i	$-0.195450\pi$
$739$	32839.5	1.63467	0.817335	+	0.576163i	$0.195450\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−25350.6	−1.25172	−0.625858	−	0.779937i	$-0.715251\pi$
$743$	−25350.6	−1.25172	−0.625858	+	0.779937i	$0.715251\pi$
$744$	0	0
$745$	434.997	0.0213920
$746$	0	0
$747$	0	0
$748$	0	0
$749$	11563.8	0.564126
$750$	0	0
$751$	3471.46	0.168675	0.0843377	−	0.996437i	$-0.473123\pi$
$751$	3471.46	0.168675	0.0843377	+	0.996437i	$0.473123\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	6741.27	0.324954
$756$	0	0
$757$	−33545.9	−1.61063	−0.805316	−	0.592846i	$-0.798004\pi$
$757$	−33545.9	−1.61063	−0.805316	+	0.592846i	$0.798004\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	5998.42	0.285733	0.142866	−	0.989742i	$-0.454368\pi$
$761$	5998.42	0.285733	0.142866	+	0.989742i	$0.454368\pi$
$762$	0	0
$763$	12884.5	0.611339
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−7607.71	−0.358146
$768$	0	0
$769$	24208.1	1.13520	0.567598	−	0.823306i	$-0.307873\pi$
$769$	24208.1	1.13520	0.567598	+	0.823306i	$0.307873\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	2134.55	0.0993199	0.0496600	−	0.998766i	$-0.484186\pi$
$773$	2134.55	0.0993199	0.0496600	+	0.998766i	$0.484186\pi$
$774$	0	0
$775$	843.221	0.0390831
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−5479.72	−0.252030
$780$	0	0
$781$	4671.18	0.214018
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−2581.84	−0.117388
$786$	0	0
$787$	15744.3	0.713117	0.356558	−	0.934273i	$-0.383950\pi$
$787$	15744.3	0.713117	0.356558	+	0.934273i	$0.383950\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	9957.27	0.447585
$792$	0	0
$793$	9253.84	0.414393
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−30160.5	−1.34045	−0.670226	−	0.742157i	$-0.733803\pi$
$797$	−30160.5	−1.34045	−0.670226	+	0.742157i	$0.733803\pi$
$798$	0	0
$799$	−5959.79	−0.263883
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−11332.6	−0.498032
$804$	0	0
$805$	239.478	0.0104851
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−31677.9	−1.37668	−0.688340	−	0.725388i	$-0.741660\pi$
$809$	−31677.9	−1.37668	−0.688340	+	0.725388i	$0.741660\pi$
$810$	0	0
$811$	14294.2	0.618912	0.309456	−	0.950914i	$-0.399853\pi$
$811$	14294.2	0.618912	0.309456	+	0.950914i	$0.399853\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−12324.4	−0.529701
$816$	0	0
$817$	6383.41	0.273350
$818$	0	0
$819$	0	0
$820$	0	0
$821$	25274.3	1.07440	0.537198	−	0.843456i	$-0.319483\pi$
$821$	25274.3	1.07440	0.537198	+	0.843456i	$0.319483\pi$
$822$	0	0
$823$	−1414.08	−0.0598928	−0.0299464	−	0.999552i	$-0.509534\pi$
$823$	−1414.08	−0.0598928	−0.0299464	+	0.999552i	$0.509534\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	19366.4	0.814311	0.407155	−	0.913359i	$-0.366521\pi$
$827$	19366.4	0.814311	0.407155	+	0.913359i	$0.366521\pi$
$828$	0	0
$829$	−14280.3	−0.598283	−0.299141	−	0.954209i	$-0.596700\pi$
$829$	−14280.3	−0.598283	−0.299141	+	0.954209i	$0.596700\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−6456.20	−0.268540
$834$	0	0
$835$	−14905.1	−0.617739
$836$	0	0
$837$	0	0
$838$	0	0
$839$	26112.9	1.07451	0.537257	−	0.843418i	$-0.319460\pi$
$839$	26112.9	1.07451	0.537257	+	0.843418i	$0.319460\pi$
$840$	0	0
$841$	18443.4	0.756220
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−8912.73	−0.362849
$846$	0	0
$847$	−8927.44	−0.362161
$848$	0	0
$849$	0	0
$850$	0	0
$851$	84.7765	0.00341493
$852$	0	0
$853$	21239.0	0.852534	0.426267	−	0.904597i	$-0.359828\pi$
$853$	21239.0	0.852534	0.426267	+	0.904597i	$0.359828\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	41563.4	1.65669	0.828343	−	0.560222i	$-0.189284\pi$
$857$	41563.4	1.65669	0.828343	+	0.560222i	$0.189284\pi$
$858$	0	0
$859$	−5885.90	−0.233788	−0.116894	−	0.993144i	$-0.537294\pi$
$859$	−5885.90	−0.233788	−0.116894	+	0.993144i	$0.537294\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−11237.9	−0.443272	−0.221636	−	0.975130i	$-0.571140\pi$
$863$	−11237.9	−0.443272	−0.221636	+	0.975130i	$0.571140\pi$
$864$	0	0
$865$	−5132.25	−0.201736
$866$	0	0
$867$	0	0
$868$	0	0
$869$	13012.6	0.507966
$870$	0	0
$871$	−9010.96	−0.350545
$872$	0	0
$873$	0	0
$874$	0	0
$875$	951.451	0.0367599
$876$	0	0
$877$	−9433.27	−0.363214	−0.181607	−	0.983371i	$-0.558130\pi$
$877$	−9433.27	−0.363214	−0.181607	+	0.983371i	$0.558130\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	29845.4	1.14134	0.570668	−	0.821181i	$-0.306685\pi$
$881$	29845.4	1.14134	0.570668	+	0.821181i	$0.306685\pi$
$882$	0	0
$883$	10785.9	0.411070	0.205535	−	0.978650i	$-0.434107\pi$
$883$	10785.9	0.411070	0.205535	+	0.978650i	$0.434107\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	30705.7	1.16234	0.581171	−	0.813782i	$-0.302595\pi$
$887$	30705.7	1.16234	0.581171	+	0.813782i	$0.302595\pi$
$888$	0	0
$889$	−15053.4	−0.567914
$890$	0	0
$891$	0	0
$892$	0	0
$893$	4962.72	0.185970
$894$	0	0
$895$	−2462.15	−0.0919561
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−6980.52	−0.258969
$900$	0	0
$901$	−5443.26	−0.201267
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−17618.7	−0.647144
$906$	0	0
$907$	7062.97	0.258569	0.129285	−	0.991608i	$-0.458732\pi$
$907$	7062.97	0.258569	0.129285	+	0.991608i	$0.458732\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	5057.09	0.183918	0.0919588	−	0.995763i	$-0.470687\pi$
$911$	5057.09	0.183918	0.0919588	+	0.995763i	$0.470687\pi$
$912$	0	0
$913$	435.760	0.0157958
$914$	0	0
$915$	0	0
$916$	0	0
$917$	14824.6	0.533862
$918$	0	0
$919$	48440.1	1.73873	0.869364	−	0.494173i	$-0.164529\pi$
$919$	48440.1	1.73873	0.869364	+	0.494173i	$0.164529\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	7562.38	0.269684
$924$	0	0
$925$	336.819	0.0119725
$926$	0	0
$927$	0	0
$928$	0	0
$929$	10086.5	0.356220	0.178110	−	0.984011i	$-0.443002\pi$
$929$	10086.5	0.356220	0.178110	+	0.984011i	$0.443002\pi$
$930$	0	0
$931$	5376.08	0.189252
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−1424.00	−0.0498074
$936$	0	0
$937$	−22271.7	−0.776504	−0.388252	−	0.921553i	$-0.626921\pi$
$937$	−22271.7	−0.776504	−0.388252	+	0.921553i	$0.626921\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−18339.1	−0.635322	−0.317661	−	0.948204i	$-0.602897\pi$
$941$	−18339.1	−0.635322	−0.317661	+	0.948204i	$0.602897\pi$
$942$	0	0
$943$	1828.33	0.0631374
$944$	0	0
$945$	0	0
$946$	0	0
$947$	10141.9	0.348012	0.174006	−	0.984745i	$-0.444329\pi$
$947$	10141.9	0.348012	0.174006	+	0.984745i	$0.444329\pi$
$948$	0	0
$949$	−18346.9	−0.627573
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−46973.4	−1.59666	−0.798331	−	0.602219i	$-0.794283\pi$
$953$	−46973.4	−1.59666	−0.798331	+	0.602219i	$0.794283\pi$
$954$	0	0
$955$	14909.3	0.505187
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−9597.53	−0.323171
$960$	0	0
$961$	−28653.4	−0.961813
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−16262.1	−0.542482
$966$	0	0
$967$	−3021.79	−0.100490	−0.0502452	−	0.998737i	$-0.516000\pi$
$967$	−3021.79	−0.100490	−0.0502452	+	0.998737i	$0.516000\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	13426.4	0.443743	0.221871	−	0.975076i	$-0.428783\pi$
$971$	13426.4	0.443743	0.221871	+	0.975076i	$0.428783\pi$
$972$	0	0
$973$	−5031.82	−0.165789
$974$	0	0
$975$	0	0
$976$	0	0
$977$	56527.9	1.85106	0.925531	−	0.378672i	$-0.123619\pi$
$977$	56527.9	1.85106	0.925531	+	0.378672i	$0.123619\pi$
$978$	0	0
$979$	13458.2	0.439352
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−45626.5	−1.48043	−0.740213	−	0.672372i	$-0.765275\pi$
$983$	−45626.5	−1.48043	−0.740213	+	0.672372i	$0.765275\pi$
$984$	0	0
$985$	21502.3	0.695554
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−2129.85	−0.0684784
$990$	0	0
$991$	55864.3	1.79071	0.895353	−	0.445358i	$-0.146924\pi$
$991$	55864.3	1.79071	0.895353	+	0.445358i	$0.146924\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−7747.94	−0.246860
$996$	0	0
$997$	19012.0	0.603927	0.301963	−	0.953320i	$-0.402358\pi$
$997$	19012.0	0.603927	0.301963	+	0.953320i	$0.402358\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1620.4.a.h.1.3		4
3.2	odd	2		1620.4.a.g.1.3		4
9.2	odd	6		180.4.i.b.121.4	yes	8
9.4	even	3		540.4.i.b.181.2		8
9.5	odd	6		180.4.i.b.61.4	✓	8
9.7	even	3		540.4.i.b.361.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
180.4.i.b.61.4	✓	8	9.5	odd	6
180.4.i.b.121.4	yes	8	9.2	odd	6
540.4.i.b.181.2		8	9.4	even	3
540.4.i.b.361.2		8	9.7	even	3
1620.4.a.g.1.3		4	3.2	odd	2
1620.4.a.h.1.3		4	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:	\( N \)	\(=\)	\( 1620 = 2^{2} \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1620.a (trivial)

\(f(q)\)	\(=\)	\(q+5.00000 q^{5} +7.61161 q^{7} -12.5749 q^{11} -20.3581 q^{13} +22.6483 q^{17} -18.8592 q^{19} +6.29244 q^{23} +25.0000 q^{25} -206.960 q^{29} +33.7288 q^{31} +38.0580 q^{35} +13.4727 q^{37} +290.559 q^{41} -338.477 q^{43} -263.145 q^{47} -285.063 q^{49} -240.338 q^{53} -62.8746 q^{55} +373.694 q^{59} -454.553 q^{61} -101.791 q^{65} +442.622 q^{67} -371.467 q^{71} +901.209 q^{73} -95.7154 q^{77} -1034.81 q^{79} -34.6531 q^{83} +113.241 q^{85} -1070.24 q^{89} -154.958 q^{91} -94.2962 q^{95} -1382.50 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 20 q^{5} - 13 q^{7} - 57 q^{11} + 14 q^{13} - 3 q^{17} - 31 q^{19} + 69 q^{23} + 100 q^{25} + 69 q^{29} - 58 q^{31} - 65 q^{35} - 388 q^{37} - 396 q^{41} + 371 q^{43} - 129 q^{47} + 111 q^{49} - 1356 q^{53}+ \cdots - 1495 q^{97}+O(q^{100}) \) 4 * q + 20 * q^5 - 13 * q^7 - 57 * q^11 + 14 * q^13 - 3 * q^17 - 31 * q^19 + 69 * q^23 + 100 * q^25 + 69 * q^29 - 58 * q^31 - 65 * q^35 - 388 * q^37 - 396 * q^41 + 371 * q^43 - 129 * q^47 + 111 * q^49 - 1356 * q^53 - 285 * q^55 - 15 * q^59 - 1441 * q^61 + 70 * q^65 + 368 * q^67 - 168 * q^71 - 955 * q^73 - 342 * q^77 - 1408 * q^79 + 789 * q^83 - 15 * q^85 - 1617 * q^89 - 1406 * q^91 - 155 * q^95 - 1495 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more