LMFDB - Embedded newform 1620.4.a.g.1.2

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			1.2
Root			$-1.78794$ 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)

$f(q)$	$=$	$q-5.00000 q^{5} -5.78863 q^{7} +O(q^{10})$	$q-5.00000 q^{5} -5.78863 q^{7} -11.2419 q^{11} +39.4377 q^{13} -19.6646 q^{17} -102.113 q^{19} +175.505 q^{23} +25.0000 q^{25} +64.6531 q^{29} +193.744 q^{31} +28.9432 q^{35} -261.480 q^{37} +207.837 q^{41} +41.1946 q^{43} -577.877 q^{47} -309.492 q^{49} +322.043 q^{53} +56.2093 q^{55} -188.109 q^{59} +225.225 q^{61} -197.189 q^{65} +419.831 q^{67} +361.341 q^{71} -779.176 q^{73} +65.0750 q^{77} +696.903 q^{79} -333.455 q^{83} +98.3229 q^{85} +570.655 q^{89} -228.291 q^{91} +510.566 q^{95} -1104.42 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 20 q^{5} - 13 q^{7}+O(q^{10}) $ 4 * q - 20 * q^5 - 13 * q^7	$ 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}+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

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
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	−5.78863	−0.312557	−0.156278	−	0.987713i	$-0.549950\pi$
$7$	−5.78863	−0.312557	−0.156278	+	0.987713i	$0.549950\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−11.2419	−0.308140	−0.154070	−	0.988060i	$-0.549238\pi$
$11$	−11.2419	−0.308140	−0.154070	+	0.988060i	$0.549238\pi$
$12$	0	0
$13$	39.4377	0.841389	0.420695	−	0.907202i	$-0.361786\pi$
$13$	39.4377	0.841389	0.420695	+	0.907202i	$0.361786\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−19.6646	−0.280551	−0.140275	−	0.990113i	$-0.544799\pi$
$17$	−19.6646	−0.280551	−0.140275	+	0.990113i	$0.544799\pi$
$18$	0	0
$19$	−102.113	−1.23297	−0.616483	−	0.787368i	$-0.711443\pi$
$19$	−102.113	−1.23297	−0.616483	+	0.787368i	$0.711443\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	175.505	1.59110	0.795552	−	0.605885i	$-0.207181\pi$
$23$	175.505	1.59110	0.795552	+	0.605885i	$0.207181\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	64.6531	0.413992	0.206996	−	0.978342i	$-0.433631\pi$
$29$	64.6531	0.413992	0.206996	+	0.978342i	$0.433631\pi$
$30$	0	0
$31$	193.744	1.12250	0.561248	−	0.827648i	$-0.310321\pi$
$31$	193.744	1.12250	0.561248	+	0.827648i	$0.310321\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	28.9432	0.139780
$36$	0	0
$37$	−261.480	−1.16181	−0.580906	−	0.813970i	$-0.697302\pi$
$37$	−261.480	−1.16181	−0.580906	+	0.813970i	$0.697302\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	207.837	0.791677	0.395838	−	0.918320i	$-0.370454\pi$
$41$	207.837	0.791677	0.395838	+	0.918320i	$0.370454\pi$
$42$	0	0
$43$	41.1946	0.146096	0.0730478	−	0.997328i	$-0.476727\pi$
$43$	41.1946	0.146096	0.0730478	+	0.997328i	$0.476727\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−577.877	−1.79345	−0.896724	−	0.442589i	$-0.854060\pi$
$47$	−577.877	−1.79345	−0.896724	+	0.442589i	$0.854060\pi$
$48$	0	0
$49$	−309.492	−0.902308
$50$	0	0
$51$	0	0
$52$	0	0
$53$	322.043	0.834643	0.417321	−	0.908759i	$-0.362969\pi$
$53$	322.043	0.834643	0.417321	+	0.908759i	$0.362969\pi$
$54$	0	0
$55$	56.2093	0.137805
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−188.109	−0.415079	−0.207540	−	0.978227i	$-0.566546\pi$
$59$	−188.109	−0.415079	−0.207540	+	0.978227i	$0.566546\pi$
$60$	0	0
$61$	225.225	0.472739	0.236369	−	0.971663i	$-0.424042\pi$
$61$	225.225	0.472739	0.236369	+	0.971663i	$0.424042\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−197.189	−0.376281
$66$	0	0
$67$	419.831	0.765531	0.382765	−	0.923846i	$-0.374972\pi$
$67$	419.831	0.765531	0.382765	+	0.923846i	$0.374972\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	361.341	0.603990	0.301995	−	0.953309i	$-0.402347\pi$
$71$	361.341	0.603990	0.301995	+	0.953309i	$0.402347\pi$
$72$	0	0
$73$	−779.176	−1.24926	−0.624628	−	0.780923i	$-0.714749\pi$
$73$	−779.176	−1.24926	−0.624628	+	0.780923i	$0.714749\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	65.0750	0.0963114
$78$	0	0
$79$	696.903	0.992503	0.496252	−	0.868179i	$-0.334709\pi$
$79$	696.903	0.992503	0.496252	+	0.868179i	$0.334709\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−333.455	−0.440981	−0.220491	−	0.975389i	$-0.570766\pi$
$83$	−333.455	−0.440981	−0.220491	+	0.975389i	$0.570766\pi$
$84$	0	0
$85$	98.3229	0.125466
$86$	0	0
$87$	0	0
$88$	0	0
$89$	570.655	0.679655	0.339828	−	0.940488i	$-0.389631\pi$
$89$	570.655	0.679655	0.339828	+	0.940488i	$0.389631\pi$
$90$	0	0
$91$	−228.291	−0.262982
$92$	0	0
$93$	0	0
$94$	0	0
$95$	510.566	0.551399
$96$	0	0
$97$	−1104.42	−1.15605	−0.578025	−	0.816019i	$-0.696176\pi$
$97$	−1104.42	−1.15605	−0.578025	+	0.816019i	$0.696176\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−291.101	−0.286789	−0.143394	−	0.989666i	$-0.545802\pi$
$101$	−291.101	−0.286789	−0.143394	+	0.989666i	$0.545802\pi$
$102$	0	0
$103$	−994.988	−0.951836	−0.475918	−	0.879490i	$-0.657884\pi$
$103$	−994.988	−0.951836	−0.475918	+	0.879490i	$0.657884\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	90.9192	0.0821448	0.0410724	−	0.999156i	$-0.486923\pi$
$107$	90.9192	0.0821448	0.0410724	+	0.999156i	$0.486923\pi$
$108$	0	0
$109$	−1720.69	−1.51204	−0.756018	−	0.654551i	$-0.772858\pi$
$109$	−1720.69	−1.51204	−0.756018	+	0.654551i	$0.772858\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−658.685	−0.548353	−0.274177	−	0.961679i	$-0.588405\pi$
$113$	−658.685	−0.548353	−0.274177	+	0.961679i	$0.588405\pi$
$114$	0	0
$115$	−877.527	−0.711563
$116$	0	0
$117$	0	0
$118$	0	0
$119$	113.831	0.0876880
$120$	0	0
$121$	−1204.62	−0.905049
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−125.000	−0.0894427
$126$	0	0
$127$	−1440.57	−1.00654	−0.503269	−	0.864130i	$-0.667869\pi$
$127$	−1440.57	−1.00654	−0.503269	+	0.864130i	$0.667869\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	1827.75	1.21901	0.609507	−	0.792781i	$-0.291367\pi$
$131$	1827.75	1.21901	0.609507	+	0.792781i	$0.291367\pi$
$132$	0	0
$133$	591.096	0.385372
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−1903.89	−1.18730	−0.593651	−	0.804723i	$-0.702314\pi$
$137$	−1903.89	−1.18730	−0.593651	+	0.804723i	$0.702314\pi$
$138$	0	0
$139$	−282.361	−0.172299	−0.0861494	−	0.996282i	$-0.527456\pi$
$139$	−282.361	−0.172299	−0.0861494	+	0.996282i	$0.527456\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−443.353	−0.259266
$144$	0	0
$145$	−323.265	−0.185143
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−1636.50	−0.899783	−0.449892	−	0.893083i	$-0.648537\pi$
$149$	−1636.50	−0.899783	−0.449892	+	0.893083i	$0.648537\pi$
$150$	0	0
$151$	−1682.35	−0.906672	−0.453336	−	0.891340i	$-0.649766\pi$
$151$	−1682.35	−0.906672	−0.453336	+	0.891340i	$0.649766\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−968.718	−0.501995
$156$	0	0
$157$	1959.18	0.995920	0.497960	−	0.867200i	$-0.334083\pi$
$157$	1959.18	0.995920	0.497960	+	0.867200i	$0.334083\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−1015.94	−0.497311
$162$	0	0
$163$	−540.779	−0.259859	−0.129930	−	0.991523i	$-0.541475\pi$
$163$	−540.779	−0.259859	−0.129930	+	0.991523i	$0.541475\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	401.908	0.186231	0.0931154	−	0.995655i	$-0.470317\pi$
$167$	401.908	0.186231	0.0931154	+	0.995655i	$0.470317\pi$
$168$	0	0
$169$	−641.666	−0.292065
$170$	0	0
$171$	0	0
$172$	0	0
$173$	3244.49	1.42586	0.712931	−	0.701234i	$-0.247367\pi$
$173$	3244.49	1.42586	0.712931	+	0.701234i	$0.247367\pi$
$174$	0	0
$175$	−144.716	−0.0625114
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−3845.07	−1.60555	−0.802777	−	0.596280i	$-0.796645\pi$
$179$	−3845.07	−1.60555	−0.802777	+	0.596280i	$0.796645\pi$
$180$	0	0
$181$	883.956	0.363005	0.181503	−	0.983390i	$-0.441904\pi$
$181$	883.956	0.363005	0.181503	+	0.983390i	$0.441904\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	1307.40	0.519578
$186$	0	0
$187$	221.066	0.0864490
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−409.326	−0.155067	−0.0775335	−	0.996990i	$-0.524704\pi$
$191$	−409.326	−0.155067	−0.0775335	+	0.996990i	$0.524704\pi$
$192$	0	0
$193$	−368.346	−0.137379	−0.0686894	−	0.997638i	$-0.521882\pi$
$193$	−368.346	−0.137379	−0.0686894	+	0.997638i	$0.521882\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	2994.23	1.08289	0.541447	−	0.840735i	$-0.317877\pi$
$197$	2994.23	1.08289	0.541447	+	0.840735i	$0.317877\pi$
$198$	0	0
$199$	887.223	0.316048	0.158024	−	0.987435i	$-0.449488\pi$
$199$	887.223	0.316048	0.158024	+	0.987435i	$0.449488\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−374.253	−0.129396
$204$	0	0
$205$	−1039.19	−0.354049
$206$	0	0
$207$	0	0
$208$	0	0
$209$	1147.94	0.379927
$210$	0	0
$211$	−5449.93	−1.77814	−0.889072	−	0.457767i	$-0.848649\pi$
$211$	−5449.93	−1.77814	−0.889072	+	0.457767i	$0.848649\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−205.973	−0.0653360
$216$	0	0
$217$	−1121.51	−0.350844
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−775.526	−0.236052
$222$	0	0
$223$	−3947.57	−1.18542	−0.592710	−	0.805416i	$-0.701942\pi$
$223$	−3947.57	−1.18542	−0.592710	+	0.805416i	$0.701942\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	6529.13	1.90905	0.954523	−	0.298138i	$-0.0963657\pi$
$227$	6529.13	1.90905	0.954523	+	0.298138i	$0.0963657\pi$
$228$	0	0
$229$	−3645.07	−1.05185	−0.525923	−	0.850532i	$-0.676280\pi$
$229$	−3645.07	−1.05185	−0.525923	+	0.850532i	$0.676280\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	1686.19	0.474104	0.237052	−	0.971497i	$-0.423819\pi$
$233$	1686.19	0.474104	0.237052	+	0.971497i	$0.423819\pi$
$234$	0	0
$235$	2889.39	0.802055
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−5899.02	−1.59655	−0.798276	−	0.602292i	$-0.794254\pi$
$239$	−5899.02	−1.59655	−0.798276	+	0.602292i	$0.794254\pi$
$240$	0	0
$241$	−7053.79	−1.88537	−0.942686	−	0.333681i	$-0.891709\pi$
$241$	−7053.79	−1.88537	−0.942686	+	0.333681i	$0.891709\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	1547.46	0.403524
$246$	0	0
$247$	−4027.11	−1.03740
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−6617.09	−1.66401	−0.832007	−	0.554766i	$-0.812808\pi$
$251$	−6617.09	−1.66401	−0.832007	+	0.554766i	$0.812808\pi$
$252$	0	0
$253$	−1973.00	−0.490284
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−5326.18	−1.29276	−0.646378	−	0.763018i	$-0.723717\pi$
$257$	−5326.18	−1.29276	−0.646378	+	0.763018i	$0.723717\pi$
$258$	0	0
$259$	1513.61	0.363133
$260$	0	0
$261$	0	0
$262$	0	0
$263$	1346.01	0.315584	0.157792	−	0.987472i	$-0.449562\pi$
$263$	1346.01	0.315584	0.157792	+	0.987472i	$0.449562\pi$
$264$	0	0
$265$	−1610.22	−0.373264
$266$	0	0
$267$	0	0
$268$	0	0
$269$	6359.78	1.44150	0.720748	−	0.693197i	$-0.243798\pi$
$269$	6359.78	1.44150	0.720748	+	0.693197i	$0.243798\pi$
$270$	0	0
$271$	8586.05	1.92459	0.962297	−	0.271999i	$-0.0876848\pi$
$271$	8586.05	1.92459	0.962297	+	0.271999i	$0.0876848\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−281.046	−0.0616281
$276$	0	0
$277$	−3757.45	−0.815030	−0.407515	−	0.913199i	$-0.633605\pi$
$277$	−3757.45	−0.815030	−0.407515	+	0.913199i	$0.633605\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−5480.33	−1.16345	−0.581725	−	0.813386i	$-0.697622\pi$
$281$	−5480.33	−1.16345	−0.581725	+	0.813386i	$0.697622\pi$
$282$	0	0
$283$	−5580.88	−1.17226	−0.586129	−	0.810218i	$-0.699349\pi$
$283$	−5580.88	−1.17226	−0.586129	+	0.810218i	$0.699349\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−1203.09	−0.247444
$288$	0	0
$289$	−4526.30	−0.921291
$290$	0	0
$291$	0	0
$292$	0	0
$293$	3217.92	0.641614	0.320807	−	0.947145i	$-0.396046\pi$
$293$	3217.92	0.641614	0.320807	+	0.947145i	$0.396046\pi$
$294$	0	0
$295$	940.544	0.185629
$296$	0	0
$297$	0	0
$298$	0	0
$299$	6921.53	1.33874
$300$	0	0
$301$	−238.460	−0.0456632
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−1126.12	−0.211415
$306$	0	0
$307$	−8789.12	−1.63395	−0.816974	−	0.576675i	$-0.804350\pi$
$307$	−8789.12	−1.63395	−0.816974	+	0.576675i	$0.804350\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	7334.46	1.33730	0.668648	−	0.743579i	$-0.266873\pi$
$311$	7334.46	1.33730	0.668648	+	0.743579i	$0.266873\pi$
$312$	0	0
$313$	−2702.52	−0.488036	−0.244018	−	0.969771i	$-0.578466\pi$
$313$	−2702.52	−0.488036	−0.244018	+	0.969771i	$0.578466\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	3581.86	0.634629	0.317314	−	0.948320i	$-0.397219\pi$
$317$	3581.86	0.634629	0.317314	+	0.948320i	$0.397219\pi$
$318$	0	0
$319$	−726.820	−0.127568
$320$	0	0
$321$	0	0
$322$	0	0
$323$	2008.01	0.345910
$324$	0	0
$325$	985.943	0.168278
$326$	0	0
$327$	0	0
$328$	0	0
$329$	3345.12	0.560555
$330$	0	0
$331$	1677.08	0.278491	0.139245	−	0.990258i	$-0.455532\pi$
$331$	1677.08	0.278491	0.139245	+	0.990258i	$0.455532\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−2099.16	−0.342356
$336$	0	0
$337$	481.195	0.0777814	0.0388907	−	0.999243i	$-0.487618\pi$
$337$	481.195	0.0777814	0.0388907	+	0.999243i	$0.487618\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−2178.04	−0.345886
$342$	0	0
$343$	3777.04	0.594580
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−10122.2	−1.56596	−0.782980	−	0.622047i	$-0.786301\pi$
$347$	−10122.2	−1.56596	−0.782980	+	0.622047i	$0.786301\pi$
$348$	0	0
$349$	−9700.50	−1.48784	−0.743920	−	0.668269i	$-0.767035\pi$
$349$	−9700.50	−1.48784	−0.743920	+	0.668269i	$0.767035\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−918.558	−0.138498	−0.0692492	−	0.997599i	$-0.522060\pi$
$353$	−918.558	−0.138498	−0.0692492	+	0.997599i	$0.522060\pi$
$354$	0	0
$355$	−1806.71	−0.270113
$356$	0	0
$357$	0	0
$358$	0	0
$359$	1178.66	0.173279	0.0866396	−	0.996240i	$-0.472387\pi$
$359$	1178.66	0.173279	0.0866396	+	0.996240i	$0.472387\pi$
$360$	0	0
$361$	3568.09	0.520206
$362$	0	0
$363$	0	0
$364$	0	0
$365$	3895.88	0.558684
$366$	0	0
$367$	9493.40	1.35028	0.675139	−	0.737691i	$-0.264084\pi$
$367$	9493.40	1.35028	0.675139	+	0.737691i	$0.264084\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−1864.19	−0.260873
$372$	0	0
$373$	9529.44	1.32283	0.661415	−	0.750020i	$-0.269956\pi$
$373$	9529.44	1.32283	0.661415	+	0.750020i	$0.269956\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	2549.77	0.348329
$378$	0	0
$379$	−7371.67	−0.999096	−0.499548	−	0.866286i	$-0.666501\pi$
$379$	−7371.67	−0.999096	−0.499548	+	0.866286i	$0.666501\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	2259.76	0.301484	0.150742	−	0.988573i	$-0.451834\pi$
$383$	2259.76	0.301484	0.150742	+	0.988573i	$0.451834\pi$
$384$	0	0
$385$	−325.375	−0.0430718
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−10708.9	−1.39579	−0.697896	−	0.716199i	$-0.745880\pi$
$389$	−10708.9	−1.39579	−0.697896	+	0.716199i	$0.745880\pi$
$390$	0	0
$391$	−3451.24	−0.446385
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−3484.52	−0.443861
$396$	0	0
$397$	−14817.3	−1.87320	−0.936598	−	0.350406i	$-0.886044\pi$
$397$	−14817.3	−1.87320	−0.936598	+	0.350406i	$0.886044\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	7873.06	0.980453	0.490227	−	0.871595i	$-0.336914\pi$
$401$	7873.06	0.980453	0.490227	+	0.871595i	$0.336914\pi$
$402$	0	0
$403$	7640.81	0.944456
$404$	0	0
$405$	0	0
$406$	0	0
$407$	2939.52	0.358001
$408$	0	0
$409$	13158.4	1.59081	0.795403	−	0.606081i	$-0.207259\pi$
$409$	13158.4	1.59081	0.795403	+	0.606081i	$0.207259\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	1088.89	0.129736
$414$	0	0
$415$	1667.27	0.197213
$416$	0	0
$417$	0	0
$418$	0	0
$419$	12925.7	1.50707	0.753536	−	0.657407i	$-0.228347\pi$
$419$	12925.7	1.50707	0.753536	+	0.657407i	$0.228347\pi$
$420$	0	0
$421$	−12597.0	−1.45829	−0.729144	−	0.684360i	$-0.760081\pi$
$421$	−12597.0	−1.45829	−0.729144	+	0.684360i	$0.760081\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−491.615	−0.0561101
$426$	0	0
$427$	−1303.74	−0.147758
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−11443.9	−1.27896	−0.639480	−	0.768808i	$-0.720850\pi$
$431$	−11443.9	−1.27896	−0.639480	+	0.768808i	$0.720850\pi$
$432$	0	0
$433$	−13890.5	−1.54165	−0.770826	−	0.637046i	$-0.780156\pi$
$433$	−13890.5	−1.54165	−0.770826	+	0.637046i	$0.780156\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−17921.4	−1.96178
$438$	0	0
$439$	−2052.06	−0.223097	−0.111548	−	0.993759i	$-0.535581\pi$
$439$	−2052.06	−0.223097	−0.111548	+	0.993759i	$0.535581\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−1686.91	−0.180920	−0.0904598	−	0.995900i	$-0.528834\pi$
$443$	−1686.91	−0.180920	−0.0904598	+	0.995900i	$0.528834\pi$
$444$	0	0
$445$	−2853.27	−0.303951
$446$	0	0
$447$	0	0
$448$	0	0
$449$	579.542	0.0609138	0.0304569	−	0.999536i	$-0.490304\pi$
$449$	579.542	0.0609138	0.0304569	+	0.999536i	$0.490304\pi$
$450$	0	0
$451$	−2336.48	−0.243948
$452$	0	0
$453$	0	0
$454$	0	0
$455$	1141.45	0.117609
$456$	0	0
$457$	16306.5	1.66912	0.834560	−	0.550917i	$-0.185722\pi$
$457$	16306.5	1.66912	0.834560	+	0.550917i	$0.185722\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−18894.3	−1.90888	−0.954442	−	0.298396i	$-0.903549\pi$
$461$	−18894.3	−1.90888	−0.954442	+	0.298396i	$0.903549\pi$
$462$	0	0
$463$	877.924	0.0881222	0.0440611	−	0.999029i	$-0.485970\pi$
$463$	877.924	0.0881222	0.0440611	+	0.999029i	$0.485970\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	1534.76	0.152078	0.0760388	−	0.997105i	$-0.475773\pi$
$467$	1534.76	0.152078	0.0760388	+	0.997105i	$0.475773\pi$
$468$	0	0
$469$	−2430.25	−0.239272
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−463.103	−0.0450180
$474$	0	0
$475$	−2552.83	−0.246593
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−11238.9	−1.07207	−0.536033	−	0.844197i	$-0.680078\pi$
$479$	−11238.9	−1.07207	−0.536033	+	0.844197i	$0.680078\pi$
$480$	0	0
$481$	−10312.2	−0.977536
$482$	0	0
$483$	0	0
$484$	0	0
$485$	5522.10	0.517001
$486$	0	0
$487$	−280.487	−0.0260987	−0.0130493	−	0.999915i	$-0.504154\pi$
$487$	−280.487	−0.0260987	−0.0130493	+	0.999915i	$0.504154\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	5703.07	0.524187	0.262094	−	0.965042i	$-0.415587\pi$
$491$	5703.07	0.524187	0.262094	+	0.965042i	$0.415587\pi$
$492$	0	0
$493$	−1271.38	−0.116146
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−2091.67	−0.188781
$498$	0	0
$499$	14889.4	1.33575	0.667875	−	0.744273i	$-0.267204\pi$
$499$	14889.4	1.33575	0.667875	+	0.744273i	$0.267204\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	20369.3	1.80561	0.902806	−	0.430048i	$-0.141503\pi$
$503$	20369.3	1.80561	0.902806	+	0.430048i	$0.141503\pi$
$504$	0	0
$505$	1455.51	0.128256
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−13531.2	−1.17831	−0.589154	−	0.808021i	$-0.700539\pi$
$509$	−13531.2	−1.17831	−0.589154	+	0.808021i	$0.700539\pi$
$510$	0	0
$511$	4510.37	0.390464
$512$	0	0
$513$	0	0
$514$	0	0
$515$	4974.94	0.425674
$516$	0	0
$517$	6496.41	0.552634
$518$	0	0
$519$	0	0
$520$	0	0
$521$	9222.17	0.775491	0.387745	−	0.921766i	$-0.373254\pi$
$521$	9222.17	0.775491	0.387745	+	0.921766i	$0.373254\pi$
$522$	0	0
$523$	14246.2	1.19109	0.595546	−	0.803321i	$-0.296936\pi$
$523$	14246.2	1.19109	0.595546	+	0.803321i	$0.296936\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−3809.89	−0.314917
$528$	0	0
$529$	18635.1	1.53161
$530$	0	0
$531$	0	0
$532$	0	0
$533$	8196.63	0.666108
$534$	0	0
$535$	−454.596	−0.0367363
$536$	0	0
$537$	0	0
$538$	0	0
$539$	3479.26	0.278038
$540$	0	0
$541$	19105.2	1.51829	0.759145	−	0.650922i	$-0.225617\pi$
$541$	19105.2	1.51829	0.759145	+	0.650922i	$0.225617\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	8603.44	0.676203
$546$	0	0
$547$	13200.3	1.03182	0.515909	−	0.856643i	$-0.327454\pi$
$547$	13200.3	1.03182	0.515909	+	0.856643i	$0.327454\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−6601.93	−0.510439
$552$	0	0
$553$	−4034.12	−0.310214
$554$	0	0
$555$	0	0
$556$	0	0
$557$	3808.13	0.289687	0.144844	−	0.989455i	$-0.453732\pi$
$557$	3808.13	0.289687	0.144844	+	0.989455i	$0.453732\pi$
$558$	0	0
$559$	1624.62	0.122923
$560$	0	0
$561$	0	0
$562$	0	0
$563$	18886.7	1.41381	0.706907	−	0.707306i	$-0.250090\pi$
$563$	18886.7	1.41381	0.706907	+	0.707306i	$0.250090\pi$
$564$	0	0
$565$	3293.43	0.245231
$566$	0	0
$567$	0	0
$568$	0	0
$569$	5176.45	0.381385	0.190693	−	0.981650i	$-0.438927\pi$
$569$	5176.45	0.381385	0.190693	+	0.981650i	$0.438927\pi$
$570$	0	0
$571$	3393.12	0.248683	0.124341	−	0.992240i	$-0.460318\pi$
$571$	3393.12	0.248683	0.124341	+	0.992240i	$0.460318\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	4387.63	0.318221
$576$	0	0
$577$	−3452.26	−0.249081	−0.124540	−	0.992215i	$-0.539746\pi$
$577$	−3452.26	−0.249081	−0.124540	+	0.992215i	$0.539746\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	1930.25	0.137832
$582$	0	0
$583$	−3620.36	−0.257187
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−8434.27	−0.593049	−0.296524	−	0.955025i	$-0.595828\pi$
$587$	−8434.27	−0.593049	−0.296524	+	0.955025i	$0.595828\pi$
$588$	0	0
$589$	−19783.8	−1.38400
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−15781.6	−1.09287	−0.546437	−	0.837500i	$-0.684016\pi$
$593$	−15781.6	−1.09287	−0.546437	+	0.837500i	$0.684016\pi$
$594$	0	0
$595$	−569.155	−0.0392153
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−18571.8	−1.26682	−0.633408	−	0.773818i	$-0.718344\pi$
$599$	−18571.8	−1.26682	−0.633408	+	0.773818i	$0.718344\pi$
$600$	0	0
$601$	−4244.55	−0.288084	−0.144042	−	0.989572i	$-0.546010\pi$
$601$	−4244.55	−0.288084	−0.144042	+	0.989572i	$0.546010\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	6023.10	0.404750
$606$	0	0
$607$	22200.3	1.48448	0.742241	−	0.670133i	$-0.233763\pi$
$607$	22200.3	1.48448	0.742241	+	0.670133i	$0.233763\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−22790.2	−1.50899
$612$	0	0
$613$	−20246.2	−1.33399	−0.666995	−	0.745062i	$-0.732420\pi$
$613$	−20246.2	−1.33399	−0.666995	+	0.745062i	$0.732420\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	1763.79	0.115085	0.0575425	−	0.998343i	$-0.481674\pi$
$617$	1763.79	0.115085	0.0575425	+	0.998343i	$0.481674\pi$
$618$	0	0
$619$	6638.91	0.431083	0.215542	−	0.976495i	$-0.430848\pi$
$619$	6638.91	0.431083	0.215542	+	0.976495i	$0.430848\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−3303.31	−0.212431
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	5141.90	0.325947
$630$	0	0
$631$	9459.54	0.596796	0.298398	−	0.954442i	$-0.403548\pi$
$631$	9459.54	0.596796	0.298398	+	0.954442i	$0.403548\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	7202.87	0.450137
$636$	0	0
$637$	−12205.6	−0.759192
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−6876.38	−0.423714	−0.211857	−	0.977301i	$-0.567951\pi$
$641$	−6876.38	−0.423714	−0.211857	+	0.977301i	$0.567951\pi$
$642$	0	0
$643$	23182.6	1.42183	0.710913	−	0.703280i	$-0.248282\pi$
$643$	23182.6	1.42183	0.710913	+	0.703280i	$0.248282\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−6573.94	−0.399457	−0.199728	−	0.979851i	$-0.564006\pi$
$647$	−6573.94	−0.399457	−0.199728	+	0.979851i	$0.564006\pi$
$648$	0	0
$649$	2114.69	0.127903
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−9591.02	−0.574771	−0.287386	−	0.957815i	$-0.592786\pi$
$653$	−9591.02	−0.574771	−0.287386	+	0.957815i	$0.592786\pi$
$654$	0	0
$655$	−9138.73	−0.545160
$656$	0	0
$657$	0	0
$658$	0	0
$659$	21843.5	1.29120	0.645600	−	0.763676i	$-0.276607\pi$
$659$	21843.5	1.29120	0.645600	+	0.763676i	$0.276607\pi$
$660$	0	0
$661$	15967.9	0.939604	0.469802	−	0.882772i	$-0.344325\pi$
$661$	15967.9	0.939604	0.469802	+	0.882772i	$0.344325\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−2955.48	−0.172344
$666$	0	0
$667$	11347.0	0.658705
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−2531.94	−0.145670
$672$	0	0
$673$	10292.5	0.589521	0.294760	−	0.955571i	$-0.404760\pi$
$673$	10292.5	0.589521	0.294760	+	0.955571i	$0.404760\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	10125.7	0.574831	0.287416	−	0.957806i	$-0.407204\pi$
$677$	10125.7	0.574831	0.287416	+	0.957806i	$0.407204\pi$
$678$	0	0
$679$	6393.09	0.361332
$680$	0	0
$681$	0	0
$682$	0	0
$683$	21430.9	1.20063	0.600317	−	0.799762i	$-0.295041\pi$
$683$	21430.9	1.20063	0.600317	+	0.799762i	$0.295041\pi$
$684$	0	0
$685$	9519.45	0.530977
$686$	0	0
$687$	0	0
$688$	0	0
$689$	12700.7	0.702259
$690$	0	0
$691$	19890.2	1.09502	0.547510	−	0.836799i	$-0.315576\pi$
$691$	19890.2	1.09502	0.547510	+	0.836799i	$0.315576\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	1411.80	0.0770543
$696$	0	0
$697$	−4087.04	−0.222105
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−13792.1	−0.743112	−0.371556	−	0.928410i	$-0.621176\pi$
$701$	−13792.1	−0.743112	−0.371556	+	0.928410i	$0.621176\pi$
$702$	0	0
$703$	26700.6	1.43248
$704$	0	0
$705$	0	0
$706$	0	0
$707$	1685.08	0.0896377
$708$	0	0
$709$	839.009	0.0444424	0.0222212	−	0.999753i	$-0.492926\pi$
$709$	839.009	0.0444424	0.0222212	+	0.999753i	$0.492926\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	34003.0	1.78601
$714$	0	0
$715$	2216.77	0.115947
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−13911.1	−0.721554	−0.360777	−	0.932652i	$-0.617488\pi$
$719$	−13911.1	−0.721554	−0.360777	+	0.932652i	$0.617488\pi$
$720$	0	0
$721$	5759.62	0.297503
$722$	0	0
$723$	0	0
$724$	0	0
$725$	1616.33	0.0827984
$726$	0	0
$727$	−22702.7	−1.15818	−0.579089	−	0.815264i	$-0.696592\pi$
$727$	−22702.7	−1.15818	−0.579089	+	0.815264i	$0.696592\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−810.074	−0.0409872
$732$	0	0
$733$	−10727.1	−0.540536	−0.270268	−	0.962785i	$-0.587112\pi$
$733$	−10727.1	−0.540536	−0.270268	+	0.962785i	$0.587112\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−4719.68	−0.235891
$738$	0	0
$739$	7251.82	0.360978	0.180489	−	0.983577i	$-0.442232\pi$
$739$	7251.82	0.360978	0.180489	+	0.983577i	$0.442232\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	11369.6	0.561385	0.280693	−	0.959798i	$-0.409436\pi$
$743$	11369.6	0.561385	0.280693	+	0.959798i	$0.409436\pi$
$744$	0	0
$745$	8182.52	0.402395
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−526.298	−0.0256749
$750$	0	0
$751$	7313.05	0.355336	0.177668	−	0.984091i	$-0.443145\pi$
$751$	7313.05	0.355336	0.177668	+	0.984091i	$0.443145\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	8411.73	0.405476
$756$	0	0
$757$	12759.1	0.612601	0.306301	−	0.951935i	$-0.400909\pi$
$757$	12759.1	0.612601	0.306301	+	0.951935i	$0.400909\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−16085.9	−0.766247	−0.383124	−	0.923697i	$-0.625152\pi$
$761$	−16085.9	−0.766247	−0.383124	+	0.923697i	$0.625152\pi$
$762$	0	0
$763$	9960.43	0.472597
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−7418.58	−0.349243
$768$	0	0
$769$	−14518.3	−0.680808	−0.340404	−	0.940279i	$-0.610564\pi$
$769$	−14518.3	−0.680808	−0.340404	+	0.940279i	$0.610564\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−9065.89	−0.421834	−0.210917	−	0.977504i	$-0.567645\pi$
$773$	−9065.89	−0.421834	−0.210917	+	0.977504i	$0.567645\pi$
$774$	0	0
$775$	4843.59	0.224499
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−21222.9	−0.976111
$780$	0	0
$781$	−4062.14	−0.186114
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−9795.89	−0.445389
$786$	0	0
$787$	13860.6	0.627796	0.313898	−	0.949457i	$-0.398365\pi$
$787$	13860.6	0.627796	0.313898	+	0.949457i	$0.398365\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	3812.89	0.171392
$792$	0	0
$793$	8882.35	0.397757
$794$	0	0
$795$	0	0
$796$	0	0
$797$	8384.39	0.372635	0.186318	−	0.982490i	$-0.440345\pi$
$797$	8384.39	0.372635	0.186318	+	0.982490i	$0.440345\pi$
$798$	0	0
$799$	11363.7	0.503153
$800$	0	0
$801$	0	0
$802$	0	0
$803$	8759.38	0.384946
$804$	0	0
$805$	5079.68	0.222404
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−21047.6	−0.914704	−0.457352	−	0.889286i	$-0.651202\pi$
$809$	−21047.6	−0.914704	−0.457352	+	0.889286i	$0.651202\pi$
$810$	0	0
$811$	19804.0	0.857474	0.428737	−	0.903429i	$-0.358959\pi$
$811$	19804.0	0.857474	0.428737	+	0.903429i	$0.358959\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	2703.89	0.116213
$816$	0	0
$817$	−4206.51	−0.180131
$818$	0	0
$819$	0	0
$820$	0	0
$821$	28001.8	1.19034	0.595170	−	0.803600i	$-0.297085\pi$
$821$	28001.8	1.19034	0.595170	+	0.803600i	$0.297085\pi$
$822$	0	0
$823$	−27449.3	−1.16260	−0.581302	−	0.813688i	$-0.697457\pi$
$823$	−27449.3	−1.16260	−0.581302	+	0.813688i	$0.697457\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−796.827	−0.0335047	−0.0167524	−	0.999860i	$-0.505333\pi$
$827$	−796.827	−0.0335047	−0.0167524	+	0.999860i	$0.505333\pi$
$828$	0	0
$829$	39005.0	1.63414	0.817068	−	0.576541i	$-0.195598\pi$
$829$	39005.0	1.63414	0.817068	+	0.576541i	$0.195598\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	6086.02	0.253143
$834$	0	0
$835$	−2009.54	−0.0832850
$836$	0	0
$837$	0	0
$838$	0	0
$839$	31810.5	1.30896	0.654482	−	0.756077i	$-0.272887\pi$
$839$	31810.5	1.30896	0.654482	+	0.756077i	$0.272887\pi$
$840$	0	0
$841$	−20209.0	−0.828610
$842$	0	0
$843$	0	0
$844$	0	0
$845$	3208.33	0.130615
$846$	0	0
$847$	6973.11	0.282879
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−45891.2	−1.84856
$852$	0	0
$853$	−12145.7	−0.487528	−0.243764	−	0.969835i	$-0.578382\pi$
$853$	−12145.7	−0.487528	−0.243764	+	0.969835i	$0.578382\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−37821.7	−1.50754	−0.753771	−	0.657137i	$-0.771767\pi$
$857$	−37821.7	−1.50754	−0.753771	+	0.657137i	$0.771767\pi$
$858$	0	0
$859$	29061.8	1.15434	0.577168	−	0.816625i	$-0.304158\pi$
$859$	29061.8	1.15434	0.577168	+	0.816625i	$0.304158\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	33579.7	1.32452	0.662262	−	0.749272i	$-0.269596\pi$
$863$	33579.7	1.32452	0.662262	+	0.749272i	$0.269596\pi$
$864$	0	0
$865$	−16222.5	−0.637665
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−7834.48	−0.305830
$870$	0	0
$871$	16557.2	0.644109
$872$	0	0
$873$	0	0
$874$	0	0
$875$	723.579	0.0279559
$876$	0	0
$877$	−22286.0	−0.858091	−0.429045	−	0.903283i	$-0.641150\pi$
$877$	−22286.0	−0.858091	−0.429045	+	0.903283i	$0.641150\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−29440.6	−1.12586	−0.562928	−	0.826506i	$-0.690325\pi$
$881$	−29440.6	−1.12586	−0.562928	+	0.826506i	$0.690325\pi$
$882$	0	0
$883$	25076.6	0.955713	0.477857	−	0.878438i	$-0.341414\pi$
$883$	25076.6	0.955713	0.477857	+	0.878438i	$0.341414\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−3512.01	−0.132944	−0.0664722	−	0.997788i	$-0.521174\pi$
$887$	−3512.01	−0.132944	−0.0664722	+	0.997788i	$0.521174\pi$
$888$	0	0
$889$	8338.96	0.314600
$890$	0	0
$891$	0	0
$892$	0	0
$893$	59008.9	2.21126
$894$	0	0
$895$	19225.4	0.718025
$896$	0	0
$897$	0	0
$898$	0	0
$899$	12526.1	0.464705
$900$	0	0
$901$	−6332.85	−0.234160
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−4419.78	−0.162341
$906$	0	0
$907$	9752.72	0.357038	0.178519	−	0.983936i	$-0.442869\pi$
$907$	9752.72	0.357038	0.178519	+	0.983936i	$0.442869\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−38243.9	−1.39086	−0.695432	−	0.718592i	$-0.744787\pi$
$911$	−38243.9	−1.39086	−0.695432	+	0.718592i	$0.744787\pi$
$912$	0	0
$913$	3748.65	0.135884
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−10580.1	−0.381011
$918$	0	0
$919$	−1523.05	−0.0546691	−0.0273345	−	0.999626i	$-0.508702\pi$
$919$	−1523.05	−0.0546691	−0.0273345	+	0.999626i	$0.508702\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	14250.5	0.508191
$924$	0	0
$925$	−6537.00	−0.232363
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−25605.5	−0.904294	−0.452147	−	0.891943i	$-0.649342\pi$
$929$	−25605.5	−0.904294	−0.452147	+	0.891943i	$0.649342\pi$
$930$	0	0
$931$	31603.2	1.11252
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−1105.33	−0.0386612
$936$	0	0
$937$	36738.1	1.28088	0.640439	−	0.768009i	$-0.278752\pi$
$937$	36738.1	1.28088	0.640439	+	0.768009i	$0.278752\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−46983.5	−1.62765	−0.813824	−	0.581111i	$-0.802618\pi$
$941$	−46983.5	−1.62765	−0.813824	+	0.581111i	$0.802618\pi$
$942$	0	0
$943$	36476.6	1.25964
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−12024.4	−0.412607	−0.206304	−	0.978488i	$-0.566143\pi$
$947$	−12024.4	−0.412607	−0.206304	+	0.978488i	$0.566143\pi$
$948$	0	0
$949$	−30728.9	−1.05111
$950$	0	0
$951$	0	0
$952$	0	0
$953$	35244.7	1.19799	0.598996	−	0.800752i	$-0.295567\pi$
$953$	35244.7	1.19799	0.598996	+	0.800752i	$0.295567\pi$
$954$	0	0
$955$	2046.63	0.0693480
$956$	0	0
$957$	0	0
$958$	0	0
$959$	11020.9	0.371099
$960$	0	0
$961$	7745.57	0.259997
$962$	0	0
$963$	0	0
$964$	0	0
$965$	1841.73	0.0614377
$966$	0	0
$967$	27478.2	0.913795	0.456898	−	0.889519i	$-0.348961\pi$
$967$	27478.2	0.913795	0.456898	+	0.889519i	$0.348961\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	10673.6	0.352762	0.176381	−	0.984322i	$-0.443561\pi$
$971$	10673.6	0.352762	0.176381	+	0.984322i	$0.443561\pi$
$972$	0	0
$973$	1634.48	0.0538532
$974$	0	0
$975$	0	0
$976$	0	0
$977$	22601.8	0.740120	0.370060	−	0.929008i	$-0.379337\pi$
$977$	22601.8	0.740120	0.370060	+	0.929008i	$0.379337\pi$
$978$	0	0
$979$	−6415.22	−0.209429
$980$	0	0
$981$	0	0
$982$	0	0
$983$	2121.82	0.0688458	0.0344229	−	0.999407i	$-0.489041\pi$
$983$	2121.82	0.0688458	0.0344229	+	0.999407i	$0.489041\pi$
$984$	0	0
$985$	−14971.1	−0.484285
$986$	0	0
$987$	0	0
$988$	0	0
$989$	7229.87	0.232453
$990$	0	0
$991$	−45219.3	−1.44948	−0.724741	−	0.689021i	$-0.758041\pi$
$991$	−45219.3	−1.44948	−0.724741	+	0.689021i	$0.758041\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−4436.11	−0.141341
$996$	0	0
$997$	−36545.9	−1.16090	−0.580452	−	0.814295i	$-0.697124\pi$
$997$	−36545.9	−1.16090	−0.580452	+	0.814295i	$0.697124\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1620.4.a.g.1.2		4
3.2	odd	2		1620.4.a.h.1.2		4
9.2	odd	6		540.4.i.b.361.3		8
9.4	even	3		180.4.i.b.61.1	✓	8
9.5	odd	6		540.4.i.b.181.3		8
9.7	even	3		180.4.i.b.121.1	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
180.4.i.b.61.1	✓	8	9.4	even	3
180.4.i.b.121.1	yes	8	9.7	even	3
540.4.i.b.181.3		8	9.5	odd	6
540.4.i.b.361.3		8	9.2	odd	6
1620.4.a.g.1.2		4	1.1	even	1	trivial
1620.4.a.h.1.2		4	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1620 = 2^{2} \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1620.a (trivial)

\(f(q)\)	\(=\)	\(q-5.00000 q^{5} -5.78863 q^{7} +O(q^{10})\)	\(q-5.00000 q^{5} -5.78863 q^{7} -11.2419 q^{11} +39.4377 q^{13} -19.6646 q^{17} -102.113 q^{19} +175.505 q^{23} +25.0000 q^{25} +64.6531 q^{29} +193.744 q^{31} +28.9432 q^{35} -261.480 q^{37} +207.837 q^{41} +41.1946 q^{43} -577.877 q^{47} -309.492 q^{49} +322.043 q^{53} +56.2093 q^{55} -188.109 q^{59} +225.225 q^{61} -197.189 q^{65} +419.831 q^{67} +361.341 q^{71} -779.176 q^{73} +65.0750 q^{77} +696.903 q^{79} -333.455 q^{83} +98.3229 q^{85} +570.655 q^{89} -228.291 q^{91} +510.566 q^{95} -1104.42 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 20 q^{5} - 13 q^{7}+O(q^{10}) \) 4 * q - 20 * q^5 - 13 * q^7	\( 4 q - 20 q^{5} - 13 q^{7} + 57 q^{11} + 14 q^{13} + 3 q^{17} - 31 q^{19} - 69 q^{23} + 100 q^{25} - 69 q^{29} - 58 q^{31} + 65 q^{35} - 388 q^{37} + 396 q^{41} + 371 q^{43} + 129 q^{47} + 111 q^{49} + 1356 q^{53} - 285 q^{55} + 15 q^{59} - 1441 q^{61} - 70 q^{65} + 368 q^{67} + 168 q^{71} - 955 q^{73} + 342 q^{77} - 1408 q^{79} - 789 q^{83} - 15 q^{85} + 1617 q^{89} - 1406 q^{91} + 155 q^{95} - 1495 q^{97}+O(q^{100}) \) 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