LMFDB - Embedded newform 1620.4.a.k.1.1

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:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 2x^{6} - 167x^{5} + 940x^{4} + 1153x^{3} - 13196x^{2} + 16629x + 714 $ x^7 - 2x^6 - 167x^5 + 940x^4 + 1153x^3 - 13196x^2 + 16629x + 714
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{5}\cdot 3^{7} $
Twist minimal:	no (minimal twist has level 180)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-3.82911$ 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)

$f(q)$	$=$	$q-5.00000 q^{5} -29.9233 q^{7} +O(q^{10})$	$q-5.00000 q^{5} -29.9233 q^{7} -56.7936 q^{11} +30.3736 q^{13} -99.8818 q^{17} -81.8966 q^{19} -137.407 q^{23} +25.0000 q^{25} +49.0987 q^{29} +3.50558 q^{31} +149.617 q^{35} -289.311 q^{37} +199.921 q^{41} -516.748 q^{43} +109.761 q^{47} +552.406 q^{49} -283.500 q^{53} +283.968 q^{55} +185.703 q^{59} -235.900 q^{61} -151.868 q^{65} -341.466 q^{67} -1136.54 q^{71} +1111.18 q^{73} +1699.45 q^{77} -200.600 q^{79} -205.419 q^{83} +499.409 q^{85} -350.554 q^{89} -908.879 q^{91} +409.483 q^{95} +923.606 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - 35 q^{5} + 8 q^{7}+O(q^{10}) $ 7 * q - 35 * q^5 + 8 * q^7	$ 7 q - 35 q^{5} + 8 q^{7} - 27 q^{11} + 32 q^{13} - 123 q^{17} - 67 q^{19} - 42 q^{23} + 175 q^{25} - 324 q^{29} + 98 q^{31} - 40 q^{35} + 356 q^{37} - 339 q^{41} + 119 q^{43} + 96 q^{47} + 813 q^{49} - 858 q^{53} + 135 q^{55} - 549 q^{59} + 260 q^{61} - 160 q^{65} + 881 q^{67} - 342 q^{71} + 737 q^{73} + 456 q^{77} + 1886 q^{79} + 132 q^{83} + 615 q^{85} - 396 q^{89} + 1264 q^{91} + 335 q^{95} + 1991 q^{97}+O(q^{100}) $ 7 * q - 35 * q^5 + 8 * q^7 - 27 * q^11 + 32 * q^13 - 123 * q^17 - 67 * q^19 - 42 * q^23 + 175 * q^25 - 324 * q^29 + 98 * q^31 - 40 * q^35 + 356 * q^37 - 339 * q^41 + 119 * q^43 + 96 * q^47 + 813 * q^49 - 858 * q^53 + 135 * q^55 - 549 * q^59 + 260 * q^61 - 160 * q^65 + 881 * q^67 - 342 * q^71 + 737 * q^73 + 456 * q^77 + 1886 * q^79 + 132 * q^83 + 615 * q^85 - 396 * q^89 + 1264 * q^91 + 335 * q^95 + 1991 * 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
$7$	−29.9233	−1.61571	−0.807854	−	0.589382i	$-0.799371\pi$
$7$	−29.9233	−1.61571	−0.807854	+	0.589382i	$0.799371\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−56.7936	−1.55672	−0.778360	−	0.627818i	$-0.783948\pi$
$11$	−56.7936	−1.55672	−0.778360	+	0.627818i	$0.783948\pi$
$12$	0	0
$13$	30.3736	0.648009	0.324004	−	0.946056i	$-0.394971\pi$
$13$	30.3736	0.648009	0.324004	+	0.946056i	$0.394971\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−99.8818	−1.42499	−0.712497	−	0.701675i	$-0.752436\pi$
$17$	−99.8818	−1.42499	−0.712497	+	0.701675i	$0.752436\pi$
$18$	0	0
$19$	−81.8966	−0.988862	−0.494431	−	0.869217i	$-0.664624\pi$
$19$	−81.8966	−0.988862	−0.494431	+	0.869217i	$0.664624\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−137.407	−1.24571	−0.622856	−	0.782336i	$-0.714028\pi$
$23$	−137.407	−1.24571	−0.622856	+	0.782336i	$0.714028\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	49.0987	0.314393	0.157197	−	0.987567i	$-0.449754\pi$
$29$	49.0987	0.314393	0.157197	+	0.987567i	$0.449754\pi$
$30$	0	0
$31$	3.50558	0.0203103	0.0101552	−	0.999948i	$-0.496767\pi$
$31$	3.50558	0.0203103	0.0101552	+	0.999948i	$0.496767\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	149.617	0.722567
$36$	0	0
$37$	−289.311	−1.28547	−0.642736	−	0.766088i	$-0.722201\pi$
$37$	−289.311	−1.28547	−0.642736	+	0.766088i	$0.722201\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	199.921	0.761523	0.380761	−	0.924673i	$-0.375662\pi$
$41$	199.921	0.761523	0.380761	+	0.924673i	$0.375662\pi$
$42$	0	0
$43$	−516.748	−1.83264	−0.916318	−	0.400452i	$-0.868853\pi$
$43$	−516.748	−1.83264	−0.916318	+	0.400452i	$0.868853\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	109.761	0.340646	0.170323	−	0.985388i	$-0.445519\pi$
$47$	109.761	0.340646	0.170323	+	0.985388i	$0.445519\pi$
$48$	0	0
$49$	552.406	1.61051
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−283.500	−0.734748	−0.367374	−	0.930073i	$-0.619743\pi$
$53$	−283.500	−0.734748	−0.367374	+	0.930073i	$0.619743\pi$
$54$	0	0
$55$	283.968	0.696186
$56$	0	0
$57$	0	0
$58$	0	0
$59$	185.703	0.409771	0.204885	−	0.978786i	$-0.434318\pi$
$59$	185.703	0.409771	0.204885	+	0.978786i	$0.434318\pi$
$60$	0	0
$61$	−235.900	−0.495146	−0.247573	−	0.968869i	$-0.579633\pi$
$61$	−235.900	−0.495146	−0.247573	+	0.968869i	$0.579633\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−151.868	−0.289798
$66$	0	0
$67$	−341.466	−0.622637	−0.311319	−	0.950306i	$-0.600771\pi$
$67$	−341.466	−0.622637	−0.311319	+	0.950306i	$0.600771\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−1136.54	−1.89976	−0.949878	−	0.312621i	$-0.898793\pi$
$71$	−1136.54	−1.89976	−0.949878	+	0.312621i	$0.898793\pi$
$72$	0	0
$73$	1111.18	1.78156	0.890782	−	0.454431i	$-0.150157\pi$
$73$	1111.18	1.78156	0.890782	+	0.454431i	$0.150157\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	1699.45	2.51520
$78$	0	0
$79$	−200.600	−0.285687	−0.142844	−	0.989745i	$-0.545625\pi$
$79$	−200.600	−0.285687	−0.142844	+	0.989745i	$0.545625\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−205.419	−0.271658	−0.135829	−	0.990732i	$-0.543370\pi$
$83$	−205.419	−0.271658	−0.135829	+	0.990732i	$0.543370\pi$
$84$	0	0
$85$	499.409	0.637277
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−350.554	−0.417513	−0.208757	−	0.977968i	$-0.566942\pi$
$89$	−350.554	−0.417513	−0.208757	+	0.977968i	$0.566942\pi$
$90$	0	0
$91$	−908.879	−1.04699
$92$	0	0
$93$	0	0
$94$	0	0
$95$	409.483	0.442232
$96$	0	0
$97$	923.606	0.966783	0.483391	−	0.875404i	$-0.339405\pi$
$97$	923.606	0.966783	0.483391	+	0.875404i	$0.339405\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−746.924	−0.735859	−0.367930	−	0.929854i	$-0.619933\pi$
$101$	−746.924	−0.735859	−0.367930	+	0.929854i	$0.619933\pi$
$102$	0	0
$103$	1984.54	1.89847	0.949233	−	0.314573i	$-0.101861\pi$
$103$	1984.54	1.89847	0.949233	+	0.314573i	$0.101861\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−150.201	−0.135705	−0.0678525	−	0.997695i	$-0.521615\pi$
$107$	−150.201	−0.135705	−0.0678525	+	0.997695i	$0.521615\pi$
$108$	0	0
$109$	1832.32	1.61013	0.805066	−	0.593185i	$-0.202130\pi$
$109$	1832.32	1.61013	0.805066	+	0.593185i	$0.202130\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−824.140	−0.686094	−0.343047	−	0.939318i	$-0.611459\pi$
$113$	−824.140	−0.686094	−0.343047	+	0.939318i	$0.611459\pi$
$114$	0	0
$115$	687.036	0.557100
$116$	0	0
$117$	0	0
$118$	0	0
$119$	2988.80	2.30237
$120$	0	0
$121$	1894.51	1.42338
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−125.000	−0.0894427
$126$	0	0
$127$	1262.02	0.881781	0.440891	−	0.897561i	$-0.354663\pi$
$127$	1262.02	0.881781	0.440891	+	0.897561i	$0.354663\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−900.088	−0.600313	−0.300157	−	0.953890i	$-0.597039\pi$
$131$	−900.088	−0.600313	−0.300157	+	0.953890i	$0.597039\pi$
$132$	0	0
$133$	2450.62	1.59771
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−1929.62	−1.20335	−0.601674	−	0.798742i	$-0.705499\pi$
$137$	−1929.62	−1.20335	−0.601674	+	0.798742i	$0.705499\pi$
$138$	0	0
$139$	1495.38	0.912490	0.456245	−	0.889854i	$-0.349194\pi$
$139$	1495.38	0.912490	0.456245	+	0.889854i	$0.349194\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−1725.02	−1.00877
$144$	0	0
$145$	−245.493	−0.140601
$146$	0	0
$147$	0	0
$148$	0	0
$149$	1234.07	0.678514	0.339257	−	0.940694i	$-0.389824\pi$
$149$	1234.07	0.678514	0.339257	+	0.940694i	$0.389824\pi$
$150$	0	0
$151$	−3350.55	−1.80572	−0.902860	−	0.429934i	$-0.858537\pi$
$151$	−3350.55	−1.80572	−0.902860	+	0.429934i	$0.858537\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−17.5279	−0.00908306
$156$	0	0
$157$	−1543.84	−0.784789	−0.392394	−	0.919797i	$-0.628353\pi$
$157$	−1543.84	−0.784789	−0.392394	+	0.919797i	$0.628353\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	4111.68	2.01271
$162$	0	0
$163$	−1650.49	−0.793105	−0.396553	−	0.918012i	$-0.629794\pi$
$163$	−1650.49	−0.793105	−0.396553	+	0.918012i	$0.629794\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−2494.57	−1.15590	−0.577950	−	0.816072i	$-0.696147\pi$
$167$	−2494.57	−1.15590	−0.577950	+	0.816072i	$0.696147\pi$
$168$	0	0
$169$	−1274.45	−0.580085
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−1068.00	−0.469356	−0.234678	−	0.972073i	$-0.575404\pi$
$173$	−1068.00	−0.469356	−0.234678	+	0.972073i	$0.575404\pi$
$174$	0	0
$175$	−748.083	−0.323142
$176$	0	0
$177$	0	0
$178$	0	0
$179$	3372.01	1.40802	0.704010	−	0.710190i	$-0.251391\pi$
$179$	3372.01	1.40802	0.704010	+	0.710190i	$0.251391\pi$
$180$	0	0
$181$	2998.84	1.23150	0.615752	−	0.787940i	$-0.288852\pi$
$181$	2998.84	1.23150	0.615752	+	0.787940i	$0.288852\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	1446.56	0.574880
$186$	0	0
$187$	5672.65	2.21832
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−2025.20	−0.767217	−0.383609	−	0.923496i	$-0.625319\pi$
$191$	−2025.20	−0.767217	−0.383609	+	0.923496i	$0.625319\pi$
$192$	0	0
$193$	966.885	0.360611	0.180305	−	0.983611i	$-0.442291\pi$
$193$	966.885	0.360611	0.180305	+	0.983611i	$0.442291\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−1396.27	−0.504977	−0.252488	−	0.967600i	$-0.581249\pi$
$197$	−1396.27	−0.504977	−0.252488	+	0.967600i	$0.581249\pi$
$198$	0	0
$199$	−2644.97	−0.942195	−0.471098	−	0.882081i	$-0.656142\pi$
$199$	−2644.97	−0.942195	−0.471098	+	0.882081i	$0.656142\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−1469.20	−0.507967
$204$	0	0
$205$	−999.606	−0.340563
$206$	0	0
$207$	0	0
$208$	0	0
$209$	4651.20	1.53938
$210$	0	0
$211$	−653.657	−0.213268	−0.106634	−	0.994298i	$-0.534007\pi$
$211$	−653.657	−0.213268	−0.106634	+	0.994298i	$0.534007\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	2583.74	0.819579
$216$	0	0
$217$	−104.899	−0.0328156
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−3033.77	−0.923408
$222$	0	0
$223$	4058.86	1.21884	0.609420	−	0.792848i	$-0.291403\pi$
$223$	4058.86	1.21884	0.609420	+	0.792848i	$0.291403\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	5535.47	1.61851	0.809256	−	0.587457i	$-0.199871\pi$
$227$	5535.47	1.61851	0.809256	+	0.587457i	$0.199871\pi$
$228$	0	0
$229$	−3251.07	−0.938152	−0.469076	−	0.883158i	$-0.655413\pi$
$229$	−3251.07	−0.938152	−0.469076	+	0.883158i	$0.655413\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−4577.57	−1.28707	−0.643534	−	0.765418i	$-0.722532\pi$
$233$	−4577.57	−1.28707	−0.643534	+	0.765418i	$0.722532\pi$
$234$	0	0
$235$	−548.807	−0.152341
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−90.0622	−0.0243751	−0.0121875	−	0.999926i	$-0.503880\pi$
$239$	−90.0622	−0.0243751	−0.0121875	+	0.999926i	$0.503880\pi$
$240$	0	0
$241$	3387.16	0.905337	0.452668	−	0.891679i	$-0.350472\pi$
$241$	3387.16	0.905337	0.452668	+	0.891679i	$0.350472\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−2762.03	−0.720243
$246$	0	0
$247$	−2487.49	−0.640791
$248$	0	0
$249$	0	0
$250$	0	0
$251$	5537.80	1.39260	0.696301	−	0.717750i	$-0.254828\pi$
$251$	5537.80	1.39260	0.696301	+	0.717750i	$0.254828\pi$
$252$	0	0
$253$	7803.85	1.93923
$254$	0	0
$255$	0	0
$256$	0	0
$257$	1805.55	0.438237	0.219118	−	0.975698i	$-0.429682\pi$
$257$	1805.55	0.438237	0.219118	+	0.975698i	$0.429682\pi$
$258$	0	0
$259$	8657.15	2.07695
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−1828.43	−0.428692	−0.214346	−	0.976758i	$-0.568762\pi$
$263$	−1828.43	−0.428692	−0.214346	+	0.976758i	$0.568762\pi$
$264$	0	0
$265$	1417.50	0.328590
$266$	0	0
$267$	0	0
$268$	0	0
$269$	865.586	0.196192	0.0980961	−	0.995177i	$-0.468725\pi$
$269$	865.586	0.196192	0.0980961	+	0.995177i	$0.468725\pi$
$270$	0	0
$271$	3835.39	0.859717	0.429858	−	0.902896i	$-0.358563\pi$
$271$	3835.39	0.859717	0.429858	+	0.902896i	$0.358563\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−1419.84	−0.311344
$276$	0	0
$277$	3228.42	0.700279	0.350139	−	0.936698i	$-0.386134\pi$
$277$	3228.42	0.700279	0.350139	+	0.936698i	$0.386134\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−7341.12	−1.55849	−0.779243	−	0.626722i	$-0.784396\pi$
$281$	−7341.12	−1.55849	−0.779243	+	0.626722i	$0.784396\pi$
$282$	0	0
$283$	−5095.34	−1.07027	−0.535135	−	0.844767i	$-0.679739\pi$
$283$	−5095.34	−1.07027	−0.535135	+	0.844767i	$0.679739\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−5982.31	−1.23040
$288$	0	0
$289$	5063.38	1.03061
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−5035.60	−1.00404	−0.502018	−	0.864857i	$-0.667409\pi$
$293$	−5035.60	−1.00404	−0.502018	+	0.864857i	$0.667409\pi$
$294$	0	0
$295$	−928.515	−0.183255
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−4173.55	−0.807233
$300$	0	0
$301$	15462.8	2.96100
$302$	0	0
$303$	0	0
$304$	0	0
$305$	1179.50	0.221436
$306$	0	0
$307$	4904.43	0.911760	0.455880	−	0.890041i	$-0.349325\pi$
$307$	4904.43	0.911760	0.455880	+	0.890041i	$0.349325\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	6573.99	1.19864	0.599319	−	0.800510i	$-0.295438\pi$
$311$	6573.99	1.19864	0.599319	+	0.800510i	$0.295438\pi$
$312$	0	0
$313$	−5848.37	−1.05613	−0.528066	−	0.849203i	$-0.677083\pi$
$313$	−5848.37	−1.05613	−0.528066	+	0.849203i	$0.677083\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	1408.56	0.249566	0.124783	−	0.992184i	$-0.460177\pi$
$317$	1408.56	0.249566	0.124783	+	0.992184i	$0.460177\pi$
$318$	0	0
$319$	−2788.49	−0.489422
$320$	0	0
$321$	0	0
$322$	0	0
$323$	8179.98	1.40912
$324$	0	0
$325$	759.339	0.129602
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−3284.43	−0.550384
$330$	0	0
$331$	2636.62	0.437830	0.218915	−	0.975744i	$-0.429748\pi$
$331$	2636.62	0.437830	0.218915	+	0.975744i	$0.429748\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	1707.33	0.278452
$336$	0	0
$337$	−1401.10	−0.226478	−0.113239	−	0.993568i	$-0.536123\pi$
$337$	−1401.10	−0.226478	−0.113239	+	0.993568i	$0.536123\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−199.094	−0.0316175
$342$	0	0
$343$	−6266.13	−0.986411
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−7591.66	−1.17447	−0.587236	−	0.809416i	$-0.699784\pi$
$347$	−7591.66	−1.17447	−0.587236	+	0.809416i	$0.699784\pi$
$348$	0	0
$349$	7639.58	1.17174	0.585870	−	0.810405i	$-0.300753\pi$
$349$	7639.58	1.17174	0.585870	+	0.810405i	$0.300753\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	769.756	0.116062	0.0580312	−	0.998315i	$-0.481518\pi$
$353$	769.756	0.116062	0.0580312	+	0.998315i	$0.481518\pi$
$354$	0	0
$355$	5682.71	0.849597
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−10590.6	−1.55696	−0.778482	−	0.627667i	$-0.784010\pi$
$359$	−10590.6	−1.55696	−0.778482	+	0.627667i	$0.784010\pi$
$360$	0	0
$361$	−151.943	−0.0221523
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−5555.92	−0.796740
$366$	0	0
$367$	8951.59	1.27321	0.636607	−	0.771189i	$-0.280338\pi$
$367$	8951.59	1.27321	0.636607	+	0.771189i	$0.280338\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	8483.26	1.18714
$372$	0	0
$373$	48.0298	0.00666726	0.00333363	−	0.999994i	$-0.498939\pi$
$373$	48.0298	0.00666726	0.00333363	+	0.999994i	$0.498939\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	1491.30	0.203729
$378$	0	0
$379$	−1079.70	−0.146334	−0.0731670	−	0.997320i	$-0.523311\pi$
$379$	−1079.70	−0.146334	−0.0731670	+	0.997320i	$0.523311\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−7059.37	−0.941820	−0.470910	−	0.882181i	$-0.656074\pi$
$383$	−7059.37	−0.941820	−0.470910	+	0.882181i	$0.656074\pi$
$384$	0	0
$385$	−8497.27	−1.12483
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−6004.88	−0.782672	−0.391336	−	0.920248i	$-0.627987\pi$
$389$	−6004.88	−0.782672	−0.391336	+	0.920248i	$0.627987\pi$
$390$	0	0
$391$	13724.5	1.77513
$392$	0	0
$393$	0	0
$394$	0	0
$395$	1003.00	0.127763
$396$	0	0
$397$	−1805.81	−0.228290	−0.114145	−	0.993464i	$-0.536413\pi$
$397$	−1805.81	−0.228290	−0.114145	+	0.993464i	$0.536413\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	11255.4	1.40167	0.700835	−	0.713323i	$-0.252811\pi$
$401$	11255.4	1.40167	0.700835	+	0.713323i	$0.252811\pi$
$402$	0	0
$403$	106.477	0.0131613
$404$	0	0
$405$	0	0
$406$	0	0
$407$	16431.0	2.00112
$408$	0	0
$409$	3246.31	0.392469	0.196234	−	0.980557i	$-0.437129\pi$
$409$	3246.31	0.392469	0.196234	+	0.980557i	$0.437129\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−5556.85	−0.662070
$414$	0	0
$415$	1027.09	0.121489
$416$	0	0
$417$	0	0
$418$	0	0
$419$	9390.58	1.09489	0.547446	−	0.836841i	$-0.315600\pi$
$419$	9390.58	1.09489	0.547446	+	0.836841i	$0.315600\pi$
$420$	0	0
$421$	−9342.87	−1.08158	−0.540788	−	0.841159i	$-0.681874\pi$
$421$	−9342.87	−1.08158	−0.540788	+	0.841159i	$0.681874\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−2497.05	−0.284999
$426$	0	0
$427$	7058.92	0.800012
$428$	0	0
$429$	0	0
$430$	0	0
$431$	16878.0	1.88628	0.943139	−	0.332399i	$-0.107858\pi$
$431$	16878.0	1.88628	0.943139	+	0.332399i	$0.107858\pi$
$432$	0	0
$433$	−3033.14	−0.336636	−0.168318	−	0.985733i	$-0.553833\pi$
$433$	−3033.14	−0.336636	−0.168318	+	0.985733i	$0.553833\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	11253.2	1.23184
$438$	0	0
$439$	−697.181	−0.0757964	−0.0378982	−	0.999282i	$-0.512066\pi$
$439$	−697.181	−0.0757964	−0.0378982	+	0.999282i	$0.512066\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−9247.35	−0.991772	−0.495886	−	0.868388i	$-0.665157\pi$
$443$	−9247.35	−0.991772	−0.495886	+	0.868388i	$0.665157\pi$
$444$	0	0
$445$	1752.77	0.186718
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−2635.50	−0.277009	−0.138504	−	0.990362i	$-0.544230\pi$
$449$	−2635.50	−0.277009	−0.138504	+	0.990362i	$0.544230\pi$
$450$	0	0
$451$	−11354.2	−1.18548
$452$	0	0
$453$	0	0
$454$	0	0
$455$	4544.39	0.468229
$456$	0	0
$457$	−12233.3	−1.25218	−0.626092	−	0.779749i	$-0.715346\pi$
$457$	−12233.3	−1.25218	−0.626092	+	0.779749i	$0.715346\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−18373.8	−1.85630	−0.928148	−	0.372212i	$-0.878599\pi$
$461$	−18373.8	−1.85630	−0.928148	+	0.372212i	$0.878599\pi$
$462$	0	0
$463$	−6761.84	−0.678725	−0.339362	−	0.940656i	$-0.610211\pi$
$463$	−6761.84	−0.678725	−0.339362	+	0.940656i	$0.610211\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−15691.3	−1.55483	−0.777416	−	0.628987i	$-0.783470\pi$
$467$	−15691.3	−1.55483	−0.777416	+	0.628987i	$0.783470\pi$
$468$	0	0
$469$	10217.8	1.00600
$470$	0	0
$471$	0	0
$472$	0	0
$473$	29348.0	2.85290
$474$	0	0
$475$	−2047.42	−0.197772
$476$	0	0
$477$	0	0
$478$	0	0
$479$	4638.43	0.442454	0.221227	−	0.975222i	$-0.428994\pi$
$479$	4638.43	0.442454	0.221227	+	0.975222i	$0.428994\pi$
$480$	0	0
$481$	−8787.41	−0.832997
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−4618.03	−0.432358
$486$	0	0
$487$	6602.00	0.614302	0.307151	−	0.951661i	$-0.400624\pi$
$487$	6602.00	0.614302	0.307151	+	0.951661i	$0.400624\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−4911.13	−0.451397	−0.225699	−	0.974197i	$-0.572466\pi$
$491$	−4911.13	−0.451397	−0.225699	+	0.974197i	$0.572466\pi$
$492$	0	0
$493$	−4904.07	−0.448008
$494$	0	0
$495$	0	0
$496$	0	0
$497$	34009.1	3.06945
$498$	0	0
$499$	−7347.00	−0.659112	−0.329556	−	0.944136i	$-0.606899\pi$
$499$	−7347.00	−0.659112	−0.329556	+	0.944136i	$0.606899\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−8360.36	−0.741094	−0.370547	−	0.928814i	$-0.620830\pi$
$503$	−8360.36	−0.741094	−0.370547	+	0.928814i	$0.620830\pi$
$504$	0	0
$505$	3734.62	0.329086
$506$	0	0
$507$	0	0
$508$	0	0
$509$	1711.27	0.149019	0.0745094	−	0.997220i	$-0.476261\pi$
$509$	1711.27	0.149019	0.0745094	+	0.997220i	$0.476261\pi$
$510$	0	0
$511$	−33250.3	−2.87849
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−9922.68	−0.849020
$516$	0	0
$517$	−6233.74	−0.530290
$518$	0	0
$519$	0	0
$520$	0	0
$521$	18452.7	1.55168	0.775841	−	0.630929i	$-0.217326\pi$
$521$	18452.7	1.55168	0.775841	+	0.630929i	$0.217326\pi$
$522$	0	0
$523$	−12259.0	−1.02495	−0.512474	−	0.858703i	$-0.671271\pi$
$523$	−12259.0	−1.02495	−0.512474	+	0.858703i	$0.671271\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−350.144	−0.0289421
$528$	0	0
$529$	6713.75	0.551800
$530$	0	0
$531$	0	0
$532$	0	0
$533$	6072.32	0.493473
$534$	0	0
$535$	751.003	0.0606891
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−31373.1	−2.50712
$540$	0	0
$541$	8844.75	0.702894	0.351447	−	0.936208i	$-0.385690\pi$
$541$	8844.75	0.702894	0.351447	+	0.936208i	$0.385690\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−9161.60	−0.720073
$546$	0	0
$547$	−4293.80	−0.335630	−0.167815	−	0.985819i	$-0.553671\pi$
$547$	−4293.80	−0.335630	−0.167815	+	0.985819i	$0.553671\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−4021.02	−0.310891
$552$	0	0
$553$	6002.63	0.461587
$554$	0	0
$555$	0	0
$556$	0	0
$557$	834.868	0.0635090	0.0317545	−	0.999496i	$-0.489891\pi$
$557$	834.868	0.0635090	0.0317545	+	0.999496i	$0.489891\pi$
$558$	0	0
$559$	−15695.5	−1.18756
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−7684.60	−0.575253	−0.287626	−	0.957743i	$-0.592866\pi$
$563$	−7684.60	−0.575253	−0.287626	+	0.957743i	$0.592866\pi$
$564$	0	0
$565$	4120.70	0.306830
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−5205.87	−0.383552	−0.191776	−	0.981439i	$-0.561425\pi$
$569$	−5205.87	−0.383552	−0.191776	+	0.981439i	$0.561425\pi$
$570$	0	0
$571$	−4747.91	−0.347975	−0.173987	−	0.984748i	$-0.555665\pi$
$571$	−4747.91	−0.347975	−0.173987	+	0.984748i	$0.555665\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−3435.18	−0.249143
$576$	0	0
$577$	−15968.8	−1.15215	−0.576073	−	0.817398i	$-0.695416\pi$
$577$	−15968.8	−1.15215	−0.576073	+	0.817398i	$0.695416\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	6146.81	0.438920
$582$	0	0
$583$	16101.0	1.14380
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−27491.7	−1.93305	−0.966527	−	0.256563i	$-0.917410\pi$
$587$	−27491.7	−1.93305	−0.966527	+	0.256563i	$0.917410\pi$
$588$	0	0
$589$	−287.095	−0.0200841
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−20632.4	−1.42879	−0.714393	−	0.699744i	$-0.753297\pi$
$593$	−20632.4	−1.42879	−0.714393	+	0.699744i	$0.753297\pi$
$594$	0	0
$595$	−14944.0	−1.02965
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−16889.7	−1.15208	−0.576040	−	0.817421i	$-0.695403\pi$
$599$	−16889.7	−1.15208	−0.576040	+	0.817421i	$0.695403\pi$
$600$	0	0
$601$	−11561.1	−0.784669	−0.392334	−	0.919823i	$-0.628332\pi$
$601$	−11561.1	−0.784669	−0.392334	+	0.919823i	$0.628332\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−9472.57	−0.636553
$606$	0	0
$607$	−15520.2	−1.03780	−0.518902	−	0.854834i	$-0.673659\pi$
$607$	−15520.2	−1.03780	−0.518902	+	0.854834i	$0.673659\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	3333.84	0.220741
$612$	0	0
$613$	12999.7	0.856530	0.428265	−	0.903653i	$-0.359125\pi$
$613$	12999.7	0.856530	0.428265	+	0.903653i	$0.359125\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−3775.70	−0.246360	−0.123180	−	0.992384i	$-0.539309\pi$
$617$	−3775.70	−0.246360	−0.123180	+	0.992384i	$0.539309\pi$
$618$	0	0
$619$	−20325.1	−1.31977	−0.659884	−	0.751368i	$-0.729395\pi$
$619$	−20325.1	−1.31977	−0.659884	+	0.751368i	$0.729395\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	10489.7	0.674579
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	28896.9	1.83179
$630$	0	0
$631$	16252.6	1.02537	0.512683	−	0.858578i	$-0.328652\pi$
$631$	16252.6	1.02537	0.512683	+	0.858578i	$0.328652\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−6310.10	−0.394345
$636$	0	0
$637$	16778.5	1.04363
$638$	0	0
$639$	0	0
$640$	0	0
$641$	8670.76	0.534281	0.267141	−	0.963658i	$-0.413921\pi$
$641$	8670.76	0.534281	0.267141	+	0.963658i	$0.413921\pi$
$642$	0	0
$643$	14797.6	0.907557	0.453779	−	0.891114i	$-0.350076\pi$
$643$	14797.6	0.907557	0.453779	+	0.891114i	$0.350076\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	7105.11	0.431732	0.215866	−	0.976423i	$-0.430743\pi$
$647$	7105.11	0.431732	0.215866	+	0.976423i	$0.430743\pi$
$648$	0	0
$649$	−10546.7	−0.637898
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−5675.92	−0.340147	−0.170073	−	0.985431i	$-0.554401\pi$
$653$	−5675.92	−0.340147	−0.170073	+	0.985431i	$0.554401\pi$
$654$	0	0
$655$	4500.44	0.268468
$656$	0	0
$657$	0	0
$658$	0	0
$659$	709.292	0.0419273	0.0209636	−	0.999780i	$-0.493327\pi$
$659$	709.292	0.0419273	0.0209636	+	0.999780i	$0.493327\pi$
$660$	0	0
$661$	13681.7	0.805075	0.402538	−	0.915403i	$-0.368128\pi$
$661$	13681.7	0.805075	0.402538	+	0.915403i	$0.368128\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−12253.1	−0.714519
$666$	0	0
$667$	−6746.52	−0.391643
$668$	0	0
$669$	0	0
$670$	0	0
$671$	13397.6	0.770804
$672$	0	0
$673$	28643.1	1.64058	0.820291	−	0.571947i	$-0.193812\pi$
$673$	28643.1	1.64058	0.820291	+	0.571947i	$0.193812\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−3820.45	−0.216886	−0.108443	−	0.994103i	$-0.534587\pi$
$677$	−3820.45	−0.216886	−0.108443	+	0.994103i	$0.534587\pi$
$678$	0	0
$679$	−27637.4	−1.56204
$680$	0	0
$681$	0	0
$682$	0	0
$683$	16575.9	0.928640	0.464320	−	0.885668i	$-0.346299\pi$
$683$	16575.9	0.928640	0.464320	+	0.885668i	$0.346299\pi$
$684$	0	0
$685$	9648.11	0.538154
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−8610.90	−0.476123
$690$	0	0
$691$	−7364.80	−0.405456	−0.202728	−	0.979235i	$-0.564981\pi$
$691$	−7364.80	−0.405456	−0.202728	+	0.979235i	$0.564981\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−7476.88	−0.408078
$696$	0	0
$697$	−19968.5	−1.08517
$698$	0	0
$699$	0	0
$700$	0	0
$701$	7789.14	0.419674	0.209837	−	0.977736i	$-0.432707\pi$
$701$	7789.14	0.419674	0.209837	+	0.977736i	$0.432707\pi$
$702$	0	0
$703$	23693.6	1.27115
$704$	0	0
$705$	0	0
$706$	0	0
$707$	22350.5	1.18893
$708$	0	0
$709$	−2897.22	−0.153466	−0.0767331	−	0.997052i	$-0.524449\pi$
$709$	−2897.22	−0.153466	−0.0767331	+	0.997052i	$0.524449\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−481.692	−0.0253009
$714$	0	0
$715$	8625.12	0.451135
$716$	0	0
$717$	0	0
$718$	0	0
$719$	24981.1	1.29574	0.647871	−	0.761750i	$-0.275660\pi$
$719$	24981.1	1.29574	0.647871	+	0.761750i	$0.275660\pi$
$720$	0	0
$721$	−59383.9	−3.06737
$722$	0	0
$723$	0	0
$724$	0	0
$725$	1227.47	0.0628786
$726$	0	0
$727$	−11419.9	−0.582585	−0.291293	−	0.956634i	$-0.594085\pi$
$727$	−11419.9	−0.582585	−0.291293	+	0.956634i	$0.594085\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	51613.7	2.61149
$732$	0	0
$733$	−5059.59	−0.254953	−0.127476	−	0.991842i	$-0.540688\pi$
$733$	−5059.59	−0.254953	−0.127476	+	0.991842i	$0.540688\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	19393.1	0.969271
$738$	0	0
$739$	−19395.5	−0.965460	−0.482730	−	0.875769i	$-0.660355\pi$
$739$	−19395.5	−0.965460	−0.482730	+	0.875769i	$0.660355\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−36435.8	−1.79906	−0.899530	−	0.436860i	$-0.856091\pi$
$743$	−36435.8	−1.79906	−0.899530	+	0.436860i	$0.856091\pi$
$744$	0	0
$745$	−6170.33	−0.303441
$746$	0	0
$747$	0	0
$748$	0	0
$749$	4494.50	0.219260
$750$	0	0
$751$	−11701.6	−0.568573	−0.284286	−	0.958739i	$-0.591757\pi$
$751$	−11701.6	−0.568573	−0.284286	+	0.958739i	$0.591757\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	16752.7	0.807543
$756$	0	0
$757$	17852.7	0.857157	0.428579	−	0.903504i	$-0.359014\pi$
$757$	17852.7	0.857157	0.428579	+	0.903504i	$0.359014\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−12295.5	−0.585693	−0.292846	−	0.956160i	$-0.594602\pi$
$761$	−12295.5	−0.585693	−0.292846	+	0.956160i	$0.594602\pi$
$762$	0	0
$763$	−54829.1	−2.60150
$764$	0	0
$765$	0	0
$766$	0	0
$767$	5640.46	0.265535
$768$	0	0
$769$	24649.6	1.15590	0.577949	−	0.816073i	$-0.303853\pi$
$769$	24649.6	1.15590	0.577949	+	0.816073i	$0.303853\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	3791.57	0.176421	0.0882103	−	0.996102i	$-0.471885\pi$
$773$	3791.57	0.176421	0.0882103	+	0.996102i	$0.471885\pi$
$774$	0	0
$775$	87.6395	0.00406207
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−16372.9	−0.753041
$780$	0	0
$781$	64548.3	2.95739
$782$	0	0
$783$	0	0
$784$	0	0
$785$	7719.20	0.350968
$786$	0	0
$787$	−34666.8	−1.57019	−0.785094	−	0.619377i	$-0.787385\pi$
$787$	−34666.8	−1.57019	−0.785094	+	0.619377i	$0.787385\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	24661.0	1.10853
$792$	0	0
$793$	−7165.13	−0.320859
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−21645.1	−0.961993	−0.480996	−	0.876723i	$-0.659725\pi$
$797$	−21645.1	−0.961993	−0.480996	+	0.876723i	$0.659725\pi$
$798$	0	0
$799$	−10963.2	−0.485418
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−63108.1	−2.77340
$804$	0	0
$805$	−20558.4	−0.900111
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−35629.0	−1.54839	−0.774196	−	0.632946i	$-0.781845\pi$
$809$	−35629.0	−1.54839	−0.774196	+	0.632946i	$0.781845\pi$
$810$	0	0
$811$	−4363.91	−0.188949	−0.0944745	−	0.995527i	$-0.530117\pi$
$811$	−4363.91	−0.188949	−0.0944745	+	0.995527i	$0.530117\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	8252.44	0.354688
$816$	0	0
$817$	42319.9	1.81222
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−12584.7	−0.534970	−0.267485	−	0.963562i	$-0.586193\pi$
$821$	−12584.7	−0.534970	−0.267485	+	0.963562i	$0.586193\pi$
$822$	0	0
$823$	−31273.0	−1.32455	−0.662277	−	0.749259i	$-0.730410\pi$
$823$	−31273.0	−1.32455	−0.662277	+	0.749259i	$0.730410\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−84.4299	−0.00355008	−0.00177504	−	0.999998i	$-0.500565\pi$
$827$	−84.4299	−0.00355008	−0.00177504	+	0.999998i	$0.500565\pi$
$828$	0	0
$829$	27605.8	1.15656	0.578281	−	0.815838i	$-0.303724\pi$
$829$	27605.8	1.15656	0.578281	+	0.815838i	$0.303724\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−55175.3	−2.29497
$834$	0	0
$835$	12472.8	0.516934
$836$	0	0
$837$	0	0
$838$	0	0
$839$	27890.0	1.14764	0.573819	−	0.818982i	$-0.305461\pi$
$839$	27890.0	1.14764	0.573819	+	0.818982i	$0.305461\pi$
$840$	0	0
$841$	−21978.3	−0.901157
$842$	0	0
$843$	0	0
$844$	0	0
$845$	6372.23	0.259422
$846$	0	0
$847$	−56690.2	−2.29976
$848$	0	0
$849$	0	0
$850$	0	0
$851$	39753.4	1.60133
$852$	0	0
$853$	−10149.9	−0.407416	−0.203708	−	0.979032i	$-0.565299\pi$
$853$	−10149.9	−0.407416	−0.203708	+	0.979032i	$0.565299\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−29795.0	−1.18761	−0.593803	−	0.804611i	$-0.702374\pi$
$857$	−29795.0	−1.18761	−0.593803	+	0.804611i	$0.702374\pi$
$858$	0	0
$859$	−38121.7	−1.51420	−0.757099	−	0.653300i	$-0.773384\pi$
$859$	−38121.7	−1.51420	−0.757099	+	0.653300i	$0.773384\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−25067.7	−0.988775	−0.494388	−	0.869242i	$-0.664608\pi$
$863$	−25067.7	−0.988775	−0.494388	+	0.869242i	$0.664608\pi$
$864$	0	0
$865$	5340.01	0.209903
$866$	0	0
$867$	0	0
$868$	0	0
$869$	11392.8	0.444735
$870$	0	0
$871$	−10371.5	−0.403474
$872$	0	0
$873$	0	0
$874$	0	0
$875$	3740.42	0.144513
$876$	0	0
$877$	−13693.1	−0.527232	−0.263616	−	0.964628i	$-0.584915\pi$
$877$	−13693.1	−0.527232	−0.263616	+	0.964628i	$0.584915\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−37926.8	−1.45038	−0.725191	−	0.688548i	$-0.758248\pi$
$881$	−37926.8	−1.45038	−0.725191	+	0.688548i	$0.758248\pi$
$882$	0	0
$883$	−16737.7	−0.637902	−0.318951	−	0.947771i	$-0.603331\pi$
$883$	−16737.7	−0.637902	−0.318951	+	0.947771i	$0.603331\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	44213.6	1.67367	0.836837	−	0.547453i	$-0.184402\pi$
$887$	44213.6	1.67367	0.836837	+	0.547453i	$0.184402\pi$
$888$	0	0
$889$	−37763.9	−1.42470
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−8989.09	−0.336851
$894$	0	0
$895$	−16860.0	−0.629686
$896$	0	0
$897$	0	0
$898$	0	0
$899$	172.119	0.00638543
$900$	0	0
$901$	28316.5	1.04701
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−14994.2	−0.550746
$906$	0	0
$907$	−27295.6	−0.999268	−0.499634	−	0.866237i	$-0.666532\pi$
$907$	−27295.6	−0.999268	−0.499634	+	0.866237i	$0.666532\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−8988.41	−0.326893	−0.163446	−	0.986552i	$-0.552261\pi$
$911$	−8988.41	−0.326893	−0.163446	+	0.986552i	$0.552261\pi$
$912$	0	0
$913$	11666.5	0.422895
$914$	0	0
$915$	0	0
$916$	0	0
$917$	26933.6	0.969931
$918$	0	0
$919$	−14112.0	−0.506543	−0.253271	−	0.967395i	$-0.581507\pi$
$919$	−14112.0	−0.506543	−0.253271	+	0.967395i	$0.581507\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−34520.8	−1.23106
$924$	0	0
$925$	−7232.78	−0.257094
$926$	0	0
$927$	0	0
$928$	0	0
$929$	33721.8	1.19093	0.595466	−	0.803381i	$-0.296968\pi$
$929$	33721.8	1.19093	0.595466	+	0.803381i	$0.296968\pi$
$930$	0	0
$931$	−45240.2	−1.59258
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−28363.2	−0.992061
$936$	0	0
$937$	20706.5	0.721933	0.360966	−	0.932579i	$-0.382447\pi$
$937$	20706.5	0.721933	0.360966	+	0.932579i	$0.382447\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−3927.55	−0.136062	−0.0680310	−	0.997683i	$-0.521672\pi$
$941$	−3927.55	−0.136062	−0.0680310	+	0.997683i	$0.521672\pi$
$942$	0	0
$943$	−27470.6	−0.948639
$944$	0	0
$945$	0	0
$946$	0	0
$947$	5651.43	0.193925	0.0969625	−	0.995288i	$-0.469087\pi$
$947$	5651.43	0.193925	0.0969625	+	0.995288i	$0.469087\pi$
$948$	0	0
$949$	33750.6	1.15447
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−26437.4	−0.898626	−0.449313	−	0.893374i	$-0.648331\pi$
$953$	−26437.4	−0.898626	−0.449313	+	0.893374i	$0.648331\pi$
$954$	0	0
$955$	10126.0	0.343110
$956$	0	0
$957$	0	0
$958$	0	0
$959$	57740.7	1.94426
$960$	0	0
$961$	−29778.7	−0.999587
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−4834.42	−0.161270
$966$	0	0
$967$	51240.2	1.70401	0.852003	−	0.523536i	$-0.175388\pi$
$967$	51240.2	1.70401	0.852003	+	0.523536i	$0.175388\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	6795.21	0.224581	0.112291	−	0.993675i	$-0.464181\pi$
$971$	6795.21	0.224581	0.112291	+	0.993675i	$0.464181\pi$
$972$	0	0
$973$	−44746.6	−1.47432
$974$	0	0
$975$	0	0
$976$	0	0
$977$	41923.0	1.37281	0.686405	−	0.727219i	$-0.259188\pi$
$977$	41923.0	1.37281	0.686405	+	0.727219i	$0.259188\pi$
$978$	0	0
$979$	19909.2	0.649951
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−18450.8	−0.598667	−0.299334	−	0.954149i	$-0.596764\pi$
$983$	−18450.8	−0.598667	−0.299334	+	0.954149i	$0.596764\pi$
$984$	0	0
$985$	6981.37	0.225832
$986$	0	0
$987$	0	0
$988$	0	0
$989$	71004.9	2.28294
$990$	0	0
$991$	13429.0	0.430460	0.215230	−	0.976563i	$-0.430950\pi$
$991$	13429.0	0.430460	0.215230	+	0.976563i	$0.430950\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	13224.8	0.421362
$996$	0	0
$997$	−20869.4	−0.662928	−0.331464	−	0.943468i	$-0.607543\pi$
$997$	−20869.4	−0.662928	−0.331464	+	0.943468i	$0.607543\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1620.4.a.k.1.1		7
3.2	odd	2		1620.4.a.l.1.1		7
9.2	odd	6		180.4.i.c.121.4	yes	14
9.4	even	3		540.4.i.c.181.7		14
9.5	odd	6		180.4.i.c.61.4	✓	14
9.7	even	3		540.4.i.c.361.7		14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
180.4.i.c.61.4	✓	14	9.5	odd	6
180.4.i.c.121.4	yes	14	9.2	odd	6
540.4.i.c.181.7		14	9.4	even	3
540.4.i.c.361.7		14	9.7	even	3
1620.4.a.k.1.1		7	1.1	even	1	trivial
1620.4.a.l.1.1		7	3.2	odd	2

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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)

\(f(q)\)	\(=\)	\(q-5.00000 q^{5} -29.9233 q^{7} +O(q^{10})\)	\(q-5.00000 q^{5} -29.9233 q^{7} -56.7936 q^{11} +30.3736 q^{13} -99.8818 q^{17} -81.8966 q^{19} -137.407 q^{23} +25.0000 q^{25} +49.0987 q^{29} +3.50558 q^{31} +149.617 q^{35} -289.311 q^{37} +199.921 q^{41} -516.748 q^{43} +109.761 q^{47} +552.406 q^{49} -283.500 q^{53} +283.968 q^{55} +185.703 q^{59} -235.900 q^{61} -151.868 q^{65} -341.466 q^{67} -1136.54 q^{71} +1111.18 q^{73} +1699.45 q^{77} -200.600 q^{79} -205.419 q^{83} +499.409 q^{85} -350.554 q^{89} -908.879 q^{91} +409.483 q^{95} +923.606 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 35 q^{5} + 8 q^{7}+O(q^{10}) \) 7 * q - 35 * q^5 + 8 * q^7	\( 7 q - 35 q^{5} + 8 q^{7} - 27 q^{11} + 32 q^{13} - 123 q^{17} - 67 q^{19} - 42 q^{23} + 175 q^{25} - 324 q^{29} + 98 q^{31} - 40 q^{35} + 356 q^{37} - 339 q^{41} + 119 q^{43} + 96 q^{47} + 813 q^{49} - 858 q^{53} + 135 q^{55} - 549 q^{59} + 260 q^{61} - 160 q^{65} + 881 q^{67} - 342 q^{71} + 737 q^{73} + 456 q^{77} + 1886 q^{79} + 132 q^{83} + 615 q^{85} - 396 q^{89} + 1264 q^{91} + 335 q^{95} + 1991 q^{97}+O(q^{100}) \) 7 * q - 35 * q^5 + 8 * q^7 - 27 * q^11 + 32 * q^13 - 123 * q^17 - 67 * q^19 - 42 * q^23 + 175 * q^25 - 324 * q^29 + 98 * q^31 - 40 * q^35 + 356 * q^37 - 339 * q^41 + 119 * q^43 + 96 * q^47 + 813 * q^49 - 858 * q^53 + 135 * q^55 - 549 * q^59 + 260 * q^61 - 160 * q^65 + 881 * q^67 - 342 * q^71 + 737 * q^73 + 456 * q^77 + 1886 * q^79 + 132 * q^83 + 615 * q^85 - 396 * q^89 + 1264 * q^91 + 335 * q^95 + 1991 * q^97

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