LMFDB - Embedded newform 2160.4.a.bi.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2160,4,Mod(1,2160)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2160, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("2160.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2160 = 2^{4} \cdot 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2160.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:	$127.444125612$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.1772.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 12x + 8 $ x^3 - x^2 - 12*x + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2}\cdot 3 $
Twist minimal:	no (minimal twist has level 135)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-3.32803$ of defining polynomial
Character	$\chi$	$=$	2160.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.00000 q^{5} -30.7000 q^{7} +O(q^{10})$	$q-5.00000 q^{5} -30.7000 q^{7} +50.1548 q^{11} -15.9592 q^{13} -105.668 q^{17} +21.3040 q^{19} +136.137 q^{23} +25.0000 q^{25} +224.323 q^{29} +225.982 q^{31} +153.500 q^{35} -416.386 q^{37} -76.1411 q^{41} -31.7372 q^{43} +60.8026 q^{47} +599.493 q^{49} +466.532 q^{53} -250.774 q^{55} +95.4239 q^{59} -357.174 q^{61} +79.7962 q^{65} -87.8344 q^{67} +412.693 q^{71} -331.133 q^{73} -1539.75 q^{77} +248.123 q^{79} -552.505 q^{83} +528.341 q^{85} +291.478 q^{89} +489.949 q^{91} -106.520 q^{95} +198.606 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 15 q^{5} + 4 q^{7}+O(q^{10}) $ 3 * q - 15 * q^5 + 4 * q^7	$ 3 q - 15 q^{5} + 4 q^{7} + 5 q^{11} + 7 q^{13} - 155 q^{17} + 50 q^{19} + 285 q^{23} + 75 q^{25} - 115 q^{29} + 115 q^{31} - 20 q^{35} - 384 q^{37} - 580 q^{41} + 797 q^{43} - 145 q^{47} + 577 q^{49} + 400 q^{53} - 25 q^{55} + 380 q^{59} - 152 q^{61} - 35 q^{65} - 2 q^{67} + 40 q^{71} - 980 q^{73} - 1950 q^{77} - 1013 q^{79} + 270 q^{83} + 775 q^{85} - 1020 q^{89} + 632 q^{91} - 250 q^{95} + 720 q^{97}+O(q^{100}) $ 3 * q - 15 * q^5 + 4 * q^7 + 5 * q^11 + 7 * q^13 - 155 * q^17 + 50 * q^19 + 285 * q^23 + 75 * q^25 - 115 * q^29 + 115 * q^31 - 20 * q^35 - 384 * q^37 - 580 * q^41 + 797 * q^43 - 145 * q^47 + 577 * q^49 + 400 * q^53 - 25 * q^55 + 380 * q^59 - 152 * q^61 - 35 * q^65 - 2 * q^67 + 40 * q^71 - 980 * q^73 - 1950 * q^77 - 1013 * q^79 + 270 * q^83 + 775 * q^85 - 1020 * q^89 + 632 * q^91 - 250 * q^95 + 720 * 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
$7$	−30.7000	−1.65765	−0.828823	−	0.559511i	$-0.810989\pi$
$7$	−30.7000	−1.65765	−0.828823	+	0.559511i	$0.810989\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	50.1548	1.37475	0.687375	−	0.726303i	$-0.258763\pi$
$11$	50.1548	1.37475	0.687375	+	0.726303i	$0.258763\pi$
$12$	0	0
$13$	−15.9592	−0.340484	−0.170242	−	0.985402i	$-0.554455\pi$
$13$	−15.9592	−0.340484	−0.170242	+	0.985402i	$0.554455\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−105.668	−1.50755	−0.753774	−	0.657134i	$-0.771769\pi$
$17$	−105.668	−1.50755	−0.753774	+	0.657134i	$0.771769\pi$
$18$	0	0
$19$	21.3040	0.257235	0.128618	−	0.991694i	$-0.458946\pi$
$19$	21.3040	0.257235	0.128618	+	0.991694i	$0.458946\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	136.137	1.23420	0.617098	−	0.786886i	$-0.288308\pi$
$23$	136.137	1.23420	0.617098	+	0.786886i	$0.288308\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	224.323	1.43641	0.718203	−	0.695834i	$-0.244965\pi$
$29$	224.323	1.43641	0.718203	+	0.695834i	$0.244965\pi$
$30$	0	0
$31$	225.982	1.30928	0.654638	−	0.755943i	$-0.272821\pi$
$31$	225.982	1.30928	0.654638	+	0.755943i	$0.272821\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	153.500	0.741322
$36$	0	0
$37$	−416.386	−1.85009	−0.925046	−	0.379854i	$-0.875974\pi$
$37$	−416.386	−1.85009	−0.925046	+	0.379854i	$0.875974\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−76.1411	−0.290030	−0.145015	−	0.989429i	$-0.546323\pi$
$41$	−76.1411	−0.290030	−0.145015	+	0.989429i	$0.546323\pi$
$42$	0	0
$43$	−31.7372	−0.112555	−0.0562777	−	0.998415i	$-0.517923\pi$
$43$	−31.7372	−0.112555	−0.0562777	+	0.998415i	$0.517923\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	60.8026	0.188701	0.0943507	−	0.995539i	$-0.469922\pi$
$47$	60.8026	0.188701	0.0943507	+	0.995539i	$0.469922\pi$
$48$	0	0
$49$	599.493	1.74779
$50$	0	0
$51$	0	0
$52$	0	0
$53$	466.532	1.20911	0.604557	−	0.796562i	$-0.293350\pi$
$53$	466.532	1.20911	0.604557	+	0.796562i	$0.293350\pi$
$54$	0	0
$55$	−250.774	−0.614806
$56$	0	0
$57$	0	0
$58$	0	0
$59$	95.4239	0.210562	0.105281	−	0.994443i	$-0.466426\pi$
$59$	95.4239	0.210562	0.105281	+	0.994443i	$0.466426\pi$
$60$	0	0
$61$	−357.174	−0.749696	−0.374848	−	0.927086i	$-0.622305\pi$
$61$	−357.174	−0.749696	−0.374848	+	0.927086i	$0.622305\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	79.7962	0.152269
$66$	0	0
$67$	−87.8344	−0.160159	−0.0800797	−	0.996788i	$-0.525517\pi$
$67$	−87.8344	−0.160159	−0.0800797	+	0.996788i	$0.525517\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	412.693	0.689826	0.344913	−	0.938635i	$-0.387908\pi$
$71$	412.693	0.689826	0.344913	+	0.938635i	$0.387908\pi$
$72$	0	0
$73$	−331.133	−0.530906	−0.265453	−	0.964124i	$-0.585522\pi$
$73$	−331.133	−0.530906	−0.265453	+	0.964124i	$0.585522\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−1539.75	−2.27885
$78$	0	0
$79$	248.123	0.353368	0.176684	−	0.984268i	$-0.443463\pi$
$79$	248.123	0.353368	0.176684	+	0.984268i	$0.443463\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−552.505	−0.730667	−0.365333	−	0.930877i	$-0.619045\pi$
$83$	−552.505	−0.730667	−0.365333	+	0.930877i	$0.619045\pi$
$84$	0	0
$85$	528.341	0.674196
$86$	0	0
$87$	0	0
$88$	0	0
$89$	291.478	0.347153	0.173577	−	0.984820i	$-0.444468\pi$
$89$	291.478	0.347153	0.173577	+	0.984820i	$0.444468\pi$
$90$	0	0
$91$	489.949	0.564402
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−106.520	−0.115039
$96$	0	0
$97$	198.606	0.207891	0.103946	−	0.994583i	$-0.466853\pi$
$97$	198.606	0.207891	0.103946	+	0.994583i	$0.466853\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−816.235	−0.804143	−0.402071	−	0.915608i	$-0.631710\pi$
$101$	−816.235	−0.804143	−0.402071	+	0.915608i	$0.631710\pi$
$102$	0	0
$103$	1402.37	1.34155	0.670776	−	0.741660i	$-0.265961\pi$
$103$	1402.37	1.34155	0.670776	+	0.741660i	$0.265961\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−978.996	−0.884515	−0.442258	−	0.896888i	$-0.645822\pi$
$107$	−978.996	−0.884515	−0.442258	+	0.896888i	$0.645822\pi$
$108$	0	0
$109$	2122.96	1.86553	0.932766	−	0.360484i	$-0.117388\pi$
$109$	2122.96	1.86553	0.932766	+	0.360484i	$0.117388\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−1794.80	−1.49416	−0.747082	−	0.664732i	$-0.768546\pi$
$113$	−1794.80	−1.49416	−0.747082	+	0.664732i	$0.768546\pi$
$114$	0	0
$115$	−680.684	−0.551949
$116$	0	0
$117$	0	0
$118$	0	0
$119$	3244.02	2.49898
$120$	0	0
$121$	1184.50	0.889935
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−125.000	−0.0894427
$126$	0	0
$127$	748.894	0.523257	0.261628	−	0.965169i	$-0.415741\pi$
$127$	748.894	0.523257	0.261628	+	0.965169i	$0.415741\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−2396.11	−1.59808	−0.799042	−	0.601275i	$-0.794660\pi$
$131$	−2396.11	−1.59808	−0.799042	+	0.601275i	$0.794660\pi$
$132$	0	0
$133$	−654.033	−0.426405
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−1004.48	−0.626409	−0.313205	−	0.949686i	$-0.601403\pi$
$137$	−1004.48	−0.626409	−0.313205	+	0.949686i	$0.601403\pi$
$138$	0	0
$139$	−2403.02	−1.46634	−0.733172	−	0.680043i	$-0.761961\pi$
$139$	−2403.02	−1.46634	−0.733172	+	0.680043i	$0.761961\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−800.432	−0.468080
$144$	0	0
$145$	−1121.62	−0.642380
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−509.648	−0.280215	−0.140107	−	0.990136i	$-0.544745\pi$
$149$	−509.648	−0.280215	−0.140107	+	0.990136i	$0.544745\pi$
$150$	0	0
$151$	−1443.30	−0.777840	−0.388920	−	0.921272i	$-0.627152\pi$
$151$	−1443.30	−0.777840	−0.388920	+	0.921272i	$0.627152\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−1129.91	−0.585526
$156$	0	0
$157$	−2155.64	−1.09579	−0.547895	−	0.836547i	$-0.684571\pi$
$157$	−2155.64	−1.09579	−0.547895	+	0.836547i	$0.684571\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−4179.41	−2.04586
$162$	0	0
$163$	529.909	0.254636	0.127318	−	0.991862i	$-0.459363\pi$
$163$	529.909	0.254636	0.127318	+	0.991862i	$0.459363\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−2979.28	−1.38050	−0.690250	−	0.723571i	$-0.742499\pi$
$167$	−2979.28	−1.38050	−0.690250	+	0.723571i	$0.742499\pi$
$168$	0	0
$169$	−1942.30	−0.884071
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−1779.88	−0.782205	−0.391103	−	0.920347i	$-0.627906\pi$
$173$	−1779.88	−0.782205	−0.391103	+	0.920347i	$0.627906\pi$
$174$	0	0
$175$	−767.501	−0.331529
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−2836.26	−1.18431	−0.592157	−	0.805823i	$-0.701724\pi$
$179$	−2836.26	−1.18431	−0.592157	+	0.805823i	$0.701724\pi$
$180$	0	0
$181$	811.890	0.333410	0.166705	−	0.986007i	$-0.446687\pi$
$181$	811.890	0.333410	0.166705	+	0.986007i	$0.446687\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	2081.93	0.827387
$186$	0	0
$187$	−5299.77	−2.07250
$188$	0	0
$189$	0	0
$190$	0	0
$191$	1148.46	0.435077	0.217539	−	0.976052i	$-0.430197\pi$
$191$	1148.46	0.435077	0.217539	+	0.976052i	$0.430197\pi$
$192$	0	0
$193$	−2150.93	−0.802215	−0.401107	−	0.916031i	$-0.631375\pi$
$193$	−2150.93	−0.802215	−0.401107	+	0.916031i	$0.631375\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	5057.51	1.82910	0.914551	−	0.404471i	$-0.132544\pi$
$197$	5057.51	1.82910	0.914551	+	0.404471i	$0.132544\pi$
$198$	0	0
$199$	3554.12	1.26605	0.633027	−	0.774130i	$-0.281812\pi$
$199$	3554.12	1.26605	0.633027	+	0.774130i	$0.281812\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−6886.73	−2.38105
$204$	0	0
$205$	380.706	0.129706
$206$	0	0
$207$	0	0
$208$	0	0
$209$	1068.50	0.353634
$210$	0	0
$211$	107.909	0.0352075	0.0176038	−	0.999845i	$-0.494396\pi$
$211$	107.909	0.0352075	0.0176038	+	0.999845i	$0.494396\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	158.686	0.0503363
$216$	0	0
$217$	−6937.65	−2.17032
$218$	0	0
$219$	0	0
$220$	0	0
$221$	1686.38	0.513296
$222$	0	0
$223$	1942.03	0.583174	0.291587	−	0.956544i	$-0.405817\pi$
$223$	1942.03	0.583174	0.291587	+	0.956544i	$0.405817\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	3443.99	1.00698	0.503492	−	0.864000i	$-0.332048\pi$
$227$	3443.99	1.00698	0.503492	+	0.864000i	$0.332048\pi$
$228$	0	0
$229$	2069.39	0.597159	0.298579	−	0.954385i	$-0.403487\pi$
$229$	2069.39	0.597159	0.298579	+	0.954385i	$0.403487\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	2769.24	0.778622	0.389311	−	0.921106i	$-0.372713\pi$
$233$	2769.24	0.778622	0.389311	+	0.921106i	$0.372713\pi$
$234$	0	0
$235$	−304.013	−0.0843898
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−2943.84	−0.796742	−0.398371	−	0.917224i	$-0.630424\pi$
$239$	−2943.84	−0.796742	−0.398371	+	0.917224i	$0.630424\pi$
$240$	0	0
$241$	4474.54	1.19598	0.597988	−	0.801505i	$-0.295967\pi$
$241$	4474.54	1.19598	0.597988	+	0.801505i	$0.295967\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−2997.46	−0.781636
$246$	0	0
$247$	−339.995	−0.0875845
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−2121.41	−0.533474	−0.266737	−	0.963769i	$-0.585945\pi$
$251$	−2121.41	−0.533474	−0.266737	+	0.963769i	$0.585945\pi$
$252$	0	0
$253$	6827.92	1.69671
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−4548.24	−1.10393	−0.551967	−	0.833866i	$-0.686123\pi$
$257$	−4548.24	−1.10393	−0.551967	+	0.833866i	$0.686123\pi$
$258$	0	0
$259$	12783.1	3.06680
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−3499.84	−0.820567	−0.410284	−	0.911958i	$-0.634570\pi$
$263$	−3499.84	−0.820567	−0.410284	+	0.911958i	$0.634570\pi$
$264$	0	0
$265$	−2332.66	−0.540732
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−2594.25	−0.588008	−0.294004	−	0.955804i	$-0.594988\pi$
$269$	−2594.25	−0.588008	−0.294004	+	0.955804i	$0.594988\pi$
$270$	0	0
$271$	−8518.20	−1.90939	−0.954693	−	0.297593i	$-0.903816\pi$
$271$	−8518.20	−1.90939	−0.954693	+	0.297593i	$0.903816\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	1253.87	0.274950
$276$	0	0
$277$	−1890.50	−0.410068	−0.205034	−	0.978755i	$-0.565731\pi$
$277$	−1890.50	−0.410068	−0.205034	+	0.978755i	$0.565731\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−3138.68	−0.666328	−0.333164	−	0.942869i	$-0.608116\pi$
$281$	−3138.68	−0.666328	−0.333164	+	0.942869i	$0.608116\pi$
$282$	0	0
$283$	−4069.09	−0.854708	−0.427354	−	0.904084i	$-0.640554\pi$
$283$	−4069.09	−0.854708	−0.427354	+	0.904084i	$0.640554\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	2337.54	0.480768
$288$	0	0
$289$	6252.77	1.27270
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−3637.95	−0.725362	−0.362681	−	0.931913i	$-0.618139\pi$
$293$	−3637.95	−0.725362	−0.362681	+	0.931913i	$0.618139\pi$
$294$	0	0
$295$	−477.119	−0.0941660
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−2172.64	−0.420224
$300$	0	0
$301$	974.334	0.186577
$302$	0	0
$303$	0	0
$304$	0	0
$305$	1785.87	0.335274
$306$	0	0
$307$	6829.07	1.26956	0.634781	−	0.772692i	$-0.281090\pi$
$307$	6829.07	1.26956	0.634781	+	0.772692i	$0.281090\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	6601.03	1.20357	0.601785	−	0.798659i	$-0.294457\pi$
$311$	6601.03	1.20357	0.601785	+	0.798659i	$0.294457\pi$
$312$	0	0
$313$	2766.59	0.499607	0.249803	−	0.968297i	$-0.419634\pi$
$313$	2766.59	0.499607	0.249803	+	0.968297i	$0.419634\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−4564.41	−0.808716	−0.404358	−	0.914601i	$-0.632505\pi$
$317$	−4564.41	−0.808716	−0.404358	+	0.914601i	$0.632505\pi$
$318$	0	0
$319$	11250.9	1.97470
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−2251.15	−0.387794
$324$	0	0
$325$	−398.981	−0.0680968
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−1866.64	−0.312800
$330$	0	0
$331$	4665.62	0.774760	0.387380	−	0.921920i	$-0.373380\pi$
$331$	4665.62	0.774760	0.387380	+	0.921920i	$0.373380\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	439.172	0.0716254
$336$	0	0
$337$	−3807.06	−0.615382	−0.307691	−	0.951486i	$-0.599556\pi$
$337$	−3807.06	−0.615382	−0.307691	+	0.951486i	$0.599556\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	11334.1	1.79993
$342$	0	0
$343$	−7874.33	−1.23957
$344$	0	0
$345$	0	0
$346$	0	0
$347$	2311.26	0.357565	0.178783	−	0.983889i	$-0.442784\pi$
$347$	2311.26	0.357565	0.178783	+	0.983889i	$0.442784\pi$
$348$	0	0
$349$	−10873.2	−1.66771	−0.833854	−	0.551984i	$-0.813871\pi$
$349$	−10873.2	−1.66771	−0.833854	+	0.551984i	$0.813871\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	2893.74	0.436312	0.218156	−	0.975914i	$-0.429996\pi$
$353$	2893.74	0.436312	0.218156	+	0.975914i	$0.429996\pi$
$354$	0	0
$355$	−2063.46	−0.308500
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−9875.47	−1.45183	−0.725915	−	0.687784i	$-0.758584\pi$
$359$	−9875.47	−1.45183	−0.725915	+	0.687784i	$0.758584\pi$
$360$	0	0
$361$	−6405.14	−0.933830
$362$	0	0
$363$	0	0
$364$	0	0
$365$	1655.66	0.237429
$366$	0	0
$367$	−10722.2	−1.52505	−0.762523	−	0.646961i	$-0.776040\pi$
$367$	−10722.2	−1.52505	−0.762523	+	0.646961i	$0.776040\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−14322.5	−2.00428
$372$	0	0
$373$	4175.90	0.579678	0.289839	−	0.957075i	$-0.406398\pi$
$373$	4175.90	0.579678	0.289839	+	0.957075i	$0.406398\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−3580.03	−0.489074
$378$	0	0
$379$	−1715.14	−0.232457	−0.116228	−	0.993223i	$-0.537080\pi$
$379$	−1715.14	−0.232457	−0.116228	+	0.993223i	$0.537080\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	5139.06	0.685624	0.342812	−	0.939404i	$-0.388621\pi$
$383$	5139.06	0.685624	0.342812	+	0.939404i	$0.388621\pi$
$384$	0	0
$385$	7698.77	1.01913
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−8996.53	−1.17260	−0.586301	−	0.810093i	$-0.699416\pi$
$389$	−8996.53	−1.17260	−0.586301	+	0.810093i	$0.699416\pi$
$390$	0	0
$391$	−14385.3	−1.86061
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−1240.62	−0.158031
$396$	0	0
$397$	−105.823	−0.0133781	−0.00668905	−	0.999978i	$-0.502129\pi$
$397$	−105.823	−0.0133781	−0.00668905	+	0.999978i	$0.502129\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−14481.8	−1.80346	−0.901730	−	0.432300i	$-0.857702\pi$
$401$	−14481.8	−1.80346	−0.901730	+	0.432300i	$0.857702\pi$
$402$	0	0
$403$	−3606.50	−0.445788
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−20883.8	−2.54341
$408$	0	0
$409$	−861.065	−0.104100	−0.0520500	−	0.998644i	$-0.516576\pi$
$409$	−861.065	−0.104100	−0.0520500	+	0.998644i	$0.516576\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−2929.52	−0.349037
$414$	0	0
$415$	2762.53	0.326764
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−13017.7	−1.51780	−0.758900	−	0.651207i	$-0.774263\pi$
$419$	−13017.7	−1.51780	−0.758900	+	0.651207i	$0.774263\pi$
$420$	0	0
$421$	14546.6	1.68398	0.841991	−	0.539492i	$-0.181384\pi$
$421$	14546.6	1.68398	0.841991	+	0.539492i	$0.181384\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−2641.71	−0.301510
$426$	0	0
$427$	10965.3	1.24273
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−3539.94	−0.395622	−0.197811	−	0.980240i	$-0.563383\pi$
$431$	−3539.94	−0.395622	−0.197811	+	0.980240i	$0.563383\pi$
$432$	0	0
$433$	669.471	0.0743019	0.0371509	−	0.999310i	$-0.488172\pi$
$433$	669.471	0.0743019	0.0371509	+	0.999310i	$0.488172\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	2900.26	0.317478
$438$	0	0
$439$	−12568.5	−1.36643	−0.683216	−	0.730216i	$-0.739419\pi$
$439$	−12568.5	−1.36643	−0.683216	+	0.730216i	$0.739419\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	11060.3	1.18621	0.593106	−	0.805124i	$-0.297901\pi$
$443$	11060.3	1.18621	0.593106	+	0.805124i	$0.297901\pi$
$444$	0	0
$445$	−1457.39	−0.155252
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−18553.9	−1.95014	−0.975072	−	0.221891i	$-0.928777\pi$
$449$	−18553.9	−1.95014	−0.975072	+	0.221891i	$0.928777\pi$
$450$	0	0
$451$	−3818.84	−0.398719
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−2449.75	−0.252408
$456$	0	0
$457$	−6802.26	−0.696272	−0.348136	−	0.937444i	$-0.613185\pi$
$457$	−6802.26	−0.696272	−0.348136	+	0.937444i	$0.613185\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−14894.4	−1.50477	−0.752386	−	0.658722i	$-0.771097\pi$
$461$	−14894.4	−1.50477	−0.752386	+	0.658722i	$0.771097\pi$
$462$	0	0
$463$	14288.7	1.43423	0.717117	−	0.696952i	$-0.245461\pi$
$463$	14288.7	1.43423	0.717117	+	0.696952i	$0.245461\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	13115.1	1.29956	0.649780	−	0.760122i	$-0.274861\pi$
$467$	13115.1	1.29956	0.649780	+	0.760122i	$0.274861\pi$
$468$	0	0
$469$	2696.52	0.265488
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−1591.77	−0.154735
$474$	0	0
$475$	532.599	0.0514470
$476$	0	0
$477$	0	0
$478$	0	0
$479$	7919.79	0.755458	0.377729	−	0.925916i	$-0.376705\pi$
$479$	7919.79	0.755458	0.377729	+	0.925916i	$0.376705\pi$
$480$	0	0
$481$	6645.20	0.629927
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−993.032	−0.0929717
$486$	0	0
$487$	17003.1	1.58210	0.791051	−	0.611751i	$-0.209534\pi$
$487$	17003.1	1.58210	0.791051	+	0.611751i	$0.209534\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	6391.74	0.587485	0.293743	−	0.955885i	$-0.405099\pi$
$491$	6391.74	0.587485	0.293743	+	0.955885i	$0.405099\pi$
$492$	0	0
$493$	−23703.8	−2.16545
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−12669.7	−1.14349
$498$	0	0
$499$	2674.12	0.239900	0.119950	−	0.992780i	$-0.461727\pi$
$499$	2674.12	0.239900	0.119950	+	0.992780i	$0.461727\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	1263.13	0.111969	0.0559843	−	0.998432i	$-0.482170\pi$
$503$	1263.13	0.111969	0.0559843	+	0.998432i	$0.482170\pi$
$504$	0	0
$505$	4081.17	0.359624
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−1380.84	−0.120245	−0.0601226	−	0.998191i	$-0.519149\pi$
$509$	−1380.84	−0.120245	−0.0601226	+	0.998191i	$0.519149\pi$
$510$	0	0
$511$	10165.8	0.880055
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−7011.86	−0.599960
$516$	0	0
$517$	3049.54	0.259417
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−2689.80	−0.226185	−0.113092	−	0.993584i	$-0.536076\pi$
$521$	−2689.80	−0.226185	−0.113092	+	0.993584i	$0.536076\pi$
$522$	0	0
$523$	−7144.18	−0.597310	−0.298655	−	0.954361i	$-0.596538\pi$
$523$	−7144.18	−0.597310	−0.298655	+	0.954361i	$0.596538\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−23879.1	−1.97379
$528$	0	0
$529$	6366.25	0.523239
$530$	0	0
$531$	0	0
$532$	0	0
$533$	1215.15	0.0987508
$534$	0	0
$535$	4894.98	0.395567
$536$	0	0
$537$	0	0
$538$	0	0
$539$	30067.4	2.40278
$540$	0	0
$541$	−3310.57	−0.263091	−0.131546	−	0.991310i	$-0.541994\pi$
$541$	−3310.57	−0.263091	−0.131546	+	0.991310i	$0.541994\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−10614.8	−0.834291
$546$	0	0
$547$	−2286.52	−0.178729	−0.0893643	−	0.995999i	$-0.528484\pi$
$547$	−2286.52	−0.178729	−0.0893643	+	0.995999i	$0.528484\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	4778.98	0.369494
$552$	0	0
$553$	−7617.39	−0.585758
$554$	0	0
$555$	0	0
$556$	0	0
$557$	13846.6	1.05332	0.526658	−	0.850077i	$-0.323445\pi$
$557$	13846.6	1.05332	0.526658	+	0.850077i	$0.323445\pi$
$558$	0	0
$559$	506.502	0.0383233
$560$	0	0
$561$	0	0
$562$	0	0
$563$	16164.4	1.21003	0.605017	−	0.796213i	$-0.293166\pi$
$563$	16164.4	1.21003	0.605017	+	0.796213i	$0.293166\pi$
$564$	0	0
$565$	8973.99	0.668210
$566$	0	0
$567$	0	0
$568$	0	0
$569$	496.334	0.0365684	0.0182842	−	0.999833i	$-0.494180\pi$
$569$	496.334	0.0365684	0.0182842	+	0.999833i	$0.494180\pi$
$570$	0	0
$571$	−5971.18	−0.437629	−0.218815	−	0.975766i	$-0.570219\pi$
$571$	−5971.18	−0.437629	−0.218815	+	0.975766i	$0.570219\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	3403.42	0.246839
$576$	0	0
$577$	9571.82	0.690607	0.345304	−	0.938491i	$-0.387776\pi$
$577$	9571.82	0.690607	0.345304	+	0.938491i	$0.387776\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	16961.9	1.21119
$582$	0	0
$583$	23398.8	1.66223
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−5322.18	−0.374225	−0.187112	−	0.982339i	$-0.559913\pi$
$587$	−5322.18	−0.374225	−0.187112	+	0.982339i	$0.559913\pi$
$588$	0	0
$589$	4814.31	0.336791
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−11066.5	−0.766354	−0.383177	−	0.923675i	$-0.625170\pi$
$593$	−11066.5	−0.766354	−0.383177	+	0.923675i	$0.625170\pi$
$594$	0	0
$595$	−16220.1	−1.11758
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−18385.0	−1.25407	−0.627037	−	0.778989i	$-0.715733\pi$
$599$	−18385.0	−1.25407	−0.627037	+	0.778989i	$0.715733\pi$
$600$	0	0
$601$	−617.830	−0.0419331	−0.0209666	−	0.999780i	$-0.506674\pi$
$601$	−617.830	−0.0419331	−0.0209666	+	0.999780i	$0.506674\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−5922.52	−0.397991
$606$	0	0
$607$	−13839.8	−0.925436	−0.462718	−	0.886506i	$-0.653126\pi$
$607$	−13839.8	−0.925436	−0.462718	+	0.886506i	$0.653126\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−970.362	−0.0642498
$612$	0	0
$613$	−25655.4	−1.69039	−0.845196	−	0.534457i	$-0.820516\pi$
$613$	−25655.4	−1.69039	−0.845196	+	0.534457i	$0.820516\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	4218.26	0.275236	0.137618	−	0.990485i	$-0.456055\pi$
$617$	4218.26	0.275236	0.137618	+	0.990485i	$0.456055\pi$
$618$	0	0
$619$	−12182.6	−0.791047	−0.395524	−	0.918456i	$-0.629437\pi$
$619$	−12182.6	−0.791047	−0.395524	+	0.918456i	$0.629437\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−8948.40	−0.575458
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	43998.8	2.78910
$630$	0	0
$631$	−16673.2	−1.05190	−0.525951	−	0.850515i	$-0.676291\pi$
$631$	−16673.2	−1.05190	−0.525951	+	0.850515i	$0.676291\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−3744.47	−0.234007
$636$	0	0
$637$	−9567.44	−0.595096
$638$	0	0
$639$	0	0
$640$	0	0
$641$	26270.8	1.61878	0.809388	−	0.587275i	$-0.199799\pi$
$641$	26270.8	1.61878	0.809388	+	0.587275i	$0.199799\pi$
$642$	0	0
$643$	−11556.3	−0.708767	−0.354384	−	0.935100i	$-0.615309\pi$
$643$	−11556.3	−0.708767	−0.354384	+	0.935100i	$0.615309\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	25395.2	1.54310	0.771552	−	0.636166i	$-0.219481\pi$
$647$	25395.2	1.54310	0.771552	+	0.636166i	$0.219481\pi$
$648$	0	0
$649$	4785.96	0.289469
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−6182.84	−0.370526	−0.185263	−	0.982689i	$-0.559314\pi$
$653$	−6182.84	−0.370526	−0.185263	+	0.982689i	$0.559314\pi$
$654$	0	0
$655$	11980.5	0.714685
$656$	0	0
$657$	0	0
$658$	0	0
$659$	8885.32	0.525224	0.262612	−	0.964901i	$-0.415416\pi$
$659$	8885.32	0.525224	0.262612	+	0.964901i	$0.415416\pi$
$660$	0	0
$661$	−7171.60	−0.422001	−0.211001	−	0.977486i	$-0.567672\pi$
$661$	−7171.60	−0.422001	−0.211001	+	0.977486i	$0.567672\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	3270.16	0.190694
$666$	0	0
$667$	30538.7	1.77281
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−17914.0	−1.03064
$672$	0	0
$673$	−4250.52	−0.243455	−0.121728	−	0.992564i	$-0.538843\pi$
$673$	−4250.52	−0.243455	−0.121728	+	0.992564i	$0.538843\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	2233.65	0.126804	0.0634019	−	0.997988i	$-0.479805\pi$
$677$	2233.65	0.126804	0.0634019	+	0.997988i	$0.479805\pi$
$678$	0	0
$679$	−6097.23	−0.344610
$680$	0	0
$681$	0	0
$682$	0	0
$683$	6071.62	0.340153	0.170076	−	0.985431i	$-0.445599\pi$
$683$	6071.62	0.340153	0.170076	+	0.985431i	$0.445599\pi$
$684$	0	0
$685$	5022.38	0.280139
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−7445.49	−0.411684
$690$	0	0
$691$	765.784	0.0421589	0.0210794	−	0.999778i	$-0.493290\pi$
$691$	765.784	0.0421589	0.0210794	+	0.999778i	$0.493290\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	12015.1	0.655769
$696$	0	0
$697$	8045.70	0.437235
$698$	0	0
$699$	0	0
$700$	0	0
$701$	23564.3	1.26963	0.634814	−	0.772665i	$-0.281077\pi$
$701$	23564.3	1.26963	0.634814	+	0.772665i	$0.281077\pi$
$702$	0	0
$703$	−8870.67	−0.475909
$704$	0	0
$705$	0	0
$706$	0	0
$707$	25058.4	1.33298
$708$	0	0
$709$	24172.3	1.28041	0.640204	−	0.768205i	$-0.278850\pi$
$709$	24172.3	1.28041	0.640204	+	0.768205i	$0.278850\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	30764.5	1.61590
$714$	0	0
$715$	4002.16	0.209332
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−12829.0	−0.665424	−0.332712	−	0.943028i	$-0.607964\pi$
$719$	−12829.0	−0.665424	−0.332712	+	0.943028i	$0.607964\pi$
$720$	0	0
$721$	−43052.9	−2.22382
$722$	0	0
$723$	0	0
$724$	0	0
$725$	5608.08	0.287281
$726$	0	0
$727$	24724.7	1.26133	0.630666	−	0.776054i	$-0.282782\pi$
$727$	24724.7	1.26133	0.630666	+	0.776054i	$0.282782\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	3353.62	0.169683
$732$	0	0
$733$	−20172.2	−1.01648	−0.508239	−	0.861216i	$-0.669703\pi$
$733$	−20172.2	−1.01648	−0.508239	+	0.861216i	$0.669703\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−4405.32	−0.220179
$738$	0	0
$739$	16452.8	0.818979	0.409489	−	0.912315i	$-0.365707\pi$
$739$	16452.8	0.818979	0.409489	+	0.912315i	$0.365707\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−20864.6	−1.03021	−0.515107	−	0.857126i	$-0.672248\pi$
$743$	−20864.6	−1.03021	−0.515107	+	0.857126i	$0.672248\pi$
$744$	0	0
$745$	2548.24	0.125316
$746$	0	0
$747$	0	0
$748$	0	0
$749$	30055.2	1.46621
$750$	0	0
$751$	−15525.7	−0.754381	−0.377191	−	0.926136i	$-0.623110\pi$
$751$	−15525.7	−0.754381	−0.377191	+	0.926136i	$0.623110\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	7216.48	0.347861
$756$	0	0
$757$	30105.7	1.44546	0.722729	−	0.691132i	$-0.242888\pi$
$757$	30105.7	1.44546	0.722729	+	0.691132i	$0.242888\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−21739.9	−1.03557	−0.517786	−	0.855510i	$-0.673244\pi$
$761$	−21739.9	−1.03557	−0.517786	+	0.855510i	$0.673244\pi$
$762$	0	0
$763$	−65175.0	−3.09239
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−1522.89	−0.0716929
$768$	0	0
$769$	1942.22	0.0910772	0.0455386	−	0.998963i	$-0.485500\pi$
$769$	1942.22	0.0910772	0.0455386	+	0.998963i	$0.485500\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	7921.02	0.368563	0.184282	−	0.982873i	$-0.441004\pi$
$773$	7921.02	0.368563	0.184282	+	0.982873i	$0.441004\pi$
$774$	0	0
$775$	5649.55	0.261855
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−1622.11	−0.0746060
$780$	0	0
$781$	20698.5	0.948338
$782$	0	0
$783$	0	0
$784$	0	0
$785$	10778.2	0.490052
$786$	0	0
$787$	1361.40	0.0616630	0.0308315	−	0.999525i	$-0.490184\pi$
$787$	1361.40	0.0616630	0.0308315	+	0.999525i	$0.490184\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	55100.4	2.47679
$792$	0	0
$793$	5700.22	0.255260
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−27917.5	−1.24076	−0.620381	−	0.784300i	$-0.713022\pi$
$797$	−27917.5	−1.24076	−0.620381	+	0.784300i	$0.713022\pi$
$798$	0	0
$799$	−6424.90	−0.284476
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−16607.9	−0.729863
$804$	0	0
$805$	20897.0	0.914937
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−5275.87	−0.229283	−0.114641	−	0.993407i	$-0.536572\pi$
$809$	−5275.87	−0.229283	−0.114641	+	0.993407i	$0.536572\pi$
$810$	0	0
$811$	23769.1	1.02916	0.514579	−	0.857443i	$-0.327948\pi$
$811$	23769.1	1.02916	0.514579	+	0.857443i	$0.327948\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−2649.54	−0.113877
$816$	0	0
$817$	−676.129	−0.0289532
$818$	0	0
$819$	0	0
$820$	0	0
$821$	34209.3	1.45422	0.727108	−	0.686523i	$-0.240864\pi$
$821$	34209.3	1.45422	0.727108	+	0.686523i	$0.240864\pi$
$822$	0	0
$823$	−1240.13	−0.0525252	−0.0262626	−	0.999655i	$-0.508361\pi$
$823$	−1240.13	−0.0525252	−0.0262626	+	0.999655i	$0.508361\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−26971.0	−1.13407	−0.567034	−	0.823694i	$-0.691909\pi$
$827$	−26971.0	−1.13407	−0.567034	+	0.823694i	$0.691909\pi$
$828$	0	0
$829$	−26743.1	−1.12042	−0.560208	−	0.828352i	$-0.689279\pi$
$829$	−26743.1	−1.12042	−0.560208	+	0.828352i	$0.689279\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−63347.3	−2.63488
$834$	0	0
$835$	14896.4	0.617378
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−5273.14	−0.216983	−0.108492	−	0.994097i	$-0.534602\pi$
$839$	−5273.14	−0.216983	−0.108492	+	0.994097i	$0.534602\pi$
$840$	0	0
$841$	25931.9	1.06326
$842$	0	0
$843$	0	0
$844$	0	0
$845$	9711.51	0.395368
$846$	0	0
$847$	−36364.3	−1.47520
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−56685.5	−2.28338
$852$	0	0
$853$	5043.46	0.202444	0.101222	−	0.994864i	$-0.467725\pi$
$853$	5043.46	0.202444	0.101222	+	0.994864i	$0.467725\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	14478.5	0.577103	0.288552	−	0.957464i	$-0.406826\pi$
$857$	14478.5	0.577103	0.288552	+	0.957464i	$0.406826\pi$
$858$	0	0
$859$	7873.88	0.312751	0.156376	−	0.987698i	$-0.450019\pi$
$859$	7873.88	0.312751	0.156376	+	0.987698i	$0.450019\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	24166.0	0.953209	0.476604	−	0.879118i	$-0.341867\pi$
$863$	24166.0	0.953209	0.476604	+	0.879118i	$0.341867\pi$
$864$	0	0
$865$	8899.39	0.349813
$866$	0	0
$867$	0	0
$868$	0	0
$869$	12444.6	0.485792
$870$	0	0
$871$	1401.77	0.0545317
$872$	0	0
$873$	0	0
$874$	0	0
$875$	3837.51	0.148264
$876$	0	0
$877$	6411.82	0.246878	0.123439	−	0.992352i	$-0.460608\pi$
$877$	6411.82	0.246878	0.123439	+	0.992352i	$0.460608\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−22908.5	−0.876059	−0.438029	−	0.898961i	$-0.644323\pi$
$881$	−22908.5	−0.876059	−0.438029	+	0.898961i	$0.644323\pi$
$882$	0	0
$883$	−39198.4	−1.49392	−0.746960	−	0.664869i	$-0.768487\pi$
$883$	−39198.4	−1.49392	−0.746960	+	0.664869i	$0.768487\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	5422.19	0.205253	0.102626	−	0.994720i	$-0.467275\pi$
$887$	5422.19	0.205253	0.102626	+	0.994720i	$0.467275\pi$
$888$	0	0
$889$	−22991.1	−0.867375
$890$	0	0
$891$	0	0
$892$	0	0
$893$	1295.34	0.0485406
$894$	0	0
$895$	14181.3	0.529641
$896$	0	0
$897$	0	0
$898$	0	0
$899$	50693.0	1.88065
$900$	0	0
$901$	−49297.6	−1.82280
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−4059.45	−0.149106
$906$	0	0
$907$	30727.0	1.12489	0.562444	−	0.826836i	$-0.309861\pi$
$907$	30727.0	1.12489	0.562444	+	0.826836i	$0.309861\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	4101.34	0.149159	0.0745793	−	0.997215i	$-0.476239\pi$
$911$	4101.34	0.149159	0.0745793	+	0.997215i	$0.476239\pi$
$912$	0	0
$913$	−27710.8	−1.00448
$914$	0	0
$915$	0	0
$916$	0	0
$917$	73560.7	2.64906
$918$	0	0
$919$	50926.3	1.82797	0.913984	−	0.405750i	$-0.132990\pi$
$919$	50926.3	1.82797	0.913984	+	0.405750i	$0.132990\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−6586.26	−0.234875
$924$	0	0
$925$	−10409.6	−0.370019
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−22040.8	−0.778402	−0.389201	−	0.921153i	$-0.627249\pi$
$929$	−22040.8	−0.778402	−0.389201	+	0.921153i	$0.627249\pi$
$930$	0	0
$931$	12771.6	0.449593
$932$	0	0
$933$	0	0
$934$	0	0
$935$	26498.8	0.926850
$936$	0	0
$937$	28111.2	0.980097	0.490049	−	0.871695i	$-0.336979\pi$
$937$	28111.2	0.980097	0.490049	+	0.871695i	$0.336979\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−52811.3	−1.82954	−0.914770	−	0.403974i	$-0.867629\pi$
$941$	−52811.3	−1.82954	−0.914770	+	0.403974i	$0.867629\pi$
$942$	0	0
$943$	−10365.6	−0.357954
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−1826.23	−0.0626657	−0.0313329	−	0.999509i	$-0.509975\pi$
$947$	−1826.23	−0.0626657	−0.0313329	+	0.999509i	$0.509975\pi$
$948$	0	0
$949$	5284.63	0.180765
$950$	0	0
$951$	0	0
$952$	0	0
$953$	41164.2	1.39920	0.699601	−	0.714534i	$-0.253361\pi$
$953$	41164.2	1.39920	0.699601	+	0.714534i	$0.253361\pi$
$954$	0	0
$955$	−5742.31	−0.194573
$956$	0	0
$957$	0	0
$958$	0	0
$959$	30837.4	1.03837
$960$	0	0
$961$	21276.8	0.714202
$962$	0	0
$963$	0	0
$964$	0	0
$965$	10754.7	0.358761
$966$	0	0
$967$	7088.78	0.235739	0.117870	−	0.993029i	$-0.462394\pi$
$967$	7088.78	0.235739	0.117870	+	0.993029i	$0.462394\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	2355.90	0.0778625	0.0389312	−	0.999242i	$-0.487605\pi$
$971$	2355.90	0.0778625	0.0389312	+	0.999242i	$0.487605\pi$
$972$	0	0
$973$	73773.0	2.43068
$974$	0	0
$975$	0	0
$976$	0	0
$977$	16795.1	0.549972	0.274986	−	0.961448i	$-0.411327\pi$
$977$	16795.1	0.549972	0.274986	+	0.961448i	$0.411327\pi$
$978$	0	0
$979$	14619.0	0.477249
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−14733.8	−0.478063	−0.239032	−	0.971012i	$-0.576830\pi$
$983$	−14733.8	−0.478063	−0.239032	+	0.971012i	$0.576830\pi$
$984$	0	0
$985$	−25287.6	−0.817999
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−4320.61	−0.138915
$990$	0	0
$991$	−36809.2	−1.17990	−0.589950	−	0.807440i	$-0.700853\pi$
$991$	−36809.2	−1.17990	−0.589950	+	0.807440i	$0.700853\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−17770.6	−0.566196
$996$	0	0
$997$	−21210.0	−0.673749	−0.336875	−	0.941550i	$-0.609370\pi$
$997$	−21210.0	−0.673749	−0.336875	+	0.941550i	$0.609370\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2160.4.a.bi.1.1		3
3.2	odd	2		2160.4.a.bq.1.1		3
4.3	odd	2		135.4.a.e.1.2	✓	3
12.11	even	2		135.4.a.h.1.2	yes	3
20.3	even	4		675.4.b.m.649.4		6
20.7	even	4		675.4.b.m.649.3		6
20.19	odd	2		675.4.a.s.1.2		3
36.7	odd	6		405.4.e.v.271.2		6
36.11	even	6		405.4.e.q.271.2		6
36.23	even	6		405.4.e.q.136.2		6
36.31	odd	6		405.4.e.v.136.2		6
60.23	odd	4		675.4.b.n.649.3		6
60.47	odd	4		675.4.b.n.649.4		6
60.59	even	2		675.4.a.p.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
135.4.a.e.1.2	✓	3	4.3	odd	2
135.4.a.h.1.2	yes	3	12.11	even	2
405.4.e.q.136.2		6	36.23	even	6
405.4.e.q.271.2		6	36.11	even	6
405.4.e.v.136.2		6	36.31	odd	6
405.4.e.v.271.2		6	36.7	odd	6
675.4.a.p.1.2		3	60.59	even	2
675.4.a.s.1.2		3	20.19	odd	2
675.4.b.m.649.3		6	20.7	even	4
675.4.b.m.649.4		6	20.3	even	4
675.4.b.n.649.3		6	60.23	odd	4
675.4.b.n.649.4		6	60.47	odd	4
2160.4.a.bi.1.1		3	1.1	even	1	trivial
2160.4.a.bq.1.1		3	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:	\( N \)	\(=\)	\( 2160 = 2^{4} \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2160.a (trivial)

\(f(q)\)	\(=\)	\(q-5.00000 q^{5} -30.7000 q^{7} +O(q^{10})\)	\(q-5.00000 q^{5} -30.7000 q^{7} +50.1548 q^{11} -15.9592 q^{13} -105.668 q^{17} +21.3040 q^{19} +136.137 q^{23} +25.0000 q^{25} +224.323 q^{29} +225.982 q^{31} +153.500 q^{35} -416.386 q^{37} -76.1411 q^{41} -31.7372 q^{43} +60.8026 q^{47} +599.493 q^{49} +466.532 q^{53} -250.774 q^{55} +95.4239 q^{59} -357.174 q^{61} +79.7962 q^{65} -87.8344 q^{67} +412.693 q^{71} -331.133 q^{73} -1539.75 q^{77} +248.123 q^{79} -552.505 q^{83} +528.341 q^{85} +291.478 q^{89} +489.949 q^{91} -106.520 q^{95} +198.606 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 15 q^{5} + 4 q^{7}+O(q^{10}) \) 3 * q - 15 * q^5 + 4 * q^7	\( 3 q - 15 q^{5} + 4 q^{7} + 5 q^{11} + 7 q^{13} - 155 q^{17} + 50 q^{19} + 285 q^{23} + 75 q^{25} - 115 q^{29} + 115 q^{31} - 20 q^{35} - 384 q^{37} - 580 q^{41} + 797 q^{43} - 145 q^{47} + 577 q^{49} + 400 q^{53} - 25 q^{55} + 380 q^{59} - 152 q^{61} - 35 q^{65} - 2 q^{67} + 40 q^{71} - 980 q^{73} - 1950 q^{77} - 1013 q^{79} + 270 q^{83} + 775 q^{85} - 1020 q^{89} + 632 q^{91} - 250 q^{95} + 720 q^{97}+O(q^{100}) \) 3 * q - 15 * q^5 + 4 * q^7 + 5 * q^11 + 7 * q^13 - 155 * q^17 + 50 * q^19 + 285 * q^23 + 75 * q^25 - 115 * q^29 + 115 * q^31 - 20 * q^35 - 384 * q^37 - 580 * q^41 + 797 * q^43 - 145 * q^47 + 577 * q^49 + 400 * q^53 - 25 * q^55 + 380 * q^59 - 152 * q^61 - 35 * q^65 - 2 * q^67 + 40 * q^71 - 980 * q^73 - 1950 * q^77 - 1013 * q^79 + 270 * q^83 + 775 * q^85 - 1020 * q^89 + 632 * q^91 - 250 * q^95 + 720 * q^97

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