LMFDB - Embedded newform 2475.4.a.by.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2475,4,Mod(1,2475)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2475, 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("2475.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2475 = 3^{2} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2475.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:	$146.029727264$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 4 x^{9} - 49 x^{8} + 180 x^{7} + 753 x^{6} - 2420 x^{5} - 4059 x^{4} + 9796 x^{3} + 7226 x^{2} + \cdots - 2112 $ x^10 - 4x^9 - 49x^8 + 180x^7 + 753x^6 - 2420x^5 - 4059x^4 + 9796x^3 + 7226x^2 - 9120*x - 2112
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-3.94490$ of defining polynomial
Character	$\chi$	$=$	2475.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.94490 q^{2} +7.56227 q^{4} -16.6840 q^{7} +1.72681 q^{8} +O(q^{10})$	$q-3.94490 q^{2} +7.56227 q^{4} -16.6840 q^{7} +1.72681 q^{8} +11.0000 q^{11} +79.9090 q^{13} +65.8169 q^{14} -67.3103 q^{16} +5.54304 q^{17} +24.4157 q^{19} -43.3939 q^{22} -120.982 q^{23} -315.233 q^{26} -126.169 q^{28} +125.418 q^{29} -139.618 q^{31} +251.718 q^{32} -21.8668 q^{34} +394.195 q^{37} -96.3178 q^{38} +314.803 q^{41} +261.662 q^{43} +83.1849 q^{44} +477.262 q^{46} +156.210 q^{47} -64.6432 q^{49} +604.293 q^{52} +489.894 q^{53} -28.8102 q^{56} -494.760 q^{58} +247.848 q^{59} -774.596 q^{61} +550.781 q^{62} -454.521 q^{64} -687.119 q^{67} +41.9179 q^{68} -942.247 q^{71} -655.534 q^{73} -1555.06 q^{74} +184.638 q^{76} -183.524 q^{77} +611.627 q^{79} -1241.87 q^{82} +1223.67 q^{83} -1032.23 q^{86} +18.9950 q^{88} +20.7239 q^{89} -1333.20 q^{91} -914.897 q^{92} -616.234 q^{94} +1321.78 q^{97} +255.011 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 4 q^{2} + 34 q^{4} + 2 q^{7} + 48 q^{8}+O(q^{10}) $ 10 * q + 4 * q^2 + 34 * q^4 + 2 * q^7 + 48 * q^8	$ 10 q + 4 q^{2} + 34 q^{4} + 2 q^{7} + 48 q^{8} + 110 q^{11} + 26 q^{13} - 72 q^{14} + 206 q^{16} + 148 q^{17} - 114 q^{19} + 44 q^{22} - 34 q^{23} - 100 q^{26} - 86 q^{28} + 38 q^{29} + 232 q^{31} + 448 q^{32} - 20 q^{34} + 754 q^{37} + 780 q^{38} - 160 q^{41} - 66 q^{43} + 374 q^{44} + 682 q^{46} + 450 q^{47} + 590 q^{49} + 200 q^{52} + 1068 q^{53} - 268 q^{56} - 138 q^{58} + 838 q^{59} - 566 q^{61} + 1230 q^{62} + 462 q^{64} + 430 q^{67} + 2234 q^{68} - 518 q^{71} - 184 q^{73} - 402 q^{74} + 386 q^{76} + 22 q^{77} + 956 q^{79} - 2180 q^{82} + 2094 q^{83} - 892 q^{86} + 528 q^{88} + 512 q^{89} - 858 q^{91} + 4476 q^{92} - 294 q^{94} - 1006 q^{97} + 2226 q^{98}+O(q^{100}) $ 10 * q + 4 * q^2 + 34 * q^4 + 2 * q^7 + 48 * q^8 + 110 * q^11 + 26 * q^13 - 72 * q^14 + 206 * q^16 + 148 * q^17 - 114 * q^19 + 44 * q^22 - 34 * q^23 - 100 * q^26 - 86 * q^28 + 38 * q^29 + 232 * q^31 + 448 * q^32 - 20 * q^34 + 754 * q^37 + 780 * q^38 - 160 * q^41 - 66 * q^43 + 374 * q^44 + 682 * q^46 + 450 * q^47 + 590 * q^49 + 200 * q^52 + 1068 * q^53 - 268 * q^56 - 138 * q^58 + 838 * q^59 - 566 * q^61 + 1230 * q^62 + 462 * q^64 + 430 * q^67 + 2234 * q^68 - 518 * q^71 - 184 * q^73 - 402 * q^74 + 386 * q^76 + 22 * q^77 + 956 * q^79 - 2180 * q^82 + 2094 * q^83 - 892 * q^86 + 528 * q^88 + 512 * q^89 - 858 * q^91 + 4476 * q^92 - 294 * q^94 - 1006 * q^97 + 2226 * q^98

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 $

$2$	−3.94490	−1.39473	−0.697367	−	0.716714i	$-0.745645\pi$
$2$	−3.94490	−1.39473	−0.697367	+	0.716714i	$0.745645\pi$
$3$	0	0
$3$	0	0
$4$	7.56227	0.945283
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−16.6840	−0.900853	−0.450426	−	0.892814i	$-0.648728\pi$
$7$	−16.6840	−0.900853	−0.450426	+	0.892814i	$0.648728\pi$
$8$	1.72681	0.0763152
$9$	0	0
$10$	0	0
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	0	0
$13$	79.9090	1.70483	0.852414	−	0.522867i	$-0.175138\pi$
$13$	79.9090	1.70483	0.852414	+	0.522867i	$0.175138\pi$
$14$	65.8169	1.25645
$15$	0	0
$16$	−67.3103	−1.05172
$17$	5.54304	0.0790814	0.0395407	−	0.999218i	$-0.487411\pi$
$17$	5.54304	0.0790814	0.0395407	+	0.999218i	$0.487411\pi$
$18$	0	0
$19$	24.4157	0.294808	0.147404	−	0.989076i	$-0.452908\pi$
$19$	24.4157	0.294808	0.147404	+	0.989076i	$0.452908\pi$
$20$	0	0
$21$	0	0
$22$	−43.3939	−0.420528
$23$	−120.982	−1.09680	−0.548401	−	0.836215i	$-0.684763\pi$
$23$	−120.982	−1.09680	−0.548401	+	0.836215i	$0.684763\pi$
$24$	0	0
$25$	0	0
$26$	−315.233	−2.37778
$27$	0	0
$28$	−126.169	−0.851561
$29$	125.418	0.803085	0.401543	−	0.915840i	$-0.368474\pi$
$29$	125.418	0.803085	0.401543	+	0.915840i	$0.368474\pi$
$30$	0	0
$31$	−139.618	−0.808909	−0.404455	−	0.914558i	$-0.632539\pi$
$31$	−139.618	−0.808909	−0.404455	+	0.914558i	$0.632539\pi$
$32$	251.718	1.39056
$33$	0	0
$34$	−21.8668	−0.110298
$35$	0	0
$36$	0	0
$37$	394.195	1.75149	0.875747	−	0.482770i	$-0.160369\pi$
$37$	394.195	1.75149	0.875747	+	0.482770i	$0.160369\pi$
$38$	−96.3178	−0.411179
$39$	0	0
$40$	0	0
$41$	314.803	1.19912	0.599560	−	0.800330i	$-0.295342\pi$
$41$	314.803	1.19912	0.599560	+	0.800330i	$0.295342\pi$
$42$	0	0
$43$	261.662	0.927978	0.463989	−	0.885841i	$-0.346418\pi$
$43$	261.662	0.927978	0.463989	+	0.885841i	$0.346418\pi$
$44$	83.1849	0.285014
$45$	0	0
$46$	477.262	1.52975
$47$	156.210	0.484800	0.242400	−	0.970176i	$-0.422065\pi$
$47$	156.210	0.484800	0.242400	+	0.970176i	$0.422065\pi$
$48$	0	0
$49$	−64.6432	−0.188464
$50$	0	0
$51$	0	0
$52$	604.293	1.61155
$53$	489.894	1.26966	0.634831	−	0.772651i	$-0.281070\pi$
$53$	489.894	1.26966	0.634831	+	0.772651i	$0.281070\pi$
$54$	0	0
$55$	0	0
$56$	−28.8102	−0.0687487
$57$	0	0
$58$	−494.760	−1.12009
$59$	247.848	0.546900	0.273450	−	0.961886i	$-0.411835\pi$
$59$	247.848	0.546900	0.273450	+	0.961886i	$0.411835\pi$
$60$	0	0
$61$	−774.596	−1.62585	−0.812925	−	0.582368i	$-0.802126\pi$
$61$	−774.596	−1.62585	−0.812925	+	0.582368i	$0.802126\pi$
$62$	550.781	1.12821
$63$	0	0
$64$	−454.521	−0.887737
$65$	0	0
$66$	0	0
$67$	−687.119	−1.25291	−0.626454	−	0.779458i	$-0.715495\pi$
$67$	−687.119	−1.25291	−0.626454	+	0.779458i	$0.715495\pi$
$68$	41.9179	0.0747544
$69$	0	0
$70$	0	0
$71$	−942.247	−1.57499	−0.787494	−	0.616323i	$-0.788622\pi$
$71$	−942.247	−1.57499	−0.787494	+	0.616323i	$0.788622\pi$
$72$	0	0
$73$	−655.534	−1.05102	−0.525510	−	0.850787i	$-0.676125\pi$
$73$	−655.534	−1.05102	−0.525510	+	0.850787i	$0.676125\pi$
$74$	−1555.06	−2.44287
$75$	0	0
$76$	184.638	0.278677
$77$	−183.524	−0.271617
$78$	0	0
$79$	611.627	0.871056	0.435528	−	0.900175i	$-0.356562\pi$
$79$	611.627	0.871056	0.435528	+	0.900175i	$0.356562\pi$
$80$	0	0
$81$	0	0
$82$	−1241.87	−1.67245
$83$	1223.67	1.61825	0.809126	−	0.587635i	$-0.199941\pi$
$83$	1223.67	1.61825	0.809126	+	0.587635i	$0.199941\pi$
$84$	0	0
$85$	0	0
$86$	−1032.23	−1.29428
$87$	0	0
$88$	18.9950	0.0230099
$89$	20.7239	0.0246823	0.0123412	−	0.999924i	$-0.496072\pi$
$89$	20.7239	0.0246823	0.0123412	+	0.999924i	$0.496072\pi$
$90$	0	0
$91$	−1333.20	−1.53580
$92$	−914.897	−1.03679
$93$	0	0
$94$	−616.234	−0.676166
$95$	0	0
$96$	0	0
$97$	1321.78	1.38357	0.691786	−	0.722103i	$-0.256824\pi$
$97$	1321.78	1.38357	0.691786	+	0.722103i	$0.256824\pi$
$98$	255.011	0.262857
$99$	0	0
$100$	0	0
$101$	630.618	0.621275	0.310638	−	0.950528i	$-0.399457\pi$
$101$	630.618	0.621275	0.310638	+	0.950528i	$0.399457\pi$
$102$	0	0
$103$	−1309.80	−1.25300	−0.626498	−	0.779423i	$-0.715512\pi$
$103$	−1309.80	−1.25300	−0.626498	+	0.779423i	$0.715512\pi$
$104$	137.988	0.130104
$105$	0	0
$106$	−1932.58	−1.77084
$107$	1966.97	1.77714	0.888572	−	0.458736i	$-0.151698\pi$
$107$	1966.97	1.77714	0.888572	+	0.458736i	$0.151698\pi$
$108$	0	0
$109$	−615.843	−0.541166	−0.270583	−	0.962697i	$-0.587216\pi$
$109$	−615.843	−0.541166	−0.270583	+	0.962697i	$0.587216\pi$
$110$	0	0
$111$	0	0
$112$	1123.01	0.947447
$113$	1672.07	1.39199	0.695994	−	0.718048i	$-0.254964\pi$
$113$	1672.07	1.39199	0.695994	+	0.718048i	$0.254964\pi$
$114$	0	0
$115$	0	0
$116$	948.442	0.759143
$117$	0	0
$118$	−977.737	−0.762780
$119$	−92.4802	−0.0712407
$120$	0	0
$121$	121.000	0.0909091
$122$	3055.71	2.26763
$123$	0	0
$124$	−1055.83	−0.764648
$125$	0	0
$126$	0	0
$127$	280.481	0.195974	0.0979869	−	0.995188i	$-0.468760\pi$
$127$	280.481	0.195974	0.0979869	+	0.995188i	$0.468760\pi$
$128$	−220.701	−0.152402
$129$	0	0
$130$	0	0
$131$	−834.801	−0.556770	−0.278385	−	0.960470i	$-0.589799\pi$
$131$	−834.801	−0.556770	−0.278385	+	0.960470i	$0.589799\pi$
$132$	0	0
$133$	−407.353	−0.265579
$134$	2710.62	1.74747
$135$	0	0
$136$	9.57180	0.00603511
$137$	−3035.90	−1.89325	−0.946623	−	0.322343i	$-0.895530\pi$
$137$	−3035.90	−1.89325	−0.946623	+	0.322343i	$0.895530\pi$
$138$	0	0
$139$	−2013.73	−1.22880	−0.614398	−	0.788996i	$-0.710601\pi$
$139$	−2013.73	−1.22880	−0.614398	+	0.788996i	$0.710601\pi$
$140$	0	0
$141$	0	0
$142$	3717.07	2.19669
$143$	878.999	0.514025
$144$	0	0
$145$	0	0
$146$	2586.02	1.46589
$147$	0	0
$148$	2981.01	1.65566
$149$	−135.311	−0.0743969	−0.0371984	−	0.999308i	$-0.511843\pi$
$149$	−135.311	−0.0743969	−0.0371984	+	0.999308i	$0.511843\pi$
$150$	0	0
$151$	−1134.88	−0.611622	−0.305811	−	0.952092i	$-0.598928\pi$
$151$	−1134.88	−0.611622	−0.305811	+	0.952092i	$0.598928\pi$
$152$	42.1615	0.0224983
$153$	0	0
$154$	723.986	0.378834
$155$	0	0
$156$	0	0
$157$	26.3188	0.0133788	0.00668940	−	0.999978i	$-0.497871\pi$
$157$	26.3188	0.0133788	0.00668940	+	0.999978i	$0.497871\pi$
$158$	−2412.81	−1.21489
$159$	0	0
$160$	0	0
$161$	2018.46	0.988057
$162$	0	0
$163$	−2118.62	−1.01805	−0.509027	−	0.860751i	$-0.669995\pi$
$163$	−2118.62	−1.01805	−0.509027	+	0.860751i	$0.669995\pi$
$164$	2380.62	1.13351
$165$	0	0
$166$	−4827.25	−2.25703
$167$	−2927.24	−1.35639	−0.678194	−	0.734883i	$-0.737237\pi$
$167$	−2927.24	−1.35639	−0.678194	+	0.734883i	$0.737237\pi$
$168$	0	0
$169$	4188.45	1.90644
$170$	0	0
$171$	0	0
$172$	1978.76	0.877203
$173$	2594.71	1.14030	0.570151	−	0.821540i	$-0.306885\pi$
$173$	2594.71	1.14030	0.570151	+	0.821540i	$0.306885\pi$
$174$	0	0
$175$	0	0
$176$	−740.413	−0.317106
$177$	0	0
$178$	−81.7538	−0.0344253
$179$	−867.958	−0.362426	−0.181213	−	0.983444i	$-0.558002\pi$
$179$	−867.958	−0.362426	−0.181213	+	0.983444i	$0.558002\pi$
$180$	0	0
$181$	−1197.87	−0.491916	−0.245958	−	0.969280i	$-0.579103\pi$
$181$	−1197.87	−0.491916	−0.245958	+	0.969280i	$0.579103\pi$
$182$	5259.36	2.14203
$183$	0	0
$184$	−208.913	−0.0837026
$185$	0	0
$186$	0	0
$187$	60.9734	0.0238440
$188$	1181.30	0.458273
$189$	0	0
$190$	0	0
$191$	2411.57	0.913587	0.456794	−	0.889573i	$-0.348998\pi$
$191$	2411.57	0.913587	0.456794	+	0.889573i	$0.348998\pi$
$192$	0	0
$193$	−2704.23	−1.00858	−0.504288	−	0.863536i	$-0.668245\pi$
$193$	−2704.23	−1.00858	−0.504288	+	0.863536i	$0.668245\pi$
$194$	−5214.29	−1.92971
$195$	0	0
$196$	−488.849	−0.178152
$197$	4435.53	1.60415	0.802076	−	0.597221i	$-0.203729\pi$
$197$	4435.53	1.60415	0.802076	+	0.597221i	$0.203729\pi$
$198$	0	0
$199$	−406.246	−0.144713	−0.0723567	−	0.997379i	$-0.523052\pi$
$199$	−406.246	−0.144713	−0.0723567	+	0.997379i	$0.523052\pi$
$200$	0	0
$201$	0	0
$202$	−2487.73	−0.866514
$203$	−2092.47	−0.723462
$204$	0	0
$205$	0	0
$206$	5167.04	1.74760
$207$	0	0
$208$	−5378.70	−1.79301
$209$	268.573	0.0888880
$210$	0	0
$211$	−3765.92	−1.22871	−0.614353	−	0.789032i	$-0.710583\pi$
$211$	−3765.92	−1.22871	−0.614353	+	0.789032i	$0.710583\pi$
$212$	3704.71	1.20019
$213$	0	0
$214$	−7759.52	−2.47864
$215$	0	0
$216$	0	0
$217$	2329.40	0.728708
$218$	2429.44	0.754782
$219$	0	0
$220$	0	0
$221$	442.939	0.134820
$222$	0	0
$223$	1325.10	0.397915	0.198957	−	0.980008i	$-0.436244\pi$
$223$	1325.10	0.397915	0.198957	+	0.980008i	$0.436244\pi$
$224$	−4199.67	−1.25269
$225$	0	0
$226$	−6596.14	−1.94145
$227$	1614.74	0.472133	0.236066	−	0.971737i	$-0.424142\pi$
$227$	1614.74	0.472133	0.236066	+	0.971737i	$0.424142\pi$
$228$	0	0
$229$	1526.38	0.440463	0.220232	−	0.975448i	$-0.429319\pi$
$229$	1526.38	0.440463	0.220232	+	0.975448i	$0.429319\pi$
$230$	0	0
$231$	0	0
$232$	216.573	0.0612876
$233$	−4469.11	−1.25657	−0.628285	−	0.777983i	$-0.716243\pi$
$233$	−4469.11	−1.25657	−0.628285	+	0.777983i	$0.716243\pi$
$234$	0	0
$235$	0	0
$236$	1874.29	0.516975
$237$	0	0
$238$	364.826	0.0993619
$239$	−5290.82	−1.43194	−0.715972	−	0.698129i	$-0.754016\pi$
$239$	−5290.82	−1.43194	−0.715972	+	0.698129i	$0.754016\pi$
$240$	0	0
$241$	278.820	0.0745243	0.0372621	−	0.999306i	$-0.488136\pi$
$241$	278.820	0.0745243	0.0372621	+	0.999306i	$0.488136\pi$
$242$	−477.333	−0.126794
$243$	0	0
$244$	−5857.70	−1.53689
$245$	0	0
$246$	0	0
$247$	1951.04	0.502597
$248$	−241.095	−0.0617320
$249$	0	0
$250$	0	0
$251$	−2232.97	−0.561529	−0.280764	−	0.959777i	$-0.590588\pi$
$251$	−2232.97	−0.561529	−0.280764	+	0.959777i	$0.590588\pi$
$252$	0	0
$253$	−1330.80	−0.330698
$254$	−1106.47	−0.273331
$255$	0	0
$256$	4506.82	1.10030
$257$	2309.03	0.560442	0.280221	−	0.959936i	$-0.409592\pi$
$257$	2309.03	0.560442	0.280221	+	0.959936i	$0.409592\pi$
$258$	0	0
$259$	−6576.76	−1.57784
$260$	0	0
$261$	0	0
$262$	3293.21	0.776547
$263$	849.390	0.199147	0.0995735	−	0.995030i	$-0.468252\pi$
$263$	849.390	0.199147	0.0995735	+	0.995030i	$0.468252\pi$
$264$	0	0
$265$	0	0
$266$	1606.97	0.370412
$267$	0	0
$268$	−5196.17	−1.18435
$269$	−341.785	−0.0774684	−0.0387342	−	0.999250i	$-0.512333\pi$
$269$	−341.785	−0.0774684	−0.0387342	+	0.999250i	$0.512333\pi$
$270$	0	0
$271$	5548.74	1.24377	0.621885	−	0.783109i	$-0.286367\pi$
$271$	5548.74	1.24377	0.621885	+	0.783109i	$0.286367\pi$
$272$	−373.103	−0.0831718
$273$	0	0
$274$	11976.3	2.64057
$275$	0	0
$276$	0	0
$277$	5821.54	1.26275	0.631377	−	0.775476i	$-0.282490\pi$
$277$	5821.54	1.26275	0.631377	+	0.775476i	$0.282490\pi$
$278$	7943.98	1.71384
$279$	0	0
$280$	0	0
$281$	4809.06	1.02094	0.510470	−	0.859895i	$-0.329471\pi$
$281$	4809.06	1.02094	0.510470	+	0.859895i	$0.329471\pi$
$282$	0	0
$283$	−3397.76	−0.713695	−0.356847	−	0.934163i	$-0.616148\pi$
$283$	−3397.76	−0.713695	−0.356847	+	0.934163i	$0.616148\pi$
$284$	−7125.52	−1.48881
$285$	0	0
$286$	−3467.57	−0.716928
$287$	−5252.18	−1.08023
$288$	0	0
$289$	−4882.27	−0.993746
$290$	0	0
$291$	0	0
$292$	−4957.32	−0.993512
$293$	−3016.92	−0.601538	−0.300769	−	0.953697i	$-0.597243\pi$
$293$	−3016.92	−0.601538	−0.300769	+	0.953697i	$0.597243\pi$
$294$	0	0
$295$	0	0
$296$	680.702	0.133666
$297$	0	0
$298$	533.790	0.103764
$299$	−9667.53	−1.86986
$300$	0	0
$301$	−4365.58	−0.835972
$302$	4476.98	0.853050
$303$	0	0
$304$	−1643.43	−0.310056
$305$	0	0
$306$	0	0
$307$	2529.37	0.470225	0.235112	−	0.971968i	$-0.424454\pi$
$307$	2529.37	0.470225	0.235112	+	0.971968i	$0.424454\pi$
$308$	−1387.86	−0.256755
$309$	0	0
$310$	0	0
$311$	−7873.07	−1.43550	−0.717750	−	0.696300i	$-0.754828\pi$
$311$	−7873.07	−1.43550	−0.717750	+	0.696300i	$0.754828\pi$
$312$	0	0
$313$	1781.41	0.321698	0.160849	−	0.986979i	$-0.448577\pi$
$313$	1781.41	0.321698	0.160849	+	0.986979i	$0.448577\pi$
$314$	−103.825	−0.0186599
$315$	0	0
$316$	4625.29	0.823394
$317$	6490.18	1.14992	0.574960	−	0.818181i	$-0.305018\pi$
$317$	6490.18	1.14992	0.574960	+	0.818181i	$0.305018\pi$
$318$	0	0
$319$	1379.59	0.242139
$320$	0	0
$321$	0	0
$322$	−7962.65	−1.37808
$323$	135.337	0.0233139
$324$	0	0
$325$	0	0
$326$	8357.73	1.41991
$327$	0	0
$328$	543.606	0.0915110
$329$	−2606.21	−0.436733
$330$	0	0
$331$	2704.34	0.449075	0.224538	−	0.974465i	$-0.427913\pi$
$331$	2704.34	0.449075	0.224538	+	0.974465i	$0.427913\pi$
$332$	9253.70	1.52971
$333$	0	0
$334$	11547.7	1.89180
$335$	0	0
$336$	0	0
$337$	1760.69	0.284601	0.142301	−	0.989823i	$-0.454550\pi$
$337$	1760.69	0.284601	0.142301	+	0.989823i	$0.454550\pi$
$338$	−16523.0	−2.65898
$339$	0	0
$340$	0	0
$341$	−1535.80	−0.243895
$342$	0	0
$343$	6801.13	1.07063
$344$	451.842	0.0708188
$345$	0	0
$346$	−10235.9	−1.59042
$347$	−3022.89	−0.467658	−0.233829	−	0.972278i	$-0.575126\pi$
$347$	−3022.89	−0.467658	−0.233829	+	0.972278i	$0.575126\pi$
$348$	0	0
$349$	9815.51	1.50548	0.752739	−	0.658319i	$-0.228732\pi$
$349$	9815.51	1.50548	0.752739	+	0.658319i	$0.228732\pi$
$350$	0	0
$351$	0	0
$352$	2768.90	0.419269
$353$	−2692.64	−0.405991	−0.202995	−	0.979180i	$-0.565068\pi$
$353$	−2692.64	−0.405991	−0.202995	+	0.979180i	$0.565068\pi$
$354$	0	0
$355$	0	0
$356$	156.720	0.0233318
$357$	0	0
$358$	3424.01	0.505488
$359$	−1291.60	−0.189883	−0.0949414	−	0.995483i	$-0.530266\pi$
$359$	−1291.60	−0.189883	−0.0949414	+	0.995483i	$0.530266\pi$
$360$	0	0
$361$	−6262.87	−0.913088
$362$	4725.48	0.686093
$363$	0	0
$364$	−10082.0	−1.45177
$365$	0	0
$366$	0	0
$367$	4779.71	0.679833	0.339917	−	0.940456i	$-0.389601\pi$
$367$	4779.71	0.679833	0.339917	+	0.940456i	$0.389601\pi$
$368$	8143.32	1.15353
$369$	0	0
$370$	0	0
$371$	−8173.40	−1.14378
$372$	0	0
$373$	−1184.17	−0.164381	−0.0821905	−	0.996617i	$-0.526192\pi$
$373$	−1184.17	−0.164381	−0.0821905	+	0.996617i	$0.526192\pi$
$374$	−240.534	−0.0332560
$375$	0	0
$376$	269.746	0.0369976
$377$	10022.0	1.36912
$378$	0	0
$379$	9722.93	1.31777	0.658883	−	0.752245i	$-0.271029\pi$
$379$	9722.93	1.31777	0.658883	+	0.752245i	$0.271029\pi$
$380$	0	0
$381$	0	0
$382$	−9513.42	−1.27421
$383$	8188.96	1.09252	0.546262	−	0.837615i	$-0.316050\pi$
$383$	8188.96	1.09252	0.546262	+	0.837615i	$0.316050\pi$
$384$	0	0
$385$	0	0
$386$	10667.9	1.40669
$387$	0	0
$388$	9995.65	1.30787
$389$	−1152.50	−0.150216	−0.0751082	−	0.997175i	$-0.523930\pi$
$389$	−1152.50	−0.150216	−0.0751082	+	0.997175i	$0.523930\pi$
$390$	0	0
$391$	−670.607	−0.0867367
$392$	−111.627	−0.0143827
$393$	0	0
$394$	−17497.7	−2.23737
$395$	0	0
$396$	0	0
$397$	−5195.42	−0.656803	−0.328401	−	0.944538i	$-0.606510\pi$
$397$	−5195.42	−0.656803	−0.328401	+	0.944538i	$0.606510\pi$
$398$	1602.60	0.201837
$399$	0	0
$400$	0	0
$401$	14789.6	1.84179	0.920894	−	0.389813i	$-0.127460\pi$
$401$	14789.6	1.84179	0.920894	+	0.389813i	$0.127460\pi$
$402$	0	0
$403$	−11156.8	−1.37905
$404$	4768.90	0.587281
$405$	0	0
$406$	8254.60	1.00904
$407$	4336.15	0.528095
$408$	0	0
$409$	1850.70	0.223744	0.111872	−	0.993723i	$-0.464315\pi$
$409$	1850.70	0.223744	0.111872	+	0.993723i	$0.464315\pi$
$410$	0	0
$411$	0	0
$412$	−9905.08	−1.18444
$413$	−4135.11	−0.492676
$414$	0	0
$415$	0	0
$416$	20114.5	2.37066
$417$	0	0
$418$	−1059.50	−0.123975
$419$	4255.34	0.496150	0.248075	−	0.968741i	$-0.420202\pi$
$419$	4255.34	0.496150	0.248075	+	0.968741i	$0.420202\pi$
$420$	0	0
$421$	9138.40	1.05791	0.528953	−	0.848651i	$-0.322585\pi$
$421$	9138.40	1.05791	0.528953	+	0.848651i	$0.322585\pi$
$422$	14856.2	1.71372
$423$	0	0
$424$	845.956	0.0968944
$425$	0	0
$426$	0	0
$427$	12923.4	1.46465
$428$	14874.8	1.67991
$429$	0	0
$430$	0	0
$431$	6025.32	0.673387	0.336693	−	0.941614i	$-0.390691\pi$
$431$	6025.32	0.673387	0.336693	+	0.941614i	$0.390691\pi$
$432$	0	0
$433$	1357.15	0.150625	0.0753126	−	0.997160i	$-0.476005\pi$
$433$	1357.15	0.150625	0.0753126	+	0.997160i	$0.476005\pi$
$434$	−9189.24	−1.01635
$435$	0	0
$436$	−4657.17	−0.511555
$437$	−2953.86	−0.323346
$438$	0	0
$439$	5127.82	0.557488	0.278744	−	0.960365i	$-0.410082\pi$
$439$	5127.82	0.557488	0.278744	+	0.960365i	$0.410082\pi$
$440$	0	0
$441$	0	0
$442$	−1747.35	−0.188038
$443$	−735.724	−0.0789059	−0.0394530	−	0.999221i	$-0.512562\pi$
$443$	−735.724	−0.0789059	−0.0394530	+	0.999221i	$0.512562\pi$
$444$	0	0
$445$	0	0
$446$	−5227.38	−0.554985
$447$	0	0
$448$	7583.24	0.799720
$449$	14652.7	1.54010	0.770050	−	0.637984i	$-0.220231\pi$
$449$	14652.7	1.54010	0.770050	+	0.637984i	$0.220231\pi$
$450$	0	0
$451$	3462.83	0.361548
$452$	12644.6	1.31582
$453$	0	0
$454$	−6370.00	−0.658500
$455$	0	0
$456$	0	0
$457$	−11152.6	−1.14157	−0.570784	−	0.821100i	$-0.693361\pi$
$457$	−11152.6	−1.14157	−0.570784	+	0.821100i	$0.693361\pi$
$458$	−6021.43	−0.614329
$459$	0	0
$460$	0	0
$461$	−9082.82	−0.917633	−0.458817	−	0.888531i	$-0.651727\pi$
$461$	−9082.82	−0.917633	−0.458817	+	0.888531i	$0.651727\pi$
$462$	0	0
$463$	10646.9	1.06869	0.534345	−	0.845266i	$-0.320558\pi$
$463$	10646.9	1.06869	0.534345	+	0.845266i	$0.320558\pi$
$464$	−8441.89	−0.844623
$465$	0	0
$466$	17630.2	1.75258
$467$	14995.1	1.48585	0.742924	−	0.669376i	$-0.233438\pi$
$467$	14995.1	1.48585	0.742924	+	0.669376i	$0.233438\pi$
$468$	0	0
$469$	11463.9	1.12869
$470$	0	0
$471$	0	0
$472$	427.988	0.0417367
$473$	2878.28	0.279796
$474$	0	0
$475$	0	0
$476$	−699.360	−0.0673427
$477$	0	0
$478$	20871.8	1.99718
$479$	−10561.7	−1.00747	−0.503734	−	0.863859i	$-0.668041\pi$
$479$	−10561.7	−1.00747	−0.503734	+	0.863859i	$0.668041\pi$
$480$	0	0
$481$	31499.7	2.98600
$482$	−1099.92	−0.103942
$483$	0	0
$484$	915.034	0.0859349
$485$	0	0
$486$	0	0
$487$	−5363.60	−0.499072	−0.249536	−	0.968366i	$-0.580278\pi$
$487$	−5363.60	−0.499072	−0.249536	+	0.968366i	$0.580278\pi$
$488$	−1337.58	−0.124077
$489$	0	0
$490$	0	0
$491$	−5816.05	−0.534571	−0.267286	−	0.963617i	$-0.586127\pi$
$491$	−5816.05	−0.534571	−0.267286	+	0.963617i	$0.586127\pi$
$492$	0	0
$493$	695.195	0.0635091
$494$	−7696.66	−0.700990
$495$	0	0
$496$	9397.74	0.850748
$497$	15720.5	1.41883
$498$	0	0
$499$	−3222.56	−0.289102	−0.144551	−	0.989497i	$-0.546174\pi$
$499$	−3222.56	−0.289102	−0.144551	+	0.989497i	$0.546174\pi$
$500$	0	0
$501$	0	0
$502$	8808.84	0.783183
$503$	20819.7	1.84554	0.922770	−	0.385352i	$-0.125920\pi$
$503$	20819.7	1.84554	0.922770	+	0.385352i	$0.125920\pi$
$504$	0	0
$505$	0	0
$506$	5249.88	0.461236
$507$	0	0
$508$	2121.07	0.185251
$509$	15456.1	1.34593	0.672964	−	0.739675i	$-0.265021\pi$
$509$	15456.1	1.34593	0.672964	+	0.739675i	$0.265021\pi$
$510$	0	0
$511$	10936.9	0.946814
$512$	−16013.3	−1.38222
$513$	0	0
$514$	−9108.91	−0.781667
$515$	0	0
$516$	0	0
$517$	1718.31	0.146173
$518$	25944.7	2.20067
$519$	0	0
$520$	0	0
$521$	1637.23	0.137674	0.0688370	−	0.997628i	$-0.478071\pi$
$521$	1637.23	0.137674	0.0688370	+	0.997628i	$0.478071\pi$
$522$	0	0
$523$	−22820.6	−1.90798	−0.953990	−	0.299839i	$-0.903067\pi$
$523$	−22820.6	−1.90798	−0.953990	+	0.299839i	$0.903067\pi$
$524$	−6312.99	−0.526306
$525$	0	0
$526$	−3350.76	−0.277757
$527$	−773.910	−0.0639697
$528$	0	0
$529$	2469.59	0.202975
$530$	0	0
$531$	0	0
$532$	−3080.51	−0.251047
$533$	25155.6	2.04429
$534$	0	0
$535$	0	0
$536$	−1186.53	−0.0956159
$537$	0	0
$538$	1348.31	0.108048
$539$	−711.075	−0.0568240
$540$	0	0
$541$	−6722.21	−0.534215	−0.267108	−	0.963667i	$-0.586068\pi$
$541$	−6722.21	−0.534215	−0.267108	+	0.963667i	$0.586068\pi$
$542$	−21889.2	−1.73473
$543$	0	0
$544$	1395.28	0.109967
$545$	0	0
$546$	0	0
$547$	−2895.01	−0.226292	−0.113146	−	0.993578i	$-0.536093\pi$
$547$	−2895.01	−0.226292	−0.113146	+	0.993578i	$0.536093\pi$
$548$	−22958.3	−1.78965
$549$	0	0
$550$	0	0
$551$	3062.16	0.236756
$552$	0	0
$553$	−10204.4	−0.784693
$554$	−22965.4	−1.76121
$555$	0	0
$556$	−15228.4	−1.16156
$557$	−10513.1	−0.799736	−0.399868	−	0.916573i	$-0.630944\pi$
$557$	−10513.1	−0.799736	−0.399868	+	0.916573i	$0.630944\pi$
$558$	0	0
$559$	20909.1	1.58204
$560$	0	0
$561$	0	0
$562$	−18971.3	−1.42394
$563$	17562.1	1.31466	0.657332	−	0.753601i	$-0.271685\pi$
$563$	17562.1	1.31466	0.657332	+	0.753601i	$0.271685\pi$
$564$	0	0
$565$	0	0
$566$	13403.8	0.995415
$567$	0	0
$568$	−1627.09	−0.120195
$569$	17126.9	1.26186	0.630929	−	0.775840i	$-0.282674\pi$
$569$	17126.9	1.26186	0.630929	+	0.775840i	$0.282674\pi$
$570$	0	0
$571$	10083.4	0.739012	0.369506	−	0.929228i	$-0.379527\pi$
$571$	10083.4	0.739012	0.369506	+	0.929228i	$0.379527\pi$
$572$	6647.23	0.485899
$573$	0	0
$574$	20719.3	1.50663
$575$	0	0
$576$	0	0
$577$	−10866.9	−0.784046	−0.392023	−	0.919955i	$-0.628225\pi$
$577$	−10866.9	−0.784046	−0.392023	+	0.919955i	$0.628225\pi$
$578$	19260.1	1.38601
$579$	0	0
$580$	0	0
$581$	−20415.7	−1.45781
$582$	0	0
$583$	5388.83	0.382817
$584$	−1131.99	−0.0802087
$585$	0	0
$586$	11901.5	0.838986
$587$	1901.42	0.133697	0.0668484	−	0.997763i	$-0.478706\pi$
$587$	1901.42	0.133697	0.0668484	+	0.997763i	$0.478706\pi$
$588$	0	0
$589$	−3408.88	−0.238473
$590$	0	0
$591$	0	0
$592$	−26533.4	−1.84209
$593$	−13835.1	−0.958074	−0.479037	−	0.877795i	$-0.659014\pi$
$593$	−13835.1	−0.958074	−0.479037	+	0.877795i	$0.659014\pi$
$594$	0	0
$595$	0	0
$596$	−1023.26	−0.0703261
$597$	0	0
$598$	38137.5	2.60796
$599$	6975.10	0.475785	0.237892	−	0.971292i	$-0.423543\pi$
$599$	6975.10	0.475785	0.237892	+	0.971292i	$0.423543\pi$
$600$	0	0
$601$	15359.8	1.04249	0.521247	−	0.853406i	$-0.325467\pi$
$601$	15359.8	1.04249	0.521247	+	0.853406i	$0.325467\pi$
$602$	17221.8	1.16596
$603$	0	0
$604$	−8582.23	−0.578156
$605$	0	0
$606$	0	0
$607$	−13472.7	−0.900893	−0.450446	−	0.892804i	$-0.648735\pi$
$607$	−13472.7	−0.900893	−0.450446	+	0.892804i	$0.648735\pi$
$608$	6145.88	0.409948
$609$	0	0
$610$	0	0
$611$	12482.6	0.826500
$612$	0	0
$613$	−25040.1	−1.64985	−0.824925	−	0.565242i	$-0.808783\pi$
$613$	−25040.1	−1.64985	−0.824925	+	0.565242i	$0.808783\pi$
$614$	−9978.14	−0.655839
$615$	0	0
$616$	−316.913	−0.0207285
$617$	−15631.7	−1.01995	−0.509973	−	0.860190i	$-0.670345\pi$
$617$	−15631.7	−1.01995	−0.509973	+	0.860190i	$0.670345\pi$
$618$	0	0
$619$	25770.6	1.67335	0.836677	−	0.547697i	$-0.184495\pi$
$619$	25770.6	1.67335	0.836677	+	0.547697i	$0.184495\pi$
$620$	0	0
$621$	0	0
$622$	31058.5	2.00214
$623$	−345.758	−0.0222352
$624$	0	0
$625$	0	0
$626$	−7027.50	−0.448683
$627$	0	0
$628$	199.030	0.0126468
$629$	2185.04	0.138511
$630$	0	0
$631$	6300.88	0.397518	0.198759	−	0.980048i	$-0.436309\pi$
$631$	6300.88	0.397518	0.198759	+	0.980048i	$0.436309\pi$
$632$	1056.17	0.0664747
$633$	0	0
$634$	−25603.1	−1.60383
$635$	0	0
$636$	0	0
$637$	−5165.57	−0.321299
$638$	−5442.37	−0.337720
$639$	0	0
$640$	0	0
$641$	19439.9	1.19786	0.598930	−	0.800801i	$-0.295593\pi$
$641$	19439.9	1.19786	0.598930	+	0.800801i	$0.295593\pi$
$642$	0	0
$643$	31093.7	1.90702	0.953511	−	0.301358i	$-0.0974400\pi$
$643$	31093.7	1.90702	0.953511	+	0.301358i	$0.0974400\pi$
$644$	15264.2	0.933994
$645$	0	0
$646$	−533.893	−0.0325166
$647$	17299.7	1.05119	0.525595	−	0.850735i	$-0.323843\pi$
$647$	17299.7	1.05119	0.525595	+	0.850735i	$0.323843\pi$
$648$	0	0
$649$	2726.33	0.164896
$650$	0	0
$651$	0	0
$652$	−16021.5	−0.962349
$653$	4012.58	0.240466	0.120233	−	0.992746i	$-0.461636\pi$
$653$	4012.58	0.240466	0.120233	+	0.992746i	$0.461636\pi$
$654$	0	0
$655$	0	0
$656$	−21189.5	−1.26114
$657$	0	0
$658$	10281.3	0.609127
$659$	−16636.8	−0.983428	−0.491714	−	0.870757i	$-0.663629\pi$
$659$	−16636.8	−0.983428	−0.491714	+	0.870757i	$0.663629\pi$
$660$	0	0
$661$	−15785.0	−0.928841	−0.464421	−	0.885615i	$-0.653737\pi$
$661$	−15785.0	−0.928841	−0.464421	+	0.885615i	$0.653737\pi$
$662$	−10668.4	−0.626340
$663$	0	0
$664$	2113.05	0.123497
$665$	0	0
$666$	0	0
$667$	−15173.2	−0.880825
$668$	−22136.6	−1.28217
$669$	0	0
$670$	0	0
$671$	−8520.56	−0.490212
$672$	0	0
$673$	27335.7	1.56570	0.782848	−	0.622213i	$-0.213766\pi$
$673$	27335.7	1.56570	0.782848	+	0.622213i	$0.213766\pi$
$674$	−6945.74	−0.396943
$675$	0	0
$676$	31674.2	1.80213
$677$	21712.6	1.23262	0.616308	−	0.787505i	$-0.288628\pi$
$677$	21712.6	1.23262	0.616308	+	0.787505i	$0.288628\pi$
$678$	0	0
$679$	−22052.6	−1.24639
$680$	0	0
$681$	0	0
$682$	6058.59	0.340169
$683$	−10884.7	−0.609800	−0.304900	−	0.952384i	$-0.598623\pi$
$683$	−10884.7	−0.609800	−0.304900	+	0.952384i	$0.598623\pi$
$684$	0	0
$685$	0	0
$686$	−26829.8	−1.49325
$687$	0	0
$688$	−17612.5	−0.975976
$689$	39146.9	2.16456
$690$	0	0
$691$	−1686.89	−0.0928688	−0.0464344	−	0.998921i	$-0.514786\pi$
$691$	−1686.89	−0.0928688	−0.0464344	+	0.998921i	$0.514786\pi$
$692$	19621.9	1.07791
$693$	0	0
$694$	11925.0	0.652259
$695$	0	0
$696$	0	0
$697$	1744.96	0.0948282
$698$	−38721.2	−2.09974
$699$	0	0
$700$	0	0
$701$	382.044	0.0205843	0.0102922	−	0.999947i	$-0.496724\pi$
$701$	382.044	0.0205843	0.0102922	+	0.999947i	$0.496724\pi$
$702$	0	0
$703$	9624.57	0.516355
$704$	−4999.73	−0.267663
$705$	0	0
$706$	10622.2	0.566249
$707$	−10521.2	−0.559678
$708$	0	0
$709$	33000.8	1.74805	0.874027	−	0.485878i	$-0.161500\pi$
$709$	33000.8	1.74805	0.874027	+	0.485878i	$0.161500\pi$
$710$	0	0
$711$	0	0
$712$	35.7863	0.00188364
$713$	16891.3	0.887213
$714$	0	0
$715$	0	0
$716$	−6563.73	−0.342595
$717$	0	0
$718$	5095.23	0.264836
$719$	25624.9	1.32913	0.664567	−	0.747229i	$-0.268616\pi$
$719$	25624.9	1.32913	0.664567	+	0.747229i	$0.268616\pi$
$720$	0	0
$721$	21852.8	1.12877
$722$	24706.4	1.27352
$723$	0	0
$724$	−9058.60	−0.465000
$725$	0	0
$726$	0	0
$727$	25412.1	1.29640	0.648200	−	0.761470i	$-0.275522\pi$
$727$	25412.1	1.29640	0.648200	+	0.761470i	$0.275522\pi$
$728$	−2302.20	−0.117205
$729$	0	0
$730$	0	0
$731$	1450.40	0.0733859
$732$	0	0
$733$	9796.96	0.493668	0.246834	−	0.969058i	$-0.420610\pi$
$733$	9796.96	0.493668	0.246834	+	0.969058i	$0.420610\pi$
$734$	−18855.5	−0.948187
$735$	0	0
$736$	−30453.3	−1.52517
$737$	−7558.30	−0.377766
$738$	0	0
$739$	25193.8	1.25409	0.627043	−	0.778984i	$-0.284265\pi$
$739$	25193.8	1.25409	0.627043	+	0.778984i	$0.284265\pi$
$740$	0	0
$741$	0	0
$742$	32243.3	1.59527
$743$	−1834.61	−0.0905859	−0.0452930	−	0.998974i	$-0.514422\pi$
$743$	−1834.61	−0.0905859	−0.0452930	+	0.998974i	$0.514422\pi$
$744$	0	0
$745$	0	0
$746$	4671.44	0.229268
$747$	0	0
$748$	461.097	0.0225393
$749$	−32817.0	−1.60095
$750$	0	0
$751$	29768.3	1.44642	0.723209	−	0.690629i	$-0.242666\pi$
$751$	29768.3	1.44642	0.723209	+	0.690629i	$0.242666\pi$
$752$	−10514.5	−0.509875
$753$	0	0
$754$	−39535.8	−1.90956
$755$	0	0
$756$	0	0
$757$	30022.9	1.44148	0.720739	−	0.693206i	$-0.243803\pi$
$757$	30022.9	1.44148	0.720739	+	0.693206i	$0.243803\pi$
$758$	−38356.0	−1.83793
$759$	0	0
$760$	0	0
$761$	24166.6	1.15117	0.575583	−	0.817743i	$-0.304775\pi$
$761$	24166.6	1.15117	0.575583	+	0.817743i	$0.304775\pi$
$762$	0	0
$763$	10274.7	0.487511
$764$	18236.9	0.863599
$765$	0	0
$766$	−32304.7	−1.52378
$767$	19805.3	0.932370
$768$	0	0
$769$	15459.0	0.724922	0.362461	−	0.931999i	$-0.381937\pi$
$769$	15459.0	0.724922	0.362461	+	0.931999i	$0.381937\pi$
$770$	0	0
$771$	0	0
$772$	−20450.1	−0.953389
$773$	−4725.65	−0.219884	−0.109942	−	0.993938i	$-0.535066\pi$
$773$	−4725.65	−0.219884	−0.109942	+	0.993938i	$0.535066\pi$
$774$	0	0
$775$	0	0
$776$	2282.47	0.105587
$777$	0	0
$778$	4546.51	0.209512
$779$	7686.14	0.353510
$780$	0	0
$781$	−10364.7	−0.474877
$782$	2645.48	0.120975
$783$	0	0
$784$	4351.15	0.198212
$785$	0	0
$786$	0	0
$787$	18551.1	0.840249	0.420124	−	0.907467i	$-0.361986\pi$
$787$	18551.1	0.840249	0.420124	+	0.907467i	$0.361986\pi$
$788$	33542.6	1.51638
$789$	0	0
$790$	0	0
$791$	−27896.8	−1.25398
$792$	0	0
$793$	−61897.2	−2.77180
$794$	20495.4	0.916065
$795$	0	0
$796$	−3072.14	−0.136795
$797$	−29553.3	−1.31347	−0.656734	−	0.754123i	$-0.728062\pi$
$797$	−29553.3	−1.31347	−0.656734	+	0.754123i	$0.728062\pi$
$798$	0	0
$799$	865.878	0.0383386
$800$	0	0
$801$	0	0
$802$	−58343.5	−2.56880
$803$	−7210.87	−0.316894
$804$	0	0
$805$	0	0
$806$	44012.3	1.92341
$807$	0	0
$808$	1088.96	0.0474127
$809$	6255.09	0.271838	0.135919	−	0.990720i	$-0.456601\pi$
$809$	6255.09	0.271838	0.135919	+	0.990720i	$0.456601\pi$
$810$	0	0
$811$	−6611.89	−0.286282	−0.143141	−	0.989702i	$-0.545720\pi$
$811$	−6611.89	−0.286282	−0.143141	+	0.989702i	$0.545720\pi$
$812$	−15823.8	−0.683876
$813$	0	0
$814$	−17105.7	−0.736553
$815$	0	0
$816$	0	0
$817$	6388.67	0.273576
$818$	−7300.82	−0.312063
$819$	0	0
$820$	0	0
$821$	4082.86	0.173560	0.0867800	−	0.996227i	$-0.472342\pi$
$821$	4082.86	0.173560	0.0867800	+	0.996227i	$0.472342\pi$
$822$	0	0
$823$	−29714.5	−1.25855	−0.629273	−	0.777184i	$-0.716647\pi$
$823$	−29714.5	−1.25855	−0.629273	+	0.777184i	$0.716647\pi$
$824$	−2261.79	−0.0956226
$825$	0	0
$826$	16312.6	0.687152
$827$	−35650.7	−1.49903	−0.749514	−	0.661989i	$-0.769713\pi$
$827$	−35650.7	−1.49903	−0.749514	+	0.661989i	$0.769713\pi$
$828$	0	0
$829$	9713.38	0.406948	0.203474	−	0.979080i	$-0.434777\pi$
$829$	9713.38	0.406948	0.203474	+	0.979080i	$0.434777\pi$
$830$	0	0
$831$	0	0
$832$	−36320.3	−1.51344
$833$	−358.320	−0.0149040
$834$	0	0
$835$	0	0
$836$	2031.02	0.0840244
$837$	0	0
$838$	−16786.9	−0.691997
$839$	−32124.1	−1.32187	−0.660934	−	0.750444i	$-0.729840\pi$
$839$	−32124.1	−1.32187	−0.660934	+	0.750444i	$0.729840\pi$
$840$	0	0
$841$	−8659.42	−0.355054
$842$	−36050.1	−1.47550
$843$	0	0
$844$	−28478.9	−1.16147
$845$	0	0
$846$	0	0
$847$	−2018.77	−0.0818957
$848$	−32974.9	−1.33533
$849$	0	0
$850$	0	0
$851$	−47690.4	−1.92104
$852$	0	0
$853$	15453.1	0.620286	0.310143	−	0.950690i	$-0.399623\pi$
$853$	15453.1	0.620286	0.310143	+	0.950690i	$0.399623\pi$
$854$	−50981.5	−2.04280
$855$	0	0
$856$	3396.60	0.135623
$857$	17782.4	0.708792	0.354396	−	0.935096i	$-0.384687\pi$
$857$	17782.4	0.708792	0.354396	+	0.935096i	$0.384687\pi$
$858$	0	0
$859$	6582.50	0.261457	0.130729	−	0.991418i	$-0.458268\pi$
$859$	6582.50	0.261457	0.130729	+	0.991418i	$0.458268\pi$
$860$	0	0
$861$	0	0
$862$	−23769.3	−0.939195
$863$	25883.0	1.02094	0.510469	−	0.859896i	$-0.329472\pi$
$863$	25883.0	1.02094	0.510469	+	0.859896i	$0.329472\pi$
$864$	0	0
$865$	0	0
$866$	−5353.84	−0.210082
$867$	0	0
$868$	17615.5	0.688836
$869$	6727.89	0.262633
$870$	0	0
$871$	−54907.0	−2.13599
$872$	−1063.45	−0.0412991
$873$	0	0
$874$	11652.7	0.450982
$875$	0	0
$876$	0	0
$877$	−9905.59	−0.381400	−0.190700	−	0.981648i	$-0.561076\pi$
$877$	−9905.59	−0.381400	−0.190700	+	0.981648i	$0.561076\pi$
$878$	−20228.7	−0.777548
$879$	0	0
$880$	0	0
$881$	3211.27	0.122804	0.0614021	−	0.998113i	$-0.480443\pi$
$881$	3211.27	0.122804	0.0614021	+	0.998113i	$0.480443\pi$
$882$	0	0
$883$	−38119.8	−1.45281	−0.726407	−	0.687265i	$-0.758811\pi$
$883$	−38119.8	−1.45281	−0.726407	+	0.687265i	$0.758811\pi$
$884$	3349.62	0.127443
$885$	0	0
$886$	2902.36	0.110053
$887$	12463.1	0.471780	0.235890	−	0.971780i	$-0.424199\pi$
$887$	12463.1	0.471780	0.235890	+	0.971780i	$0.424199\pi$
$888$	0	0
$889$	−4679.56	−0.176544
$890$	0	0
$891$	0	0
$892$	10020.7	0.376142
$893$	3813.98	0.142923
$894$	0	0
$895$	0	0
$896$	3682.19	0.137292
$897$	0	0
$898$	−57803.6	−2.14803
$899$	−17510.6	−0.649623
$900$	0	0
$901$	2715.50	0.100407
$902$	−13660.5	−0.504264
$903$	0	0
$904$	2887.35	0.106230
$905$	0	0
$906$	0	0
$907$	47187.2	1.72748	0.863741	−	0.503936i	$-0.168115\pi$
$907$	47187.2	1.72748	0.863741	+	0.503936i	$0.168115\pi$
$908$	12211.1	0.446299
$909$	0	0
$910$	0	0
$911$	−7401.01	−0.269162	−0.134581	−	0.990903i	$-0.542969\pi$
$911$	−7401.01	−0.269162	−0.134581	+	0.990903i	$0.542969\pi$
$912$	0	0
$913$	13460.3	0.487921
$914$	43995.9	1.59218
$915$	0	0
$916$	11542.9	0.416363
$917$	13927.8	0.501568
$918$	0	0
$919$	25127.2	0.901927	0.450964	−	0.892542i	$-0.351080\pi$
$919$	25127.2	0.901927	0.450964	+	0.892542i	$0.351080\pi$
$920$	0	0
$921$	0	0
$922$	35830.8	1.27985
$923$	−75294.0	−2.68508
$924$	0	0
$925$	0	0
$926$	−42001.0	−1.49054
$927$	0	0
$928$	31569.9	1.11674
$929$	27340.9	0.965583	0.482791	−	0.875735i	$-0.339623\pi$
$929$	27340.9	0.965583	0.482791	+	0.875735i	$0.339623\pi$
$930$	0	0
$931$	−1578.31	−0.0555607
$932$	−33796.6	−1.18782
$933$	0	0
$934$	−59154.3	−2.07236
$935$	0	0
$936$	0	0
$937$	53125.7	1.85223	0.926116	−	0.377239i	$-0.123127\pi$
$937$	53125.7	1.85223	0.926116	+	0.377239i	$0.123127\pi$
$938$	−45224.0	−1.57422
$939$	0	0
$940$	0	0
$941$	44737.6	1.54984	0.774922	−	0.632057i	$-0.217789\pi$
$941$	44737.6	1.54984	0.774922	+	0.632057i	$0.217789\pi$
$942$	0	0
$943$	−38085.4	−1.31520
$944$	−16682.7	−0.575187
$945$	0	0
$946$	−11354.5	−0.390241
$947$	−40351.5	−1.38463	−0.692317	−	0.721594i	$-0.743410\pi$
$947$	−40351.5	−1.38463	−0.692317	+	0.721594i	$0.743410\pi$
$948$	0	0
$949$	−52383.1	−1.79181
$950$	0	0
$951$	0	0
$952$	−159.696	−0.00543675
$953$	33025.8	1.12257	0.561285	−	0.827622i	$-0.310307\pi$
$953$	33025.8	1.12257	0.561285	+	0.827622i	$0.310307\pi$
$954$	0	0
$955$	0	0
$956$	−40010.6	−1.35359
$957$	0	0
$958$	41665.0	1.40515
$959$	50651.1	1.70554
$960$	0	0
$961$	−10297.7	−0.345666
$962$	−124263.	−4.16467
$963$	0	0
$964$	2108.51	0.0704465
$965$	0	0
$966$	0	0
$967$	−13842.5	−0.460337	−0.230168	−	0.973151i	$-0.573928\pi$
$967$	−13842.5	−0.460337	−0.230168	+	0.973151i	$0.573928\pi$
$968$	208.945	0.00693774
$969$	0	0
$970$	0	0
$971$	8077.80	0.266971	0.133486	−	0.991051i	$-0.457383\pi$
$971$	8077.80	0.266971	0.133486	+	0.991051i	$0.457383\pi$
$972$	0	0
$973$	33597.2	1.10696
$974$	21158.9	0.696072
$975$	0	0
$976$	52138.3	1.70994
$977$	−32544.4	−1.06570	−0.532850	−	0.846210i	$-0.678879\pi$
$977$	−32544.4	−1.06570	−0.532850	+	0.846210i	$0.678879\pi$
$978$	0	0
$979$	227.963	0.00744200
$980$	0	0
$981$	0	0
$982$	22943.7	0.745585
$983$	−45044.9	−1.46156	−0.730779	−	0.682615i	$-0.760843\pi$
$983$	−45044.9	−1.46156	−0.730779	+	0.682615i	$0.760843\pi$
$984$	0	0
$985$	0	0
$986$	−2742.48	−0.0885784
$987$	0	0
$988$	14754.3	0.475097
$989$	−31656.3	−1.01781
$990$	0	0
$991$	29561.1	0.947568	0.473784	−	0.880641i	$-0.342888\pi$
$991$	29561.1	0.947568	0.473784	+	0.880641i	$0.342888\pi$
$992$	−35144.4	−1.12484
$993$	0	0
$994$	−62015.7	−1.97889
$995$	0	0
$996$	0	0
$997$	−17165.6	−0.545276	−0.272638	−	0.962117i	$-0.587896\pi$
$997$	−17165.6	−0.545276	−0.272638	+	0.962117i	$0.587896\pi$
$998$	12712.7	0.403220
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2475.4.a.by.1.2	yes	10
3.2	odd	2		2475.4.a.bv.1.9	yes	10
5.4	even	2		2475.4.a.bu.1.9	✓	10
15.14	odd	2		2475.4.a.bx.1.2	yes	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2475.4.a.bu.1.9	✓	10	5.4	even	2
2475.4.a.bv.1.9	yes	10	3.2	odd	2
2475.4.a.bx.1.2	yes	10	15.14	odd	2
2475.4.a.by.1.2	yes	10	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2475 = 3^{2} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2475.a (trivial)

\(f(q)\)	\(=\)	\(q-3.94490 q^{2} +7.56227 q^{4} -16.6840 q^{7} +1.72681 q^{8} +O(q^{10})\)	\(q-3.94490 q^{2} +7.56227 q^{4} -16.6840 q^{7} +1.72681 q^{8} +11.0000 q^{11} +79.9090 q^{13} +65.8169 q^{14} -67.3103 q^{16} +5.54304 q^{17} +24.4157 q^{19} -43.3939 q^{22} -120.982 q^{23} -315.233 q^{26} -126.169 q^{28} +125.418 q^{29} -139.618 q^{31} +251.718 q^{32} -21.8668 q^{34} +394.195 q^{37} -96.3178 q^{38} +314.803 q^{41} +261.662 q^{43} +83.1849 q^{44} +477.262 q^{46} +156.210 q^{47} -64.6432 q^{49} +604.293 q^{52} +489.894 q^{53} -28.8102 q^{56} -494.760 q^{58} +247.848 q^{59} -774.596 q^{61} +550.781 q^{62} -454.521 q^{64} -687.119 q^{67} +41.9179 q^{68} -942.247 q^{71} -655.534 q^{73} -1555.06 q^{74} +184.638 q^{76} -183.524 q^{77} +611.627 q^{79} -1241.87 q^{82} +1223.67 q^{83} -1032.23 q^{86} +18.9950 q^{88} +20.7239 q^{89} -1333.20 q^{91} -914.897 q^{92} -616.234 q^{94} +1321.78 q^{97} +255.011 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 4 q^{2} + 34 q^{4} + 2 q^{7} + 48 q^{8}+O(q^{10}) \) 10 * q + 4 * q^2 + 34 * q^4 + 2 * q^7 + 48 * q^8	\( 10 q + 4 q^{2} + 34 q^{4} + 2 q^{7} + 48 q^{8} + 110 q^{11} + 26 q^{13} - 72 q^{14} + 206 q^{16} + 148 q^{17} - 114 q^{19} + 44 q^{22} - 34 q^{23} - 100 q^{26} - 86 q^{28} + 38 q^{29} + 232 q^{31} + 448 q^{32} - 20 q^{34} + 754 q^{37} + 780 q^{38} - 160 q^{41} - 66 q^{43} + 374 q^{44} + 682 q^{46} + 450 q^{47} + 590 q^{49} + 200 q^{52} + 1068 q^{53} - 268 q^{56} - 138 q^{58} + 838 q^{59} - 566 q^{61} + 1230 q^{62} + 462 q^{64} + 430 q^{67} + 2234 q^{68} - 518 q^{71} - 184 q^{73} - 402 q^{74} + 386 q^{76} + 22 q^{77} + 956 q^{79} - 2180 q^{82} + 2094 q^{83} - 892 q^{86} + 528 q^{88} + 512 q^{89} - 858 q^{91} + 4476 q^{92} - 294 q^{94} - 1006 q^{97} + 2226 q^{98}+O(q^{100}) \) 10 * q + 4 * q^2 + 34 * q^4 + 2 * q^7 + 48 * q^8 + 110 * q^11 + 26 * q^13 - 72 * q^14 + 206 * q^16 + 148 * q^17 - 114 * q^19 + 44 * q^22 - 34 * q^23 - 100 * q^26 - 86 * q^28 + 38 * q^29 + 232 * q^31 + 448 * q^32 - 20 * q^34 + 754 * q^37 + 780 * q^38 - 160 * q^41 - 66 * q^43 + 374 * q^44 + 682 * q^46 + 450 * q^47 + 590 * q^49 + 200 * q^52 + 1068 * q^53 - 268 * q^56 - 138 * q^58 + 838 * q^59 - 566 * q^61 + 1230 * q^62 + 462 * q^64 + 430 * q^67 + 2234 * q^68 - 518 * q^71 - 184 * q^73 - 402 * q^74 + 386 * q^76 + 22 * q^77 + 956 * q^79 - 2180 * q^82 + 2094 * q^83 - 892 * q^86 + 528 * q^88 + 512 * q^89 - 858 * q^91 + 4476 * q^92 - 294 * q^94 - 1006 * q^97 + 2226 * q^98

Introduction

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