LMFDB - Embedded newform 2475.4.a.bt.1.7

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:	$7$
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} - 41x^{5} + 40x^{4} + 424x^{3} - 168x^{2} - 1042x - 388 $ x^7 - 2x^6 - 41x^5 + 40x^4 + 424x^3 - 168x^2 - 1042x - 388
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 495)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$-4.54090$ 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)

$f(q)$	$=$	$q+5.54090 q^{2} +22.7016 q^{4} -17.7319 q^{7} +81.4600 q^{8} +O(q^{10})$	$q+5.54090 q^{2} +22.7016 q^{4} -17.7319 q^{7} +81.4600 q^{8} -11.0000 q^{11} +53.5704 q^{13} -98.2508 q^{14} +269.749 q^{16} -112.431 q^{17} -1.24467 q^{19} -60.9499 q^{22} +78.4188 q^{23} +296.828 q^{26} -402.543 q^{28} +174.127 q^{29} +82.5240 q^{31} +842.973 q^{32} -622.968 q^{34} +149.081 q^{37} -6.89659 q^{38} +414.603 q^{41} +182.535 q^{43} -249.717 q^{44} +434.511 q^{46} +438.738 q^{47} -28.5788 q^{49} +1216.13 q^{52} +490.962 q^{53} -1444.44 q^{56} +964.819 q^{58} -6.02762 q^{59} +434.789 q^{61} +457.257 q^{62} +2512.84 q^{64} -935.150 q^{67} -2552.36 q^{68} -510.132 q^{71} +1045.70 q^{73} +826.042 q^{74} -28.2560 q^{76} +195.051 q^{77} +226.439 q^{79} +2297.27 q^{82} -1185.72 q^{83} +1011.41 q^{86} -896.060 q^{88} +1443.15 q^{89} -949.907 q^{91} +1780.23 q^{92} +2431.01 q^{94} -825.703 q^{97} -158.352 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 5 q^{2} + 33 q^{4} - 30 q^{7} + 45 q^{8}+O(q^{10}) $ 7 * q + 5 * q^2 + 33 * q^4 - 30 * q^7 + 45 * q^8	$ 7 q + 5 q^{2} + 33 q^{4} - 30 q^{7} + 45 q^{8} - 77 q^{11} - 38 q^{13} - 20 q^{14} + 309 q^{16} + 12 q^{17} + 226 q^{19} - 55 q^{22} + 334 q^{23} + 372 q^{26} - 812 q^{28} + 258 q^{29} + 336 q^{31} + 485 q^{32} + 78 q^{34} - 466 q^{37} - 494 q^{38} + 258 q^{41} - 308 q^{43} - 363 q^{44} + 98 q^{46} + 546 q^{47} + 735 q^{49} - 512 q^{52} + 110 q^{53} - 20 q^{56} - 1362 q^{58} + 68 q^{59} + 1096 q^{61} + 356 q^{62} + 2761 q^{64} - 2268 q^{67} - 1186 q^{68} + 166 q^{71} - 200 q^{73} + 1710 q^{74} + 3310 q^{76} + 330 q^{77} + 2152 q^{79} + 1006 q^{82} + 370 q^{83} - 106 q^{86} - 495 q^{88} + 252 q^{89} + 2768 q^{91} + 3774 q^{92} + 2218 q^{94} - 3698 q^{97} + 697 q^{98}+O(q^{100}) $ 7 * q + 5 * q^2 + 33 * q^4 - 30 * q^7 + 45 * q^8 - 77 * q^11 - 38 * q^13 - 20 * q^14 + 309 * q^16 + 12 * q^17 + 226 * q^19 - 55 * q^22 + 334 * q^23 + 372 * q^26 - 812 * q^28 + 258 * q^29 + 336 * q^31 + 485 * q^32 + 78 * q^34 - 466 * q^37 - 494 * q^38 + 258 * q^41 - 308 * q^43 - 363 * q^44 + 98 * q^46 + 546 * q^47 + 735 * q^49 - 512 * q^52 + 110 * q^53 - 20 * q^56 - 1362 * q^58 + 68 * q^59 + 1096 * q^61 + 356 * q^62 + 2761 * q^64 - 2268 * q^67 - 1186 * q^68 + 166 * q^71 - 200 * q^73 + 1710 * q^74 + 3310 * q^76 + 330 * q^77 + 2152 * q^79 + 1006 * q^82 + 370 * q^83 - 106 * q^86 - 495 * q^88 + 252 * q^89 + 2768 * q^91 + 3774 * q^92 + 2218 * q^94 - 3698 * q^97 + 697 * 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 $

$2$	5.54090	1.95900	0.979502	−	0.201434i	$-0.0645601\pi$
$2$	5.54090	1.95900	0.979502	+	0.201434i	$0.0645601\pi$
$3$	0	0
$3$	0	0
$4$	22.7016	2.83770
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−17.7319	−0.957434	−0.478717	−	0.877969i	$-0.658898\pi$
$7$	−17.7319	−0.957434	−0.478717	+	0.877969i	$0.658898\pi$
$8$	81.4600	3.60006
$9$	0	0
$10$	0	0
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	0	0
$13$	53.5704	1.14290	0.571452	−	0.820635i	$-0.306380\pi$
$13$	53.5704	1.14290	0.571452	+	0.820635i	$0.306380\pi$
$14$	−98.2508	−1.87562
$15$	0	0
$16$	269.749	4.21483
$17$	−112.431	−1.60403	−0.802014	−	0.597305i	$-0.796238\pi$
$17$	−112.431	−1.60403	−0.802014	+	0.597305i	$0.796238\pi$
$18$	0	0
$19$	−1.24467	−0.0150288	−0.00751439	−	0.999972i	$-0.502392\pi$
$19$	−1.24467	−0.0150288	−0.00751439	+	0.999972i	$0.502392\pi$
$20$	0	0
$21$	0	0
$22$	−60.9499	−0.590662
$23$	78.4188	0.710933	0.355466	−	0.934689i	$-0.384322\pi$
$23$	78.4188	0.710933	0.355466	+	0.934689i	$0.384322\pi$
$24$	0	0
$25$	0	0
$26$	296.828	2.23896
$27$	0	0
$28$	−402.543	−2.71691
$29$	174.127	1.11498	0.557492	−	0.830182i	$-0.311764\pi$
$29$	174.127	1.11498	0.557492	+	0.830182i	$0.311764\pi$
$30$	0	0
$31$	82.5240	0.478121	0.239060	−	0.971005i	$-0.423161\pi$
$31$	82.5240	0.478121	0.239060	+	0.971005i	$0.423161\pi$
$32$	842.973	4.65681
$33$	0	0
$34$	−622.968	−3.14230
$35$	0	0
$36$	0	0
$37$	149.081	0.662398	0.331199	−	0.943561i	$-0.392547\pi$
$37$	149.081	0.662398	0.331199	+	0.943561i	$0.392547\pi$
$38$	−6.89659	−0.0294414
$39$	0	0
$40$	0	0
$41$	414.603	1.57927	0.789635	−	0.613577i	$-0.210270\pi$
$41$	414.603	1.57927	0.789635	+	0.613577i	$0.210270\pi$
$42$	0	0
$43$	182.535	0.647357	0.323679	−	0.946167i	$-0.395080\pi$
$43$	182.535	0.647357	0.323679	+	0.946167i	$0.395080\pi$
$44$	−249.717	−0.855598
$45$	0	0
$46$	434.511	1.39272
$47$	438.738	1.36163	0.680815	−	0.732456i	$-0.261626\pi$
$47$	438.738	1.36163	0.680815	+	0.732456i	$0.261626\pi$
$48$	0	0
$49$	−28.5788	−0.0833201
$50$	0	0
$51$	0	0
$52$	1216.13	3.24322
$53$	490.962	1.27243	0.636216	−	0.771511i	$-0.280499\pi$
$53$	490.962	1.27243	0.636216	+	0.771511i	$0.280499\pi$
$54$	0	0
$55$	0	0
$56$	−1444.44	−3.44682
$57$	0	0
$58$	964.819	2.18426
$59$	−6.02762	−0.0133005	−0.00665025	−	0.999978i	$-0.502117\pi$
$59$	−6.02762	−0.0133005	−0.00665025	+	0.999978i	$0.502117\pi$
$60$	0	0
$61$	434.789	0.912606	0.456303	−	0.889824i	$-0.349173\pi$
$61$	434.789	0.912606	0.456303	+	0.889824i	$0.349173\pi$
$62$	457.257	0.936640
$63$	0	0
$64$	2512.84	4.90789
$65$	0	0
$66$	0	0
$67$	−935.150	−1.70518	−0.852588	−	0.522584i	$-0.824968\pi$
$67$	−935.150	−1.70518	−0.852588	+	0.522584i	$0.824968\pi$
$68$	−2552.36	−4.55175
$69$	0	0
$70$	0	0
$71$	−510.132	−0.852698	−0.426349	−	0.904559i	$-0.640200\pi$
$71$	−510.132	−0.852698	−0.426349	+	0.904559i	$0.640200\pi$
$72$	0	0
$73$	1045.70	1.67658	0.838290	−	0.545225i	$-0.183556\pi$
$73$	1045.70	1.67658	0.838290	+	0.545225i	$0.183556\pi$
$74$	826.042	1.29764
$75$	0	0
$76$	−28.2560	−0.0426471
$77$	195.051	0.288677
$78$	0	0
$79$	226.439	0.322485	0.161243	−	0.986915i	$-0.448450\pi$
$79$	226.439	0.322485	0.161243	+	0.986915i	$0.448450\pi$
$80$	0	0
$81$	0	0
$82$	2297.27	3.09380
$83$	−1185.72	−1.56807	−0.784035	−	0.620716i	$-0.786842\pi$
$83$	−1185.72	−1.56807	−0.784035	+	0.620716i	$0.786842\pi$
$84$	0	0
$85$	0	0
$86$	1011.41	1.26818
$87$	0	0
$88$	−896.060	−1.08546
$89$	1443.15	1.71880	0.859400	−	0.511303i	$-0.170837\pi$
$89$	1443.15	1.71880	0.859400	+	0.511303i	$0.170837\pi$
$90$	0	0
$91$	−949.907	−1.09426
$92$	1780.23	2.01741
$93$	0	0
$94$	2431.01	2.66744
$95$	0	0
$96$	0	0
$97$	−825.703	−0.864303	−0.432152	−	0.901801i	$-0.642245\pi$
$97$	−825.703	−0.864303	−0.432152	+	0.901801i	$0.642245\pi$
$98$	−158.352	−0.163224
$99$	0	0
$100$	0	0
$101$	−18.3996	−0.0181270	−0.00906349	−	0.999959i	$-0.502885\pi$
$101$	−18.3996	−0.0181270	−0.00906349	+	0.999959i	$0.502885\pi$
$102$	0	0
$103$	−955.858	−0.914402	−0.457201	−	0.889363i	$-0.651148\pi$
$103$	−955.858	−0.914402	−0.457201	+	0.889363i	$0.651148\pi$
$104$	4363.85	4.11452
$105$	0	0
$106$	2720.37	2.49270
$107$	−1500.76	−1.35593	−0.677963	−	0.735096i	$-0.737137\pi$
$107$	−1500.76	−1.35593	−0.677963	+	0.735096i	$0.737137\pi$
$108$	0	0
$109$	−727.915	−0.639648	−0.319824	−	0.947477i	$-0.603624\pi$
$109$	−727.915	−0.639648	−0.319824	+	0.947477i	$0.603624\pi$
$110$	0	0
$111$	0	0
$112$	−4783.17	−4.03542
$113$	−1341.55	−1.11684	−0.558419	−	0.829559i	$-0.688592\pi$
$113$	−1341.55	−1.11684	−0.558419	+	0.829559i	$0.688592\pi$
$114$	0	0
$115$	0	0
$116$	3952.95	3.16399
$117$	0	0
$118$	−33.3985	−0.0260557
$119$	1993.62	1.53575
$120$	0	0
$121$	121.000	0.0909091
$122$	2409.12	1.78780
$123$	0	0
$124$	1873.42	1.35676
$125$	0	0
$126$	0	0
$127$	−1685.73	−1.17783	−0.588914	−	0.808196i	$-0.700444\pi$
$127$	−1685.73	−1.17783	−0.588914	+	0.808196i	$0.700444\pi$
$128$	7179.60	4.95776
$129$	0	0
$130$	0	0
$131$	473.852	0.316035	0.158018	−	0.987436i	$-0.449490\pi$
$131$	473.852	0.316035	0.158018	+	0.987436i	$0.449490\pi$
$132$	0	0
$133$	22.0704	0.0143891
$134$	−5181.57	−3.34045
$135$	0	0
$136$	−9158.62	−5.77460
$137$	724.768	0.451979	0.225990	−	0.974130i	$-0.427438\pi$
$137$	724.768	0.451979	0.225990	+	0.974130i	$0.427438\pi$
$138$	0	0
$139$	2396.73	1.46250	0.731251	−	0.682108i	$-0.238937\pi$
$139$	2396.73	1.46250	0.731251	+	0.682108i	$0.238937\pi$
$140$	0	0
$141$	0	0
$142$	−2826.59	−1.67044
$143$	−589.275	−0.344599
$144$	0	0
$145$	0	0
$146$	5794.14	3.28443
$147$	0	0
$148$	3384.37	1.87969
$149$	693.282	0.381180	0.190590	−	0.981670i	$-0.438960\pi$
$149$	693.282	0.381180	0.190590	+	0.981670i	$0.438960\pi$
$150$	0	0
$151$	−2682.28	−1.44557	−0.722785	−	0.691073i	$-0.757138\pi$
$151$	−2682.28	−1.44557	−0.722785	+	0.691073i	$0.757138\pi$
$152$	−101.391	−0.0541045
$153$	0	0
$154$	1080.76	0.565520
$155$	0	0
$156$	0	0
$157$	1209.54	0.614852	0.307426	−	0.951572i	$-0.400532\pi$
$157$	1209.54	0.614852	0.307426	+	0.951572i	$0.400532\pi$
$158$	1254.67	0.631750
$159$	0	0
$160$	0	0
$161$	−1390.52	−0.680671
$162$	0	0
$163$	3429.67	1.64805	0.824026	−	0.566553i	$-0.191723\pi$
$163$	3429.67	1.64805	0.824026	+	0.566553i	$0.191723\pi$
$164$	9412.14	4.48149
$165$	0	0
$166$	−6569.97	−3.07186
$167$	−1523.60	−0.705986	−0.352993	−	0.935626i	$-0.614836\pi$
$167$	−1523.60	−0.705986	−0.352993	+	0.935626i	$0.614836\pi$
$168$	0	0
$169$	672.791	0.306232
$170$	0	0
$171$	0	0
$172$	4143.84	1.83700
$173$	−739.726	−0.325089	−0.162544	−	0.986701i	$-0.551970\pi$
$173$	−739.726	−0.325089	−0.162544	+	0.986701i	$0.551970\pi$
$174$	0	0
$175$	0	0
$176$	−2967.24	−1.27082
$177$	0	0
$178$	7996.33	3.36714
$179$	−2090.21	−0.872791	−0.436395	−	0.899755i	$-0.643745\pi$
$179$	−2090.21	−0.872791	−0.436395	+	0.899755i	$0.643745\pi$
$180$	0	0
$181$	521.612	0.214205	0.107103	−	0.994248i	$-0.465843\pi$
$181$	521.612	0.214205	0.107103	+	0.994248i	$0.465843\pi$
$182$	−5263.34	−2.14365
$183$	0	0
$184$	6388.00	2.55940
$185$	0	0
$186$	0	0
$187$	1236.74	0.483633
$188$	9960.06	3.86389
$189$	0	0
$190$	0	0
$191$	1336.98	0.506494	0.253247	−	0.967402i	$-0.418501\pi$
$191$	1336.98	0.506494	0.253247	+	0.967402i	$0.418501\pi$
$192$	0	0
$193$	−4650.76	−1.73455	−0.867277	−	0.497825i	$-0.834132\pi$
$193$	−4650.76	−1.73455	−0.867277	+	0.497825i	$0.834132\pi$
$194$	−4575.14	−1.69317
$195$	0	0
$196$	−648.784	−0.236437
$197$	−3293.66	−1.19118	−0.595592	−	0.803287i	$-0.703083\pi$
$197$	−3293.66	−1.19118	−0.595592	+	0.803287i	$0.703083\pi$
$198$	0	0
$199$	3613.76	1.28730	0.643650	−	0.765320i	$-0.277419\pi$
$199$	3613.76	1.28730	0.643650	+	0.765320i	$0.277419\pi$
$200$	0	0
$201$	0	0
$202$	−101.950	−0.0355108
$203$	−3087.60	−1.06752
$204$	0	0
$205$	0	0
$206$	−5296.31	−1.79132
$207$	0	0
$208$	14450.6	4.81715
$209$	13.6914	0.00453135
$210$	0	0
$211$	5661.64	1.84722	0.923609	−	0.383335i	$-0.125225\pi$
$211$	5661.64	1.84722	0.923609	+	0.383335i	$0.125225\pi$
$212$	11145.6	3.61078
$213$	0	0
$214$	−8315.57	−2.65626
$215$	0	0
$216$	0	0
$217$	−1463.31	−0.457769
$218$	−4033.31	−1.25307
$219$	0	0
$220$	0	0
$221$	−6022.97	−1.83325
$222$	0	0
$223$	−5866.64	−1.76170	−0.880850	−	0.473396i	$-0.843028\pi$
$223$	−5866.64	−1.76170	−0.880850	+	0.473396i	$0.843028\pi$
$224$	−14947.5	−4.45859
$225$	0	0
$226$	−7433.41	−2.18789
$227$	671.916	0.196461	0.0982304	−	0.995164i	$-0.468682\pi$
$227$	671.916	0.196461	0.0982304	+	0.995164i	$0.468682\pi$
$228$	0	0
$229$	−1265.84	−0.365281	−0.182640	−	0.983180i	$-0.558464\pi$
$229$	−1265.84	−0.365281	−0.182640	+	0.983180i	$0.558464\pi$
$230$	0	0
$231$	0	0
$232$	14184.4	4.01401
$233$	−173.516	−0.0487873	−0.0243936	−	0.999702i	$-0.507766\pi$
$233$	−173.516	−0.0487873	−0.0243936	+	0.999702i	$0.507766\pi$
$234$	0	0
$235$	0	0
$236$	−136.837	−0.0377428
$237$	0	0
$238$	11046.4	3.00854
$239$	2261.08	0.611955	0.305977	−	0.952039i	$-0.401017\pi$
$239$	2261.08	0.611955	0.305977	+	0.952039i	$0.401017\pi$
$240$	0	0
$241$	2069.83	0.553233	0.276617	−	0.960980i	$-0.410787\pi$
$241$	2069.83	0.553233	0.276617	+	0.960980i	$0.410787\pi$
$242$	670.449	0.178091
$243$	0	0
$244$	9870.39	2.58970
$245$	0	0
$246$	0	0
$247$	−66.6775	−0.0171765
$248$	6722.40	1.72126
$249$	0	0
$250$	0	0
$251$	2511.60	0.631597	0.315798	−	0.948826i	$-0.397728\pi$
$251$	2511.60	0.631597	0.315798	+	0.948826i	$0.397728\pi$
$252$	0	0
$253$	−862.607	−0.214354
$254$	−9340.45	−2.30737
$255$	0	0
$256$	19678.8	4.80438
$257$	261.418	0.0634506	0.0317253	−	0.999497i	$-0.489900\pi$
$257$	261.418	0.0634506	0.0317253	+	0.999497i	$0.489900\pi$
$258$	0	0
$259$	−2643.49	−0.634202
$260$	0	0
$261$	0	0
$262$	2625.56	0.619114
$263$	2015.73	0.472605	0.236303	−	0.971680i	$-0.424064\pi$
$263$	2015.73	0.472605	0.236303	+	0.971680i	$0.424064\pi$
$264$	0	0
$265$	0	0
$266$	122.290	0.0281882
$267$	0	0
$268$	−21229.4	−4.83877
$269$	1638.25	0.371324	0.185662	−	0.982614i	$-0.440557\pi$
$269$	1638.25	0.371324	0.185662	+	0.982614i	$0.440557\pi$
$270$	0	0
$271$	2200.32	0.493209	0.246605	−	0.969116i	$-0.420685\pi$
$271$	2200.32	0.493209	0.246605	+	0.969116i	$0.420685\pi$
$272$	−30328.1	−6.76071
$273$	0	0
$274$	4015.87	0.885429
$275$	0	0
$276$	0	0
$277$	−6953.40	−1.50827	−0.754133	−	0.656722i	$-0.771943\pi$
$277$	−6953.40	−1.50827	−0.754133	+	0.656722i	$0.771943\pi$
$278$	13280.0	2.86505
$279$	0	0
$280$	0	0
$281$	6425.61	1.36413	0.682063	−	0.731293i	$-0.261083\pi$
$281$	6425.61	1.36413	0.682063	+	0.731293i	$0.261083\pi$
$282$	0	0
$283$	1799.64	0.378013	0.189006	−	0.981976i	$-0.439473\pi$
$283$	1799.64	0.378013	0.189006	+	0.981976i	$0.439473\pi$
$284$	−11580.8	−2.41970
$285$	0	0
$286$	−3265.11	−0.675071
$287$	−7351.70	−1.51205
$288$	0	0
$289$	7727.70	1.57291
$290$	0	0
$291$	0	0
$292$	23739.1	4.75763
$293$	4715.83	0.940279	0.470140	−	0.882592i	$-0.344204\pi$
$293$	4715.83	0.940279	0.470140	+	0.882592i	$0.344204\pi$
$294$	0	0
$295$	0	0
$296$	12144.1	2.38467
$297$	0	0
$298$	3841.41	0.746734
$299$	4200.93	0.812529
$300$	0	0
$301$	−3236.70	−0.619802
$302$	−14862.3	−2.83188
$303$	0	0
$304$	−335.749	−0.0633438
$305$	0	0
$306$	0	0
$307$	3963.76	0.736885	0.368442	−	0.929651i	$-0.379891\pi$
$307$	3963.76	0.736885	0.368442	+	0.929651i	$0.379891\pi$
$308$	4427.97	0.819179
$309$	0	0
$310$	0	0
$311$	−6788.34	−1.23772	−0.618861	−	0.785501i	$-0.712405\pi$
$311$	−6788.34	−1.23772	−0.618861	+	0.785501i	$0.712405\pi$
$312$	0	0
$313$	−3894.31	−0.703256	−0.351628	−	0.936140i	$-0.614372\pi$
$313$	−3894.31	−0.703256	−0.351628	+	0.936140i	$0.614372\pi$
$314$	6701.94	1.20450
$315$	0	0
$316$	5140.52	0.915116
$317$	7108.82	1.25953	0.629766	−	0.776785i	$-0.283151\pi$
$317$	7108.82	1.25953	0.629766	+	0.776785i	$0.283151\pi$
$318$	0	0
$319$	−1915.39	−0.336180
$320$	0	0
$321$	0	0
$322$	−7704.72	−1.33344
$323$	139.939	0.0241066
$324$	0	0
$325$	0	0
$326$	19003.5	3.22854
$327$	0	0
$328$	33773.5	5.68546
$329$	−7779.68	−1.30367
$330$	0	0
$331$	−7241.67	−1.20253	−0.601266	−	0.799049i	$-0.705337\pi$
$331$	−7241.67	−1.20253	−0.601266	+	0.799049i	$0.705337\pi$
$332$	−26917.8	−4.44971
$333$	0	0
$334$	−8442.11	−1.38303
$335$	0	0
$336$	0	0
$337$	−7982.45	−1.29030	−0.645151	−	0.764055i	$-0.723206\pi$
$337$	−7982.45	−1.29030	−0.645151	+	0.764055i	$0.723206\pi$
$338$	3727.87	0.599909
$339$	0	0
$340$	0	0
$341$	−907.764	−0.144159
$342$	0	0
$343$	6588.81	1.03721
$344$	14869.3	2.33052
$345$	0	0
$346$	−4098.75	−0.636850
$347$	3956.38	0.612074	0.306037	−	0.952020i	$-0.400997\pi$
$347$	3956.38	0.612074	0.306037	+	0.952020i	$0.400997\pi$
$348$	0	0
$349$	−1242.91	−0.190635	−0.0953174	−	0.995447i	$-0.530387\pi$
$349$	−1242.91	−0.190635	−0.0953174	+	0.995447i	$0.530387\pi$
$350$	0	0
$351$	0	0
$352$	−9272.71	−1.40408
$353$	−9549.18	−1.43981	−0.719903	−	0.694075i	$-0.755814\pi$
$353$	−9549.18	−1.43981	−0.719903	+	0.694075i	$0.755814\pi$
$354$	0	0
$355$	0	0
$356$	32761.7	4.87744
$357$	0	0
$358$	−11581.6	−1.70980
$359$	−8171.34	−1.20130	−0.600650	−	0.799512i	$-0.705091\pi$
$359$	−8171.34	−1.20130	−0.600650	+	0.799512i	$0.705091\pi$
$360$	0	0
$361$	−6857.45	−0.999774
$362$	2890.20	0.419629
$363$	0	0
$364$	−21564.4	−3.10517
$365$	0	0
$366$	0	0
$367$	−7101.30	−1.01004	−0.505020	−	0.863107i	$-0.668515\pi$
$367$	−7101.30	−1.01004	−0.505020	+	0.863107i	$0.668515\pi$
$368$	21153.4	2.99646
$369$	0	0
$370$	0	0
$371$	−8705.71	−1.21827
$372$	0	0
$373$	−12150.0	−1.68661	−0.843304	−	0.537437i	$-0.819393\pi$
$373$	−12150.0	−1.68661	−0.843304	+	0.537437i	$0.819393\pi$
$374$	6852.65	0.947439
$375$	0	0
$376$	35739.6	4.90194
$377$	9328.05	1.27432
$378$	0	0
$379$	1034.28	0.140177	0.0700887	−	0.997541i	$-0.477672\pi$
$379$	1034.28	0.140177	0.0700887	+	0.997541i	$0.477672\pi$
$380$	0	0
$381$	0	0
$382$	7408.07	0.992224
$383$	−5508.99	−0.734977	−0.367489	−	0.930028i	$-0.619782\pi$
$383$	−5508.99	−0.734977	−0.367489	+	0.930028i	$0.619782\pi$
$384$	0	0
$385$	0	0
$386$	−25769.4	−3.39800
$387$	0	0
$388$	−18744.8	−2.45263
$389$	−8347.36	−1.08799	−0.543995	−	0.839089i	$-0.683089\pi$
$389$	−8347.36	−1.08799	−0.543995	+	0.839089i	$0.683089\pi$
$390$	0	0
$391$	−8816.70	−1.14036
$392$	−2328.03	−0.299957
$393$	0	0
$394$	−18249.8	−2.33354
$395$	0	0
$396$	0	0
$397$	1233.32	0.155916	0.0779581	−	0.996957i	$-0.475160\pi$
$397$	1233.32	0.155916	0.0779581	+	0.996957i	$0.475160\pi$
$398$	20023.5	2.52182
$399$	0	0
$400$	0	0
$401$	3405.69	0.424120	0.212060	−	0.977257i	$-0.431983\pi$
$401$	3405.69	0.424120	0.212060	+	0.977257i	$0.431983\pi$
$402$	0	0
$403$	4420.84	0.546446
$404$	−417.699	−0.0514389
$405$	0	0
$406$	−17108.1	−2.09128
$407$	−1639.89	−0.199720
$408$	0	0
$409$	−2468.78	−0.298468	−0.149234	−	0.988802i	$-0.547681\pi$
$409$	−2468.78	−0.298468	−0.149234	+	0.988802i	$0.547681\pi$
$410$	0	0
$411$	0	0
$412$	−21699.5	−2.59480
$413$	106.881	0.0127344
$414$	0	0
$415$	0	0
$416$	45158.4	5.32230
$417$	0	0
$418$	75.8625	0.00887693
$419$	−1712.51	−0.199670	−0.0998349	−	0.995004i	$-0.531831\pi$
$419$	−1712.51	−0.199670	−0.0998349	+	0.995004i	$0.531831\pi$
$420$	0	0
$421$	−6612.82	−0.765533	−0.382766	−	0.923845i	$-0.625029\pi$
$421$	−6612.82	−0.765533	−0.382766	+	0.923845i	$0.625029\pi$
$422$	31370.6	3.61871
$423$	0	0
$424$	39993.8	4.58083
$425$	0	0
$426$	0	0
$427$	−7709.64	−0.873760
$428$	−34069.7	−3.84771
$429$	0	0
$430$	0	0
$431$	−15380.0	−1.71886	−0.859432	−	0.511250i	$-0.829183\pi$
$431$	−15380.0	−1.71886	−0.859432	+	0.511250i	$0.829183\pi$
$432$	0	0
$433$	9228.58	1.02424	0.512121	−	0.858913i	$-0.328860\pi$
$433$	9228.58	1.02424	0.512121	+	0.858913i	$0.328860\pi$
$434$	−8108.05	−0.896771
$435$	0	0
$436$	−16524.8	−1.81513
$437$	−97.6055	−0.0106845
$438$	0	0
$439$	3483.80	0.378753	0.189376	−	0.981905i	$-0.439353\pi$
$439$	3483.80	0.378753	0.189376	+	0.981905i	$0.439353\pi$
$440$	0	0
$441$	0	0
$442$	−33372.7	−3.59135
$443$	−4979.97	−0.534099	−0.267049	−	0.963683i	$-0.586049\pi$
$443$	−4979.97	−0.534099	−0.267049	+	0.963683i	$0.586049\pi$
$444$	0	0
$445$	0	0
$446$	−32506.5	−3.45118
$447$	0	0
$448$	−44557.5	−4.69898
$449$	4167.11	0.437991	0.218996	−	0.975726i	$-0.429722\pi$
$449$	4167.11	0.437991	0.218996	+	0.975726i	$0.429722\pi$
$450$	0	0
$451$	−4560.63	−0.476168
$452$	−30455.4	−3.16925
$453$	0	0
$454$	3723.02	0.384868
$455$	0	0
$456$	0	0
$457$	−865.844	−0.0886269	−0.0443134	−	0.999018i	$-0.514110\pi$
$457$	−865.844	−0.0886269	−0.0443134	+	0.999018i	$0.514110\pi$
$458$	−7013.91	−0.715587
$459$	0	0
$460$	0	0
$461$	−5354.67	−0.540980	−0.270490	−	0.962723i	$-0.587186\pi$
$461$	−5354.67	−0.540980	−0.270490	+	0.962723i	$0.587186\pi$
$462$	0	0
$463$	−2232.71	−0.224110	−0.112055	−	0.993702i	$-0.535743\pi$
$463$	−2232.71	−0.224110	−0.112055	+	0.993702i	$0.535743\pi$
$464$	46970.6	4.69947
$465$	0	0
$466$	−961.437	−0.0955744
$467$	18874.0	1.87020	0.935102	−	0.354377i	$-0.115307\pi$
$467$	18874.0	1.87020	0.935102	+	0.354377i	$0.115307\pi$
$468$	0	0
$469$	16582.0	1.63259
$470$	0	0
$471$	0	0
$472$	−491.010	−0.0478826
$473$	−2007.89	−0.195186
$474$	0	0
$475$	0	0
$476$	45258.2	4.35800
$477$	0	0
$478$	12528.4	1.19882
$479$	−4981.32	−0.475161	−0.237580	−	0.971368i	$-0.576354\pi$
$479$	−4981.32	−0.475161	−0.237580	+	0.971368i	$0.576354\pi$
$480$	0	0
$481$	7986.32	0.757058
$482$	11468.7	1.08379
$483$	0	0
$484$	2746.89	0.257973
$485$	0	0
$486$	0	0
$487$	−3704.78	−0.344722	−0.172361	−	0.985034i	$-0.555140\pi$
$487$	−3704.78	−0.344722	−0.172361	+	0.985034i	$0.555140\pi$
$488$	35417.9	3.28543
$489$	0	0
$490$	0	0
$491$	−1746.76	−0.160550	−0.0802751	−	0.996773i	$-0.525580\pi$
$491$	−1746.76	−0.160550	−0.0802751	+	0.996773i	$0.525580\pi$
$492$	0	0
$493$	−19577.2	−1.78847
$494$	−369.453	−0.0336488
$495$	0	0
$496$	22260.8	2.01520
$497$	9045.63	0.816402
$498$	0	0
$499$	8368.30	0.750735	0.375367	−	0.926876i	$-0.377517\pi$
$499$	8368.30	0.750735	0.375367	+	0.926876i	$0.377517\pi$
$500$	0	0
$501$	0	0
$502$	13916.5	1.23730
$503$	−2647.07	−0.234646	−0.117323	−	0.993094i	$-0.537431\pi$
$503$	−2647.07	−0.234646	−0.117323	+	0.993094i	$0.537431\pi$
$504$	0	0
$505$	0	0
$506$	−4779.62	−0.419921
$507$	0	0
$508$	−38268.7	−3.34232
$509$	9531.20	0.829987	0.414993	−	0.909824i	$-0.363784\pi$
$509$	9531.20	0.829987	0.414993	+	0.909824i	$0.363784\pi$
$510$	0	0
$511$	−18542.3	−1.60521
$512$	51601.2	4.45405
$513$	0	0
$514$	1448.49	0.124300
$515$	0	0
$516$	0	0
$517$	−4826.12	−0.410547
$518$	−14647.3	−1.24241
$519$	0	0
$520$	0	0
$521$	3791.36	0.318815	0.159407	−	0.987213i	$-0.449042\pi$
$521$	3791.36	0.318815	0.159407	+	0.987213i	$0.449042\pi$
$522$	0	0
$523$	−5578.17	−0.466379	−0.233190	−	0.972431i	$-0.574916\pi$
$523$	−5578.17	−0.466379	−0.233190	+	0.972431i	$0.574916\pi$
$524$	10757.2	0.896812
$525$	0	0
$526$	11169.0	0.925836
$527$	−9278.24	−0.766919
$528$	0	0
$529$	−6017.49	−0.494574
$530$	0	0
$531$	0	0
$532$	501.033	0.0408318
$533$	22210.4	1.80495
$534$	0	0
$535$	0	0
$536$	−76177.3	−6.13873
$537$	0	0
$538$	9077.40	0.727425
$539$	314.367	0.0251220
$540$	0	0
$541$	−11942.4	−0.949061	−0.474531	−	0.880239i	$-0.657382\pi$
$541$	−11942.4	−0.949061	−0.474531	+	0.880239i	$0.657382\pi$
$542$	12191.7	0.966200
$543$	0	0
$544$	−94776.2	−7.46966
$545$	0	0
$546$	0	0
$547$	6873.38	0.537266	0.268633	−	0.963243i	$-0.413428\pi$
$547$	6873.38	0.537266	0.268633	+	0.963243i	$0.413428\pi$
$548$	16453.4	1.28258
$549$	0	0
$550$	0	0
$551$	−216.730	−0.0167568
$552$	0	0
$553$	−4015.19	−0.308758
$554$	−38528.1	−2.95470
$555$	0	0
$556$	54409.5	4.15014
$557$	26189.4	1.99225	0.996123	−	0.0879714i	$-0.0280384\pi$
$557$	26189.4	1.99225	0.996123	+	0.0879714i	$0.0280384\pi$
$558$	0	0
$559$	9778.49	0.739868
$560$	0	0
$561$	0	0
$562$	35603.7	2.67233
$563$	−2732.90	−0.204579	−0.102289	−	0.994755i	$-0.532617\pi$
$563$	−2732.90	−0.204579	−0.102289	+	0.994755i	$0.532617\pi$
$564$	0	0
$565$	0	0
$566$	9971.64	0.740529
$567$	0	0
$568$	−41555.4	−3.06976
$569$	23590.4	1.73807	0.869034	−	0.494753i	$-0.164741\pi$
$569$	23590.4	1.73807	0.869034	+	0.494753i	$0.164741\pi$
$570$	0	0
$571$	−24477.2	−1.79394	−0.896969	−	0.442093i	$-0.854236\pi$
$571$	−24477.2	−1.79394	−0.896969	+	0.442093i	$0.854236\pi$
$572$	−13377.5	−0.977867
$573$	0	0
$574$	−40735.1	−2.96211
$575$	0	0
$576$	0	0
$577$	−23152.4	−1.67044	−0.835221	−	0.549914i	$-0.814660\pi$
$577$	−23152.4	−1.67044	−0.835221	+	0.549914i	$0.814660\pi$
$578$	42818.4	3.08133
$579$	0	0
$580$	0	0
$581$	21025.1	1.50132
$582$	0	0
$583$	−5400.59	−0.383653
$584$	85183.0	6.03578
$585$	0	0
$586$	26130.0	1.84201
$587$	−4917.15	−0.345746	−0.172873	−	0.984944i	$-0.555305\pi$
$587$	−4917.15	−0.345746	−0.172873	+	0.984944i	$0.555305\pi$
$588$	0	0
$589$	−102.715	−0.00718557
$590$	0	0
$591$	0	0
$592$	40214.4	2.79190
$593$	24489.7	1.69591	0.847953	−	0.530071i	$-0.177835\pi$
$593$	24489.7	1.69591	0.847953	+	0.530071i	$0.177835\pi$
$594$	0	0
$595$	0	0
$596$	15738.6	1.08167
$597$	0	0
$598$	23276.9	1.59175
$599$	5515.59	0.376229	0.188114	−	0.982147i	$-0.439762\pi$
$599$	5515.59	0.376229	0.188114	+	0.982147i	$0.439762\pi$
$600$	0	0
$601$	17496.2	1.18750	0.593749	−	0.804651i	$-0.297647\pi$
$601$	17496.2	1.18750	0.593749	+	0.804651i	$0.297647\pi$
$602$	−17934.2	−1.21419
$603$	0	0
$604$	−60892.0	−4.10209
$605$	0	0
$606$	0	0
$607$	−5146.06	−0.344106	−0.172053	−	0.985088i	$-0.555040\pi$
$607$	−5146.06	−0.344106	−0.172053	+	0.985088i	$0.555040\pi$
$608$	−1049.22	−0.0699862
$609$	0	0
$610$	0	0
$611$	23503.4	1.55621
$612$	0	0
$613$	16329.5	1.07593	0.537963	−	0.842969i	$-0.319194\pi$
$613$	16329.5	1.07593	0.537963	+	0.842969i	$0.319194\pi$
$614$	21962.8	1.44356
$615$	0	0
$616$	15888.9	1.03925
$617$	−7734.93	−0.504695	−0.252347	−	0.967637i	$-0.581203\pi$
$617$	−7734.93	−0.504695	−0.252347	+	0.967637i	$0.581203\pi$
$618$	0	0
$619$	−966.773	−0.0627753	−0.0313876	−	0.999507i	$-0.509993\pi$
$619$	−966.773	−0.0627753	−0.0313876	+	0.999507i	$0.509993\pi$
$620$	0	0
$621$	0	0
$622$	−37613.5	−2.42470
$623$	−25589.8	−1.64564
$624$	0	0
$625$	0	0
$626$	−21578.0	−1.37768
$627$	0	0
$628$	27458.5	1.74477
$629$	−16761.3	−1.06251
$630$	0	0
$631$	22262.9	1.40455	0.702275	−	0.711906i	$-0.252168\pi$
$631$	22262.9	1.40455	0.702275	+	0.711906i	$0.252168\pi$
$632$	18445.7	1.16097
$633$	0	0
$634$	39389.3	2.46743
$635$	0	0
$636$	0	0
$637$	−1530.98	−0.0952270
$638$	−10613.0	−0.658579
$639$	0	0
$640$	0	0
$641$	9081.43	0.559587	0.279793	−	0.960060i	$-0.409734\pi$
$641$	9081.43	0.559587	0.279793	+	0.960060i	$0.409734\pi$
$642$	0	0
$643$	26891.9	1.64932	0.824661	−	0.565628i	$-0.191366\pi$
$643$	26891.9	1.64932	0.824661	+	0.565628i	$0.191366\pi$
$644$	−31566.9	−1.93154
$645$	0	0
$646$	775.390	0.0472249
$647$	30371.4	1.84548	0.922738	−	0.385429i	$-0.125947\pi$
$647$	30371.4	1.84548	0.922738	+	0.385429i	$0.125947\pi$
$648$	0	0
$649$	66.3039	0.00401025
$650$	0	0
$651$	0	0
$652$	77858.9	4.67667
$653$	−16502.1	−0.988941	−0.494471	−	0.869194i	$-0.664638\pi$
$653$	−16502.1	−0.988941	−0.494471	+	0.869194i	$0.664638\pi$
$654$	0	0
$655$	0	0
$656$	111839.	6.65635
$657$	0	0
$658$	−43106.4	−2.55390
$659$	−18549.5	−1.09649	−0.548244	−	0.836319i	$-0.684703\pi$
$659$	−18549.5	−1.09649	−0.548244	+	0.836319i	$0.684703\pi$
$660$	0	0
$661$	−13963.5	−0.821662	−0.410831	−	0.911711i	$-0.634761\pi$
$661$	−13963.5	−0.821662	−0.410831	+	0.911711i	$0.634761\pi$
$662$	−40125.4	−2.35577
$663$	0	0
$664$	−96588.9	−5.64514
$665$	0	0
$666$	0	0
$667$	13654.8	0.792679
$668$	−34588.1	−2.00337
$669$	0	0
$670$	0	0
$671$	−4782.67	−0.275161
$672$	0	0
$673$	10248.0	0.586972	0.293486	−	0.955963i	$-0.405185\pi$
$673$	10248.0	0.586972	0.293486	+	0.955963i	$0.405185\pi$
$674$	−44230.0	−2.52771
$675$	0	0
$676$	15273.4	0.868993
$677$	−1595.91	−0.0905994	−0.0452997	−	0.998973i	$-0.514424\pi$
$677$	−1595.91	−0.0905994	−0.0452997	+	0.998973i	$0.514424\pi$
$678$	0	0
$679$	14641.3	0.827513
$680$	0	0
$681$	0	0
$682$	−5029.83	−0.282408
$683$	6165.40	0.345406	0.172703	−	0.984974i	$-0.444750\pi$
$683$	6165.40	0.345406	0.172703	+	0.984974i	$0.444750\pi$
$684$	0	0
$685$	0	0
$686$	36507.9	2.03189
$687$	0	0
$688$	49238.7	2.72850
$689$	26301.1	1.45427
$690$	0	0
$691$	−13398.1	−0.737609	−0.368804	−	0.929507i	$-0.620233\pi$
$691$	−13398.1	−0.737609	−0.368804	+	0.929507i	$0.620233\pi$
$692$	−16792.9	−0.922503
$693$	0	0
$694$	21921.9	1.19906
$695$	0	0
$696$	0	0
$697$	−46614.1	−2.53319
$698$	−6886.85	−0.373455
$699$	0	0
$700$	0	0
$701$	−19182.6	−1.03354	−0.516772	−	0.856123i	$-0.672867\pi$
$701$	−19182.6	−1.03354	−0.516772	+	0.856123i	$0.672867\pi$
$702$	0	0
$703$	−185.556	−0.00995503
$704$	−27641.2	−1.47978
$705$	0	0
$706$	−52911.0	−2.82059
$707$	326.260	0.0173554
$708$	0	0
$709$	−14076.7	−0.745641	−0.372821	−	0.927903i	$-0.621609\pi$
$709$	−14076.7	−0.745641	−0.372821	+	0.927903i	$0.621609\pi$
$710$	0	0
$711$	0	0
$712$	117559.	6.18778
$713$	6471.43	0.339912
$714$	0	0
$715$	0	0
$716$	−47451.0	−2.47672
$717$	0	0
$718$	−45276.6	−2.35335
$719$	30477.5	1.58083	0.790417	−	0.612569i	$-0.209864\pi$
$719$	30477.5	1.58083	0.790417	+	0.612569i	$0.209864\pi$
$720$	0	0
$721$	16949.2	0.875480
$722$	−37996.5	−1.95856
$723$	0	0
$724$	11841.4	0.607849
$725$	0	0
$726$	0	0
$727$	−16141.1	−0.823440	−0.411720	−	0.911310i	$-0.635072\pi$
$727$	−16141.1	−0.823440	−0.411720	+	0.911310i	$0.635072\pi$
$728$	−77379.4	−3.93939
$729$	0	0
$730$	0	0
$731$	−20522.6	−1.03838
$732$	0	0
$733$	−32962.3	−1.66097	−0.830484	−	0.557043i	$-0.811936\pi$
$733$	−32962.3	−1.66097	−0.830484	+	0.557043i	$0.811936\pi$
$734$	−39347.6	−1.97867
$735$	0	0
$736$	66105.0	3.31068
$737$	10286.7	0.514130
$738$	0	0
$739$	1509.25	0.0751267	0.0375634	−	0.999294i	$-0.488040\pi$
$739$	1509.25	0.0751267	0.0375634	+	0.999294i	$0.488040\pi$
$740$	0	0
$741$	0	0
$742$	−48237.5	−2.38659
$743$	14601.9	0.720984	0.360492	−	0.932762i	$-0.382609\pi$
$743$	14601.9	0.720984	0.360492	+	0.932762i	$0.382609\pi$
$744$	0	0
$745$	0	0
$746$	−67322.1	−3.30407
$747$	0	0
$748$	28075.9	1.37240
$749$	26611.4	1.29821
$750$	0	0
$751$	−17830.7	−0.866381	−0.433191	−	0.901302i	$-0.642612\pi$
$751$	−17830.7	−0.866381	−0.433191	+	0.901302i	$0.642612\pi$
$752$	118349.	5.73904
$753$	0	0
$754$	51685.8	2.49640
$755$	0	0
$756$	0	0
$757$	−15800.6	−0.758632	−0.379316	−	0.925267i	$-0.623841\pi$
$757$	−15800.6	−0.758632	−0.379316	+	0.925267i	$0.623841\pi$
$758$	5730.83	0.274608
$759$	0	0
$760$	0	0
$761$	1707.35	0.0813292	0.0406646	−	0.999173i	$-0.487052\pi$
$761$	1707.35	0.0813292	0.0406646	+	0.999173i	$0.487052\pi$
$762$	0	0
$763$	12907.3	0.612421
$764$	30351.5	1.43728
$765$	0	0
$766$	−30524.8	−1.43982
$767$	−322.902	−0.0152012
$768$	0	0
$769$	−13359.8	−0.626486	−0.313243	−	0.949673i	$-0.601415\pi$
$769$	−13359.8	−0.626486	−0.313243	+	0.949673i	$0.601415\pi$
$770$	0	0
$771$	0	0
$772$	−105580.	−4.92214
$773$	−28990.9	−1.34894	−0.674469	−	0.738303i	$-0.735628\pi$
$773$	−28990.9	−1.34894	−0.674469	+	0.738303i	$0.735628\pi$
$774$	0	0
$775$	0	0
$776$	−67261.8	−3.11154
$777$	0	0
$778$	−46251.9	−2.13138
$779$	−516.043	−0.0237345
$780$	0	0
$781$	5611.45	0.257098
$782$	−48852.4	−2.23396
$783$	0	0
$784$	−7709.11	−0.351180
$785$	0	0
$786$	0	0
$787$	22010.7	0.996947	0.498473	−	0.866905i	$-0.333894\pi$
$787$	22010.7	0.996947	0.498473	+	0.866905i	$0.333894\pi$
$788$	−74771.2	−3.38022
$789$	0	0
$790$	0	0
$791$	23788.3	1.06930
$792$	0	0
$793$	23291.8	1.04302
$794$	6833.73	0.305441
$795$	0	0
$796$	82038.0	3.65297
$797$	−26709.2	−1.18706	−0.593531	−	0.804811i	$-0.702267\pi$
$797$	−26709.2	−1.18706	−0.593531	+	0.804811i	$0.702267\pi$
$798$	0	0
$799$	−49327.7	−2.18409
$800$	0	0
$801$	0	0
$802$	18870.6	0.830853
$803$	−11502.7	−0.505508
$804$	0	0
$805$	0	0
$806$	24495.5	1.07049
$807$	0	0
$808$	−1498.83	−0.0652582
$809$	−11202.0	−0.486823	−0.243412	−	0.969923i	$-0.578267\pi$
$809$	−11202.0	−0.486823	−0.243412	+	0.969923i	$0.578267\pi$
$810$	0	0
$811$	−34979.8	−1.51456	−0.757280	−	0.653091i	$-0.773472\pi$
$811$	−34979.8	−1.51456	−0.757280	+	0.653091i	$0.773472\pi$
$812$	−70093.5	−3.02931
$813$	0	0
$814$	−9086.46	−0.391253
$815$	0	0
$816$	0	0
$817$	−227.196	−0.00972899
$818$	−13679.3	−0.584700
$819$	0	0
$820$	0	0
$821$	3802.17	0.161628	0.0808140	−	0.996729i	$-0.474248\pi$
$821$	3802.17	0.161628	0.0808140	+	0.996729i	$0.474248\pi$
$822$	0	0
$823$	−4717.91	−0.199825	−0.0999126	−	0.994996i	$-0.531856\pi$
$823$	−4717.91	−0.199825	−0.0999126	+	0.994996i	$0.531856\pi$
$824$	−77864.2	−3.29190
$825$	0	0
$826$	592.219	0.0249467
$827$	−30244.0	−1.27169	−0.635845	−	0.771817i	$-0.719348\pi$
$827$	−30244.0	−1.27169	−0.635845	+	0.771817i	$0.719348\pi$
$828$	0	0
$829$	26008.5	1.08964	0.544821	−	0.838552i	$-0.316597\pi$
$829$	26008.5	1.08964	0.544821	+	0.838552i	$0.316597\pi$
$830$	0	0
$831$	0	0
$832$	134614.	5.60925
$833$	3213.14	0.133648
$834$	0	0
$835$	0	0
$836$	310.816	0.0128586
$837$	0	0
$838$	−9488.86	−0.391154
$839$	−5360.85	−0.220593	−0.110296	−	0.993899i	$-0.535180\pi$
$839$	−5360.85	−0.220593	−0.110296	+	0.993899i	$0.535180\pi$
$840$	0	0
$841$	5931.14	0.243189
$842$	−36641.0	−1.49968
$843$	0	0
$844$	128528.	5.24185
$845$	0	0
$846$	0	0
$847$	−2145.56	−0.0870395
$848$	132437.	5.36308
$849$	0	0
$850$	0	0
$851$	11690.7	0.470920
$852$	0	0
$853$	31301.1	1.25642	0.628212	−	0.778043i	$-0.283787\pi$
$853$	31301.1	1.25642	0.628212	+	0.778043i	$0.283787\pi$
$854$	−42718.3	−1.71170
$855$	0	0
$856$	−122252.	−4.88141
$857$	1445.51	0.0576167	0.0288084	−	0.999585i	$-0.490829\pi$
$857$	1445.51	0.0576167	0.0288084	+	0.999585i	$0.490829\pi$
$858$	0	0
$859$	−27178.3	−1.07953	−0.539763	−	0.841817i	$-0.681486\pi$
$859$	−27178.3	−1.07953	−0.539763	+	0.841817i	$0.681486\pi$
$860$	0	0
$861$	0	0
$862$	−85219.3	−3.36726
$863$	1232.36	0.0486097	0.0243049	−	0.999705i	$-0.492263\pi$
$863$	1232.36	0.0486097	0.0243049	+	0.999705i	$0.492263\pi$
$864$	0	0
$865$	0	0
$866$	51134.6	2.00650
$867$	0	0
$868$	−33219.4	−1.29901
$869$	−2490.82	−0.0972330
$870$	0	0
$871$	−50096.4	−1.94885
$872$	−59296.0	−2.30277
$873$	0	0
$874$	−540.823	−0.0209309
$875$	0	0
$876$	0	0
$877$	22320.0	0.859399	0.429699	−	0.902972i	$-0.358620\pi$
$877$	22320.0	0.859399	0.429699	+	0.902972i	$0.358620\pi$
$878$	19303.4	0.741979
$879$	0	0
$880$	0	0
$881$	−22244.7	−0.850674	−0.425337	−	0.905035i	$-0.639844\pi$
$881$	−22244.7	−0.850674	−0.425337	+	0.905035i	$0.639844\pi$
$882$	0	0
$883$	−35958.1	−1.37043	−0.685214	−	0.728342i	$-0.740291\pi$
$883$	−35958.1	−1.37043	−0.685214	+	0.728342i	$0.740291\pi$
$884$	−136731.	−5.20222
$885$	0	0
$886$	−27593.5	−1.04630
$887$	26277.0	0.994696	0.497348	−	0.867551i	$-0.334307\pi$
$887$	26277.0	0.994696	0.497348	+	0.867551i	$0.334307\pi$
$888$	0	0
$889$	29891.2	1.12769
$890$	0	0
$891$	0	0
$892$	−133182.	−4.99917
$893$	−546.084	−0.0204636
$894$	0	0
$895$	0	0
$896$	−127308.	−4.74673
$897$	0	0
$898$	23089.5	0.858027
$899$	14369.6	0.533097
$900$	0	0
$901$	−55199.3	−2.04102
$902$	−25270.0	−0.932815
$903$	0	0
$904$	−109283.	−4.02068
$905$	0	0
$906$	0	0
$907$	−30458.2	−1.11505	−0.557523	−	0.830161i	$-0.688248\pi$
$907$	−30458.2	−1.11505	−0.557523	+	0.830161i	$0.688248\pi$
$908$	15253.5	0.557496
$909$	0	0
$910$	0	0
$911$	44396.7	1.61463	0.807315	−	0.590120i	$-0.200920\pi$
$911$	44396.7	1.61463	0.807315	+	0.590120i	$0.200920\pi$
$912$	0	0
$913$	13042.9	0.472791
$914$	−4797.56	−0.173620
$915$	0	0
$916$	−28736.7	−1.03656
$917$	−8402.30	−0.302583
$918$	0	0
$919$	23498.2	0.843454	0.421727	−	0.906723i	$-0.361424\pi$
$919$	23498.2	0.843454	0.421727	+	0.906723i	$0.361424\pi$
$920$	0	0
$921$	0	0
$922$	−29669.7	−1.05978
$923$	−27328.0	−0.974553
$924$	0	0
$925$	0	0
$926$	−12371.2	−0.439032
$927$	0	0
$928$	146784.	5.19227
$929$	40096.3	1.41606	0.708029	−	0.706184i	$-0.249585\pi$
$929$	40096.3	1.41606	0.708029	+	0.706184i	$0.249585\pi$
$930$	0	0
$931$	35.5712	0.00125220
$932$	−3939.09	−0.138443
$933$	0	0
$934$	104579.	3.66374
$935$	0	0
$936$	0	0
$937$	36306.5	1.26583	0.632914	−	0.774222i	$-0.281859\pi$
$937$	36306.5	1.26583	0.632914	+	0.774222i	$0.281859\pi$
$938$	91879.3	3.19826
$939$	0	0
$940$	0	0
$941$	25929.1	0.898262	0.449131	−	0.893466i	$-0.351734\pi$
$941$	25929.1	0.898262	0.449131	+	0.893466i	$0.351734\pi$
$942$	0	0
$943$	32512.7	1.12275
$944$	−1625.95	−0.0560594
$945$	0	0
$946$	−11125.5	−0.382369
$947$	13367.3	0.458689	0.229345	−	0.973345i	$-0.426342\pi$
$947$	13367.3	0.458689	0.229345	+	0.973345i	$0.426342\pi$
$948$	0	0
$949$	56018.8	1.91617
$950$	0	0
$951$	0	0
$952$	162400.	5.52880
$953$	35619.6	1.21074	0.605368	−	0.795946i	$-0.293026\pi$
$953$	35619.6	1.21074	0.605368	+	0.795946i	$0.293026\pi$
$954$	0	0
$955$	0	0
$956$	51330.1	1.73654
$957$	0	0
$958$	−27601.0	−0.930842
$959$	−12851.5	−0.432740
$960$	0	0
$961$	−22980.8	−0.771401
$962$	44251.4	1.48308
$963$	0	0
$964$	46988.3	1.56991
$965$	0	0
$966$	0	0
$967$	−14921.5	−0.496217	−0.248109	−	0.968732i	$-0.579809\pi$
$967$	−14921.5	−0.496217	−0.248109	+	0.968732i	$0.579809\pi$
$968$	9856.66	0.327278
$969$	0	0
$970$	0	0
$971$	−32520.5	−1.07480	−0.537401	−	0.843327i	$-0.680594\pi$
$971$	−32520.5	−1.07480	−0.537401	+	0.843327i	$0.680594\pi$
$972$	0	0
$973$	−42498.6	−1.40025
$974$	−20527.8	−0.675313
$975$	0	0
$976$	117284.	3.84648
$977$	28946.6	0.947884	0.473942	−	0.880556i	$-0.342831\pi$
$977$	28946.6	0.947884	0.473942	+	0.880556i	$0.342831\pi$
$978$	0	0
$979$	−15874.6	−0.518238
$980$	0	0
$981$	0	0
$982$	−9678.63	−0.314519
$983$	−1920.56	−0.0623158	−0.0311579	−	0.999514i	$-0.509919\pi$
$983$	−1920.56	−0.0623158	−0.0311579	+	0.999514i	$0.509919\pi$
$984$	0	0
$985$	0	0
$986$	−108475.	−3.50361
$987$	0	0
$988$	−1513.68	−0.0487416
$989$	14314.2	0.460228
$990$	0	0
$991$	−30472.4	−0.976780	−0.488390	−	0.872625i	$-0.662416\pi$
$991$	−30472.4	−0.976780	−0.488390	+	0.872625i	$0.662416\pi$
$992$	69565.5	2.22652
$993$	0	0
$994$	50120.9	1.59934
$995$	0	0
$996$	0	0
$997$	−52854.5	−1.67895	−0.839477	−	0.543395i	$-0.817139\pi$
$997$	−52854.5	−1.67895	−0.839477	+	0.543395i	$0.817139\pi$
$998$	46367.9	1.47069
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2475.4.a.bt.1.7		7
3.2	odd	2		2475.4.a.bp.1.1		7
5.4	even	2		495.4.a.o.1.1	✓	7
15.14	odd	2		495.4.a.p.1.7	yes	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
495.4.a.o.1.1	✓	7	5.4	even	2
495.4.a.p.1.7	yes	7	15.14	odd	2
2475.4.a.bp.1.1		7	3.2	odd	2
2475.4.a.bt.1.7		7	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)

\(f(q)\)	\(=\)	\(q+5.54090 q^{2} +22.7016 q^{4} -17.7319 q^{7} +81.4600 q^{8} +O(q^{10})\)	\(q+5.54090 q^{2} +22.7016 q^{4} -17.7319 q^{7} +81.4600 q^{8} -11.0000 q^{11} +53.5704 q^{13} -98.2508 q^{14} +269.749 q^{16} -112.431 q^{17} -1.24467 q^{19} -60.9499 q^{22} +78.4188 q^{23} +296.828 q^{26} -402.543 q^{28} +174.127 q^{29} +82.5240 q^{31} +842.973 q^{32} -622.968 q^{34} +149.081 q^{37} -6.89659 q^{38} +414.603 q^{41} +182.535 q^{43} -249.717 q^{44} +434.511 q^{46} +438.738 q^{47} -28.5788 q^{49} +1216.13 q^{52} +490.962 q^{53} -1444.44 q^{56} +964.819 q^{58} -6.02762 q^{59} +434.789 q^{61} +457.257 q^{62} +2512.84 q^{64} -935.150 q^{67} -2552.36 q^{68} -510.132 q^{71} +1045.70 q^{73} +826.042 q^{74} -28.2560 q^{76} +195.051 q^{77} +226.439 q^{79} +2297.27 q^{82} -1185.72 q^{83} +1011.41 q^{86} -896.060 q^{88} +1443.15 q^{89} -949.907 q^{91} +1780.23 q^{92} +2431.01 q^{94} -825.703 q^{97} -158.352 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 5 q^{2} + 33 q^{4} - 30 q^{7} + 45 q^{8}+O(q^{10}) \) 7 * q + 5 * q^2 + 33 * q^4 - 30 * q^7 + 45 * q^8	\( 7 q + 5 q^{2} + 33 q^{4} - 30 q^{7} + 45 q^{8} - 77 q^{11} - 38 q^{13} - 20 q^{14} + 309 q^{16} + 12 q^{17} + 226 q^{19} - 55 q^{22} + 334 q^{23} + 372 q^{26} - 812 q^{28} + 258 q^{29} + 336 q^{31} + 485 q^{32} + 78 q^{34} - 466 q^{37} - 494 q^{38} + 258 q^{41} - 308 q^{43} - 363 q^{44} + 98 q^{46} + 546 q^{47} + 735 q^{49} - 512 q^{52} + 110 q^{53} - 20 q^{56} - 1362 q^{58} + 68 q^{59} + 1096 q^{61} + 356 q^{62} + 2761 q^{64} - 2268 q^{67} - 1186 q^{68} + 166 q^{71} - 200 q^{73} + 1710 q^{74} + 3310 q^{76} + 330 q^{77} + 2152 q^{79} + 1006 q^{82} + 370 q^{83} - 106 q^{86} - 495 q^{88} + 252 q^{89} + 2768 q^{91} + 3774 q^{92} + 2218 q^{94} - 3698 q^{97} + 697 q^{98}+O(q^{100}) \) 7 * q + 5 * q^2 + 33 * q^4 - 30 * q^7 + 45 * q^8 - 77 * q^11 - 38 * q^13 - 20 * q^14 + 309 * q^16 + 12 * q^17 + 226 * q^19 - 55 * q^22 + 334 * q^23 + 372 * q^26 - 812 * q^28 + 258 * q^29 + 336 * q^31 + 485 * q^32 + 78 * q^34 - 466 * q^37 - 494 * q^38 + 258 * q^41 - 308 * q^43 - 363 * q^44 + 98 * q^46 + 546 * q^47 + 735 * q^49 - 512 * q^52 + 110 * q^53 - 20 * q^56 - 1362 * q^58 + 68 * q^59 + 1096 * q^61 + 356 * q^62 + 2761 * q^64 - 2268 * q^67 - 1186 * q^68 + 166 * q^71 - 200 * q^73 + 1710 * q^74 + 3310 * q^76 + 330 * q^77 + 2152 * q^79 + 1006 * q^82 + 370 * q^83 - 106 * q^86 - 495 * q^88 + 252 * q^89 + 2768 * q^91 + 3774 * q^92 + 2218 * q^94 - 3698 * q^97 + 697 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more