LMFDB - Embedded newform 2475.4.a.bw.1.1

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} - 72x^{8} + 1771x^{6} - 17056x^{4} + 52892x^{2} - 3136 $ x^10 - 72x^8 + 1771x^6 - 17056x^4 + 52892x^2 - 3136
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2^{4}\cdot 5^{2} $
Twist minimal:	no (minimal twist has level 55)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-5.44091$ 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)

$f(q)$	$=$	$q-5.44091 q^{2} +21.6035 q^{4} +18.8252 q^{7} -74.0156 q^{8} -11.0000 q^{11} +1.72915 q^{13} -102.426 q^{14} +229.884 q^{16} -3.07229 q^{17} +123.822 q^{19} +59.8500 q^{22} +92.3179 q^{23} -9.40815 q^{26} +406.690 q^{28} +119.388 q^{29} +292.648 q^{31} -658.654 q^{32} +16.7161 q^{34} -235.769 q^{37} -673.704 q^{38} +51.2730 q^{41} -229.869 q^{43} -237.639 q^{44} -502.293 q^{46} +356.494 q^{47} +11.3869 q^{49} +37.3557 q^{52} +117.848 q^{53} -1393.36 q^{56} -649.579 q^{58} +205.210 q^{59} +490.921 q^{61} -1592.27 q^{62} +1744.61 q^{64} -890.724 q^{67} -66.3723 q^{68} +655.118 q^{71} +80.8916 q^{73} +1282.80 q^{74} +2674.99 q^{76} -207.077 q^{77} -335.730 q^{79} -278.972 q^{82} +432.146 q^{83} +1250.69 q^{86} +814.171 q^{88} -455.024 q^{89} +32.5515 q^{91} +1994.39 q^{92} -1939.65 q^{94} -842.417 q^{97} -61.9550 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 64 q^{4} - 110 q^{11} - 34 q^{14} + 468 q^{16} + 90 q^{19} - 392 q^{26} + 58 q^{29} + 1242 q^{31} - 66 q^{34} - 416 q^{41} - 704 q^{44} + 1816 q^{46} + 1980 q^{49} - 2626 q^{56} - 476 q^{59} + 1650 q^{61}+ \cdots - 2860 q^{94}+O(q^{100}) $ 10 * q + 64 * q^4 - 110 * q^11 - 34 * q^14 + 468 * q^16 + 90 * q^19 - 392 * q^26 + 58 * q^29 + 1242 * q^31 - 66 * q^34 - 416 * q^41 - 704 * q^44 + 1816 * q^46 + 1980 * q^49 - 2626 * q^56 - 476 * q^59 + 1650 * q^61 + 2576 * q^64 + 498 * q^71 + 5374 * q^74 + 1410 * q^76 - 416 * q^79 + 6872 * q^86 - 1918 * q^89 + 1384 * q^91 - 2860 * q^94

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 $

$2$	−5.44091	−1.92365	−0.961826	−	0.273660i	$-0.911766\pi$
$2$	−5.44091	−1.92365	−0.961826	+	0.273660i	$0.911766\pi$
$3$	0	0
$3$	0	0
$4$	21.6035	2.70044
$5$	0	0
$5$	0	0
$6$	0	0
$7$	18.8252	1.01646	0.508232	−	0.861220i	$-0.330299\pi$
$7$	18.8252	1.01646	0.508232	+	0.861220i	$0.330299\pi$
$8$	−74.0156	−3.27106
$9$	0	0
$10$	0	0
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	0	0
$13$	1.72915	0.0368907	0.0184454	−	0.999830i	$-0.494128\pi$
$13$	1.72915	0.0368907	0.0184454	+	0.999830i	$0.494128\pi$
$14$	−102.426	−1.95532
$15$	0	0
$16$	229.884	3.59194
$17$	−3.07229	−0.0438317	−0.0219159	−	0.999760i	$-0.506977\pi$
$17$	−3.07229	−0.0438317	−0.0219159	+	0.999760i	$0.506977\pi$
$18$	0	0
$19$	123.822	1.49509	0.747545	−	0.664211i	$-0.231232\pi$
$19$	123.822	1.49509	0.747545	+	0.664211i	$0.231232\pi$
$20$	0	0
$21$	0	0
$22$	59.8500	0.580003
$23$	92.3179	0.836939	0.418470	−	0.908231i	$-0.362567\pi$
$23$	92.3179	0.836939	0.418470	+	0.908231i	$0.362567\pi$
$24$	0	0
$25$	0	0
$26$	−9.40815	−0.0709650
$27$	0	0
$28$	406.690	2.74490
$29$	119.388	0.764475	0.382237	−	0.924064i	$-0.375154\pi$
$29$	119.388	0.764475	0.382237	+	0.924064i	$0.375154\pi$
$30$	0	0
$31$	292.648	1.69552	0.847761	−	0.530379i	$-0.177950\pi$
$31$	292.648	1.69552	0.847761	+	0.530379i	$0.177950\pi$
$32$	−658.654	−3.63858
$33$	0	0
$34$	16.7161	0.0843170
$35$	0	0
$36$	0	0
$37$	−235.769	−1.04757	−0.523787	−	0.851849i	$-0.675481\pi$
$37$	−235.769	−1.04757	−0.523787	+	0.851849i	$0.675481\pi$
$38$	−673.704	−2.87603
$39$	0	0
$40$	0	0
$41$	51.2730	0.195305	0.0976524	−	0.995221i	$-0.468867\pi$
$41$	51.2730	0.195305	0.0976524	+	0.995221i	$0.468867\pi$
$42$	0	0
$43$	−229.869	−0.815224	−0.407612	−	0.913155i	$-0.633638\pi$
$43$	−229.869	−0.815224	−0.407612	+	0.913155i	$0.633638\pi$
$44$	−237.639	−0.814213
$45$	0	0
$46$	−502.293	−1.60998
$47$	356.494	1.10638	0.553191	−	0.833055i	$-0.313410\pi$
$47$	356.494	1.10638	0.553191	+	0.833055i	$0.313410\pi$
$48$	0	0
$49$	11.3869	0.0331979
$50$	0	0
$51$	0	0
$52$	37.3557	0.0996212
$53$	117.848	0.305427	0.152713	−	0.988271i	$-0.451199\pi$
$53$	117.848	0.305427	0.152713	+	0.988271i	$0.451199\pi$
$54$	0	0
$55$	0	0
$56$	−1393.36	−3.32491
$57$	0	0
$58$	−649.579	−1.47058
$59$	205.210	0.452815	0.226407	−	0.974033i	$-0.427302\pi$
$59$	205.210	0.452815	0.226407	+	0.974033i	$0.427302\pi$
$60$	0	0
$61$	490.921	1.03043	0.515213	−	0.857062i	$-0.327713\pi$
$61$	490.921	1.03043	0.515213	+	0.857062i	$0.327713\pi$
$62$	−1592.27	−3.26159
$63$	0	0
$64$	1744.61	3.40744
$65$	0	0
$66$	0	0
$67$	−890.724	−1.62417	−0.812084	−	0.583540i	$-0.801667\pi$
$67$	−890.724	−1.62417	−0.812084	+	0.583540i	$0.801667\pi$
$68$	−66.3723	−0.118365
$69$	0	0
$70$	0	0
$71$	655.118	1.09504	0.547522	−	0.836791i	$-0.315571\pi$
$71$	655.118	1.09504	0.547522	+	0.836791i	$0.315571\pi$
$72$	0	0
$73$	80.8916	0.129694	0.0648469	−	0.997895i	$-0.479344\pi$
$73$	80.8916	0.129694	0.0648469	+	0.997895i	$0.479344\pi$
$74$	1282.80	2.01517
$75$	0	0
$76$	2674.99	4.03740
$77$	−207.077	−0.306475
$78$	0	0
$79$	−335.730	−0.478134	−0.239067	−	0.971003i	$-0.576842\pi$
$79$	−335.730	−0.478134	−0.239067	+	0.971003i	$0.576842\pi$
$80$	0	0
$81$	0	0
$82$	−278.972	−0.375699
$83$	432.146	0.571497	0.285748	−	0.958305i	$-0.407758\pi$
$83$	432.146	0.571497	0.285748	+	0.958305i	$0.407758\pi$
$84$	0	0
$85$	0	0
$86$	1250.69	1.56821
$87$	0	0
$88$	814.171	0.986261
$89$	−455.024	−0.541937	−0.270969	−	0.962588i	$-0.587344\pi$
$89$	−455.024	−0.541937	−0.270969	+	0.962588i	$0.587344\pi$
$90$	0	0
$91$	32.5515	0.0374981
$92$	1994.39	2.26010
$93$	0	0
$94$	−1939.65	−2.12829
$95$	0	0
$96$	0	0
$97$	−842.417	−0.881798	−0.440899	−	0.897557i	$-0.645340\pi$
$97$	−842.417	−0.881798	−0.440899	+	0.897557i	$0.645340\pi$
$98$	−61.9550	−0.0638612
$99$	0	0
$100$	0	0
$101$	524.725	0.516952	0.258476	−	0.966018i	$-0.416780\pi$
$101$	524.725	0.516952	0.258476	+	0.966018i	$0.416780\pi$
$102$	0	0
$103$	239.172	0.228799	0.114400	−	0.993435i	$-0.463506\pi$
$103$	239.172	0.228799	0.114400	+	0.993435i	$0.463506\pi$
$104$	−127.984	−0.120672
$105$	0	0
$106$	−641.199	−0.587535
$107$	891.407	0.805379	0.402690	−	0.915337i	$-0.368075\pi$
$107$	891.407	0.805379	0.402690	+	0.915337i	$0.368075\pi$
$108$	0	0
$109$	1423.96	1.25129	0.625645	−	0.780108i	$-0.284836\pi$
$109$	1423.96	1.25129	0.625645	+	0.780108i	$0.284836\pi$
$110$	0	0
$111$	0	0
$112$	4327.61	3.65107
$113$	−102.586	−0.0854022	−0.0427011	−	0.999088i	$-0.513596\pi$
$113$	−102.586	−0.0854022	−0.0427011	+	0.999088i	$0.513596\pi$
$114$	0	0
$115$	0	0
$116$	2579.20	2.06442
$117$	0	0
$118$	−1116.53	−0.871058
$119$	−57.8363	−0.0445534
$120$	0	0
$121$	121.000	0.0909091
$122$	−2671.06	−1.98218
$123$	0	0
$124$	6322.23	4.57865
$125$	0	0
$126$	0	0
$127$	238.080	0.166348	0.0831740	−	0.996535i	$-0.473494\pi$
$127$	238.080	0.166348	0.0831740	+	0.996535i	$0.473494\pi$
$128$	−4223.02	−2.91614
$129$	0	0
$130$	0	0
$131$	−532.630	−0.355238	−0.177619	−	0.984099i	$-0.556839\pi$
$131$	−532.630	−0.355238	−0.177619	+	0.984099i	$0.556839\pi$
$132$	0	0
$133$	2330.97	1.51970
$134$	4846.35	3.12434
$135$	0	0
$136$	227.397	0.143376
$137$	−478.506	−0.298405	−0.149203	−	0.988807i	$-0.547671\pi$
$137$	−478.506	−0.298405	−0.149203	+	0.988807i	$0.547671\pi$
$138$	0	0
$139$	354.237	0.216158	0.108079	−	0.994142i	$-0.465530\pi$
$139$	354.237	0.216158	0.108079	+	0.994142i	$0.465530\pi$
$140$	0	0
$141$	0	0
$142$	−3564.44	−2.10649
$143$	−19.0206	−0.0111230
$144$	0	0
$145$	0	0
$146$	−440.124	−0.249486
$147$	0	0
$148$	−5093.45	−2.82891
$149$	−3002.16	−1.65065	−0.825324	−	0.564660i	$-0.809007\pi$
$149$	−3002.16	−1.65065	−0.825324	+	0.564660i	$0.809007\pi$
$150$	0	0
$151$	2231.65	1.20271	0.601356	−	0.798981i	$-0.294627\pi$
$151$	2231.65	1.20271	0.601356	+	0.798981i	$0.294627\pi$
$152$	−9164.75	−4.89052
$153$	0	0
$154$	1126.69	0.589552
$155$	0	0
$156$	0	0
$157$	283.657	0.144193	0.0720965	−	0.997398i	$-0.477031\pi$
$157$	283.657	0.144193	0.0720965	+	0.997398i	$0.477031\pi$
$158$	1826.68	0.919764
$159$	0	0
$160$	0	0
$161$	1737.90	0.850718
$162$	0	0
$163$	3474.38	1.66953	0.834767	−	0.550603i	$-0.185602\pi$
$163$	3474.38	1.66953	0.834767	+	0.550603i	$0.185602\pi$
$164$	1107.68	0.527409
$165$	0	0
$166$	−2351.27	−1.09936
$167$	1276.60	0.591536	0.295768	−	0.955260i	$-0.404424\pi$
$167$	1276.60	0.591536	0.295768	+	0.955260i	$0.404424\pi$
$168$	0	0
$169$	−2194.01	−0.998639
$170$	0	0
$171$	0	0
$172$	−4965.97	−2.20146
$173$	421.138	0.185078	0.0925392	−	0.995709i	$-0.470502\pi$
$173$	421.138	0.185078	0.0925392	+	0.995709i	$0.470502\pi$
$174$	0	0
$175$	0	0
$176$	−2528.72	−1.08301
$177$	0	0
$178$	2475.74	1.04250
$179$	2631.95	1.09900	0.549501	−	0.835493i	$-0.314818\pi$
$179$	2631.95	1.09900	0.549501	+	0.835493i	$0.314818\pi$
$180$	0	0
$181$	−2427.32	−0.996804	−0.498402	−	0.866946i	$-0.666080\pi$
$181$	−2427.32	−0.996804	−0.498402	+	0.866946i	$0.666080\pi$
$182$	−177.110	−0.0721333
$183$	0	0
$184$	−6832.96	−2.73768
$185$	0	0
$186$	0	0
$187$	33.7952	0.0132158
$188$	7701.52	2.98772
$189$	0	0
$190$	0	0
$191$	−927.802	−0.351484	−0.175742	−	0.984436i	$-0.556232\pi$
$191$	−927.802	−0.351484	−0.175742	+	0.984436i	$0.556232\pi$
$192$	0	0
$193$	1353.07	0.504643	0.252322	−	0.967643i	$-0.418806\pi$
$193$	1353.07	0.504643	0.252322	+	0.967643i	$0.418806\pi$
$194$	4583.51	1.69627
$195$	0	0
$196$	245.997	0.0896489
$197$	−2697.55	−0.975596	−0.487798	−	0.872957i	$-0.662200\pi$
$197$	−2697.55	−0.975596	−0.487798	+	0.872957i	$0.662200\pi$
$198$	0	0
$199$	1796.87	0.640084	0.320042	−	0.947403i	$-0.396303\pi$
$199$	1796.87	0.640084	0.320042	+	0.947403i	$0.396303\pi$
$200$	0	0
$201$	0	0
$202$	−2854.98	−0.994436
$203$	2247.50	0.777061
$204$	0	0
$205$	0	0
$206$	−1301.31	−0.440130
$207$	0	0
$208$	397.504	0.132509
$209$	−1362.04	−0.450787
$210$	0	0
$211$	−235.773	−0.0769254	−0.0384627	−	0.999260i	$-0.512246\pi$
$211$	−235.773	−0.0769254	−0.0384627	+	0.999260i	$0.512246\pi$
$212$	2545.92	0.824787
$213$	0	0
$214$	−4850.07	−1.54927
$215$	0	0
$216$	0	0
$217$	5509.15	1.72344
$218$	−7747.64	−2.40705
$219$	0	0
$220$	0	0
$221$	−5.31244	−0.00161699
$222$	0	0
$223$	−4355.88	−1.30803	−0.654017	−	0.756480i	$-0.726918\pi$
$223$	−4355.88	−1.30803	−0.654017	+	0.756480i	$0.726918\pi$
$224$	−12399.3	−3.69849
$225$	0	0
$226$	558.160	0.164284
$227$	−3029.90	−0.885910	−0.442955	−	0.896544i	$-0.646070\pi$
$227$	−3029.90	−0.885910	−0.442955	+	0.896544i	$0.646070\pi$
$228$	0	0
$229$	−6092.66	−1.75814	−0.879070	−	0.476693i	$-0.841835\pi$
$229$	−6092.66	−1.75814	−0.879070	+	0.476693i	$0.841835\pi$
$230$	0	0
$231$	0	0
$232$	−8836.56	−2.50064
$233$	−1632.51	−0.459008	−0.229504	−	0.973308i	$-0.573710\pi$
$233$	−1632.51	−0.459008	−0.229504	+	0.973308i	$0.573710\pi$
$234$	0	0
$235$	0	0
$236$	4433.26	1.22280
$237$	0	0
$238$	314.682	0.0857052
$239$	−5627.21	−1.52299	−0.761494	−	0.648172i	$-0.775534\pi$
$239$	−5627.21	−1.52299	−0.761494	+	0.648172i	$0.775534\pi$
$240$	0	0
$241$	3486.63	0.931925	0.465962	−	0.884805i	$-0.345708\pi$
$241$	3486.63	0.931925	0.465962	+	0.884805i	$0.345708\pi$
$242$	−658.350	−0.174878
$243$	0	0
$244$	10605.6	2.78260
$245$	0	0
$246$	0	0
$247$	214.107	0.0551550
$248$	−21660.5	−5.54615
$249$	0	0
$250$	0	0
$251$	5704.48	1.43452	0.717258	−	0.696808i	$-0.245397\pi$
$251$	5704.48	1.43452	0.717258	+	0.696808i	$0.245397\pi$
$252$	0	0
$253$	−1015.50	−0.252347
$254$	−1295.37	−0.319996
$255$	0	0
$256$	9020.23	2.20220
$257$	−2633.52	−0.639200	−0.319600	−	0.947553i	$-0.603549\pi$
$257$	−2633.52	−0.639200	−0.319600	+	0.947553i	$0.603549\pi$
$258$	0	0
$259$	−4438.40	−1.06482
$260$	0	0
$261$	0	0
$262$	2898.00	0.683354
$263$	4713.39	1.10509	0.552547	−	0.833482i	$-0.313656\pi$
$263$	4713.39	1.10509	0.552547	+	0.833482i	$0.313656\pi$
$264$	0	0
$265$	0	0
$266$	−12682.6	−2.92338
$267$	0	0
$268$	−19242.8	−4.38597
$269$	−4145.66	−0.939649	−0.469825	−	0.882760i	$-0.655683\pi$
$269$	−4145.66	−0.939649	−0.469825	+	0.882760i	$0.655683\pi$
$270$	0	0
$271$	−7155.58	−1.60395	−0.801975	−	0.597357i	$-0.796217\pi$
$271$	−7155.58	−1.60395	−0.801975	+	0.597357i	$0.796217\pi$
$272$	−706.270	−0.157441
$273$	0	0
$274$	2603.51	0.574028
$275$	0	0
$276$	0	0
$277$	−8076.45	−1.75187	−0.875933	−	0.482433i	$-0.839753\pi$
$277$	−8076.45	−1.75187	−0.875933	+	0.482433i	$0.839753\pi$
$278$	−1927.37	−0.415813
$279$	0	0
$280$	0	0
$281$	−1445.70	−0.306915	−0.153458	−	0.988155i	$-0.549041\pi$
$281$	−1445.70	−0.306915	−0.153458	+	0.988155i	$0.549041\pi$
$282$	0	0
$283$	8260.14	1.73503	0.867517	−	0.497408i	$-0.165715\pi$
$283$	8260.14	1.73503	0.867517	+	0.497408i	$0.165715\pi$
$284$	14152.8	2.95710
$285$	0	0
$286$	103.490	0.0213967
$287$	965.222	0.198520
$288$	0	0
$289$	−4903.56	−0.998079
$290$	0	0
$291$	0	0
$292$	1747.54	0.350230
$293$	−5784.63	−1.15338	−0.576692	−	0.816961i	$-0.695657\pi$
$293$	−5784.63	−1.15338	−0.576692	+	0.816961i	$0.695657\pi$
$294$	0	0
$295$	0	0
$296$	17450.6	3.42667
$297$	0	0
$298$	16334.5	3.17527
$299$	159.631	0.0308753
$300$	0	0
$301$	−4327.31	−0.828645
$302$	−12142.2	−2.31360
$303$	0	0
$304$	28464.7	5.37027
$305$	0	0
$306$	0	0
$307$	9458.25	1.75834	0.879171	−	0.476506i	$-0.158097\pi$
$307$	9458.25	1.75834	0.879171	+	0.476506i	$0.158097\pi$
$308$	−4473.59	−0.827618
$309$	0	0
$310$	0	0
$311$	5984.54	1.09116	0.545582	−	0.838057i	$-0.316309\pi$
$311$	5984.54	1.09116	0.545582	+	0.838057i	$0.316309\pi$
$312$	0	0
$313$	−1282.31	−0.231568	−0.115784	−	0.993274i	$-0.536938\pi$
$313$	−1282.31	−0.231568	−0.115784	+	0.993274i	$0.536938\pi$
$314$	−1543.35	−0.277377
$315$	0	0
$316$	−7252.95	−1.29117
$317$	−1727.90	−0.306147	−0.153073	−	0.988215i	$-0.548917\pi$
$317$	−1727.90	−0.306147	−0.153073	+	0.988215i	$0.548917\pi$
$318$	0	0
$319$	−1313.27	−0.230498
$320$	0	0
$321$	0	0
$322$	−9455.76	−1.63649
$323$	−380.417	−0.0655324
$324$	0	0
$325$	0	0
$326$	−18903.8	−3.21160
$327$	0	0
$328$	−3795.00	−0.638853
$329$	6711.05	1.12460
$330$	0	0
$331$	−2047.24	−0.339959	−0.169980	−	0.985448i	$-0.554370\pi$
$331$	−2047.24	−0.339959	−0.169980	+	0.985448i	$0.554370\pi$
$332$	9335.89	1.54329
$333$	0	0
$334$	−6945.89	−1.13791
$335$	0	0
$336$	0	0
$337$	7259.71	1.17348	0.586738	−	0.809777i	$-0.300412\pi$
$337$	7259.71	1.17348	0.586738	+	0.809777i	$0.300412\pi$
$338$	11937.4	1.92103
$339$	0	0
$340$	0	0
$341$	−3219.13	−0.511219
$342$	0	0
$343$	−6242.67	−0.982719
$344$	17013.9	2.66664
$345$	0	0
$346$	−2291.38	−0.356027
$347$	6537.39	1.01137	0.505685	−	0.862718i	$-0.331240\pi$
$347$	6537.39	1.01137	0.505685	+	0.862718i	$0.331240\pi$
$348$	0	0
$349$	10911.5	1.67357	0.836787	−	0.547529i	$-0.184431\pi$
$349$	10911.5	1.67357	0.836787	+	0.547529i	$0.184431\pi$
$350$	0	0
$351$	0	0
$352$	7245.20	1.09707
$353$	4974.07	0.749980	0.374990	−	0.927029i	$-0.377646\pi$
$353$	4974.07	0.749980	0.374990	+	0.927029i	$0.377646\pi$
$354$	0	0
$355$	0	0
$356$	−9830.11	−1.46347
$357$	0	0
$358$	−14320.2	−2.11410
$359$	−2468.77	−0.362943	−0.181471	−	0.983396i	$-0.558086\pi$
$359$	−2468.77	−0.362943	−0.181471	+	0.983396i	$0.558086\pi$
$360$	0	0
$361$	8472.88	1.23529
$362$	13206.9	1.91750
$363$	0	0
$364$	703.227	0.101261
$365$	0	0
$366$	0	0
$367$	−7504.87	−1.06744	−0.533721	−	0.845661i	$-0.679207\pi$
$367$	−7504.87	−1.06744	−0.533721	+	0.845661i	$0.679207\pi$
$368$	21222.4	3.00623
$369$	0	0
$370$	0	0
$371$	2218.50	0.310455
$372$	0	0
$373$	7696.58	1.06840	0.534201	−	0.845357i	$-0.320613\pi$
$373$	7696.58	1.06840	0.534201	+	0.845357i	$0.320613\pi$
$374$	−183.877	−0.0254225
$375$	0	0
$376$	−26386.1	−3.61904
$377$	206.439	0.0282020
$378$	0	0
$379$	6248.82	0.846913	0.423457	−	0.905916i	$-0.360817\pi$
$379$	6248.82	0.846913	0.423457	+	0.905916i	$0.360817\pi$
$380$	0	0
$381$	0	0
$382$	5048.09	0.676133
$383$	4505.13	0.601048	0.300524	−	0.953774i	$-0.402838\pi$
$383$	4505.13	0.601048	0.300524	+	0.953774i	$0.402838\pi$
$384$	0	0
$385$	0	0
$386$	−7361.94	−0.970759
$387$	0	0
$388$	−18199.2	−2.38124
$389$	−5426.65	−0.707305	−0.353653	−	0.935377i	$-0.615061\pi$
$389$	−5426.65	−0.707305	−0.353653	+	0.935377i	$0.615061\pi$
$390$	0	0
$391$	−283.627	−0.0366845
$392$	−842.806	−0.108592
$393$	0	0
$394$	14677.1	1.87671
$395$	0	0
$396$	0	0
$397$	11783.1	1.48961	0.744805	−	0.667282i	$-0.232542\pi$
$397$	11783.1	1.48961	0.744805	+	0.667282i	$0.232542\pi$
$398$	−9776.60	−1.23130
$399$	0	0
$400$	0	0
$401$	−10743.2	−1.33788	−0.668941	−	0.743315i	$-0.733252\pi$
$401$	−10743.2	−1.33788	−0.668941	+	0.743315i	$0.733252\pi$
$402$	0	0
$403$	506.032	0.0625490
$404$	11335.9	1.39600
$405$	0	0
$406$	−12228.4	−1.49479
$407$	2593.46	0.315855
$408$	0	0
$409$	5156.54	0.623409	0.311705	−	0.950179i	$-0.399100\pi$
$409$	5156.54	0.623409	0.311705	+	0.950179i	$0.399100\pi$
$410$	0	0
$411$	0	0
$412$	5166.95	0.617858
$413$	3863.11	0.460270
$414$	0	0
$415$	0	0
$416$	−1138.91	−0.134230
$417$	0	0
$418$	7410.75	0.867157
$419$	−9993.25	−1.16516	−0.582580	−	0.812773i	$-0.697957\pi$
$419$	−9993.25	−1.16516	−0.582580	+	0.812773i	$0.697957\pi$
$420$	0	0
$421$	−1823.62	−0.211111	−0.105555	−	0.994413i	$-0.533662\pi$
$421$	−1823.62	−0.211111	−0.105555	+	0.994413i	$0.533662\pi$
$422$	1282.82	0.147978
$423$	0	0
$424$	−8722.56	−0.999069
$425$	0	0
$426$	0	0
$427$	9241.66	1.04739
$428$	19257.5	2.17488
$429$	0	0
$430$	0	0
$431$	6639.30	0.742005	0.371002	−	0.928632i	$-0.379014\pi$
$431$	6639.30	0.742005	0.371002	+	0.928632i	$0.379014\pi$
$432$	0	0
$433$	−17034.9	−1.89063	−0.945316	−	0.326156i	$-0.894247\pi$
$433$	−17034.9	−1.89063	−0.945316	+	0.326156i	$0.894247\pi$
$434$	−29974.8	−3.31529
$435$	0	0
$436$	30762.6	3.37904
$437$	11431.0	1.25130
$438$	0	0
$439$	−8175.20	−0.888795	−0.444397	−	0.895830i	$-0.646582\pi$
$439$	−8175.20	−0.888795	−0.444397	+	0.895830i	$0.646582\pi$
$440$	0	0
$441$	0	0
$442$	28.9045	0.00311052
$443$	−3336.90	−0.357880	−0.178940	−	0.983860i	$-0.557267\pi$
$443$	−3336.90	−0.357880	−0.178940	+	0.983860i	$0.557267\pi$
$444$	0	0
$445$	0	0
$446$	23700.0	2.51620
$447$	0	0
$448$	32842.5	3.46353
$449$	8068.30	0.848033	0.424016	−	0.905654i	$-0.360620\pi$
$449$	8068.30	0.848033	0.424016	+	0.905654i	$0.360620\pi$
$450$	0	0
$451$	−564.003	−0.0588866
$452$	−2216.21	−0.230624
$453$	0	0
$454$	16485.4	1.70418
$455$	0	0
$456$	0	0
$457$	−6937.49	−0.710114	−0.355057	−	0.934845i	$-0.615538\pi$
$457$	−6937.49	−0.710114	−0.355057	+	0.934845i	$0.615538\pi$
$458$	33149.6	3.38205
$459$	0	0
$460$	0	0
$461$	−7068.73	−0.714151	−0.357075	−	0.934076i	$-0.616226\pi$
$461$	−7068.73	−0.714151	−0.357075	+	0.934076i	$0.616226\pi$
$462$	0	0
$463$	5832.99	0.585490	0.292745	−	0.956191i	$-0.405431\pi$
$463$	5832.99	0.585490	0.292745	+	0.956191i	$0.405431\pi$
$464$	27445.4	2.74595
$465$	0	0
$466$	8882.32	0.882973
$467$	6157.77	0.610166	0.305083	−	0.952326i	$-0.401316\pi$
$467$	6157.77	0.610166	0.305083	+	0.952326i	$0.401316\pi$
$468$	0	0
$469$	−16768.0	−1.65091
$470$	0	0
$471$	0	0
$472$	−15188.7	−1.48118
$473$	2528.55	0.245799
$474$	0	0
$475$	0	0
$476$	−1249.47	−0.120314
$477$	0	0
$478$	30617.2	2.92970
$479$	7042.80	0.671803	0.335901	−	0.941897i	$-0.390959\pi$
$479$	7042.80	0.671803	0.335901	+	0.941897i	$0.390959\pi$
$480$	0	0
$481$	−407.680	−0.0386458
$482$	−18970.5	−1.79270
$483$	0	0
$484$	2614.03	0.245495
$485$	0	0
$486$	0	0
$487$	−4698.83	−0.437216	−0.218608	−	0.975813i	$-0.570152\pi$
$487$	−4698.83	−0.437216	−0.218608	+	0.975813i	$0.570152\pi$
$488$	−36335.8	−3.37058
$489$	0	0
$490$	0	0
$491$	18714.1	1.72007	0.860037	−	0.510232i	$-0.170440\pi$
$491$	18714.1	1.72007	0.860037	+	0.510232i	$0.170440\pi$
$492$	0	0
$493$	−366.794	−0.0335082
$494$	−1164.94	−0.106099
$495$	0	0
$496$	67275.1	6.09021
$497$	12332.7	1.11307
$498$	0	0
$499$	−6019.34	−0.540005	−0.270003	−	0.962860i	$-0.587025\pi$
$499$	−6019.34	−0.540005	−0.270003	+	0.962860i	$0.587025\pi$
$500$	0	0
$501$	0	0
$502$	−31037.6	−2.75951
$503$	−171.433	−0.0151965	−0.00759823	−	0.999971i	$-0.502419\pi$
$503$	−171.433	−0.0151965	−0.00759823	+	0.999971i	$0.502419\pi$
$504$	0	0
$505$	0	0
$506$	5525.23	0.485427
$507$	0	0
$508$	5143.37	0.449213
$509$	5719.31	0.498043	0.249022	−	0.968498i	$-0.419891\pi$
$509$	5719.31	0.498043	0.249022	+	0.968498i	$0.419891\pi$
$510$	0	0
$511$	1522.80	0.131829
$512$	−15294.1	−1.32014
$513$	0	0
$514$	14328.7	1.22960
$515$	0	0
$516$	0	0
$517$	−3921.43	−0.333587
$518$	24148.9	2.04834
$519$	0	0
$520$	0	0
$521$	−3500.52	−0.294358	−0.147179	−	0.989110i	$-0.547019\pi$
$521$	−3500.52	−0.294358	−0.147179	+	0.989110i	$0.547019\pi$
$522$	0	0
$523$	7103.24	0.593887	0.296943	−	0.954895i	$-0.404033\pi$
$523$	7103.24	0.593887	0.296943	+	0.954895i	$0.404033\pi$
$524$	−11506.7	−0.959298
$525$	0	0
$526$	−25645.1	−2.12582
$527$	−899.100	−0.0743176
$528$	0	0
$529$	−3644.41	−0.299533
$530$	0	0
$531$	0	0
$532$	50357.1	4.10387
$533$	88.6586	0.00720494
$534$	0	0
$535$	0	0
$536$	65927.5	5.31275
$537$	0	0
$538$	22556.2	1.80756
$539$	−125.256	−0.0100095
$540$	0	0
$541$	2035.89	0.161792	0.0808961	−	0.996723i	$-0.474222\pi$
$541$	2035.89	0.161792	0.0808961	+	0.996723i	$0.474222\pi$
$542$	38932.9	3.08544
$543$	0	0
$544$	2023.58	0.159485
$545$	0	0
$546$	0	0
$547$	24546.4	1.91870	0.959348	−	0.282226i	$-0.0910728\pi$
$547$	24546.4	1.91870	0.959348	+	0.282226i	$0.0910728\pi$
$548$	−10337.4	−0.805825
$549$	0	0
$550$	0	0
$551$	14782.8	1.14296
$552$	0	0
$553$	−6320.17	−0.486006
$554$	43943.3	3.36998
$555$	0	0
$556$	7652.76	0.583722
$557$	17398.0	1.32348	0.661740	−	0.749733i	$-0.269818\pi$
$557$	17398.0	1.32348	0.661740	+	0.749733i	$0.269818\pi$
$558$	0	0
$559$	−397.477	−0.0300742
$560$	0	0
$561$	0	0
$562$	7865.93	0.590399
$563$	13198.2	0.987992	0.493996	−	0.869464i	$-0.335536\pi$
$563$	13198.2	0.987992	0.493996	+	0.869464i	$0.335536\pi$
$564$	0	0
$565$	0	0
$566$	−44942.7	−3.33760
$567$	0	0
$568$	−48488.9	−3.58195
$569$	15124.2	1.11431	0.557153	−	0.830410i	$-0.311894\pi$
$569$	15124.2	1.11431	0.557153	+	0.830410i	$0.311894\pi$
$570$	0	0
$571$	8485.10	0.621875	0.310937	−	0.950430i	$-0.399357\pi$
$571$	8485.10	0.621875	0.310937	+	0.950430i	$0.399357\pi$
$572$	−410.913	−0.0300369
$573$	0	0
$574$	−5251.69	−0.381884
$575$	0	0
$576$	0	0
$577$	−14482.9	−1.04494	−0.522471	−	0.852657i	$-0.674990\pi$
$577$	−14482.9	−1.04494	−0.522471	+	0.852657i	$0.674990\pi$
$578$	26679.8	1.91996
$579$	0	0
$580$	0	0
$581$	8135.23	0.580906
$582$	0	0
$583$	−1296.32	−0.0920896
$584$	−5987.24	−0.424236
$585$	0	0
$586$	31473.7	2.21871
$587$	−6985.26	−0.491163	−0.245581	−	0.969376i	$-0.578979\pi$
$587$	−6985.26	−0.491163	−0.245581	+	0.969376i	$0.578979\pi$
$588$	0	0
$589$	36236.3	2.53496
$590$	0	0
$591$	0	0
$592$	−54199.6	−3.76282
$593$	585.572	0.0405507	0.0202753	−	0.999794i	$-0.493546\pi$
$593$	585.572	0.0405507	0.0202753	+	0.999794i	$0.493546\pi$
$594$	0	0
$595$	0	0
$596$	−64857.2	−4.45748
$597$	0	0
$598$	−868.540	−0.0593934
$599$	19170.0	1.30762	0.653809	−	0.756659i	$-0.273170\pi$
$599$	19170.0	1.30762	0.653809	+	0.756659i	$0.273170\pi$
$600$	0	0
$601$	22098.9	1.49989	0.749943	−	0.661502i	$-0.230081\pi$
$601$	22098.9	1.49989	0.749943	+	0.661502i	$0.230081\pi$
$602$	23544.5	1.59403
$603$	0	0
$604$	48211.6	3.24785
$605$	0	0
$606$	0	0
$607$	−3717.96	−0.248612	−0.124306	−	0.992244i	$-0.539670\pi$
$607$	−3717.96	−0.248612	−0.124306	+	0.992244i	$0.539670\pi$
$608$	−81555.9	−5.44001
$609$	0	0
$610$	0	0
$611$	616.430	0.0408152
$612$	0	0
$613$	25728.4	1.69520	0.847602	−	0.530632i	$-0.178045\pi$
$613$	25728.4	1.69520	0.847602	+	0.530632i	$0.178045\pi$
$614$	−51461.5	−3.38244
$615$	0	0
$616$	15326.9	1.00250
$617$	−5085.97	−0.331853	−0.165927	−	0.986138i	$-0.553061\pi$
$617$	−5085.97	−0.331853	−0.165927	+	0.986138i	$0.553061\pi$
$618$	0	0
$619$	770.194	0.0500108	0.0250054	−	0.999687i	$-0.492040\pi$
$619$	770.194	0.0500108	0.0250054	+	0.999687i	$0.492040\pi$
$620$	0	0
$621$	0	0
$622$	−32561.4	−2.09902
$623$	−8565.90	−0.550859
$624$	0	0
$625$	0	0
$626$	6976.96	0.445456
$627$	0	0
$628$	6127.99	0.389385
$629$	724.351	0.0459170
$630$	0	0
$631$	−12396.1	−0.782063	−0.391031	−	0.920377i	$-0.627882\pi$
$631$	−12396.1	−0.782063	−0.391031	+	0.920377i	$0.627882\pi$
$632$	24849.3	1.56400
$633$	0	0
$634$	9401.35	0.588920
$635$	0	0
$636$	0	0
$637$	19.6896	0.00122469
$638$	7145.37	0.443398
$639$	0	0
$640$	0	0
$641$	−5728.50	−0.352983	−0.176491	−	0.984302i	$-0.556475\pi$
$641$	−5728.50	−0.352983	−0.176491	+	0.984302i	$0.556475\pi$
$642$	0	0
$643$	−1536.24	−0.0942202	−0.0471101	−	0.998890i	$-0.515001\pi$
$643$	−1536.24	−0.0942202	−0.0471101	+	0.998890i	$0.515001\pi$
$644$	37544.7	2.29731
$645$	0	0
$646$	2069.81	0.126062
$647$	−26647.1	−1.61917	−0.809587	−	0.586999i	$-0.800309\pi$
$647$	−26647.1	−1.61917	−0.809587	+	0.586999i	$0.800309\pi$
$648$	0	0
$649$	−2257.31	−0.136529
$650$	0	0
$651$	0	0
$652$	75058.8	4.50848
$653$	28405.1	1.70226	0.851131	−	0.524954i	$-0.175917\pi$
$653$	28405.1	1.70226	0.851131	+	0.524954i	$0.175917\pi$
$654$	0	0
$655$	0	0
$656$	11786.8	0.701522
$657$	0	0
$658$	−36514.2	−2.16333
$659$	−12108.5	−0.715752	−0.357876	−	0.933769i	$-0.616499\pi$
$659$	−12108.5	−0.715752	−0.357876	+	0.933769i	$0.616499\pi$
$660$	0	0
$661$	9862.98	0.580371	0.290186	−	0.956970i	$-0.406283\pi$
$661$	9862.98	0.580371	0.290186	+	0.956970i	$0.406283\pi$
$662$	11138.9	0.653964
$663$	0	0
$664$	−31985.6	−1.86940
$665$	0	0
$666$	0	0
$667$	11021.6	0.639819
$668$	27579.1	1.59741
$669$	0	0
$670$	0	0
$671$	−5400.13	−0.310685
$672$	0	0
$673$	2282.66	0.130743	0.0653714	−	0.997861i	$-0.479177\pi$
$673$	2282.66	0.130743	0.0653714	+	0.997861i	$0.479177\pi$
$674$	−39499.4	−2.25736
$675$	0	0
$676$	−47398.3	−2.69677
$677$	−10499.2	−0.596038	−0.298019	−	0.954560i	$-0.596326\pi$
$677$	−10499.2	−0.596038	−0.298019	+	0.954560i	$0.596326\pi$
$678$	0	0
$679$	−15858.6	−0.896316
$680$	0	0
$681$	0	0
$682$	17515.0	0.983408
$683$	−26681.2	−1.49477	−0.747386	−	0.664391i	$-0.768691\pi$
$683$	−26681.2	−1.49477	−0.747386	+	0.664391i	$0.768691\pi$
$684$	0	0
$685$	0	0
$686$	33965.8	1.89041
$687$	0	0
$688$	−52843.1	−2.92823
$689$	203.776	0.0112674
$690$	0	0
$691$	−1530.22	−0.0842434	−0.0421217	−	0.999112i	$-0.513412\pi$
$691$	−1530.22	−0.0842434	−0.0421217	+	0.999112i	$0.513412\pi$
$692$	9098.07	0.499793
$693$	0	0
$694$	−35569.4	−1.94553
$695$	0	0
$696$	0	0
$697$	−157.525	−0.00856054
$698$	−59368.3	−3.21937
$699$	0	0
$700$	0	0
$701$	−25470.4	−1.37233	−0.686165	−	0.727446i	$-0.740707\pi$
$701$	−25470.4	−1.37233	−0.686165	+	0.727446i	$0.740707\pi$
$702$	0	0
$703$	−29193.4	−1.56622
$704$	−19190.7	−1.02738
$705$	0	0
$706$	−27063.5	−1.44270
$707$	9878.04	0.525463
$708$	0	0
$709$	36906.7	1.95495	0.977475	−	0.211052i	$-0.0676891\pi$
$709$	36906.7	1.95495	0.977475	+	0.211052i	$0.0676891\pi$
$710$	0	0
$711$	0	0
$712$	33678.8	1.77271
$713$	27016.7	1.41905
$714$	0	0
$715$	0	0
$716$	56859.4	2.96779
$717$	0	0
$718$	13432.3	0.698176
$719$	−13606.1	−0.705734	−0.352867	−	0.935673i	$-0.614793\pi$
$719$	−13606.1	−0.705734	−0.352867	+	0.935673i	$0.614793\pi$
$720$	0	0
$721$	4502.45	0.232566
$722$	−46100.2	−2.37628
$723$	0	0
$724$	−52438.7	−2.69181
$725$	0	0
$726$	0	0
$727$	25132.2	1.28212	0.641060	−	0.767491i	$-0.278495\pi$
$727$	25132.2	1.28212	0.641060	+	0.767491i	$0.278495\pi$
$728$	−2409.32	−0.122658
$729$	0	0
$730$	0	0
$731$	706.223	0.0357327
$732$	0	0
$733$	10789.5	0.543681	0.271841	−	0.962342i	$-0.412368\pi$
$733$	10789.5	0.543681	0.271841	+	0.962342i	$0.412368\pi$
$734$	40833.3	2.05339
$735$	0	0
$736$	−60805.5	−3.04527
$737$	9797.97	0.489705
$738$	0	0
$739$	−34785.9	−1.73156	−0.865778	−	0.500428i	$-0.833176\pi$
$739$	−34785.9	−1.73156	−0.865778	+	0.500428i	$0.833176\pi$
$740$	0	0
$741$	0	0
$742$	−12070.7	−0.597208
$743$	19790.8	0.977195	0.488597	−	0.872509i	$-0.337509\pi$
$743$	19790.8	0.977195	0.488597	+	0.872509i	$0.337509\pi$
$744$	0	0
$745$	0	0
$746$	−41876.4	−2.05523
$747$	0	0
$748$	730.095	0.0356884
$749$	16780.9	0.818639
$750$	0	0
$751$	17836.9	0.866681	0.433341	−	0.901230i	$-0.357335\pi$
$751$	17836.9	0.866681	0.433341	+	0.901230i	$0.357335\pi$
$752$	81952.2	3.97405
$753$	0	0
$754$	−1123.22	−0.0542509
$755$	0	0
$756$	0	0
$757$	−26599.8	−1.27713	−0.638564	−	0.769569i	$-0.720471\pi$
$757$	−26599.8	−1.27713	−0.638564	+	0.769569i	$0.720471\pi$
$758$	−33999.3	−1.62917
$759$	0	0
$760$	0	0
$761$	−8248.59	−0.392919	−0.196459	−	0.980512i	$-0.562944\pi$
$761$	−8248.59	−0.392919	−0.196459	+	0.980512i	$0.562944\pi$
$762$	0	0
$763$	26806.3	1.27189
$764$	−20043.8	−0.949161
$765$	0	0
$766$	−24512.0	−1.15621
$767$	354.839	0.0167047
$768$	0	0
$769$	1496.76	0.0701880	0.0350940	−	0.999384i	$-0.488827\pi$
$769$	1496.76	0.0701880	0.0350940	+	0.999384i	$0.488827\pi$
$770$	0	0
$771$	0	0
$772$	29231.1	1.36276
$773$	459.376	0.0213746	0.0106873	−	0.999943i	$-0.496598\pi$
$773$	459.376	0.0213746	0.0106873	+	0.999943i	$0.496598\pi$
$774$	0	0
$775$	0	0
$776$	62351.9	2.88441
$777$	0	0
$778$	29525.9	1.36061
$779$	6348.72	0.291998
$780$	0	0
$781$	−7206.29	−0.330168
$782$	1543.19	0.0705682
$783$	0	0
$784$	2617.66	0.119245
$785$	0	0
$786$	0	0
$787$	21110.4	0.956170	0.478085	−	0.878314i	$-0.341331\pi$
$787$	21110.4	0.956170	0.478085	+	0.878314i	$0.341331\pi$
$788$	−58276.5	−2.63454
$789$	0	0
$790$	0	0
$791$	−1931.19	−0.0868082
$792$	0	0
$793$	848.875	0.0380132
$794$	−64110.7	−2.86549
$795$	0	0
$796$	38818.7	1.72851
$797$	−44355.5	−1.97133	−0.985666	−	0.168711i	$-0.946040\pi$
$797$	−44355.5	−1.97133	−0.985666	+	0.168711i	$0.946040\pi$
$798$	0	0
$799$	−1095.25	−0.0484946
$800$	0	0
$801$	0	0
$802$	58452.9	2.57362
$803$	−889.807	−0.0391041
$804$	0	0
$805$	0	0
$806$	−2753.28	−0.120323
$807$	0	0
$808$	−38837.8	−1.69098
$809$	15021.8	0.652830	0.326415	−	0.945227i	$-0.394159\pi$
$809$	15021.8	0.652830	0.326415	+	0.945227i	$0.394159\pi$
$810$	0	0
$811$	20178.0	0.873667	0.436833	−	0.899542i	$-0.356100\pi$
$811$	20178.0	0.873667	0.436833	+	0.899542i	$0.356100\pi$
$812$	48553.8	2.09841
$813$	0	0
$814$	−14110.8	−0.607596
$815$	0	0
$816$	0	0
$817$	−28462.8	−1.21883
$818$	−28056.3	−1.19922
$819$	0	0
$820$	0	0
$821$	−8507.06	−0.361630	−0.180815	−	0.983517i	$-0.557874\pi$
$821$	−8507.06	−0.361630	−0.180815	+	0.983517i	$0.557874\pi$
$822$	0	0
$823$	13562.9	0.574451	0.287225	−	0.957863i	$-0.407267\pi$
$823$	13562.9	0.574451	0.287225	+	0.957863i	$0.407267\pi$
$824$	−17702.4	−0.748415
$825$	0	0
$826$	−21018.9	−0.885399
$827$	42090.2	1.76979	0.884897	−	0.465786i	$-0.154229\pi$
$827$	42090.2	1.76979	0.884897	+	0.465786i	$0.154229\pi$
$828$	0	0
$829$	19524.7	0.817998	0.408999	−	0.912535i	$-0.365878\pi$
$829$	19524.7	0.817998	0.408999	+	0.912535i	$0.365878\pi$
$830$	0	0
$831$	0	0
$832$	3016.69	0.125703
$833$	−34.9838	−0.00145512
$834$	0	0
$835$	0	0
$836$	−29424.9	−1.21732
$837$	0	0
$838$	54372.4	2.24136
$839$	−40733.8	−1.67615	−0.838073	−	0.545559i	$-0.816317\pi$
$839$	−40733.8	−1.67615	−0.838073	+	0.545559i	$0.816317\pi$
$840$	0	0
$841$	−10135.5	−0.415578
$842$	9922.13	0.406104
$843$	0	0
$844$	−5093.52	−0.207732
$845$	0	0
$846$	0	0
$847$	2277.85	0.0924058
$848$	27091.3	1.09707
$849$	0	0
$850$	0	0
$851$	−21765.7	−0.876756
$852$	0	0
$853$	7476.59	0.300110	0.150055	−	0.988678i	$-0.452055\pi$
$853$	7476.59	0.300110	0.150055	+	0.988678i	$0.452055\pi$
$854$	−50283.1	−2.01481
$855$	0	0
$856$	−65978.0	−2.63444
$857$	−46353.7	−1.84762	−0.923812	−	0.382847i	$-0.874944\pi$
$857$	−46353.7	−1.84762	−0.923812	+	0.382847i	$0.874944\pi$
$858$	0	0
$859$	−3254.04	−0.129251	−0.0646254	−	0.997910i	$-0.520585\pi$
$859$	−3254.04	−0.129251	−0.0646254	+	0.997910i	$0.520585\pi$
$860$	0	0
$861$	0	0
$862$	−36123.9	−1.42736
$863$	38864.6	1.53299	0.766493	−	0.642252i	$-0.222000\pi$
$863$	38864.6	1.53299	0.766493	+	0.642252i	$0.222000\pi$
$864$	0	0
$865$	0	0
$866$	92685.2	3.63692
$867$	0	0
$868$	119017.	4.65404
$869$	3693.03	0.144163
$870$	0	0
$871$	−1540.19	−0.0599168
$872$	−105395.	−4.09304
$873$	0	0
$874$	−62195.0	−2.40707
$875$	0	0
$876$	0	0
$877$	−35144.3	−1.35318	−0.676589	−	0.736361i	$-0.736543\pi$
$877$	−35144.3	−1.35318	−0.676589	+	0.736361i	$0.736543\pi$
$878$	44480.5	1.70973
$879$	0	0
$880$	0	0
$881$	−31530.7	−1.20578	−0.602892	−	0.797823i	$-0.705985\pi$
$881$	−31530.7	−1.20578	−0.602892	+	0.797823i	$0.705985\pi$
$882$	0	0
$883$	35263.1	1.34394	0.671970	−	0.740579i	$-0.265448\pi$
$883$	35263.1	1.34394	0.671970	+	0.740579i	$0.265448\pi$
$884$	−114.768	−0.00436657
$885$	0	0
$886$	18155.8	0.688436
$887$	−10912.5	−0.413086	−0.206543	−	0.978438i	$-0.566221\pi$
$887$	−10912.5	−0.413086	−0.206543	+	0.978438i	$0.566221\pi$
$888$	0	0
$889$	4481.90	0.169087
$890$	0	0
$891$	0	0
$892$	−94102.4	−3.53227
$893$	44141.7	1.65414
$894$	0	0
$895$	0	0
$896$	−79499.1	−2.96415
$897$	0	0
$898$	−43898.9	−1.63132
$899$	34938.6	1.29618
$900$	0	0
$901$	−362.062	−0.0133874
$902$	3068.69	0.113277
$903$	0	0
$904$	7592.94	0.279356
$905$	0	0
$906$	0	0
$907$	6918.62	0.253285	0.126642	−	0.991948i	$-0.459580\pi$
$907$	6918.62	0.253285	0.126642	+	0.991948i	$0.459580\pi$
$908$	−65456.5	−2.39235
$909$	0	0
$910$	0	0
$911$	−9101.44	−0.331003	−0.165502	−	0.986210i	$-0.552924\pi$
$911$	−9101.44	−0.331003	−0.165502	+	0.986210i	$0.552924\pi$
$912$	0	0
$913$	−4753.61	−0.172313
$914$	37746.3	1.36601
$915$	0	0
$916$	−131623.	−4.74775
$917$	−10026.9	−0.361086
$918$	0	0
$919$	34392.5	1.23450	0.617248	−	0.786768i	$-0.288247\pi$
$919$	34392.5	1.23450	0.617248	+	0.786768i	$0.288247\pi$
$920$	0	0
$921$	0	0
$922$	38460.3	1.37378
$923$	1132.80	0.0403970
$924$	0	0
$925$	0	0
$926$	−31736.8	−1.12628
$927$	0	0
$928$	−78635.3	−2.78161
$929$	−4364.57	−0.154141	−0.0770704	−	0.997026i	$-0.524557\pi$
$929$	−4364.57	−0.154141	−0.0770704	+	0.997026i	$0.524557\pi$
$930$	0	0
$931$	1409.95	0.0496338
$932$	−35267.9	−1.23952
$933$	0	0
$934$	−33503.9	−1.17375
$935$	0	0
$936$	0	0
$937$	1816.07	0.0633174	0.0316587	−	0.999499i	$-0.489921\pi$
$937$	1816.07	0.0633174	0.0316587	+	0.999499i	$0.489921\pi$
$938$	91233.4	3.17577
$939$	0	0
$940$	0	0
$941$	35113.5	1.21644	0.608218	−	0.793770i	$-0.291885\pi$
$941$	35113.5	1.21644	0.608218	+	0.793770i	$0.291885\pi$
$942$	0	0
$943$	4733.41	0.163458
$944$	47174.5	1.62648
$945$	0	0
$946$	−13757.6	−0.472833
$947$	10104.6	0.346731	0.173365	−	0.984858i	$-0.444536\pi$
$947$	10104.6	0.346731	0.173365	+	0.984858i	$0.444536\pi$
$948$	0	0
$949$	139.874	0.00478450
$950$	0	0
$951$	0	0
$952$	4280.79	0.145737
$953$	46422.4	1.57793	0.788965	−	0.614438i	$-0.210617\pi$
$953$	46422.4	1.57793	0.788965	+	0.614438i	$0.210617\pi$
$954$	0	0
$955$	0	0
$956$	−121568.	−4.11274
$957$	0	0
$958$	−38319.2	−1.29232
$959$	−9007.95	−0.303318
$960$	0	0
$961$	55852.0	1.87479
$962$	2218.15	0.0743410
$963$	0	0
$964$	75323.6	2.51661
$965$	0	0
$966$	0	0
$967$	928.679	0.0308835	0.0154417	−	0.999881i	$-0.495085\pi$
$967$	928.679	0.0308835	0.0154417	+	0.999881i	$0.495085\pi$
$968$	−8955.88	−0.297369
$969$	0	0
$970$	0	0
$971$	51195.8	1.69202	0.846011	−	0.533166i	$-0.178998\pi$
$971$	51195.8	1.69202	0.846011	+	0.533166i	$0.178998\pi$
$972$	0	0
$973$	6668.56	0.219717
$974$	25565.9	0.841052
$975$	0	0
$976$	112855.	3.70122
$977$	31543.9	1.03294	0.516469	−	0.856306i	$-0.327246\pi$
$977$	31543.9	1.03294	0.516469	+	0.856306i	$0.327246\pi$
$978$	0	0
$979$	5005.26	0.163400
$980$	0	0
$981$	0	0
$982$	−101822.	−3.30883
$983$	30743.8	0.997532	0.498766	−	0.866737i	$-0.333787\pi$
$983$	30743.8	0.997532	0.498766	+	0.866737i	$0.333787\pi$
$984$	0	0
$985$	0	0
$986$	1995.69	0.0644582
$987$	0	0
$988$	4625.46	0.148943
$989$	−21221.0	−0.682293
$990$	0	0
$991$	−31389.1	−1.00616	−0.503082	−	0.864239i	$-0.667801\pi$
$991$	−31389.1	−1.00616	−0.503082	+	0.864239i	$0.667801\pi$
$992$	−192754.	−6.16930
$993$	0	0
$994$	−67101.1	−2.14117
$995$	0	0
$996$	0	0
$997$	10343.4	0.328565	0.164282	−	0.986413i	$-0.447469\pi$
$997$	10343.4	0.328565	0.164282	+	0.986413i	$0.447469\pi$
$998$	32750.7	1.03878
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2475.4.a.bw.1.1		10
3.2	odd	2		275.4.a.k.1.10		10
5.2	odd	4		495.4.c.b.199.1		10
5.3	odd	4		495.4.c.b.199.10		10
5.4	even	2	inner	2475.4.a.bw.1.10		10
15.2	even	4		55.4.b.b.34.10	yes	10
15.8	even	4		55.4.b.b.34.1	✓	10
15.14	odd	2		275.4.a.k.1.1		10
60.23	odd	4		880.4.b.i.529.8		10
60.47	odd	4		880.4.b.i.529.3		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
55.4.b.b.34.1	✓	10	15.8	even	4
55.4.b.b.34.10	yes	10	15.2	even	4
275.4.a.k.1.1		10	15.14	odd	2
275.4.a.k.1.10		10	3.2	odd	2
495.4.c.b.199.1		10	5.2	odd	4
495.4.c.b.199.10		10	5.3	odd	4
880.4.b.i.529.3		10	60.47	odd	4
880.4.b.i.529.8		10	60.23	odd	4
2475.4.a.bw.1.1		10	1.1	even	1	trivial
2475.4.a.bw.1.10		10	5.4	even	2	inner

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)

\(f(q)\)	\(=\)	\(q-5.44091 q^{2} +21.6035 q^{4} +18.8252 q^{7} -74.0156 q^{8} -11.0000 q^{11} +1.72915 q^{13} -102.426 q^{14} +229.884 q^{16} -3.07229 q^{17} +123.822 q^{19} +59.8500 q^{22} +92.3179 q^{23} -9.40815 q^{26} +406.690 q^{28} +119.388 q^{29} +292.648 q^{31} -658.654 q^{32} +16.7161 q^{34} -235.769 q^{37} -673.704 q^{38} +51.2730 q^{41} -229.869 q^{43} -237.639 q^{44} -502.293 q^{46} +356.494 q^{47} +11.3869 q^{49} +37.3557 q^{52} +117.848 q^{53} -1393.36 q^{56} -649.579 q^{58} +205.210 q^{59} +490.921 q^{61} -1592.27 q^{62} +1744.61 q^{64} -890.724 q^{67} -66.3723 q^{68} +655.118 q^{71} +80.8916 q^{73} +1282.80 q^{74} +2674.99 q^{76} -207.077 q^{77} -335.730 q^{79} -278.972 q^{82} +432.146 q^{83} +1250.69 q^{86} +814.171 q^{88} -455.024 q^{89} +32.5515 q^{91} +1994.39 q^{92} -1939.65 q^{94} -842.417 q^{97} -61.9550 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 64 q^{4} - 110 q^{11} - 34 q^{14} + 468 q^{16} + 90 q^{19} - 392 q^{26} + 58 q^{29} + 1242 q^{31} - 66 q^{34} - 416 q^{41} - 704 q^{44} + 1816 q^{46} + 1980 q^{49} - 2626 q^{56} - 476 q^{59} + 1650 q^{61}+ \cdots - 2860 q^{94}+O(q^{100}) \) 10 * q + 64 * q^4 - 110 * q^11 - 34 * q^14 + 468 * q^16 + 90 * q^19 - 392 * q^26 + 58 * q^29 + 1242 * q^31 - 66 * q^34 - 416 * q^41 - 704 * q^44 + 1816 * q^46 + 1980 * q^49 - 2626 * q^56 - 476 * q^59 + 1650 * q^61 + 2576 * q^64 + 498 * q^71 + 5374 * q^74 + 1410 * q^76 - 416 * q^79 + 6872 * q^86 - 1918 * q^89 + 1384 * q^91 - 2860 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more