LMFDB - Embedded newform 2205.4.a.z.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2205,4,Mod(1,2205)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2205, 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("2205.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2205 = 3^{2} \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2205.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:	$130.099211563$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{65}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 16 $ x^2 - x - 16
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 105)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-3.53113$ of defining polynomial
Character	$\chi$	$=$	2205.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.53113 q^{2} +4.46887 q^{4} +5.00000 q^{5} -12.4689 q^{8} +O(q^{10})$	$q+3.53113 q^{2} +4.46887 q^{4} +5.00000 q^{5} -12.4689 q^{8} +17.6556 q^{10} +2.93774 q^{11} +19.0623 q^{13} -79.7802 q^{16} +122.498 q^{17} -107.436 q^{19} +22.3444 q^{20} +10.3735 q^{22} -210.623 q^{23} +25.0000 q^{25} +67.3113 q^{26} -95.4942 q^{29} +94.3074 q^{31} -181.963 q^{32} +432.556 q^{34} +97.1206 q^{37} -379.370 q^{38} -62.3444 q^{40} -491.113 q^{41} -43.0039 q^{43} +13.1284 q^{44} -743.735 q^{46} +473.494 q^{47} +88.2782 q^{50} +85.1868 q^{52} +183.677 q^{53} +14.6887 q^{55} -337.202 q^{58} -760.615 q^{59} +198.747 q^{61} +333.012 q^{62} -4.29373 q^{64} +95.3113 q^{65} -309.992 q^{67} +547.428 q^{68} -665.693 q^{71} -621.288 q^{73} +342.945 q^{74} -480.117 q^{76} -24.7626 q^{79} -398.901 q^{80} -1734.18 q^{82} -406.724 q^{83} +612.490 q^{85} -151.852 q^{86} -36.6303 q^{88} +261.751 q^{89} -941.245 q^{92} +1671.97 q^{94} -537.179 q^{95} +1004.77 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{2} + 17 q^{4} + 10 q^{5} - 33 q^{8}+O(q^{10}) $ 2 * q - q^2 + 17 * q^4 + 10 * q^5 - 33 * q^8	$ 2 q - q^{2} + 17 q^{4} + 10 q^{5} - 33 q^{8} - 5 q^{10} + 22 q^{11} + 22 q^{13} - 87 q^{16} + 116 q^{17} - 102 q^{19} + 85 q^{20} - 76 q^{22} - 260 q^{23} + 50 q^{25} + 54 q^{26} + 196 q^{29} - 150 q^{31} + 15 q^{32} + 462 q^{34} - 96 q^{37} - 404 q^{38} - 165 q^{40} - 176 q^{41} - 344 q^{43} + 252 q^{44} - 520 q^{46} + 560 q^{47} - 25 q^{50} + 122 q^{52} - 326 q^{53} + 110 q^{55} - 1658 q^{58} - 844 q^{59} + 204 q^{61} + 1440 q^{62} - 839 q^{64} + 110 q^{65} - 104 q^{67} + 466 q^{68} - 1670 q^{71} + 386 q^{73} + 1218 q^{74} - 412 q^{76} - 888 q^{79} - 435 q^{80} - 3162 q^{82} + 928 q^{83} + 580 q^{85} + 1212 q^{86} - 428 q^{88} + 588 q^{89} - 1560 q^{92} + 1280 q^{94} - 510 q^{95} - 522 q^{97}+O(q^{100}) $ 2 * q - q^2 + 17 * q^4 + 10 * q^5 - 33 * q^8 - 5 * q^10 + 22 * q^11 + 22 * q^13 - 87 * q^16 + 116 * q^17 - 102 * q^19 + 85 * q^20 - 76 * q^22 - 260 * q^23 + 50 * q^25 + 54 * q^26 + 196 * q^29 - 150 * q^31 + 15 * q^32 + 462 * q^34 - 96 * q^37 - 404 * q^38 - 165 * q^40 - 176 * q^41 - 344 * q^43 + 252 * q^44 - 520 * q^46 + 560 * q^47 - 25 * q^50 + 122 * q^52 - 326 * q^53 + 110 * q^55 - 1658 * q^58 - 844 * q^59 + 204 * q^61 + 1440 * q^62 - 839 * q^64 + 110 * q^65 - 104 * q^67 + 466 * q^68 - 1670 * q^71 + 386 * q^73 + 1218 * q^74 - 412 * q^76 - 888 * q^79 - 435 * q^80 - 3162 * q^82 + 928 * q^83 + 580 * q^85 + 1212 * q^86 - 428 * q^88 + 588 * q^89 - 1560 * q^92 + 1280 * q^94 - 510 * q^95 - 522 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	3.53113	1.24844	0.624221	−	0.781248i	$-0.285416\pi$
$2$	3.53113	1.24844	0.624221	+	0.781248i	$0.285416\pi$
$3$	0	0
$3$	0	0
$4$	4.46887	0.558609
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−12.4689	−0.551051
$9$	0	0
$10$	17.6556	0.558320
$11$	2.93774	0.0805239	0.0402619	−	0.999189i	$-0.487181\pi$
$11$	2.93774	0.0805239	0.0402619	+	0.999189i	$0.487181\pi$
$12$	0	0
$13$	19.0623	0.406686	0.203343	−	0.979108i	$-0.434819\pi$
$13$	19.0623	0.406686	0.203343	+	0.979108i	$0.434819\pi$
$14$	0	0
$15$	0	0
$16$	−79.7802	−1.24656
$17$	122.498	1.74766	0.873828	−	0.486236i	$-0.161630\pi$
$17$	122.498	1.74766	0.873828	+	0.486236i	$0.161630\pi$
$18$	0	0
$19$	−107.436	−1.29723	−0.648617	−	0.761115i	$-0.724652\pi$
$19$	−107.436	−1.29723	−0.648617	+	0.761115i	$0.724652\pi$
$20$	22.3444	0.249817
$21$	0	0
$22$	10.3735	0.100529
$23$	−210.623	−1.90947	−0.954736	−	0.297455i	$-0.903862\pi$
$23$	−210.623	−1.90947	−0.954736	+	0.297455i	$0.903862\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	67.3113	0.507724
$27$	0	0
$28$	0	0
$29$	−95.4942	−0.611477	−0.305738	−	0.952116i	$-0.598903\pi$
$29$	−95.4942	−0.611477	−0.305738	+	0.952116i	$0.598903\pi$
$30$	0	0
$31$	94.3074	0.546391	0.273195	−	0.961959i	$-0.411919\pi$
$31$	94.3074	0.546391	0.273195	+	0.961959i	$0.411919\pi$
$32$	−181.963	−1.00521
$33$	0	0
$34$	432.556	2.18185
$35$	0	0
$36$	0	0
$37$	97.1206	0.431528	0.215764	−	0.976446i	$-0.430776\pi$
$37$	97.1206	0.431528	0.215764	+	0.976446i	$0.430776\pi$
$38$	−379.370	−1.61952
$39$	0	0
$40$	−62.3444	−0.246438
$41$	−491.113	−1.87071	−0.935353	−	0.353716i	$-0.884918\pi$
$41$	−491.113	−1.87071	−0.935353	+	0.353716i	$0.884918\pi$
$42$	0	0
$43$	−43.0039	−0.152512	−0.0762562	−	0.997088i	$-0.524297\pi$
$43$	−43.0039	−0.152512	−0.0762562	+	0.997088i	$0.524297\pi$
$44$	13.1284	0.0449814
$45$	0	0
$46$	−743.735	−2.38387
$47$	473.494	1.46949	0.734747	−	0.678341i	$-0.237301\pi$
$47$	473.494	1.46949	0.734747	+	0.678341i	$0.237301\pi$
$48$	0	0
$49$	0	0
$50$	88.2782	0.249689
$51$	0	0
$52$	85.1868	0.227178
$53$	183.677	0.476038	0.238019	−	0.971261i	$-0.423502\pi$
$53$	183.677	0.476038	0.238019	+	0.971261i	$0.423502\pi$
$54$	0	0
$55$	14.6887	0.0360114
$56$	0	0
$57$	0	0
$58$	−337.202	−0.763394
$59$	−760.615	−1.67837	−0.839183	−	0.543849i	$-0.816966\pi$
$59$	−760.615	−1.67837	−0.839183	+	0.543849i	$0.816966\pi$
$60$	0	0
$61$	198.747	0.417163	0.208582	−	0.978005i	$-0.433115\pi$
$61$	198.747	0.417163	0.208582	+	0.978005i	$0.433115\pi$
$62$	333.012	0.682137
$63$	0	0
$64$	−4.29373	−0.00838618
$65$	95.3113	0.181876
$66$	0	0
$67$	−309.992	−0.565247	−0.282624	−	0.959231i	$-0.591205\pi$
$67$	−309.992	−0.565247	−0.282624	+	0.959231i	$0.591205\pi$
$68$	547.428	0.976256
$69$	0	0
$70$	0	0
$71$	−665.693	−1.11272	−0.556360	−	0.830941i	$-0.687803\pi$
$71$	−665.693	−1.11272	−0.556360	+	0.830941i	$0.687803\pi$
$72$	0	0
$73$	−621.288	−0.996113	−0.498057	−	0.867145i	$-0.665953\pi$
$73$	−621.288	−0.996113	−0.498057	+	0.867145i	$0.665953\pi$
$74$	342.945	0.538738
$75$	0	0
$76$	−480.117	−0.724647
$77$	0	0
$78$	0	0
$79$	−24.7626	−0.0352659	−0.0176330	−	0.999845i	$-0.505613\pi$
$79$	−24.7626	−0.0352659	−0.0176330	+	0.999845i	$0.505613\pi$
$80$	−398.901	−0.557481
$81$	0	0
$82$	−1734.18	−2.33547
$83$	−406.724	−0.537876	−0.268938	−	0.963157i	$-0.586673\pi$
$83$	−406.724	−0.537876	−0.268938	+	0.963157i	$0.586673\pi$
$84$	0	0
$85$	612.490	0.781575
$86$	−151.852	−0.190403
$87$	0	0
$88$	−36.6303	−0.0443728
$89$	261.751	0.311748	0.155874	−	0.987777i	$-0.450181\pi$
$89$	261.751	0.311748	0.155874	+	0.987777i	$0.450181\pi$
$90$	0	0
$91$	0	0
$92$	−941.245	−1.06665
$93$	0	0
$94$	1671.97	1.83458
$95$	−537.179	−0.580141
$96$	0	0
$97$	1004.77	1.05175	0.525873	−	0.850563i	$-0.323739\pi$
$97$	1004.77	1.05175	0.525873	+	0.850563i	$0.323739\pi$
$98$	0	0
$99$	0	0
$100$	111.722	0.111722
$101$	−128.872	−0.126962	−0.0634812	−	0.997983i	$-0.520220\pi$
$101$	−128.872	−0.126962	−0.0634812	+	0.997983i	$0.520220\pi$
$102$	0	0
$103$	−806.008	−0.771051	−0.385526	−	0.922697i	$-0.625980\pi$
$103$	−806.008	−0.771051	−0.385526	+	0.922697i	$0.625980\pi$
$104$	−237.685	−0.224105
$105$	0	0
$106$	648.587	0.594305
$107$	769.712	0.695429	0.347714	−	0.937600i	$-0.386958\pi$
$107$	769.712	0.695429	0.347714	+	0.937600i	$0.386958\pi$
$108$	0	0
$109$	−780.856	−0.686169	−0.343085	−	0.939304i	$-0.611472\pi$
$109$	−780.856	−0.686169	−0.343085	+	0.939304i	$0.611472\pi$
$110$	51.8677	0.0449581
$111$	0	0
$112$	0	0
$113$	1115.65	0.928771	0.464386	−	0.885633i	$-0.346275\pi$
$113$	1115.65	0.928771	0.464386	+	0.885633i	$0.346275\pi$
$114$	0	0
$115$	−1053.11	−0.853942
$116$	−426.751	−0.341576
$117$	0	0
$118$	−2685.83	−2.09534
$119$	0	0
$120$	0	0
$121$	−1322.37	−0.993516
$122$	701.802	0.520804
$123$	0	0
$124$	421.448	0.305219
$125$	125.000	0.0894427
$126$	0	0
$127$	−1875.98	−1.31076	−0.655381	−	0.755299i	$-0.727492\pi$
$127$	−1875.98	−1.31076	−0.655381	+	0.755299i	$0.727492\pi$
$128$	1440.54	0.994744
$129$	0	0
$130$	336.556	0.227061
$131$	364.203	0.242905	0.121452	−	0.992597i	$-0.461245\pi$
$131$	364.203	0.242905	0.121452	+	0.992597i	$0.461245\pi$
$132$	0	0
$133$	0	0
$134$	−1094.62	−0.705679
$135$	0	0
$136$	−1527.41	−0.963048
$137$	−1603.13	−0.999743	−0.499872	−	0.866099i	$-0.666620\pi$
$137$	−1603.13	−0.999743	−0.499872	+	0.866099i	$0.666620\pi$
$138$	0	0
$139$	−2431.12	−1.48349	−0.741746	−	0.670681i	$-0.766002\pi$
$139$	−2431.12	−1.48349	−0.741746	+	0.670681i	$0.766002\pi$
$140$	0	0
$141$	0	0
$142$	−2350.65	−1.38917
$143$	56.0000	0.0327479
$144$	0	0
$145$	−477.471	−0.273461
$146$	−2193.85	−1.24359
$147$	0	0
$148$	434.020	0.241055
$149$	−2341.57	−1.28744	−0.643722	−	0.765260i	$-0.722611\pi$
$149$	−2341.57	−1.28744	−0.643722	+	0.765260i	$0.722611\pi$
$150$	0	0
$151$	−2104.07	−1.13395	−0.566976	−	0.823734i	$-0.691887\pi$
$151$	−2104.07	−1.13395	−0.566976	+	0.823734i	$0.691887\pi$
$152$	1339.60	0.714843
$153$	0	0
$154$	0	0
$155$	471.537	0.244353
$156$	0	0
$157$	593.467	0.301680	0.150840	−	0.988558i	$-0.451802\pi$
$157$	593.467	0.301680	0.150840	+	0.988558i	$0.451802\pi$
$158$	−87.4399	−0.0440275
$159$	0	0
$160$	−909.815	−0.449545
$161$	0	0
$162$	0	0
$163$	2178.71	1.04693	0.523465	−	0.852047i	$-0.324639\pi$
$163$	2178.71	1.04693	0.523465	+	0.852047i	$0.324639\pi$
$164$	−2194.72	−1.04499
$165$	0	0
$166$	−1436.19	−0.671508
$167$	799.502	0.370463	0.185231	−	0.982695i	$-0.440697\pi$
$167$	799.502	0.370463	0.185231	+	0.982695i	$0.440697\pi$
$168$	0	0
$169$	−1833.63	−0.834606
$170$	2162.78	0.975752
$171$	0	0
$172$	−192.179	−0.0851947
$173$	−1444.36	−0.634754	−0.317377	−	0.948299i	$-0.602802\pi$
$173$	−1444.36	−0.634754	−0.317377	+	0.948299i	$0.602802\pi$
$174$	0	0
$175$	0	0
$176$	−234.374	−0.100378
$177$	0	0
$178$	924.276	0.389199
$179$	−3343.49	−1.39611	−0.698056	−	0.716043i	$-0.745952\pi$
$179$	−3343.49	−1.39611	−0.698056	+	0.716043i	$0.745952\pi$
$180$	0	0
$181$	−2251.81	−0.924729	−0.462365	−	0.886690i	$-0.652999\pi$
$181$	−2251.81	−0.924729	−0.462365	+	0.886690i	$0.652999\pi$
$182$	0	0
$183$	0	0
$184$	2626.23	1.05222
$185$	485.603	0.192985
$186$	0	0
$187$	359.868	0.140728
$188$	2115.98	0.820873
$189$	0	0
$190$	−1896.85	−0.724273
$191$	1001.93	0.379565	0.189782	−	0.981826i	$-0.439222\pi$
$191$	1001.93	0.379565	0.189782	+	0.981826i	$0.439222\pi$
$192$	0	0
$193$	−4054.97	−1.51235	−0.756173	−	0.654372i	$-0.772933\pi$
$193$	−4054.97	−1.51235	−0.756173	+	0.654372i	$0.772933\pi$
$194$	3547.99	1.31304
$195$	0	0
$196$	0	0
$197$	5140.23	1.85902	0.929508	−	0.368802i	$-0.120232\pi$
$197$	5140.23	1.85902	0.929508	+	0.368802i	$0.120232\pi$
$198$	0	0
$199$	−585.631	−0.208614	−0.104307	−	0.994545i	$-0.533263\pi$
$199$	−585.631	−0.208614	−0.104307	+	0.994545i	$0.533263\pi$
$200$	−311.722	−0.110210
$201$	0	0
$202$	−455.062	−0.158505
$203$	0	0
$204$	0	0
$205$	−2455.56	−0.836605
$206$	−2846.12	−0.962614
$207$	0	0
$208$	−1520.79	−0.506961
$209$	−315.619	−0.104458
$210$	0	0
$211$	−1055.16	−0.344266	−0.172133	−	0.985074i	$-0.555066\pi$
$211$	−1055.16	−0.344266	−0.172133	+	0.985074i	$0.555066\pi$
$212$	820.829	0.265919
$213$	0	0
$214$	2717.95	0.868203
$215$	−215.019	−0.0682056
$216$	0	0
$217$	0	0
$218$	−2757.30	−0.856643
$219$	0	0
$220$	65.6420	0.0201163
$221$	2335.09	0.710747
$222$	0	0
$223$	−4675.85	−1.40412	−0.702059	−	0.712119i	$-0.747736\pi$
$223$	−4675.85	−1.40412	−0.702059	+	0.712119i	$0.747736\pi$
$224$	0	0
$225$	0	0
$226$	3939.49	1.15952
$227$	5443.11	1.59151	0.795754	−	0.605621i	$-0.207075\pi$
$227$	5443.11	1.59151	0.795754	+	0.605621i	$0.207075\pi$
$228$	0	0
$229$	536.303	0.154759	0.0773797	−	0.997002i	$-0.475345\pi$
$229$	536.303	0.154759	0.0773797	+	0.997002i	$0.475345\pi$
$230$	−3718.68	−1.06610
$231$	0	0
$232$	1190.70	0.336955
$233$	183.490	0.0515916	0.0257958	−	0.999667i	$-0.491788\pi$
$233$	183.490	0.0515916	0.0257958	+	0.999667i	$0.491788\pi$
$234$	0	0
$235$	2367.47	0.657178
$236$	−3399.09	−0.937550
$237$	0	0
$238$	0	0
$239$	−643.218	−0.174085	−0.0870425	−	0.996205i	$-0.527742\pi$
$239$	−643.218	−0.174085	−0.0870425	+	0.996205i	$0.527742\pi$
$240$	0	0
$241$	5755.61	1.53839	0.769194	−	0.639015i	$-0.220658\pi$
$241$	5755.61	1.53839	0.769194	+	0.639015i	$0.220658\pi$
$242$	−4669.46	−1.24035
$243$	0	0
$244$	888.175	0.233031
$245$	0	0
$246$	0	0
$247$	−2047.97	−0.527567
$248$	−1175.91	−0.301089
$249$	0	0
$250$	441.391	0.111664
$251$	−5132.27	−1.29062	−0.645311	−	0.763920i	$-0.723272\pi$
$251$	−5132.27	−1.29062	−0.645311	+	0.763920i	$0.723272\pi$
$252$	0	0
$253$	−618.755	−0.153758
$254$	−6624.34	−1.63641
$255$	0	0
$256$	5121.09	1.25027
$257$	5041.74	1.22372	0.611859	−	0.790967i	$-0.290422\pi$
$257$	5041.74	1.22372	0.611859	+	0.790967i	$0.290422\pi$
$258$	0	0
$259$	0	0
$260$	425.934	0.101597
$261$	0	0
$262$	1286.05	0.303253
$263$	−7577.00	−1.77649	−0.888246	−	0.459367i	$-0.848076\pi$
$263$	−7577.00	−1.77649	−0.888246	+	0.459367i	$0.848076\pi$
$264$	0	0
$265$	918.385	0.212890
$266$	0	0
$267$	0	0
$268$	−1385.32	−0.315752
$269$	1023.10	0.231893	0.115947	−	0.993255i	$-0.463010\pi$
$269$	1023.10	0.231893	0.115947	+	0.993255i	$0.463010\pi$
$270$	0	0
$271$	2251.98	0.504790	0.252395	−	0.967624i	$-0.418782\pi$
$271$	2251.98	0.504790	0.252395	+	0.967624i	$0.418782\pi$
$272$	−9772.91	−2.17857
$273$	0	0
$274$	−5660.87	−1.24812
$275$	73.4436	0.0161048
$276$	0	0
$277$	−8630.72	−1.87209	−0.936047	−	0.351875i	$-0.885544\pi$
$277$	−8630.72	−1.87209	−0.936047	+	0.351875i	$0.885544\pi$
$278$	−8584.61	−1.85205
$279$	0	0
$280$	0	0
$281$	7521.62	1.59680	0.798402	−	0.602124i	$-0.205679\pi$
$281$	7521.62	1.59680	0.798402	+	0.602124i	$0.205679\pi$
$282$	0	0
$283$	−14.8169	−0.00311226	−0.00155613	−	0.999999i	$-0.500495\pi$
$283$	−14.8169	−0.00311226	−0.00155613	+	0.999999i	$0.500495\pi$
$284$	−2974.89	−0.621576
$285$	0	0
$286$	197.743	0.0408839
$287$	0	0
$288$	0	0
$289$	10092.8	2.05430
$290$	−1686.01	−0.341400
$291$	0	0
$292$	−2776.46	−0.556438
$293$	6913.39	1.37844	0.689222	−	0.724550i	$-0.257952\pi$
$293$	6913.39	1.37844	0.689222	+	0.724550i	$0.257952\pi$
$294$	0	0
$295$	−3803.07	−0.750588
$296$	−1210.98	−0.237794
$297$	0	0
$298$	−8268.39	−1.60730
$299$	−4014.94	−0.776555
$300$	0	0
$301$	0	0
$302$	−7429.74	−1.41567
$303$	0	0
$304$	8571.25	1.61709
$305$	993.735	0.186561
$306$	0	0
$307$	7644.12	1.42108	0.710542	−	0.703655i	$-0.248450\pi$
$307$	7644.12	1.42108	0.710542	+	0.703655i	$0.248450\pi$
$308$	0	0
$309$	0	0
$310$	1665.06	0.305061
$311$	7593.99	1.38462	0.692308	−	0.721602i	$-0.256594\pi$
$311$	7593.99	1.38462	0.692308	+	0.721602i	$0.256594\pi$
$312$	0	0
$313$	9127.84	1.64836	0.824179	−	0.566329i	$-0.191637\pi$
$313$	9127.84	1.64836	0.824179	+	0.566329i	$0.191637\pi$
$314$	2095.61	0.376631
$315$	0	0
$316$	−110.661	−0.0196999
$317$	4929.81	0.873456	0.436728	−	0.899593i	$-0.356137\pi$
$317$	4929.81	0.873456	0.436728	+	0.899593i	$0.356137\pi$
$318$	0	0
$319$	−280.537	−0.0492385
$320$	−21.4686	−0.00375042
$321$	0	0
$322$	0	0
$323$	−13160.7	−2.26712
$324$	0	0
$325$	476.556	0.0813372
$326$	7693.30	1.30703
$327$	0	0
$328$	6123.62	1.03086
$329$	0	0
$330$	0	0
$331$	1221.67	0.202867	0.101433	−	0.994842i	$-0.467657\pi$
$331$	1221.67	0.202867	0.101433	+	0.994842i	$0.467657\pi$
$332$	−1817.60	−0.300463
$333$	0	0
$334$	2823.14	0.462502
$335$	−1549.96	−0.252786
$336$	0	0
$337$	−8744.83	−1.41354	−0.706768	−	0.707446i	$-0.749847\pi$
$337$	−8744.83	−1.41354	−0.706768	+	0.707446i	$0.749847\pi$
$338$	−6474.79	−1.04196
$339$	0	0
$340$	2737.14	0.436595
$341$	277.051	0.0439975
$342$	0	0
$343$	0	0
$344$	536.210	0.0840421
$345$	0	0
$346$	−5100.21	−0.792454
$347$	−4589.56	−0.710031	−0.355015	−	0.934860i	$-0.615524\pi$
$347$	−4589.56	−0.710031	−0.355015	+	0.934860i	$0.615524\pi$
$348$	0	0
$349$	3989.89	0.611960	0.305980	−	0.952038i	$-0.401016\pi$
$349$	3989.89	0.611960	0.305980	+	0.952038i	$0.401016\pi$
$350$	0	0
$351$	0	0
$352$	−534.561	−0.0809437
$353$	2416.35	0.364333	0.182166	−	0.983268i	$-0.441689\pi$
$353$	2416.35	0.364333	0.182166	+	0.983268i	$0.441689\pi$
$354$	0	0
$355$	−3328.46	−0.497624
$356$	1169.73	0.174145
$357$	0	0
$358$	−11806.3	−1.74297
$359$	2756.24	0.405206	0.202603	−	0.979261i	$-0.435060\pi$
$359$	2756.24	0.405206	0.202603	+	0.979261i	$0.435060\pi$
$360$	0	0
$361$	4683.45	0.682818
$362$	−7951.44	−1.15447
$363$	0	0
$364$	0	0
$365$	−3106.44	−0.445475
$366$	0	0
$367$	−11112.8	−1.58061	−0.790307	−	0.612711i	$-0.790079\pi$
$367$	−11112.8	−1.58061	−0.790307	+	0.612711i	$0.790079\pi$
$368$	16803.5	2.38028
$369$	0	0
$370$	1714.73	0.240931
$371$	0	0
$372$	0	0
$373$	6091.09	0.845535	0.422768	−	0.906238i	$-0.361059\pi$
$373$	6091.09	0.845535	0.422768	+	0.906238i	$0.361059\pi$
$374$	1270.74	0.175691
$375$	0	0
$376$	−5903.94	−0.809767
$377$	−1820.33	−0.248679
$378$	0	0
$379$	3984.29	0.539998	0.269999	−	0.962861i	$-0.412977\pi$
$379$	3984.29	0.539998	0.269999	+	0.962861i	$0.412977\pi$
$380$	−2400.58	−0.324072
$381$	0	0
$382$	3537.93	0.473865
$383$	318.475	0.0424890	0.0212445	−	0.999774i	$-0.493237\pi$
$383$	318.475	0.0424890	0.0212445	+	0.999774i	$0.493237\pi$
$384$	0	0
$385$	0	0
$386$	−14318.6	−1.88808
$387$	0	0
$388$	4490.21	0.587515
$389$	−3885.46	−0.506429	−0.253214	−	0.967410i	$-0.581488\pi$
$389$	−3885.46	−0.506429	−0.253214	+	0.967410i	$0.581488\pi$
$390$	0	0
$391$	−25800.9	−3.33710
$392$	0	0
$393$	0	0
$394$	18150.8	2.32088
$395$	−123.813	−0.0157714
$396$	0	0
$397$	−4806.04	−0.607578	−0.303789	−	0.952739i	$-0.598252\pi$
$397$	−4806.04	−0.607578	−0.303789	+	0.952739i	$0.598252\pi$
$398$	−2067.94	−0.260443
$399$	0	0
$400$	−1994.50	−0.249313
$401$	−3618.59	−0.450633	−0.225316	−	0.974286i	$-0.572342\pi$
$401$	−3618.59	−0.450633	−0.225316	+	0.974286i	$0.572342\pi$
$402$	0	0
$403$	1797.71	0.222209
$404$	−575.911	−0.0709223
$405$	0	0
$406$	0	0
$407$	285.315	0.0347483
$408$	0	0
$409$	2109.05	0.254978	0.127489	−	0.991840i	$-0.459308\pi$
$409$	2109.05	0.254978	0.127489	+	0.991840i	$0.459308\pi$
$410$	−8670.91	−1.04445
$411$	0	0
$412$	−3601.94	−0.430716
$413$	0	0
$414$	0	0
$415$	−2033.62	−0.240546
$416$	−3468.63	−0.408806
$417$	0	0
$418$	−1114.49	−0.130410
$419$	−6905.91	−0.805193	−0.402597	−	0.915377i	$-0.631892\pi$
$419$	−6905.91	−0.805193	−0.402597	+	0.915377i	$0.631892\pi$
$420$	0	0
$421$	−9647.54	−1.11685	−0.558423	−	0.829556i	$-0.688593\pi$
$421$	−9647.54	−1.11685	−0.558423	+	0.829556i	$0.688593\pi$
$422$	−3725.91	−0.429797
$423$	0	0
$424$	−2290.25	−0.262321
$425$	3062.45	0.349531
$426$	0	0
$427$	0	0
$428$	3439.74	0.388473
$429$	0	0
$430$	−759.261	−0.0851508
$431$	13002.7	1.45318	0.726589	−	0.687073i	$-0.241105\pi$
$431$	13002.7	1.45318	0.726589	+	0.687073i	$0.241105\pi$
$432$	0	0
$433$	−7356.07	−0.816420	−0.408210	−	0.912888i	$-0.633847\pi$
$433$	−7356.07	−0.816420	−0.408210	+	0.912888i	$0.633847\pi$
$434$	0	0
$435$	0	0
$436$	−3489.55	−0.383300
$437$	22628.4	2.47703
$438$	0	0
$439$	−6909.21	−0.751159	−0.375579	−	0.926790i	$-0.622556\pi$
$439$	−6909.21	−0.751159	−0.375579	+	0.926790i	$0.622556\pi$
$440$	−183.152	−0.0198441
$441$	0	0
$442$	8245.50	0.887327
$443$	14812.6	1.58864	0.794318	−	0.607502i	$-0.207828\pi$
$443$	14812.6	1.58864	0.794318	+	0.607502i	$0.207828\pi$
$444$	0	0
$445$	1308.75	0.139418
$446$	−16511.0	−1.75296
$447$	0	0
$448$	0	0
$449$	10654.5	1.11986	0.559932	−	0.828538i	$-0.310827\pi$
$449$	10654.5	1.11986	0.559932	+	0.828538i	$0.310827\pi$
$450$	0	0
$451$	−1442.76	−0.150636
$452$	4985.68	0.518820
$453$	0	0
$454$	19220.3	1.98691
$455$	0	0
$456$	0	0
$457$	−5855.16	−0.599328	−0.299664	−	0.954045i	$-0.596875\pi$
$457$	−5855.16	−0.599328	−0.299664	+	0.954045i	$0.596875\pi$
$458$	1893.76	0.193208
$459$	0	0
$460$	−4706.23	−0.477019
$461$	3204.74	0.323774	0.161887	−	0.986809i	$-0.448242\pi$
$461$	3204.74	0.323774	0.161887	+	0.986809i	$0.448242\pi$
$462$	0	0
$463$	371.658	0.0373054	0.0186527	−	0.999826i	$-0.494062\pi$
$463$	371.658	0.0373054	0.0186527	+	0.999826i	$0.494062\pi$
$464$	7618.54	0.762245
$465$	0	0
$466$	647.927	0.0644091
$467$	−19752.3	−1.95723	−0.978614	−	0.205703i	$-0.934052\pi$
$467$	−19752.3	−1.95723	−0.978614	+	0.205703i	$0.934052\pi$
$468$	0	0
$469$	0	0
$470$	8359.84	0.820449
$471$	0	0
$472$	9484.01	0.924866
$473$	−126.334	−0.0122809
$474$	0	0
$475$	−2685.90	−0.259447
$476$	0	0
$477$	0	0
$478$	−2271.28	−0.217335
$479$	−20762.0	−1.98046	−0.990232	−	0.139433i	$-0.955472\pi$
$479$	−20762.0	−1.98046	−0.990232	+	0.139433i	$0.955472\pi$
$480$	0	0
$481$	1851.34	0.175496
$482$	20323.8	1.92059
$483$	0	0
$484$	−5909.50	−0.554987
$485$	5023.87	0.470355
$486$	0	0
$487$	17647.6	1.64207	0.821035	−	0.570878i	$-0.193397\pi$
$487$	17647.6	1.64207	0.821035	+	0.570878i	$0.193397\pi$
$488$	−2478.15	−0.229878
$489$	0	0
$490$	0	0
$491$	5637.46	0.518157	0.259078	−	0.965856i	$-0.416581\pi$
$491$	5637.46	0.518157	0.259078	+	0.965856i	$0.416581\pi$
$492$	0	0
$493$	−11697.9	−1.06865
$494$	−7231.64	−0.658638
$495$	0	0
$496$	−7523.86	−0.681112
$497$	0	0
$498$	0	0
$499$	−17474.1	−1.56764	−0.783818	−	0.620991i	$-0.786730\pi$
$499$	−17474.1	−1.56764	−0.783818	+	0.620991i	$0.786730\pi$
$500$	558.609	0.0499635
$501$	0	0
$502$	−18122.7	−1.61127
$503$	7444.81	0.659936	0.329968	−	0.943992i	$-0.392962\pi$
$503$	7444.81	0.659936	0.329968	+	0.943992i	$0.392962\pi$
$504$	0	0
$505$	−644.358	−0.0567793
$506$	−2184.90	−0.191958
$507$	0	0
$508$	−8383.53	−0.732203
$509$	−3384.48	−0.294724	−0.147362	−	0.989083i	$-0.547078\pi$
$509$	−3384.48	−0.294724	−0.147362	+	0.989083i	$0.547078\pi$
$510$	0	0
$511$	0	0
$512$	6558.89	0.566142
$513$	0	0
$514$	17803.0	1.52774
$515$	−4030.04	−0.344825
$516$	0	0
$517$	1391.00	0.118329
$518$	0	0
$519$	0	0
$520$	−1188.42	−0.100223
$521$	2973.12	0.250009	0.125005	−	0.992156i	$-0.460105\pi$
$521$	2973.12	0.250009	0.125005	+	0.992156i	$0.460105\pi$
$522$	0	0
$523$	−2689.02	−0.224823	−0.112412	−	0.993662i	$-0.535858\pi$
$523$	−2689.02	−0.224823	−0.112412	+	0.993662i	$0.535858\pi$
$524$	1627.57	0.135689
$525$	0	0
$526$	−26755.3	−2.21785
$527$	11552.5	0.954903
$528$	0	0
$529$	32194.9	2.64608
$530$	3242.94	0.265781
$531$	0	0
$532$	0	0
$533$	−9361.72	−0.760790
$534$	0	0
$535$	3848.56	0.311005
$536$	3865.25	0.311480
$537$	0	0
$538$	3612.69	0.289506
$539$	0	0
$540$	0	0
$541$	−14429.5	−1.14671	−0.573356	−	0.819306i	$-0.694359\pi$
$541$	−14429.5	−1.14671	−0.573356	+	0.819306i	$0.694359\pi$
$542$	7952.03	0.630201
$543$	0	0
$544$	−22290.1	−1.75677
$545$	−3904.28	−0.306864
$546$	0	0
$547$	13811.2	1.07957	0.539784	−	0.841804i	$-0.318506\pi$
$547$	13811.2	1.07957	0.539784	+	0.841804i	$0.318506\pi$
$548$	−7164.19	−0.558466
$549$	0	0
$550$	259.339	0.0201059
$551$	10259.5	0.793229
$552$	0	0
$553$	0	0
$554$	−30476.2	−2.33720
$555$	0	0
$556$	−10864.4	−0.828692
$557$	6033.26	0.458954	0.229477	−	0.973314i	$-0.426298\pi$
$557$	6033.26	0.458954	0.229477	+	0.973314i	$0.426298\pi$
$558$	0	0
$559$	−819.751	−0.0620246
$560$	0	0
$561$	0	0
$562$	26559.8	1.99352
$563$	−6958.47	−0.520896	−0.260448	−	0.965488i	$-0.583870\pi$
$563$	−6958.47	−0.520896	−0.260448	+	0.965488i	$0.583870\pi$
$564$	0	0
$565$	5578.23	0.415359
$566$	−52.3202	−0.00388548
$567$	0	0
$568$	8300.44	0.613166
$569$	13396.4	0.987009	0.493505	−	0.869743i	$-0.335716\pi$
$569$	13396.4	0.987009	0.493505	+	0.869743i	$0.335716\pi$
$570$	0	0
$571$	−8055.84	−0.590414	−0.295207	−	0.955433i	$-0.595389\pi$
$571$	−8055.84	−0.590414	−0.295207	+	0.955433i	$0.595389\pi$
$572$	250.257	0.0182933
$573$	0	0
$574$	0	0
$575$	−5265.56	−0.381894
$576$	0	0
$577$	21456.9	1.54812	0.774059	−	0.633114i	$-0.218223\pi$
$577$	21456.9	1.54812	0.774059	+	0.633114i	$0.218223\pi$
$578$	35638.9	2.56468
$579$	0	0
$580$	−2133.76	−0.152758
$581$	0	0
$582$	0	0
$583$	539.596	0.0383324
$584$	7746.76	0.548910
$585$	0	0
$586$	24412.1	1.72091
$587$	20156.3	1.41728	0.708638	−	0.705572i	$-0.249310\pi$
$587$	20156.3	1.41728	0.708638	+	0.705572i	$0.249310\pi$
$588$	0	0
$589$	−10132.0	−0.708797
$590$	−13429.1	−0.937066
$591$	0	0
$592$	−7748.30	−0.537928
$593$	599.307	0.0415018	0.0207509	−	0.999785i	$-0.493394\pi$
$593$	599.307	0.0415018	0.0207509	+	0.999785i	$0.493394\pi$
$594$	0	0
$595$	0	0
$596$	−10464.2	−0.719177
$597$	0	0
$598$	−14177.3	−0.969485
$599$	5493.05	0.374691	0.187346	−	0.982294i	$-0.440012\pi$
$599$	5493.05	0.374691	0.187346	+	0.982294i	$0.440012\pi$
$600$	0	0
$601$	−24292.8	−1.64879	−0.824396	−	0.566014i	$-0.808485\pi$
$601$	−24292.8	−1.64879	−0.824396	+	0.566014i	$0.808485\pi$
$602$	0	0
$603$	0	0
$604$	−9402.82	−0.633436
$605$	−6611.85	−0.444314
$606$	0	0
$607$	−3029.50	−0.202576	−0.101288	−	0.994857i	$-0.532296\pi$
$607$	−3029.50	−0.202576	−0.101288	+	0.994857i	$0.532296\pi$
$608$	19549.3	1.30400
$609$	0	0
$610$	3509.01	0.232911
$611$	9025.87	0.597623
$612$	0	0
$613$	−19339.6	−1.27426	−0.637129	−	0.770757i	$-0.719878\pi$
$613$	−19339.6	−1.27426	−0.637129	+	0.770757i	$0.719878\pi$
$614$	26992.4	1.77414
$615$	0	0
$616$	0	0
$617$	5743.91	0.374783	0.187391	−	0.982285i	$-0.439997\pi$
$617$	5743.91	0.374783	0.187391	+	0.982285i	$0.439997\pi$
$618$	0	0
$619$	8243.35	0.535264	0.267632	−	0.963521i	$-0.413759\pi$
$619$	8243.35	0.535264	0.267632	+	0.963521i	$0.413759\pi$
$620$	2107.24	0.136498
$621$	0	0
$622$	26815.4	1.72861
$623$	0	0
$624$	0	0
$625$	625.000	0.0400000
$626$	32231.6	2.05788
$627$	0	0
$628$	2652.13	0.168521
$629$	11897.1	0.754162
$630$	0	0
$631$	−4376.56	−0.276114	−0.138057	−	0.990424i	$-0.544086\pi$
$631$	−4376.56	−0.276114	−0.138057	+	0.990424i	$0.544086\pi$
$632$	308.762	0.0194334
$633$	0	0
$634$	17407.8	1.09046
$635$	−9379.92	−0.586190
$636$	0	0
$637$	0	0
$638$	−990.613	−0.0614714
$639$	0	0
$640$	7202.71	0.444863
$641$	−11836.6	−0.729357	−0.364678	−	0.931133i	$-0.618821\pi$
$641$	−11836.6	−0.729357	−0.364678	+	0.931133i	$0.618821\pi$
$642$	0	0
$643$	−1448.21	−0.0888209	−0.0444104	−	0.999013i	$-0.514141\pi$
$643$	−1448.21	−0.0888209	−0.0444104	+	0.999013i	$0.514141\pi$
$644$	0	0
$645$	0	0
$646$	−46472.0	−2.83037
$647$	−8732.95	−0.530646	−0.265323	−	0.964160i	$-0.585479\pi$
$647$	−8732.95	−0.530646	−0.265323	+	0.964160i	$0.585479\pi$
$648$	0	0
$649$	−2234.49	−0.135149
$650$	1682.78	0.101545
$651$	0	0
$652$	9736.37	0.584824
$653$	21978.4	1.31712	0.658562	−	0.752527i	$-0.271165\pi$
$653$	21978.4	1.31712	0.658562	+	0.752527i	$0.271165\pi$
$654$	0	0
$655$	1821.01	0.108630
$656$	39181.1	2.33196
$657$	0	0
$658$	0	0
$659$	27761.7	1.64103	0.820516	−	0.571623i	$-0.193686\pi$
$659$	27761.7	1.64103	0.820516	+	0.571623i	$0.193686\pi$
$660$	0	0
$661$	8573.72	0.504507	0.252254	−	0.967661i	$-0.418828\pi$
$661$	8573.72	0.504507	0.252254	+	0.967661i	$0.418828\pi$
$662$	4313.86	0.253267
$663$	0	0
$664$	5071.39	0.296398
$665$	0	0
$666$	0	0
$667$	20113.2	1.16760
$668$	3572.87	0.206944
$669$	0	0
$670$	−5473.11	−0.315589
$671$	583.868	0.0335916
$672$	0	0
$673$	27159.2	1.55559	0.777795	−	0.628518i	$-0.216338\pi$
$673$	27159.2	1.55559	0.777795	+	0.628518i	$0.216338\pi$
$674$	−30879.1	−1.76472
$675$	0	0
$676$	−8194.26	−0.466219
$677$	−1392.30	−0.0790404	−0.0395202	−	0.999219i	$-0.512583\pi$
$677$	−1392.30	−0.0790404	−0.0395202	+	0.999219i	$0.512583\pi$
$678$	0	0
$679$	0	0
$680$	−7637.06	−0.430688
$681$	0	0
$682$	978.302	0.0549283
$683$	8675.09	0.486007	0.243004	−	0.970025i	$-0.421867\pi$
$683$	8675.09	0.486007	0.243004	+	0.970025i	$0.421867\pi$
$684$	0	0
$685$	−8015.66	−0.447099
$686$	0	0
$687$	0	0
$688$	3430.86	0.190117
$689$	3501.30	0.193598
$690$	0	0
$691$	21426.0	1.17957	0.589785	−	0.807561i	$-0.299213\pi$
$691$	21426.0	1.17957	0.589785	+	0.807561i	$0.299213\pi$
$692$	−6454.65	−0.354579
$693$	0	0
$694$	−16206.3	−0.886433
$695$	−12155.6	−0.663438
$696$	0	0
$697$	−60160.4	−3.26935
$698$	14088.8	0.763997
$699$	0	0
$700$	0	0
$701$	−24840.5	−1.33839	−0.669197	−	0.743085i	$-0.733362\pi$
$701$	−24840.5	−1.33839	−0.669197	+	0.743085i	$0.733362\pi$
$702$	0	0
$703$	−10434.2	−0.559793
$704$	−12.6139	−0.000675288 0
$705$	0	0
$706$	8532.45	0.454848
$707$	0	0
$708$	0	0
$709$	12525.0	0.663450	0.331725	−	0.943376i	$-0.392369\pi$
$709$	12525.0	0.663450	0.331725	+	0.943376i	$0.392369\pi$
$710$	−11753.2	−0.621255
$711$	0	0
$712$	−3263.74	−0.171789
$713$	−19863.3	−1.04332
$714$	0	0
$715$	280.000	0.0146453
$716$	−14941.6	−0.779881
$717$	0	0
$718$	9732.66	0.505877
$719$	28085.0	1.45674	0.728369	−	0.685185i	$-0.240279\pi$
$719$	28085.0	1.45674	0.728369	+	0.685185i	$0.240279\pi$
$720$	0	0
$721$	0	0
$722$	16537.9	0.852460
$723$	0	0
$724$	−10063.1	−0.516562
$725$	−2387.35	−0.122295
$726$	0	0
$727$	14326.2	0.730851	0.365426	−	0.930841i	$-0.380923\pi$
$727$	14326.2	0.730851	0.365426	+	0.930841i	$0.380923\pi$
$728$	0	0
$729$	0	0
$730$	−10969.2	−0.556150
$731$	−5267.89	−0.266539
$732$	0	0
$733$	−6727.85	−0.339016	−0.169508	−	0.985529i	$-0.554218\pi$
$733$	−6727.85	−0.339016	−0.169508	+	0.985529i	$0.554218\pi$
$734$	−39240.8	−1.97331
$735$	0	0
$736$	38325.5	1.91943
$737$	−910.677	−0.0455159
$738$	0	0
$739$	−3418.51	−0.170165	−0.0850826	−	0.996374i	$-0.527115\pi$
$739$	−3418.51	−0.170165	−0.0850826	+	0.996374i	$0.527115\pi$
$740$	2170.10	0.107803
$741$	0	0
$742$	0	0
$743$	−8095.50	−0.399724	−0.199862	−	0.979824i	$-0.564049\pi$
$743$	−8095.50	−0.399724	−0.199862	+	0.979824i	$0.564049\pi$
$744$	0	0
$745$	−11707.9	−0.575762
$746$	21508.4	1.05560
$747$	0	0
$748$	1608.20	0.0786119
$749$	0	0
$750$	0	0
$751$	13446.8	0.653371	0.326686	−	0.945133i	$-0.394068\pi$
$751$	13446.8	0.653371	0.326686	+	0.945133i	$0.394068\pi$
$752$	−37775.4	−1.83182
$753$	0	0
$754$	−6427.84	−0.310462
$755$	−10520.4	−0.507119
$756$	0	0
$757$	−2593.24	−0.124508	−0.0622541	−	0.998060i	$-0.519829\pi$
$757$	−2593.24	−0.124508	−0.0622541	+	0.998060i	$0.519829\pi$
$758$	14069.0	0.674156
$759$	0	0
$760$	6698.02	0.319688
$761$	27079.4	1.28992	0.644959	−	0.764217i	$-0.276875\pi$
$761$	27079.4	1.28992	0.644959	+	0.764217i	$0.276875\pi$
$762$	0	0
$763$	0	0
$764$	4477.48	0.212028
$765$	0	0
$766$	1124.58	0.0530451
$767$	−14499.0	−0.682568
$768$	0	0
$769$	−2138.72	−0.100292	−0.0501458	−	0.998742i	$-0.515969\pi$
$769$	−2138.72	−0.100292	−0.0501458	+	0.998742i	$0.515969\pi$
$770$	0	0
$771$	0	0
$772$	−18121.1	−0.844810
$773$	25864.0	1.20345	0.601724	−	0.798704i	$-0.294481\pi$
$773$	25864.0	1.20345	0.601724	+	0.798704i	$0.294481\pi$
$774$	0	0
$775$	2357.69	0.109278
$776$	−12528.4	−0.579566
$777$	0	0
$778$	−13720.1	−0.632247
$779$	52763.1	2.42675
$780$	0	0
$781$	−1955.63	−0.0896006
$782$	−91106.2	−4.16618
$783$	0	0
$784$	0	0
$785$	2967.33	0.134916
$786$	0	0
$787$	32371.3	1.46621	0.733107	−	0.680113i	$-0.238069\pi$
$787$	32371.3	1.46621	0.733107	+	0.680113i	$0.238069\pi$
$788$	22971.0	1.03846
$789$	0	0
$790$	−437.200	−0.0196897
$791$	0	0
$792$	0	0
$793$	3788.57	0.169654
$794$	−16970.8	−0.758526
$795$	0	0
$796$	−2617.11	−0.116534
$797$	2024.33	0.0899691	0.0449845	−	0.998988i	$-0.485676\pi$
$797$	2024.33	0.0899691	0.0449845	+	0.998988i	$0.485676\pi$
$798$	0	0
$799$	58002.1	2.56817
$800$	−4549.08	−0.201043
$801$	0	0
$802$	−12777.7	−0.562589
$803$	−1825.18	−0.0802109
$804$	0	0
$805$	0	0
$806$	6347.95	0.277416
$807$	0	0
$808$	1606.88	0.0699628
$809$	−12391.7	−0.538526	−0.269263	−	0.963067i	$-0.586780\pi$
$809$	−12391.7	−0.538526	−0.269263	+	0.963067i	$0.586780\pi$
$810$	0	0
$811$	−14654.5	−0.634511	−0.317256	−	0.948340i	$-0.602761\pi$
$811$	−14654.5	−0.634511	−0.317256	+	0.948340i	$0.602761\pi$
$812$	0	0
$813$	0	0
$814$	1007.49	0.0433813
$815$	10893.5	0.468201
$816$	0	0
$817$	4620.16	0.197844
$818$	7447.33	0.318325
$819$	0	0
$820$	−10973.6	−0.467335
$821$	−23887.9	−1.01546	−0.507731	−	0.861516i	$-0.669515\pi$
$821$	−23887.9	−1.01546	−0.507731	+	0.861516i	$0.669515\pi$
$822$	0	0
$823$	−4008.41	−0.169774	−0.0848871	−	0.996391i	$-0.527053\pi$
$823$	−4008.41	−0.169774	−0.0848871	+	0.996391i	$0.527053\pi$
$824$	10050.0	0.424889
$825$	0	0
$826$	0	0
$827$	45110.4	1.89679	0.948394	−	0.317096i	$-0.102708\pi$
$827$	45110.4	1.89679	0.948394	+	0.317096i	$0.102708\pi$
$828$	0	0
$829$	−16165.4	−0.677260	−0.338630	−	0.940920i	$-0.609964\pi$
$829$	−16165.4	−0.677260	−0.338630	+	0.940920i	$0.609964\pi$
$830$	−7180.97	−0.300307
$831$	0	0
$832$	−81.8481	−0.00341054
$833$	0	0
$834$	0	0
$835$	3997.51	0.165676
$836$	−1410.46	−0.0583514
$837$	0	0
$838$	−24385.7	−1.00524
$839$	−25244.4	−1.03878	−0.519388	−	0.854538i	$-0.673840\pi$
$839$	−25244.4	−1.03878	−0.519388	+	0.854538i	$0.673840\pi$
$840$	0	0
$841$	−15269.9	−0.626096
$842$	−34066.7	−1.39432
$843$	0	0
$844$	−4715.37	−0.192310
$845$	−9168.15	−0.373247
$846$	0	0
$847$	0	0
$848$	−14653.8	−0.593412
$849$	0	0
$850$	10813.9	0.436370
$851$	−20455.8	−0.823990
$852$	0	0
$853$	30168.1	1.21094	0.605472	−	0.795867i	$-0.292984\pi$
$853$	30168.1	1.21094	0.605472	+	0.795867i	$0.292984\pi$
$854$	0	0
$855$	0	0
$856$	−9597.44	−0.383217
$857$	−13393.6	−0.533857	−0.266929	−	0.963716i	$-0.586009\pi$
$857$	−13393.6	−0.533857	−0.266929	+	0.963716i	$0.586009\pi$
$858$	0	0
$859$	−19060.4	−0.757081	−0.378541	−	0.925585i	$-0.623574\pi$
$859$	−19060.4	−0.757081	−0.378541	+	0.925585i	$0.623574\pi$
$860$	−960.894	−0.0381002
$861$	0	0
$862$	45914.3	1.81421
$863$	9466.86	0.373413	0.186707	−	0.982416i	$-0.440219\pi$
$863$	9466.86	0.373413	0.186707	+	0.982416i	$0.440219\pi$
$864$	0	0
$865$	−7221.79	−0.283871
$866$	−25975.2	−1.01925
$867$	0	0
$868$	0	0
$869$	−72.7461	−0.00283975
$870$	0	0
$871$	−5909.15	−0.229878
$872$	9736.39	0.378115
$873$	0	0
$874$	79903.8	3.09243
$875$	0	0
$876$	0	0
$877$	37740.6	1.45315	0.726573	−	0.687090i	$-0.241112\pi$
$877$	37740.6	1.45315	0.726573	+	0.687090i	$0.241112\pi$
$878$	−24397.3	−0.937778
$879$	0	0
$880$	−1171.87	−0.0448905
$881$	25991.5	0.993957	0.496979	−	0.867763i	$-0.334443\pi$
$881$	25991.5	0.993957	0.496979	+	0.867763i	$0.334443\pi$
$882$	0	0
$883$	39420.3	1.50238	0.751189	−	0.660087i	$-0.229481\pi$
$883$	39420.3	1.50238	0.751189	+	0.660087i	$0.229481\pi$
$884$	10435.2	0.397030
$885$	0	0
$886$	52305.0	1.98332
$887$	46005.2	1.74149	0.870745	−	0.491735i	$-0.163637\pi$
$887$	46005.2	1.74149	0.870745	+	0.491735i	$0.163637\pi$
$888$	0	0
$889$	0	0
$890$	4621.38	0.174055
$891$	0	0
$892$	−20895.8	−0.784353
$893$	−50870.2	−1.90628
$894$	0	0
$895$	−16717.5	−0.624361
$896$	0	0
$897$	0	0
$898$	37622.6	1.39809
$899$	−9005.81	−0.334105
$900$	0	0
$901$	22500.1	0.831950
$902$	−5094.58	−0.188061
$903$	0	0
$904$	−13910.8	−0.511801
$905$	−11259.1	−0.413551
$906$	0	0
$907$	−2838.97	−0.103932	−0.0519661	−	0.998649i	$-0.516549\pi$
$907$	−2838.97	−0.103932	−0.0519661	+	0.998649i	$0.516549\pi$
$908$	24324.6	0.889030
$909$	0	0
$910$	0	0
$911$	−39890.9	−1.45076	−0.725382	−	0.688347i	$-0.758337\pi$
$911$	−39890.9	−1.45076	−0.725382	+	0.688347i	$0.758337\pi$
$912$	0	0
$913$	−1194.85	−0.0433119
$914$	−20675.3	−0.748226
$915$	0	0
$916$	2396.67	0.0864500
$917$	0	0
$918$	0	0
$919$	−646.475	−0.0232048	−0.0116024	−	0.999933i	$-0.503693\pi$
$919$	−646.475	−0.0232048	−0.0116024	+	0.999933i	$0.503693\pi$
$920$	13131.1	0.470566
$921$	0	0
$922$	11316.3	0.404213
$923$	−12689.6	−0.452528
$924$	0	0
$925$	2428.02	0.0863056
$926$	1312.37	0.0465737
$927$	0	0
$928$	17376.4	0.614665
$929$	51188.2	1.80778	0.903892	−	0.427760i	$-0.140697\pi$
$929$	51188.2	1.80778	0.903892	+	0.427760i	$0.140697\pi$
$930$	0	0
$931$	0	0
$932$	819.993	0.0288195
$933$	0	0
$934$	−69747.8	−2.44349
$935$	1799.34	0.0629355
$936$	0	0
$937$	−29786.1	−1.03849	−0.519247	−	0.854624i	$-0.673788\pi$
$937$	−29786.1	−1.03849	−0.519247	+	0.854624i	$0.673788\pi$
$938$	0	0
$939$	0	0
$940$	10579.9	0.367105
$941$	−44817.4	−1.55261	−0.776304	−	0.630358i	$-0.782908\pi$
$941$	−44817.4	−1.55261	−0.776304	+	0.630358i	$0.782908\pi$
$942$	0	0
$943$	103439.	3.57206
$944$	60682.0	2.09219
$945$	0	0
$946$	−446.103	−0.0153320
$947$	−54697.1	−1.87689	−0.938446	−	0.345425i	$-0.887735\pi$
$947$	−54697.1	−1.87689	−0.938446	+	0.345425i	$0.887735\pi$
$948$	0	0
$949$	−11843.2	−0.405105
$950$	−9484.24	−0.323905
$951$	0	0
$952$	0	0
$953$	7577.51	0.257565	0.128783	−	0.991673i	$-0.458893\pi$
$953$	7577.51	0.257565	0.128783	+	0.991673i	$0.458893\pi$
$954$	0	0
$955$	5009.63	0.169746
$956$	−2874.46	−0.0972454
$957$	0	0
$958$	−73313.4	−2.47249
$959$	0	0
$960$	0	0
$961$	−20897.1	−0.701457
$962$	6537.32	0.219097
$963$	0	0
$964$	25721.1	0.859357
$965$	−20274.8	−0.676342
$966$	0	0
$967$	50779.0	1.68867	0.844334	−	0.535817i	$-0.179996\pi$
$967$	50779.0	1.68867	0.844334	+	0.535817i	$0.179996\pi$
$968$	16488.5	0.547478
$969$	0	0
$970$	17739.9	0.587212
$971$	−15313.2	−0.506102	−0.253051	−	0.967453i	$-0.581434\pi$
$971$	−15313.2	−0.506102	−0.253051	+	0.967453i	$0.581434\pi$
$972$	0	0
$973$	0	0
$974$	62315.9	2.05003
$975$	0	0
$976$	−15856.1	−0.520021
$977$	−46620.4	−1.52663	−0.763316	−	0.646025i	$-0.776430\pi$
$977$	−46620.4	−1.52663	−0.763316	+	0.646025i	$0.776430\pi$
$978$	0	0
$979$	768.957	0.0251031
$980$	0	0
$981$	0	0
$982$	19906.6	0.646889
$983$	−2824.37	−0.0916414	−0.0458207	−	0.998950i	$-0.514590\pi$
$983$	−2824.37	−0.0916414	−0.0458207	+	0.998950i	$0.514590\pi$
$984$	0	0
$985$	25701.1	0.831377
$986$	−41306.6	−1.33415
$987$	0	0
$988$	−9152.11	−0.294704
$989$	9057.59	0.291218
$990$	0	0
$991$	16951.4	0.543370	0.271685	−	0.962386i	$-0.412419\pi$
$991$	16951.4	0.543370	0.271685	+	0.962386i	$0.412419\pi$
$992$	−17160.5	−0.549239
$993$	0	0
$994$	0	0
$995$	−2928.15	−0.0932952
$996$	0	0
$997$	−23847.8	−0.757540	−0.378770	−	0.925491i	$-0.623653\pi$
$997$	−23847.8	−0.757540	−0.378770	+	0.925491i	$0.623653\pi$
$998$	−61703.4	−1.95710
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2205.4.a.z.1.2		2
3.2	odd	2		735.4.a.p.1.1		2
7.6	odd	2		315.4.a.i.1.2		2
21.20	even	2		105.4.a.f.1.1	✓	2
35.34	odd	2		1575.4.a.w.1.1		2
84.83	odd	2		1680.4.a.bg.1.1		2
105.62	odd	4		525.4.d.h.274.2		4
105.83	odd	4		525.4.d.h.274.3		4
105.104	even	2		525.4.a.k.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.4.a.f.1.1	✓	2	21.20	even	2
315.4.a.i.1.2		2	7.6	odd	2
525.4.a.k.1.2		2	105.104	even	2
525.4.d.h.274.2		4	105.62	odd	4
525.4.d.h.274.3		4	105.83	odd	4
735.4.a.p.1.1		2	3.2	odd	2
1575.4.a.w.1.1		2	35.34	odd	2
1680.4.a.bg.1.1		2	84.83	odd	2
2205.4.a.z.1.2		2	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:	\( N \)	\(=\)	\( 2205 = 3^{2} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2205.a (trivial)

\(f(q)\)	\(=\)	\(q+3.53113 q^{2} +4.46887 q^{4} +5.00000 q^{5} -12.4689 q^{8} +O(q^{10})\)	\(q+3.53113 q^{2} +4.46887 q^{4} +5.00000 q^{5} -12.4689 q^{8} +17.6556 q^{10} +2.93774 q^{11} +19.0623 q^{13} -79.7802 q^{16} +122.498 q^{17} -107.436 q^{19} +22.3444 q^{20} +10.3735 q^{22} -210.623 q^{23} +25.0000 q^{25} +67.3113 q^{26} -95.4942 q^{29} +94.3074 q^{31} -181.963 q^{32} +432.556 q^{34} +97.1206 q^{37} -379.370 q^{38} -62.3444 q^{40} -491.113 q^{41} -43.0039 q^{43} +13.1284 q^{44} -743.735 q^{46} +473.494 q^{47} +88.2782 q^{50} +85.1868 q^{52} +183.677 q^{53} +14.6887 q^{55} -337.202 q^{58} -760.615 q^{59} +198.747 q^{61} +333.012 q^{62} -4.29373 q^{64} +95.3113 q^{65} -309.992 q^{67} +547.428 q^{68} -665.693 q^{71} -621.288 q^{73} +342.945 q^{74} -480.117 q^{76} -24.7626 q^{79} -398.901 q^{80} -1734.18 q^{82} -406.724 q^{83} +612.490 q^{85} -151.852 q^{86} -36.6303 q^{88} +261.751 q^{89} -941.245 q^{92} +1671.97 q^{94} -537.179 q^{95} +1004.77 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} + 17 q^{4} + 10 q^{5} - 33 q^{8}+O(q^{10}) \) 2 * q - q^2 + 17 * q^4 + 10 * q^5 - 33 * q^8	\( 2 q - q^{2} + 17 q^{4} + 10 q^{5} - 33 q^{8} - 5 q^{10} + 22 q^{11} + 22 q^{13} - 87 q^{16} + 116 q^{17} - 102 q^{19} + 85 q^{20} - 76 q^{22} - 260 q^{23} + 50 q^{25} + 54 q^{26} + 196 q^{29} - 150 q^{31} + 15 q^{32} + 462 q^{34} - 96 q^{37} - 404 q^{38} - 165 q^{40} - 176 q^{41} - 344 q^{43} + 252 q^{44} - 520 q^{46} + 560 q^{47} - 25 q^{50} + 122 q^{52} - 326 q^{53} + 110 q^{55} - 1658 q^{58} - 844 q^{59} + 204 q^{61} + 1440 q^{62} - 839 q^{64} + 110 q^{65} - 104 q^{67} + 466 q^{68} - 1670 q^{71} + 386 q^{73} + 1218 q^{74} - 412 q^{76} - 888 q^{79} - 435 q^{80} - 3162 q^{82} + 928 q^{83} + 580 q^{85} + 1212 q^{86} - 428 q^{88} + 588 q^{89} - 1560 q^{92} + 1280 q^{94} - 510 q^{95} - 522 q^{97}+O(q^{100}) \) 2 * q - q^2 + 17 * q^4 + 10 * q^5 - 33 * q^8 - 5 * q^10 + 22 * q^11 + 22 * q^13 - 87 * q^16 + 116 * q^17 - 102 * q^19 + 85 * q^20 - 76 * q^22 - 260 * q^23 + 50 * q^25 + 54 * q^26 + 196 * q^29 - 150 * q^31 + 15 * q^32 + 462 * q^34 - 96 * q^37 - 404 * q^38 - 165 * q^40 - 176 * q^41 - 344 * q^43 + 252 * q^44 - 520 * q^46 + 560 * q^47 - 25 * q^50 + 122 * q^52 - 326 * q^53 + 110 * q^55 - 1658 * q^58 - 844 * q^59 + 204 * q^61 + 1440 * q^62 - 839 * q^64 + 110 * q^65 - 104 * q^67 + 466 * q^68 - 1670 * q^71 + 386 * q^73 + 1218 * q^74 - 412 * q^76 - 888 * q^79 - 435 * q^80 - 3162 * q^82 + 928 * q^83 + 580 * q^85 + 1212 * q^86 - 428 * q^88 + 588 * q^89 - 1560 * q^92 + 1280 * q^94 - 510 * q^95 - 522 * q^97

Introduction

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