LMFDB - Embedded newform 2205.4.a.bp.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:	$4$
Coefficient field:	4.4.51264.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 15x^{2} + 16x + 46 $ x^4 - 2x^3 - 15x^2 + 16*x + 46
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 735)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.50184$ 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)

$f(q)$	$=$	$q+0.912375 q^{2} -7.16757 q^{4} -5.00000 q^{5} -13.8385 q^{8} +O(q^{10})$	$q+0.912375 q^{2} -7.16757 q^{4} -5.00000 q^{5} -13.8385 q^{8} -4.56187 q^{10} +21.7933 q^{11} -34.4151 q^{13} +44.7147 q^{16} -5.70794 q^{17} +24.6906 q^{19} +35.8379 q^{20} +19.8837 q^{22} -92.6544 q^{23} +25.0000 q^{25} -31.3995 q^{26} +55.2788 q^{29} +116.651 q^{31} +151.505 q^{32} -5.20778 q^{34} +90.6006 q^{37} +22.5271 q^{38} +69.1925 q^{40} -38.1301 q^{41} +229.920 q^{43} -156.205 q^{44} -84.5355 q^{46} +19.6129 q^{47} +22.8094 q^{50} +246.673 q^{52} +111.356 q^{53} -108.967 q^{55} +50.4350 q^{58} -402.344 q^{59} +359.954 q^{61} +106.429 q^{62} -219.488 q^{64} +172.075 q^{65} -864.308 q^{67} +40.9120 q^{68} +1078.94 q^{71} +596.587 q^{73} +82.6617 q^{74} -176.972 q^{76} -723.054 q^{79} -223.573 q^{80} -34.7889 q^{82} +1043.84 q^{83} +28.5397 q^{85} +209.773 q^{86} -301.587 q^{88} +243.768 q^{89} +664.107 q^{92} +17.8943 q^{94} -123.453 q^{95} -609.252 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 6 q^{2} + 18 q^{4} - 20 q^{5} + 30 q^{8}+O(q^{10}) $ 4 * q + 6 * q^2 + 18 * q^4 - 20 * q^5 + 30 * q^8	$ 4 q + 6 q^{2} + 18 q^{4} - 20 q^{5} + 30 q^{8} - 30 q^{10} + 24 q^{11} - 46 q^{16} - 88 q^{17} - 72 q^{19} - 90 q^{20} - 28 q^{22} + 100 q^{25} + 112 q^{26} + 636 q^{29} - 228 q^{31} + 126 q^{32} + 48 q^{34} - 68 q^{37} - 408 q^{38} - 150 q^{40} - 56 q^{41} + 20 q^{43} - 276 q^{44} - 596 q^{46} - 744 q^{47} + 150 q^{50} + 768 q^{52} + 360 q^{53} - 120 q^{55} + 1352 q^{58} - 1828 q^{59} - 1212 q^{61} - 1044 q^{62} - 718 q^{64} - 1036 q^{67} - 1056 q^{68} + 876 q^{71} - 420 q^{73} + 1584 q^{74} - 2400 q^{76} - 700 q^{79} + 230 q^{80} - 492 q^{82} - 1020 q^{83} + 440 q^{85} - 636 q^{86} - 748 q^{88} + 1192 q^{89} + 540 q^{92} - 3732 q^{94} + 360 q^{95} - 588 q^{97}+O(q^{100}) $ 4 * q + 6 * q^2 + 18 * q^4 - 20 * q^5 + 30 * q^8 - 30 * q^10 + 24 * q^11 - 46 * q^16 - 88 * q^17 - 72 * q^19 - 90 * q^20 - 28 * q^22 + 100 * q^25 + 112 * q^26 + 636 * q^29 - 228 * q^31 + 126 * q^32 + 48 * q^34 - 68 * q^37 - 408 * q^38 - 150 * q^40 - 56 * q^41 + 20 * q^43 - 276 * q^44 - 596 * q^46 - 744 * q^47 + 150 * q^50 + 768 * q^52 + 360 * q^53 - 120 * q^55 + 1352 * q^58 - 1828 * q^59 - 1212 * q^61 - 1044 * q^62 - 718 * q^64 - 1036 * q^67 - 1056 * q^68 + 876 * q^71 - 420 * q^73 + 1584 * q^74 - 2400 * q^76 - 700 * q^79 + 230 * q^80 - 492 * q^82 - 1020 * q^83 + 440 * q^85 - 636 * q^86 - 748 * q^88 + 1192 * q^89 + 540 * q^92 - 3732 * q^94 + 360 * q^95 - 588 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0.912375	0.322573	0.161287	−	0.986908i	$-0.448436\pi$
$2$	0.912375	0.322573	0.161287	+	0.986908i	$0.448436\pi$
$3$	0	0
$3$	0	0
$4$	−7.16757	−0.895947
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−13.8385	−0.611581
$9$	0	0
$10$	−4.56187	−0.144259
$11$	21.7933	0.597358	0.298679	−	0.954354i	$-0.403454\pi$
$11$	21.7933	0.597358	0.298679	+	0.954354i	$0.403454\pi$
$12$	0	0
$13$	−34.4151	−0.734233	−0.367117	−	0.930175i	$-0.619655\pi$
$13$	−34.4151	−0.734233	−0.367117	+	0.930175i	$0.619655\pi$
$14$	0	0
$15$	0	0
$16$	44.7147	0.698667
$17$	−5.70794	−0.0814340	−0.0407170	−	0.999171i	$-0.512964\pi$
$17$	−5.70794	−0.0814340	−0.0407170	+	0.999171i	$0.512964\pi$
$18$	0	0
$19$	24.6906	0.298127	0.149064	−	0.988828i	$-0.452374\pi$
$19$	24.6906	0.298127	0.149064	+	0.988828i	$0.452374\pi$
$20$	35.8379	0.400679
$21$	0	0
$22$	19.8837	0.192692
$23$	−92.6544	−0.839990	−0.419995	−	0.907526i	$-0.637968\pi$
$23$	−92.6544	−0.839990	−0.419995	+	0.907526i	$0.637968\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	−31.3995	−0.236844
$27$	0	0
$28$	0	0
$29$	55.2788	0.353966	0.176983	−	0.984214i	$-0.443366\pi$
$29$	55.2788	0.353966	0.176983	+	0.984214i	$0.443366\pi$
$30$	0	0
$31$	116.651	0.675841	0.337920	−	0.941175i	$-0.390277\pi$
$31$	116.651	0.675841	0.337920	+	0.941175i	$0.390277\pi$
$32$	151.505	0.836953
$33$	0	0
$34$	−5.20778	−0.0262684
$35$	0	0
$36$	0	0
$37$	90.6006	0.402558	0.201279	−	0.979534i	$-0.435490\pi$
$37$	90.6006	0.402558	0.201279	+	0.979534i	$0.435490\pi$
$38$	22.5271	0.0961679
$39$	0	0
$40$	69.1925	0.273508
$41$	−38.1301	−0.145242	−0.0726209	−	0.997360i	$-0.523136\pi$
$41$	−38.1301	−0.145242	−0.0726209	+	0.997360i	$0.523136\pi$
$42$	0	0
$43$	229.920	0.815407	0.407703	−	0.913114i	$-0.366330\pi$
$43$	229.920	0.815407	0.407703	+	0.913114i	$0.366330\pi$
$44$	−156.205	−0.535201
$45$	0	0
$46$	−84.5355	−0.270958
$47$	19.6129	0.0608689	0.0304345	−	0.999537i	$-0.490311\pi$
$47$	19.6129	0.0608689	0.0304345	+	0.999537i	$0.490311\pi$
$48$	0	0
$49$	0	0
$50$	22.8094	0.0645146
$51$	0	0
$52$	246.673	0.657834
$53$	111.356	0.288603	0.144302	−	0.989534i	$-0.453906\pi$
$53$	111.356	0.288603	0.144302	+	0.989534i	$0.453906\pi$
$54$	0	0
$55$	−108.967	−0.267147
$56$	0	0
$57$	0	0
$58$	50.4350	0.114180
$59$	−402.344	−0.887809	−0.443905	−	0.896074i	$-0.646407\pi$
$59$	−402.344	−0.887809	−0.443905	+	0.896074i	$0.646407\pi$
$60$	0	0
$61$	359.954	0.755530	0.377765	−	0.925901i	$-0.376693\pi$
$61$	359.954	0.755530	0.377765	+	0.925901i	$0.376693\pi$
$62$	106.429	0.218008
$63$	0	0
$64$	−219.488	−0.428688
$65$	172.075	0.328359
$66$	0	0
$67$	−864.308	−1.57600	−0.788000	−	0.615675i	$-0.788883\pi$
$67$	−864.308	−1.57600	−0.788000	+	0.615675i	$0.788883\pi$
$68$	40.9120	0.0729605
$69$	0	0
$70$	0	0
$71$	1078.94	1.80348	0.901739	−	0.432280i	$-0.142291\pi$
$71$	1078.94	1.80348	0.901739	+	0.432280i	$0.142291\pi$
$72$	0	0
$73$	596.587	0.956509	0.478255	−	0.878221i	$-0.341270\pi$
$73$	596.587	0.956509	0.478255	+	0.878221i	$0.341270\pi$
$74$	82.6617	0.129854
$75$	0	0
$76$	−176.972	−0.267106
$77$	0	0
$78$	0	0
$79$	−723.054	−1.02975	−0.514873	−	0.857267i	$-0.672161\pi$
$79$	−723.054	−1.02975	−0.514873	+	0.857267i	$0.672161\pi$
$80$	−223.573	−0.312453
$81$	0	0
$82$	−34.7889	−0.0468511
$83$	1043.84	1.38044	0.690219	−	0.723601i	$-0.257514\pi$
$83$	1043.84	1.38044	0.690219	+	0.723601i	$0.257514\pi$
$84$	0	0
$85$	28.5397	0.0364184
$86$	209.773	0.263028
$87$	0	0
$88$	−301.587	−0.365333
$89$	243.768	0.290329	0.145165	−	0.989408i	$-0.453629\pi$
$89$	243.768	0.290329	0.145165	+	0.989408i	$0.453629\pi$
$90$	0	0
$91$	0	0
$92$	664.107	0.752586
$93$	0	0
$94$	17.8943	0.0196347
$95$	−123.453	−0.133327
$96$	0	0
$97$	−609.252	−0.637734	−0.318867	−	0.947800i	$-0.603302\pi$
$97$	−609.252	−0.637734	−0.318867	+	0.947800i	$0.603302\pi$
$98$	0	0
$99$	0	0
$100$	−179.189	−0.179189
$101$	−441.823	−0.435278	−0.217639	−	0.976029i	$-0.569836\pi$
$101$	−441.823	−0.435278	−0.217639	+	0.976029i	$0.569836\pi$
$102$	0	0
$103$	−806.414	−0.771440	−0.385720	−	0.922616i	$-0.626047\pi$
$103$	−806.414	−0.771440	−0.385720	+	0.922616i	$0.626047\pi$
$104$	476.254	0.449043
$105$	0	0
$106$	101.599	0.0930956
$107$	618.019	0.558375	0.279188	−	0.960237i	$-0.409935\pi$
$107$	618.019	0.558375	0.279188	+	0.960237i	$0.409935\pi$
$108$	0	0
$109$	−2147.07	−1.88671	−0.943357	−	0.331779i	$-0.892351\pi$
$109$	−2147.07	−1.88671	−0.943357	+	0.331779i	$0.892351\pi$
$110$	−99.4184	−0.0861743
$111$	0	0
$112$	0	0
$113$	1235.34	1.02842	0.514209	−	0.857665i	$-0.328085\pi$
$113$	1235.34	1.02842	0.514209	+	0.857665i	$0.328085\pi$
$114$	0	0
$115$	463.272	0.375655
$116$	−396.215	−0.317135
$117$	0	0
$118$	−367.089	−0.286383
$119$	0	0
$120$	0	0
$121$	−856.051	−0.643164
$122$	328.413	0.243714
$123$	0	0
$124$	−836.102	−0.605517
$125$	−125.000	−0.0894427
$126$	0	0
$127$	2200.67	1.53762	0.768811	−	0.639475i	$-0.220848\pi$
$127$	2200.67	1.53762	0.768811	+	0.639475i	$0.220848\pi$
$128$	−1412.29	−0.975236
$129$	0	0
$130$	156.997	0.105920
$131$	−2656.55	−1.77179	−0.885893	−	0.463890i	$-0.846453\pi$
$131$	−2656.55	−1.77179	−0.885893	+	0.463890i	$0.846453\pi$
$132$	0	0
$133$	0	0
$134$	−788.572	−0.508375
$135$	0	0
$136$	78.9893	0.0498035
$137$	2143.69	1.33684	0.668422	−	0.743782i	$-0.266970\pi$
$137$	2143.69	1.33684	0.668422	+	0.743782i	$0.266970\pi$
$138$	0	0
$139$	−2827.76	−1.72552	−0.862761	−	0.505612i	$-0.831267\pi$
$139$	−2827.76	−1.72552	−0.862761	+	0.505612i	$0.831267\pi$
$140$	0	0
$141$	0	0
$142$	984.400	0.581754
$143$	−750.020	−0.438600
$144$	0	0
$145$	−276.394	−0.158298
$146$	544.311	0.308544
$147$	0	0
$148$	−649.386	−0.360670
$149$	922.724	0.507332	0.253666	−	0.967292i	$-0.418364\pi$
$149$	922.724	0.507332	0.253666	+	0.967292i	$0.418364\pi$
$150$	0	0
$151$	−835.324	−0.450183	−0.225092	−	0.974338i	$-0.572268\pi$
$151$	−835.324	−0.450183	−0.225092	+	0.974338i	$0.572268\pi$
$152$	−341.682	−0.182329
$153$	0	0
$154$	0	0
$155$	−583.253	−0.302245
$156$	0	0
$157$	−2055.05	−1.04465	−0.522327	−	0.852745i	$-0.674936\pi$
$157$	−2055.05	−1.04465	−0.522327	+	0.852745i	$0.674936\pi$
$158$	−659.696	−0.332168
$159$	0	0
$160$	−757.523	−0.374297
$161$	0	0
$162$	0	0
$163$	1361.53	0.654251	0.327126	−	0.944981i	$-0.393920\pi$
$163$	1361.53	0.654251	0.327126	+	0.944981i	$0.393920\pi$
$164$	273.300	0.130129
$165$	0	0
$166$	952.373	0.445292
$167$	−1042.90	−0.483247	−0.241623	−	0.970370i	$-0.577680\pi$
$167$	−1042.90	−0.483247	−0.241623	+	0.970370i	$0.577680\pi$
$168$	0	0
$169$	−1012.60	−0.460902
$170$	26.0389	0.0117476
$171$	0	0
$172$	−1647.97	−0.730561
$173$	−3397.23	−1.49299	−0.746494	−	0.665392i	$-0.768264\pi$
$173$	−3397.23	−1.49299	−0.746494	+	0.665392i	$0.768264\pi$
$174$	0	0
$175$	0	0
$176$	974.482	0.417354
$177$	0	0
$178$	222.407	0.0936524
$179$	−125.381	−0.0523543	−0.0261771	−	0.999657i	$-0.508333\pi$
$179$	−125.381	−0.0523543	−0.0261771	+	0.999657i	$0.508333\pi$
$180$	0	0
$181$	988.234	0.405828	0.202914	−	0.979197i	$-0.434959\pi$
$181$	988.234	0.405828	0.202914	+	0.979197i	$0.434959\pi$
$182$	0	0
$183$	0	0
$184$	1282.20	0.513722
$185$	−453.003	−0.180029
$186$	0	0
$187$	−124.395	−0.0486452
$188$	−140.577	−0.0545353
$189$	0	0
$190$	−112.636	−0.0430076
$191$	2669.26	1.01121	0.505604	−	0.862766i	$-0.331270\pi$
$191$	2669.26	1.01121	0.505604	+	0.862766i	$0.331270\pi$
$192$	0	0
$193$	−2228.29	−0.831067	−0.415534	−	0.909578i	$-0.636405\pi$
$193$	−2228.29	−0.831067	−0.415534	+	0.909578i	$0.636405\pi$
$194$	−555.866	−0.205716
$195$	0	0
$196$	0	0
$197$	365.525	0.132196	0.0660979	−	0.997813i	$-0.478945\pi$
$197$	365.525	0.132196	0.0660979	+	0.997813i	$0.478945\pi$
$198$	0	0
$199$	−2789.70	−0.993751	−0.496875	−	0.867822i	$-0.665519\pi$
$199$	−2789.70	−0.993751	−0.496875	+	0.867822i	$0.665519\pi$
$200$	−345.963	−0.122316
$201$	0	0
$202$	−403.108	−0.140409
$203$	0	0
$204$	0	0
$205$	190.650	0.0649541
$206$	−735.752	−0.248846
$207$	0	0
$208$	−1538.86	−0.512984
$209$	538.091	0.178089
$210$	0	0
$211$	−578.631	−0.188789	−0.0943947	−	0.995535i	$-0.530092\pi$
$211$	−578.631	−0.188789	−0.0943947	+	0.995535i	$0.530092\pi$
$212$	−798.155	−0.258573
$213$	0	0
$214$	563.865	0.180117
$215$	−1149.60	−0.364661
$216$	0	0
$217$	0	0
$218$	−1958.93	−0.608603
$219$	0	0
$220$	781.026	0.239349
$221$	196.439	0.0597915
$222$	0	0
$223$	4664.97	1.40085	0.700424	−	0.713727i	$-0.252994\pi$
$223$	4664.97	1.40085	0.700424	+	0.713727i	$0.252994\pi$
$224$	0	0
$225$	0	0
$226$	1127.10	0.331740
$227$	−2575.43	−0.753028	−0.376514	−	0.926411i	$-0.622877\pi$
$227$	−2575.43	−0.753028	−0.376514	+	0.926411i	$0.622877\pi$
$228$	0	0
$229$	−4924.10	−1.42093	−0.710467	−	0.703731i	$-0.751516\pi$
$229$	−4924.10	−1.42093	−0.710467	+	0.703731i	$0.751516\pi$
$230$	422.678	0.121176
$231$	0	0
$232$	−764.976	−0.216479
$233$	−6013.17	−1.69071	−0.845356	−	0.534203i	$-0.820612\pi$
$233$	−6013.17	−1.69071	−0.845356	+	0.534203i	$0.820612\pi$
$234$	0	0
$235$	−98.0646	−0.0272214
$236$	2883.83	0.795430
$237$	0	0
$238$	0	0
$239$	−302.198	−0.0817890	−0.0408945	−	0.999163i	$-0.513021\pi$
$239$	−302.198	−0.0817890	−0.0408945	+	0.999163i	$0.513021\pi$
$240$	0	0
$241$	−4196.17	−1.12157	−0.560786	−	0.827961i	$-0.689501\pi$
$241$	−4196.17	−1.12157	−0.560786	+	0.827961i	$0.689501\pi$
$242$	−781.039	−0.207467
$243$	0	0
$244$	−2579.99	−0.676915
$245$	0	0
$246$	0	0
$247$	−849.731	−0.218895
$248$	−1614.27	−0.413332
$249$	0	0
$250$	−114.047	−0.0288518
$251$	−7388.87	−1.85809	−0.929046	−	0.369964i	$-0.879370\pi$
$251$	−7388.87	−1.85809	−0.929046	+	0.369964i	$0.879370\pi$
$252$	0	0
$253$	−2019.25	−0.501775
$254$	2007.84	0.495996
$255$	0	0
$256$	467.368	0.114104
$257$	290.777	0.0705766	0.0352883	−	0.999377i	$-0.488765\pi$
$257$	290.777	0.0705766	0.0352883	+	0.999377i	$0.488765\pi$
$258$	0	0
$259$	0	0
$260$	−1233.36	−0.294192
$261$	0	0
$262$	−2423.77	−0.571530
$263$	−1165.41	−0.273241	−0.136620	−	0.990623i	$-0.543624\pi$
$263$	−1165.41	−0.273241	−0.136620	+	0.990623i	$0.543624\pi$
$264$	0	0
$265$	−556.782	−0.129067
$266$	0	0
$267$	0	0
$268$	6194.99	1.41201
$269$	4355.80	0.987278	0.493639	−	0.869667i	$-0.335666\pi$
$269$	4355.80	0.987278	0.493639	+	0.869667i	$0.335666\pi$
$270$	0	0
$271$	−4132.57	−0.926330	−0.463165	−	0.886272i	$-0.653286\pi$
$271$	−4132.57	−0.926330	−0.463165	+	0.886272i	$0.653286\pi$
$272$	−255.229	−0.0568952
$273$	0	0
$274$	1955.85	0.431230
$275$	544.833	0.119472
$276$	0	0
$277$	−2253.23	−0.488749	−0.244374	−	0.969681i	$-0.578583\pi$
$277$	−2253.23	−0.488749	−0.244374	+	0.969681i	$0.578583\pi$
$278$	−2579.98	−0.556607
$279$	0	0
$280$	0	0
$281$	4130.62	0.876912	0.438456	−	0.898753i	$-0.355526\pi$
$281$	4130.62	0.876912	0.438456	+	0.898753i	$0.355526\pi$
$282$	0	0
$283$	−5688.49	−1.19486	−0.597430	−	0.801921i	$-0.703811\pi$
$283$	−5688.49	−1.19486	−0.597430	+	0.801921i	$0.703811\pi$
$284$	−7733.40	−1.61582
$285$	0	0
$286$	−684.299	−0.141481
$287$	0	0
$288$	0	0
$289$	−4880.42	−0.993369
$290$	−252.175	−0.0510628
$291$	0	0
$292$	−4276.08	−0.856981
$293$	−4506.38	−0.898518	−0.449259	−	0.893402i	$-0.648312\pi$
$293$	−4506.38	−0.898518	−0.449259	+	0.893402i	$0.648312\pi$
$294$	0	0
$295$	2011.72	0.397040
$296$	−1253.78	−0.246197
$297$	0	0
$298$	841.870	0.163652
$299$	3188.71	0.616749
$300$	0	0
$301$	0	0
$302$	−762.128	−0.145217
$303$	0	0
$304$	1104.03	0.208292
$305$	−1799.77	−0.337883
$306$	0	0
$307$	5380.42	1.00025	0.500125	−	0.865953i	$-0.333287\pi$
$307$	5380.42	1.00025	0.500125	+	0.865953i	$0.333287\pi$
$308$	0	0
$309$	0	0
$310$	−532.145	−0.0974962
$311$	1862.37	0.339568	0.169784	−	0.985481i	$-0.445693\pi$
$311$	1862.37	0.339568	0.169784	+	0.985481i	$0.445693\pi$
$312$	0	0
$313$	−7756.59	−1.40073	−0.700364	−	0.713785i	$-0.746979\pi$
$313$	−7756.59	−1.40073	−0.700364	+	0.713785i	$0.746979\pi$
$314$	−1874.97	−0.336977
$315$	0	0
$316$	5182.54	0.922597
$317$	4108.22	0.727888	0.363944	−	0.931421i	$-0.381430\pi$
$317$	4108.22	0.727888	0.363944	+	0.931421i	$0.381430\pi$
$318$	0	0
$319$	1204.71	0.211444
$320$	1097.44	0.191715
$321$	0	0
$322$	0	0
$323$	−140.933	−0.0242777
$324$	0	0
$325$	−860.377	−0.146847
$326$	1242.22	0.211044
$327$	0	0
$328$	527.663	0.0888272
$329$	0	0
$330$	0	0
$331$	10738.0	1.78312	0.891561	−	0.452901i	$-0.149611\pi$
$331$	10738.0	1.78312	0.891561	+	0.452901i	$0.149611\pi$
$332$	−7481.80	−1.23680
$333$	0	0
$334$	−951.518	−0.155882
$335$	4321.54	0.704809
$336$	0	0
$337$	6108.83	0.987446	0.493723	−	0.869619i	$-0.335636\pi$
$337$	6108.83	0.987446	0.493723	+	0.869619i	$0.335636\pi$
$338$	−923.872	−0.148675
$339$	0	0
$340$	−204.560	−0.0326289
$341$	2542.21	0.403719
$342$	0	0
$343$	0	0
$344$	−3181.75	−0.498688
$345$	0	0
$346$	−3099.55	−0.481598
$347$	−4083.12	−0.631680	−0.315840	−	0.948812i	$-0.602286\pi$
$347$	−4083.12	−0.631680	−0.315840	+	0.948812i	$0.602286\pi$
$348$	0	0
$349$	−6984.77	−1.07131	−0.535653	−	0.844438i	$-0.679935\pi$
$349$	−6984.77	−1.07131	−0.535653	+	0.844438i	$0.679935\pi$
$350$	0	0
$351$	0	0
$352$	3301.79	0.499960
$353$	5150.68	0.776610	0.388305	−	0.921531i	$-0.373061\pi$
$353$	5150.68	0.776610	0.388305	+	0.921531i	$0.373061\pi$
$354$	0	0
$355$	−5394.71	−0.806540
$356$	−1747.22	−0.260120
$357$	0	0
$358$	−114.394	−0.0168881
$359$	9534.37	1.40169	0.700843	−	0.713316i	$-0.252808\pi$
$359$	9534.37	1.40169	0.700843	+	0.713316i	$0.252808\pi$
$360$	0	0
$361$	−6249.37	−0.911120
$362$	901.640	0.130909
$363$	0	0
$364$	0	0
$365$	−2982.93	−0.427764
$366$	0	0
$367$	−2838.67	−0.403752	−0.201876	−	0.979411i	$-0.564704\pi$
$367$	−2838.67	−0.403752	−0.201876	+	0.979411i	$0.564704\pi$
$368$	−4143.01	−0.586873
$369$	0	0
$370$	−413.308	−0.0580727
$371$	0	0
$372$	0	0
$373$	5050.10	0.701030	0.350515	−	0.936557i	$-0.386007\pi$
$373$	5050.10	0.701030	0.350515	+	0.936557i	$0.386007\pi$
$374$	−113.495	−0.0156916
$375$	0	0
$376$	−271.414	−0.0372263
$377$	−1902.42	−0.259893
$378$	0	0
$379$	10250.5	1.38927	0.694633	−	0.719365i	$-0.255567\pi$
$379$	10250.5	1.38927	0.694633	+	0.719365i	$0.255567\pi$
$380$	884.860	0.119454
$381$	0	0
$382$	2435.36	0.326189
$383$	−10788.3	−1.43931	−0.719656	−	0.694330i	$-0.755701\pi$
$383$	−10788.3	−1.43931	−0.719656	+	0.694330i	$0.755701\pi$
$384$	0	0
$385$	0	0
$386$	−2033.04	−0.268080
$387$	0	0
$388$	4366.86	0.571375
$389$	−13666.5	−1.78129	−0.890644	−	0.454701i	$-0.849746\pi$
$389$	−13666.5	−1.78129	−0.890644	+	0.454701i	$0.849746\pi$
$390$	0	0
$391$	528.865	0.0684038
$392$	0	0
$393$	0	0
$394$	333.496	0.0426428
$395$	3615.27	0.460516
$396$	0	0
$397$	4595.66	0.580982	0.290491	−	0.956878i	$-0.406181\pi$
$397$	4595.66	0.580982	0.290491	+	0.956878i	$0.406181\pi$
$398$	−2545.25	−0.320557
$399$	0	0
$400$	1117.87	0.139733
$401$	−743.095	−0.0925396	−0.0462698	−	0.998929i	$-0.514733\pi$
$401$	−743.095	−0.0925396	−0.0462698	+	0.998929i	$0.514733\pi$
$402$	0	0
$403$	−4014.54	−0.496225
$404$	3166.80	0.389986
$405$	0	0
$406$	0	0
$407$	1974.49	0.240471
$408$	0	0
$409$	−10501.4	−1.26959	−0.634793	−	0.772683i	$-0.718914\pi$
$409$	−10501.4	−1.26959	−0.634793	+	0.772683i	$0.718914\pi$
$410$	173.944	0.0209525
$411$	0	0
$412$	5780.03	0.691169
$413$	0	0
$414$	0	0
$415$	−5219.20	−0.617350
$416$	−5214.04	−0.614518
$417$	0	0
$418$	490.941	0.0574467
$419$	5007.49	0.583847	0.291924	−	0.956442i	$-0.405705\pi$
$419$	5007.49	0.583847	0.291924	+	0.956442i	$0.405705\pi$
$420$	0	0
$421$	3845.77	0.445205	0.222603	−	0.974909i	$-0.428545\pi$
$421$	3845.77	0.445205	0.222603	+	0.974909i	$0.428545\pi$
$422$	−527.928	−0.0608984
$423$	0	0
$424$	−1541.01	−0.176504
$425$	−142.698	−0.0162868
$426$	0	0
$427$	0	0
$428$	−4429.70	−0.500275
$429$	0	0
$430$	−1048.87	−0.117630
$431$	−2164.42	−0.241894	−0.120947	−	0.992659i	$-0.538593\pi$
$431$	−2164.42	−0.241894	−0.120947	+	0.992659i	$0.538593\pi$
$432$	0	0
$433$	9645.06	1.07047	0.535233	−	0.844704i	$-0.320224\pi$
$433$	9645.06	1.07047	0.535233	+	0.844704i	$0.320224\pi$
$434$	0	0
$435$	0	0
$436$	15389.3	1.69040
$437$	−2287.70	−0.250424
$438$	0	0
$439$	−3346.93	−0.363873	−0.181936	−	0.983310i	$-0.558236\pi$
$439$	−3346.93	−0.363873	−0.181936	+	0.983310i	$0.558236\pi$
$440$	1507.94	0.163382
$441$	0	0
$442$	179.226	0.0192871
$443$	1595.52	0.171118	0.0855592	−	0.996333i	$-0.472732\pi$
$443$	1595.52	0.171118	0.0855592	+	0.996333i	$0.472732\pi$
$444$	0	0
$445$	−1218.84	−0.129839
$446$	4256.20	0.451876
$447$	0	0
$448$	0	0
$449$	4120.69	0.433112	0.216556	−	0.976270i	$-0.430518\pi$
$449$	4120.69	0.433112	0.216556	+	0.976270i	$0.430518\pi$
$450$	0	0
$451$	−830.981	−0.0867613
$452$	−8854.41	−0.921408
$453$	0	0
$454$	−2349.76	−0.242907
$455$	0	0
$456$	0	0
$457$	−2387.24	−0.244355	−0.122177	−	0.992508i	$-0.538988\pi$
$457$	−2387.24	−0.244355	−0.122177	+	0.992508i	$0.538988\pi$
$458$	−4492.62	−0.458355
$459$	0	0
$460$	−3320.54	−0.336567
$461$	−5144.27	−0.519724	−0.259862	−	0.965646i	$-0.583677\pi$
$461$	−5144.27	−0.519724	−0.259862	+	0.965646i	$0.583677\pi$
$462$	0	0
$463$	−5168.77	−0.518818	−0.259409	−	0.965768i	$-0.583528\pi$
$463$	−5168.77	−0.518818	−0.259409	+	0.965768i	$0.583528\pi$
$464$	2471.77	0.247304
$465$	0	0
$466$	−5486.26	−0.545378
$467$	−13240.9	−1.31202	−0.656011	−	0.754751i	$-0.727757\pi$
$467$	−13240.9	−1.31202	−0.656011	+	0.754751i	$0.727757\pi$
$468$	0	0
$469$	0	0
$470$	−89.4717	−0.00878090
$471$	0	0
$472$	5567.84	0.542968
$473$	5010.73	0.487090
$474$	0	0
$475$	617.266	0.0596255
$476$	0	0
$477$	0	0
$478$	−275.718	−0.0263829
$479$	5317.53	0.507232	0.253616	−	0.967305i	$-0.418380\pi$
$479$	5317.53	0.507232	0.253616	+	0.967305i	$0.418380\pi$
$480$	0	0
$481$	−3118.03	−0.295571
$482$	−3828.48	−0.361789
$483$	0	0
$484$	6135.81	0.576240
$485$	3046.26	0.285203
$486$	0	0
$487$	−6179.43	−0.574983	−0.287491	−	0.957783i	$-0.592821\pi$
$487$	−6179.43	−0.574983	−0.287491	+	0.957783i	$0.592821\pi$
$488$	−4981.22	−0.462068
$489$	0	0
$490$	0	0
$491$	14637.1	1.34534	0.672670	−	0.739943i	$-0.265148\pi$
$491$	14637.1	1.34534	0.672670	+	0.739943i	$0.265148\pi$
$492$	0	0
$493$	−315.528	−0.0288249
$494$	−775.273	−0.0706096
$495$	0	0
$496$	5215.99	0.472188
$497$	0	0
$498$	0	0
$499$	−19822.8	−1.77833	−0.889167	−	0.457582i	$-0.848716\pi$
$499$	−19822.8	−1.77833	−0.889167	+	0.457582i	$0.848716\pi$
$500$	895.947	0.0801359
$501$	0	0
$502$	−6741.41	−0.599371
$503$	−11114.5	−0.985228	−0.492614	−	0.870248i	$-0.663959\pi$
$503$	−11114.5	−0.985228	−0.492614	+	0.870248i	$0.663959\pi$
$504$	0	0
$505$	2209.12	0.194662
$506$	−1842.31	−0.161859
$507$	0	0
$508$	−15773.5	−1.37763
$509$	16458.2	1.43320	0.716599	−	0.697485i	$-0.245698\pi$
$509$	16458.2	1.43320	0.716599	+	0.697485i	$0.245698\pi$
$510$	0	0
$511$	0	0
$512$	11724.8	1.01204
$513$	0	0
$514$	265.298	0.0227661
$515$	4032.07	0.344999
$516$	0	0
$517$	427.431	0.0363605
$518$	0	0
$519$	0	0
$520$	−2381.27	−0.200818
$521$	957.425	0.0805097	0.0402548	−	0.999189i	$-0.487183\pi$
$521$	957.425	0.0805097	0.0402548	+	0.999189i	$0.487183\pi$
$522$	0	0
$523$	17562.5	1.46836	0.734181	−	0.678954i	$-0.237566\pi$
$523$	17562.5	1.46836	0.734181	+	0.678954i	$0.237566\pi$
$524$	19041.0	1.58743
$525$	0	0
$526$	−1063.29	−0.0881402
$527$	−665.834	−0.0550364
$528$	0	0
$529$	−3582.16	−0.294416
$530$	−507.993	−0.0416336
$531$	0	0
$532$	0	0
$533$	1312.25	0.106641
$534$	0	0
$535$	−3090.10	−0.249713
$536$	11960.7	0.963852
$537$	0	0
$538$	3974.12	0.318469
$539$	0	0
$540$	0	0
$541$	154.715	0.0122953	0.00614763	−	0.999981i	$-0.498043\pi$
$541$	154.715	0.0122953	0.00614763	+	0.999981i	$0.498043\pi$
$542$	−3770.45	−0.298809
$543$	0	0
$544$	−864.779	−0.0681564
$545$	10735.3	0.843764
$546$	0	0
$547$	−1178.60	−0.0921264	−0.0460632	−	0.998939i	$-0.514668\pi$
$547$	−1178.60	−0.0921264	−0.0460632	+	0.998939i	$0.514668\pi$
$548$	−15365.0	−1.19774
$549$	0	0
$550$	497.092	0.0385383
$551$	1364.87	0.105527
$552$	0	0
$553$	0	0
$554$	−2055.79	−0.157657
$555$	0	0
$556$	20268.2	1.54598
$557$	22450.8	1.70785	0.853924	−	0.520398i	$-0.174216\pi$
$557$	22450.8	1.70785	0.853924	+	0.520398i	$0.174216\pi$
$558$	0	0
$559$	−7912.72	−0.598699
$560$	0	0
$561$	0	0
$562$	3768.67	0.282868
$563$	16467.4	1.23271	0.616357	−	0.787467i	$-0.288608\pi$
$563$	16467.4	1.23271	0.616357	+	0.787467i	$0.288608\pi$
$564$	0	0
$565$	−6176.72	−0.459923
$566$	−5190.03	−0.385430
$567$	0	0
$568$	−14931.0	−1.10297
$569$	18743.5	1.38096	0.690480	−	0.723351i	$-0.257399\pi$
$569$	18743.5	1.38096	0.690480	+	0.723351i	$0.257399\pi$
$570$	0	0
$571$	−6688.98	−0.490236	−0.245118	−	0.969493i	$-0.578827\pi$
$571$	−6688.98	−0.490236	−0.245118	+	0.969493i	$0.578827\pi$
$572$	5375.82	0.392962
$573$	0	0
$574$	0	0
$575$	−2316.36	−0.167998
$576$	0	0
$577$	−6065.16	−0.437601	−0.218800	−	0.975770i	$-0.570214\pi$
$577$	−6065.16	−0.437601	−0.218800	+	0.975770i	$0.570214\pi$
$578$	−4452.77	−0.320434
$579$	0	0
$580$	1981.07	0.141827
$581$	0	0
$582$	0	0
$583$	2426.83	0.172399
$584$	−8255.87	−0.584983
$585$	0	0
$586$	−4111.51	−0.289838
$587$	24343.0	1.71166	0.855828	−	0.517260i	$-0.173048\pi$
$587$	24343.0	1.71166	0.855828	+	0.517260i	$0.173048\pi$
$588$	0	0
$589$	2880.18	0.201487
$590$	1835.44	0.128075
$591$	0	0
$592$	4051.18	0.281254
$593$	−13866.9	−0.960278	−0.480139	−	0.877193i	$-0.659414\pi$
$593$	−13866.9	−0.960278	−0.480139	+	0.877193i	$0.659414\pi$
$594$	0	0
$595$	0	0
$596$	−6613.69	−0.454543
$597$	0	0
$598$	2909.30	0.198947
$599$	−6098.42	−0.415985	−0.207992	−	0.978130i	$-0.566693\pi$
$599$	−6098.42	−0.415985	−0.207992	+	0.978130i	$0.566693\pi$
$600$	0	0
$601$	−17556.7	−1.19160	−0.595802	−	0.803131i	$-0.703166\pi$
$601$	−17556.7	−1.19160	−0.595802	+	0.803131i	$0.703166\pi$
$602$	0	0
$603$	0	0
$604$	5987.24	0.403340
$605$	4280.25	0.287631
$606$	0	0
$607$	8367.17	0.559494	0.279747	−	0.960074i	$-0.409749\pi$
$607$	8367.17	0.559494	0.279747	+	0.960074i	$0.409749\pi$
$608$	3740.75	0.249519
$609$	0	0
$610$	−1642.06	−0.108992
$611$	−674.981	−0.0446920
$612$	0	0
$613$	−26940.0	−1.77503	−0.887516	−	0.460776i	$-0.847571\pi$
$613$	−26940.0	−1.77503	−0.887516	+	0.460776i	$0.847571\pi$
$614$	4908.96	0.322654
$615$	0	0
$616$	0	0
$617$	13320.3	0.869136	0.434568	−	0.900639i	$-0.356901\pi$
$617$	13320.3	0.869136	0.434568	+	0.900639i	$0.356901\pi$
$618$	0	0
$619$	7336.05	0.476350	0.238175	−	0.971222i	$-0.423451\pi$
$619$	7336.05	0.476350	0.238175	+	0.971222i	$0.423451\pi$
$620$	4180.51	0.270796
$621$	0	0
$622$	1699.18	0.109535
$623$	0	0
$624$	0	0
$625$	625.000	0.0400000
$626$	−7076.91	−0.451838
$627$	0	0
$628$	14729.7	0.935954
$629$	−517.142	−0.0327819
$630$	0	0
$631$	103.633	0.00653815	0.00326908	−	0.999995i	$-0.498959\pi$
$631$	103.633	0.00653815	0.00326908	+	0.999995i	$0.498959\pi$
$632$	10006.0	0.629773
$633$	0	0
$634$	3748.23	0.234797
$635$	−11003.4	−0.687646
$636$	0	0
$637$	0	0
$638$	1099.15	0.0682063
$639$	0	0
$640$	7061.46	0.436139
$641$	−9950.30	−0.613125	−0.306562	−	0.951851i	$-0.599179\pi$
$641$	−9950.30	−0.613125	−0.306562	+	0.951851i	$0.599179\pi$
$642$	0	0
$643$	−1176.12	−0.0721334	−0.0360667	−	0.999349i	$-0.511483\pi$
$643$	−1176.12	−0.0721334	−0.0360667	+	0.999349i	$0.511483\pi$
$644$	0	0
$645$	0	0
$646$	−128.583	−0.00783134
$647$	−5590.22	−0.339682	−0.169841	−	0.985471i	$-0.554325\pi$
$647$	−5590.22	−0.339682	−0.169841	+	0.985471i	$0.554325\pi$
$648$	0	0
$649$	−8768.42	−0.530340
$650$	−784.986	−0.0473688
$651$	0	0
$652$	−9758.83	−0.586174
$653$	−24889.8	−1.49160	−0.745799	−	0.666171i	$-0.767932\pi$
$653$	−24889.8	−1.49160	−0.745799	+	0.666171i	$0.767932\pi$
$654$	0	0
$655$	13282.8	0.792367
$656$	−1704.97	−0.101476
$657$	0	0
$658$	0	0
$659$	−30597.0	−1.80863	−0.904317	−	0.426861i	$-0.859619\pi$
$659$	−30597.0	−1.80863	−0.904317	+	0.426861i	$0.859619\pi$
$660$	0	0
$661$	11335.6	0.667024	0.333512	−	0.942746i	$-0.391766\pi$
$661$	11335.6	0.667024	0.333512	+	0.942746i	$0.391766\pi$
$662$	9797.07	0.575187
$663$	0	0
$664$	−14445.2	−0.844250
$665$	0	0
$666$	0	0
$667$	−5121.82	−0.297328
$668$	7475.08	0.432963
$669$	0	0
$670$	3942.86	0.227352
$671$	7844.59	0.451322
$672$	0	0
$673$	−7919.80	−0.453619	−0.226810	−	0.973939i	$-0.572830\pi$
$673$	−7919.80	−0.453619	−0.226810	+	0.973939i	$0.572830\pi$
$674$	5573.54	0.318524
$675$	0	0
$676$	7257.89	0.412943
$677$	1189.67	0.0675373	0.0337686	−	0.999430i	$-0.489249\pi$
$677$	1189.67	0.0675373	0.0337686	+	0.999430i	$0.489249\pi$
$678$	0	0
$679$	0	0
$680$	−394.947	−0.0222728
$681$	0	0
$682$	2319.44	0.130229
$683$	15705.9	0.879895	0.439947	−	0.898024i	$-0.354997\pi$
$683$	15705.9	0.879895	0.439947	+	0.898024i	$0.354997\pi$
$684$	0	0
$685$	−10718.4	−0.597855
$686$	0	0
$687$	0	0
$688$	10280.8	0.569698
$689$	−3832.34	−0.211902
$690$	0	0
$691$	21910.8	1.20626	0.603131	−	0.797642i	$-0.293920\pi$
$691$	21910.8	1.20626	0.603131	+	0.797642i	$0.293920\pi$
$692$	24349.9	1.33764
$693$	0	0
$694$	−3725.33	−0.203763
$695$	14138.8	0.771677
$696$	0	0
$697$	217.644	0.0118276
$698$	−6372.72	−0.345575
$699$	0	0
$700$	0	0
$701$	11148.3	0.600666	0.300333	−	0.953834i	$-0.402902\pi$
$701$	11148.3	0.600666	0.300333	+	0.953834i	$0.402902\pi$
$702$	0	0
$703$	2236.99	0.120014
$704$	−4783.39	−0.256080
$705$	0	0
$706$	4699.35	0.250513
$707$	0	0
$708$	0	0
$709$	−32409.1	−1.71671	−0.858357	−	0.513053i	$-0.828514\pi$
$709$	−32409.1	−1.71671	−0.858357	+	0.513053i	$0.828514\pi$
$710$	−4922.00	−0.260168
$711$	0	0
$712$	−3373.38	−0.177560
$713$	−10808.2	−0.567700
$714$	0	0
$715$	3750.10	0.196148
$716$	898.677	0.0469066
$717$	0	0
$718$	8698.92	0.452146
$719$	−6557.21	−0.340115	−0.170058	−	0.985434i	$-0.554395\pi$
$719$	−6557.21	−0.340115	−0.170058	+	0.985434i	$0.554395\pi$
$720$	0	0
$721$	0	0
$722$	−5701.77	−0.293903
$723$	0	0
$724$	−7083.24	−0.363600
$725$	1381.97	0.0707932
$726$	0	0
$727$	−13303.4	−0.678673	−0.339337	−	0.940665i	$-0.610203\pi$
$727$	−13303.4	−0.678673	−0.339337	+	0.940665i	$0.610203\pi$
$728$	0	0
$729$	0	0
$730$	−2721.55	−0.137985
$731$	−1312.37	−0.0664018
$732$	0	0
$733$	−4179.54	−0.210607	−0.105304	−	0.994440i	$-0.533581\pi$
$733$	−4179.54	−0.210607	−0.105304	+	0.994440i	$0.533581\pi$
$734$	−2589.93	−0.130240
$735$	0	0
$736$	−14037.6	−0.703032
$737$	−18836.1	−0.941436
$738$	0	0
$739$	15706.6	0.781835	0.390918	−	0.920426i	$-0.372158\pi$
$739$	15706.6	0.781835	0.390918	+	0.920426i	$0.372158\pi$
$740$	3246.93	0.161297
$741$	0	0
$742$	0	0
$743$	4706.99	0.232413	0.116206	−	0.993225i	$-0.462927\pi$
$743$	4706.99	0.232413	0.116206	+	0.993225i	$0.462927\pi$
$744$	0	0
$745$	−4613.62	−0.226886
$746$	4607.58	0.226134
$747$	0	0
$748$	891.610	0.0435835
$749$	0	0
$750$	0	0
$751$	10206.7	0.495934	0.247967	−	0.968768i	$-0.420238\pi$
$751$	10206.7	0.495934	0.247967	+	0.968768i	$0.420238\pi$
$752$	876.986	0.0425271
$753$	0	0
$754$	−1735.72	−0.0838346
$755$	4176.62	0.201328
$756$	0	0
$757$	−2215.43	−0.106369	−0.0531843	−	0.998585i	$-0.516937\pi$
$757$	−2215.43	−0.106369	−0.0531843	+	0.998585i	$0.516937\pi$
$758$	9352.27	0.448140
$759$	0	0
$760$	1708.41	0.0815401
$761$	−13185.5	−0.628085	−0.314043	−	0.949409i	$-0.601683\pi$
$761$	−13185.5	−0.628085	−0.314043	+	0.949409i	$0.601683\pi$
$762$	0	0
$763$	0	0
$764$	−19132.1	−0.905988
$765$	0	0
$766$	−9842.98	−0.464284
$767$	13846.7	0.651859
$768$	0	0
$769$	−20178.7	−0.946247	−0.473123	−	0.880996i	$-0.656874\pi$
$769$	−20178.7	−0.946247	−0.473123	+	0.880996i	$0.656874\pi$
$770$	0	0
$771$	0	0
$772$	15971.4	0.744592
$773$	−24715.2	−1.14999	−0.574996	−	0.818156i	$-0.694997\pi$
$773$	−24715.2	−1.14999	−0.574996	+	0.818156i	$0.694997\pi$
$774$	0	0
$775$	2916.27	0.135168
$776$	8431.14	0.390026
$777$	0	0
$778$	−12469.0	−0.574596
$779$	−941.455	−0.0433006
$780$	0	0
$781$	23513.8	1.07732
$782$	482.523	0.0220652
$783$	0	0
$784$	0	0
$785$	10275.2	0.467183
$786$	0	0
$787$	−19544.5	−0.885244	−0.442622	−	0.896708i	$-0.645952\pi$
$787$	−19544.5	−0.885244	−0.442622	+	0.896708i	$0.645952\pi$
$788$	−2619.93	−0.118440
$789$	0	0
$790$	3298.48	0.148550
$791$	0	0
$792$	0	0
$793$	−12387.8	−0.554735
$794$	4192.97	0.187409
$795$	0	0
$796$	19995.4	0.890348
$797$	9401.10	0.417822	0.208911	−	0.977935i	$-0.433008\pi$
$797$	9401.10	0.417822	0.208911	+	0.977935i	$0.433008\pi$
$798$	0	0
$799$	−111.949	−0.00495680
$800$	3787.61	0.167391
$801$	0	0
$802$	−677.981	−0.0298508
$803$	13001.6	0.571378
$804$	0	0
$805$	0	0
$806$	−3662.77	−0.160069
$807$	0	0
$808$	6114.18	0.266208
$809$	6808.77	0.295901	0.147950	−	0.988995i	$-0.452732\pi$
$809$	6808.77	0.295901	0.147950	+	0.988995i	$0.452732\pi$
$810$	0	0
$811$	−23645.6	−1.02381	−0.511905	−	0.859042i	$-0.671060\pi$
$811$	−23645.6	−1.02381	−0.511905	+	0.859042i	$0.671060\pi$
$812$	0	0
$813$	0	0
$814$	1801.47	0.0775696
$815$	−6807.63	−0.292590
$816$	0	0
$817$	5676.88	0.243095
$818$	−9581.20	−0.409534
$819$	0	0
$820$	−1366.50	−0.0581954
$821$	29.4211	0.00125067	0.000625337	−	1.00000i	$-0.499801\pi$
$821$	29.4211	0.00125067	0.000625337		1.00000i	$0.499801\pi$
$822$	0	0
$823$	1489.88	0.0631031	0.0315516	−	0.999502i	$-0.489955\pi$
$823$	1489.88	0.0631031	0.0315516	+	0.999502i	$0.489955\pi$
$824$	11159.6	0.471799
$825$	0	0
$826$	0	0
$827$	−3903.20	−0.164121	−0.0820603	−	0.996627i	$-0.526150\pi$
$827$	−3903.20	−0.164121	−0.0820603	+	0.996627i	$0.526150\pi$
$828$	0	0
$829$	15812.3	0.662464	0.331232	−	0.943549i	$-0.392536\pi$
$829$	15812.3	0.662464	0.331232	+	0.943549i	$0.392536\pi$
$830$	−4761.87	−0.199141
$831$	0	0
$832$	7553.72	0.314757
$833$	0	0
$834$	0	0
$835$	5214.51	0.216115
$836$	−3856.81	−0.159558
$837$	0	0
$838$	4568.71	0.188333
$839$	−14730.3	−0.606136	−0.303068	−	0.952969i	$-0.598011\pi$
$839$	−14730.3	−0.606136	−0.303068	+	0.952969i	$0.598011\pi$
$840$	0	0
$841$	−21333.3	−0.874708
$842$	3508.78	0.143611
$843$	0	0
$844$	4147.38	0.169145
$845$	5063.01	0.206122
$846$	0	0
$847$	0	0
$848$	4979.26	0.201637
$849$	0	0
$850$	−130.194	−0.00525368
$851$	−8394.54	−0.338145
$852$	0	0
$853$	12630.7	0.506996	0.253498	−	0.967336i	$-0.418419\pi$
$853$	12630.7	0.506996	0.253498	+	0.967336i	$0.418419\pi$
$854$	0	0
$855$	0	0
$856$	−8552.46	−0.341492
$857$	−11507.3	−0.458671	−0.229335	−	0.973347i	$-0.573655\pi$
$857$	−11507.3	−0.458671	−0.229335	+	0.973347i	$0.573655\pi$
$858$	0	0
$859$	−4264.27	−0.169377	−0.0846886	−	0.996407i	$-0.526990\pi$
$859$	−4264.27	−0.169377	−0.0846886	+	0.996407i	$0.526990\pi$
$860$	8239.85	0.326717
$861$	0	0
$862$	−1974.76	−0.0780284
$863$	−17273.3	−0.681333	−0.340666	−	0.940184i	$-0.610653\pi$
$863$	−17273.3	−0.681333	−0.340666	+	0.940184i	$0.610653\pi$
$864$	0	0
$865$	16986.2	0.667685
$866$	8799.91	0.345304
$867$	0	0
$868$	0	0
$869$	−15757.8	−0.615127
$870$	0	0
$871$	29745.2	1.15715
$872$	29712.2	1.15388
$873$	0	0
$874$	−2087.24	−0.0807801
$875$	0	0
$876$	0	0
$877$	−17372.6	−0.668906	−0.334453	−	0.942412i	$-0.608552\pi$
$877$	−17372.6	−0.668906	−0.334453	+	0.942412i	$0.608552\pi$
$878$	−3053.65	−0.117376
$879$	0	0
$880$	−4872.41	−0.186646
$881$	−49105.7	−1.87788	−0.938941	−	0.344078i	$-0.888191\pi$
$881$	−49105.7	−1.87788	−0.938941	+	0.344078i	$0.888191\pi$
$882$	0	0
$883$	−1915.84	−0.0730161	−0.0365081	−	0.999333i	$-0.511623\pi$
$883$	−1915.84	−0.0730161	−0.0365081	+	0.999333i	$0.511623\pi$
$884$	−1407.99	−0.0535700
$885$	0	0
$886$	1455.71	0.0551982
$887$	175.503	0.00664354	0.00332177	−	0.999994i	$-0.498943\pi$
$887$	175.503	0.00664354	0.00332177	+	0.999994i	$0.498943\pi$
$888$	0	0
$889$	0	0
$890$	−1112.04	−0.0418826
$891$	0	0
$892$	−33436.5	−1.25509
$893$	484.256	0.0181467
$894$	0	0
$895$	626.905	0.0234135
$896$	0	0
$897$	0	0
$898$	3759.61	0.139710
$899$	6448.30	0.239225
$900$	0	0
$901$	−635.615	−0.0235021
$902$	−758.166	−0.0279869
$903$	0	0
$904$	−17095.3	−0.628962
$905$	−4941.17	−0.181492
$906$	0	0
$907$	35638.4	1.30469	0.652346	−	0.757922i	$-0.273785\pi$
$907$	35638.4	1.30469	0.652346	+	0.757922i	$0.273785\pi$
$908$	18459.6	0.674673
$909$	0	0
$910$	0	0
$911$	6181.19	0.224799	0.112400	−	0.993663i	$-0.464146\pi$
$911$	6181.19	0.224799	0.112400	+	0.993663i	$0.464146\pi$
$912$	0	0
$913$	22748.8	0.824615
$914$	−2178.05	−0.0788223
$915$	0	0
$916$	35293.9	1.27308
$917$	0	0
$918$	0	0
$919$	−21118.6	−0.758040	−0.379020	−	0.925388i	$-0.623739\pi$
$919$	−21118.6	−0.758040	−0.379020	+	0.925388i	$0.623739\pi$
$920$	−6410.99	−0.229744
$921$	0	0
$922$	−4693.50	−0.167649
$923$	−37131.9	−1.32417
$924$	0	0
$925$	2265.01	0.0805116
$926$	−4715.85	−0.167357
$927$	0	0
$928$	8374.99	0.296253
$929$	38221.8	1.34986	0.674928	−	0.737884i	$-0.264175\pi$
$929$	38221.8	1.34986	0.674928	+	0.737884i	$0.264175\pi$
$930$	0	0
$931$	0	0
$932$	43099.8	1.51479
$933$	0	0
$934$	−12080.6	−0.423223
$935$	621.975	0.0217548
$936$	0	0
$937$	51676.0	1.80169	0.900844	−	0.434142i	$-0.142948\pi$
$937$	51676.0	1.80169	0.900844	+	0.434142i	$0.142948\pi$
$938$	0	0
$939$	0	0
$940$	702.885	0.0243889
$941$	37184.2	1.28817	0.644086	−	0.764953i	$-0.277238\pi$
$941$	37184.2	1.28817	0.644086	+	0.764953i	$0.277238\pi$
$942$	0	0
$943$	3532.92	0.122002
$944$	−17990.7	−0.620283
$945$	0	0
$946$	4571.66	0.157122
$947$	−31923.3	−1.09543	−0.547713	−	0.836666i	$-0.684502\pi$
$947$	−31923.3	−1.09543	−0.547713	+	0.836666i	$0.684502\pi$
$948$	0	0
$949$	−20531.6	−0.702301
$950$	563.178	0.0192336
$951$	0	0
$952$	0	0
$953$	−2088.54	−0.0709912	−0.0354956	−	0.999370i	$-0.511301\pi$
$953$	−2088.54	−0.0709912	−0.0354956	+	0.999370i	$0.511301\pi$
$954$	0	0
$955$	−13346.3	−0.452226
$956$	2166.03	0.0732786
$957$	0	0
$958$	4851.58	0.163619
$959$	0	0
$960$	0	0
$961$	−16183.6	−0.543239
$962$	−2844.81	−0.0953434
$963$	0	0
$964$	30076.3	1.00487
$965$	11141.5	0.371664
$966$	0	0
$967$	46995.4	1.56285	0.781423	−	0.624002i	$-0.214494\pi$
$967$	46995.4	1.56285	0.781423	+	0.624002i	$0.214494\pi$
$968$	11846.5	0.393347
$969$	0	0
$970$	2779.33	0.0919989
$971$	−27063.6	−0.894452	−0.447226	−	0.894421i	$-0.647588\pi$
$971$	−27063.6	−0.894452	−0.447226	+	0.894421i	$0.647588\pi$
$972$	0	0
$973$	0	0
$974$	−5637.95	−0.185474
$975$	0	0
$976$	16095.2	0.527864
$977$	15115.3	0.494964	0.247482	−	0.968892i	$-0.420397\pi$
$977$	15115.3	0.494964	0.247482	+	0.968892i	$0.420397\pi$
$978$	0	0
$979$	5312.51	0.173431
$980$	0	0
$981$	0	0
$982$	13354.5	0.433970
$983$	31800.7	1.03182	0.515912	−	0.856641i	$-0.327453\pi$
$983$	31800.7	1.03182	0.515912	+	0.856641i	$0.327453\pi$
$984$	0	0
$985$	−1827.62	−0.0591198
$986$	−287.879	−0.00929812
$987$	0	0
$988$	6090.51	0.196118
$989$	−21303.1	−0.684934
$990$	0	0
$991$	59005.2	1.89139	0.945693	−	0.325061i	$-0.105385\pi$
$991$	59005.2	1.89139	0.945693	+	0.325061i	$0.105385\pi$
$992$	17673.1	0.565647
$993$	0	0
$994$	0	0
$995$	13948.5	0.444419
$996$	0	0
$997$	−3071.30	−0.0975618	−0.0487809	−	0.998810i	$-0.515534\pi$
$997$	−3071.30	−0.0975618	−0.0487809	+	0.998810i	$0.515534\pi$
$998$	−18085.8	−0.573643
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2205.4.a.bp.1.2		4
3.2	odd	2		735.4.a.t.1.3	✓	4
7.6	odd	2		2205.4.a.bq.1.2		4
21.20	even	2		735.4.a.u.1.3	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
735.4.a.t.1.3	✓	4	3.2	odd	2
735.4.a.u.1.3	yes	4	21.20	even	2
2205.4.a.bp.1.2		4	1.1	even	1	trivial
2205.4.a.bq.1.2		4	7.6	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2205 = 3^{2} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2205.a (trivial)

\(f(q)\)	\(=\)	\(q+0.912375 q^{2} -7.16757 q^{4} -5.00000 q^{5} -13.8385 q^{8} +O(q^{10})\)	\(q+0.912375 q^{2} -7.16757 q^{4} -5.00000 q^{5} -13.8385 q^{8} -4.56187 q^{10} +21.7933 q^{11} -34.4151 q^{13} +44.7147 q^{16} -5.70794 q^{17} +24.6906 q^{19} +35.8379 q^{20} +19.8837 q^{22} -92.6544 q^{23} +25.0000 q^{25} -31.3995 q^{26} +55.2788 q^{29} +116.651 q^{31} +151.505 q^{32} -5.20778 q^{34} +90.6006 q^{37} +22.5271 q^{38} +69.1925 q^{40} -38.1301 q^{41} +229.920 q^{43} -156.205 q^{44} -84.5355 q^{46} +19.6129 q^{47} +22.8094 q^{50} +246.673 q^{52} +111.356 q^{53} -108.967 q^{55} +50.4350 q^{58} -402.344 q^{59} +359.954 q^{61} +106.429 q^{62} -219.488 q^{64} +172.075 q^{65} -864.308 q^{67} +40.9120 q^{68} +1078.94 q^{71} +596.587 q^{73} +82.6617 q^{74} -176.972 q^{76} -723.054 q^{79} -223.573 q^{80} -34.7889 q^{82} +1043.84 q^{83} +28.5397 q^{85} +209.773 q^{86} -301.587 q^{88} +243.768 q^{89} +664.107 q^{92} +17.8943 q^{94} -123.453 q^{95} -609.252 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 6 q^{2} + 18 q^{4} - 20 q^{5} + 30 q^{8}+O(q^{10}) \) 4 * q + 6 * q^2 + 18 * q^4 - 20 * q^5 + 30 * q^8	\( 4 q + 6 q^{2} + 18 q^{4} - 20 q^{5} + 30 q^{8} - 30 q^{10} + 24 q^{11} - 46 q^{16} - 88 q^{17} - 72 q^{19} - 90 q^{20} - 28 q^{22} + 100 q^{25} + 112 q^{26} + 636 q^{29} - 228 q^{31} + 126 q^{32} + 48 q^{34} - 68 q^{37} - 408 q^{38} - 150 q^{40} - 56 q^{41} + 20 q^{43} - 276 q^{44} - 596 q^{46} - 744 q^{47} + 150 q^{50} + 768 q^{52} + 360 q^{53} - 120 q^{55} + 1352 q^{58} - 1828 q^{59} - 1212 q^{61} - 1044 q^{62} - 718 q^{64} - 1036 q^{67} - 1056 q^{68} + 876 q^{71} - 420 q^{73} + 1584 q^{74} - 2400 q^{76} - 700 q^{79} + 230 q^{80} - 492 q^{82} - 1020 q^{83} + 440 q^{85} - 636 q^{86} - 748 q^{88} + 1192 q^{89} + 540 q^{92} - 3732 q^{94} + 360 q^{95} - 588 q^{97}+O(q^{100}) \) 4 * q + 6 * q^2 + 18 * q^4 - 20 * q^5 + 30 * q^8 - 30 * q^10 + 24 * q^11 - 46 * q^16 - 88 * q^17 - 72 * q^19 - 90 * q^20 - 28 * q^22 + 100 * q^25 + 112 * q^26 + 636 * q^29 - 228 * q^31 + 126 * q^32 + 48 * q^34 - 68 * q^37 - 408 * q^38 - 150 * q^40 - 56 * q^41 + 20 * q^43 - 276 * q^44 - 596 * q^46 - 744 * q^47 + 150 * q^50 + 768 * q^52 + 360 * q^53 - 120 * q^55 + 1352 * q^58 - 1828 * q^59 - 1212 * q^61 - 1044 * q^62 - 718 * q^64 - 1036 * q^67 - 1056 * q^68 + 876 * q^71 - 420 * q^73 + 1584 * q^74 - 2400 * q^76 - 700 * q^79 + 230 * q^80 - 492 * q^82 - 1020 * q^83 + 440 * q^85 - 636 * q^86 - 748 * q^88 + 1192 * q^89 + 540 * q^92 - 3732 * q^94 + 360 * q^95 - 588 * q^97

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

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