LMFDB - Embedded newform 2205.4.a.bb.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:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 105)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ 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)

$f(q)$	$=$	$q+3.82843 q^{2} +6.65685 q^{4} +5.00000 q^{5} -5.14214 q^{8} +O(q^{10})$	$q+3.82843 q^{2} +6.65685 q^{4} +5.00000 q^{5} -5.14214 q^{8} +19.1421 q^{10} -48.5685 q^{11} +43.6569 q^{13} -72.9411 q^{16} -67.6569 q^{17} +93.2548 q^{19} +33.2843 q^{20} -185.941 q^{22} +104.167 q^{23} +25.0000 q^{25} +167.137 q^{26} +58.7351 q^{29} +9.08831 q^{31} -238.113 q^{32} -259.019 q^{34} -252.676 q^{37} +357.019 q^{38} -25.7107 q^{40} +276.274 q^{41} -92.6375 q^{43} -323.314 q^{44} +398.794 q^{46} -582.794 q^{47} +95.7107 q^{50} +290.617 q^{52} -623.019 q^{53} -242.843 q^{55} +224.863 q^{58} -524.999 q^{59} +352.794 q^{61} +34.7939 q^{62} -328.068 q^{64} +218.284 q^{65} -736.520 q^{67} -450.382 q^{68} +492.264 q^{71} -1164.75 q^{73} -967.352 q^{74} +620.784 q^{76} -872.195 q^{79} -364.706 q^{80} +1057.70 q^{82} -529.588 q^{83} -338.284 q^{85} -354.656 q^{86} +249.746 q^{88} -385.216 q^{89} +693.421 q^{92} -2231.18 q^{94} +466.274 q^{95} +463.892 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{4} + 10 q^{5} + 18 q^{8}+O(q^{10}) $ 2 * q + 2 * q^2 + 2 * q^4 + 10 * q^5 + 18 * q^8	$ 2 q + 2 q^{2} + 2 q^{4} + 10 q^{5} + 18 q^{8} + 10 q^{10} + 16 q^{11} + 76 q^{13} - 78 q^{16} - 124 q^{17} + 96 q^{19} + 10 q^{20} - 304 q^{22} + 16 q^{23} + 50 q^{25} + 108 q^{26} - 188 q^{29} + 120 q^{31} - 414 q^{32} - 156 q^{34} - 132 q^{37} + 352 q^{38} + 90 q^{40} + 100 q^{41} - 536 q^{43} - 624 q^{44} + 560 q^{46} - 928 q^{47} + 50 q^{50} + 140 q^{52} - 884 q^{53} + 80 q^{55} + 676 q^{58} + 104 q^{59} + 468 q^{61} - 168 q^{62} + 34 q^{64} + 380 q^{65} - 1688 q^{67} - 188 q^{68} + 136 q^{71} - 508 q^{73} - 1188 q^{74} + 608 q^{76} - 432 q^{79} - 390 q^{80} + 1380 q^{82} - 584 q^{83} - 620 q^{85} + 456 q^{86} + 1744 q^{88} - 1404 q^{89} + 1104 q^{92} - 1600 q^{94} + 480 q^{95} + 1188 q^{97}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^4 + 10 * q^5 + 18 * q^8 + 10 * q^10 + 16 * q^11 + 76 * q^13 - 78 * q^16 - 124 * q^17 + 96 * q^19 + 10 * q^20 - 304 * q^22 + 16 * q^23 + 50 * q^25 + 108 * q^26 - 188 * q^29 + 120 * q^31 - 414 * q^32 - 156 * q^34 - 132 * q^37 + 352 * q^38 + 90 * q^40 + 100 * q^41 - 536 * q^43 - 624 * q^44 + 560 * q^46 - 928 * q^47 + 50 * q^50 + 140 * q^52 - 884 * q^53 + 80 * q^55 + 676 * q^58 + 104 * q^59 + 468 * q^61 - 168 * q^62 + 34 * q^64 + 380 * q^65 - 1688 * q^67 - 188 * q^68 + 136 * q^71 - 508 * q^73 - 1188 * q^74 + 608 * q^76 - 432 * q^79 - 390 * q^80 + 1380 * q^82 - 584 * q^83 - 620 * q^85 + 456 * q^86 + 1744 * q^88 - 1404 * q^89 + 1104 * q^92 - 1600 * q^94 + 480 * q^95 + 1188 * 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 $

$2$	3.82843	1.35355	0.676777	−	0.736188i	$-0.263376\pi$
$2$	3.82843	1.35355	0.676777	+	0.736188i	$0.263376\pi$
$3$	0	0
$3$	0	0
$4$	6.65685	0.832107
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−5.14214	−0.227252
$9$	0	0
$10$	19.1421	0.605327
$11$	−48.5685	−1.33127	−0.665635	−	0.746278i	$-0.731839\pi$
$11$	−48.5685	−1.33127	−0.665635	+	0.746278i	$0.731839\pi$
$12$	0	0
$13$	43.6569	0.931403	0.465701	−	0.884942i	$-0.345802\pi$
$13$	43.6569	0.931403	0.465701	+	0.884942i	$0.345802\pi$
$14$	0	0
$15$	0	0
$16$	−72.9411	−1.13971
$17$	−67.6569	−0.965247	−0.482623	−	0.875828i	$-0.660316\pi$
$17$	−67.6569	−0.965247	−0.482623	+	0.875828i	$0.660316\pi$
$18$	0	0
$19$	93.2548	1.12601	0.563003	−	0.826455i	$-0.309646\pi$
$19$	93.2548	1.12601	0.563003	+	0.826455i	$0.309646\pi$
$20$	33.2843	0.372129
$21$	0	0
$22$	−185.941	−1.80194
$23$	104.167	0.944357	0.472179	−	0.881503i	$-0.343468\pi$
$23$	104.167	0.944357	0.472179	+	0.881503i	$0.343468\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	167.137	1.26070
$27$	0	0
$28$	0	0
$29$	58.7351	0.376098	0.188049	−	0.982160i	$-0.439784\pi$
$29$	58.7351	0.376098	0.188049	+	0.982160i	$0.439784\pi$
$30$	0	0
$31$	9.08831	0.0526551	0.0263276	−	0.999653i	$-0.491619\pi$
$31$	9.08831	0.0526551	0.0263276	+	0.999653i	$0.491619\pi$
$32$	−238.113	−1.31540
$33$	0	0
$34$	−259.019	−1.30651
$35$	0	0
$36$	0	0
$37$	−252.676	−1.12269	−0.561347	−	0.827580i	$-0.689717\pi$
$37$	−252.676	−1.12269	−0.561347	+	0.827580i	$0.689717\pi$
$38$	357.019	1.52411
$39$	0	0
$40$	−25.7107	−0.101630
$41$	276.274	1.05236	0.526180	−	0.850373i	$-0.323624\pi$
$41$	276.274	1.05236	0.526180	+	0.850373i	$0.323624\pi$
$42$	0	0
$43$	−92.6375	−0.328537	−0.164268	−	0.986416i	$-0.552526\pi$
$43$	−92.6375	−0.328537	−0.164268	+	0.986416i	$0.552526\pi$
$44$	−323.314	−1.10776
$45$	0	0
$46$	398.794	1.27824
$47$	−582.794	−1.80871	−0.904354	−	0.426784i	$-0.859646\pi$
$47$	−582.794	−1.80871	−0.904354	+	0.426784i	$0.859646\pi$
$48$	0	0
$49$	0	0
$50$	95.7107	0.270711
$51$	0	0
$52$	290.617	0.775026
$53$	−623.019	−1.61468	−0.807342	−	0.590083i	$-0.799095\pi$
$53$	−623.019	−1.61468	−0.807342	+	0.590083i	$0.799095\pi$
$54$	0	0
$55$	−242.843	−0.595362
$56$	0	0
$57$	0	0
$58$	224.863	0.509068
$59$	−524.999	−1.15846	−0.579229	−	0.815165i	$-0.696646\pi$
$59$	−524.999	−1.15846	−0.579229	+	0.815165i	$0.696646\pi$
$60$	0	0
$61$	352.794	0.740502	0.370251	−	0.928932i	$-0.379272\pi$
$61$	352.794	0.740502	0.370251	+	0.928932i	$0.379272\pi$
$62$	34.7939	0.0712715
$63$	0	0
$64$	−328.068	−0.640758
$65$	218.284	0.416536
$66$	0	0
$67$	−736.520	−1.34299	−0.671494	−	0.741010i	$-0.734347\pi$
$67$	−736.520	−1.34299	−0.671494	+	0.741010i	$0.734347\pi$
$68$	−450.382	−0.803188
$69$	0	0
$70$	0	0
$71$	492.264	0.822831	0.411415	−	0.911448i	$-0.365035\pi$
$71$	492.264	0.822831	0.411415	+	0.911448i	$0.365035\pi$
$72$	0	0
$73$	−1164.75	−1.86745	−0.933727	−	0.357987i	$-0.883463\pi$
$73$	−1164.75	−1.86745	−0.933727	+	0.357987i	$0.883463\pi$
$74$	−967.352	−1.51963
$75$	0	0
$76$	620.784	0.936958
$77$	0	0
$78$	0	0
$79$	−872.195	−1.24215	−0.621074	−	0.783752i	$-0.713303\pi$
$79$	−872.195	−1.24215	−0.621074	+	0.783752i	$0.713303\pi$
$80$	−364.706	−0.509692
$81$	0	0
$82$	1057.70	1.42443
$83$	−529.588	−0.700359	−0.350180	−	0.936683i	$-0.613879\pi$
$83$	−529.588	−0.700359	−0.350180	+	0.936683i	$0.613879\pi$
$84$	0	0
$85$	−338.284	−0.431672
$86$	−354.656	−0.444692
$87$	0	0
$88$	249.746	0.302534
$89$	−385.216	−0.458796	−0.229398	−	0.973333i	$-0.573676\pi$
$89$	−385.216	−0.458796	−0.229398	+	0.973333i	$0.573676\pi$
$90$	0	0
$91$	0	0
$92$	693.421	0.785806
$93$	0	0
$94$	−2231.18	−2.44818
$95$	466.274	0.503565
$96$	0	0
$97$	463.892	0.485579	0.242789	−	0.970079i	$-0.421938\pi$
$97$	463.892	0.485579	0.242789	+	0.970079i	$0.421938\pi$
$98$	0	0
$99$	0	0
$100$	166.421	0.166421
$101$	−432.725	−0.426314	−0.213157	−	0.977018i	$-0.568375\pi$
$101$	−432.725	−0.426314	−0.213157	+	0.977018i	$0.568375\pi$
$102$	0	0
$103$	−512.626	−0.490393	−0.245197	−	0.969473i	$-0.578853\pi$
$103$	−512.626	−0.490393	−0.245197	+	0.969473i	$0.578853\pi$
$104$	−224.489	−0.211663
$105$	0	0
$106$	−2385.18	−2.18556
$107$	−1963.09	−1.77363	−0.886817	−	0.462122i	$-0.847088\pi$
$107$	−1963.09	−1.77363	−0.886817	+	0.462122i	$0.847088\pi$
$108$	0	0
$109$	545.176	0.479068	0.239534	−	0.970888i	$-0.423005\pi$
$109$	545.176	0.479068	0.239534	+	0.970888i	$0.423005\pi$
$110$	−929.706	−0.805854
$111$	0	0
$112$	0	0
$113$	231.823	0.192992	0.0964961	−	0.995333i	$-0.469236\pi$
$113$	231.823	0.192992	0.0964961	+	0.995333i	$0.469236\pi$
$114$	0	0
$115$	520.833	0.422329
$116$	390.991	0.312953
$117$	0	0
$118$	−2009.92	−1.56804
$119$	0	0
$120$	0	0
$121$	1027.90	0.772279
$122$	1350.65	1.00231
$123$	0	0
$124$	60.4996	0.0438147
$125$	125.000	0.0894427
$126$	0	0
$127$	2372.90	1.65796	0.828979	−	0.559280i	$-0.188922\pi$
$127$	2372.90	1.65796	0.828979	+	0.559280i	$0.188922\pi$
$128$	648.917	0.448099
$129$	0	0
$130$	835.685	0.563804
$131$	1200.04	0.800364	0.400182	−	0.916436i	$-0.368947\pi$
$131$	1200.04	0.800364	0.400182	+	0.916436i	$0.368947\pi$
$132$	0	0
$133$	0	0
$134$	−2819.71	−1.81781
$135$	0	0
$136$	347.901	0.219355
$137$	−2781.25	−1.73444	−0.867221	−	0.497924i	$-0.834096\pi$
$137$	−2781.25	−1.73444	−0.867221	+	0.497924i	$0.834096\pi$
$138$	0	0
$139$	1245.60	0.760078	0.380039	−	0.924971i	$-0.375911\pi$
$139$	1245.60	0.760078	0.380039	+	0.924971i	$0.375911\pi$
$140$	0	0
$141$	0	0
$142$	1884.60	1.11375
$143$	−2120.35	−1.23995
$144$	0	0
$145$	293.675	0.168196
$146$	−4459.17	−2.52770
$147$	0	0
$148$	−1682.03	−0.934202
$149$	−19.4046	−0.0106690	−0.00533452	−	0.999986i	$-0.501698\pi$
$149$	−19.4046	−0.0106690	−0.00533452	+	0.999986i	$0.501698\pi$
$150$	0	0
$151$	−2349.80	−1.26638	−0.633192	−	0.773995i	$-0.718256\pi$
$151$	−2349.80	−1.26638	−0.633192	+	0.773995i	$0.718256\pi$
$152$	−479.529	−0.255888
$153$	0	0
$154$	0	0
$155$	45.4416	0.0235481
$156$	0	0
$157$	3898.46	1.98172	0.990862	−	0.134880i	$-0.0430650\pi$
$157$	3898.46	1.98172	0.990862	+	0.134880i	$0.0430650\pi$
$158$	−3339.14	−1.68131
$159$	0	0
$160$	−1190.56	−0.588264
$161$	0	0
$162$	0	0
$163$	1527.54	0.734024	0.367012	−	0.930216i	$-0.380381\pi$
$163$	1527.54	0.734024	0.367012	+	0.930216i	$0.380381\pi$
$164$	1839.12	0.875676
$165$	0	0
$166$	−2027.49	−0.947974
$167$	−998.518	−0.462681	−0.231340	−	0.972873i	$-0.574311\pi$
$167$	−998.518	−0.462681	−0.231340	+	0.972873i	$0.574311\pi$
$168$	0	0
$169$	−291.079	−0.132489
$170$	−1295.10	−0.584290
$171$	0	0
$172$	−616.674	−0.273378
$173$	685.253	0.301149	0.150575	−	0.988599i	$-0.451888\pi$
$173$	685.253	0.301149	0.150575	+	0.988599i	$0.451888\pi$
$174$	0	0
$175$	0	0
$176$	3542.64	1.51725
$177$	0	0
$178$	−1474.77	−0.621005
$179$	−1025.58	−0.428245	−0.214122	−	0.976807i	$-0.568689\pi$
$179$	−1025.58	−0.428245	−0.214122	+	0.976807i	$0.568689\pi$
$180$	0	0
$181$	−2899.40	−1.19067	−0.595333	−	0.803479i	$-0.702980\pi$
$181$	−2899.40	−1.19067	−0.595333	+	0.803479i	$0.702980\pi$
$182$	0	0
$183$	0	0
$184$	−535.638	−0.214608
$185$	−1263.38	−0.502084
$186$	0	0
$187$	3285.99	1.28500
$188$	−3879.57	−1.50504
$189$	0	0
$190$	1785.10	0.681603
$191$	−1074.18	−0.406939	−0.203469	−	0.979081i	$-0.565222\pi$
$191$	−1074.18	−0.406939	−0.203469	+	0.979081i	$0.565222\pi$
$192$	0	0
$193$	898.999	0.335292	0.167646	−	0.985847i	$-0.446383\pi$
$193$	898.999	0.335292	0.167646	+	0.985847i	$0.446383\pi$
$194$	1775.98	0.657257
$195$	0	0
$196$	0	0
$197$	−3063.63	−1.10799	−0.553996	−	0.832519i	$-0.686898\pi$
$197$	−3063.63	−1.10799	−0.553996	+	0.832519i	$0.686898\pi$
$198$	0	0
$199$	949.522	0.338240	0.169120	−	0.985595i	$-0.445907\pi$
$199$	949.522	0.338240	0.169120	+	0.985595i	$0.445907\pi$
$200$	−128.553	−0.0454505
$201$	0	0
$202$	−1656.66	−0.577039
$203$	0	0
$204$	0	0
$205$	1381.37	0.470630
$206$	−1962.55	−0.663774
$207$	0	0
$208$	−3184.38	−1.06152
$209$	−4529.25	−1.49902
$210$	0	0
$211$	2306.64	0.752587	0.376294	−	0.926500i	$-0.377198\pi$
$211$	2306.64	0.752587	0.376294	+	0.926500i	$0.377198\pi$
$212$	−4147.35	−1.34359
$213$	0	0
$214$	−7515.53	−2.40071
$215$	−463.188	−0.146926
$216$	0	0
$217$	0	0
$218$	2087.17	0.648444
$219$	0	0
$220$	−1616.57	−0.495405
$221$	−2953.69	−0.899033
$222$	0	0
$223$	3227.61	0.969222	0.484611	−	0.874730i	$-0.338961\pi$
$223$	3227.61	0.969222	0.484611	+	0.874730i	$0.338961\pi$
$224$	0	0
$225$	0	0
$226$	887.519	0.261225
$227$	−637.820	−0.186492	−0.0932458	−	0.995643i	$-0.529724\pi$
$227$	−637.820	−0.186492	−0.0932458	+	0.995643i	$0.529724\pi$
$228$	0	0
$229$	−544.774	−0.157204	−0.0786019	−	0.996906i	$-0.525046\pi$
$229$	−544.774	−0.157204	−0.0786019	+	0.996906i	$0.525046\pi$
$230$	1993.97	0.571646
$231$	0	0
$232$	−302.024	−0.0854691
$233$	5748.54	1.61631	0.808154	−	0.588972i	$-0.200467\pi$
$233$	5748.54	1.61631	0.808154	+	0.588972i	$0.200467\pi$
$234$	0	0
$235$	−2913.97	−0.808878
$236$	−3494.84	−0.963961
$237$	0	0
$238$	0	0
$239$	2678.10	0.724820	0.362410	−	0.932019i	$-0.381954\pi$
$239$	2678.10	0.724820	0.362410	+	0.932019i	$0.381954\pi$
$240$	0	0
$241$	2202.16	0.588604	0.294302	−	0.955713i	$-0.404913\pi$
$241$	2202.16	0.588604	0.294302	+	0.955713i	$0.404913\pi$
$242$	3935.25	1.04532
$243$	0	0
$244$	2348.50	0.616177
$245$	0	0
$246$	0	0
$247$	4071.21	1.04877
$248$	−46.7333	−0.0119660
$249$	0	0
$250$	478.553	0.121065
$251$	−5716.90	−1.43764	−0.718820	−	0.695196i	$-0.755317\pi$
$251$	−5716.90	−1.43764	−0.718820	+	0.695196i	$0.755317\pi$
$252$	0	0
$253$	−5059.22	−1.25719
$254$	9084.47	2.24413
$255$	0	0
$256$	5108.88	1.24728
$257$	4724.29	1.14666	0.573332	−	0.819323i	$-0.305650\pi$
$257$	4724.29	1.14666	0.573332	+	0.819323i	$0.305650\pi$
$258$	0	0
$259$	0	0
$260$	1453.09	0.346602
$261$	0	0
$262$	4594.25	1.08334
$263$	−5975.36	−1.40097	−0.700487	−	0.713665i	$-0.747034\pi$
$263$	−5975.36	−1.40097	−0.700487	+	0.713665i	$0.747034\pi$
$264$	0	0
$265$	−3115.10	−0.722109
$266$	0	0
$267$	0	0
$268$	−4902.90	−1.11751
$269$	4486.11	1.01681	0.508407	−	0.861117i	$-0.330234\pi$
$269$	4486.11	1.01681	0.508407	+	0.861117i	$0.330234\pi$
$270$	0	0
$271$	−3827.68	−0.857989	−0.428994	−	0.903307i	$-0.641132\pi$
$271$	−3827.68	−0.857989	−0.428994	+	0.903307i	$0.641132\pi$
$272$	4934.97	1.10010
$273$	0	0
$274$	−10647.8	−2.34766
$275$	−1214.21	−0.266254
$276$	0	0
$277$	−3420.54	−0.741950	−0.370975	−	0.928643i	$-0.620976\pi$
$277$	−3420.54	−0.741950	−0.370975	+	0.928643i	$0.620976\pi$
$278$	4768.71	1.02881
$279$	0	0
$280$	0	0
$281$	−5235.92	−1.11156	−0.555781	−	0.831329i	$-0.687581\pi$
$281$	−5235.92	−1.11156	−0.555781	+	0.831329i	$0.687581\pi$
$282$	0	0
$283$	6985.88	1.46738	0.733688	−	0.679486i	$-0.237797\pi$
$283$	6985.88	1.46738	0.733688	+	0.679486i	$0.237797\pi$
$284$	3276.93	0.684683
$285$	0	0
$286$	−8117.60	−1.67834
$287$	0	0
$288$	0	0
$289$	−335.550	−0.0682984
$290$	1124.31	0.227662
$291$	0	0
$292$	−7753.59	−1.55392
$293$	7399.70	1.47541	0.737705	−	0.675123i	$-0.235910\pi$
$293$	7399.70	1.47541	0.737705	+	0.675123i	$0.235910\pi$
$294$	0	0
$295$	−2625.00	−0.518078
$296$	1299.30	0.255135
$297$	0	0
$298$	−74.2892	−0.0144411
$299$	4547.58	0.879577
$300$	0	0
$301$	0	0
$302$	−8996.04	−1.71412
$303$	0	0
$304$	−6802.11	−1.28332
$305$	1763.97	0.331163
$306$	0	0
$307$	−2668.64	−0.496116	−0.248058	−	0.968745i	$-0.579792\pi$
$307$	−2668.64	−0.496116	−0.248058	+	0.968745i	$0.579792\pi$
$308$	0	0
$309$	0	0
$310$	173.970	0.0318736
$311$	6189.25	1.12849	0.564244	−	0.825608i	$-0.309168\pi$
$311$	6189.25	1.12849	0.564244	+	0.825608i	$0.309168\pi$
$312$	0	0
$313$	2921.59	0.527598	0.263799	−	0.964578i	$-0.415024\pi$
$313$	2921.59	0.527598	0.263799	+	0.964578i	$0.415024\pi$
$314$	14925.0	2.68237
$315$	0	0
$316$	−5806.08	−1.03360
$317$	−9825.56	−1.74088	−0.870439	−	0.492276i	$-0.836165\pi$
$317$	−9825.56	−1.74088	−0.870439	+	0.492276i	$0.836165\pi$
$318$	0	0
$319$	−2852.68	−0.500687
$320$	−1640.34	−0.286556
$321$	0	0
$322$	0	0
$323$	−6309.33	−1.08687
$324$	0	0
$325$	1091.42	0.186281
$326$	5848.07	0.993541
$327$	0	0
$328$	−1420.64	−0.239151
$329$	0	0
$330$	0	0
$331$	−9258.17	−1.53739	−0.768693	−	0.639618i	$-0.779093\pi$
$331$	−9258.17	−1.53739	−0.768693	+	0.639618i	$0.779093\pi$
$332$	−3525.39	−0.582774
$333$	0	0
$334$	−3822.75	−0.626263
$335$	−3682.60	−0.600603
$336$	0	0
$337$	−3693.98	−0.597103	−0.298552	−	0.954394i	$-0.596503\pi$
$337$	−3693.98	−0.597103	−0.298552	+	0.954394i	$0.596503\pi$
$338$	−1114.38	−0.179331
$339$	0	0
$340$	−2251.91	−0.359197
$341$	−441.406	−0.0700982
$342$	0	0
$343$	0	0
$344$	476.355	0.0746608
$345$	0	0
$346$	2623.44	0.407622
$347$	−3832.83	−0.592960	−0.296480	−	0.955039i	$-0.595813\pi$
$347$	−3832.83	−0.592960	−0.296480	+	0.955039i	$0.595813\pi$
$348$	0	0
$349$	−8325.22	−1.27690	−0.638451	−	0.769662i	$-0.720425\pi$
$349$	−8325.22	−1.27690	−0.638451	+	0.769662i	$0.720425\pi$
$350$	0	0
$351$	0	0
$352$	11564.8	1.75115
$353$	−8991.52	−1.35572	−0.677862	−	0.735189i	$-0.737093\pi$
$353$	−8991.52	−1.35572	−0.677862	+	0.735189i	$0.737093\pi$
$354$	0	0
$355$	2461.32	0.367981
$356$	−2564.33	−0.381767
$357$	0	0
$358$	−3926.38	−0.579652
$359$	12893.8	1.89557	0.947783	−	0.318917i	$-0.103319\pi$
$359$	12893.8	1.89557	0.947783	+	0.318917i	$0.103319\pi$
$360$	0	0
$361$	1837.46	0.267891
$362$	−11100.1	−1.61163
$363$	0	0
$364$	0	0
$365$	−5823.77	−0.835150
$366$	0	0
$367$	7480.17	1.06393	0.531964	−	0.846767i	$-0.321454\pi$
$367$	7480.17	1.06393	0.531964	+	0.846767i	$0.321454\pi$
$368$	−7598.02	−1.07629
$369$	0	0
$370$	−4836.76	−0.679598
$371$	0	0
$372$	0	0
$373$	−3523.32	−0.489090	−0.244545	−	0.969638i	$-0.578639\pi$
$373$	−3523.32	−0.489090	−0.244545	+	0.969638i	$0.578639\pi$
$374$	12580.2	1.73932
$375$	0	0
$376$	2996.81	0.411033
$377$	2564.19	0.350298
$378$	0	0
$379$	13515.4	1.83177	0.915886	−	0.401438i	$-0.131490\pi$
$379$	13515.4	1.83177	0.915886	+	0.401438i	$0.131490\pi$
$380$	3103.92	0.419020
$381$	0	0
$382$	−4112.44	−0.550813
$383$	−657.182	−0.0876774	−0.0438387	−	0.999039i	$-0.513959\pi$
$383$	−657.182	−0.0876774	−0.0438387	+	0.999039i	$0.513959\pi$
$384$	0	0
$385$	0	0
$386$	3441.75	0.453836
$387$	0	0
$388$	3088.06	0.404053
$389$	9741.87	1.26975	0.634875	−	0.772615i	$-0.281052\pi$
$389$	9741.87	1.26975	0.634875	+	0.772615i	$0.281052\pi$
$390$	0	0
$391$	−7047.58	−0.911538
$392$	0	0
$393$	0	0
$394$	−11728.9	−1.49973
$395$	−4360.98	−0.555505
$396$	0	0
$397$	−4407.42	−0.557184	−0.278592	−	0.960410i	$-0.589868\pi$
$397$	−4407.42	−0.557184	−0.278592	+	0.960410i	$0.589868\pi$
$398$	3635.18	0.457827
$399$	0	0
$400$	−1823.53	−0.227941
$401$	11569.5	1.44078	0.720391	−	0.693568i	$-0.243963\pi$
$401$	11569.5	1.44078	0.720391	+	0.693568i	$0.243963\pi$
$402$	0	0
$403$	396.767	0.0490431
$404$	−2880.59	−0.354739
$405$	0	0
$406$	0	0
$407$	12272.1	1.49461
$408$	0	0
$409$	3083.03	0.372729	0.186364	−	0.982481i	$-0.440330\pi$
$409$	3083.03	0.372729	0.186364	+	0.982481i	$0.440330\pi$
$410$	5288.48	0.637023
$411$	0	0
$412$	−3412.47	−0.408060
$413$	0	0
$414$	0	0
$415$	−2647.94	−0.313210
$416$	−10395.3	−1.22517
$417$	0	0
$418$	−17339.9	−2.02900
$419$	−5415.21	−0.631385	−0.315692	−	0.948862i	$-0.602237\pi$
$419$	−5415.21	−0.631385	−0.315692	+	0.948862i	$0.602237\pi$
$420$	0	0
$421$	4188.34	0.484863	0.242432	−	0.970168i	$-0.422055\pi$
$421$	4188.34	0.484863	0.242432	+	0.970168i	$0.422055\pi$
$422$	8830.82	1.01867
$423$	0	0
$424$	3203.65	0.366941
$425$	−1691.42	−0.193049
$426$	0	0
$427$	0	0
$428$	−13068.0	−1.47585
$429$	0	0
$430$	−1773.28	−0.198872
$431$	9108.41	1.01795	0.508975	−	0.860781i	$-0.330024\pi$
$431$	9108.41	1.01795	0.508975	+	0.860781i	$0.330024\pi$
$432$	0	0
$433$	16847.0	1.86979	0.934893	−	0.354930i	$-0.115495\pi$
$433$	16847.0	1.86979	0.934893	+	0.354930i	$0.115495\pi$
$434$	0	0
$435$	0	0
$436$	3629.16	0.398635
$437$	9714.03	1.06335
$438$	0	0
$439$	−8434.14	−0.916946	−0.458473	−	0.888708i	$-0.651603\pi$
$439$	−8434.14	−0.916946	−0.458473	+	0.888708i	$0.651603\pi$
$440$	1248.73	0.135297
$441$	0	0
$442$	−11308.0	−1.21689
$443$	4298.49	0.461010	0.230505	−	0.973071i	$-0.425962\pi$
$443$	4298.49	0.461010	0.230505	+	0.973071i	$0.425962\pi$
$444$	0	0
$445$	−1926.08	−0.205180
$446$	12356.7	1.31189
$447$	0	0
$448$	0	0
$449$	−10545.0	−1.10835	−0.554173	−	0.832402i	$-0.686965\pi$
$449$	−10545.0	−1.10835	−0.554173	+	0.832402i	$0.686965\pi$
$450$	0	0
$451$	−13418.2	−1.40098
$452$	1543.21	0.160590
$453$	0	0
$454$	−2441.85	−0.252426
$455$	0	0
$456$	0	0
$457$	11952.4	1.22344	0.611719	−	0.791075i	$-0.290478\pi$
$457$	11952.4	1.22344	0.611719	+	0.791075i	$0.290478\pi$
$458$	−2085.63	−0.212784
$459$	0	0
$460$	3467.11	0.351423
$461$	−17200.9	−1.73780	−0.868900	−	0.494988i	$-0.835173\pi$
$461$	−17200.9	−1.73780	−0.868900	+	0.494988i	$0.835173\pi$
$462$	0	0
$463$	−10368.7	−1.04076	−0.520381	−	0.853934i	$-0.674210\pi$
$463$	−10368.7	−1.04076	−0.520381	+	0.853934i	$0.674210\pi$
$464$	−4284.20	−0.428640
$465$	0	0
$466$	22007.9	2.18776
$467$	−16879.5	−1.67257	−0.836284	−	0.548296i	$-0.815277\pi$
$467$	−16879.5	−1.67257	−0.836284	+	0.548296i	$0.815277\pi$
$468$	0	0
$469$	0	0
$470$	−11155.9	−1.09486
$471$	0	0
$472$	2699.62	0.263263
$473$	4499.27	0.437371
$474$	0	0
$475$	2331.37	0.225201
$476$	0	0
$477$	0	0
$478$	10252.9	0.981082
$479$	7329.12	0.699115	0.349558	−	0.936915i	$-0.386332\pi$
$479$	7329.12	0.699115	0.349558	+	0.936915i	$0.386332\pi$
$480$	0	0
$481$	−11031.0	−1.04568
$482$	8430.80	0.796706
$483$	0	0
$484$	6842.60	0.642619
$485$	2319.46	0.217157
$486$	0	0
$487$	−17209.5	−1.60131	−0.800655	−	0.599125i	$-0.795515\pi$
$487$	−17209.5	−1.60131	−0.800655	+	0.599125i	$0.795515\pi$
$488$	−1814.11	−0.168281
$489$	0	0
$490$	0	0
$491$	−11392.5	−1.04712	−0.523560	−	0.851989i	$-0.675396\pi$
$491$	−11392.5	−1.04712	−0.523560	+	0.851989i	$0.675396\pi$
$492$	0	0
$493$	−3973.83	−0.363027
$494$	15586.3	1.41956
$495$	0	0
$496$	−662.912	−0.0600113
$497$	0	0
$498$	0	0
$499$	19079.4	1.71164	0.855822	−	0.517271i	$-0.173052\pi$
$499$	19079.4	1.71164	0.855822	+	0.517271i	$0.173052\pi$
$500$	832.107	0.0744259
$501$	0	0
$502$	−21886.7	−1.94592
$503$	−13499.4	−1.19663	−0.598317	−	0.801259i	$-0.704164\pi$
$503$	−13499.4	−1.19663	−0.598317	+	0.801259i	$0.704164\pi$
$504$	0	0
$505$	−2163.62	−0.190654
$506$	−19368.8	−1.70168
$507$	0	0
$508$	15796.0	1.37960
$509$	−4328.85	−0.376960	−0.188480	−	0.982077i	$-0.560356\pi$
$509$	−4328.85	−0.376960	−0.188480	+	0.982077i	$0.560356\pi$
$510$	0	0
$511$	0	0
$512$	14367.6	1.24017
$513$	0	0
$514$	18086.6	1.55207
$515$	−2563.13	−0.219311
$516$	0	0
$517$	28305.5	2.40788
$518$	0	0
$519$	0	0
$520$	−1122.45	−0.0946588
$521$	19395.7	1.63098	0.815490	−	0.578771i	$-0.196468\pi$
$521$	19395.7	1.63098	0.815490	+	0.578771i	$0.196468\pi$
$522$	0	0
$523$	−20413.8	−1.70675	−0.853377	−	0.521294i	$-0.825450\pi$
$523$	−20413.8	−1.70675	−0.853377	+	0.521294i	$0.825450\pi$
$524$	7988.47	0.665989
$525$	0	0
$526$	−22876.2	−1.89629
$527$	−614.887	−0.0508252
$528$	0	0
$529$	−1316.34	−0.108189
$530$	−11925.9	−0.977413
$531$	0	0
$532$	0	0
$533$	12061.3	0.980171
$534$	0	0
$535$	−9815.43	−0.793193
$536$	3787.28	0.305197
$537$	0	0
$538$	17174.8	1.37631
$539$	0	0
$540$	0	0
$541$	−4919.18	−0.390928	−0.195464	−	0.980711i	$-0.562621\pi$
$541$	−4919.18	−0.390928	−0.195464	+	0.980711i	$0.562621\pi$
$542$	−14654.0	−1.16133
$543$	0	0
$544$	16110.0	1.26969
$545$	2725.88	0.214246
$546$	0	0
$547$	15334.2	1.19862	0.599308	−	0.800518i	$-0.295442\pi$
$547$	15334.2	1.19862	0.599308	+	0.800518i	$0.295442\pi$
$548$	−18514.4	−1.44324
$549$	0	0
$550$	−4648.53	−0.360389
$551$	5477.33	0.423488
$552$	0	0
$553$	0	0
$554$	−13095.3	−1.00427
$555$	0	0
$556$	8291.81	0.632466
$557$	8613.78	0.655256	0.327628	−	0.944807i	$-0.393751\pi$
$557$	8613.78	0.655256	0.327628	+	0.944807i	$0.393751\pi$
$558$	0	0
$559$	−4044.26	−0.306000
$560$	0	0
$561$	0	0
$562$	−20045.3	−1.50456
$563$	−2320.81	−0.173731	−0.0868654	−	0.996220i	$-0.527685\pi$
$563$	−2320.81	−0.173731	−0.0868654	+	0.996220i	$0.527685\pi$
$564$	0	0
$565$	1159.12	0.0863087
$566$	26744.9	1.98617
$567$	0	0
$568$	−2531.29	−0.186990
$569$	1736.04	0.127906	0.0639529	−	0.997953i	$-0.479629\pi$
$569$	1736.04	0.127906	0.0639529	+	0.997953i	$0.479629\pi$
$570$	0	0
$571$	23897.8	1.75148	0.875738	−	0.482786i	$-0.160375\pi$
$571$	23897.8	1.75148	0.875738	+	0.482786i	$0.160375\pi$
$572$	−14114.9	−1.03177
$573$	0	0
$574$	0	0
$575$	2604.16	0.188871
$576$	0	0
$577$	−8029.26	−0.579311	−0.289655	−	0.957131i	$-0.593541\pi$
$577$	−8029.26	−0.579311	−0.289655	+	0.957131i	$0.593541\pi$
$578$	−1284.63	−0.0924455
$579$	0	0
$580$	1954.95	0.139957
$581$	0	0
$582$	0	0
$583$	30259.1	2.14958
$584$	5989.32	0.424383
$585$	0	0
$586$	28329.2	1.99705
$587$	−8015.14	−0.563578	−0.281789	−	0.959476i	$-0.590928\pi$
$587$	−8015.14	−0.563578	−0.281789	+	0.959476i	$0.590928\pi$
$588$	0	0
$589$	847.529	0.0592900
$590$	−10049.6	−0.701247
$591$	0	0
$592$	18430.5	1.27954
$593$	12820.3	0.887801	0.443901	−	0.896076i	$-0.353594\pi$
$593$	12820.3	0.887801	0.443901	+	0.896076i	$0.353594\pi$
$594$	0	0
$595$	0	0
$596$	−129.174	−0.00887779
$597$	0	0
$598$	17410.1	1.19055
$599$	15330.5	1.04572	0.522860	−	0.852418i	$-0.324865\pi$
$599$	15330.5	1.04572	0.522860	+	0.852418i	$0.324865\pi$
$600$	0	0
$601$	−107.658	−0.00730694	−0.00365347	−	0.999993i	$-0.501163\pi$
$601$	−107.658	−0.00730694	−0.00365347	+	0.999993i	$0.501163\pi$
$602$	0	0
$603$	0	0
$604$	−15642.3	−1.05377
$605$	5139.52	0.345374
$606$	0	0
$607$	8213.06	0.549189	0.274595	−	0.961560i	$-0.411456\pi$
$607$	8213.06	0.549189	0.274595	+	0.961560i	$0.411456\pi$
$608$	−22205.2	−1.48115
$609$	0	0
$610$	6753.23	0.448246
$611$	−25443.0	−1.68463
$612$	0	0
$613$	1242.60	0.0818732	0.0409366	−	0.999162i	$-0.486966\pi$
$613$	1242.60	0.0818732	0.0409366	+	0.999162i	$0.486966\pi$
$614$	−10216.7	−0.671519
$615$	0	0
$616$	0	0
$617$	13170.6	0.859367	0.429683	−	0.902980i	$-0.358625\pi$
$617$	13170.6	0.859367	0.429683	+	0.902980i	$0.358625\pi$
$618$	0	0
$619$	−19774.1	−1.28399	−0.641993	−	0.766711i	$-0.721892\pi$
$619$	−19774.1	−1.28399	−0.641993	+	0.766711i	$0.721892\pi$
$620$	302.498	0.0195945
$621$	0	0
$622$	23695.1	1.52747
$623$	0	0
$624$	0	0
$625$	625.000	0.0400000
$626$	11185.1	0.714132
$627$	0	0
$628$	25951.5	1.64901
$629$	17095.3	1.08368
$630$	0	0
$631$	−14308.8	−0.902735	−0.451367	−	0.892338i	$-0.649064\pi$
$631$	−14308.8	−0.902735	−0.451367	+	0.892338i	$0.649064\pi$
$632$	4484.95	0.282281
$633$	0	0
$634$	−37616.4	−2.35637
$635$	11864.5	0.741461
$636$	0	0
$637$	0	0
$638$	−10921.3	−0.677707
$639$	0	0
$640$	3244.58	0.200396
$641$	−11537.5	−0.710925	−0.355463	−	0.934691i	$-0.615677\pi$
$641$	−11537.5	−0.710925	−0.355463	+	0.934691i	$0.615677\pi$
$642$	0	0
$643$	−19603.0	−1.20228	−0.601139	−	0.799144i	$-0.705286\pi$
$643$	−19603.0	−1.20228	−0.601139	+	0.799144i	$0.705286\pi$
$644$	0	0
$645$	0	0
$646$	−24154.8	−1.47114
$647$	21650.0	1.31553	0.657765	−	0.753223i	$-0.271502\pi$
$647$	21650.0	1.31553	0.657765	+	0.753223i	$0.271502\pi$
$648$	0	0
$649$	25498.4	1.54222
$650$	4178.43	0.252141
$651$	0	0
$652$	10168.6	0.610787
$653$	2927.33	0.175429	0.0877145	−	0.996146i	$-0.472044\pi$
$653$	2927.33	0.175429	0.0877145	+	0.996146i	$0.472044\pi$
$654$	0	0
$655$	6000.18	0.357934
$656$	−20151.7	−1.19938
$657$	0	0
$658$	0	0
$659$	4778.76	0.282480	0.141240	−	0.989975i	$-0.454891\pi$
$659$	4778.76	0.282480	0.141240	+	0.989975i	$0.454891\pi$
$660$	0	0
$661$	31510.3	1.85417	0.927086	−	0.374849i	$-0.122305\pi$
$661$	31510.3	1.85417	0.927086	+	0.374849i	$0.122305\pi$
$662$	−35444.2	−2.08093
$663$	0	0
$664$	2723.21	0.159158
$665$	0	0
$666$	0	0
$667$	6118.23	0.355170
$668$	−6646.99	−0.385000
$669$	0	0
$670$	−14098.6	−0.812948
$671$	−17134.7	−0.985808
$672$	0	0
$673$	−8992.15	−0.515040	−0.257520	−	0.966273i	$-0.582905\pi$
$673$	−8992.15	−0.515040	−0.257520	+	0.966273i	$0.582905\pi$
$674$	−14142.1	−0.808211
$675$	0	0
$676$	−1937.67	−0.110245
$677$	19340.8	1.09797	0.548985	−	0.835832i	$-0.315014\pi$
$677$	19340.8	1.09797	0.548985	+	0.835832i	$0.315014\pi$
$678$	0	0
$679$	0	0
$680$	1739.50	0.0980984
$681$	0	0
$682$	−1689.89	−0.0948816
$683$	4255.14	0.238387	0.119194	−	0.992871i	$-0.461969\pi$
$683$	4255.14	0.238387	0.119194	+	0.992871i	$0.461969\pi$
$684$	0	0
$685$	−13906.3	−0.775666
$686$	0	0
$687$	0	0
$688$	6757.08	0.374435
$689$	−27199.1	−1.50392
$690$	0	0
$691$	17505.5	0.963733	0.481867	−	0.876245i	$-0.339959\pi$
$691$	17505.5	0.963733	0.481867	+	0.876245i	$0.339959\pi$
$692$	4561.63	0.250588
$693$	0	0
$694$	−14673.7	−0.802603
$695$	6228.02	0.339917
$696$	0	0
$697$	−18691.8	−1.01579
$698$	−31872.5	−1.72836
$699$	0	0
$700$	0	0
$701$	3240.77	0.174611	0.0873054	−	0.996182i	$-0.472174\pi$
$701$	3240.77	0.174611	0.0873054	+	0.996182i	$0.472174\pi$
$702$	0	0
$703$	−23563.3	−1.26416
$704$	15933.8	0.853022
$705$	0	0
$706$	−34423.4	−1.83504
$707$	0	0
$708$	0	0
$709$	19949.3	1.05672	0.528358	−	0.849022i	$-0.322808\pi$
$709$	19949.3	1.05672	0.528358	+	0.849022i	$0.322808\pi$
$710$	9422.99	0.498082
$711$	0	0
$712$	1980.83	0.104262
$713$	946.698	0.0497253
$714$	0	0
$715$	−10601.7	−0.554522
$716$	−6827.17	−0.356345
$717$	0	0
$718$	49362.9	2.56575
$719$	−11259.4	−0.584011	−0.292006	−	0.956417i	$-0.594323\pi$
$719$	−11259.4	−0.584011	−0.292006	+	0.956417i	$0.594323\pi$
$720$	0	0
$721$	0	0
$722$	7034.60	0.362605
$723$	0	0
$724$	−19300.9	−0.990761
$725$	1468.38	0.0752195
$726$	0	0
$727$	12228.5	0.623840	0.311920	−	0.950108i	$-0.399028\pi$
$727$	12228.5	0.623840	0.311920	+	0.950108i	$0.399028\pi$
$728$	0	0
$729$	0	0
$730$	−22295.9	−1.13042
$731$	6267.56	0.317119
$732$	0	0
$733$	26635.1	1.34214	0.671072	−	0.741392i	$-0.265834\pi$
$733$	26635.1	1.34214	0.671072	+	0.741392i	$0.265834\pi$
$734$	28637.3	1.44008
$735$	0	0
$736$	−24803.4	−1.24221
$737$	35771.7	1.78788
$738$	0	0
$739$	−6074.00	−0.302349	−0.151174	−	0.988507i	$-0.548306\pi$
$739$	−6074.00	−0.302349	−0.151174	+	0.988507i	$0.548306\pi$
$740$	−8410.14	−0.417788
$741$	0	0
$742$	0	0
$743$	4016.87	0.198337	0.0991686	−	0.995071i	$-0.468382\pi$
$743$	4016.87	0.198337	0.0991686	+	0.995071i	$0.468382\pi$
$744$	0	0
$745$	−97.0231	−0.00477134
$746$	−13488.8	−0.662010
$747$	0	0
$748$	21874.4	1.06926
$749$	0	0
$750$	0	0
$751$	−23913.2	−1.16192	−0.580962	−	0.813931i	$-0.697324\pi$
$751$	−23913.2	−1.16192	−0.580962	+	0.813931i	$0.697324\pi$
$752$	42509.6	2.06139
$753$	0	0
$754$	9816.81	0.474147
$755$	−11749.0	−0.566344
$756$	0	0
$757$	31044.9	1.49055	0.745275	−	0.666758i	$-0.232318\pi$
$757$	31044.9	1.49055	0.745275	+	0.666758i	$0.232318\pi$
$758$	51742.9	2.47940
$759$	0	0
$760$	−2397.65	−0.114436
$761$	14011.8	0.667446	0.333723	−	0.942671i	$-0.391695\pi$
$761$	14011.8	0.667446	0.333723	+	0.942671i	$0.391695\pi$
$762$	0	0
$763$	0	0
$764$	−7150.69	−0.338616
$765$	0	0
$766$	−2515.97	−0.118676
$767$	−22919.8	−1.07899
$768$	0	0
$769$	−3342.49	−0.156740	−0.0783701	−	0.996924i	$-0.524972\pi$
$769$	−3342.49	−0.156740	−0.0783701	+	0.996924i	$0.524972\pi$
$770$	0	0
$771$	0	0
$772$	5984.51	0.278999
$773$	−21074.6	−0.980594	−0.490297	−	0.871555i	$-0.663112\pi$
$773$	−21074.6	−0.980594	−0.490297	+	0.871555i	$0.663112\pi$
$774$	0	0
$775$	227.208	0.0105310
$776$	−2385.40	−0.110349
$777$	0	0
$778$	37296.0	1.71867
$779$	25763.9	1.18496
$780$	0	0
$781$	−23908.5	−1.09541
$782$	−26981.1	−1.23382
$783$	0	0
$784$	0	0
$785$	19492.3	0.886254
$786$	0	0
$787$	−21394.8	−0.969048	−0.484524	−	0.874778i	$-0.661007\pi$
$787$	−21394.8	−0.969048	−0.484524	+	0.874778i	$0.661007\pi$
$788$	−20394.1	−0.921968
$789$	0	0
$790$	−16695.7	−0.751906
$791$	0	0
$792$	0	0
$793$	15401.9	0.689706
$794$	−16873.5	−0.754179
$795$	0	0
$796$	6320.83	0.281452
$797$	20645.0	0.917547	0.458773	−	0.888553i	$-0.348289\pi$
$797$	20645.0	0.917547	0.458773	+	0.888553i	$0.348289\pi$
$798$	0	0
$799$	39430.0	1.74585
$800$	−5952.82	−0.263080
$801$	0	0
$802$	44293.0	1.95017
$803$	56570.4	2.48608
$804$	0	0
$805$	0	0
$806$	1518.99	0.0663825
$807$	0	0
$808$	2225.13	0.0968810
$809$	15939.0	0.692688	0.346344	−	0.938108i	$-0.387423\pi$
$809$	15939.0	0.692688	0.346344	+	0.938108i	$0.387423\pi$
$810$	0	0
$811$	−22829.2	−0.988460	−0.494230	−	0.869331i	$-0.664550\pi$
$811$	−22829.2	−0.988460	−0.494230	+	0.869331i	$0.664550\pi$
$812$	0	0
$813$	0	0
$814$	46982.9	2.02303
$815$	7637.69	0.328266
$816$	0	0
$817$	−8638.90	−0.369935
$818$	11803.2	0.504508
$819$	0	0
$820$	9195.58	0.391614
$821$	5700.22	0.242313	0.121157	−	0.992633i	$-0.461340\pi$
$821$	5700.22	0.242313	0.121157	+	0.992633i	$0.461340\pi$
$822$	0	0
$823$	−32438.0	−1.37390	−0.686948	−	0.726707i	$-0.741050\pi$
$823$	−32438.0	−1.37390	−0.686948	+	0.726707i	$0.741050\pi$
$824$	2635.99	0.111443
$825$	0	0
$826$	0	0
$827$	12762.6	0.536638	0.268319	−	0.963330i	$-0.413532\pi$
$827$	12762.6	0.536638	0.268319	+	0.963330i	$0.413532\pi$
$828$	0	0
$829$	30766.7	1.28899	0.644494	−	0.764609i	$-0.277068\pi$
$829$	30766.7	1.28899	0.644494	+	0.764609i	$0.277068\pi$
$830$	−10137.4	−0.423947
$831$	0	0
$832$	−14322.4	−0.596804
$833$	0	0
$834$	0	0
$835$	−4992.59	−0.206917
$836$	−30150.6	−1.24734
$837$	0	0
$838$	−20731.7	−0.854613
$839$	−9779.71	−0.402423	−0.201212	−	0.979548i	$-0.564488\pi$
$839$	−9779.71	−0.402423	−0.201212	+	0.979548i	$0.564488\pi$
$840$	0	0
$841$	−20939.2	−0.858551
$842$	16034.8	0.656288
$843$	0	0
$844$	15355.0	0.626233
$845$	−1455.40	−0.0592510
$846$	0	0
$847$	0	0
$848$	45443.7	1.84026
$849$	0	0
$850$	−6475.48	−0.261303
$851$	−26320.4	−1.06023
$852$	0	0
$853$	−24201.5	−0.971445	−0.485723	−	0.874113i	$-0.661444\pi$
$853$	−24201.5	−0.971445	−0.485723	+	0.874113i	$0.661444\pi$
$854$	0	0
$855$	0	0
$856$	10094.5	0.403062
$857$	21036.7	0.838507	0.419254	−	0.907869i	$-0.362292\pi$
$857$	21036.7	0.838507	0.419254	+	0.907869i	$0.362292\pi$
$858$	0	0
$859$	6179.19	0.245438	0.122719	−	0.992441i	$-0.460839\pi$
$859$	6179.19	0.245438	0.122719	+	0.992441i	$0.460839\pi$
$860$	−3083.37	−0.122258
$861$	0	0
$862$	34870.9	1.37785
$863$	−50256.2	−1.98232	−0.991160	−	0.132671i	$-0.957644\pi$
$863$	−50256.2	−1.98232	−0.991160	+	0.132671i	$0.957644\pi$
$864$	0	0
$865$	3426.27	0.134678
$866$	64497.7	2.53085
$867$	0	0
$868$	0	0
$869$	42361.2	1.65363
$870$	0	0
$871$	−32154.1	−1.25086
$872$	−2803.37	−0.108869
$873$	0	0
$874$	37189.5	1.43930
$875$	0	0
$876$	0	0
$877$	−9175.95	−0.353306	−0.176653	−	0.984273i	$-0.556527\pi$
$877$	−9175.95	−0.353306	−0.176653	+	0.984273i	$0.556527\pi$
$878$	−32289.5	−1.24114
$879$	0	0
$880$	17713.2	0.678537
$881$	−26172.7	−1.00089	−0.500444	−	0.865769i	$-0.666830\pi$
$881$	−26172.7	−1.00089	−0.500444	+	0.865769i	$0.666830\pi$
$882$	0	0
$883$	18615.5	0.709471	0.354736	−	0.934967i	$-0.384571\pi$
$883$	18615.5	0.709471	0.354736	+	0.934967i	$0.384571\pi$
$884$	−19662.3	−0.748092
$885$	0	0
$886$	16456.4	0.624001
$887$	−12837.8	−0.485964	−0.242982	−	0.970031i	$-0.578126\pi$
$887$	−12837.8	−0.485964	−0.242982	+	0.970031i	$0.578126\pi$
$888$	0	0
$889$	0	0
$890$	−7373.86	−0.277722
$891$	0	0
$892$	21485.7	0.806496
$893$	−54348.4	−2.03662
$894$	0	0
$895$	−5127.92	−0.191517
$896$	0	0
$897$	0	0
$898$	−40370.6	−1.50020
$899$	533.803	0.0198035
$900$	0	0
$901$	42151.5	1.55857
$902$	−51370.7	−1.89630
$903$	0	0
$904$	−1192.07	−0.0438579
$905$	−14497.0	−0.532482
$906$	0	0
$907$	−26766.1	−0.979885	−0.489942	−	0.871755i	$-0.662982\pi$
$907$	−26766.1	−0.979885	−0.489942	+	0.871755i	$0.662982\pi$
$908$	−4245.87	−0.155181
$909$	0	0
$910$	0	0
$911$	−5022.67	−0.182666	−0.0913328	−	0.995820i	$-0.529113\pi$
$911$	−5022.67	−0.182666	−0.0913328	+	0.995820i	$0.529113\pi$
$912$	0	0
$913$	25721.3	0.932367
$914$	45759.0	1.65599
$915$	0	0
$916$	−3626.48	−0.130810
$917$	0	0
$918$	0	0
$919$	9541.79	0.342497	0.171248	−	0.985228i	$-0.445220\pi$
$919$	9541.79	0.342497	0.171248	+	0.985228i	$0.445220\pi$
$920$	−2678.19	−0.0959754
$921$	0	0
$922$	−65852.4	−2.35220
$923$	21490.7	0.766387
$924$	0	0
$925$	−6316.90	−0.224539
$926$	−39695.7	−1.40873
$927$	0	0
$928$	−13985.6	−0.494718
$929$	−25479.6	−0.899846	−0.449923	−	0.893067i	$-0.648549\pi$
$929$	−25479.6	−0.899846	−0.449923	+	0.893067i	$0.648549\pi$
$930$	0	0
$931$	0	0
$932$	38267.2	1.34494
$933$	0	0
$934$	−64621.9	−2.26391
$935$	16430.0	0.574671
$936$	0	0
$937$	33608.3	1.17176	0.585878	−	0.810399i	$-0.300750\pi$
$937$	33608.3	1.17176	0.585878	+	0.810399i	$0.300750\pi$
$938$	0	0
$939$	0	0
$940$	−19397.9	−0.673073
$941$	−19173.6	−0.664232	−0.332116	−	0.943239i	$-0.607762\pi$
$941$	−19173.6	−0.664232	−0.332116	+	0.943239i	$0.607762\pi$
$942$	0	0
$943$	28778.5	0.993804
$944$	38294.0	1.32030
$945$	0	0
$946$	17225.1	0.592005
$947$	979.315	0.0336045	0.0168023	−	0.999859i	$-0.494651\pi$
$947$	979.315	0.0336045	0.0168023	+	0.999859i	$0.494651\pi$
$948$	0	0
$949$	−50849.5	−1.73935
$950$	8925.48	0.304822
$951$	0	0
$952$	0	0
$953$	−3048.61	−0.103624	−0.0518122	−	0.998657i	$-0.516500\pi$
$953$	−3048.61	−0.103624	−0.0518122	+	0.998657i	$0.516500\pi$
$954$	0	0
$955$	−5370.92	−0.181989
$956$	17827.7	0.603127
$957$	0	0
$958$	28059.0	0.946290
$959$	0	0
$960$	0	0
$961$	−29708.4	−0.997227
$962$	−42231.6	−1.41538
$963$	0	0
$964$	14659.4	0.489781
$965$	4495.00	0.149947
$966$	0	0
$967$	−14467.9	−0.481133	−0.240567	−	0.970633i	$-0.577333\pi$
$967$	−14467.9	−0.481133	−0.240567	+	0.970633i	$0.577333\pi$
$968$	−5285.62	−0.175502
$969$	0	0
$970$	8879.89	0.293934
$971$	12952.8	0.428090	0.214045	−	0.976824i	$-0.431336\pi$
$971$	12952.8	0.428090	0.214045	+	0.976824i	$0.431336\pi$
$972$	0	0
$973$	0	0
$974$	−65885.4	−2.16746
$975$	0	0
$976$	−25733.2	−0.843954
$977$	−47244.2	−1.54706	−0.773529	−	0.633760i	$-0.781511\pi$
$977$	−47244.2	−1.54706	−0.773529	+	0.633760i	$0.781511\pi$
$978$	0	0
$979$	18709.4	0.610781
$980$	0	0
$981$	0	0
$982$	−43615.3	−1.41733
$983$	−1536.84	−0.0498651	−0.0249326	−	0.999689i	$-0.507937\pi$
$983$	−1536.84	−0.0498651	−0.0249326	+	0.999689i	$0.507937\pi$
$984$	0	0
$985$	−15318.1	−0.495509
$986$	−15213.5	−0.491376
$987$	0	0
$988$	27101.5	0.872685
$989$	−9649.73	−0.310256
$990$	0	0
$991$	3785.22	0.121334	0.0606668	−	0.998158i	$-0.480677\pi$
$991$	3785.22	0.121334	0.0606668	+	0.998158i	$0.480677\pi$
$992$	−2164.04	−0.0692625
$993$	0	0
$994$	0	0
$995$	4747.61	0.151266
$996$	0	0
$997$	25894.9	0.822566	0.411283	−	0.911508i	$-0.365081\pi$
$997$	25894.9	0.822566	0.411283	+	0.911508i	$0.365081\pi$
$998$	73044.0	2.31680
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2205.4.a.bb.1.2		2
3.2	odd	2		735.4.a.o.1.1		2
7.6	odd	2		315.4.a.k.1.2		2
21.20	even	2		105.4.a.e.1.1	✓	2
35.34	odd	2		1575.4.a.q.1.1		2
84.83	odd	2		1680.4.a.bo.1.1		2
105.62	odd	4		525.4.d.l.274.1		4
105.83	odd	4		525.4.d.l.274.4		4
105.104	even	2		525.4.a.l.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.4.a.e.1.1	✓	2	21.20	even	2
315.4.a.k.1.2		2	7.6	odd	2
525.4.a.l.1.2		2	105.104	even	2
525.4.d.l.274.1		4	105.62	odd	4
525.4.d.l.274.4		4	105.83	odd	4
735.4.a.o.1.1		2	3.2	odd	2
1575.4.a.q.1.1		2	35.34	odd	2
1680.4.a.bo.1.1		2	84.83	odd	2
2205.4.a.bb.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)

\(f(q)\)	\(=\)	\(q+3.82843 q^{2} +6.65685 q^{4} +5.00000 q^{5} -5.14214 q^{8} +O(q^{10})\)	\(q+3.82843 q^{2} +6.65685 q^{4} +5.00000 q^{5} -5.14214 q^{8} +19.1421 q^{10} -48.5685 q^{11} +43.6569 q^{13} -72.9411 q^{16} -67.6569 q^{17} +93.2548 q^{19} +33.2843 q^{20} -185.941 q^{22} +104.167 q^{23} +25.0000 q^{25} +167.137 q^{26} +58.7351 q^{29} +9.08831 q^{31} -238.113 q^{32} -259.019 q^{34} -252.676 q^{37} +357.019 q^{38} -25.7107 q^{40} +276.274 q^{41} -92.6375 q^{43} -323.314 q^{44} +398.794 q^{46} -582.794 q^{47} +95.7107 q^{50} +290.617 q^{52} -623.019 q^{53} -242.843 q^{55} +224.863 q^{58} -524.999 q^{59} +352.794 q^{61} +34.7939 q^{62} -328.068 q^{64} +218.284 q^{65} -736.520 q^{67} -450.382 q^{68} +492.264 q^{71} -1164.75 q^{73} -967.352 q^{74} +620.784 q^{76} -872.195 q^{79} -364.706 q^{80} +1057.70 q^{82} -529.588 q^{83} -338.284 q^{85} -354.656 q^{86} +249.746 q^{88} -385.216 q^{89} +693.421 q^{92} -2231.18 q^{94} +466.274 q^{95} +463.892 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} + 10 q^{5} + 18 q^{8}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^4 + 10 * q^5 + 18 * q^8	\( 2 q + 2 q^{2} + 2 q^{4} + 10 q^{5} + 18 q^{8} + 10 q^{10} + 16 q^{11} + 76 q^{13} - 78 q^{16} - 124 q^{17} + 96 q^{19} + 10 q^{20} - 304 q^{22} + 16 q^{23} + 50 q^{25} + 108 q^{26} - 188 q^{29} + 120 q^{31} - 414 q^{32} - 156 q^{34} - 132 q^{37} + 352 q^{38} + 90 q^{40} + 100 q^{41} - 536 q^{43} - 624 q^{44} + 560 q^{46} - 928 q^{47} + 50 q^{50} + 140 q^{52} - 884 q^{53} + 80 q^{55} + 676 q^{58} + 104 q^{59} + 468 q^{61} - 168 q^{62} + 34 q^{64} + 380 q^{65} - 1688 q^{67} - 188 q^{68} + 136 q^{71} - 508 q^{73} - 1188 q^{74} + 608 q^{76} - 432 q^{79} - 390 q^{80} + 1380 q^{82} - 584 q^{83} - 620 q^{85} + 456 q^{86} + 1744 q^{88} - 1404 q^{89} + 1104 q^{92} - 1600 q^{94} + 480 q^{95} + 1188 q^{97}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 + 10 * q^5 + 18 * q^8 + 10 * q^10 + 16 * q^11 + 76 * q^13 - 78 * q^16 - 124 * q^17 + 96 * q^19 + 10 * q^20 - 304 * q^22 + 16 * q^23 + 50 * q^25 + 108 * q^26 - 188 * q^29 + 120 * q^31 - 414 * q^32 - 156 * q^34 - 132 * q^37 + 352 * q^38 + 90 * q^40 + 100 * q^41 - 536 * q^43 - 624 * q^44 + 560 * q^46 - 928 * q^47 + 50 * q^50 + 140 * q^52 - 884 * q^53 + 80 * q^55 + 676 * q^58 + 104 * q^59 + 468 * q^61 - 168 * q^62 + 34 * q^64 + 380 * q^65 - 1688 * q^67 - 188 * q^68 + 136 * q^71 - 508 * q^73 - 1188 * q^74 + 608 * q^76 - 432 * q^79 - 390 * q^80 + 1380 * q^82 - 584 * q^83 - 620 * q^85 + 456 * q^86 + 1744 * q^88 - 1404 * q^89 + 1104 * q^92 - 1600 * q^94 + 480 * q^95 + 1188 * q^97

Introduction

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