LMFDB - Embedded newform 2205.4.a.bx.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:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 34x^{4} + 241x^{2} - 64 $ x^6 - 34x^4 + 241x^2 - 64
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{9} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-3.09945$ 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)

$f(q)$	$=$	$q-3.09945 q^{2} +1.60662 q^{4} -5.00000 q^{5} +19.8160 q^{8} +O(q^{10})$	$q-3.09945 q^{2} +1.60662 q^{4} -5.00000 q^{5} +19.8160 q^{8} +15.4973 q^{10} +31.1878 q^{11} -54.1032 q^{13} -74.2717 q^{16} +81.8784 q^{17} +37.3653 q^{19} -8.03310 q^{20} -96.6651 q^{22} -116.264 q^{23} +25.0000 q^{25} +167.691 q^{26} -22.5290 q^{29} -89.9535 q^{31} +71.6739 q^{32} -253.778 q^{34} +344.782 q^{37} -115.812 q^{38} -99.0800 q^{40} -245.889 q^{41} +41.4519 q^{43} +50.1069 q^{44} +360.356 q^{46} -431.524 q^{47} -77.4864 q^{50} -86.9233 q^{52} +263.673 q^{53} -155.939 q^{55} +69.8276 q^{58} -627.005 q^{59} +715.666 q^{61} +278.807 q^{62} +372.024 q^{64} +270.516 q^{65} -809.809 q^{67} +131.547 q^{68} -54.8762 q^{71} +508.548 q^{73} -1068.64 q^{74} +60.0319 q^{76} +61.8168 q^{79} +371.359 q^{80} +762.122 q^{82} -560.154 q^{83} -409.392 q^{85} -128.478 q^{86} +618.017 q^{88} -102.286 q^{89} -186.792 q^{92} +1337.49 q^{94} -186.827 q^{95} -253.910 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 20 q^{4} - 30 q^{5}+O(q^{10}) $ 6 * q + 20 * q^4 - 30 * q^5	$ 6 q + 20 q^{4} - 30 q^{5} + 100 q^{16} - 44 q^{17} - 100 q^{20} - 24 q^{22} + 150 q^{25} - 168 q^{26} + 380 q^{37} - 56 q^{38} - 612 q^{41} - 328 q^{43} + 432 q^{46} - 120 q^{47} + 1120 q^{58} - 136 q^{59} - 2264 q^{62} - 1052 q^{64} + 1112 q^{67} - 264 q^{68} + 1400 q^{79} - 500 q^{80} - 2912 q^{83} + 220 q^{85} + 1384 q^{88} - 372 q^{89}+O(q^{100}) $ 6 * q + 20 * q^4 - 30 * q^5 + 100 * q^16 - 44 * q^17 - 100 * q^20 - 24 * q^22 + 150 * q^25 - 168 * q^26 + 380 * q^37 - 56 * q^38 - 612 * q^41 - 328 * q^43 + 432 * q^46 - 120 * q^47 + 1120 * q^58 - 136 * q^59 - 2264 * q^62 - 1052 * q^64 + 1112 * q^67 - 264 * q^68 + 1400 * q^79 - 500 * q^80 - 2912 * q^83 + 220 * q^85 + 1384 * q^88 - 372 * q^89

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 $

$2$	−3.09945	−1.09582	−0.547911	−	0.836536i	$-0.684577\pi$
$2$	−3.09945	−1.09582	−0.547911	+	0.836536i	$0.684577\pi$
$3$	0	0
$3$	0	0
$4$	1.60662	0.200827
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	0	0
$7$	0	0
$8$	19.8160	0.875751
$9$	0	0
$10$	15.4973	0.490067
$11$	31.1878	0.854861	0.427430	−	0.904048i	$-0.359419\pi$
$11$	31.1878	0.854861	0.427430	+	0.904048i	$0.359419\pi$
$12$	0	0
$13$	−54.1032	−1.15427	−0.577136	−	0.816648i	$-0.695830\pi$
$13$	−54.1032	−1.15427	−0.577136	+	0.816648i	$0.695830\pi$
$14$	0	0
$15$	0	0
$16$	−74.2717	−1.16050
$17$	81.8784	1.16814	0.584071	−	0.811703i	$-0.301459\pi$
$17$	81.8784	1.16814	0.584071	+	0.811703i	$0.301459\pi$
$18$	0	0
$19$	37.3653	0.451168	0.225584	−	0.974224i	$-0.427571\pi$
$19$	37.3653	0.451168	0.225584	+	0.974224i	$0.427571\pi$
$20$	−8.03310	−0.0898128
$21$	0	0
$22$	−96.6651	−0.936776
$23$	−116.264	−1.05403	−0.527017	−	0.849855i	$-0.676689\pi$
$23$	−116.264	−1.05403	−0.527017	+	0.849855i	$0.676689\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	167.691	1.26488
$27$	0	0
$28$	0	0
$29$	−22.5290	−0.144260	−0.0721298	−	0.997395i	$-0.522980\pi$
$29$	−22.5290	−0.144260	−0.0721298	+	0.997395i	$0.522980\pi$
$30$	0	0
$31$	−89.9535	−0.521165	−0.260583	−	0.965452i	$-0.583915\pi$
$31$	−89.9535	−0.521165	−0.260583	+	0.965452i	$0.583915\pi$
$32$	71.6739	0.395946
$33$	0	0
$34$	−253.778	−1.28008
$35$	0	0
$36$	0	0
$37$	344.782	1.53194	0.765971	−	0.642876i	$-0.222259\pi$
$37$	344.782	1.53194	0.765971	+	0.642876i	$0.222259\pi$
$38$	−115.812	−0.494400
$39$	0	0
$40$	−99.0800	−0.391648
$41$	−245.889	−0.936620	−0.468310	−	0.883564i	$-0.655137\pi$
$41$	−245.889	−0.936620	−0.468310	+	0.883564i	$0.655137\pi$
$42$	0	0
$43$	41.4519	0.147008	0.0735041	−	0.997295i	$-0.476582\pi$
$43$	41.4519	0.147008	0.0735041	+	0.997295i	$0.476582\pi$
$44$	50.1069	0.171680
$45$	0	0
$46$	360.356	1.15503
$47$	−431.524	−1.33924	−0.669620	−	0.742704i	$-0.733543\pi$
$47$	−431.524	−1.33924	−0.669620	+	0.742704i	$0.733543\pi$
$48$	0	0
$49$	0	0
$50$	−77.4864	−0.219165
$51$	0	0
$52$	−86.9233	−0.231810
$53$	263.673	0.683365	0.341682	−	0.939815i	$-0.389003\pi$
$53$	263.673	0.683365	0.341682	+	0.939815i	$0.389003\pi$
$54$	0	0
$55$	−155.939	−0.382305
$56$	0	0
$57$	0	0
$58$	69.8276	0.158083
$59$	−627.005	−1.38354	−0.691772	−	0.722116i	$-0.743170\pi$
$59$	−627.005	−1.38354	−0.691772	+	0.722116i	$0.743170\pi$
$60$	0	0
$61$	715.666	1.50216	0.751079	−	0.660212i	$-0.229534\pi$
$61$	715.666	1.50216	0.751079	+	0.660212i	$0.229534\pi$
$62$	278.807	0.571105
$63$	0	0
$64$	372.024	0.726609
$65$	270.516	0.516206
$66$	0	0
$67$	−809.809	−1.47663	−0.738313	−	0.674459i	$-0.764377\pi$
$67$	−809.809	−1.47663	−0.738313	+	0.674459i	$0.764377\pi$
$68$	131.547	0.234595
$69$	0	0
$70$	0	0
$71$	−54.8762	−0.0917268	−0.0458634	−	0.998948i	$-0.514604\pi$
$71$	−54.8762	−0.0917268	−0.0458634	+	0.998948i	$0.514604\pi$
$72$	0	0
$73$	508.548	0.815356	0.407678	−	0.913126i	$-0.366339\pi$
$73$	508.548	0.815356	0.407678	+	0.913126i	$0.366339\pi$
$74$	−1068.64	−1.67874
$75$	0	0
$76$	60.0319	0.0906070
$77$	0	0
$78$	0	0
$79$	61.8168	0.0880372	0.0440186	−	0.999031i	$-0.485984\pi$
$79$	61.8168	0.0880372	0.0440186	+	0.999031i	$0.485984\pi$
$80$	371.359	0.518989
$81$	0	0
$82$	762.122	1.02637
$83$	−560.154	−0.740782	−0.370391	−	0.928876i	$-0.620776\pi$
$83$	−560.154	−0.740782	−0.370391	+	0.928876i	$0.620776\pi$
$84$	0	0
$85$	−409.392	−0.522409
$86$	−128.478	−0.161095
$87$	0	0
$88$	618.017	0.748646
$89$	−102.286	−0.121824	−0.0609119	−	0.998143i	$-0.519401\pi$
$89$	−102.286	−0.121824	−0.0609119	+	0.998143i	$0.519401\pi$
$90$	0	0
$91$	0	0
$92$	−186.792	−0.211679
$93$	0	0
$94$	1337.49	1.46757
$95$	−186.827	−0.201769
$96$	0	0
$97$	−253.910	−0.265780	−0.132890	−	0.991131i	$-0.542426\pi$
$97$	−253.910	−0.265780	−0.132890	+	0.991131i	$0.542426\pi$
$98$	0	0
$99$	0	0
$100$	40.1655	0.0401655
$101$	612.694	0.603617	0.301809	−	0.953369i	$-0.402410\pi$
$101$	612.694	0.603617	0.301809	+	0.953369i	$0.402410\pi$
$102$	0	0
$103$	1865.47	1.78457	0.892283	−	0.451477i	$-0.149103\pi$
$103$	1865.47	1.78457	0.892283	+	0.451477i	$0.149103\pi$
$104$	−1072.11	−1.01086
$105$	0	0
$106$	−817.244	−0.748847
$107$	997.858	0.901557	0.450779	−	0.892636i	$-0.351146\pi$
$107$	997.858	0.901557	0.450779	+	0.892636i	$0.351146\pi$
$108$	0	0
$109$	1553.97	1.36554	0.682768	−	0.730635i	$-0.260776\pi$
$109$	1553.97	1.36554	0.682768	+	0.730635i	$0.260776\pi$
$110$	483.326	0.418939
$111$	0	0
$112$	0	0
$113$	−268.484	−0.223512	−0.111756	−	0.993736i	$-0.535647\pi$
$113$	−268.484	−0.223512	−0.111756	+	0.993736i	$0.535647\pi$
$114$	0	0
$115$	581.321	0.471378
$116$	−36.1955	−0.0289713
$117$	0	0
$118$	1943.37	1.51612
$119$	0	0
$120$	0	0
$121$	−358.322	−0.269213
$122$	−2218.18	−1.64610
$123$	0	0
$124$	−144.521	−0.104664
$125$	−125.000	−0.0894427
$126$	0	0
$127$	−43.7474	−0.0305665	−0.0152833	−	0.999883i	$-0.504865\pi$
$127$	−43.7474	−0.0305665	−0.0152833	+	0.999883i	$0.504865\pi$
$128$	−1726.46	−1.19218
$129$	0	0
$130$	−838.453	−0.565670
$131$	1641.52	1.09481	0.547407	−	0.836867i	$-0.315615\pi$
$131$	1641.52	1.09481	0.547407	+	0.836867i	$0.315615\pi$
$132$	0	0
$133$	0	0
$134$	2509.97	1.61812
$135$	0	0
$136$	1622.50	1.02300
$137$	2465.15	1.53731	0.768656	−	0.639662i	$-0.220926\pi$
$137$	2465.15	1.53731	0.768656	+	0.639662i	$0.220926\pi$
$138$	0	0
$139$	−2954.45	−1.80283	−0.901416	−	0.432955i	$-0.857471\pi$
$139$	−2954.45	−1.80283	−0.901416	+	0.432955i	$0.857471\pi$
$140$	0	0
$141$	0	0
$142$	170.086	0.100516
$143$	−1687.36	−0.986742
$144$	0	0
$145$	112.645	0.0645148
$146$	−1576.22	−0.893486
$147$	0	0
$148$	553.934	0.307656
$149$	16.8760	0.00927875	0.00463938	−	0.999989i	$-0.498523\pi$
$149$	16.8760	0.00927875	0.00463938	+	0.999989i	$0.498523\pi$
$150$	0	0
$151$	−1792.06	−0.965797	−0.482899	−	0.875676i	$-0.660416\pi$
$151$	−1792.06	−0.965797	−0.482899	+	0.875676i	$0.660416\pi$
$152$	740.431	0.395111
$153$	0	0
$154$	0	0
$155$	449.767	0.233072
$156$	0	0
$157$	3543.34	1.80121	0.900603	−	0.434643i	$-0.143125\pi$
$157$	3543.34	1.80121	0.900603	+	0.434643i	$0.143125\pi$
$158$	−191.598	−0.0964731
$159$	0	0
$160$	−358.370	−0.177073
$161$	0	0
$162$	0	0
$163$	−70.4005	−0.0338294	−0.0169147	−	0.999857i	$-0.505384\pi$
$163$	−70.4005	−0.0338294	−0.0169147	+	0.999857i	$0.505384\pi$
$164$	−395.050	−0.188099
$165$	0	0
$166$	1736.17	0.811765
$167$	−2493.09	−1.15522	−0.577609	−	0.816313i	$-0.696014\pi$
$167$	−2493.09	−1.15522	−0.577609	+	0.816313i	$0.696014\pi$
$168$	0	0
$169$	730.159	0.332344
$170$	1268.89	0.572468
$171$	0	0
$172$	66.5974	0.0295233
$173$	−487.259	−0.214137	−0.107068	−	0.994252i	$-0.534146\pi$
$173$	−487.259	−0.214137	−0.107068	+	0.994252i	$0.534146\pi$
$174$	0	0
$175$	0	0
$176$	−2316.37	−0.992062
$177$	0	0
$178$	317.032	0.133497
$179$	340.024	0.141981	0.0709904	−	0.997477i	$-0.477384\pi$
$179$	340.024	0.141981	0.0709904	+	0.997477i	$0.477384\pi$
$180$	0	0
$181$	−1863.23	−0.765152	−0.382576	−	0.923924i	$-0.624963\pi$
$181$	−1863.23	−0.765152	−0.382576	+	0.923924i	$0.624963\pi$
$182$	0	0
$183$	0	0
$184$	−2303.89	−0.923071
$185$	−1723.91	−0.685105
$186$	0	0
$187$	2553.60	0.998599
$188$	−693.295	−0.268956
$189$	0	0
$190$	579.061	0.221103
$191$	−1557.97	−0.590214	−0.295107	−	0.955464i	$-0.595355\pi$
$191$	−1557.97	−0.590214	−0.295107	+	0.955464i	$0.595355\pi$
$192$	0	0
$193$	−338.441	−0.126225	−0.0631127	−	0.998006i	$-0.520103\pi$
$193$	−338.441	−0.126225	−0.0631127	+	0.998006i	$0.520103\pi$
$194$	786.982	0.291248
$195$	0	0
$196$	0	0
$197$	−903.950	−0.326923	−0.163461	−	0.986550i	$-0.552266\pi$
$197$	−903.950	−0.326923	−0.163461	+	0.986550i	$0.552266\pi$
$198$	0	0
$199$	1987.44	0.707971	0.353985	−	0.935251i	$-0.384826\pi$
$199$	1987.44	0.707971	0.353985	+	0.935251i	$0.384826\pi$
$200$	495.400	0.175150
$201$	0	0
$202$	−1899.02	−0.661457
$203$	0	0
$204$	0	0
$205$	1229.45	0.418869
$206$	−5781.94	−1.95557
$207$	0	0
$208$	4018.34	1.33953
$209$	1165.34	0.385686
$210$	0	0
$211$	3749.01	1.22319	0.611594	−	0.791172i	$-0.290529\pi$
$211$	3749.01	1.22319	0.611594	+	0.791172i	$0.290529\pi$
$212$	423.623	0.137238
$213$	0	0
$214$	−3092.82	−0.987947
$215$	−207.259	−0.0657441
$216$	0	0
$217$	0	0
$218$	−4816.47	−1.49639
$219$	0	0
$220$	−250.534	−0.0767774
$221$	−4429.88	−1.34835
$222$	0	0
$223$	−3939.34	−1.18295	−0.591475	−	0.806324i	$-0.701454\pi$
$223$	−3939.34	−1.18295	−0.591475	+	0.806324i	$0.701454\pi$
$224$	0	0
$225$	0	0
$226$	832.154	0.244929
$227$	−2808.27	−0.821107	−0.410554	−	0.911836i	$-0.634665\pi$
$227$	−2808.27	−0.821107	−0.410554	+	0.911836i	$0.634665\pi$
$228$	0	0
$229$	1809.85	0.522264	0.261132	−	0.965303i	$-0.415904\pi$
$229$	1809.85	0.522264	0.261132	+	0.965303i	$0.415904\pi$
$230$	−1801.78	−0.516547
$231$	0	0
$232$	−446.434	−0.126336
$233$	−973.995	−0.273856	−0.136928	−	0.990581i	$-0.543723\pi$
$233$	−973.995	−0.273856	−0.136928	+	0.990581i	$0.543723\pi$
$234$	0	0
$235$	2157.62	0.598926
$236$	−1007.36	−0.277854
$237$	0	0
$238$	0	0
$239$	6015.41	1.62805	0.814027	−	0.580828i	$-0.197271\pi$
$239$	6015.41	1.62805	0.814027	+	0.580828i	$0.197271\pi$
$240$	0	0
$241$	384.989	0.102902	0.0514509	−	0.998676i	$-0.483615\pi$
$241$	384.989	0.102902	0.0514509	+	0.998676i	$0.483615\pi$
$242$	1110.60	0.295010
$243$	0	0
$244$	1149.80	0.301675
$245$	0	0
$246$	0	0
$247$	−2021.59	−0.520771
$248$	−1782.52	−0.456411
$249$	0	0
$250$	387.432	0.0980134
$251$	−1857.04	−0.466994	−0.233497	−	0.972357i	$-0.575017\pi$
$251$	−1857.04	−0.466994	−0.233497	+	0.972357i	$0.575017\pi$
$252$	0	0
$253$	−3626.02	−0.901052
$254$	135.593	0.0334955
$255$	0	0
$256$	2374.90	0.579810
$257$	4825.03	1.17112	0.585559	−	0.810630i	$-0.300875\pi$
$257$	4825.03	1.17112	0.585559	+	0.810630i	$0.300875\pi$
$258$	0	0
$259$	0	0
$260$	434.617	0.103668
$261$	0	0
$262$	−5087.83	−1.19972
$263$	−836.796	−0.196194	−0.0980971	−	0.995177i	$-0.531276\pi$
$263$	−836.796	−0.196194	−0.0980971	+	0.995177i	$0.531276\pi$
$264$	0	0
$265$	−1318.37	−0.305610
$266$	0	0
$267$	0	0
$268$	−1301.05	−0.296547
$269$	−5454.11	−1.23622	−0.618110	−	0.786092i	$-0.712101\pi$
$269$	−5454.11	−1.23622	−0.618110	+	0.786092i	$0.712101\pi$
$270$	0	0
$271$	2580.16	0.578352	0.289176	−	0.957276i	$-0.406619\pi$
$271$	2580.16	0.578352	0.289176	+	0.957276i	$0.406619\pi$
$272$	−6081.25	−1.35562
$273$	0	0
$274$	−7640.61	−1.68462
$275$	779.694	0.170972
$276$	0	0
$277$	591.932	0.128396	0.0641981	−	0.997937i	$-0.479551\pi$
$277$	591.932	0.128396	0.0641981	+	0.997937i	$0.479551\pi$
$278$	9157.20	1.97558
$279$	0	0
$280$	0	0
$281$	−6365.13	−1.35129	−0.675644	−	0.737228i	$-0.736134\pi$
$281$	−6365.13	−1.35129	−0.675644	+	0.737228i	$0.736134\pi$
$282$	0	0
$283$	1074.93	0.225787	0.112894	−	0.993607i	$-0.463988\pi$
$283$	1074.93	0.225787	0.112894	+	0.993607i	$0.463988\pi$
$284$	−88.1651	−0.0184213
$285$	0	0
$286$	5229.89	1.08129
$287$	0	0
$288$	0	0
$289$	1791.06	0.364556
$290$	−349.138	−0.0706968
$291$	0	0
$292$	817.042	0.163746
$293$	−5914.66	−1.17931	−0.589656	−	0.807655i	$-0.700737\pi$
$293$	−5914.66	−1.17931	−0.589656	+	0.807655i	$0.700737\pi$
$294$	0	0
$295$	3135.03	0.618740
$296$	6832.20	1.34160
$297$	0	0
$298$	−52.3063	−0.0101679
$299$	6290.27	1.21664
$300$	0	0
$301$	0	0
$302$	5554.39	1.05834
$303$	0	0
$304$	−2775.19	−0.523579
$305$	−3578.33	−0.671786
$306$	0	0
$307$	3162.52	0.587929	0.293965	−	0.955816i	$-0.405025\pi$
$307$	3162.52	0.587929	0.293965	+	0.955816i	$0.405025\pi$
$308$	0	0
$309$	0	0
$310$	−1394.03	−0.255406
$311$	−6941.22	−1.26560	−0.632798	−	0.774317i	$-0.718094\pi$
$311$	−6941.22	−1.26560	−0.632798	+	0.774317i	$0.718094\pi$
$312$	0	0
$313$	−8789.17	−1.58720	−0.793600	−	0.608440i	$-0.791795\pi$
$313$	−8789.17	−1.58720	−0.793600	+	0.608440i	$0.791795\pi$
$314$	−10982.4	−1.97380
$315$	0	0
$316$	99.3161	0.0176803
$317$	−4837.69	−0.857135	−0.428568	−	0.903510i	$-0.640982\pi$
$317$	−4837.69	−0.857135	−0.428568	+	0.903510i	$0.640982\pi$
$318$	0	0
$319$	−702.629	−0.123322
$320$	−1860.12	−0.324949
$321$	0	0
$322$	0	0
$323$	3059.41	0.527029
$324$	0	0
$325$	−1352.58	−0.230854
$326$	218.203	0.0370710
$327$	0	0
$328$	−4872.54	−0.820246
$329$	0	0
$330$	0	0
$331$	−4122.43	−0.684560	−0.342280	−	0.939598i	$-0.611199\pi$
$331$	−4122.43	−0.684560	−0.342280	+	0.939598i	$0.611199\pi$
$332$	−899.954	−0.148769
$333$	0	0
$334$	7727.23	1.26591
$335$	4049.04	0.660367
$336$	0	0
$337$	9671.74	1.56336	0.781681	−	0.623678i	$-0.214362\pi$
$337$	9671.74	1.56336	0.781681	+	0.623678i	$0.214362\pi$
$338$	−2263.10	−0.364190
$339$	0	0
$340$	−657.737	−0.104914
$341$	−2805.45	−0.445524
$342$	0	0
$343$	0	0
$344$	821.410	0.128743
$345$	0	0
$346$	1510.24	0.234656
$347$	6169.21	0.954411	0.477205	−	0.878792i	$-0.341650\pi$
$347$	6169.21	0.954411	0.477205	+	0.878792i	$0.341650\pi$
$348$	0	0
$349$	7346.44	1.12678	0.563390	−	0.826191i	$-0.309497\pi$
$349$	7346.44	1.12678	0.563390	+	0.826191i	$0.309497\pi$
$350$	0	0
$351$	0	0
$352$	2235.35	0.338479
$353$	−10850.5	−1.63602	−0.818008	−	0.575206i	$-0.804922\pi$
$353$	−10850.5	−1.63602	−0.818008	+	0.575206i	$0.804922\pi$
$354$	0	0
$355$	274.381	0.0410215
$356$	−164.335	−0.0244656
$357$	0	0
$358$	−1053.89	−0.155586
$359$	4365.52	0.641792	0.320896	−	0.947114i	$-0.396016\pi$
$359$	4365.52	0.641792	0.320896	+	0.947114i	$0.396016\pi$
$360$	0	0
$361$	−5462.83	−0.796447
$362$	5774.99	0.838471
$363$	0	0
$364$	0	0
$365$	−2542.74	−0.364638
$366$	0	0
$367$	7197.13	1.02367	0.511835	−	0.859084i	$-0.328966\pi$
$367$	7197.13	1.02367	0.511835	+	0.859084i	$0.328966\pi$
$368$	8635.14	1.22320
$369$	0	0
$370$	5343.18	0.750754
$371$	0	0
$372$	0	0
$373$	12300.6	1.70751	0.853753	−	0.520678i	$-0.174321\pi$
$373$	12300.6	1.70751	0.853753	+	0.520678i	$0.174321\pi$
$374$	−7914.78	−1.09429
$375$	0	0
$376$	−8551.08	−1.17284
$377$	1218.89	0.166515
$378$	0	0
$379$	−888.207	−0.120380	−0.0601901	−	0.998187i	$-0.519171\pi$
$379$	−888.207	−0.120380	−0.0601901	+	0.998187i	$0.519171\pi$
$380$	−300.159	−0.0405207
$381$	0	0
$382$	4828.86	0.646770
$383$	−10524.1	−1.40406	−0.702031	−	0.712147i	$-0.747723\pi$
$383$	−10524.1	−1.40406	−0.702031	+	0.712147i	$0.747723\pi$
$384$	0	0
$385$	0	0
$386$	1048.98	0.138321
$387$	0	0
$388$	−407.937	−0.0533759
$389$	6006.81	0.782924	0.391462	−	0.920194i	$-0.371970\pi$
$389$	6006.81	0.782924	0.391462	+	0.920194i	$0.371970\pi$
$390$	0	0
$391$	−9519.52	−1.23126
$392$	0	0
$393$	0	0
$394$	2801.75	0.358249
$395$	−309.084	−0.0393714
$396$	0	0
$397$	−14706.6	−1.85920	−0.929602	−	0.368565i	$-0.879849\pi$
$397$	−14706.6	−1.85920	−0.929602	+	0.368565i	$0.879849\pi$
$398$	−6159.99	−0.775810
$399$	0	0
$400$	−1856.79	−0.232099
$401$	−2513.22	−0.312978	−0.156489	−	0.987680i	$-0.550018\pi$
$401$	−2513.22	−0.312978	−0.156489	+	0.987680i	$0.550018\pi$
$402$	0	0
$403$	4866.77	0.601566
$404$	984.366	0.121223
$405$	0	0
$406$	0	0
$407$	10753.0	1.30960
$408$	0	0
$409$	−4288.99	−0.518526	−0.259263	−	0.965807i	$-0.583480\pi$
$409$	−4288.99	−0.518526	−0.259263	+	0.965807i	$0.583480\pi$
$410$	−3810.61	−0.459006
$411$	0	0
$412$	2997.10	0.358390
$413$	0	0
$414$	0	0
$415$	2800.77	0.331288
$416$	−3877.79	−0.457030
$417$	0	0
$418$	−3611.92	−0.422643
$419$	−6467.11	−0.754031	−0.377016	−	0.926207i	$-0.623050\pi$
$419$	−6467.11	−0.754031	−0.377016	+	0.926207i	$0.623050\pi$
$420$	0	0
$421$	−13647.0	−1.57984	−0.789921	−	0.613209i	$-0.789878\pi$
$421$	−13647.0	−1.57984	−0.789921	+	0.613209i	$0.789878\pi$
$422$	−11619.9	−1.34040
$423$	0	0
$424$	5224.95	0.598458
$425$	2046.96	0.233628
$426$	0	0
$427$	0	0
$428$	1603.18	0.181057
$429$	0	0
$430$	642.391	0.0720438
$431$	−17252.7	−1.92815	−0.964074	−	0.265634i	$-0.914419\pi$
$431$	−17252.7	−1.92815	−0.964074	+	0.265634i	$0.914419\pi$
$432$	0	0
$433$	4033.03	0.447610	0.223805	−	0.974634i	$-0.428152\pi$
$433$	4033.03	0.447610	0.223805	+	0.974634i	$0.428152\pi$
$434$	0	0
$435$	0	0
$436$	2496.64	0.274237
$437$	−4344.25	−0.475546
$438$	0	0
$439$	−16593.8	−1.80405	−0.902025	−	0.431684i	$-0.857920\pi$
$439$	−16593.8	−1.80405	−0.902025	+	0.431684i	$0.857920\pi$
$440$	−3090.08	−0.334804
$441$	0	0
$442$	13730.2	1.47756
$443$	5587.14	0.599216	0.299608	−	0.954062i	$-0.403144\pi$
$443$	5587.14	0.599216	0.299608	+	0.954062i	$0.403144\pi$
$444$	0	0
$445$	511.431	0.0544813
$446$	12209.8	1.29630
$447$	0	0
$448$	0	0
$449$	−13090.2	−1.37587	−0.687935	−	0.725772i	$-0.741483\pi$
$449$	−13090.2	−1.37587	−0.687935	+	0.725772i	$0.741483\pi$
$450$	0	0
$451$	−7668.73	−0.800680
$452$	−431.352	−0.0448873
$453$	0	0
$454$	8704.10	0.899788
$455$	0	0
$456$	0	0
$457$	7594.11	0.777324	0.388662	−	0.921380i	$-0.372937\pi$
$457$	7594.11	0.777324	0.388662	+	0.921380i	$0.372937\pi$
$458$	−5609.55	−0.572308
$459$	0	0
$460$	933.962	0.0946656
$461$	−15260.2	−1.54173	−0.770865	−	0.636998i	$-0.780176\pi$
$461$	−15260.2	−1.54173	−0.770865	+	0.636998i	$0.780176\pi$
$462$	0	0
$463$	−4278.64	−0.429472	−0.214736	−	0.976672i	$-0.568889\pi$
$463$	−4278.64	−0.429472	−0.214736	+	0.976672i	$0.568889\pi$
$464$	1673.27	0.167413
$465$	0	0
$466$	3018.85	0.300098
$467$	−9102.32	−0.901938	−0.450969	−	0.892540i	$-0.648922\pi$
$467$	−9102.32	−0.901938	−0.450969	+	0.892540i	$0.648922\pi$
$468$	0	0
$469$	0	0
$470$	−6687.45	−0.656317
$471$	0	0
$472$	−12424.7	−1.21164
$473$	1292.79	0.125672
$474$	0	0
$475$	934.133	0.0902336
$476$	0	0
$477$	0	0
$478$	−18644.5	−1.78406
$479$	−4226.58	−0.403167	−0.201584	−	0.979471i	$-0.564609\pi$
$479$	−4226.58	−0.403167	−0.201584	+	0.979471i	$0.564609\pi$
$480$	0	0
$481$	−18653.8	−1.76828
$482$	−1193.26	−0.112762
$483$	0	0
$484$	−575.688	−0.0540653
$485$	1269.55	0.118860
$486$	0	0
$487$	−9576.48	−0.891071	−0.445536	−	0.895264i	$-0.646987\pi$
$487$	−9576.48	−0.891071	−0.445536	+	0.895264i	$0.646987\pi$
$488$	14181.6	1.31552
$489$	0	0
$490$	0	0
$491$	−21412.2	−1.96806	−0.984030	−	0.178000i	$-0.943037\pi$
$491$	−21412.2	−1.96806	−0.984030	+	0.178000i	$0.943037\pi$
$492$	0	0
$493$	−1844.64	−0.168516
$494$	6265.81	0.570672
$495$	0	0
$496$	6681.00	0.604810
$497$	0	0
$498$	0	0
$499$	−19706.9	−1.76794	−0.883970	−	0.467543i	$-0.845139\pi$
$499$	−19706.9	−1.76794	−0.883970	+	0.467543i	$0.845139\pi$
$500$	−200.827	−0.0179626
$501$	0	0
$502$	5755.82	0.511743
$503$	−176.759	−0.0156685	−0.00783427	−	0.999969i	$-0.502494\pi$
$503$	−176.759	−0.0156685	−0.00783427	+	0.999969i	$0.502494\pi$
$504$	0	0
$505$	−3063.47	−0.269946
$506$	11238.7	0.987393
$507$	0	0
$508$	−70.2854	−0.00613860
$509$	−776.451	−0.0676142	−0.0338071	−	0.999428i	$-0.510763\pi$
$509$	−776.451	−0.0676142	−0.0338071	+	0.999428i	$0.510763\pi$
$510$	0	0
$511$	0	0
$512$	6450.80	0.556812
$513$	0	0
$514$	−14955.0	−1.28334
$515$	−9327.35	−0.798082
$516$	0	0
$517$	−13458.3	−1.14486
$518$	0	0
$519$	0	0
$520$	5360.55	0.452068
$521$	−4248.01	−0.357215	−0.178607	−	0.983920i	$-0.557159\pi$
$521$	−4248.01	−0.357215	−0.178607	+	0.983920i	$0.557159\pi$
$522$	0	0
$523$	−13490.8	−1.12794	−0.563968	−	0.825797i	$-0.690726\pi$
$523$	−13490.8	−1.12794	−0.563968	+	0.825797i	$0.690726\pi$
$524$	2637.30	0.219869
$525$	0	0
$526$	2593.61	0.214994
$527$	−7365.24	−0.608795
$528$	0	0
$529$	1350.36	0.110986
$530$	4086.22	0.334894
$531$	0	0
$532$	0	0
$533$	13303.4	1.08111
$534$	0	0
$535$	−4989.29	−0.403189
$536$	−16047.2	−1.29316
$537$	0	0
$538$	16904.8	1.35468
$539$	0	0
$540$	0	0
$541$	−10582.4	−0.840984	−0.420492	−	0.907296i	$-0.638143\pi$
$541$	−10582.4	−0.840984	−0.420492	+	0.907296i	$0.638143\pi$
$542$	−7997.08	−0.633771
$543$	0	0
$544$	5868.54	0.462521
$545$	−7769.86	−0.610687
$546$	0	0
$547$	−6182.43	−0.483257	−0.241629	−	0.970369i	$-0.577682\pi$
$547$	−6182.43	−0.483257	−0.241629	+	0.970369i	$0.577682\pi$
$548$	3960.56	0.308735
$549$	0	0
$550$	−2416.63	−0.187355
$551$	−841.803	−0.0650853
$552$	0	0
$553$	0	0
$554$	−1834.67	−0.140699
$555$	0	0
$556$	−4746.68	−0.362058
$557$	20848.5	1.58596	0.792979	−	0.609249i	$-0.208529\pi$
$557$	20848.5	1.58596	0.792979	+	0.609249i	$0.208529\pi$
$558$	0	0
$559$	−2242.68	−0.169687
$560$	0	0
$561$	0	0
$562$	19728.4	1.48077
$563$	13326.0	0.997557	0.498779	−	0.866729i	$-0.333782\pi$
$563$	13326.0	0.997557	0.498779	+	0.866729i	$0.333782\pi$
$564$	0	0
$565$	1342.42	0.0999576
$566$	−3331.69	−0.247423
$567$	0	0
$568$	−1087.43	−0.0803299
$569$	−11592.9	−0.854126	−0.427063	−	0.904222i	$-0.640452\pi$
$569$	−11592.9	−0.854126	−0.427063	+	0.904222i	$0.640452\pi$
$570$	0	0
$571$	1603.77	0.117541	0.0587704	−	0.998272i	$-0.481282\pi$
$571$	1603.77	0.117541	0.0587704	+	0.998272i	$0.481282\pi$
$572$	−2710.94	−0.198165
$573$	0	0
$574$	0	0
$575$	−2906.60	−0.210807
$576$	0	0
$577$	−12475.4	−0.900097	−0.450048	−	0.893004i	$-0.648593\pi$
$577$	−12475.4	−0.900097	−0.450048	+	0.893004i	$0.648593\pi$
$578$	−5551.32	−0.399489
$579$	0	0
$580$	180.978	0.0129564
$581$	0	0
$582$	0	0
$583$	8223.39	0.584182
$584$	10077.4	0.714049
$585$	0	0
$586$	18332.2	1.29232
$587$	−11367.0	−0.799264	−0.399632	−	0.916676i	$-0.630862\pi$
$587$	−11367.0	−0.799264	−0.399632	+	0.916676i	$0.630862\pi$
$588$	0	0
$589$	−3361.14	−0.235133
$590$	−9716.87	−0.678029
$591$	0	0
$592$	−25607.6	−1.77781
$593$	−2463.42	−0.170591	−0.0852955	−	0.996356i	$-0.527183\pi$
$593$	−2463.42	−0.170591	−0.0852955	+	0.996356i	$0.527183\pi$
$594$	0	0
$595$	0	0
$596$	27.1133	0.00186343
$597$	0	0
$598$	−19496.4	−1.33322
$599$	26493.7	1.80719	0.903593	−	0.428393i	$-0.140920\pi$
$599$	26493.7	1.80719	0.903593	+	0.428393i	$0.140920\pi$
$600$	0	0
$601$	−5703.89	−0.387132	−0.193566	−	0.981087i	$-0.562005\pi$
$601$	−5703.89	−0.387132	−0.193566	+	0.981087i	$0.562005\pi$
$602$	0	0
$603$	0	0
$604$	−2879.15	−0.193959
$605$	1791.61	0.120396
$606$	0	0
$607$	−6111.92	−0.408691	−0.204345	−	0.978899i	$-0.565507\pi$
$607$	−6111.92	−0.408691	−0.204345	+	0.978899i	$0.565507\pi$
$608$	2678.12	0.178638
$609$	0	0
$610$	11090.9	0.736158
$611$	23346.8	1.54585
$612$	0	0
$613$	15496.0	1.02101	0.510504	−	0.859875i	$-0.329459\pi$
$613$	15496.0	1.02101	0.510504	+	0.859875i	$0.329459\pi$
$614$	−9802.08	−0.644266
$615$	0	0
$616$	0	0
$617$	23071.0	1.50535	0.752676	−	0.658391i	$-0.228763\pi$
$617$	23071.0	1.50535	0.752676	+	0.658391i	$0.228763\pi$
$618$	0	0
$619$	9160.95	0.594846	0.297423	−	0.954746i	$-0.403873\pi$
$619$	9160.95	0.594846	0.297423	+	0.954746i	$0.403873\pi$
$620$	722.605	0.0468073
$621$	0	0
$622$	21514.0	1.38687
$623$	0	0
$624$	0	0
$625$	625.000	0.0400000
$626$	27241.6	1.73929
$627$	0	0
$628$	5692.80	0.361732
$629$	28230.2	1.78953
$630$	0	0
$631$	26685.9	1.68360	0.841798	−	0.539793i	$-0.181497\pi$
$631$	26685.9	1.68360	0.841798	+	0.539793i	$0.181497\pi$
$632$	1224.96	0.0770987
$633$	0	0
$634$	14994.2	0.939268
$635$	218.737	0.0136698
$636$	0	0
$637$	0	0
$638$	2177.77	0.135139
$639$	0	0
$640$	8632.31	0.533159
$641$	−22763.0	−1.40263	−0.701314	−	0.712852i	$-0.747403\pi$
$641$	−22763.0	−1.40263	−0.701314	+	0.712852i	$0.747403\pi$
$642$	0	0
$643$	−13560.1	−0.831660	−0.415830	−	0.909442i	$-0.636509\pi$
$643$	−13560.1	−0.831660	−0.415830	+	0.909442i	$0.636509\pi$
$644$	0	0
$645$	0	0
$646$	−9482.51	−0.577530
$647$	−24534.7	−1.49082	−0.745408	−	0.666609i	$-0.767745\pi$
$647$	−24534.7	−1.49082	−0.745408	+	0.666609i	$0.767745\pi$
$648$	0	0
$649$	−19554.9	−1.18274
$650$	4192.26	0.252975
$651$	0	0
$652$	−113.107	−0.00679388
$653$	14176.0	0.849539	0.424769	−	0.905302i	$-0.360355\pi$
$653$	14176.0	0.849539	0.424769	+	0.905302i	$0.360355\pi$
$654$	0	0
$655$	−8207.62	−0.489616
$656$	18262.6	1.08694
$657$	0	0
$658$	0	0
$659$	20212.3	1.19478	0.597391	−	0.801950i	$-0.296204\pi$
$659$	20212.3	1.19478	0.597391	+	0.801950i	$0.296204\pi$
$660$	0	0
$661$	−23850.7	−1.40346	−0.701728	−	0.712445i	$-0.747588\pi$
$661$	−23850.7	−1.40346	−0.701728	+	0.712445i	$0.747588\pi$
$662$	12777.3	0.750156
$663$	0	0
$664$	−11100.0	−0.648741
$665$	0	0
$666$	0	0
$667$	2619.31	0.152054
$668$	−4005.45	−0.232000
$669$	0	0
$670$	−12549.8	−0.723645
$671$	22320.0	1.28414
$672$	0	0
$673$	395.817	0.0226711	0.0113355	−	0.999936i	$-0.496392\pi$
$673$	395.817	0.0226711	0.0113355	+	0.999936i	$0.496392\pi$
$674$	−29977.1	−1.71317
$675$	0	0
$676$	1173.09	0.0667437
$677$	−8729.25	−0.495557	−0.247779	−	0.968817i	$-0.579701\pi$
$677$	−8729.25	−0.495557	−0.247779	+	0.968817i	$0.579701\pi$
$678$	0	0
$679$	0	0
$680$	−8112.50	−0.457500
$681$	0	0
$682$	8695.36	0.488215
$683$	−7531.85	−0.421959	−0.210980	−	0.977490i	$-0.567665\pi$
$683$	−7531.85	−0.421959	−0.210980	+	0.977490i	$0.567665\pi$
$684$	0	0
$685$	−12325.7	−0.687507
$686$	0	0
$687$	0	0
$688$	−3078.70	−0.170602
$689$	−14265.6	−0.788789
$690$	0	0
$691$	−163.381	−0.00899465	−0.00449732	−	0.999990i	$-0.501432\pi$
$691$	−163.381	−0.00899465	−0.00449732	+	0.999990i	$0.501432\pi$
$692$	−782.840	−0.0430045
$693$	0	0
$694$	−19121.2	−1.04587
$695$	14772.3	0.806251
$696$	0	0
$697$	−20133.0	−1.09411
$698$	−22770.0	−1.23475
$699$	0	0
$700$	0	0
$701$	−29759.6	−1.60343	−0.801716	−	0.597705i	$-0.796079\pi$
$701$	−29759.6	−1.60343	−0.801716	+	0.597705i	$0.796079\pi$
$702$	0	0
$703$	12882.9	0.691163
$704$	11602.6	0.621150
$705$	0	0
$706$	33630.6	1.79278
$707$	0	0
$708$	0	0
$709$	−15454.6	−0.818633	−0.409316	−	0.912393i	$-0.634233\pi$
$709$	−15454.6	−0.818633	−0.409316	+	0.912393i	$0.634233\pi$
$710$	−850.431	−0.0449523
$711$	0	0
$712$	−2026.90	−0.106687
$713$	10458.4	0.549325
$714$	0	0
$715$	8436.80	0.441284
$716$	546.289	0.0285136
$717$	0	0
$718$	−13530.7	−0.703290
$719$	−13452.1	−0.697744	−0.348872	−	0.937170i	$-0.613435\pi$
$719$	−13452.1	−0.697744	−0.348872	+	0.937170i	$0.613435\pi$
$720$	0	0
$721$	0	0
$722$	16931.8	0.872765
$723$	0	0
$724$	−2993.50	−0.153664
$725$	−563.225	−0.0288519
$726$	0	0
$727$	6644.23	0.338956	0.169478	−	0.985534i	$-0.445792\pi$
$727$	6644.23	0.338956	0.169478	+	0.985534i	$0.445792\pi$
$728$	0	0
$729$	0	0
$730$	7881.10	0.399579
$731$	3394.01	0.171726
$732$	0	0
$733$	12532.8	0.631525	0.315762	−	0.948838i	$-0.397740\pi$
$733$	12532.8	0.631525	0.315762	+	0.948838i	$0.397740\pi$
$734$	−22307.2	−1.12176
$735$	0	0
$736$	−8333.11	−0.417340
$737$	−25256.1	−1.26231
$738$	0	0
$739$	16509.4	0.821799	0.410900	−	0.911681i	$-0.365215\pi$
$739$	16509.4	0.821799	0.410900	+	0.911681i	$0.365215\pi$
$740$	−2769.67	−0.137588
$741$	0	0
$742$	0	0
$743$	11768.9	0.581102	0.290551	−	0.956859i	$-0.406161\pi$
$743$	11768.9	0.581102	0.290551	+	0.956859i	$0.406161\pi$
$744$	0	0
$745$	−84.3799	−0.00414958
$746$	−38125.1	−1.87112
$747$	0	0
$748$	4102.67	0.200546
$749$	0	0
$750$	0	0
$751$	18336.1	0.890938	0.445469	−	0.895297i	$-0.353037\pi$
$751$	18336.1	0.890938	0.445469	+	0.895297i	$0.353037\pi$
$752$	32050.0	1.55418
$753$	0	0
$754$	−3777.90	−0.182471
$755$	8960.28	0.431918
$756$	0	0
$757$	24340.5	1.16865	0.584326	−	0.811519i	$-0.301359\pi$
$757$	24340.5	1.16865	0.584326	+	0.811519i	$0.301359\pi$
$758$	2752.96	0.131915
$759$	0	0
$760$	−3702.16	−0.176699
$761$	−28022.5	−1.33484	−0.667421	−	0.744681i	$-0.732602\pi$
$761$	−28022.5	−1.33484	−0.667421	+	0.744681i	$0.732602\pi$
$762$	0	0
$763$	0	0
$764$	−2503.07	−0.118531
$765$	0	0
$766$	32618.9	1.53860
$767$	33923.0	1.59699
$768$	0	0
$769$	−33595.9	−1.57542	−0.787712	−	0.616044i	$-0.788734\pi$
$769$	−33595.9	−1.57542	−0.787712	+	0.616044i	$0.788734\pi$
$770$	0	0
$771$	0	0
$772$	−543.745	−0.0253495
$773$	−668.213	−0.0310918	−0.0155459	−	0.999879i	$-0.504949\pi$
$773$	−668.213	−0.0310918	−0.0155459	+	0.999879i	$0.504949\pi$
$774$	0	0
$775$	−2248.84	−0.104233
$776$	−5031.48	−0.232757
$777$	0	0
$778$	−18617.8	−0.857945
$779$	−9187.73	−0.422573
$780$	0	0
$781$	−1711.47	−0.0784136
$782$	29505.3	1.34924
$783$	0	0
$784$	0	0
$785$	−17716.7	−0.805524
$786$	0	0
$787$	16719.9	0.757305	0.378653	−	0.925539i	$-0.376388\pi$
$787$	16719.9	0.757305	0.378653	+	0.925539i	$0.376388\pi$
$788$	−1452.30	−0.0656550
$789$	0	0
$790$	957.992	0.0431441
$791$	0	0
$792$	0	0
$793$	−38719.9	−1.73390
$794$	45582.5	2.03736
$795$	0	0
$796$	3193.07	0.142180
$797$	40821.1	1.81425	0.907126	−	0.420860i	$-0.138272\pi$
$797$	40821.1	1.81425	0.907126	+	0.420860i	$0.138272\pi$
$798$	0	0
$799$	−35332.5	−1.56442
$800$	1791.85	0.0791892
$801$	0	0
$802$	7789.60	0.342968
$803$	15860.5	0.697016
$804$	0	0
$805$	0	0
$806$	−15084.3	−0.659210
$807$	0	0
$808$	12141.1	0.528619
$809$	10822.7	0.470340	0.235170	−	0.971954i	$-0.424435\pi$
$809$	10822.7	0.470340	0.235170	+	0.971954i	$0.424435\pi$
$810$	0	0
$811$	−19650.6	−0.850833	−0.425417	−	0.904998i	$-0.639872\pi$
$811$	−19650.6	−0.850833	−0.425417	+	0.904998i	$0.639872\pi$
$812$	0	0
$813$	0	0
$814$	−33328.4	−1.43509
$815$	352.003	0.0151290
$816$	0	0
$817$	1548.86	0.0663254
$818$	13293.5	0.568212
$819$	0	0
$820$	1975.25	0.0841204
$821$	24495.9	1.04130	0.520652	−	0.853769i	$-0.325689\pi$
$821$	24495.9	1.04130	0.520652	+	0.853769i	$0.325689\pi$
$822$	0	0
$823$	21651.7	0.917048	0.458524	−	0.888682i	$-0.348378\pi$
$823$	21651.7	0.917048	0.458524	+	0.888682i	$0.348378\pi$
$824$	36966.1	1.56284
$825$	0	0
$826$	0	0
$827$	4550.97	0.191358	0.0956788	−	0.995412i	$-0.469498\pi$
$827$	4550.97	0.191358	0.0956788	+	0.995412i	$0.469498\pi$
$828$	0	0
$829$	43235.4	1.81137	0.905686	−	0.423949i	$-0.139356\pi$
$829$	43235.4	1.81137	0.905686	+	0.423949i	$0.139356\pi$
$830$	−8680.86	−0.363033
$831$	0	0
$832$	−20127.7	−0.838704
$833$	0	0
$834$	0	0
$835$	12465.5	0.516629
$836$	1872.26	0.0774563
$837$	0	0
$838$	20044.5	0.826285
$839$	139.264	0.00573053	0.00286527	−	0.999996i	$-0.499088\pi$
$839$	139.264	0.00573053	0.00286527	+	0.999996i	$0.499088\pi$
$840$	0	0
$841$	−23881.4	−0.979189
$842$	42298.2	1.73123
$843$	0	0
$844$	6023.24	0.245650
$845$	−3650.80	−0.148629
$846$	0	0
$847$	0	0
$848$	−19583.5	−0.793042
$849$	0	0
$850$	−6344.46	−0.256015
$851$	−40085.8	−1.61472
$852$	0	0
$853$	13239.6	0.531437	0.265718	−	0.964051i	$-0.414391\pi$
$853$	13239.6	0.531437	0.265718	+	0.964051i	$0.414391\pi$
$854$	0	0
$855$	0	0
$856$	19773.6	0.789540
$857$	44722.7	1.78261	0.891305	−	0.453404i	$-0.149791\pi$
$857$	44722.7	1.78261	0.891305	+	0.453404i	$0.149791\pi$
$858$	0	0
$859$	19350.8	0.768617	0.384308	−	0.923205i	$-0.374440\pi$
$859$	19350.8	0.768617	0.384308	+	0.923205i	$0.374440\pi$
$860$	−332.987	−0.0132032
$861$	0	0
$862$	53473.9	2.11291
$863$	−34706.8	−1.36899	−0.684493	−	0.729020i	$-0.739976\pi$
$863$	−34706.8	−1.36899	−0.684493	+	0.729020i	$0.739976\pi$
$864$	0	0
$865$	2436.30	0.0957648
$866$	−12500.2	−0.490501
$867$	0	0
$868$	0	0
$869$	1927.93	0.0752595
$870$	0	0
$871$	43813.3	1.70443
$872$	30793.5	1.19587
$873$	0	0
$874$	13464.8	0.521114
$875$	0	0
$876$	0	0
$877$	33566.2	1.29242	0.646209	−	0.763160i	$-0.276353\pi$
$877$	33566.2	1.29242	0.646209	+	0.763160i	$0.276353\pi$
$878$	51431.7	1.97692
$879$	0	0
$880$	11581.9	0.443664
$881$	32706.1	1.25074	0.625368	−	0.780330i	$-0.284949\pi$
$881$	32706.1	1.25074	0.625368	+	0.780330i	$0.284949\pi$
$882$	0	0
$883$	−19854.3	−0.756683	−0.378342	−	0.925666i	$-0.623506\pi$
$883$	−19854.3	−0.756683	−0.378342	+	0.925666i	$0.623506\pi$
$884$	−7117.14	−0.270786
$885$	0	0
$886$	−17317.1	−0.656635
$887$	−29902.1	−1.13192	−0.565960	−	0.824433i	$-0.691494\pi$
$887$	−29902.1	−1.13192	−0.565960	+	0.824433i	$0.691494\pi$
$888$	0	0
$889$	0	0
$890$	−1585.16	−0.0597018
$891$	0	0
$892$	−6329.02	−0.237569
$893$	−16124.0	−0.604222
$894$	0	0
$895$	−1700.12	−0.0634957
$896$	0	0
$897$	0	0
$898$	40572.5	1.50771
$899$	2026.56	0.0751831
$900$	0	0
$901$	21589.1	0.798267
$902$	23768.9	0.877403
$903$	0	0
$904$	−5320.28	−0.195741
$905$	9316.14	0.342187
$906$	0	0
$907$	38973.5	1.42679	0.713393	−	0.700764i	$-0.247158\pi$
$907$	38973.5	1.42679	0.713393	+	0.700764i	$0.247158\pi$
$908$	−4511.82	−0.164901
$909$	0	0
$910$	0	0
$911$	−41356.0	−1.50404	−0.752022	−	0.659138i	$-0.770922\pi$
$911$	−41356.0	−1.50404	−0.752022	+	0.659138i	$0.770922\pi$
$912$	0	0
$913$	−17470.0	−0.633265
$914$	−23537.6	−0.851810
$915$	0	0
$916$	2907.74	0.104885
$917$	0	0
$918$	0	0
$919$	14009.2	0.502852	0.251426	−	0.967876i	$-0.419100\pi$
$919$	14009.2	0.502852	0.251426	+	0.967876i	$0.419100\pi$
$920$	11519.5	0.412810
$921$	0	0
$922$	47298.3	1.68946
$923$	2968.98	0.105878
$924$	0	0
$925$	8619.55	0.306388
$926$	13261.5	0.470625
$927$	0	0
$928$	−1614.74	−0.0571190
$929$	−39996.1	−1.41252	−0.706259	−	0.707954i	$-0.749619\pi$
$929$	−39996.1	−1.41252	−0.706259	+	0.707954i	$0.749619\pi$
$930$	0	0
$931$	0	0
$932$	−1564.84	−0.0549979
$933$	0	0
$934$	28212.2	0.988365
$935$	−12768.0	−0.446587
$936$	0	0
$937$	−14964.6	−0.521743	−0.260872	−	0.965374i	$-0.584010\pi$
$937$	−14964.6	−0.521743	−0.260872	+	0.965374i	$0.584010\pi$
$938$	0	0
$939$	0	0
$940$	3466.48	0.120281
$941$	−31615.3	−1.09525	−0.547624	−	0.836725i	$-0.684468\pi$
$941$	−31615.3	−1.09525	−0.547624	+	0.836725i	$0.684468\pi$
$942$	0	0
$943$	28588.1	0.987229
$944$	46568.8	1.60560
$945$	0	0
$946$	−4006.95	−0.137714
$947$	−29872.2	−1.02504	−0.512522	−	0.858674i	$-0.671289\pi$
$947$	−29872.2	−1.02504	−0.512522	+	0.858674i	$0.671289\pi$
$948$	0	0
$949$	−27514.1	−0.941143
$950$	−2895.30	−0.0988801
$951$	0	0
$952$	0	0
$953$	41530.2	1.41164	0.705821	−	0.708390i	$-0.250578\pi$
$953$	41530.2	1.41164	0.705821	+	0.708390i	$0.250578\pi$
$954$	0	0
$955$	7789.85	0.263952
$956$	9664.48	0.326958
$957$	0	0
$958$	13100.1	0.441800
$959$	0	0
$960$	0	0
$961$	−21699.4	−0.728387
$962$	57816.7	1.93772
$963$	0	0
$964$	618.531	0.0206655
$965$	1692.20	0.0564497
$966$	0	0
$967$	53414.9	1.77633	0.888164	−	0.459527i	$-0.151981\pi$
$967$	53414.9	1.77633	0.888164	+	0.459527i	$0.151981\pi$
$968$	−7100.51	−0.235764
$969$	0	0
$970$	−3934.91	−0.130250
$971$	−38471.8	−1.27149	−0.635747	−	0.771898i	$-0.719308\pi$
$971$	−38471.8	−1.27149	−0.635747	+	0.771898i	$0.719308\pi$
$972$	0	0
$973$	0	0
$974$	29681.9	0.976456
$975$	0	0
$976$	−53153.8	−1.74325
$977$	53414.8	1.74912	0.874561	−	0.484916i	$-0.161150\pi$
$977$	53414.8	1.74912	0.874561	+	0.484916i	$0.161150\pi$
$978$	0	0
$979$	−3190.08	−0.104142
$980$	0	0
$981$	0	0
$982$	66366.1	2.15665
$983$	−13893.2	−0.450788	−0.225394	−	0.974268i	$-0.572367\pi$
$983$	−13893.2	−0.450788	−0.225394	+	0.974268i	$0.572367\pi$
$984$	0	0
$985$	4519.75	0.146204
$986$	5717.37	0.184663
$987$	0	0
$988$	−3247.92	−0.104585
$989$	−4819.37	−0.154951
$990$	0	0
$991$	10389.5	0.333031	0.166516	−	0.986039i	$-0.446748\pi$
$991$	10389.5	0.333031	0.166516	+	0.986039i	$0.446748\pi$
$992$	−6447.32	−0.206353
$993$	0	0
$994$	0	0
$995$	−9937.22	−0.316614
$996$	0	0
$997$	36396.7	1.15616	0.578082	−	0.815979i	$-0.303801\pi$
$997$	36396.7	1.15616	0.578082	+	0.815979i	$0.303801\pi$
$998$	61080.7	1.93735
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2205.4.a.bx.1.2	✓	6
3.2	odd	2		2205.4.a.by.1.5	yes	6
7.6	odd	2		2205.4.a.by.1.2	yes	6
21.20	even	2	inner	2205.4.a.bx.1.5	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2205.4.a.bx.1.2	✓	6	1.1	even	1	trivial
2205.4.a.bx.1.5	yes	6	21.20	even	2	inner
2205.4.a.by.1.2	yes	6	7.6	odd	2
2205.4.a.by.1.5	yes	6	3.2	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)

\(f(q)\)	\(=\)	\(q-3.09945 q^{2} +1.60662 q^{4} -5.00000 q^{5} +19.8160 q^{8} +O(q^{10})\)	\(q-3.09945 q^{2} +1.60662 q^{4} -5.00000 q^{5} +19.8160 q^{8} +15.4973 q^{10} +31.1878 q^{11} -54.1032 q^{13} -74.2717 q^{16} +81.8784 q^{17} +37.3653 q^{19} -8.03310 q^{20} -96.6651 q^{22} -116.264 q^{23} +25.0000 q^{25} +167.691 q^{26} -22.5290 q^{29} -89.9535 q^{31} +71.6739 q^{32} -253.778 q^{34} +344.782 q^{37} -115.812 q^{38} -99.0800 q^{40} -245.889 q^{41} +41.4519 q^{43} +50.1069 q^{44} +360.356 q^{46} -431.524 q^{47} -77.4864 q^{50} -86.9233 q^{52} +263.673 q^{53} -155.939 q^{55} +69.8276 q^{58} -627.005 q^{59} +715.666 q^{61} +278.807 q^{62} +372.024 q^{64} +270.516 q^{65} -809.809 q^{67} +131.547 q^{68} -54.8762 q^{71} +508.548 q^{73} -1068.64 q^{74} +60.0319 q^{76} +61.8168 q^{79} +371.359 q^{80} +762.122 q^{82} -560.154 q^{83} -409.392 q^{85} -128.478 q^{86} +618.017 q^{88} -102.286 q^{89} -186.792 q^{92} +1337.49 q^{94} -186.827 q^{95} -253.910 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 20 q^{4} - 30 q^{5}+O(q^{10}) \) 6 * q + 20 * q^4 - 30 * q^5	\( 6 q + 20 q^{4} - 30 q^{5} + 100 q^{16} - 44 q^{17} - 100 q^{20} - 24 q^{22} + 150 q^{25} - 168 q^{26} + 380 q^{37} - 56 q^{38} - 612 q^{41} - 328 q^{43} + 432 q^{46} - 120 q^{47} + 1120 q^{58} - 136 q^{59} - 2264 q^{62} - 1052 q^{64} + 1112 q^{67} - 264 q^{68} + 1400 q^{79} - 500 q^{80} - 2912 q^{83} + 220 q^{85} + 1384 q^{88} - 372 q^{89}+O(q^{100}) \) 6 * q + 20 * q^4 - 30 * q^5 + 100 * q^16 - 44 * q^17 - 100 * q^20 - 24 * q^22 + 150 * q^25 - 168 * q^26 + 380 * q^37 - 56 * q^38 - 612 * q^41 - 328 * q^43 + 432 * q^46 - 120 * q^47 + 1120 * q^58 - 136 * q^59 - 2264 * q^62 - 1052 * q^64 + 1112 * q^67 - 264 * q^68 + 1400 * q^79 - 500 * q^80 - 2912 * q^83 + 220 * q^85 + 1384 * q^88 - 372 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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