LMFDB - Embedded newform 6028.2.a.e.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6028,2,Mod(1,6028)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6028, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("6028.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6028 = 2^{2} \cdot 11 \cdot 137 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6028.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:	$48.1338223384$
Analytic rank:	$0$
Dimension:	$27$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Character	$\chi$	$=$	6028.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.12719 q^{3} -0.932078 q^{5} -0.358800 q^{7} +1.52492 q^{9} +O(q^{10})$	$q-2.12719 q^{3} -0.932078 q^{5} -0.358800 q^{7} +1.52492 q^{9} -1.00000 q^{11} -0.852915 q^{13} +1.98270 q^{15} -5.36656 q^{17} -6.45871 q^{19} +0.763234 q^{21} -4.52936 q^{23} -4.13123 q^{25} +3.13778 q^{27} -6.95063 q^{29} +3.01636 q^{31} +2.12719 q^{33} +0.334430 q^{35} +1.01302 q^{37} +1.81431 q^{39} +5.66565 q^{41} -6.95926 q^{43} -1.42134 q^{45} +5.16209 q^{47} -6.87126 q^{49} +11.4157 q^{51} +8.47098 q^{53} +0.932078 q^{55} +13.7389 q^{57} -7.89557 q^{59} -2.89200 q^{61} -0.547140 q^{63} +0.794983 q^{65} +3.50753 q^{67} +9.63480 q^{69} -5.50927 q^{71} -9.53318 q^{73} +8.78789 q^{75} +0.358800 q^{77} +8.14298 q^{79} -11.2494 q^{81} -3.87466 q^{83} +5.00205 q^{85} +14.7853 q^{87} -2.84809 q^{89} +0.306026 q^{91} -6.41635 q^{93} +6.02002 q^{95} -5.54535 q^{97} -1.52492 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 27 q + 5 q^{3} - 2 q^{5} + 7 q^{7} + 28 q^{9}+O(q^{10}) $ 27 * q + 5 * q^3 - 2 * q^5 + 7 * q^7 + 28 * q^9	$ 27 q + 5 q^{3} - 2 q^{5} + 7 q^{7} + 28 q^{9} - 27 q^{11} + 2 q^{13} + 10 q^{15} + 17 q^{17} + 4 q^{19} - 4 q^{21} + 25 q^{23} + 21 q^{25} + 32 q^{27} - q^{29} + 8 q^{31} - 5 q^{33} + 30 q^{35} - 24 q^{37} + 10 q^{39} + 31 q^{41} + 17 q^{43} - 15 q^{45} + 27 q^{47} + 40 q^{49} + 30 q^{51} + 2 q^{55} + 13 q^{57} + 31 q^{59} + 2 q^{61} + 61 q^{63} + 2 q^{67} - 15 q^{69} + 40 q^{71} + 17 q^{73} + 30 q^{75} - 7 q^{77} + 33 q^{79} + 31 q^{81} + 66 q^{83} + 26 q^{85} + 43 q^{87} + 22 q^{89} + 18 q^{91} - 9 q^{93} + 95 q^{95} - 21 q^{97} - 28 q^{99}+O(q^{100}) $ 27 * q + 5 * q^3 - 2 * q^5 + 7 * q^7 + 28 * q^9 - 27 * q^11 + 2 * q^13 + 10 * q^15 + 17 * q^17 + 4 * q^19 - 4 * q^21 + 25 * q^23 + 21 * q^25 + 32 * q^27 - q^29 + 8 * q^31 - 5 * q^33 + 30 * q^35 - 24 * q^37 + 10 * q^39 + 31 * q^41 + 17 * q^43 - 15 * q^45 + 27 * q^47 + 40 * q^49 + 30 * q^51 + 2 * q^55 + 13 * q^57 + 31 * q^59 + 2 * q^61 + 61 * q^63 + 2 * q^67 - 15 * q^69 + 40 * q^71 + 17 * q^73 + 30 * q^75 - 7 * q^77 + 33 * q^79 + 31 * q^81 + 66 * q^83 + 26 * q^85 + 43 * q^87 + 22 * q^89 + 18 * q^91 - 9 * q^93 + 95 * q^95 - 21 * q^97 - 28 * q^99

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	0
$2$	0	0
$3$	−2.12719	−1.22813	−0.614065	−	0.789255i	$-0.710467\pi$
$3$	−2.12719	−1.22813	−0.614065	+	0.789255i	$0.710467\pi$
$4$	0	0
$5$	−0.932078	−0.416838	−0.208419	−	0.978040i	$-0.566832\pi$
$5$	−0.932078	−0.416838	−0.208419	+	0.978040i	$0.566832\pi$
$6$	0	0
$7$	−0.358800	−0.135614	−0.0678068	−	0.997698i	$-0.521600\pi$
$7$	−0.358800	−0.135614	−0.0678068	+	0.997698i	$0.521600\pi$
$8$	0	0
$9$	1.52492	0.508305
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−0.852915	−0.236556	−0.118278	−	0.992981i	$-0.537737\pi$
$13$	−0.852915	−0.236556	−0.118278	+	0.992981i	$0.537737\pi$
$14$	0	0
$15$	1.98270	0.511932
$16$	0	0
$17$	−5.36656	−1.30158	−0.650791	−	0.759257i	$-0.725562\pi$
$17$	−5.36656	−1.30158	−0.650791	+	0.759257i	$0.725562\pi$
$18$	0	0
$19$	−6.45871	−1.48173	−0.740865	−	0.671654i	$-0.765584\pi$
$19$	−6.45871	−1.48173	−0.740865	+	0.671654i	$0.765584\pi$
$20$	0	0
$21$	0.763234	0.166551
$22$	0	0
$23$	−4.52936	−0.944438	−0.472219	−	0.881481i	$-0.656547\pi$
$23$	−4.52936	−0.944438	−0.472219	+	0.881481i	$0.656547\pi$
$24$	0	0
$25$	−4.13123	−0.826246
$26$	0	0
$27$	3.13778	0.603865
$28$	0	0
$29$	−6.95063	−1.29070	−0.645350	−	0.763887i	$-0.723288\pi$
$29$	−6.95063	−1.29070	−0.645350	+	0.763887i	$0.723288\pi$
$30$	0	0
$31$	3.01636	0.541753	0.270877	−	0.962614i	$-0.412686\pi$
$31$	3.01636	0.541753	0.270877	+	0.962614i	$0.412686\pi$
$32$	0	0
$33$	2.12719	0.370295
$34$	0	0
$35$	0.334430	0.0565289
$36$	0	0
$37$	1.01302	0.166539	0.0832697	−	0.996527i	$-0.473464\pi$
$37$	1.01302	0.166539	0.0832697	+	0.996527i	$0.473464\pi$
$38$	0	0
$39$	1.81431	0.290522
$40$	0	0
$41$	5.66565	0.884825	0.442413	−	0.896812i	$-0.354123\pi$
$41$	5.66565	0.884825	0.442413	+	0.896812i	$0.354123\pi$
$42$	0	0
$43$	−6.95926	−1.06128	−0.530639	−	0.847598i	$-0.678048\pi$
$43$	−6.95926	−1.06128	−0.530639	+	0.847598i	$0.678048\pi$
$44$	0	0
$45$	−1.42134	−0.211881
$46$	0	0
$47$	5.16209	0.752968	0.376484	−	0.926423i	$-0.377133\pi$
$47$	5.16209	0.752968	0.376484	+	0.926423i	$0.377133\pi$
$48$	0	0
$49$	−6.87126	−0.981609
$50$	0	0
$51$	11.4157	1.59851
$52$	0	0
$53$	8.47098	1.16358	0.581789	−	0.813339i	$-0.302353\pi$
$53$	8.47098	1.16358	0.581789	+	0.813339i	$0.302353\pi$
$54$	0	0
$55$	0.932078	0.125681
$56$	0	0
$57$	13.7389	1.81976
$58$	0	0
$59$	−7.89557	−1.02792	−0.513958	−	0.857816i	$-0.671821\pi$
$59$	−7.89557	−1.02792	−0.513958	+	0.857816i	$0.671821\pi$
$60$	0	0
$61$	−2.89200	−0.370283	−0.185142	−	0.982712i	$-0.559274\pi$
$61$	−2.89200	−0.370283	−0.185142	+	0.982712i	$0.559274\pi$
$62$	0	0
$63$	−0.547140	−0.0689332
$64$	0	0
$65$	0.794983	0.0986056
$66$	0	0
$67$	3.50753	0.428512	0.214256	−	0.976777i	$-0.431267\pi$
$67$	3.50753	0.428512	0.214256	+	0.976777i	$0.431267\pi$
$68$	0	0
$69$	9.63480	1.15989
$70$	0	0
$71$	−5.50927	−0.653830	−0.326915	−	0.945054i	$-0.606009\pi$
$71$	−5.50927	−0.653830	−0.326915	+	0.945054i	$0.606009\pi$
$72$	0	0
$73$	−9.53318	−1.11577	−0.557887	−	0.829917i	$-0.688388\pi$
$73$	−9.53318	−1.11577	−0.557887	+	0.829917i	$0.688388\pi$
$74$	0	0
$75$	8.78789	1.01474
$76$	0	0
$77$	0.358800	0.0408891
$78$	0	0
$79$	8.14298	0.916157	0.458078	−	0.888912i	$-0.348538\pi$
$79$	8.14298	0.916157	0.458078	+	0.888912i	$0.348538\pi$
$80$	0	0
$81$	−11.2494	−1.24993
$82$	0	0
$83$	−3.87466	−0.425300	−0.212650	−	0.977128i	$-0.568209\pi$
$83$	−3.87466	−0.425300	−0.212650	+	0.977128i	$0.568209\pi$
$84$	0	0
$85$	5.00205	0.542549
$86$	0	0
$87$	14.7853	1.58515
$88$	0	0
$89$	−2.84809	−0.301897	−0.150948	−	0.988542i	$-0.548233\pi$
$89$	−2.84809	−0.301897	−0.150948	+	0.988542i	$0.548233\pi$
$90$	0	0
$91$	0.306026	0.0320802
$92$	0	0
$93$	−6.41635	−0.665344
$94$	0	0
$95$	6.02002	0.617641
$96$	0	0
$97$	−5.54535	−0.563045	−0.281522	−	0.959555i	$-0.590839\pi$
$97$	−5.54535	−0.563045	−0.281522	+	0.959555i	$0.590839\pi$
$98$	0	0
$99$	−1.52492	−0.153260
$100$	0	0
$101$	−2.45200	−0.243983	−0.121991	−	0.992531i	$-0.538928\pi$
$101$	−2.45200	−0.243983	−0.121991	+	0.992531i	$0.538928\pi$
$102$	0	0
$103$	−12.4722	−1.22892	−0.614459	−	0.788949i	$-0.710626\pi$
$103$	−12.4722	−1.22892	−0.614459	+	0.788949i	$0.710626\pi$
$104$	0	0
$105$	−0.711394	−0.0694249
$106$	0	0
$107$	−15.8019	−1.52763	−0.763815	−	0.645436i	$-0.776676\pi$
$107$	−15.8019	−1.52763	−0.763815	+	0.645436i	$0.776676\pi$
$108$	0	0
$109$	0.175368	0.0167972	0.00839862	−	0.999965i	$-0.497327\pi$
$109$	0.175368	0.0167972	0.00839862	+	0.999965i	$0.497327\pi$
$110$	0	0
$111$	−2.15488	−0.204532
$112$	0	0
$113$	−13.8611	−1.30394	−0.651972	−	0.758243i	$-0.726058\pi$
$113$	−13.8611	−1.30394	−0.651972	+	0.758243i	$0.726058\pi$
$114$	0	0
$115$	4.22172	0.393678
$116$	0	0
$117$	−1.30062	−0.120243
$118$	0	0
$119$	1.92552	0.176512
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−12.0519	−1.08668
$124$	0	0
$125$	8.51102	0.761249
$126$	0	0
$127$	11.9784	1.06291	0.531454	−	0.847087i	$-0.321646\pi$
$127$	11.9784	1.06291	0.531454	+	0.847087i	$0.321646\pi$
$128$	0	0
$129$	14.8036	1.30339
$130$	0	0
$131$	5.43792	0.475114	0.237557	−	0.971374i	$-0.423653\pi$
$131$	5.43792	0.475114	0.237557	+	0.971374i	$0.423653\pi$
$132$	0	0
$133$	2.31738	0.200943
$134$	0	0
$135$	−2.92465	−0.251714
$136$	0	0
$137$	−1.00000	−0.0854358
$137$	−1.00000	−0.0854358
$138$	0	0
$139$	13.4297	1.13909	0.569547	−	0.821959i	$-0.307119\pi$
$139$	13.4297	1.13909	0.569547	+	0.821959i	$0.307119\pi$
$140$	0	0
$141$	−10.9807	−0.924744
$142$	0	0
$143$	0.852915	0.0713243
$144$	0	0
$145$	6.47853	0.538013
$146$	0	0
$147$	14.6164	1.20554
$148$	0	0
$149$	−3.17904	−0.260437	−0.130219	−	0.991485i	$-0.541568\pi$
$149$	−3.17904	−0.260437	−0.130219	+	0.991485i	$0.541568\pi$
$150$	0	0
$151$	−8.92000	−0.725900	−0.362950	−	0.931809i	$-0.618230\pi$
$151$	−8.92000	−0.725900	−0.362950	+	0.931809i	$0.618230\pi$
$152$	0	0
$153$	−8.18355	−0.661601
$154$	0	0
$155$	−2.81148	−0.225823
$156$	0	0
$157$	−3.43938	−0.274493	−0.137246	−	0.990537i	$-0.543825\pi$
$157$	−3.43938	−0.274493	−0.137246	+	0.990537i	$0.543825\pi$
$158$	0	0
$159$	−18.0193	−1.42903
$160$	0	0
$161$	1.62514	0.128079
$162$	0	0
$163$	−13.7915	−1.08023	−0.540117	−	0.841590i	$-0.681620\pi$
$163$	−13.7915	−1.08023	−0.540117	+	0.841590i	$0.681620\pi$
$164$	0	0
$165$	−1.98270	−0.154353
$166$	0	0
$167$	15.8055	1.22307	0.611534	−	0.791218i	$-0.290553\pi$
$167$	15.8055	1.22307	0.611534	+	0.791218i	$0.290553\pi$
$168$	0	0
$169$	−12.2725	−0.944041
$170$	0	0
$171$	−9.84899	−0.753171
$172$	0	0
$173$	−14.7915	−1.12458	−0.562289	−	0.826941i	$-0.690079\pi$
$173$	−14.7915	−1.12458	−0.562289	+	0.826941i	$0.690079\pi$
$174$	0	0
$175$	1.48229	0.112050
$176$	0	0
$177$	16.7953	1.26241
$178$	0	0
$179$	4.42781	0.330950	0.165475	−	0.986214i	$-0.447084\pi$
$179$	4.42781	0.330950	0.165475	+	0.986214i	$0.447084\pi$
$180$	0	0
$181$	−15.5049	−1.15247	−0.576234	−	0.817285i	$-0.695478\pi$
$181$	−15.5049	−1.15247	−0.576234	+	0.817285i	$0.695478\pi$
$182$	0	0
$183$	6.15183	0.454756
$184$	0	0
$185$	−0.944214	−0.0694200
$186$	0	0
$187$	5.36656	0.392441
$188$	0	0
$189$	−1.12583	−0.0818924
$190$	0	0
$191$	18.2697	1.32195	0.660976	−	0.750407i	$-0.270143\pi$
$191$	18.2697	1.32195	0.660976	+	0.750407i	$0.270143\pi$
$192$	0	0
$193$	16.5268	1.18963	0.594814	−	0.803863i	$-0.297226\pi$
$193$	16.5268	1.18963	0.594814	+	0.803863i	$0.297226\pi$
$194$	0	0
$195$	−1.69108	−0.121101
$196$	0	0
$197$	2.89246	0.206079	0.103040	−	0.994677i	$-0.467143\pi$
$197$	2.89246	0.206079	0.103040	+	0.994677i	$0.467143\pi$
$198$	0	0
$199$	9.04325	0.641059	0.320529	−	0.947239i	$-0.396139\pi$
$199$	9.04325	0.641059	0.320529	+	0.947239i	$0.396139\pi$
$200$	0	0
$201$	−7.46116	−0.526269
$202$	0	0
$203$	2.49389	0.175036
$204$	0	0
$205$	−5.28083	−0.368829
$206$	0	0
$207$	−6.90690	−0.480063
$208$	0	0
$209$	6.45871	0.446758
$210$	0	0
$211$	−9.68350	−0.666639	−0.333320	−	0.942814i	$-0.608169\pi$
$211$	−9.68350	−0.666639	−0.333320	+	0.942814i	$0.608169\pi$
$212$	0	0
$213$	11.7192	0.802989
$214$	0	0
$215$	6.48658	0.442381
$216$	0	0
$217$	−1.08227	−0.0734692
$218$	0	0
$219$	20.2788	1.37032
$220$	0	0
$221$	4.57722	0.307897
$222$	0	0
$223$	15.5451	1.04098	0.520488	−	0.853869i	$-0.325750\pi$
$223$	15.5451	1.04098	0.520488	+	0.853869i	$0.325750\pi$
$224$	0	0
$225$	−6.29978	−0.419985
$226$	0	0
$227$	28.7343	1.90716	0.953581	−	0.301138i	$-0.0973664\pi$
$227$	28.7343	1.90716	0.953581	+	0.301138i	$0.0973664\pi$
$228$	0	0
$229$	−3.88830	−0.256946	−0.128473	−	0.991713i	$-0.541008\pi$
$229$	−3.88830	−0.256946	−0.128473	+	0.991713i	$0.541008\pi$
$230$	0	0
$231$	−0.763234	−0.0502171
$232$	0	0
$233$	27.6661	1.81247	0.906233	−	0.422780i	$-0.138946\pi$
$233$	27.6661	1.81247	0.906233	+	0.422780i	$0.138946\pi$
$234$	0	0
$235$	−4.81147	−0.313866
$236$	0	0
$237$	−17.3216	−1.12516
$238$	0	0
$239$	21.7793	1.40879	0.704394	−	0.709809i	$-0.251219\pi$
$239$	21.7793	1.40879	0.704394	+	0.709809i	$0.251219\pi$
$240$	0	0
$241$	−13.2841	−0.855704	−0.427852	−	0.903849i	$-0.640730\pi$
$241$	−13.2841	−0.855704	−0.427852	+	0.903849i	$0.640730\pi$
$242$	0	0
$243$	14.5162	0.931214
$244$	0	0
$245$	6.40455	0.409172
$246$	0	0
$247$	5.50873	0.350512
$248$	0	0
$249$	8.24213	0.522323
$250$	0	0
$251$	−17.1839	−1.08464	−0.542320	−	0.840172i	$-0.682454\pi$
$251$	−17.1839	−1.08464	−0.542320	+	0.840172i	$0.682454\pi$
$252$	0	0
$253$	4.52936	0.284759
$254$	0	0
$255$	−10.6403	−0.666321
$256$	0	0
$257$	3.06430	0.191146	0.0955730	−	0.995422i	$-0.469532\pi$
$257$	3.06430	0.191146	0.0955730	+	0.995422i	$0.469532\pi$
$258$	0	0
$259$	−0.363472	−0.0225850
$260$	0	0
$261$	−10.5991	−0.656069
$262$	0	0
$263$	25.4738	1.57078	0.785392	−	0.618999i	$-0.212462\pi$
$263$	25.4738	1.57078	0.785392	+	0.618999i	$0.212462\pi$
$264$	0	0
$265$	−7.89562	−0.485024
$266$	0	0
$267$	6.05842	0.370769
$268$	0	0
$269$	9.30594	0.567393	0.283696	−	0.958914i	$-0.408439\pi$
$269$	9.30594	0.567393	0.283696	+	0.958914i	$0.408439\pi$
$270$	0	0
$271$	23.0791	1.40195	0.700977	−	0.713184i	$-0.252748\pi$
$271$	23.0791	1.40195	0.700977	+	0.713184i	$0.252748\pi$
$272$	0	0
$273$	−0.650974	−0.0393987
$274$	0	0
$275$	4.13123	0.249123
$276$	0	0
$277$	−24.7693	−1.48825	−0.744123	−	0.668043i	$-0.767132\pi$
$277$	−24.7693	−1.48825	−0.744123	+	0.668043i	$0.767132\pi$
$278$	0	0
$279$	4.59969	0.275376
$280$	0	0
$281$	−4.92988	−0.294092	−0.147046	−	0.989130i	$-0.546977\pi$
$281$	−4.92988	−0.294092	−0.147046	+	0.989130i	$0.546977\pi$
$282$	0	0
$283$	−20.3813	−1.21154	−0.605771	−	0.795639i	$-0.707135\pi$
$283$	−20.3813	−1.21154	−0.605771	+	0.795639i	$0.707135\pi$
$284$	0	0
$285$	−12.8057	−0.758544
$286$	0	0
$287$	−2.03283	−0.119994
$288$	0	0
$289$	11.7999	0.694114
$290$	0	0
$291$	11.7960	0.691492
$292$	0	0
$293$	7.65799	0.447385	0.223692	−	0.974660i	$-0.428189\pi$
$293$	7.65799	0.447385	0.223692	+	0.974660i	$0.428189\pi$
$294$	0	0
$295$	7.35929	0.428474
$296$	0	0
$297$	−3.13778	−0.182072
$298$	0	0
$299$	3.86316	0.223412
$300$	0	0
$301$	2.49698	0.143924
$302$	0	0
$303$	5.21585	0.299643
$304$	0	0
$305$	2.69557	0.154348
$306$	0	0
$307$	11.3599	0.648341	0.324171	−	0.945999i	$-0.394915\pi$
$307$	11.3599	0.648341	0.324171	+	0.945999i	$0.394915\pi$
$308$	0	0
$309$	26.5306	1.50927
$310$	0	0
$311$	−8.18249	−0.463986	−0.231993	−	0.972717i	$-0.574525\pi$
$311$	−8.18249	−0.463986	−0.231993	+	0.972717i	$0.574525\pi$
$312$	0	0
$313$	−9.92481	−0.560984	−0.280492	−	0.959856i	$-0.590498\pi$
$313$	−9.92481	−0.560984	−0.280492	+	0.959856i	$0.590498\pi$
$314$	0	0
$315$	0.509977	0.0287340
$316$	0	0
$317$	1.73117	0.0972323	0.0486161	−	0.998818i	$-0.484519\pi$
$317$	1.73117	0.0972323	0.0486161	+	0.998818i	$0.484519\pi$
$318$	0	0
$319$	6.95063	0.389160
$320$	0	0
$321$	33.6136	1.87613
$322$	0	0
$323$	34.6610	1.92859
$324$	0	0
$325$	3.52359	0.195453
$326$	0	0
$327$	−0.373041	−0.0206292
$328$	0	0
$329$	−1.85216	−0.102113
$330$	0	0
$331$	5.45031	0.299576	0.149788	−	0.988718i	$-0.452141\pi$
$331$	5.45031	0.299576	0.149788	+	0.988718i	$0.452141\pi$
$332$	0	0
$333$	1.54477	0.0846529
$334$	0	0
$335$	−3.26929	−0.178620
$336$	0	0
$337$	33.3801	1.81833	0.909164	−	0.416439i	$-0.136722\pi$
$337$	33.3801	1.81833	0.909164	+	0.416439i	$0.136722\pi$
$338$	0	0
$339$	29.4852	1.60141
$340$	0	0
$341$	−3.01636	−0.163345
$342$	0	0
$343$	4.97701	0.268733
$344$	0	0
$345$	−8.98038	−0.483488
$346$	0	0
$347$	−18.3509	−0.985128	−0.492564	−	0.870276i	$-0.663940\pi$
$347$	−18.3509	−0.985128	−0.492564	+	0.870276i	$0.663940\pi$
$348$	0	0
$349$	14.7484	0.789461	0.394731	−	0.918797i	$-0.370838\pi$
$349$	14.7484	0.789461	0.394731	+	0.918797i	$0.370838\pi$
$350$	0	0
$351$	−2.67626	−0.142848
$352$	0	0
$353$	−9.65266	−0.513759	−0.256880	−	0.966443i	$-0.582694\pi$
$353$	−9.65266	−0.513759	−0.256880	+	0.966443i	$0.582694\pi$
$354$	0	0
$355$	5.13507	0.272541
$356$	0	0
$357$	−4.09594	−0.216780
$358$	0	0
$359$	32.5281	1.71677	0.858383	−	0.513009i	$-0.171469\pi$
$359$	32.5281	1.71677	0.858383	+	0.513009i	$0.171469\pi$
$360$	0	0
$361$	22.7149	1.19552
$362$	0	0
$363$	−2.12719	−0.111648
$364$	0	0
$365$	8.88567	0.465097
$366$	0	0
$367$	12.4270	0.648685	0.324343	−	0.945940i	$-0.394857\pi$
$367$	12.4270	0.648685	0.324343	+	0.945940i	$0.394857\pi$
$368$	0	0
$369$	8.63964	0.449762
$370$	0	0
$371$	−3.03939	−0.157797
$372$	0	0
$373$	−0.879272	−0.0455270	−0.0227635	−	0.999741i	$-0.507246\pi$
$373$	−0.879272	−0.0455270	−0.0227635	+	0.999741i	$0.507246\pi$
$374$	0	0
$375$	−18.1045	−0.934913
$376$	0	0
$377$	5.92829	0.305323
$378$	0	0
$379$	−9.29945	−0.477681	−0.238840	−	0.971059i	$-0.576767\pi$
$379$	−9.29945	−0.477681	−0.238840	+	0.971059i	$0.576767\pi$
$380$	0	0
$381$	−25.4802	−1.30539
$382$	0	0
$383$	−5.42695	−0.277304	−0.138652	−	0.990341i	$-0.544277\pi$
$383$	−5.42695	−0.277304	−0.138652	+	0.990341i	$0.544277\pi$
$384$	0	0
$385$	−0.334430	−0.0170441
$386$	0	0
$387$	−10.6123	−0.539453
$388$	0	0
$389$	−19.2656	−0.976805	−0.488402	−	0.872619i	$-0.662420\pi$
$389$	−19.2656	−0.976805	−0.488402	+	0.872619i	$0.662420\pi$
$390$	0	0
$391$	24.3071	1.22926
$392$	0	0
$393$	−11.5675	−0.583502
$394$	0	0
$395$	−7.58989	−0.381889
$396$	0	0
$397$	−21.2544	−1.06673	−0.533363	−	0.845887i	$-0.679072\pi$
$397$	−21.2544	−1.06673	−0.533363	+	0.845887i	$0.679072\pi$
$398$	0	0
$399$	−4.92951	−0.246784
$400$	0	0
$401$	5.07938	0.253652	0.126826	−	0.991925i	$-0.459521\pi$
$401$	5.07938	0.253652	0.126826	+	0.991925i	$0.459521\pi$
$402$	0	0
$403$	−2.57269	−0.128155
$404$	0	0
$405$	10.4853	0.521019
$406$	0	0
$407$	−1.01302	−0.0502135
$408$	0	0
$409$	−30.3100	−1.49873	−0.749367	−	0.662155i	$-0.769642\pi$
$409$	−30.3100	−1.49873	−0.749367	+	0.662155i	$0.769642\pi$
$410$	0	0
$411$	2.12719	0.104926
$412$	0	0
$413$	2.83293	0.139399
$414$	0	0
$415$	3.61149	0.177281
$416$	0	0
$417$	−28.5675	−1.39896
$418$	0	0
$419$	22.7899	1.11336	0.556679	−	0.830728i	$-0.312075\pi$
$419$	22.7899	1.11336	0.556679	+	0.830728i	$0.312075\pi$
$420$	0	0
$421$	7.22333	0.352043	0.176022	−	0.984386i	$-0.443677\pi$
$421$	7.22333	0.352043	0.176022	+	0.984386i	$0.443677\pi$
$422$	0	0
$423$	7.87176	0.382738
$424$	0	0
$425$	22.1705	1.07543
$426$	0	0
$427$	1.03765	0.0502155
$428$	0	0
$429$	−1.81431	−0.0875956
$430$	0	0
$431$	−14.1928	−0.683644	−0.341822	−	0.939765i	$-0.611044\pi$
$431$	−14.1928	−0.683644	−0.341822	+	0.939765i	$0.611044\pi$
$432$	0	0
$433$	30.2983	1.45604	0.728021	−	0.685555i	$-0.240440\pi$
$433$	30.2983	1.45604	0.728021	+	0.685555i	$0.240440\pi$
$434$	0	0
$435$	−13.7810	−0.660750
$436$	0	0
$437$	29.2538	1.39940
$438$	0	0
$439$	−22.1385	−1.05661	−0.528305	−	0.849054i	$-0.677172\pi$
$439$	−22.1385	−1.05661	−0.528305	+	0.849054i	$0.677172\pi$
$440$	0	0
$441$	−10.4781	−0.498957
$442$	0	0
$443$	−35.7237	−1.69729	−0.848643	−	0.528966i	$-0.822580\pi$
$443$	−35.7237	−1.69729	−0.848643	+	0.528966i	$0.822580\pi$
$444$	0	0
$445$	2.65464	0.125842
$446$	0	0
$447$	6.76241	0.319851
$448$	0	0
$449$	−22.9282	−1.08205	−0.541024	−	0.841007i	$-0.681963\pi$
$449$	−22.9282	−1.08205	−0.541024	+	0.841007i	$0.681963\pi$
$450$	0	0
$451$	−5.66565	−0.266785
$452$	0	0
$453$	18.9745	0.891500
$454$	0	0
$455$	−0.285240	−0.0133723
$456$	0	0
$457$	−1.62817	−0.0761627	−0.0380814	−	0.999275i	$-0.512125\pi$
$457$	−1.62817	−0.0761627	−0.0380814	+	0.999275i	$0.512125\pi$
$458$	0	0
$459$	−16.8391	−0.785980
$460$	0	0
$461$	10.6177	0.494515	0.247257	−	0.968950i	$-0.420471\pi$
$461$	10.6177	0.494515	0.247257	+	0.968950i	$0.420471\pi$
$462$	0	0
$463$	−15.8062	−0.734578	−0.367289	−	0.930107i	$-0.619714\pi$
$463$	−15.8062	−0.734578	−0.367289	+	0.930107i	$0.619714\pi$
$464$	0	0
$465$	5.98054	0.277341
$466$	0	0
$467$	32.1798	1.48910	0.744552	−	0.667565i	$-0.232663\pi$
$467$	32.1798	1.48910	0.744552	+	0.667565i	$0.232663\pi$
$468$	0	0
$469$	−1.25850	−0.0581121
$470$	0	0
$471$	7.31620	0.337113
$472$	0	0
$473$	6.95926	0.319987
$474$	0	0
$475$	26.6824	1.22427
$476$	0	0
$477$	12.9175	0.591453
$478$	0	0
$479$	16.7864	0.766989	0.383494	−	0.923543i	$-0.374721\pi$
$479$	16.7864	0.766989	0.383494	+	0.923543i	$0.374721\pi$
$480$	0	0
$481$	−0.864020	−0.0393959
$482$	0	0
$483$	−3.45697	−0.157297
$484$	0	0
$485$	5.16870	0.234698
$486$	0	0
$487$	−1.01477	−0.0459836	−0.0229918	−	0.999736i	$-0.507319\pi$
$487$	−1.01477	−0.0459836	−0.0229918	+	0.999736i	$0.507319\pi$
$488$	0	0
$489$	29.3371	1.32667
$490$	0	0
$491$	0.523957	0.0236459	0.0118229	−	0.999930i	$-0.496237\pi$
$491$	0.523957	0.0236459	0.0118229	+	0.999930i	$0.496237\pi$
$492$	0	0
$493$	37.3009	1.67995
$494$	0	0
$495$	1.42134	0.0638845
$496$	0	0
$497$	1.97673	0.0886683
$498$	0	0
$499$	−6.38401	−0.285787	−0.142894	−	0.989738i	$-0.545641\pi$
$499$	−6.38401	−0.285787	−0.142894	+	0.989738i	$0.545641\pi$
$500$	0	0
$501$	−33.6213	−1.50209
$502$	0	0
$503$	−5.55050	−0.247485	−0.123742	−	0.992314i	$-0.539490\pi$
$503$	−5.55050	−0.247485	−0.123742	+	0.992314i	$0.539490\pi$
$504$	0	0
$505$	2.28545	0.101701
$506$	0	0
$507$	26.1060	1.15941
$508$	0	0
$509$	−18.5139	−0.820615	−0.410307	−	0.911947i	$-0.634579\pi$
$509$	−18.5139	−0.820615	−0.410307	+	0.911947i	$0.634579\pi$
$510$	0	0
$511$	3.42051	0.151314
$512$	0	0
$513$	−20.2660	−0.894765
$514$	0	0
$515$	11.6250	0.512260
$516$	0	0
$517$	−5.16209	−0.227028
$518$	0	0
$519$	31.4643	1.38113
$520$	0	0
$521$	−34.0167	−1.49030	−0.745149	−	0.666898i	$-0.767622\pi$
$521$	−34.0167	−1.49030	−0.745149	+	0.666898i	$0.767622\pi$
$522$	0	0
$523$	−44.5358	−1.94741	−0.973707	−	0.227803i	$-0.926846\pi$
$523$	−44.5358	−1.94741	−0.973707	+	0.227803i	$0.926846\pi$
$524$	0	0
$525$	−3.15310	−0.137612
$526$	0	0
$527$	−16.1874	−0.705136
$528$	0	0
$529$	−2.48486	−0.108037
$530$	0	0
$531$	−12.0401	−0.522495
$532$	0	0
$533$	−4.83231	−0.209311
$534$	0	0
$535$	14.7286	0.636774
$536$	0	0
$537$	−9.41877	−0.406450
$538$	0	0
$539$	6.87126	0.295966
$540$	0	0
$541$	25.7699	1.10793	0.553967	−	0.832539i	$-0.313113\pi$
$541$	25.7699	1.10793	0.553967	+	0.832539i	$0.313113\pi$
$542$	0	0
$543$	32.9817	1.41538
$544$	0	0
$545$	−0.163457	−0.00700173
$546$	0	0
$547$	14.2121	0.607667	0.303833	−	0.952725i	$-0.401733\pi$
$547$	14.2121	0.607667	0.303833	+	0.952725i	$0.401733\pi$
$548$	0	0
$549$	−4.41006	−0.188217
$550$	0	0
$551$	44.8921	1.91247
$552$	0	0
$553$	−2.92170	−0.124243
$554$	0	0
$555$	2.00852	0.0852568
$556$	0	0
$557$	−13.3225	−0.564492	−0.282246	−	0.959342i	$-0.591079\pi$
$557$	−13.3225	−0.564492	−0.282246	+	0.959342i	$0.591079\pi$
$558$	0	0
$559$	5.93566	0.251052
$560$	0	0
$561$	−11.4157	−0.481970
$562$	0	0
$563$	24.1908	1.01952	0.509761	−	0.860316i	$-0.329734\pi$
$563$	24.1908	1.01952	0.509761	+	0.860316i	$0.329734\pi$
$564$	0	0
$565$	12.9196	0.543533
$566$	0	0
$567$	4.03628	0.169508
$568$	0	0
$569$	−1.36314	−0.0571457	−0.0285729	−	0.999592i	$-0.509096\pi$
$569$	−1.36314	−0.0571457	−0.0285729	+	0.999592i	$0.509096\pi$
$570$	0	0
$571$	−5.56335	−0.232819	−0.116409	−	0.993201i	$-0.537138\pi$
$571$	−5.56335	−0.232819	−0.116409	+	0.993201i	$0.537138\pi$
$572$	0	0
$573$	−38.8631	−1.62353
$574$	0	0
$575$	18.7118	0.780338
$576$	0	0
$577$	24.6052	1.02433	0.512163	−	0.858888i	$-0.328844\pi$
$577$	24.6052	1.02433	0.512163	+	0.858888i	$0.328844\pi$
$578$	0	0
$579$	−35.1556	−1.46102
$580$	0	0
$581$	1.39023	0.0576764
$582$	0	0
$583$	−8.47098	−0.350832
$584$	0	0
$585$	1.21228	0.0501217
$586$	0	0
$587$	24.1152	0.995342	0.497671	−	0.867366i	$-0.334189\pi$
$587$	24.1152	0.995342	0.497671	+	0.867366i	$0.334189\pi$
$588$	0	0
$589$	−19.4818	−0.802732
$590$	0	0
$591$	−6.15281	−0.253093
$592$	0	0
$593$	−21.6784	−0.890227	−0.445114	−	0.895474i	$-0.646837\pi$
$593$	−21.6784	−0.890227	−0.445114	+	0.895474i	$0.646837\pi$
$594$	0	0
$595$	−1.79474	−0.0735770
$596$	0	0
$597$	−19.2367	−0.787304
$598$	0	0
$599$	−7.85477	−0.320937	−0.160469	−	0.987041i	$-0.551301\pi$
$599$	−7.85477	−0.320937	−0.160469	+	0.987041i	$0.551301\pi$
$600$	0	0
$601$	−2.66118	−0.108552	−0.0542760	−	0.998526i	$-0.517285\pi$
$601$	−2.66118	−0.108552	−0.0542760	+	0.998526i	$0.517285\pi$
$602$	0	0
$603$	5.34868	0.217815
$604$	0	0
$605$	−0.932078	−0.0378944
$606$	0	0
$607$	43.4265	1.76263	0.881314	−	0.472531i	$-0.156659\pi$
$607$	43.4265	1.76263	0.881314	+	0.472531i	$0.156659\pi$
$608$	0	0
$609$	−5.30496	−0.214968
$610$	0	0
$611$	−4.40282	−0.178119
$612$	0	0
$613$	24.7224	0.998529	0.499265	−	0.866450i	$-0.333604\pi$
$613$	24.7224	0.998529	0.499265	+	0.866450i	$0.333604\pi$
$614$	0	0
$615$	11.2333	0.452970
$616$	0	0
$617$	12.4407	0.500846	0.250423	−	0.968137i	$-0.419430\pi$
$617$	12.4407	0.500846	0.250423	+	0.968137i	$0.419430\pi$
$618$	0	0
$619$	−22.7837	−0.915753	−0.457876	−	0.889016i	$-0.651390\pi$
$619$	−22.7837	−0.915753	−0.457876	+	0.889016i	$0.651390\pi$
$620$	0	0
$621$	−14.2121	−0.570313
$622$	0	0
$623$	1.02189	0.0409414
$624$	0	0
$625$	12.7232	0.508929
$626$	0	0
$627$	−13.7389	−0.548678
$628$	0	0
$629$	−5.43643	−0.216765
$630$	0	0
$631$	−27.2271	−1.08389	−0.541947	−	0.840413i	$-0.682313\pi$
$631$	−27.2271	−1.08389	−0.541947	+	0.840413i	$0.682313\pi$
$632$	0	0
$633$	20.5986	0.818720
$634$	0	0
$635$	−11.1648	−0.443061
$636$	0	0
$637$	5.86060	0.232206
$638$	0	0
$639$	−8.40118	−0.332346
$640$	0	0
$641$	−42.2435	−1.66852	−0.834259	−	0.551373i	$-0.814104\pi$
$641$	−42.2435	−1.66852	−0.834259	+	0.551373i	$0.814104\pi$
$642$	0	0
$643$	22.0214	0.868439	0.434220	−	0.900807i	$-0.357024\pi$
$643$	22.0214	0.868439	0.434220	+	0.900807i	$0.357024\pi$
$644$	0	0
$645$	−13.7982	−0.543302
$646$	0	0
$647$	14.0410	0.552008	0.276004	−	0.961156i	$-0.410990\pi$
$647$	14.0410	0.552008	0.276004	+	0.961156i	$0.410990\pi$
$648$	0	0
$649$	7.89557	0.309928
$650$	0	0
$651$	2.30219	0.0902297
$652$	0	0
$653$	18.1496	0.710249	0.355124	−	0.934819i	$-0.384438\pi$
$653$	18.1496	0.710249	0.355124	+	0.934819i	$0.384438\pi$
$654$	0	0
$655$	−5.06857	−0.198045
$656$	0	0
$657$	−14.5373	−0.567154
$658$	0	0
$659$	−6.98762	−0.272199	−0.136099	−	0.990695i	$-0.543457\pi$
$659$	−6.98762	−0.272199	−0.136099	+	0.990695i	$0.543457\pi$
$660$	0	0
$661$	34.8284	1.35467	0.677334	−	0.735676i	$-0.263135\pi$
$661$	34.8284	1.35467	0.677334	+	0.735676i	$0.263135\pi$
$662$	0	0
$663$	−9.73659	−0.378138
$664$	0	0
$665$	−2.15998	−0.0837606
$666$	0	0
$667$	31.4819	1.21898
$668$	0	0
$669$	−33.0673	−1.27846
$670$	0	0
$671$	2.89200	0.111645
$672$	0	0
$673$	−4.58500	−0.176739	−0.0883694	−	0.996088i	$-0.528166\pi$
$673$	−4.58500	−0.176739	−0.0883694	+	0.996088i	$0.528166\pi$
$674$	0	0
$675$	−12.9629	−0.498941
$676$	0	0
$677$	8.10797	0.311615	0.155807	−	0.987787i	$-0.450202\pi$
$677$	8.10797	0.311615	0.155807	+	0.987787i	$0.450202\pi$
$678$	0	0
$679$	1.98967	0.0763565
$680$	0	0
$681$	−61.1231	−2.34224
$682$	0	0
$683$	43.4938	1.66425	0.832123	−	0.554592i	$-0.187125\pi$
$683$	43.4938	1.66425	0.832123	+	0.554592i	$0.187125\pi$
$684$	0	0
$685$	0.932078	0.0356129
$686$	0	0
$687$	8.27114	0.315564
$688$	0	0
$689$	−7.22503	−0.275252
$690$	0	0
$691$	−12.3660	−0.470423	−0.235212	−	0.971944i	$-0.575578\pi$
$691$	−12.3660	−0.470423	−0.235212	+	0.971944i	$0.575578\pi$
$692$	0	0
$693$	0.547140	0.0207841
$694$	0	0
$695$	−12.5175	−0.474817
$696$	0	0
$697$	−30.4050	−1.15167
$698$	0	0
$699$	−58.8509	−2.22594
$700$	0	0
$701$	3.05381	0.115341	0.0576705	−	0.998336i	$-0.481633\pi$
$701$	3.05381	0.115341	0.0576705	+	0.998336i	$0.481633\pi$
$702$	0	0
$703$	−6.54280	−0.246766
$704$	0	0
$705$	10.2349	0.385468
$706$	0	0
$707$	0.879776	0.0330874
$708$	0	0
$709$	−37.7743	−1.41864	−0.709322	−	0.704884i	$-0.750999\pi$
$709$	−37.7743	−1.41864	−0.709322	+	0.704884i	$0.750999\pi$
$710$	0	0
$711$	12.4174	0.465687
$712$	0	0
$713$	−13.6622	−0.511652
$714$	0	0
$715$	−0.794983	−0.0297307
$716$	0	0
$717$	−46.3287	−1.73018
$718$	0	0
$719$	21.3004	0.794370	0.397185	−	0.917739i	$-0.369987\pi$
$719$	21.3004	0.794370	0.397185	+	0.917739i	$0.369987\pi$
$720$	0	0
$721$	4.47501	0.166658
$722$	0	0
$723$	28.2577	1.05092
$724$	0	0
$725$	28.7146	1.06644
$726$	0	0
$727$	36.8130	1.36532	0.682660	−	0.730736i	$-0.260823\pi$
$727$	36.8130	1.36532	0.682660	+	0.730736i	$0.260823\pi$
$728$	0	0
$729$	2.86953	0.106279
$730$	0	0
$731$	37.3473	1.38134
$732$	0	0
$733$	36.2764	1.33990	0.669949	−	0.742407i	$-0.266316\pi$
$733$	36.2764	1.33990	0.669949	+	0.742407i	$0.266316\pi$
$734$	0	0
$735$	−13.6237	−0.502517
$736$	0	0
$737$	−3.50753	−0.129201
$738$	0	0
$739$	6.27302	0.230757	0.115378	−	0.993322i	$-0.463192\pi$
$739$	6.27302	0.230757	0.115378	+	0.993322i	$0.463192\pi$
$740$	0	0
$741$	−11.7181	−0.430475
$742$	0	0
$743$	−12.6283	−0.463286	−0.231643	−	0.972801i	$-0.574410\pi$
$743$	−12.6283	−0.463286	−0.231643	+	0.972801i	$0.574410\pi$
$744$	0	0
$745$	2.96312	0.108560
$746$	0	0
$747$	−5.90854	−0.216182
$748$	0	0
$749$	5.66973	0.207167
$750$	0	0
$751$	−21.7427	−0.793403	−0.396702	−	0.917948i	$-0.629845\pi$
$751$	−21.7427	−0.793403	−0.396702	+	0.917948i	$0.629845\pi$
$752$	0	0
$753$	36.5534	1.33208
$754$	0	0
$755$	8.31414	0.302583
$756$	0	0
$757$	−28.1904	−1.02460	−0.512298	−	0.858808i	$-0.671206\pi$
$757$	−28.1904	−1.02460	−0.512298	+	0.858808i	$0.671206\pi$
$758$	0	0
$759$	−9.63480	−0.349721
$760$	0	0
$761$	7.90152	0.286430	0.143215	−	0.989692i	$-0.454256\pi$
$761$	7.90152	0.286430	0.143215	+	0.989692i	$0.454256\pi$
$762$	0	0
$763$	−0.0629222	−0.00227794
$764$	0	0
$765$	7.62771	0.275780
$766$	0	0
$767$	6.73425	0.243160
$768$	0	0
$769$	−1.13424	−0.0409018	−0.0204509	−	0.999791i	$-0.506510\pi$
$769$	−1.13424	−0.0409018	−0.0204509	+	0.999791i	$0.506510\pi$
$770$	0	0
$771$	−6.51834	−0.234752
$772$	0	0
$773$	1.07501	0.0386655	0.0193328	−	0.999813i	$-0.493846\pi$
$773$	1.07501	0.0386655	0.0193328	+	0.999813i	$0.493846\pi$
$774$	0	0
$775$	−12.4613	−0.447622
$776$	0	0
$777$	0.773171	0.0277374
$778$	0	0
$779$	−36.5928	−1.31107
$780$	0	0
$781$	5.50927	0.197137
$782$	0	0
$783$	−21.8095	−0.779408
$784$	0	0
$785$	3.20577	0.114419
$786$	0	0
$787$	−33.0214	−1.17709	−0.588543	−	0.808466i	$-0.700298\pi$
$787$	−33.0214	−1.17709	−0.588543	+	0.808466i	$0.700298\pi$
$788$	0	0
$789$	−54.1875	−1.92913
$790$	0	0
$791$	4.97337	0.176833
$792$	0	0
$793$	2.46663	0.0875927
$794$	0	0
$795$	16.7954	0.595673
$796$	0	0
$797$	15.9343	0.564423	0.282211	−	0.959352i	$-0.408932\pi$
$797$	15.9343	0.564423	0.282211	+	0.959352i	$0.408932\pi$
$798$	0	0
$799$	−27.7027	−0.980049
$800$	0	0
$801$	−4.34310	−0.153456
$802$	0	0
$803$	9.53318	0.336419
$804$	0	0
$805$	−1.51475	−0.0533881
$806$	0	0
$807$	−19.7954	−0.696833
$808$	0	0
$809$	−29.6998	−1.04419	−0.522095	−	0.852888i	$-0.674849\pi$
$809$	−29.6998	−1.04419	−0.522095	+	0.852888i	$0.674849\pi$
$810$	0	0
$811$	10.6473	0.373878	0.186939	−	0.982372i	$-0.440143\pi$
$811$	10.6473	0.373878	0.186939	+	0.982372i	$0.440143\pi$
$812$	0	0
$813$	−49.0934	−1.72178
$814$	0	0
$815$	12.8548	0.450283
$816$	0	0
$817$	44.9479	1.57253
$818$	0	0
$819$	0.466664	0.0163066
$820$	0	0
$821$	−30.6111	−1.06834	−0.534168	−	0.845378i	$-0.679375\pi$
$821$	−30.6111	−1.06834	−0.534168	+	0.845378i	$0.679375\pi$
$822$	0	0
$823$	−22.1016	−0.770414	−0.385207	−	0.922830i	$-0.625870\pi$
$823$	−22.1016	−0.770414	−0.385207	+	0.922830i	$0.625870\pi$
$824$	0	0
$825$	−8.78789	−0.305955
$826$	0	0
$827$	−38.9876	−1.35573	−0.677866	−	0.735186i	$-0.737095\pi$
$827$	−38.9876	−1.35573	−0.677866	+	0.735186i	$0.737095\pi$
$828$	0	0
$829$	−20.8778	−0.725114	−0.362557	−	0.931962i	$-0.618096\pi$
$829$	−20.8778	−0.725114	−0.362557	+	0.931962i	$0.618096\pi$
$830$	0	0
$831$	52.6890	1.82776
$832$	0	0
$833$	36.8750	1.27764
$834$	0	0
$835$	−14.7320	−0.509821
$836$	0	0
$837$	9.46465	0.327146
$838$	0	0
$839$	23.9778	0.827807	0.413904	−	0.910321i	$-0.364165\pi$
$839$	23.9778	0.827807	0.413904	+	0.910321i	$0.364165\pi$
$840$	0	0
$841$	19.3112	0.665904
$842$	0	0
$843$	10.4868	0.361183
$844$	0	0
$845$	11.4390	0.393512
$846$	0	0
$847$	−0.358800	−0.0123285
$848$	0	0
$849$	43.3548	1.48793
$850$	0	0
$851$	−4.58834	−0.157286
$852$	0	0
$853$	−2.41363	−0.0826411	−0.0413205	−	0.999146i	$-0.513156\pi$
$853$	−2.41363	−0.0826411	−0.0413205	+	0.999146i	$0.513156\pi$
$854$	0	0
$855$	9.18003	0.313950
$856$	0	0
$857$	19.7412	0.674348	0.337174	−	0.941442i	$-0.390529\pi$
$857$	19.7412	0.674348	0.337174	+	0.941442i	$0.390529\pi$
$858$	0	0
$859$	−21.7483	−0.742042	−0.371021	−	0.928624i	$-0.620992\pi$
$859$	−21.7483	−0.742042	−0.371021	+	0.928624i	$0.620992\pi$
$860$	0	0
$861$	4.32421	0.147369
$862$	0	0
$863$	42.8970	1.46023	0.730115	−	0.683324i	$-0.239466\pi$
$863$	42.8970	1.46023	0.730115	+	0.683324i	$0.239466\pi$
$864$	0	0
$865$	13.7868	0.468767
$866$	0	0
$867$	−25.1006	−0.852462
$868$	0	0
$869$	−8.14298	−0.276232
$870$	0	0
$871$	−2.99162	−0.101367
$872$	0	0
$873$	−8.45619	−0.286199
$874$	0	0
$875$	−3.05375	−0.103236
$876$	0	0
$877$	4.70615	0.158915	0.0794577	−	0.996838i	$-0.474681\pi$
$877$	4.70615	0.158915	0.0794577	+	0.996838i	$0.474681\pi$
$878$	0	0
$879$	−16.2900	−0.549447
$880$	0	0
$881$	35.6727	1.20184	0.600922	−	0.799308i	$-0.294800\pi$
$881$	35.6727	1.20184	0.600922	+	0.799308i	$0.294800\pi$
$882$	0	0
$883$	2.54967	0.0858031	0.0429016	−	0.999079i	$-0.486340\pi$
$883$	2.54967	0.0858031	0.0429016	+	0.999079i	$0.486340\pi$
$884$	0	0
$885$	−15.6546	−0.526222
$886$	0	0
$887$	23.5468	0.790625	0.395313	−	0.918547i	$-0.370636\pi$
$887$	23.5468	0.790625	0.395313	+	0.918547i	$0.370636\pi$
$888$	0	0
$889$	−4.29784	−0.144145
$890$	0	0
$891$	11.2494	0.376868
$892$	0	0
$893$	−33.3404	−1.11570
$894$	0	0
$895$	−4.12706	−0.137953
$896$	0	0
$897$	−8.21766	−0.274380
$898$	0	0
$899$	−20.9656	−0.699241
$900$	0	0
$901$	−45.4600	−1.51449
$902$	0	0
$903$	−5.31155	−0.176757
$904$	0	0
$905$	14.4517	0.480392
$906$	0	0
$907$	−12.1790	−0.404395	−0.202198	−	0.979345i	$-0.564808\pi$
$907$	−12.1790	−0.404395	−0.202198	+	0.979345i	$0.564808\pi$
$908$	0	0
$909$	−3.73909	−0.124018
$910$	0	0
$911$	−23.4313	−0.776315	−0.388158	−	0.921593i	$-0.626888\pi$
$911$	−23.4313	−0.776315	−0.388158	+	0.921593i	$0.626888\pi$
$912$	0	0
$913$	3.87466	0.128233
$914$	0	0
$915$	−5.73399	−0.189560
$916$	0	0
$917$	−1.95113	−0.0644319
$918$	0	0
$919$	−48.4994	−1.59985	−0.799923	−	0.600102i	$-0.795126\pi$
$919$	−48.4994	−1.59985	−0.799923	+	0.600102i	$0.795126\pi$
$920$	0	0
$921$	−24.1645	−0.796248
$922$	0	0
$923$	4.69894	0.154668
$924$	0	0
$925$	−4.18502	−0.137603
$926$	0	0
$927$	−19.0190	−0.624666
$928$	0	0
$929$	−29.4699	−0.966875	−0.483438	−	0.875379i	$-0.660612\pi$
$929$	−29.4699	−0.966875	−0.483438	+	0.875379i	$0.660612\pi$
$930$	0	0
$931$	44.3795	1.45448
$932$	0	0
$933$	17.4057	0.569836
$934$	0	0
$935$	−5.00205	−0.163585
$936$	0	0
$937$	36.6347	1.19680	0.598402	−	0.801196i	$-0.295803\pi$
$937$	36.6347	1.19680	0.598402	+	0.801196i	$0.295803\pi$
$938$	0	0
$939$	21.1119	0.688961
$940$	0	0
$941$	−15.9663	−0.520485	−0.260243	−	0.965543i	$-0.583803\pi$
$941$	−15.9663	−0.520485	−0.260243	+	0.965543i	$0.583803\pi$
$942$	0	0
$943$	−25.6618	−0.835662
$944$	0	0
$945$	1.04937	0.0341359
$946$	0	0
$947$	54.9411	1.78535	0.892673	−	0.450706i	$-0.148828\pi$
$947$	54.9411	1.78535	0.892673	+	0.450706i	$0.148828\pi$
$948$	0	0
$949$	8.13099	0.263943
$950$	0	0
$951$	−3.68252	−0.119414
$952$	0	0
$953$	18.2471	0.591083	0.295542	−	0.955330i	$-0.404500\pi$
$953$	18.2471	0.591083	0.295542	+	0.955330i	$0.404500\pi$
$954$	0	0
$955$	−17.0288	−0.551040
$956$	0	0
$957$	−14.7853	−0.477940
$958$	0	0
$959$	0.358800	0.0115863
$960$	0	0
$961$	−21.9016	−0.706503
$962$	0	0
$963$	−24.0966	−0.776502
$964$	0	0
$965$	−15.4043	−0.495882
$966$	0	0
$967$	−48.2314	−1.55102	−0.775508	−	0.631338i	$-0.782506\pi$
$967$	−48.2314	−1.55102	−0.775508	+	0.631338i	$0.782506\pi$
$968$	0	0
$969$	−73.7304	−2.36856
$970$	0	0
$971$	−44.8843	−1.44041	−0.720204	−	0.693763i	$-0.755952\pi$
$971$	−44.8843	−1.44041	−0.720204	+	0.693763i	$0.755952\pi$
$972$	0	0
$973$	−4.81858	−0.154477
$974$	0	0
$975$	−7.49532	−0.240042
$976$	0	0
$977$	−20.2533	−0.647960	−0.323980	−	0.946064i	$-0.605021\pi$
$977$	−20.2533	−0.647960	−0.323980	+	0.946064i	$0.605021\pi$
$978$	0	0
$979$	2.84809	0.0910254
$980$	0	0
$981$	0.267422	0.00853813
$982$	0	0
$983$	29.6621	0.946075	0.473038	−	0.881042i	$-0.343158\pi$
$983$	29.6621	0.946075	0.473038	+	0.881042i	$0.343158\pi$
$984$	0	0
$985$	−2.69600	−0.0859018
$986$	0	0
$987$	3.93988	0.125408
$988$	0	0
$989$	31.5210	1.00231
$990$	0	0
$991$	24.6924	0.784381	0.392190	−	0.919884i	$-0.371717\pi$
$991$	24.6924	0.784381	0.392190	+	0.919884i	$0.371717\pi$
$992$	0	0
$993$	−11.5938	−0.367919
$994$	0	0
$995$	−8.42901	−0.267218
$996$	0	0
$997$	−38.0388	−1.20470	−0.602350	−	0.798232i	$-0.705769\pi$
$997$	−38.0388	−1.20470	−0.602350	+	0.798232i	$0.705769\pi$
$998$	0	0
$999$	3.17863	0.100567

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6028.2.a.e.1.5	✓	27

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6028.2.a.e.1.5	✓	27	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:	\( N \)	\(=\)	\( 6028 = 2^{2} \cdot 11 \cdot 137 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6028.a (trivial)

\(f(q)\)	\(=\)	\(q-2.12719 q^{3} -0.932078 q^{5} -0.358800 q^{7} +1.52492 q^{9} +O(q^{10})\)	\(q-2.12719 q^{3} -0.932078 q^{5} -0.358800 q^{7} +1.52492 q^{9} -1.00000 q^{11} -0.852915 q^{13} +1.98270 q^{15} -5.36656 q^{17} -6.45871 q^{19} +0.763234 q^{21} -4.52936 q^{23} -4.13123 q^{25} +3.13778 q^{27} -6.95063 q^{29} +3.01636 q^{31} +2.12719 q^{33} +0.334430 q^{35} +1.01302 q^{37} +1.81431 q^{39} +5.66565 q^{41} -6.95926 q^{43} -1.42134 q^{45} +5.16209 q^{47} -6.87126 q^{49} +11.4157 q^{51} +8.47098 q^{53} +0.932078 q^{55} +13.7389 q^{57} -7.89557 q^{59} -2.89200 q^{61} -0.547140 q^{63} +0.794983 q^{65} +3.50753 q^{67} +9.63480 q^{69} -5.50927 q^{71} -9.53318 q^{73} +8.78789 q^{75} +0.358800 q^{77} +8.14298 q^{79} -11.2494 q^{81} -3.87466 q^{83} +5.00205 q^{85} +14.7853 q^{87} -2.84809 q^{89} +0.306026 q^{91} -6.41635 q^{93} +6.02002 q^{95} -5.54535 q^{97} -1.52492 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 27 q + 5 q^{3} - 2 q^{5} + 7 q^{7} + 28 q^{9}+O(q^{10}) \) 27 * q + 5 * q^3 - 2 * q^5 + 7 * q^7 + 28 * q^9	\( 27 q + 5 q^{3} - 2 q^{5} + 7 q^{7} + 28 q^{9} - 27 q^{11} + 2 q^{13} + 10 q^{15} + 17 q^{17} + 4 q^{19} - 4 q^{21} + 25 q^{23} + 21 q^{25} + 32 q^{27} - q^{29} + 8 q^{31} - 5 q^{33} + 30 q^{35} - 24 q^{37} + 10 q^{39} + 31 q^{41} + 17 q^{43} - 15 q^{45} + 27 q^{47} + 40 q^{49} + 30 q^{51} + 2 q^{55} + 13 q^{57} + 31 q^{59} + 2 q^{61} + 61 q^{63} + 2 q^{67} - 15 q^{69} + 40 q^{71} + 17 q^{73} + 30 q^{75} - 7 q^{77} + 33 q^{79} + 31 q^{81} + 66 q^{83} + 26 q^{85} + 43 q^{87} + 22 q^{89} + 18 q^{91} - 9 q^{93} + 95 q^{95} - 21 q^{97} - 28 q^{99}+O(q^{100}) \) 27 * q + 5 * q^3 - 2 * q^5 + 7 * q^7 + 28 * q^9 - 27 * q^11 + 2 * q^13 + 10 * q^15 + 17 * q^17 + 4 * q^19 - 4 * q^21 + 25 * q^23 + 21 * q^25 + 32 * q^27 - q^29 + 8 * q^31 - 5 * q^33 + 30 * q^35 - 24 * q^37 + 10 * q^39 + 31 * q^41 + 17 * q^43 - 15 * q^45 + 27 * q^47 + 40 * q^49 + 30 * q^51 + 2 * q^55 + 13 * q^57 + 31 * q^59 + 2 * q^61 + 61 * q^63 + 2 * q^67 - 15 * q^69 + 40 * q^71 + 17 * q^73 + 30 * q^75 - 7 * q^77 + 33 * q^79 + 31 * q^81 + 66 * q^83 + 26 * q^85 + 43 * q^87 + 22 * q^89 + 18 * q^91 - 9 * q^93 + 95 * q^95 - 21 * q^97 - 28 * q^99

Introduction

L-functions

Modular forms

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Database

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