LMFDB - Embedded newform 6028.2.a.c.1.10

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:	$1$
Dimension:	$25$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
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)

$f(q)$	$=$	$q-1.44202 q^{3} +3.24879 q^{5} -4.91695 q^{7} -0.920583 q^{9} +O(q^{10})$	$q-1.44202 q^{3} +3.24879 q^{5} -4.91695 q^{7} -0.920583 q^{9} +1.00000 q^{11} -3.35934 q^{13} -4.68482 q^{15} -2.78284 q^{17} +6.83249 q^{19} +7.09033 q^{21} -2.33086 q^{23} +5.55467 q^{25} +5.65355 q^{27} +6.81200 q^{29} -0.976993 q^{31} -1.44202 q^{33} -15.9742 q^{35} +3.16459 q^{37} +4.84422 q^{39} +3.93968 q^{41} +4.92668 q^{43} -2.99078 q^{45} +3.13226 q^{47} +17.1764 q^{49} +4.01291 q^{51} -4.65984 q^{53} +3.24879 q^{55} -9.85258 q^{57} -3.54476 q^{59} +13.1079 q^{61} +4.52646 q^{63} -10.9138 q^{65} -9.75423 q^{67} +3.36115 q^{69} -13.8417 q^{71} +7.37784 q^{73} -8.00993 q^{75} -4.91695 q^{77} -13.3747 q^{79} -5.39078 q^{81} -10.1170 q^{83} -9.04088 q^{85} -9.82304 q^{87} +12.0143 q^{89} +16.5177 q^{91} +1.40884 q^{93} +22.1974 q^{95} -13.5932 q^{97} -0.920583 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 25 q - 11 q^{3} - 2 q^{5} - 9 q^{7} + 18 q^{9}+O(q^{10}) $ 25 * q - 11 * q^3 - 2 * q^5 - 9 * q^7 + 18 * q^9	$ 25 q - 11 q^{3} - 2 q^{5} - 9 q^{7} + 18 q^{9} + 25 q^{11} - 4 q^{13} - 10 q^{15} - 19 q^{17} - 12 q^{19} + 8 q^{21} - 31 q^{23} - q^{25} - 44 q^{27} - q^{29} - 8 q^{31} - 11 q^{33} - 16 q^{35} - 14 q^{37} - 18 q^{39} - 5 q^{41} - 15 q^{43} - 15 q^{45} - 41 q^{47} + 2 q^{49} + 10 q^{51} + 4 q^{53} - 2 q^{55} - 3 q^{57} - 35 q^{59} - 4 q^{61} - 45 q^{63} - 28 q^{65} - 30 q^{67} - 3 q^{69} + 4 q^{71} - 7 q^{73} - 18 q^{75} - 9 q^{77} - 9 q^{79} + 29 q^{81} - 72 q^{83} - 33 q^{87} - 30 q^{89} - 10 q^{91} + 7 q^{93} + 9 q^{95} - 37 q^{97} + 18 q^{99}+O(q^{100}) $ 25 * q - 11 * q^3 - 2 * q^5 - 9 * q^7 + 18 * q^9 + 25 * q^11 - 4 * q^13 - 10 * q^15 - 19 * q^17 - 12 * q^19 + 8 * q^21 - 31 * q^23 - q^25 - 44 * q^27 - q^29 - 8 * q^31 - 11 * q^33 - 16 * q^35 - 14 * q^37 - 18 * q^39 - 5 * q^41 - 15 * q^43 - 15 * q^45 - 41 * q^47 + 2 * q^49 + 10 * q^51 + 4 * q^53 - 2 * q^55 - 3 * q^57 - 35 * q^59 - 4 * q^61 - 45 * q^63 - 28 * q^65 - 30 * q^67 - 3 * q^69 + 4 * q^71 - 7 * q^73 - 18 * q^75 - 9 * q^77 - 9 * q^79 + 29 * q^81 - 72 * q^83 - 33 * q^87 - 30 * q^89 - 10 * q^91 + 7 * q^93 + 9 * q^95 - 37 * q^97 + 18 * 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
$3$	−1.44202	−0.832550	−0.416275	−	0.909239i	$-0.636665\pi$
$3$	−1.44202	−0.832550	−0.416275	+	0.909239i	$0.636665\pi$
$4$	0	0
$5$	3.24879	1.45291	0.726453	−	0.687217i	$-0.241168\pi$
$5$	3.24879	1.45291	0.726453	+	0.687217i	$0.241168\pi$
$6$	0	0
$7$	−4.91695	−1.85843	−0.929216	−	0.369537i	$-0.879516\pi$
$7$	−4.91695	−1.85843	−0.929216	+	0.369537i	$0.879516\pi$
$8$	0	0
$9$	−0.920583	−0.306861
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−3.35934	−0.931712	−0.465856	−	0.884861i	$-0.654254\pi$
$13$	−3.35934	−0.931712	−0.465856	+	0.884861i	$0.654254\pi$
$14$	0	0
$15$	−4.68482	−1.20962
$16$	0	0
$17$	−2.78284	−0.674938	−0.337469	−	0.941337i	$-0.609571\pi$
$17$	−2.78284	−0.674938	−0.337469	+	0.941337i	$0.609571\pi$
$18$	0	0
$19$	6.83249	1.56748	0.783740	−	0.621089i	$-0.213309\pi$
$19$	6.83249	1.56748	0.783740	+	0.621089i	$0.213309\pi$
$20$	0	0
$21$	7.09033	1.54724
$22$	0	0
$23$	−2.33086	−0.486018	−0.243009	−	0.970024i	$-0.578135\pi$
$23$	−2.33086	−0.486018	−0.243009	+	0.970024i	$0.578135\pi$
$24$	0	0
$25$	5.55467	1.11093
$26$	0	0
$27$	5.65355	1.08803
$28$	0	0
$29$	6.81200	1.26496	0.632479	−	0.774578i	$-0.282038\pi$
$29$	6.81200	1.26496	0.632479	+	0.774578i	$0.282038\pi$
$30$	0	0
$31$	−0.976993	−0.175473	−0.0877366	−	0.996144i	$-0.527963\pi$
$31$	−0.976993	−0.175473	−0.0877366	+	0.996144i	$0.527963\pi$
$32$	0	0
$33$	−1.44202	−0.251023
$34$	0	0
$35$	−15.9742	−2.70013
$36$	0	0
$37$	3.16459	0.520256	0.260128	−	0.965574i	$-0.416235\pi$
$37$	3.16459	0.520256	0.260128	+	0.965574i	$0.416235\pi$
$38$	0	0
$39$	4.84422	0.775697
$40$	0	0
$41$	3.93968	0.615274	0.307637	−	0.951504i	$-0.400462\pi$
$41$	3.93968	0.615274	0.307637	+	0.951504i	$0.400462\pi$
$42$	0	0
$43$	4.92668	0.751311	0.375655	−	0.926759i	$-0.377418\pi$
$43$	4.92668	0.751311	0.375655	+	0.926759i	$0.377418\pi$
$44$	0	0
$45$	−2.99078	−0.445840
$46$	0	0
$47$	3.13226	0.456888	0.228444	−	0.973557i	$-0.426636\pi$
$47$	3.13226	0.456888	0.228444	+	0.973557i	$0.426636\pi$
$48$	0	0
$49$	17.1764	2.45377
$50$	0	0
$51$	4.01291	0.561920
$52$	0	0
$53$	−4.65984	−0.640078	−0.320039	−	0.947404i	$-0.603696\pi$
$53$	−4.65984	−0.640078	−0.320039	+	0.947404i	$0.603696\pi$
$54$	0	0
$55$	3.24879	0.438067
$56$	0	0
$57$	−9.85258	−1.30501
$58$	0	0
$59$	−3.54476	−0.461488	−0.230744	−	0.973014i	$-0.574116\pi$
$59$	−3.54476	−0.461488	−0.230744	+	0.973014i	$0.574116\pi$
$60$	0	0
$61$	13.1079	1.67829	0.839145	−	0.543908i	$-0.183056\pi$
$61$	13.1079	1.67829	0.839145	+	0.543908i	$0.183056\pi$
$62$	0	0
$63$	4.52646	0.570280
$64$	0	0
$65$	−10.9138	−1.35369
$66$	0	0
$67$	−9.75423	−1.19167	−0.595834	−	0.803108i	$-0.703178\pi$
$67$	−9.75423	−1.19167	−0.595834	+	0.803108i	$0.703178\pi$
$68$	0	0
$69$	3.36115	0.404634
$70$	0	0
$71$	−13.8417	−1.64271	−0.821353	−	0.570421i	$-0.806780\pi$
$71$	−13.8417	−1.64271	−0.821353	+	0.570421i	$0.806780\pi$
$72$	0	0
$73$	7.37784	0.863511	0.431755	−	0.901991i	$-0.357894\pi$
$73$	7.37784	0.863511	0.431755	+	0.901991i	$0.357894\pi$
$74$	0	0
$75$	−8.00993	−0.924907
$76$	0	0
$77$	−4.91695	−0.560338
$78$	0	0
$79$	−13.3747	−1.50477	−0.752386	−	0.658722i	$-0.771097\pi$
$79$	−13.3747	−1.50477	−0.752386	+	0.658722i	$0.771097\pi$
$80$	0	0
$81$	−5.39078	−0.598976
$82$	0	0
$83$	−10.1170	−1.11048	−0.555242	−	0.831689i	$-0.687374\pi$
$83$	−10.1170	−1.11048	−0.555242	+	0.831689i	$0.687374\pi$
$84$	0	0
$85$	−9.04088	−0.980621
$86$	0	0
$87$	−9.82304	−1.05314
$88$	0	0
$89$	12.0143	1.27351	0.636757	−	0.771064i	$-0.280275\pi$
$89$	12.0143	1.27351	0.636757	+	0.771064i	$0.280275\pi$
$90$	0	0
$91$	16.5177	1.73152
$92$	0	0
$93$	1.40884	0.146090
$94$	0	0
$95$	22.1974	2.27740
$96$	0	0
$97$	−13.5932	−1.38018	−0.690091	−	0.723722i	$-0.742430\pi$
$97$	−13.5932	−1.38018	−0.690091	+	0.723722i	$0.742430\pi$
$98$	0	0
$99$	−0.920583	−0.0925220
$100$	0	0
$101$	−6.81482	−0.678100	−0.339050	−	0.940768i	$-0.610106\pi$
$101$	−6.81482	−0.678100	−0.339050	+	0.940768i	$0.610106\pi$
$102$	0	0
$103$	−14.2393	−1.40304	−0.701520	−	0.712650i	$-0.747495\pi$
$103$	−14.2393	−1.40304	−0.701520	+	0.712650i	$0.747495\pi$
$104$	0	0
$105$	23.0350	2.24799
$106$	0	0
$107$	−9.38562	−0.907342	−0.453671	−	0.891169i	$-0.649886\pi$
$107$	−9.38562	−0.907342	−0.453671	+	0.891169i	$0.649886\pi$
$108$	0	0
$109$	−2.53156	−0.242479	−0.121240	−	0.992623i	$-0.538687\pi$
$109$	−2.53156	−0.242479	−0.121240	+	0.992623i	$0.538687\pi$
$110$	0	0
$111$	−4.56340	−0.433139
$112$	0	0
$113$	−13.4464	−1.26493	−0.632467	−	0.774587i	$-0.717958\pi$
$113$	−13.4464	−1.26493	−0.632467	+	0.774587i	$0.717958\pi$
$114$	0	0
$115$	−7.57249	−0.706139
$116$	0	0
$117$	3.09255	0.285906
$118$	0	0
$119$	13.6831	1.25433
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−5.68109	−0.512246
$124$	0	0
$125$	1.80200	0.161175
$126$	0	0
$127$	13.4611	1.19448	0.597238	−	0.802064i	$-0.296265\pi$
$127$	13.4611	1.19448	0.597238	+	0.802064i	$0.296265\pi$
$128$	0	0
$129$	−7.10436	−0.625504
$130$	0	0
$131$	−16.7053	−1.45955	−0.729775	−	0.683687i	$-0.760375\pi$
$131$	−16.7053	−1.45955	−0.729775	+	0.683687i	$0.760375\pi$
$132$	0	0
$133$	−33.5950	−2.91306
$134$	0	0
$135$	18.3672	1.58080
$136$	0	0
$137$	−1.00000	−0.0854358
$137$	−1.00000	−0.0854358
$138$	0	0
$139$	−9.61191	−0.815272	−0.407636	−	0.913145i	$-0.633647\pi$
$139$	−9.61191	−0.815272	−0.407636	+	0.913145i	$0.633647\pi$
$140$	0	0
$141$	−4.51678	−0.380382
$142$	0	0
$143$	−3.35934	−0.280922
$144$	0	0
$145$	22.1308	1.83786
$146$	0	0
$147$	−24.7687	−2.04289
$148$	0	0
$149$	12.2985	1.00753	0.503764	−	0.863841i	$-0.331948\pi$
$149$	12.2985	1.00753	0.503764	+	0.863841i	$0.331948\pi$
$150$	0	0
$151$	−21.4635	−1.74667	−0.873336	−	0.487118i	$-0.838048\pi$
$151$	−21.4635	−1.74667	−0.873336	+	0.487118i	$0.838048\pi$
$152$	0	0
$153$	2.56184	0.207112
$154$	0	0
$155$	−3.17405	−0.254946
$156$	0	0
$157$	−3.32307	−0.265210	−0.132605	−	0.991169i	$-0.542334\pi$
$157$	−3.32307	−0.265210	−0.132605	+	0.991169i	$0.542334\pi$
$158$	0	0
$159$	6.71957	0.532897
$160$	0	0
$161$	11.4607	0.903232
$162$	0	0
$163$	−9.04537	−0.708488	−0.354244	−	0.935153i	$-0.615262\pi$
$163$	−9.04537	−0.708488	−0.354244	+	0.935153i	$0.615262\pi$
$164$	0	0
$165$	−4.68482	−0.364713
$166$	0	0
$167$	15.3875	1.19072	0.595361	−	0.803458i	$-0.297009\pi$
$167$	15.3875	1.19072	0.595361	+	0.803458i	$0.297009\pi$
$168$	0	0
$169$	−1.71487	−0.131913
$170$	0	0
$171$	−6.28987	−0.480998
$172$	0	0
$173$	−2.34489	−0.178279	−0.0891393	−	0.996019i	$-0.528412\pi$
$173$	−2.34489	−0.178279	−0.0891393	+	0.996019i	$0.528412\pi$
$174$	0	0
$175$	−27.3120	−2.06459
$176$	0	0
$177$	5.11161	0.384212
$178$	0	0
$179$	−10.0602	−0.751935	−0.375968	−	0.926633i	$-0.622690\pi$
$179$	−10.0602	−0.751935	−0.375968	+	0.926633i	$0.622690\pi$
$180$	0	0
$181$	−8.20378	−0.609782	−0.304891	−	0.952387i	$-0.598620\pi$
$181$	−8.20378	−0.609782	−0.304891	+	0.952387i	$0.598620\pi$
$182$	0	0
$183$	−18.9018	−1.39726
$184$	0	0
$185$	10.2811	0.755882
$186$	0	0
$187$	−2.78284	−0.203502
$188$	0	0
$189$	−27.7982	−2.02202
$190$	0	0
$191$	1.14970	0.0831894	0.0415947	−	0.999135i	$-0.486756\pi$
$191$	1.14970	0.0831894	0.0415947	+	0.999135i	$0.486756\pi$
$192$	0	0
$193$	−23.4385	−1.68714	−0.843570	−	0.537020i	$-0.819550\pi$
$193$	−23.4385	−1.68714	−0.843570	+	0.537020i	$0.819550\pi$
$194$	0	0
$195$	15.7379	1.12701
$196$	0	0
$197$	−5.50669	−0.392335	−0.196168	−	0.980570i	$-0.562850\pi$
$197$	−5.50669	−0.392335	−0.196168	+	0.980570i	$0.562850\pi$
$198$	0	0
$199$	−4.08376	−0.289490	−0.144745	−	0.989469i	$-0.546236\pi$
$199$	−4.08376	−0.289490	−0.144745	+	0.989469i	$0.546236\pi$
$200$	0	0
$201$	14.0658	0.992123
$202$	0	0
$203$	−33.4943	−2.35084
$204$	0	0
$205$	12.7992	0.893935
$206$	0	0
$207$	2.14575	0.149140
$208$	0	0
$209$	6.83249	0.472613
$210$	0	0
$211$	9.59704	0.660688	0.330344	−	0.943861i	$-0.392835\pi$
$211$	9.59704	0.660688	0.330344	+	0.943861i	$0.392835\pi$
$212$	0	0
$213$	19.9600	1.36763
$214$	0	0
$215$	16.0058	1.09158
$216$	0	0
$217$	4.80382	0.326105
$218$	0	0
$219$	−10.6390	−0.718916
$220$	0	0
$221$	9.34850	0.628848
$222$	0	0
$223$	7.55568	0.505966	0.252983	−	0.967471i	$-0.418588\pi$
$223$	7.55568	0.505966	0.252983	+	0.967471i	$0.418588\pi$
$224$	0	0
$225$	−5.11353	−0.340902
$226$	0	0
$227$	−9.36142	−0.621340	−0.310670	−	0.950518i	$-0.600553\pi$
$227$	−9.36142	−0.621340	−0.310670	+	0.950518i	$0.600553\pi$
$228$	0	0
$229$	3.57772	0.236423	0.118211	−	0.992988i	$-0.462284\pi$
$229$	3.57772	0.236423	0.118211	+	0.992988i	$0.462284\pi$
$230$	0	0
$231$	7.09033	0.466510
$232$	0	0
$233$	15.4575	1.01266	0.506328	−	0.862341i	$-0.331002\pi$
$233$	15.4575	1.01266	0.506328	+	0.862341i	$0.331002\pi$
$234$	0	0
$235$	10.1761	0.663814
$236$	0	0
$237$	19.2866	1.25280
$238$	0	0
$239$	25.5399	1.65204	0.826018	−	0.563643i	$-0.190601\pi$
$239$	25.5399	1.65204	0.826018	+	0.563643i	$0.190601\pi$
$240$	0	0
$241$	7.58820	0.488799	0.244399	−	0.969675i	$-0.421409\pi$
$241$	7.58820	0.488799	0.244399	+	0.969675i	$0.421409\pi$
$242$	0	0
$243$	−9.18705	−0.589350
$244$	0	0
$245$	55.8026	3.56509
$246$	0	0
$247$	−22.9526	−1.46044
$248$	0	0
$249$	14.5889	0.924533
$250$	0	0
$251$	25.5888	1.61515	0.807575	−	0.589764i	$-0.200779\pi$
$251$	25.5888	1.61515	0.807575	+	0.589764i	$0.200779\pi$
$252$	0	0
$253$	−2.33086	−0.146540
$254$	0	0
$255$	13.0371	0.816416
$256$	0	0
$257$	13.9649	0.871105	0.435553	−	0.900163i	$-0.356553\pi$
$257$	13.9649	0.871105	0.435553	+	0.900163i	$0.356553\pi$
$258$	0	0
$259$	−15.5601	−0.966860
$260$	0	0
$261$	−6.27101	−0.388166
$262$	0	0
$263$	11.9002	0.733796	0.366898	−	0.930261i	$-0.380420\pi$
$263$	11.9002	0.733796	0.366898	+	0.930261i	$0.380420\pi$
$264$	0	0
$265$	−15.1389	−0.929973
$266$	0	0
$267$	−17.3249	−1.06026
$268$	0	0
$269$	−17.5213	−1.06829	−0.534146	−	0.845392i	$-0.679367\pi$
$269$	−17.5213	−1.06829	−0.534146	+	0.845392i	$0.679367\pi$
$270$	0	0
$271$	12.2158	0.742059	0.371029	−	0.928621i	$-0.379005\pi$
$271$	12.2158	0.742059	0.371029	+	0.928621i	$0.379005\pi$
$272$	0	0
$273$	−23.8188	−1.44158
$274$	0	0
$275$	5.55467	0.334959
$276$	0	0
$277$	5.91500	0.355398	0.177699	−	0.984085i	$-0.443135\pi$
$277$	5.91500	0.355398	0.177699	+	0.984085i	$0.443135\pi$
$278$	0	0
$279$	0.899403	0.0538458
$280$	0	0
$281$	−6.65921	−0.397255	−0.198628	−	0.980075i	$-0.563648\pi$
$281$	−6.65921	−0.397255	−0.198628	+	0.980075i	$0.563648\pi$
$282$	0	0
$283$	−21.0499	−1.25129	−0.625644	−	0.780109i	$-0.715164\pi$
$283$	−21.0499	−1.25129	−0.625644	+	0.780109i	$0.715164\pi$
$284$	0	0
$285$	−32.0090	−1.89605
$286$	0	0
$287$	−19.3712	−1.14345
$288$	0	0
$289$	−9.25579	−0.544458
$290$	0	0
$291$	19.6017	1.14907
$292$	0	0
$293$	23.1776	1.35405	0.677024	−	0.735961i	$-0.263269\pi$
$293$	23.1776	1.35405	0.677024	+	0.735961i	$0.263269\pi$
$294$	0	0
$295$	−11.5162	−0.670499
$296$	0	0
$297$	5.65355	0.328052
$298$	0	0
$299$	7.83015	0.452829
$300$	0	0
$301$	−24.2242	−1.39626
$302$	0	0
$303$	9.82710	0.564552
$304$	0	0
$305$	42.5848	2.43840
$306$	0	0
$307$	−15.7998	−0.901740	−0.450870	−	0.892590i	$-0.648886\pi$
$307$	−15.7998	−0.901740	−0.450870	+	0.892590i	$0.648886\pi$
$308$	0	0
$309$	20.5333	1.16810
$310$	0	0
$311$	0.208389	0.0118166	0.00590832	−	0.999983i	$-0.498119\pi$
$311$	0.208389	0.0118166	0.00590832	+	0.999983i	$0.498119\pi$
$312$	0	0
$313$	23.6574	1.33720	0.668598	−	0.743624i	$-0.266895\pi$
$313$	23.6574	1.33720	0.668598	+	0.743624i	$0.266895\pi$
$314$	0	0
$315$	14.7055	0.828563
$316$	0	0
$317$	−2.61083	−0.146639	−0.0733194	−	0.997309i	$-0.523359\pi$
$317$	−2.61083	−0.146639	−0.0733194	+	0.997309i	$0.523359\pi$
$318$	0	0
$319$	6.81200	0.381399
$320$	0	0
$321$	13.5342	0.755407
$322$	0	0
$323$	−19.0137	−1.05795
$324$	0	0
$325$	−18.6600	−1.03507
$326$	0	0
$327$	3.65056	0.201876
$328$	0	0
$329$	−15.4012	−0.849095
$330$	0	0
$331$	1.24528	0.0684467	0.0342234	−	0.999414i	$-0.489104\pi$
$331$	1.24528	0.0684467	0.0342234	+	0.999414i	$0.489104\pi$
$332$	0	0
$333$	−2.91327	−0.159646
$334$	0	0
$335$	−31.6895	−1.73138
$336$	0	0
$337$	−29.6230	−1.61367	−0.806834	−	0.590778i	$-0.798821\pi$
$337$	−29.6230	−1.61367	−0.806834	+	0.590778i	$0.798821\pi$
$338$	0	0
$339$	19.3900	1.05312
$340$	0	0
$341$	−0.976993	−0.0529071
$342$	0	0
$343$	−50.0368	−2.70173
$344$	0	0
$345$	10.9197	0.587895
$346$	0	0
$347$	−30.8502	−1.65613	−0.828063	−	0.560635i	$-0.810557\pi$
$347$	−30.8502	−1.65613	−0.828063	+	0.560635i	$0.810557\pi$
$348$	0	0
$349$	−12.8636	−0.688575	−0.344288	−	0.938864i	$-0.611880\pi$
$349$	−12.8636	−0.688575	−0.344288	+	0.938864i	$0.611880\pi$
$350$	0	0
$351$	−18.9922	−1.01373
$352$	0	0
$353$	−11.5910	−0.616928	−0.308464	−	0.951236i	$-0.599815\pi$
$353$	−11.5910	−0.616928	−0.308464	+	0.951236i	$0.599815\pi$
$354$	0	0
$355$	−44.9688	−2.38670
$356$	0	0
$357$	−19.7313	−1.04429
$358$	0	0
$359$	−16.8315	−0.888334	−0.444167	−	0.895944i	$-0.646500\pi$
$359$	−16.8315	−0.888334	−0.444167	+	0.895944i	$0.646500\pi$
$360$	0	0
$361$	27.6829	1.45699
$362$	0	0
$363$	−1.44202	−0.0756863
$364$	0	0
$365$	23.9691	1.25460
$366$	0	0
$367$	2.10327	0.109790	0.0548948	−	0.998492i	$-0.482518\pi$
$367$	2.10327	0.109790	0.0548948	+	0.998492i	$0.482518\pi$
$368$	0	0
$369$	−3.62680	−0.188804
$370$	0	0
$371$	22.9122	1.18954
$372$	0	0
$373$	32.2447	1.66957	0.834783	−	0.550580i	$-0.185593\pi$
$373$	32.2447	1.66957	0.834783	+	0.550580i	$0.185593\pi$
$374$	0	0
$375$	−2.59851	−0.134187
$376$	0	0
$377$	−22.8838	−1.17858
$378$	0	0
$379$	3.52987	0.181317	0.0906587	−	0.995882i	$-0.471103\pi$
$379$	3.52987	0.181317	0.0906587	+	0.995882i	$0.471103\pi$
$380$	0	0
$381$	−19.4111	−0.994461
$382$	0	0
$383$	9.18551	0.469358	0.234679	−	0.972073i	$-0.424596\pi$
$383$	9.18551	0.469358	0.234679	+	0.972073i	$0.424596\pi$
$384$	0	0
$385$	−15.9742	−0.814118
$386$	0	0
$387$	−4.53541	−0.230548
$388$	0	0
$389$	9.31025	0.472049	0.236024	−	0.971747i	$-0.424156\pi$
$389$	9.31025	0.472049	0.236024	+	0.971747i	$0.424156\pi$
$390$	0	0
$391$	6.48642	0.328032
$392$	0	0
$393$	24.0894	1.21515
$394$	0	0
$395$	−43.4517	−2.18629
$396$	0	0
$397$	7.40812	0.371803	0.185901	−	0.982568i	$-0.440479\pi$
$397$	7.40812	0.371803	0.185901	+	0.982568i	$0.440479\pi$
$398$	0	0
$399$	48.4446	2.42526
$400$	0	0
$401$	−19.0254	−0.950082	−0.475041	−	0.879964i	$-0.657567\pi$
$401$	−19.0254	−0.950082	−0.475041	+	0.879964i	$0.657567\pi$
$402$	0	0
$403$	3.28205	0.163490
$404$	0	0
$405$	−17.5135	−0.870255
$406$	0	0
$407$	3.16459	0.156863
$408$	0	0
$409$	−11.2591	−0.556725	−0.278362	−	0.960476i	$-0.589792\pi$
$409$	−11.2591	−0.556725	−0.278362	+	0.960476i	$0.589792\pi$
$410$	0	0
$411$	1.44202	0.0711295
$412$	0	0
$413$	17.4294	0.857645
$414$	0	0
$415$	−32.8680	−1.61343
$416$	0	0
$417$	13.8606	0.678755
$418$	0	0
$419$	21.5539	1.05298	0.526488	−	0.850182i	$-0.323508\pi$
$419$	21.5539	1.05298	0.526488	+	0.850182i	$0.323508\pi$
$420$	0	0
$421$	−1.66849	−0.0813175	−0.0406587	−	0.999173i	$-0.512946\pi$
$421$	−1.66849	−0.0813175	−0.0406587	+	0.999173i	$0.512946\pi$
$422$	0	0
$423$	−2.88351	−0.140201
$424$	0	0
$425$	−15.4578	−0.749811
$426$	0	0
$427$	−64.4507	−3.11899
$428$	0	0
$429$	4.84422	0.233881
$430$	0	0
$431$	−21.0300	−1.01298	−0.506491	−	0.862245i	$-0.669058\pi$
$431$	−21.0300	−1.01298	−0.506491	+	0.862245i	$0.669058\pi$
$432$	0	0
$433$	9.68170	0.465273	0.232637	−	0.972564i	$-0.425265\pi$
$433$	9.68170	0.465273	0.232637	+	0.972564i	$0.425265\pi$
$434$	0	0
$435$	−31.9130	−1.53011
$436$	0	0
$437$	−15.9256	−0.761824
$438$	0	0
$439$	−23.5748	−1.12517	−0.562583	−	0.826741i	$-0.690192\pi$
$439$	−23.5748	−1.12517	−0.562583	+	0.826741i	$0.690192\pi$
$440$	0	0
$441$	−15.8123	−0.752966
$442$	0	0
$443$	−30.7429	−1.46064	−0.730319	−	0.683106i	$-0.760629\pi$
$443$	−30.7429	−1.46064	−0.730319	+	0.683106i	$0.760629\pi$
$444$	0	0
$445$	39.0320	1.85030
$446$	0	0
$447$	−17.7346	−0.838818
$448$	0	0
$449$	−33.2625	−1.56976	−0.784878	−	0.619650i	$-0.787274\pi$
$449$	−33.2625	−1.56976	−0.784878	+	0.619650i	$0.787274\pi$
$450$	0	0
$451$	3.93968	0.185512
$452$	0	0
$453$	30.9507	1.45419
$454$	0	0
$455$	53.6625	2.51574
$456$	0	0
$457$	40.8590	1.91130	0.955652	−	0.294498i	$-0.0951523\pi$
$457$	40.8590	1.91130	0.955652	+	0.294498i	$0.0951523\pi$
$458$	0	0
$459$	−15.7329	−0.734351
$460$	0	0
$461$	9.75003	0.454104	0.227052	−	0.973883i	$-0.427091\pi$
$461$	9.75003	0.454104	0.227052	+	0.973883i	$0.427091\pi$
$462$	0	0
$463$	17.7534	0.825070	0.412535	−	0.910942i	$-0.364643\pi$
$463$	17.7534	0.825070	0.412535	+	0.910942i	$0.364643\pi$
$464$	0	0
$465$	4.57704	0.212255
$466$	0	0
$467$	−20.3673	−0.942486	−0.471243	−	0.882003i	$-0.656195\pi$
$467$	−20.3673	−0.942486	−0.471243	+	0.882003i	$0.656195\pi$
$468$	0	0
$469$	47.9610	2.21463
$470$	0	0
$471$	4.79192	0.220800
$472$	0	0
$473$	4.92668	0.226529
$474$	0	0
$475$	37.9522	1.74137
$476$	0	0
$477$	4.28977	0.196415
$478$	0	0
$479$	27.0617	1.23648	0.618240	−	0.785990i	$-0.287846\pi$
$479$	27.0617	1.23648	0.618240	+	0.785990i	$0.287846\pi$
$480$	0	0
$481$	−10.6309	−0.484728
$482$	0	0
$483$	−16.5266	−0.751986
$484$	0	0
$485$	−44.1616	−2.00527
$486$	0	0
$487$	−26.6467	−1.20748	−0.603738	−	0.797183i	$-0.706323\pi$
$487$	−26.6467	−1.20748	−0.603738	+	0.797183i	$0.706323\pi$
$488$	0	0
$489$	13.0436	0.589852
$490$	0	0
$491$	8.95080	0.403944	0.201972	−	0.979391i	$-0.435265\pi$
$491$	8.95080	0.403944	0.201972	+	0.979391i	$0.435265\pi$
$492$	0	0
$493$	−18.9567	−0.853768
$494$	0	0
$495$	−2.99078	−0.134426
$496$	0	0
$497$	68.0588	3.05286
$498$	0	0
$499$	−39.6537	−1.77514	−0.887571	−	0.460671i	$-0.847609\pi$
$499$	−39.6537	−1.77514	−0.887571	+	0.460671i	$0.847609\pi$
$500$	0	0
$501$	−22.1891	−0.991335
$502$	0	0
$503$	31.9406	1.42416	0.712080	−	0.702099i	$-0.247754\pi$
$503$	31.9406	1.42416	0.712080	+	0.702099i	$0.247754\pi$
$504$	0	0
$505$	−22.1399	−0.985215
$506$	0	0
$507$	2.47287	0.109824
$508$	0	0
$509$	−25.8468	−1.14564	−0.572819	−	0.819682i	$-0.694150\pi$
$509$	−25.8468	−1.14564	−0.572819	+	0.819682i	$0.694150\pi$
$510$	0	0
$511$	−36.2765	−1.60478
$512$	0	0
$513$	38.6278	1.70546
$514$	0	0
$515$	−46.2606	−2.03848
$516$	0	0
$517$	3.13226	0.137757
$518$	0	0
$519$	3.38137	0.148426
$520$	0	0
$521$	−28.9819	−1.26972	−0.634861	−	0.772627i	$-0.718943\pi$
$521$	−28.9819	−1.26972	−0.634861	+	0.772627i	$0.718943\pi$
$522$	0	0
$523$	12.4051	0.542435	0.271218	−	0.962518i	$-0.412574\pi$
$523$	12.4051	0.542435	0.271218	+	0.962518i	$0.412574\pi$
$524$	0	0
$525$	39.3844	1.71888
$526$	0	0
$527$	2.71882	0.118434
$528$	0	0
$529$	−17.5671	−0.763786
$530$	0	0
$531$	3.26324	0.141613
$532$	0	0
$533$	−13.2347	−0.573258
$534$	0	0
$535$	−30.4919	−1.31828
$536$	0	0
$537$	14.5070	0.626024
$538$	0	0
$539$	17.1764	0.739839
$540$	0	0
$541$	4.01642	0.172679	0.0863396	−	0.996266i	$-0.472483\pi$
$541$	4.01642	0.172679	0.0863396	+	0.996266i	$0.472483\pi$
$542$	0	0
$543$	11.8300	0.507674
$544$	0	0
$545$	−8.22452	−0.352300
$546$	0	0
$547$	17.5723	0.751338	0.375669	−	0.926754i	$-0.377413\pi$
$547$	17.5723	0.751338	0.375669	+	0.926754i	$0.377413\pi$
$548$	0	0
$549$	−12.0669	−0.515002
$550$	0	0
$551$	46.5429	1.98280
$552$	0	0
$553$	65.7627	2.79652
$554$	0	0
$555$	−14.8255	−0.629310
$556$	0	0
$557$	44.1419	1.87035	0.935177	−	0.354180i	$-0.115240\pi$
$557$	44.1419	1.87035	0.935177	+	0.354180i	$0.115240\pi$
$558$	0	0
$559$	−16.5504	−0.700005
$560$	0	0
$561$	4.01291	0.169425
$562$	0	0
$563$	−27.7598	−1.16994	−0.584969	−	0.811056i	$-0.698893\pi$
$563$	−27.7598	−1.16994	−0.584969	+	0.811056i	$0.698893\pi$
$564$	0	0
$565$	−43.6847	−1.83783
$566$	0	0
$567$	26.5062	1.11316
$568$	0	0
$569$	13.4495	0.563832	0.281916	−	0.959439i	$-0.409030\pi$
$569$	13.4495	0.563832	0.281916	+	0.959439i	$0.409030\pi$
$570$	0	0
$571$	24.3742	1.02003	0.510015	−	0.860166i	$-0.329640\pi$
$571$	24.3742	1.02003	0.510015	+	0.860166i	$0.329640\pi$
$572$	0	0
$573$	−1.65789	−0.0692593
$574$	0	0
$575$	−12.9472	−0.539934
$576$	0	0
$577$	−32.4372	−1.35038	−0.675189	−	0.737645i	$-0.735938\pi$
$577$	−32.4372	−1.35038	−0.675189	+	0.737645i	$0.735938\pi$
$578$	0	0
$579$	33.7987	1.40463
$580$	0	0
$581$	49.7447	2.06376
$582$	0	0
$583$	−4.65984	−0.192991
$584$	0	0
$585$	10.0470	0.415394
$586$	0	0
$587$	−35.7555	−1.47579	−0.737894	−	0.674917i	$-0.764180\pi$
$587$	−35.7555	−1.47579	−0.737894	+	0.674917i	$0.764180\pi$
$588$	0	0
$589$	−6.67529	−0.275051
$590$	0	0
$591$	7.94074	0.326639
$592$	0	0
$593$	−36.9217	−1.51619	−0.758097	−	0.652142i	$-0.773871\pi$
$593$	−36.9217	−1.51619	−0.758097	+	0.652142i	$0.773871\pi$
$594$	0	0
$595$	44.4535	1.82242
$596$	0	0
$597$	5.88885	0.241015
$598$	0	0
$599$	25.1108	1.02600	0.513001	−	0.858388i	$-0.328534\pi$
$599$	25.1108	1.02600	0.513001	+	0.858388i	$0.328534\pi$
$600$	0	0
$601$	19.1894	0.782752	0.391376	−	0.920231i	$-0.371999\pi$
$601$	19.1894	0.782752	0.391376	+	0.920231i	$0.371999\pi$
$602$	0	0
$603$	8.97957	0.365676
$604$	0	0
$605$	3.24879	0.132082
$606$	0	0
$607$	29.3559	1.19152	0.595759	−	0.803163i	$-0.296851\pi$
$607$	29.3559	1.19152	0.595759	+	0.803163i	$0.296851\pi$
$608$	0	0
$609$	48.2994	1.95719
$610$	0	0
$611$	−10.5223	−0.425688
$612$	0	0
$613$	−32.2536	−1.30271	−0.651355	−	0.758773i	$-0.725799\pi$
$613$	−32.2536	−1.30271	−0.651355	+	0.758773i	$0.725799\pi$
$614$	0	0
$615$	−18.4567	−0.744245
$616$	0	0
$617$	−29.2530	−1.17768	−0.588840	−	0.808249i	$-0.700415\pi$
$617$	−29.2530	−1.17768	−0.588840	+	0.808249i	$0.700415\pi$
$618$	0	0
$619$	31.9187	1.28292	0.641460	−	0.767156i	$-0.278329\pi$
$619$	31.9187	1.28292	0.641460	+	0.767156i	$0.278329\pi$
$620$	0	0
$621$	−13.1777	−0.528801
$622$	0	0
$623$	−59.0737	−2.36674
$624$	0	0
$625$	−21.9190	−0.876761
$626$	0	0
$627$	−9.85258	−0.393474
$628$	0	0
$629$	−8.80656	−0.351140
$630$	0	0
$631$	18.7635	0.746963	0.373481	−	0.927638i	$-0.378164\pi$
$631$	18.7635	0.746963	0.373481	+	0.927638i	$0.378164\pi$
$632$	0	0
$633$	−13.8391	−0.550055
$634$	0	0
$635$	43.7322	1.73546
$636$	0	0
$637$	−57.7012	−2.28621
$638$	0	0
$639$	12.7424	0.504082
$640$	0	0
$641$	32.1843	1.27120	0.635601	−	0.772017i	$-0.280752\pi$
$641$	32.1843	1.27120	0.635601	+	0.772017i	$0.280752\pi$
$642$	0	0
$643$	−28.0379	−1.10571	−0.552853	−	0.833279i	$-0.686461\pi$
$643$	−28.0379	−1.10571	−0.552853	+	0.833279i	$0.686461\pi$
$644$	0	0
$645$	−23.0806	−0.908798
$646$	0	0
$647$	8.12955	0.319606	0.159803	−	0.987149i	$-0.448914\pi$
$647$	8.12955	0.319606	0.159803	+	0.987149i	$0.448914\pi$
$648$	0	0
$649$	−3.54476	−0.139144
$650$	0	0
$651$	−6.92720	−0.271499
$652$	0	0
$653$	−26.3502	−1.03116	−0.515582	−	0.856840i	$-0.672424\pi$
$653$	−26.3502	−1.03116	−0.515582	+	0.856840i	$0.672424\pi$
$654$	0	0
$655$	−54.2721	−2.12059
$656$	0	0
$657$	−6.79191	−0.264978
$658$	0	0
$659$	−10.2384	−0.398833	−0.199417	−	0.979915i	$-0.563905\pi$
$659$	−10.2384	−0.398833	−0.199417	+	0.979915i	$0.563905\pi$
$660$	0	0
$661$	−21.9045	−0.851988	−0.425994	−	0.904726i	$-0.640076\pi$
$661$	−21.9045	−0.851988	−0.425994	+	0.904726i	$0.640076\pi$
$662$	0	0
$663$	−13.4807	−0.523547
$664$	0	0
$665$	−109.143	−4.23239
$666$	0	0
$667$	−15.8778	−0.614793
$668$	0	0
$669$	−10.8954	−0.421242
$670$	0	0
$671$	13.1079	0.506024
$672$	0	0
$673$	−33.9201	−1.30752	−0.653762	−	0.756700i	$-0.726810\pi$
$673$	−33.9201	−1.30752	−0.653762	+	0.756700i	$0.726810\pi$
$674$	0	0
$675$	31.4036	1.20872
$676$	0	0
$677$	−35.7028	−1.37217	−0.686084	−	0.727522i	$-0.740672\pi$
$677$	−35.7028	−1.37217	−0.686084	+	0.727522i	$0.740672\pi$
$678$	0	0
$679$	66.8372	2.56498
$680$	0	0
$681$	13.4993	0.517296
$682$	0	0
$683$	36.1264	1.38234	0.691168	−	0.722694i	$-0.257096\pi$
$683$	36.1264	1.38234	0.691168	+	0.722694i	$0.257096\pi$
$684$	0	0
$685$	−3.24879	−0.124130
$686$	0	0
$687$	−5.15914	−0.196834
$688$	0	0
$689$	15.6540	0.596368
$690$	0	0
$691$	−39.7844	−1.51347	−0.756734	−	0.653723i	$-0.773206\pi$
$691$	−39.7844	−1.51347	−0.756734	+	0.653723i	$0.773206\pi$
$692$	0	0
$693$	4.52646	0.171946
$694$	0	0
$695$	−31.2271	−1.18451
$696$	0	0
$697$	−10.9635	−0.415272
$698$	0	0
$699$	−22.2901	−0.843087
$700$	0	0
$701$	−37.1589	−1.40347	−0.701737	−	0.712436i	$-0.747592\pi$
$701$	−37.1589	−1.40347	−0.701737	+	0.712436i	$0.747592\pi$
$702$	0	0
$703$	21.6220	0.815491
$704$	0	0
$705$	−14.6741	−0.552659
$706$	0	0
$707$	33.5081	1.26020
$708$	0	0
$709$	−2.01355	−0.0756205	−0.0378102	−	0.999285i	$-0.512038\pi$
$709$	−2.01355	−0.0756205	−0.0378102	+	0.999285i	$0.512038\pi$
$710$	0	0
$711$	12.3125	0.461756
$712$	0	0
$713$	2.27724	0.0852832
$714$	0	0
$715$	−10.9138	−0.408153
$716$	0	0
$717$	−36.8290	−1.37540
$718$	0	0
$719$	18.1005	0.675033	0.337516	−	0.941320i	$-0.390413\pi$
$719$	18.1005	0.675033	0.337516	+	0.941320i	$0.390413\pi$
$720$	0	0
$721$	70.0139	2.60745
$722$	0	0
$723$	−10.9423	−0.406949
$724$	0	0
$725$	37.8384	1.40528
$726$	0	0
$727$	40.6626	1.50809	0.754046	−	0.656821i	$-0.228099\pi$
$727$	40.6626	1.50809	0.754046	+	0.656821i	$0.228099\pi$
$728$	0	0
$729$	29.4202	1.08964
$730$	0	0
$731$	−13.7102	−0.507088
$732$	0	0
$733$	32.5166	1.20103	0.600515	−	0.799614i	$-0.294962\pi$
$733$	32.5166	1.20103	0.600515	+	0.799614i	$0.294962\pi$
$734$	0	0
$735$	−80.4683	−2.96812
$736$	0	0
$737$	−9.75423	−0.359302
$738$	0	0
$739$	−50.2169	−1.84726	−0.923630	−	0.383285i	$-0.874793\pi$
$739$	−50.2169	−1.84726	−0.923630	+	0.383285i	$0.874793\pi$
$740$	0	0
$741$	33.0981	1.21589
$742$	0	0
$743$	1.11757	0.0409996	0.0204998	−	0.999790i	$-0.493474\pi$
$743$	1.11757	0.0409996	0.0204998	+	0.999790i	$0.493474\pi$
$744$	0	0
$745$	39.9552	1.46384
$746$	0	0
$747$	9.31353	0.340764
$748$	0	0
$749$	46.1486	1.68623
$750$	0	0
$751$	−15.8052	−0.576740	−0.288370	−	0.957519i	$-0.593113\pi$
$751$	−15.8052	−0.576740	−0.288370	+	0.957519i	$0.593113\pi$
$752$	0	0
$753$	−36.8995	−1.34469
$754$	0	0
$755$	−69.7304	−2.53775
$756$	0	0
$757$	−40.3600	−1.46691	−0.733455	−	0.679738i	$-0.762093\pi$
$757$	−40.3600	−1.46691	−0.733455	+	0.679738i	$0.762093\pi$
$758$	0	0
$759$	3.36115	0.122002
$760$	0	0
$761$	20.4019	0.739569	0.369785	−	0.929117i	$-0.379431\pi$
$761$	20.4019	0.739569	0.369785	+	0.929117i	$0.379431\pi$
$762$	0	0
$763$	12.4475	0.450631
$764$	0	0
$765$	8.32288	0.300914
$766$	0	0
$767$	11.9080	0.429974
$768$	0	0
$769$	−8.73806	−0.315102	−0.157551	−	0.987511i	$-0.550360\pi$
$769$	−8.73806	−0.315102	−0.157551	+	0.987511i	$0.550360\pi$
$770$	0	0
$771$	−20.1376	−0.725239
$772$	0	0
$773$	7.02864	0.252803	0.126401	−	0.991979i	$-0.459657\pi$
$773$	7.02864	0.252803	0.126401	+	0.991979i	$0.459657\pi$
$774$	0	0
$775$	−5.42687	−0.194939
$776$	0	0
$777$	22.4380	0.804959
$778$	0	0
$779$	26.9178	0.964430
$780$	0	0
$781$	−13.8417	−0.495294
$782$	0	0
$783$	38.5120	1.37631
$784$	0	0
$785$	−10.7960	−0.385324
$786$	0	0
$787$	1.82969	0.0652213	0.0326107	−	0.999468i	$-0.489618\pi$
$787$	1.82969	0.0652213	0.0326107	+	0.999468i	$0.489618\pi$
$788$	0	0
$789$	−17.1603	−0.610921
$790$	0	0
$791$	66.1154	2.35079
$792$	0	0
$793$	−44.0337	−1.56368
$794$	0	0
$795$	21.8305	0.774249
$796$	0	0
$797$	−19.5308	−0.691816	−0.345908	−	0.938269i	$-0.612429\pi$
$797$	−19.5308	−0.691816	−0.345908	+	0.938269i	$0.612429\pi$
$798$	0	0
$799$	−8.71659	−0.308371
$800$	0	0
$801$	−11.0602	−0.390792
$802$	0	0
$803$	7.37784	0.260358
$804$	0	0
$805$	37.2336	1.31231
$806$	0	0
$807$	25.2660	0.889407
$808$	0	0
$809$	−11.3530	−0.399150	−0.199575	−	0.979883i	$-0.563956\pi$
$809$	−11.3530	−0.399150	−0.199575	+	0.979883i	$0.563956\pi$
$810$	0	0
$811$	56.3775	1.97968	0.989841	−	0.142178i	$-0.0454105\pi$
$811$	56.3775	1.97968	0.989841	+	0.142178i	$0.0454105\pi$
$812$	0	0
$813$	−17.6155	−0.617801
$814$	0	0
$815$	−29.3865	−1.02937
$816$	0	0
$817$	33.6615	1.17767
$818$	0	0
$819$	−15.2059	−0.531337
$820$	0	0
$821$	35.7685	1.24833	0.624165	−	0.781293i	$-0.285439\pi$
$821$	35.7685	1.24833	0.624165	+	0.781293i	$0.285439\pi$
$822$	0	0
$823$	38.3131	1.33551	0.667755	−	0.744382i	$-0.267256\pi$
$823$	38.3131	1.33551	0.667755	+	0.744382i	$0.267256\pi$
$824$	0	0
$825$	−8.00993	−0.278870
$826$	0	0
$827$	17.6260	0.612917	0.306458	−	0.951884i	$-0.400856\pi$
$827$	17.6260	0.612917	0.306458	+	0.951884i	$0.400856\pi$
$828$	0	0
$829$	−47.3850	−1.64575	−0.822875	−	0.568222i	$-0.807631\pi$
$829$	−47.3850	−1.64575	−0.822875	+	0.568222i	$0.807631\pi$
$830$	0	0
$831$	−8.52953	−0.295886
$832$	0	0
$833$	−47.7992	−1.65614
$834$	0	0
$835$	49.9909	1.73001
$836$	0	0
$837$	−5.52348	−0.190919
$838$	0	0
$839$	−12.0922	−0.417468	−0.208734	−	0.977973i	$-0.566934\pi$
$839$	−12.0922	−0.417468	−0.208734	+	0.977973i	$0.566934\pi$
$840$	0	0
$841$	17.4034	0.600118
$842$	0	0
$843$	9.60271	0.330735
$844$	0	0
$845$	−5.57125	−0.191657
$846$	0	0
$847$	−4.91695	−0.168948
$848$	0	0
$849$	30.3544	1.04176
$850$	0	0
$851$	−7.37623	−0.252854
$852$	0	0
$853$	−37.1430	−1.27175	−0.635876	−	0.771791i	$-0.719361\pi$
$853$	−37.1430	−1.27175	−0.635876	+	0.771791i	$0.719361\pi$
$854$	0	0
$855$	−20.4345	−0.698845
$856$	0	0
$857$	−21.1406	−0.722148	−0.361074	−	0.932537i	$-0.617590\pi$
$857$	−21.1406	−0.722148	−0.361074	+	0.932537i	$0.617590\pi$
$858$	0	0
$859$	−34.0781	−1.16273	−0.581364	−	0.813644i	$-0.697481\pi$
$859$	−34.0781	−1.16273	−0.581364	+	0.813644i	$0.697481\pi$
$860$	0	0
$861$	27.9336	0.951975
$862$	0	0
$863$	27.8915	0.949439	0.474720	−	0.880137i	$-0.342549\pi$
$863$	27.8915	0.949439	0.474720	+	0.880137i	$0.342549\pi$
$864$	0	0
$865$	−7.61806	−0.259022
$866$	0	0
$867$	13.3470	0.453289
$868$	0	0
$869$	−13.3747	−0.453706
$870$	0	0
$871$	32.7677	1.11029
$872$	0	0
$873$	12.5137	0.423524
$874$	0	0
$875$	−8.86032	−0.299533
$876$	0	0
$877$	12.5194	0.422751	0.211375	−	0.977405i	$-0.432206\pi$
$877$	12.5194	0.422751	0.211375	+	0.977405i	$0.432206\pi$
$878$	0	0
$879$	−33.4225	−1.12731
$880$	0	0
$881$	−27.8319	−0.937682	−0.468841	−	0.883283i	$-0.655328\pi$
$881$	−27.8319	−0.937682	−0.468841	+	0.883283i	$0.655328\pi$
$882$	0	0
$883$	−6.41401	−0.215849	−0.107924	−	0.994159i	$-0.534420\pi$
$883$	−6.41401	−0.215849	−0.107924	+	0.994159i	$0.534420\pi$
$884$	0	0
$885$	16.6066	0.558224
$886$	0	0
$887$	18.1599	0.609748	0.304874	−	0.952393i	$-0.401386\pi$
$887$	18.1599	0.609748	0.304874	+	0.952393i	$0.401386\pi$
$888$	0	0
$889$	−66.1874	−2.21985
$890$	0	0
$891$	−5.39078	−0.180598
$892$	0	0
$893$	21.4012	0.716162
$894$	0	0
$895$	−32.6836	−1.09249
$896$	0	0
$897$	−11.2912	−0.377003
$898$	0	0
$899$	−6.65528	−0.221966
$900$	0	0
$901$	12.9676	0.432013
$902$	0	0
$903$	34.9318	1.16246
$904$	0	0
$905$	−26.6524	−0.885955
$906$	0	0
$907$	−12.9156	−0.428857	−0.214428	−	0.976740i	$-0.568789\pi$
$907$	−12.9156	−0.428857	−0.214428	+	0.976740i	$0.568789\pi$
$908$	0	0
$909$	6.27360	0.208082
$910$	0	0
$911$	−53.5389	−1.77382	−0.886912	−	0.461939i	$-0.847154\pi$
$911$	−53.5389	−1.77382	−0.886912	+	0.461939i	$0.847154\pi$
$912$	0	0
$913$	−10.1170	−0.334824
$914$	0	0
$915$	−61.4080	−2.03009
$916$	0	0
$917$	82.1392	2.71247
$918$	0	0
$919$	−10.3405	−0.341102	−0.170551	−	0.985349i	$-0.554555\pi$
$919$	−10.3405	−0.341102	−0.170551	+	0.985349i	$0.554555\pi$
$920$	0	0
$921$	22.7836	0.750744
$922$	0	0
$923$	46.4989	1.53053
$924$	0	0
$925$	17.5782	0.577969
$926$	0	0
$927$	13.1085	0.430538
$928$	0	0
$929$	36.0130	1.18155	0.590774	−	0.806837i	$-0.298823\pi$
$929$	36.0130	1.18155	0.590774	+	0.806837i	$0.298823\pi$
$930$	0	0
$931$	117.357	3.84624
$932$	0	0
$933$	−0.300500	−0.00983794
$934$	0	0
$935$	−9.04088	−0.295668
$936$	0	0
$937$	−33.6509	−1.09933	−0.549664	−	0.835386i	$-0.685244\pi$
$937$	−33.6509	−1.09933	−0.549664	+	0.835386i	$0.685244\pi$
$938$	0	0
$939$	−34.1144	−1.11328
$940$	0	0
$941$	14.3501	0.467801	0.233901	−	0.972261i	$-0.424851\pi$
$941$	14.3501	0.467801	0.233901	+	0.972261i	$0.424851\pi$
$942$	0	0
$943$	−9.18284	−0.299035
$944$	0	0
$945$	−90.3107	−2.93781
$946$	0	0
$947$	26.3227	0.855374	0.427687	−	0.903927i	$-0.359329\pi$
$947$	26.3227	0.855374	0.427687	+	0.903927i	$0.359329\pi$
$948$	0	0
$949$	−24.7846	−0.804543
$950$	0	0
$951$	3.76487	0.122084
$952$	0	0
$953$	2.47254	0.0800935	0.0400467	−	0.999198i	$-0.487249\pi$
$953$	2.47254	0.0800935	0.0400467	+	0.999198i	$0.487249\pi$
$954$	0	0
$955$	3.73514	0.120866
$956$	0	0
$957$	−9.82304	−0.317534
$958$	0	0
$959$	4.91695	0.158777
$960$	0	0
$961$	−30.0455	−0.969209
$962$	0	0
$963$	8.64024	0.278428
$964$	0	0
$965$	−76.1468	−2.45125
$966$	0	0
$967$	−1.72140	−0.0553564	−0.0276782	−	0.999617i	$-0.508811\pi$
$967$	−1.72140	−0.0553564	−0.0276782	+	0.999617i	$0.508811\pi$
$968$	0	0
$969$	27.4182	0.880798
$970$	0	0
$971$	10.7963	0.346470	0.173235	−	0.984881i	$-0.444578\pi$
$971$	10.7963	0.346470	0.173235	+	0.984881i	$0.444578\pi$
$972$	0	0
$973$	47.2613	1.51513
$974$	0	0
$975$	26.9080	0.861747
$976$	0	0
$977$	−13.0732	−0.418249	−0.209124	−	0.977889i	$-0.567061\pi$
$977$	−13.0732	−0.418249	−0.209124	+	0.977889i	$0.567061\pi$
$978$	0	0
$979$	12.0143	0.383979
$980$	0	0
$981$	2.33051	0.0744074
$982$	0	0
$983$	−16.7777	−0.535127	−0.267563	−	0.963540i	$-0.586218\pi$
$983$	−16.7777	−0.535127	−0.267563	+	0.963540i	$0.586218\pi$
$984$	0	0
$985$	−17.8901	−0.570026
$986$	0	0
$987$	22.2088	0.706914
$988$	0	0
$989$	−11.4834	−0.365151
$990$	0	0
$991$	−21.0312	−0.668078	−0.334039	−	0.942559i	$-0.608412\pi$
$991$	−21.0312	−0.668078	−0.334039	+	0.942559i	$0.608412\pi$
$992$	0	0
$993$	−1.79572	−0.0569853
$994$	0	0
$995$	−13.2673	−0.420601
$996$	0	0
$997$	37.0625	1.17378	0.586891	−	0.809666i	$-0.300352\pi$
$997$	37.0625	1.17378	0.586891	+	0.809666i	$0.300352\pi$
$998$	0	0
$999$	17.8912	0.566052

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6028.2.a.c.1.10	✓	25

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6028.2.a.c.1.10	✓	25	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)

\(f(q)\)	\(=\)	\(q-1.44202 q^{3} +3.24879 q^{5} -4.91695 q^{7} -0.920583 q^{9} +O(q^{10})\)	\(q-1.44202 q^{3} +3.24879 q^{5} -4.91695 q^{7} -0.920583 q^{9} +1.00000 q^{11} -3.35934 q^{13} -4.68482 q^{15} -2.78284 q^{17} +6.83249 q^{19} +7.09033 q^{21} -2.33086 q^{23} +5.55467 q^{25} +5.65355 q^{27} +6.81200 q^{29} -0.976993 q^{31} -1.44202 q^{33} -15.9742 q^{35} +3.16459 q^{37} +4.84422 q^{39} +3.93968 q^{41} +4.92668 q^{43} -2.99078 q^{45} +3.13226 q^{47} +17.1764 q^{49} +4.01291 q^{51} -4.65984 q^{53} +3.24879 q^{55} -9.85258 q^{57} -3.54476 q^{59} +13.1079 q^{61} +4.52646 q^{63} -10.9138 q^{65} -9.75423 q^{67} +3.36115 q^{69} -13.8417 q^{71} +7.37784 q^{73} -8.00993 q^{75} -4.91695 q^{77} -13.3747 q^{79} -5.39078 q^{81} -10.1170 q^{83} -9.04088 q^{85} -9.82304 q^{87} +12.0143 q^{89} +16.5177 q^{91} +1.40884 q^{93} +22.1974 q^{95} -13.5932 q^{97} -0.920583 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 25 q - 11 q^{3} - 2 q^{5} - 9 q^{7} + 18 q^{9}+O(q^{10}) \) 25 * q - 11 * q^3 - 2 * q^5 - 9 * q^7 + 18 * q^9	\( 25 q - 11 q^{3} - 2 q^{5} - 9 q^{7} + 18 q^{9} + 25 q^{11} - 4 q^{13} - 10 q^{15} - 19 q^{17} - 12 q^{19} + 8 q^{21} - 31 q^{23} - q^{25} - 44 q^{27} - q^{29} - 8 q^{31} - 11 q^{33} - 16 q^{35} - 14 q^{37} - 18 q^{39} - 5 q^{41} - 15 q^{43} - 15 q^{45} - 41 q^{47} + 2 q^{49} + 10 q^{51} + 4 q^{53} - 2 q^{55} - 3 q^{57} - 35 q^{59} - 4 q^{61} - 45 q^{63} - 28 q^{65} - 30 q^{67} - 3 q^{69} + 4 q^{71} - 7 q^{73} - 18 q^{75} - 9 q^{77} - 9 q^{79} + 29 q^{81} - 72 q^{83} - 33 q^{87} - 30 q^{89} - 10 q^{91} + 7 q^{93} + 9 q^{95} - 37 q^{97} + 18 q^{99}+O(q^{100}) \) 25 * q - 11 * q^3 - 2 * q^5 - 9 * q^7 + 18 * q^9 + 25 * q^11 - 4 * q^13 - 10 * q^15 - 19 * q^17 - 12 * q^19 + 8 * q^21 - 31 * q^23 - q^25 - 44 * q^27 - q^29 - 8 * q^31 - 11 * q^33 - 16 * q^35 - 14 * q^37 - 18 * q^39 - 5 * q^41 - 15 * q^43 - 15 * q^45 - 41 * q^47 + 2 * q^49 + 10 * q^51 + 4 * q^53 - 2 * q^55 - 3 * q^57 - 35 * q^59 - 4 * q^61 - 45 * q^63 - 28 * q^65 - 30 * q^67 - 3 * q^69 + 4 * q^71 - 7 * q^73 - 18 * q^75 - 9 * q^77 - 9 * q^79 + 29 * q^81 - 72 * q^83 - 33 * q^87 - 30 * q^89 - 10 * q^91 + 7 * q^93 + 9 * q^95 - 37 * q^97 + 18 * q^99

Introduction

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