LMFDB - Embedded newform 6028.2.a.c.1.3

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			1.3
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)

$f(q)$	$=$	$q-3.18883 q^{3} +2.53247 q^{5} -2.00866 q^{7} +7.16863 q^{9} +O(q^{10})$	$q-3.18883 q^{3} +2.53247 q^{5} -2.00866 q^{7} +7.16863 q^{9} +1.00000 q^{11} +1.17991 q^{13} -8.07562 q^{15} -6.92771 q^{17} +0.00144675 q^{19} +6.40527 q^{21} +5.89949 q^{23} +1.41341 q^{25} -13.2930 q^{27} +8.10817 q^{29} -4.27213 q^{31} -3.18883 q^{33} -5.08687 q^{35} -2.62778 q^{37} -3.76252 q^{39} +11.8323 q^{41} -11.0907 q^{43} +18.1544 q^{45} -8.51122 q^{47} -2.96529 q^{49} +22.0913 q^{51} +0.619922 q^{53} +2.53247 q^{55} -0.00461343 q^{57} -0.786387 q^{59} -2.03018 q^{61} -14.3993 q^{63} +2.98808 q^{65} -14.0982 q^{67} -18.8125 q^{69} +13.4659 q^{71} -5.27357 q^{73} -4.50714 q^{75} -2.00866 q^{77} +3.88035 q^{79} +20.8833 q^{81} -9.82672 q^{83} -17.5442 q^{85} -25.8556 q^{87} -12.2151 q^{89} -2.37003 q^{91} +13.6231 q^{93} +0.00366385 q^{95} +10.9613 q^{97} +7.16863 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−3.18883	−1.84107	−0.920536	−	0.390659i	$-0.872247\pi$
$3$	−3.18883	−1.84107	−0.920536	+	0.390659i	$0.872247\pi$
$4$	0	0
$5$	2.53247	1.13256	0.566278	−	0.824214i	$-0.308383\pi$
$5$	2.53247	1.13256	0.566278	+	0.824214i	$0.308383\pi$
$6$	0	0
$7$	−2.00866	−0.759201	−0.379601	−	0.925150i	$-0.623939\pi$
$7$	−2.00866	−0.759201	−0.379601	+	0.925150i	$0.623939\pi$
$8$	0	0
$9$	7.16863	2.38954
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	1.17991	0.327247	0.163624	−	0.986523i	$-0.447682\pi$
$13$	1.17991	0.327247	0.163624	+	0.986523i	$0.447682\pi$
$14$	0	0
$15$	−8.07562	−2.08512
$16$	0	0
$17$	−6.92771	−1.68022	−0.840108	−	0.542419i	$-0.817509\pi$
$17$	−6.92771	−1.68022	−0.840108	+	0.542419i	$0.817509\pi$
$18$	0	0
$19$	0.00144675	0.000331907 0	0.000165953	−	1.00000i	$-0.499947\pi$
$19$	0.00144675	0.000331907 0	0.000165953		1.00000i	$0.499947\pi$
$20$	0	0
$21$	6.40527	1.39774
$22$	0	0
$23$	5.89949	1.23013	0.615065	−	0.788477i	$-0.289130\pi$
$23$	5.89949	1.23013	0.615065	+	0.788477i	$0.289130\pi$
$24$	0	0
$25$	1.41341	0.282683
$26$	0	0
$27$	−13.2930	−2.55825
$28$	0	0
$29$	8.10817	1.50565	0.752825	−	0.658221i	$-0.228691\pi$
$29$	8.10817	1.50565	0.752825	+	0.658221i	$0.228691\pi$
$30$	0	0
$31$	−4.27213	−0.767297	−0.383649	−	0.923479i	$-0.625333\pi$
$31$	−4.27213	−0.767297	−0.383649	+	0.923479i	$0.625333\pi$
$32$	0	0
$33$	−3.18883	−0.555104
$34$	0	0
$35$	−5.08687	−0.859838
$36$	0	0
$37$	−2.62778	−0.432005	−0.216002	−	0.976393i	$-0.569302\pi$
$37$	−2.62778	−0.432005	−0.216002	+	0.976393i	$0.569302\pi$
$38$	0	0
$39$	−3.76252	−0.602486
$40$	0	0
$41$	11.8323	1.84789	0.923944	−	0.382527i	$-0.124946\pi$
$41$	11.8323	1.84789	0.923944	+	0.382527i	$0.124946\pi$
$42$	0	0
$43$	−11.0907	−1.69132	−0.845659	−	0.533724i	$-0.820792\pi$
$43$	−11.0907	−1.69132	−0.845659	+	0.533724i	$0.820792\pi$
$44$	0	0
$45$	18.1544	2.70629
$46$	0	0
$47$	−8.51122	−1.24149	−0.620744	−	0.784013i	$-0.713169\pi$
$47$	−8.51122	−1.24149	−0.620744	+	0.784013i	$0.713169\pi$
$48$	0	0
$49$	−2.96529	−0.423613
$50$	0	0
$51$	22.0913	3.09340
$52$	0	0
$53$	0.619922	0.0851528	0.0425764	−	0.999093i	$-0.486443\pi$
$53$	0.619922	0.0851528	0.0425764	+	0.999093i	$0.486443\pi$
$54$	0	0
$55$	2.53247	0.341478
$56$	0	0
$57$	−0.00461343	−0.000611064 0
$58$	0	0
$59$	−0.786387	−0.102379	−0.0511894	−	0.998689i	$-0.516301\pi$
$59$	−0.786387	−0.102379	−0.0511894	+	0.998689i	$0.516301\pi$
$60$	0	0
$61$	−2.03018	−0.259938	−0.129969	−	0.991518i	$-0.541488\pi$
$61$	−2.03018	−0.259938	−0.129969	+	0.991518i	$0.541488\pi$
$62$	0	0
$63$	−14.3993	−1.81414
$64$	0	0
$65$	2.98808	0.370626
$66$	0	0
$67$	−14.0982	−1.72236	−0.861182	−	0.508296i	$-0.830276\pi$
$67$	−14.0982	−1.72236	−0.861182	+	0.508296i	$0.830276\pi$
$68$	0	0
$69$	−18.8125	−2.26476
$70$	0	0
$71$	13.4659	1.59811	0.799056	−	0.601257i	$-0.205333\pi$
$71$	13.4659	1.59811	0.799056	+	0.601257i	$0.205333\pi$
$72$	0	0
$73$	−5.27357	−0.617225	−0.308612	−	0.951188i	$-0.599865\pi$
$73$	−5.27357	−0.617225	−0.308612	+	0.951188i	$0.599865\pi$
$74$	0	0
$75$	−4.50714	−0.520439
$76$	0	0
$77$	−2.00866	−0.228908
$78$	0	0
$79$	3.88035	0.436574	0.218287	−	0.975885i	$-0.429953\pi$
$79$	3.88035	0.436574	0.218287	+	0.975885i	$0.429953\pi$
$80$	0	0
$81$	20.8833	2.32037
$82$	0	0
$83$	−9.82672	−1.07862	−0.539311	−	0.842107i	$-0.681315\pi$
$83$	−9.82672	−1.07862	−0.539311	+	0.842107i	$0.681315\pi$
$84$	0	0
$85$	−17.5442	−1.90294
$86$	0	0
$87$	−25.8556	−2.77201
$88$	0	0
$89$	−12.2151	−1.29480	−0.647398	−	0.762152i	$-0.724143\pi$
$89$	−12.2151	−1.29480	−0.647398	+	0.762152i	$0.724143\pi$
$90$	0	0
$91$	−2.37003	−0.248447
$92$	0	0
$93$	13.6231	1.41265
$94$	0	0
$95$	0.00366385	0.000375903 0
$96$	0	0
$97$	10.9613	1.11296	0.556478	−	0.830862i	$-0.312152\pi$
$97$	10.9613	1.11296	0.556478	+	0.830862i	$0.312152\pi$
$98$	0	0
$99$	7.16863	0.720474
$100$	0	0
$101$	19.4618	1.93652	0.968262	−	0.249936i	$-0.0804097\pi$
$101$	19.4618	1.93652	0.968262	+	0.249936i	$0.0804097\pi$
$102$	0	0
$103$	0.471945	0.0465022	0.0232511	−	0.999730i	$-0.492598\pi$
$103$	0.471945	0.0465022	0.0232511	+	0.999730i	$0.492598\pi$
$104$	0	0
$105$	16.2212	1.58302
$106$	0	0
$107$	−0.921535	−0.0890882	−0.0445441	−	0.999007i	$-0.514184\pi$
$107$	−0.921535	−0.0890882	−0.0445441	+	0.999007i	$0.514184\pi$
$108$	0	0
$109$	−0.788848	−0.0755579	−0.0377789	−	0.999286i	$-0.512028\pi$
$109$	−0.788848	−0.0755579	−0.0377789	+	0.999286i	$0.512028\pi$
$110$	0	0
$111$	8.37955	0.795351
$112$	0	0
$113$	17.2604	1.62372	0.811860	−	0.583852i	$-0.198455\pi$
$113$	17.2604	1.62372	0.811860	+	0.583852i	$0.198455\pi$
$114$	0	0
$115$	14.9403	1.39319
$116$	0	0
$117$	8.45832	0.781972
$118$	0	0
$119$	13.9154	1.27562
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−37.7311	−3.40209
$124$	0	0
$125$	−9.08293	−0.812402
$126$	0	0
$127$	9.69846	0.860600	0.430300	−	0.902686i	$-0.358408\pi$
$127$	9.69846	0.860600	0.430300	+	0.902686i	$0.358408\pi$
$128$	0	0
$129$	35.3664	3.11384
$130$	0	0
$131$	13.5629	1.18499	0.592497	−	0.805573i	$-0.298142\pi$
$131$	13.5629	1.18499	0.592497	+	0.805573i	$0.298142\pi$
$132$	0	0
$133$	−0.00290602	−0.000251984 0
$134$	0	0
$135$	−33.6643	−2.89736
$136$	0	0
$137$	−1.00000	−0.0854358
$137$	−1.00000	−0.0854358
$138$	0	0
$139$	11.5395	0.978770	0.489385	−	0.872068i	$-0.337221\pi$
$139$	11.5395	0.978770	0.489385	+	0.872068i	$0.337221\pi$
$140$	0	0
$141$	27.1408	2.28567
$142$	0	0
$143$	1.17991	0.0986688
$144$	0	0
$145$	20.5337	1.70523
$146$	0	0
$147$	9.45581	0.779902
$148$	0	0
$149$	−19.8746	−1.62819	−0.814097	−	0.580729i	$-0.802768\pi$
$149$	−19.8746	−1.62819	−0.814097	+	0.580729i	$0.802768\pi$
$150$	0	0
$151$	−20.1572	−1.64037	−0.820184	−	0.572100i	$-0.806129\pi$
$151$	−20.1572	−1.64037	−0.820184	+	0.572100i	$0.806129\pi$
$152$	0	0
$153$	−49.6622	−4.01495
$154$	0	0
$155$	−10.8191	−0.869007
$156$	0	0
$157$	−8.82923	−0.704650	−0.352325	−	0.935878i	$-0.614609\pi$
$157$	−8.82923	−0.704650	−0.352325	+	0.935878i	$0.614609\pi$
$158$	0	0
$159$	−1.97682	−0.156772
$160$	0	0
$161$	−11.8501	−0.933916
$162$	0	0
$163$	25.1836	1.97253	0.986265	−	0.165173i	$-0.0528182\pi$
$163$	25.1836	1.97253	0.986265	+	0.165173i	$0.0528182\pi$
$164$	0	0
$165$	−8.07562	−0.628686
$166$	0	0
$167$	−5.14838	−0.398393	−0.199197	−	0.979960i	$-0.563833\pi$
$167$	−5.14838	−0.398393	−0.199197	+	0.979960i	$0.563833\pi$
$168$	0	0
$169$	−11.6078	−0.892909
$170$	0	0
$171$	0.0103712	0.000793105 0
$172$	0	0
$173$	−14.0018	−1.06454	−0.532270	−	0.846575i	$-0.678661\pi$
$173$	−14.0018	−1.06454	−0.532270	+	0.846575i	$0.678661\pi$
$174$	0	0
$175$	−2.83907	−0.214613
$176$	0	0
$177$	2.50765	0.188487
$178$	0	0
$179$	−2.54517	−0.190235	−0.0951174	−	0.995466i	$-0.530323\pi$
$179$	−2.54517	−0.190235	−0.0951174	+	0.995466i	$0.530323\pi$
$180$	0	0
$181$	6.94273	0.516049	0.258024	−	0.966138i	$-0.416929\pi$
$181$	6.94273	0.516049	0.258024	+	0.966138i	$0.416929\pi$
$182$	0	0
$183$	6.47390	0.478564
$184$	0	0
$185$	−6.65478	−0.489269
$186$	0	0
$187$	−6.92771	−0.506604
$188$	0	0
$189$	26.7012	1.94222
$190$	0	0
$191$	15.7836	1.14206	0.571032	−	0.820928i	$-0.306543\pi$
$191$	15.7836	1.14206	0.571032	+	0.820928i	$0.306543\pi$
$192$	0	0
$193$	7.69919	0.554200	0.277100	−	0.960841i	$-0.410627\pi$
$193$	7.69919	0.554200	0.277100	+	0.960841i	$0.410627\pi$
$194$	0	0
$195$	−9.52848	−0.682349
$196$	0	0
$197$	7.52901	0.536419	0.268210	−	0.963361i	$-0.413568\pi$
$197$	7.52901	0.536419	0.268210	+	0.963361i	$0.413568\pi$
$198$	0	0
$199$	−20.1285	−1.42687	−0.713434	−	0.700722i	$-0.752861\pi$
$199$	−20.1285	−1.42687	−0.713434	+	0.700722i	$0.752861\pi$
$200$	0	0
$201$	44.9566	3.17100
$202$	0	0
$203$	−16.2865	−1.14309
$204$	0	0
$205$	29.9649	2.09284
$206$	0	0
$207$	42.2913	2.93945
$208$	0	0
$209$	0.00144675	0.000100074 0
$210$	0	0
$211$	9.11450	0.627468	0.313734	−	0.949511i	$-0.398420\pi$
$211$	9.11450	0.627468	0.313734	+	0.949511i	$0.398420\pi$
$212$	0	0
$213$	−42.9406	−2.94224
$214$	0	0
$215$	−28.0869	−1.91551
$216$	0	0
$217$	8.58125	0.582533
$218$	0	0
$219$	16.8165	1.13635
$220$	0	0
$221$	−8.17405	−0.549846
$222$	0	0
$223$	−13.1541	−0.880863	−0.440432	−	0.897786i	$-0.645175\pi$
$223$	−13.1541	−0.880863	−0.440432	+	0.897786i	$0.645175\pi$
$224$	0	0
$225$	10.1322	0.675483
$226$	0	0
$227$	12.6211	0.837692	0.418846	−	0.908057i	$-0.362435\pi$
$227$	12.6211	0.837692	0.418846	+	0.908057i	$0.362435\pi$
$228$	0	0
$229$	−25.6800	−1.69698	−0.848492	−	0.529208i	$-0.822489\pi$
$229$	−25.6800	−1.69698	−0.848492	+	0.529208i	$0.822489\pi$
$230$	0	0
$231$	6.40527	0.421436
$232$	0	0
$233$	−2.46413	−0.161431	−0.0807154	−	0.996737i	$-0.525720\pi$
$233$	−2.46413	−0.161431	−0.0807154	+	0.996737i	$0.525720\pi$
$234$	0	0
$235$	−21.5544	−1.40606
$236$	0	0
$237$	−12.3738	−0.803764
$238$	0	0
$239$	−8.50818	−0.550348	−0.275174	−	0.961394i	$-0.588736\pi$
$239$	−8.50818	−0.550348	−0.275174	+	0.961394i	$0.588736\pi$
$240$	0	0
$241$	−27.1661	−1.74992	−0.874962	−	0.484191i	$-0.839114\pi$
$241$	−27.1661	−1.74992	−0.874962	+	0.484191i	$0.839114\pi$
$242$	0	0
$243$	−26.7143	−1.71372
$244$	0	0
$245$	−7.50952	−0.479766
$246$	0	0
$247$	0.00170703	0.000108616 0
$248$	0	0
$249$	31.3357	1.98582
$250$	0	0
$251$	−14.3715	−0.907119	−0.453559	−	0.891226i	$-0.649846\pi$
$251$	−14.3715	−0.907119	−0.453559	+	0.891226i	$0.649846\pi$
$252$	0	0
$253$	5.89949	0.370898
$254$	0	0
$255$	55.9455	3.50345
$256$	0	0
$257$	11.6369	0.725888	0.362944	−	0.931811i	$-0.381772\pi$
$257$	11.6369	0.725888	0.362944	+	0.931811i	$0.381772\pi$
$258$	0	0
$259$	5.27832	0.327979
$260$	0	0
$261$	58.1244	3.59781
$262$	0	0
$263$	−3.31677	−0.204521	−0.102260	−	0.994758i	$-0.532608\pi$
$263$	−3.31677	−0.204521	−0.102260	+	0.994758i	$0.532608\pi$
$264$	0	0
$265$	1.56993	0.0964403
$266$	0	0
$267$	38.9518	2.38381
$268$	0	0
$269$	−8.50557	−0.518594	−0.259297	−	0.965798i	$-0.583491\pi$
$269$	−8.50557	−0.518594	−0.259297	+	0.965798i	$0.583491\pi$
$270$	0	0
$271$	−10.2656	−0.623592	−0.311796	−	0.950149i	$-0.600930\pi$
$271$	−10.2656	−0.623592	−0.311796	+	0.950149i	$0.600930\pi$
$272$	0	0
$273$	7.55762	0.457408
$274$	0	0
$275$	1.41341	0.0852321
$276$	0	0
$277$	−30.4137	−1.82738	−0.913691	−	0.406411i	$-0.866780\pi$
$277$	−30.4137	−1.82738	−0.913691	+	0.406411i	$0.866780\pi$
$278$	0	0
$279$	−30.6253	−1.83349
$280$	0	0
$281$	−1.29153	−0.0770460	−0.0385230	−	0.999258i	$-0.512265\pi$
$281$	−1.29153	−0.0770460	−0.0385230	+	0.999258i	$0.512265\pi$
$282$	0	0
$283$	−12.4034	−0.737308	−0.368654	−	0.929567i	$-0.620181\pi$
$283$	−12.4034	−0.737308	−0.368654	+	0.929567i	$0.620181\pi$
$284$	0	0
$285$	−0.0116834	−0.000692064 0
$286$	0	0
$287$	−23.7670	−1.40292
$288$	0	0
$289$	30.9932	1.82313
$290$	0	0
$291$	−34.9539	−2.04903
$292$	0	0
$293$	29.5048	1.72369	0.861845	−	0.507172i	$-0.169309\pi$
$293$	29.5048	1.72369	0.861845	+	0.507172i	$0.169309\pi$
$294$	0	0
$295$	−1.99150	−0.115950
$296$	0	0
$297$	−13.2930	−0.771341
$298$	0	0
$299$	6.96085	0.402557
$300$	0	0
$301$	22.2774	1.28405
$302$	0	0
$303$	−62.0604	−3.56528
$304$	0	0
$305$	−5.14138	−0.294394
$306$	0	0
$307$	−17.2243	−0.983043	−0.491522	−	0.870865i	$-0.663559\pi$
$307$	−17.2243	−0.983043	−0.491522	+	0.870865i	$0.663559\pi$
$308$	0	0
$309$	−1.50495	−0.0856138
$310$	0	0
$311$	−6.36695	−0.361037	−0.180518	−	0.983572i	$-0.557778\pi$
$311$	−6.36695	−0.361037	−0.180518	+	0.983572i	$0.557778\pi$
$312$	0	0
$313$	−17.8907	−1.01124	−0.505622	−	0.862755i	$-0.668737\pi$
$313$	−17.8907	−1.01124	−0.505622	+	0.862755i	$0.668737\pi$
$314$	0	0
$315$	−36.4659	−2.05462
$316$	0	0
$317$	−12.3582	−0.694103	−0.347051	−	0.937846i	$-0.612817\pi$
$317$	−12.3582	−0.694103	−0.347051	+	0.937846i	$0.612817\pi$
$318$	0	0
$319$	8.10817	0.453970
$320$	0	0
$321$	2.93862	0.164018
$322$	0	0
$323$	−0.0100226	−0.000557675 0
$324$	0	0
$325$	1.66770	0.0925073
$326$	0	0
$327$	2.51550	0.139107
$328$	0	0
$329$	17.0961	0.942540
$330$	0	0
$331$	−32.8058	−1.80317	−0.901585	−	0.432602i	$-0.857595\pi$
$331$	−32.8058	−1.80317	−0.901585	+	0.432602i	$0.857595\pi$
$332$	0	0
$333$	−18.8376	−1.03229
$334$	0	0
$335$	−35.7032	−1.95067
$336$	0	0
$337$	12.0167	0.654591	0.327295	−	0.944922i	$-0.393863\pi$
$337$	12.0167	0.654591	0.327295	+	0.944922i	$0.393863\pi$
$338$	0	0
$339$	−55.0404	−2.98938
$340$	0	0
$341$	−4.27213	−0.231349
$342$	0	0
$343$	20.0169	1.08081
$344$	0	0
$345$	−47.6421	−2.56496
$346$	0	0
$347$	11.2252	0.602599	0.301300	−	0.953530i	$-0.402580\pi$
$347$	11.2252	0.602599	0.301300	+	0.953530i	$0.402580\pi$
$348$	0	0
$349$	−5.54490	−0.296812	−0.148406	−	0.988927i	$-0.547414\pi$
$349$	−5.54490	−0.296812	−0.148406	+	0.988927i	$0.547414\pi$
$350$	0	0
$351$	−15.6846	−0.837180
$352$	0	0
$353$	18.8474	1.00315	0.501573	−	0.865115i	$-0.332755\pi$
$353$	18.8474	1.00315	0.501573	+	0.865115i	$0.332755\pi$
$354$	0	0
$355$	34.1021	1.80995
$356$	0	0
$357$	−44.3738	−2.34851
$358$	0	0
$359$	−26.2873	−1.38739	−0.693696	−	0.720268i	$-0.744019\pi$
$359$	−26.2873	−1.38739	−0.693696	+	0.720268i	$0.744019\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	0	0
$363$	−3.18883	−0.167370
$364$	0	0
$365$	−13.3552	−0.699042
$366$	0	0
$367$	−23.6363	−1.23380	−0.616902	−	0.787040i	$-0.711612\pi$
$367$	−23.6363	−1.23380	−0.616902	+	0.787040i	$0.711612\pi$
$368$	0	0
$369$	84.8211	4.41561
$370$	0	0
$371$	−1.24521	−0.0646481
$372$	0	0
$373$	−8.61850	−0.446249	−0.223124	−	0.974790i	$-0.571626\pi$
$373$	−8.61850	−0.446249	−0.223124	+	0.974790i	$0.571626\pi$
$374$	0	0
$375$	28.9639	1.49569
$376$	0	0
$377$	9.56689	0.492720
$378$	0	0
$379$	−14.5898	−0.749430	−0.374715	−	0.927140i	$-0.622259\pi$
$379$	−14.5898	−0.749430	−0.374715	+	0.927140i	$0.622259\pi$
$380$	0	0
$381$	−30.9267	−1.58442
$382$	0	0
$383$	22.4891	1.14914	0.574571	−	0.818455i	$-0.305169\pi$
$383$	22.4891	1.14914	0.574571	+	0.818455i	$0.305169\pi$
$384$	0	0
$385$	−5.08687	−0.259251
$386$	0	0
$387$	−79.5052	−4.04147
$388$	0	0
$389$	7.09692	0.359828	0.179914	−	0.983682i	$-0.442418\pi$
$389$	7.09692	0.359828	0.179914	+	0.983682i	$0.442418\pi$
$390$	0	0
$391$	−40.8700	−2.06688
$392$	0	0
$393$	−43.2497	−2.18166
$394$	0	0
$395$	9.82689	0.494445
$396$	0	0
$397$	−2.44396	−0.122659	−0.0613294	−	0.998118i	$-0.519534\pi$
$397$	−2.44396	−0.122659	−0.0613294	+	0.998118i	$0.519534\pi$
$398$	0	0
$399$	0.00926680	0.000463920 0
$400$	0	0
$401$	−18.6969	−0.933680	−0.466840	−	0.884342i	$-0.654608\pi$
$401$	−18.6969	−0.933680	−0.466840	+	0.884342i	$0.654608\pi$
$402$	0	0
$403$	−5.04072	−0.251096
$404$	0	0
$405$	52.8865	2.62795
$406$	0	0
$407$	−2.62778	−0.130254
$408$	0	0
$409$	10.0158	0.495251	0.247626	−	0.968856i	$-0.420350\pi$
$409$	10.0158	0.495251	0.247626	+	0.968856i	$0.420350\pi$
$410$	0	0
$411$	3.18883	0.157293
$412$	0	0
$413$	1.57958	0.0777261
$414$	0	0
$415$	−24.8859	−1.22160
$416$	0	0
$417$	−36.7976	−1.80198
$418$	0	0
$419$	−32.9907	−1.61170	−0.805852	−	0.592117i	$-0.798292\pi$
$419$	−32.9907	−1.61170	−0.805852	+	0.592117i	$0.798292\pi$
$420$	0	0
$421$	9.59037	0.467406	0.233703	−	0.972308i	$-0.424916\pi$
$421$	9.59037	0.467406	0.233703	+	0.972308i	$0.424916\pi$
$422$	0	0
$423$	−61.0137	−2.96659
$424$	0	0
$425$	−9.79173	−0.474969
$426$	0	0
$427$	4.07794	0.197345
$428$	0	0
$429$	−3.76252	−0.181656
$430$	0	0
$431$	−38.5184	−1.85536	−0.927682	−	0.373370i	$-0.878202\pi$
$431$	−38.5184	−1.85536	−0.927682	+	0.373370i	$0.878202\pi$
$432$	0	0
$433$	−30.3014	−1.45619	−0.728096	−	0.685476i	$-0.759594\pi$
$433$	−30.3014	−1.45619	−0.728096	+	0.685476i	$0.759594\pi$
$434$	0	0
$435$	−65.4785	−3.13945
$436$	0	0
$437$	0.00853508	0.000408288 0
$438$	0	0
$439$	−11.6859	−0.557738	−0.278869	−	0.960329i	$-0.589959\pi$
$439$	−11.6859	−0.557738	−0.278869	+	0.960329i	$0.589959\pi$
$440$	0	0
$441$	−21.2571	−1.01224
$442$	0	0
$443$	12.9914	0.617239	0.308620	−	0.951186i	$-0.400133\pi$
$443$	12.9914	0.617239	0.308620	+	0.951186i	$0.400133\pi$
$444$	0	0
$445$	−30.9343	−1.46643
$446$	0	0
$447$	63.3768	2.99762
$448$	0	0
$449$	−14.2516	−0.672573	−0.336287	−	0.941760i	$-0.609171\pi$
$449$	−14.2516	−0.672573	−0.336287	+	0.941760i	$0.609171\pi$
$450$	0	0
$451$	11.8323	0.557159
$452$	0	0
$453$	64.2778	3.02003
$454$	0	0
$455$	−6.00204	−0.281380
$456$	0	0
$457$	−16.5160	−0.772585	−0.386293	−	0.922376i	$-0.626245\pi$
$457$	−16.5160	−0.772585	−0.386293	+	0.922376i	$0.626245\pi$
$458$	0	0
$459$	92.0903	4.29841
$460$	0	0
$461$	14.1868	0.660745	0.330372	−	0.943851i	$-0.392826\pi$
$461$	14.1868	0.660745	0.330372	+	0.943851i	$0.392826\pi$
$462$	0	0
$463$	14.4619	0.672103	0.336051	−	0.941844i	$-0.390908\pi$
$463$	14.4619	0.672103	0.336051	+	0.941844i	$0.390908\pi$
$464$	0	0
$465$	34.5001	1.59990
$466$	0	0
$467$	−20.3473	−0.941559	−0.470780	−	0.882251i	$-0.656027\pi$
$467$	−20.3473	−0.941559	−0.470780	+	0.882251i	$0.656027\pi$
$468$	0	0
$469$	28.3184	1.30762
$470$	0	0
$471$	28.1549	1.29731
$472$	0	0
$473$	−11.0907	−0.509951
$474$	0	0
$475$	0.00204485	9.38244e−5 0
$476$	0	0
$477$	4.44399	0.203476
$478$	0	0
$479$	−0.0693037	−0.00316657	−0.00158328	−	0.999999i	$-0.500504\pi$
$479$	−0.0693037	−0.00316657	−0.00158328	+	0.999999i	$0.500504\pi$
$480$	0	0
$481$	−3.10054	−0.141372
$482$	0	0
$483$	37.7878	1.71941
$484$	0	0
$485$	27.7593	1.26048
$486$	0	0
$487$	6.32928	0.286807	0.143404	−	0.989664i	$-0.454195\pi$
$487$	6.32928	0.286807	0.143404	+	0.989664i	$0.454195\pi$
$488$	0	0
$489$	−80.3061	−3.63157
$490$	0	0
$491$	−17.3140	−0.781370	−0.390685	−	0.920525i	$-0.627762\pi$
$491$	−17.3140	−0.781370	−0.390685	+	0.920525i	$0.627762\pi$
$492$	0	0
$493$	−56.1710	−2.52982
$494$	0	0
$495$	18.1544	0.815977
$496$	0	0
$497$	−27.0485	−1.21329
$498$	0	0
$499$	25.1974	1.12799	0.563996	−	0.825778i	$-0.309264\pi$
$499$	25.1974	1.12799	0.563996	+	0.825778i	$0.309264\pi$
$500$	0	0
$501$	16.4173	0.733471
$502$	0	0
$503$	21.3612	0.952447	0.476224	−	0.879324i	$-0.342005\pi$
$503$	21.3612	0.952447	0.476224	+	0.879324i	$0.342005\pi$
$504$	0	0
$505$	49.2865	2.19322
$506$	0	0
$507$	37.0153	1.64391
$508$	0	0
$509$	−24.9733	−1.10692	−0.553462	−	0.832874i	$-0.686694\pi$
$509$	−24.9733	−1.10692	−0.553462	+	0.832874i	$0.686694\pi$
$510$	0	0
$511$	10.5928	0.468598
$512$	0	0
$513$	−0.0192317	−0.000849099 0
$514$	0	0
$515$	1.19519	0.0526663
$516$	0	0
$517$	−8.51122	−0.374323
$518$	0	0
$519$	44.6494	1.95989
$520$	0	0
$521$	−28.3191	−1.24068	−0.620341	−	0.784332i	$-0.713006\pi$
$521$	−28.3191	−1.24068	−0.620341	+	0.784332i	$0.713006\pi$
$522$	0	0
$523$	−25.5320	−1.11644	−0.558219	−	0.829694i	$-0.688515\pi$
$523$	−25.5320	−1.11644	−0.558219	+	0.829694i	$0.688515\pi$
$524$	0	0
$525$	9.05330	0.395118
$526$	0	0
$527$	29.5961	1.28923
$528$	0	0
$529$	11.8040	0.513219
$530$	0	0
$531$	−5.63731	−0.244639
$532$	0	0
$533$	13.9610	0.604717
$534$	0	0
$535$	−2.33376	−0.100897
$536$	0	0
$537$	8.11610	0.350236
$538$	0	0
$539$	−2.96529	−0.127724
$540$	0	0
$541$	13.6100	0.585138	0.292569	−	0.956244i	$-0.405490\pi$
$541$	13.6100	0.585138	0.292569	+	0.956244i	$0.405490\pi$
$542$	0	0
$543$	−22.1392	−0.950083
$544$	0	0
$545$	−1.99773	−0.0855735
$546$	0	0
$547$	9.56754	0.409079	0.204539	−	0.978858i	$-0.434430\pi$
$547$	9.56754	0.409079	0.204539	+	0.978858i	$0.434430\pi$
$548$	0	0
$549$	−14.5536	−0.621133
$550$	0	0
$551$	0.0117305	0.000499735 0
$552$	0	0
$553$	−7.79431	−0.331448
$554$	0	0
$555$	21.2210	0.900780
$556$	0	0
$557$	36.3197	1.53891	0.769457	−	0.638699i	$-0.220527\pi$
$557$	36.3197	1.53891	0.769457	+	0.638699i	$0.220527\pi$
$558$	0	0
$559$	−13.0860	−0.553479
$560$	0	0
$561$	22.0913	0.932695
$562$	0	0
$563$	−11.2400	−0.473708	−0.236854	−	0.971545i	$-0.576116\pi$
$563$	−11.2400	−0.473708	−0.236854	+	0.971545i	$0.576116\pi$
$564$	0	0
$565$	43.7114	1.83895
$566$	0	0
$567$	−41.9475	−1.76163
$568$	0	0
$569$	31.4972	1.32043	0.660216	−	0.751076i	$-0.270465\pi$
$569$	31.4972	1.32043	0.660216	+	0.751076i	$0.270465\pi$
$570$	0	0
$571$	37.9692	1.58896	0.794481	−	0.607289i	$-0.207743\pi$
$571$	37.9692	1.58896	0.794481	+	0.607289i	$0.207743\pi$
$572$	0	0
$573$	−50.3314	−2.10262
$574$	0	0
$575$	8.33843	0.347737
$576$	0	0
$577$	−36.5144	−1.52012	−0.760058	−	0.649855i	$-0.774829\pi$
$577$	−36.5144	−1.52012	−0.760058	+	0.649855i	$0.774829\pi$
$578$	0	0
$579$	−24.5514	−1.02032
$580$	0	0
$581$	19.7385	0.818892
$582$	0	0
$583$	0.619922	0.0256745
$584$	0	0
$585$	21.4204	0.885627
$586$	0	0
$587$	29.6265	1.22282	0.611408	−	0.791315i	$-0.290603\pi$
$587$	29.6265	1.22282	0.611408	+	0.791315i	$0.290603\pi$
$588$	0	0
$589$	−0.00618069	−0.000254671 0
$590$	0	0
$591$	−24.0087	−0.987586
$592$	0	0
$593$	−23.8538	−0.979558	−0.489779	−	0.871847i	$-0.662923\pi$
$593$	−23.8538	−0.979558	−0.489779	+	0.871847i	$0.662923\pi$
$594$	0	0
$595$	35.2404	1.44471
$596$	0	0
$597$	64.1862	2.62697
$598$	0	0
$599$	−24.0883	−0.984222	−0.492111	−	0.870533i	$-0.663775\pi$
$599$	−24.0883	−0.984222	−0.492111	+	0.870533i	$0.663775\pi$
$600$	0	0
$601$	14.8227	0.604629	0.302314	−	0.953208i	$-0.402241\pi$
$601$	14.8227	0.604629	0.302314	+	0.953208i	$0.402241\pi$
$602$	0	0
$603$	−101.065	−4.11566
$604$	0	0
$605$	2.53247	0.102960
$606$	0	0
$607$	−0.0495075	−0.00200945	−0.00100472	−	0.999999i	$-0.500320\pi$
$607$	−0.0495075	−0.00200945	−0.00100472	+	0.999999i	$0.500320\pi$
$608$	0	0
$609$	51.9350	2.10451
$610$	0	0
$611$	−10.0424	−0.406274
$612$	0	0
$613$	−42.7373	−1.72614	−0.863071	−	0.505083i	$-0.831462\pi$
$613$	−42.7373	−1.72614	−0.863071	+	0.505083i	$0.831462\pi$
$614$	0	0
$615$	−95.5528	−3.85306
$616$	0	0
$617$	35.7233	1.43816	0.719082	−	0.694925i	$-0.244562\pi$
$617$	35.7233	1.43816	0.719082	+	0.694925i	$0.244562\pi$
$618$	0	0
$619$	26.5074	1.06542	0.532711	−	0.846297i	$-0.321173\pi$
$619$	26.5074	1.06542	0.532711	+	0.846297i	$0.321173\pi$
$620$	0	0
$621$	−78.4222	−3.14698
$622$	0	0
$623$	24.5359	0.983010
$624$	0	0
$625$	−30.0693	−1.20277
$626$	0	0
$627$	−0.00461343	−0.000184243 0
$628$	0	0
$629$	18.2045	0.725861
$630$	0	0
$631$	−30.3977	−1.21011	−0.605056	−	0.796183i	$-0.706849\pi$
$631$	−30.3977	−1.21011	−0.605056	+	0.796183i	$0.706849\pi$
$632$	0	0
$633$	−29.0646	−1.15521
$634$	0	0
$635$	24.5611	0.974677
$636$	0	0
$637$	−3.49877	−0.138626
$638$	0	0
$639$	96.5323	3.81876
$640$	0	0
$641$	−37.5268	−1.48222	−0.741109	−	0.671385i	$-0.765700\pi$
$641$	−37.5268	−1.48222	−0.741109	+	0.671385i	$0.765700\pi$
$642$	0	0
$643$	−0.646114	−0.0254802	−0.0127401	−	0.999919i	$-0.504055\pi$
$643$	−0.646114	−0.0254802	−0.0127401	+	0.999919i	$0.504055\pi$
$644$	0	0
$645$	89.5643	3.52659
$646$	0	0
$647$	2.32508	0.0914085	0.0457042	−	0.998955i	$-0.485447\pi$
$647$	2.32508	0.0914085	0.0457042	+	0.998955i	$0.485447\pi$
$648$	0	0
$649$	−0.786387	−0.0308684
$650$	0	0
$651$	−27.3641	−1.07249
$652$	0	0
$653$	0.676102	0.0264579	0.0132290	−	0.999912i	$-0.495789\pi$
$653$	0.676102	0.0264579	0.0132290	+	0.999912i	$0.495789\pi$
$654$	0	0
$655$	34.3476	1.34207
$656$	0	0
$657$	−37.8043	−1.47489
$658$	0	0
$659$	5.18343	0.201918	0.100959	−	0.994891i	$-0.467809\pi$
$659$	5.18343	0.201918	0.100959	+	0.994891i	$0.467809\pi$
$660$	0	0
$661$	−30.1140	−1.17130	−0.585649	−	0.810565i	$-0.699160\pi$
$661$	−30.1140	−1.17130	−0.585649	+	0.810565i	$0.699160\pi$
$662$	0	0
$663$	26.0657	1.01231
$664$	0	0
$665$	−0.00735942	−0.000285386 0
$666$	0	0
$667$	47.8341	1.85214
$668$	0	0
$669$	41.9461	1.62173
$670$	0	0
$671$	−2.03018	−0.0783743
$672$	0	0
$673$	2.46060	0.0948491	0.0474245	−	0.998875i	$-0.484899\pi$
$673$	2.46060	0.0948491	0.0474245	+	0.998875i	$0.484899\pi$
$674$	0	0
$675$	−18.7886	−0.723173
$676$	0	0
$677$	35.4991	1.36434	0.682170	−	0.731193i	$-0.261036\pi$
$677$	35.4991	1.36434	0.682170	+	0.731193i	$0.261036\pi$
$678$	0	0
$679$	−22.0176	−0.844958
$680$	0	0
$681$	−40.2465	−1.54225
$682$	0	0
$683$	42.2267	1.61576	0.807879	−	0.589348i	$-0.200615\pi$
$683$	42.2267	1.61576	0.807879	+	0.589348i	$0.200615\pi$
$684$	0	0
$685$	−2.53247	−0.0967608
$686$	0	0
$687$	81.8892	3.12427
$688$	0	0
$689$	0.731450	0.0278660
$690$	0	0
$691$	36.3510	1.38286	0.691429	−	0.722445i	$-0.256982\pi$
$691$	36.3510	1.38286	0.691429	+	0.722445i	$0.256982\pi$
$692$	0	0
$693$	−14.3993	−0.546985
$694$	0	0
$695$	29.2235	1.10851
$696$	0	0
$697$	−81.9705	−3.10485
$698$	0	0
$699$	7.85770	0.297205
$700$	0	0
$701$	22.8493	0.863006	0.431503	−	0.902112i	$-0.357983\pi$
$701$	22.8493	0.863006	0.431503	+	0.902112i	$0.357983\pi$
$702$	0	0
$703$	−0.00380174	−0.000143385 0
$704$	0	0
$705$	68.7333	2.58865
$706$	0	0
$707$	−39.0922	−1.47021
$708$	0	0
$709$	−12.7970	−0.480600	−0.240300	−	0.970699i	$-0.577246\pi$
$709$	−12.7970	−0.480600	−0.240300	+	0.970699i	$0.577246\pi$
$710$	0	0
$711$	27.8168	1.04321
$712$	0	0
$713$	−25.2034	−0.943875
$714$	0	0
$715$	2.98808	0.111748
$716$	0	0
$717$	27.1311	1.01323
$718$	0	0
$719$	−45.2362	−1.68702	−0.843512	−	0.537110i	$-0.819516\pi$
$719$	−45.2362	−1.68702	−0.843512	+	0.537110i	$0.819516\pi$
$720$	0	0
$721$	−0.947977	−0.0353045
$722$	0	0
$723$	86.6282	3.22174
$724$	0	0
$725$	11.4602	0.425621
$726$	0	0
$727$	12.5216	0.464399	0.232200	−	0.972668i	$-0.425408\pi$
$727$	12.5216	0.464399	0.232200	+	0.972668i	$0.425408\pi$
$728$	0	0
$729$	22.5373	0.834713
$730$	0	0
$731$	76.8332	2.84178
$732$	0	0
$733$	15.9194	0.587997	0.293999	−	0.955806i	$-0.405014\pi$
$733$	15.9194	0.587997	0.293999	+	0.955806i	$0.405014\pi$
$734$	0	0
$735$	23.9466	0.883282
$736$	0	0
$737$	−14.0982	−0.519313
$738$	0	0
$739$	13.8114	0.508061	0.254030	−	0.967196i	$-0.418244\pi$
$739$	13.8114	0.508061	0.254030	+	0.967196i	$0.418244\pi$
$740$	0	0
$741$	−0.00544342	−0.000199969 0
$742$	0	0
$743$	−20.5525	−0.753997	−0.376998	−	0.926214i	$-0.623044\pi$
$743$	−20.5525	−0.753997	−0.376998	+	0.926214i	$0.623044\pi$
$744$	0	0
$745$	−50.3320	−1.84402
$746$	0	0
$747$	−70.4441	−2.57741
$748$	0	0
$749$	1.85105	0.0676359
$750$	0	0
$751$	−49.9005	−1.82090	−0.910448	−	0.413624i	$-0.864263\pi$
$751$	−49.9005	−1.82090	−0.910448	+	0.413624i	$0.864263\pi$
$752$	0	0
$753$	45.8281	1.67007
$754$	0	0
$755$	−51.0475	−1.85781
$756$	0	0
$757$	−33.2474	−1.20840	−0.604198	−	0.796834i	$-0.706506\pi$
$757$	−33.2474	−1.20840	−0.604198	+	0.796834i	$0.706506\pi$
$758$	0	0
$759$	−18.8125	−0.682850
$760$	0	0
$761$	19.4053	0.703440	0.351720	−	0.936105i	$-0.385597\pi$
$761$	19.4053	0.703440	0.351720	+	0.936105i	$0.385597\pi$
$762$	0	0
$763$	1.58453	0.0573637
$764$	0	0
$765$	−125.768	−4.54715
$766$	0	0
$767$	−0.927863	−0.0335032
$768$	0	0
$769$	37.6537	1.35783	0.678914	−	0.734218i	$-0.262451\pi$
$769$	37.6537	1.35783	0.678914	+	0.734218i	$0.262451\pi$
$770$	0	0
$771$	−37.1080	−1.33641
$772$	0	0
$773$	−16.4487	−0.591619	−0.295810	−	0.955247i	$-0.595589\pi$
$773$	−16.4487	−0.591619	−0.295810	+	0.955247i	$0.595589\pi$
$774$	0	0
$775$	−6.03829	−0.216902
$776$	0	0
$777$	−16.8316	−0.603832
$778$	0	0
$779$	0.0171183	0.000613327 0
$780$	0	0
$781$	13.4659	0.481849
$782$	0	0
$783$	−107.782	−3.85182
$784$	0	0
$785$	−22.3598	−0.798055
$786$	0	0
$787$	−12.4653	−0.444342	−0.222171	−	0.975008i	$-0.571314\pi$
$787$	−12.4653	−0.444342	−0.222171	+	0.975008i	$0.571314\pi$
$788$	0	0
$789$	10.5766	0.376538
$790$	0	0
$791$	−34.6702	−1.23273
$792$	0	0
$793$	−2.39542	−0.0850640
$794$	0	0
$795$	−5.00625	−0.177553
$796$	0	0
$797$	4.62584	0.163856	0.0819278	−	0.996638i	$-0.473892\pi$
$797$	4.62584	0.163856	0.0819278	+	0.996638i	$0.473892\pi$
$798$	0	0
$799$	58.9632	2.08597
$800$	0	0
$801$	−87.5653	−3.09397
$802$	0	0
$803$	−5.27357	−0.186100
$804$	0	0
$805$	−30.0100	−1.05771
$806$	0	0
$807$	27.1228	0.954768
$808$	0	0
$809$	−47.8627	−1.68276	−0.841381	−	0.540442i	$-0.818257\pi$
$809$	−47.8627	−1.68276	−0.841381	+	0.540442i	$0.818257\pi$
$810$	0	0
$811$	−34.3111	−1.20483	−0.602414	−	0.798184i	$-0.705794\pi$
$811$	−34.3111	−1.20483	−0.602414	+	0.798184i	$0.705794\pi$
$812$	0	0
$813$	32.7353	1.14808
$814$	0	0
$815$	63.7767	2.23400
$816$	0	0
$817$	−0.0160455	−0.000561359 0
$818$	0	0
$819$	−16.9899	−0.593674
$820$	0	0
$821$	32.0162	1.11737	0.558686	−	0.829379i	$-0.311306\pi$
$821$	32.0162	1.11737	0.558686	+	0.829379i	$0.311306\pi$
$822$	0	0
$823$	−18.6018	−0.648418	−0.324209	−	0.945986i	$-0.605098\pi$
$823$	−18.6018	−0.648418	−0.324209	+	0.945986i	$0.605098\pi$
$824$	0	0
$825$	−4.50714	−0.156918
$826$	0	0
$827$	−22.4610	−0.781045	−0.390522	−	0.920593i	$-0.627706\pi$
$827$	−22.4610	−0.781045	−0.390522	+	0.920593i	$0.627706\pi$
$828$	0	0
$829$	−28.2795	−0.982189	−0.491095	−	0.871106i	$-0.663403\pi$
$829$	−28.2795	−0.982189	−0.491095	+	0.871106i	$0.663403\pi$
$830$	0	0
$831$	96.9840	3.36434
$832$	0	0
$833$	20.5427	0.711762
$834$	0	0
$835$	−13.0381	−0.451203
$836$	0	0
$837$	56.7896	1.96294
$838$	0	0
$839$	−47.7127	−1.64722	−0.823612	−	0.567153i	$-0.808045\pi$
$839$	−47.7127	−1.64722	−0.823612	+	0.567153i	$0.808045\pi$
$840$	0	0
$841$	36.7424	1.26698
$842$	0	0
$843$	4.11846	0.141847
$844$	0	0
$845$	−29.3965	−1.01127
$846$	0	0
$847$	−2.00866	−0.0690183
$848$	0	0
$849$	39.5524	1.35744
$850$	0	0
$851$	−15.5026	−0.531422
$852$	0	0
$853$	−12.4622	−0.426696	−0.213348	−	0.976976i	$-0.568437\pi$
$853$	−12.4622	−0.426696	−0.213348	+	0.976976i	$0.568437\pi$
$854$	0	0
$855$	0.0262648	0.000898236 0
$856$	0	0
$857$	−54.9882	−1.87836	−0.939180	−	0.343426i	$-0.888413\pi$
$857$	−54.9882	−1.87836	−0.939180	+	0.343426i	$0.888413\pi$
$858$	0	0
$859$	5.48267	0.187066	0.0935331	−	0.995616i	$-0.470184\pi$
$859$	5.48267	0.187066	0.0935331	+	0.995616i	$0.470184\pi$
$860$	0	0
$861$	75.7888	2.58288
$862$	0	0
$863$	21.5435	0.733349	0.366674	−	0.930349i	$-0.380496\pi$
$863$	21.5435	0.733349	0.366674	+	0.930349i	$0.380496\pi$
$864$	0	0
$865$	−35.4592	−1.20565
$866$	0	0
$867$	−98.8319	−3.35651
$868$	0	0
$869$	3.88035	0.131632
$870$	0	0
$871$	−16.6345	−0.563639
$872$	0	0
$873$	78.5778	2.65946
$874$	0	0
$875$	18.2445	0.616777
$876$	0	0
$877$	14.0203	0.473431	0.236715	−	0.971579i	$-0.423929\pi$
$877$	14.0203	0.473431	0.236715	+	0.971579i	$0.423929\pi$
$878$	0	0
$879$	−94.0858	−3.17343
$880$	0	0
$881$	−27.5610	−0.928555	−0.464277	−	0.885690i	$-0.653686\pi$
$881$	−27.5610	−0.928555	−0.464277	+	0.885690i	$0.653686\pi$
$882$	0	0
$883$	−39.6401	−1.33400	−0.666999	−	0.745059i	$-0.732421\pi$
$883$	−39.6401	−1.33400	−0.666999	+	0.745059i	$0.732421\pi$
$884$	0	0
$885$	6.35056	0.213472
$886$	0	0
$887$	−13.1634	−0.441983	−0.220992	−	0.975276i	$-0.570929\pi$
$887$	−13.1634	−0.441983	−0.220992	+	0.975276i	$0.570929\pi$
$888$	0	0
$889$	−19.4809	−0.653368
$890$	0	0
$891$	20.8833	0.699618
$892$	0	0
$893$	−0.0123136	−0.000412058 0
$894$	0	0
$895$	−6.44557	−0.215451
$896$	0	0
$897$	−22.1970	−0.741135
$898$	0	0
$899$	−34.6392	−1.15528
$900$	0	0
$901$	−4.29464	−0.143075
$902$	0	0
$903$	−71.0390	−2.36403
$904$	0	0
$905$	17.5823	0.584454
$906$	0	0
$907$	28.1326	0.934128	0.467064	−	0.884223i	$-0.345312\pi$
$907$	28.1326	0.934128	0.467064	+	0.884223i	$0.345312\pi$
$908$	0	0
$909$	139.515	4.62741
$910$	0	0
$911$	38.1728	1.26472	0.632360	−	0.774675i	$-0.282086\pi$
$911$	38.1728	1.26472	0.632360	+	0.774675i	$0.282086\pi$
$912$	0	0
$913$	−9.82672	−0.325217
$914$	0	0
$915$	16.3950	0.542001
$916$	0	0
$917$	−27.2432	−0.899649
$918$	0	0
$919$	19.5539	0.645024	0.322512	−	0.946565i	$-0.395473\pi$
$919$	19.5539	0.645024	0.322512	+	0.946565i	$0.395473\pi$
$920$	0	0
$921$	54.9254	1.80985
$922$	0	0
$923$	15.8885	0.522978
$924$	0	0
$925$	−3.71415	−0.122120
$926$	0	0
$927$	3.38320	0.111119
$928$	0	0
$929$	−20.0801	−0.658807	−0.329404	−	0.944189i	$-0.606848\pi$
$929$	−20.0801	−0.658807	−0.329404	+	0.944189i	$0.606848\pi$
$930$	0	0
$931$	−0.00429003	−0.000140600 0
$932$	0	0
$933$	20.3031	0.664695
$934$	0	0
$935$	−17.5442	−0.573758
$936$	0	0
$937$	−12.4150	−0.405580	−0.202790	−	0.979222i	$-0.565001\pi$
$937$	−12.4150	−0.405580	−0.202790	+	0.979222i	$0.565001\pi$
$938$	0	0
$939$	57.0505	1.86177
$940$	0	0
$941$	29.7263	0.969049	0.484525	−	0.874778i	$-0.338993\pi$
$941$	29.7263	0.969049	0.484525	+	0.874778i	$0.338993\pi$
$942$	0	0
$943$	69.8044	2.27314
$944$	0	0
$945$	67.6200	2.19968
$946$	0	0
$947$	24.5041	0.796276	0.398138	−	0.917326i	$-0.369657\pi$
$947$	24.5041	0.796276	0.398138	+	0.917326i	$0.369657\pi$
$948$	0	0
$949$	−6.22232	−0.201985
$950$	0	0
$951$	39.4080	1.27789
$952$	0	0
$953$	−35.5470	−1.15148	−0.575741	−	0.817632i	$-0.695286\pi$
$953$	−35.5470	−1.15148	−0.575741	+	0.817632i	$0.695286\pi$
$954$	0	0
$955$	39.9717	1.29345
$956$	0	0
$957$	−25.8556	−0.835792
$958$	0	0
$959$	2.00866	0.0648630
$960$	0	0
$961$	−12.7489	−0.411255
$962$	0	0
$963$	−6.60615	−0.212880
$964$	0	0
$965$	19.4980	0.627662
$966$	0	0
$967$	−30.9481	−0.995223	−0.497612	−	0.867400i	$-0.665789\pi$
$967$	−30.9481	−0.995223	−0.497612	+	0.867400i	$0.665789\pi$
$968$	0	0
$969$	0.0319605	0.00102672
$970$	0	0
$971$	29.6323	0.950945	0.475473	−	0.879730i	$-0.342277\pi$
$971$	29.6323	0.950945	0.475473	+	0.879730i	$0.342277\pi$
$972$	0	0
$973$	−23.1790	−0.743083
$974$	0	0
$975$	−5.31800	−0.170312
$976$	0	0
$977$	5.04766	0.161489	0.0807445	−	0.996735i	$-0.474270\pi$
$977$	5.04766	0.161489	0.0807445	+	0.996735i	$0.474270\pi$
$978$	0	0
$979$	−12.2151	−0.390395
$980$	0	0
$981$	−5.65496	−0.180549
$982$	0	0
$983$	37.3435	1.19107	0.595536	−	0.803328i	$-0.296940\pi$
$983$	37.3435	1.19107	0.595536	+	0.803328i	$0.296940\pi$
$984$	0	0
$985$	19.0670	0.607525
$986$	0	0
$987$	−54.5166	−1.73528
$988$	0	0
$989$	−65.4296	−2.08054
$990$	0	0
$991$	39.0784	1.24137	0.620684	−	0.784061i	$-0.286855\pi$
$991$	39.0784	1.24137	0.620684	+	0.784061i	$0.286855\pi$
$992$	0	0
$993$	104.612	3.31976
$994$	0	0
$995$	−50.9748	−1.61601
$996$	0	0
$997$	−28.3714	−0.898532	−0.449266	−	0.893398i	$-0.648315\pi$
$997$	−28.3714	−0.898532	−0.449266	+	0.893398i	$0.648315\pi$
$998$	0	0
$999$	34.9312	1.10517

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6028.2.a.c.1.3	✓	25

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6028.2.a.c.1.3	✓	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)

\(f(q)\)	\(=\)	\(q-3.18883 q^{3} +2.53247 q^{5} -2.00866 q^{7} +7.16863 q^{9} +O(q^{10})\)	\(q-3.18883 q^{3} +2.53247 q^{5} -2.00866 q^{7} +7.16863 q^{9} +1.00000 q^{11} +1.17991 q^{13} -8.07562 q^{15} -6.92771 q^{17} +0.00144675 q^{19} +6.40527 q^{21} +5.89949 q^{23} +1.41341 q^{25} -13.2930 q^{27} +8.10817 q^{29} -4.27213 q^{31} -3.18883 q^{33} -5.08687 q^{35} -2.62778 q^{37} -3.76252 q^{39} +11.8323 q^{41} -11.0907 q^{43} +18.1544 q^{45} -8.51122 q^{47} -2.96529 q^{49} +22.0913 q^{51} +0.619922 q^{53} +2.53247 q^{55} -0.00461343 q^{57} -0.786387 q^{59} -2.03018 q^{61} -14.3993 q^{63} +2.98808 q^{65} -14.0982 q^{67} -18.8125 q^{69} +13.4659 q^{71} -5.27357 q^{73} -4.50714 q^{75} -2.00866 q^{77} +3.88035 q^{79} +20.8833 q^{81} -9.82672 q^{83} -17.5442 q^{85} -25.8556 q^{87} -12.2151 q^{89} -2.37003 q^{91} +13.6231 q^{93} +0.00366385 q^{95} +10.9613 q^{97} +7.16863 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 25 q - 11 q^{3} - 2 q^{5} - 9 q^{7} + 18 q^{9}+O(q^{10}) \) 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

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more