LMFDB - Embedded newform 6030.2.a.bt.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6030,2,Mod(1,6030)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6030, 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("6030.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6030 = 2 \cdot 3^{2} \cdot 5 \cdot 67 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6030.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:	$48.1497924188$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.15188.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 7x^{2} + x + 2 $ x^4 - x^3 - 7*x^2 + x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 670)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.599159$ of defining polynomial
Character	$\chi$	$=$	6030.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -1.84982 q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -1.84982 q^{7} +1.00000 q^{8} +1.00000 q^{10} -5.13969 q^{11} -2.74934 q^{13} -1.84982 q^{14} +1.00000 q^{16} +3.68651 q^{17} +8.48033 q^{19} +1.00000 q^{20} -5.13969 q^{22} -8.67602 q^{23} +1.00000 q^{25} -2.74934 q^{26} -1.84982 q^{28} -4.05234 q^{29} -0.448975 q^{31} +1.00000 q^{32} +3.68651 q^{34} -1.84982 q^{35} +6.28567 q^{37} +8.48033 q^{38} +1.00000 q^{40} +8.16685 q^{41} -8.18783 q^{43} -5.13969 q^{44} -8.67602 q^{46} +3.02873 q^{47} -3.57818 q^{49} +1.00000 q^{50} -2.74934 q^{52} +5.73099 q^{53} -5.13969 q^{55} -1.84982 q^{56} -4.05234 q^{58} -7.09419 q^{59} -12.5651 q^{61} -0.448975 q^{62} +1.00000 q^{64} -2.74934 q^{65} +1.00000 q^{67} +3.68651 q^{68} -1.84982 q^{70} -2.60702 q^{71} -9.66553 q^{73} +6.28567 q^{74} +8.48033 q^{76} +9.50749 q^{77} -12.4803 q^{79} +1.00000 q^{80} +8.16685 q^{82} -16.0559 q^{83} +3.68651 q^{85} -8.18783 q^{86} -5.13969 q^{88} -7.13969 q^{89} +5.08578 q^{91} -8.67602 q^{92} +3.02873 q^{94} +8.48033 q^{95} -11.6200 q^{97} -3.57818 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{2} + 4 q^{4} + 4 q^{5} - 5 q^{7} + 4 q^{8}+O(q^{10}) $ 4 * q + 4 * q^2 + 4 * q^4 + 4 * q^5 - 5 * q^7 + 4 * q^8	$ 4 q + 4 q^{2} + 4 q^{4} + 4 q^{5} - 5 q^{7} + 4 q^{8} + 4 q^{10} - 9 q^{11} - 12 q^{13} - 5 q^{14} + 4 q^{16} + 4 q^{19} + 4 q^{20} - 9 q^{22} - 6 q^{23} + 4 q^{25} - 12 q^{26} - 5 q^{28} - 18 q^{29} + 2 q^{31} + 4 q^{32} - 5 q^{35} + 9 q^{37} + 4 q^{38} + 4 q^{40} - 12 q^{41} - 16 q^{43} - 9 q^{44} - 6 q^{46} - 10 q^{47} + 15 q^{49} + 4 q^{50} - 12 q^{52} - 8 q^{53} - 9 q^{55} - 5 q^{56} - 18 q^{58} - 18 q^{59} - 11 q^{61} + 2 q^{62} + 4 q^{64} - 12 q^{65} + 4 q^{67} - 5 q^{70} - 27 q^{71} + 4 q^{73} + 9 q^{74} + 4 q^{76} - 25 q^{77} - 20 q^{79} + 4 q^{80} - 12 q^{82} - 9 q^{83} - 16 q^{86} - 9 q^{88} - 17 q^{89} - 4 q^{91} - 6 q^{92} - 10 q^{94} + 4 q^{95} - 5 q^{97} + 15 q^{98}+O(q^{100}) $ 4 * q + 4 * q^2 + 4 * q^4 + 4 * q^5 - 5 * q^7 + 4 * q^8 + 4 * q^10 - 9 * q^11 - 12 * q^13 - 5 * q^14 + 4 * q^16 + 4 * q^19 + 4 * q^20 - 9 * q^22 - 6 * q^23 + 4 * q^25 - 12 * q^26 - 5 * q^28 - 18 * q^29 + 2 * q^31 + 4 * q^32 - 5 * q^35 + 9 * q^37 + 4 * q^38 + 4 * q^40 - 12 * q^41 - 16 * q^43 - 9 * q^44 - 6 * q^46 - 10 * q^47 + 15 * q^49 + 4 * q^50 - 12 * q^52 - 8 * q^53 - 9 * q^55 - 5 * q^56 - 18 * q^58 - 18 * q^59 - 11 * q^61 + 2 * q^62 + 4 * q^64 - 12 * q^65 + 4 * q^67 - 5 * q^70 - 27 * q^71 + 4 * q^73 + 9 * q^74 + 4 * q^76 - 25 * q^77 - 20 * q^79 + 4 * q^80 - 12 * q^82 - 9 * q^83 - 16 * q^86 - 9 * q^88 - 17 * q^89 - 4 * q^91 - 6 * q^92 - 10 * q^94 + 4 * q^95 - 5 * q^97 + 15 * q^98

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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−1.84982	−0.699165	−0.349582	−	0.936906i	$-0.613677\pi$
$7$	−1.84982	−0.699165	−0.349582	+	0.936906i	$0.613677\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	1.00000	0.316228
$11$	−5.13969	−1.54968	−0.774838	−	0.632160i	$-0.782169\pi$
$11$	−5.13969	−1.54968	−0.774838	+	0.632160i	$0.782169\pi$
$12$	0	0
$13$	−2.74934	−0.762530	−0.381265	−	0.924466i	$-0.624512\pi$
$13$	−2.74934	−0.762530	−0.381265	+	0.924466i	$0.624512\pi$
$14$	−1.84982	−0.494384
$15$	0	0
$16$	1.00000	0.250000
$17$	3.68651	0.894111	0.447055	−	0.894506i	$-0.352473\pi$
$17$	3.68651	0.894111	0.447055	+	0.894506i	$0.352473\pi$
$18$	0	0
$19$	8.48033	1.94552	0.972761	−	0.231810i	$-0.0744647\pi$
$19$	8.48033	1.94552	0.972761	+	0.231810i	$0.0744647\pi$
$20$	1.00000	0.223607
$21$	0	0
$22$	−5.13969	−1.09579
$23$	−8.67602	−1.80908	−0.904538	−	0.426393i	$-0.859784\pi$
$23$	−8.67602	−1.80908	−0.904538	+	0.426393i	$0.859784\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−2.74934	−0.539190
$27$	0	0
$28$	−1.84982	−0.349582
$29$	−4.05234	−0.752501	−0.376250	−	0.926518i	$-0.622787\pi$
$29$	−4.05234	−0.752501	−0.376250	+	0.926518i	$0.622787\pi$
$30$	0	0
$31$	−0.448975	−0.0806384	−0.0403192	−	0.999187i	$-0.512837\pi$
$31$	−0.448975	−0.0806384	−0.0403192	+	0.999187i	$0.512837\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	3.68651	0.632232
$35$	−1.84982	−0.312676
$36$	0	0
$37$	6.28567	1.03336	0.516679	−	0.856179i	$-0.327168\pi$
$37$	6.28567	1.03336	0.516679	+	0.856179i	$0.327168\pi$
$38$	8.48033	1.37569
$39$	0	0
$40$	1.00000	0.158114
$41$	8.16685	1.27545	0.637724	−	0.770265i	$-0.279876\pi$
$41$	8.16685	1.27545	0.637724	+	0.770265i	$0.279876\pi$
$42$	0	0
$43$	−8.18783	−1.24863	−0.624316	−	0.781172i	$-0.714622\pi$
$43$	−8.18783	−1.24863	−0.624316	+	0.781172i	$0.714622\pi$
$44$	−5.13969	−0.774838
$45$	0	0
$46$	−8.67602	−1.27921
$47$	3.02873	0.441786	0.220893	−	0.975298i	$-0.429103\pi$
$47$	3.02873	0.441786	0.220893	+	0.975298i	$0.429103\pi$
$48$	0	0
$49$	−3.57818	−0.511168
$50$	1.00000	0.141421
$51$	0	0
$52$	−2.74934	−0.381265
$53$	5.73099	0.787212	0.393606	−	0.919279i	$-0.371227\pi$
$53$	5.73099	0.787212	0.393606	+	0.919279i	$0.371227\pi$
$54$	0	0
$55$	−5.13969	−0.693036
$56$	−1.84982	−0.247192
$57$	0	0
$58$	−4.05234	−0.532098
$59$	−7.09419	−0.923585	−0.461792	−	0.886988i	$-0.652793\pi$
$59$	−7.09419	−0.923585	−0.461792	+	0.886988i	$0.652793\pi$
$60$	0	0
$61$	−12.5651	−1.60879	−0.804396	−	0.594094i	$-0.797511\pi$
$61$	−12.5651	−1.60879	−0.804396	+	0.594094i	$0.797511\pi$
$62$	−0.448975	−0.0570199
$63$	0	0
$64$	1.00000	0.125000
$65$	−2.74934	−0.341014
$66$	0	0
$67$	1.00000	0.122169
$67$	1.00000	0.122169
$68$	3.68651	0.447055
$69$	0	0
$70$	−1.84982	−0.221095
$71$	−2.60702	−0.309396	−0.154698	−	0.987962i	$-0.549441\pi$
$71$	−2.60702	−0.309396	−0.154698	+	0.987962i	$0.549441\pi$
$72$	0	0
$73$	−9.66553	−1.13127	−0.565633	−	0.824657i	$-0.691368\pi$
$73$	−9.66553	−1.13127	−0.565633	+	0.824657i	$0.691368\pi$
$74$	6.28567	0.730695
$75$	0	0
$76$	8.48033	0.972761
$77$	9.50749	1.08348
$78$	0	0
$79$	−12.4803	−1.40415	−0.702074	−	0.712104i	$-0.747742\pi$
$79$	−12.4803	−1.40415	−0.702074	+	0.712104i	$0.747742\pi$
$80$	1.00000	0.111803
$81$	0	0
$82$	8.16685	0.901878
$83$	−16.0559	−1.76236	−0.881181	−	0.472779i	$-0.843251\pi$
$83$	−16.0559	−1.76236	−0.881181	+	0.472779i	$0.843251\pi$
$84$	0	0
$85$	3.68651	0.399859
$86$	−8.18783	−0.882916
$87$	0	0
$88$	−5.13969	−0.547893
$89$	−7.13969	−0.756806	−0.378403	−	0.925641i	$-0.623527\pi$
$89$	−7.13969	−0.756806	−0.378403	+	0.925641i	$0.623527\pi$
$90$	0	0
$91$	5.08578	0.533135
$92$	−8.67602	−0.904538
$93$	0	0
$94$	3.02873	0.312390
$95$	8.48033	0.870064
$96$	0	0
$97$	−11.6200	−1.17984	−0.589918	−	0.807463i	$-0.700840\pi$
$97$	−11.6200	−1.17984	−0.589918	+	0.807463i	$0.700840\pi$
$98$	−3.57818	−0.361451
$99$	0	0
$100$	1.00000	0.100000
$101$	6.42957	0.639766	0.319883	−	0.947457i	$-0.396356\pi$
$101$	6.42957	0.639766	0.319883	+	0.947457i	$0.396356\pi$
$102$	0	0
$103$	−12.4490	−1.22663	−0.613317	−	0.789837i	$-0.710165\pi$
$103$	−12.4490	−1.22663	−0.613317	+	0.789837i	$0.710165\pi$
$104$	−2.74934	−0.269595
$105$	0	0
$106$	5.73099	0.556643
$107$	−8.44111	−0.816033	−0.408017	−	0.912974i	$-0.633779\pi$
$107$	−8.44111	−0.816033	−0.408017	+	0.912974i	$0.633779\pi$
$108$	0	0
$109$	−0.0455051	−0.00435860	−0.00217930	−	0.999998i	$-0.500694\pi$
$109$	−0.0455051	−0.00435860	−0.00217930	+	0.999998i	$0.500694\pi$
$110$	−5.13969	−0.490051
$111$	0	0
$112$	−1.84982	−0.174791
$113$	−2.00683	−0.188787	−0.0943935	−	0.995535i	$-0.530091\pi$
$113$	−2.00683	−0.188787	−0.0943935	+	0.995535i	$0.530091\pi$
$114$	0	0
$115$	−8.67602	−0.809043
$116$	−4.05234	−0.376250
$117$	0	0
$118$	−7.09419	−0.653073
$119$	−6.81937	−0.625131
$120$	0	0
$121$	15.4165	1.40150
$122$	−12.5651	−1.13759
$123$	0	0
$124$	−0.448975	−0.0403192
$125$	1.00000	0.0894427
$126$	0	0
$127$	−11.5216	−1.02238	−0.511190	−	0.859468i	$-0.670795\pi$
$127$	−11.5216	−1.02238	−0.511190	+	0.859468i	$0.670795\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	−2.74934	−0.241133
$131$	−4.97639	−0.434789	−0.217395	−	0.976084i	$-0.569756\pi$
$131$	−4.97639	−0.434789	−0.217395	+	0.976084i	$0.569756\pi$
$132$	0	0
$133$	−15.6871	−1.36024
$134$	1.00000	0.0863868
$135$	0	0
$136$	3.68651	0.316116
$137$	−2.31977	−0.198191	−0.0990957	−	0.995078i	$-0.531595\pi$
$137$	−2.31977	−0.198191	−0.0990957	+	0.995078i	$0.531595\pi$
$138$	0	0
$139$	10.8115	0.917021	0.458510	−	0.888689i	$-0.348383\pi$
$139$	10.8115	0.917021	0.458510	+	0.888689i	$0.348383\pi$
$140$	−1.84982	−0.156338
$141$	0	0
$142$	−2.60702	−0.218776
$143$	14.1308	1.18168
$144$	0	0
$145$	−4.05234	−0.336528
$146$	−9.66553	−0.799925
$147$	0	0
$148$	6.28567	0.516679
$149$	13.3730	1.09556	0.547781	−	0.836622i	$-0.315473\pi$
$149$	13.3730	1.09556	0.547781	+	0.836622i	$0.315473\pi$
$150$	0	0
$151$	14.7450	1.19993	0.599967	−	0.800025i	$-0.295180\pi$
$151$	14.7450	1.19993	0.599967	+	0.800025i	$0.295180\pi$
$152$	8.48033	0.687846
$153$	0	0
$154$	9.50749	0.766135
$155$	−0.448975	−0.0360626
$156$	0	0
$157$	−9.13812	−0.729301	−0.364651	−	0.931144i	$-0.618812\pi$
$157$	−9.13812	−0.729301	−0.364651	+	0.931144i	$0.618812\pi$
$158$	−12.4803	−0.992882
$159$	0	0
$160$	1.00000	0.0790569
$161$	16.0491	1.26484
$162$	0	0
$163$	3.53633	0.276987	0.138493	−	0.990363i	$-0.455774\pi$
$163$	3.53633	0.276987	0.138493	+	0.990363i	$0.455774\pi$
$164$	8.16685	0.637724
$165$	0	0
$166$	−16.0559	−1.24618
$167$	12.4541	0.963727	0.481863	−	0.876246i	$-0.339960\pi$
$167$	12.4541	0.963727	0.481863	+	0.876246i	$0.339960\pi$
$168$	0	0
$169$	−5.44111	−0.418547
$170$	3.68651	0.282743
$171$	0	0
$172$	−8.18783	−0.624316
$173$	−23.3248	−1.77335	−0.886675	−	0.462393i	$-0.846991\pi$
$173$	−23.3248	−1.77335	−0.886675	+	0.462393i	$0.846991\pi$
$174$	0	0
$175$	−1.84982	−0.139833
$176$	−5.13969	−0.387419
$177$	0	0
$178$	−7.13969	−0.535143
$179$	1.14653	0.0856955	0.0428478	−	0.999082i	$-0.486357\pi$
$179$	1.14653	0.0856955	0.0428478	+	0.999082i	$0.486357\pi$
$180$	0	0
$181$	−18.4980	−1.37495	−0.687474	−	0.726209i	$-0.741280\pi$
$181$	−18.4980	−1.37495	−0.687474	+	0.726209i	$0.741280\pi$
$182$	5.08578	0.376983
$183$	0	0
$184$	−8.67602	−0.639605
$185$	6.28567	0.462132
$186$	0	0
$187$	−18.9476	−1.38558
$188$	3.02873	0.220893
$189$	0	0
$190$	8.48033	0.615228
$191$	−2.84561	−0.205901	−0.102951	−	0.994686i	$-0.532828\pi$
$191$	−2.84561	−0.205901	−0.102951	+	0.994686i	$0.532828\pi$
$192$	0	0
$193$	24.6472	1.77414	0.887072	−	0.461632i	$-0.152736\pi$
$193$	24.6472	1.77414	0.887072	+	0.461632i	$0.152736\pi$
$194$	−11.6200	−0.834269
$195$	0	0
$196$	−3.57818	−0.255584
$197$	−8.68128	−0.618516	−0.309258	−	0.950978i	$-0.600081\pi$
$197$	−8.68128	−0.618516	−0.309258	+	0.950978i	$0.600081\pi$
$198$	0	0
$199$	−3.60691	−0.255687	−0.127844	−	0.991794i	$-0.540806\pi$
$199$	−3.60691	−0.255687	−0.127844	+	0.991794i	$0.540806\pi$
$200$	1.00000	0.0707107
$201$	0	0
$202$	6.42957	0.452383
$203$	7.49608	0.526122
$204$	0	0
$205$	8.16685	0.570397
$206$	−12.4490	−0.867361
$207$	0	0
$208$	−2.74934	−0.190633
$209$	−43.5863	−3.01493
$210$	0	0
$211$	−14.0496	−0.967214	−0.483607	−	0.875285i	$-0.660674\pi$
$211$	−14.0496	−0.967214	−0.483607	+	0.875285i	$0.660674\pi$
$212$	5.73099	0.393606
$213$	0	0
$214$	−8.44111	−0.577023
$215$	−8.18783	−0.558405
$216$	0	0
$217$	0.830522	0.0563795
$218$	−0.0455051	−0.00308200
$219$	0	0
$220$	−5.13969	−0.346518
$221$	−10.1355	−0.681787
$222$	0	0
$223$	−21.6787	−1.45171	−0.725855	−	0.687848i	$-0.758556\pi$
$223$	−21.6787	−1.45171	−0.725855	+	0.687848i	$0.758556\pi$
$224$	−1.84982	−0.123596
$225$	0	0
$226$	−2.00683	−0.133493
$227$	23.8114	1.58042	0.790209	−	0.612837i	$-0.209972\pi$
$227$	23.8114	1.58042	0.790209	+	0.612837i	$0.209972\pi$
$228$	0	0
$229$	20.6403	1.36395	0.681976	−	0.731374i	$-0.261121\pi$
$229$	20.6403	1.36395	0.681976	+	0.731374i	$0.261121\pi$
$230$	−8.67602	−0.572080
$231$	0	0
$232$	−4.05234	−0.266049
$233$	13.4944	0.884046	0.442023	−	0.897004i	$-0.354261\pi$
$233$	13.4944	0.884046	0.442023	+	0.897004i	$0.354261\pi$
$234$	0	0
$235$	3.02873	0.197573
$236$	−7.09419	−0.461792
$237$	0	0
$238$	−6.81937	−0.442034
$239$	6.66761	0.431292	0.215646	−	0.976472i	$-0.430814\pi$
$239$	6.66761	0.431292	0.215646	+	0.976472i	$0.430814\pi$
$240$	0	0
$241$	−3.95962	−0.255061	−0.127531	−	0.991835i	$-0.540705\pi$
$241$	−3.95962	−0.255061	−0.127531	+	0.991835i	$0.540705\pi$
$242$	15.4165	0.991007
$243$	0	0
$244$	−12.5651	−0.804396
$245$	−3.57818	−0.228601
$246$	0	0
$247$	−23.3153	−1.48352
$248$	−0.448975	−0.0285100
$249$	0	0
$250$	1.00000	0.0632456
$251$	0.620139	0.0391428	0.0195714	−	0.999808i	$-0.493770\pi$
$251$	0.620139	0.0391428	0.0195714	+	0.999808i	$0.493770\pi$
$252$	0	0
$253$	44.5921	2.80348
$254$	−11.5216	−0.722932
$255$	0	0
$256$	1.00000	0.0625000
$257$	15.8539	0.988940	0.494470	−	0.869195i	$-0.335362\pi$
$257$	15.8539	0.988940	0.494470	+	0.869195i	$0.335362\pi$
$258$	0	0
$259$	−11.6273	−0.722488
$260$	−2.74934	−0.170507
$261$	0	0
$262$	−4.97639	−0.307442
$263$	−16.4227	−1.01267	−0.506335	−	0.862337i	$-0.669000\pi$
$263$	−16.4227	−1.01267	−0.506335	+	0.862337i	$0.669000\pi$
$264$	0	0
$265$	5.73099	0.352052
$266$	−15.6871	−0.961836
$267$	0	0
$268$	1.00000	0.0610847
$269$	2.52229	0.153787	0.0768935	−	0.997039i	$-0.475500\pi$
$269$	2.52229	0.153787	0.0768935	+	0.997039i	$0.475500\pi$
$270$	0	0
$271$	1.42025	0.0862738	0.0431369	−	0.999069i	$-0.486265\pi$
$271$	1.42025	0.0862738	0.0431369	+	0.999069i	$0.486265\pi$
$272$	3.68651	0.223528
$273$	0	0
$274$	−2.31977	−0.140143
$275$	−5.13969	−0.309935
$276$	0	0
$277$	6.53425	0.392605	0.196303	−	0.980543i	$-0.437107\pi$
$277$	6.53425	0.392605	0.196303	+	0.980543i	$0.437107\pi$
$278$	10.8115	0.648431
$279$	0	0
$280$	−1.84982	−0.110548
$281$	21.8670	1.30448	0.652239	−	0.758014i	$-0.273830\pi$
$281$	21.8670	1.30448	0.652239	+	0.758014i	$0.273830\pi$
$282$	0	0
$283$	−15.4552	−0.918713	−0.459357	−	0.888252i	$-0.651920\pi$
$283$	−15.4552	−0.918713	−0.459357	+	0.888252i	$0.651920\pi$
$284$	−2.60702	−0.154698
$285$	0	0
$286$	14.1308	0.835570
$287$	−15.1072	−0.891748
$288$	0	0
$289$	−3.40962	−0.200566
$290$	−4.05234	−0.237962
$291$	0	0
$292$	−9.66553	−0.565633
$293$	−11.7032	−0.683707	−0.341853	−	0.939753i	$-0.611055\pi$
$293$	−11.7032	−0.683707	−0.341853	+	0.939753i	$0.611055\pi$
$294$	0	0
$295$	−7.09419	−0.413040
$296$	6.28567	0.365347
$297$	0	0
$298$	13.3730	0.774679
$299$	23.8534	1.37948
$300$	0	0
$301$	15.1460	0.873000
$302$	14.7450	0.848481
$303$	0	0
$304$	8.48033	0.486381
$305$	−12.5651	−0.719473
$306$	0	0
$307$	12.5362	0.715480	0.357740	−	0.933821i	$-0.383547\pi$
$307$	12.5362	0.715480	0.357740	+	0.933821i	$0.383547\pi$
$308$	9.50749	0.541740
$309$	0	0
$310$	−0.448975	−0.0255001
$311$	7.34167	0.416308	0.208154	−	0.978096i	$-0.433255\pi$
$311$	7.34167	0.416308	0.208154	+	0.978096i	$0.433255\pi$
$312$	0	0
$313$	−3.44989	−0.194999	−0.0974997	−	0.995236i	$-0.531085\pi$
$313$	−3.44989	−0.194999	−0.0974997	+	0.995236i	$0.531085\pi$
$314$	−9.13812	−0.515694
$315$	0	0
$316$	−12.4803	−0.702074
$317$	1.43575	0.0806395	0.0403198	−	0.999187i	$-0.487162\pi$
$317$	1.43575	0.0806395	0.0403198	+	0.999187i	$0.487162\pi$
$318$	0	0
$319$	20.8278	1.16613
$320$	1.00000	0.0559017
$321$	0	0
$322$	16.0491	0.894379
$323$	31.2629	1.73951
$324$	0	0
$325$	−2.74934	−0.152506
$326$	3.53633	0.195859
$327$	0	0
$328$	8.16685	0.450939
$329$	−5.60259	−0.308881
$330$	0	0
$331$	29.5879	1.62630	0.813149	−	0.582056i	$-0.197751\pi$
$331$	29.5879	1.62630	0.813149	+	0.582056i	$0.197751\pi$
$332$	−16.0559	−0.881181
$333$	0	0
$334$	12.4541	0.681458
$335$	1.00000	0.0546358
$336$	0	0
$337$	5.64784	0.307658	0.153829	−	0.988098i	$-0.450840\pi$
$337$	5.64784	0.307658	0.153829	+	0.988098i	$0.450840\pi$
$338$	−5.44111	−0.295958
$339$	0	0
$340$	3.68651	0.199929
$341$	2.30760	0.124963
$342$	0	0
$343$	19.5677	1.05656
$344$	−8.18783	−0.441458
$345$	0	0
$346$	−23.3248	−1.25395
$347$	−9.83238	−0.527830	−0.263915	−	0.964546i	$-0.585014\pi$
$347$	−9.83238	−0.527830	−0.263915	+	0.964546i	$0.585014\pi$
$348$	0	0
$349$	1.38670	0.0742281	0.0371141	−	0.999311i	$-0.488184\pi$
$349$	1.38670	0.0742281	0.0371141	+	0.999311i	$0.488184\pi$
$350$	−1.84982	−0.0988768
$351$	0	0
$352$	−5.13969	−0.273947
$353$	33.1609	1.76498	0.882489	−	0.470333i	$-0.155866\pi$
$353$	33.1609	1.76498	0.882489	+	0.470333i	$0.155866\pi$
$354$	0	0
$355$	−2.60702	−0.138366
$356$	−7.13969	−0.378403
$357$	0	0
$358$	1.14653	0.0605959
$359$	3.56769	0.188295	0.0941477	−	0.995558i	$-0.469987\pi$
$359$	3.56769	0.188295	0.0941477	+	0.995558i	$0.469987\pi$
$360$	0	0
$361$	52.9161	2.78506
$362$	−18.4980	−0.972235
$363$	0	0
$364$	5.08578	0.266567
$365$	−9.66553	−0.505917
$366$	0	0
$367$	5.02818	0.262469	0.131234	−	0.991351i	$-0.458106\pi$
$367$	5.02818	0.262469	0.131234	+	0.991351i	$0.458106\pi$
$368$	−8.67602	−0.452269
$369$	0	0
$370$	6.28567	0.326777
$371$	−10.6013	−0.550391
$372$	0	0
$373$	29.7099	1.53832	0.769160	−	0.639057i	$-0.220675\pi$
$373$	29.7099	1.53832	0.769160	+	0.639057i	$0.220675\pi$
$374$	−18.9476	−0.979755
$375$	0	0
$376$	3.02873	0.156195
$377$	11.1413	0.573805
$378$	0	0
$379$	34.8115	1.78815	0.894074	−	0.447920i	$-0.147835\pi$
$379$	34.8115	1.78815	0.894074	+	0.447920i	$0.147835\pi$
$380$	8.48033	0.435032
$381$	0	0
$382$	−2.84561	−0.145594
$383$	14.8235	0.757444	0.378722	−	0.925511i	$-0.376364\pi$
$383$	14.8235	0.757444	0.378722	+	0.925511i	$0.376364\pi$
$384$	0	0
$385$	9.50749	0.484547
$386$	24.6472	1.25451
$387$	0	0
$388$	−11.6200	−0.589918
$389$	−9.37097	−0.475127	−0.237564	−	0.971372i	$-0.576349\pi$
$389$	−9.37097	−0.475127	−0.237564	+	0.971372i	$0.576349\pi$
$390$	0	0
$391$	−31.9843	−1.61751
$392$	−3.57818	−0.180725
$393$	0	0
$394$	−8.68128	−0.437357
$395$	−12.4803	−0.627954
$396$	0	0
$397$	35.1024	1.76174	0.880871	−	0.473357i	$-0.156958\pi$
$397$	35.1024	1.76174	0.880871	+	0.473357i	$0.156958\pi$
$398$	−3.60691	−0.180798
$399$	0	0
$400$	1.00000	0.0500000
$401$	−21.4698	−1.07215	−0.536076	−	0.844169i	$-0.680094\pi$
$401$	−21.4698	−1.07215	−0.536076	+	0.844169i	$0.680094\pi$
$402$	0	0
$403$	1.23439	0.0614892
$404$	6.42957	0.319883
$405$	0	0
$406$	7.49608	0.372024
$407$	−32.3064	−1.60137
$408$	0	0
$409$	−34.5090	−1.70636	−0.853179	−	0.521618i	$-0.825329\pi$
$409$	−34.5090	−1.70636	−0.853179	+	0.521618i	$0.825329\pi$
$410$	8.16685	0.403332
$411$	0	0
$412$	−12.4490	−0.613317
$413$	13.1229	0.645738
$414$	0	0
$415$	−16.0559	−0.788152
$416$	−2.74934	−0.134798
$417$	0	0
$418$	−43.5863	−2.13188
$419$	10.6813	0.521815	0.260907	−	0.965364i	$-0.415978\pi$
$419$	10.6813	0.521815	0.260907	+	0.965364i	$0.415978\pi$
$420$	0	0
$421$	−5.85139	−0.285179	−0.142590	−	0.989782i	$-0.545543\pi$
$421$	−5.85139	−0.285179	−0.142590	+	0.989782i	$0.545543\pi$
$422$	−14.0496	−0.683924
$423$	0	0
$424$	5.73099	0.278322
$425$	3.68651	0.178822
$426$	0	0
$427$	23.2431	1.12481
$428$	−8.44111	−0.408017
$429$	0	0
$430$	−8.18783	−0.394852
$431$	−8.89692	−0.428550	−0.214275	−	0.976773i	$-0.568739\pi$
$431$	−8.89692	−0.428550	−0.214275	+	0.976773i	$0.568739\pi$
$432$	0	0
$433$	7.83512	0.376532	0.188266	−	0.982118i	$-0.439713\pi$
$433$	7.83512	0.376532	0.188266	+	0.982118i	$0.439713\pi$
$434$	0.830522	0.0398663
$435$	0	0
$436$	−0.0455051	−0.00217930
$437$	−73.5756	−3.51960
$438$	0	0
$439$	15.5022	0.739881	0.369941	−	0.929055i	$-0.379378\pi$
$439$	15.5022	0.739881	0.369941	+	0.929055i	$0.379378\pi$
$440$	−5.13969	−0.245025
$441$	0	0
$442$	−10.1355	−0.482096
$443$	0.871712	0.0414163	0.0207082	−	0.999786i	$-0.493408\pi$
$443$	0.871712	0.0414163	0.0207082	+	0.999786i	$0.493408\pi$
$444$	0	0
$445$	−7.13969	−0.338454
$446$	−21.6787	−1.02651
$447$	0	0
$448$	−1.84982	−0.0873956
$449$	−11.4656	−0.541096	−0.270548	−	0.962707i	$-0.587205\pi$
$449$	−11.4656	−0.541096	−0.270548	+	0.962707i	$0.587205\pi$
$450$	0	0
$451$	−41.9751	−1.97653
$452$	−2.00683	−0.0943935
$453$	0	0
$454$	23.8114	1.11752
$455$	5.08578	0.238425
$456$	0	0
$457$	−22.8298	−1.06793	−0.533965	−	0.845506i	$-0.679299\pi$
$457$	−22.8298	−1.06793	−0.533965	+	0.845506i	$0.679299\pi$
$458$	20.6403	0.964460
$459$	0	0
$460$	−8.67602	−0.404522
$461$	−10.5294	−0.490402	−0.245201	−	0.969472i	$-0.578854\pi$
$461$	−10.5294	−0.490402	−0.245201	+	0.969472i	$0.578854\pi$
$462$	0	0
$463$	9.18575	0.426898	0.213449	−	0.976954i	$-0.431530\pi$
$463$	9.18575	0.426898	0.213449	+	0.976954i	$0.431530\pi$
$464$	−4.05234	−0.188125
$465$	0	0
$466$	13.4944	0.625115
$467$	−1.81846	−0.0841481	−0.0420741	−	0.999114i	$-0.513397\pi$
$467$	−1.81846	−0.0841481	−0.0420741	+	0.999114i	$0.513397\pi$
$468$	0	0
$469$	−1.84982	−0.0854166
$470$	3.02873	0.139705
$471$	0	0
$472$	−7.09419	−0.326536
$473$	42.0829	1.93498
$474$	0	0
$475$	8.48033	0.389104
$476$	−6.81937	−0.312565
$477$	0	0
$478$	6.66761	0.304970
$479$	6.93089	0.316680	0.158340	−	0.987385i	$-0.449386\pi$
$479$	6.93089	0.316680	0.158340	+	0.987385i	$0.449386\pi$
$480$	0	0
$481$	−17.2815	−0.787967
$482$	−3.95962	−0.180356
$483$	0	0
$484$	15.4165	0.700748
$485$	−11.6200	−0.527638
$486$	0	0
$487$	−3.75355	−0.170089	−0.0850447	−	0.996377i	$-0.527103\pi$
$487$	−3.75355	−0.170089	−0.0850447	+	0.996377i	$0.527103\pi$
$488$	−12.5651	−0.568794
$489$	0	0
$490$	−3.57818	−0.161646
$491$	25.9586	1.17149	0.585747	−	0.810494i	$-0.300801\pi$
$491$	25.9586	1.17149	0.585747	+	0.810494i	$0.300801\pi$
$492$	0	0
$493$	−14.9390	−0.672819
$494$	−23.3153	−1.04901
$495$	0	0
$496$	−0.448975	−0.0201596
$497$	4.82251	0.216319
$498$	0	0
$499$	−7.87866	−0.352697	−0.176349	−	0.984328i	$-0.556429\pi$
$499$	−7.87866	−0.352697	−0.176349	+	0.984328i	$0.556429\pi$
$500$	1.00000	0.0447214
$501$	0	0
$502$	0.620139	0.0276781
$503$	−41.2671	−1.84001	−0.920005	−	0.391908i	$-0.871815\pi$
$503$	−41.2671	−1.84001	−0.920005	+	0.391908i	$0.871815\pi$
$504$	0	0
$505$	6.42957	0.286112
$506$	44.5921	1.98236
$507$	0	0
$508$	−11.5216	−0.511190
$509$	−14.7858	−0.655370	−0.327685	−	0.944787i	$-0.606268\pi$
$509$	−14.7858	−0.655370	−0.327685	+	0.944787i	$0.606268\pi$
$510$	0	0
$511$	17.8795	0.790941
$512$	1.00000	0.0441942
$513$	0	0
$514$	15.8539	0.699286
$515$	−12.4490	−0.548567
$516$	0	0
$517$	−15.5667	−0.684625
$518$	−11.6273	−0.510876
$519$	0	0
$520$	−2.74934	−0.120567
$521$	−15.7650	−0.690675	−0.345338	−	0.938479i	$-0.612236\pi$
$521$	−15.7650	−0.690675	−0.345338	+	0.938479i	$0.612236\pi$
$522$	0	0
$523$	−32.9091	−1.43902	−0.719508	−	0.694484i	$-0.755633\pi$
$523$	−32.9091	−1.43902	−0.719508	+	0.694484i	$0.755633\pi$
$524$	−4.97639	−0.217395
$525$	0	0
$526$	−16.4227	−0.716065
$527$	−1.65515	−0.0720996
$528$	0	0
$529$	52.2734	2.27276
$530$	5.73099	0.248938
$531$	0	0
$532$	−15.6871	−0.680120
$533$	−22.4535	−0.972568
$534$	0	0
$535$	−8.44111	−0.364941
$536$	1.00000	0.0431934
$537$	0	0
$538$	2.52229	0.108744
$539$	18.3907	0.792146
$540$	0	0
$541$	−41.7334	−1.79426	−0.897129	−	0.441769i	$-0.854351\pi$
$541$	−41.7334	−1.79426	−0.897129	+	0.441769i	$0.854351\pi$
$542$	1.42025	0.0610048
$543$	0	0
$544$	3.68651	0.158058
$545$	−0.0455051	−0.00194923
$546$	0	0
$547$	−29.7434	−1.27174	−0.635869	−	0.771797i	$-0.719358\pi$
$547$	−29.7434	−1.27174	−0.635869	+	0.771797i	$0.719358\pi$
$548$	−2.31977	−0.0990957
$549$	0	0
$550$	−5.13969	−0.219157
$551$	−34.3652	−1.46401
$552$	0	0
$553$	23.0863	0.981731
$554$	6.53425	0.277614
$555$	0	0
$556$	10.8115	0.458510
$557$	−0.354677	−0.0150281	−0.00751407	−	0.999972i	$-0.502392\pi$
$557$	−0.354677	−0.0150281	−0.00751407	+	0.999972i	$0.502392\pi$
$558$	0	0
$559$	22.5111	0.952120
$560$	−1.84982	−0.0781690
$561$	0	0
$562$	21.8670	0.922405
$563$	−37.2395	−1.56946	−0.784729	−	0.619839i	$-0.787198\pi$
$563$	−37.2395	−1.56946	−0.784729	+	0.619839i	$0.787198\pi$
$564$	0	0
$565$	−2.00683	−0.0844281
$566$	−15.4552	−0.649628
$567$	0	0
$568$	−2.60702	−0.109388
$569$	37.4289	1.56910	0.784551	−	0.620065i	$-0.212894\pi$
$569$	37.4289	1.56910	0.784551	+	0.620065i	$0.212894\pi$
$570$	0	0
$571$	−43.1149	−1.80430	−0.902152	−	0.431418i	$-0.858013\pi$
$571$	−43.1149	−1.80430	−0.902152	+	0.431418i	$0.858013\pi$
$572$	14.1308	0.590838
$573$	0	0
$574$	−15.1072	−0.630561
$575$	−8.67602	−0.361815
$576$	0	0
$577$	1.97376	0.0821688	0.0410844	−	0.999156i	$-0.486919\pi$
$577$	1.97376	0.0821688	0.0410844	+	0.999156i	$0.486919\pi$
$578$	−3.40962	−0.141821
$579$	0	0
$580$	−4.05234	−0.168264
$581$	29.7004	1.23218
$582$	0	0
$583$	−29.4555	−1.21992
$584$	−9.66553	−0.399963
$585$	0	0
$586$	−11.7032	−0.483454
$587$	7.42550	0.306483	0.153242	−	0.988189i	$-0.451029\pi$
$587$	7.42550	0.306483	0.153242	+	0.988189i	$0.451029\pi$
$588$	0	0
$589$	−3.80746	−0.156884
$590$	−7.09419	−0.292063
$591$	0	0
$592$	6.28567	0.258340
$593$	−11.9988	−0.492733	−0.246367	−	0.969177i	$-0.579237\pi$
$593$	−11.9988	−0.492733	−0.246367	+	0.969177i	$0.579237\pi$
$594$	0	0
$595$	−6.81937	−0.279567
$596$	13.3730	0.547781
$597$	0	0
$598$	23.8534	0.975437
$599$	−21.4325	−0.875706	−0.437853	−	0.899046i	$-0.644261\pi$
$599$	−21.4325	−0.875706	−0.437853	+	0.899046i	$0.644261\pi$
$600$	0	0
$601$	40.5310	1.65329	0.826646	−	0.562722i	$-0.190246\pi$
$601$	40.5310	1.65329	0.826646	+	0.562722i	$0.190246\pi$
$602$	15.1460	0.617304
$603$	0	0
$604$	14.7450	0.599967
$605$	15.4165	0.626768
$606$	0	0
$607$	−39.1091	−1.58739	−0.793696	−	0.608315i	$-0.791846\pi$
$607$	−39.1091	−1.58739	−0.793696	+	0.608315i	$0.791846\pi$
$608$	8.48033	0.343923
$609$	0	0
$610$	−12.5651	−0.508744
$611$	−8.32702	−0.336875
$612$	0	0
$613$	27.6552	1.11698	0.558491	−	0.829511i	$-0.311381\pi$
$613$	27.6552	1.11698	0.558491	+	0.829511i	$0.311381\pi$
$614$	12.5362	0.505921
$615$	0	0
$616$	9.50749	0.383068
$617$	30.5215	1.22875	0.614375	−	0.789014i	$-0.289408\pi$
$617$	30.5215	1.22875	0.614375	+	0.789014i	$0.289408\pi$
$618$	0	0
$619$	32.0596	1.28858	0.644291	−	0.764780i	$-0.277152\pi$
$619$	32.0596	1.28858	0.644291	+	0.764780i	$0.277152\pi$
$620$	−0.448975	−0.0180313
$621$	0	0
$622$	7.34167	0.294374
$623$	13.2071	0.529132
$624$	0	0
$625$	1.00000	0.0400000
$626$	−3.44989	−0.137885
$627$	0	0
$628$	−9.13812	−0.364651
$629$	23.1722	0.923937
$630$	0	0
$631$	−19.7048	−0.784434	−0.392217	−	0.919873i	$-0.628292\pi$
$631$	−19.7048	−0.784434	−0.392217	+	0.919873i	$0.628292\pi$
$632$	−12.4803	−0.496441
$633$	0	0
$634$	1.43575	0.0570208
$635$	−11.5216	−0.457222
$636$	0	0
$637$	9.83764	0.389782
$638$	20.8278	0.824580
$639$	0	0
$640$	1.00000	0.0395285
$641$	3.35991	0.132708	0.0663542	−	0.997796i	$-0.478863\pi$
$641$	3.35991	0.132708	0.0663542	+	0.997796i	$0.478863\pi$
$642$	0	0
$643$	−4.63995	−0.182982	−0.0914910	−	0.995806i	$-0.529163\pi$
$643$	−4.63995	−0.182982	−0.0914910	+	0.995806i	$0.529163\pi$
$644$	16.0491	0.632421
$645$	0	0
$646$	31.2629	1.23002
$647$	−16.1789	−0.636060	−0.318030	−	0.948081i	$-0.603021\pi$
$647$	−16.1789	−0.636060	−0.318030	+	0.948081i	$0.603021\pi$
$648$	0	0
$649$	36.4620	1.43126
$650$	−2.74934	−0.107838
$651$	0	0
$652$	3.53633	0.138493
$653$	−21.3624	−0.835976	−0.417988	−	0.908452i	$-0.637265\pi$
$653$	−21.3624	−0.835976	−0.417988	+	0.908452i	$0.637265\pi$
$654$	0	0
$655$	−4.97639	−0.194444
$656$	8.16685	0.318862
$657$	0	0
$658$	−5.60259	−0.218412
$659$	4.99477	0.194569	0.0972843	−	0.995257i	$-0.468984\pi$
$659$	4.99477	0.194569	0.0972843	+	0.995257i	$0.468984\pi$
$660$	0	0
$661$	2.37566	0.0924023	0.0462012	−	0.998932i	$-0.485288\pi$
$661$	2.37566	0.0924023	0.0462012	+	0.998932i	$0.485288\pi$
$662$	29.5879	1.14997
$663$	0	0
$664$	−16.0559	−0.623089
$665$	−15.6871	−0.608318
$666$	0	0
$667$	35.1582	1.36133
$668$	12.4541	0.481863
$669$	0	0
$670$	1.00000	0.0386334
$671$	64.5806	2.49310
$672$	0	0
$673$	11.2866	0.435066	0.217533	−	0.976053i	$-0.430199\pi$
$673$	11.2866	0.435066	0.217533	+	0.976053i	$0.430199\pi$
$674$	5.64784	0.217547
$675$	0	0
$676$	−5.44111	−0.209274
$677$	−18.6498	−0.716769	−0.358385	−	0.933574i	$-0.616672\pi$
$677$	−18.6498	−0.716769	−0.358385	+	0.933574i	$0.616672\pi$
$678$	0	0
$679$	21.4949	0.824899
$680$	3.68651	0.141371
$681$	0	0
$682$	2.30760	0.0883624
$683$	−4.14664	−0.158667	−0.0793333	−	0.996848i	$-0.525279\pi$
$683$	−4.14664	−0.158667	−0.0793333	+	0.996848i	$0.525279\pi$
$684$	0	0
$685$	−2.31977	−0.0886339
$686$	19.5677	0.747098
$687$	0	0
$688$	−8.18783	−0.312158
$689$	−15.7565	−0.600273
$690$	0	0
$691$	−31.5032	−1.19844	−0.599219	−	0.800585i	$-0.704522\pi$
$691$	−31.5032	−1.19844	−0.599219	+	0.800585i	$0.704522\pi$
$692$	−23.3248	−0.886675
$693$	0	0
$694$	−9.83238	−0.373232
$695$	10.8115	0.410104
$696$	0	0
$697$	30.1072	1.14039
$698$	1.38670	0.0524872
$699$	0	0
$700$	−1.84982	−0.0699165
$701$	−10.9715	−0.414389	−0.207194	−	0.978300i	$-0.566433\pi$
$701$	−10.9715	−0.414389	−0.207194	+	0.978300i	$0.566433\pi$
$702$	0	0
$703$	53.3046	2.01042
$704$	−5.13969	−0.193709
$705$	0	0
$706$	33.1609	1.24803
$707$	−11.8935	−0.447302
$708$	0	0
$709$	32.0594	1.20402	0.602009	−	0.798490i	$-0.294367\pi$
$709$	32.0594	1.20402	0.602009	+	0.798490i	$0.294367\pi$
$710$	−2.60702	−0.0978397
$711$	0	0
$712$	−7.13969	−0.267571
$713$	3.89532	0.145881
$714$	0	0
$715$	14.1308	0.528461
$716$	1.14653	0.0428478
$717$	0	0
$718$	3.56769	0.133145
$719$	−27.2789	−1.01733	−0.508665	−	0.860965i	$-0.669861\pi$
$719$	−27.2789	−1.01733	−0.508665	+	0.860965i	$0.669861\pi$
$720$	0	0
$721$	23.0283	0.857619
$722$	52.9161	1.96933
$723$	0	0
$724$	−18.4980	−0.687474
$725$	−4.05234	−0.150500
$726$	0	0
$727$	13.0555	0.484202	0.242101	−	0.970251i	$-0.422163\pi$
$727$	13.0555	0.484202	0.242101	+	0.970251i	$0.422163\pi$
$728$	5.08578	0.188492
$729$	0	0
$730$	−9.66553	−0.357737
$731$	−30.1845	−1.11642
$732$	0	0
$733$	4.14335	0.153038	0.0765191	−	0.997068i	$-0.475619\pi$
$733$	4.14335	0.153038	0.0765191	+	0.997068i	$0.475619\pi$
$734$	5.02818	0.185594
$735$	0	0
$736$	−8.67602	−0.319802
$737$	−5.13969	−0.189323
$738$	0	0
$739$	31.5851	1.16188	0.580939	−	0.813947i	$-0.302686\pi$
$739$	31.5851	1.16188	0.580939	+	0.813947i	$0.302686\pi$
$740$	6.28567	0.231066
$741$	0	0
$742$	−10.6013	−0.389185
$743$	−26.3202	−0.965593	−0.482797	−	0.875733i	$-0.660379\pi$
$743$	−26.3202	−0.965593	−0.482797	+	0.875733i	$0.660379\pi$
$744$	0	0
$745$	13.3730	0.489950
$746$	29.7099	1.08776
$747$	0	0
$748$	−18.9476	−0.692791
$749$	15.6145	0.570542
$750$	0	0
$751$	−18.4594	−0.673591	−0.336796	−	0.941578i	$-0.609343\pi$
$751$	−18.4594	−0.673591	−0.336796	+	0.941578i	$0.609343\pi$
$752$	3.02873	0.110446
$753$	0	0
$754$	11.1413	0.405741
$755$	14.7450	0.536627
$756$	0	0
$757$	2.64466	0.0961220	0.0480610	−	0.998844i	$-0.484696\pi$
$757$	2.64466	0.0961220	0.0480610	+	0.998844i	$0.484696\pi$
$758$	34.8115	1.26441
$759$	0	0
$760$	8.48033	0.307614
$761$	2.87581	0.104248	0.0521240	−	0.998641i	$-0.483401\pi$
$761$	2.87581	0.104248	0.0521240	+	0.998641i	$0.483401\pi$
$762$	0	0
$763$	0.0841761	0.00304738
$764$	−2.84561	−0.102951
$765$	0	0
$766$	14.8235	0.535594
$767$	19.5044	0.704261
$768$	0	0
$769$	2.90896	0.104900	0.0524499	−	0.998624i	$-0.483297\pi$
$769$	2.90896	0.104900	0.0524499	+	0.998624i	$0.483297\pi$
$770$	9.50749	0.342626
$771$	0	0
$772$	24.6472	0.887072
$773$	−18.4394	−0.663219	−0.331610	−	0.943417i	$-0.607592\pi$
$773$	−18.4394	−0.663219	−0.331610	+	0.943417i	$0.607592\pi$
$774$	0	0
$775$	−0.448975	−0.0161277
$776$	−11.6200	−0.417135
$777$	0	0
$778$	−9.37097	−0.335966
$779$	69.2576	2.48141
$780$	0	0
$781$	13.3993	0.479464
$782$	−31.9843	−1.14376
$783$	0	0
$784$	−3.57818	−0.127792
$785$	−9.13812	−0.326153
$786$	0	0
$787$	20.8345	0.742668	0.371334	−	0.928499i	$-0.378900\pi$
$787$	20.8345	0.742668	0.371334	+	0.928499i	$0.378900\pi$
$788$	−8.68128	−0.309258
$789$	0	0
$790$	−12.4803	−0.444030
$791$	3.71227	0.131993
$792$	0	0
$793$	34.5457	1.22675
$794$	35.1024	1.24574
$795$	0	0
$796$	−3.60691	−0.127844
$797$	−39.6377	−1.40404	−0.702020	−	0.712157i	$-0.747718\pi$
$797$	−39.6377	−1.40404	−0.702020	+	0.712157i	$0.747718\pi$
$798$	0	0
$799$	11.1655	0.395005
$800$	1.00000	0.0353553
$801$	0	0
$802$	−21.4698	−0.758127
$803$	49.6779	1.75309
$804$	0	0
$805$	16.0491	0.565655
$806$	1.23439	0.0434794
$807$	0	0
$808$	6.42957	0.226192
$809$	−0.744771	−0.0261847	−0.0130924	−	0.999914i	$-0.504168\pi$
$809$	−0.744771	−0.0261847	−0.0130924	+	0.999914i	$0.504168\pi$
$810$	0	0
$811$	14.7109	0.516571	0.258285	−	0.966069i	$-0.416843\pi$
$811$	14.7109	0.516571	0.258285	+	0.966069i	$0.416843\pi$
$812$	7.49608	0.263061
$813$	0	0
$814$	−32.3064	−1.13234
$815$	3.53633	0.123872
$816$	0	0
$817$	−69.4355	−2.42924
$818$	−34.5090	−1.20658
$819$	0	0
$820$	8.16685	0.285199
$821$	25.4023	0.886548	0.443274	−	0.896386i	$-0.353817\pi$
$821$	25.4023	0.886548	0.443274	+	0.896386i	$0.353817\pi$
$822$	0	0
$823$	−30.5327	−1.06430	−0.532151	−	0.846649i	$-0.678616\pi$
$823$	−30.5327	−1.06430	−0.532151	+	0.846649i	$0.678616\pi$
$824$	−12.4490	−0.433681
$825$	0	0
$826$	13.1229	0.456606
$827$	−16.4217	−0.571039	−0.285519	−	0.958373i	$-0.592166\pi$
$827$	−16.4217	−0.571039	−0.285519	+	0.958373i	$0.592166\pi$
$828$	0	0
$829$	−22.4373	−0.779279	−0.389639	−	0.920967i	$-0.627400\pi$
$829$	−22.4373	−0.779279	−0.389639	+	0.920967i	$0.627400\pi$
$830$	−16.0559	−0.557308
$831$	0	0
$832$	−2.74934	−0.0953163
$833$	−13.1910	−0.457041
$834$	0	0
$835$	12.4541	0.430992
$836$	−43.5863	−1.50746
$837$	0	0
$838$	10.6813	0.368979
$839$	44.8662	1.54895	0.774477	−	0.632602i	$-0.218013\pi$
$839$	44.8662	1.54895	0.774477	+	0.632602i	$0.218013\pi$
$840$	0	0
$841$	−12.5785	−0.433743
$842$	−5.85139	−0.201652
$843$	0	0
$844$	−14.0496	−0.483607
$845$	−5.44111	−0.187180
$846$	0	0
$847$	−28.5176	−0.979877
$848$	5.73099	0.196803
$849$	0	0
$850$	3.68651	0.126446
$851$	−54.5346	−1.86942
$852$	0	0
$853$	−14.2290	−0.487192	−0.243596	−	0.969877i	$-0.578327\pi$
$853$	−14.2290	−0.487192	−0.243596	+	0.969877i	$0.578327\pi$
$854$	23.2431	0.795361
$855$	0	0
$856$	−8.44111	−0.288511
$857$	36.5282	1.24778	0.623890	−	0.781512i	$-0.285551\pi$
$857$	36.5282	1.24778	0.623890	+	0.781512i	$0.285551\pi$
$858$	0	0
$859$	−21.1556	−0.721819	−0.360910	−	0.932601i	$-0.617534\pi$
$859$	−21.1556	−0.721819	−0.360910	+	0.932601i	$0.617534\pi$
$860$	−8.18783	−0.279203
$861$	0	0
$862$	−8.89692	−0.303030
$863$	−41.0202	−1.39635	−0.698173	−	0.715930i	$-0.746003\pi$
$863$	−41.0202	−1.39635	−0.698173	+	0.715930i	$0.746003\pi$
$864$	0	0
$865$	−23.3248	−0.793066
$866$	7.83512	0.266248
$867$	0	0
$868$	0.830522	0.0281898
$869$	64.1451	2.17597
$870$	0	0
$871$	−2.74934	−0.0931579
$872$	−0.0455051	−0.00154100
$873$	0	0
$874$	−73.5756	−2.48873
$875$	−1.84982	−0.0625352
$876$	0	0
$877$	−45.5247	−1.53726	−0.768630	−	0.639694i	$-0.779061\pi$
$877$	−45.5247	−1.53726	−0.768630	+	0.639694i	$0.779061\pi$
$878$	15.5022	0.523175
$879$	0	0
$880$	−5.13969	−0.173259
$881$	14.8628	0.500741	0.250371	−	0.968150i	$-0.419448\pi$
$881$	14.8628	0.500741	0.250371	+	0.968150i	$0.419448\pi$
$882$	0	0
$883$	−31.1385	−1.04789	−0.523947	−	0.851751i	$-0.675541\pi$
$883$	−31.1385	−1.04789	−0.523947	+	0.851751i	$0.675541\pi$
$884$	−10.1355	−0.340893
$885$	0	0
$886$	0.871712	0.0292858
$887$	57.0744	1.91637	0.958186	−	0.286148i	$-0.0923747\pi$
$887$	57.0744	1.91637	0.958186	+	0.286148i	$0.0923747\pi$
$888$	0	0
$889$	21.3129	0.714812
$890$	−7.13969	−0.239323
$891$	0	0
$892$	−21.6787	−0.725855
$893$	25.6846	0.859504
$894$	0	0
$895$	1.14653	0.0383242
$896$	−1.84982	−0.0617980
$897$	0	0
$898$	−11.4656	−0.382612
$899$	1.81940	0.0606804
$900$	0	0
$901$	21.1274	0.703855
$902$	−41.9751	−1.39762
$903$	0	0
$904$	−2.00683	−0.0667463
$905$	−18.4980	−0.614895
$906$	0	0
$907$	18.9203	0.628238	0.314119	−	0.949384i	$-0.398291\pi$
$907$	18.9203	0.628238	0.314119	+	0.949384i	$0.398291\pi$
$908$	23.8114	0.790209
$909$	0	0
$910$	5.08578	0.168592
$911$	−20.1841	−0.668731	−0.334365	−	0.942444i	$-0.608522\pi$
$911$	−20.1841	−0.668731	−0.334365	+	0.942444i	$0.608522\pi$
$912$	0	0
$913$	82.5223	2.73109
$914$	−22.8298	−0.755141
$915$	0	0
$916$	20.6403	0.681976
$917$	9.20541	0.303989
$918$	0	0
$919$	16.9627	0.559549	0.279774	−	0.960066i	$-0.409740\pi$
$919$	16.9627	0.559549	0.279774	+	0.960066i	$0.409740\pi$
$920$	−8.67602	−0.286040
$921$	0	0
$922$	−10.5294	−0.346767
$923$	7.16759	0.235924
$924$	0	0
$925$	6.28567	0.206672
$926$	9.18575	0.301862
$927$	0	0
$928$	−4.05234	−0.133025
$929$	−21.8114	−0.715609	−0.357804	−	0.933797i	$-0.616475\pi$
$929$	−21.8114	−0.715609	−0.357804	+	0.933797i	$0.616475\pi$
$930$	0	0
$931$	−30.3442	−0.994490
$932$	13.4944	0.442023
$933$	0	0
$934$	−1.81846	−0.0595017
$935$	−18.9476	−0.619651
$936$	0	0
$937$	38.4813	1.25713	0.628564	−	0.777758i	$-0.283643\pi$
$937$	38.4813	1.25713	0.628564	+	0.777758i	$0.283643\pi$
$938$	−1.84982	−0.0603986
$939$	0	0
$940$	3.02873	0.0987863
$941$	18.3494	0.598174	0.299087	−	0.954226i	$-0.403318\pi$
$941$	18.3494	0.598174	0.299087	+	0.954226i	$0.403318\pi$
$942$	0	0
$943$	−70.8558	−2.30738
$944$	−7.09419	−0.230896
$945$	0	0
$946$	42.0829	1.36823
$947$	42.2356	1.37247	0.686237	−	0.727378i	$-0.259261\pi$
$947$	42.2356	1.37247	0.686237	+	0.727378i	$0.259261\pi$
$948$	0	0
$949$	26.5739	0.862624
$950$	8.48033	0.275138
$951$	0	0
$952$	−6.81937	−0.221017
$953$	−18.4561	−0.597853	−0.298927	−	0.954276i	$-0.596629\pi$
$953$	−18.4561	−0.597853	−0.298927	+	0.954276i	$0.596629\pi$
$954$	0	0
$955$	−2.84561	−0.0920818
$956$	6.66761	0.215646
$957$	0	0
$958$	6.93089	0.223927
$959$	4.29115	0.138569
$960$	0	0
$961$	−30.7984	−0.993497
$962$	−17.2815	−0.557177
$963$	0	0
$964$	−3.95962	−0.127531
$965$	24.6472	0.793421
$966$	0	0
$967$	4.70738	0.151379	0.0756896	−	0.997131i	$-0.475884\pi$
$967$	4.70738	0.151379	0.0756896	+	0.997131i	$0.475884\pi$
$968$	15.4165	0.495504
$969$	0	0
$970$	−11.6200	−0.373097
$971$	−18.7754	−0.602533	−0.301266	−	0.953540i	$-0.597409\pi$
$971$	−18.7754	−0.602533	−0.301266	+	0.953540i	$0.597409\pi$
$972$	0	0
$973$	−19.9993	−0.641149
$974$	−3.75355	−0.120271
$975$	0	0
$976$	−12.5651	−0.402198
$977$	−25.7177	−0.822783	−0.411392	−	0.911459i	$-0.634957\pi$
$977$	−25.7177	−0.822783	−0.411392	+	0.911459i	$0.634957\pi$
$978$	0	0
$979$	36.6958	1.17280
$980$	−3.57818	−0.114301
$981$	0	0
$982$	25.9586	0.828372
$983$	26.5064	0.845421	0.422711	−	0.906265i	$-0.361079\pi$
$983$	26.5064	0.845421	0.422711	+	0.906265i	$0.361079\pi$
$984$	0	0
$985$	−8.68128	−0.276609
$986$	−14.9390	−0.475755
$987$	0	0
$988$	−23.3153	−0.741760
$989$	71.0378	2.25887
$990$	0	0
$991$	14.3390	0.455492	0.227746	−	0.973721i	$-0.426864\pi$
$991$	14.3390	0.455492	0.227746	+	0.973721i	$0.426864\pi$
$992$	−0.448975	−0.0142550
$993$	0	0
$994$	4.82251	0.152961
$995$	−3.60691	−0.114347
$996$	0	0
$997$	−56.7266	−1.79655	−0.898274	−	0.439435i	$-0.855179\pi$
$997$	−56.7266	−1.79655	−0.898274	+	0.439435i	$0.855179\pi$
$998$	−7.87866	−0.249395
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6030.2.a.bt.1.3		4
3.2	odd	2		670.2.a.j.1.4	✓	4
12.11	even	2		5360.2.a.be.1.1		4
15.2	even	4		3350.2.c.m.2949.1		8
15.8	even	4		3350.2.c.m.2949.8		8
15.14	odd	2		3350.2.a.n.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
670.2.a.j.1.4	✓	4	3.2	odd	2
3350.2.a.n.1.1		4	15.14	odd	2
3350.2.c.m.2949.1		8	15.2	even	4
3350.2.c.m.2949.8		8	15.8	even	4
5360.2.a.be.1.1		4	12.11	even	2
6030.2.a.bt.1.3		4	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:	\( N \)	\(=\)	\( 6030 = 2 \cdot 3^{2} \cdot 5 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6030.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -1.84982 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -1.84982 q^{7} +1.00000 q^{8} +1.00000 q^{10} -5.13969 q^{11} -2.74934 q^{13} -1.84982 q^{14} +1.00000 q^{16} +3.68651 q^{17} +8.48033 q^{19} +1.00000 q^{20} -5.13969 q^{22} -8.67602 q^{23} +1.00000 q^{25} -2.74934 q^{26} -1.84982 q^{28} -4.05234 q^{29} -0.448975 q^{31} +1.00000 q^{32} +3.68651 q^{34} -1.84982 q^{35} +6.28567 q^{37} +8.48033 q^{38} +1.00000 q^{40} +8.16685 q^{41} -8.18783 q^{43} -5.13969 q^{44} -8.67602 q^{46} +3.02873 q^{47} -3.57818 q^{49} +1.00000 q^{50} -2.74934 q^{52} +5.73099 q^{53} -5.13969 q^{55} -1.84982 q^{56} -4.05234 q^{58} -7.09419 q^{59} -12.5651 q^{61} -0.448975 q^{62} +1.00000 q^{64} -2.74934 q^{65} +1.00000 q^{67} +3.68651 q^{68} -1.84982 q^{70} -2.60702 q^{71} -9.66553 q^{73} +6.28567 q^{74} +8.48033 q^{76} +9.50749 q^{77} -12.4803 q^{79} +1.00000 q^{80} +8.16685 q^{82} -16.0559 q^{83} +3.68651 q^{85} -8.18783 q^{86} -5.13969 q^{88} -7.13969 q^{89} +5.08578 q^{91} -8.67602 q^{92} +3.02873 q^{94} +8.48033 q^{95} -11.6200 q^{97} -3.57818 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{2} + 4 q^{4} + 4 q^{5} - 5 q^{7} + 4 q^{8}+O(q^{10}) \) 4 * q + 4 * q^2 + 4 * q^4 + 4 * q^5 - 5 * q^7 + 4 * q^8	\( 4 q + 4 q^{2} + 4 q^{4} + 4 q^{5} - 5 q^{7} + 4 q^{8} + 4 q^{10} - 9 q^{11} - 12 q^{13} - 5 q^{14} + 4 q^{16} + 4 q^{19} + 4 q^{20} - 9 q^{22} - 6 q^{23} + 4 q^{25} - 12 q^{26} - 5 q^{28} - 18 q^{29} + 2 q^{31} + 4 q^{32} - 5 q^{35} + 9 q^{37} + 4 q^{38} + 4 q^{40} - 12 q^{41} - 16 q^{43} - 9 q^{44} - 6 q^{46} - 10 q^{47} + 15 q^{49} + 4 q^{50} - 12 q^{52} - 8 q^{53} - 9 q^{55} - 5 q^{56} - 18 q^{58} - 18 q^{59} - 11 q^{61} + 2 q^{62} + 4 q^{64} - 12 q^{65} + 4 q^{67} - 5 q^{70} - 27 q^{71} + 4 q^{73} + 9 q^{74} + 4 q^{76} - 25 q^{77} - 20 q^{79} + 4 q^{80} - 12 q^{82} - 9 q^{83} - 16 q^{86} - 9 q^{88} - 17 q^{89} - 4 q^{91} - 6 q^{92} - 10 q^{94} + 4 q^{95} - 5 q^{97} + 15 q^{98}+O(q^{100}) \) 4 * q + 4 * q^2 + 4 * q^4 + 4 * q^5 - 5 * q^7 + 4 * q^8 + 4 * q^10 - 9 * q^11 - 12 * q^13 - 5 * q^14 + 4 * q^16 + 4 * q^19 + 4 * q^20 - 9 * q^22 - 6 * q^23 + 4 * q^25 - 12 * q^26 - 5 * q^28 - 18 * q^29 + 2 * q^31 + 4 * q^32 - 5 * q^35 + 9 * q^37 + 4 * q^38 + 4 * q^40 - 12 * q^41 - 16 * q^43 - 9 * q^44 - 6 * q^46 - 10 * q^47 + 15 * q^49 + 4 * q^50 - 12 * q^52 - 8 * q^53 - 9 * q^55 - 5 * q^56 - 18 * q^58 - 18 * q^59 - 11 * q^61 + 2 * q^62 + 4 * q^64 - 12 * q^65 + 4 * q^67 - 5 * q^70 - 27 * q^71 + 4 * q^73 + 9 * q^74 + 4 * q^76 - 25 * q^77 - 20 * q^79 + 4 * q^80 - 12 * q^82 - 9 * q^83 - 16 * q^86 - 9 * q^88 - 17 * q^89 - 4 * q^91 - 6 * q^92 - 10 * q^94 + 4 * q^95 - 5 * q^97 + 15 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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