LMFDB - Embedded newform 3020.2.a.f.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3020,2,Mod(1,3020)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3020, 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("3020.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3020 = 2^{2} \cdot 5 \cdot 151 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3020.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:	$24.1148214104$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.40310669.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 7x^{4} + 10x^{3} + 11x^{2} - 11x + 2 $ x^6 - 2x^5 - 7x^4 + 10x^3 + 11x^2 - 11*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$0.266886$ of defining polynomial
Character	$\chi$	$=$	3020.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.19566 q^{3} +1.00000 q^{5} +0.489165 q^{7} +7.21223 q^{9} +O(q^{10})$	$q+3.19566 q^{3} +1.00000 q^{5} +0.489165 q^{7} +7.21223 q^{9} -1.28160 q^{11} -5.44922 q^{13} +3.19566 q^{15} +7.20528 q^{17} +7.11355 q^{19} +1.56320 q^{21} +5.63768 q^{23} +1.00000 q^{25} +13.4608 q^{27} +3.37333 q^{29} -10.5941 q^{31} -4.09556 q^{33} +0.489165 q^{35} -10.6571 q^{37} -17.4139 q^{39} +5.09966 q^{41} -10.6520 q^{43} +7.21223 q^{45} -2.69954 q^{47} -6.76072 q^{49} +23.0256 q^{51} +2.51223 q^{53} -1.28160 q^{55} +22.7325 q^{57} -0.657219 q^{59} +3.41215 q^{61} +3.52797 q^{63} -5.44922 q^{65} +12.9207 q^{67} +18.0161 q^{69} +4.23257 q^{71} -0.126042 q^{73} +3.19566 q^{75} -0.626915 q^{77} -6.21108 q^{79} +21.3796 q^{81} +11.9371 q^{83} +7.20528 q^{85} +10.7800 q^{87} -6.55743 q^{89} -2.66557 q^{91} -33.8550 q^{93} +7.11355 q^{95} +8.27554 q^{97} -9.24321 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 2 q^{3} + 6 q^{5} + 3 q^{7} + 20 q^{9}+O(q^{10}) $ 6 * q + 2 * q^3 + 6 * q^5 + 3 * q^7 + 20 * q^9	$ 6 q + 2 q^{3} + 6 q^{5} + 3 q^{7} + 20 q^{9} + 3 q^{11} + 2 q^{13} + 2 q^{15} + 9 q^{17} + 16 q^{19} - 12 q^{21} + 6 q^{23} + 6 q^{25} + 23 q^{27} + 2 q^{29} + 15 q^{31} - 14 q^{33} + 3 q^{35} + 9 q^{37} - 17 q^{39} - 16 q^{41} + 15 q^{43} + 20 q^{45} + 12 q^{47} - 7 q^{49} + 11 q^{51} - 14 q^{53} + 3 q^{55} + 27 q^{57} - 20 q^{59} + 17 q^{61} + 17 q^{63} + 2 q^{65} + 29 q^{67} - 40 q^{69} + 13 q^{71} - 6 q^{73} + 2 q^{75} + 13 q^{77} - 8 q^{79} + 30 q^{81} + 36 q^{83} + 9 q^{85} + 2 q^{87} - 15 q^{89} - 14 q^{91} - 31 q^{93} + 16 q^{95} - 23 q^{97} + 38 q^{99}+O(q^{100}) $ 6 * q + 2 * q^3 + 6 * q^5 + 3 * q^7 + 20 * q^9 + 3 * q^11 + 2 * q^13 + 2 * q^15 + 9 * q^17 + 16 * q^19 - 12 * q^21 + 6 * q^23 + 6 * q^25 + 23 * q^27 + 2 * q^29 + 15 * q^31 - 14 * q^33 + 3 * q^35 + 9 * q^37 - 17 * q^39 - 16 * q^41 + 15 * q^43 + 20 * q^45 + 12 * q^47 - 7 * q^49 + 11 * q^51 - 14 * q^53 + 3 * q^55 + 27 * q^57 - 20 * q^59 + 17 * q^61 + 17 * q^63 + 2 * q^65 + 29 * q^67 - 40 * q^69 + 13 * q^71 - 6 * q^73 + 2 * q^75 + 13 * q^77 - 8 * q^79 + 30 * q^81 + 36 * q^83 + 9 * q^85 + 2 * q^87 - 15 * q^89 - 14 * q^91 - 31 * q^93 + 16 * q^95 - 23 * q^97 + 38 * 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
$3$	3.19566	1.84501	0.922507	−	0.385980i	$-0.126137\pi$
$3$	3.19566	1.84501	0.922507	+	0.385980i	$0.126137\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	0.489165	0.184887	0.0924435	−	0.995718i	$-0.470532\pi$
$7$	0.489165	0.184887	0.0924435	+	0.995718i	$0.470532\pi$
$8$	0	0
$9$	7.21223	2.40408
$10$	0	0
$11$	−1.28160	−0.386417	−0.193209	−	0.981158i	$-0.561889\pi$
$11$	−1.28160	−0.386417	−0.193209	+	0.981158i	$0.561889\pi$
$12$	0	0
$13$	−5.44922	−1.51134	−0.755671	−	0.654951i	$-0.772689\pi$
$13$	−5.44922	−1.51134	−0.755671	+	0.654951i	$0.772689\pi$
$14$	0	0
$15$	3.19566	0.825115
$16$	0	0
$17$	7.20528	1.74754	0.873768	−	0.486343i	$-0.161669\pi$
$17$	7.20528	1.74754	0.873768	+	0.486343i	$0.161669\pi$
$18$	0	0
$19$	7.11355	1.63196	0.815980	−	0.578080i	$-0.196198\pi$
$19$	7.11355	1.63196	0.815980	+	0.578080i	$0.196198\pi$
$20$	0	0
$21$	1.56320	0.341119
$22$	0	0
$23$	5.63768	1.17554	0.587769	−	0.809029i	$-0.300006\pi$
$23$	5.63768	1.17554	0.587769	+	0.809029i	$0.300006\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	13.4608	2.59054
$28$	0	0
$29$	3.37333	0.626412	0.313206	−	0.949685i	$-0.398597\pi$
$29$	3.37333	0.626412	0.313206	+	0.949685i	$0.398597\pi$
$30$	0	0
$31$	−10.5941	−1.90275	−0.951375	−	0.308036i	$-0.900328\pi$
$31$	−10.5941	−1.90275	−0.951375	+	0.308036i	$0.900328\pi$
$32$	0	0
$33$	−4.09556	−0.712946
$34$	0	0
$35$	0.489165	0.0826840
$36$	0	0
$37$	−10.6571	−1.75201	−0.876005	−	0.482301i	$-0.839801\pi$
$37$	−10.6571	−1.75201	−0.876005	+	0.482301i	$0.839801\pi$
$38$	0	0
$39$	−17.4139	−2.78845
$40$	0	0
$41$	5.09966	0.796434	0.398217	−	0.917291i	$-0.369629\pi$
$41$	5.09966	0.796434	0.398217	+	0.917291i	$0.369629\pi$
$42$	0	0
$43$	−10.6520	−1.62441	−0.812205	−	0.583372i	$-0.801733\pi$
$43$	−10.6520	−1.62441	−0.812205	+	0.583372i	$0.801733\pi$
$44$	0	0
$45$	7.21223	1.07514
$46$	0	0
$47$	−2.69954	−0.393768	−0.196884	−	0.980427i	$-0.563082\pi$
$47$	−2.69954	−0.393768	−0.196884	+	0.980427i	$0.563082\pi$
$48$	0	0
$49$	−6.76072	−0.965817
$50$	0	0
$51$	23.0256	3.22423
$52$	0	0
$53$	2.51223	0.345081	0.172541	−	0.985002i	$-0.444802\pi$
$53$	2.51223	0.345081	0.172541	+	0.985002i	$0.444802\pi$
$54$	0	0
$55$	−1.28160	−0.172811
$56$	0	0
$57$	22.7325	3.01099
$58$	0	0
$59$	−0.657219	−0.0855626	−0.0427813	−	0.999084i	$-0.513622\pi$
$59$	−0.657219	−0.0855626	−0.0427813	+	0.999084i	$0.513622\pi$
$60$	0	0
$61$	3.41215	0.436881	0.218441	−	0.975850i	$-0.429903\pi$
$61$	3.41215	0.436881	0.218441	+	0.975850i	$0.429903\pi$
$62$	0	0
$63$	3.52797	0.444482
$64$	0	0
$65$	−5.44922	−0.675893
$66$	0	0
$67$	12.9207	1.57851	0.789257	−	0.614063i	$-0.210466\pi$
$67$	12.9207	1.57851	0.789257	+	0.614063i	$0.210466\pi$
$68$	0	0
$69$	18.0161	2.16888
$70$	0	0
$71$	4.23257	0.502314	0.251157	−	0.967946i	$-0.419189\pi$
$71$	4.23257	0.502314	0.251157	+	0.967946i	$0.419189\pi$
$72$	0	0
$73$	−0.126042	−0.0147521	−0.00737605	−	0.999973i	$-0.502348\pi$
$73$	−0.126042	−0.0147521	−0.00737605	+	0.999973i	$0.502348\pi$
$74$	0	0
$75$	3.19566	0.369003
$76$	0	0
$77$	−0.626915	−0.0714436
$78$	0	0
$79$	−6.21108	−0.698801	−0.349400	−	0.936973i	$-0.613615\pi$
$79$	−6.21108	−0.698801	−0.349400	+	0.936973i	$0.613615\pi$
$80$	0	0
$81$	21.3796	2.37551
$82$	0	0
$83$	11.9371	1.31027	0.655135	−	0.755512i	$-0.272612\pi$
$83$	11.9371	1.31027	0.655135	+	0.755512i	$0.272612\pi$
$84$	0	0
$85$	7.20528	0.781522
$86$	0	0
$87$	10.7800	1.15574
$88$	0	0
$89$	−6.55743	−0.695086	−0.347543	−	0.937664i	$-0.612984\pi$
$89$	−6.55743	−0.695086	−0.347543	+	0.937664i	$0.612984\pi$
$90$	0	0
$91$	−2.66557	−0.279428
$92$	0	0
$93$	−33.8550	−3.51060
$94$	0	0
$95$	7.11355	0.729835
$96$	0	0
$97$	8.27554	0.840254	0.420127	−	0.907465i	$-0.361985\pi$
$97$	8.27554	0.840254	0.420127	+	0.907465i	$0.361985\pi$
$98$	0	0
$99$	−9.24321	−0.928977
$100$	0	0
$101$	−11.4028	−1.13462	−0.567310	−	0.823504i	$-0.692016\pi$
$101$	−11.4028	−1.13462	−0.567310	+	0.823504i	$0.692016\pi$
$102$	0	0
$103$	12.5753	1.23908	0.619539	−	0.784966i	$-0.287320\pi$
$103$	12.5753	1.23908	0.619539	+	0.784966i	$0.287320\pi$
$104$	0	0
$105$	1.56320	0.152553
$106$	0	0
$107$	0.146973	0.0142084	0.00710420	−	0.999975i	$-0.497739\pi$
$107$	0.146973	0.0142084	0.00710420	+	0.999975i	$0.497739\pi$
$108$	0	0
$109$	10.4298	0.998991	0.499496	−	0.866316i	$-0.333519\pi$
$109$	10.4298	0.998991	0.499496	+	0.866316i	$0.333519\pi$
$110$	0	0
$111$	−34.0563	−3.23248
$112$	0	0
$113$	0.436946	0.0411045	0.0205522	−	0.999789i	$-0.493458\pi$
$113$	0.436946	0.0411045	0.0205522	+	0.999789i	$0.493458\pi$
$114$	0	0
$115$	5.63768	0.525717
$116$	0	0
$117$	−39.3011	−3.63338
$118$	0	0
$119$	3.52457	0.323097
$120$	0	0
$121$	−9.35750	−0.850682
$122$	0	0
$123$	16.2968	1.46943
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−9.29959	−0.825205	−0.412603	−	0.910911i	$-0.635380\pi$
$127$	−9.29959	−0.825205	−0.412603	+	0.910911i	$0.635380\pi$
$128$	0	0
$129$	−34.0400	−2.99706
$130$	0	0
$131$	6.48249	0.566378	0.283189	−	0.959064i	$-0.408608\pi$
$131$	6.48249	0.566378	0.283189	+	0.959064i	$0.408608\pi$
$132$	0	0
$133$	3.47970	0.301728
$134$	0	0
$135$	13.4608	1.15853
$136$	0	0
$137$	−2.25309	−0.192494	−0.0962472	−	0.995357i	$-0.530684\pi$
$137$	−2.25309	−0.192494	−0.0962472	+	0.995357i	$0.530684\pi$
$138$	0	0
$139$	−14.3560	−1.21766	−0.608832	−	0.793299i	$-0.708362\pi$
$139$	−14.3560	−1.21766	−0.608832	+	0.793299i	$0.708362\pi$
$140$	0	0
$141$	−8.62680	−0.726508
$142$	0	0
$143$	6.98374	0.584009
$144$	0	0
$145$	3.37333	0.280140
$146$	0	0
$147$	−21.6049	−1.78195
$148$	0	0
$149$	1.71142	0.140205	0.0701023	−	0.997540i	$-0.477667\pi$
$149$	1.71142	0.140205	0.0701023	+	0.997540i	$0.477667\pi$
$150$	0	0
$151$	1.00000	0.0813788
$151$	1.00000	0.0813788
$152$	0	0
$153$	51.9661	4.20121
$154$	0	0
$155$	−10.5941	−0.850935
$156$	0	0
$157$	2.15785	0.172216	0.0861078	−	0.996286i	$-0.472557\pi$
$157$	2.15785	0.172216	0.0861078	+	0.996286i	$0.472557\pi$
$158$	0	0
$159$	8.02823	0.636680
$160$	0	0
$161$	2.75776	0.217342
$162$	0	0
$163$	−5.49893	−0.430709	−0.215355	−	0.976536i	$-0.569091\pi$
$163$	−5.49893	−0.430709	−0.215355	+	0.976536i	$0.569091\pi$
$164$	0	0
$165$	−4.09556	−0.318839
$166$	0	0
$167$	−15.2376	−1.17912	−0.589560	−	0.807725i	$-0.700699\pi$
$167$	−15.2376	−1.17912	−0.589560	+	0.807725i	$0.700699\pi$
$168$	0	0
$169$	16.6940	1.28416
$170$	0	0
$171$	51.3045	3.92336
$172$	0	0
$173$	−4.84860	−0.368633	−0.184316	−	0.982867i	$-0.559007\pi$
$173$	−4.84860	−0.368633	−0.184316	+	0.982867i	$0.559007\pi$
$174$	0	0
$175$	0.489165	0.0369774
$176$	0	0
$177$	−2.10025	−0.157864
$178$	0	0
$179$	−21.6999	−1.62193	−0.810963	−	0.585098i	$-0.801056\pi$
$179$	−21.6999	−1.62193	−0.810963	+	0.585098i	$0.801056\pi$
$180$	0	0
$181$	25.3122	1.88144	0.940720	−	0.339185i	$-0.110151\pi$
$181$	25.3122	1.88144	0.940720	+	0.339185i	$0.110151\pi$
$182$	0	0
$183$	10.9041	0.806052
$184$	0	0
$185$	−10.6571	−0.783523
$186$	0	0
$187$	−9.23429	−0.675278
$188$	0	0
$189$	6.58457	0.478957
$190$	0	0
$191$	1.38992	0.100571	0.0502855	−	0.998735i	$-0.483987\pi$
$191$	1.38992	0.100571	0.0502855	+	0.998735i	$0.483987\pi$
$192$	0	0
$193$	6.83508	0.491999	0.246000	−	0.969270i	$-0.420884\pi$
$193$	6.83508	0.491999	0.246000	+	0.969270i	$0.420884\pi$
$194$	0	0
$195$	−17.4139	−1.24703
$196$	0	0
$197$	−25.5591	−1.82101	−0.910505	−	0.413498i	$-0.864307\pi$
$197$	−25.5591	−1.82101	−0.910505	+	0.413498i	$0.864307\pi$
$198$	0	0
$199$	16.2607	1.15269	0.576345	−	0.817207i	$-0.304479\pi$
$199$	16.2607	1.15269	0.576345	+	0.817207i	$0.304479\pi$
$200$	0	0
$201$	41.2901	2.91238
$202$	0	0
$203$	1.65011	0.115815
$204$	0	0
$205$	5.09966	0.356176
$206$	0	0
$207$	40.6603	2.82608
$208$	0	0
$209$	−9.11674	−0.630618
$210$	0	0
$211$	5.12951	0.353130	0.176565	−	0.984289i	$-0.443501\pi$
$211$	5.12951	0.353130	0.176565	+	0.984289i	$0.443501\pi$
$212$	0	0
$213$	13.5259	0.926776
$214$	0	0
$215$	−10.6520	−0.726458
$216$	0	0
$217$	−5.18224	−0.351794
$218$	0	0
$219$	−0.402787	−0.0272178
$220$	0	0
$221$	−39.2632	−2.64113
$222$	0	0
$223$	−19.6284	−1.31442	−0.657209	−	0.753709i	$-0.728263\pi$
$223$	−19.6284	−1.31442	−0.657209	+	0.753709i	$0.728263\pi$
$224$	0	0
$225$	7.21223	0.480815
$226$	0	0
$227$	−9.45225	−0.627368	−0.313684	−	0.949527i	$-0.601563\pi$
$227$	−9.45225	−0.627368	−0.313684	+	0.949527i	$0.601563\pi$
$228$	0	0
$229$	−9.05466	−0.598349	−0.299174	−	0.954198i	$-0.596711\pi$
$229$	−9.05466	−0.598349	−0.299174	+	0.954198i	$0.596711\pi$
$230$	0	0
$231$	−2.00340	−0.131814
$232$	0	0
$233$	−10.5577	−0.691655	−0.345827	−	0.938298i	$-0.612402\pi$
$233$	−10.5577	−0.691655	−0.345827	+	0.938298i	$0.612402\pi$
$234$	0	0
$235$	−2.69954	−0.176098
$236$	0	0
$237$	−19.8485	−1.28930
$238$	0	0
$239$	−7.54552	−0.488079	−0.244040	−	0.969765i	$-0.578473\pi$
$239$	−7.54552	−0.488079	−0.244040	+	0.969765i	$0.578473\pi$
$240$	0	0
$241$	−10.3078	−0.663985	−0.331993	−	0.943282i	$-0.607721\pi$
$241$	−10.3078	−0.663985	−0.331993	+	0.943282i	$0.607721\pi$
$242$	0	0
$243$	27.9393	1.79230
$244$	0	0
$245$	−6.76072	−0.431926
$246$	0	0
$247$	−38.7633	−2.46645
$248$	0	0
$249$	38.1470	2.41747
$250$	0	0
$251$	−27.1567	−1.71412	−0.857059	−	0.515218i	$-0.827711\pi$
$251$	−27.1567	−1.71412	−0.857059	+	0.515218i	$0.827711\pi$
$252$	0	0
$253$	−7.22527	−0.454249
$254$	0	0
$255$	23.0256	1.44192
$256$	0	0
$257$	19.4899	1.21575	0.607874	−	0.794034i	$-0.292023\pi$
$257$	19.4899	1.21575	0.607874	+	0.794034i	$0.292023\pi$
$258$	0	0
$259$	−5.21306	−0.323924
$260$	0	0
$261$	24.3292	1.50594
$262$	0	0
$263$	26.0998	1.60938	0.804690	−	0.593695i	$-0.202331\pi$
$263$	26.0998	1.60938	0.804690	+	0.593695i	$0.202331\pi$
$264$	0	0
$265$	2.51223	0.154325
$266$	0	0
$267$	−20.9553	−1.28244
$268$	0	0
$269$	14.8827	0.907413	0.453706	−	0.891151i	$-0.350101\pi$
$269$	14.8827	0.907413	0.453706	+	0.891151i	$0.350101\pi$
$270$	0	0
$271$	13.8962	0.844131	0.422066	−	0.906565i	$-0.361305\pi$
$271$	13.8962	0.844131	0.422066	+	0.906565i	$0.361305\pi$
$272$	0	0
$273$	−8.51825	−0.515548
$274$	0	0
$275$	−1.28160	−0.0772835
$276$	0	0
$277$	−9.97811	−0.599527	−0.299763	−	0.954014i	$-0.596908\pi$
$277$	−9.97811	−0.599527	−0.299763	+	0.954014i	$0.596908\pi$
$278$	0	0
$279$	−76.4068	−4.57435
$280$	0	0
$281$	4.02517	0.240121	0.120061	−	0.992767i	$-0.461691\pi$
$281$	4.02517	0.240121	0.120061	+	0.992767i	$0.461691\pi$
$282$	0	0
$283$	−15.3196	−0.910656	−0.455328	−	0.890324i	$-0.650478\pi$
$283$	−15.3196	−0.910656	−0.455328	+	0.890324i	$0.650478\pi$
$284$	0	0
$285$	22.7325	1.34656
$286$	0	0
$287$	2.49458	0.147250
$288$	0	0
$289$	34.9160	2.05388
$290$	0	0
$291$	26.4458	1.55028
$292$	0	0
$293$	10.4029	0.607742	0.303871	−	0.952713i	$-0.401721\pi$
$293$	10.4029	0.607742	0.303871	+	0.952713i	$0.401721\pi$
$294$	0	0
$295$	−0.657219	−0.0382648
$296$	0	0
$297$	−17.2514	−1.00103
$298$	0	0
$299$	−30.7210	−1.77664
$300$	0	0
$301$	−5.21057	−0.300332
$302$	0	0
$303$	−36.4394	−2.09339
$304$	0	0
$305$	3.41215	0.195379
$306$	0	0
$307$	−31.7083	−1.80969	−0.904845	−	0.425741i	$-0.860013\pi$
$307$	−31.7083	−1.80969	−0.904845	+	0.425741i	$0.860013\pi$
$308$	0	0
$309$	40.1862	2.28612
$310$	0	0
$311$	25.7919	1.46252	0.731262	−	0.682097i	$-0.238932\pi$
$311$	25.7919	1.46252	0.731262	+	0.682097i	$0.238932\pi$
$312$	0	0
$313$	−2.19158	−0.123875	−0.0619376	−	0.998080i	$-0.519728\pi$
$313$	−2.19158	−0.123875	−0.0619376	+	0.998080i	$0.519728\pi$
$314$	0	0
$315$	3.52797	0.198779
$316$	0	0
$317$	−24.5280	−1.37763	−0.688816	−	0.724936i	$-0.741869\pi$
$317$	−24.5280	−1.37763	−0.688816	+	0.724936i	$0.741869\pi$
$318$	0	0
$319$	−4.32326	−0.242056
$320$	0	0
$321$	0.469675	0.0262147
$322$	0	0
$323$	51.2551	2.85191
$324$	0	0
$325$	−5.44922	−0.302269
$326$	0	0
$327$	33.3300	1.84315
$328$	0	0
$329$	−1.32052	−0.0728026
$330$	0	0
$331$	1.59740	0.0878009	0.0439004	−	0.999036i	$-0.486022\pi$
$331$	1.59740	0.0878009	0.0439004	+	0.999036i	$0.486022\pi$
$332$	0	0
$333$	−76.8612	−4.21197
$334$	0	0
$335$	12.9207	0.705933
$336$	0	0
$337$	−23.1356	−1.26027	−0.630137	−	0.776484i	$-0.717001\pi$
$337$	−23.1356	−1.26027	−0.630137	+	0.776484i	$0.717001\pi$
$338$	0	0
$339$	1.39633	0.0758383
$340$	0	0
$341$	13.5774	0.735256
$342$	0	0
$343$	−6.73126	−0.363454
$344$	0	0
$345$	18.0161	0.969955
$346$	0	0
$347$	−16.2864	−0.874301	−0.437151	−	0.899388i	$-0.644012\pi$
$347$	−16.2864	−0.874301	−0.437151	+	0.899388i	$0.644012\pi$
$348$	0	0
$349$	−21.4003	−1.14553	−0.572767	−	0.819718i	$-0.694130\pi$
$349$	−21.4003	−1.14553	−0.572767	+	0.819718i	$0.694130\pi$
$350$	0	0
$351$	−73.3512	−3.91520
$352$	0	0
$353$	−32.8761	−1.74982	−0.874909	−	0.484287i	$-0.839079\pi$
$353$	−32.8761	−1.74982	−0.874909	+	0.484287i	$0.839079\pi$
$354$	0	0
$355$	4.23257	0.224642
$356$	0	0
$357$	11.2633	0.596118
$358$	0	0
$359$	−4.82159	−0.254474	−0.127237	−	0.991872i	$-0.540611\pi$
$359$	−4.82159	−0.254474	−0.127237	+	0.991872i	$0.540611\pi$
$360$	0	0
$361$	31.6026	1.66329
$362$	0	0
$363$	−29.9034	−1.56952
$364$	0	0
$365$	−0.126042	−0.00659734
$366$	0	0
$367$	7.85339	0.409944	0.204972	−	0.978768i	$-0.434290\pi$
$367$	7.85339	0.409944	0.204972	+	0.978768i	$0.434290\pi$
$368$	0	0
$369$	36.7800	1.91469
$370$	0	0
$371$	1.22890	0.0638011
$372$	0	0
$373$	5.09612	0.263867	0.131934	−	0.991259i	$-0.457881\pi$
$373$	5.09612	0.263867	0.131934	+	0.991259i	$0.457881\pi$
$374$	0	0
$375$	3.19566	0.165023
$376$	0	0
$377$	−18.3820	−0.946723
$378$	0	0
$379$	−12.7958	−0.657275	−0.328637	−	0.944456i	$-0.606589\pi$
$379$	−12.7958	−0.657275	−0.328637	+	0.944456i	$0.606589\pi$
$380$	0	0
$381$	−29.7183	−1.52251
$382$	0	0
$383$	38.5717	1.97092	0.985461	−	0.169902i	$-0.0543450\pi$
$383$	38.5717	1.97092	0.985461	+	0.169902i	$0.0543450\pi$
$384$	0	0
$385$	−0.626915	−0.0319505
$386$	0	0
$387$	−76.8244	−3.90521
$388$	0	0
$389$	−24.3442	−1.23430	−0.617150	−	0.786845i	$-0.711713\pi$
$389$	−24.3442	−1.23430	−0.617150	+	0.786845i	$0.711713\pi$
$390$	0	0
$391$	40.6211	2.05430
$392$	0	0
$393$	20.7158	1.04497
$394$	0	0
$395$	−6.21108	−0.312513
$396$	0	0
$397$	21.9880	1.10355	0.551773	−	0.833995i	$-0.313952\pi$
$397$	21.9880	1.10355	0.551773	+	0.833995i	$0.313952\pi$
$398$	0	0
$399$	11.1199	0.556693
$400$	0	0
$401$	−9.74949	−0.486866	−0.243433	−	0.969918i	$-0.578274\pi$
$401$	−9.74949	−0.486866	−0.243433	+	0.969918i	$0.578274\pi$
$402$	0	0
$403$	57.7294	2.87571
$404$	0	0
$405$	21.3796	1.06236
$406$	0	0
$407$	13.6581	0.677008
$408$	0	0
$409$	−12.5289	−0.619512	−0.309756	−	0.950816i	$-0.600247\pi$
$409$	−12.5289	−0.619512	−0.309756	+	0.950816i	$0.600247\pi$
$410$	0	0
$411$	−7.20011	−0.355155
$412$	0	0
$413$	−0.321488	−0.0158194
$414$	0	0
$415$	11.9371	0.585970
$416$	0	0
$417$	−45.8770	−2.24661
$418$	0	0
$419$	−10.8570	−0.530397	−0.265199	−	0.964194i	$-0.585438\pi$
$419$	−10.8570	−0.530397	−0.265199	+	0.964194i	$0.585438\pi$
$420$	0	0
$421$	−24.8992	−1.21351	−0.606756	−	0.794888i	$-0.707530\pi$
$421$	−24.8992	−1.21351	−0.606756	+	0.794888i	$0.707530\pi$
$422$	0	0
$423$	−19.4697	−0.946649
$424$	0	0
$425$	7.20528	0.349507
$426$	0	0
$427$	1.66911	0.0807737
$428$	0	0
$429$	22.3176	1.07751
$430$	0	0
$431$	−5.31535	−0.256032	−0.128016	−	0.991772i	$-0.540861\pi$
$431$	−5.31535	−0.256032	−0.128016	+	0.991772i	$0.540861\pi$
$432$	0	0
$433$	2.41577	0.116095	0.0580473	−	0.998314i	$-0.481513\pi$
$433$	2.41577	0.116095	0.0580473	+	0.998314i	$0.481513\pi$
$434$	0	0
$435$	10.7800	0.516862
$436$	0	0
$437$	40.1039	1.91843
$438$	0	0
$439$	−18.1076	−0.864228	−0.432114	−	0.901819i	$-0.642232\pi$
$439$	−18.1076	−0.864228	−0.432114	+	0.901819i	$0.642232\pi$
$440$	0	0
$441$	−48.7598	−2.32190
$442$	0	0
$443$	3.44198	0.163533	0.0817667	−	0.996651i	$-0.473944\pi$
$443$	3.44198	0.163533	0.0817667	+	0.996651i	$0.473944\pi$
$444$	0	0
$445$	−6.55743	−0.310852
$446$	0	0
$447$	5.46910	0.258679
$448$	0	0
$449$	−0.228766	−0.0107962	−0.00539808	−	0.999985i	$-0.501718\pi$
$449$	−0.228766	−0.0107962	−0.00539808	+	0.999985i	$0.501718\pi$
$450$	0	0
$451$	−6.53574	−0.307756
$452$	0	0
$453$	3.19566	0.150145
$454$	0	0
$455$	−2.66557	−0.124964
$456$	0	0
$457$	−6.73685	−0.315136	−0.157568	−	0.987508i	$-0.550365\pi$
$457$	−6.73685	−0.315136	−0.157568	+	0.987508i	$0.550365\pi$
$458$	0	0
$459$	96.9891	4.52706
$460$	0	0
$461$	−31.9167	−1.48651	−0.743255	−	0.669008i	$-0.766719\pi$
$461$	−31.9167	−1.48651	−0.743255	+	0.669008i	$0.766719\pi$
$462$	0	0
$463$	1.12960	0.0524968	0.0262484	−	0.999655i	$-0.491644\pi$
$463$	1.12960	0.0524968	0.0262484	+	0.999655i	$0.491644\pi$
$464$	0	0
$465$	−33.8550	−1.56999
$466$	0	0
$467$	25.3122	1.17131	0.585654	−	0.810561i	$-0.300838\pi$
$467$	25.3122	1.17131	0.585654	+	0.810561i	$0.300838\pi$
$468$	0	0
$469$	6.32035	0.291847
$470$	0	0
$471$	6.89577	0.317740
$472$	0	0
$473$	13.6516	0.627700
$474$	0	0
$475$	7.11355	0.326392
$476$	0	0
$477$	18.1188	0.829602
$478$	0	0
$479$	−16.5518	−0.756271	−0.378135	−	0.925750i	$-0.623435\pi$
$479$	−16.5518	−0.756271	−0.378135	+	0.925750i	$0.623435\pi$
$480$	0	0
$481$	58.0727	2.64789
$482$	0	0
$483$	8.81285	0.400998
$484$	0	0
$485$	8.27554	0.375773
$486$	0	0
$487$	−17.3540	−0.786386	−0.393193	−	0.919456i	$-0.628630\pi$
$487$	−17.3540	−0.786386	−0.393193	+	0.919456i	$0.628630\pi$
$488$	0	0
$489$	−17.5727	−0.794665
$490$	0	0
$491$	11.8352	0.534116	0.267058	−	0.963680i	$-0.413948\pi$
$491$	11.8352	0.534116	0.267058	+	0.963680i	$0.413948\pi$
$492$	0	0
$493$	24.3058	1.09468
$494$	0	0
$495$	−9.24321	−0.415451
$496$	0	0
$497$	2.07043	0.0928713
$498$	0	0
$499$	15.3209	0.685856	0.342928	−	0.939362i	$-0.388581\pi$
$499$	15.3209	0.685856	0.342928	+	0.939362i	$0.388581\pi$
$500$	0	0
$501$	−48.6941	−2.17549
$502$	0	0
$503$	27.0461	1.20593	0.602963	−	0.797769i	$-0.293986\pi$
$503$	27.0461	1.20593	0.602963	+	0.797769i	$0.293986\pi$
$504$	0	0
$505$	−11.4028	−0.507417
$506$	0	0
$507$	53.3485	2.36929
$508$	0	0
$509$	39.5157	1.75150	0.875750	−	0.482764i	$-0.160367\pi$
$509$	39.5157	1.75150	0.875750	+	0.482764i	$0.160367\pi$
$510$	0	0
$511$	−0.0616553	−0.00272747
$512$	0	0
$513$	95.7544	4.22766
$514$	0	0
$515$	12.5753	0.554132
$516$	0	0
$517$	3.45973	0.152159
$518$	0	0
$519$	−15.4945	−0.680132
$520$	0	0
$521$	−6.06359	−0.265651	−0.132825	−	0.991139i	$-0.542405\pi$
$521$	−6.06359	−0.265651	−0.132825	+	0.991139i	$0.542405\pi$
$522$	0	0
$523$	30.3117	1.32544	0.662720	−	0.748867i	$-0.269402\pi$
$523$	30.3117	1.32544	0.662720	+	0.748867i	$0.269402\pi$
$524$	0	0
$525$	1.56320	0.0682238
$526$	0	0
$527$	−76.3331	−3.32512
$528$	0	0
$529$	8.78348	0.381890
$530$	0	0
$531$	−4.74001	−0.205699
$532$	0	0
$533$	−27.7892	−1.20368
$534$	0	0
$535$	0.146973	0.00635419
$536$	0	0
$537$	−69.3454	−2.99248
$538$	0	0
$539$	8.66455	0.373209
$540$	0	0
$541$	2.12605	0.0914060	0.0457030	−	0.998955i	$-0.485447\pi$
$541$	2.12605	0.0914060	0.0457030	+	0.998955i	$0.485447\pi$
$542$	0	0
$543$	80.8890	3.47128
$544$	0	0
$545$	10.4298	0.446762
$546$	0	0
$547$	21.7759	0.931070	0.465535	−	0.885029i	$-0.345862\pi$
$547$	21.7759	0.931070	0.465535	+	0.885029i	$0.345862\pi$
$548$	0	0
$549$	24.6092	1.05030
$550$	0	0
$551$	23.9963	1.02228
$552$	0	0
$553$	−3.03824	−0.129199
$554$	0	0
$555$	−34.0563	−1.44561
$556$	0	0
$557$	31.8326	1.34879	0.674395	−	0.738370i	$-0.264404\pi$
$557$	31.8326	1.34879	0.674395	+	0.738370i	$0.264404\pi$
$558$	0	0
$559$	58.0450	2.45504
$560$	0	0
$561$	−29.5096	−1.24590
$562$	0	0
$563$	32.3361	1.36280	0.681402	−	0.731909i	$-0.261371\pi$
$563$	32.3361	1.36280	0.681402	+	0.731909i	$0.261371\pi$
$564$	0	0
$565$	0.436946	0.0183825
$566$	0	0
$567$	10.4581	0.439200
$568$	0	0
$569$	−34.5176	−1.44705	−0.723527	−	0.690296i	$-0.757480\pi$
$569$	−34.5176	−1.44705	−0.723527	+	0.690296i	$0.757480\pi$
$570$	0	0
$571$	−3.29298	−0.137807	−0.0689035	−	0.997623i	$-0.521950\pi$
$571$	−3.29298	−0.137807	−0.0689035	+	0.997623i	$0.521950\pi$
$572$	0	0
$573$	4.44171	0.185555
$574$	0	0
$575$	5.63768	0.235108
$576$	0	0
$577$	11.1055	0.462329	0.231164	−	0.972915i	$-0.425746\pi$
$577$	11.1055	0.462329	0.231164	+	0.972915i	$0.425746\pi$
$578$	0	0
$579$	21.8426	0.907746
$580$	0	0
$581$	5.83922	0.242252
$582$	0	0
$583$	−3.21968	−0.133346
$584$	0	0
$585$	−39.3011	−1.62490
$586$	0	0
$587$	−24.1086	−0.995069	−0.497535	−	0.867444i	$-0.665761\pi$
$587$	−24.1086	−0.995069	−0.497535	+	0.867444i	$0.665761\pi$
$588$	0	0
$589$	−75.3613	−3.10521
$590$	0	0
$591$	−81.6781	−3.35979
$592$	0	0
$593$	−41.9717	−1.72357	−0.861785	−	0.507274i	$-0.830653\pi$
$593$	−41.9717	−1.72357	−0.861785	+	0.507274i	$0.830653\pi$
$594$	0	0
$595$	3.52457	0.144493
$596$	0	0
$597$	51.9636	2.12673
$598$	0	0
$599$	1.39492	0.0569949	0.0284975	−	0.999594i	$-0.490928\pi$
$599$	1.39492	0.0569949	0.0284975	+	0.999594i	$0.490928\pi$
$600$	0	0
$601$	−45.2390	−1.84534	−0.922669	−	0.385593i	$-0.873997\pi$
$601$	−45.2390	−1.84534	−0.922669	+	0.385593i	$0.873997\pi$
$602$	0	0
$603$	93.1871	3.79487
$604$	0	0
$605$	−9.35750	−0.380436
$606$	0	0
$607$	−16.5943	−0.673543	−0.336771	−	0.941586i	$-0.609335\pi$
$607$	−16.5943	−0.673543	−0.336771	+	0.941586i	$0.609335\pi$
$608$	0	0
$609$	5.27320	0.213681
$610$	0	0
$611$	14.7104	0.595119
$612$	0	0
$613$	−40.1550	−1.62185	−0.810923	−	0.585152i	$-0.801035\pi$
$613$	−40.1550	−1.62185	−0.810923	+	0.585152i	$0.801035\pi$
$614$	0	0
$615$	16.2968	0.657150
$616$	0	0
$617$	47.8011	1.92440	0.962199	−	0.272346i	$-0.0877996\pi$
$617$	47.8011	1.92440	0.962199	+	0.272346i	$0.0877996\pi$
$618$	0	0
$619$	−27.3522	−1.09938	−0.549688	−	0.835370i	$-0.685253\pi$
$619$	−27.3522	−1.09938	−0.549688	+	0.835370i	$0.685253\pi$
$620$	0	0
$621$	75.8880	3.04528
$622$	0	0
$623$	−3.20766	−0.128512
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−29.1340	−1.16350
$628$	0	0
$629$	−76.7871	−3.06170
$630$	0	0
$631$	−2.18133	−0.0868373	−0.0434187	−	0.999057i	$-0.513825\pi$
$631$	−2.18133	−0.0868373	−0.0434187	+	0.999057i	$0.513825\pi$
$632$	0	0
$633$	16.3922	0.651530
$634$	0	0
$635$	−9.29959	−0.369043
$636$	0	0
$637$	36.8407	1.45968
$638$	0	0
$639$	30.5263	1.20760
$640$	0	0
$641$	−27.0946	−1.07017	−0.535086	−	0.844798i	$-0.679721\pi$
$641$	−27.0946	−1.07017	−0.535086	+	0.844798i	$0.679721\pi$
$642$	0	0
$643$	−8.08913	−0.319004	−0.159502	−	0.987198i	$-0.550989\pi$
$643$	−8.08913	−0.319004	−0.159502	+	0.987198i	$0.550989\pi$
$644$	0	0
$645$	−34.0400	−1.34033
$646$	0	0
$647$	40.1324	1.57777	0.788883	−	0.614543i	$-0.210660\pi$
$647$	40.1324	1.57777	0.788883	+	0.614543i	$0.210660\pi$
$648$	0	0
$649$	0.842293	0.0330629
$650$	0	0
$651$	−16.5607	−0.649064
$652$	0	0
$653$	3.94740	0.154474	0.0772368	−	0.997013i	$-0.475390\pi$
$653$	3.94740	0.154474	0.0772368	+	0.997013i	$0.475390\pi$
$654$	0	0
$655$	6.48249	0.253292
$656$	0	0
$657$	−0.909043	−0.0354652
$658$	0	0
$659$	−27.2382	−1.06105	−0.530525	−	0.847669i	$-0.678005\pi$
$659$	−27.2382	−1.06105	−0.530525	+	0.847669i	$0.678005\pi$
$660$	0	0
$661$	22.8556	0.888980	0.444490	−	0.895784i	$-0.353385\pi$
$661$	22.8556	0.888980	0.444490	+	0.895784i	$0.353385\pi$
$662$	0	0
$663$	−125.472	−4.87291
$664$	0	0
$665$	3.47970	0.134937
$666$	0	0
$667$	19.0178	0.736371
$668$	0	0
$669$	−62.7258	−2.42512
$670$	0	0
$671$	−4.37302	−0.168819
$672$	0	0
$673$	43.0705	1.66025	0.830123	−	0.557580i	$-0.188270\pi$
$673$	43.0705	1.66025	0.830123	+	0.557580i	$0.188270\pi$
$674$	0	0
$675$	13.4608	0.518108
$676$	0	0
$677$	−25.4804	−0.979292	−0.489646	−	0.871921i	$-0.662874\pi$
$677$	−25.4804	−0.979292	−0.489646	+	0.871921i	$0.662874\pi$
$678$	0	0
$679$	4.04811	0.155352
$680$	0	0
$681$	−30.2062	−1.15750
$682$	0	0
$683$	−26.6582	−1.02005	−0.510024	−	0.860160i	$-0.670364\pi$
$683$	−26.6582	−1.02005	−0.510024	+	0.860160i	$0.670364\pi$
$684$	0	0
$685$	−2.25309	−0.0860861
$686$	0	0
$687$	−28.9356	−1.10396
$688$	0	0
$689$	−13.6897	−0.521536
$690$	0	0
$691$	39.3575	1.49723	0.748614	−	0.663006i	$-0.230720\pi$
$691$	39.3575	1.49723	0.748614	+	0.663006i	$0.230720\pi$
$692$	0	0
$693$	−4.52145	−0.171756
$694$	0	0
$695$	−14.3560	−0.544556
$696$	0	0
$697$	36.7445	1.39180
$698$	0	0
$699$	−33.7386	−1.27611
$700$	0	0
$701$	−2.90095	−0.109567	−0.0547836	−	0.998498i	$-0.517447\pi$
$701$	−2.90095	−0.109567	−0.0547836	+	0.998498i	$0.517447\pi$
$702$	0	0
$703$	−75.8095	−2.85921
$704$	0	0
$705$	−8.62680	−0.324904
$706$	0	0
$707$	−5.57784	−0.209776
$708$	0	0
$709$	−16.0560	−0.602995	−0.301497	−	0.953467i	$-0.597486\pi$
$709$	−16.0560	−0.602995	−0.301497	+	0.953467i	$0.597486\pi$
$710$	0	0
$711$	−44.7957	−1.67997
$712$	0	0
$713$	−59.7260	−2.23675
$714$	0	0
$715$	6.98374	0.261177
$716$	0	0
$717$	−24.1129	−0.900513
$718$	0	0
$719$	−35.4188	−1.32090	−0.660450	−	0.750870i	$-0.729634\pi$
$719$	−35.4188	−1.32090	−0.660450	+	0.750870i	$0.729634\pi$
$720$	0	0
$721$	6.15138	0.229089
$722$	0	0
$723$	−32.9403	−1.22506
$724$	0	0
$725$	3.37333	0.125282
$726$	0	0
$727$	13.3585	0.495439	0.247720	−	0.968832i	$-0.420319\pi$
$727$	13.3585	0.495439	0.247720	+	0.968832i	$0.420319\pi$
$728$	0	0
$729$	25.1456	0.931318
$730$	0	0
$731$	−76.7504	−2.83871
$732$	0	0
$733$	−21.5966	−0.797690	−0.398845	−	0.917018i	$-0.630589\pi$
$733$	−21.5966	−0.797690	−0.398845	+	0.917018i	$0.630589\pi$
$734$	0	0
$735$	−21.6049	−0.796910
$736$	0	0
$737$	−16.5592	−0.609966
$738$	0	0
$739$	50.2828	1.84968	0.924842	−	0.380352i	$-0.124197\pi$
$739$	50.2828	1.84968	0.924842	+	0.380352i	$0.124197\pi$
$740$	0	0
$741$	−123.874	−4.55064
$742$	0	0
$743$	22.6999	0.832778	0.416389	−	0.909187i	$-0.363296\pi$
$743$	22.6999	0.832778	0.416389	+	0.909187i	$0.363296\pi$
$744$	0	0
$745$	1.71142	0.0627014
$746$	0	0
$747$	86.0933	3.14999
$748$	0	0
$749$	0.0718940	0.00262695
$750$	0	0
$751$	24.5824	0.897026	0.448513	−	0.893776i	$-0.351954\pi$
$751$	24.5824	0.897026	0.448513	+	0.893776i	$0.351954\pi$
$752$	0	0
$753$	−86.7836	−3.16257
$754$	0	0
$755$	1.00000	0.0363937
$756$	0	0
$757$	−12.9398	−0.470306	−0.235153	−	0.971958i	$-0.575559\pi$
$757$	−12.9398	−0.470306	−0.235153	+	0.971958i	$0.575559\pi$
$758$	0	0
$759$	−23.0895	−0.838095
$760$	0	0
$761$	0.566721	0.0205436	0.0102718	−	0.999947i	$-0.496730\pi$
$761$	0.566721	0.0205436	0.0102718	+	0.999947i	$0.496730\pi$
$762$	0	0
$763$	5.10188	0.184700
$764$	0	0
$765$	51.9661	1.87884
$766$	0	0
$767$	3.58133	0.129314
$768$	0	0
$769$	38.7941	1.39895	0.699475	−	0.714657i	$-0.253417\pi$
$769$	38.7941	1.39895	0.699475	+	0.714657i	$0.253417\pi$
$770$	0	0
$771$	62.2831	2.24307
$772$	0	0
$773$	−47.4490	−1.70662	−0.853311	−	0.521402i	$-0.825409\pi$
$773$	−47.4490	−1.70662	−0.853311	+	0.521402i	$0.825409\pi$
$774$	0	0
$775$	−10.5941	−0.380550
$776$	0	0
$777$	−16.6592	−0.597644
$778$	0	0
$779$	36.2767	1.29975
$780$	0	0
$781$	−5.42447	−0.194103
$782$	0	0
$783$	45.4079	1.62274
$784$	0	0
$785$	2.15785	0.0770171
$786$	0	0
$787$	28.6545	1.02142	0.510711	−	0.859752i	$-0.329382\pi$
$787$	28.6545	1.02142	0.510711	+	0.859752i	$0.329382\pi$
$788$	0	0
$789$	83.4059	2.96933
$790$	0	0
$791$	0.213739	0.00759968
$792$	0	0
$793$	−18.5936	−0.660278
$794$	0	0
$795$	8.02823	0.284732
$796$	0	0
$797$	25.4525	0.901575	0.450787	−	0.892631i	$-0.351143\pi$
$797$	25.4525	0.901575	0.450787	+	0.892631i	$0.351143\pi$
$798$	0	0
$799$	−19.4509	−0.688124
$800$	0	0
$801$	−47.2937	−1.67104
$802$	0	0
$803$	0.161536	0.00570047
$804$	0	0
$805$	2.75776	0.0971982
$806$	0	0
$807$	47.5599	1.67419
$808$	0	0
$809$	23.5072	0.826469	0.413234	−	0.910625i	$-0.364399\pi$
$809$	23.5072	0.826469	0.413234	+	0.910625i	$0.364399\pi$
$810$	0	0
$811$	−0.677577	−0.0237929	−0.0118965	−	0.999929i	$-0.503787\pi$
$811$	−0.677577	−0.0237929	−0.0118965	+	0.999929i	$0.503787\pi$
$812$	0	0
$813$	44.4074	1.55743
$814$	0	0
$815$	−5.49893	−0.192619
$816$	0	0
$817$	−75.7733	−2.65097
$818$	0	0
$819$	−19.2247	−0.671765
$820$	0	0
$821$	24.5955	0.858388	0.429194	−	0.903212i	$-0.358798\pi$
$821$	24.5955	0.858388	0.429194	+	0.903212i	$0.358798\pi$
$822$	0	0
$823$	35.6680	1.24331	0.621655	−	0.783291i	$-0.286461\pi$
$823$	35.6680	1.24331	0.621655	+	0.783291i	$0.286461\pi$
$824$	0	0
$825$	−4.09556	−0.142589
$826$	0	0
$827$	2.98698	0.103867	0.0519337	−	0.998651i	$-0.483462\pi$
$827$	2.98698	0.103867	0.0519337	+	0.998651i	$0.483462\pi$
$828$	0	0
$829$	0.171191	0.00594569	0.00297285	−	0.999996i	$-0.499054\pi$
$829$	0.171191	0.00594569	0.00297285	+	0.999996i	$0.499054\pi$
$830$	0	0
$831$	−31.8866	−1.10614
$832$	0	0
$833$	−48.7128	−1.68780
$834$	0	0
$835$	−15.2376	−0.527318
$836$	0	0
$837$	−142.605	−4.92915
$838$	0	0
$839$	−30.2601	−1.04470	−0.522348	−	0.852732i	$-0.674944\pi$
$839$	−30.2601	−1.04470	−0.522348	+	0.852732i	$0.674944\pi$
$840$	0	0
$841$	−17.6206	−0.607609
$842$	0	0
$843$	12.8631	0.443027
$844$	0	0
$845$	16.6940	0.574293
$846$	0	0
$847$	−4.57736	−0.157280
$848$	0	0
$849$	−48.9562	−1.68017
$850$	0	0
$851$	−60.0812	−2.05956
$852$	0	0
$853$	8.70832	0.298167	0.149084	−	0.988825i	$-0.452368\pi$
$853$	8.70832	0.298167	0.149084	+	0.988825i	$0.452368\pi$
$854$	0	0
$855$	51.3045	1.75458
$856$	0	0
$857$	23.8027	0.813086	0.406543	−	0.913632i	$-0.366734\pi$
$857$	23.8027	0.813086	0.406543	+	0.913632i	$0.366734\pi$
$858$	0	0
$859$	12.9610	0.442224	0.221112	−	0.975248i	$-0.429031\pi$
$859$	12.9610	0.442224	0.221112	+	0.975248i	$0.429031\pi$
$860$	0	0
$861$	7.97182	0.271679
$862$	0	0
$863$	52.4700	1.78610	0.893049	−	0.449959i	$-0.148561\pi$
$863$	52.4700	1.78610	0.893049	+	0.449959i	$0.148561\pi$
$864$	0	0
$865$	−4.84860	−0.164857
$866$	0	0
$867$	111.580	3.78944
$868$	0	0
$869$	7.96013	0.270029
$870$	0	0
$871$	−70.4078	−2.38568
$872$	0	0
$873$	59.6851	2.02004
$874$	0	0
$875$	0.489165	0.0165368
$876$	0	0
$877$	−7.85060	−0.265096	−0.132548	−	0.991177i	$-0.542316\pi$
$877$	−7.85060	−0.265096	−0.132548	+	0.991177i	$0.542316\pi$
$878$	0	0
$879$	33.2440	1.12129
$880$	0	0
$881$	26.3404	0.887431	0.443715	−	0.896168i	$-0.353660\pi$
$881$	26.3404	0.887431	0.443715	+	0.896168i	$0.353660\pi$
$882$	0	0
$883$	58.1013	1.95526	0.977632	−	0.210322i	$-0.0674512\pi$
$883$	58.1013	1.95526	0.977632	+	0.210322i	$0.0674512\pi$
$884$	0	0
$885$	−2.10025	−0.0705990
$886$	0	0
$887$	18.1730	0.610188	0.305094	−	0.952322i	$-0.401312\pi$
$887$	18.1730	0.610188	0.305094	+	0.952322i	$0.401312\pi$
$888$	0	0
$889$	−4.54903	−0.152570
$890$	0	0
$891$	−27.4001	−0.917938
$892$	0	0
$893$	−19.2033	−0.642614
$894$	0	0
$895$	−21.6999	−0.725347
$896$	0	0
$897$	−98.1738	−3.27793
$898$	0	0
$899$	−35.7372	−1.19190
$900$	0	0
$901$	18.1013	0.603042
$902$	0	0
$903$	−16.6512	−0.554117
$904$	0	0
$905$	25.3122	0.841405
$906$	0	0
$907$	20.2945	0.673869	0.336935	−	0.941528i	$-0.390610\pi$
$907$	20.2945	0.673869	0.336935	+	0.941528i	$0.390610\pi$
$908$	0	0
$909$	−82.2395	−2.72771
$910$	0	0
$911$	57.8911	1.91802	0.959008	−	0.283379i	$-0.0914553\pi$
$911$	57.8911	1.91802	0.959008	+	0.283379i	$0.0914553\pi$
$912$	0	0
$913$	−15.2986	−0.506311
$914$	0	0
$915$	10.9041	0.360478
$916$	0	0
$917$	3.17101	0.104716
$918$	0	0
$919$	32.1632	1.06097	0.530484	−	0.847695i	$-0.322010\pi$
$919$	32.1632	1.06097	0.530484	+	0.847695i	$0.322010\pi$
$920$	0	0
$921$	−101.329	−3.33890
$922$	0	0
$923$	−23.0642	−0.759168
$924$	0	0
$925$	−10.6571	−0.350402
$926$	0	0
$927$	90.6957	2.97884
$928$	0	0
$929$	−56.1669	−1.84278	−0.921388	−	0.388643i	$-0.872944\pi$
$929$	−56.1669	−1.84278	−0.921388	+	0.388643i	$0.872944\pi$
$930$	0	0
$931$	−48.0927	−1.57617
$932$	0	0
$933$	82.4220	2.69838
$934$	0	0
$935$	−9.23429	−0.301994
$936$	0	0
$937$	−14.6670	−0.479150	−0.239575	−	0.970878i	$-0.577008\pi$
$937$	−14.6670	−0.479150	−0.239575	+	0.970878i	$0.577008\pi$
$938$	0	0
$939$	−7.00353	−0.228552
$940$	0	0
$941$	12.6515	0.412429	0.206214	−	0.978507i	$-0.433886\pi$
$941$	12.6515	0.412429	0.206214	+	0.978507i	$0.433886\pi$
$942$	0	0
$943$	28.7503	0.936239
$944$	0	0
$945$	6.58457	0.214196
$946$	0	0
$947$	52.0398	1.69107	0.845533	−	0.533924i	$-0.179283\pi$
$947$	52.0398	1.69107	0.845533	+	0.533924i	$0.179283\pi$
$948$	0	0
$949$	0.686831	0.0222955
$950$	0	0
$951$	−78.3832	−2.54175
$952$	0	0
$953$	21.0364	0.681435	0.340718	−	0.940166i	$-0.389330\pi$
$953$	21.0364	0.681435	0.340718	+	0.940166i	$0.389330\pi$
$954$	0	0
$955$	1.38992	0.0449768
$956$	0	0
$957$	−13.8157	−0.446597
$958$	0	0
$959$	−1.10213	−0.0355897
$960$	0	0
$961$	81.2341	2.62045
$962$	0	0
$963$	1.06000	0.0341581
$964$	0	0
$965$	6.83508	0.220029
$966$	0	0
$967$	−40.3295	−1.29691	−0.648455	−	0.761253i	$-0.724584\pi$
$967$	−40.3295	−1.29691	−0.648455	+	0.761253i	$0.724584\pi$
$968$	0	0
$969$	163.794	5.26181
$970$	0	0
$971$	30.2316	0.970179	0.485089	−	0.874465i	$-0.338787\pi$
$971$	30.2316	0.970179	0.485089	+	0.874465i	$0.338787\pi$
$972$	0	0
$973$	−7.02247	−0.225130
$974$	0	0
$975$	−17.4139	−0.557690
$976$	0	0
$977$	11.5852	0.370644	0.185322	−	0.982678i	$-0.440667\pi$
$977$	11.5852	0.370644	0.185322	+	0.982678i	$0.440667\pi$
$978$	0	0
$979$	8.40401	0.268593
$980$	0	0
$981$	75.2219	2.40165
$982$	0	0
$983$	−12.2434	−0.390504	−0.195252	−	0.980753i	$-0.562552\pi$
$983$	−12.2434	−0.390504	−0.195252	+	0.980753i	$0.562552\pi$
$984$	0	0
$985$	−25.5591	−0.814380
$986$	0	0
$987$	−4.21993	−0.134322
$988$	0	0
$989$	−60.0524	−1.90956
$990$	0	0
$991$	27.1306	0.861833	0.430916	−	0.902392i	$-0.358190\pi$
$991$	27.1306	0.861833	0.430916	+	0.902392i	$0.358190\pi$
$992$	0	0
$993$	5.10473	0.161994
$994$	0	0
$995$	16.2607	0.515498
$996$	0	0
$997$	−3.29154	−0.104244	−0.0521220	−	0.998641i	$-0.516598\pi$
$997$	−3.29154	−0.104244	−0.0521220	+	0.998641i	$0.516598\pi$
$998$	0	0
$999$	−143.453	−4.53866

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3020.2.a.f.1.5	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3020.2.a.f.1.5	✓	6	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:	\( N \)	\(=\)	\( 3020 = 2^{2} \cdot 5 \cdot 151 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3020.a (trivial)

\(f(q)\)	\(=\)	\(q+3.19566 q^{3} +1.00000 q^{5} +0.489165 q^{7} +7.21223 q^{9} +O(q^{10})\)	\(q+3.19566 q^{3} +1.00000 q^{5} +0.489165 q^{7} +7.21223 q^{9} -1.28160 q^{11} -5.44922 q^{13} +3.19566 q^{15} +7.20528 q^{17} +7.11355 q^{19} +1.56320 q^{21} +5.63768 q^{23} +1.00000 q^{25} +13.4608 q^{27} +3.37333 q^{29} -10.5941 q^{31} -4.09556 q^{33} +0.489165 q^{35} -10.6571 q^{37} -17.4139 q^{39} +5.09966 q^{41} -10.6520 q^{43} +7.21223 q^{45} -2.69954 q^{47} -6.76072 q^{49} +23.0256 q^{51} +2.51223 q^{53} -1.28160 q^{55} +22.7325 q^{57} -0.657219 q^{59} +3.41215 q^{61} +3.52797 q^{63} -5.44922 q^{65} +12.9207 q^{67} +18.0161 q^{69} +4.23257 q^{71} -0.126042 q^{73} +3.19566 q^{75} -0.626915 q^{77} -6.21108 q^{79} +21.3796 q^{81} +11.9371 q^{83} +7.20528 q^{85} +10.7800 q^{87} -6.55743 q^{89} -2.66557 q^{91} -33.8550 q^{93} +7.11355 q^{95} +8.27554 q^{97} -9.24321 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 2 q^{3} + 6 q^{5} + 3 q^{7} + 20 q^{9}+O(q^{10}) \) 6 * q + 2 * q^3 + 6 * q^5 + 3 * q^7 + 20 * q^9	\( 6 q + 2 q^{3} + 6 q^{5} + 3 q^{7} + 20 q^{9} + 3 q^{11} + 2 q^{13} + 2 q^{15} + 9 q^{17} + 16 q^{19} - 12 q^{21} + 6 q^{23} + 6 q^{25} + 23 q^{27} + 2 q^{29} + 15 q^{31} - 14 q^{33} + 3 q^{35} + 9 q^{37} - 17 q^{39} - 16 q^{41} + 15 q^{43} + 20 q^{45} + 12 q^{47} - 7 q^{49} + 11 q^{51} - 14 q^{53} + 3 q^{55} + 27 q^{57} - 20 q^{59} + 17 q^{61} + 17 q^{63} + 2 q^{65} + 29 q^{67} - 40 q^{69} + 13 q^{71} - 6 q^{73} + 2 q^{75} + 13 q^{77} - 8 q^{79} + 30 q^{81} + 36 q^{83} + 9 q^{85} + 2 q^{87} - 15 q^{89} - 14 q^{91} - 31 q^{93} + 16 q^{95} - 23 q^{97} + 38 q^{99}+O(q^{100}) \) 6 * q + 2 * q^3 + 6 * q^5 + 3 * q^7 + 20 * q^9 + 3 * q^11 + 2 * q^13 + 2 * q^15 + 9 * q^17 + 16 * q^19 - 12 * q^21 + 6 * q^23 + 6 * q^25 + 23 * q^27 + 2 * q^29 + 15 * q^31 - 14 * q^33 + 3 * q^35 + 9 * q^37 - 17 * q^39 - 16 * q^41 + 15 * q^43 + 20 * q^45 + 12 * q^47 - 7 * q^49 + 11 * q^51 - 14 * q^53 + 3 * q^55 + 27 * q^57 - 20 * q^59 + 17 * q^61 + 17 * q^63 + 2 * q^65 + 29 * q^67 - 40 * q^69 + 13 * q^71 - 6 * q^73 + 2 * q^75 + 13 * q^77 - 8 * q^79 + 30 * q^81 + 36 * q^83 + 9 * q^85 + 2 * q^87 - 15 * q^89 - 14 * q^91 - 31 * q^93 + 16 * q^95 - 23 * q^97 + 38 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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