LMFDB - Embedded newform 6020.2.a.f.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6020,2,Mod(1,6020)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6020, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("6020.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6020 = 2^{2} \cdot 5 \cdot 7 \cdot 43 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6020.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:	$48.0699420168$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{7} - 12x^{6} + 26x^{5} + 55x^{4} - 52x^{3} - 82x^{2} + 22x + 27 $ x^8 - 3x^7 - 12x^6 + 26x^5 + 55x^4 - 52x^3 - 82x^2 + 22*x + 27
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			1.6
Root			$1.41294$ of defining polynomial
Character	$\chi$	$=$	6020.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.412945 q^{3} -1.00000 q^{5} +1.00000 q^{7} -2.82948 q^{9} +O(q^{10})$	$q+0.412945 q^{3} -1.00000 q^{5} +1.00000 q^{7} -2.82948 q^{9} +3.52290 q^{11} +0.621509 q^{13} -0.412945 q^{15} +0.334344 q^{17} +1.41653 q^{19} +0.412945 q^{21} -0.654973 q^{23} +1.00000 q^{25} -2.40725 q^{27} -5.77605 q^{29} -8.74351 q^{31} +1.45477 q^{33} -1.00000 q^{35} -4.03355 q^{37} +0.256649 q^{39} -4.62928 q^{41} -1.00000 q^{43} +2.82948 q^{45} -4.99470 q^{47} +1.00000 q^{49} +0.138065 q^{51} +7.86325 q^{53} -3.52290 q^{55} +0.584949 q^{57} +1.66609 q^{59} +9.92345 q^{61} -2.82948 q^{63} -0.621509 q^{65} -11.9800 q^{67} -0.270468 q^{69} -6.43902 q^{71} -13.7141 q^{73} +0.412945 q^{75} +3.52290 q^{77} +16.6260 q^{79} +7.49437 q^{81} +13.7502 q^{83} -0.334344 q^{85} -2.38519 q^{87} -3.91990 q^{89} +0.621509 q^{91} -3.61059 q^{93} -1.41653 q^{95} +3.63045 q^{97} -9.96798 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 5 q^{3} - 8 q^{5} + 8 q^{7} + 11 q^{9}+O(q^{10}) $ 8 * q - 5 * q^3 - 8 * q^5 + 8 * q^7 + 11 * q^9	$ 8 q - 5 q^{3} - 8 q^{5} + 8 q^{7} + 11 q^{9} + 6 q^{11} - 13 q^{13} + 5 q^{15} - 2 q^{17} - 14 q^{19} - 5 q^{21} + 6 q^{23} + 8 q^{25} - 11 q^{27} - 7 q^{29} - 18 q^{31} - q^{33} - 8 q^{35} + 2 q^{37} + 9 q^{39} - 18 q^{41} - 8 q^{43} - 11 q^{45} - q^{47} + 8 q^{49} - 19 q^{51} + 5 q^{53} - 6 q^{55} - 4 q^{57} - 12 q^{59} - 23 q^{61} + 11 q^{63} + 13 q^{65} + 8 q^{67} - 18 q^{69} + 20 q^{71} - 4 q^{73} - 5 q^{75} + 6 q^{77} + 24 q^{79} - 8 q^{81} - 14 q^{83} + 2 q^{85} + 10 q^{87} - 21 q^{89} - 13 q^{91} + q^{93} + 14 q^{95} + 7 q^{97} + 4 q^{99}+O(q^{100}) $ 8 * q - 5 * q^3 - 8 * q^5 + 8 * q^7 + 11 * q^9 + 6 * q^11 - 13 * q^13 + 5 * q^15 - 2 * q^17 - 14 * q^19 - 5 * q^21 + 6 * q^23 + 8 * q^25 - 11 * q^27 - 7 * q^29 - 18 * q^31 - q^33 - 8 * q^35 + 2 * q^37 + 9 * q^39 - 18 * q^41 - 8 * q^43 - 11 * q^45 - q^47 + 8 * q^49 - 19 * q^51 + 5 * q^53 - 6 * q^55 - 4 * q^57 - 12 * q^59 - 23 * q^61 + 11 * q^63 + 13 * q^65 + 8 * q^67 - 18 * q^69 + 20 * q^71 - 4 * q^73 - 5 * q^75 + 6 * q^77 + 24 * q^79 - 8 * q^81 - 14 * q^83 + 2 * q^85 + 10 * q^87 - 21 * q^89 - 13 * q^91 + q^93 + 14 * q^95 + 7 * q^97 + 4 * 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
$3$	0.412945	0.238414	0.119207	−	0.992869i	$-0.461965\pi$
$3$	0.412945	0.238414	0.119207	+	0.992869i	$0.461965\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	−2.82948	−0.943159
$10$	0	0
$11$	3.52290	1.06220	0.531098	−	0.847311i	$-0.321780\pi$
$11$	3.52290	1.06220	0.531098	+	0.847311i	$0.321780\pi$
$12$	0	0
$13$	0.621509	0.172375	0.0861877	−	0.996279i	$-0.472532\pi$
$13$	0.621509	0.172375	0.0861877	+	0.996279i	$0.472532\pi$
$14$	0	0
$15$	−0.412945	−0.106622
$16$	0	0
$17$	0.334344	0.0810902	0.0405451	−	0.999178i	$-0.487091\pi$
$17$	0.334344	0.0810902	0.0405451	+	0.999178i	$0.487091\pi$
$18$	0	0
$19$	1.41653	0.324975	0.162487	−	0.986711i	$-0.448048\pi$
$19$	1.41653	0.324975	0.162487	+	0.986711i	$0.448048\pi$
$20$	0	0
$21$	0.412945	0.0901120
$22$	0	0
$23$	−0.654973	−0.136571	−0.0682856	−	0.997666i	$-0.521753\pi$
$23$	−0.654973	−0.136571	−0.0682856	+	0.997666i	$0.521753\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−2.40725	−0.463276
$28$	0	0
$29$	−5.77605	−1.07259	−0.536293	−	0.844032i	$-0.680176\pi$
$29$	−5.77605	−1.07259	−0.536293	+	0.844032i	$0.680176\pi$
$30$	0	0
$31$	−8.74351	−1.57038	−0.785190	−	0.619255i	$-0.787435\pi$
$31$	−8.74351	−1.57038	−0.785190	+	0.619255i	$0.787435\pi$
$32$	0	0
$33$	1.45477	0.253242
$34$	0	0
$35$	−1.00000	−0.169031
$36$	0	0
$37$	−4.03355	−0.663111	−0.331556	−	0.943436i	$-0.607573\pi$
$37$	−4.03355	−0.663111	−0.331556	+	0.943436i	$0.607573\pi$
$38$	0	0
$39$	0.256649	0.0410967
$40$	0	0
$41$	−4.62928	−0.722972	−0.361486	−	0.932378i	$-0.617730\pi$
$41$	−4.62928	−0.722972	−0.361486	+	0.932378i	$0.617730\pi$
$42$	0	0
$43$	−1.00000	−0.152499
$43$	−1.00000	−0.152499
$44$	0	0
$45$	2.82948	0.421793
$46$	0	0
$47$	−4.99470	−0.728552	−0.364276	−	0.931291i	$-0.618684\pi$
$47$	−4.99470	−0.728552	−0.364276	+	0.931291i	$0.618684\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0.138065	0.0193330
$52$	0	0
$53$	7.86325	1.08010	0.540050	−	0.841633i	$-0.318405\pi$
$53$	7.86325	1.08010	0.540050	+	0.841633i	$0.318405\pi$
$54$	0	0
$55$	−3.52290	−0.475028
$56$	0	0
$57$	0.584949	0.0774785
$58$	0	0
$59$	1.66609	0.216906	0.108453	−	0.994102i	$-0.465410\pi$
$59$	1.66609	0.216906	0.108453	+	0.994102i	$0.465410\pi$
$60$	0	0
$61$	9.92345	1.27057	0.635284	−	0.772279i	$-0.280883\pi$
$61$	9.92345	1.27057	0.635284	+	0.772279i	$0.280883\pi$
$62$	0	0
$63$	−2.82948	−0.356481
$64$	0	0
$65$	−0.621509	−0.0770887
$66$	0	0
$67$	−11.9800	−1.46359	−0.731796	−	0.681524i	$-0.761318\pi$
$67$	−11.9800	−1.46359	−0.731796	+	0.681524i	$0.761318\pi$
$68$	0	0
$69$	−0.270468	−0.0325605
$70$	0	0
$71$	−6.43902	−0.764171	−0.382086	−	0.924127i	$-0.624794\pi$
$71$	−6.43902	−0.764171	−0.382086	+	0.924127i	$0.624794\pi$
$72$	0	0
$73$	−13.7141	−1.60511	−0.802555	−	0.596578i	$-0.796527\pi$
$73$	−13.7141	−1.60511	−0.802555	+	0.596578i	$0.796527\pi$
$74$	0	0
$75$	0.412945	0.0476828
$76$	0	0
$77$	3.52290	0.401472
$78$	0	0
$79$	16.6260	1.87057	0.935284	−	0.353897i	$-0.115144\pi$
$79$	16.6260	1.87057	0.935284	+	0.353897i	$0.115144\pi$
$80$	0	0
$81$	7.49437	0.832707
$82$	0	0
$83$	13.7502	1.50928	0.754641	−	0.656138i	$-0.227811\pi$
$83$	13.7502	1.50928	0.754641	+	0.656138i	$0.227811\pi$
$84$	0	0
$85$	−0.334344	−0.0362647
$86$	0	0
$87$	−2.38519	−0.255719
$88$	0	0
$89$	−3.91990	−0.415508	−0.207754	−	0.978181i	$-0.566615\pi$
$89$	−3.91990	−0.415508	−0.207754	+	0.978181i	$0.566615\pi$
$90$	0	0
$91$	0.621509	0.0651518
$92$	0	0
$93$	−3.61059	−0.374400
$94$	0	0
$95$	−1.41653	−0.145333
$96$	0	0
$97$	3.63045	0.368617	0.184308	−	0.982868i	$-0.440995\pi$
$97$	3.63045	0.368617	0.184308	+	0.982868i	$0.440995\pi$
$98$	0	0
$99$	−9.96798	−1.00182
$100$	0	0
$101$	2.83013	0.281609	0.140804	−	0.990037i	$-0.455031\pi$
$101$	2.83013	0.281609	0.140804	+	0.990037i	$0.455031\pi$
$102$	0	0
$103$	2.45335	0.241736	0.120868	−	0.992669i	$-0.461432\pi$
$103$	2.45335	0.241736	0.120868	+	0.992669i	$0.461432\pi$
$104$	0	0
$105$	−0.412945	−0.0402993
$106$	0	0
$107$	−5.67904	−0.549013	−0.274507	−	0.961585i	$-0.588515\pi$
$107$	−5.67904	−0.549013	−0.274507	+	0.961585i	$0.588515\pi$
$108$	0	0
$109$	14.3409	1.37361	0.686803	−	0.726843i	$-0.259013\pi$
$109$	14.3409	1.37361	0.686803	+	0.726843i	$0.259013\pi$
$110$	0	0
$111$	−1.66563	−0.158095
$112$	0	0
$113$	−2.11632	−0.199086	−0.0995431	−	0.995033i	$-0.531738\pi$
$113$	−2.11632	−0.199086	−0.0995431	+	0.995033i	$0.531738\pi$
$114$	0	0
$115$	0.654973	0.0610765
$116$	0	0
$117$	−1.75854	−0.162577
$118$	0	0
$119$	0.334344	0.0306492
$120$	0	0
$121$	1.41086	0.128260
$122$	0	0
$123$	−1.91164	−0.172366
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−9.58285	−0.850341	−0.425170	−	0.905113i	$-0.639786\pi$
$127$	−9.58285	−0.850341	−0.425170	+	0.905113i	$0.639786\pi$
$128$	0	0
$129$	−0.412945	−0.0363578
$130$	0	0
$131$	−18.4759	−1.61425	−0.807124	−	0.590382i	$-0.798977\pi$
$131$	−18.4759	−1.61425	−0.807124	+	0.590382i	$0.798977\pi$
$132$	0	0
$133$	1.41653	0.122829
$134$	0	0
$135$	2.40725	0.207183
$136$	0	0
$137$	1.32239	0.112980	0.0564898	−	0.998403i	$-0.482009\pi$
$137$	1.32239	0.112980	0.0564898	+	0.998403i	$0.482009\pi$
$138$	0	0
$139$	−7.66019	−0.649729	−0.324865	−	0.945761i	$-0.605319\pi$
$139$	−7.66019	−0.649729	−0.324865	+	0.945761i	$0.605319\pi$
$140$	0	0
$141$	−2.06254	−0.173697
$142$	0	0
$143$	2.18952	0.183096
$144$	0	0
$145$	5.77605	0.479675
$146$	0	0
$147$	0.412945	0.0340591
$148$	0	0
$149$	14.2244	1.16531	0.582654	−	0.812720i	$-0.302014\pi$
$149$	14.2244	1.16531	0.582654	+	0.812720i	$0.302014\pi$
$150$	0	0
$151$	−20.6030	−1.67665	−0.838324	−	0.545173i	$-0.816464\pi$
$151$	−20.6030	−1.67665	−0.838324	+	0.545173i	$0.816464\pi$
$152$	0	0
$153$	−0.946017	−0.0764810
$154$	0	0
$155$	8.74351	0.702295
$156$	0	0
$157$	−13.1511	−1.04957	−0.524785	−	0.851235i	$-0.675854\pi$
$157$	−13.1511	−1.04957	−0.524785	+	0.851235i	$0.675854\pi$
$158$	0	0
$159$	3.24709	0.257511
$160$	0	0
$161$	−0.654973	−0.0516191
$162$	0	0
$163$	6.24108	0.488839	0.244420	−	0.969670i	$-0.421403\pi$
$163$	6.24108	0.488839	0.244420	+	0.969670i	$0.421403\pi$
$164$	0	0
$165$	−1.45477	−0.113253
$166$	0	0
$167$	−10.9954	−0.850850	−0.425425	−	0.904994i	$-0.639875\pi$
$167$	−10.9954	−0.850850	−0.425425	+	0.904994i	$0.639875\pi$
$168$	0	0
$169$	−12.6137	−0.970287
$170$	0	0
$171$	−4.00804	−0.306503
$172$	0	0
$173$	11.5825	0.880605	0.440302	−	0.897850i	$-0.354871\pi$
$173$	11.5825	0.880605	0.440302	+	0.897850i	$0.354871\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	0	0
$177$	0.688003	0.0517135
$178$	0	0
$179$	−18.4230	−1.37700	−0.688501	−	0.725235i	$-0.741731\pi$
$179$	−18.4230	−1.37700	−0.688501	+	0.725235i	$0.741731\pi$
$180$	0	0
$181$	−26.7685	−1.98968	−0.994842	−	0.101434i	$-0.967657\pi$
$181$	−26.7685	−1.98968	−0.994842	+	0.101434i	$0.967657\pi$
$182$	0	0
$183$	4.09784	0.302921
$184$	0	0
$185$	4.03355	0.296552
$186$	0	0
$187$	1.17786	0.0861337
$188$	0	0
$189$	−2.40725	−0.175102
$190$	0	0
$191$	−23.7291	−1.71698	−0.858489	−	0.512832i	$-0.828596\pi$
$191$	−23.7291	−1.71698	−0.858489	+	0.512832i	$0.828596\pi$
$192$	0	0
$193$	−15.4419	−1.11153	−0.555765	−	0.831340i	$-0.687574\pi$
$193$	−15.4419	−1.11153	−0.555765	+	0.831340i	$0.687574\pi$
$194$	0	0
$195$	−0.256649	−0.0183790
$196$	0	0
$197$	−15.5492	−1.10783	−0.553916	−	0.832573i	$-0.686867\pi$
$197$	−15.5492	−1.10783	−0.553916	+	0.832573i	$0.686867\pi$
$198$	0	0
$199$	−13.6977	−0.971005	−0.485503	−	0.874235i	$-0.661363\pi$
$199$	−13.6977	−0.971005	−0.485503	+	0.874235i	$0.661363\pi$
$200$	0	0
$201$	−4.94709	−0.348941
$202$	0	0
$203$	−5.77605	−0.405399
$204$	0	0
$205$	4.62928	0.323323
$206$	0	0
$207$	1.85323	0.128808
$208$	0	0
$209$	4.99031	0.345187
$210$	0	0
$211$	16.7550	1.15346	0.576731	−	0.816934i	$-0.304328\pi$
$211$	16.7550	1.15346	0.576731	+	0.816934i	$0.304328\pi$
$212$	0	0
$213$	−2.65896	−0.182189
$214$	0	0
$215$	1.00000	0.0681994
$216$	0	0
$217$	−8.74351	−0.593548
$218$	0	0
$219$	−5.66315	−0.382680
$220$	0	0
$221$	0.207797	0.0139780
$222$	0	0
$223$	6.92630	0.463819	0.231910	−	0.972737i	$-0.425503\pi$
$223$	6.92630	0.463819	0.231910	+	0.972737i	$0.425503\pi$
$224$	0	0
$225$	−2.82948	−0.188632
$226$	0	0
$227$	−4.61975	−0.306624	−0.153312	−	0.988178i	$-0.548994\pi$
$227$	−4.61975	−0.306624	−0.153312	+	0.988178i	$0.548994\pi$
$228$	0	0
$229$	12.6009	0.832694	0.416347	−	0.909206i	$-0.363310\pi$
$229$	12.6009	0.832694	0.416347	+	0.909206i	$0.363310\pi$
$230$	0	0
$231$	1.45477	0.0957165
$232$	0	0
$233$	1.89335	0.124037	0.0620187	−	0.998075i	$-0.480246\pi$
$233$	1.89335	0.124037	0.0620187	+	0.998075i	$0.480246\pi$
$234$	0	0
$235$	4.99470	0.325818
$236$	0	0
$237$	6.86561	0.445969
$238$	0	0
$239$	−2.18765	−0.141508	−0.0707538	−	0.997494i	$-0.522540\pi$
$239$	−2.18765	−0.141508	−0.0707538	+	0.997494i	$0.522540\pi$
$240$	0	0
$241$	−11.9918	−0.772458	−0.386229	−	0.922403i	$-0.626223\pi$
$241$	−11.9918	−0.772458	−0.386229	+	0.922403i	$0.626223\pi$
$242$	0	0
$243$	10.3165	0.661805
$244$	0	0
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	0.880387	0.0560177
$248$	0	0
$249$	5.67808	0.359834
$250$	0	0
$251$	−28.7937	−1.81744	−0.908722	−	0.417403i	$-0.862940\pi$
$251$	−28.7937	−1.81744	−0.908722	+	0.417403i	$0.862940\pi$
$252$	0	0
$253$	−2.30741	−0.145065
$254$	0	0
$255$	−0.138065	−0.00864600
$256$	0	0
$257$	−0.322525	−0.0201186	−0.0100593	−	0.999949i	$-0.503202\pi$
$257$	−0.322525	−0.0201186	−0.0100593	+	0.999949i	$0.503202\pi$
$258$	0	0
$259$	−4.03355	−0.250632
$260$	0	0
$261$	16.3432	1.01162
$262$	0	0
$263$	28.2979	1.74493	0.872463	−	0.488680i	$-0.162521\pi$
$263$	28.2979	1.74493	0.872463	+	0.488680i	$0.162521\pi$
$264$	0	0
$265$	−7.86325	−0.483035
$266$	0	0
$267$	−1.61870	−0.0990629
$268$	0	0
$269$	−3.94613	−0.240600	−0.120300	−	0.992738i	$-0.538386\pi$
$269$	−3.94613	−0.240600	−0.120300	+	0.992738i	$0.538386\pi$
$270$	0	0
$271$	17.3740	1.05540	0.527699	−	0.849431i	$-0.323055\pi$
$271$	17.3740	1.05540	0.527699	+	0.849431i	$0.323055\pi$
$272$	0	0
$273$	0.256649	0.0155331
$274$	0	0
$275$	3.52290	0.212439
$276$	0	0
$277$	−21.1969	−1.27360	−0.636800	−	0.771029i	$-0.719742\pi$
$277$	−21.1969	−1.27360	−0.636800	+	0.771029i	$0.719742\pi$
$278$	0	0
$279$	24.7396	1.48112
$280$	0	0
$281$	−6.42415	−0.383233	−0.191616	−	0.981470i	$-0.561373\pi$
$281$	−6.42415	−0.383233	−0.191616	+	0.981470i	$0.561373\pi$
$282$	0	0
$283$	−19.7028	−1.17121	−0.585606	−	0.810596i	$-0.699143\pi$
$283$	−19.7028	−1.17121	−0.585606	+	0.810596i	$0.699143\pi$
$284$	0	0
$285$	−0.584949	−0.0346494
$286$	0	0
$287$	−4.62928	−0.273258
$288$	0	0
$289$	−16.8882	−0.993424
$290$	0	0
$291$	1.49918	0.0878833
$292$	0	0
$293$	11.9181	0.696261	0.348131	−	0.937446i	$-0.386817\pi$
$293$	11.9181	0.696261	0.348131	+	0.937446i	$0.386817\pi$
$294$	0	0
$295$	−1.66609	−0.0970035
$296$	0	0
$297$	−8.48052	−0.492090
$298$	0	0
$299$	−0.407071	−0.0235415
$300$	0	0
$301$	−1.00000	−0.0576390
$302$	0	0
$303$	1.16869	0.0671394
$304$	0	0
$305$	−9.92345	−0.568215
$306$	0	0
$307$	2.92038	0.166675	0.0833375	−	0.996521i	$-0.473442\pi$
$307$	2.92038	0.166675	0.0833375	+	0.996521i	$0.473442\pi$
$308$	0	0
$309$	1.01310	0.0576332
$310$	0	0
$311$	31.4885	1.78555	0.892774	−	0.450506i	$-0.148756\pi$
$311$	31.4885	1.78555	0.892774	+	0.450506i	$0.148756\pi$
$312$	0	0
$313$	−33.7100	−1.90540	−0.952700	−	0.303911i	$-0.901707\pi$
$313$	−33.7100	−1.90540	−0.952700	+	0.303911i	$0.901707\pi$
$314$	0	0
$315$	2.82948	0.159423
$316$	0	0
$317$	20.8734	1.17237	0.586183	−	0.810179i	$-0.300630\pi$
$317$	20.8734	1.17237	0.586183	+	0.810179i	$0.300630\pi$
$318$	0	0
$319$	−20.3485	−1.13930
$320$	0	0
$321$	−2.34513	−0.130892
$322$	0	0
$323$	0.473608	0.0263523
$324$	0	0
$325$	0.621509	0.0344751
$326$	0	0
$327$	5.92199	0.327487
$328$	0	0
$329$	−4.99470	−0.275367
$330$	0	0
$331$	3.30324	0.181562	0.0907812	−	0.995871i	$-0.471064\pi$
$331$	3.30324	0.181562	0.0907812	+	0.995871i	$0.471064\pi$
$332$	0	0
$333$	11.4128	0.625419
$334$	0	0
$335$	11.9800	0.654538
$336$	0	0
$337$	17.2987	0.942321	0.471160	−	0.882048i	$-0.343835\pi$
$337$	17.2987	0.942321	0.471160	+	0.882048i	$0.343835\pi$
$338$	0	0
$339$	−0.873922	−0.0474649
$340$	0	0
$341$	−30.8025	−1.66805
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	0.270468	0.0145615
$346$	0	0
$347$	−34.3912	−1.84622	−0.923108	−	0.384540i	$-0.874360\pi$
$347$	−34.3912	−1.84622	−0.923108	+	0.384540i	$0.874360\pi$
$348$	0	0
$349$	−9.43403	−0.504992	−0.252496	−	0.967598i	$-0.581251\pi$
$349$	−9.43403	−0.504992	−0.252496	+	0.967598i	$0.581251\pi$
$350$	0	0
$351$	−1.49613	−0.0798574
$352$	0	0
$353$	−4.95801	−0.263888	−0.131944	−	0.991257i	$-0.542122\pi$
$353$	−4.95801	−0.263888	−0.131944	+	0.991257i	$0.542122\pi$
$354$	0	0
$355$	6.43902	0.341748
$356$	0	0
$357$	0.138065	0.00730720
$358$	0	0
$359$	−14.8986	−0.786321	−0.393160	−	0.919470i	$-0.628618\pi$
$359$	−14.8986	−0.786321	−0.393160	+	0.919470i	$0.628618\pi$
$360$	0	0
$361$	−16.9934	−0.894391
$362$	0	0
$363$	0.582606	0.0305789
$364$	0	0
$365$	13.7141	0.717827
$366$	0	0
$367$	2.08608	0.108892	0.0544461	−	0.998517i	$-0.482661\pi$
$367$	2.08608	0.108892	0.0544461	+	0.998517i	$0.482661\pi$
$368$	0	0
$369$	13.0984	0.681877
$370$	0	0
$371$	7.86325	0.408239
$372$	0	0
$373$	21.5472	1.11567	0.557835	−	0.829952i	$-0.311632\pi$
$373$	21.5472	1.11567	0.557835	+	0.829952i	$0.311632\pi$
$374$	0	0
$375$	−0.412945	−0.0213244
$376$	0	0
$377$	−3.58986	−0.184887
$378$	0	0
$379$	−15.9414	−0.818855	−0.409428	−	0.912343i	$-0.634272\pi$
$379$	−15.9414	−0.818855	−0.409428	+	0.912343i	$0.634272\pi$
$380$	0	0
$381$	−3.95719	−0.202733
$382$	0	0
$383$	−24.6472	−1.25941	−0.629707	−	0.776833i	$-0.716825\pi$
$383$	−24.6472	−1.25941	−0.629707	+	0.776833i	$0.716825\pi$
$384$	0	0
$385$	−3.52290	−0.179544
$386$	0	0
$387$	2.82948	0.143830
$388$	0	0
$389$	−12.6478	−0.641267	−0.320634	−	0.947203i	$-0.603896\pi$
$389$	−12.6478	−0.641267	−0.320634	+	0.947203i	$0.603896\pi$
$390$	0	0
$391$	−0.218986	−0.0110746
$392$	0	0
$393$	−7.62953	−0.384859
$394$	0	0
$395$	−16.6260	−0.836544
$396$	0	0
$397$	−28.3001	−1.42034	−0.710170	−	0.704030i	$-0.751382\pi$
$397$	−28.3001	−1.42034	−0.710170	+	0.704030i	$0.751382\pi$
$398$	0	0
$399$	0.584949	0.0292841
$400$	0	0
$401$	28.0708	1.40179	0.700895	−	0.713264i	$-0.252784\pi$
$401$	28.0708	1.40179	0.700895	+	0.713264i	$0.252784\pi$
$402$	0	0
$403$	−5.43417	−0.270695
$404$	0	0
$405$	−7.49437	−0.372398
$406$	0	0
$407$	−14.2098	−0.704354
$408$	0	0
$409$	22.9313	1.13388	0.566940	−	0.823759i	$-0.308127\pi$
$409$	22.9313	1.13388	0.566940	+	0.823759i	$0.308127\pi$
$410$	0	0
$411$	0.546075	0.0269359
$412$	0	0
$413$	1.66609	0.0819829
$414$	0	0
$415$	−13.7502	−0.674971
$416$	0	0
$417$	−3.16324	−0.154904
$418$	0	0
$419$	8.64578	0.422374	0.211187	−	0.977446i	$-0.432267\pi$
$419$	8.64578	0.422374	0.211187	+	0.977446i	$0.432267\pi$
$420$	0	0
$421$	10.6691	0.519982	0.259991	−	0.965611i	$-0.416280\pi$
$421$	10.6691	0.519982	0.259991	+	0.965611i	$0.416280\pi$
$422$	0	0
$423$	14.1324	0.687140
$424$	0	0
$425$	0.334344	0.0162180
$426$	0	0
$427$	9.92345	0.480230
$428$	0	0
$429$	0.904149	0.0436527
$430$	0	0
$431$	20.7559	0.999778	0.499889	−	0.866090i	$-0.333374\pi$
$431$	20.7559	0.999778	0.499889	+	0.866090i	$0.333374\pi$
$432$	0	0
$433$	−3.76466	−0.180918	−0.0904590	−	0.995900i	$-0.528833\pi$
$433$	−3.76466	−0.180918	−0.0904590	+	0.995900i	$0.528833\pi$
$434$	0	0
$435$	2.38519	0.114361
$436$	0	0
$437$	−0.927790	−0.0443822
$438$	0	0
$439$	8.01176	0.382380	0.191190	−	0.981553i	$-0.438765\pi$
$439$	8.01176	0.382380	0.191190	+	0.981553i	$0.438765\pi$
$440$	0	0
$441$	−2.82948	−0.134737
$442$	0	0
$443$	30.5849	1.45313	0.726567	−	0.687096i	$-0.241115\pi$
$443$	30.5849	1.45313	0.726567	+	0.687096i	$0.241115\pi$
$444$	0	0
$445$	3.91990	0.185821
$446$	0	0
$447$	5.87389	0.277826
$448$	0	0
$449$	21.9300	1.03494	0.517471	−	0.855701i	$-0.326874\pi$
$449$	21.9300	1.03494	0.517471	+	0.855701i	$0.326874\pi$
$450$	0	0
$451$	−16.3085	−0.767937
$452$	0	0
$453$	−8.50790	−0.399736
$454$	0	0
$455$	−0.621509	−0.0291368
$456$	0	0
$457$	24.8111	1.16061	0.580307	−	0.814398i	$-0.302933\pi$
$457$	24.8111	1.16061	0.580307	+	0.814398i	$0.302933\pi$
$458$	0	0
$459$	−0.804849	−0.0375672
$460$	0	0
$461$	−6.82233	−0.317748	−0.158874	−	0.987299i	$-0.550786\pi$
$461$	−6.82233	−0.317748	−0.158874	+	0.987299i	$0.550786\pi$
$462$	0	0
$463$	−20.4906	−0.952280	−0.476140	−	0.879370i	$-0.657964\pi$
$463$	−20.4906	−0.952280	−0.476140	+	0.879370i	$0.657964\pi$
$464$	0	0
$465$	3.61059	0.167437
$466$	0	0
$467$	3.72473	0.172360	0.0861801	−	0.996280i	$-0.472534\pi$
$467$	3.72473	0.172360	0.0861801	+	0.996280i	$0.472534\pi$
$468$	0	0
$469$	−11.9800	−0.553186
$470$	0	0
$471$	−5.43067	−0.250232
$472$	0	0
$473$	−3.52290	−0.161983
$474$	0	0
$475$	1.41653	0.0649949
$476$	0	0
$477$	−22.2489	−1.01871
$478$	0	0
$479$	14.4716	0.661225	0.330612	−	0.943767i	$-0.392745\pi$
$479$	14.4716	0.661225	0.330612	+	0.943767i	$0.392745\pi$
$480$	0	0
$481$	−2.50688	−0.114304
$482$	0	0
$483$	−0.270468	−0.0123067
$484$	0	0
$485$	−3.63045	−0.164850
$486$	0	0
$487$	29.7625	1.34867	0.674334	−	0.738427i	$-0.264431\pi$
$487$	29.7625	1.34867	0.674334	+	0.738427i	$0.264431\pi$
$488$	0	0
$489$	2.57722	0.116546
$490$	0	0
$491$	23.2937	1.05123	0.525615	−	0.850723i	$-0.323835\pi$
$491$	23.2937	1.05123	0.525615	+	0.850723i	$0.323835\pi$
$492$	0	0
$493$	−1.93119	−0.0869762
$494$	0	0
$495$	9.96798	0.448027
$496$	0	0
$497$	−6.43902	−0.288830
$498$	0	0
$499$	17.3881	0.778399	0.389200	−	0.921153i	$-0.372752\pi$
$499$	17.3881	0.778399	0.389200	+	0.921153i	$0.372752\pi$
$500$	0	0
$501$	−4.54049	−0.202854
$502$	0	0
$503$	−29.0564	−1.29556	−0.647781	−	0.761827i	$-0.724303\pi$
$503$	−29.0564	−1.29556	−0.647781	+	0.761827i	$0.724303\pi$
$504$	0	0
$505$	−2.83013	−0.125939
$506$	0	0
$507$	−5.20877	−0.231330
$508$	0	0
$509$	−17.9370	−0.795044	−0.397522	−	0.917593i	$-0.630130\pi$
$509$	−17.9370	−0.795044	−0.397522	+	0.917593i	$0.630130\pi$
$510$	0	0
$511$	−13.7141	−0.606675
$512$	0	0
$513$	−3.40995	−0.150553
$514$	0	0
$515$	−2.45335	−0.108108
$516$	0	0
$517$	−17.5959	−0.773865
$518$	0	0
$519$	4.78295	0.209948
$520$	0	0
$521$	−24.6266	−1.07891	−0.539456	−	0.842014i	$-0.681370\pi$
$521$	−24.6266	−1.07891	−0.539456	+	0.842014i	$0.681370\pi$
$522$	0	0
$523$	40.8268	1.78523	0.892616	−	0.450818i	$-0.148868\pi$
$523$	40.8268	1.78523	0.892616	+	0.450818i	$0.148868\pi$
$524$	0	0
$525$	0.412945	0.0180224
$526$	0	0
$527$	−2.92334	−0.127343
$528$	0	0
$529$	−22.5710	−0.981348
$530$	0	0
$531$	−4.71416	−0.204577
$532$	0	0
$533$	−2.87714	−0.124623
$534$	0	0
$535$	5.67904	0.245526
$536$	0	0
$537$	−7.60770	−0.328296
$538$	0	0
$539$	3.52290	0.151742
$540$	0	0
$541$	18.6976	0.803874	0.401937	−	0.915667i	$-0.368337\pi$
$541$	18.6976	0.803874	0.401937	+	0.915667i	$0.368337\pi$
$542$	0	0
$543$	−11.0539	−0.474368
$544$	0	0
$545$	−14.3409	−0.614296
$546$	0	0
$547$	−35.5815	−1.52135	−0.760677	−	0.649131i	$-0.775133\pi$
$547$	−35.5815	−1.52135	−0.760677	+	0.649131i	$0.775133\pi$
$548$	0	0
$549$	−28.0782	−1.19835
$550$	0	0
$551$	−8.18196	−0.348563
$552$	0	0
$553$	16.6260	0.707009
$554$	0	0
$555$	1.66563	0.0707022
$556$	0	0
$557$	3.73351	0.158194	0.0790969	−	0.996867i	$-0.474796\pi$
$557$	3.73351	0.158194	0.0790969	+	0.996867i	$0.474796\pi$
$558$	0	0
$559$	−0.621509	−0.0262870
$560$	0	0
$561$	0.486391	0.0205355
$562$	0	0
$563$	−7.36417	−0.310363	−0.155181	−	0.987886i	$-0.549596\pi$
$563$	−7.36417	−0.310363	−0.155181	+	0.987886i	$0.549596\pi$
$564$	0	0
$565$	2.11632	0.0890341
$566$	0	0
$567$	7.49437	0.314734
$568$	0	0
$569$	0.717635	0.0300848	0.0150424	−	0.999887i	$-0.495212\pi$
$569$	0.717635	0.0300848	0.0150424	+	0.999887i	$0.495212\pi$
$570$	0	0
$571$	34.0708	1.42582	0.712910	−	0.701256i	$-0.247377\pi$
$571$	34.0708	1.42582	0.712910	+	0.701256i	$0.247377\pi$
$572$	0	0
$573$	−9.79881	−0.409351
$574$	0	0
$575$	−0.654973	−0.0273142
$576$	0	0
$577$	12.7941	0.532626	0.266313	−	0.963887i	$-0.414194\pi$
$577$	12.7941	0.532626	0.266313	+	0.963887i	$0.414194\pi$
$578$	0	0
$579$	−6.37664	−0.265004
$580$	0	0
$581$	13.7502	0.570455
$582$	0	0
$583$	27.7015	1.14728
$584$	0	0
$585$	1.75854	0.0727069
$586$	0	0
$587$	45.5549	1.88025	0.940126	−	0.340826i	$-0.110707\pi$
$587$	45.5549	1.88025	0.940126	+	0.340826i	$0.110707\pi$
$588$	0	0
$589$	−12.3855	−0.510334
$590$	0	0
$591$	−6.42094	−0.264122
$592$	0	0
$593$	−15.1052	−0.620297	−0.310149	−	0.950688i	$-0.600379\pi$
$593$	−15.1052	−0.620297	−0.310149	+	0.950688i	$0.600379\pi$
$594$	0	0
$595$	−0.334344	−0.0137068
$596$	0	0
$597$	−5.65640	−0.231501
$598$	0	0
$599$	−28.9791	−1.18405	−0.592026	−	0.805919i	$-0.701672\pi$
$599$	−28.9791	−1.18405	−0.592026	+	0.805919i	$0.701672\pi$
$600$	0	0
$601$	9.85348	0.401932	0.200966	−	0.979598i	$-0.435592\pi$
$601$	9.85348	0.401932	0.200966	+	0.979598i	$0.435592\pi$
$602$	0	0
$603$	33.8972	1.38040
$604$	0	0
$605$	−1.41086	−0.0573594
$606$	0	0
$607$	28.8032	1.16909	0.584544	−	0.811362i	$-0.301274\pi$
$607$	28.8032	1.16909	0.584544	+	0.811362i	$0.301274\pi$
$608$	0	0
$609$	−2.38519	−0.0966528
$610$	0	0
$611$	−3.10425	−0.125585
$612$	0	0
$613$	−36.3112	−1.46660	−0.733298	−	0.679908i	$-0.762020\pi$
$613$	−36.3112	−1.46660	−0.733298	+	0.679908i	$0.762020\pi$
$614$	0	0
$615$	1.91164	0.0770846
$616$	0	0
$617$	−33.3172	−1.34130	−0.670649	−	0.741775i	$-0.733984\pi$
$617$	−33.3172	−1.34130	−0.670649	+	0.741775i	$0.733984\pi$
$618$	0	0
$619$	12.6359	0.507880	0.253940	−	0.967220i	$-0.418273\pi$
$619$	12.6359	0.507880	0.253940	+	0.967220i	$0.418273\pi$
$620$	0	0
$621$	1.57668	0.0632702
$622$	0	0
$623$	−3.91990	−0.157047
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	2.06072	0.0822973
$628$	0	0
$629$	−1.34859	−0.0537718
$630$	0	0
$631$	16.4751	0.655863	0.327932	−	0.944701i	$-0.393648\pi$
$631$	16.4751	0.655863	0.327932	+	0.944701i	$0.393648\pi$
$632$	0	0
$633$	6.91890	0.275001
$634$	0	0
$635$	9.58285	0.380284
$636$	0	0
$637$	0.621509	0.0246251
$638$	0	0
$639$	18.2191	0.720735
$640$	0	0
$641$	−8.81572	−0.348200	−0.174100	−	0.984728i	$-0.555702\pi$
$641$	−8.81572	−0.348200	−0.174100	+	0.984728i	$0.555702\pi$
$642$	0	0
$643$	−33.8441	−1.33468	−0.667340	−	0.744753i	$-0.732567\pi$
$643$	−33.8441	−1.33468	−0.667340	+	0.744753i	$0.732567\pi$
$644$	0	0
$645$	0.412945	0.0162597
$646$	0	0
$647$	30.9059	1.21504	0.607519	−	0.794305i	$-0.292165\pi$
$647$	30.9059	1.21504	0.607519	+	0.794305i	$0.292165\pi$
$648$	0	0
$649$	5.86948	0.230397
$650$	0	0
$651$	−3.61059	−0.141510
$652$	0	0
$653$	47.8697	1.87329	0.936643	−	0.350286i	$-0.113916\pi$
$653$	47.8697	1.87329	0.936643	+	0.350286i	$0.113916\pi$
$654$	0	0
$655$	18.4759	0.721914
$656$	0	0
$657$	38.8036	1.51387
$658$	0	0
$659$	50.4143	1.96386	0.981931	−	0.189240i	$-0.0606023\pi$
$659$	50.4143	1.96386	0.981931	+	0.189240i	$0.0606023\pi$
$660$	0	0
$661$	−41.6360	−1.61945	−0.809727	−	0.586807i	$-0.800385\pi$
$661$	−41.6360	−1.61945	−0.809727	+	0.586807i	$0.800385\pi$
$662$	0	0
$663$	0.0858089	0.00333254
$664$	0	0
$665$	−1.41653	−0.0549307
$666$	0	0
$667$	3.78315	0.146484
$668$	0	0
$669$	2.86018	0.110581
$670$	0	0
$671$	34.9594	1.34959
$672$	0	0
$673$	−3.59398	−0.138538	−0.0692689	−	0.997598i	$-0.522067\pi$
$673$	−3.59398	−0.138538	−0.0692689	+	0.997598i	$0.522067\pi$
$674$	0	0
$675$	−2.40725	−0.0926552
$676$	0	0
$677$	35.4133	1.36104	0.680521	−	0.732729i	$-0.261754\pi$
$677$	35.4133	1.36104	0.680521	+	0.732729i	$0.261754\pi$
$678$	0	0
$679$	3.63045	0.139324
$680$	0	0
$681$	−1.90770	−0.0731033
$682$	0	0
$683$	40.5703	1.55238	0.776190	−	0.630499i	$-0.217150\pi$
$683$	40.5703	1.55238	0.776190	+	0.630499i	$0.217150\pi$
$684$	0	0
$685$	−1.32239	−0.0505260
$686$	0	0
$687$	5.20349	0.198526
$688$	0	0
$689$	4.88708	0.186183
$690$	0	0
$691$	−42.2031	−1.60548	−0.802740	−	0.596329i	$-0.796625\pi$
$691$	−42.2031	−1.60548	−0.802740	+	0.596329i	$0.796625\pi$
$692$	0	0
$693$	−9.96798	−0.378652
$694$	0	0
$695$	7.66019	0.290568
$696$	0	0
$697$	−1.54777	−0.0586259
$698$	0	0
$699$	0.781849	0.0295722
$700$	0	0
$701$	17.9625	0.678433	0.339217	−	0.940708i	$-0.389838\pi$
$701$	17.9625	0.678433	0.339217	+	0.940708i	$0.389838\pi$
$702$	0	0
$703$	−5.71365	−0.215494
$704$	0	0
$705$	2.06254	0.0776796
$706$	0	0
$707$	2.83013	0.106438
$708$	0	0
$709$	−49.6612	−1.86507	−0.932533	−	0.361084i	$-0.882407\pi$
$709$	−49.6612	−1.86507	−0.932533	+	0.361084i	$0.882407\pi$
$710$	0	0
$711$	−47.0428	−1.76424
$712$	0	0
$713$	5.72676	0.214469
$714$	0	0
$715$	−2.18952	−0.0818832
$716$	0	0
$717$	−0.903380	−0.0337374
$718$	0	0
$719$	0.243665	0.00908719	0.00454359	−	0.999990i	$-0.498554\pi$
$719$	0.243665	0.00908719	0.00454359	+	0.999990i	$0.498554\pi$
$720$	0	0
$721$	2.45335	0.0913676
$722$	0	0
$723$	−4.95194	−0.184165
$724$	0	0
$725$	−5.77605	−0.214517
$726$	0	0
$727$	−12.0426	−0.446634	−0.223317	−	0.974746i	$-0.571688\pi$
$727$	−12.0426	−0.446634	−0.223317	+	0.974746i	$0.571688\pi$
$728$	0	0
$729$	−18.2229	−0.674924
$730$	0	0
$731$	−0.334344	−0.0123661
$732$	0	0
$733$	−1.80361	−0.0666179	−0.0333090	−	0.999445i	$-0.510605\pi$
$733$	−1.80361	−0.0666179	−0.0333090	+	0.999445i	$0.510605\pi$
$734$	0	0
$735$	−0.412945	−0.0152317
$736$	0	0
$737$	−42.2045	−1.55462
$738$	0	0
$739$	−4.03467	−0.148418	−0.0742088	−	0.997243i	$-0.523643\pi$
$739$	−4.03467	−0.148418	−0.0742088	+	0.997243i	$0.523643\pi$
$740$	0	0
$741$	0.363551	0.0133554
$742$	0	0
$743$	45.5522	1.67115	0.835573	−	0.549379i	$-0.185136\pi$
$743$	45.5522	1.67115	0.835573	+	0.549379i	$0.185136\pi$
$744$	0	0
$745$	−14.2244	−0.521142
$746$	0	0
$747$	−38.9059	−1.42349
$748$	0	0
$749$	−5.67904	−0.207507
$750$	0	0
$751$	−41.8592	−1.52746	−0.763732	−	0.645533i	$-0.776635\pi$
$751$	−41.8592	−1.52746	−0.763732	+	0.645533i	$0.776635\pi$
$752$	0	0
$753$	−11.8902	−0.433304
$754$	0	0
$755$	20.6030	0.749820
$756$	0	0
$757$	−4.19586	−0.152501	−0.0762505	−	0.997089i	$-0.524295\pi$
$757$	−4.19586	−0.152501	−0.0762505	+	0.997089i	$0.524295\pi$
$758$	0	0
$759$	−0.952832	−0.0345856
$760$	0	0
$761$	−44.9961	−1.63111	−0.815554	−	0.578682i	$-0.803567\pi$
$761$	−44.9961	−1.63111	−0.815554	+	0.578682i	$0.803567\pi$
$762$	0	0
$763$	14.3409	0.519174
$764$	0	0
$765$	0.946017	0.0342033
$766$	0	0
$767$	1.03549	0.0373894
$768$	0	0
$769$	−50.9907	−1.83877	−0.919385	−	0.393358i	$-0.871313\pi$
$769$	−50.9907	−1.83877	−0.919385	+	0.393358i	$0.871313\pi$
$770$	0	0
$771$	−0.133185	−0.00479654
$772$	0	0
$773$	27.9213	1.00426	0.502129	−	0.864793i	$-0.332550\pi$
$773$	27.9213	1.00426	0.502129	+	0.864793i	$0.332550\pi$
$774$	0	0
$775$	−8.74351	−0.314076
$776$	0	0
$777$	−1.66563	−0.0597542
$778$	0	0
$779$	−6.55752	−0.234947
$780$	0	0
$781$	−22.6841	−0.811699
$782$	0	0
$783$	13.9044	0.496903
$784$	0	0
$785$	13.1511	0.469382
$786$	0	0
$787$	−7.47885	−0.266592	−0.133296	−	0.991076i	$-0.542556\pi$
$787$	−7.47885	−0.266592	−0.133296	+	0.991076i	$0.542556\pi$
$788$	0	0
$789$	11.6855	0.416014
$790$	0	0
$791$	−2.11632	−0.0752475
$792$	0	0
$793$	6.16751	0.219015
$794$	0	0
$795$	−3.24709	−0.115162
$796$	0	0
$797$	11.9066	0.421753	0.210876	−	0.977513i	$-0.432368\pi$
$797$	11.9066	0.421753	0.210876	+	0.977513i	$0.432368\pi$
$798$	0	0
$799$	−1.66995	−0.0590785
$800$	0	0
$801$	11.0913	0.391890
$802$	0	0
$803$	−48.3133	−1.70494
$804$	0	0
$805$	0.654973	0.0230848
$806$	0	0
$807$	−1.62953	−0.0573623
$808$	0	0
$809$	10.5979	0.372602	0.186301	−	0.982493i	$-0.440350\pi$
$809$	10.5979	0.372602	0.186301	+	0.982493i	$0.440350\pi$
$810$	0	0
$811$	15.0764	0.529403	0.264701	−	0.964330i	$-0.414727\pi$
$811$	15.0764	0.529403	0.264701	+	0.964330i	$0.414727\pi$
$812$	0	0
$813$	7.17452	0.251622
$814$	0	0
$815$	−6.24108	−0.218616
$816$	0	0
$817$	−1.41653	−0.0495582
$818$	0	0
$819$	−1.75854	−0.0614485
$820$	0	0
$821$	8.90401	0.310752	0.155376	−	0.987855i	$-0.450341\pi$
$821$	8.90401	0.310752	0.155376	+	0.987855i	$0.450341\pi$
$822$	0	0
$823$	35.3442	1.23202	0.616011	−	0.787738i	$-0.288748\pi$
$823$	35.3442	1.23202	0.616011	+	0.787738i	$0.288748\pi$
$824$	0	0
$825$	1.45477	0.0506484
$826$	0	0
$827$	9.33223	0.324514	0.162257	−	0.986749i	$-0.448123\pi$
$827$	9.33223	0.324514	0.162257	+	0.986749i	$0.448123\pi$
$828$	0	0
$829$	−38.5400	−1.33855	−0.669275	−	0.743015i	$-0.733395\pi$
$829$	−38.5400	−1.33855	−0.669275	+	0.743015i	$0.733395\pi$
$830$	0	0
$831$	−8.75317	−0.303644
$832$	0	0
$833$	0.334344	0.0115843
$834$	0	0
$835$	10.9954	0.380512
$836$	0	0
$837$	21.0478	0.727519
$838$	0	0
$839$	−1.57078	−0.0542293	−0.0271146	−	0.999632i	$-0.508632\pi$
$839$	−1.57078	−0.0542293	−0.0271146	+	0.999632i	$0.508632\pi$
$840$	0	0
$841$	4.36275	0.150440
$842$	0	0
$843$	−2.65282	−0.0913680
$844$	0	0
$845$	12.6137	0.433925
$846$	0	0
$847$	1.41086	0.0484776
$848$	0	0
$849$	−8.13618	−0.279233
$850$	0	0
$851$	2.64186	0.0905619
$852$	0	0
$853$	38.1898	1.30760	0.653798	−	0.756669i	$-0.273175\pi$
$853$	38.1898	1.30760	0.653798	+	0.756669i	$0.273175\pi$
$854$	0	0
$855$	4.00804	0.137072
$856$	0	0
$857$	−5.23771	−0.178917	−0.0894584	−	0.995991i	$-0.528514\pi$
$857$	−5.23771	−0.178917	−0.0894584	+	0.995991i	$0.528514\pi$
$858$	0	0
$859$	−16.0188	−0.546553	−0.273276	−	0.961936i	$-0.588107\pi$
$859$	−16.0188	−0.546553	−0.273276	+	0.961936i	$0.588107\pi$
$860$	0	0
$861$	−1.91164	−0.0651484
$862$	0	0
$863$	−52.6941	−1.79373	−0.896863	−	0.442307i	$-0.854160\pi$
$863$	−52.6941	−1.79373	−0.896863	+	0.442307i	$0.854160\pi$
$864$	0	0
$865$	−11.5825	−0.393818
$866$	0	0
$867$	−6.97390	−0.236846
$868$	0	0
$869$	58.5717	1.98691
$870$	0	0
$871$	−7.44568	−0.252287
$872$	0	0
$873$	−10.2723	−0.347664
$874$	0	0
$875$	−1.00000	−0.0338062
$876$	0	0
$877$	25.0101	0.844532	0.422266	−	0.906472i	$-0.361235\pi$
$877$	25.0101	0.844532	0.422266	+	0.906472i	$0.361235\pi$
$878$	0	0
$879$	4.92151	0.165998
$880$	0	0
$881$	44.1170	1.48634	0.743170	−	0.669103i	$-0.233321\pi$
$881$	44.1170	1.48634	0.743170	+	0.669103i	$0.233321\pi$
$882$	0	0
$883$	51.2067	1.72324	0.861621	−	0.507552i	$-0.169450\pi$
$883$	51.2067	1.72324	0.861621	+	0.507552i	$0.169450\pi$
$884$	0	0
$885$	−0.688003	−0.0231270
$886$	0	0
$887$	−45.0221	−1.51169	−0.755847	−	0.654748i	$-0.772775\pi$
$887$	−45.0221	−1.51169	−0.755847	+	0.654748i	$0.772775\pi$
$888$	0	0
$889$	−9.58285	−0.321399
$890$	0	0
$891$	26.4019	0.884498
$892$	0	0
$893$	−7.07515	−0.236761
$894$	0	0
$895$	18.4230	0.615814
$896$	0	0
$897$	−0.168098	−0.00561263
$898$	0	0
$899$	50.5029	1.68437
$900$	0	0
$901$	2.62903	0.0875855
$902$	0	0
$903$	−0.412945	−0.0137419
$904$	0	0
$905$	26.7685	0.889814
$906$	0	0
$907$	13.4483	0.446543	0.223271	−	0.974756i	$-0.428326\pi$
$907$	13.4483	0.446543	0.223271	+	0.974756i	$0.428326\pi$
$908$	0	0
$909$	−8.00779	−0.265602
$910$	0	0
$911$	31.5729	1.04606	0.523028	−	0.852316i	$-0.324802\pi$
$911$	31.5729	1.04606	0.523028	+	0.852316i	$0.324802\pi$
$912$	0	0
$913$	48.4407	1.60315
$914$	0	0
$915$	−4.09784	−0.135470
$916$	0	0
$917$	−18.4759	−0.610129
$918$	0	0
$919$	−18.8240	−0.620947	−0.310474	−	0.950582i	$-0.600488\pi$
$919$	−18.8240	−0.620947	−0.310474	+	0.950582i	$0.600488\pi$
$920$	0	0
$921$	1.20596	0.0397376
$922$	0	0
$923$	−4.00191	−0.131724
$924$	0	0
$925$	−4.03355	−0.132622
$926$	0	0
$927$	−6.94170	−0.227995
$928$	0	0
$929$	22.4188	0.735538	0.367769	−	0.929917i	$-0.380122\pi$
$929$	22.4188	0.735538	0.367769	+	0.929917i	$0.380122\pi$
$930$	0	0
$931$	1.41653	0.0464250
$932$	0	0
$933$	13.0030	0.425699
$934$	0	0
$935$	−1.17786	−0.0385202
$936$	0	0
$937$	−9.46028	−0.309054	−0.154527	−	0.987989i	$-0.549385\pi$
$937$	−9.46028	−0.309054	−0.154527	+	0.987989i	$0.549385\pi$
$938$	0	0
$939$	−13.9204	−0.454274
$940$	0	0
$941$	−47.2607	−1.54065	−0.770327	−	0.637649i	$-0.779907\pi$
$941$	−47.2607	−1.54065	−0.770327	+	0.637649i	$0.779907\pi$
$942$	0	0
$943$	3.03205	0.0987371
$944$	0	0
$945$	2.40725	0.0783079
$946$	0	0
$947$	−6.05707	−0.196828	−0.0984142	−	0.995146i	$-0.531377\pi$
$947$	−6.05707	−0.196828	−0.0984142	+	0.995146i	$0.531377\pi$
$948$	0	0
$949$	−8.52341	−0.276682
$950$	0	0
$951$	8.61955	0.279508
$952$	0	0
$953$	−11.8417	−0.383589	−0.191795	−	0.981435i	$-0.561431\pi$
$953$	−11.8417	−0.383589	−0.191795	+	0.981435i	$0.561431\pi$
$954$	0	0
$955$	23.7291	0.767856
$956$	0	0
$957$	−8.40280	−0.271624
$958$	0	0
$959$	1.32239	0.0427023
$960$	0	0
$961$	45.4489	1.46609
$962$	0	0
$963$	16.0687	0.517807
$964$	0	0
$965$	15.4419	0.497091
$966$	0	0
$967$	56.9104	1.83011	0.915057	−	0.403324i	$-0.132145\pi$
$967$	56.9104	1.83011	0.915057	+	0.403324i	$0.132145\pi$
$968$	0	0
$969$	0.195574	0.00628275
$970$	0	0
$971$	−17.6861	−0.567572	−0.283786	−	0.958888i	$-0.591591\pi$
$971$	−17.6861	−0.567572	−0.283786	+	0.958888i	$0.591591\pi$
$972$	0	0
$973$	−7.66019	−0.245574
$974$	0	0
$975$	0.256649	0.00821934
$976$	0	0
$977$	−42.9605	−1.37443	−0.687214	−	0.726455i	$-0.741166\pi$
$977$	−42.9605	−1.37443	−0.687214	+	0.726455i	$0.741166\pi$
$978$	0	0
$979$	−13.8094	−0.441351
$980$	0	0
$981$	−40.5772	−1.29553
$982$	0	0
$983$	−3.56526	−0.113714	−0.0568571	−	0.998382i	$-0.518108\pi$
$983$	−3.56526	−0.113714	−0.0568571	+	0.998382i	$0.518108\pi$
$984$	0	0
$985$	15.5492	0.495437
$986$	0	0
$987$	−2.06254	−0.0656512
$988$	0	0
$989$	0.654973	0.0208269
$990$	0	0
$991$	−47.9781	−1.52407	−0.762037	−	0.647534i	$-0.775800\pi$
$991$	−47.9781	−1.52407	−0.762037	+	0.647534i	$0.775800\pi$
$992$	0	0
$993$	1.36406	0.0432870
$994$	0	0
$995$	13.6977	0.434247
$996$	0	0
$997$	33.0376	1.04631	0.523155	−	0.852238i	$-0.324755\pi$
$997$	33.0376	1.04631	0.523155	+	0.852238i	$0.324755\pi$
$998$	0	0
$999$	9.70977	0.307203

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6020.2.a.f.1.6	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6020.2.a.f.1.6	✓	8	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:	\( N \)	\(=\)	\( 6020 = 2^{2} \cdot 5 \cdot 7 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6020.a (trivial)

\(f(q)\)	\(=\)	\(q+0.412945 q^{3} -1.00000 q^{5} +1.00000 q^{7} -2.82948 q^{9} +O(q^{10})\)	\(q+0.412945 q^{3} -1.00000 q^{5} +1.00000 q^{7} -2.82948 q^{9} +3.52290 q^{11} +0.621509 q^{13} -0.412945 q^{15} +0.334344 q^{17} +1.41653 q^{19} +0.412945 q^{21} -0.654973 q^{23} +1.00000 q^{25} -2.40725 q^{27} -5.77605 q^{29} -8.74351 q^{31} +1.45477 q^{33} -1.00000 q^{35} -4.03355 q^{37} +0.256649 q^{39} -4.62928 q^{41} -1.00000 q^{43} +2.82948 q^{45} -4.99470 q^{47} +1.00000 q^{49} +0.138065 q^{51} +7.86325 q^{53} -3.52290 q^{55} +0.584949 q^{57} +1.66609 q^{59} +9.92345 q^{61} -2.82948 q^{63} -0.621509 q^{65} -11.9800 q^{67} -0.270468 q^{69} -6.43902 q^{71} -13.7141 q^{73} +0.412945 q^{75} +3.52290 q^{77} +16.6260 q^{79} +7.49437 q^{81} +13.7502 q^{83} -0.334344 q^{85} -2.38519 q^{87} -3.91990 q^{89} +0.621509 q^{91} -3.61059 q^{93} -1.41653 q^{95} +3.63045 q^{97} -9.96798 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 5 q^{3} - 8 q^{5} + 8 q^{7} + 11 q^{9}+O(q^{10}) \) 8 * q - 5 * q^3 - 8 * q^5 + 8 * q^7 + 11 * q^9	\( 8 q - 5 q^{3} - 8 q^{5} + 8 q^{7} + 11 q^{9} + 6 q^{11} - 13 q^{13} + 5 q^{15} - 2 q^{17} - 14 q^{19} - 5 q^{21} + 6 q^{23} + 8 q^{25} - 11 q^{27} - 7 q^{29} - 18 q^{31} - q^{33} - 8 q^{35} + 2 q^{37} + 9 q^{39} - 18 q^{41} - 8 q^{43} - 11 q^{45} - q^{47} + 8 q^{49} - 19 q^{51} + 5 q^{53} - 6 q^{55} - 4 q^{57} - 12 q^{59} - 23 q^{61} + 11 q^{63} + 13 q^{65} + 8 q^{67} - 18 q^{69} + 20 q^{71} - 4 q^{73} - 5 q^{75} + 6 q^{77} + 24 q^{79} - 8 q^{81} - 14 q^{83} + 2 q^{85} + 10 q^{87} - 21 q^{89} - 13 q^{91} + q^{93} + 14 q^{95} + 7 q^{97} + 4 q^{99}+O(q^{100}) \) 8 * q - 5 * q^3 - 8 * q^5 + 8 * q^7 + 11 * q^9 + 6 * q^11 - 13 * q^13 + 5 * q^15 - 2 * q^17 - 14 * q^19 - 5 * q^21 + 6 * q^23 + 8 * q^25 - 11 * q^27 - 7 * q^29 - 18 * q^31 - q^33 - 8 * q^35 + 2 * q^37 + 9 * q^39 - 18 * q^41 - 8 * q^43 - 11 * q^45 - q^47 + 8 * q^49 - 19 * q^51 + 5 * q^53 - 6 * q^55 - 4 * q^57 - 12 * q^59 - 23 * q^61 + 11 * q^63 + 13 * q^65 + 8 * q^67 - 18 * q^69 + 20 * q^71 - 4 * q^73 - 5 * q^75 + 6 * q^77 + 24 * q^79 - 8 * q^81 - 14 * q^83 + 2 * q^85 + 10 * q^87 - 21 * q^89 - 13 * q^91 + q^93 + 14 * q^95 + 7 * q^97 + 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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