LMFDB - Embedded newform 6020.2.a.f.1.4

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			1.4
Root			$-0.604219$ 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)

$f(q)$	$=$	$q-1.60422 q^{3} -1.00000 q^{5} +1.00000 q^{7} -0.426481 q^{9} +O(q^{10})$	$q-1.60422 q^{3} -1.00000 q^{5} +1.00000 q^{7} -0.426481 q^{9} -3.70626 q^{11} -4.52797 q^{13} +1.60422 q^{15} +3.84013 q^{17} +1.03070 q^{19} -1.60422 q^{21} -0.0720190 q^{23} +1.00000 q^{25} +5.49683 q^{27} +5.78889 q^{29} +1.81124 q^{31} +5.94565 q^{33} -1.00000 q^{35} +1.11159 q^{37} +7.26386 q^{39} +9.44322 q^{41} -1.00000 q^{43} +0.426481 q^{45} +6.74658 q^{47} +1.00000 q^{49} -6.16040 q^{51} -8.80481 q^{53} +3.70626 q^{55} -1.65347 q^{57} -4.68685 q^{59} -6.51000 q^{61} -0.426481 q^{63} +4.52797 q^{65} -0.238127 q^{67} +0.115534 q^{69} -0.589762 q^{71} +2.62304 q^{73} -1.60422 q^{75} -3.70626 q^{77} +15.0117 q^{79} -7.53867 q^{81} +5.63248 q^{83} -3.84013 q^{85} -9.28665 q^{87} -17.3432 q^{89} -4.52797 q^{91} -2.90562 q^{93} -1.03070 q^{95} +5.76530 q^{97} +1.58065 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−1.60422	−0.926196	−0.463098	−	0.886307i	$-0.653262\pi$
$3$	−1.60422	−0.926196	−0.463098	+	0.886307i	$0.653262\pi$
$4$	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
$9$	−0.426481	−0.142160
$10$	0	0
$11$	−3.70626	−1.11748	−0.558739	−	0.829343i	$-0.688715\pi$
$11$	−3.70626	−1.11748	−0.558739	+	0.829343i	$0.688715\pi$
$12$	0	0
$13$	−4.52797	−1.25583	−0.627917	−	0.778280i	$-0.716092\pi$
$13$	−4.52797	−1.25583	−0.627917	+	0.778280i	$0.716092\pi$
$14$	0	0
$15$	1.60422	0.414208
$16$	0	0
$17$	3.84013	0.931367	0.465684	−	0.884951i	$-0.345808\pi$
$17$	3.84013	0.931367	0.465684	+	0.884951i	$0.345808\pi$
$18$	0	0
$19$	1.03070	0.236459	0.118229	−	0.992986i	$-0.462278\pi$
$19$	1.03070	0.236459	0.118229	+	0.992986i	$0.462278\pi$
$20$	0	0
$21$	−1.60422	−0.350069
$22$	0	0
$23$	−0.0720190	−0.0150170	−0.00750850	−	0.999972i	$-0.502390\pi$
$23$	−0.0720190	−0.0150170	−0.00750850	+	0.999972i	$0.502390\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	5.49683	1.05786
$28$	0	0
$29$	5.78889	1.07497	0.537485	−	0.843273i	$-0.319374\pi$
$29$	5.78889	1.07497	0.537485	+	0.843273i	$0.319374\pi$
$30$	0	0
$31$	1.81124	0.325308	0.162654	−	0.986683i	$-0.447995\pi$
$31$	1.81124	0.325308	0.162654	+	0.986683i	$0.447995\pi$
$32$	0	0
$33$	5.94565	1.03500
$34$	0	0
$35$	−1.00000	−0.169031
$36$	0	0
$37$	1.11159	0.182744	0.0913718	−	0.995817i	$-0.470875\pi$
$37$	1.11159	0.182744	0.0913718	+	0.995817i	$0.470875\pi$
$38$	0	0
$39$	7.26386	1.16315
$40$	0	0
$41$	9.44322	1.47478	0.737391	−	0.675466i	$-0.236057\pi$
$41$	9.44322	1.47478	0.737391	+	0.675466i	$0.236057\pi$
$42$	0	0
$43$	−1.00000	−0.152499
$43$	−1.00000	−0.152499
$44$	0	0
$45$	0.426481	0.0635761
$46$	0	0
$47$	6.74658	0.984090	0.492045	−	0.870570i	$-0.336249\pi$
$47$	6.74658	0.984090	0.492045	+	0.870570i	$0.336249\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−6.16040	−0.862629
$52$	0	0
$53$	−8.80481	−1.20943	−0.604717	−	0.796440i	$-0.706714\pi$
$53$	−8.80481	−1.20943	−0.604717	+	0.796440i	$0.706714\pi$
$54$	0	0
$55$	3.70626	0.499752
$56$	0	0
$57$	−1.65347	−0.219007
$58$	0	0
$59$	−4.68685	−0.610177	−0.305088	−	0.952324i	$-0.598686\pi$
$59$	−4.68685	−0.610177	−0.305088	+	0.952324i	$0.598686\pi$
$60$	0	0
$61$	−6.51000	−0.833520	−0.416760	−	0.909017i	$-0.636834\pi$
$61$	−6.51000	−0.833520	−0.416760	+	0.909017i	$0.636834\pi$
$62$	0	0
$63$	−0.426481	−0.0537316
$64$	0	0
$65$	4.52797	0.561626
$66$	0	0
$67$	−0.238127	−0.0290918	−0.0145459	−	0.999894i	$-0.504630\pi$
$67$	−0.238127	−0.0290918	−0.0145459	+	0.999894i	$0.504630\pi$
$68$	0	0
$69$	0.115534	0.0139087
$70$	0	0
$71$	−0.589762	−0.0699919	−0.0349959	−	0.999387i	$-0.511142\pi$
$71$	−0.589762	−0.0699919	−0.0349959	+	0.999387i	$0.511142\pi$
$72$	0	0
$73$	2.62304	0.307004	0.153502	−	0.988148i	$-0.450945\pi$
$73$	2.62304	0.307004	0.153502	+	0.988148i	$0.450945\pi$
$74$	0	0
$75$	−1.60422	−0.185239
$76$	0	0
$77$	−3.70626	−0.422367
$78$	0	0
$79$	15.0117	1.68895	0.844477	−	0.535592i	$-0.179912\pi$
$79$	15.0117	1.68895	0.844477	+	0.535592i	$0.179912\pi$
$80$	0	0
$81$	−7.53867	−0.837630
$82$	0	0
$83$	5.63248	0.618245	0.309123	−	0.951022i	$-0.399965\pi$
$83$	5.63248	0.618245	0.309123	+	0.951022i	$0.399965\pi$
$84$	0	0
$85$	−3.84013	−0.416520
$86$	0	0
$87$	−9.28665	−0.995634
$88$	0	0
$89$	−17.3432	−1.83837	−0.919186	−	0.393824i	$-0.871152\pi$
$89$	−17.3432	−1.83837	−0.919186	+	0.393824i	$0.871152\pi$
$90$	0	0
$91$	−4.52797	−0.474661
$92$	0	0
$93$	−2.90562	−0.301299
$94$	0	0
$95$	−1.03070	−0.105748
$96$	0	0
$97$	5.76530	0.585378	0.292689	−	0.956208i	$-0.405450\pi$
$97$	5.76530	0.585378	0.292689	+	0.956208i	$0.405450\pi$
$98$	0	0
$99$	1.58065	0.158861
$100$	0	0
$101$	−11.5400	−1.14827	−0.574134	−	0.818761i	$-0.694661\pi$
$101$	−11.5400	−1.14827	−0.574134	+	0.818761i	$0.694661\pi$
$102$	0	0
$103$	−6.95627	−0.685421	−0.342711	−	0.939441i	$-0.611345\pi$
$103$	−6.95627	−0.685421	−0.342711	+	0.939441i	$0.611345\pi$
$104$	0	0
$105$	1.60422	0.156556
$106$	0	0
$107$	2.88869	0.279260	0.139630	−	0.990204i	$-0.455409\pi$
$107$	2.88869	0.279260	0.139630	+	0.990204i	$0.455409\pi$
$108$	0	0
$109$	−0.0586034	−0.00561319	−0.00280659	−	0.999996i	$-0.500893\pi$
$109$	−0.0586034	−0.00561319	−0.00280659	+	0.999996i	$0.500893\pi$
$110$	0	0
$111$	−1.78323	−0.169256
$112$	0	0
$113$	−15.1657	−1.42667	−0.713335	−	0.700823i	$-0.752816\pi$
$113$	−15.1657	−1.42667	−0.713335	+	0.700823i	$0.752816\pi$
$114$	0	0
$115$	0.0720190	0.00671581
$116$	0	0
$117$	1.93110	0.178530
$118$	0	0
$119$	3.84013	0.352024
$120$	0	0
$121$	2.73635	0.248759
$122$	0	0
$123$	−15.1490	−1.36594
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−8.70102	−0.772091	−0.386045	−	0.922480i	$-0.626159\pi$
$127$	−8.70102	−0.772091	−0.386045	+	0.922480i	$0.626159\pi$
$128$	0	0
$129$	1.60422	0.141244
$130$	0	0
$131$	−10.6345	−0.929145	−0.464572	−	0.885535i	$-0.653792\pi$
$131$	−10.6345	−0.929145	−0.464572	+	0.885535i	$0.653792\pi$
$132$	0	0
$133$	1.03070	0.0893731
$134$	0	0
$135$	−5.49683	−0.473092
$136$	0	0
$137$	17.9987	1.53773	0.768865	−	0.639412i	$-0.220822\pi$
$137$	17.9987	1.53773	0.768865	+	0.639412i	$0.220822\pi$
$138$	0	0
$139$	−12.1385	−1.02957	−0.514787	−	0.857318i	$-0.672129\pi$
$139$	−12.1385	−1.02957	−0.514787	+	0.857318i	$0.672129\pi$
$140$	0	0
$141$	−10.8230	−0.911461
$142$	0	0
$143$	16.7818	1.40337
$144$	0	0
$145$	−5.78889	−0.480742
$146$	0	0
$147$	−1.60422	−0.132314
$148$	0	0
$149$	18.8029	1.54040	0.770198	−	0.637805i	$-0.220158\pi$
$149$	18.8029	1.54040	0.770198	+	0.637805i	$0.220158\pi$
$150$	0	0
$151$	6.51088	0.529848	0.264924	−	0.964269i	$-0.414653\pi$
$151$	6.51088	0.529848	0.264924	+	0.964269i	$0.414653\pi$
$152$	0	0
$153$	−1.63774	−0.132404
$154$	0	0
$155$	−1.81124	−0.145482
$156$	0	0
$157$	16.5411	1.32012	0.660061	−	0.751212i	$-0.270530\pi$
$157$	16.5411	1.32012	0.660061	+	0.751212i	$0.270530\pi$
$158$	0	0
$159$	14.1248	1.12017
$160$	0	0
$161$	−0.0720190	−0.00567589
$162$	0	0
$163$	−6.70060	−0.524832	−0.262416	−	0.964955i	$-0.584519\pi$
$163$	−6.70060	−0.524832	−0.262416	+	0.964955i	$0.584519\pi$
$164$	0	0
$165$	−5.94565	−0.462868
$166$	0	0
$167$	−13.9508	−1.07954	−0.539772	−	0.841811i	$-0.681489\pi$
$167$	−13.9508	−1.07954	−0.539772	+	0.841811i	$0.681489\pi$
$168$	0	0
$169$	7.50255	0.577119
$170$	0	0
$171$	−0.439575	−0.0336151
$172$	0	0
$173$	−9.89991	−0.752676	−0.376338	−	0.926482i	$-0.622817\pi$
$173$	−9.89991	−0.752676	−0.376338	+	0.926482i	$0.622817\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	0	0
$177$	7.51874	0.565143
$178$	0	0
$179$	−3.75532	−0.280686	−0.140343	−	0.990103i	$-0.544821\pi$
$179$	−3.75532	−0.280686	−0.140343	+	0.990103i	$0.544821\pi$
$180$	0	0
$181$	3.60713	0.268116	0.134058	−	0.990973i	$-0.457199\pi$
$181$	3.60713	0.268116	0.134058	+	0.990973i	$0.457199\pi$
$182$	0	0
$183$	10.4435	0.772003
$184$	0	0
$185$	−1.11159	−0.0817254
$186$	0	0
$187$	−14.2325	−1.04078
$188$	0	0
$189$	5.49683	0.399835
$190$	0	0
$191$	0.381764	0.0276235	0.0138117	−	0.999905i	$-0.495603\pi$
$191$	0.381764	0.0276235	0.0138117	+	0.999905i	$0.495603\pi$
$192$	0	0
$193$	3.17639	0.228642	0.114321	−	0.993444i	$-0.463531\pi$
$193$	3.17639	0.228642	0.114321	+	0.993444i	$0.463531\pi$
$194$	0	0
$195$	−7.26386	−0.520176
$196$	0	0
$197$	−12.4076	−0.884007	−0.442003	−	0.897013i	$-0.645732\pi$
$197$	−12.4076	−0.884007	−0.442003	+	0.897013i	$0.645732\pi$
$198$	0	0
$199$	−13.4620	−0.954298	−0.477149	−	0.878822i	$-0.658330\pi$
$199$	−13.4620	−0.954298	−0.477149	+	0.878822i	$0.658330\pi$
$200$	0	0
$201$	0.382008	0.0269448
$202$	0	0
$203$	5.78889	0.406301
$204$	0	0
$205$	−9.44322	−0.659543
$206$	0	0
$207$	0.0307148	0.00213482
$208$	0	0
$209$	−3.82004	−0.264238
$210$	0	0
$211$	24.1440	1.66214	0.831069	−	0.556169i	$-0.187729\pi$
$211$	24.1440	1.66214	0.831069	+	0.556169i	$0.187729\pi$
$212$	0	0
$213$	0.946108	0.0648262
$214$	0	0
$215$	1.00000	0.0681994
$216$	0	0
$217$	1.81124	0.122955
$218$	0	0
$219$	−4.20793	−0.284346
$220$	0	0
$221$	−17.3880	−1.16964
$222$	0	0
$223$	13.8029	0.924313	0.462157	−	0.886798i	$-0.347076\pi$
$223$	13.8029	0.924313	0.462157	+	0.886798i	$0.347076\pi$
$224$	0	0
$225$	−0.426481	−0.0284321
$226$	0	0
$227$	−1.39415	−0.0925329	−0.0462665	−	0.998929i	$-0.514732\pi$
$227$	−1.39415	−0.0925329	−0.0462665	+	0.998929i	$0.514732\pi$
$228$	0	0
$229$	−25.5418	−1.68785	−0.843926	−	0.536460i	$-0.819761\pi$
$229$	−25.5418	−1.68785	−0.843926	+	0.536460i	$0.819761\pi$
$230$	0	0
$231$	5.94565	0.391195
$232$	0	0
$233$	7.25116	0.475039	0.237520	−	0.971383i	$-0.423666\pi$
$233$	7.25116	0.475039	0.237520	+	0.971383i	$0.423666\pi$
$234$	0	0
$235$	−6.74658	−0.440099
$236$	0	0
$237$	−24.0821	−1.56430
$238$	0	0
$239$	19.9160	1.28826	0.644129	−	0.764917i	$-0.277220\pi$
$239$	19.9160	1.28826	0.644129	+	0.764917i	$0.277220\pi$
$240$	0	0
$241$	−29.0380	−1.87050	−0.935252	−	0.353983i	$-0.884827\pi$
$241$	−29.0380	−1.87050	−0.935252	+	0.353983i	$0.884827\pi$
$242$	0	0
$243$	−4.39680	−0.282055
$244$	0	0
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	−4.66699	−0.296953
$248$	0	0
$249$	−9.03573	−0.572616
$250$	0	0
$251$	−1.15176	−0.0726986	−0.0363493	−	0.999339i	$-0.511573\pi$
$251$	−1.15176	−0.0726986	−0.0363493	+	0.999339i	$0.511573\pi$
$252$	0	0
$253$	0.266921	0.0167812
$254$	0	0
$255$	6.16040	0.385779
$256$	0	0
$257$	5.09836	0.318027	0.159013	−	0.987276i	$-0.449169\pi$
$257$	5.09836	0.318027	0.159013	+	0.987276i	$0.449169\pi$
$258$	0	0
$259$	1.11159	0.0690706
$260$	0	0
$261$	−2.46886	−0.152818
$262$	0	0
$263$	2.35304	0.145095	0.0725473	−	0.997365i	$-0.476887\pi$
$263$	2.35304	0.145095	0.0725473	+	0.997365i	$0.476887\pi$
$264$	0	0
$265$	8.80481	0.540875
$266$	0	0
$267$	27.8222	1.70269
$268$	0	0
$269$	−14.3747	−0.876440	−0.438220	−	0.898868i	$-0.644391\pi$
$269$	−14.3747	−0.876440	−0.438220	+	0.898868i	$0.644391\pi$
$270$	0	0
$271$	−18.8019	−1.14213	−0.571067	−	0.820904i	$-0.693470\pi$
$271$	−18.8019	−1.14213	−0.571067	+	0.820904i	$0.693470\pi$
$272$	0	0
$273$	7.26386	0.439629
$274$	0	0
$275$	−3.70626	−0.223496
$276$	0	0
$277$	8.47908	0.509459	0.254729	−	0.967012i	$-0.418014\pi$
$277$	8.47908	0.509459	0.254729	+	0.967012i	$0.418014\pi$
$278$	0	0
$279$	−0.772459	−0.0462459
$280$	0	0
$281$	−17.9211	−1.06908	−0.534541	−	0.845142i	$-0.679516\pi$
$281$	−17.9211	−1.06908	−0.534541	+	0.845142i	$0.679516\pi$
$282$	0	0
$283$	−28.5648	−1.69800	−0.848999	−	0.528394i	$-0.822795\pi$
$283$	−28.5648	−1.69800	−0.848999	+	0.528394i	$0.822795\pi$
$284$	0	0
$285$	1.65347	0.0979431
$286$	0	0
$287$	9.44322	0.557415
$288$	0	0
$289$	−2.25343	−0.132555
$290$	0	0
$291$	−9.24881	−0.542175
$292$	0	0
$293$	10.1487	0.592895	0.296448	−	0.955049i	$-0.404198\pi$
$293$	10.1487	0.592895	0.296448	+	0.955049i	$0.404198\pi$
$294$	0	0
$295$	4.68685	0.272879
$296$	0	0
$297$	−20.3727	−1.18214
$298$	0	0
$299$	0.326100	0.0188589
$300$	0	0
$301$	−1.00000	−0.0576390
$302$	0	0
$303$	18.5126	1.06352
$304$	0	0
$305$	6.51000	0.372761
$306$	0	0
$307$	−9.65885	−0.551260	−0.275630	−	0.961264i	$-0.588886\pi$
$307$	−9.65885	−0.551260	−0.275630	+	0.961264i	$0.588886\pi$
$308$	0	0
$309$	11.1594	0.634835
$310$	0	0
$311$	9.69409	0.549701	0.274851	−	0.961487i	$-0.411372\pi$
$311$	9.69409	0.549701	0.274851	+	0.961487i	$0.411372\pi$
$312$	0	0
$313$	−12.4236	−0.702223	−0.351112	−	0.936334i	$-0.614196\pi$
$313$	−12.4236	−0.702223	−0.351112	+	0.936334i	$0.614196\pi$
$314$	0	0
$315$	0.426481	0.0240295
$316$	0	0
$317$	−1.56900	−0.0881240	−0.0440620	−	0.999029i	$-0.514030\pi$
$317$	−1.56900	−0.0881240	−0.0440620	+	0.999029i	$0.514030\pi$
$318$	0	0
$319$	−21.4551	−1.20126
$320$	0	0
$321$	−4.63408	−0.258649
$322$	0	0
$323$	3.95802	0.220230
$324$	0	0
$325$	−4.52797	−0.251167
$326$	0	0
$327$	0.0940127	0.00519892
$328$	0	0
$329$	6.74658	0.371951
$330$	0	0
$331$	4.70615	0.258673	0.129337	−	0.991601i	$-0.458715\pi$
$331$	4.70615	0.258673	0.129337	+	0.991601i	$0.458715\pi$
$332$	0	0
$333$	−0.474071	−0.0259789
$334$	0	0
$335$	0.238127	0.0130103
$336$	0	0
$337$	17.8568	0.972724	0.486362	−	0.873757i	$-0.338324\pi$
$337$	17.8568	0.972724	0.486362	+	0.873757i	$0.338324\pi$
$338$	0	0
$339$	24.3291	1.32138
$340$	0	0
$341$	−6.71291	−0.363524
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−0.115534	−0.00622016
$346$	0	0
$347$	16.5245	0.887084	0.443542	−	0.896254i	$-0.353722\pi$
$347$	16.5245	0.887084	0.443542	+	0.896254i	$0.353722\pi$
$348$	0	0
$349$	18.5145	0.991059	0.495530	−	0.868591i	$-0.334974\pi$
$349$	18.5145	0.991059	0.495530	+	0.868591i	$0.334974\pi$
$350$	0	0
$351$	−24.8895	−1.32850
$352$	0	0
$353$	16.5531	0.881033	0.440516	−	0.897745i	$-0.354795\pi$
$353$	16.5531	0.881033	0.440516	+	0.897745i	$0.354795\pi$
$354$	0	0
$355$	0.589762	0.0313013
$356$	0	0
$357$	−6.16040	−0.326043
$358$	0	0
$359$	19.6879	1.03909	0.519543	−	0.854444i	$-0.326102\pi$
$359$	19.6879	1.03909	0.519543	+	0.854444i	$0.326102\pi$
$360$	0	0
$361$	−17.9377	−0.944087
$362$	0	0
$363$	−4.38971	−0.230400
$364$	0	0
$365$	−2.62304	−0.137296
$366$	0	0
$367$	−8.61893	−0.449905	−0.224952	−	0.974370i	$-0.572223\pi$
$367$	−8.61893	−0.449905	−0.224952	+	0.974370i	$0.572223\pi$
$368$	0	0
$369$	−4.02736	−0.209656
$370$	0	0
$371$	−8.80481	−0.457123
$372$	0	0
$373$	−26.3744	−1.36562	−0.682808	−	0.730598i	$-0.739242\pi$
$373$	−26.3744	−1.36562	−0.682808	+	0.730598i	$0.739242\pi$
$374$	0	0
$375$	1.60422	0.0828415
$376$	0	0
$377$	−26.2120	−1.34998
$378$	0	0
$379$	−8.12124	−0.417160	−0.208580	−	0.978005i	$-0.566884\pi$
$379$	−8.12124	−0.417160	−0.208580	+	0.978005i	$0.566884\pi$
$380$	0	0
$381$	13.9583	0.715108
$382$	0	0
$383$	24.0376	1.22826	0.614131	−	0.789204i	$-0.289507\pi$
$383$	24.0376	1.22826	0.614131	+	0.789204i	$0.289507\pi$
$384$	0	0
$385$	3.70626	0.188888
$386$	0	0
$387$	0.426481	0.0216793
$388$	0	0
$389$	−30.8216	−1.56272	−0.781358	−	0.624084i	$-0.785472\pi$
$389$	−30.8216	−1.56272	−0.781358	+	0.624084i	$0.785472\pi$
$390$	0	0
$391$	−0.276562	−0.0139863
$392$	0	0
$393$	17.0601	0.860570
$394$	0	0
$395$	−15.0117	−0.755323
$396$	0	0
$397$	−38.7342	−1.94402	−0.972008	−	0.234947i	$-0.924508\pi$
$397$	−38.7342	−1.94402	−0.972008	+	0.234947i	$0.924508\pi$
$398$	0	0
$399$	−1.65347	−0.0827770
$400$	0	0
$401$	−17.9061	−0.894187	−0.447093	−	0.894487i	$-0.647541\pi$
$401$	−17.9061	−0.894187	−0.447093	+	0.894487i	$0.647541\pi$
$402$	0	0
$403$	−8.20123	−0.408532
$404$	0	0
$405$	7.53867	0.374599
$406$	0	0
$407$	−4.11982	−0.204212
$408$	0	0
$409$	−38.9650	−1.92669	−0.963346	−	0.268260i	$-0.913551\pi$
$409$	−38.9650	−1.92669	−0.963346	+	0.268260i	$0.913551\pi$
$410$	0	0
$411$	−28.8738	−1.42424
$412$	0	0
$413$	−4.68685	−0.230625
$414$	0	0
$415$	−5.63248	−0.276488
$416$	0	0
$417$	19.4728	0.953588
$418$	0	0
$419$	−9.16709	−0.447842	−0.223921	−	0.974607i	$-0.571886\pi$
$419$	−9.16709	−0.447842	−0.223921	+	0.974607i	$0.571886\pi$
$420$	0	0
$421$	−28.5882	−1.39331	−0.696653	−	0.717409i	$-0.745328\pi$
$421$	−28.5882	−1.39331	−0.696653	+	0.717409i	$0.745328\pi$
$422$	0	0
$423$	−2.87729	−0.139899
$424$	0	0
$425$	3.84013	0.186273
$426$	0	0
$427$	−6.51000	−0.315041
$428$	0	0
$429$	−26.9218	−1.29979
$430$	0	0
$431$	4.72585	0.227636	0.113818	−	0.993502i	$-0.463692\pi$
$431$	4.72585	0.227636	0.113818	+	0.993502i	$0.463692\pi$
$432$	0	0
$433$	15.6050	0.749927	0.374964	−	0.927040i	$-0.377655\pi$
$433$	15.6050	0.749927	0.374964	+	0.927040i	$0.377655\pi$
$434$	0	0
$435$	9.28665	0.445261
$436$	0	0
$437$	−0.0742300	−0.00355090
$438$	0	0
$439$	23.8615	1.13885	0.569423	−	0.822045i	$-0.307167\pi$
$439$	23.8615	1.13885	0.569423	+	0.822045i	$0.307167\pi$
$440$	0	0
$441$	−0.426481	−0.0203086
$442$	0	0
$443$	40.3716	1.91811	0.959055	−	0.283218i	$-0.0914021\pi$
$443$	40.3716	1.91811	0.959055	+	0.283218i	$0.0914021\pi$
$444$	0	0
$445$	17.3432	0.822145
$446$	0	0
$447$	−30.1640	−1.42671
$448$	0	0
$449$	−31.9407	−1.50737	−0.753686	−	0.657234i	$-0.771726\pi$
$449$	−31.9407	−1.50737	−0.753686	+	0.657234i	$0.771726\pi$
$450$	0	0
$451$	−34.9990	−1.64804
$452$	0	0
$453$	−10.4449	−0.490743
$454$	0	0
$455$	4.52797	0.212275
$456$	0	0
$457$	−12.7377	−0.595846	−0.297923	−	0.954590i	$-0.596294\pi$
$457$	−12.7377	−0.595846	−0.297923	+	0.954590i	$0.596294\pi$
$458$	0	0
$459$	21.1085	0.985261
$460$	0	0
$461$	12.0684	0.562082	0.281041	−	0.959696i	$-0.409320\pi$
$461$	12.0684	0.562082	0.281041	+	0.959696i	$0.409320\pi$
$462$	0	0
$463$	−24.6884	−1.14737	−0.573684	−	0.819076i	$-0.694486\pi$
$463$	−24.6884	−1.14737	−0.573684	+	0.819076i	$0.694486\pi$
$464$	0	0
$465$	2.90562	0.134745
$466$	0	0
$467$	28.2887	1.30905	0.654523	−	0.756042i	$-0.272869\pi$
$467$	28.2887	1.30905	0.654523	+	0.756042i	$0.272869\pi$
$468$	0	0
$469$	−0.238127	−0.0109957
$470$	0	0
$471$	−26.5355	−1.22269
$472$	0	0
$473$	3.70626	0.170414
$474$	0	0
$475$	1.03070	0.0472918
$476$	0	0
$477$	3.75509	0.171934
$478$	0	0
$479$	−35.2403	−1.61017	−0.805084	−	0.593161i	$-0.797880\pi$
$479$	−35.2403	−1.61017	−0.805084	+	0.593161i	$0.797880\pi$
$480$	0	0
$481$	−5.03323	−0.229496
$482$	0	0
$483$	0.115534	0.00525699
$484$	0	0
$485$	−5.76530	−0.261789
$486$	0	0
$487$	−19.7726	−0.895984	−0.447992	−	0.894038i	$-0.647861\pi$
$487$	−19.7726	−0.895984	−0.447992	+	0.894038i	$0.647861\pi$
$488$	0	0
$489$	10.7492	0.486097
$490$	0	0
$491$	24.6981	1.11461	0.557305	−	0.830308i	$-0.311835\pi$
$491$	24.6981	1.11461	0.557305	+	0.830308i	$0.311835\pi$
$492$	0	0
$493$	22.2301	1.00119
$494$	0	0
$495$	−1.58065	−0.0710449
$496$	0	0
$497$	−0.589762	−0.0264544
$498$	0	0
$499$	−19.5886	−0.876905	−0.438453	−	0.898754i	$-0.644473\pi$
$499$	−19.5886	−0.876905	−0.438453	+	0.898754i	$0.644473\pi$
$500$	0	0
$501$	22.3801	0.999869
$502$	0	0
$503$	34.5011	1.53833	0.769164	−	0.639051i	$-0.220673\pi$
$503$	34.5011	1.53833	0.769164	+	0.639051i	$0.220673\pi$
$504$	0	0
$505$	11.5400	0.513521
$506$	0	0
$507$	−12.0357	−0.534526
$508$	0	0
$509$	−0.858509	−0.0380527	−0.0190264	−	0.999819i	$-0.506057\pi$
$509$	−0.858509	−0.0380527	−0.0190264	+	0.999819i	$0.506057\pi$
$510$	0	0
$511$	2.62304	0.116036
$512$	0	0
$513$	5.66558	0.250142
$514$	0	0
$515$	6.95627	0.306530
$516$	0	0
$517$	−25.0046	−1.09970
$518$	0	0
$519$	15.8816	0.697126
$520$	0	0
$521$	−45.3171	−1.98538	−0.992690	−	0.120696i	$-0.961487\pi$
$521$	−45.3171	−1.98538	−0.992690	+	0.120696i	$0.961487\pi$
$522$	0	0
$523$	7.72898	0.337965	0.168982	−	0.985619i	$-0.445952\pi$
$523$	7.72898	0.337965	0.168982	+	0.985619i	$0.445952\pi$
$524$	0	0
$525$	−1.60422	−0.0700139
$526$	0	0
$527$	6.95538	0.302981
$528$	0	0
$529$	−22.9948	−0.999774
$530$	0	0
$531$	1.99886	0.0867430
$532$	0	0
$533$	−42.7586	−1.85208
$534$	0	0
$535$	−2.88869	−0.124889
$536$	0	0
$537$	6.02436	0.259970
$538$	0	0
$539$	−3.70626	−0.159640
$540$	0	0
$541$	−5.99221	−0.257625	−0.128813	−	0.991669i	$-0.541117\pi$
$541$	−5.99221	−0.257625	−0.128813	+	0.991669i	$0.541117\pi$
$542$	0	0
$543$	−5.78663	−0.248328
$544$	0	0
$545$	0.0586034	0.00251029
$546$	0	0
$547$	15.2742	0.653078	0.326539	−	0.945184i	$-0.394117\pi$
$547$	15.2742	0.653078	0.326539	+	0.945184i	$0.394117\pi$
$548$	0	0
$549$	2.77639	0.118494
$550$	0	0
$551$	5.96662	0.254186
$552$	0	0
$553$	15.0117	0.638364
$554$	0	0
$555$	1.78323	0.0756938
$556$	0	0
$557$	−23.7601	−1.00675	−0.503373	−	0.864069i	$-0.667908\pi$
$557$	−23.7601	−1.00675	−0.503373	+	0.864069i	$0.667908\pi$
$558$	0	0
$559$	4.52797	0.191513
$560$	0	0
$561$	22.8320	0.963970
$562$	0	0
$563$	−32.4975	−1.36961	−0.684803	−	0.728728i	$-0.740112\pi$
$563$	−32.4975	−1.36961	−0.684803	+	0.728728i	$0.740112\pi$
$564$	0	0
$565$	15.1657	0.638027
$566$	0	0
$567$	−7.53867	−0.316594
$568$	0	0
$569$	−23.4082	−0.981324	−0.490662	−	0.871350i	$-0.663245\pi$
$569$	−23.4082	−0.981324	−0.490662	+	0.871350i	$0.663245\pi$
$570$	0	0
$571$	−13.0779	−0.547294	−0.273647	−	0.961830i	$-0.588230\pi$
$571$	−13.0779	−0.547294	−0.273647	+	0.961830i	$0.588230\pi$
$572$	0	0
$573$	−0.612433	−0.0255847
$574$	0	0
$575$	−0.0720190	−0.00300340
$576$	0	0
$577$	−7.09914	−0.295541	−0.147771	−	0.989022i	$-0.547210\pi$
$577$	−7.09914	−0.295541	−0.147771	+	0.989022i	$0.547210\pi$
$578$	0	0
$579$	−5.09563	−0.211767
$580$	0	0
$581$	5.63248	0.233675
$582$	0	0
$583$	32.6329	1.35152
$584$	0	0
$585$	−1.93110	−0.0798410
$586$	0	0
$587$	23.6040	0.974242	0.487121	−	0.873335i	$-0.338047\pi$
$587$	23.6040	0.974242	0.487121	+	0.873335i	$0.338047\pi$
$588$	0	0
$589$	1.86684	0.0769219
$590$	0	0
$591$	19.9046	0.818764
$592$	0	0
$593$	−9.73849	−0.399912	−0.199956	−	0.979805i	$-0.564080\pi$
$593$	−9.73849	−0.399912	−0.199956	+	0.979805i	$0.564080\pi$
$594$	0	0
$595$	−3.84013	−0.157430
$596$	0	0
$597$	21.5961	0.883868
$598$	0	0
$599$	−4.51399	−0.184436	−0.0922182	−	0.995739i	$-0.529396\pi$
$599$	−4.51399	−0.184436	−0.0922182	+	0.995739i	$0.529396\pi$
$600$	0	0
$601$	−28.8724	−1.17773	−0.588865	−	0.808232i	$-0.700425\pi$
$601$	−28.8724	−1.17773	−0.588865	+	0.808232i	$0.700425\pi$
$602$	0	0
$603$	0.101557	0.00413571
$604$	0	0
$605$	−2.73635	−0.111249
$606$	0	0
$607$	−21.3230	−0.865474	−0.432737	−	0.901520i	$-0.642452\pi$
$607$	−21.3230	−0.865474	−0.432737	+	0.901520i	$0.642452\pi$
$608$	0	0
$609$	−9.28665	−0.376314
$610$	0	0
$611$	−30.5484	−1.23585
$612$	0	0
$613$	7.46577	0.301540	0.150770	−	0.988569i	$-0.451825\pi$
$613$	7.46577	0.301540	0.150770	+	0.988569i	$0.451825\pi$
$614$	0	0
$615$	15.1490	0.610866
$616$	0	0
$617$	−27.1257	−1.09204	−0.546020	−	0.837772i	$-0.683858\pi$
$617$	−27.1257	−1.09204	−0.546020	+	0.837772i	$0.683858\pi$
$618$	0	0
$619$	−34.3296	−1.37982	−0.689911	−	0.723894i	$-0.742350\pi$
$619$	−34.3296	−1.37982	−0.689911	+	0.723894i	$0.742350\pi$
$620$	0	0
$621$	−0.395876	−0.0158860
$622$	0	0
$623$	−17.3432	−0.694839
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	6.12818	0.244736
$628$	0	0
$629$	4.26863	0.170201
$630$	0	0
$631$	36.2047	1.44129	0.720644	−	0.693306i	$-0.243846\pi$
$631$	36.2047	1.44129	0.720644	+	0.693306i	$0.243846\pi$
$632$	0	0
$633$	−38.7322	−1.53947
$634$	0	0
$635$	8.70102	0.345289
$636$	0	0
$637$	−4.52797	−0.179405
$638$	0	0
$639$	0.251523	0.00995008
$640$	0	0
$641$	37.0538	1.46354	0.731768	−	0.681554i	$-0.238695\pi$
$641$	37.0538	1.46354	0.731768	+	0.681554i	$0.238695\pi$
$642$	0	0
$643$	45.1919	1.78220	0.891098	−	0.453811i	$-0.149936\pi$
$643$	45.1919	1.78220	0.891098	+	0.453811i	$0.149936\pi$
$644$	0	0
$645$	−1.60422	−0.0631661
$646$	0	0
$647$	−25.8518	−1.01634	−0.508169	−	0.861257i	$-0.669678\pi$
$647$	−25.8518	−1.01634	−0.508169	+	0.861257i	$0.669678\pi$
$648$	0	0
$649$	17.3707	0.681859
$650$	0	0
$651$	−2.90562	−0.113880
$652$	0	0
$653$	−6.62413	−0.259222	−0.129611	−	0.991565i	$-0.541373\pi$
$653$	−6.62413	−0.259222	−0.129611	+	0.991565i	$0.541373\pi$
$654$	0	0
$655$	10.6345	0.415526
$656$	0	0
$657$	−1.11868	−0.0436438
$658$	0	0
$659$	16.6362	0.648054	0.324027	−	0.946048i	$-0.394963\pi$
$659$	16.6362	0.648054	0.324027	+	0.946048i	$0.394963\pi$
$660$	0	0
$661$	−39.9161	−1.55256	−0.776278	−	0.630391i	$-0.782895\pi$
$661$	−39.9161	−1.55256	−0.776278	+	0.630391i	$0.782895\pi$
$662$	0	0
$663$	27.8941	1.08332
$664$	0	0
$665$	−1.03070	−0.0399688
$666$	0	0
$667$	−0.416910	−0.0161428
$668$	0	0
$669$	−22.1429	−0.856095
$670$	0	0
$671$	24.1277	0.931441
$672$	0	0
$673$	18.6003	0.716987	0.358494	−	0.933532i	$-0.383291\pi$
$673$	18.6003	0.716987	0.358494	+	0.933532i	$0.383291\pi$
$674$	0	0
$675$	5.49683	0.211573
$676$	0	0
$677$	37.3416	1.43515	0.717577	−	0.696479i	$-0.245251\pi$
$677$	37.3416	1.43515	0.717577	+	0.696479i	$0.245251\pi$
$678$	0	0
$679$	5.76530	0.221252
$680$	0	0
$681$	2.23652	0.0857037
$682$	0	0
$683$	−8.72352	−0.333796	−0.166898	−	0.985974i	$-0.553375\pi$
$683$	−8.72352	−0.333796	−0.166898	+	0.985974i	$0.553375\pi$
$684$	0	0
$685$	−17.9987	−0.687693
$686$	0	0
$687$	40.9747	1.56328
$688$	0	0
$689$	39.8680	1.51885
$690$	0	0
$691$	−38.7571	−1.47439	−0.737194	−	0.675681i	$-0.763850\pi$
$691$	−38.7571	−1.47439	−0.737194	+	0.675681i	$0.763850\pi$
$692$	0	0
$693$	1.58065	0.0600439
$694$	0	0
$695$	12.1385	0.460440
$696$	0	0
$697$	36.2631	1.37356
$698$	0	0
$699$	−11.6325	−0.439980
$700$	0	0
$701$	5.86902	0.221670	0.110835	−	0.993839i	$-0.464648\pi$
$701$	5.86902	0.221670	0.110835	+	0.993839i	$0.464648\pi$
$702$	0	0
$703$	1.14571	0.0432113
$704$	0	0
$705$	10.8230	0.407618
$706$	0	0
$707$	−11.5400	−0.434005
$708$	0	0
$709$	−13.0492	−0.490072	−0.245036	−	0.969514i	$-0.578800\pi$
$709$	−13.0492	−0.490072	−0.245036	+	0.969514i	$0.578800\pi$
$710$	0	0
$711$	−6.40223	−0.240102
$712$	0	0
$713$	−0.130443	−0.00488515
$714$	0	0
$715$	−16.7818	−0.627605
$716$	0	0
$717$	−31.9496	−1.19318
$718$	0	0
$719$	7.44554	0.277672	0.138836	−	0.990315i	$-0.455664\pi$
$719$	7.44554	0.277672	0.138836	+	0.990315i	$0.455664\pi$
$720$	0	0
$721$	−6.95627	−0.259065
$722$	0	0
$723$	46.5833	1.73245
$724$	0	0
$725$	5.78889	0.214994
$726$	0	0
$727$	−37.3913	−1.38677	−0.693383	−	0.720569i	$-0.743881\pi$
$727$	−37.3913	−1.38677	−0.693383	+	0.720569i	$0.743881\pi$
$728$	0	0
$729$	29.6694	1.09887
$730$	0	0
$731$	−3.84013	−0.142032
$732$	0	0
$733$	15.7018	0.579959	0.289979	−	0.957033i	$-0.406352\pi$
$733$	15.7018	0.579959	0.289979	+	0.957033i	$0.406352\pi$
$734$	0	0
$735$	1.60422	0.0591725
$736$	0	0
$737$	0.882560	0.0325095
$738$	0	0
$739$	35.6258	1.31052	0.655259	−	0.755405i	$-0.272560\pi$
$739$	35.6258	1.31052	0.655259	+	0.755405i	$0.272560\pi$
$740$	0	0
$741$	7.48687	0.275037
$742$	0	0
$743$	39.4746	1.44818	0.724092	−	0.689703i	$-0.242259\pi$
$743$	39.4746	1.44818	0.724092	+	0.689703i	$0.242259\pi$
$744$	0	0
$745$	−18.8029	−0.688886
$746$	0	0
$747$	−2.40215	−0.0878900
$748$	0	0
$749$	2.88869	0.105550
$750$	0	0
$751$	13.7550	0.501925	0.250963	−	0.967997i	$-0.419253\pi$
$751$	13.7550	0.501925	0.250963	+	0.967997i	$0.419253\pi$
$752$	0	0
$753$	1.84768	0.0673331
$754$	0	0
$755$	−6.51088	−0.236955
$756$	0	0
$757$	12.4683	0.453169	0.226584	−	0.973992i	$-0.427244\pi$
$757$	12.4683	0.453169	0.226584	+	0.973992i	$0.427244\pi$
$758$	0	0
$759$	−0.428200	−0.0155427
$760$	0	0
$761$	10.0241	0.363372	0.181686	−	0.983357i	$-0.441844\pi$
$761$	10.0241	0.363372	0.181686	+	0.983357i	$0.441844\pi$
$762$	0	0
$763$	−0.0586034	−0.00212159
$764$	0	0
$765$	1.63774	0.0592127
$766$	0	0
$767$	21.2220	0.766281
$768$	0	0
$769$	27.1603	0.979425	0.489713	−	0.871884i	$-0.337102\pi$
$769$	27.1603	0.979425	0.489713	+	0.871884i	$0.337102\pi$
$770$	0	0
$771$	−8.17889	−0.294555
$772$	0	0
$773$	−10.2495	−0.368647	−0.184324	−	0.982866i	$-0.559009\pi$
$773$	−10.2495	−0.368647	−0.184324	+	0.982866i	$0.559009\pi$
$774$	0	0
$775$	1.81124	0.0650615
$776$	0	0
$777$	−1.78323	−0.0639729
$778$	0	0
$779$	9.73313	0.348725
$780$	0	0
$781$	2.18581	0.0782145
$782$	0	0
$783$	31.8205	1.13717
$784$	0	0
$785$	−16.5411	−0.590377
$786$	0	0
$787$	−24.5216	−0.874102	−0.437051	−	0.899437i	$-0.643977\pi$
$787$	−24.5216	−0.874102	−0.437051	+	0.899437i	$0.643977\pi$
$788$	0	0
$789$	−3.77479	−0.134386
$790$	0	0
$791$	−15.1657	−0.539231
$792$	0	0
$793$	29.4771	1.04676
$794$	0	0
$795$	−14.1248	−0.500957
$796$	0	0
$797$	51.1635	1.81230	0.906151	−	0.422953i	$-0.139007\pi$
$797$	51.1635	1.81230	0.906151	+	0.422953i	$0.139007\pi$
$798$	0	0
$799$	25.9077	0.916550
$800$	0	0
$801$	7.39654	0.261344
$802$	0	0
$803$	−9.72166	−0.343070
$804$	0	0
$805$	0.0720190	0.00253834
$806$	0	0
$807$	23.0601	0.811755
$808$	0	0
$809$	37.3654	1.31370	0.656849	−	0.754022i	$-0.271889\pi$
$809$	37.3654	1.31370	0.656849	+	0.754022i	$0.271889\pi$
$810$	0	0
$811$	−22.9400	−0.805532	−0.402766	−	0.915303i	$-0.631951\pi$
$811$	−22.9400	−0.805532	−0.402766	+	0.915303i	$0.631951\pi$
$812$	0	0
$813$	30.1623	1.05784
$814$	0	0
$815$	6.70060	0.234712
$816$	0	0
$817$	−1.03070	−0.0360596
$818$	0	0
$819$	1.93110	0.0674780
$820$	0	0
$821$	−21.9515	−0.766112	−0.383056	−	0.923725i	$-0.625128\pi$
$821$	−21.9515	−0.766112	−0.383056	+	0.923725i	$0.625128\pi$
$822$	0	0
$823$	−24.7691	−0.863396	−0.431698	−	0.902018i	$-0.642085\pi$
$823$	−24.7691	−0.863396	−0.431698	+	0.902018i	$0.642085\pi$
$824$	0	0
$825$	5.94565	0.207001
$826$	0	0
$827$	52.2858	1.81816	0.909079	−	0.416625i	$-0.136787\pi$
$827$	52.2858	1.81816	0.909079	+	0.416625i	$0.136787\pi$
$828$	0	0
$829$	33.8328	1.17506	0.587531	−	0.809202i	$-0.300100\pi$
$829$	33.8328	1.17506	0.587531	+	0.809202i	$0.300100\pi$
$830$	0	0
$831$	−13.6023	−0.471859
$832$	0	0
$833$	3.84013	0.133052
$834$	0	0
$835$	13.9508	0.482787
$836$	0	0
$837$	9.95605	0.344132
$838$	0	0
$839$	6.92889	0.239212	0.119606	−	0.992821i	$-0.461837\pi$
$839$	6.92889	0.239212	0.119606	+	0.992821i	$0.461837\pi$
$840$	0	0
$841$	4.51130	0.155562
$842$	0	0
$843$	28.7494	0.990180
$844$	0	0
$845$	−7.50255	−0.258096
$846$	0	0
$847$	2.73635	0.0940222
$848$	0	0
$849$	45.8241	1.57268
$850$	0	0
$851$	−0.0800553	−0.00274426
$852$	0	0
$853$	−42.6062	−1.45881	−0.729404	−	0.684083i	$-0.760203\pi$
$853$	−42.6062	−1.45881	−0.729404	+	0.684083i	$0.760203\pi$
$854$	0	0
$855$	0.439575	0.0150331
$856$	0	0
$857$	48.9125	1.67082	0.835409	−	0.549628i	$-0.185231\pi$
$857$	48.9125	1.67082	0.835409	+	0.549628i	$0.185231\pi$
$858$	0	0
$859$	32.7030	1.11581	0.557907	−	0.829904i	$-0.311605\pi$
$859$	32.7030	1.11581	0.557907	+	0.829904i	$0.311605\pi$
$860$	0	0
$861$	−15.1490	−0.516276
$862$	0	0
$863$	47.1805	1.60604	0.803022	−	0.595949i	$-0.203224\pi$
$863$	47.1805	1.60604	0.803022	+	0.595949i	$0.203224\pi$
$864$	0	0
$865$	9.89991	0.336607
$866$	0	0
$867$	3.61500	0.122772
$868$	0	0
$869$	−55.6374	−1.88737
$870$	0	0
$871$	1.07823	0.0365345
$872$	0	0
$873$	−2.45879	−0.0832176
$874$	0	0
$875$	−1.00000	−0.0338062
$876$	0	0
$877$	−19.7446	−0.666729	−0.333365	−	0.942798i	$-0.608184\pi$
$877$	−19.7446	−0.666729	−0.333365	+	0.942798i	$0.608184\pi$
$878$	0	0
$879$	−16.2808	−0.549137
$880$	0	0
$881$	15.5825	0.524989	0.262495	−	0.964933i	$-0.415455\pi$
$881$	15.5825	0.524989	0.262495	+	0.964933i	$0.415455\pi$
$882$	0	0
$883$	23.8649	0.803117	0.401559	−	0.915833i	$-0.368469\pi$
$883$	23.8649	0.803117	0.401559	+	0.915833i	$0.368469\pi$
$884$	0	0
$885$	−7.51874	−0.252740
$886$	0	0
$887$	34.0560	1.14349	0.571744	−	0.820432i	$-0.306267\pi$
$887$	34.0560	1.14349	0.571744	+	0.820432i	$0.306267\pi$
$888$	0	0
$889$	−8.70102	−0.291823
$890$	0	0
$891$	27.9403	0.936034
$892$	0	0
$893$	6.95371	0.232697
$894$	0	0
$895$	3.75532	0.125527
$896$	0	0
$897$	−0.523136	−0.0174670
$898$	0	0
$899$	10.4851	0.349696
$900$	0	0
$901$	−33.8116	−1.12643
$902$	0	0
$903$	1.60422	0.0533851
$904$	0	0
$905$	−3.60713	−0.119905
$906$	0	0
$907$	−21.4767	−0.713123	−0.356562	−	0.934272i	$-0.616051\pi$
$907$	−21.4767	−0.713123	−0.356562	+	0.934272i	$0.616051\pi$
$908$	0	0
$909$	4.92158	0.163238
$910$	0	0
$911$	−55.6311	−1.84314	−0.921570	−	0.388213i	$-0.873092\pi$
$911$	−55.6311	−1.84314	−0.921570	+	0.388213i	$0.873092\pi$
$912$	0	0
$913$	−20.8754	−0.690876
$914$	0	0
$915$	−10.4435	−0.345250
$916$	0	0
$917$	−10.6345	−0.351184
$918$	0	0
$919$	18.9670	0.625665	0.312833	−	0.949808i	$-0.398722\pi$
$919$	18.9670	0.625665	0.312833	+	0.949808i	$0.398722\pi$
$920$	0	0
$921$	15.4949	0.510575
$922$	0	0
$923$	2.67043	0.0878982
$924$	0	0
$925$	1.11159	0.0365487
$926$	0	0
$927$	2.96672	0.0974398
$928$	0	0
$929$	−22.8426	−0.749441	−0.374721	−	0.927138i	$-0.622261\pi$
$929$	−22.8426	−0.749441	−0.374721	+	0.927138i	$0.622261\pi$
$930$	0	0
$931$	1.03070	0.0337798
$932$	0	0
$933$	−15.5514	−0.509131
$934$	0	0
$935$	14.2325	0.465452
$936$	0	0
$937$	−26.4435	−0.863871	−0.431936	−	0.901904i	$-0.642169\pi$
$937$	−26.4435	−0.863871	−0.431936	+	0.901904i	$0.642169\pi$
$938$	0	0
$939$	19.9302	0.650397
$940$	0	0
$941$	0.969434	0.0316026	0.0158013	−	0.999875i	$-0.494970\pi$
$941$	0.969434	0.0316026	0.0158013	+	0.999875i	$0.494970\pi$
$942$	0	0
$943$	−0.680091	−0.0221468
$944$	0	0
$945$	−5.49683	−0.178812
$946$	0	0
$947$	30.0208	0.975545	0.487772	−	0.872971i	$-0.337810\pi$
$947$	30.0208	0.975545	0.487772	+	0.872971i	$0.337810\pi$
$948$	0	0
$949$	−11.8771	−0.385546
$950$	0	0
$951$	2.51702	0.0816201
$952$	0	0
$953$	46.0364	1.49126	0.745632	−	0.666358i	$-0.232148\pi$
$953$	46.0364	1.49126	0.745632	+	0.666358i	$0.232148\pi$
$954$	0	0
$955$	−0.381764	−0.0123536
$956$	0	0
$957$	34.4187	1.11260
$958$	0	0
$959$	17.9987	0.581207
$960$	0	0
$961$	−27.7194	−0.894175
$962$	0	0
$963$	−1.23197	−0.0396997
$964$	0	0
$965$	−3.17639	−0.102252
$966$	0	0
$967$	17.1938	0.552915	0.276457	−	0.961026i	$-0.410840\pi$
$967$	17.1938	0.552915	0.276457	+	0.961026i	$0.410840\pi$
$968$	0	0
$969$	−6.34953	−0.203976
$970$	0	0
$971$	33.4388	1.07310	0.536551	−	0.843868i	$-0.319727\pi$
$971$	33.4388	1.07310	0.536551	+	0.843868i	$0.319727\pi$
$972$	0	0
$973$	−12.1385	−0.389143
$974$	0	0
$975$	7.26386	0.232630
$976$	0	0
$977$	4.08294	0.130625	0.0653124	−	0.997865i	$-0.479196\pi$
$977$	4.08294	0.130625	0.0653124	+	0.997865i	$0.479196\pi$
$978$	0	0
$979$	64.2782	2.05434
$980$	0	0
$981$	0.0249933	0.000797974 0
$982$	0	0
$983$	−4.19043	−0.133654	−0.0668270	−	0.997765i	$-0.521288\pi$
$983$	−4.19043	−0.133654	−0.0668270	+	0.997765i	$0.521288\pi$
$984$	0	0
$985$	12.4076	0.395340
$986$	0	0
$987$	−10.8230	−0.344500
$988$	0	0
$989$	0.0720190	0.00229007
$990$	0	0
$991$	46.6265	1.48114	0.740569	−	0.671980i	$-0.234556\pi$
$991$	46.6265	1.48114	0.740569	+	0.671980i	$0.234556\pi$
$992$	0	0
$993$	−7.54969	−0.239582
$994$	0	0
$995$	13.4620	0.426775
$996$	0	0
$997$	−2.49414	−0.0789901	−0.0394951	−	0.999220i	$-0.512575\pi$
$997$	−2.49414	−0.0789901	−0.0394951	+	0.999220i	$0.512575\pi$
$998$	0	0
$999$	6.11019	0.193318

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6020.2.a.f.1.4	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6020.2.a.f.1.4	✓	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)

\(f(q)\)	\(=\)	\(q-1.60422 q^{3} -1.00000 q^{5} +1.00000 q^{7} -0.426481 q^{9} +O(q^{10})\)	\(q-1.60422 q^{3} -1.00000 q^{5} +1.00000 q^{7} -0.426481 q^{9} -3.70626 q^{11} -4.52797 q^{13} +1.60422 q^{15} +3.84013 q^{17} +1.03070 q^{19} -1.60422 q^{21} -0.0720190 q^{23} +1.00000 q^{25} +5.49683 q^{27} +5.78889 q^{29} +1.81124 q^{31} +5.94565 q^{33} -1.00000 q^{35} +1.11159 q^{37} +7.26386 q^{39} +9.44322 q^{41} -1.00000 q^{43} +0.426481 q^{45} +6.74658 q^{47} +1.00000 q^{49} -6.16040 q^{51} -8.80481 q^{53} +3.70626 q^{55} -1.65347 q^{57} -4.68685 q^{59} -6.51000 q^{61} -0.426481 q^{63} +4.52797 q^{65} -0.238127 q^{67} +0.115534 q^{69} -0.589762 q^{71} +2.62304 q^{73} -1.60422 q^{75} -3.70626 q^{77} +15.0117 q^{79} -7.53867 q^{81} +5.63248 q^{83} -3.84013 q^{85} -9.28665 q^{87} -17.3432 q^{89} -4.52797 q^{91} -2.90562 q^{93} -1.03070 q^{95} +5.76530 q^{97} +1.58065 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 5 q^{3} - 8 q^{5} + 8 q^{7} + 11 q^{9}+O(q^{10}) \) 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

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