LMFDB - Embedded newform 6020.2.a.k.1.7

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:	$0$
Dimension:	$13$
Coefficient field:	$\mathbb{Q}[x]/(x^{13} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{13} - 27 x^{11} - 2 x^{10} + 268 x^{9} + 37 x^{8} - 1201 x^{7} - 189 x^{6} + 2384 x^{5} + 231 x^{4} + \cdots - 4 $ x^13 - 27x^11 - 2x^10 + 268x^9 + 37x^8 - 1201x^7 - 189x^6 + 2384x^5 + 231x^4 - 1729x^3 + 20x^2 + 105*x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$0.0387611$ 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)

$f(q)$	$=$	$q+0.0387611 q^{3} +1.00000 q^{5} -1.00000 q^{7} -2.99850 q^{9} +O(q^{10})$	$q+0.0387611 q^{3} +1.00000 q^{5} -1.00000 q^{7} -2.99850 q^{9} -4.92253 q^{11} +1.04616 q^{13} +0.0387611 q^{15} +2.87792 q^{17} -2.41705 q^{19} -0.0387611 q^{21} -6.95829 q^{23} +1.00000 q^{25} -0.232508 q^{27} +3.44515 q^{29} -4.67505 q^{31} -0.190803 q^{33} -1.00000 q^{35} +9.21150 q^{37} +0.0405502 q^{39} +10.2387 q^{41} +1.00000 q^{43} -2.99850 q^{45} -5.98033 q^{47} +1.00000 q^{49} +0.111551 q^{51} -5.06597 q^{53} -4.92253 q^{55} -0.0936876 q^{57} +8.66802 q^{59} +4.21737 q^{61} +2.99850 q^{63} +1.04616 q^{65} -10.4781 q^{67} -0.269711 q^{69} -11.3521 q^{71} -0.593172 q^{73} +0.0387611 q^{75} +4.92253 q^{77} +17.4617 q^{79} +8.98648 q^{81} +6.82289 q^{83} +2.87792 q^{85} +0.133538 q^{87} +16.0341 q^{89} -1.04616 q^{91} -0.181210 q^{93} -2.41705 q^{95} +2.99877 q^{97} +14.7602 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 13 q + 13 q^{5} - 13 q^{7} + 15 q^{9}+O(q^{10}) $ 13 * q + 13 * q^5 - 13 * q^7 + 15 * q^9	$ 13 q + 13 q^{5} - 13 q^{7} + 15 q^{9} + 11 q^{13} + 16 q^{17} + 3 q^{23} + 13 q^{25} + 6 q^{27} + 10 q^{29} - q^{31} + 14 q^{33} - 13 q^{35} + 16 q^{37} - 14 q^{39} + 23 q^{41} + 13 q^{43} + 15 q^{45} + 2 q^{47} + 13 q^{49} + 4 q^{51} + 20 q^{53} + 22 q^{57} + 2 q^{59} + 5 q^{61} - 15 q^{63} + 11 q^{65} + 19 q^{67} + 16 q^{69} + 4 q^{71} + 34 q^{73} - 15 q^{79} + 17 q^{81} + 27 q^{83} + 16 q^{85} - 5 q^{87} + 3 q^{89} - 11 q^{91} + 35 q^{93} + 45 q^{97} + q^{99}+O(q^{100}) $ 13 * q + 13 * q^5 - 13 * q^7 + 15 * q^9 + 11 * q^13 + 16 * q^17 + 3 * q^23 + 13 * q^25 + 6 * q^27 + 10 * q^29 - q^31 + 14 * q^33 - 13 * q^35 + 16 * q^37 - 14 * q^39 + 23 * q^41 + 13 * q^43 + 15 * q^45 + 2 * q^47 + 13 * q^49 + 4 * q^51 + 20 * q^53 + 22 * q^57 + 2 * q^59 + 5 * q^61 - 15 * q^63 + 11 * q^65 + 19 * q^67 + 16 * q^69 + 4 * q^71 + 34 * q^73 - 15 * q^79 + 17 * q^81 + 27 * q^83 + 16 * q^85 - 5 * q^87 + 3 * q^89 - 11 * q^91 + 35 * q^93 + 45 * q^97 + 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
$3$	0.0387611	0.0223787	0.0111894	−	0.999937i	$-0.496438\pi$
$3$	0.0387611	0.0223787	0.0111894	+	0.999937i	$0.496438\pi$
$4$	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
$9$	−2.99850	−0.999499
$10$	0	0
$11$	−4.92253	−1.48420	−0.742100	−	0.670289i	$-0.766170\pi$
$11$	−4.92253	−1.48420	−0.742100	+	0.670289i	$0.766170\pi$
$12$	0	0
$13$	1.04616	0.290152	0.145076	−	0.989420i	$-0.453657\pi$
$13$	1.04616	0.290152	0.145076	+	0.989420i	$0.453657\pi$
$14$	0	0
$15$	0.0387611	0.0100081
$16$	0	0
$17$	2.87792	0.697997	0.348999	−	0.937123i	$-0.386522\pi$
$17$	2.87792	0.697997	0.348999	+	0.937123i	$0.386522\pi$
$18$	0	0
$19$	−2.41705	−0.554510	−0.277255	−	0.960796i	$-0.589425\pi$
$19$	−2.41705	−0.554510	−0.277255	+	0.960796i	$0.589425\pi$
$20$	0	0
$21$	−0.0387611	−0.00845836
$22$	0	0
$23$	−6.95829	−1.45090	−0.725452	−	0.688273i	$-0.758369\pi$
$23$	−6.95829	−1.45090	−0.725452	+	0.688273i	$0.758369\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−0.232508	−0.0447462
$28$	0	0
$29$	3.44515	0.639748	0.319874	−	0.947460i	$-0.396359\pi$
$29$	3.44515	0.639748	0.319874	+	0.947460i	$0.396359\pi$
$30$	0	0
$31$	−4.67505	−0.839664	−0.419832	−	0.907602i	$-0.637911\pi$
$31$	−4.67505	−0.839664	−0.419832	+	0.907602i	$0.637911\pi$
$32$	0	0
$33$	−0.190803	−0.0332145
$34$	0	0
$35$	−1.00000	−0.169031
$36$	0	0
$37$	9.21150	1.51436	0.757180	−	0.653206i	$-0.226576\pi$
$37$	9.21150	1.51436	0.757180	+	0.653206i	$0.226576\pi$
$38$	0	0
$39$	0.0405502	0.00649324
$40$	0	0
$41$	10.2387	1.59901	0.799506	−	0.600658i	$-0.205095\pi$
$41$	10.2387	1.59901	0.799506	+	0.600658i	$0.205095\pi$
$42$	0	0
$43$	1.00000	0.152499
$43$	1.00000	0.152499
$44$	0	0
$45$	−2.99850	−0.446990
$46$	0	0
$47$	−5.98033	−0.872321	−0.436160	−	0.899869i	$-0.643662\pi$
$47$	−5.98033	−0.872321	−0.436160	+	0.899869i	$0.643662\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0.111551	0.0156203
$52$	0	0
$53$	−5.06597	−0.695864	−0.347932	−	0.937520i	$-0.613116\pi$
$53$	−5.06597	−0.695864	−0.347932	+	0.937520i	$0.613116\pi$
$54$	0	0
$55$	−4.92253	−0.663754
$56$	0	0
$57$	−0.0936876	−0.0124092
$58$	0	0
$59$	8.66802	1.12848	0.564240	−	0.825611i	$-0.309169\pi$
$59$	8.66802	1.12848	0.564240	+	0.825611i	$0.309169\pi$
$60$	0	0
$61$	4.21737	0.539979	0.269990	−	0.962863i	$-0.412980\pi$
$61$	4.21737	0.539979	0.269990	+	0.962863i	$0.412980\pi$
$62$	0	0
$63$	2.99850	0.377775
$64$	0	0
$65$	1.04616	0.129760
$66$	0	0
$67$	−10.4781	−1.28011	−0.640054	−	0.768330i	$-0.721088\pi$
$67$	−10.4781	−1.28011	−0.640054	+	0.768330i	$0.721088\pi$
$68$	0	0
$69$	−0.269711	−0.0324694
$70$	0	0
$71$	−11.3521	−1.34725	−0.673624	−	0.739074i	$-0.735263\pi$
$71$	−11.3521	−1.34725	−0.673624	+	0.739074i	$0.735263\pi$
$72$	0	0
$73$	−0.593172	−0.0694256	−0.0347128	−	0.999397i	$-0.511052\pi$
$73$	−0.593172	−0.0694256	−0.0347128	+	0.999397i	$0.511052\pi$
$74$	0	0
$75$	0.0387611	0.00447574
$76$	0	0
$77$	4.92253	0.560975
$78$	0	0
$79$	17.4617	1.96459	0.982296	−	0.187335i	$-0.0599850\pi$
$79$	17.4617	1.96459	0.982296	+	0.187335i	$0.0599850\pi$
$80$	0	0
$81$	8.98648	0.998498
$82$	0	0
$83$	6.82289	0.748909	0.374455	−	0.927245i	$-0.377830\pi$
$83$	6.82289	0.748909	0.374455	+	0.927245i	$0.377830\pi$
$84$	0	0
$85$	2.87792	0.312154
$86$	0	0
$87$	0.133538	0.0143167
$88$	0	0
$89$	16.0341	1.69961	0.849807	−	0.527094i	$-0.176718\pi$
$89$	16.0341	1.69961	0.849807	+	0.527094i	$0.176718\pi$
$90$	0	0
$91$	−1.04616	−0.109667
$92$	0	0
$93$	−0.181210	−0.0187906
$94$	0	0
$95$	−2.41705	−0.247985
$96$	0	0
$97$	2.99877	0.304479	0.152240	−	0.988344i	$-0.451351\pi$
$97$	2.99877	0.304479	0.152240	+	0.988344i	$0.451351\pi$
$98$	0	0
$99$	14.7602	1.48346
$100$	0	0
$101$	−2.04580	−0.203565	−0.101782	−	0.994807i	$-0.532455\pi$
$101$	−2.04580	−0.203565	−0.101782	+	0.994807i	$0.532455\pi$
$102$	0	0
$103$	18.7048	1.84304	0.921521	−	0.388328i	$-0.126947\pi$
$103$	18.7048	1.84304	0.921521	+	0.388328i	$0.126947\pi$
$104$	0	0
$105$	−0.0387611	−0.00378269
$106$	0	0
$107$	5.93525	0.573783	0.286891	−	0.957963i	$-0.407378\pi$
$107$	5.93525	0.573783	0.286891	+	0.957963i	$0.407378\pi$
$108$	0	0
$109$	−9.75409	−0.934273	−0.467136	−	0.884185i	$-0.654714\pi$
$109$	−9.75409	−0.934273	−0.467136	+	0.884185i	$0.654714\pi$
$110$	0	0
$111$	0.357047	0.0338894
$112$	0	0
$113$	2.30490	0.216827	0.108413	−	0.994106i	$-0.465423\pi$
$113$	2.30490	0.216827	0.108413	+	0.994106i	$0.465423\pi$
$114$	0	0
$115$	−6.95829	−0.648864
$116$	0	0
$117$	−3.13691	−0.290007
$118$	0	0
$119$	−2.87792	−0.263818
$120$	0	0
$121$	13.2313	1.20285
$122$	0	0
$123$	0.396862	0.0357838
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−10.7635	−0.955106	−0.477553	−	0.878603i	$-0.658476\pi$
$127$	−10.7635	−0.955106	−0.477553	+	0.878603i	$0.658476\pi$
$128$	0	0
$129$	0.0387611	0.00341272
$130$	0	0
$131$	1.95860	0.171124	0.0855620	−	0.996333i	$-0.472731\pi$
$131$	1.95860	0.171124	0.0855620	+	0.996333i	$0.472731\pi$
$132$	0	0
$133$	2.41705	0.209585
$134$	0	0
$135$	−0.232508	−0.0200111
$136$	0	0
$137$	−3.51345	−0.300174	−0.150087	−	0.988673i	$-0.547955\pi$
$137$	−3.51345	−0.300174	−0.150087	+	0.988673i	$0.547955\pi$
$138$	0	0
$139$	14.6886	1.24587	0.622936	−	0.782273i	$-0.285940\pi$
$139$	14.6886	1.24587	0.622936	+	0.782273i	$0.285940\pi$
$140$	0	0
$141$	−0.231804	−0.0195214
$142$	0	0
$143$	−5.14976	−0.430644
$144$	0	0
$145$	3.44515	0.286104
$146$	0	0
$147$	0.0387611	0.00319696
$148$	0	0
$149$	−11.8540	−0.971114	−0.485557	−	0.874205i	$-0.661383\pi$
$149$	−11.8540	−0.971114	−0.485557	+	0.874205i	$0.661383\pi$
$150$	0	0
$151$	8.26240	0.672385	0.336192	−	0.941793i	$-0.390861\pi$
$151$	8.26240	0.672385	0.336192	+	0.941793i	$0.390861\pi$
$152$	0	0
$153$	−8.62943	−0.697648
$154$	0	0
$155$	−4.67505	−0.375509
$156$	0	0
$157$	7.43764	0.593588	0.296794	−	0.954941i	$-0.404082\pi$
$157$	7.43764	0.593588	0.296794	+	0.954941i	$0.404082\pi$
$158$	0	0
$159$	−0.196362	−0.0155725
$160$	0	0
$161$	6.95829	0.548390
$162$	0	0
$163$	−6.59740	−0.516748	−0.258374	−	0.966045i	$-0.583187\pi$
$163$	−6.59740	−0.516748	−0.258374	+	0.966045i	$0.583187\pi$
$164$	0	0
$165$	−0.190803	−0.0148540
$166$	0	0
$167$	16.4759	1.27494	0.637472	−	0.770473i	$-0.279980\pi$
$167$	16.4759	1.27494	0.637472	+	0.770473i	$0.279980\pi$
$168$	0	0
$169$	−11.9056	−0.915812
$170$	0	0
$171$	7.24753	0.554233
$172$	0	0
$173$	0.468104	0.0355893	0.0177946	−	0.999842i	$-0.494335\pi$
$173$	0.468104	0.0355893	0.0177946	+	0.999842i	$0.494335\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	0	0
$177$	0.335982	0.0252539
$178$	0	0
$179$	−12.3043	−0.919665	−0.459832	−	0.888006i	$-0.652091\pi$
$179$	−12.3043	−0.919665	−0.459832	+	0.888006i	$0.652091\pi$
$180$	0	0
$181$	24.7884	1.84251	0.921255	−	0.388960i	$-0.127165\pi$
$181$	24.7884	1.84251	0.921255	+	0.388960i	$0.127165\pi$
$182$	0	0
$183$	0.163470	0.0120840
$184$	0	0
$185$	9.21150	0.677243
$186$	0	0
$187$	−14.1666	−1.03597
$188$	0	0
$189$	0.232508	0.0169125
$190$	0	0
$191$	22.7995	1.64972	0.824858	−	0.565340i	$-0.191255\pi$
$191$	22.7995	1.64972	0.824858	+	0.565340i	$0.191255\pi$
$192$	0	0
$193$	−4.93553	−0.355268	−0.177634	−	0.984097i	$-0.556844\pi$
$193$	−4.93553	−0.355268	−0.177634	+	0.984097i	$0.556844\pi$
$194$	0	0
$195$	0.0405502	0.00290386
$196$	0	0
$197$	−6.44372	−0.459096	−0.229548	−	0.973297i	$-0.573725\pi$
$197$	−6.44372	−0.459096	−0.229548	+	0.973297i	$0.573725\pi$
$198$	0	0
$199$	7.46874	0.529445	0.264723	−	0.964325i	$-0.414720\pi$
$199$	7.46874	0.529445	0.264723	+	0.964325i	$0.414720\pi$
$200$	0	0
$201$	−0.406144	−0.0286472
$202$	0	0
$203$	−3.44515	−0.241802
$204$	0	0
$205$	10.2387	0.715100
$206$	0	0
$207$	20.8644	1.45018
$208$	0	0
$209$	11.8980	0.823004
$210$	0	0
$211$	10.4178	0.717188	0.358594	−	0.933494i	$-0.383256\pi$
$211$	10.4178	0.717188	0.358594	+	0.933494i	$0.383256\pi$
$212$	0	0
$213$	−0.440020	−0.0301497
$214$	0	0
$215$	1.00000	0.0681994
$216$	0	0
$217$	4.67505	0.317363
$218$	0	0
$219$	−0.0229920	−0.00155365
$220$	0	0
$221$	3.01076	0.202526
$222$	0	0
$223$	28.7432	1.92479	0.962394	−	0.271658i	$-0.0875719\pi$
$223$	28.7432	1.92479	0.962394	+	0.271658i	$0.0875719\pi$
$224$	0	0
$225$	−2.99850	−0.199900
$226$	0	0
$227$	1.29964	0.0862603	0.0431302	−	0.999069i	$-0.486267\pi$
$227$	1.29964	0.0862603	0.0431302	+	0.999069i	$0.486267\pi$
$228$	0	0
$229$	−1.84861	−0.122160	−0.0610799	−	0.998133i	$-0.519454\pi$
$229$	−1.84861	−0.122160	−0.0610799	+	0.998133i	$0.519454\pi$
$230$	0	0
$231$	0.190803	0.0125539
$232$	0	0
$233$	12.8564	0.842253	0.421127	−	0.907002i	$-0.361635\pi$
$233$	12.8564	0.842253	0.421127	+	0.907002i	$0.361635\pi$
$234$	0	0
$235$	−5.98033	−0.390114
$236$	0	0
$237$	0.676833	0.0439650
$238$	0	0
$239$	−22.3813	−1.44773	−0.723863	−	0.689944i	$-0.757635\pi$
$239$	−22.3813	−1.44773	−0.723863	+	0.689944i	$0.757635\pi$
$240$	0	0
$241$	−23.5558	−1.51736	−0.758680	−	0.651463i	$-0.774156\pi$
$241$	−23.5558	−1.51736	−0.758680	+	0.651463i	$0.774156\pi$
$242$	0	0
$243$	1.04585	0.0670913
$244$	0	0
$245$	1.00000	0.0638877
$246$	0	0
$247$	−2.52862	−0.160892
$248$	0	0
$249$	0.264462	0.0167596
$250$	0	0
$251$	−14.1656	−0.894125	−0.447062	−	0.894503i	$-0.647530\pi$
$251$	−14.1656	−0.894125	−0.447062	+	0.894503i	$0.647530\pi$
$252$	0	0
$253$	34.2524	2.15343
$254$	0	0
$255$	0.111551	0.00698560
$256$	0	0
$257$	−7.91720	−0.493861	−0.246931	−	0.969033i	$-0.579422\pi$
$257$	−7.91720	−0.493861	−0.246931	+	0.969033i	$0.579422\pi$
$258$	0	0
$259$	−9.21150	−0.572375
$260$	0	0
$261$	−10.3303	−0.639428
$262$	0	0
$263$	12.1215	0.747444	0.373722	−	0.927541i	$-0.378081\pi$
$263$	12.1215	0.747444	0.373722	+	0.927541i	$0.378081\pi$
$264$	0	0
$265$	−5.06597	−0.311200
$266$	0	0
$267$	0.621500	0.0380352
$268$	0	0
$269$	28.4648	1.73553	0.867765	−	0.496974i	$-0.165556\pi$
$269$	28.4648	1.73553	0.867765	+	0.496974i	$0.165556\pi$
$270$	0	0
$271$	−17.6430	−1.07173	−0.535867	−	0.844302i	$-0.680015\pi$
$271$	−17.6430	−1.07173	−0.535867	+	0.844302i	$0.680015\pi$
$272$	0	0
$273$	−0.0405502	−0.00245421
$274$	0	0
$275$	−4.92253	−0.296840
$276$	0	0
$277$	−2.24899	−0.135129	−0.0675643	−	0.997715i	$-0.521523\pi$
$277$	−2.24899	−0.135129	−0.0675643	+	0.997715i	$0.521523\pi$
$278$	0	0
$279$	14.0181	0.839243
$280$	0	0
$281$	−27.2109	−1.62326	−0.811632	−	0.584169i	$-0.801420\pi$
$281$	−27.2109	−1.62326	−0.811632	+	0.584169i	$0.801420\pi$
$282$	0	0
$283$	10.3847	0.617307	0.308653	−	0.951175i	$-0.400122\pi$
$283$	10.3847	0.617307	0.308653	+	0.951175i	$0.400122\pi$
$284$	0	0
$285$	−0.0936876	−0.00554957
$286$	0	0
$287$	−10.2387	−0.604370
$288$	0	0
$289$	−8.71759	−0.512800
$290$	0	0
$291$	0.116236	0.00681385
$292$	0	0
$293$	3.78580	0.221169	0.110584	−	0.993867i	$-0.464728\pi$
$293$	3.78580	0.221169	0.110584	+	0.993867i	$0.464728\pi$
$294$	0	0
$295$	8.66802	0.504672
$296$	0	0
$297$	1.14453	0.0664123
$298$	0	0
$299$	−7.27948	−0.420983
$300$	0	0
$301$	−1.00000	−0.0576390
$302$	0	0
$303$	−0.0792973	−0.00455551
$304$	0	0
$305$	4.21737	0.241486
$306$	0	0
$307$	−6.80831	−0.388571	−0.194286	−	0.980945i	$-0.562239\pi$
$307$	−6.80831	−0.388571	−0.194286	+	0.980945i	$0.562239\pi$
$308$	0	0
$309$	0.725019	0.0412449
$310$	0	0
$311$	5.84972	0.331707	0.165854	−	0.986150i	$-0.446962\pi$
$311$	5.84972	0.331707	0.165854	+	0.986150i	$0.446962\pi$
$312$	0	0
$313$	12.7526	0.720820	0.360410	−	0.932794i	$-0.382637\pi$
$313$	12.7526	0.720820	0.360410	+	0.932794i	$0.382637\pi$
$314$	0	0
$315$	2.99850	0.168946
$316$	0	0
$317$	−19.7393	−1.10867	−0.554334	−	0.832295i	$-0.687027\pi$
$317$	−19.7393	−1.10867	−0.554334	+	0.832295i	$0.687027\pi$
$318$	0	0
$319$	−16.9589	−0.949514
$320$	0	0
$321$	0.230057	0.0128405
$322$	0	0
$323$	−6.95608	−0.387047
$324$	0	0
$325$	1.04616	0.0580305
$326$	0	0
$327$	−0.378079	−0.0209078
$328$	0	0
$329$	5.98033	0.329706
$330$	0	0
$331$	19.8026	1.08845	0.544226	−	0.838939i	$-0.316823\pi$
$331$	19.8026	1.08845	0.544226	+	0.838939i	$0.316823\pi$
$332$	0	0
$333$	−27.6206	−1.51360
$334$	0	0
$335$	−10.4781	−0.572482
$336$	0	0
$337$	15.5426	0.846659	0.423330	−	0.905976i	$-0.360861\pi$
$337$	15.5426	0.846659	0.423330	+	0.905976i	$0.360861\pi$
$338$	0	0
$339$	0.0893404	0.00485230
$340$	0	0
$341$	23.0131	1.24623
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−0.269711	−0.0145207
$346$	0	0
$347$	−10.1621	−0.545529	−0.272764	−	0.962081i	$-0.587938\pi$
$347$	−10.1621	−0.545529	−0.272764	+	0.962081i	$0.587938\pi$
$348$	0	0
$349$	−11.8889	−0.636398	−0.318199	−	0.948024i	$-0.603078\pi$
$349$	−11.8889	−0.636398	−0.318199	+	0.948024i	$0.603078\pi$
$350$	0	0
$351$	−0.243241	−0.0129832
$352$	0	0
$353$	−3.29265	−0.175250	−0.0876249	−	0.996154i	$-0.527928\pi$
$353$	−3.29265	−0.175250	−0.0876249	+	0.996154i	$0.527928\pi$
$354$	0	0
$355$	−11.3521	−0.602507
$356$	0	0
$357$	−0.111551	−0.00590391
$358$	0	0
$359$	−26.2834	−1.38719	−0.693593	−	0.720367i	$-0.743973\pi$
$359$	−26.2834	−1.38719	−0.693593	+	0.720367i	$0.743973\pi$
$360$	0	0
$361$	−13.1578	−0.692518
$362$	0	0
$363$	0.512861	0.0269182
$364$	0	0
$365$	−0.593172	−0.0310481
$366$	0	0
$367$	−0.423635	−0.0221136	−0.0110568	−	0.999939i	$-0.503520\pi$
$367$	−0.423635	−0.0221136	−0.0110568	+	0.999939i	$0.503520\pi$
$368$	0	0
$369$	−30.7006	−1.59821
$370$	0	0
$371$	5.06597	0.263012
$372$	0	0
$373$	11.7945	0.610695	0.305348	−	0.952241i	$-0.401227\pi$
$373$	11.7945	0.610695	0.305348	+	0.952241i	$0.401227\pi$
$374$	0	0
$375$	0.0387611	0.00200161
$376$	0	0
$377$	3.60417	0.185624
$378$	0	0
$379$	4.18602	0.215022	0.107511	−	0.994204i	$-0.465712\pi$
$379$	4.18602	0.215022	0.107511	+	0.994204i	$0.465712\pi$
$380$	0	0
$381$	−0.417205	−0.0213740
$382$	0	0
$383$	22.6070	1.15516	0.577582	−	0.816332i	$-0.303996\pi$
$383$	22.6070	1.15516	0.577582	+	0.816332i	$0.303996\pi$
$384$	0	0
$385$	4.92253	0.250876
$386$	0	0
$387$	−2.99850	−0.152422
$388$	0	0
$389$	27.2564	1.38195	0.690977	−	0.722877i	$-0.257181\pi$
$389$	27.2564	1.38195	0.690977	+	0.722877i	$0.257181\pi$
$390$	0	0
$391$	−20.0254	−1.01273
$392$	0	0
$393$	0.0759176	0.00382954
$394$	0	0
$395$	17.4617	0.878592
$396$	0	0
$397$	−22.7277	−1.14067	−0.570337	−	0.821411i	$-0.693187\pi$
$397$	−22.7277	−1.14067	−0.570337	+	0.821411i	$0.693187\pi$
$398$	0	0
$399$	0.0936876	0.00469025
$400$	0	0
$401$	11.1738	0.557993	0.278997	−	0.960292i	$-0.409998\pi$
$401$	11.1738	0.557993	0.278997	+	0.960292i	$0.409998\pi$
$402$	0	0
$403$	−4.89085	−0.243630
$404$	0	0
$405$	8.98648	0.446542
$406$	0	0
$407$	−45.3439	−2.24761
$408$	0	0
$409$	−7.76946	−0.384175	−0.192088	−	0.981378i	$-0.561526\pi$
$409$	−7.76946	−0.384175	−0.192088	+	0.981378i	$0.561526\pi$
$410$	0	0
$411$	−0.136185	−0.00671751
$412$	0	0
$413$	−8.66802	−0.426525
$414$	0	0
$415$	6.82289	0.334922
$416$	0	0
$417$	0.569346	0.0278810
$418$	0	0
$419$	12.5760	0.614380	0.307190	−	0.951648i	$-0.400611\pi$
$419$	12.5760	0.614380	0.307190	+	0.951648i	$0.400611\pi$
$420$	0	0
$421$	−19.5650	−0.953540	−0.476770	−	0.879028i	$-0.658193\pi$
$421$	−19.5650	−0.953540	−0.476770	+	0.879028i	$0.658193\pi$
$422$	0	0
$423$	17.9320	0.871884
$424$	0	0
$425$	2.87792	0.139599
$426$	0	0
$427$	−4.21737	−0.204093
$428$	0	0
$429$	−0.199610	−0.00963726
$430$	0	0
$431$	−23.8850	−1.15050	−0.575251	−	0.817977i	$-0.695096\pi$
$431$	−23.8850	−1.15050	−0.575251	+	0.817977i	$0.695096\pi$
$432$	0	0
$433$	27.5398	1.32348	0.661739	−	0.749735i	$-0.269819\pi$
$433$	27.5398	1.32348	0.661739	+	0.749735i	$0.269819\pi$
$434$	0	0
$435$	0.133538	0.00640264
$436$	0	0
$437$	16.8186	0.804541
$438$	0	0
$439$	8.46543	0.404033	0.202016	−	0.979382i	$-0.435251\pi$
$439$	8.46543	0.404033	0.202016	+	0.979382i	$0.435251\pi$
$440$	0	0
$441$	−2.99850	−0.142786
$442$	0	0
$443$	2.42524	0.115227	0.0576134	−	0.998339i	$-0.481651\pi$
$443$	2.42524	0.115227	0.0576134	+	0.998339i	$0.481651\pi$
$444$	0	0
$445$	16.0341	0.760091
$446$	0	0
$447$	−0.459472	−0.0217323
$448$	0	0
$449$	27.3860	1.29242	0.646211	−	0.763158i	$-0.276352\pi$
$449$	27.3860	1.29242	0.646211	+	0.763158i	$0.276352\pi$
$450$	0	0
$451$	−50.4002	−2.37325
$452$	0	0
$453$	0.320259	0.0150471
$454$	0	0
$455$	−1.04616	−0.0490447
$456$	0	0
$457$	19.1066	0.893771	0.446885	−	0.894591i	$-0.352533\pi$
$457$	19.1066	0.893771	0.446885	+	0.894591i	$0.352533\pi$
$458$	0	0
$459$	−0.669139	−0.0312327
$460$	0	0
$461$	−18.2141	−0.848314	−0.424157	−	0.905589i	$-0.639430\pi$
$461$	−18.2141	−0.848314	−0.424157	+	0.905589i	$0.639430\pi$
$462$	0	0
$463$	25.1120	1.16705	0.583527	−	0.812094i	$-0.301672\pi$
$463$	25.1120	1.16705	0.583527	+	0.812094i	$0.301672\pi$
$464$	0	0
$465$	−0.181210	−0.00840341
$466$	0	0
$467$	4.05268	0.187536	0.0937679	−	0.995594i	$-0.470109\pi$
$467$	4.05268	0.187536	0.0937679	+	0.995594i	$0.470109\pi$
$468$	0	0
$469$	10.4781	0.483836
$470$	0	0
$471$	0.288291	0.0132837
$472$	0	0
$473$	−4.92253	−0.226338
$474$	0	0
$475$	−2.41705	−0.110902
$476$	0	0
$477$	15.1903	0.695516
$478$	0	0
$479$	−22.0556	−1.00775	−0.503873	−	0.863778i	$-0.668092\pi$
$479$	−22.0556	−1.00775	−0.503873	+	0.863778i	$0.668092\pi$
$480$	0	0
$481$	9.63669	0.439395
$482$	0	0
$483$	0.269711	0.0122723
$484$	0	0
$485$	2.99877	0.136167
$486$	0	0
$487$	22.2716	1.00922	0.504612	−	0.863346i	$-0.331635\pi$
$487$	22.2716	1.00922	0.504612	+	0.863346i	$0.331635\pi$
$488$	0	0
$489$	−0.255722	−0.0115642
$490$	0	0
$491$	37.4078	1.68819	0.844095	−	0.536194i	$-0.180139\pi$
$491$	37.4078	1.68819	0.844095	+	0.536194i	$0.180139\pi$
$492$	0	0
$493$	9.91485	0.446543
$494$	0	0
$495$	14.7602	0.663422
$496$	0	0
$497$	11.3521	0.509212
$498$	0	0
$499$	10.9157	0.488653	0.244326	−	0.969693i	$-0.421433\pi$
$499$	10.9157	0.488653	0.244326	+	0.969693i	$0.421433\pi$
$500$	0	0
$501$	0.638624	0.0285316
$502$	0	0
$503$	25.0665	1.11766	0.558830	−	0.829282i	$-0.311250\pi$
$503$	25.0665	1.11766	0.558830	+	0.829282i	$0.311250\pi$
$504$	0	0
$505$	−2.04580	−0.0910368
$506$	0	0
$507$	−0.461472	−0.0204947
$508$	0	0
$509$	−11.3246	−0.501956	−0.250978	−	0.967993i	$-0.580752\pi$
$509$	−11.3246	−0.501956	−0.250978	+	0.967993i	$0.580752\pi$
$510$	0	0
$511$	0.593172	0.0262404
$512$	0	0
$513$	0.561985	0.0248122
$514$	0	0
$515$	18.7048	0.824234
$516$	0	0
$517$	29.4384	1.29470
$518$	0	0
$519$	0.0181442	0.000796442 0
$520$	0	0
$521$	16.8834	0.739677	0.369838	−	0.929096i	$-0.379413\pi$
$521$	16.8834	0.739677	0.369838	+	0.929096i	$0.379413\pi$
$522$	0	0
$523$	−13.9353	−0.609348	−0.304674	−	0.952457i	$-0.598548\pi$
$523$	−13.9353	−0.609348	−0.304674	+	0.952457i	$0.598548\pi$
$524$	0	0
$525$	−0.0387611	−0.00169167
$526$	0	0
$527$	−13.4544	−0.586083
$528$	0	0
$529$	25.4178	1.10512
$530$	0	0
$531$	−25.9910	−1.12791
$532$	0	0
$533$	10.7113	0.463957
$534$	0	0
$535$	5.93525	0.256603
$536$	0	0
$537$	−0.476927	−0.0205809
$538$	0	0
$539$	−4.92253	−0.212029
$540$	0	0
$541$	23.9639	1.03029	0.515143	−	0.857104i	$-0.327739\pi$
$541$	23.9639	1.03029	0.515143	+	0.857104i	$0.327739\pi$
$542$	0	0
$543$	0.960826	0.0412330
$544$	0	0
$545$	−9.75409	−0.417819
$546$	0	0
$547$	34.2682	1.46520	0.732601	−	0.680658i	$-0.238306\pi$
$547$	34.2682	1.46520	0.732601	+	0.680658i	$0.238306\pi$
$548$	0	0
$549$	−12.6458	−0.539709
$550$	0	0
$551$	−8.32711	−0.354747
$552$	0	0
$553$	−17.4617	−0.742546
$554$	0	0
$555$	0.357047	0.0151558
$556$	0	0
$557$	35.2616	1.49408	0.747042	−	0.664777i	$-0.231473\pi$
$557$	35.2616	1.49408	0.747042	+	0.664777i	$0.231473\pi$
$558$	0	0
$559$	1.04616	0.0442478
$560$	0	0
$561$	−0.549114	−0.0231836
$562$	0	0
$563$	−36.4749	−1.53723	−0.768616	−	0.639710i	$-0.779054\pi$
$563$	−36.4749	−1.53723	−0.768616	+	0.639710i	$0.779054\pi$
$564$	0	0
$565$	2.30490	0.0969679
$566$	0	0
$567$	−8.98648	−0.377397
$568$	0	0
$569$	−3.40831	−0.142884	−0.0714418	−	0.997445i	$-0.522760\pi$
$569$	−3.40831	−0.142884	−0.0714418	+	0.997445i	$0.522760\pi$
$570$	0	0
$571$	−14.7526	−0.617379	−0.308689	−	0.951163i	$-0.599890\pi$
$571$	−14.7526	−0.617379	−0.308689	+	0.951163i	$0.599890\pi$
$572$	0	0
$573$	0.883734	0.0369185
$574$	0	0
$575$	−6.95829	−0.290181
$576$	0	0
$577$	−31.8179	−1.32460	−0.662299	−	0.749240i	$-0.730419\pi$
$577$	−31.8179	−1.32460	−0.662299	+	0.749240i	$0.730419\pi$
$578$	0	0
$579$	−0.191307	−0.00795043
$580$	0	0
$581$	−6.82289	−0.283061
$582$	0	0
$583$	24.9374	1.03280
$584$	0	0
$585$	−3.13691	−0.129695
$586$	0	0
$587$	45.0101	1.85777	0.928883	−	0.370373i	$-0.120770\pi$
$587$	45.0101	1.85777	0.928883	+	0.370373i	$0.120770\pi$
$588$	0	0
$589$	11.2998	0.465602
$590$	0	0
$591$	−0.249765	−0.0102740
$592$	0	0
$593$	−10.9227	−0.448541	−0.224271	−	0.974527i	$-0.572000\pi$
$593$	−10.9227	−0.448541	−0.224271	+	0.974527i	$0.572000\pi$
$594$	0	0
$595$	−2.87792	−0.117983
$596$	0	0
$597$	0.289496	0.0118483
$598$	0	0
$599$	9.19284	0.375609	0.187805	−	0.982206i	$-0.439863\pi$
$599$	9.19284	0.375609	0.187805	+	0.982206i	$0.439863\pi$
$600$	0	0
$601$	−5.01338	−0.204500	−0.102250	−	0.994759i	$-0.532604\pi$
$601$	−5.01338	−0.204500	−0.102250	+	0.994759i	$0.532604\pi$
$602$	0	0
$603$	31.4187	1.27947
$604$	0	0
$605$	13.2313	0.537931
$606$	0	0
$607$	−25.8168	−1.04787	−0.523936	−	0.851758i	$-0.675537\pi$
$607$	−25.8168	−1.04787	−0.523936	+	0.851758i	$0.675537\pi$
$608$	0	0
$609$	−0.133538	−0.00541122
$610$	0	0
$611$	−6.25638	−0.253106
$612$	0	0
$613$	32.5524	1.31478	0.657389	−	0.753551i	$-0.271661\pi$
$613$	32.5524	1.31478	0.657389	+	0.753551i	$0.271661\pi$
$614$	0	0
$615$	0.396862	0.0160030
$616$	0	0
$617$	16.4221	0.661130	0.330565	−	0.943783i	$-0.392761\pi$
$617$	16.4221	0.661130	0.330565	+	0.943783i	$0.392761\pi$
$618$	0	0
$619$	−25.6478	−1.03087	−0.515436	−	0.856928i	$-0.672370\pi$
$619$	−25.6478	−1.03087	−0.515436	+	0.856928i	$0.672370\pi$
$620$	0	0
$621$	1.61786	0.0649225
$622$	0	0
$623$	−16.0341	−0.642394
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0.461180	0.0184178
$628$	0	0
$629$	26.5099	1.05702
$630$	0	0
$631$	18.5386	0.738008	0.369004	−	0.929428i	$-0.379699\pi$
$631$	18.5386	0.738008	0.369004	+	0.929428i	$0.379699\pi$
$632$	0	0
$633$	0.403803	0.0160497
$634$	0	0
$635$	−10.7635	−0.427136
$636$	0	0
$637$	1.04616	0.0414503
$638$	0	0
$639$	34.0393	1.34657
$640$	0	0
$641$	−41.8895	−1.65454	−0.827268	−	0.561808i	$-0.810106\pi$
$641$	−41.8895	−1.65454	−0.827268	+	0.561808i	$0.810106\pi$
$642$	0	0
$643$	11.4001	0.449575	0.224787	−	0.974408i	$-0.427831\pi$
$643$	11.4001	0.449575	0.224787	+	0.974408i	$0.427831\pi$
$644$	0	0
$645$	0.0387611	0.00152622
$646$	0	0
$647$	−12.2036	−0.479773	−0.239886	−	0.970801i	$-0.577110\pi$
$647$	−12.2036	−0.479773	−0.239886	+	0.970801i	$0.577110\pi$
$648$	0	0
$649$	−42.6686	−1.67489
$650$	0	0
$651$	0.181210	0.00710217
$652$	0	0
$653$	−27.1791	−1.06360	−0.531801	−	0.846869i	$-0.678485\pi$
$653$	−27.1791	−1.06360	−0.531801	+	0.846869i	$0.678485\pi$
$654$	0	0
$655$	1.95860	0.0765290
$656$	0	0
$657$	1.77863	0.0693908
$658$	0	0
$659$	−11.6585	−0.454151	−0.227075	−	0.973877i	$-0.572916\pi$
$659$	−11.6585	−0.454151	−0.227075	+	0.973877i	$0.572916\pi$
$660$	0	0
$661$	2.65248	0.103169	0.0515847	−	0.998669i	$-0.483573\pi$
$661$	2.65248	0.103169	0.0515847	+	0.998669i	$0.483573\pi$
$662$	0	0
$663$	0.116700	0.00453226
$664$	0	0
$665$	2.41705	0.0937293
$666$	0	0
$667$	−23.9723	−0.928213
$668$	0	0
$669$	1.11412	0.0430743
$670$	0	0
$671$	−20.7602	−0.801437
$672$	0	0
$673$	23.4083	0.902324	0.451162	−	0.892442i	$-0.351010\pi$
$673$	23.4083	0.902324	0.451162	+	0.892442i	$0.351010\pi$
$674$	0	0
$675$	−0.232508	−0.00894924
$676$	0	0
$677$	4.17204	0.160345	0.0801723	−	0.996781i	$-0.474453\pi$
$677$	4.17204	0.160345	0.0801723	+	0.996781i	$0.474453\pi$
$678$	0	0
$679$	−2.99877	−0.115082
$680$	0	0
$681$	0.0503755	0.00193040
$682$	0	0
$683$	25.2857	0.967532	0.483766	−	0.875198i	$-0.339269\pi$
$683$	25.2857	0.967532	0.483766	+	0.875198i	$0.339269\pi$
$684$	0	0
$685$	−3.51345	−0.134242
$686$	0	0
$687$	−0.0716542	−0.00273378
$688$	0	0
$689$	−5.29981	−0.201907
$690$	0	0
$691$	22.8688	0.869969	0.434984	−	0.900438i	$-0.356754\pi$
$691$	22.8688	0.869969	0.434984	+	0.900438i	$0.356754\pi$
$692$	0	0
$693$	−14.7602	−0.560694
$694$	0	0
$695$	14.6886	0.557171
$696$	0	0
$697$	29.4660	1.11611
$698$	0	0
$699$	0.498329	0.0188485
$700$	0	0
$701$	−30.1638	−1.13927	−0.569635	−	0.821898i	$-0.692915\pi$
$701$	−30.1638	−1.13927	−0.569635	+	0.821898i	$0.692915\pi$
$702$	0	0
$703$	−22.2647	−0.839729
$704$	0	0
$705$	−0.231804	−0.00873024
$706$	0	0
$707$	2.04580	0.0769402
$708$	0	0
$709$	9.35776	0.351438	0.175719	−	0.984440i	$-0.443775\pi$
$709$	9.35776	0.351438	0.175719	+	0.984440i	$0.443775\pi$
$710$	0	0
$711$	−52.3588	−1.96361
$712$	0	0
$713$	32.5304	1.21827
$714$	0	0
$715$	−5.14976	−0.192590
$716$	0	0
$717$	−0.867523	−0.0323982
$718$	0	0
$719$	10.4253	0.388800	0.194400	−	0.980922i	$-0.437724\pi$
$719$	10.4253	0.388800	0.194400	+	0.980922i	$0.437724\pi$
$720$	0	0
$721$	−18.7048	−0.696605
$722$	0	0
$723$	−0.913047	−0.0339566
$724$	0	0
$725$	3.44515	0.127950
$726$	0	0
$727$	1.93824	0.0718852	0.0359426	−	0.999354i	$-0.488557\pi$
$727$	1.93824	0.0718852	0.0359426	+	0.999354i	$0.488557\pi$
$728$	0	0
$729$	−26.9189	−0.996996
$730$	0	0
$731$	2.87792	0.106444
$732$	0	0
$733$	45.3772	1.67605	0.838023	−	0.545635i	$-0.183711\pi$
$733$	45.3772	1.67605	0.838023	+	0.545635i	$0.183711\pi$
$734$	0	0
$735$	0.0387611	0.00142972
$736$	0	0
$737$	51.5790	1.89994
$738$	0	0
$739$	22.6894	0.834644	0.417322	−	0.908759i	$-0.362969\pi$
$739$	22.6894	0.834644	0.417322	+	0.908759i	$0.362969\pi$
$740$	0	0
$741$	−0.0980121	−0.00360057
$742$	0	0
$743$	40.5443	1.48743	0.743713	−	0.668499i	$-0.233063\pi$
$743$	40.5443	1.48743	0.743713	+	0.668499i	$0.233063\pi$
$744$	0	0
$745$	−11.8540	−0.434296
$746$	0	0
$747$	−20.4584	−0.748534
$748$	0	0
$749$	−5.93525	−0.216869
$750$	0	0
$751$	18.9196	0.690384	0.345192	−	0.938532i	$-0.387814\pi$
$751$	18.9196	0.690384	0.345192	+	0.938532i	$0.387814\pi$
$752$	0	0
$753$	−0.549074	−0.0200094
$754$	0	0
$755$	8.26240	0.300700
$756$	0	0
$757$	−18.9774	−0.689746	−0.344873	−	0.938649i	$-0.612078\pi$
$757$	−18.9774	−0.689746	−0.344873	+	0.938649i	$0.612078\pi$
$758$	0	0
$759$	1.32766	0.0481910
$760$	0	0
$761$	13.1488	0.476644	0.238322	−	0.971186i	$-0.423403\pi$
$761$	13.1488	0.476644	0.238322	+	0.971186i	$0.423403\pi$
$762$	0	0
$763$	9.75409	0.353122
$764$	0	0
$765$	−8.62943	−0.311998
$766$	0	0
$767$	9.06813	0.327431
$768$	0	0
$769$	−44.7180	−1.61257	−0.806286	−	0.591525i	$-0.798526\pi$
$769$	−44.7180	−1.61257	−0.806286	+	0.591525i	$0.798526\pi$
$770$	0	0
$771$	−0.306879	−0.0110520
$772$	0	0
$773$	14.2988	0.514293	0.257146	−	0.966372i	$-0.417218\pi$
$773$	14.2988	0.514293	0.257146	+	0.966372i	$0.417218\pi$
$774$	0	0
$775$	−4.67505	−0.167933
$776$	0	0
$777$	−0.357047	−0.0128090
$778$	0	0
$779$	−24.7474	−0.886668
$780$	0	0
$781$	55.8812	1.99959
$782$	0	0
$783$	−0.801025	−0.0286263
$784$	0	0
$785$	7.43764	0.265461
$786$	0	0
$787$	15.0828	0.537644	0.268822	−	0.963190i	$-0.413366\pi$
$787$	15.0828	0.537644	0.268822	+	0.963190i	$0.413366\pi$
$788$	0	0
$789$	0.469843	0.0167268
$790$	0	0
$791$	−2.30490	−0.0819528
$792$	0	0
$793$	4.41204	0.156676
$794$	0	0
$795$	−0.196362	−0.00696426
$796$	0	0
$797$	−16.4393	−0.582310	−0.291155	−	0.956676i	$-0.594040\pi$
$797$	−16.4393	−0.582310	−0.291155	+	0.956676i	$0.594040\pi$
$798$	0	0
$799$	−17.2109	−0.608878
$800$	0	0
$801$	−48.0783	−1.69876
$802$	0	0
$803$	2.91991	0.103041
$804$	0	0
$805$	6.95829	0.245248
$806$	0	0
$807$	1.10333	0.0388389
$808$	0	0
$809$	30.4243	1.06966	0.534832	−	0.844959i	$-0.320375\pi$
$809$	30.4243	1.06966	0.534832	+	0.844959i	$0.320375\pi$
$810$	0	0
$811$	−40.4708	−1.42112	−0.710562	−	0.703635i	$-0.751559\pi$
$811$	−40.4708	−1.42112	−0.710562	+	0.703635i	$0.751559\pi$
$812$	0	0
$813$	−0.683860	−0.0239840
$814$	0	0
$815$	−6.59740	−0.231097
$816$	0	0
$817$	−2.41705	−0.0845620
$818$	0	0
$819$	3.13691	0.109612
$820$	0	0
$821$	13.2774	0.463384	0.231692	−	0.972789i	$-0.425574\pi$
$821$	13.2774	0.463384	0.231692	+	0.972789i	$0.425574\pi$
$822$	0	0
$823$	1.94383	0.0677575	0.0338788	−	0.999426i	$-0.489214\pi$
$823$	1.94383	0.0677575	0.0338788	+	0.999426i	$0.489214\pi$
$824$	0	0
$825$	−0.190803	−0.00664290
$826$	0	0
$827$	−34.1419	−1.18723	−0.593615	−	0.804749i	$-0.702300\pi$
$827$	−34.1419	−1.18723	−0.593615	+	0.804749i	$0.702300\pi$
$828$	0	0
$829$	25.9144	0.900045	0.450023	−	0.893017i	$-0.351416\pi$
$829$	25.9144	0.900045	0.450023	+	0.893017i	$0.351416\pi$
$830$	0	0
$831$	−0.0871732	−0.00302400
$832$	0	0
$833$	2.87792	0.0997139
$834$	0	0
$835$	16.4759	0.570173
$836$	0	0
$837$	1.08699	0.0375718
$838$	0	0
$839$	16.0794	0.555124	0.277562	−	0.960708i	$-0.410474\pi$
$839$	16.0794	0.555124	0.277562	+	0.960708i	$0.410474\pi$
$840$	0	0
$841$	−17.1310	−0.590722
$842$	0	0
$843$	−1.05472	−0.0363265
$844$	0	0
$845$	−11.9056	−0.409563
$846$	0	0
$847$	−13.2313	−0.454634
$848$	0	0
$849$	0.402522	0.0138145
$850$	0	0
$851$	−64.0963	−2.19719
$852$	0	0
$853$	−27.8007	−0.951877	−0.475939	−	0.879479i	$-0.657892\pi$
$853$	−27.8007	−0.951877	−0.475939	+	0.879479i	$0.657892\pi$
$854$	0	0
$855$	7.24753	0.247860
$856$	0	0
$857$	−58.0243	−1.98207	−0.991036	−	0.133595i	$-0.957348\pi$
$857$	−58.0243	−1.98207	−0.991036	+	0.133595i	$0.957348\pi$
$858$	0	0
$859$	−1.35345	−0.0461792	−0.0230896	−	0.999733i	$-0.507350\pi$
$859$	−1.35345	−0.0461792	−0.0230896	+	0.999733i	$0.507350\pi$
$860$	0	0
$861$	−0.396862	−0.0135250
$862$	0	0
$863$	4.28612	0.145901	0.0729506	−	0.997336i	$-0.476758\pi$
$863$	4.28612	0.145901	0.0729506	+	0.997336i	$0.476758\pi$
$864$	0	0
$865$	0.468104	0.0159160
$866$	0	0
$867$	−0.337903	−0.0114758
$868$	0	0
$869$	−85.9557	−2.91585
$870$	0	0
$871$	−10.9618	−0.371427
$872$	0	0
$873$	−8.99182	−0.304327
$874$	0	0
$875$	−1.00000	−0.0338062
$876$	0	0
$877$	−41.3198	−1.39527	−0.697636	−	0.716453i	$-0.745764\pi$
$877$	−41.3198	−1.39527	−0.697636	+	0.716453i	$0.745764\pi$
$878$	0	0
$879$	0.146742	0.00494947
$880$	0	0
$881$	−3.38514	−0.114048	−0.0570241	−	0.998373i	$-0.518161\pi$
$881$	−3.38514	−0.114048	−0.0570241	+	0.998373i	$0.518161\pi$
$882$	0	0
$883$	2.17698	0.0732611	0.0366306	−	0.999329i	$-0.488338\pi$
$883$	2.17698	0.0732611	0.0366306	+	0.999329i	$0.488338\pi$
$884$	0	0
$885$	0.335982	0.0112939
$886$	0	0
$887$	−54.9507	−1.84506	−0.922532	−	0.385921i	$-0.873884\pi$
$887$	−54.9507	−1.84506	−0.922532	+	0.385921i	$0.873884\pi$
$888$	0	0
$889$	10.7635	0.360996
$890$	0	0
$891$	−44.2363	−1.48197
$892$	0	0
$893$	14.4548	0.483711
$894$	0	0
$895$	−12.3043	−0.411287
$896$	0	0
$897$	−0.282160	−0.00942106
$898$	0	0
$899$	−16.1062	−0.537173
$900$	0	0
$901$	−14.5794	−0.485712
$902$	0	0
$903$	−0.0387611	−0.00128989
$904$	0	0
$905$	24.7884	0.823995
$906$	0	0
$907$	−25.5068	−0.846939	−0.423470	−	0.905910i	$-0.639188\pi$
$907$	−25.5068	−0.846939	−0.423470	+	0.905910i	$0.639188\pi$
$908$	0	0
$909$	6.13432	0.203463
$910$	0	0
$911$	−1.33111	−0.0441017	−0.0220509	−	0.999757i	$-0.507020\pi$
$911$	−1.33111	−0.0441017	−0.0220509	+	0.999757i	$0.507020\pi$
$912$	0	0
$913$	−33.5859	−1.11153
$914$	0	0
$915$	0.163470	0.00540414
$916$	0	0
$917$	−1.95860	−0.0646788
$918$	0	0
$919$	45.7731	1.50992	0.754958	−	0.655773i	$-0.227657\pi$
$919$	45.7731	1.50992	0.754958	+	0.655773i	$0.227657\pi$
$920$	0	0
$921$	−0.263897	−0.00869572
$922$	0	0
$923$	−11.8761	−0.390907
$924$	0	0
$925$	9.21150	0.302872
$926$	0	0
$927$	−56.0864	−1.84212
$928$	0	0
$929$	18.4308	0.604696	0.302348	−	0.953198i	$-0.402229\pi$
$929$	18.4308	0.604696	0.302348	+	0.953198i	$0.402229\pi$
$930$	0	0
$931$	−2.41705	−0.0792158
$932$	0	0
$933$	0.226741	0.00742318
$934$	0	0
$935$	−14.1666	−0.463299
$936$	0	0
$937$	32.7186	1.06887	0.534435	−	0.845210i	$-0.320524\pi$
$937$	32.7186	1.06887	0.534435	+	0.845210i	$0.320524\pi$
$938$	0	0
$939$	0.494304	0.0161310
$940$	0	0
$941$	8.89362	0.289924	0.144962	−	0.989437i	$-0.453694\pi$
$941$	8.89362	0.289924	0.144962	+	0.989437i	$0.453694\pi$
$942$	0	0
$943$	−71.2437	−2.32001
$944$	0	0
$945$	0.232508	0.00756349
$946$	0	0
$947$	22.1053	0.718326	0.359163	−	0.933275i	$-0.383062\pi$
$947$	22.1053	0.718326	0.359163	+	0.933275i	$0.383062\pi$
$948$	0	0
$949$	−0.620553	−0.0201440
$950$	0	0
$951$	−0.765114	−0.0248105
$952$	0	0
$953$	1.43866	0.0466026	0.0233013	−	0.999728i	$-0.492582\pi$
$953$	1.43866	0.0466026	0.0233013	+	0.999728i	$0.492582\pi$
$954$	0	0
$955$	22.7995	0.737775
$956$	0	0
$957$	−0.657344	−0.0212489
$958$	0	0
$959$	3.51345	0.113455
$960$	0	0
$961$	−9.14391	−0.294965
$962$	0	0
$963$	−17.7968	−0.573495
$964$	0	0
$965$	−4.93553	−0.158880
$966$	0	0
$967$	−52.3807	−1.68445	−0.842224	−	0.539127i	$-0.818754\pi$
$967$	−52.3807	−1.68445	−0.842224	+	0.539127i	$0.818754\pi$
$968$	0	0
$969$	−0.269625	−0.00866161
$970$	0	0
$971$	−30.7394	−0.986474	−0.493237	−	0.869895i	$-0.664186\pi$
$971$	−30.7394	−0.986474	−0.493237	+	0.869895i	$0.664186\pi$
$972$	0	0
$973$	−14.6886	−0.470895
$974$	0	0
$975$	0.0405502	0.00129865
$976$	0	0
$977$	39.4898	1.26339	0.631695	−	0.775217i	$-0.282360\pi$
$977$	39.4898	1.26339	0.631695	+	0.775217i	$0.282360\pi$
$978$	0	0
$979$	−78.9286	−2.52257
$980$	0	0
$981$	29.2476	0.933805
$982$	0	0
$983$	−7.69955	−0.245578	−0.122789	−	0.992433i	$-0.539184\pi$
$983$	−7.69955	−0.245578	−0.122789	+	0.992433i	$0.539184\pi$
$984$	0	0
$985$	−6.44372	−0.205314
$986$	0	0
$987$	0.231804	0.00737840
$988$	0	0
$989$	−6.95829	−0.221261
$990$	0	0
$991$	−45.2163	−1.43634	−0.718172	−	0.695866i	$-0.755021\pi$
$991$	−45.2163	−1.43634	−0.718172	+	0.695866i	$0.755021\pi$
$992$	0	0
$993$	0.767571	0.0243581
$994$	0	0
$995$	7.46874	0.236775
$996$	0	0
$997$	31.9785	1.01277	0.506385	−	0.862307i	$-0.330981\pi$
$997$	31.9785	1.01277	0.506385	+	0.862307i	$0.330981\pi$
$998$	0	0
$999$	−2.14175	−0.0677619

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6020.2.a.k.1.7	✓	13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6020.2.a.k.1.7	✓	13	1.1	even	1	trivial

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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)

\(f(q)\)	\(=\)	\(q+0.0387611 q^{3} +1.00000 q^{5} -1.00000 q^{7} -2.99850 q^{9} +O(q^{10})\)	\(q+0.0387611 q^{3} +1.00000 q^{5} -1.00000 q^{7} -2.99850 q^{9} -4.92253 q^{11} +1.04616 q^{13} +0.0387611 q^{15} +2.87792 q^{17} -2.41705 q^{19} -0.0387611 q^{21} -6.95829 q^{23} +1.00000 q^{25} -0.232508 q^{27} +3.44515 q^{29} -4.67505 q^{31} -0.190803 q^{33} -1.00000 q^{35} +9.21150 q^{37} +0.0405502 q^{39} +10.2387 q^{41} +1.00000 q^{43} -2.99850 q^{45} -5.98033 q^{47} +1.00000 q^{49} +0.111551 q^{51} -5.06597 q^{53} -4.92253 q^{55} -0.0936876 q^{57} +8.66802 q^{59} +4.21737 q^{61} +2.99850 q^{63} +1.04616 q^{65} -10.4781 q^{67} -0.269711 q^{69} -11.3521 q^{71} -0.593172 q^{73} +0.0387611 q^{75} +4.92253 q^{77} +17.4617 q^{79} +8.98648 q^{81} +6.82289 q^{83} +2.87792 q^{85} +0.133538 q^{87} +16.0341 q^{89} -1.04616 q^{91} -0.181210 q^{93} -2.41705 q^{95} +2.99877 q^{97} +14.7602 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q + 13 q^{5} - 13 q^{7} + 15 q^{9}+O(q^{10}) \) 13 * q + 13 * q^5 - 13 * q^7 + 15 * q^9	\( 13 q + 13 q^{5} - 13 q^{7} + 15 q^{9} + 11 q^{13} + 16 q^{17} + 3 q^{23} + 13 q^{25} + 6 q^{27} + 10 q^{29} - q^{31} + 14 q^{33} - 13 q^{35} + 16 q^{37} - 14 q^{39} + 23 q^{41} + 13 q^{43} + 15 q^{45} + 2 q^{47} + 13 q^{49} + 4 q^{51} + 20 q^{53} + 22 q^{57} + 2 q^{59} + 5 q^{61} - 15 q^{63} + 11 q^{65} + 19 q^{67} + 16 q^{69} + 4 q^{71} + 34 q^{73} - 15 q^{79} + 17 q^{81} + 27 q^{83} + 16 q^{85} - 5 q^{87} + 3 q^{89} - 11 q^{91} + 35 q^{93} + 45 q^{97} + q^{99}+O(q^{100}) \) 13 * q + 13 * q^5 - 13 * q^7 + 15 * q^9 + 11 * q^13 + 16 * q^17 + 3 * q^23 + 13 * q^25 + 6 * q^27 + 10 * q^29 - q^31 + 14 * q^33 - 13 * q^35 + 16 * q^37 - 14 * q^39 + 23 * q^41 + 13 * q^43 + 15 * q^45 + 2 * q^47 + 13 * q^49 + 4 * q^51 + 20 * q^53 + 22 * q^57 + 2 * q^59 + 5 * q^61 - 15 * q^63 + 11 * q^65 + 19 * q^67 + 16 * q^69 + 4 * q^71 + 34 * q^73 - 15 * q^79 + 17 * q^81 + 27 * q^83 + 16 * q^85 - 5 * q^87 + 3 * q^89 - 11 * q^91 + 35 * q^93 + 45 * q^97 + q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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