LMFDB - Embedded newform 6020.2.a.j.1.10

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} - x^{12} - 28 x^{11} + 26 x^{10} + 286 x^{9} - 235 x^{8} - 1298 x^{7} + 895 x^{6} + 2571 x^{5} + \cdots - 216 $ x^13 - x^12 - 28x^11 + 26x^10 + 286x^9 - 235x^8 - 1298x^7 + 895x^6 + 2571x^5 - 1548x^4 - 1833x^3 + 1119x^2 + 342*x - 216
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			1.10
Root			$-1.84160$ 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)

$f(q)$	$=$	$q+1.84160 q^{3} -1.00000 q^{5} -1.00000 q^{7} +0.391491 q^{9} +O(q^{10})$	$q+1.84160 q^{3} -1.00000 q^{5} -1.00000 q^{7} +0.391491 q^{9} -1.47566 q^{11} +6.74267 q^{13} -1.84160 q^{15} -5.81313 q^{17} +1.33183 q^{19} -1.84160 q^{21} -3.05218 q^{23} +1.00000 q^{25} -4.80383 q^{27} -0.185024 q^{29} +3.25967 q^{31} -2.71758 q^{33} +1.00000 q^{35} +11.1891 q^{37} +12.4173 q^{39} -3.49671 q^{41} -1.00000 q^{43} -0.391491 q^{45} +3.86398 q^{47} +1.00000 q^{49} -10.7055 q^{51} +4.12550 q^{53} +1.47566 q^{55} +2.45270 q^{57} +6.32037 q^{59} +12.7854 q^{61} -0.391491 q^{63} -6.74267 q^{65} +3.94432 q^{67} -5.62089 q^{69} +1.77796 q^{71} +11.3861 q^{73} +1.84160 q^{75} +1.47566 q^{77} +14.5712 q^{79} -10.0212 q^{81} -8.32132 q^{83} +5.81313 q^{85} -0.340740 q^{87} +14.4983 q^{89} -6.74267 q^{91} +6.00300 q^{93} -1.33183 q^{95} +17.4278 q^{97} -0.577708 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 13 q - q^{3} - 13 q^{5} - 13 q^{7} + 18 q^{9}+O(q^{10}) $ 13 * q - q^3 - 13 * q^5 - 13 * q^7 + 18 * q^9	$ 13 q - q^{3} - 13 q^{5} - 13 q^{7} + 18 q^{9} + 6 q^{11} + q^{13} + q^{15} - 6 q^{17} + 2 q^{19} + q^{21} - 6 q^{23} + 13 q^{25} - q^{27} + 19 q^{29} - 24 q^{31} + 17 q^{33} + 13 q^{35} + 15 q^{39} + 4 q^{41} - 13 q^{43} - 18 q^{45} - 3 q^{47} + 13 q^{49} + 7 q^{51} - 5 q^{53} - 6 q^{55} + 6 q^{57} - 6 q^{59} + 3 q^{61} - 18 q^{63} - q^{65} - 2 q^{67} + 20 q^{69} + 18 q^{71} + 14 q^{73} - q^{75} - 6 q^{77} + 12 q^{79} + 37 q^{81} + 2 q^{83} + 6 q^{85} - 2 q^{87} + 17 q^{89} - q^{91} + 15 q^{93} - 2 q^{95} + 17 q^{97} + 80 q^{99}+O(q^{100}) $ 13 * q - q^3 - 13 * q^5 - 13 * q^7 + 18 * q^9 + 6 * q^11 + q^13 + q^15 - 6 * q^17 + 2 * q^19 + q^21 - 6 * q^23 + 13 * q^25 - q^27 + 19 * q^29 - 24 * q^31 + 17 * q^33 + 13 * q^35 + 15 * q^39 + 4 * q^41 - 13 * q^43 - 18 * q^45 - 3 * q^47 + 13 * q^49 + 7 * q^51 - 5 * q^53 - 6 * q^55 + 6 * q^57 - 6 * q^59 + 3 * q^61 - 18 * q^63 - q^65 - 2 * q^67 + 20 * q^69 + 18 * q^71 + 14 * q^73 - q^75 - 6 * q^77 + 12 * q^79 + 37 * q^81 + 2 * q^83 + 6 * q^85 - 2 * q^87 + 17 * q^89 - q^91 + 15 * q^93 - 2 * q^95 + 17 * q^97 + 80 * 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
$3$	1.84160	1.06325	0.531624	−	0.846980i	$-0.321582\pi$
$3$	1.84160	1.06325	0.531624	+	0.846980i	$0.321582\pi$
$4$	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
$9$	0.391491	0.130497
$10$	0	0
$11$	−1.47566	−0.444929	−0.222464	−	0.974941i	$-0.571410\pi$
$11$	−1.47566	−0.444929	−0.222464	+	0.974941i	$0.571410\pi$
$12$	0	0
$13$	6.74267	1.87008	0.935040	−	0.354543i	$-0.115363\pi$
$13$	6.74267	1.87008	0.935040	+	0.354543i	$0.115363\pi$
$14$	0	0
$15$	−1.84160	−0.475499
$16$	0	0
$17$	−5.81313	−1.40989	−0.704946	−	0.709261i	$-0.749029\pi$
$17$	−5.81313	−1.40989	−0.704946	+	0.709261i	$0.749029\pi$
$18$	0	0
$19$	1.33183	0.305543	0.152771	−	0.988262i	$-0.451180\pi$
$19$	1.33183	0.305543	0.152771	+	0.988262i	$0.451180\pi$
$20$	0	0
$21$	−1.84160	−0.401870
$22$	0	0
$23$	−3.05218	−0.636423	−0.318211	−	0.948020i	$-0.603082\pi$
$23$	−3.05218	−0.636423	−0.318211	+	0.948020i	$0.603082\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−4.80383	−0.924498
$28$	0	0
$29$	−0.185024	−0.0343581	−0.0171790	−	0.999852i	$-0.505469\pi$
$29$	−0.185024	−0.0343581	−0.0171790	+	0.999852i	$0.505469\pi$
$30$	0	0
$31$	3.25967	0.585453	0.292727	−	0.956196i	$-0.405437\pi$
$31$	3.25967	0.585453	0.292727	+	0.956196i	$0.405437\pi$
$32$	0	0
$33$	−2.71758	−0.473070
$34$	0	0
$35$	1.00000	0.169031
$36$	0	0
$37$	11.1891	1.83948	0.919740	−	0.392529i	$-0.128400\pi$
$37$	11.1891	1.83948	0.919740	+	0.392529i	$0.128400\pi$
$38$	0	0
$39$	12.4173	1.98836
$40$	0	0
$41$	−3.49671	−0.546094	−0.273047	−	0.962001i	$-0.588031\pi$
$41$	−3.49671	−0.546094	−0.273047	+	0.962001i	$0.588031\pi$
$42$	0	0
$43$	−1.00000	−0.152499
$43$	−1.00000	−0.152499
$44$	0	0
$45$	−0.391491	−0.0583600
$46$	0	0
$47$	3.86398	0.563619	0.281810	−	0.959470i	$-0.409065\pi$
$47$	3.86398	0.563619	0.281810	+	0.959470i	$0.409065\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−10.7055	−1.49906
$52$	0	0
$53$	4.12550	0.566681	0.283340	−	0.959019i	$-0.408557\pi$
$53$	4.12550	0.566681	0.283340	+	0.959019i	$0.408557\pi$
$54$	0	0
$55$	1.47566	0.198978
$56$	0	0
$57$	2.45270	0.324868
$58$	0	0
$59$	6.32037	0.822842	0.411421	−	0.911446i	$-0.365033\pi$
$59$	6.32037	0.822842	0.411421	+	0.911446i	$0.365033\pi$
$60$	0	0
$61$	12.7854	1.63701	0.818503	−	0.574503i	$-0.194805\pi$
$61$	12.7854	1.63701	0.818503	+	0.574503i	$0.194805\pi$
$62$	0	0
$63$	−0.391491	−0.0493232
$64$	0	0
$65$	−6.74267	−0.836325
$66$	0	0
$67$	3.94432	0.481876	0.240938	−	0.970541i	$-0.422545\pi$
$67$	3.94432	0.481876	0.240938	+	0.970541i	$0.422545\pi$
$68$	0	0
$69$	−5.62089	−0.676675
$70$	0	0
$71$	1.77796	0.211005	0.105503	−	0.994419i	$-0.466355\pi$
$71$	1.77796	0.211005	0.105503	+	0.994419i	$0.466355\pi$
$72$	0	0
$73$	11.3861	1.33264	0.666321	−	0.745665i	$-0.267868\pi$
$73$	11.3861	1.33264	0.666321	+	0.745665i	$0.267868\pi$
$74$	0	0
$75$	1.84160	0.212650
$76$	0	0
$77$	1.47566	0.168167
$78$	0	0
$79$	14.5712	1.63939	0.819696	−	0.572799i	$-0.194143\pi$
$79$	14.5712	1.63939	0.819696	+	0.572799i	$0.194143\pi$
$80$	0	0
$81$	−10.0212	−1.11347
$82$	0	0
$83$	−8.32132	−0.913383	−0.456692	−	0.889625i	$-0.650966\pi$
$83$	−8.32132	−0.913383	−0.456692	+	0.889625i	$0.650966\pi$
$84$	0	0
$85$	5.81313	0.630523
$86$	0	0
$87$	−0.340740	−0.0365312
$88$	0	0
$89$	14.4983	1.53682	0.768410	−	0.639958i	$-0.221048\pi$
$89$	14.4983	1.53682	0.768410	+	0.639958i	$0.221048\pi$
$90$	0	0
$91$	−6.74267	−0.706824
$92$	0	0
$93$	6.00300	0.622482
$94$	0	0
$95$	−1.33183	−0.136643
$96$	0	0
$97$	17.4278	1.76953	0.884765	−	0.466038i	$-0.154319\pi$
$97$	17.4278	1.76953	0.884765	+	0.466038i	$0.154319\pi$
$98$	0	0
$99$	−0.577708	−0.0580619
$100$	0	0
$101$	−18.1511	−1.80611	−0.903053	−	0.429529i	$-0.858679\pi$
$101$	−18.1511	−1.80611	−0.903053	+	0.429529i	$0.858679\pi$
$102$	0	0
$103$	−17.2442	−1.69913	−0.849563	−	0.527487i	$-0.823134\pi$
$103$	−17.2442	−1.69913	−0.849563	+	0.527487i	$0.823134\pi$
$104$	0	0
$105$	1.84160	0.179722
$106$	0	0
$107$	11.6179	1.12315	0.561574	−	0.827427i	$-0.310196\pi$
$107$	11.6179	1.12315	0.561574	+	0.827427i	$0.310196\pi$
$108$	0	0
$109$	7.41900	0.710611	0.355306	−	0.934750i	$-0.384377\pi$
$109$	7.41900	0.710611	0.355306	+	0.934750i	$0.384377\pi$
$110$	0	0
$111$	20.6059	1.95582
$112$	0	0
$113$	−0.518613	−0.0487870	−0.0243935	−	0.999702i	$-0.507765\pi$
$113$	−0.518613	−0.0487870	−0.0243935	+	0.999702i	$0.507765\pi$
$114$	0	0
$115$	3.05218	0.284617
$116$	0	0
$117$	2.63969	0.244040
$118$	0	0
$119$	5.81313	0.532889
$120$	0	0
$121$	−8.82242	−0.802039
$122$	0	0
$123$	−6.43953	−0.580633
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	0.908152	0.0805855	0.0402927	−	0.999188i	$-0.487171\pi$
$127$	0.908152	0.0805855	0.0402927	+	0.999188i	$0.487171\pi$
$128$	0	0
$129$	−1.84160	−0.162144
$130$	0	0
$131$	4.35573	0.380562	0.190281	−	0.981730i	$-0.439060\pi$
$131$	4.35573	0.380562	0.190281	+	0.981730i	$0.439060\pi$
$132$	0	0
$133$	−1.33183	−0.115484
$134$	0	0
$135$	4.80383	0.413448
$136$	0	0
$137$	−18.0021	−1.53802	−0.769012	−	0.639235i	$-0.779251\pi$
$137$	−18.0021	−1.53802	−0.769012	+	0.639235i	$0.779251\pi$
$138$	0	0
$139$	−11.6020	−0.984069	−0.492034	−	0.870576i	$-0.663747\pi$
$139$	−11.6020	−0.984069	−0.492034	+	0.870576i	$0.663747\pi$
$140$	0	0
$141$	7.11591	0.599267
$142$	0	0
$143$	−9.94989	−0.832052
$144$	0	0
$145$	0.185024	0.0153654
$146$	0	0
$147$	1.84160	0.151893
$148$	0	0
$149$	−12.8330	−1.05132	−0.525661	−	0.850694i	$-0.676182\pi$
$149$	−12.8330	−1.05132	−0.525661	+	0.850694i	$0.676182\pi$
$150$	0	0
$151$	3.22950	0.262813	0.131406	−	0.991329i	$-0.458051\pi$
$151$	3.22950	0.262813	0.131406	+	0.991329i	$0.458051\pi$
$152$	0	0
$153$	−2.27579	−0.183987
$154$	0	0
$155$	−3.25967	−0.261823
$156$	0	0
$157$	7.74649	0.618237	0.309118	−	0.951024i	$-0.399966\pi$
$157$	7.74649	0.618237	0.309118	+	0.951024i	$0.399966\pi$
$158$	0	0
$159$	7.59752	0.602522
$160$	0	0
$161$	3.05218	0.240545
$162$	0	0
$163$	−10.4775	−0.820659	−0.410329	−	0.911937i	$-0.634586\pi$
$163$	−10.4775	−0.820659	−0.410329	+	0.911937i	$0.634586\pi$
$164$	0	0
$165$	2.71758	0.211563
$166$	0	0
$167$	8.46712	0.655205	0.327603	−	0.944816i	$-0.393759\pi$
$167$	8.46712	0.655205	0.327603	+	0.944816i	$0.393759\pi$
$168$	0	0
$169$	32.4636	2.49720
$170$	0	0
$171$	0.521400	0.0398724
$172$	0	0
$173$	11.7531	0.893575	0.446787	−	0.894640i	$-0.352568\pi$
$173$	11.7531	0.893575	0.446787	+	0.894640i	$0.352568\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	0	0
$177$	11.6396	0.874885
$178$	0	0
$179$	−5.08908	−0.380376	−0.190188	−	0.981748i	$-0.560910\pi$
$179$	−5.08908	−0.380376	−0.190188	+	0.981748i	$0.560910\pi$
$180$	0	0
$181$	−0.314710	−0.0233922	−0.0116961	−	0.999932i	$-0.503723\pi$
$181$	−0.314710	−0.0233922	−0.0116961	+	0.999932i	$0.503723\pi$
$182$	0	0
$183$	23.5456	1.74054
$184$	0	0
$185$	−11.1891	−0.822640
$186$	0	0
$187$	8.57821	0.627301
$188$	0	0
$189$	4.80383	0.349427
$190$	0	0
$191$	−25.1504	−1.81982	−0.909910	−	0.414806i	$-0.863850\pi$
$191$	−25.1504	−1.81982	−0.909910	+	0.414806i	$0.863850\pi$
$192$	0	0
$193$	4.16540	0.299832	0.149916	−	0.988699i	$-0.452100\pi$
$193$	4.16540	0.299832	0.149916	+	0.988699i	$0.452100\pi$
$194$	0	0
$195$	−12.4173	−0.889221
$196$	0	0
$197$	6.02221	0.429065	0.214532	−	0.976717i	$-0.431177\pi$
$197$	6.02221	0.429065	0.214532	+	0.976717i	$0.431177\pi$
$198$	0	0
$199$	−1.18219	−0.0838034	−0.0419017	−	0.999122i	$-0.513342\pi$
$199$	−1.18219	−0.0838034	−0.0419017	+	0.999122i	$0.513342\pi$
$200$	0	0
$201$	7.26386	0.512353
$202$	0	0
$203$	0.185024	0.0129861
$204$	0	0
$205$	3.49671	0.244220
$206$	0	0
$207$	−1.19490	−0.0830513
$208$	0	0
$209$	−1.96533	−0.135945
$210$	0	0
$211$	−16.4597	−1.13313	−0.566566	−	0.824017i	$-0.691728\pi$
$211$	−16.4597	−1.13313	−0.566566	+	0.824017i	$0.691728\pi$
$212$	0	0
$213$	3.27430	0.224351
$214$	0	0
$215$	1.00000	0.0681994
$216$	0	0
$217$	−3.25967	−0.221281
$218$	0	0
$219$	20.9686	1.41693
$220$	0	0
$221$	−39.1960	−2.63661
$222$	0	0
$223$	19.6193	1.31381	0.656904	−	0.753974i	$-0.271866\pi$
$223$	19.6193	1.31381	0.656904	+	0.753974i	$0.271866\pi$
$224$	0	0
$225$	0.391491	0.0260994
$226$	0	0
$227$	16.2944	1.08150	0.540750	−	0.841183i	$-0.318140\pi$
$227$	16.2944	1.08150	0.540750	+	0.841183i	$0.318140\pi$
$228$	0	0
$229$	11.6858	0.772220	0.386110	−	0.922453i	$-0.373818\pi$
$229$	11.6858	0.772220	0.386110	+	0.922453i	$0.373818\pi$
$230$	0	0
$231$	2.71758	0.178803
$232$	0	0
$233$	14.8423	0.972349	0.486174	−	0.873862i	$-0.338392\pi$
$233$	14.8423	0.972349	0.486174	+	0.873862i	$0.338392\pi$
$234$	0	0
$235$	−3.86398	−0.252058
$236$	0	0
$237$	26.8344	1.74308
$238$	0	0
$239$	15.3449	0.992581	0.496290	−	0.868157i	$-0.334695\pi$
$239$	15.3449	0.992581	0.496290	+	0.868157i	$0.334695\pi$
$240$	0	0
$241$	11.7712	0.758249	0.379125	−	0.925346i	$-0.376225\pi$
$241$	11.7712	0.758249	0.379125	+	0.925346i	$0.376225\pi$
$242$	0	0
$243$	−4.04357	−0.259395
$244$	0	0
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	8.98009	0.571389
$248$	0	0
$249$	−15.3245	−0.971153
$250$	0	0
$251$	7.62491	0.481280	0.240640	−	0.970614i	$-0.422643\pi$
$251$	7.62491	0.481280	0.240640	+	0.970614i	$0.422643\pi$
$252$	0	0
$253$	4.50398	0.283163
$254$	0	0
$255$	10.7055	0.670402
$256$	0	0
$257$	13.5151	0.843047	0.421524	−	0.906817i	$-0.361495\pi$
$257$	13.5151	0.843047	0.421524	+	0.906817i	$0.361495\pi$
$258$	0	0
$259$	−11.1891	−0.695258
$260$	0	0
$261$	−0.0724352	−0.00448363
$262$	0	0
$263$	27.5584	1.69933	0.849663	−	0.527326i	$-0.176806\pi$
$263$	27.5584	1.69933	0.849663	+	0.527326i	$0.176806\pi$
$264$	0	0
$265$	−4.12550	−0.253427
$266$	0	0
$267$	26.7001	1.63402
$268$	0	0
$269$	27.6262	1.68440	0.842201	−	0.539164i	$-0.181260\pi$
$269$	27.6262	1.68440	0.842201	+	0.539164i	$0.181260\pi$
$270$	0	0
$271$	14.3991	0.874681	0.437340	−	0.899296i	$-0.355920\pi$
$271$	14.3991	0.874681	0.437340	+	0.899296i	$0.355920\pi$
$272$	0	0
$273$	−12.4173	−0.751529
$274$	0	0
$275$	−1.47566	−0.0889857
$276$	0	0
$277$	−33.2142	−1.99565	−0.997824	−	0.0659412i	$-0.978995\pi$
$277$	−33.2142	−1.99565	−0.997824	+	0.0659412i	$0.978995\pi$
$278$	0	0
$279$	1.27613	0.0763999
$280$	0	0
$281$	21.8264	1.30205	0.651027	−	0.759055i	$-0.274339\pi$
$281$	21.8264	1.30205	0.651027	+	0.759055i	$0.274339\pi$
$282$	0	0
$283$	2.19946	0.130745	0.0653723	−	0.997861i	$-0.479176\pi$
$283$	2.19946	0.130745	0.0653723	+	0.997861i	$0.479176\pi$
$284$	0	0
$285$	−2.45270	−0.145285
$286$	0	0
$287$	3.49671	0.206404
$288$	0	0
$289$	16.7925	0.987794
$290$	0	0
$291$	32.0951	1.88145
$292$	0	0
$293$	−20.5568	−1.20094	−0.600470	−	0.799647i	$-0.705020\pi$
$293$	−20.5568	−1.20094	−0.600470	+	0.799647i	$0.705020\pi$
$294$	0	0
$295$	−6.32037	−0.367986
$296$	0	0
$297$	7.08883	0.411335
$298$	0	0
$299$	−20.5798	−1.19016
$300$	0	0
$301$	1.00000	0.0576390
$302$	0	0
$303$	−33.4271	−1.92034
$304$	0	0
$305$	−12.7854	−0.732091
$306$	0	0
$307$	17.3749	0.991639	0.495819	−	0.868426i	$-0.334868\pi$
$307$	17.3749	0.991639	0.495819	+	0.868426i	$0.334868\pi$
$308$	0	0
$309$	−31.7570	−1.80659
$310$	0	0
$311$	−15.2430	−0.864349	−0.432174	−	0.901790i	$-0.642253\pi$
$311$	−15.2430	−0.864349	−0.432174	+	0.901790i	$0.642253\pi$
$312$	0	0
$313$	−28.8084	−1.62835	−0.814174	−	0.580622i	$-0.802810\pi$
$313$	−28.8084	−1.62835	−0.814174	+	0.580622i	$0.802810\pi$
$314$	0	0
$315$	0.391491	0.0220580
$316$	0	0
$317$	−7.83619	−0.440124	−0.220062	−	0.975486i	$-0.570626\pi$
$317$	−7.83619	−0.440124	−0.220062	+	0.975486i	$0.570626\pi$
$318$	0	0
$319$	0.273033	0.0152869
$320$	0	0
$321$	21.3956	1.19418
$322$	0	0
$323$	−7.74210	−0.430782
$324$	0	0
$325$	6.74267	0.374016
$326$	0	0
$327$	13.6628	0.755556
$328$	0	0
$329$	−3.86398	−0.213028
$330$	0	0
$331$	34.9754	1.92242	0.961211	−	0.275812i	$-0.0889469\pi$
$331$	34.9754	1.92242	0.961211	+	0.275812i	$0.0889469\pi$
$332$	0	0
$333$	4.38044	0.240047
$334$	0	0
$335$	−3.94432	−0.215501
$336$	0	0
$337$	−6.85846	−0.373604	−0.186802	−	0.982398i	$-0.559812\pi$
$337$	−6.85846	−0.373604	−0.186802	+	0.982398i	$0.559812\pi$
$338$	0	0
$339$	−0.955077	−0.0518727
$340$	0	0
$341$	−4.81016	−0.260485
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	5.62089	0.302618
$346$	0	0
$347$	−14.2339	−0.764116	−0.382058	−	0.924138i	$-0.624785\pi$
$347$	−14.2339	−0.764116	−0.382058	+	0.924138i	$0.624785\pi$
$348$	0	0
$349$	1.59932	0.0856096	0.0428048	−	0.999083i	$-0.486371\pi$
$349$	1.59932	0.0856096	0.0428048	+	0.999083i	$0.486371\pi$
$350$	0	0
$351$	−32.3906	−1.72888
$352$	0	0
$353$	6.38812	0.340005	0.170003	−	0.985444i	$-0.445622\pi$
$353$	6.38812	0.340005	0.170003	+	0.985444i	$0.445622\pi$
$354$	0	0
$355$	−1.77796	−0.0943644
$356$	0	0
$357$	10.7055	0.566593
$358$	0	0
$359$	8.00778	0.422634	0.211317	−	0.977418i	$-0.432225\pi$
$359$	8.00778	0.422634	0.211317	+	0.977418i	$0.432225\pi$
$360$	0	0
$361$	−17.2262	−0.906644
$362$	0	0
$363$	−16.2474	−0.852766
$364$	0	0
$365$	−11.3861	−0.595976
$366$	0	0
$367$	1.98706	0.103724	0.0518618	−	0.998654i	$-0.483484\pi$
$367$	1.98706	0.103724	0.0518618	+	0.998654i	$0.483484\pi$
$368$	0	0
$369$	−1.36893	−0.0712636
$370$	0	0
$371$	−4.12550	−0.214185
$372$	0	0
$373$	15.8226	0.819264	0.409632	−	0.912251i	$-0.365657\pi$
$373$	15.8226	0.819264	0.409632	+	0.912251i	$0.365657\pi$
$374$	0	0
$375$	−1.84160	−0.0950998
$376$	0	0
$377$	−1.24755	−0.0642523
$378$	0	0
$379$	−28.7474	−1.47666	−0.738328	−	0.674442i	$-0.764384\pi$
$379$	−28.7474	−1.47666	−0.738328	+	0.674442i	$0.764384\pi$
$380$	0	0
$381$	1.67245	0.0856824
$382$	0	0
$383$	21.8599	1.11699	0.558494	−	0.829509i	$-0.311380\pi$
$383$	21.8599	1.11699	0.558494	+	0.829509i	$0.311380\pi$
$384$	0	0
$385$	−1.47566	−0.0752067
$386$	0	0
$387$	−0.391491	−0.0199006
$388$	0	0
$389$	−9.95793	−0.504887	−0.252444	−	0.967612i	$-0.581234\pi$
$389$	−9.95793	−0.504887	−0.252444	+	0.967612i	$0.581234\pi$
$390$	0	0
$391$	17.7427	0.897287
$392$	0	0
$393$	8.02152	0.404632
$394$	0	0
$395$	−14.5712	−0.733158
$396$	0	0
$397$	0.158655	0.00796266	0.00398133	−	0.999992i	$-0.498733\pi$
$397$	0.158655	0.00796266	0.00398133	+	0.999992i	$0.498733\pi$
$398$	0	0
$399$	−2.45270	−0.122788
$400$	0	0
$401$	37.4988	1.87260	0.936299	−	0.351203i	$-0.114227\pi$
$401$	37.4988	1.87260	0.936299	+	0.351203i	$0.114227\pi$
$402$	0	0
$403$	21.9788	1.09484
$404$	0	0
$405$	10.0212	0.497958
$406$	0	0
$407$	−16.5113	−0.818437
$408$	0	0
$409$	2.24652	0.111083	0.0555417	−	0.998456i	$-0.482311\pi$
$409$	2.24652	0.111083	0.0555417	+	0.998456i	$0.482311\pi$
$410$	0	0
$411$	−33.1527	−1.63530
$412$	0	0
$413$	−6.32037	−0.311005
$414$	0	0
$415$	8.32132	0.408477
$416$	0	0
$417$	−21.3662	−1.04631
$418$	0	0
$419$	−10.2686	−0.501652	−0.250826	−	0.968032i	$-0.580702\pi$
$419$	−10.2686	−0.501652	−0.250826	+	0.968032i	$0.580702\pi$
$420$	0	0
$421$	−35.0468	−1.70808	−0.854039	−	0.520210i	$-0.825854\pi$
$421$	−35.0468	−1.70808	−0.854039	+	0.520210i	$0.825854\pi$
$422$	0	0
$423$	1.51271	0.0735507
$424$	0	0
$425$	−5.81313	−0.281978
$426$	0	0
$427$	−12.7854	−0.618730
$428$	0	0
$429$	−18.3237	−0.884678
$430$	0	0
$431$	12.8926	0.621016	0.310508	−	0.950571i	$-0.399501\pi$
$431$	12.8926	0.621016	0.310508	+	0.950571i	$0.399501\pi$
$432$	0	0
$433$	−19.0918	−0.917494	−0.458747	−	0.888567i	$-0.651702\pi$
$433$	−19.0918	−0.917494	−0.458747	+	0.888567i	$0.651702\pi$
$434$	0	0
$435$	0.340740	0.0163372
$436$	0	0
$437$	−4.06498	−0.194454
$438$	0	0
$439$	−5.22821	−0.249529	−0.124764	−	0.992186i	$-0.539818\pi$
$439$	−5.22821	−0.249529	−0.124764	+	0.992186i	$0.539818\pi$
$440$	0	0
$441$	0.391491	0.0186424
$442$	0	0
$443$	35.7715	1.69955	0.849776	−	0.527143i	$-0.176737\pi$
$443$	35.7715	1.69955	0.849776	+	0.527143i	$0.176737\pi$
$444$	0	0
$445$	−14.4983	−0.687287
$446$	0	0
$447$	−23.6333	−1.11782
$448$	0	0
$449$	−8.27153	−0.390358	−0.195179	−	0.980768i	$-0.562529\pi$
$449$	−8.27153	−0.390358	−0.195179	+	0.980768i	$0.562529\pi$
$450$	0	0
$451$	5.15995	0.242973
$452$	0	0
$453$	5.94744	0.279435
$454$	0	0
$455$	6.74267	0.316101
$456$	0	0
$457$	28.7578	1.34523	0.672616	−	0.739992i	$-0.265171\pi$
$457$	28.7578	1.34523	0.672616	+	0.739992i	$0.265171\pi$
$458$	0	0
$459$	27.9253	1.30344
$460$	0	0
$461$	−10.1656	−0.473458	−0.236729	−	0.971576i	$-0.576075\pi$
$461$	−10.1656	−0.473458	−0.236729	+	0.971576i	$0.576075\pi$
$462$	0	0
$463$	−12.0489	−0.559959	−0.279979	−	0.960006i	$-0.590328\pi$
$463$	−12.0489	−0.559959	−0.279979	+	0.960006i	$0.590328\pi$
$464$	0	0
$465$	−6.00300	−0.278382
$466$	0	0
$467$	−32.1605	−1.48821	−0.744105	−	0.668063i	$-0.767124\pi$
$467$	−32.1605	−1.48821	−0.744105	+	0.668063i	$0.767124\pi$
$468$	0	0
$469$	−3.94432	−0.182132
$470$	0	0
$471$	14.2659	0.657339
$472$	0	0
$473$	1.47566	0.0678510
$474$	0	0
$475$	1.33183	0.0611085
$476$	0	0
$477$	1.61510	0.0739502
$478$	0	0
$479$	−24.9314	−1.13914	−0.569572	−	0.821942i	$-0.692891\pi$
$479$	−24.9314	−1.13914	−0.569572	+	0.821942i	$0.692891\pi$
$480$	0	0
$481$	75.4445	3.43997
$482$	0	0
$483$	5.62089	0.255759
$484$	0	0
$485$	−17.4278	−0.791358
$486$	0	0
$487$	−33.7851	−1.53095	−0.765474	−	0.643467i	$-0.777495\pi$
$487$	−33.7851	−1.53095	−0.765474	+	0.643467i	$0.777495\pi$
$488$	0	0
$489$	−19.2953	−0.872564
$490$	0	0
$491$	5.53312	0.249706	0.124853	−	0.992175i	$-0.460154\pi$
$491$	5.53312	0.249706	0.124853	+	0.992175i	$0.460154\pi$
$492$	0	0
$493$	1.07557	0.0484412
$494$	0	0
$495$	0.577708	0.0259661
$496$	0	0
$497$	−1.77796	−0.0797525
$498$	0	0
$499$	41.1460	1.84195	0.920974	−	0.389625i	$-0.127395\pi$
$499$	41.1460	1.84195	0.920974	+	0.389625i	$0.127395\pi$
$500$	0	0
$501$	15.5930	0.696646
$502$	0	0
$503$	−8.72162	−0.388878	−0.194439	−	0.980915i	$-0.562289\pi$
$503$	−8.72162	−0.388878	−0.194439	+	0.980915i	$0.562289\pi$
$504$	0	0
$505$	18.1511	0.807715
$506$	0	0
$507$	59.7849	2.65514
$508$	0	0
$509$	−15.3862	−0.681979	−0.340990	−	0.940067i	$-0.610762\pi$
$509$	−15.3862	−0.681979	−0.340990	+	0.940067i	$0.610762\pi$
$510$	0	0
$511$	−11.3861	−0.503691
$512$	0	0
$513$	−6.39788	−0.282474
$514$	0	0
$515$	17.2442	0.759872
$516$	0	0
$517$	−5.70193	−0.250770
$518$	0	0
$519$	21.6446	0.950092
$520$	0	0
$521$	4.04686	0.177296	0.0886481	−	0.996063i	$-0.471745\pi$
$521$	4.04686	0.177296	0.0886481	+	0.996063i	$0.471745\pi$
$522$	0	0
$523$	−27.4022	−1.19821	−0.599106	−	0.800669i	$-0.704477\pi$
$523$	−27.4022	−1.19821	−0.599106	+	0.800669i	$0.704477\pi$
$524$	0	0
$525$	−1.84160	−0.0803740
$526$	0	0
$527$	−18.9489	−0.825426
$528$	0	0
$529$	−13.6842	−0.594966
$530$	0	0
$531$	2.47437	0.107378
$532$	0	0
$533$	−23.5771	−1.02124
$534$	0	0
$535$	−11.6179	−0.502287
$536$	0	0
$537$	−9.37205	−0.404434
$538$	0	0
$539$	−1.47566	−0.0635612
$540$	0	0
$541$	−18.6026	−0.799786	−0.399893	−	0.916562i	$-0.630953\pi$
$541$	−18.6026	−0.799786	−0.399893	+	0.916562i	$0.630953\pi$
$542$	0	0
$543$	−0.579569	−0.0248717
$544$	0	0
$545$	−7.41900	−0.317795
$546$	0	0
$547$	28.7297	1.22839	0.614197	−	0.789153i	$-0.289480\pi$
$547$	28.7297	1.22839	0.614197	+	0.789153i	$0.289480\pi$
$548$	0	0
$549$	5.00538	0.213624
$550$	0	0
$551$	−0.246420	−0.0104979
$552$	0	0
$553$	−14.5712	−0.619632
$554$	0	0
$555$	−20.6059	−0.874671
$556$	0	0
$557$	−26.0464	−1.10362	−0.551810	−	0.833970i	$-0.686063\pi$
$557$	−26.0464	−1.10362	−0.551810	+	0.833970i	$0.686063\pi$
$558$	0	0
$559$	−6.74267	−0.285184
$560$	0	0
$561$	15.7976	0.666977
$562$	0	0
$563$	11.9614	0.504111	0.252056	−	0.967713i	$-0.418893\pi$
$563$	11.9614	0.504111	0.252056	+	0.967713i	$0.418893\pi$
$564$	0	0
$565$	0.518613	0.0218182
$566$	0	0
$567$	10.0212	0.420851
$568$	0	0
$569$	13.5403	0.567641	0.283820	−	0.958877i	$-0.408398\pi$
$569$	13.5403	0.567641	0.283820	+	0.958877i	$0.408398\pi$
$570$	0	0
$571$	−36.2429	−1.51672	−0.758358	−	0.651838i	$-0.773998\pi$
$571$	−36.2429	−1.51672	−0.758358	+	0.651838i	$0.773998\pi$
$572$	0	0
$573$	−46.3170	−1.93492
$574$	0	0
$575$	−3.05218	−0.127285
$576$	0	0
$577$	8.59567	0.357842	0.178921	−	0.983863i	$-0.442739\pi$
$577$	8.59567	0.357842	0.178921	+	0.983863i	$0.442739\pi$
$578$	0	0
$579$	7.67100	0.318796
$580$	0	0
$581$	8.32132	0.345226
$582$	0	0
$583$	−6.08784	−0.252132
$584$	0	0
$585$	−2.63969	−0.109138
$586$	0	0
$587$	18.5018	0.763652	0.381826	−	0.924234i	$-0.375295\pi$
$587$	18.5018	0.763652	0.381826	+	0.924234i	$0.375295\pi$
$588$	0	0
$589$	4.34132	0.178881
$590$	0	0
$591$	11.0905	0.456202
$592$	0	0
$593$	11.3860	0.467567	0.233783	−	0.972289i	$-0.424889\pi$
$593$	11.3860	0.467567	0.233783	+	0.972289i	$0.424889\pi$
$594$	0	0
$595$	−5.81313	−0.238315
$596$	0	0
$597$	−2.17713	−0.0891038
$598$	0	0
$599$	38.2660	1.56351	0.781754	−	0.623587i	$-0.214325\pi$
$599$	38.2660	1.56351	0.781754	+	0.623587i	$0.214325\pi$
$600$	0	0
$601$	−11.6231	−0.474116	−0.237058	−	0.971495i	$-0.576183\pi$
$601$	−11.6231	−0.474116	−0.237058	+	0.971495i	$0.576183\pi$
$602$	0	0
$603$	1.54417	0.0628833
$604$	0	0
$605$	8.82242	0.358683
$606$	0	0
$607$	−19.3280	−0.784498	−0.392249	−	0.919859i	$-0.628303\pi$
$607$	−19.3280	−0.784498	−0.392249	+	0.919859i	$0.628303\pi$
$608$	0	0
$609$	0.340740	0.0138075
$610$	0	0
$611$	26.0535	1.05401
$612$	0	0
$613$	−24.7126	−0.998131	−0.499066	−	0.866564i	$-0.666323\pi$
$613$	−24.7126	−0.998131	−0.499066	+	0.866564i	$0.666323\pi$
$614$	0	0
$615$	6.43953	0.259667
$616$	0	0
$617$	11.4515	0.461022	0.230511	−	0.973070i	$-0.425960\pi$
$617$	11.4515	0.461022	0.230511	+	0.973070i	$0.425960\pi$
$618$	0	0
$619$	−34.1223	−1.37149	−0.685746	−	0.727841i	$-0.740524\pi$
$619$	−34.1223	−1.37149	−0.685746	+	0.727841i	$0.740524\pi$
$620$	0	0
$621$	14.6621	0.588371
$622$	0	0
$623$	−14.4983	−0.580863
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−3.61935	−0.144543
$628$	0	0
$629$	−65.0438	−2.59347
$630$	0	0
$631$	−20.7132	−0.824581	−0.412290	−	0.911052i	$-0.635271\pi$
$631$	−20.7132	−0.824581	−0.412290	+	0.911052i	$0.635271\pi$
$632$	0	0
$633$	−30.3121	−1.20480
$634$	0	0
$635$	−0.908152	−0.0360389
$636$	0	0
$637$	6.74267	0.267154
$638$	0	0
$639$	0.696056	0.0275356
$640$	0	0
$641$	24.7298	0.976767	0.488384	−	0.872629i	$-0.337587\pi$
$641$	24.7298	0.976767	0.488384	+	0.872629i	$0.337587\pi$
$642$	0	0
$643$	−31.4373	−1.23976	−0.619882	−	0.784695i	$-0.712820\pi$
$643$	−31.4373	−1.23976	−0.619882	+	0.784695i	$0.712820\pi$
$644$	0	0
$645$	1.84160	0.0725129
$646$	0	0
$647$	−8.44694	−0.332083	−0.166042	−	0.986119i	$-0.553099\pi$
$647$	−8.44694	−0.332083	−0.166042	+	0.986119i	$0.553099\pi$
$648$	0	0
$649$	−9.32672	−0.366106
$650$	0	0
$651$	−6.00300	−0.235276
$652$	0	0
$653$	27.7850	1.08731	0.543655	−	0.839309i	$-0.317040\pi$
$653$	27.7850	1.08731	0.543655	+	0.839309i	$0.317040\pi$
$654$	0	0
$655$	−4.35573	−0.170193
$656$	0	0
$657$	4.45756	0.173906
$658$	0	0
$659$	32.3815	1.26140	0.630702	−	0.776025i	$-0.282767\pi$
$659$	32.3815	1.26140	0.630702	+	0.776025i	$0.282767\pi$
$660$	0	0
$661$	−17.7294	−0.689594	−0.344797	−	0.938677i	$-0.612052\pi$
$661$	−17.7294	−0.689594	−0.344797	+	0.938677i	$0.612052\pi$
$662$	0	0
$663$	−72.1834	−2.80337
$664$	0	0
$665$	1.33183	0.0516461
$666$	0	0
$667$	0.564726	0.0218663
$668$	0	0
$669$	36.1310	1.39690
$670$	0	0
$671$	−18.8669	−0.728350
$672$	0	0
$673$	−41.3608	−1.59434	−0.797171	−	0.603754i	$-0.793671\pi$
$673$	−41.3608	−1.59434	−0.797171	+	0.603754i	$0.793671\pi$
$674$	0	0
$675$	−4.80383	−0.184900
$676$	0	0
$677$	38.6984	1.48730	0.743649	−	0.668570i	$-0.233093\pi$
$677$	38.6984	1.48730	0.743649	+	0.668570i	$0.233093\pi$
$678$	0	0
$679$	−17.4278	−0.668819
$680$	0	0
$681$	30.0078	1.14990
$682$	0	0
$683$	−0.802417	−0.0307036	−0.0153518	−	0.999882i	$-0.504887\pi$
$683$	−0.802417	−0.0307036	−0.0153518	+	0.999882i	$0.504887\pi$
$684$	0	0
$685$	18.0021	0.687825
$686$	0	0
$687$	21.5206	0.821062
$688$	0	0
$689$	27.8169	1.05974
$690$	0	0
$691$	−34.6446	−1.31794	−0.658971	−	0.752168i	$-0.729008\pi$
$691$	−34.6446	−1.31794	−0.658971	+	0.752168i	$0.729008\pi$
$692$	0	0
$693$	0.577708	0.0219453
$694$	0	0
$695$	11.6020	0.440089
$696$	0	0
$697$	20.3268	0.769933
$698$	0	0
$699$	27.3335	1.03385
$700$	0	0
$701$	9.26487	0.349929	0.174965	−	0.984575i	$-0.444019\pi$
$701$	9.26487	0.349929	0.174965	+	0.984575i	$0.444019\pi$
$702$	0	0
$703$	14.9020	0.562040
$704$	0	0
$705$	−7.11591	−0.268001
$706$	0	0
$707$	18.1511	0.682644
$708$	0	0
$709$	30.9001	1.16048	0.580239	−	0.814446i	$-0.302959\pi$
$709$	30.9001	1.16048	0.580239	+	0.814446i	$0.302959\pi$
$710$	0	0
$711$	5.70451	0.213936
$712$	0	0
$713$	−9.94907	−0.372596
$714$	0	0
$715$	9.94989	0.372105
$716$	0	0
$717$	28.2592	1.05536
$718$	0	0
$719$	−15.4835	−0.577438	−0.288719	−	0.957414i	$-0.593229\pi$
$719$	−15.4835	−0.577438	−0.288719	+	0.957414i	$0.593229\pi$
$720$	0	0
$721$	17.2442	0.642209
$722$	0	0
$723$	21.6778	0.806207
$724$	0	0
$725$	−0.185024	−0.00687162
$726$	0	0
$727$	−12.5633	−0.465948	−0.232974	−	0.972483i	$-0.574846\pi$
$727$	−12.5633	−0.465948	−0.232974	+	0.972483i	$0.574846\pi$
$728$	0	0
$729$	22.6170	0.837666
$730$	0	0
$731$	5.81313	0.215006
$732$	0	0
$733$	−15.2181	−0.562094	−0.281047	−	0.959694i	$-0.590682\pi$
$733$	−15.2181	−0.562094	−0.281047	+	0.959694i	$0.590682\pi$
$734$	0	0
$735$	−1.84160	−0.0679284
$736$	0	0
$737$	−5.82048	−0.214400
$738$	0	0
$739$	−27.3622	−1.00653	−0.503267	−	0.864131i	$-0.667869\pi$
$739$	−27.3622	−1.00653	−0.503267	+	0.864131i	$0.667869\pi$
$740$	0	0
$741$	16.5377	0.607529
$742$	0	0
$743$	−50.7759	−1.86279	−0.931394	−	0.364012i	$-0.881407\pi$
$743$	−50.7759	−1.86279	−0.931394	+	0.364012i	$0.881407\pi$
$744$	0	0
$745$	12.8330	0.470166
$746$	0	0
$747$	−3.25772	−0.119194
$748$	0	0
$749$	−11.6179	−0.424510
$750$	0	0
$751$	48.2070	1.75910	0.879550	−	0.475807i	$-0.157844\pi$
$751$	48.2070	1.75910	0.879550	+	0.475807i	$0.157844\pi$
$752$	0	0
$753$	14.0420	0.511720
$754$	0	0
$755$	−3.22950	−0.117533
$756$	0	0
$757$	−3.80724	−0.138376	−0.0691882	−	0.997604i	$-0.522041\pi$
$757$	−3.80724	−0.138376	−0.0691882	+	0.997604i	$0.522041\pi$
$758$	0	0
$759$	8.29453	0.301072
$760$	0	0
$761$	−36.1800	−1.31152	−0.655762	−	0.754968i	$-0.727652\pi$
$761$	−36.1800	−1.31152	−0.655762	+	0.754968i	$0.727652\pi$
$762$	0	0
$763$	−7.41900	−0.268586
$764$	0	0
$765$	2.27579	0.0822813
$766$	0	0
$767$	42.6161	1.53878
$768$	0	0
$769$	0.948099	0.0341893	0.0170947	−	0.999854i	$-0.494558\pi$
$769$	0.948099	0.0341893	0.0170947	+	0.999854i	$0.494558\pi$
$770$	0	0
$771$	24.8894	0.896369
$772$	0	0
$773$	−36.7493	−1.32178	−0.660890	−	0.750482i	$-0.729821\pi$
$773$	−36.7493	−1.32178	−0.660890	+	0.750482i	$0.729821\pi$
$774$	0	0
$775$	3.25967	0.117091
$776$	0	0
$777$	−20.6059	−0.739232
$778$	0	0
$779$	−4.65702	−0.166855
$780$	0	0
$781$	−2.62367	−0.0938823
$782$	0	0
$783$	0.888824	0.0317640
$784$	0	0
$785$	−7.74649	−0.276484
$786$	0	0
$787$	−5.46586	−0.194837	−0.0974184	−	0.995244i	$-0.531059\pi$
$787$	−5.46586	−0.194837	−0.0974184	+	0.995244i	$0.531059\pi$
$788$	0	0
$789$	50.7516	1.80681
$790$	0	0
$791$	0.518613	0.0184397
$792$	0	0
$793$	86.2078	3.06133
$794$	0	0
$795$	−7.59752	−0.269456
$796$	0	0
$797$	27.6798	0.980467	0.490234	−	0.871591i	$-0.336911\pi$
$797$	27.6798	0.980467	0.490234	+	0.871591i	$0.336911\pi$
$798$	0	0
$799$	−22.4618	−0.794642
$800$	0	0
$801$	5.67597	0.200550
$802$	0	0
$803$	−16.8020	−0.592930
$804$	0	0
$805$	−3.05218	−0.107575
$806$	0	0
$807$	50.8765	1.79094
$808$	0	0
$809$	−2.47954	−0.0871761	−0.0435880	−	0.999050i	$-0.513879\pi$
$809$	−2.47954	−0.0871761	−0.0435880	+	0.999050i	$0.513879\pi$
$810$	0	0
$811$	33.2121	1.16623	0.583117	−	0.812388i	$-0.301833\pi$
$811$	33.2121	1.16623	0.583117	+	0.812388i	$0.301833\pi$
$812$	0	0
$813$	26.5173	0.930003
$814$	0	0
$815$	10.4775	0.367010
$816$	0	0
$817$	−1.33183	−0.0465948
$818$	0	0
$819$	−2.63969	−0.0922384
$820$	0	0
$821$	31.8908	1.11300	0.556498	−	0.830849i	$-0.312145\pi$
$821$	31.8908	1.11300	0.556498	+	0.830849i	$0.312145\pi$
$822$	0	0
$823$	−19.6875	−0.686263	−0.343131	−	0.939287i	$-0.611488\pi$
$823$	−19.6875	−0.686263	−0.343131	+	0.939287i	$0.611488\pi$
$824$	0	0
$825$	−2.71758	−0.0946139
$826$	0	0
$827$	−40.4005	−1.40486	−0.702431	−	0.711752i	$-0.747902\pi$
$827$	−40.4005	−1.40486	−0.702431	+	0.711752i	$0.747902\pi$
$828$	0	0
$829$	32.5062	1.12899	0.564493	−	0.825438i	$-0.309072\pi$
$829$	32.5062	1.12899	0.564493	+	0.825438i	$0.309072\pi$
$830$	0	0
$831$	−61.1672	−2.12187
$832$	0	0
$833$	−5.81313	−0.201413
$834$	0	0
$835$	−8.46712	−0.293017
$836$	0	0
$837$	−15.6589	−0.541250
$838$	0	0
$839$	18.8349	0.650255	0.325127	−	0.945670i	$-0.394593\pi$
$839$	18.8349	0.650255	0.325127	+	0.945670i	$0.394593\pi$
$840$	0	0
$841$	−28.9658	−0.998820
$842$	0	0
$843$	40.1955	1.38441
$844$	0	0
$845$	−32.4636	−1.11678
$846$	0	0
$847$	8.82242	0.303142
$848$	0	0
$849$	4.05053	0.139014
$850$	0	0
$851$	−34.1512	−1.17069
$852$	0	0
$853$	−36.2620	−1.24159	−0.620793	−	0.783974i	$-0.713189\pi$
$853$	−36.2620	−1.24159	−0.620793	+	0.783974i	$0.713189\pi$
$854$	0	0
$855$	−0.521400	−0.0178315
$856$	0	0
$857$	55.0644	1.88096	0.940481	−	0.339846i	$-0.110375\pi$
$857$	55.0644	1.88096	0.940481	+	0.339846i	$0.110375\pi$
$858$	0	0
$859$	5.34486	0.182364	0.0911821	−	0.995834i	$-0.470935\pi$
$859$	5.34486	0.182364	0.0911821	+	0.995834i	$0.470935\pi$
$860$	0	0
$861$	6.43953	0.219459
$862$	0	0
$863$	−45.3904	−1.54511	−0.772553	−	0.634950i	$-0.781021\pi$
$863$	−45.3904	−1.54511	−0.772553	+	0.634950i	$0.781021\pi$
$864$	0	0
$865$	−11.7531	−0.399619
$866$	0	0
$867$	30.9251	1.05027
$868$	0	0
$869$	−21.5022	−0.729412
$870$	0	0
$871$	26.5953	0.901146
$872$	0	0
$873$	6.82285	0.230918
$874$	0	0
$875$	1.00000	0.0338062
$876$	0	0
$877$	−16.7179	−0.564524	−0.282262	−	0.959337i	$-0.591085\pi$
$877$	−16.7179	−0.564524	−0.282262	+	0.959337i	$0.591085\pi$
$878$	0	0
$879$	−37.8574	−1.27690
$880$	0	0
$881$	−7.34611	−0.247497	−0.123748	−	0.992314i	$-0.539492\pi$
$881$	−7.34611	−0.247497	−0.123748	+	0.992314i	$0.539492\pi$
$882$	0	0
$883$	25.1006	0.844702	0.422351	−	0.906432i	$-0.361205\pi$
$883$	25.1006	0.844702	0.422351	+	0.906432i	$0.361205\pi$
$884$	0	0
$885$	−11.6396	−0.391260
$886$	0	0
$887$	−12.7086	−0.426713	−0.213356	−	0.976974i	$-0.568440\pi$
$887$	−12.7086	−0.426713	−0.213356	+	0.976974i	$0.568440\pi$
$888$	0	0
$889$	−0.908152	−0.0304585
$890$	0	0
$891$	14.7879	0.495414
$892$	0	0
$893$	5.14616	0.172210
$894$	0	0
$895$	5.08908	0.170109
$896$	0	0
$897$	−37.8998	−1.26544
$898$	0	0
$899$	−0.603116	−0.0201150
$900$	0	0
$901$	−23.9821	−0.798958
$902$	0	0
$903$	1.84160	0.0612846
$904$	0	0
$905$	0.314710	0.0104613
$906$	0	0
$907$	−22.6083	−0.750696	−0.375348	−	0.926884i	$-0.622477\pi$
$907$	−22.6083	−0.750696	−0.375348	+	0.926884i	$0.622477\pi$
$908$	0	0
$909$	−7.10601	−0.235691
$910$	0	0
$911$	9.26792	0.307060	0.153530	−	0.988144i	$-0.450936\pi$
$911$	9.26792	0.307060	0.153530	+	0.988144i	$0.450936\pi$
$912$	0	0
$913$	12.2794	0.406390
$914$	0	0
$915$	−23.5456	−0.778394
$916$	0	0
$917$	−4.35573	−0.143839
$918$	0	0
$919$	−19.3626	−0.638714	−0.319357	−	0.947635i	$-0.603467\pi$
$919$	−19.3626	−0.638714	−0.319357	+	0.947635i	$0.603467\pi$
$920$	0	0
$921$	31.9976	1.05436
$922$	0	0
$923$	11.9882	0.394597
$924$	0	0
$925$	11.1891	0.367896
$926$	0	0
$927$	−6.75097	−0.221731
$928$	0	0
$929$	17.7022	0.580790	0.290395	−	0.956907i	$-0.406213\pi$
$929$	17.7022	0.580790	0.290395	+	0.956907i	$0.406213\pi$
$930$	0	0
$931$	1.33183	0.0436490
$932$	0	0
$933$	−28.0714	−0.919017
$934$	0	0
$935$	−8.57821	−0.280538
$936$	0	0
$937$	7.23535	0.236369	0.118184	−	0.992992i	$-0.462293\pi$
$937$	7.23535	0.236369	0.118184	+	0.992992i	$0.462293\pi$
$938$	0	0
$939$	−53.0536	−1.73134
$940$	0	0
$941$	−40.3090	−1.31404	−0.657018	−	0.753875i	$-0.728182\pi$
$941$	−40.3090	−1.31404	−0.657018	+	0.753875i	$0.728182\pi$
$942$	0	0
$943$	10.6726	0.347546
$944$	0	0
$945$	−4.80383	−0.156269
$946$	0	0
$947$	28.1967	0.916270	0.458135	−	0.888883i	$-0.348518\pi$
$947$	28.1967	0.916270	0.458135	+	0.888883i	$0.348518\pi$
$948$	0	0
$949$	76.7727	2.49215
$950$	0	0
$951$	−14.4311	−0.467962
$952$	0	0
$953$	22.0303	0.713631	0.356816	−	0.934175i	$-0.383862\pi$
$953$	22.0303	0.713631	0.356816	+	0.934175i	$0.383862\pi$
$954$	0	0
$955$	25.1504	0.813848
$956$	0	0
$957$	0.502817	0.0162538
$958$	0	0
$959$	18.0021	0.581318
$960$	0	0
$961$	−20.3746	−0.657245
$962$	0	0
$963$	4.54831	0.146567
$964$	0	0
$965$	−4.16540	−0.134089
$966$	0	0
$967$	−37.7037	−1.21247	−0.606235	−	0.795286i	$-0.707321\pi$
$967$	−37.7037	−1.21247	−0.606235	+	0.795286i	$0.707321\pi$
$968$	0	0
$969$	−14.2579	−0.458028
$970$	0	0
$971$	−27.2888	−0.875739	−0.437869	−	0.899039i	$-0.644267\pi$
$971$	−27.2888	−0.875739	−0.437869	+	0.899039i	$0.644267\pi$
$972$	0	0
$973$	11.6020	0.371943
$974$	0	0
$975$	12.4173	0.397672
$976$	0	0
$977$	−49.4776	−1.58293	−0.791465	−	0.611214i	$-0.790681\pi$
$977$	−49.4776	−1.58293	−0.791465	+	0.611214i	$0.790681\pi$
$978$	0	0
$979$	−21.3946	−0.683775
$980$	0	0
$981$	2.90447	0.0927327
$982$	0	0
$983$	30.9033	0.985663	0.492832	−	0.870125i	$-0.335962\pi$
$983$	30.9033	0.985663	0.492832	+	0.870125i	$0.335962\pi$
$984$	0	0
$985$	−6.02221	−0.191884
$986$	0	0
$987$	−7.11591	−0.226502
$988$	0	0
$989$	3.05218	0.0970536
$990$	0	0
$991$	−24.1769	−0.768005	−0.384003	−	0.923332i	$-0.625455\pi$
$991$	−24.1769	−0.768005	−0.384003	+	0.923332i	$0.625455\pi$
$992$	0	0
$993$	64.4107	2.04401
$994$	0	0
$995$	1.18219	0.0374780
$996$	0	0
$997$	−54.6709	−1.73144	−0.865722	−	0.500525i	$-0.833140\pi$
$997$	−54.6709	−1.73144	−0.865722	+	0.500525i	$0.833140\pi$
$998$	0	0
$999$	−53.7506	−1.70059

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6020.2.a.j.1.10	✓	13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6020.2.a.j.1.10	✓	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)

\(f(q)\)	\(=\)	\(q+1.84160 q^{3} -1.00000 q^{5} -1.00000 q^{7} +0.391491 q^{9} +O(q^{10})\)	\(q+1.84160 q^{3} -1.00000 q^{5} -1.00000 q^{7} +0.391491 q^{9} -1.47566 q^{11} +6.74267 q^{13} -1.84160 q^{15} -5.81313 q^{17} +1.33183 q^{19} -1.84160 q^{21} -3.05218 q^{23} +1.00000 q^{25} -4.80383 q^{27} -0.185024 q^{29} +3.25967 q^{31} -2.71758 q^{33} +1.00000 q^{35} +11.1891 q^{37} +12.4173 q^{39} -3.49671 q^{41} -1.00000 q^{43} -0.391491 q^{45} +3.86398 q^{47} +1.00000 q^{49} -10.7055 q^{51} +4.12550 q^{53} +1.47566 q^{55} +2.45270 q^{57} +6.32037 q^{59} +12.7854 q^{61} -0.391491 q^{63} -6.74267 q^{65} +3.94432 q^{67} -5.62089 q^{69} +1.77796 q^{71} +11.3861 q^{73} +1.84160 q^{75} +1.47566 q^{77} +14.5712 q^{79} -10.0212 q^{81} -8.32132 q^{83} +5.81313 q^{85} -0.340740 q^{87} +14.4983 q^{89} -6.74267 q^{91} +6.00300 q^{93} -1.33183 q^{95} +17.4278 q^{97} -0.577708 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q - q^{3} - 13 q^{5} - 13 q^{7} + 18 q^{9}+O(q^{10}) \) 13 * q - q^3 - 13 * q^5 - 13 * q^7 + 18 * q^9	\( 13 q - q^{3} - 13 q^{5} - 13 q^{7} + 18 q^{9} + 6 q^{11} + q^{13} + q^{15} - 6 q^{17} + 2 q^{19} + q^{21} - 6 q^{23} + 13 q^{25} - q^{27} + 19 q^{29} - 24 q^{31} + 17 q^{33} + 13 q^{35} + 15 q^{39} + 4 q^{41} - 13 q^{43} - 18 q^{45} - 3 q^{47} + 13 q^{49} + 7 q^{51} - 5 q^{53} - 6 q^{55} + 6 q^{57} - 6 q^{59} + 3 q^{61} - 18 q^{63} - q^{65} - 2 q^{67} + 20 q^{69} + 18 q^{71} + 14 q^{73} - q^{75} - 6 q^{77} + 12 q^{79} + 37 q^{81} + 2 q^{83} + 6 q^{85} - 2 q^{87} + 17 q^{89} - q^{91} + 15 q^{93} - 2 q^{95} + 17 q^{97} + 80 q^{99}+O(q^{100}) \) 13 * q - q^3 - 13 * q^5 - 13 * q^7 + 18 * q^9 + 6 * q^11 + q^13 + q^15 - 6 * q^17 + 2 * q^19 + q^21 - 6 * q^23 + 13 * q^25 - q^27 + 19 * q^29 - 24 * q^31 + 17 * q^33 + 13 * q^35 + 15 * q^39 + 4 * q^41 - 13 * q^43 - 18 * q^45 - 3 * q^47 + 13 * q^49 + 7 * q^51 - 5 * q^53 - 6 * q^55 + 6 * q^57 - 6 * q^59 + 3 * q^61 - 18 * q^63 - q^65 - 2 * q^67 + 20 * q^69 + 18 * q^71 + 14 * q^73 - q^75 - 6 * q^77 + 12 * q^79 + 37 * q^81 + 2 * q^83 + 6 * q^85 - 2 * q^87 + 17 * q^89 - q^91 + 15 * q^93 - 2 * q^95 + 17 * q^97 + 80 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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