LMFDB - Embedded newform 6040.2.a.p.1.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6040,2,Mod(1,6040)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6040, 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("6040.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6040 = 2^{3} \cdot 5 \cdot 151 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6040.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:	$48.2296428209$
Analytic rank:	$1$
Dimension:	$19$
Coefficient field:	$\mathbb{Q}[x]/(x^{19} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{19} - 5 x^{18} - 29 x^{17} + 165 x^{16} + 325 x^{15} - 2208 x^{14} - 1891 x^{13} + 15895 x^{12} + \cdots - 5628 $ x^19 - 5x^18 - 29x^17 + 165x^16 + 325x^15 - 2208x^14 - 1891x^13 + 15895x^12 + 6652x^11 - 67665x^10 - 17345x^9 + 174105x^8 + 41499x^7 - 262172x^6 - 80919x^5 + 206783x^4 + 91643x^3 - 59750x^2 - 40224x - 5628
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{7} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Root			$-0.237601$ of defining polynomial
Character	$\chi$	$=$	6040.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.237601 q^{3} +1.00000 q^{5} +4.27350 q^{7} -2.94355 q^{9} +O(q^{10})$	$q+0.237601 q^{3} +1.00000 q^{5} +4.27350 q^{7} -2.94355 q^{9} -2.02953 q^{11} -4.81969 q^{13} +0.237601 q^{15} -5.46461 q^{17} +3.56063 q^{19} +1.01539 q^{21} -0.164087 q^{23} +1.00000 q^{25} -1.41219 q^{27} -7.17670 q^{29} +8.23553 q^{31} -0.482219 q^{33} +4.27350 q^{35} +7.02949 q^{37} -1.14516 q^{39} -7.73949 q^{41} +3.47560 q^{43} -2.94355 q^{45} +7.39739 q^{47} +11.2628 q^{49} -1.29840 q^{51} -5.18979 q^{53} -2.02953 q^{55} +0.846010 q^{57} -13.8741 q^{59} -11.9562 q^{61} -12.5792 q^{63} -4.81969 q^{65} -8.08848 q^{67} -0.0389872 q^{69} -0.838605 q^{71} -4.52854 q^{73} +0.237601 q^{75} -8.67319 q^{77} +11.6713 q^{79} +8.49510 q^{81} -6.71143 q^{83} -5.46461 q^{85} -1.70519 q^{87} -7.26146 q^{89} -20.5969 q^{91} +1.95677 q^{93} +3.56063 q^{95} +3.11104 q^{97} +5.97401 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 19 q - 5 q^{3} + 19 q^{5} - 8 q^{7} + 26 q^{9}+O(q^{10}) $ 19 * q - 5 * q^3 + 19 * q^5 - 8 * q^7 + 26 * q^9	$ 19 q - 5 q^{3} + 19 q^{5} - 8 q^{7} + 26 q^{9} - 18 q^{11} + 5 q^{13} - 5 q^{15} - 4 q^{17} - 27 q^{19} - 18 q^{21} - 25 q^{23} + 19 q^{25} - 35 q^{27} - 35 q^{29} - 26 q^{31} - 8 q^{35} - 10 q^{37} - 48 q^{39} - 14 q^{41} - 21 q^{43} + 26 q^{45} - 40 q^{47} + 23 q^{49} - 32 q^{51} - 3 q^{53} - 18 q^{55} - 13 q^{57} - 28 q^{59} - 46 q^{61} - 53 q^{63} + 5 q^{65} - 42 q^{67} - 31 q^{69} - 46 q^{71} + 31 q^{73} - 5 q^{75} + 15 q^{77} - 56 q^{79} + 31 q^{81} - 25 q^{83} - 4 q^{85} - 20 q^{87} - 7 q^{89} - 61 q^{91} + 29 q^{93} - 27 q^{95} + 39 q^{97} - 52 q^{99}+O(q^{100}) $ 19 * q - 5 * q^3 + 19 * q^5 - 8 * q^7 + 26 * q^9 - 18 * q^11 + 5 * q^13 - 5 * q^15 - 4 * q^17 - 27 * q^19 - 18 * q^21 - 25 * q^23 + 19 * q^25 - 35 * q^27 - 35 * q^29 - 26 * q^31 - 8 * q^35 - 10 * q^37 - 48 * q^39 - 14 * q^41 - 21 * q^43 + 26 * q^45 - 40 * q^47 + 23 * q^49 - 32 * q^51 - 3 * q^53 - 18 * q^55 - 13 * q^57 - 28 * q^59 - 46 * q^61 - 53 * q^63 + 5 * q^65 - 42 * q^67 - 31 * q^69 - 46 * q^71 + 31 * q^73 - 5 * q^75 + 15 * q^77 - 56 * q^79 + 31 * q^81 - 25 * q^83 - 4 * q^85 - 20 * q^87 - 7 * q^89 - 61 * q^91 + 29 * q^93 - 27 * q^95 + 39 * q^97 - 52 * 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
$3$	0.237601	0.137179	0.0685896	−	0.997645i	$-0.478150\pi$
$3$	0.237601	0.137179	0.0685896	+	0.997645i	$0.478150\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	4.27350	1.61523	0.807615	−	0.589709i	$-0.200758\pi$
$7$	4.27350	1.61523	0.807615	+	0.589709i	$0.200758\pi$
$8$	0	0
$9$	−2.94355	−0.981182
$10$	0	0
$11$	−2.02953	−0.611926	−0.305963	−	0.952043i	$-0.598978\pi$
$11$	−2.02953	−0.611926	−0.305963	+	0.952043i	$0.598978\pi$
$12$	0	0
$13$	−4.81969	−1.33674	−0.668371	−	0.743828i	$-0.733008\pi$
$13$	−4.81969	−1.33674	−0.668371	+	0.743828i	$0.733008\pi$
$14$	0	0
$15$	0.237601	0.0613484
$16$	0	0
$17$	−5.46461	−1.32536	−0.662681	−	0.748902i	$-0.730581\pi$
$17$	−5.46461	−1.32536	−0.662681	+	0.748902i	$0.730581\pi$
$18$	0	0
$19$	3.56063	0.816864	0.408432	−	0.912789i	$-0.366076\pi$
$19$	3.56063	0.816864	0.408432	+	0.912789i	$0.366076\pi$
$20$	0	0
$21$	1.01539	0.221576
$22$	0	0
$23$	−0.164087	−0.0342144	−0.0171072	−	0.999854i	$-0.505446\pi$
$23$	−0.164087	−0.0342144	−0.0171072	+	0.999854i	$0.505446\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.41219	−0.271777
$28$	0	0
$29$	−7.17670	−1.33268	−0.666340	−	0.745648i	$-0.732140\pi$
$29$	−7.17670	−1.33268	−0.666340	+	0.745648i	$0.732140\pi$
$30$	0	0
$31$	8.23553	1.47914	0.739572	−	0.673077i	$-0.235028\pi$
$31$	8.23553	1.47914	0.739572	+	0.673077i	$0.235028\pi$
$32$	0	0
$33$	−0.482219	−0.0839435
$34$	0	0
$35$	4.27350	0.722353
$36$	0	0
$37$	7.02949	1.15564	0.577821	−	0.816164i	$-0.303903\pi$
$37$	7.02949	1.15564	0.577821	+	0.816164i	$0.303903\pi$
$38$	0	0
$39$	−1.14516	−0.183373
$40$	0	0
$41$	−7.73949	−1.20871	−0.604353	−	0.796717i	$-0.706568\pi$
$41$	−7.73949	−1.20871	−0.604353	+	0.796717i	$0.706568\pi$
$42$	0	0
$43$	3.47560	0.530023	0.265012	−	0.964245i	$-0.414624\pi$
$43$	3.47560	0.530023	0.265012	+	0.964245i	$0.414624\pi$
$44$	0	0
$45$	−2.94355	−0.438798
$46$	0	0
$47$	7.39739	1.07902	0.539510	−	0.841979i	$-0.318609\pi$
$47$	7.39739	1.07902	0.539510	+	0.841979i	$0.318609\pi$
$48$	0	0
$49$	11.2628	1.60897
$50$	0	0
$51$	−1.29840	−0.181812
$52$	0	0
$53$	−5.18979	−0.712872	−0.356436	−	0.934320i	$-0.616008\pi$
$53$	−5.18979	−0.712872	−0.356436	+	0.934320i	$0.616008\pi$
$54$	0	0
$55$	−2.02953	−0.273662
$56$	0	0
$57$	0.846010	0.112057
$58$	0	0
$59$	−13.8741	−1.80625	−0.903126	−	0.429377i	$-0.858733\pi$
$59$	−13.8741	−1.80625	−0.903126	+	0.429377i	$0.858733\pi$
$60$	0	0
$61$	−11.9562	−1.53084	−0.765420	−	0.643531i	$-0.777469\pi$
$61$	−11.9562	−1.53084	−0.765420	+	0.643531i	$0.777469\pi$
$62$	0	0
$63$	−12.5792	−1.58484
$64$	0	0
$65$	−4.81969	−0.597809
$66$	0	0
$67$	−8.08848	−0.988165	−0.494082	−	0.869415i	$-0.664496\pi$
$67$	−8.08848	−0.988165	−0.494082	+	0.869415i	$0.664496\pi$
$68$	0	0
$69$	−0.0389872	−0.00469351
$70$	0	0
$71$	−0.838605	−0.0995241	−0.0497621	−	0.998761i	$-0.515846\pi$
$71$	−0.838605	−0.0995241	−0.0497621	+	0.998761i	$0.515846\pi$
$72$	0	0
$73$	−4.52854	−0.530026	−0.265013	−	0.964245i	$-0.585376\pi$
$73$	−4.52854	−0.530026	−0.265013	+	0.964245i	$0.585376\pi$
$74$	0	0
$75$	0.237601	0.0274358
$76$	0	0
$77$	−8.67319	−0.988402
$78$	0	0
$79$	11.6713	1.31312	0.656561	−	0.754273i	$-0.272011\pi$
$79$	11.6713	1.31312	0.656561	+	0.754273i	$0.272011\pi$
$80$	0	0
$81$	8.49510	0.943900
$82$	0	0
$83$	−6.71143	−0.736675	−0.368337	−	0.929692i	$-0.620073\pi$
$83$	−6.71143	−0.736675	−0.368337	+	0.929692i	$0.620073\pi$
$84$	0	0
$85$	−5.46461	−0.592720
$86$	0	0
$87$	−1.70519	−0.182816
$88$	0	0
$89$	−7.26146	−0.769713	−0.384856	−	0.922976i	$-0.625749\pi$
$89$	−7.26146	−0.769713	−0.384856	+	0.922976i	$0.625749\pi$
$90$	0	0
$91$	−20.5969	−2.15915
$92$	0	0
$93$	1.95677	0.202908
$94$	0	0
$95$	3.56063	0.365313
$96$	0	0
$97$	3.11104	0.315878	0.157939	−	0.987449i	$-0.449515\pi$
$97$	3.11104	0.315878	0.157939	+	0.987449i	$0.449515\pi$
$98$	0	0
$99$	5.97401	0.600411
$100$	0	0
$101$	0.834076	0.0829937	0.0414969	−	0.999139i	$-0.486787\pi$
$101$	0.834076	0.0829937	0.0414969	+	0.999139i	$0.486787\pi$
$102$	0	0
$103$	−10.2189	−1.00690	−0.503448	−	0.864025i	$-0.667936\pi$
$103$	−10.2189	−1.00690	−0.503448	+	0.864025i	$0.667936\pi$
$104$	0	0
$105$	1.01539	0.0990918
$106$	0	0
$107$	−12.3395	−1.19291	−0.596453	−	0.802648i	$-0.703424\pi$
$107$	−12.3395	−1.19291	−0.596453	+	0.802648i	$0.703424\pi$
$108$	0	0
$109$	−6.18989	−0.592884	−0.296442	−	0.955051i	$-0.595800\pi$
$109$	−6.18989	−0.592884	−0.296442	+	0.955051i	$0.595800\pi$
$110$	0	0
$111$	1.67022	0.158530
$112$	0	0
$113$	−19.4597	−1.83061	−0.915305	−	0.402761i	$-0.868050\pi$
$113$	−19.4597	−1.83061	−0.915305	+	0.402761i	$0.868050\pi$
$114$	0	0
$115$	−0.164087	−0.0153012
$116$	0	0
$117$	14.1870	1.31159
$118$	0	0
$119$	−23.3530	−2.14077
$120$	0	0
$121$	−6.88101	−0.625547
$122$	0	0
$123$	−1.83891	−0.165809
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−7.01836	−0.622779	−0.311389	−	0.950282i	$-0.600794\pi$
$127$	−7.01836	−0.622779	−0.311389	+	0.950282i	$0.600794\pi$
$128$	0	0
$129$	0.825806	0.0727082
$130$	0	0
$131$	2.02148	0.176618	0.0883090	−	0.996093i	$-0.471854\pi$
$131$	2.02148	0.176618	0.0883090	+	0.996093i	$0.471854\pi$
$132$	0	0
$133$	15.2163	1.31942
$134$	0	0
$135$	−1.41219	−0.121542
$136$	0	0
$137$	−5.48501	−0.468616	−0.234308	−	0.972162i	$-0.575282\pi$
$137$	−5.48501	−0.468616	−0.234308	+	0.972162i	$0.575282\pi$
$138$	0	0
$139$	21.0354	1.78420	0.892099	−	0.451840i	$-0.149232\pi$
$139$	21.0354	1.78420	0.892099	+	0.451840i	$0.149232\pi$
$140$	0	0
$141$	1.75763	0.148019
$142$	0	0
$143$	9.78170	0.817987
$144$	0	0
$145$	−7.17670	−0.595993
$146$	0	0
$147$	2.67606	0.220717
$148$	0	0
$149$	6.61132	0.541620	0.270810	−	0.962633i	$-0.412708\pi$
$149$	6.61132	0.541620	0.270810	+	0.962633i	$0.412708\pi$
$150$	0	0
$151$	1.00000	0.0813788
$151$	1.00000	0.0813788
$152$	0	0
$153$	16.0853	1.30042
$154$	0	0
$155$	8.23553	0.661494
$156$	0	0
$157$	18.5833	1.48311	0.741555	−	0.670892i	$-0.234088\pi$
$157$	18.5833	1.48311	0.741555	+	0.670892i	$0.234088\pi$
$158$	0	0
$159$	−1.23310	−0.0977912
$160$	0	0
$161$	−0.701225	−0.0552642
$162$	0	0
$163$	2.12360	0.166333	0.0831666	−	0.996536i	$-0.473497\pi$
$163$	2.12360	0.166333	0.0831666	+	0.996536i	$0.473497\pi$
$164$	0	0
$165$	−0.482219	−0.0375407
$166$	0	0
$167$	−11.9362	−0.923650	−0.461825	−	0.886971i	$-0.652805\pi$
$167$	−11.9362	−0.923650	−0.461825	+	0.886971i	$0.652805\pi$
$168$	0	0
$169$	10.2294	0.786878
$170$	0	0
$171$	−10.4809	−0.801492
$172$	0	0
$173$	−4.81892	−0.366376	−0.183188	−	0.983078i	$-0.558642\pi$
$173$	−4.81892	−0.366376	−0.183188	+	0.983078i	$0.558642\pi$
$174$	0	0
$175$	4.27350	0.323046
$176$	0	0
$177$	−3.29650	−0.247780
$178$	0	0
$179$	−12.2951	−0.918975	−0.459488	−	0.888184i	$-0.651967\pi$
$179$	−12.2951	−0.918975	−0.459488	+	0.888184i	$0.651967\pi$
$180$	0	0
$181$	−6.56893	−0.488264	−0.244132	−	0.969742i	$-0.578503\pi$
$181$	−6.56893	−0.488264	−0.244132	+	0.969742i	$0.578503\pi$
$182$	0	0
$183$	−2.84082	−0.209999
$184$	0	0
$185$	7.02949	0.516818
$186$	0	0
$187$	11.0906	0.811023
$188$	0	0
$189$	−6.03501	−0.438983
$190$	0	0
$191$	17.1704	1.24241	0.621205	−	0.783648i	$-0.286644\pi$
$191$	17.1704	1.24241	0.621205	+	0.783648i	$0.286644\pi$
$192$	0	0
$193$	−15.9716	−1.14966	−0.574829	−	0.818274i	$-0.694931\pi$
$193$	−15.9716	−1.14966	−0.574829	+	0.818274i	$0.694931\pi$
$194$	0	0
$195$	−1.14516	−0.0820070
$196$	0	0
$197$	5.95356	0.424173	0.212087	−	0.977251i	$-0.431974\pi$
$197$	5.95356	0.424173	0.212087	+	0.977251i	$0.431974\pi$
$198$	0	0
$199$	−18.7828	−1.33148	−0.665739	−	0.746184i	$-0.731884\pi$
$199$	−18.7828	−1.33148	−0.665739	+	0.746184i	$0.731884\pi$
$200$	0	0
$201$	−1.92183	−0.135556
$202$	0	0
$203$	−30.6696	−2.15259
$204$	0	0
$205$	−7.73949	−0.540550
$206$	0	0
$207$	0.482997	0.0335706
$208$	0	0
$209$	−7.22639	−0.499860
$210$	0	0
$211$	−15.9244	−1.09628	−0.548141	−	0.836386i	$-0.684664\pi$
$211$	−15.9244	−1.09628	−0.548141	+	0.836386i	$0.684664\pi$
$212$	0	0
$213$	−0.199254	−0.0136526
$214$	0	0
$215$	3.47560	0.237034
$216$	0	0
$217$	35.1945	2.38916
$218$	0	0
$219$	−1.07599	−0.0727085
$220$	0	0
$221$	26.3377	1.77167
$222$	0	0
$223$	7.63255	0.511113	0.255557	−	0.966794i	$-0.417741\pi$
$223$	7.63255	0.511113	0.255557	+	0.966794i	$0.417741\pi$
$224$	0	0
$225$	−2.94355	−0.196236
$226$	0	0
$227$	−10.2914	−0.683062	−0.341531	−	0.939871i	$-0.610945\pi$
$227$	−10.2914	−0.683062	−0.341531	+	0.939871i	$0.610945\pi$
$228$	0	0
$229$	−14.5034	−0.958414	−0.479207	−	0.877702i	$-0.659076\pi$
$229$	−14.5034	−0.958414	−0.479207	+	0.877702i	$0.659076\pi$
$230$	0	0
$231$	−2.06076	−0.135588
$232$	0	0
$233$	−27.1081	−1.77591	−0.887954	−	0.459932i	$-0.847874\pi$
$233$	−27.1081	−1.77591	−0.887954	+	0.459932i	$0.847874\pi$
$234$	0	0
$235$	7.39739	0.482553
$236$	0	0
$237$	2.77311	0.180133
$238$	0	0
$239$	18.9948	1.22867	0.614335	−	0.789045i	$-0.289424\pi$
$239$	18.9948	1.22867	0.614335	+	0.789045i	$0.289424\pi$
$240$	0	0
$241$	−7.70051	−0.496033	−0.248017	−	0.968756i	$-0.579779\pi$
$241$	−7.70051	−0.496033	−0.248017	+	0.968756i	$0.579779\pi$
$242$	0	0
$243$	6.25503	0.401260
$244$	0	0
$245$	11.2628	0.719554
$246$	0	0
$247$	−17.1611	−1.09194
$248$	0	0
$249$	−1.59464	−0.101056
$250$	0	0
$251$	−1.17780	−0.0743422	−0.0371711	−	0.999309i	$-0.511835\pi$
$251$	−1.17780	−0.0743422	−0.0371711	+	0.999309i	$0.511835\pi$
$252$	0	0
$253$	0.333019	0.0209367
$254$	0	0
$255$	−1.29840	−0.0813088
$256$	0	0
$257$	20.8865	1.30286	0.651432	−	0.758707i	$-0.274169\pi$
$257$	20.8865	1.30286	0.651432	+	0.758707i	$0.274169\pi$
$258$	0	0
$259$	30.0405	1.86663
$260$	0	0
$261$	21.1250	1.30760
$262$	0	0
$263$	11.2224	0.692003	0.346001	−	0.938234i	$-0.387539\pi$
$263$	11.2224	0.692003	0.346001	+	0.938234i	$0.387539\pi$
$264$	0	0
$265$	−5.18979	−0.318806
$266$	0	0
$267$	−1.72533	−0.105589
$268$	0	0
$269$	12.8636	0.784307	0.392154	−	0.919900i	$-0.371730\pi$
$269$	12.8636	0.784307	0.392154	+	0.919900i	$0.371730\pi$
$270$	0	0
$271$	30.7540	1.86817	0.934085	−	0.357050i	$-0.116218\pi$
$271$	30.7540	1.86817	0.934085	+	0.357050i	$0.116218\pi$
$272$	0	0
$273$	−4.89386	−0.296190
$274$	0	0
$275$	−2.02953	−0.122385
$276$	0	0
$277$	0.904791	0.0543636	0.0271818	−	0.999631i	$-0.491347\pi$
$277$	0.904791	0.0543636	0.0271818	+	0.999631i	$0.491347\pi$
$278$	0	0
$279$	−24.2417	−1.45131
$280$	0	0
$281$	13.5608	0.808970	0.404485	−	0.914545i	$-0.367451\pi$
$281$	13.5608	0.808970	0.404485	+	0.914545i	$0.367451\pi$
$282$	0	0
$283$	−3.26259	−0.193941	−0.0969705	−	0.995287i	$-0.530915\pi$
$283$	−3.26259	−0.193941	−0.0969705	+	0.995287i	$0.530915\pi$
$284$	0	0
$285$	0.846010	0.0501133
$286$	0	0
$287$	−33.0747	−1.95234
$288$	0	0
$289$	12.8619	0.756584
$290$	0	0
$291$	0.739186	0.0433319
$292$	0	0
$293$	16.8087	0.981975	0.490987	−	0.871167i	$-0.336636\pi$
$293$	16.8087	0.981975	0.490987	+	0.871167i	$0.336636\pi$
$294$	0	0
$295$	−13.8741	−0.807780
$296$	0	0
$297$	2.86609	0.166307
$298$	0	0
$299$	0.790847	0.0457359
$300$	0	0
$301$	14.8530	0.856110
$302$	0	0
$303$	0.198178	0.0113850
$304$	0	0
$305$	−11.9562	−0.684613
$306$	0	0
$307$	−19.9288	−1.13740	−0.568699	−	0.822546i	$-0.692553\pi$
$307$	−19.9288	−1.13740	−0.568699	+	0.822546i	$0.692553\pi$
$308$	0	0
$309$	−2.42802	−0.138125
$310$	0	0
$311$	4.71747	0.267503	0.133752	−	0.991015i	$-0.457298\pi$
$311$	4.71747	0.267503	0.133752	+	0.991015i	$0.457298\pi$
$312$	0	0
$313$	18.7915	1.06216	0.531080	−	0.847322i	$-0.321786\pi$
$313$	18.7915	1.06216	0.531080	+	0.847322i	$0.321786\pi$
$314$	0	0
$315$	−12.5792	−0.708760
$316$	0	0
$317$	24.1265	1.35508	0.677540	−	0.735486i	$-0.263046\pi$
$317$	24.1265	1.35508	0.677540	+	0.735486i	$0.263046\pi$
$318$	0	0
$319$	14.5653	0.815502
$320$	0	0
$321$	−2.93189	−0.163642
$322$	0	0
$323$	−19.4574	−1.08264
$324$	0	0
$325$	−4.81969	−0.267348
$326$	0	0
$327$	−1.47073	−0.0813313
$328$	0	0
$329$	31.6128	1.74287
$330$	0	0
$331$	−23.6036	−1.29737	−0.648686	−	0.761056i	$-0.724681\pi$
$331$	−23.6036	−1.29737	−0.648686	+	0.761056i	$0.724681\pi$
$332$	0	0
$333$	−20.6916	−1.13389
$334$	0	0
$335$	−8.08848	−0.441921
$336$	0	0
$337$	2.47693	0.134927	0.0674634	−	0.997722i	$-0.478509\pi$
$337$	2.47693	0.134927	0.0674634	+	0.997722i	$0.478509\pi$
$338$	0	0
$339$	−4.62364	−0.251122
$340$	0	0
$341$	−16.7142	−0.905127
$342$	0	0
$343$	18.2171	0.983629
$344$	0	0
$345$	−0.0389872	−0.00209900
$346$	0	0
$347$	−15.0865	−0.809884	−0.404942	−	0.914342i	$-0.632708\pi$
$347$	−15.0865	−0.809884	−0.404942	+	0.914342i	$0.632708\pi$
$348$	0	0
$349$	−32.7748	−1.75439	−0.877197	−	0.480131i	$-0.840589\pi$
$349$	−32.7748	−1.75439	−0.877197	+	0.480131i	$0.840589\pi$
$350$	0	0
$351$	6.80634	0.363296
$352$	0	0
$353$	−12.4770	−0.664081	−0.332041	−	0.943265i	$-0.607737\pi$
$353$	−12.4770	−0.664081	−0.332041	+	0.943265i	$0.607737\pi$
$354$	0	0
$355$	−0.838605	−0.0445085
$356$	0	0
$357$	−5.54870	−0.293668
$358$	0	0
$359$	30.8290	1.62709	0.813546	−	0.581501i	$-0.197534\pi$
$359$	30.8290	1.62709	0.813546	+	0.581501i	$0.197534\pi$
$360$	0	0
$361$	−6.32194	−0.332734
$362$	0	0
$363$	−1.63494	−0.0858120
$364$	0	0
$365$	−4.52854	−0.237035
$366$	0	0
$367$	8.44434	0.440791	0.220395	−	0.975411i	$-0.429265\pi$
$367$	8.44434	0.440791	0.220395	+	0.975411i	$0.429265\pi$
$368$	0	0
$369$	22.7815	1.18596
$370$	0	0
$371$	−22.1786	−1.15145
$372$	0	0
$373$	−30.2385	−1.56569	−0.782845	−	0.622217i	$-0.786232\pi$
$373$	−30.2385	−1.56569	−0.782845	+	0.622217i	$0.786232\pi$
$374$	0	0
$375$	0.237601	0.0122697
$376$	0	0
$377$	34.5895	1.78145
$378$	0	0
$379$	−35.8112	−1.83950	−0.919748	−	0.392509i	$-0.871607\pi$
$379$	−35.8112	−1.83950	−0.919748	+	0.392509i	$0.871607\pi$
$380$	0	0
$381$	−1.66757	−0.0854323
$382$	0	0
$383$	−9.15450	−0.467773	−0.233887	−	0.972264i	$-0.575144\pi$
$383$	−9.15450	−0.467773	−0.233887	+	0.972264i	$0.575144\pi$
$384$	0	0
$385$	−8.67319	−0.442027
$386$	0	0
$387$	−10.2306	−0.520049
$388$	0	0
$389$	−10.9144	−0.553382	−0.276691	−	0.960959i	$-0.589238\pi$
$389$	−10.9144	−0.553382	−0.276691	+	0.960959i	$0.589238\pi$
$390$	0	0
$391$	0.896669	0.0453465
$392$	0	0
$393$	0.480307	0.0242283
$394$	0	0
$395$	11.6713	0.587246
$396$	0	0
$397$	6.97111	0.349870	0.174935	−	0.984580i	$-0.444028\pi$
$397$	6.97111	0.349870	0.174935	+	0.984580i	$0.444028\pi$
$398$	0	0
$399$	3.61542	0.180997
$400$	0	0
$401$	13.2016	0.659256	0.329628	−	0.944111i	$-0.393077\pi$
$401$	13.2016	0.659256	0.329628	+	0.944111i	$0.393077\pi$
$402$	0	0
$403$	−39.6927	−1.97723
$404$	0	0
$405$	8.49510	0.422125
$406$	0	0
$407$	−14.2666	−0.707167
$408$	0	0
$409$	−17.5141	−0.866018	−0.433009	−	0.901390i	$-0.642548\pi$
$409$	−17.5141	−0.866018	−0.433009	+	0.901390i	$0.642548\pi$
$410$	0	0
$411$	−1.30325	−0.0642844
$412$	0	0
$413$	−59.2909	−2.91751
$414$	0	0
$415$	−6.71143	−0.329451
$416$	0	0
$417$	4.99803	0.244755
$418$	0	0
$419$	15.8073	0.772237	0.386119	−	0.922449i	$-0.373816\pi$
$419$	15.8073	0.772237	0.386119	+	0.922449i	$0.373816\pi$
$420$	0	0
$421$	17.6010	0.857819	0.428909	−	0.903347i	$-0.358898\pi$
$421$	17.6010	0.857819	0.428909	+	0.903347i	$0.358898\pi$
$422$	0	0
$423$	−21.7746	−1.05872
$424$	0	0
$425$	−5.46461	−0.265072
$426$	0	0
$427$	−51.0950	−2.47266
$428$	0	0
$429$	2.32414	0.112211
$430$	0	0
$431$	−3.38829	−0.163208	−0.0816041	−	0.996665i	$-0.526004\pi$
$431$	−3.38829	−0.163208	−0.0816041	+	0.996665i	$0.526004\pi$
$432$	0	0
$433$	9.89980	0.475754	0.237877	−	0.971295i	$-0.423548\pi$
$433$	9.89980	0.475754	0.237877	+	0.971295i	$0.423548\pi$
$434$	0	0
$435$	−1.70519	−0.0817578
$436$	0	0
$437$	−0.584251	−0.0279485
$438$	0	0
$439$	26.6328	1.27112	0.635558	−	0.772053i	$-0.280770\pi$
$439$	26.6328	1.27112	0.635558	+	0.772053i	$0.280770\pi$
$440$	0	0
$441$	−33.1526	−1.57869
$442$	0	0
$443$	11.4998	0.546374	0.273187	−	0.961961i	$-0.411922\pi$
$443$	11.4998	0.546374	0.273187	+	0.961961i	$0.411922\pi$
$444$	0	0
$445$	−7.26146	−0.344226
$446$	0	0
$447$	1.57086	0.0742990
$448$	0	0
$449$	25.0921	1.18417	0.592085	−	0.805876i	$-0.298305\pi$
$449$	25.0921	1.18417	0.592085	+	0.805876i	$0.298305\pi$
$450$	0	0
$451$	15.7075	0.739638
$452$	0	0
$453$	0.237601	0.0111635
$454$	0	0
$455$	−20.5969	−0.965600
$456$	0	0
$457$	4.49466	0.210251	0.105126	−	0.994459i	$-0.466476\pi$
$457$	4.49466	0.210251	0.105126	+	0.994459i	$0.466476\pi$
$458$	0	0
$459$	7.71709	0.360203
$460$	0	0
$461$	−29.1580	−1.35802	−0.679011	−	0.734128i	$-0.737591\pi$
$461$	−29.1580	−1.35802	−0.679011	+	0.734128i	$0.737591\pi$
$462$	0	0
$463$	6.97403	0.324110	0.162055	−	0.986782i	$-0.448188\pi$
$463$	6.97403	0.324110	0.162055	+	0.986782i	$0.448188\pi$
$464$	0	0
$465$	1.95677	0.0907432
$466$	0	0
$467$	31.6575	1.46493	0.732467	−	0.680802i	$-0.238369\pi$
$467$	31.6575	1.46493	0.732467	+	0.680802i	$0.238369\pi$
$468$	0	0
$469$	−34.5661	−1.59611
$470$	0	0
$471$	4.41542	0.203452
$472$	0	0
$473$	−7.05382	−0.324335
$474$	0	0
$475$	3.56063	0.163373
$476$	0	0
$477$	15.2764	0.699457
$478$	0	0
$479$	−29.8506	−1.36391	−0.681954	−	0.731395i	$-0.738870\pi$
$479$	−29.8506	−1.36391	−0.681954	+	0.731395i	$0.738870\pi$
$480$	0	0
$481$	−33.8800	−1.54479
$482$	0	0
$483$	−0.166612	−0.00758110
$484$	0	0
$485$	3.11104	0.141265
$486$	0	0
$487$	17.7584	0.804709	0.402355	−	0.915484i	$-0.368192\pi$
$487$	17.7584	0.804709	0.402355	+	0.915484i	$0.368192\pi$
$488$	0	0
$489$	0.504570	0.0228174
$490$	0	0
$491$	3.08870	0.139391	0.0696955	−	0.997568i	$-0.477797\pi$
$491$	3.08870	0.139391	0.0696955	+	0.997568i	$0.477797\pi$
$492$	0	0
$493$	39.2179	1.76628
$494$	0	0
$495$	5.97401	0.268512
$496$	0	0
$497$	−3.58378	−0.160754
$498$	0	0
$499$	38.0808	1.70473	0.852366	−	0.522945i	$-0.175167\pi$
$499$	38.0808	1.70473	0.852366	+	0.522945i	$0.175167\pi$
$500$	0	0
$501$	−2.83605	−0.126706
$502$	0	0
$503$	−37.5152	−1.67272	−0.836361	−	0.548180i	$-0.815321\pi$
$503$	−37.5152	−1.67272	−0.836361	+	0.548180i	$0.815321\pi$
$504$	0	0
$505$	0.834076	0.0371159
$506$	0	0
$507$	2.43052	0.107943
$508$	0	0
$509$	34.8821	1.54612	0.773061	−	0.634331i	$-0.218724\pi$
$509$	34.8821	1.54612	0.773061	+	0.634331i	$0.218724\pi$
$510$	0	0
$511$	−19.3527	−0.856114
$512$	0	0
$513$	−5.02830	−0.222005
$514$	0	0
$515$	−10.2189	−0.450298
$516$	0	0
$517$	−15.0132	−0.660281
$518$	0	0
$519$	−1.14498	−0.0502591
$520$	0	0
$521$	−44.4217	−1.94615	−0.973075	−	0.230490i	$-0.925967\pi$
$521$	−44.4217	−1.94615	−0.973075	+	0.230490i	$0.925967\pi$
$522$	0	0
$523$	23.2976	1.01873	0.509366	−	0.860550i	$-0.329880\pi$
$523$	23.2976	1.01873	0.509366	+	0.860550i	$0.329880\pi$
$524$	0	0
$525$	1.01539	0.0443152
$526$	0	0
$527$	−45.0039	−1.96040
$528$	0	0
$529$	−22.9731	−0.998829
$530$	0	0
$531$	40.8390	1.77226
$532$	0	0
$533$	37.3020	1.61573
$534$	0	0
$535$	−12.3395	−0.533484
$536$	0	0
$537$	−2.92132	−0.126064
$538$	0	0
$539$	−22.8582	−0.984571
$540$	0	0
$541$	−37.2569	−1.60180	−0.800899	−	0.598799i	$-0.795645\pi$
$541$	−37.2569	−1.60180	−0.800899	+	0.598799i	$0.795645\pi$
$542$	0	0
$543$	−1.56079	−0.0669797
$544$	0	0
$545$	−6.18989	−0.265146
$546$	0	0
$547$	−9.59085	−0.410075	−0.205038	−	0.978754i	$-0.565732\pi$
$547$	−9.59085	−0.410075	−0.205038	+	0.978754i	$0.565732\pi$
$548$	0	0
$549$	35.1938	1.50203
$550$	0	0
$551$	−25.5536	−1.08862
$552$	0	0
$553$	49.8772	2.12099
$554$	0	0
$555$	1.67022	0.0708967
$556$	0	0
$557$	18.3090	0.775777	0.387889	−	0.921706i	$-0.373205\pi$
$557$	18.3090	0.775777	0.387889	+	0.921706i	$0.373205\pi$
$558$	0	0
$559$	−16.7513	−0.708504
$560$	0	0
$561$	2.63514	0.111256
$562$	0	0
$563$	−13.2883	−0.560034	−0.280017	−	0.959995i	$-0.590340\pi$
$563$	−13.2883	−0.560034	−0.280017	+	0.959995i	$0.590340\pi$
$564$	0	0
$565$	−19.4597	−0.818674
$566$	0	0
$567$	36.3038	1.52462
$568$	0	0
$569$	4.45598	0.186804	0.0934022	−	0.995628i	$-0.470226\pi$
$569$	4.45598	0.186804	0.0934022	+	0.995628i	$0.470226\pi$
$570$	0	0
$571$	−35.1551	−1.47120	−0.735598	−	0.677418i	$-0.763099\pi$
$571$	−35.1551	−1.47120	−0.735598	+	0.677418i	$0.763099\pi$
$572$	0	0
$573$	4.07972	0.170433
$574$	0	0
$575$	−0.164087	−0.00684289
$576$	0	0
$577$	−25.5308	−1.06286	−0.531430	−	0.847102i	$-0.678345\pi$
$577$	−25.5308	−1.06286	−0.531430	+	0.847102i	$0.678345\pi$
$578$	0	0
$579$	−3.79486	−0.157709
$580$	0	0
$581$	−28.6813	−1.18990
$582$	0	0
$583$	10.5328	0.436225
$584$	0	0
$585$	14.1870	0.586559
$586$	0	0
$587$	−25.1426	−1.03775	−0.518873	−	0.854852i	$-0.673648\pi$
$587$	−25.1426	−1.03775	−0.518873	+	0.854852i	$0.673648\pi$
$588$	0	0
$589$	29.3236	1.20826
$590$	0	0
$591$	1.41457	0.0581878
$592$	0	0
$593$	19.2215	0.789333	0.394667	−	0.918824i	$-0.370860\pi$
$593$	19.2215	0.789333	0.394667	+	0.918824i	$0.370860\pi$
$594$	0	0
$595$	−23.3530	−0.957379
$596$	0	0
$597$	−4.46282	−0.182651
$598$	0	0
$599$	−46.5042	−1.90011	−0.950054	−	0.312084i	$-0.898973\pi$
$599$	−46.5042	−1.90011	−0.950054	+	0.312084i	$0.898973\pi$
$600$	0	0
$601$	−14.9252	−0.608814	−0.304407	−	0.952542i	$-0.598458\pi$
$601$	−14.9252	−0.608814	−0.304407	+	0.952542i	$0.598458\pi$
$602$	0	0
$603$	23.8088	0.969569
$604$	0	0
$605$	−6.88101	−0.279753
$606$	0	0
$607$	28.4843	1.15614	0.578070	−	0.815987i	$-0.303806\pi$
$607$	28.4843	1.15614	0.578070	+	0.815987i	$0.303806\pi$
$608$	0	0
$609$	−7.28715	−0.295290
$610$	0	0
$611$	−35.6531	−1.44237
$612$	0	0
$613$	3.66993	0.148227	0.0741135	−	0.997250i	$-0.476387\pi$
$613$	3.66993	0.148227	0.0741135	+	0.997250i	$0.476387\pi$
$614$	0	0
$615$	−1.83891	−0.0741522
$616$	0	0
$617$	−7.29051	−0.293505	−0.146752	−	0.989173i	$-0.546882\pi$
$617$	−7.29051	−0.293505	−0.146752	+	0.989173i	$0.546882\pi$
$618$	0	0
$619$	8.93223	0.359017	0.179508	−	0.983756i	$-0.442549\pi$
$619$	8.93223	0.359017	0.179508	+	0.983756i	$0.442549\pi$
$620$	0	0
$621$	0.231722	0.00929870
$622$	0	0
$623$	−31.0318	−1.24326
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−1.71700	−0.0685704
$628$	0	0
$629$	−38.4134	−1.53164
$630$	0	0
$631$	5.16658	0.205678	0.102839	−	0.994698i	$-0.467207\pi$
$631$	5.16658	0.205678	0.102839	+	0.994698i	$0.467207\pi$
$632$	0	0
$633$	−3.78366	−0.150387
$634$	0	0
$635$	−7.01836	−0.278515
$636$	0	0
$637$	−54.2832	−2.15078
$638$	0	0
$639$	2.46847	0.0976513
$640$	0	0
$641$	−19.5346	−0.771570	−0.385785	−	0.922589i	$-0.626069\pi$
$641$	−19.5346	−0.771570	−0.385785	+	0.922589i	$0.626069\pi$
$642$	0	0
$643$	30.9124	1.21907	0.609533	−	0.792761i	$-0.291357\pi$
$643$	30.9124	1.21907	0.609533	+	0.792761i	$0.291357\pi$
$644$	0	0
$645$	0.825806	0.0325161
$646$	0	0
$647$	33.2644	1.30776	0.653879	−	0.756599i	$-0.273140\pi$
$647$	33.2644	1.30776	0.653879	+	0.756599i	$0.273140\pi$
$648$	0	0
$649$	28.1578	1.10529
$650$	0	0
$651$	8.36227	0.327743
$652$	0	0
$653$	48.1848	1.88562	0.942808	−	0.333336i	$-0.108174\pi$
$653$	48.1848	1.88562	0.942808	+	0.333336i	$0.108174\pi$
$654$	0	0
$655$	2.02148	0.0789859
$656$	0	0
$657$	13.3300	0.520052
$658$	0	0
$659$	18.0500	0.703129	0.351565	−	0.936164i	$-0.385650\pi$
$659$	18.0500	0.703129	0.351565	+	0.936164i	$0.385650\pi$
$660$	0	0
$661$	14.7920	0.575342	0.287671	−	0.957729i	$-0.407119\pi$
$661$	14.7920	0.575342	0.287671	+	0.957729i	$0.407119\pi$
$662$	0	0
$663$	6.25788	0.243036
$664$	0	0
$665$	15.2163	0.590064
$666$	0	0
$667$	1.17760	0.0455969
$668$	0	0
$669$	1.81350	0.0701141
$670$	0	0
$671$	24.2655	0.936761
$672$	0	0
$673$	12.5320	0.483073	0.241537	−	0.970392i	$-0.422349\pi$
$673$	12.5320	0.483073	0.241537	+	0.970392i	$0.422349\pi$
$674$	0	0
$675$	−1.41219	−0.0543554
$676$	0	0
$677$	13.5499	0.520765	0.260382	−	0.965506i	$-0.416151\pi$
$677$	13.5499	0.520765	0.260382	+	0.965506i	$0.416151\pi$
$678$	0	0
$679$	13.2950	0.510216
$680$	0	0
$681$	−2.44524	−0.0937018
$682$	0	0
$683$	−2.48253	−0.0949914	−0.0474957	−	0.998871i	$-0.515124\pi$
$683$	−2.48253	−0.0949914	−0.0474957	+	0.998871i	$0.515124\pi$
$684$	0	0
$685$	−5.48501	−0.209572
$686$	0	0
$687$	−3.44604	−0.131475
$688$	0	0
$689$	25.0132	0.952926
$690$	0	0
$691$	−11.2056	−0.426282	−0.213141	−	0.977021i	$-0.568369\pi$
$691$	−11.2056	−0.426282	−0.213141	+	0.977021i	$0.568369\pi$
$692$	0	0
$693$	25.5299	0.969802
$694$	0	0
$695$	21.0354	0.797917
$696$	0	0
$697$	42.2933	1.60197
$698$	0	0
$699$	−6.44091	−0.243618
$700$	0	0
$701$	49.3595	1.86428	0.932141	−	0.362095i	$-0.117938\pi$
$701$	49.3595	1.86428	0.932141	+	0.362095i	$0.117938\pi$
$702$	0	0
$703$	25.0294	0.944001
$704$	0	0
$705$	1.75763	0.0661962
$706$	0	0
$707$	3.56442	0.134054
$708$	0	0
$709$	27.1001	1.01777	0.508883	−	0.860836i	$-0.330058\pi$
$709$	27.1001	1.01777	0.508883	+	0.860836i	$0.330058\pi$
$710$	0	0
$711$	−34.3549	−1.28841
$712$	0	0
$713$	−1.35134	−0.0506081
$714$	0	0
$715$	9.78170	0.365815
$716$	0	0
$717$	4.51319	0.168548
$718$	0	0
$719$	36.1601	1.34854	0.674272	−	0.738483i	$-0.264457\pi$
$719$	36.1601	1.34854	0.674272	+	0.738483i	$0.264457\pi$
$720$	0	0
$721$	−43.6704	−1.62637
$722$	0	0
$723$	−1.82965	−0.0680455
$724$	0	0
$725$	−7.17670	−0.266536
$726$	0	0
$727$	19.4147	0.720053	0.360026	−	0.932942i	$-0.382768\pi$
$727$	19.4147	0.720053	0.360026	+	0.932942i	$0.382768\pi$
$728$	0	0
$729$	−23.9991	−0.888855
$730$	0	0
$731$	−18.9928	−0.702473
$732$	0	0
$733$	19.2990	0.712825	0.356413	−	0.934329i	$-0.384000\pi$
$733$	19.2990	0.712825	0.356413	+	0.934329i	$0.384000\pi$
$734$	0	0
$735$	2.67606	0.0987078
$736$	0	0
$737$	16.4158	0.604683
$738$	0	0
$739$	35.2703	1.29744	0.648720	−	0.761027i	$-0.275305\pi$
$739$	35.2703	1.29744	0.648720	+	0.761027i	$0.275305\pi$
$740$	0	0
$741$	−4.07750	−0.149791
$742$	0	0
$743$	3.91163	0.143504	0.0717520	−	0.997423i	$-0.477141\pi$
$743$	3.91163	0.143504	0.0717520	+	0.997423i	$0.477141\pi$
$744$	0	0
$745$	6.61132	0.242220
$746$	0	0
$747$	19.7554	0.722812
$748$	0	0
$749$	−52.7330	−1.92682
$750$	0	0
$751$	−10.9269	−0.398727	−0.199364	−	0.979926i	$-0.563888\pi$
$751$	−10.9269	−0.398727	−0.199364	+	0.979926i	$0.563888\pi$
$752$	0	0
$753$	−0.279847	−0.0101982
$754$	0	0
$755$	1.00000	0.0363937
$756$	0	0
$757$	−0.733083	−0.0266444	−0.0133222	−	0.999911i	$-0.504241\pi$
$757$	−0.733083	−0.0266444	−0.0133222	+	0.999911i	$0.504241\pi$
$758$	0	0
$759$	0.0791257	0.00287208
$760$	0	0
$761$	0.956193	0.0346620	0.0173310	−	0.999850i	$-0.494483\pi$
$761$	0.956193	0.0346620	0.0173310	+	0.999850i	$0.494483\pi$
$762$	0	0
$763$	−26.4525	−0.957644
$764$	0	0
$765$	16.0853	0.581566
$766$	0	0
$767$	66.8688	2.41449
$768$	0	0
$769$	23.4489	0.845590	0.422795	−	0.906225i	$-0.361049\pi$
$769$	23.4489	0.845590	0.422795	+	0.906225i	$0.361049\pi$
$770$	0	0
$771$	4.96266	0.178726
$772$	0	0
$773$	11.0156	0.396205	0.198103	−	0.980181i	$-0.436522\pi$
$773$	11.0156	0.396205	0.198103	+	0.980181i	$0.436522\pi$
$774$	0	0
$775$	8.23553	0.295829
$776$	0	0
$777$	7.13767	0.256062
$778$	0	0
$779$	−27.5574	−0.987348
$780$	0	0
$781$	1.70197	0.0609014
$782$	0	0
$783$	10.1349	0.362192
$784$	0	0
$785$	18.5833	0.663267
$786$	0	0
$787$	−10.1718	−0.362584	−0.181292	−	0.983429i	$-0.558028\pi$
$787$	−10.1718	−0.362584	−0.181292	+	0.983429i	$0.558028\pi$
$788$	0	0
$789$	2.66646	0.0949284
$790$	0	0
$791$	−83.1608	−2.95686
$792$	0	0
$793$	57.6254	2.04634
$794$	0	0
$795$	−1.23310	−0.0437336
$796$	0	0
$797$	6.05676	0.214542	0.107271	−	0.994230i	$-0.465789\pi$
$797$	6.05676	0.214542	0.107271	+	0.994230i	$0.465789\pi$
$798$	0	0
$799$	−40.4238	−1.43009
$800$	0	0
$801$	21.3744	0.755228
$802$	0	0
$803$	9.19081	0.324336
$804$	0	0
$805$	−0.701225	−0.0247149
$806$	0	0
$807$	3.05641	0.107591
$808$	0	0
$809$	9.09326	0.319702	0.159851	−	0.987141i	$-0.448899\pi$
$809$	9.09326	0.319702	0.159851	+	0.987141i	$0.448899\pi$
$810$	0	0
$811$	−6.31290	−0.221676	−0.110838	−	0.993838i	$-0.535353\pi$
$811$	−6.31290	−0.221676	−0.110838	+	0.993838i	$0.535353\pi$
$812$	0	0
$813$	7.30718	0.256274
$814$	0	0
$815$	2.12360	0.0743864
$816$	0	0
$817$	12.3753	0.432957
$818$	0	0
$819$	60.6280	2.11852
$820$	0	0
$821$	−37.6187	−1.31290	−0.656450	−	0.754369i	$-0.727943\pi$
$821$	−37.6187	−1.31290	−0.656450	+	0.754369i	$0.727943\pi$
$822$	0	0
$823$	42.1888	1.47061	0.735305	−	0.677737i	$-0.237039\pi$
$823$	42.1888	1.47061	0.735305	+	0.677737i	$0.237039\pi$
$824$	0	0
$825$	−0.482219	−0.0167887
$826$	0	0
$827$	4.72001	0.164131	0.0820655	−	0.996627i	$-0.473848\pi$
$827$	4.72001	0.164131	0.0820655	+	0.996627i	$0.473848\pi$
$828$	0	0
$829$	−45.3958	−1.57666	−0.788330	−	0.615253i	$-0.789054\pi$
$829$	−45.3958	−1.57666	−0.788330	+	0.615253i	$0.789054\pi$
$830$	0	0
$831$	0.214980	0.00745756
$832$	0	0
$833$	−61.5467	−2.13247
$834$	0	0
$835$	−11.9362	−0.413069
$836$	0	0
$837$	−11.6302	−0.401997
$838$	0	0
$839$	6.32046	0.218207	0.109103	−	0.994030i	$-0.465202\pi$
$839$	6.32046	0.218207	0.109103	+	0.994030i	$0.465202\pi$
$840$	0	0
$841$	22.5051	0.776037
$842$	0	0
$843$	3.22207	0.110974
$844$	0	0
$845$	10.2294	0.351903
$846$	0	0
$847$	−29.4060	−1.01040
$848$	0	0
$849$	−0.775197	−0.0266047
$850$	0	0
$851$	−1.15345	−0.0395396
$852$	0	0
$853$	−38.5429	−1.31968	−0.659842	−	0.751404i	$-0.729377\pi$
$853$	−38.5429	−1.31968	−0.659842	+	0.751404i	$0.729377\pi$
$854$	0	0
$855$	−10.4809	−0.358438
$856$	0	0
$857$	38.4681	1.31405	0.657023	−	0.753871i	$-0.271816\pi$
$857$	38.4681	1.31405	0.657023	+	0.753871i	$0.271816\pi$
$858$	0	0
$859$	−4.61818	−0.157570	−0.0787851	−	0.996892i	$-0.525104\pi$
$859$	−4.61818	−0.157570	−0.0787851	+	0.996892i	$0.525104\pi$
$860$	0	0
$861$	−7.85860	−0.267820
$862$	0	0
$863$	2.05034	0.0697943	0.0348971	−	0.999391i	$-0.488890\pi$
$863$	2.05034	0.0697943	0.0348971	+	0.999391i	$0.488890\pi$
$864$	0	0
$865$	−4.81892	−0.163848
$866$	0	0
$867$	3.05601	0.103788
$868$	0	0
$869$	−23.6872	−0.803533
$870$	0	0
$871$	38.9839	1.32092
$872$	0	0
$873$	−9.15748	−0.309934
$874$	0	0
$875$	4.27350	0.144471
$876$	0	0
$877$	−55.4576	−1.87267	−0.936335	−	0.351107i	$-0.885805\pi$
$877$	−55.4576	−1.87267	−0.936335	+	0.351107i	$0.885805\pi$
$878$	0	0
$879$	3.99377	0.134707
$880$	0	0
$881$	−28.3974	−0.956734	−0.478367	−	0.878160i	$-0.658771\pi$
$881$	−28.3974	−0.956734	−0.478367	+	0.878160i	$0.658771\pi$
$882$	0	0
$883$	24.6982	0.831160	0.415580	−	0.909557i	$-0.363579\pi$
$883$	24.6982	0.831160	0.415580	+	0.909557i	$0.363579\pi$
$884$	0	0
$885$	−3.29650	−0.110811
$886$	0	0
$887$	−38.3374	−1.28724	−0.643622	−	0.765343i	$-0.722569\pi$
$887$	−38.3374	−1.28724	−0.643622	+	0.765343i	$0.722569\pi$
$888$	0	0
$889$	−29.9929	−1.00593
$890$	0	0
$891$	−17.2410	−0.577597
$892$	0	0
$893$	26.3393	0.881413
$894$	0	0
$895$	−12.2951	−0.410978
$896$	0	0
$897$	0.187906	0.00627401
$898$	0	0
$899$	−59.1039	−1.97123
$900$	0	0
$901$	28.3601	0.944813
$902$	0	0
$903$	3.52908	0.117440
$904$	0	0
$905$	−6.56893	−0.218358
$906$	0	0
$907$	7.77151	0.258049	0.129024	−	0.991641i	$-0.458815\pi$
$907$	7.77151	0.258049	0.129024	+	0.991641i	$0.458815\pi$
$908$	0	0
$909$	−2.45514	−0.0814319
$910$	0	0
$911$	19.3165	0.639984	0.319992	−	0.947420i	$-0.396320\pi$
$911$	19.3165	0.639984	0.319992	+	0.947420i	$0.396320\pi$
$912$	0	0
$913$	13.6210	0.450790
$914$	0	0
$915$	−2.84082	−0.0939146
$916$	0	0
$917$	8.63881	0.285279
$918$	0	0
$919$	−37.3484	−1.23201	−0.616005	−	0.787742i	$-0.711250\pi$
$919$	−37.3484	−1.23201	−0.616005	+	0.787742i	$0.711250\pi$
$920$	0	0
$921$	−4.73512	−0.156027
$922$	0	0
$923$	4.04182	0.133038
$924$	0	0
$925$	7.02949	0.231128
$926$	0	0
$927$	30.0797	0.987948
$928$	0	0
$929$	13.9336	0.457146	0.228573	−	0.973527i	$-0.426594\pi$
$929$	13.9336	0.457146	0.228573	+	0.973527i	$0.426594\pi$
$930$	0	0
$931$	40.1026	1.31431
$932$	0	0
$933$	1.12088	0.0366959
$934$	0	0
$935$	11.0906	0.362701
$936$	0	0
$937$	−32.3733	−1.05759	−0.528794	−	0.848750i	$-0.677356\pi$
$937$	−32.3733	−1.05759	−0.528794	+	0.848750i	$0.677356\pi$
$938$	0	0
$939$	4.46489	0.145706
$940$	0	0
$941$	23.7927	0.775621	0.387811	−	0.921739i	$-0.373231\pi$
$941$	23.7927	0.775621	0.387811	+	0.921739i	$0.373231\pi$
$942$	0	0
$943$	1.26995	0.0413552
$944$	0	0
$945$	−6.03501	−0.196319
$946$	0	0
$947$	−26.6485	−0.865961	−0.432981	−	0.901403i	$-0.642538\pi$
$947$	−26.6485	−0.865961	−0.432981	+	0.901403i	$0.642538\pi$
$948$	0	0
$949$	21.8262	0.708507
$950$	0	0
$951$	5.73249	0.185889
$952$	0	0
$953$	26.4232	0.855932	0.427966	−	0.903795i	$-0.359230\pi$
$953$	26.4232	0.855932	0.427966	+	0.903795i	$0.359230\pi$
$954$	0	0
$955$	17.1704	0.555622
$956$	0	0
$957$	3.46074	0.111870
$958$	0	0
$959$	−23.4402	−0.756923
$960$	0	0
$961$	36.8240	1.18787
$962$	0	0
$963$	36.3220	1.17046
$964$	0	0
$965$	−15.9716	−0.514143
$966$	0	0
$967$	50.5130	1.62439	0.812195	−	0.583386i	$-0.198273\pi$
$967$	50.5130	1.62439	0.812195	+	0.583386i	$0.198273\pi$
$968$	0	0
$969$	−4.62311	−0.148516
$970$	0	0
$971$	33.7689	1.08370	0.541848	−	0.840476i	$-0.317725\pi$
$971$	33.7689	1.08370	0.541848	+	0.840476i	$0.317725\pi$
$972$	0	0
$973$	89.8947	2.88189
$974$	0	0
$975$	−1.14516	−0.0366746
$976$	0	0
$977$	2.09686	0.0670844	0.0335422	−	0.999437i	$-0.489321\pi$
$977$	2.09686	0.0670844	0.0335422	+	0.999437i	$0.489321\pi$
$978$	0	0
$979$	14.7373	0.471007
$980$	0	0
$981$	18.2202	0.581727
$982$	0	0
$983$	54.3369	1.73308	0.866540	−	0.499108i	$-0.166339\pi$
$983$	54.3369	1.73308	0.866540	+	0.499108i	$0.166339\pi$
$984$	0	0
$985$	5.95356	0.189696
$986$	0	0
$987$	7.51123	0.239085
$988$	0	0
$989$	−0.570299	−0.0181345
$990$	0	0
$991$	−57.9400	−1.84053	−0.920263	−	0.391300i	$-0.872025\pi$
$991$	−57.9400	−1.84053	−0.920263	+	0.391300i	$0.872025\pi$
$992$	0	0
$993$	−5.60825	−0.177972
$994$	0	0
$995$	−18.7828	−0.595455
$996$	0	0
$997$	−28.5055	−0.902779	−0.451390	−	0.892327i	$-0.649072\pi$
$997$	−28.5055	−0.902779	−0.451390	+	0.892327i	$0.649072\pi$
$998$	0	0
$999$	−9.92701	−0.314077

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6040.2.a.p.1.10	✓	19

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6040.2.a.p.1.10	✓	19	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:	\( N \)	\(=\)	\( 6040 = 2^{3} \cdot 5 \cdot 151 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6040.a (trivial)

\(f(q)\)	\(=\)	\(q+0.237601 q^{3} +1.00000 q^{5} +4.27350 q^{7} -2.94355 q^{9} +O(q^{10})\)	\(q+0.237601 q^{3} +1.00000 q^{5} +4.27350 q^{7} -2.94355 q^{9} -2.02953 q^{11} -4.81969 q^{13} +0.237601 q^{15} -5.46461 q^{17} +3.56063 q^{19} +1.01539 q^{21} -0.164087 q^{23} +1.00000 q^{25} -1.41219 q^{27} -7.17670 q^{29} +8.23553 q^{31} -0.482219 q^{33} +4.27350 q^{35} +7.02949 q^{37} -1.14516 q^{39} -7.73949 q^{41} +3.47560 q^{43} -2.94355 q^{45} +7.39739 q^{47} +11.2628 q^{49} -1.29840 q^{51} -5.18979 q^{53} -2.02953 q^{55} +0.846010 q^{57} -13.8741 q^{59} -11.9562 q^{61} -12.5792 q^{63} -4.81969 q^{65} -8.08848 q^{67} -0.0389872 q^{69} -0.838605 q^{71} -4.52854 q^{73} +0.237601 q^{75} -8.67319 q^{77} +11.6713 q^{79} +8.49510 q^{81} -6.71143 q^{83} -5.46461 q^{85} -1.70519 q^{87} -7.26146 q^{89} -20.5969 q^{91} +1.95677 q^{93} +3.56063 q^{95} +3.11104 q^{97} +5.97401 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 19 q - 5 q^{3} + 19 q^{5} - 8 q^{7} + 26 q^{9}+O(q^{10}) \) 19 * q - 5 * q^3 + 19 * q^5 - 8 * q^7 + 26 * q^9	\( 19 q - 5 q^{3} + 19 q^{5} - 8 q^{7} + 26 q^{9} - 18 q^{11} + 5 q^{13} - 5 q^{15} - 4 q^{17} - 27 q^{19} - 18 q^{21} - 25 q^{23} + 19 q^{25} - 35 q^{27} - 35 q^{29} - 26 q^{31} - 8 q^{35} - 10 q^{37} - 48 q^{39} - 14 q^{41} - 21 q^{43} + 26 q^{45} - 40 q^{47} + 23 q^{49} - 32 q^{51} - 3 q^{53} - 18 q^{55} - 13 q^{57} - 28 q^{59} - 46 q^{61} - 53 q^{63} + 5 q^{65} - 42 q^{67} - 31 q^{69} - 46 q^{71} + 31 q^{73} - 5 q^{75} + 15 q^{77} - 56 q^{79} + 31 q^{81} - 25 q^{83} - 4 q^{85} - 20 q^{87} - 7 q^{89} - 61 q^{91} + 29 q^{93} - 27 q^{95} + 39 q^{97} - 52 q^{99}+O(q^{100}) \) 19 * q - 5 * q^3 + 19 * q^5 - 8 * q^7 + 26 * q^9 - 18 * q^11 + 5 * q^13 - 5 * q^15 - 4 * q^17 - 27 * q^19 - 18 * q^21 - 25 * q^23 + 19 * q^25 - 35 * q^27 - 35 * q^29 - 26 * q^31 - 8 * q^35 - 10 * q^37 - 48 * q^39 - 14 * q^41 - 21 * q^43 + 26 * q^45 - 40 * q^47 + 23 * q^49 - 32 * q^51 - 3 * q^53 - 18 * q^55 - 13 * q^57 - 28 * q^59 - 46 * q^61 - 53 * q^63 + 5 * q^65 - 42 * q^67 - 31 * q^69 - 46 * q^71 + 31 * q^73 - 5 * q^75 + 15 * q^77 - 56 * q^79 + 31 * q^81 - 25 * q^83 - 4 * q^85 - 20 * q^87 - 7 * q^89 - 61 * q^91 + 29 * q^93 - 27 * q^95 + 39 * q^97 - 52 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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