LMFDB - Embedded newform 1380.4.a.j.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1380,4,Mod(1,1380)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1380, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1380.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1380 = 2^{2} \cdot 3 \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1380.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:	$81.4226358079$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 1420x^{5} - 7866x^{4} + 519199x^{3} + 5329890x^{2} - 8528484x - 84125016 $ x^7 - 1420x^5 - 7866x^4 + 519199x^3 + 5329890x^2 - 8528484*x - 84125016
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-3.94389$ of defining polynomial
Character	$\chi$	$=$	1380.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.00000 q^{3} +5.00000 q^{5} +8.94389 q^{7} +9.00000 q^{9} +O(q^{10})$	$q+3.00000 q^{3} +5.00000 q^{5} +8.94389 q^{7} +9.00000 q^{9} +55.8775 q^{11} +43.0658 q^{13} +15.0000 q^{15} -75.2623 q^{17} +26.7676 q^{19} +26.8317 q^{21} +23.0000 q^{23} +25.0000 q^{25} +27.0000 q^{27} +57.1622 q^{29} +150.649 q^{31} +167.632 q^{33} +44.7194 q^{35} +317.031 q^{37} +129.197 q^{39} -121.969 q^{41} -313.610 q^{43} +45.0000 q^{45} +386.090 q^{47} -263.007 q^{49} -225.787 q^{51} -249.005 q^{53} +279.387 q^{55} +80.3029 q^{57} -144.437 q^{59} -673.564 q^{61} +80.4950 q^{63} +215.329 q^{65} +629.461 q^{67} +69.0000 q^{69} -873.585 q^{71} -522.460 q^{73} +75.0000 q^{75} +499.762 q^{77} +708.680 q^{79} +81.0000 q^{81} +946.192 q^{83} -376.312 q^{85} +171.487 q^{87} +569.069 q^{89} +385.176 q^{91} +451.947 q^{93} +133.838 q^{95} -816.579 q^{97} +502.897 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 21 q^{3} + 35 q^{5} + 35 q^{7} + 63 q^{9}+O(q^{10}) $ 7 * q + 21 * q^3 + 35 * q^5 + 35 * q^7 + 63 * q^9	$ 7 q + 21 q^{3} + 35 q^{5} + 35 q^{7} + 63 q^{9} + 48 q^{11} + 90 q^{13} + 105 q^{15} + 163 q^{17} + 200 q^{19} + 105 q^{21} + 161 q^{23} + 175 q^{25} + 189 q^{27} + 81 q^{29} + 125 q^{31} + 144 q^{33} + 175 q^{35} + 5 q^{37} + 270 q^{39} + 369 q^{41} + 462 q^{43} + 315 q^{45} + 134 q^{47} + 614 q^{49} + 489 q^{51} + 561 q^{53} + 240 q^{55} + 600 q^{57} + 951 q^{59} + 860 q^{61} + 315 q^{63} + 450 q^{65} + 447 q^{67} + 483 q^{69} + 735 q^{71} + 1460 q^{73} + 525 q^{75} + 496 q^{77} + 18 q^{79} + 567 q^{81} + 261 q^{83} + 815 q^{85} + 243 q^{87} + 2024 q^{89} + 692 q^{91} + 375 q^{93} + 1000 q^{95} + 668 q^{97} + 432 q^{99}+O(q^{100}) $ 7 * q + 21 * q^3 + 35 * q^5 + 35 * q^7 + 63 * q^9 + 48 * q^11 + 90 * q^13 + 105 * q^15 + 163 * q^17 + 200 * q^19 + 105 * q^21 + 161 * q^23 + 175 * q^25 + 189 * q^27 + 81 * q^29 + 125 * q^31 + 144 * q^33 + 175 * q^35 + 5 * q^37 + 270 * q^39 + 369 * q^41 + 462 * q^43 + 315 * q^45 + 134 * q^47 + 614 * q^49 + 489 * q^51 + 561 * q^53 + 240 * q^55 + 600 * q^57 + 951 * q^59 + 860 * q^61 + 315 * q^63 + 450 * q^65 + 447 * q^67 + 483 * q^69 + 735 * q^71 + 1460 * q^73 + 525 * q^75 + 496 * q^77 + 18 * q^79 + 567 * q^81 + 261 * q^83 + 815 * q^85 + 243 * q^87 + 2024 * q^89 + 692 * q^91 + 375 * q^93 + 1000 * q^95 + 668 * q^97 + 432 * 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
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	0	0
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	8.94389	0.482925	0.241462	−	0.970410i	$-0.422373\pi$
$7$	8.94389	0.482925	0.241462	+	0.970410i	$0.422373\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	55.8775	1.53161	0.765804	−	0.643074i	$-0.222341\pi$
$11$	55.8775	1.53161	0.765804	+	0.643074i	$0.222341\pi$
$12$	0	0
$13$	43.0658	0.918792	0.459396	−	0.888231i	$-0.348066\pi$
$13$	43.0658	0.918792	0.459396	+	0.888231i	$0.348066\pi$
$14$	0	0
$15$	15.0000	0.258199
$16$	0	0
$17$	−75.2623	−1.07375	−0.536876	−	0.843661i	$-0.680396\pi$
$17$	−75.2623	−1.07375	−0.536876	+	0.843661i	$0.680396\pi$
$18$	0	0
$19$	26.7676	0.323206	0.161603	−	0.986856i	$-0.448334\pi$
$19$	26.7676	0.323206	0.161603	+	0.986856i	$0.448334\pi$
$20$	0	0
$21$	26.8317	0.278817
$22$	0	0
$23$	23.0000	0.208514
$23$	23.0000	0.208514
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	27.0000	0.192450
$28$	0	0
$29$	57.1622	0.366026	0.183013	−	0.983110i	$-0.441415\pi$
$29$	57.1622	0.366026	0.183013	+	0.983110i	$0.441415\pi$
$30$	0	0
$31$	150.649	0.872819	0.436409	−	0.899748i	$-0.356250\pi$
$31$	150.649	0.872819	0.436409	+	0.899748i	$0.356250\pi$
$32$	0	0
$33$	167.632	0.884275
$34$	0	0
$35$	44.7194	0.215970
$36$	0	0
$37$	317.031	1.40864	0.704318	−	0.709884i	$-0.251253\pi$
$37$	317.031	1.40864	0.704318	+	0.709884i	$0.251253\pi$
$38$	0	0
$39$	129.197	0.530465
$40$	0	0
$41$	−121.969	−0.464596	−0.232298	−	0.972645i	$-0.574624\pi$
$41$	−121.969	−0.464596	−0.232298	+	0.972645i	$0.574624\pi$
$42$	0	0
$43$	−313.610	−1.11221	−0.556106	−	0.831112i	$-0.687705\pi$
$43$	−313.610	−1.11221	−0.556106	+	0.831112i	$0.687705\pi$
$44$	0	0
$45$	45.0000	0.149071
$46$	0	0
$47$	386.090	1.19823	0.599117	−	0.800662i	$-0.295518\pi$
$47$	386.090	1.19823	0.599117	+	0.800662i	$0.295518\pi$
$48$	0	0
$49$	−263.007	−0.766784
$50$	0	0
$51$	−225.787	−0.619931
$52$	0	0
$53$	−249.005	−0.645349	−0.322674	−	0.946510i	$-0.604582\pi$
$53$	−249.005	−0.645349	−0.322674	+	0.946510i	$0.604582\pi$
$54$	0	0
$55$	279.387	0.684956
$56$	0	0
$57$	80.3029	0.186603
$58$	0	0
$59$	−144.437	−0.318714	−0.159357	−	0.987221i	$-0.550942\pi$
$59$	−144.437	−0.318714	−0.159357	+	0.987221i	$0.550942\pi$
$60$	0	0
$61$	−673.564	−1.41379	−0.706894	−	0.707320i	$-0.749904\pi$
$61$	−673.564	−1.41379	−0.706894	+	0.707320i	$0.749904\pi$
$62$	0	0
$63$	80.4950	0.160975
$64$	0	0
$65$	215.329	0.410896
$66$	0	0
$67$	629.461	1.14778	0.573888	−	0.818934i	$-0.305435\pi$
$67$	629.461	1.14778	0.573888	+	0.818934i	$0.305435\pi$
$68$	0	0
$69$	69.0000	0.120386
$70$	0	0
$71$	−873.585	−1.46022	−0.730109	−	0.683331i	$-0.760531\pi$
$71$	−873.585	−1.46022	−0.730109	+	0.683331i	$0.760531\pi$
$72$	0	0
$73$	−522.460	−0.837661	−0.418831	−	0.908064i	$-0.637560\pi$
$73$	−522.460	−0.837661	−0.418831	+	0.908064i	$0.637560\pi$
$74$	0	0
$75$	75.0000	0.115470
$76$	0	0
$77$	499.762	0.739651
$78$	0	0
$79$	708.680	1.00927	0.504637	−	0.863331i	$-0.331626\pi$
$79$	708.680	1.00927	0.504637	+	0.863331i	$0.331626\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	946.192	1.25130	0.625651	−	0.780103i	$-0.284834\pi$
$83$	946.192	1.25130	0.625651	+	0.780103i	$0.284834\pi$
$84$	0	0
$85$	−376.312	−0.480197
$86$	0	0
$87$	171.487	0.211325
$88$	0	0
$89$	569.069	0.677766	0.338883	−	0.940829i	$-0.389951\pi$
$89$	569.069	0.677766	0.338883	+	0.940829i	$0.389951\pi$
$90$	0	0
$91$	385.176	0.443707
$92$	0	0
$93$	451.947	0.503922
$94$	0	0
$95$	133.838	0.144542
$96$	0	0
$97$	−816.579	−0.854753	−0.427376	−	0.904074i	$-0.640562\pi$
$97$	−816.579	−0.854753	−0.427376	+	0.904074i	$0.640562\pi$
$98$	0	0
$99$	502.897	0.510536
$100$	0	0
$101$	1935.39	1.90672	0.953360	−	0.301837i	$-0.0975998\pi$
$101$	1935.39	1.90672	0.953360	+	0.301837i	$0.0975998\pi$
$102$	0	0
$103$	−70.7807	−0.0677110	−0.0338555	−	0.999427i	$-0.510779\pi$
$103$	−70.7807	−0.0677110	−0.0338555	+	0.999427i	$0.510779\pi$
$104$	0	0
$105$	134.158	0.124691
$106$	0	0
$107$	−327.492	−0.295886	−0.147943	−	0.988996i	$-0.547265\pi$
$107$	−327.492	−0.295886	−0.147943	+	0.988996i	$0.547265\pi$
$108$	0	0
$109$	193.410	0.169957	0.0849787	−	0.996383i	$-0.472918\pi$
$109$	193.410	0.169957	0.0849787	+	0.996383i	$0.472918\pi$
$110$	0	0
$111$	951.093	0.813277
$112$	0	0
$113$	−772.682	−0.643255	−0.321628	−	0.946866i	$-0.604230\pi$
$113$	−772.682	−0.643255	−0.321628	+	0.946866i	$0.604230\pi$
$114$	0	0
$115$	115.000	0.0932505
$116$	0	0
$117$	387.592	0.306264
$118$	0	0
$119$	−673.138	−0.518541
$120$	0	0
$121$	1791.29	1.34582
$122$	0	0
$123$	−365.908	−0.268235
$124$	0	0
$125$	125.000	0.0894427
$126$	0	0
$127$	1941.33	1.35642	0.678209	−	0.734869i	$-0.262756\pi$
$127$	1941.33	1.35642	0.678209	+	0.734869i	$0.262756\pi$
$128$	0	0
$129$	−940.830	−0.642136
$130$	0	0
$131$	−2570.29	−1.71426	−0.857128	−	0.515103i	$-0.827754\pi$
$131$	−2570.29	−1.71426	−0.857128	+	0.515103i	$0.827754\pi$
$132$	0	0
$133$	239.407	0.156084
$134$	0	0
$135$	135.000	0.0860663
$136$	0	0
$137$	1261.81	0.786888	0.393444	−	0.919349i	$-0.371284\pi$
$137$	1261.81	0.786888	0.393444	+	0.919349i	$0.371284\pi$
$138$	0	0
$139$	−45.4353	−0.0277250	−0.0138625	−	0.999904i	$-0.504413\pi$
$139$	−45.4353	−0.0277250	−0.0138625	+	0.999904i	$0.504413\pi$
$140$	0	0
$141$	1158.27	0.691801
$142$	0	0
$143$	2406.41	1.40723
$144$	0	0
$145$	285.811	0.163692
$146$	0	0
$147$	−789.021	−0.442703
$148$	0	0
$149$	−198.706	−0.109253	−0.0546264	−	0.998507i	$-0.517397\pi$
$149$	−198.706	−0.109253	−0.0546264	+	0.998507i	$0.517397\pi$
$150$	0	0
$151$	−5.83391	−0.00314408	−0.00157204	−	0.999999i	$-0.500500\pi$
$151$	−5.83391	−0.00314408	−0.00157204	+	0.999999i	$0.500500\pi$
$152$	0	0
$153$	−677.361	−0.357917
$154$	0	0
$155$	753.246	0.390336
$156$	0	0
$157$	−272.163	−0.138350	−0.0691750	−	0.997605i	$-0.522037\pi$
$157$	−272.163	−0.138350	−0.0691750	+	0.997605i	$0.522037\pi$
$158$	0	0
$159$	−747.015	−0.372592
$160$	0	0
$161$	205.709	0.100697
$162$	0	0
$163$	−1398.24	−0.671894	−0.335947	−	0.941881i	$-0.609056\pi$
$163$	−1398.24	−0.671894	−0.335947	+	0.941881i	$0.609056\pi$
$164$	0	0
$165$	838.162	0.395460
$166$	0	0
$167$	1002.35	0.464458	0.232229	−	0.972661i	$-0.425398\pi$
$167$	1002.35	0.464458	0.232229	+	0.972661i	$0.425398\pi$
$168$	0	0
$169$	−342.338	−0.155821
$170$	0	0
$171$	240.909	0.107735
$172$	0	0
$173$	−781.097	−0.343270	−0.171635	−	0.985161i	$-0.554905\pi$
$173$	−781.097	−0.343270	−0.171635	+	0.985161i	$0.554905\pi$
$174$	0	0
$175$	223.597	0.0965849
$176$	0	0
$177$	−433.312	−0.184010
$178$	0	0
$179$	−262.503	−0.109611	−0.0548056	−	0.998497i	$-0.517454\pi$
$179$	−262.503	−0.109611	−0.0548056	+	0.998497i	$0.517454\pi$
$180$	0	0
$181$	990.365	0.406703	0.203351	−	0.979106i	$-0.434817\pi$
$181$	990.365	0.406703	0.203351	+	0.979106i	$0.434817\pi$
$182$	0	0
$183$	−2020.69	−0.816250
$184$	0	0
$185$	1585.15	0.629962
$186$	0	0
$187$	−4205.47	−1.64457
$188$	0	0
$189$	241.485	0.0929389
$190$	0	0
$191$	−1218.75	−0.461705	−0.230853	−	0.972989i	$-0.574152\pi$
$191$	−1218.75	−0.461705	−0.230853	+	0.972989i	$0.574152\pi$
$192$	0	0
$193$	2958.30	1.10333	0.551667	−	0.834065i	$-0.313992\pi$
$193$	2958.30	1.10333	0.551667	+	0.834065i	$0.313992\pi$
$194$	0	0
$195$	645.987	0.237231
$196$	0	0
$197$	−3343.36	−1.20916	−0.604580	−	0.796544i	$-0.706659\pi$
$197$	−3343.36	−1.20916	−0.604580	+	0.796544i	$0.706659\pi$
$198$	0	0
$199$	4693.50	1.67193	0.835963	−	0.548785i	$-0.184910\pi$
$199$	4693.50	1.67193	0.835963	+	0.548785i	$0.184910\pi$
$200$	0	0
$201$	1888.38	0.662668
$202$	0	0
$203$	511.252	0.176763
$204$	0	0
$205$	−609.847	−0.207774
$206$	0	0
$207$	207.000	0.0695048
$208$	0	0
$209$	1495.71	0.495025
$210$	0	0
$211$	4070.56	1.32810	0.664050	−	0.747688i	$-0.268836\pi$
$211$	4070.56	1.32810	0.664050	+	0.747688i	$0.268836\pi$
$212$	0	0
$213$	−2620.75	−0.843057
$214$	0	0
$215$	−1568.05	−0.497396
$216$	0	0
$217$	1347.39	0.421506
$218$	0	0
$219$	−1567.38	−0.483624
$220$	0	0
$221$	−3241.23	−0.986555
$222$	0	0
$223$	3342.25	1.00365	0.501825	−	0.864969i	$-0.332662\pi$
$223$	3342.25	1.00365	0.501825	+	0.864969i	$0.332662\pi$
$224$	0	0
$225$	225.000	0.0666667
$226$	0	0
$227$	3095.37	0.905052	0.452526	−	0.891751i	$-0.350523\pi$
$227$	3095.37	0.905052	0.452526	+	0.891751i	$0.350523\pi$
$228$	0	0
$229$	−2598.89	−0.749953	−0.374976	−	0.927034i	$-0.622349\pi$
$229$	−2598.89	−0.749953	−0.374976	+	0.927034i	$0.622349\pi$
$230$	0	0
$231$	1499.29	0.427038
$232$	0	0
$233$	2739.04	0.770132	0.385066	−	0.922889i	$-0.374179\pi$
$233$	2739.04	0.770132	0.385066	+	0.922889i	$0.374179\pi$
$234$	0	0
$235$	1930.45	0.535866
$236$	0	0
$237$	2126.04	0.582705
$238$	0	0
$239$	2822.08	0.763787	0.381893	−	0.924206i	$-0.375272\pi$
$239$	2822.08	0.763787	0.381893	+	0.924206i	$0.375272\pi$
$240$	0	0
$241$	6175.17	1.65053	0.825265	−	0.564746i	$-0.191026\pi$
$241$	6175.17	1.65053	0.825265	+	0.564746i	$0.191026\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	0	0
$245$	−1315.03	−0.342916
$246$	0	0
$247$	1152.77	0.296959
$248$	0	0
$249$	2838.58	0.722439
$250$	0	0
$251$	−2995.72	−0.753339	−0.376669	−	0.926348i	$-0.622931\pi$
$251$	−2995.72	−0.753339	−0.376669	+	0.926348i	$0.622931\pi$
$252$	0	0
$253$	1285.18	0.319362
$254$	0	0
$255$	−1128.93	−0.277242
$256$	0	0
$257$	3650.50	0.886038	0.443019	−	0.896512i	$-0.353907\pi$
$257$	3650.50	0.886038	0.443019	+	0.896512i	$0.353907\pi$
$258$	0	0
$259$	2835.49	0.680265
$260$	0	0
$261$	514.460	0.122009
$262$	0	0
$263$	−2525.93	−0.592226	−0.296113	−	0.955153i	$-0.595691\pi$
$263$	−2525.93	−0.592226	−0.296113	+	0.955153i	$0.595691\pi$
$264$	0	0
$265$	−1245.03	−0.288609
$266$	0	0
$267$	1707.21	0.391308
$268$	0	0
$269$	−503.473	−0.114116	−0.0570581	−	0.998371i	$-0.518172\pi$
$269$	−503.473	−0.114116	−0.0570581	+	0.998371i	$0.518172\pi$
$270$	0	0
$271$	114.841	0.0257420	0.0128710	−	0.999917i	$-0.495903\pi$
$271$	114.841	0.0257420	0.0128710	+	0.999917i	$0.495903\pi$
$272$	0	0
$273$	1155.53	0.256175
$274$	0	0
$275$	1396.94	0.306322
$276$	0	0
$277$	1211.26	0.262734	0.131367	−	0.991334i	$-0.458063\pi$
$277$	1211.26	0.262734	0.131367	+	0.991334i	$0.458063\pi$
$278$	0	0
$279$	1355.84	0.290940
$280$	0	0
$281$	1101.83	0.233913	0.116957	−	0.993137i	$-0.462686\pi$
$281$	1101.83	0.233913	0.116957	+	0.993137i	$0.462686\pi$
$282$	0	0
$283$	−1064.53	−0.223603	−0.111801	−	0.993731i	$-0.535662\pi$
$283$	−1064.53	−0.223603	−0.111801	+	0.993731i	$0.535662\pi$
$284$	0	0
$285$	401.515	0.0834515
$286$	0	0
$287$	−1090.88	−0.224365
$288$	0	0
$289$	751.414	0.152944
$290$	0	0
$291$	−2449.74	−0.493492
$292$	0	0
$293$	−3036.12	−0.605364	−0.302682	−	0.953092i	$-0.597882\pi$
$293$	−3036.12	−0.605364	−0.302682	+	0.953092i	$0.597882\pi$
$294$	0	0
$295$	−722.187	−0.142533
$296$	0	0
$297$	1508.69	0.294758
$298$	0	0
$299$	990.513	0.191581
$300$	0	0
$301$	−2804.89	−0.537114
$302$	0	0
$303$	5806.17	1.10084
$304$	0	0
$305$	−3367.82	−0.632265
$306$	0	0
$307$	−7740.57	−1.43902	−0.719508	−	0.694484i	$-0.755633\pi$
$307$	−7740.57	−1.43902	−0.719508	+	0.694484i	$0.755633\pi$
$308$	0	0
$309$	−212.342	−0.0390929
$310$	0	0
$311$	−9517.91	−1.73540	−0.867702	−	0.497084i	$-0.834404\pi$
$311$	−9517.91	−1.73540	−0.867702	+	0.497084i	$0.834404\pi$
$312$	0	0
$313$	−3536.47	−0.638636	−0.319318	−	0.947648i	$-0.603454\pi$
$313$	−3536.47	−0.638636	−0.319318	+	0.947648i	$0.603454\pi$
$314$	0	0
$315$	402.475	0.0719902
$316$	0	0
$317$	2255.28	0.399587	0.199793	−	0.979838i	$-0.435973\pi$
$317$	2255.28	0.399587	0.199793	+	0.979838i	$0.435973\pi$
$318$	0	0
$319$	3194.08	0.560608
$320$	0	0
$321$	−982.475	−0.170830
$322$	0	0
$323$	−2014.59	−0.347043
$324$	0	0
$325$	1076.64	0.183758
$326$	0	0
$327$	580.231	0.0981249
$328$	0	0
$329$	3453.14	0.578657
$330$	0	0
$331$	8514.37	1.41387	0.706937	−	0.707277i	$-0.250077\pi$
$331$	8514.37	1.41387	0.706937	+	0.707277i	$0.250077\pi$
$332$	0	0
$333$	2853.28	0.469546
$334$	0	0
$335$	3147.31	0.513301
$336$	0	0
$337$	−1058.10	−0.171034	−0.0855172	−	0.996337i	$-0.527254\pi$
$337$	−1058.10	−0.171034	−0.0855172	+	0.996337i	$0.527254\pi$
$338$	0	0
$339$	−2318.05	−0.371384
$340$	0	0
$341$	8417.89	1.33682
$342$	0	0
$343$	−5420.06	−0.853223
$344$	0	0
$345$	345.000	0.0538382
$346$	0	0
$347$	104.409	0.0161527	0.00807635	−	0.999967i	$-0.497429\pi$
$347$	104.409	0.0161527	0.00807635	+	0.999967i	$0.497429\pi$
$348$	0	0
$349$	−3408.97	−0.522859	−0.261429	−	0.965223i	$-0.584194\pi$
$349$	−3408.97	−0.522859	−0.261429	+	0.965223i	$0.584194\pi$
$350$	0	0
$351$	1162.78	0.176822
$352$	0	0
$353$	9624.72	1.45120	0.725598	−	0.688119i	$-0.241563\pi$
$353$	9624.72	1.45120	0.725598	+	0.688119i	$0.241563\pi$
$354$	0	0
$355$	−4367.92	−0.653029
$356$	0	0
$357$	−2019.41	−0.299380
$358$	0	0
$359$	−3619.56	−0.532126	−0.266063	−	0.963956i	$-0.585723\pi$
$359$	−3619.56	−0.532126	−0.266063	+	0.963956i	$0.585723\pi$
$360$	0	0
$361$	−6142.49	−0.895538
$362$	0	0
$363$	5373.88	0.777012
$364$	0	0
$365$	−2612.30	−0.374614
$366$	0	0
$367$	−2338.87	−0.332666	−0.166333	−	0.986070i	$-0.553193\pi$
$367$	−2338.87	−0.332666	−0.166333	+	0.986070i	$0.553193\pi$
$368$	0	0
$369$	−1097.73	−0.154865
$370$	0	0
$371$	−2227.07	−0.311655
$372$	0	0
$373$	−3723.18	−0.516834	−0.258417	−	0.966033i	$-0.583201\pi$
$373$	−3723.18	−0.516834	−0.258417	+	0.966033i	$0.583201\pi$
$374$	0	0
$375$	375.000	0.0516398
$376$	0	0
$377$	2461.73	0.336302
$378$	0	0
$379$	7346.34	0.995662	0.497831	−	0.867274i	$-0.334130\pi$
$379$	7346.34	0.995662	0.497831	+	0.867274i	$0.334130\pi$
$380$	0	0
$381$	5823.99	0.783129
$382$	0	0
$383$	−10821.2	−1.44370	−0.721848	−	0.692052i	$-0.756707\pi$
$383$	−10821.2	−1.44370	−0.721848	+	0.692052i	$0.756707\pi$
$384$	0	0
$385$	2498.81	0.330782
$386$	0	0
$387$	−2822.49	−0.370737
$388$	0	0
$389$	−2552.19	−0.332651	−0.166326	−	0.986071i	$-0.553190\pi$
$389$	−2552.19	−0.332651	−0.166326	+	0.986071i	$0.553190\pi$
$390$	0	0
$391$	−1731.03	−0.223893
$392$	0	0
$393$	−7710.88	−0.989727
$394$	0	0
$395$	3543.40	0.451361
$396$	0	0
$397$	12949.2	1.63703	0.818515	−	0.574486i	$-0.194798\pi$
$397$	12949.2	1.63703	0.818515	+	0.574486i	$0.194798\pi$
$398$	0	0
$399$	718.221	0.0901153
$400$	0	0
$401$	−15180.4	−1.89046	−0.945228	−	0.326410i	$-0.894161\pi$
$401$	−15180.4	−1.89046	−0.945228	+	0.326410i	$0.894161\pi$
$402$	0	0
$403$	6487.82	0.801939
$404$	0	0
$405$	405.000	0.0496904
$406$	0	0
$407$	17714.9	2.15748
$408$	0	0
$409$	2096.83	0.253500	0.126750	−	0.991935i	$-0.459545\pi$
$409$	2096.83	0.253500	0.126750	+	0.991935i	$0.459545\pi$
$410$	0	0
$411$	3785.43	0.454310
$412$	0	0
$413$	−1291.83	−0.153915
$414$	0	0
$415$	4730.96	0.559599
$416$	0	0
$417$	−136.306	−0.0160070
$418$	0	0
$419$	1161.84	0.135465	0.0677324	−	0.997704i	$-0.478424\pi$
$419$	1161.84	0.135465	0.0677324	+	0.997704i	$0.478424\pi$
$420$	0	0
$421$	−822.548	−0.0952222	−0.0476111	−	0.998866i	$-0.515161\pi$
$421$	−822.548	−0.0952222	−0.0476111	+	0.998866i	$0.515161\pi$
$422$	0	0
$423$	3474.81	0.399411
$424$	0	0
$425$	−1881.56	−0.214750
$426$	0	0
$427$	−6024.28	−0.682753
$428$	0	0
$429$	7219.22	0.812465
$430$	0	0
$431$	−11832.2	−1.32236	−0.661181	−	0.750227i	$-0.729944\pi$
$431$	−11832.2	−1.32236	−0.661181	+	0.750227i	$0.729944\pi$
$432$	0	0
$433$	4138.15	0.459277	0.229639	−	0.973276i	$-0.426246\pi$
$433$	4138.15	0.459277	0.229639	+	0.973276i	$0.426246\pi$
$434$	0	0
$435$	857.433	0.0945075
$436$	0	0
$437$	615.656	0.0673932
$438$	0	0
$439$	9456.48	1.02809	0.514047	−	0.857762i	$-0.328146\pi$
$439$	9456.48	1.02809	0.514047	+	0.857762i	$0.328146\pi$
$440$	0	0
$441$	−2367.06	−0.255595
$442$	0	0
$443$	−13853.9	−1.48583	−0.742913	−	0.669388i	$-0.766556\pi$
$443$	−13853.9	−1.48583	−0.742913	+	0.669388i	$0.766556\pi$
$444$	0	0
$445$	2845.34	0.303106
$446$	0	0
$447$	−596.119	−0.0630771
$448$	0	0
$449$	−1712.20	−0.179964	−0.0899818	−	0.995943i	$-0.528681\pi$
$449$	−1712.20	−0.179964	−0.0899818	+	0.995943i	$0.528681\pi$
$450$	0	0
$451$	−6815.35	−0.711579
$452$	0	0
$453$	−17.5017	−0.00181524
$454$	0	0
$455$	1925.88	0.198432
$456$	0	0
$457$	13920.1	1.42485	0.712423	−	0.701750i	$-0.247598\pi$
$457$	13920.1	1.42485	0.712423	+	0.701750i	$0.247598\pi$
$458$	0	0
$459$	−2032.08	−0.206644
$460$	0	0
$461$	−332.510	−0.0335933	−0.0167967	−	0.999859i	$-0.505347\pi$
$461$	−332.510	−0.0335933	−0.0167967	+	0.999859i	$0.505347\pi$
$462$	0	0
$463$	−14582.0	−1.46367	−0.731837	−	0.681480i	$-0.761337\pi$
$463$	−14582.0	−1.46367	−0.731837	+	0.681480i	$0.761337\pi$
$464$	0	0
$465$	2259.74	0.225361
$466$	0	0
$467$	2138.10	0.211862	0.105931	−	0.994373i	$-0.466218\pi$
$467$	2138.10	0.211862	0.105931	+	0.994373i	$0.466218\pi$
$468$	0	0
$469$	5629.83	0.554289
$470$	0	0
$471$	−816.488	−0.0798764
$472$	0	0
$473$	−17523.7	−1.70347
$474$	0	0
$475$	669.191	0.0646413
$476$	0	0
$477$	−2241.05	−0.215116
$478$	0	0
$479$	5490.55	0.523736	0.261868	−	0.965104i	$-0.415661\pi$
$479$	5490.55	0.523736	0.261868	+	0.965104i	$0.415661\pi$
$480$	0	0
$481$	13653.2	1.29424
$482$	0	0
$483$	617.128	0.0581373
$484$	0	0
$485$	−4082.89	−0.382257
$486$	0	0
$487$	11341.8	1.05533	0.527663	−	0.849454i	$-0.323068\pi$
$487$	11341.8	1.05533	0.527663	+	0.849454i	$0.323068\pi$
$488$	0	0
$489$	−4194.72	−0.387918
$490$	0	0
$491$	−6857.37	−0.630283	−0.315141	−	0.949045i	$-0.602052\pi$
$491$	−6857.37	−0.630283	−0.315141	+	0.949045i	$0.602052\pi$
$492$	0	0
$493$	−4302.16	−0.393021
$494$	0	0
$495$	2514.49	0.228319
$496$	0	0
$497$	−7813.25	−0.705175
$498$	0	0
$499$	−16420.9	−1.47314	−0.736571	−	0.676360i	$-0.763557\pi$
$499$	−16420.9	−1.47314	−0.736571	+	0.676360i	$0.763557\pi$
$500$	0	0
$501$	3007.06	0.268155
$502$	0	0
$503$	−3985.61	−0.353300	−0.176650	−	0.984274i	$-0.556526\pi$
$503$	−3985.61	−0.353300	−0.176650	+	0.984274i	$0.556526\pi$
$504$	0	0
$505$	9676.96	0.852711
$506$	0	0
$507$	−1027.01	−0.0899631
$508$	0	0
$509$	−21264.2	−1.85170	−0.925852	−	0.377885i	$-0.876651\pi$
$509$	−21264.2	−1.85170	−0.925852	+	0.377885i	$0.876651\pi$
$510$	0	0
$511$	−4672.82	−0.404527
$512$	0	0
$513$	722.726	0.0622011
$514$	0	0
$515$	−353.903	−0.0302813
$516$	0	0
$517$	21573.7	1.83522
$518$	0	0
$519$	−2343.29	−0.198187
$520$	0	0
$521$	5680.89	0.477705	0.238852	−	0.971056i	$-0.423229\pi$
$521$	5680.89	0.477705	0.238852	+	0.971056i	$0.423229\pi$
$522$	0	0
$523$	−10728.7	−0.897002	−0.448501	−	0.893782i	$-0.648042\pi$
$523$	−10728.7	−0.897002	−0.448501	+	0.893782i	$0.648042\pi$
$524$	0	0
$525$	670.792	0.0557633
$526$	0	0
$527$	−11338.2	−0.937191
$528$	0	0
$529$	529.000	0.0434783
$530$	0	0
$531$	−1299.94	−0.106238
$532$	0	0
$533$	−5252.71	−0.426867
$534$	0	0
$535$	−1637.46	−0.132324
$536$	0	0
$537$	−787.509	−0.0632841
$538$	0	0
$539$	−14696.2	−1.17441
$540$	0	0
$541$	−12935.0	−1.02794	−0.513972	−	0.857807i	$-0.671827\pi$
$541$	−12935.0	−1.02794	−0.513972	+	0.857807i	$0.671827\pi$
$542$	0	0
$543$	2971.09	0.234810
$544$	0	0
$545$	967.052	0.0760072
$546$	0	0
$547$	22877.3	1.78823	0.894114	−	0.447839i	$-0.147806\pi$
$547$	22877.3	1.78823	0.894114	+	0.447839i	$0.147806\pi$
$548$	0	0
$549$	−6062.08	−0.471262
$550$	0	0
$551$	1530.10	0.118302
$552$	0	0
$553$	6338.35	0.487404
$554$	0	0
$555$	4755.46	0.363708
$556$	0	0
$557$	−8486.40	−0.645566	−0.322783	−	0.946473i	$-0.604618\pi$
$557$	−8486.40	−0.645566	−0.322783	+	0.946473i	$0.604618\pi$
$558$	0	0
$559$	−13505.9	−1.02189
$560$	0	0
$561$	−12616.4	−0.949492
$562$	0	0
$563$	11587.1	0.867389	0.433694	−	0.901060i	$-0.357210\pi$
$563$	11587.1	0.867389	0.433694	+	0.901060i	$0.357210\pi$
$564$	0	0
$565$	−3863.41	−0.287672
$566$	0	0
$567$	724.455	0.0536583
$568$	0	0
$569$	−5097.61	−0.375576	−0.187788	−	0.982210i	$-0.560132\pi$
$569$	−5097.61	−0.375576	−0.187788	+	0.982210i	$0.560132\pi$
$570$	0	0
$571$	−22458.3	−1.64597	−0.822987	−	0.568060i	$-0.807694\pi$
$571$	−22458.3	−1.64597	−0.822987	+	0.568060i	$0.807694\pi$
$572$	0	0
$573$	−3656.25	−0.266566
$574$	0	0
$575$	575.000	0.0417029
$576$	0	0
$577$	5191.74	0.374584	0.187292	−	0.982304i	$-0.440029\pi$
$577$	5191.74	0.374584	0.187292	+	0.982304i	$0.440029\pi$
$578$	0	0
$579$	8874.91	0.637010
$580$	0	0
$581$	8462.63	0.604284
$582$	0	0
$583$	−13913.8	−0.988422
$584$	0	0
$585$	1937.96	0.136965
$586$	0	0
$587$	−12502.1	−0.879078	−0.439539	−	0.898224i	$-0.644858\pi$
$587$	−12502.1	−0.879078	−0.439539	+	0.898224i	$0.644858\pi$
$588$	0	0
$589$	4032.52	0.282100
$590$	0	0
$591$	−10030.1	−0.698109
$592$	0	0
$593$	−28712.8	−1.98836	−0.994178	−	0.107754i	$-0.965634\pi$
$593$	−28712.8	−1.98836	−0.994178	+	0.107754i	$0.965634\pi$
$594$	0	0
$595$	−3365.69	−0.231899
$596$	0	0
$597$	14080.5	0.965287
$598$	0	0
$599$	−9707.41	−0.662160	−0.331080	−	0.943603i	$-0.607413\pi$
$599$	−9707.41	−0.662160	−0.331080	+	0.943603i	$0.607413\pi$
$600$	0	0
$601$	25629.5	1.73951	0.869757	−	0.493480i	$-0.164276\pi$
$601$	25629.5	1.73951	0.869757	+	0.493480i	$0.164276\pi$
$602$	0	0
$603$	5665.15	0.382592
$604$	0	0
$605$	8956.46	0.601871
$606$	0	0
$607$	11864.0	0.793318	0.396659	−	0.917966i	$-0.370170\pi$
$607$	11864.0	0.793318	0.396659	+	0.917966i	$0.370170\pi$
$608$	0	0
$609$	1533.76	0.102054
$610$	0	0
$611$	16627.3	1.10093
$612$	0	0
$613$	1819.62	0.119892	0.0599460	−	0.998202i	$-0.480907\pi$
$613$	1819.62	0.119892	0.0599460	+	0.998202i	$0.480907\pi$
$614$	0	0
$615$	−1829.54	−0.119958
$616$	0	0
$617$	−7604.82	−0.496205	−0.248102	−	0.968734i	$-0.579807\pi$
$617$	−7604.82	−0.496205	−0.248102	+	0.968734i	$0.579807\pi$
$618$	0	0
$619$	−21311.7	−1.38383	−0.691913	−	0.721981i	$-0.743232\pi$
$619$	−21311.7	−1.38383	−0.691913	+	0.721981i	$0.743232\pi$
$620$	0	0
$621$	621.000	0.0401286
$622$	0	0
$623$	5089.69	0.327310
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	4487.13	0.285803
$628$	0	0
$629$	−23860.5	−1.51253
$630$	0	0
$631$	11479.4	0.724227	0.362114	−	0.932134i	$-0.382055\pi$
$631$	11479.4	0.724227	0.362114	+	0.932134i	$0.382055\pi$
$632$	0	0
$633$	12211.7	0.766779
$634$	0	0
$635$	9706.65	0.606609
$636$	0	0
$637$	−11326.6	−0.704515
$638$	0	0
$639$	−7862.26	−0.486739
$640$	0	0
$641$	23.6048	0.00145450	0.000727248	−	1.00000i	$-0.499769\pi$
$641$	23.6048	0.00145450	0.000727248		1.00000i	$0.499769\pi$
$642$	0	0
$643$	10651.2	0.653254	0.326627	−	0.945153i	$-0.394088\pi$
$643$	10651.2	0.653254	0.326627	+	0.945153i	$0.394088\pi$
$644$	0	0
$645$	−4704.15	−0.287172
$646$	0	0
$647$	−27326.3	−1.66044	−0.830221	−	0.557434i	$-0.811786\pi$
$647$	−27326.3	−1.66044	−0.830221	+	0.557434i	$0.811786\pi$
$648$	0	0
$649$	−8070.80	−0.488146
$650$	0	0
$651$	4042.17	0.243356
$652$	0	0
$653$	5133.44	0.307637	0.153818	−	0.988099i	$-0.450843\pi$
$653$	5133.44	0.307637	0.153818	+	0.988099i	$0.450843\pi$
$654$	0	0
$655$	−12851.5	−0.766639
$656$	0	0
$657$	−4702.14	−0.279220
$658$	0	0
$659$	−20095.2	−1.18786	−0.593928	−	0.804518i	$-0.702424\pi$
$659$	−20095.2	−1.18786	−0.593928	+	0.804518i	$0.702424\pi$
$660$	0	0
$661$	−14542.5	−0.855729	−0.427864	−	0.903843i	$-0.640734\pi$
$661$	−14542.5	−0.855729	−0.427864	+	0.903843i	$0.640734\pi$
$662$	0	0
$663$	−9723.69	−0.569588
$664$	0	0
$665$	1197.03	0.0698030
$666$	0	0
$667$	1314.73	0.0763217
$668$	0	0
$669$	10026.8	0.579457
$670$	0	0
$671$	−37637.1	−2.16537
$672$	0	0
$673$	−17763.0	−1.01740	−0.508701	−	0.860943i	$-0.669874\pi$
$673$	−17763.0	−1.01740	−0.508701	+	0.860943i	$0.669874\pi$
$674$	0	0
$675$	675.000	0.0384900
$676$	0	0
$677$	−4917.79	−0.279182	−0.139591	−	0.990209i	$-0.544579\pi$
$677$	−4917.79	−0.279182	−0.139591	+	0.990209i	$0.544579\pi$
$678$	0	0
$679$	−7303.39	−0.412781
$680$	0	0
$681$	9286.10	0.522532
$682$	0	0
$683$	−31286.7	−1.75279	−0.876394	−	0.481595i	$-0.840058\pi$
$683$	−31286.7	−1.75279	−0.876394	+	0.481595i	$0.840058\pi$
$684$	0	0
$685$	6309.05	0.351907
$686$	0	0
$687$	−7796.66	−0.432985
$688$	0	0
$689$	−10723.6	−0.592941
$690$	0	0
$691$	−12558.9	−0.691408	−0.345704	−	0.938344i	$-0.612360\pi$
$691$	−12558.9	−0.691408	−0.345704	+	0.938344i	$0.612360\pi$
$692$	0	0
$693$	4497.86	0.246550
$694$	0	0
$695$	−227.177	−0.0123990
$696$	0	0
$697$	9179.70	0.498861
$698$	0	0
$699$	8217.13	0.444636
$700$	0	0
$701$	−21917.1	−1.18088	−0.590441	−	0.807080i	$-0.701046\pi$
$701$	−21917.1	−1.18088	−0.590441	+	0.807080i	$0.701046\pi$
$702$	0	0
$703$	8486.17	0.455280
$704$	0	0
$705$	5791.35	0.309383
$706$	0	0
$707$	17309.9	0.920802
$708$	0	0
$709$	7113.37	0.376796	0.188398	−	0.982093i	$-0.439671\pi$
$709$	7113.37	0.376796	0.188398	+	0.982093i	$0.439671\pi$
$710$	0	0
$711$	6378.12	0.336425
$712$	0	0
$713$	3464.93	0.181995
$714$	0	0
$715$	12032.0	0.629332
$716$	0	0
$717$	8466.23	0.440972
$718$	0	0
$719$	−9089.96	−0.471486	−0.235743	−	0.971815i	$-0.575752\pi$
$719$	−9089.96	−0.471486	−0.235743	+	0.971815i	$0.575752\pi$
$720$	0	0
$721$	−633.055	−0.0326993
$722$	0	0
$723$	18525.5	0.952934
$724$	0	0
$725$	1429.05	0.0732052
$726$	0	0
$727$	−7284.39	−0.371614	−0.185807	−	0.982586i	$-0.559490\pi$
$727$	−7284.39	−0.371614	−0.185807	+	0.982586i	$0.559490\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	23603.0	1.19424
$732$	0	0
$733$	−1140.56	−0.0574727	−0.0287363	−	0.999587i	$-0.509148\pi$
$733$	−1140.56	−0.0574727	−0.0287363	+	0.999587i	$0.509148\pi$
$734$	0	0
$735$	−3945.10	−0.197983
$736$	0	0
$737$	35172.7	1.75794
$738$	0	0
$739$	−31186.1	−1.55237	−0.776183	−	0.630508i	$-0.782847\pi$
$739$	−31186.1	−1.55237	−0.776183	+	0.630508i	$0.782847\pi$
$740$	0	0
$741$	3458.31	0.171450
$742$	0	0
$743$	−7381.14	−0.364452	−0.182226	−	0.983257i	$-0.558330\pi$
$743$	−7381.14	−0.364452	−0.182226	+	0.983257i	$0.558330\pi$
$744$	0	0
$745$	−993.532	−0.0488593
$746$	0	0
$747$	8515.73	0.417101
$748$	0	0
$749$	−2929.05	−0.142891
$750$	0	0
$751$	−2020.69	−0.0981839	−0.0490920	−	0.998794i	$-0.515633\pi$
$751$	−2020.69	−0.0981839	−0.0490920	+	0.998794i	$0.515633\pi$
$752$	0	0
$753$	−8987.15	−0.434940
$754$	0	0
$755$	−29.1695	−0.00140608
$756$	0	0
$757$	12060.7	0.579068	0.289534	−	0.957168i	$-0.406500\pi$
$757$	12060.7	0.579068	0.289534	+	0.957168i	$0.406500\pi$
$758$	0	0
$759$	3855.55	0.184384
$760$	0	0
$761$	−7121.01	−0.339207	−0.169604	−	0.985512i	$-0.554249\pi$
$761$	−7121.01	−0.339207	−0.169604	+	0.985512i	$0.554249\pi$
$762$	0	0
$763$	1729.84	0.0820766
$764$	0	0
$765$	−3386.80	−0.160066
$766$	0	0
$767$	−6220.31	−0.292832
$768$	0	0
$769$	19585.3	0.918421	0.459210	−	0.888328i	$-0.348132\pi$
$769$	19585.3	0.918421	0.459210	+	0.888328i	$0.348132\pi$
$770$	0	0
$771$	10951.5	0.511554
$772$	0	0
$773$	25869.9	1.20372	0.601860	−	0.798602i	$-0.294426\pi$
$773$	25869.9	1.20372	0.601860	+	0.798602i	$0.294426\pi$
$774$	0	0
$775$	3766.23	0.174564
$776$	0	0
$777$	8506.47	0.392751
$778$	0	0
$779$	−3264.84	−0.150160
$780$	0	0
$781$	−48813.7	−2.23648
$782$	0	0
$783$	1543.38	0.0704417
$784$	0	0
$785$	−1360.81	−0.0618720
$786$	0	0
$787$	−32371.2	−1.46621	−0.733107	−	0.680113i	$-0.761931\pi$
$787$	−32371.2	−1.46621	−0.733107	+	0.680113i	$0.761931\pi$
$788$	0	0
$789$	−7577.79	−0.341922
$790$	0	0
$791$	−6910.79	−0.310644
$792$	0	0
$793$	−29007.6	−1.29898
$794$	0	0
$795$	−3735.08	−0.166628
$796$	0	0
$797$	−2428.68	−0.107940	−0.0539700	−	0.998543i	$-0.517188\pi$
$797$	−2428.68	−0.107940	−0.0539700	+	0.998543i	$0.517188\pi$
$798$	0	0
$799$	−29058.0	−1.28661
$800$	0	0
$801$	5121.62	0.225922
$802$	0	0
$803$	−29193.7	−1.28297
$804$	0	0
$805$	1028.55	0.0450330
$806$	0	0
$807$	−1510.42	−0.0658851
$808$	0	0
$809$	−1617.10	−0.0702771	−0.0351385	−	0.999382i	$-0.511187\pi$
$809$	−1617.10	−0.0702771	−0.0351385	+	0.999382i	$0.511187\pi$
$810$	0	0
$811$	−516.984	−0.0223844	−0.0111922	−	0.999937i	$-0.503563\pi$
$811$	−516.984	−0.0223844	−0.0111922	+	0.999937i	$0.503563\pi$
$812$	0	0
$813$	344.522	0.0148622
$814$	0	0
$815$	−6991.20	−0.300480
$816$	0	0
$817$	−8394.60	−0.359474
$818$	0	0
$819$	3466.58	0.147902
$820$	0	0
$821$	38255.2	1.62621	0.813104	−	0.582119i	$-0.197776\pi$
$821$	38255.2	1.62621	0.813104	+	0.582119i	$0.197776\pi$
$822$	0	0
$823$	6327.35	0.267992	0.133996	−	0.990982i	$-0.457219\pi$
$823$	6327.35	0.267992	0.133996	+	0.990982i	$0.457219\pi$
$824$	0	0
$825$	4190.81	0.176855
$826$	0	0
$827$	−8650.32	−0.363726	−0.181863	−	0.983324i	$-0.558213\pi$
$827$	−8650.32	−0.363726	−0.181863	+	0.983324i	$0.558213\pi$
$828$	0	0
$829$	−24937.7	−1.04478	−0.522390	−	0.852707i	$-0.674959\pi$
$829$	−24937.7	−1.04478	−0.522390	+	0.852707i	$0.674959\pi$
$830$	0	0
$831$	3633.77	0.151690
$832$	0	0
$833$	19794.5	0.823336
$834$	0	0
$835$	5011.77	0.207712
$836$	0	0
$837$	4067.53	0.167974
$838$	0	0
$839$	−46153.2	−1.89915	−0.949574	−	0.313542i	$-0.898484\pi$
$839$	−46153.2	−1.89915	−0.949574	+	0.313542i	$0.898484\pi$
$840$	0	0
$841$	−21121.5	−0.866025
$842$	0	0
$843$	3305.48	0.135050
$844$	0	0
$845$	−1711.69	−0.0696851
$846$	0	0
$847$	16021.1	0.649932
$848$	0	0
$849$	−3193.58	−0.129097
$850$	0	0
$851$	7291.71	0.293721
$852$	0	0
$853$	44951.7	1.80436	0.902179	−	0.431363i	$-0.141967\pi$
$853$	44951.7	1.80436	0.902179	+	0.431363i	$0.141967\pi$
$854$	0	0
$855$	1204.54	0.0481807
$856$	0	0
$857$	45393.1	1.80933	0.904667	−	0.426119i	$-0.140119\pi$
$857$	45393.1	1.80933	0.904667	+	0.426119i	$0.140119\pi$
$858$	0	0
$859$	6777.11	0.269188	0.134594	−	0.990901i	$-0.457027\pi$
$859$	6777.11	0.269188	0.134594	+	0.990901i	$0.457027\pi$
$860$	0	0
$861$	−3272.64	−0.129537
$862$	0	0
$863$	12954.2	0.510967	0.255483	−	0.966813i	$-0.417765\pi$
$863$	12954.2	0.510967	0.255483	+	0.966813i	$0.417765\pi$
$864$	0	0
$865$	−3905.49	−0.153515
$866$	0	0
$867$	2254.24	0.0883023
$868$	0	0
$869$	39599.2	1.54581
$870$	0	0
$871$	27108.2	1.05457
$872$	0	0
$873$	−7349.21	−0.284918
$874$	0	0
$875$	1117.99	0.0431941
$876$	0	0
$877$	43377.4	1.67018	0.835091	−	0.550112i	$-0.185415\pi$
$877$	43377.4	1.67018	0.835091	+	0.550112i	$0.185415\pi$
$878$	0	0
$879$	−9108.35	−0.349507
$880$	0	0
$881$	15461.5	0.591271	0.295636	−	0.955301i	$-0.404469\pi$
$881$	15461.5	0.591271	0.295636	+	0.955301i	$0.404469\pi$
$882$	0	0
$883$	−29374.7	−1.11952	−0.559761	−	0.828654i	$-0.689107\pi$
$883$	−29374.7	−1.11952	−0.559761	+	0.828654i	$0.689107\pi$
$884$	0	0
$885$	−2166.56	−0.0822917
$886$	0	0
$887$	−8361.52	−0.316519	−0.158260	−	0.987398i	$-0.550588\pi$
$887$	−8361.52	−0.316519	−0.158260	+	0.987398i	$0.550588\pi$
$888$	0	0
$889$	17363.0	0.655048
$890$	0	0
$891$	4526.08	0.170179
$892$	0	0
$893$	10334.7	0.387277
$894$	0	0
$895$	−1312.52	−0.0490196
$896$	0	0
$897$	2971.54	0.110610
$898$	0	0
$899$	8611.43	0.319474
$900$	0	0
$901$	18740.7	0.692945
$902$	0	0
$903$	−8414.68	−0.310103
$904$	0	0
$905$	4951.82	0.181883
$906$	0	0
$907$	16319.5	0.597442	0.298721	−	0.954340i	$-0.403440\pi$
$907$	16319.5	0.597442	0.298721	+	0.954340i	$0.403440\pi$
$908$	0	0
$909$	17418.5	0.635573
$910$	0	0
$911$	−28764.2	−1.04610	−0.523051	−	0.852301i	$-0.675206\pi$
$911$	−28764.2	−1.04610	−0.523051	+	0.852301i	$0.675206\pi$
$912$	0	0
$913$	52870.8	1.91650
$914$	0	0
$915$	−10103.5	−0.365038
$916$	0	0
$917$	−22988.4	−0.827857
$918$	0	0
$919$	−12794.8	−0.459261	−0.229631	−	0.973278i	$-0.573752\pi$
$919$	−12794.8	−0.459261	−0.229631	+	0.973278i	$0.573752\pi$
$920$	0	0
$921$	−23221.7	−0.830816
$922$	0	0
$923$	−37621.6	−1.34164
$924$	0	0
$925$	7925.77	0.281727
$926$	0	0
$927$	−637.026	−0.0225703
$928$	0	0
$929$	−36551.7	−1.29087	−0.645437	−	0.763814i	$-0.723325\pi$
$929$	−36551.7	−1.29087	−0.645437	+	0.763814i	$0.723325\pi$
$930$	0	0
$931$	−7040.07	−0.247829
$932$	0	0
$933$	−28553.7	−1.00194
$934$	0	0
$935$	−21027.3	−0.735473
$936$	0	0
$937$	12813.9	0.446758	0.223379	−	0.974732i	$-0.428291\pi$
$937$	12813.9	0.446758	0.223379	+	0.974732i	$0.428291\pi$
$938$	0	0
$939$	−10609.4	−0.368717
$940$	0	0
$941$	13038.1	0.451680	0.225840	−	0.974164i	$-0.427487\pi$
$941$	13038.1	0.451680	0.225840	+	0.974164i	$0.427487\pi$
$942$	0	0
$943$	−2805.30	−0.0968749
$944$	0	0
$945$	1207.43	0.0415635
$946$	0	0
$947$	−47934.9	−1.64485	−0.822427	−	0.568871i	$-0.807380\pi$
$947$	−47934.9	−1.64485	−0.822427	+	0.568871i	$0.807380\pi$
$948$	0	0
$949$	−22500.1	−0.769637
$950$	0	0
$951$	6765.83	0.230701
$952$	0	0
$953$	3479.48	0.118270	0.0591351	−	0.998250i	$-0.481166\pi$
$953$	3479.48	0.118270	0.0591351	+	0.998250i	$0.481166\pi$
$954$	0	0
$955$	−6093.76	−0.206481
$956$	0	0
$957$	9582.24	0.323667
$958$	0	0
$959$	11285.5	0.380008
$960$	0	0
$961$	−7095.85	−0.238188
$962$	0	0
$963$	−2947.43	−0.0986287
$964$	0	0
$965$	14791.5	0.493426
$966$	0	0
$967$	17259.0	0.573953	0.286976	−	0.957938i	$-0.407350\pi$
$967$	17259.0	0.573953	0.286976	+	0.957938i	$0.407350\pi$
$968$	0	0
$969$	−6043.78	−0.200366
$970$	0	0
$971$	33581.1	1.10986	0.554928	−	0.831899i	$-0.312746\pi$
$971$	33581.1	1.10986	0.554928	+	0.831899i	$0.312746\pi$
$972$	0	0
$973$	−406.369	−0.0133891
$974$	0	0
$975$	3229.93	0.106093
$976$	0	0
$977$	−32513.5	−1.06469	−0.532343	−	0.846529i	$-0.678688\pi$
$977$	−32513.5	−1.06469	−0.532343	+	0.846529i	$0.678688\pi$
$978$	0	0
$979$	31798.1	1.03807
$980$	0	0
$981$	1740.69	0.0566525
$982$	0	0
$983$	−23679.9	−0.768333	−0.384166	−	0.923264i	$-0.625511\pi$
$983$	−23679.9	−0.768333	−0.384166	+	0.923264i	$0.625511\pi$
$984$	0	0
$985$	−16716.8	−0.540753
$986$	0	0
$987$	10359.4	0.334088
$988$	0	0
$989$	−7213.03	−0.231912
$990$	0	0
$991$	−30301.5	−0.971300	−0.485650	−	0.874153i	$-0.661417\pi$
$991$	−30301.5	−0.971300	−0.485650	+	0.874153i	$0.661417\pi$
$992$	0	0
$993$	25543.1	0.816300
$994$	0	0
$995$	23467.5	0.747708
$996$	0	0
$997$	52418.7	1.66511	0.832556	−	0.553942i	$-0.186877\pi$
$997$	52418.7	1.66511	0.832556	+	0.553942i	$0.186877\pi$
$998$	0	0
$999$	8559.83	0.271092

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1380.4.a.j.1.4	✓	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1380.4.a.j.1.4	✓	7	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:	\( N \)	\(=\)	\( 1380 = 2^{2} \cdot 3 \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1380.a (trivial)

\(f(q)\)	\(=\)	\(q+3.00000 q^{3} +5.00000 q^{5} +8.94389 q^{7} +9.00000 q^{9} +O(q^{10})\)	\(q+3.00000 q^{3} +5.00000 q^{5} +8.94389 q^{7} +9.00000 q^{9} +55.8775 q^{11} +43.0658 q^{13} +15.0000 q^{15} -75.2623 q^{17} +26.7676 q^{19} +26.8317 q^{21} +23.0000 q^{23} +25.0000 q^{25} +27.0000 q^{27} +57.1622 q^{29} +150.649 q^{31} +167.632 q^{33} +44.7194 q^{35} +317.031 q^{37} +129.197 q^{39} -121.969 q^{41} -313.610 q^{43} +45.0000 q^{45} +386.090 q^{47} -263.007 q^{49} -225.787 q^{51} -249.005 q^{53} +279.387 q^{55} +80.3029 q^{57} -144.437 q^{59} -673.564 q^{61} +80.4950 q^{63} +215.329 q^{65} +629.461 q^{67} +69.0000 q^{69} -873.585 q^{71} -522.460 q^{73} +75.0000 q^{75} +499.762 q^{77} +708.680 q^{79} +81.0000 q^{81} +946.192 q^{83} -376.312 q^{85} +171.487 q^{87} +569.069 q^{89} +385.176 q^{91} +451.947 q^{93} +133.838 q^{95} -816.579 q^{97} +502.897 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 21 q^{3} + 35 q^{5} + 35 q^{7} + 63 q^{9}+O(q^{10}) \) 7 * q + 21 * q^3 + 35 * q^5 + 35 * q^7 + 63 * q^9	\( 7 q + 21 q^{3} + 35 q^{5} + 35 q^{7} + 63 q^{9} + 48 q^{11} + 90 q^{13} + 105 q^{15} + 163 q^{17} + 200 q^{19} + 105 q^{21} + 161 q^{23} + 175 q^{25} + 189 q^{27} + 81 q^{29} + 125 q^{31} + 144 q^{33} + 175 q^{35} + 5 q^{37} + 270 q^{39} + 369 q^{41} + 462 q^{43} + 315 q^{45} + 134 q^{47} + 614 q^{49} + 489 q^{51} + 561 q^{53} + 240 q^{55} + 600 q^{57} + 951 q^{59} + 860 q^{61} + 315 q^{63} + 450 q^{65} + 447 q^{67} + 483 q^{69} + 735 q^{71} + 1460 q^{73} + 525 q^{75} + 496 q^{77} + 18 q^{79} + 567 q^{81} + 261 q^{83} + 815 q^{85} + 243 q^{87} + 2024 q^{89} + 692 q^{91} + 375 q^{93} + 1000 q^{95} + 668 q^{97} + 432 q^{99}+O(q^{100}) \) 7 * q + 21 * q^3 + 35 * q^5 + 35 * q^7 + 63 * q^9 + 48 * q^11 + 90 * q^13 + 105 * q^15 + 163 * q^17 + 200 * q^19 + 105 * q^21 + 161 * q^23 + 175 * q^25 + 189 * q^27 + 81 * q^29 + 125 * q^31 + 144 * q^33 + 175 * q^35 + 5 * q^37 + 270 * q^39 + 369 * q^41 + 462 * q^43 + 315 * q^45 + 134 * q^47 + 614 * q^49 + 489 * q^51 + 561 * q^53 + 240 * q^55 + 600 * q^57 + 951 * q^59 + 860 * q^61 + 315 * q^63 + 450 * q^65 + 447 * q^67 + 483 * q^69 + 735 * q^71 + 1460 * q^73 + 525 * q^75 + 496 * q^77 + 18 * q^79 + 567 * q^81 + 261 * q^83 + 815 * q^85 + 243 * q^87 + 2024 * q^89 + 692 * q^91 + 375 * q^93 + 1000 * q^95 + 668 * q^97 + 432 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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