LMFDB - Embedded newform 20.4.c.a.9.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [20,4,Mod(9,20)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(20, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

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

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

chi := DirichletCharacter("20.9");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 20 = 2^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	20.c (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$1.18003820011$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-19}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 5 $ x^2 - x + 5
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			9.2
Root			$0.500000 - 2.17945i$ of defining polynomial
Character	$\chi$	$=$	20.9
Dual form			20.4.c.a.9.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+8.71780i q^{3} +(7.00000 - 8.71780i) q^{5} -8.71780i q^{7} -49.0000 q^{9} +O(q^{10})$	$q+8.71780i q^{3} +(7.00000 - 8.71780i) q^{5} -8.71780i q^{7} -49.0000 q^{9} +20.0000 q^{11} -52.3068i q^{13} +(76.0000 + 61.0246i) q^{15} +69.7424i q^{17} -84.0000 q^{19} +76.0000 q^{21} +61.0246i q^{23} +(-27.0000 - 122.049i) q^{25} -191.792i q^{27} +6.00000 q^{29} -224.000 q^{31} +174.356i q^{33} +(-76.0000 - 61.0246i) q^{35} +122.049i q^{37} +456.000 q^{39} +266.000 q^{41} -305.123i q^{43} +(-343.000 + 427.172i) q^{45} +374.865i q^{47} +267.000 q^{49} -608.000 q^{51} -366.148i q^{53} +(140.000 - 174.356i) q^{55} -732.295i q^{57} -28.0000 q^{59} +182.000 q^{61} +427.172i q^{63} +(-456.000 - 366.148i) q^{65} +427.172i q^{67} -532.000 q^{69} +408.000 q^{71} +1081.01i q^{73} +(1064.00 - 235.381i) q^{75} -174.356i q^{77} +48.0000 q^{79} +349.000 q^{81} -200.509i q^{83} +(608.000 + 488.197i) q^{85} +52.3068i q^{87} -1526.00 q^{89} -456.000 q^{91} -1952.79i q^{93} +(-588.000 + 732.295i) q^{95} +557.939i q^{97} -980.000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 14 q^{5} - 98 q^{9}+O(q^{10}) $ 2 * q + 14 * q^5 - 98 * q^9	$ 2 q + 14 q^{5} - 98 q^{9} + 40 q^{11} + 152 q^{15} - 168 q^{19} + 152 q^{21} - 54 q^{25} + 12 q^{29} - 448 q^{31} - 152 q^{35} + 912 q^{39} + 532 q^{41} - 686 q^{45} + 534 q^{49} - 1216 q^{51} + 280 q^{55} - 56 q^{59} + 364 q^{61} - 912 q^{65} - 1064 q^{69} + 816 q^{71} + 2128 q^{75} + 96 q^{79} + 698 q^{81} + 1216 q^{85} - 3052 q^{89} - 912 q^{91} - 1176 q^{95} - 1960 q^{99}+O(q^{100}) $ 2 * q + 14 * q^5 - 98 * q^9 + 40 * q^11 + 152 * q^15 - 168 * q^19 + 152 * q^21 - 54 * q^25 + 12 * q^29 - 448 * q^31 - 152 * q^35 + 912 * q^39 + 532 * q^41 - 686 * q^45 + 534 * q^49 - 1216 * q^51 + 280 * q^55 - 56 * q^59 + 364 * q^61 - 912 * q^65 - 1064 * q^69 + 816 * q^71 + 2128 * q^75 + 96 * q^79 + 698 * q^81 + 1216 * q^85 - 3052 * q^89 - 912 * q^91 - 1176 * q^95 - 1960 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/20\mathbb{Z}\right)^\times$.

$n$	$11$	$17$
$\chi(n)$	$1$	$-1$

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
$3$			8.71780i			1.67774i	0.544331	+	0.838870i	$0.316784\pi$
$3$			8.71780i			1.67774i	−0.544331	+	0.838870i	$0.683216\pi$
$4$		0			0
$5$	7.00000	−	8.71780i	0.626099	−	0.779744i
$5$	7.00000	−	8.71780i	0.626099	−	0.779744i
$6$		0			0
$7$		−	8.71780i		−	0.470717i	−0.971909	−	0.235358i	$-0.924374\pi$
$7$		−	8.71780i		−	0.470717i	0.971909	−	0.235358i	$-0.0756264\pi$
$8$		0			0
$9$	−49.0000			−1.81481
$10$		0			0
$11$	20.0000			0.548202			0.274101	−	0.961701i	$-0.411620\pi$
$11$	20.0000			0.548202			0.274101	+	0.961701i	$0.411620\pi$
$12$		0			0
$13$		−	52.3068i		−	1.11595i	−0.829859	−	0.557973i	$-0.811579\pi$
$13$		−	52.3068i		−	1.11595i	0.829859	−	0.557973i	$-0.188421\pi$
$14$		0			0
$15$	76.0000	+	61.0246i	1.30821	+	1.05043i
$16$		0			0
$17$			69.7424i			0.995001i	0.867464	+	0.497500i	$0.165749\pi$
$17$			69.7424i			0.995001i	−0.867464	+	0.497500i	$0.834251\pi$
$18$		0			0
$19$	−84.0000			−1.01426			−0.507130	−	0.861870i	$-0.669293\pi$
$19$	−84.0000			−1.01426			−0.507130	+	0.861870i	$0.669293\pi$
$20$		0			0
$21$	76.0000			0.789741
$22$		0			0
$23$			61.0246i			0.553239i	0.960979	+	0.276620i	$0.0892142\pi$
$23$			61.0246i			0.553239i	−0.960979	+	0.276620i	$0.910786\pi$
$24$		0			0
$25$	−27.0000	−	122.049i	−0.216000	−	0.976393i
$26$		0			0
$27$		−	191.792i		−	1.36705i
$28$		0			0
$29$	6.00000			0.0384197			0.0192099	−	0.999815i	$-0.493885\pi$
$29$	6.00000			0.0384197			0.0192099	+	0.999815i	$0.493885\pi$
$30$		0			0
$31$	−224.000			−1.29779			−0.648897	−	0.760877i	$-0.724769\pi$
$31$	−224.000			−1.29779			−0.648897	+	0.760877i	$0.724769\pi$
$32$		0			0
$33$			174.356i			0.919742i
$34$		0			0
$35$	−76.0000	−	61.0246i	−0.367038	−	0.294715i
$36$		0			0
$37$			122.049i			0.542291i	0.962538	+	0.271145i	$0.0874024\pi$
$37$			122.049i			0.542291i	−0.962538	+	0.271145i	$0.912598\pi$
$38$		0			0
$39$	456.000			1.87227
$40$		0			0
$41$	266.000			1.01322			0.506612	−	0.862174i	$-0.330898\pi$
$41$	266.000			1.01322			0.506612	+	0.862174i	$0.330898\pi$
$42$		0			0
$43$		−	305.123i		−	1.08211i	−0.840987	−	0.541056i	$-0.818025\pi$
$43$		−	305.123i		−	1.08211i	0.840987	−	0.541056i	$-0.181975\pi$
$44$		0			0
$45$	−343.000	+	427.172i	−1.13625	+	1.41509i
$46$		0			0
$47$			374.865i			1.16340i	0.813404	+	0.581699i	$0.197612\pi$
$47$			374.865i			1.16340i	−0.813404	+	0.581699i	$0.802388\pi$
$48$		0			0
$49$	267.000			0.778426
$50$		0			0
$51$	−608.000			−1.66935
$52$		0			0
$53$		−	366.148i		−	0.948948i	−0.880270	−	0.474474i	$-0.842638\pi$
$53$		−	366.148i		−	0.948948i	0.880270	−	0.474474i	$-0.157362\pi$
$54$		0			0
$55$	140.000	−	174.356i	0.343229	−	0.427457i
$56$		0			0
$57$		−	732.295i		−	1.70166i
$58$		0			0
$59$	−28.0000			−0.0617846			−0.0308923	−	0.999523i	$-0.509835\pi$
$59$	−28.0000			−0.0617846			−0.0308923	+	0.999523i	$0.509835\pi$
$60$		0			0
$61$	182.000			0.382012			0.191006	−	0.981589i	$-0.438825\pi$
$61$	182.000			0.382012			0.191006	+	0.981589i	$0.438825\pi$
$62$		0			0
$63$			427.172i			0.854264i
$64$		0			0
$65$	−456.000	−	366.148i	−0.870151	−	0.698692i
$66$		0			0
$67$			427.172i			0.778916i	0.921044	+	0.389458i	$0.127338\pi$
$67$			427.172i			0.778916i	−0.921044	+	0.389458i	$0.872662\pi$
$68$		0			0
$69$	−532.000			−0.928192
$70$		0			0
$71$	408.000			0.681982			0.340991	−	0.940067i	$-0.389238\pi$
$71$	408.000			0.681982			0.340991	+	0.940067i	$0.389238\pi$
$72$		0			0
$73$			1081.01i			1.73318i	0.499019	+	0.866591i	$0.333694\pi$
$73$			1081.01i			1.73318i	−0.499019	+	0.866591i	$0.666306\pi$
$74$		0			0
$75$	1064.00	−	235.381i	1.63814	−	0.362392i
$76$		0			0
$77$		−	174.356i		−	0.258048i
$78$		0			0
$79$	48.0000			0.0683598			0.0341799	−	0.999416i	$-0.489118\pi$
$79$	48.0000			0.0683598			0.0341799	+	0.999416i	$0.489118\pi$
$80$		0			0
$81$	349.000			0.478738
$82$		0			0
$83$		−	200.509i		−	0.265166i	−0.991172	−	0.132583i	$-0.957673\pi$
$83$		−	200.509i		−	0.265166i	0.991172	−	0.132583i	$-0.0423271\pi$
$84$		0			0
$85$	608.000	+	488.197i	0.775845	+	0.622969i
$86$		0			0
$87$			52.3068i			0.0644583i
$88$		0			0
$89$	−1526.00			−1.81748			−0.908740	−	0.417363i	$-0.862954\pi$
$89$	−1526.00			−1.81748			−0.908740	+	0.417363i	$0.862954\pi$
$90$		0			0
$91$	−456.000			−0.525294
$92$		0			0
$93$		−	1952.79i		−	2.17736i
$94$		0			0
$95$	−588.000	+	732.295i	−0.635027	+	0.790862i
$96$		0			0
$97$			557.939i			0.584022i	0.956415	+	0.292011i	$0.0943244\pi$
$97$			557.939i			0.584022i	−0.956415	+	0.292011i	$0.905676\pi$
$98$		0			0
$99$	−980.000			−0.994886
$100$		0			0
$101$	1246.00			1.22754			0.613770	−	0.789485i	$-0.289652\pi$
$101$	1246.00			1.22754			0.613770	+	0.789485i	$0.289652\pi$
$102$		0			0
$103$		−	845.626i		−	0.808952i	−0.914549	−	0.404476i	$-0.867454\pi$
$103$		−	845.626i		−	0.808952i	0.914549	−	0.404476i	$-0.132546\pi$
$104$		0			0
$105$	532.000	−	662.553i	0.494456	−	0.615795i
$106$		0			0
$107$		−	1281.52i		−	1.15784i	−0.815384	−	0.578920i	$-0.803474\pi$
$107$		−	1281.52i		−	1.15784i	0.815384	−	0.578920i	$-0.196526\pi$
$108$		0			0
$109$	902.000			0.792623			0.396312	−	0.918116i	$-0.370290\pi$
$109$	902.000			0.792623			0.396312	+	0.918116i	$0.370290\pi$
$110$		0			0
$111$	−1064.00			−0.909824
$112$		0			0
$113$			1464.59i			1.21927i	0.792684	+	0.609633i	$0.208683\pi$
$113$			1464.59i			1.21927i	−0.792684	+	0.609633i	$0.791317\pi$
$114$		0			0
$115$	532.000	+	427.172i	0.431385	+	0.346383i
$116$		0			0
$117$			2563.03i			2.02523i
$118$		0			0
$119$	608.000			0.468364
$120$		0			0
$121$	−931.000			−0.699474
$122$		0			0
$123$			2318.93i			1.69993i
$124$		0			0
$125$	−1253.00	−	618.964i	−0.896574	−	0.442894i
$126$		0			0
$127$		−	2624.06i		−	1.83344i	−0.399525	−	0.916722i	$-0.630825\pi$
$127$		−	2624.06i		−	1.83344i	0.399525	−	0.916722i	$-0.369175\pi$
$128$		0			0
$129$	2660.00			1.81550
$130$		0			0
$131$	2940.00			1.96083			0.980416	−	0.196938i	$-0.0630997\pi$
$131$	2940.00			1.96083			0.980416	+	0.196938i	$0.0630997\pi$
$132$		0			0
$133$			732.295i			0.477429i
$134$		0			0
$135$	−1672.00	−	1342.54i	−1.06595	−	0.855908i
$136$		0			0
$137$		−	732.295i		−	0.456673i	−0.973582	−	0.228336i	$-0.926671\pi$
$137$		−	732.295i		−	0.456673i	0.973582	−	0.228336i	$-0.0733286\pi$
$138$		0			0
$139$	−364.000			−0.222116			−0.111058	−	0.993814i	$-0.535424\pi$
$139$	−364.000			−0.222116			−0.111058	+	0.993814i	$0.535424\pi$
$140$		0			0
$141$	−3268.00			−1.95188
$142$		0			0
$143$		−	1046.14i		−	0.611764i
$144$		0			0
$145$	42.0000	−	52.3068i	0.0240546	−	0.0299575i
$146$		0			0
$147$			2327.65i			1.30600i
$148$		0			0
$149$	254.000			0.139654			0.0698272	−	0.997559i	$-0.477755\pi$
$149$	254.000			0.139654			0.0698272	+	0.997559i	$0.477755\pi$
$150$		0			0
$151$	−2360.00			−1.27188			−0.635941	−	0.771738i	$-0.719388\pi$
$151$	−2360.00			−1.27188			−0.635941	+	0.771738i	$0.719388\pi$
$152$		0			0
$153$		−	3417.38i		−	1.80574i
$154$		0			0
$155$	−1568.00	+	1952.79i	−0.812547	+	1.01195i
$156$		0			0
$157$		−	2214.32i		−	1.12562i	−0.826587	−	0.562809i	$-0.809721\pi$
$157$		−	2214.32i		−	1.12562i	0.826587	−	0.562809i	$-0.190279\pi$
$158$		0			0
$159$	3192.00			1.59209
$160$		0			0
$161$	532.000			0.260419
$162$		0			0
$163$		−	1525.61i		−	0.733100i	−0.930398	−	0.366550i	$-0.880539\pi$
$163$		−	1525.61i		−	0.733100i	0.930398	−	0.366550i	$-0.119461\pi$
$164$		0			0
$165$	1520.00	+	1220.49i	0.717163	+	0.575849i
$166$		0			0
$167$			3059.95i			1.41788i	0.705269	+	0.708940i	$0.250826\pi$
$167$			3059.95i			1.41788i	−0.705269	+	0.708940i	$0.749174\pi$
$168$		0			0
$169$	−539.000			−0.245335
$170$		0			0
$171$	4116.00			1.84069
$172$		0			0
$173$			1900.48i			0.835207i	0.908629	+	0.417604i	$0.137130\pi$
$173$			1900.48i			0.835207i	−0.908629	+	0.417604i	$0.862870\pi$
$174$		0			0
$175$	−1064.00	+	235.381i	−0.459605	+	0.101675i
$176$		0			0
$177$		−	244.098i		−	0.103659i
$178$		0			0
$179$	−1972.00			−0.823431			−0.411716	−	0.911312i	$-0.635070\pi$
$179$	−1972.00			−0.823431			−0.411716	+	0.911312i	$0.635070\pi$
$180$		0			0
$181$	−1330.00			−0.546177			−0.273089	−	0.961989i	$-0.588045\pi$
$181$	−1330.00			−0.546177			−0.273089	+	0.961989i	$0.588045\pi$
$182$		0			0
$183$			1586.64i			0.640917i
$184$		0			0
$185$	1064.00	+	854.344i	0.422848	+	0.339528i
$186$		0			0
$187$			1394.85i			0.545462i
$188$		0			0
$189$	−1672.00			−0.643493
$190$		0			0
$191$	1728.00			0.654627			0.327313	−	0.944916i	$-0.393857\pi$
$191$	1728.00			0.654627			0.327313	+	0.944916i	$0.393857\pi$
$192$		0			0
$193$			976.393i			0.364157i	0.983284	+	0.182079i	$0.0582825\pi$
$193$			976.393i			0.364157i	−0.983284	+	0.182079i	$0.941717\pi$
$194$		0			0
$195$	3192.00	−	3975.32i	1.17222	−	1.45989i
$196$		0			0
$197$		−	3783.52i		−	1.36835i	−0.729318	−	0.684175i	$-0.760162\pi$
$197$		−	3783.52i		−	1.36835i	0.729318	−	0.684175i	$-0.239838\pi$
$198$		0			0
$199$	−1512.00			−0.538607			−0.269304	−	0.963055i	$-0.586794\pi$
$199$	−1512.00			−0.538607			−0.269304	+	0.963055i	$0.586794\pi$
$200$		0			0
$201$	−3724.00			−1.30682
$202$		0			0
$203$		−	52.3068i		−	0.0180848i
$204$		0			0
$205$	1862.00	−	2318.93i	0.634379	−	0.790056i
$206$		0			0
$207$		−	2990.20i		−	1.00403i
$208$		0			0
$209$	−1680.00			−0.556019
$210$		0			0
$211$	1644.00			0.536387			0.268193	−	0.963365i	$-0.413573\pi$
$211$	1644.00			0.536387			0.268193	+	0.963365i	$0.413573\pi$
$212$		0			0
$213$			3556.86i			1.14419i
$214$		0			0
$215$	−2660.00	−	2135.86i	−0.843770	−	0.677509i
$216$		0			0
$217$			1952.79i			0.610893i
$218$		0			0
$219$	−9424.00			−2.90783
$220$		0			0
$221$	3648.00			1.11037
$222$		0			0
$223$		−	1089.72i		−	0.327235i	−0.986524	−	0.163617i	$-0.947684\pi$
$223$		−	1089.72i		−	0.327235i	0.986524	−	0.163617i	$-0.0523163\pi$
$224$		0			0
$225$	1323.00	+	5980.41i	0.392000	+	1.77197i
$226$		0			0
$227$			985.111i			0.288036i	0.989575	+	0.144018i	$0.0460023\pi$
$227$			985.111i			0.288036i	−0.989575	+	0.144018i	$0.953998\pi$
$228$		0			0
$229$	3934.00			1.13522			0.567611	−	0.823297i	$-0.307868\pi$
$229$	3934.00			1.13522			0.567611	+	0.823297i	$0.307868\pi$
$230$		0			0
$231$	1520.00			0.432938
$232$		0			0
$233$		−	4637.87i		−	1.30402i	−0.758210	−	0.652010i	$-0.773926\pi$
$233$		−	4637.87i		−	1.30402i	0.758210	−	0.652010i	$-0.226074\pi$
$234$		0			0
$235$	3268.00	+	2624.06i	0.907152	+	0.728403i
$236$		0			0
$237$			418.454i			0.114690i
$238$		0			0
$239$	3856.00			1.04361			0.521807	−	0.853063i	$-0.325258\pi$
$239$	3856.00			1.04361			0.521807	+	0.853063i	$0.325258\pi$
$240$		0			0
$241$	994.000			0.265681			0.132841	−	0.991137i	$-0.457590\pi$
$241$	994.000			0.265681			0.132841	+	0.991137i	$0.457590\pi$
$242$		0			0
$243$		−	2135.86i		−	0.563850i
$244$		0			0
$245$	1869.00	−	2327.65i	0.487372	−	0.606972i
$246$		0			0
$247$			4393.77i			1.13186i
$248$		0			0
$249$	1748.00			0.444880
$250$		0			0
$251$	−6300.00			−1.58427			−0.792136	−	0.610344i	$-0.791031\pi$
$251$	−6300.00			−1.58427			−0.792136	+	0.610344i	$0.791031\pi$
$252$		0			0
$253$			1220.49i			0.303287i
$254$		0			0
$255$	−4256.00	+	5300.42i	−1.04518	+	1.30167i
$256$		0			0
$257$			6695.27i			1.62506i	0.582922	+	0.812528i	$0.301909\pi$
$257$			6695.27i			1.62506i	−0.582922	+	0.812528i	$0.698091\pi$
$258$		0			0
$259$	1064.00			0.255265
$260$		0			0
$261$	−294.000			−0.0697247
$262$		0			0
$263$		−	1403.57i		−	0.329078i	−0.986371	−	0.164539i	$-0.947386\pi$
$263$		−	1403.57i		−	0.329078i	0.986371	−	0.164539i	$-0.0526137\pi$
$264$		0			0
$265$	−3192.00	−	2563.03i	−0.739936	−	0.594135i
$266$		0			0
$267$		−	13303.4i		−	3.04926i
$268$		0			0
$269$	−5082.00			−1.15188			−0.575939	−	0.817493i	$-0.695363\pi$
$269$	−5082.00			−1.15188			−0.575939	+	0.817493i	$0.695363\pi$
$270$		0			0
$271$	2800.00			0.627631			0.313815	−	0.949484i	$-0.398393\pi$
$271$	2800.00			0.627631			0.313815	+	0.949484i	$0.398393\pi$
$272$		0			0
$273$		−	3975.32i		−	0.881308i
$274$		0			0
$275$	−540.000	−	2440.98i	−0.118412	−	0.535261i
$276$		0			0
$277$			1586.64i			0.344159i	0.985083	+	0.172079i	$0.0550485\pi$
$277$			1586.64i			0.344159i	−0.985083	+	0.172079i	$0.944951\pi$
$278$		0			0
$279$	10976.0			2.35525
$280$		0			0
$281$	−1254.00			−0.266218			−0.133109	−	0.991101i	$-0.542496\pi$
$281$	−1254.00			−0.266218			−0.133109	+	0.991101i	$0.542496\pi$
$282$		0			0
$283$			4646.59i			0.976010i	0.872841	+	0.488005i	$0.162275\pi$
$283$			4646.59i			0.976010i	−0.872841	+	0.488005i	$0.837725\pi$
$284$		0			0
$285$	−6384.00	−	5126.07i	−1.32686	−	1.06541i
$286$		0			0
$287$		−	2318.93i		−	0.476942i
$288$		0			0
$289$	49.0000			0.00997354
$290$		0			0
$291$	−4864.00			−0.979838
$292$		0			0
$293$			6538.35i			1.30367i	0.758362	+	0.651833i	$0.226000\pi$
$293$			6538.35i			1.30367i	−0.758362	+	0.651833i	$0.774000\pi$
$294$		0			0
$295$	−196.000	+	244.098i	−0.0386833	+	0.0481761i
$296$		0			0
$297$		−	3835.83i		−	0.749419i
$298$		0			0
$299$	3192.00			0.617385
$300$		0			0
$301$	−2660.00			−0.509368
$302$		0			0
$303$			10862.4i			2.05950i
$304$		0			0
$305$	1274.00	−	1586.64i	0.239177	−	0.297871i
$306$		0			0
$307$			6355.27i			1.18148i	0.806862	+	0.590741i	$0.201164\pi$
$307$			6355.27i			1.18148i	−0.806862	+	0.590741i	$0.798836\pi$
$308$		0			0
$309$	7372.00			1.35721
$310$		0			0
$311$	−5208.00			−0.949577			−0.474789	−	0.880100i	$-0.657476\pi$
$311$	−5208.00			−0.949577			−0.474789	+	0.880100i	$0.657476\pi$
$312$		0			0
$313$		−	4568.13i		−	0.824939i	−0.910972	−	0.412469i	$-0.864666\pi$
$313$		−	4568.13i		−	0.824939i	0.910972	−	0.412469i	$-0.135334\pi$
$314$		0			0
$315$	3724.00	+	2990.20i	0.666107	+	0.534854i
$316$		0			0
$317$			854.344i			0.151371i	0.997132	+	0.0756857i	$0.0241146\pi$
$317$			854.344i			0.151371i	−0.997132	+	0.0756857i	$0.975885\pi$
$318$		0			0
$319$	120.000			0.0210618
$320$		0			0
$321$	11172.0			1.94256
$322$		0			0
$323$		−	5858.36i		−	1.00919i
$324$		0			0
$325$	−6384.00	+	1412.28i	−1.08960	+	0.241044i
$326$		0			0
$327$			7863.45i			1.32982i
$328$		0			0
$329$	3268.00			0.547631
$330$		0			0
$331$	7828.00			1.29990			0.649948	−	0.759978i	$-0.274791\pi$
$331$	7828.00			1.29990			0.649948	+	0.759978i	$0.274791\pi$
$332$		0			0
$333$		−	5980.41i		−	0.984157i
$334$		0			0
$335$	3724.00	+	2990.20i	0.607355	+	0.487679i
$336$		0			0
$337$		−	5370.16i		−	0.868046i	−0.900902	−	0.434023i	$-0.857094\pi$
$337$		−	5370.16i		−	0.868046i	0.900902	−	0.434023i	$-0.142906\pi$
$338$		0			0
$339$	−12768.0			−2.04561
$340$		0			0
$341$	−4480.00			−0.711453
$342$		0			0
$343$		−	5317.86i		−	0.837135i
$344$		0			0
$345$	−3724.00	+	4637.87i	−0.581140	+	0.723752i
$346$		0			0
$347$		−	2746.11i		−	0.424838i	−0.977179	−	0.212419i	$-0.931866\pi$
$347$		−	2746.11i		−	0.424838i	0.977179	−	0.212419i	$-0.0681341\pi$
$348$		0			0
$349$	−8890.00			−1.36353			−0.681763	−	0.731573i	$-0.738787\pi$
$349$	−8890.00			−1.36353			−0.681763	+	0.731573i	$0.738787\pi$
$350$		0			0
$351$	−10032.0			−1.52555
$352$		0			0
$353$			1534.33i			0.231344i	0.993287	+	0.115672i	$0.0369021\pi$
$353$			1534.33i			0.231344i	−0.993287	+	0.115672i	$0.963098\pi$
$354$		0			0
$355$	2856.00	−	3556.86i	0.426988	−	0.531771i
$356$		0			0
$357$			5300.42i			0.785793i
$358$		0			0
$359$	9144.00			1.34429			0.672147	−	0.740417i	$-0.265372\pi$
$359$	9144.00			1.34429			0.672147	+	0.740417i	$0.265372\pi$
$360$		0			0
$361$	197.000			0.0287214
$362$		0			0
$363$		−	8116.27i		−	1.17354i
$364$		0			0
$365$	9424.00	+	7567.05i	1.35144	+	1.08514i
$366$		0			0
$367$		−	10365.5i		−	1.47431i	−0.675722	−	0.737156i	$-0.736168\pi$
$367$		−	10365.5i		−	1.47431i	0.675722	−	0.737156i	$-0.263832\pi$
$368$		0			0
$369$	−13034.0			−1.83882
$370$		0			0
$371$	−3192.00			−0.446686
$372$		0			0
$373$		−	3295.33i		−	0.457441i	−0.973492	−	0.228721i	$-0.926546\pi$
$373$		−	3295.33i		−	0.457441i	0.973492	−	0.228721i	$-0.0734542\pi$
$374$		0			0
$375$	5396.00	−	10923.4i	0.743062	−	1.50422i
$376$		0			0
$377$		−	313.841i		−	0.0428743i
$378$		0			0
$379$	−2588.00			−0.350756			−0.175378	−	0.984501i	$-0.556115\pi$
$379$	−2588.00			−0.350756			−0.175378	+	0.984501i	$0.556115\pi$
$380$		0			0
$381$	22876.0			3.07604
$382$		0			0
$383$			4071.21i			0.543157i	0.962416	+	0.271579i	$0.0875457\pi$
$383$			4071.21i			0.543157i	−0.962416	+	0.271579i	$0.912454\pi$
$384$		0			0
$385$	−1520.00	−	1220.49i	−0.201211	−	0.161564i
$386$		0			0
$387$			14951.0i			1.96383i
$388$		0			0
$389$	−5426.00			−0.707221			−0.353611	−	0.935393i	$-0.615046\pi$
$389$	−5426.00			−0.707221			−0.353611	+	0.935393i	$0.615046\pi$
$390$		0			0
$391$	−4256.00			−0.550474
$392$		0			0
$393$			25630.3i			3.28977i
$394$		0			0
$395$	336.000	−	418.454i	0.0428000	−	0.0533031i
$396$		0			0
$397$		−	9816.24i		−	1.24096i	−0.784220	−	0.620482i	$-0.786937\pi$
$397$		−	9816.24i		−	1.24096i	0.784220	−	0.620482i	$-0.213063\pi$
$398$		0			0
$399$	−6384.00			−0.801002
$400$		0			0
$401$	370.000			0.0460771			0.0230386	−	0.999735i	$-0.492666\pi$
$401$	370.000			0.0460771			0.0230386	+	0.999735i	$0.492666\pi$
$402$		0			0
$403$			11716.7i			1.44827i
$404$		0			0
$405$	2443.00	−	3042.51i	0.299737	−	0.373293i
$406$		0			0
$407$			2440.98i			0.297285i
$408$		0			0
$409$	4186.00			0.506074			0.253037	−	0.967457i	$-0.418571\pi$
$409$	4186.00			0.506074			0.253037	+	0.967457i	$0.418571\pi$
$410$		0			0
$411$	6384.00			0.766179
$412$		0			0
$413$			244.098i			0.0290830i
$414$		0			0
$415$	−1748.00	−	1403.57i	−0.206761	−	0.166020i
$416$		0			0
$417$		−	3173.28i		−	0.372653i
$418$		0			0
$419$	2716.00			0.316671			0.158336	−	0.987385i	$-0.449387\pi$
$419$	2716.00			0.316671			0.158336	+	0.987385i	$0.449387\pi$
$420$		0			0
$421$	1966.00			0.227594			0.113797	−	0.993504i	$-0.463699\pi$
$421$	1966.00			0.227594			0.113797	+	0.993504i	$0.463699\pi$
$422$		0			0
$423$		−	18368.4i		−	2.11135i
$424$		0			0
$425$	8512.00	−	1883.04i	0.971512	−	0.214920i
$426$		0			0
$427$		−	1586.64i		−	0.179819i
$428$		0			0
$429$	9120.00			1.02638
$430$		0			0
$431$	−11824.0			−1.32144			−0.660722	−	0.750631i	$-0.729750\pi$
$431$	−11824.0			−1.32144			−0.660722	+	0.750631i	$0.729750\pi$
$432$		0			0
$433$		−	2580.47i		−	0.286396i	−0.989694	−	0.143198i	$-0.954261\pi$
$433$		−	2580.47i		−	0.286396i	0.989694	−	0.143198i	$-0.0457385\pi$
$434$		0			0
$435$	456.000	+	366.148i	0.0502610	+	0.0403573i
$436$		0			0
$437$		−	5126.07i		−	0.561128i
$438$		0			0
$439$	−13272.0			−1.44291			−0.721456	−	0.692461i	$-0.756527\pi$
$439$	−13272.0			−1.44291			−0.721456	+	0.692461i	$0.756527\pi$
$440$		0			0
$441$	−13083.0			−1.41270
$442$		0			0
$443$			3112.25i			0.333787i	0.985975	+	0.166894i	$0.0533736\pi$
$443$			3112.25i			0.333787i	−0.985975	+	0.166894i	$0.946626\pi$
$444$		0			0
$445$	−10682.0	+	13303.4i	−1.13792	+	1.41717i
$446$		0			0
$447$			2214.32i			0.234304i
$448$		0			0
$449$	−2094.00			−0.220093			−0.110047	−	0.993926i	$-0.535100\pi$
$449$	−2094.00			−0.220093			−0.110047	+	0.993926i	$0.535100\pi$
$450$		0			0
$451$	5320.00			0.555452
$452$		0			0
$453$		−	20574.0i		−	2.13389i
$454$		0			0
$455$	−3192.00	+	3975.32i	−0.328886	+	0.409595i
$456$		0			0
$457$			7078.85i			0.724584i	0.932065	+	0.362292i	$0.118006\pi$
$457$			7078.85i			0.724584i	−0.932065	+	0.362292i	$0.881994\pi$
$458$		0			0
$459$	13376.0			1.36021
$460$		0			0
$461$	−9450.00			−0.954730			−0.477365	−	0.878705i	$-0.658408\pi$
$461$	−9450.00			−0.954730			−0.477365	+	0.878705i	$0.658408\pi$
$462$		0			0
$463$		−	3112.25i		−	0.312395i	−0.987726	−	0.156197i	$-0.950076\pi$
$463$		−	3112.25i		−	0.312395i	0.987726	−	0.156197i	$-0.0499236\pi$
$464$		0			0
$465$	−17024.0	−	13669.5i	−1.69778	−	1.36324i
$466$		0			0
$467$		−	5361.45i		−	0.531259i	−0.964075	−	0.265630i	$-0.914420\pi$
$467$		−	5361.45i		−	0.531259i	0.964075	−	0.265630i	$-0.0855798\pi$
$468$		0			0
$469$	3724.00			0.366649
$470$		0			0
$471$	19304.0			1.88850
$472$		0			0
$473$		−	6102.46i		−	0.593216i
$474$		0			0
$475$	2268.00	+	10252.1i	0.219080	+	0.990316i
$476$		0			0
$477$			17941.2i			1.72216i
$478$		0			0
$479$	13216.0			1.26066			0.630328	−	0.776329i	$-0.282920\pi$
$479$	13216.0			1.26066			0.630328	+	0.776329i	$0.282920\pi$
$480$		0			0
$481$	6384.00			0.605167
$482$		0			0
$483$			4637.87i			0.436916i
$484$		0			0
$485$	4864.00	+	3905.57i	0.455387	+	0.365656i
$486$		0			0
$487$		−	2379.96i		−	0.221450i	−0.993851	−	0.110725i	$-0.964683\pi$
$487$		−	2379.96i		−	0.221450i	0.993851	−	0.110725i	$-0.0353173\pi$
$488$		0			0
$489$	13300.0			1.22995
$490$		0			0
$491$	−12300.0			−1.13053			−0.565266	−	0.824909i	$-0.691226\pi$
$491$	−12300.0			−1.13053			−0.565266	+	0.824909i	$0.691226\pi$
$492$		0			0
$493$			418.454i			0.0382277i
$494$		0			0
$495$	−6860.00	+	8543.44i	−0.622897	+	0.775756i
$496$		0			0
$497$		−	3556.86i		−	0.321020i
$498$		0			0
$499$	15756.0			1.41350			0.706749	−	0.707464i	$-0.250161\pi$
$499$	15756.0			1.41350			0.706749	+	0.707464i	$0.250161\pi$
$500$		0			0
$501$	−26676.0			−2.37883
$502$		0			0
$503$			19170.4i			1.69934i	0.527316	+	0.849670i	$0.323199\pi$
$503$			19170.4i			1.69934i	−0.527316	+	0.849670i	$0.676801\pi$
$504$		0			0
$505$	8722.00	−	10862.4i	0.768562	−	0.957167i
$506$		0			0
$507$		−	4698.89i		−	0.411608i
$508$		0			0
$509$	−2394.00			−0.208472			−0.104236	−	0.994553i	$-0.533240\pi$
$509$	−2394.00			−0.208472			−0.104236	+	0.994553i	$0.533240\pi$
$510$		0			0
$511$	9424.00			0.815838
$512$		0			0
$513$			16110.5i			1.38654i
$514$		0			0
$515$	−7372.00	−	5919.38i	−0.630775	−	0.506484i
$516$		0			0
$517$			7497.31i			0.637778i
$518$		0			0
$519$	−16568.0			−1.40126
$520$		0			0
$521$	6874.00			0.578033			0.289017	−	0.957324i	$-0.406672\pi$
$521$	6874.00			0.578033			0.289017	+	0.957324i	$0.406672\pi$
$522$		0			0
$523$			16154.1i			1.35061i	0.737539	+	0.675305i	$0.235988\pi$
$523$			16154.1i			1.35061i	−0.737539	+	0.675305i	$0.764012\pi$
$524$		0			0
$525$	−2052.00	−	9275.74i	−0.170584	−	0.771098i
$526$		0			0
$527$		−	15622.3i		−	1.29131i
$528$		0			0
$529$	8443.00			0.693926
$530$		0			0
$531$	1372.00			0.112128
$532$		0			0
$533$		−	13913.6i		−	1.13070i
$534$		0			0
$535$	−11172.0	−	8970.61i	−0.902818	−	0.724922i
$536$		0			0
$537$		−	17191.5i		−	1.38150i
$538$		0			0
$539$	5340.00			0.426735
$540$		0			0
$541$	7174.00			0.570119			0.285059	−	0.958510i	$-0.407987\pi$
$541$	7174.00			0.570119			0.285059	+	0.958510i	$0.407987\pi$
$542$		0			0
$543$		−	11594.7i		−	0.916344i
$544$		0			0
$545$	6314.00	−	7863.45i	0.496261	−	0.618043i
$546$		0			0
$547$			4332.75i			0.338674i	0.985558	+	0.169337i	$0.0541627\pi$
$547$			4332.75i			0.338674i	−0.985558	+	0.169337i	$0.945837\pi$
$548$		0			0
$549$	−8918.00			−0.693280
$550$		0			0
$551$	−504.000			−0.0389676
$552$		0			0
$553$		−	418.454i		−	0.0321781i
$554$		0			0
$555$	−7448.00	+	9275.74i	−0.569640	+	0.709429i
$556$		0			0
$557$			9153.69i			0.696327i	0.937434	+	0.348164i	$0.113195\pi$
$557$			9153.69i			0.696327i	−0.937434	+	0.348164i	$0.886805\pi$
$558$		0			0
$559$	−15960.0			−1.20758
$560$		0			0
$561$	−12160.0			−0.915144
$562$		0			0
$563$		−	13870.0i		−	1.03828i	−0.854689	−	0.519140i	$-0.826252\pi$
$563$		−	13870.0i		−	1.03828i	0.854689	−	0.519140i	$-0.173748\pi$
$564$		0			0
$565$	12768.0	+	10252.1i	0.950715	+	0.763381i
$566$		0			0
$567$		−	3042.51i		−	0.225350i
$568$		0			0
$569$	−16070.0			−1.18399			−0.591994	−	0.805942i	$-0.701659\pi$
$569$	−16070.0			−1.18399			−0.591994	+	0.805942i	$0.701659\pi$
$570$		0			0
$571$	−3932.00			−0.288177			−0.144089	−	0.989565i	$-0.546025\pi$
$571$	−3932.00			−0.288177			−0.144089	+	0.989565i	$0.546025\pi$
$572$		0			0
$573$			15064.4i			1.09829i
$574$		0			0
$575$	7448.00	−	1647.66i	0.540179	−	0.119500i
$576$		0			0
$577$			16180.2i			1.16740i	0.811968	+	0.583702i	$0.198396\pi$
$577$			16180.2i			1.16740i	−0.811968	+	0.583702i	$0.801604\pi$
$578$		0			0
$579$	−8512.00			−0.610961
$580$		0			0
$581$	−1748.00			−0.124818
$582$		0			0
$583$		−	7322.95i		−	0.520215i
$584$		0			0
$585$	22344.0	+	17941.2i	1.57916	+	1.26800i
$586$		0			0
$587$			11272.1i			0.792589i	0.918123	+	0.396295i	$0.129704\pi$
$587$			11272.1i			0.792589i	−0.918123	+	0.396295i	$0.870296\pi$
$588$		0			0
$589$	18816.0			1.31630
$590$		0			0
$591$	32984.0			2.29574
$592$		0			0
$593$		−	10670.6i		−	0.738935i	−0.929244	−	0.369467i	$-0.879540\pi$
$593$		−	10670.6i		−	0.738935i	0.929244	−	0.369467i	$-0.120460\pi$
$594$		0			0
$595$	4256.00	−	5300.42i	0.293242	−	0.365203i
$596$		0			0
$597$		−	13181.3i		−	0.903643i
$598$		0			0
$599$	−6840.00			−0.466569			−0.233284	−	0.972409i	$-0.574947\pi$
$599$	−6840.00			−0.466569			−0.233284	+	0.972409i	$0.574947\pi$
$600$		0			0
$601$	−10150.0			−0.688897			−0.344449	−	0.938805i	$-0.611934\pi$
$601$	−10150.0			−0.688897			−0.344449	+	0.938805i	$0.611934\pi$
$602$		0			0
$603$		−	20931.4i		−	1.41359i
$604$		0			0
$605$	−6517.00	+	8116.27i	−0.437940	+	0.545410i
$606$		0			0
$607$			1839.46i			0.123000i	0.998107	+	0.0615002i	$0.0195885\pi$
$607$			1839.46i			0.123000i	−0.998107	+	0.0615002i	$0.980412\pi$
$608$		0			0
$609$	456.000			0.0303416
$610$		0			0
$611$	19608.0			1.29829
$612$		0			0
$613$			5004.02i			0.329707i	0.986318	+	0.164853i	$0.0527151\pi$
$613$			5004.02i			0.329707i	−0.986318	+	0.164853i	$0.947285\pi$
$614$		0			0
$615$	20216.0	+	16232.5i	1.32551	+	1.06432i
$616$		0			0
$617$			24653.9i			1.60864i	0.594197	+	0.804319i	$0.297470\pi$
$617$			24653.9i			1.60864i	−0.594197	+	0.804319i	$0.702530\pi$
$618$		0			0
$619$	−27020.0			−1.75448			−0.877242	−	0.480049i	$-0.840619\pi$
$619$	−27020.0			−1.75448			−0.877242	+	0.480049i	$0.840619\pi$
$620$		0			0
$621$	11704.0			0.756305
$622$		0			0
$623$			13303.4i			0.855518i
$624$		0			0
$625$	−14167.0	+	6590.66i	−0.906688	+	0.421802i
$626$		0			0
$627$		−	14645.9i		−	0.932856i
$628$		0			0
$629$	−8512.00			−0.539580
$630$		0			0
$631$	−10648.0			−0.671775			−0.335888	−	0.941902i	$-0.609036\pi$
$631$	−10648.0			−0.671775			−0.335888	+	0.941902i	$0.609036\pi$
$632$		0			0
$633$			14332.1i			0.899918i
$634$		0			0
$635$	−22876.0	−	18368.4i	−1.42962	−	1.14792i
$636$		0			0
$637$		−	13965.9i		−	0.868681i
$638$		0			0
$639$	−19992.0			−1.23767
$640$		0			0
$641$	1682.00			0.103643			0.0518214	−	0.998656i	$-0.483497\pi$
$641$	1682.00			0.103643			0.0518214	+	0.998656i	$0.483497\pi$
$642$		0			0
$643$		−	18542.8i		−	1.13725i	−0.822595	−	0.568627i	$-0.807475\pi$
$643$		−	18542.8i		−	1.13725i	0.822595	−	0.568627i	$-0.192525\pi$
$644$		0			0
$645$	18620.0	−	23189.3i	1.13668	−	1.41563i
$646$		0			0
$647$			30677.9i			1.86410i	0.362327	+	0.932051i	$0.381982\pi$
$647$			30677.9i			1.86410i	−0.362327	+	0.932051i	$0.618018\pi$
$648$		0			0
$649$	−560.000			−0.0338705
$650$		0			0
$651$	−17024.0			−1.02492
$652$		0			0
$653$			1342.54i			0.0804559i	0.999191	+	0.0402279i	$0.0128084\pi$
$653$			1342.54i			0.0804559i	−0.999191	+	0.0402279i	$0.987192\pi$
$654$		0			0
$655$	20580.0	−	25630.3i	1.22768	−	1.52895i
$656$		0			0
$657$		−	52969.3i		−	3.14540i
$658$		0			0
$659$	21356.0			1.26238			0.631192	−	0.775626i	$-0.282566\pi$
$659$	21356.0			1.26238			0.631192	+	0.775626i	$0.282566\pi$
$660$		0			0
$661$	−13762.0			−0.809803			−0.404901	−	0.914360i	$-0.632694\pi$
$661$	−13762.0			−0.809803			−0.404901	+	0.914360i	$0.632694\pi$
$662$		0			0
$663$			31802.5i			1.86291i
$664$		0			0
$665$	6384.00	+	5126.07i	0.372272	+	0.298918i
$666$		0			0
$667$			366.148i			0.0212553i
$668$		0			0
$669$	9500.00			0.549015
$670$		0			0
$671$	3640.00			0.209420
$672$		0			0
$673$		−	5858.36i		−	0.335547i	−0.985826	−	0.167774i	$-0.946342\pi$
$673$		−	5858.36i		−	0.335547i	0.985826	−	0.167774i	$-0.0536577\pi$
$674$		0			0
$675$	−23408.0	+	5178.37i	−1.33478	+	0.295282i
$676$		0			0
$677$		−	29588.2i		−	1.67972i	−0.542807	−	0.839858i	$-0.682638\pi$
$677$		−	29588.2i		−	1.67972i	0.542807	−	0.839858i	$-0.317362\pi$
$678$		0			0
$679$	4864.00			0.274909
$680$		0			0
$681$	−8588.00			−0.483249
$682$		0			0
$683$			17270.0i			0.967521i	0.875201	+	0.483760i	$0.160729\pi$
$683$			17270.0i			0.967521i	−0.875201	+	0.483760i	$0.839271\pi$
$684$		0			0
$685$	−6384.00	−	5126.07i	−0.356088	−	0.285922i
$686$		0			0
$687$			34295.8i			1.90461i
$688$		0			0
$689$	−19152.0			−1.05897
$690$		0			0
$691$	−1652.00			−0.0909480			−0.0454740	−	0.998966i	$-0.514480\pi$
$691$	−1652.00			−0.0909480			−0.0454740	+	0.998966i	$0.514480\pi$
$692$		0			0
$693$			8543.44i			0.468310i
$694$		0			0
$695$	−2548.00	+	3173.28i	−0.139066	+	0.173193i
$696$		0			0
$697$			18551.5i			1.00816i
$698$		0			0
$699$	40432.0			2.18781
$700$		0			0
$701$	28790.0			1.55119			0.775594	−	0.631232i	$-0.217450\pi$
$701$	28790.0			1.55119			0.775594	+	0.631232i	$0.217450\pi$
$702$		0			0
$703$		−	10252.1i		−	0.550023i
$704$		0			0
$705$	−22876.0	+	28489.8i	−1.22207	+	1.52197i
$706$		0			0
$707$		−	10862.4i		−	0.577824i
$708$		0			0
$709$	−7554.00			−0.400136			−0.200068	−	0.979782i	$-0.564116\pi$
$709$	−7554.00			−0.400136			−0.200068	+	0.979782i	$0.564116\pi$
$710$		0			0
$711$	−2352.00			−0.124060
$712$		0			0
$713$		−	13669.5i		−	0.717990i
$714$		0			0
$715$	−9120.00	−	7322.95i	−0.477019	−	0.383025i
$716$		0			0
$717$			33615.8i			1.75092i
$718$		0			0
$719$	28336.0			1.46976			0.734878	−	0.678199i	$-0.237239\pi$
$719$	28336.0			1.46976			0.734878	+	0.678199i	$0.237239\pi$
$720$		0			0
$721$	−7372.00			−0.380787
$722$		0			0
$723$			8665.49i			0.445744i
$724$		0			0
$725$	−162.000	−	732.295i	−0.00829866	−	0.0375128i
$726$		0			0
$727$		−	23511.9i		−	1.19946i	−0.800202	−	0.599730i	$-0.795274\pi$
$727$		−	23511.9i		−	1.19946i	0.800202	−	0.599730i	$-0.204726\pi$
$728$		0			0
$729$	28043.0			1.42473
$730$		0			0
$731$	21280.0			1.07670
$732$		0			0
$733$			3365.07i			0.169566i	0.996399	+	0.0847829i	$0.0270197\pi$
$733$			3365.07i			0.169566i	−0.996399	+	0.0847829i	$0.972980\pi$
$734$		0			0
$735$	20292.0	+	16293.6i	1.01834	+	0.817683i
$736$		0			0
$737$			8543.44i			0.427004i
$738$		0			0
$739$	14300.0			0.711819			0.355909	−	0.934520i	$-0.384171\pi$
$739$	14300.0			0.711819			0.355909	+	0.934520i	$0.384171\pi$
$740$		0			0
$741$	−38304.0			−1.89896
$742$		0			0
$743$		−	31671.8i		−	1.56383i	−0.623386	−	0.781914i	$-0.714244\pi$
$743$		−	31671.8i		−	1.56383i	0.623386	−	0.781914i	$-0.285756\pi$
$744$		0			0
$745$	1778.00	−	2214.32i	0.0874374	−	0.108895i
$746$		0			0
$747$			9824.96i			0.481227i
$748$		0			0
$749$	−11172.0			−0.545015
$750$		0			0
$751$	−11824.0			−0.574519			−0.287260	−	0.957853i	$-0.592744\pi$
$751$	−11824.0			−0.574519			−0.287260	+	0.957853i	$0.592744\pi$
$752$		0			0
$753$		−	54922.1i		−	2.65800i
$754$		0			0
$755$	−16520.0	+	20574.0i	−0.796324	+	0.991741i
$756$		0			0
$757$		−	1342.54i		−	0.0644590i	−0.999480	−	0.0322295i	$-0.989739\pi$
$757$		−	1342.54i		−	0.0644590i	0.999480	−	0.0322295i	$-0.0102607\pi$
$758$		0			0
$759$	−10640.0			−0.508837
$760$		0			0
$761$	6762.00			0.322106			0.161053	−	0.986946i	$-0.448511\pi$
$761$	6762.00			0.322106			0.161053	+	0.986946i	$0.448511\pi$
$762$		0			0
$763$		−	7863.45i		−	0.373101i
$764$		0			0
$765$	−29792.0	−	23921.6i	−1.40802	−	1.13057i
$766$		0			0
$767$			1464.59i			0.0689482i
$768$		0			0
$769$	29442.0			1.38063			0.690316	−	0.723508i	$-0.257472\pi$
$769$	29442.0			1.38063			0.690316	+	0.723508i	$0.257472\pi$
$770$		0			0
$771$	−58368.0			−2.72642
$772$		0			0
$773$			35830.1i			1.66717i	0.552393	+	0.833584i	$0.313715\pi$
$773$			35830.1i			1.66717i	−0.552393	+	0.833584i	$0.686285\pi$
$774$		0			0
$775$	6048.00	+	27339.0i	0.280323	+	1.26716i
$776$		0			0
$777$			9275.74i			0.428269i
$778$		0			0
$779$	−22344.0			−1.02767
$780$		0			0
$781$	8160.00			0.373864
$782$		0			0
$783$		−	1150.75i		−	0.0525216i
$784$		0			0
$785$	−19304.0	−	15500.2i	−0.877693	−	0.704748i
$786$		0			0
$787$		−	23912.9i		−	1.08310i	−0.840667	−	0.541552i	$-0.817837\pi$
$787$		−	23912.9i		−	1.08310i	0.840667	−	0.541552i	$-0.182163\pi$
$788$		0			0
$789$	12236.0			0.552108
$790$		0			0
$791$	12768.0			0.573929
$792$		0			0
$793$		−	9519.84i		−	0.426304i
$794$		0			0
$795$	22344.0	−	27827.2i	0.996805	−	1.24142i
$796$		0			0
$797$		−	17139.2i		−	0.761733i	−0.924630	−	0.380867i	$-0.875626\pi$
$797$		−	17139.2i		−	0.761733i	0.924630	−	0.380867i	$-0.124374\pi$
$798$		0			0
$799$	−26144.0			−1.15758
$800$		0			0
$801$	74774.0			3.29839
$802$		0			0
$803$			21620.1i			0.950135i
$804$		0			0
$805$	3724.00	−	4637.87i	0.163048	−	0.203060i
$806$		0			0
$807$		−	44303.8i		−	1.93255i
$808$		0			0
$809$	−8646.00			−0.375744			−0.187872	−	0.982193i	$-0.560159\pi$
$809$	−8646.00			−0.375744			−0.187872	+	0.982193i	$0.560159\pi$
$810$		0			0
$811$	−18956.0			−0.820759			−0.410379	−	0.911915i	$-0.634604\pi$
$811$	−18956.0			−0.820759			−0.410379	+	0.911915i	$0.634604\pi$
$812$		0			0
$813$			24409.8i			1.05300i
$814$		0			0
$815$	−13300.0	−	10679.3i	−0.571630	−	0.458993i
$816$		0			0
$817$			25630.3i			1.09754i
$818$		0			0
$819$	22344.0			0.953312
$820$		0			0
$821$	13726.0			0.583484			0.291742	−	0.956497i	$-0.405765\pi$
$821$	13726.0			0.583484			0.291742	+	0.956497i	$0.405765\pi$
$822$		0			0
$823$			45951.5i			1.94626i	0.230264	+	0.973128i	$0.426041\pi$
$823$			45951.5i			1.94626i	−0.230264	+	0.973128i	$0.573959\pi$
$824$		0			0
$825$	21280.0	−	4707.61i	0.898030	−	0.198664i
$826$		0			0
$827$			2135.86i			0.0898079i	0.998991	+	0.0449040i	$0.0142982\pi$
$827$			2135.86i			0.0898079i	−0.998991	+	0.0449040i	$0.985702\pi$
$828$		0			0
$829$	−8778.00			−0.367759			−0.183880	−	0.982949i	$-0.558866\pi$
$829$	−8778.00			−0.367759			−0.183880	+	0.982949i	$0.558866\pi$
$830$		0			0
$831$	−13832.0			−0.577409
$832$		0			0
$833$			18621.2i			0.774534i
$834$		0			0
$835$	26676.0	+	21419.6i	1.10558	+	0.887733i
$836$		0			0
$837$			42961.3i			1.77415i
$838$		0			0
$839$	−18088.0			−0.744299			−0.372150	−	0.928173i	$-0.621379\pi$
$839$	−18088.0			−0.744299			−0.372150	+	0.928173i	$0.621379\pi$
$840$		0			0
$841$	−24353.0			−0.998524
$842$		0			0
$843$		−	10932.1i		−	0.446646i
$844$		0			0
$845$	−3773.00	+	4698.89i	−0.153604	+	0.191298i
$846$		0			0
$847$			8116.27i			0.329254i
$848$		0			0
$849$	−40508.0			−1.63749
$850$		0			0
$851$	−7448.00			−0.300017
$852$		0			0
$853$			22927.8i			0.920320i	0.887836	+	0.460160i	$0.152208\pi$
$853$			22927.8i			0.920320i	−0.887836	+	0.460160i	$0.847792\pi$
$854$		0			0
$855$	28812.0	−	35882.5i	1.15246	−	1.43527i
$856$		0			0
$857$			4498.38i			0.179302i	0.995973	+	0.0896510i	$0.0285752\pi$
$857$			4498.38i			0.179302i	−0.995973	+	0.0896510i	$0.971425\pi$
$858$		0			0
$859$	40516.0			1.60930			0.804650	−	0.593750i	$-0.202353\pi$
$859$	40516.0			1.60930			0.804650	+	0.593750i	$0.202353\pi$
$860$		0			0
$861$	20216.0			0.800185
$862$		0			0
$863$		−	28010.3i		−	1.10484i	−0.833564	−	0.552422i	$-0.813704\pi$
$863$		−	28010.3i		−	1.10484i	0.833564	−	0.552422i	$-0.186296\pi$
$864$		0			0
$865$	16568.0	+	13303.4i	0.651247	+	0.522922i
$866$		0			0
$867$			427.172i			0.0167330i
$868$		0			0
$869$	960.000			0.0374750
$870$		0			0
$871$	22344.0			0.869228
$872$		0			0
$873$		−	27339.0i		−	1.05989i
$874$		0			0
$875$	−5396.00	+	10923.4i	−0.208478	+	0.422032i
$876$		0			0
$877$		−	33807.6i		−	1.30171i	−0.759201	−	0.650856i	$-0.774410\pi$
$877$		−	33807.6i		−	1.30171i	0.759201	−	0.650856i	$-0.225590\pi$
$878$		0			0
$879$	−57000.0			−2.18722
$880$		0			0
$881$	19362.0			0.740434			0.370217	−	0.928945i	$-0.379283\pi$
$881$	19362.0			0.740434			0.370217	+	0.928945i	$0.379283\pi$
$882$		0			0
$883$		−	4942.99i		−	0.188386i	−0.995554	−	0.0941930i	$-0.969973\pi$
$883$		−	4942.99i		−	0.188386i	0.995554	−	0.0941930i	$-0.0300271\pi$
$884$		0			0
$885$	−2128.00	−	1708.69i	−0.0808270	−	0.0649005i
$886$		0			0
$887$		−	44016.2i		−	1.66620i	−0.553124	−	0.833099i	$-0.686564\pi$
$887$		−	44016.2i		−	1.66620i	0.553124	−	0.833099i	$-0.313436\pi$
$888$		0			0
$889$	−22876.0			−0.863033
$890$		0			0
$891$	6980.00			0.262445
$892$		0			0
$893$		−	31488.7i		−	1.17999i
$894$		0			0
$895$	−13804.0	+	17191.5i	−0.515550	+	0.642065i
$896$		0			0
$897$			27827.2i			1.03581i
$898$		0			0
$899$	−1344.00			−0.0498609
$900$		0			0
$901$	25536.0			0.944204
$902$		0			0
$903$		−	23189.3i		−	0.854588i
$904$		0			0
$905$	−9310.00	+	11594.7i	−0.341961	+	0.425878i
$906$		0			0
$907$			42656.2i			1.56160i	0.624778	+	0.780802i	$0.285189\pi$
$907$			42656.2i			1.56160i	−0.624778	+	0.780802i	$0.714811\pi$
$908$		0			0
$909$	−61054.0			−2.22776
$910$		0			0
$911$	−15888.0			−0.577819			−0.288909	−	0.957356i	$-0.593293\pi$
$911$	−15888.0			−0.577819			−0.288909	+	0.957356i	$0.593293\pi$
$912$		0			0
$913$		−	4010.19i		−	0.145365i
$914$		0			0
$915$	13832.0	+	11106.5i	0.499751	+	0.401277i
$916$		0			0
$917$		−	25630.3i		−	0.922997i
$918$		0			0
$919$	7944.00			0.285145			0.142573	−	0.989784i	$-0.454463\pi$
$919$	7944.00			0.285145			0.142573	+	0.989784i	$0.454463\pi$
$920$		0			0
$921$	−55404.0			−1.98222
$922$		0			0
$923$		−	21341.2i		−	0.761054i
$924$		0			0
$925$	14896.0	−	3295.33i	0.529489	−	0.117135i
$926$		0			0
$927$			41435.7i			1.46810i
$928$		0			0
$929$	−20622.0			−0.728295			−0.364147	−	0.931341i	$-0.618640\pi$
$929$	−20622.0			−0.728295			−0.364147	+	0.931341i	$0.618640\pi$
$930$		0			0
$931$	−22428.0			−0.789525
$932$		0			0
$933$		−	45402.3i		−	1.59315i
$934$		0			0
$935$	12160.0	+	9763.93i	0.425320	+	0.341513i
$936$		0			0
$937$		−	41182.9i		−	1.43584i	−0.696123	−	0.717922i	$-0.745093\pi$
$937$		−	41182.9i		−	1.43584i	0.696123	−	0.717922i	$-0.254907\pi$
$938$		0			0
$939$	39824.0			1.38403
$940$		0			0
$941$	40502.0			1.40311			0.701556	−	0.712615i	$-0.252489\pi$
$941$	40502.0			1.40311			0.701556	+	0.712615i	$0.252489\pi$
$942$		0			0
$943$			16232.5i			0.560556i
$944$		0			0
$945$	−11704.0	+	14576.2i	−0.402890	+	0.501759i
$946$		0			0
$947$			3844.55i			0.131923i	0.997822	+	0.0659615i	$0.0210114\pi$
$947$			3844.55i			0.131923i	−0.997822	+	0.0659615i	$0.978989\pi$
$948$		0			0
$949$	56544.0			1.93414
$950$		0			0
$951$	−7448.00			−0.253962
$952$		0			0
$953$			31976.9i			1.08692i	0.839436	+	0.543459i	$0.182886\pi$
$953$			31976.9i			1.08692i	−0.839436	+	0.543459i	$0.817114\pi$
$954$		0			0
$955$	12096.0	−	15064.4i	0.409861	−	0.510441i
$956$		0			0
$957$			1046.14i			0.0353362i
$958$		0			0
$959$	−6384.00			−0.214964
$960$		0			0
$961$	20385.0			0.684267
$962$		0			0
$963$			62794.3i			2.10126i
$964$		0			0
$965$	8512.00	+	6834.75i	0.283949	+	0.227998i
$966$		0			0
$967$			19100.7i			0.635198i	0.948225	+	0.317599i	$0.102877\pi$
$967$			19100.7i			0.635198i	−0.948225	+	0.317599i	$0.897123\pi$
$968$		0			0
$969$	51072.0			1.69316
$970$		0			0
$971$	17556.0			0.580225			0.290113	−	0.956992i	$-0.406307\pi$
$971$	17556.0			0.580225			0.290113	+	0.956992i	$0.406307\pi$
$972$		0			0
$973$			3173.28i			0.104554i
$974$		0			0
$975$	−12312.0	−	55654.4i	−0.404410	−	1.82807i
$976$		0			0
$977$		−	4393.77i		−	0.143878i	−0.997409	−	0.0719392i	$-0.977081\pi$
$977$		−	4393.77i		−	0.143878i	0.997409	−	0.0719392i	$-0.0229188\pi$
$978$		0			0
$979$	−30520.0			−0.996347
$980$		0			0
$981$	−44198.0			−1.43846
$982$		0			0
$983$			9406.50i			0.305209i	0.988287	+	0.152605i	$0.0487661\pi$
$983$			9406.50i			0.305209i	−0.988287	+	0.152605i	$0.951234\pi$
$984$		0			0
$985$	−32984.0	−	26484.7i	−1.06696	−	0.856723i
$986$		0			0
$987$			28489.8i			0.918783i
$988$		0			0
$989$	18620.0			0.598667
$990$		0			0
$991$	28576.0			0.915990			0.457995	−	0.888955i	$-0.348568\pi$
$991$	28576.0			0.915990			0.457995	+	0.888955i	$0.348568\pi$
$992$		0			0
$993$			68242.9i			2.18089i
$994$		0			0
$995$	−10584.0	+	13181.3i	−0.337221	+	0.419976i
$996$		0			0
$997$		−	31541.0i		−	1.00192i	−0.865471	−	0.500960i	$-0.832981\pi$
$997$		−	31541.0i		−	1.00192i	0.865471	−	0.500960i	$-0.167019\pi$
$998$		0			0
$999$	23408.0			0.741338

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	20.4.c.a.9.2	yes	2
3.2	odd	2		180.4.d.a.109.2		2
4.3	odd	2		80.4.c.b.49.1		2
5.2	odd	4		100.4.a.d.1.2		2
5.3	odd	4		100.4.a.d.1.1		2
5.4	even	2	inner	20.4.c.a.9.1	✓	2
7.6	odd	2		980.4.e.a.589.1		2
8.3	odd	2		320.4.c.b.129.2		2
8.5	even	2		320.4.c.a.129.1		2
12.11	even	2		720.4.f.a.289.2		2
15.2	even	4		900.4.a.s.1.2		2
15.8	even	4		900.4.a.s.1.1		2
15.14	odd	2		180.4.d.a.109.1		2
20.3	even	4		400.4.a.w.1.2		2
20.7	even	4		400.4.a.w.1.1		2
20.19	odd	2		80.4.c.b.49.2		2
35.34	odd	2		980.4.e.a.589.2		2
40.3	even	4		1600.4.a.ck.1.1		2
40.13	odd	4		1600.4.a.cj.1.2		2
40.19	odd	2		320.4.c.b.129.1		2
40.27	even	4		1600.4.a.ck.1.2		2
40.29	even	2		320.4.c.a.129.2		2
40.37	odd	4		1600.4.a.cj.1.1		2
60.59	even	2		720.4.f.a.289.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
20.4.c.a.9.1	✓	2	5.4	even	2	inner
20.4.c.a.9.2	yes	2	1.1	even	1	trivial
80.4.c.b.49.1		2	4.3	odd	2
80.4.c.b.49.2		2	20.19	odd	2
100.4.a.d.1.1		2	5.3	odd	4
100.4.a.d.1.2		2	5.2	odd	4
180.4.d.a.109.1		2	15.14	odd	2
180.4.d.a.109.2		2	3.2	odd	2
320.4.c.a.129.1		2	8.5	even	2
320.4.c.a.129.2		2	40.29	even	2
320.4.c.b.129.1		2	40.19	odd	2
320.4.c.b.129.2		2	8.3	odd	2
400.4.a.w.1.1		2	20.7	even	4
400.4.a.w.1.2		2	20.3	even	4
720.4.f.a.289.1		2	60.59	even	2
720.4.f.a.289.2		2	12.11	even	2
900.4.a.s.1.1		2	15.8	even	4
900.4.a.s.1.2		2	15.2	even	4
980.4.e.a.589.1		2	7.6	odd	2
980.4.e.a.589.2		2	35.34	odd	2
1600.4.a.cj.1.1		2	40.37	odd	4
1600.4.a.cj.1.2		2	40.13	odd	4
1600.4.a.ck.1.1		2	40.3	even	4
1600.4.a.ck.1.2		2	40.27	even	4

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:	\( N \)	\(=\)	\( 20 = 2^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	20.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+8.71780i q^{3} +(7.00000 - 8.71780i) q^{5} -8.71780i q^{7} -49.0000 q^{9} +O(q^{10})\)	\(q+8.71780i q^{3} +(7.00000 - 8.71780i) q^{5} -8.71780i q^{7} -49.0000 q^{9} +20.0000 q^{11} -52.3068i q^{13} +(76.0000 + 61.0246i) q^{15} +69.7424i q^{17} -84.0000 q^{19} +76.0000 q^{21} +61.0246i q^{23} +(-27.0000 - 122.049i) q^{25} -191.792i q^{27} +6.00000 q^{29} -224.000 q^{31} +174.356i q^{33} +(-76.0000 - 61.0246i) q^{35} +122.049i q^{37} +456.000 q^{39} +266.000 q^{41} -305.123i q^{43} +(-343.000 + 427.172i) q^{45} +374.865i q^{47} +267.000 q^{49} -608.000 q^{51} -366.148i q^{53} +(140.000 - 174.356i) q^{55} -732.295i q^{57} -28.0000 q^{59} +182.000 q^{61} +427.172i q^{63} +(-456.000 - 366.148i) q^{65} +427.172i q^{67} -532.000 q^{69} +408.000 q^{71} +1081.01i q^{73} +(1064.00 - 235.381i) q^{75} -174.356i q^{77} +48.0000 q^{79} +349.000 q^{81} -200.509i q^{83} +(608.000 + 488.197i) q^{85} +52.3068i q^{87} -1526.00 q^{89} -456.000 q^{91} -1952.79i q^{93} +(-588.000 + 732.295i) q^{95} +557.939i q^{97} -980.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 14 q^{5} - 98 q^{9}+O(q^{10}) \) 2 * q + 14 * q^5 - 98 * q^9	\( 2 q + 14 q^{5} - 98 q^{9} + 40 q^{11} + 152 q^{15} - 168 q^{19} + 152 q^{21} - 54 q^{25} + 12 q^{29} - 448 q^{31} - 152 q^{35} + 912 q^{39} + 532 q^{41} - 686 q^{45} + 534 q^{49} - 1216 q^{51} + 280 q^{55} - 56 q^{59} + 364 q^{61} - 912 q^{65} - 1064 q^{69} + 816 q^{71} + 2128 q^{75} + 96 q^{79} + 698 q^{81} + 1216 q^{85} - 3052 q^{89} - 912 q^{91} - 1176 q^{95} - 1960 q^{99}+O(q^{100}) \) 2 * q + 14 * q^5 - 98 * q^9 + 40 * q^11 + 152 * q^15 - 168 * q^19 + 152 * q^21 - 54 * q^25 + 12 * q^29 - 448 * q^31 - 152 * q^35 + 912 * q^39 + 532 * q^41 - 686 * q^45 + 534 * q^49 - 1216 * q^51 + 280 * q^55 - 56 * q^59 + 364 * q^61 - 912 * q^65 - 1064 * q^69 + 816 * q^71 + 2128 * q^75 + 96 * q^79 + 698 * q^81 + 1216 * q^85 - 3052 * q^89 - 912 * q^91 - 1176 * q^95 - 1960 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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