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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [400,5,Mod(193,400)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("400.193"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(400, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 0, 3])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 400 = 2^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 400.p (of order \(4\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,-12,0,0,0,44,0,0,0,296] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(41.3479852335\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(i, \sqrt{29})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 15x^{2} + 49 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{5}\cdot 5^{2} \)
Twist minimal: no (minimal twist has level 40)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 193.2
Root \(3.19258i\) of defining polynomial
Character \(\chi\) \(=\) 400.193
Dual form 400.5.p.g.257.2

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-3.00000 - 3.00000i) q^{3} +(64.8516 - 64.8516i) q^{7} -63.0000i q^{9} -33.7033 q^{11} +(-132.852 - 132.852i) q^{13} +(270.407 - 270.407i) q^{17} +295.703i q^{19} -389.110 q^{21} +(1.44506 + 1.44506i) q^{23} +(-432.000 + 432.000i) q^{27} -297.484i q^{29} +1228.52 q^{31} +(101.110 + 101.110i) q^{33} +(-402.478 + 402.478i) q^{37} +797.110i q^{39} +860.659 q^{41} +(753.736 + 753.736i) q^{43} +(-2228.48 + 2228.48i) q^{47} -6010.47i q^{49} -1622.44 q^{51} +(-2538.77 - 2538.77i) q^{53} +(887.110 - 887.110i) q^{57} -3421.02i q^{59} -6691.91 q^{61} +(-4085.65 - 4085.65i) q^{63} +(-318.264 + 318.264i) q^{67} -8.67033i q^{69} +3427.02 q^{71} +(-3052.86 - 3052.86i) q^{73} +(-2185.71 + 2185.71i) q^{77} +2639.38i q^{79} -2511.00 q^{81} +(225.132 + 225.132i) q^{83} +(-892.451 + 892.451i) q^{87} +4989.60i q^{89} -17231.3 q^{91} +(-3685.55 - 3685.55i) q^{93} +(-4891.66 + 4891.66i) q^{97} +2123.31i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 12 q^{3} + 44 q^{7} + 296 q^{11} - 316 q^{13} + 220 q^{17} - 264 q^{21} + 652 q^{23} - 1728 q^{27} + 2760 q^{31} - 888 q^{33} + 2052 q^{37} - 4312 q^{41} - 1724 q^{43} - 5252 q^{47} - 1320 q^{51}+ \cdots - 11812 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/400\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(177\) \(351\)
\(\chi(n)\) \(1\) \(e\left(\frac{3}{4}\right)\) \(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −3.00000 − 3.00000i −0.333333 − 0.333333i 0.520518 − 0.853851i \(-0.325739\pi\)
−0.853851 + 0.520518i \(0.825739\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 64.8516 − 64.8516i 1.32350 − 1.32350i 0.412583 − 0.910920i \(-0.364627\pi\)
0.910920 − 0.412583i \(-0.135373\pi\)
\(8\) 0 0
\(9\) − 63.0000i − 0.777778i
\(10\) 0 0
\(11\) −33.7033 −0.278540 −0.139270 − 0.990254i \(-0.544476\pi\)
−0.139270 + 0.990254i \(0.544476\pi\)
\(12\) 0 0
\(13\) −132.852 − 132.852i −0.786104 − 0.786104i 0.194749 − 0.980853i \(-0.437611\pi\)
−0.980853 + 0.194749i \(0.937611\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 270.407 − 270.407i 0.935663 − 0.935663i −0.0623890 − 0.998052i \(-0.519872\pi\)
0.998052 + 0.0623890i \(0.0198719\pi\)
\(18\) 0 0
\(19\) 295.703i 0.819123i 0.912283 + 0.409561i \(0.134318\pi\)
−0.912283 + 0.409561i \(0.865682\pi\)
\(20\) 0 0
\(21\) −389.110 −0.882335
\(22\) 0 0
\(23\) 1.44506 + 1.44506i 0.00273167 + 0.00273167i 0.708471 − 0.705740i \(-0.249385\pi\)
−0.705740 + 0.708471i \(0.749385\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −432.000 + 432.000i −0.592593 + 0.592593i
\(28\) 0 0
\(29\) − 297.484i − 0.353726i −0.984235 − 0.176863i \(-0.943405\pi\)
0.984235 − 0.176863i \(-0.0565949\pi\)
\(30\) 0 0
\(31\) 1228.52 1.27837 0.639187 − 0.769052i \(-0.279271\pi\)
0.639187 + 0.769052i \(0.279271\pi\)
\(32\) 0 0
\(33\) 101.110 + 101.110i 0.0928465 + 0.0928465i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −402.478 + 402.478i −0.293994 + 0.293994i −0.838656 − 0.544662i \(-0.816658\pi\)
0.544662 + 0.838656i \(0.316658\pi\)
\(38\) 0 0
\(39\) 797.110i 0.524070i
\(40\) 0 0
\(41\) 860.659 0.511992 0.255996 − 0.966678i \(-0.417597\pi\)
0.255996 + 0.966678i \(0.417597\pi\)
\(42\) 0 0
\(43\) 753.736 + 753.736i 0.407645 + 0.407645i 0.880917 − 0.473271i \(-0.156927\pi\)
−0.473271 + 0.880917i \(0.656927\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −2228.48 + 2228.48i −1.00882 + 1.00882i −0.00885679 + 0.999961i \(0.502819\pi\)
−0.999961 + 0.00885679i \(0.997181\pi\)
\(48\) 0 0
\(49\) − 6010.47i − 2.50332i
\(50\) 0 0
\(51\) −1622.44 −0.623775
\(52\) 0 0
\(53\) −2538.77 − 2538.77i −0.903800 − 0.903800i 0.0919623 − 0.995762i \(-0.470686\pi\)
−0.995762 + 0.0919623i \(0.970686\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 887.110 − 887.110i 0.273041 − 0.273041i
\(58\) 0 0
\(59\) − 3421.02i − 0.982770i −0.870943 − 0.491385i \(-0.836491\pi\)
0.870943 − 0.491385i \(-0.163509\pi\)
\(60\) 0 0
\(61\) −6691.91 −1.79842 −0.899209 − 0.437520i \(-0.855857\pi\)
−0.899209 + 0.437520i \(0.855857\pi\)
\(62\) 0 0
\(63\) −4085.65 − 4085.65i −1.02939 − 1.02939i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −318.264 + 318.264i −0.0708986 + 0.0708986i −0.741667 − 0.670768i \(-0.765964\pi\)
0.670768 + 0.741667i \(0.265964\pi\)
\(68\) 0 0
\(69\) − 8.67033i − 0.00182112i
\(70\) 0 0
\(71\) 3427.02 0.679830 0.339915 − 0.940456i \(-0.389602\pi\)
0.339915 + 0.940456i \(0.389602\pi\)
\(72\) 0 0
\(73\) −3052.86 − 3052.86i −0.572876 − 0.572876i 0.360055 − 0.932931i \(-0.382758\pi\)
−0.932931 + 0.360055i \(0.882758\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −2185.71 + 2185.71i −0.368648 + 0.368648i
\(78\) 0 0
\(79\) 2639.38i 0.422911i 0.977388 + 0.211455i \(0.0678203\pi\)
−0.977388 + 0.211455i \(0.932180\pi\)
\(80\) 0 0
\(81\) −2511.00 −0.382716
\(82\) 0 0
\(83\) 225.132 + 225.132i 0.0326799 + 0.0326799i 0.723258 − 0.690578i \(-0.242644\pi\)
−0.690578 + 0.723258i \(0.742644\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −892.451 + 892.451i −0.117909 + 0.117909i
\(88\) 0 0
\(89\) 4989.60i 0.629921i 0.949105 + 0.314961i \(0.101991\pi\)
−0.949105 + 0.314961i \(0.898009\pi\)
\(90\) 0 0
\(91\) −17231.3 −2.08082
\(92\) 0 0
\(93\) −3685.55 − 3685.55i −0.426124 − 0.426124i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −4891.66 + 4891.66i −0.519892 + 0.519892i −0.917538 − 0.397647i \(-0.869827\pi\)
0.397647 + 0.917538i \(0.369827\pi\)
\(98\) 0 0
\(99\) 2123.31i 0.216642i
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 400.5.p.g.193.2 4
4.3 odd 2 200.5.l.c.193.1 4
5.2 odd 4 inner 400.5.p.g.257.2 4
5.3 odd 4 80.5.p.f.17.2 4
5.4 even 2 80.5.p.f.33.2 4
20.3 even 4 40.5.l.b.17.2 ✓ 4
20.7 even 4 200.5.l.c.57.1 4
20.19 odd 2 40.5.l.b.33.2 yes 4
40.3 even 4 320.5.p.n.257.1 4
40.13 odd 4 320.5.p.k.257.1 4
40.19 odd 2 320.5.p.n.193.1 4
40.29 even 2 320.5.p.k.193.1 4
60.23 odd 4 360.5.v.b.217.1 4
60.59 even 2 360.5.v.b.73.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
40.5.l.b.17.2 ✓ 4 20.3 even 4
40.5.l.b.33.2 yes 4 20.19 odd 2
80.5.p.f.17.2 4 5.3 odd 4
80.5.p.f.33.2 4 5.4 even 2
200.5.l.c.57.1 4 20.7 even 4
200.5.l.c.193.1 4 4.3 odd 2
320.5.p.k.193.1 4 40.29 even 2
320.5.p.k.257.1 4 40.13 odd 4
320.5.p.n.193.1 4 40.19 odd 2
320.5.p.n.257.1 4 40.3 even 4
360.5.v.b.73.1 4 60.59 even 2
360.5.v.b.217.1 4 60.23 odd 4
400.5.p.g.193.2 4 1.1 even 1 trivial
400.5.p.g.257.2 4 5.2 odd 4 inner