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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 = [6,0,-8,0,0,0,-40,0,0,0,-72] 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: \(6\)
Relative dimension: \(3\) over \(\Q(i)\)
Coefficient field: 6.0.313431616.3
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 27x^{4} + 145x^{2} + 144 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{6}\cdot 5^{2} \)
Twist minimal: no (minimal twist has level 40)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 257.3
Root \(4.49029i\) of defining polynomial
Character \(\chi\) \(=\) 400.257
Dual form 400.5.p.o.193.3

$q$-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)
 
\(f(q)\) \(=\) \(q+(8.28874 - 8.28874i) q^{3} +(-3.20721 - 3.20721i) q^{7} -56.4063i q^{9} +19.5694 q^{11} +(-219.840 + 219.840i) q^{13} +(-281.352 - 281.352i) q^{17} -305.648i q^{19} -53.1675 q^{21} +(-633.243 + 633.243i) q^{23} +(203.851 + 203.851i) q^{27} -885.210i q^{29} -1383.32 q^{31} +(162.205 - 162.205i) q^{33} +(-453.987 - 453.987i) q^{37} +3644.39i q^{39} -193.003 q^{41} +(-1016.66 + 1016.66i) q^{43} +(-1203.53 - 1203.53i) q^{47} -2380.43i q^{49} -4664.11 q^{51} +(530.174 - 530.174i) q^{53} +(-2533.43 - 2533.43i) q^{57} +934.801i q^{59} -1122.32 q^{61} +(-180.907 + 180.907i) q^{63} +(4760.40 + 4760.40i) q^{67} +10497.6i q^{69} +8831.10 q^{71} +(-3008.63 + 3008.63i) q^{73} +(-62.7631 - 62.7631i) q^{77} +3349.09i q^{79} +7948.24 q^{81} +(-3166.08 + 3166.08i) q^{83} +(-7337.27 - 7337.27i) q^{87} -1732.33i q^{89} +1410.15 q^{91} +(-11466.0 + 11466.0i) q^{93} +(-1659.43 - 1659.43i) q^{97} -1103.84i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 8 q^{3} - 40 q^{7} - 72 q^{11} + 62 q^{13} - 418 q^{17} + 1736 q^{21} - 1760 q^{23} + 2248 q^{27} - 2864 q^{31} + 8 q^{33} - 5366 q^{37} + 1656 q^{41} - 2760 q^{43} - 432 q^{47} - 15840 q^{51} - 534 q^{53}+ \cdots + 9418 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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
\(3\) 8.28874 − 8.28874i 0.920971 − 0.920971i −0.0761273 − 0.997098i \(-0.524256\pi\)
0.997098 + 0.0761273i \(0.0242555\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −3.20721 − 3.20721i −0.0654533 − 0.0654533i 0.673622 − 0.739076i \(-0.264737\pi\)
−0.739076 + 0.673622i \(0.764737\pi\)
\(8\) 0 0
\(9\) − 56.4063i − 0.696374i
\(10\) 0 0
\(11\) 19.5694 0.161730 0.0808652 − 0.996725i \(-0.474232\pi\)
0.0808652 + 0.996725i \(0.474232\pi\)
\(12\) 0 0
\(13\) −219.840 + 219.840i −1.30083 + 1.30083i −0.372996 + 0.927833i \(0.621669\pi\)
−0.927833 + 0.372996i \(0.878331\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −281.352 − 281.352i −0.973537 − 0.973537i 0.0261217 − 0.999659i \(-0.491684\pi\)
−0.999659 + 0.0261217i \(0.991684\pi\)
\(18\) 0 0
\(19\) − 305.648i − 0.846670i −0.905973 − 0.423335i \(-0.860859\pi\)
0.905973 − 0.423335i \(-0.139141\pi\)
\(20\) 0 0
\(21\) −53.1675 −0.120561
\(22\) 0 0
\(23\) −633.243 + 633.243i −1.19706 + 1.19706i −0.222014 + 0.975044i \(0.571263\pi\)
−0.975044 + 0.222014i \(0.928737\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 203.851 + 203.851i 0.279630 + 0.279630i
\(28\) 0 0
\(29\) − 885.210i − 1.05257i −0.850309 − 0.526284i \(-0.823585\pi\)
0.850309 − 0.526284i \(-0.176415\pi\)
\(30\) 0 0
\(31\) −1383.32 −1.43946 −0.719730 − 0.694254i \(-0.755734\pi\)
−0.719730 + 0.694254i \(0.755734\pi\)
\(32\) 0 0
\(33\) 162.205 − 162.205i 0.148949 − 0.148949i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −453.987 − 453.987i −0.331619 − 0.331619i 0.521582 − 0.853201i \(-0.325342\pi\)
−0.853201 + 0.521582i \(0.825342\pi\)
\(38\) 0 0
\(39\) 3644.39i 2.39605i
\(40\) 0 0
\(41\) −193.003 −0.114814 −0.0574071 − 0.998351i \(-0.518283\pi\)
−0.0574071 + 0.998351i \(0.518283\pi\)
\(42\) 0 0
\(43\) −1016.66 + 1016.66i −0.549846 + 0.549846i −0.926396 − 0.376550i \(-0.877110\pi\)
0.376550 + 0.926396i \(0.377110\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −1203.53 − 1203.53i −0.544828 − 0.544828i 0.380112 − 0.924940i \(-0.375885\pi\)
−0.924940 + 0.380112i \(0.875885\pi\)
\(48\) 0 0
\(49\) − 2380.43i − 0.991432i
\(50\) 0 0
\(51\) −4664.11 −1.79320
\(52\) 0 0
\(53\) 530.174 − 530.174i 0.188741 − 0.188741i −0.606410 − 0.795152i \(-0.707391\pi\)
0.795152 + 0.606410i \(0.207391\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −2533.43 − 2533.43i −0.779758 − 0.779758i
\(58\) 0 0
\(59\) 934.801i 0.268544i 0.990945 + 0.134272i \(0.0428695\pi\)
−0.990945 + 0.134272i \(0.957130\pi\)
\(60\) 0 0
\(61\) −1122.32 −0.301617 −0.150809 − 0.988563i \(-0.548188\pi\)
−0.150809 + 0.988563i \(0.548188\pi\)
\(62\) 0 0
\(63\) −180.907 + 180.907i −0.0455800 + 0.0455800i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 4760.40 + 4760.40i 1.06046 + 1.06046i 0.998051 + 0.0624093i \(0.0198784\pi\)
0.0624093 + 0.998051i \(0.480122\pi\)
\(68\) 0 0
\(69\) 10497.6i 2.20491i
\(70\) 0 0
\(71\) 8831.10 1.75185 0.875927 − 0.482443i \(-0.160250\pi\)
0.875927 + 0.482443i \(0.160250\pi\)
\(72\) 0 0
\(73\) −3008.63 + 3008.63i −0.564578 + 0.564578i −0.930604 − 0.366027i \(-0.880718\pi\)
0.366027 + 0.930604i \(0.380718\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −62.7631 − 62.7631i −0.0105858 − 0.0105858i
\(78\) 0 0
\(79\) 3349.09i 0.536627i 0.963332 + 0.268313i \(0.0864663\pi\)
−0.963332 + 0.268313i \(0.913534\pi\)
\(80\) 0 0
\(81\) 7948.24 1.21144
\(82\) 0 0
\(83\) −3166.08 + 3166.08i −0.459586 + 0.459586i −0.898519 − 0.438934i \(-0.855356\pi\)
0.438934 + 0.898519i \(0.355356\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −7337.27 − 7337.27i −0.969384 − 0.969384i
\(88\) 0 0
\(89\) − 1732.33i − 0.218700i −0.994003 − 0.109350i \(-0.965123\pi\)
0.994003 − 0.109350i \(-0.0348770\pi\)
\(90\) 0 0
\(91\) 1410.15 0.170287
\(92\) 0 0
\(93\) −11466.0 + 11466.0i −1.32570 + 1.32570i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −1659.43 − 1659.43i −0.176366 − 0.176366i 0.613404 − 0.789770i \(-0.289800\pi\)
−0.789770 + 0.613404i \(0.789800\pi\)
\(98\) 0 0
\(99\) − 1103.84i − 0.112625i
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.o.257.3 6
4.3 odd 2 200.5.l.e.57.1 6
5.2 odd 4 80.5.p.g.33.1 6
5.3 odd 4 inner 400.5.p.o.193.3 6
5.4 even 2 80.5.p.g.17.1 6
20.3 even 4 200.5.l.e.193.1 6
20.7 even 4 40.5.l.c.33.3 yes 6
20.19 odd 2 40.5.l.c.17.3 ✓ 6
40.19 odd 2 320.5.p.p.257.1 6
40.27 even 4 320.5.p.p.193.1 6
40.29 even 2 320.5.p.o.257.3 6
40.37 odd 4 320.5.p.o.193.3 6
60.47 odd 4 360.5.v.c.73.3 6
60.59 even 2 360.5.v.c.217.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
40.5.l.c.17.3 ✓ 6 20.19 odd 2
40.5.l.c.33.3 yes 6 20.7 even 4
80.5.p.g.17.1 6 5.4 even 2
80.5.p.g.33.1 6 5.2 odd 4
200.5.l.e.57.1 6 4.3 odd 2
200.5.l.e.193.1 6 20.3 even 4
320.5.p.o.193.3 6 40.37 odd 4
320.5.p.o.257.3 6 40.29 even 2
320.5.p.p.193.1 6 40.27 even 4
320.5.p.p.257.1 6 40.19 odd 2
360.5.v.c.73.3 6 60.47 odd 4
360.5.v.c.217.3 6 60.59 even 2
400.5.p.o.193.3 6 5.3 odd 4 inner
400.5.p.o.257.3 6 1.1 even 1 trivial