Show commands: Magma / Pari/GP / SageMath

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 193.1
Root \(-2.19258i\) of defining polynomial
Character \(\chi\) \(=\) 400.193
Dual form 400.5.p.g.257.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-3.00000 - 3.00000i) q^{3} +(-42.8516 + 42.8516i) q^{7} -63.0000i q^{9} +181.703 q^{11} +(-25.1484 - 25.1484i) q^{13} +(-160.407 + 160.407i) q^{17} +80.2967i q^{19} +257.110 q^{21} +(324.555 + 324.555i) q^{23} +(-432.000 + 432.000i) q^{27} -1374.52i q^{29} +151.484 q^{31} +(-545.110 - 545.110i) q^{33} +(1428.48 - 1428.48i) q^{37} +150.890i q^{39} -3016.66 q^{41} +(-1615.74 - 1615.74i) q^{43} +(-397.522 + 397.522i) q^{47} -1271.53i q^{49} +962.440 q^{51} +(-923.225 - 923.225i) q^{53} +(240.890 - 240.890i) q^{57} +4549.02i q^{59} +631.912 q^{61} +(2699.65 + 2699.65i) q^{63} +(-2687.74 + 2687.74i) q^{67} -1947.33i q^{69} -4543.02 q^{71} +(-5853.14 - 5853.14i) q^{73} +(-7786.29 + 7786.29i) q^{77} -9423.38i q^{79} -2511.00 q^{81} +(-8391.13 - 8391.13i) q^{83} +(-4123.55 + 4123.55i) q^{87} -8365.60i q^{89} +2155.30 q^{91} +(-454.451 - 454.451i) q^{93} +(-1014.34 + 1014.34i) q^{97} -11447.3i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 12 q^{3} + 44 q^{7} + 296 q^{11} - 316 q^{13} + 220 q^{17} - 264 q^{21} + 652 q^{23} - 1728 q^{27} + 2760 q^{31} - 888 q^{33} + 2052 q^{37} - 4312 q^{41} - 1724 q^{43} - 5252 q^{47} - 1320 q^{51}+ \cdots - 11812 q^{97}+O(q^{100}) \) 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
\(7\) −42.8516 + 42.8516i −0.874523 + 0.874523i −0.992961 − 0.118438i \(-0.962211\pi\)
0.118438 + 0.992961i \(0.462211\pi\)
\(8\) 0 0
\(9\) − 63.0000i − 0.777778i
\(10\) 0 0
\(11\) 181.703 1.50168 0.750840 − 0.660484i \(-0.229649\pi\)
0.750840 + 0.660484i \(0.229649\pi\)
\(12\) 0 0
\(13\) −25.1484 − 25.1484i −0.148807 − 0.148807i 0.628778 − 0.777585i \(-0.283555\pi\)
−0.777585 + 0.628778i \(0.783555\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −160.407 + 160.407i −0.555040 + 0.555040i −0.927891 − 0.372851i \(-0.878380\pi\)
0.372851 + 0.927891i \(0.378380\pi\)
\(18\) 0 0
\(19\) 80.2967i 0.222429i 0.993796 + 0.111214i \(0.0354740\pi\)
−0.993796 + 0.111214i \(0.964526\pi\)
\(20\) 0 0
\(21\) 257.110 0.583016
\(22\) 0 0
\(23\) 324.555 + 324.555i 0.613525 + 0.613525i 0.943863 − 0.330337i \(-0.107163\pi\)
−0.330337 + 0.943863i \(0.607163\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −432.000 + 432.000i −0.592593 + 0.592593i
\(28\) 0 0
\(29\) − 1374.52i − 1.63438i −0.576366 − 0.817192i \(-0.695530\pi\)
0.576366 − 0.817192i \(-0.304470\pi\)
\(30\) 0 0
\(31\) 151.484 0.157631 0.0788156 − 0.996889i \(-0.474886\pi\)
0.0788156 + 0.996889i \(0.474886\pi\)
\(32\) 0 0
\(33\) −545.110 − 545.110i −0.500560 − 0.500560i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 1428.48 − 1428.48i 1.04345 − 1.04345i 0.0444340 − 0.999012i \(-0.485852\pi\)
0.999012 − 0.0444340i \(-0.0141484\pi\)
\(38\) 0 0
\(39\) 150.890i 0.0992045i
\(40\) 0 0
\(41\) −3016.66 −1.79456 −0.897281 − 0.441459i \(-0.854461\pi\)
−0.897281 + 0.441459i \(0.854461\pi\)
\(42\) 0 0
\(43\) −1615.74 − 1615.74i −0.873843 − 0.873843i 0.119046 − 0.992889i \(-0.462017\pi\)
−0.992889 + 0.119046i \(0.962017\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −397.522 + 397.522i −0.179956 + 0.179956i −0.791336 − 0.611381i \(-0.790614\pi\)
0.611381 + 0.791336i \(0.290614\pi\)
\(48\) 0 0
\(49\) − 1271.53i − 0.529582i
\(50\) 0 0
\(51\) 962.440 0.370027
\(52\) 0 0
\(53\) −923.225 − 923.225i −0.328667 − 0.328667i 0.523413 − 0.852079i \(-0.324659\pi\)
−0.852079 + 0.523413i \(0.824659\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 240.890 − 240.890i 0.0741428 − 0.0741428i
\(58\) 0 0
\(59\) 4549.02i 1.30681i 0.757006 + 0.653407i \(0.226661\pi\)
−0.757006 + 0.653407i \(0.773339\pi\)
\(60\) 0 0
\(61\) 631.912 0.169823 0.0849116 − 0.996388i \(-0.472939\pi\)
0.0849116 + 0.996388i \(0.472939\pi\)
\(62\) 0 0
\(63\) 2699.65 + 2699.65i 0.680185 + 0.680185i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −2687.74 + 2687.74i −0.598738 + 0.598738i −0.939977 − 0.341238i \(-0.889154\pi\)
0.341238 + 0.939977i \(0.389154\pi\)
\(68\) 0 0
\(69\) − 1947.33i − 0.409017i
\(70\) 0 0
\(71\) −4543.02 −0.901214 −0.450607 − 0.892722i \(-0.648792\pi\)
−0.450607 + 0.892722i \(0.648792\pi\)
\(72\) 0 0
\(73\) −5853.14 − 5853.14i −1.09836 − 1.09836i −0.994603 − 0.103754i \(-0.966915\pi\)
−0.103754 − 0.994603i \(-0.533085\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −7786.29 + 7786.29i −1.31325 + 1.31325i
\(78\) 0 0
\(79\) − 9423.38i − 1.50992i −0.655773 − 0.754958i \(-0.727657\pi\)
0.655773 − 0.754958i \(-0.272343\pi\)
\(80\) 0 0
\(81\) −2511.00 −0.382716
\(82\) 0 0
\(83\) −8391.13 − 8391.13i −1.21805 − 1.21805i −0.968315 − 0.249733i \(-0.919657\pi\)
−0.249733 − 0.968315i \(-0.580343\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −4123.55 + 4123.55i −0.544794 + 0.544794i
\(88\) 0 0
\(89\) − 8365.60i − 1.05613i −0.849204 − 0.528065i \(-0.822918\pi\)
0.849204 − 0.528065i \(-0.177082\pi\)
\(90\) 0 0
\(91\) 2155.30 0.260270
\(92\) 0 0
\(93\) −454.451 − 454.451i −0.0525437 − 0.0525437i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −1014.34 + 1014.34i −0.107805 + 0.107805i −0.758952 − 0.651147i \(-0.774288\pi\)
0.651147 + 0.758952i \(0.274288\pi\)
\(98\) 0 0
\(99\) − 11447.3i − 1.16797i
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.193.1 4
4.3 odd 2 200.5.l.c.193.2 4
5.2 odd 4 inner 400.5.p.g.257.1 4
5.3 odd 4 80.5.p.f.17.1 4
5.4 even 2 80.5.p.f.33.1 4
20.3 even 4 40.5.l.b.17.1 ✓ 4
20.7 even 4 200.5.l.c.57.2 4
20.19 odd 2 40.5.l.b.33.1 yes 4
40.3 even 4 320.5.p.n.257.2 4
40.13 odd 4 320.5.p.k.257.2 4
40.19 odd 2 320.5.p.n.193.2 4
40.29 even 2 320.5.p.k.193.2 4
60.23 odd 4 360.5.v.b.217.2 4
60.59 even 2 360.5.v.b.73.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
40.5.l.b.17.1 ✓ 4 20.3 even 4
40.5.l.b.33.1 yes 4 20.19 odd 2
80.5.p.f.17.1 4 5.3 odd 4
80.5.p.f.33.1 4 5.4 even 2
200.5.l.c.57.2 4 20.7 even 4
200.5.l.c.193.2 4 4.3 odd 2
320.5.p.k.193.2 4 40.29 even 2
320.5.p.k.257.2 4 40.13 odd 4
320.5.p.n.193.2 4 40.19 odd 2
320.5.p.n.257.2 4 40.3 even 4
360.5.v.b.73.2 4 60.59 even 2
360.5.v.b.217.2 4 60.23 odd 4
400.5.p.g.193.1 4 1.1 even 1 trivial
400.5.p.g.257.1 4 5.2 odd 4 inner