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,-10,0,0,0,110,0,0,0,300] 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: \(\Q(i, \sqrt{241})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 121x^{2} + 3600 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 20)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 193.1
Root \(-7.26209i\) of defining polynomial
Character \(\chi\) \(=\) 400.193
Dual form 400.5.p.h.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)
 
\(f(q)\) \(=\) \(q+(-10.2621 - 10.2621i) q^{3} +(50.7863 - 50.7863i) q^{7} +129.621i q^{9} -2.62087 q^{11} +(43.4275 + 43.4275i) q^{13} +(-131.331 + 131.331i) q^{17} +403.725i q^{19} -1042.35 q^{21} +(-334.455 - 334.455i) q^{23} +(498.952 - 498.952i) q^{27} +1172.97i q^{29} -955.313 q^{31} +(26.8956 + 26.8956i) q^{33} +(-673.305 + 673.305i) q^{37} -891.313i q^{39} +818.621 q^{41} +(2.48083 + 2.48083i) q^{43} +(-1563.81 + 1563.81i) q^{47} -2757.49i q^{49} +2695.46 q^{51} +(-277.662 - 277.662i) q^{53} +(4143.06 - 4143.06i) q^{57} +6333.66i q^{59} +6518.49 q^{61} +(6582.96 + 6582.96i) q^{63} +(-713.488 + 713.488i) q^{67} +6864.42i q^{69} -288.280 q^{71} +(5564.19 + 5564.19i) q^{73} +(-133.104 + 133.104i) q^{77} -4061.80i q^{79} +258.720 q^{81} +(1284.98 + 1284.98i) q^{83} +(12037.1 - 12037.1i) q^{87} -4418.77i q^{89} +4411.04 q^{91} +(9803.51 + 9803.51i) q^{93} +(8805.47 - 8805.47i) q^{97} -339.720i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 10 q^{3} + 110 q^{7} + 300 q^{11} + 360 q^{13} - 960 q^{17} - 1996 q^{21} - 810 q^{23} + 2120 q^{27} + 836 q^{31} + 1660 q^{33} + 660 q^{37} + 2964 q^{41} + 3270 q^{43} - 2250 q^{47} - 1948 q^{51}+ \cdots + 3180 q^{97}+O(q^{100}) \) 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
\(3\) −10.2621 − 10.2621i −1.14023 − 1.14023i −0.988407 − 0.151824i \(-0.951485\pi\)
−0.151824 − 0.988407i \(-0.548515\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 50.7863 − 50.7863i 1.03645 − 1.03645i 0.0371444 − 0.999310i \(-0.488174\pi\)
0.999310 − 0.0371444i \(-0.0118262\pi\)
\(8\) 0 0
\(9\) 129.621i 1.60026i
\(10\) 0 0
\(11\) −2.62087 −0.0216601 −0.0108301 − 0.999941i \(-0.503447\pi\)
−0.0108301 + 0.999941i \(0.503447\pi\)
\(12\) 0 0
\(13\) 43.4275 + 43.4275i 0.256967 + 0.256967i 0.823820 − 0.566852i \(-0.191839\pi\)
−0.566852 + 0.823820i \(0.691839\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −131.331 + 131.331i −0.454432 + 0.454432i −0.896822 − 0.442391i \(-0.854130\pi\)
0.442391 + 0.896822i \(0.354130\pi\)
\(18\) 0 0
\(19\) 403.725i 1.11835i 0.829049 + 0.559176i \(0.188882\pi\)
−0.829049 + 0.559176i \(0.811118\pi\)
\(20\) 0 0
\(21\) −1042.35 −2.36360
\(22\) 0 0
\(23\) −334.455 − 334.455i −0.632241 − 0.632241i 0.316389 − 0.948630i \(-0.397530\pi\)
−0.948630 + 0.316389i \(0.897530\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 498.952 − 498.952i 0.684433 − 0.684433i
\(28\) 0 0
\(29\) 1172.97i 1.39473i 0.716717 + 0.697364i \(0.245644\pi\)
−0.716717 + 0.697364i \(0.754356\pi\)
\(30\) 0 0
\(31\) −955.313 −0.994082 −0.497041 − 0.867727i \(-0.665580\pi\)
−0.497041 + 0.867727i \(0.665580\pi\)
\(32\) 0 0
\(33\) 26.8956 + 26.8956i 0.0246976 + 0.0246976i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −673.305 + 673.305i −0.491823 + 0.491823i −0.908880 − 0.417057i \(-0.863061\pi\)
0.417057 + 0.908880i \(0.363061\pi\)
\(38\) 0 0
\(39\) − 891.313i − 0.586005i
\(40\) 0 0
\(41\) 818.621 0.486984 0.243492 − 0.969903i \(-0.421707\pi\)
0.243492 + 0.969903i \(0.421707\pi\)
\(42\) 0 0
\(43\) 2.48083 + 2.48083i 0.00134171 + 0.00134171i 0.707777 − 0.706436i \(-0.249698\pi\)
−0.706436 + 0.707777i \(0.749698\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −1563.81 + 1563.81i −0.707926 + 0.707926i −0.966099 − 0.258173i \(-0.916880\pi\)
0.258173 + 0.966099i \(0.416880\pi\)
\(48\) 0 0
\(49\) − 2757.49i − 1.14848i
\(50\) 0 0
\(51\) 2695.46 1.03632
\(52\) 0 0
\(53\) −277.662 − 277.662i −0.0988471 − 0.0988471i 0.655954 − 0.754801i \(-0.272267\pi\)
−0.754801 + 0.655954i \(0.772267\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 4143.06 − 4143.06i 1.27518 − 1.27518i
\(58\) 0 0
\(59\) 6333.66i 1.81949i 0.415163 + 0.909747i \(0.363725\pi\)
−0.415163 + 0.909747i \(0.636275\pi\)
\(60\) 0 0
\(61\) 6518.49 1.75181 0.875906 − 0.482483i \(-0.160265\pi\)
0.875906 + 0.482483i \(0.160265\pi\)
\(62\) 0 0
\(63\) 6582.96 + 6582.96i 1.65859 + 1.65859i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −713.488 + 713.488i −0.158942 + 0.158942i −0.782098 − 0.623156i \(-0.785850\pi\)
0.623156 + 0.782098i \(0.285850\pi\)
\(68\) 0 0
\(69\) 6864.42i 1.44180i
\(70\) 0 0
\(71\) −288.280 −0.0571871 −0.0285935 − 0.999591i \(-0.509103\pi\)
−0.0285935 + 0.999591i \(0.509103\pi\)
\(72\) 0 0
\(73\) 5564.19 + 5564.19i 1.04413 + 1.04413i 0.998980 + 0.0451542i \(0.0143779\pi\)
0.0451542 + 0.998980i \(0.485622\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −133.104 + 133.104i −0.0224497 + 0.0224497i
\(78\) 0 0
\(79\) − 4061.80i − 0.650825i −0.945572 − 0.325413i \(-0.894497\pi\)
0.945572 − 0.325413i \(-0.105503\pi\)
\(80\) 0 0
\(81\) 258.720 0.0394330
\(82\) 0 0
\(83\) 1284.98 + 1284.98i 0.186527 + 0.186527i 0.794193 − 0.607666i \(-0.207894\pi\)
−0.607666 + 0.794193i \(0.707894\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 12037.1 − 12037.1i 1.59031 − 1.59031i
\(88\) 0 0
\(89\) − 4418.77i − 0.557855i −0.960312 − 0.278927i \(-0.910021\pi\)
0.960312 − 0.278927i \(-0.0899789\pi\)
\(90\) 0 0
\(91\) 4411.04 0.532670
\(92\) 0 0
\(93\) 9803.51 + 9803.51i 1.13348 + 1.13348i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 8805.47 − 8805.47i 0.935857 − 0.935857i −0.0622067 − 0.998063i \(-0.519814\pi\)
0.998063 + 0.0622067i \(0.0198138\pi\)
\(98\) 0 0
\(99\) − 339.720i − 0.0346618i
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.h.193.1 4
4.3 odd 2 100.5.f.c.93.2 4
5.2 odd 4 inner 400.5.p.h.257.1 4
5.3 odd 4 80.5.p.e.17.2 4
5.4 even 2 80.5.p.e.33.2 4
12.11 even 2 900.5.l.a.793.1 4
20.3 even 4 20.5.f.a.17.1 yes 4
20.7 even 4 100.5.f.c.57.2 4
20.19 odd 2 20.5.f.a.13.1 ✓ 4
40.3 even 4 320.5.p.m.257.2 4
40.13 odd 4 320.5.p.l.257.1 4
40.19 odd 2 320.5.p.m.193.2 4
40.29 even 2 320.5.p.l.193.1 4
60.23 odd 4 180.5.l.a.37.2 4
60.47 odd 4 900.5.l.a.757.1 4
60.59 even 2 180.5.l.a.73.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
20.5.f.a.13.1 ✓ 4 20.19 odd 2
20.5.f.a.17.1 yes 4 20.3 even 4
80.5.p.e.17.2 4 5.3 odd 4
80.5.p.e.33.2 4 5.4 even 2
100.5.f.c.57.2 4 20.7 even 4
100.5.f.c.93.2 4 4.3 odd 2
180.5.l.a.37.2 4 60.23 odd 4
180.5.l.a.73.2 4 60.59 even 2
320.5.p.l.193.1 4 40.29 even 2
320.5.p.l.257.1 4 40.13 odd 4
320.5.p.m.193.2 4 40.19 odd 2
320.5.p.m.257.2 4 40.3 even 4
400.5.p.h.193.1 4 1.1 even 1 trivial
400.5.p.h.257.1 4 5.2 odd 4 inner
900.5.l.a.757.1 4 60.47 odd 4
900.5.l.a.793.1 4 12.11 even 2