Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [180,5,Mod(37,180)] 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("180.37"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(180, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 180 = 2^{2} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 180.l (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(18.6065933551\)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 20)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 37.1
Root \(-7.26209i\) of defining polynomial
Character \(\chi\) \(=\) 180.37
Dual form 180.5.l.a.73.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-21.7863 + 12.2621i) q^{5} +(4.21374 + 4.21374i) q^{7} +152.621 q^{11} +(-136.573 + 136.573i) q^{13} +(-348.669 - 348.669i) q^{17} -527.725i q^{19} +(70.5445 - 70.5445i) q^{23} +(324.282 - 534.290i) q^{25} -68.9670i q^{29} -1373.31 q^{31} +(-143.471 - 40.1324i) q^{35} +(-1003.31 - 1003.31i) q^{37} -663.379 q^{41} +(1632.52 - 1632.52i) q^{43} +(-438.809 - 438.809i) q^{47} -2365.49i q^{49} +(-712.338 + 712.338i) q^{53} +(-3325.04 + 1871.45i) q^{55} -2918.34i q^{59} +1395.51 q^{61} +(1300.74 - 4650.07i) q^{65} +(-1691.51 - 1691.51i) q^{67} +3282.28 q^{71} +(-1465.81 + 1465.81i) q^{73} +(643.104 + 643.104i) q^{77} +3389.80i q^{79} +(-8690.02 + 8690.02i) q^{83} +(11871.6 + 3320.79i) q^{85} +4274.77i q^{89} -1150.96 q^{91} +(6471.01 + 11497.2i) q^{95} +(7215.47 + 7215.47i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 6 q^{5} + 110 q^{7} + 300 q^{11} - 360 q^{13} - 960 q^{17} + 810 q^{23} + 1856 q^{25} - 836 q^{31} + 2562 q^{35} - 660 q^{37} - 2964 q^{41} + 3270 q^{43} + 2250 q^{47} - 1980 q^{53} - 6780 q^{55}+ \cdots - 3180 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/180\mathbb{Z}\right)^\times\).

\(n\) \(37\) \(91\) \(101\)
\(\chi(n)\) \(e\left(\frac{1}{4}\right)\) \(1\) \(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). 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
\(3\) 0 0
\(4\) 0 0
\(5\) −21.7863 + 12.2621i −0.871450 + 0.490483i
\(6\) 0 0
\(7\) 4.21374 + 4.21374i 0.0859947 + 0.0859947i 0.748796 − 0.662801i \(-0.230632\pi\)
−0.662801 + 0.748796i \(0.730632\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 152.621 1.26133 0.630665 − 0.776055i \(-0.282782\pi\)
0.630665 + 0.776055i \(0.282782\pi\)
\(12\) 0 0
\(13\) −136.573 + 136.573i −0.808121 + 0.808121i −0.984349 − 0.176228i \(-0.943610\pi\)
0.176228 + 0.984349i \(0.443610\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −348.669 − 348.669i −1.20647 − 1.20647i −0.972163 − 0.234305i \(-0.924719\pi\)
−0.234305 − 0.972163i \(-0.575281\pi\)
\(18\) 0 0
\(19\) − 527.725i − 1.46184i −0.682462 − 0.730921i \(-0.739091\pi\)
0.682462 − 0.730921i \(-0.260909\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 70.5445 − 70.5445i 0.133354 − 0.133354i −0.637279 − 0.770633i \(-0.719940\pi\)
0.770633 + 0.637279i \(0.219940\pi\)
\(24\) 0 0
\(25\) 324.282 − 534.290i 0.518852 − 0.854864i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 68.9670i − 0.0820059i −0.999159 − 0.0410030i \(-0.986945\pi\)
0.999159 − 0.0410030i \(-0.0130553\pi\)
\(30\) 0 0
\(31\) −1373.31 −1.42905 −0.714523 − 0.699612i \(-0.753356\pi\)
−0.714523 + 0.699612i \(0.753356\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −143.471 − 40.1324i −0.117119 − 0.0327611i
\(36\) 0 0
\(37\) −1003.31 − 1003.31i −0.732875 − 0.732875i 0.238314 − 0.971188i \(-0.423405\pi\)
−0.971188 + 0.238314i \(0.923405\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −663.379 −0.394634 −0.197317 − 0.980340i \(-0.563223\pi\)
−0.197317 + 0.980340i \(0.563223\pi\)
\(42\) 0 0
\(43\) 1632.52 − 1632.52i 0.882920 − 0.882920i −0.110910 − 0.993830i \(-0.535377\pi\)
0.993830 + 0.110910i \(0.0353766\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −438.809 − 438.809i −0.198646 − 0.198646i 0.600773 − 0.799419i \(-0.294859\pi\)
−0.799419 + 0.600773i \(0.794859\pi\)
\(48\) 0 0
\(49\) − 2365.49i − 0.985210i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −712.338 + 712.338i −0.253591 + 0.253591i −0.822441 − 0.568850i \(-0.807389\pi\)
0.568850 + 0.822441i \(0.307389\pi\)
\(54\) 0 0
\(55\) −3325.04 + 1871.45i −1.09919 + 0.618661i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) − 2918.34i − 0.838363i −0.907903 − 0.419181i \(-0.862317\pi\)
0.907903 − 0.419181i \(-0.137683\pi\)
\(60\) 0 0
\(61\) 1395.51 0.375037 0.187518 − 0.982261i \(-0.439956\pi\)
0.187518 + 0.982261i \(0.439956\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 1300.74 − 4650.07i 0.307868 − 1.10061i
\(66\) 0 0
\(67\) −1691.51 − 1691.51i −0.376813 − 0.376813i 0.493138 − 0.869951i \(-0.335850\pi\)
−0.869951 + 0.493138i \(0.835850\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 3282.28 0.651117 0.325558 − 0.945522i \(-0.394448\pi\)
0.325558 + 0.945522i \(0.394448\pi\)
\(72\) 0 0
\(73\) −1465.81 + 1465.81i −0.275063 + 0.275063i −0.831134 − 0.556072i \(-0.812308\pi\)
0.556072 + 0.831134i \(0.312308\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 643.104 + 643.104i 0.108468 + 0.108468i
\(78\) 0 0
\(79\) 3389.80i 0.543150i 0.962417 + 0.271575i \(0.0875446\pi\)
−0.962417 + 0.271575i \(0.912455\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −8690.02 + 8690.02i −1.26143 + 1.26143i −0.311035 + 0.950398i \(0.600676\pi\)
−0.950398 + 0.311035i \(0.899324\pi\)
\(84\) 0 0
\(85\) 11871.6 + 3320.79i 1.64313 + 0.459624i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 4274.77i 0.539675i 0.962906 + 0.269838i \(0.0869701\pi\)
−0.962906 + 0.269838i \(0.913030\pi\)
\(90\) 0 0
\(91\) −1150.96 −0.138988
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 6471.01 + 11497.2i 0.717010 + 1.27392i
\(96\) 0 0
\(97\) 7215.47 + 7215.47i 0.766869 + 0.766869i 0.977554 − 0.210685i \(-0.0675693\pi\)
−0.210685 + 0.977554i \(0.567569\pi\)
\(98\) 0 0
\(99\) 0 0
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 180.5.l.a.37.1 4
3.2 odd 2 20.5.f.a.17.2 yes 4
5.2 odd 4 900.5.l.a.793.2 4
5.3 odd 4 inner 180.5.l.a.73.1 4
5.4 even 2 900.5.l.a.757.2 4
12.11 even 2 80.5.p.e.17.1 4
15.2 even 4 100.5.f.c.93.1 4
15.8 even 4 20.5.f.a.13.2 ✓ 4
15.14 odd 2 100.5.f.c.57.1 4
24.5 odd 2 320.5.p.m.257.1 4
24.11 even 2 320.5.p.l.257.2 4
60.23 odd 4 80.5.p.e.33.1 4
60.47 odd 4 400.5.p.h.193.2 4
60.59 even 2 400.5.p.h.257.2 4
120.53 even 4 320.5.p.m.193.1 4
120.83 odd 4 320.5.p.l.193.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
20.5.f.a.13.2 ✓ 4 15.8 even 4
20.5.f.a.17.2 yes 4 3.2 odd 2
80.5.p.e.17.1 4 12.11 even 2
80.5.p.e.33.1 4 60.23 odd 4
100.5.f.c.57.1 4 15.14 odd 2
100.5.f.c.93.1 4 15.2 even 4
180.5.l.a.37.1 4 1.1 even 1 trivial
180.5.l.a.73.1 4 5.3 odd 4 inner
320.5.p.l.193.2 4 120.83 odd 4
320.5.p.l.257.2 4 24.11 even 2
320.5.p.m.193.1 4 120.53 even 4
320.5.p.m.257.1 4 24.5 odd 2
400.5.p.h.193.2 4 60.47 odd 4
400.5.p.h.257.2 4 60.59 even 2
900.5.l.a.757.2 4 5.4 even 2
900.5.l.a.793.2 4 5.2 odd 4