Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1520,2,Mod(609,1520)] 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("1520.609"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1520, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1520 = 2^{4} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1520.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,0,0,0,0,0,0,12,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(15)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(12.1372611072\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{5})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 6x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 380)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 609.4
Root \(2.28825i\) of defining polynomial
Character \(\chi\) \(=\) 1520.609
Dual form 1520.2.d.f.609.1

$q$-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)
 
\(f(q)\) \(=\) \(q+2.28825i q^{3} -2.23607 q^{5} +2.82843i q^{7} -2.23607 q^{9} +5.23607 q^{11} +4.03631i q^{13} -5.11667i q^{15} +1.08036i q^{17} -1.00000 q^{19} -6.47214 q^{21} +7.40492i q^{23} +5.00000 q^{25} +1.74806i q^{27} -4.47214 q^{29} +4.00000 q^{31} +11.9814i q^{33} -6.32456i q^{35} -6.86474i q^{37} -9.23607 q^{39} -6.00000 q^{41} -8.48528i q^{43} +5.00000 q^{45} +8.48528i q^{47} -1.00000 q^{49} -2.47214 q^{51} -6.86474i q^{53} -11.7082 q^{55} -2.28825i q^{57} -10.4721 q^{59} +1.70820 q^{61} -6.32456i q^{63} -9.02546i q^{65} +1.62054i q^{67} -16.9443 q^{69} +1.52786 q^{71} -13.7295i q^{73} +11.4412i q^{75} +14.8098i q^{77} -11.4164 q^{79} -10.7082 q^{81} +5.24419i q^{83} -2.41577i q^{85} -10.2333i q^{87} +14.9443 q^{89} -11.4164 q^{91} +9.15298i q^{93} +2.23607 q^{95} -4.44897i q^{97} -11.7082 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 12 q^{11} - 4 q^{19} - 8 q^{21} + 20 q^{25} + 16 q^{31} - 28 q^{39} - 24 q^{41} + 20 q^{45} - 4 q^{49} + 8 q^{51} - 20 q^{55} - 24 q^{59} - 20 q^{61} - 32 q^{69} + 24 q^{71} + 8 q^{79} - 16 q^{81}+ \cdots - 20 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1520\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(401\) \(1141\) \(1217\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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
\(3\) 2.28825i 1.32112i 0.750774 + 0.660560i \(0.229681\pi\)
−0.750774 + 0.660560i \(0.770319\pi\)
\(4\) 0 0
\(5\) −2.23607 −1.00000
\(6\) 0 0
\(7\) 2.82843i 1.06904i 0.845154 + 0.534522i \(0.179509\pi\)
−0.845154 + 0.534522i \(0.820491\pi\)
\(8\) 0 0
\(9\) −2.23607 −0.745356
\(10\) 0 0
\(11\) 5.23607 1.57873 0.789367 − 0.613922i \(-0.210409\pi\)
0.789367 + 0.613922i \(0.210409\pi\)
\(12\) 0 0
\(13\) 4.03631i 1.11947i 0.828671 + 0.559735i \(0.189097\pi\)
−0.828671 + 0.559735i \(0.810903\pi\)
\(14\) 0 0
\(15\) − 5.11667i − 1.32112i
\(16\) 0 0
\(17\) 1.08036i 0.262027i 0.991381 + 0.131013i \(0.0418230\pi\)
−0.991381 + 0.131013i \(0.958177\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) −6.47214 −1.41234
\(22\) 0 0
\(23\) 7.40492i 1.54403i 0.635603 + 0.772016i \(0.280752\pi\)
−0.635603 + 0.772016i \(0.719248\pi\)
\(24\) 0 0
\(25\) 5.00000 1.00000
\(26\) 0 0
\(27\) 1.74806i 0.336415i
\(28\) 0 0
\(29\) −4.47214 −0.830455 −0.415227 − 0.909718i \(-0.636298\pi\)
−0.415227 + 0.909718i \(0.636298\pi\)
\(30\) 0 0
\(31\) 4.00000 0.718421 0.359211 − 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) 0 0
\(33\) 11.9814i 2.08570i
\(34\) 0 0
\(35\) − 6.32456i − 1.06904i
\(36\) 0 0
\(37\) − 6.86474i − 1.12856i −0.825585 − 0.564278i \(-0.809155\pi\)
0.825585 − 0.564278i \(-0.190845\pi\)
\(38\) 0 0
\(39\) −9.23607 −1.47895
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 − 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) − 8.48528i − 1.29399i −0.762493 − 0.646997i \(-0.776025\pi\)
0.762493 − 0.646997i \(-0.223975\pi\)
\(44\) 0 0
\(45\) 5.00000 0.745356
\(46\) 0 0
\(47\) 8.48528i 1.23771i 0.785507 + 0.618853i \(0.212402\pi\)
−0.785507 + 0.618853i \(0.787598\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) −2.47214 −0.346168
\(52\) 0 0
\(53\) − 6.86474i − 0.942944i −0.881881 − 0.471472i \(-0.843723\pi\)
0.881881 − 0.471472i \(-0.156277\pi\)
\(54\) 0 0
\(55\) −11.7082 −1.57873
\(56\) 0 0
\(57\) − 2.28825i − 0.303086i
\(58\) 0 0
\(59\) −10.4721 −1.36336 −0.681678 − 0.731652i \(-0.738749\pi\)
−0.681678 + 0.731652i \(0.738749\pi\)
\(60\) 0 0
\(61\) 1.70820 0.218713 0.109357 − 0.994003i \(-0.465121\pi\)
0.109357 + 0.994003i \(0.465121\pi\)
\(62\) 0 0
\(63\) − 6.32456i − 0.796819i
\(64\) 0 0
\(65\) − 9.02546i − 1.11947i
\(66\) 0 0
\(67\) 1.62054i 0.197981i 0.995088 + 0.0989905i \(0.0315613\pi\)
−0.995088 + 0.0989905i \(0.968439\pi\)
\(68\) 0 0
\(69\) −16.9443 −2.03985
\(70\) 0 0
\(71\) 1.52786 0.181324 0.0906621 − 0.995882i \(-0.471102\pi\)
0.0906621 + 0.995882i \(0.471102\pi\)
\(72\) 0 0
\(73\) − 13.7295i − 1.60691i −0.595363 − 0.803457i \(-0.702992\pi\)
0.595363 − 0.803457i \(-0.297008\pi\)
\(74\) 0 0
\(75\) 11.4412i 1.32112i
\(76\) 0 0
\(77\) 14.8098i 1.68774i
\(78\) 0 0
\(79\) −11.4164 −1.28445 −0.642223 − 0.766518i \(-0.721988\pi\)
−0.642223 + 0.766518i \(0.721988\pi\)
\(80\) 0 0
\(81\) −10.7082 −1.18980
\(82\) 0 0
\(83\) 5.24419i 0.575625i 0.957687 + 0.287812i \(0.0929280\pi\)
−0.957687 + 0.287812i \(0.907072\pi\)
\(84\) 0 0
\(85\) − 2.41577i − 0.262027i
\(86\) 0 0
\(87\) − 10.2333i − 1.09713i
\(88\) 0 0
\(89\) 14.9443 1.58409 0.792045 − 0.610463i \(-0.209017\pi\)
0.792045 + 0.610463i \(0.209017\pi\)
\(90\) 0 0
\(91\) −11.4164 −1.19676
\(92\) 0 0
\(93\) 9.15298i 0.949120i
\(94\) 0 0
\(95\) 2.23607 0.229416
\(96\) 0 0
\(97\) − 4.44897i − 0.451725i −0.974159 − 0.225862i \(-0.927480\pi\)
0.974159 − 0.225862i \(-0.0725199\pi\)
\(98\) 0 0
\(99\) −11.7082 −1.17672
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 1520.2.d.f.609.4 4
4.3 odd 2 380.2.c.a.229.1 ✓ 4
5.2 odd 4 7600.2.a.ce.1.4 4
5.3 odd 4 7600.2.a.ce.1.1 4
5.4 even 2 inner 1520.2.d.f.609.1 4
12.11 even 2 3420.2.f.a.1369.3 4
20.3 even 4 1900.2.a.j.1.4 4
20.7 even 4 1900.2.a.j.1.1 4
20.19 odd 2 380.2.c.a.229.4 yes 4
60.59 even 2 3420.2.f.a.1369.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.2.c.a.229.1 ✓ 4 4.3 odd 2
380.2.c.a.229.4 yes 4 20.19 odd 2
1520.2.d.f.609.1 4 5.4 even 2 inner
1520.2.d.f.609.4 4 1.1 even 1 trivial
1900.2.a.j.1.1 4 20.7 even 4
1900.2.a.j.1.4 4 20.3 even 4
3420.2.f.a.1369.3 4 12.11 even 2
3420.2.f.a.1369.4 4 60.59 even 2
7600.2.a.ce.1.1 4 5.3 odd 4
7600.2.a.ce.1.4 4 5.2 odd 4