Properties

Label 26.5.d.a
Level $26$
Weight $5$
Character orbit 26.d
Analytic conductor $2.688$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [26,5,Mod(5,26)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(26, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([3]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("26.5");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 26 = 2 \cdot 13 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 26.d (of order \(4\), degree \(2\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.68761904018\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 234x^{4} + 13689x^{2} + 60516 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - 2 \beta_{2} - 2) q^{2} + \beta_{3} q^{3} + 8 \beta_{2} q^{4} + ( - \beta_{4} - \beta_{3} + 3 \beta_{2} + \beta_1 + 3) q^{5} + ( - 2 \beta_{3} + 2 \beta_1) q^{6} + ( - \beta_{5} + 4 \beta_{3} + 7 \beta_{2} + 4 \beta_1 - 7) q^{7} + ( - 16 \beta_{2} + 16) q^{8} + ( - \beta_{5} - \beta_{4} + 3 \beta_{3} - 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 2 \beta_{2} - 2) q^{2} + \beta_{3} q^{3} + 8 \beta_{2} q^{4} + ( - \beta_{4} - \beta_{3} + 3 \beta_{2} + \beta_1 + 3) q^{5} + ( - 2 \beta_{3} + 2 \beta_1) q^{6} + ( - \beta_{5} + 4 \beta_{3} + 7 \beta_{2} + 4 \beta_1 - 7) q^{7} + ( - 16 \beta_{2} + 16) q^{8} + ( - \beta_{5} - \beta_{4} + 3 \beta_{3} - 3) q^{9} + ( - 2 \beta_{5} + 2 \beta_{4} - 12 \beta_{2} - 4 \beta_1) q^{10} + (4 \beta_{5} + 2 \beta_{3} + 3 \beta_{2} + 2 \beta_1 - 3) q^{11} - 8 \beta_1 q^{12} + (3 \beta_{5} - 2 \beta_{4} - 13 \beta_{3} + 16 \beta_{2} + 15) q^{13} + (2 \beta_{5} + 2 \beta_{4} - 16 \beta_{3} + 28) q^{14} + (5 \beta_{4} + 15 \beta_{3} - 72 \beta_{2} - 15 \beta_1 - 72) q^{15} - 64 q^{16} + ( - 3 \beta_{5} + 3 \beta_{4} + 162 \beta_{2} - 31 \beta_1) q^{17} + (4 \beta_{4} - 6 \beta_{3} + 6 \beta_{2} + 6 \beta_1 + 6) q^{18} + ( - 4 \beta_{4} - 13 \beta_{3} - 61 \beta_{2} + 13 \beta_1 - 61) q^{19} + (8 \beta_{5} + 8 \beta_{3} + 24 \beta_{2} + 8 \beta_1 - 24) q^{20} + ( - 5 \beta_{5} + 20 \beta_{3} - 318 \beta_{2} + 20 \beta_1 + 318) q^{21} + ( - 8 \beta_{5} - 8 \beta_{4} - 8 \beta_{3} + 12) q^{22} + (\beta_{5} - \beta_{4} - 156 \beta_{2} - 34 \beta_1) q^{23} + (16 \beta_{3} + 16 \beta_1) q^{24} + ( - \beta_{5} + \beta_{4} + 659 \beta_{2} + 105 \beta_1) q^{25} + ( - 10 \beta_{5} - 2 \beta_{4} + 26 \beta_{3} - 62 \beta_{2} - 26 \beta_1 + 2) q^{26} + ( - 45 \beta_{3} + 246) q^{27} + ( - 8 \beta_{4} + 32 \beta_{3} - 56 \beta_{2} - 32 \beta_1 - 56) q^{28} + ( - \beta_{5} - \beta_{4} - 20 \beta_{3} - 1038) q^{29} + (10 \beta_{5} - 10 \beta_{4} + 288 \beta_{2} + 60 \beta_1) q^{30} + ( - 16 \beta_{4} + 21 \beta_{3} + 67 \beta_{2} - 21 \beta_1 + 67) q^{31} + (128 \beta_{2} + 128) q^{32} + ( - 16 \beta_{5} - 57 \beta_{3} - 132 \beta_{2} - 57 \beta_1 + 132) q^{33} + (12 \beta_{5} + 62 \beta_{3} - 324 \beta_{2} + 62 \beta_1 + 324) q^{34} + (36 \beta_{5} + 36 \beta_{4} + 65 \beta_{3} + 504) q^{35} + (8 \beta_{5} - 8 \beta_{4} - 24 \beta_{2} - 24 \beta_1) q^{36} + ( - 33 \beta_{5} + 35 \beta_{3} - 883 \beta_{2} + 35 \beta_1 + 883) q^{37} + ( - 8 \beta_{5} + 8 \beta_{4} + 244 \beta_{2} - 52 \beta_1) q^{38} + (4 \beta_{5} + 19 \beta_{4} - 39 \beta_{3} + 30 \beta_{2} - 91 \beta_1 - 1020) q^{39} + ( - 16 \beta_{5} - 16 \beta_{4} - 32 \beta_{3} + 96) q^{40} + ( - 16 \beta_{4} - 61 \beta_{3} - 405 \beta_{2} + 61 \beta_1 - 405) q^{41} + (10 \beta_{5} + 10 \beta_{4} - 80 \beta_{3} - 1272) q^{42} + ( - 4 \beta_{5} + 4 \beta_{4} + 930 \beta_{2} - 87 \beta_1) q^{43} + (32 \beta_{4} + 16 \beta_{3} - 24 \beta_{2} - 16 \beta_1 - 24) q^{44} + (36 \beta_{4} - 21 \beta_{3} + 897 \beta_{2} + 21 \beta_1 + 897) q^{45} + ( - 4 \beta_{5} + 68 \beta_{3} + 312 \beta_{2} + 68 \beta_1 - 312) q^{46} + (13 \beta_{5} - 102 \beta_{3} - 669 \beta_{2} - 102 \beta_1 + 669) q^{47} - 64 \beta_{3} q^{48} + (21 \beta_{5} - 21 \beta_{4} - 1423 \beta_{2} + 185 \beta_1) q^{49} + (4 \beta_{5} - 210 \beta_{3} - 1318 \beta_{2} - 210 \beta_1 + 1318) q^{50} + (40 \beta_{5} - 40 \beta_{4} + 2382 \beta_{2} - 165 \beta_1) q^{51} + (16 \beta_{5} + 24 \beta_{4} + 120 \beta_{2} + 104 \beta_1 - 128) q^{52} + ( - 3 \beta_{5} - 3 \beta_{4} + 142 \beta_{3} + 1086) q^{53} + (90 \beta_{3} - 492 \beta_{2} - 90 \beta_1 - 492) q^{54} + ( - 23 \beta_{5} - 23 \beta_{4} + 342 \beta_{3} - 4794) q^{55} + ( - 16 \beta_{5} + 16 \beta_{4} + 224 \beta_{2} + 128 \beta_1) q^{56} + (38 \beta_{4} - 40 \beta_{3} - 990 \beta_{2} + 40 \beta_1 - 990) q^{57} + (4 \beta_{4} + 40 \beta_{3} + 2076 \beta_{2} - 40 \beta_1 + 2076) q^{58} + (34 \beta_{5} + 61 \beta_{3} - 1239 \beta_{2} + 61 \beta_1 + 1239) q^{59} + ( - 40 \beta_{5} - 120 \beta_{3} - 576 \beta_{2} - 120 \beta_1 + 576) q^{60} + ( - 106 \beta_{5} - 106 \beta_{4} + 92 \beta_{3} + 1194) q^{61} + ( - 32 \beta_{5} + 32 \beta_{4} - 268 \beta_{2} + 84 \beta_1) q^{62} + (56 \beta_{5} + 129 \beta_{3} - 2157 \beta_{2} + 129 \beta_1 + 2157) q^{63} - 512 \beta_{2} q^{64} + ( - 29 \beta_{5} - 89 \beta_{4} + 13 \beta_{3} + 3273 \beta_{2} + \cdots - 2433) q^{65}+ \cdots + ( - 162 \beta_{5} + 39 \beta_{3} + 4107 \beta_{2} + 39 \beta_1 - 4107) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 12 q^{2} + 18 q^{5} - 42 q^{7} + 96 q^{8} - 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 12 q^{2} + 18 q^{5} - 42 q^{7} + 96 q^{8} - 18 q^{9} - 18 q^{11} + 90 q^{13} + 168 q^{14} - 432 q^{15} - 384 q^{16} + 36 q^{18} - 366 q^{19} - 144 q^{20} + 1908 q^{21} + 72 q^{22} + 12 q^{26} + 1476 q^{27} - 336 q^{28} - 6228 q^{29} + 402 q^{31} + 768 q^{32} + 792 q^{33} + 1944 q^{34} + 3024 q^{35} + 5298 q^{37} - 6120 q^{39} + 576 q^{40} - 2430 q^{41} - 7632 q^{42} - 144 q^{44} + 5382 q^{45} - 1872 q^{46} + 4014 q^{47} + 7908 q^{50} - 768 q^{52} + 6516 q^{53} - 2952 q^{54} - 28764 q^{55} - 5940 q^{57} + 12456 q^{58} + 7434 q^{59} + 3456 q^{60} + 7164 q^{61} + 12942 q^{63} - 14598 q^{65} - 3168 q^{66} + 5790 q^{67} - 7776 q^{68} - 6048 q^{70} + 15246 q^{71} - 288 q^{72} - 4254 q^{73} - 21192 q^{74} + 2928 q^{76} + 12600 q^{78} + 23364 q^{79} - 1152 q^{80} - 19602 q^{81} - 38034 q^{83} + 15264 q^{84} + 30672 q^{85} + 11160 q^{86} - 9288 q^{87} + 4122 q^{89} - 11922 q^{91} + 7488 q^{92} + 10404 q^{93} - 16056 q^{94} - 1362 q^{97} - 17076 q^{98} - 24642 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} + 234x^{4} + 13689x^{2} + 60516 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 117\nu ) / 246 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{4} + 117\nu^{2} ) / 246 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{5} + 3\nu^{4} + 195\nu^{3} + 597\nu^{2} + 8388\nu + 19188 ) / 492 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{5} + 3\nu^{4} - 195\nu^{3} + 597\nu^{2} - 8388\nu + 19188 ) / 492 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + \beta_{4} - 3\beta_{3} - 78 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 246\beta_{2} - 117\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -117\beta_{5} - 117\beta_{4} + 597\beta_{3} + 9126 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -246\beta_{5} + 246\beta_{4} - 47970\beta_{2} + 14427\beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/26\mathbb{Z}\right)^\times\).

\(n\) \(15\)
\(\chi(n)\) \(\beta_{2}\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1
9.55234i
2.19267i
11.7450i
9.55234i
2.19267i
11.7450i
−2.00000 + 2.00000i −9.55234 8.00000i 33.5044 33.5044i 19.1047 19.1047i −24.2573 24.2573i 16.0000 + 16.0000i 10.2471 134.018i
5.2 −2.00000 + 2.00000i −2.19267 8.00000i −28.1144 + 28.1144i 4.38533 4.38533i −49.0778 49.0778i 16.0000 + 16.0000i −76.1922 112.458i
5.3 −2.00000 + 2.00000i 11.7450 8.00000i 3.61003 3.61003i −23.4900 + 23.4900i 52.3350 + 52.3350i 16.0000 + 16.0000i 56.9451 14.4401i
21.1 −2.00000 2.00000i −9.55234 8.00000i 33.5044 + 33.5044i 19.1047 + 19.1047i −24.2573 + 24.2573i 16.0000 16.0000i 10.2471 134.018i
21.2 −2.00000 2.00000i −2.19267 8.00000i −28.1144 28.1144i 4.38533 + 4.38533i −49.0778 + 49.0778i 16.0000 16.0000i −76.1922 112.458i
21.3 −2.00000 2.00000i 11.7450 8.00000i 3.61003 + 3.61003i −23.4900 23.4900i 52.3350 52.3350i 16.0000 16.0000i 56.9451 14.4401i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.d odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 26.5.d.a 6
3.b odd 2 1 234.5.i.b 6
4.b odd 2 1 208.5.t.b 6
13.b even 2 1 338.5.d.d 6
13.d odd 4 1 inner 26.5.d.a 6
13.d odd 4 1 338.5.d.d 6
39.f even 4 1 234.5.i.b 6
52.f even 4 1 208.5.t.b 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.5.d.a 6 1.a even 1 1 trivial
26.5.d.a 6 13.d odd 4 1 inner
208.5.t.b 6 4.b odd 2 1
208.5.t.b 6 52.f even 4 1
234.5.i.b 6 3.b odd 2 1
234.5.i.b 6 39.f even 4 1
338.5.d.d 6 13.b even 2 1
338.5.d.d 6 13.d odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{3} - 117T_{3} - 246 \) acting on \(S_{5}^{\mathrm{new}}(26, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 4 T + 8)^{3} \) Copy content Toggle raw display
$3$ \( (T^{3} - 117 T - 246)^{2} \) Copy content Toggle raw display
$5$ \( T^{6} - 18 T^{5} + 162 T^{4} + \cdots + 92507202 \) Copy content Toggle raw display
$7$ \( T^{6} + 42 T^{5} + \cdots + 31054805762 \) Copy content Toggle raw display
$11$ \( T^{6} + 18 T^{5} + \cdots + 1189455693192 \) Copy content Toggle raw display
$13$ \( T^{6} - 90 T^{5} + \cdots + 23298085122481 \) Copy content Toggle raw display
$17$ \( T^{6} + 358146 T^{4} + \cdots + 11\!\cdots\!64 \) Copy content Toggle raw display
$19$ \( T^{6} + \cdots + 128090511995168 \) Copy content Toggle raw display
$23$ \( T^{6} + 352764 T^{4} + \cdots + 12\!\cdots\!36 \) Copy content Toggle raw display
$29$ \( (T^{3} + 3114 T^{2} + \cdots + 1065910092)^{2} \) Copy content Toggle raw display
$31$ \( T^{6} - 402 T^{5} + \cdots + 22\!\cdots\!88 \) Copy content Toggle raw display
$37$ \( T^{6} - 5298 T^{5} + \cdots + 30\!\cdots\!82 \) Copy content Toggle raw display
$41$ \( T^{6} + 2430 T^{5} + \cdots + 11\!\cdots\!48 \) Copy content Toggle raw display
$43$ \( T^{6} + 4449654 T^{4} + \cdots + 16\!\cdots\!96 \) Copy content Toggle raw display
$47$ \( T^{6} - 4014 T^{5} + \cdots + 12\!\cdots\!50 \) Copy content Toggle raw display
$53$ \( (T^{3} - 3258 T^{2} + 1133046 T + 235239252)^{2} \) Copy content Toggle raw display
$59$ \( T^{6} - 7434 T^{5} + \cdots + 29\!\cdots\!28 \) Copy content Toggle raw display
$61$ \( (T^{3} - 3582 T^{2} + \cdots + 141282218568)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} - 5790 T^{5} + \cdots + 27\!\cdots\!32 \) Copy content Toggle raw display
$71$ \( T^{6} - 15246 T^{5} + \cdots + 69\!\cdots\!50 \) Copy content Toggle raw display
$73$ \( T^{6} + \cdots + 511257869120000 \) Copy content Toggle raw display
$79$ \( (T^{3} - 11682 T^{2} + \cdots + 57682274340)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + 38034 T^{5} + \cdots + 45\!\cdots\!32 \) Copy content Toggle raw display
$89$ \( T^{6} - 4122 T^{5} + \cdots + 80\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{6} + 1362 T^{5} + \cdots + 24\!\cdots\!00 \) Copy content Toggle raw display
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