Properties

Label 38.9.b.a
Level $38$
Weight $9$
Character orbit 38.b
Analytic conductor $15.480$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [38,9,Mod(37,38)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(38, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("38.37");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 38 = 2 \cdot 19 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 38.b (of order \(2\), degree \(1\), 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: \(15.4803871823\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 46118 x^{10} + 738386961 x^{8} + 5214446299656 x^{6} + \cdots + 92\!\cdots\!64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{21} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{7} q^{2} + ( - \beta_{7} + \beta_{6}) q^{3} - 128 q^{4} + ( - \beta_{2} + 47) q^{5} + ( - 2 \beta_1 + 150) q^{6} + (\beta_{4} + \beta_1 - 452) q^{7} - 128 \beta_{7} q^{8} + (\beta_{5} + 2 \beta_{4} - \beta_{2} - 2 \beta_1 - 1298) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{7} q^{2} + ( - \beta_{7} + \beta_{6}) q^{3} - 128 q^{4} + ( - \beta_{2} + 47) q^{5} + ( - 2 \beta_1 + 150) q^{6} + (\beta_{4} + \beta_1 - 452) q^{7} - 128 \beta_{7} q^{8} + (\beta_{5} + 2 \beta_{4} - \beta_{2} - 2 \beta_1 - 1298) q^{9} + (2 \beta_{10} + 47 \beta_{7}) q^{10} + (\beta_{5} - \beta_{4} - 2 \beta_{3} + 3 \beta_{2} - 2 \beta_1 - 1046) q^{11} + (128 \beta_{7} - 128 \beta_{6}) q^{12} + ( - 4 \beta_{10} + 3 \beta_{9} - \beta_{8} + 269 \beta_{7} - 57 \beta_{6}) q^{13} + (\beta_{11} - 2 \beta_{8} - 439 \beta_{7} + 82 \beta_{6}) q^{14} + ( - 7 \beta_{10} + 4 \beta_{9} - 3 \beta_{8} - 48 \beta_{7} + 15 \beta_{6}) q^{15} + 16384 q^{16} + (\beta_{5} - 28 \beta_{4} - 9 \beta_{3} + 4 \beta_{2} - 16 \beta_1 + 22569) q^{17} + (2 \beta_{10} - 6 \beta_{9} - 4 \beta_{8} - 1311 \beta_{7} - 82 \beta_{6}) q^{18} + (6 \beta_{11} + \beta_{10} + 5 \beta_{9} - 3 \beta_{8} - 683 \beta_{7} - 169 \beta_{6} + \cdots + 3430) q^{19}+ \cdots + (3179 \beta_{5} + 2227 \beta_{4} + 3696 \beta_{3} + \cdots - 7164160) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 1536 q^{4} + 558 q^{5} + 1792 q^{6} - 5422 q^{7} - 15592 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 1536 q^{4} + 558 q^{5} + 1792 q^{6} - 5422 q^{7} - 15592 q^{9} - 12546 q^{11} + 196608 q^{16} + 270810 q^{17} + 41512 q^{19} - 71424 q^{20} - 823956 q^{23} - 229376 q^{24} + 865538 q^{25} - 431616 q^{26} + 694016 q^{28} + 71168 q^{30} - 1194378 q^{35} + 1995776 q^{36} + 998784 q^{38} + 5786100 q^{39} - 8383744 q^{42} + 7586646 q^{43} + 1605888 q^{44} + 2226046 q^{45} - 20260530 q^{47} - 19498842 q^{49} + 16933888 q^{54} - 14858554 q^{55} + 14430564 q^{57} - 5506560 q^{58} - 41363266 q^{61} + 32266752 q^{62} + 84235798 q^{63} - 25165824 q^{64} + 14371328 q^{66} - 34663680 q^{68} + 87906498 q^{73} - 2149632 q^{74} - 5313536 q^{76} - 78817962 q^{77} + 9142272 q^{80} - 100904812 q^{81} - 49609728 q^{82} - 55944960 q^{83} + 25440254 q^{85} + 119189604 q^{87} + 105466368 q^{92} + 105500856 q^{93} + 81396774 q^{95} + 29360128 q^{96} - 85554938 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 46118 x^{10} + 738386961 x^{8} + 5214446299656 x^{6} + \cdots + 92\!\cdots\!64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 627859595 \nu^{10} - 26324430503956 \nu^{8} + \cdots - 13\!\cdots\!44 ) / 30\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 24\!\cdots\!97 \nu^{10} + \cdots - 55\!\cdots\!76 ) / 50\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 35\!\cdots\!93 \nu^{10} + \cdots + 70\!\cdots\!64 ) / 21\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 61\!\cdots\!79 \nu^{10} + \cdots - 14\!\cdots\!92 ) / 25\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 86\!\cdots\!97 \nu^{10} + \cdots + 20\!\cdots\!56 ) / 28\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 13946998708921 \nu^{11} + \cdots + 41\!\cdots\!84 \nu ) / 11\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 13946998708921 \nu^{11} + \cdots - 29\!\cdots\!28 \nu ) / 14\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 83\!\cdots\!69 \nu^{11} + \cdots - 16\!\cdots\!04 \nu ) / 13\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 19\!\cdots\!57 \nu^{11} + \cdots + 39\!\cdots\!36 \nu ) / 59\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 42\!\cdots\!29 \nu^{11} + \cdots - 99\!\cdots\!12 \nu ) / 59\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 64\!\cdots\!57 \nu^{11} + \cdots + 15\!\cdots\!36 \nu ) / 37\!\cdots\!00 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{7} + 8\beta_{6} ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{5} + 4\beta_{4} - 2\beta_{2} - 13\beta _1 - 15367 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -120\beta_{11} - 161\beta_{10} + 15\beta_{9} - 218\beta_{8} - 96909\beta_{7} - 56459\beta_{6} ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -19769\beta_{5} - 47248\beta_{4} + 2115\beta_{3} + 37007\beta_{2} + 197287\beta _1 + 108259783 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 2409450 \beta_{11} + 2214367 \beta_{10} + 476619 \beta_{9} + 5477746 \beta_{8} + 2937881826 \beta_{7} + 1006000693 \beta_{6} ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 781327466 \beta_{5} + 1948624906 \beta_{4} - 159513309 \beta_{3} - 1389463040 \beta_{2} - 9545300437 \beta _1 - 3853054598260 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 44028806610 \beta_{11} - 24374350019 \beta_{10} - 26067322407 \beta_{9} - 118950691874 \beta_{8} - 71468966199420 \beta_{7} - 19541503881929 \beta_{6} ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 7899466534745 \beta_{5} - 19941863755858 \beta_{4} + 2150472839598 \beta_{3} + 12265474994369 \beta_{2} + 107834804471680 \beta _1 + 37\!\cdots\!47 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 824936798643090 \beta_{11} + 209168343586807 \beta_{10} + 775901785287075 \beta_{9} + \cdots + 39\!\cdots\!85 \beta_{6} ) / 4 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 32\!\cdots\!98 \beta_{5} + \cdots - 15\!\cdots\!84 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 16\!\cdots\!90 \beta_{11} + \cdots - 81\!\cdots\!57 \beta_{6} ) / 4 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/38\mathbb{Z}\right)^\times\).

\(n\) \(21\)
\(\chi(n)\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1
145.414i
64.6816i
23.4825i
61.5968i
63.2768i
111.533i
111.533i
63.2768i
61.5968i
23.4825i
64.6816i
145.414i
11.3137i 132.686i −128.000 −12.7536 −1501.17 −1217.09 1448.15i −11044.5 144.290i
37.2 11.3137i 51.9537i −128.000 629.221 −587.789 2565.52 1448.15i 3861.81 7118.82i
37.3 11.3137i 10.7546i −128.000 −919.278 −121.674 −343.629 1448.15i 6445.34 10400.4i
37.4 11.3137i 74.3247i −128.000 −154.845 840.888 1585.07 1448.15i 1036.83 1751.87i
37.5 11.3137i 76.0047i −128.000 1155.75 859.895 −2869.50 1448.15i 784.279 13075.8i
37.6 11.3137i 124.261i −128.000 −419.091 1405.85 −2431.38 1448.15i −8879.72 4741.48i
37.7 11.3137i 124.261i −128.000 −419.091 1405.85 −2431.38 1448.15i −8879.72 4741.48i
37.8 11.3137i 76.0047i −128.000 1155.75 859.895 −2869.50 1448.15i 784.279 13075.8i
37.9 11.3137i 74.3247i −128.000 −154.845 840.888 1585.07 1448.15i 1036.83 1751.87i
37.10 11.3137i 10.7546i −128.000 −919.278 −121.674 −343.629 1448.15i 6445.34 10400.4i
37.11 11.3137i 51.9537i −128.000 629.221 −587.789 2565.52 1448.15i 3861.81 7118.82i
37.12 11.3137i 132.686i −128.000 −12.7536 −1501.17 −1217.09 1448.15i −11044.5 144.290i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 37.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 38.9.b.a 12
3.b odd 2 1 342.9.d.a 12
4.b odd 2 1 304.9.e.e 12
19.b odd 2 1 inner 38.9.b.a 12
57.d even 2 1 342.9.d.a 12
76.d even 2 1 304.9.e.e 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.9.b.a 12 1.a even 1 1 trivial
38.9.b.a 12 19.b odd 2 1 inner
304.9.e.e 12 4.b odd 2 1
304.9.e.e 12 76.d even 2 1
342.9.d.a 12 3.b odd 2 1
342.9.d.a 12 57.d even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{9}^{\mathrm{new}}(38, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 128)^{6} \) Copy content Toggle raw display
$3$ \( T^{12} + 47162 T^{10} + \cdots + 27\!\cdots\!00 \) Copy content Toggle raw display
$5$ \( (T^{6} - 279 T^{5} + \cdots + 553286629221600)^{2} \) Copy content Toggle raw display
$7$ \( (T^{6} + 2711 T^{5} + \cdots + 11\!\cdots\!70)^{2} \) Copy content Toggle raw display
$11$ \( (T^{6} + 6273 T^{5} + \cdots - 77\!\cdots\!00)^{2} \) Copy content Toggle raw display
$13$ \( T^{12} + 4949101290 T^{10} + \cdots + 40\!\cdots\!84 \) Copy content Toggle raw display
$17$ \( (T^{6} - 135405 T^{5} + \cdots - 22\!\cdots\!30)^{2} \) Copy content Toggle raw display
$19$ \( T^{12} - 41512 T^{11} + \cdots + 23\!\cdots\!41 \) Copy content Toggle raw display
$23$ \( (T^{6} + 411978 T^{5} + \cdots + 32\!\cdots\!40)^{2} \) Copy content Toggle raw display
$29$ \( T^{12} + 2131908998922 T^{10} + \cdots + 35\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{12} + 2544887493528 T^{10} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{12} + 21514669315032 T^{10} + \cdots + 26\!\cdots\!84 \) Copy content Toggle raw display
$41$ \( T^{12} + 42067529126808 T^{10} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{6} - 3793323 T^{5} + \cdots - 13\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} + 10130265 T^{5} + \cdots + 36\!\cdots\!40)^{2} \) Copy content Toggle raw display
$53$ \( T^{12} + 521969657351082 T^{10} + \cdots + 11\!\cdots\!76 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 45\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{6} + 20681633 T^{5} + \cdots + 26\!\cdots\!00)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 26\!\cdots\!84 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 42\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{6} - 43953249 T^{5} + \cdots - 67\!\cdots\!50)^{2} \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 90\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( (T^{6} + 27972480 T^{5} + \cdots - 21\!\cdots\!80)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 28\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 39\!\cdots\!04 \) Copy content Toggle raw display
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