Properties

Label 380.2.l.b
Level $380$
Weight $2$
Character orbit 380.l
Analytic conductor $3.034$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [380,2,Mod(37,380)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(380, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 1, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("380.37");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 380 = 2^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 380.l (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: \(3.03431527681\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 4 x^{11} + 28 x^{10} - 64 x^{9} + 236 x^{8} - 420 x^{7} + 946 x^{6} - 1216 x^{5} + 1896 x^{4} + \cdots + 1370 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{6} \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{2} q^{3} + ( - \beta_{8} + \beta_{3}) q^{5} + \beta_{6} q^{7} + ( - \beta_{8} - \beta_{6} + \cdots + \beta_{3}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{3} + ( - \beta_{8} + \beta_{3}) q^{5} + \beta_{6} q^{7} + ( - \beta_{8} - \beta_{6} + \cdots + \beta_{3}) q^{9}+ \cdots + (\beta_{8} - \beta_{6} + \cdots + \beta_{3}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 4 q^{5} + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 4 q^{5} + 4 q^{7} + 8 q^{11} - 4 q^{17} + 4 q^{23} - 4 q^{25} + 36 q^{35} - 28 q^{43} - 40 q^{45} + 20 q^{47} - 16 q^{55} + 24 q^{57} + 16 q^{61} + 20 q^{63} - 76 q^{73} - 16 q^{77} + 4 q^{81} + 84 q^{83} - 76 q^{85} - 80 q^{87} + 8 q^{93} - 16 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 4 x^{11} + 28 x^{10} - 64 x^{9} + 236 x^{8} - 420 x^{7} + 946 x^{6} - 1216 x^{5} + 1896 x^{4} + \cdots + 1370 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 43066472536 \nu^{11} - 449979675808 \nu^{10} + 2364419690616 \nu^{9} + \cdots - 279575999635621 ) / 79197078563257 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 20904914650254 \nu^{11} + 151211473768106 \nu^{10} - 794968673642910 \nu^{9} + \cdots + 61\!\cdots\!02 ) / 22\!\cdots\!17 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 1434891834382 \nu^{11} + 4349786383365 \nu^{10} - 30559554581954 \nu^{9} + \cdots - 614674855329394 ) / 11\!\cdots\!43 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 36029778947630 \nu^{11} + 211750342123647 \nu^{10} + \cdots + 82\!\cdots\!06 ) / 22\!\cdots\!17 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 109895826 \nu^{11} - 371224964 \nu^{10} + 2654024378 \nu^{9} - 4749330090 \nu^{8} + \cdots + 32432735911 ) / 57697329013 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 46589648822398 \nu^{11} - 179791490723729 \nu^{10} + \cdots + 68\!\cdots\!68 ) / 22\!\cdots\!17 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 176200650116 \nu^{11} - 1329132544436 \nu^{10} + 6501003581929 \nu^{9} + \cdots - 625898181929096 ) / 79197078563257 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 60614681294844 \nu^{11} - 176924960309206 \nu^{10} + \cdots - 21\!\cdots\!53 ) / 22\!\cdots\!17 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 256791075958 \nu^{11} - 2213799473497 \nu^{10} + 10545830020631 \nu^{9} + \cdots - 965449441292380 ) / 79197078563257 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 115790335988434 \nu^{11} + 395417420255441 \nu^{10} + \cdots + 88\!\cdots\!30 ) / 22\!\cdots\!17 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 120425382922218 \nu^{11} + 468739736789324 \nu^{10} + \cdots - 68\!\cdots\!02 ) / 22\!\cdots\!17 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{11} + \beta_{10} + \beta_{9} + \beta_{8} - \beta_{7} + 2\beta_{6} - 3\beta_{5} + 2\beta_{2} - \beta _1 - 7 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - \beta_{10} + 3 \beta_{9} - 2 \beta_{8} + 3 \beta_{6} + 12 \beta_{5} + 10 \beta_{4} + 15 \beta_{3} + \cdots - 18 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 8 \beta_{11} - 16 \beta_{10} - 4 \beta_{9} - 30 \beta_{8} + 4 \beta_{7} - 37 \beta_{6} + 84 \beta_{5} + \cdots + 20 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 2 \beta_{11} - 10 \beta_{10} - 55 \beta_{9} - 21 \beta_{8} + 5 \beta_{7} - 150 \beta_{6} + 37 \beta_{5} + \cdots + 273 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 32 \beta_{11} + 99 \beta_{10} - 51 \beta_{9} + 183 \beta_{8} + 8 \beta_{7} + 99 \beta_{6} + \cdots + 238 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 100 \beta_{11} + 586 \beta_{10} + 532 \beta_{9} + 1074 \beta_{8} - 70 \beta_{7} + 2615 \beta_{6} + \cdots - 2648 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 176 \beta_{11} - 616 \beta_{10} + 1388 \beta_{9} - 1120 \beta_{8} - 292 \beta_{7} + 1768 \beta_{6} + \cdots - 6723 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 993 \beta_{11} - 5524 \beta_{10} - 348 \beta_{9} - 10000 \beta_{8} - 13512 \beta_{6} + 26866 \beta_{5} + \cdots + 1838 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 877 \beta_{11} - 5469 \beta_{10} - 19989 \beta_{9} - 9929 \beta_{8} + 4117 \beta_{7} - 57259 \beta_{6} + \cdots + 97170 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 23101 \beta_{11} + 124939 \beta_{10} - 88121 \beta_{9} + 226350 \beta_{8} + 17347 \beta_{7} + \cdots + 427540 ) / 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/380\mathbb{Z}\right)^\times\).

\(n\) \(21\) \(77\) \(191\)
\(\chi(n)\) \(-1\) \(\beta_{5}\) \(1\)

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
1.24060 + 1.01288i
−0.585043 + 2.22350i
0.344446 + 1.15131i
0.344446 1.84020i
−0.585043 1.05342i
1.24060 3.49408i
1.24060 1.01288i
−0.585043 2.22350i
0.344446 1.15131i
0.344446 + 1.84020i
−0.585043 + 1.05342i
1.24060 + 3.49408i
0 −2.25348 2.25348i 0 −1.48119 + 1.67513i 0 −1.48119 1.48119i 0 7.15633i 0
37.2 0 −1.63846 1.63846i 0 2.17009 + 0.539189i 0 2.17009 + 2.17009i 0 2.36910i 0
37.3 0 −1.49576 1.49576i 0 0.311108 2.21432i 0 0.311108 + 0.311108i 0 1.47457i 0
37.4 0 1.49576 + 1.49576i 0 0.311108 2.21432i 0 0.311108 + 0.311108i 0 1.47457i 0
37.5 0 1.63846 + 1.63846i 0 2.17009 + 0.539189i 0 2.17009 + 2.17009i 0 2.36910i 0
37.6 0 2.25348 + 2.25348i 0 −1.48119 + 1.67513i 0 −1.48119 1.48119i 0 7.15633i 0
113.1 0 −2.25348 + 2.25348i 0 −1.48119 1.67513i 0 −1.48119 + 1.48119i 0 7.15633i 0
113.2 0 −1.63846 + 1.63846i 0 2.17009 0.539189i 0 2.17009 2.17009i 0 2.36910i 0
113.3 0 −1.49576 + 1.49576i 0 0.311108 + 2.21432i 0 0.311108 0.311108i 0 1.47457i 0
113.4 0 1.49576 1.49576i 0 0.311108 + 2.21432i 0 0.311108 0.311108i 0 1.47457i 0
113.5 0 1.63846 1.63846i 0 2.17009 0.539189i 0 2.17009 2.17009i 0 2.36910i 0
113.6 0 2.25348 2.25348i 0 −1.48119 1.67513i 0 −1.48119 + 1.48119i 0 7.15633i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 37.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
19.b odd 2 1 inner
95.g even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 380.2.l.b 12
3.b odd 2 1 3420.2.bb.d 12
5.b even 2 1 1900.2.l.b 12
5.c odd 4 1 inner 380.2.l.b 12
5.c odd 4 1 1900.2.l.b 12
15.e even 4 1 3420.2.bb.d 12
19.b odd 2 1 inner 380.2.l.b 12
57.d even 2 1 3420.2.bb.d 12
95.d odd 2 1 1900.2.l.b 12
95.g even 4 1 inner 380.2.l.b 12
95.g even 4 1 1900.2.l.b 12
285.j odd 4 1 3420.2.bb.d 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
380.2.l.b 12 1.a even 1 1 trivial
380.2.l.b 12 5.c odd 4 1 inner
380.2.l.b 12 19.b odd 2 1 inner
380.2.l.b 12 95.g even 4 1 inner
1900.2.l.b 12 5.b even 2 1
1900.2.l.b 12 5.c odd 4 1
1900.2.l.b 12 95.d odd 2 1
1900.2.l.b 12 95.g even 4 1
3420.2.bb.d 12 3.b odd 2 1
3420.2.bb.d 12 15.e even 4 1
3420.2.bb.d 12 57.d even 2 1
3420.2.bb.d 12 285.j odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{12} + 152T_{3}^{8} + 5616T_{3}^{4} + 59536 \) acting on \(S_{2}^{\mathrm{new}}(380, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} + 152 T^{8} + \cdots + 59536 \) Copy content Toggle raw display
$5$ \( (T^{6} - 2 T^{5} + \cdots + 125)^{2} \) Copy content Toggle raw display
$7$ \( (T^{6} - 2 T^{5} + 2 T^{4} + \cdots + 8)^{2} \) Copy content Toggle raw display
$11$ \( (T^{3} - 2 T^{2} - 4 T + 4)^{4} \) Copy content Toggle raw display
$13$ \( T^{12} + 1816 T^{8} + \cdots + 59536 \) Copy content Toggle raw display
$17$ \( (T^{6} + 2 T^{5} + \cdots + 1352)^{2} \) Copy content Toggle raw display
$19$ \( T^{12} - 6 T^{10} + \cdots + 47045881 \) Copy content Toggle raw display
$23$ \( (T^{6} - 2 T^{5} + \cdots + 5000)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} - 40 T^{4} + \cdots - 1952)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} + 96 T^{4} + \cdots + 1952)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} + 20536 T^{8} + \cdots + 59536 \) Copy content Toggle raw display
$41$ \( (T^{6} + 144 T^{4} + \cdots + 7808)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} + 14 T^{5} + \cdots + 14792)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} - 10 T^{5} + \cdots + 2312)^{2} \) Copy content Toggle raw display
$53$ \( T^{12} + 29016 T^{8} + \cdots + 37210000 \) Copy content Toggle raw display
$59$ \( (T^{6} - 448 T^{4} + \cdots - 2818688)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} - 4 T^{2} - 4 T + 20)^{4} \) Copy content Toggle raw display
$67$ \( T^{12} + 16568 T^{8} + \cdots + 59536 \) Copy content Toggle raw display
$71$ \( (T^{6} + 104 T^{4} + \cdots + 1952)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} + 38 T^{5} + \cdots + 390728)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} - 184 T^{4} + \cdots - 7808)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} - 42 T^{5} + \cdots + 803912)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} - 288 T^{4} + \cdots - 329888)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 46306881767056 \) Copy content Toggle raw display
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