Properties

Label 180.12.a.c
Level $180$
Weight $12$
Character orbit 180.a
Self dual yes
Analytic conductor $138.302$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [180,12,Mod(1,180)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(180, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 12, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("180.1");
 
S:= CuspForms(chi, 12);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 180 = 2^{2} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 180.a (trivial)

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: yes
Analytic conductor: \(138.301772501\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{46729}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 11682 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2}\cdot 3 \)
Twist minimal: no (minimal twist has level 20)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(\beta = 6\sqrt{46729}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 3125 q^{5} + ( - 37 \beta + 1670) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - 3125 q^{5} + ( - 37 \beta + 1670) q^{7} + ( - 130 \beta - 149160) q^{11} + (1052 \beta + 604910) q^{13} + ( - 1676 \beta - 4528170) q^{17} + ( - 5060 \beta + 9719684) q^{19} + (2179 \beta - 27963210) q^{23} + 9765625 q^{25} + ( - 112280 \beta - 20920854) q^{29} + ( - 197750 \beta + 31115396) q^{31} + (115625 \beta - 5218750) q^{35} + (243768 \beta - 364617970) q^{37} + (209820 \beta + 304209534) q^{41} + (582905 \beta - 564720370) q^{43} + ( - 1060891 \beta + 826536450) q^{47} + ( - 123580 \beta + 328454193) q^{49} + (1305356 \beta - 2862443670) q^{53} + (406250 \beta + 466125000) q^{55} + ( - 2765480 \beta - 1878216948) q^{59} + (6912240 \beta + 2461851782) q^{61} + ( - 3287500 \beta - 1890343750) q^{65} + ( - 2305557 \beta - 2921622070) q^{67} + ( - 2955090 \beta + 21676081332) q^{71} + (3775452 \beta + 16823549810) q^{73} + (5301820 \beta + 7842496440) q^{77} + ( - 13660620 \beta - 27399712648) q^{79} + ( - 4718141 \beta + 4651516050) q^{83} + (5237500 \beta + 14150531250) q^{85} + ( - 56016840 \beta - 1844484186) q^{89} + ( - 20624830 \beta - 64469465756) q^{91} + (15812500 \beta - 30374012500) q^{95} + (37214548 \beta + 32946078890) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 6250 q^{5} + 3340 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 6250 q^{5} + 3340 q^{7} - 298320 q^{11} + 1209820 q^{13} - 9056340 q^{17} + 19439368 q^{19} - 55926420 q^{23} + 19531250 q^{25} - 41841708 q^{29} + 62230792 q^{31} - 10437500 q^{35} - 729235940 q^{37} + 608419068 q^{41} - 1129440740 q^{43} + 1653072900 q^{47} + 656908386 q^{49} - 5724887340 q^{53} + 932250000 q^{55} - 3756433896 q^{59} + 4923703564 q^{61} - 3780687500 q^{65} - 5843244140 q^{67} + 43352162664 q^{71} + 33647099620 q^{73} + 15684992880 q^{77} - 54799425296 q^{79} + 9303032100 q^{83} + 28301062500 q^{85} - 3688968372 q^{89} - 128938931512 q^{91} - 60748025000 q^{95} + 65892157780 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
108.584
−107.584
0 0 0 −3125.00 0 −46319.5 0 0 0
1.2 0 0 0 −3125.00 0 49659.5 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(5\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 180.12.a.c 2
3.b odd 2 1 20.12.a.b 2
12.b even 2 1 80.12.a.i 2
15.d odd 2 1 100.12.a.c 2
15.e even 4 2 100.12.c.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.12.a.b 2 3.b odd 2 1
80.12.a.i 2 12.b even 2 1
100.12.a.c 2 15.d odd 2 1
100.12.c.c 4 15.e even 4 2
180.12.a.c 2 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(180))\):

\( T_{7}^{2} - 3340T_{7} - 2300203136 \) Copy content Toggle raw display
\( T_{11}^{2} + 298320T_{11} - 6181218000 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( (T + 3125)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 3340 T - 2300203136 \) Copy content Toggle raw display
$11$ \( T^{2} + 298320 T - 6181218000 \) Copy content Toggle raw display
$13$ \( T^{2} - 1209820 T - 1495830055676 \) Copy content Toggle raw display
$17$ \( T^{2} + 9056340 T + 15778940526756 \) Copy content Toggle raw display
$19$ \( T^{2} - 19439368 T + 51400754581456 \) Copy content Toggle raw display
$23$ \( T^{2} + \cdots + 773953750020096 \) Copy content Toggle raw display
$29$ \( T^{2} + 41841708 T - 20\!\cdots\!84 \) Copy content Toggle raw display
$31$ \( T^{2} - 62230792 T - 64\!\cdots\!84 \) Copy content Toggle raw display
$37$ \( T^{2} + 729235940 T + 32\!\cdots\!44 \) Copy content Toggle raw display
$41$ \( T^{2} - 608419068 T + 18\!\cdots\!56 \) Copy content Toggle raw display
$43$ \( T^{2} + 1129440740 T - 25\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{2} - 1653072900 T - 12\!\cdots\!64 \) Copy content Toggle raw display
$53$ \( T^{2} + 5724887340 T + 53\!\cdots\!16 \) Copy content Toggle raw display
$59$ \( T^{2} + 3756433896 T - 93\!\cdots\!96 \) Copy content Toggle raw display
$61$ \( T^{2} - 4923703564 T - 74\!\cdots\!76 \) Copy content Toggle raw display
$67$ \( T^{2} + 5843244140 T - 40\!\cdots\!56 \) Copy content Toggle raw display
$71$ \( T^{2} - 43352162664 T + 45\!\cdots\!24 \) Copy content Toggle raw display
$73$ \( T^{2} - 33647099620 T + 25\!\cdots\!24 \) Copy content Toggle raw display
$79$ \( T^{2} + 54799425296 T + 43\!\cdots\!04 \) Copy content Toggle raw display
$83$ \( T^{2} - 9303032100 T - 15\!\cdots\!64 \) Copy content Toggle raw display
$89$ \( T^{2} + 3688968372 T - 52\!\cdots\!04 \) Copy content Toggle raw display
$97$ \( T^{2} - 65892157780 T - 12\!\cdots\!76 \) Copy content Toggle raw display
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