Properties

Label 1232.4.a.o
Level $1232$
Weight $4$
Character orbit 1232.a
Self dual yes
Analytic conductor $72.690$
Analytic rank $1$
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] = [1232,4,Mod(1,1232)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1232, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1232.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1232 = 2^{4} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1232.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: \(72.6903531271\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{57}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 14 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 154)
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 \(\beta = \frac{1}{2}(1 + \sqrt{57})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta + 3) q^{3} + ( - 3 \beta - 7) q^{5} + 7 q^{7} + ( - 5 \beta - 4) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta + 3) q^{3} + ( - 3 \beta - 7) q^{5} + 7 q^{7} + ( - 5 \beta - 4) q^{9} + 11 q^{11} + (2 \beta - 44) q^{13} + (\beta + 21) q^{15} + (26 \beta - 46) q^{17} + (4 \beta + 66) q^{19} + ( - 7 \beta + 21) q^{21} + (21 \beta - 19) q^{23} + (51 \beta + 50) q^{25} + (21 \beta - 23) q^{27} + (48 \beta + 38) q^{29} + (9 \beta + 31) q^{31} + ( - 11 \beta + 33) q^{33} + ( - 21 \beta - 49) q^{35} + ( - 17 \beta - 59) q^{37} + (48 \beta - 160) q^{39} + ( - 46 \beta - 286) q^{41} + ( - 52 \beta - 276) q^{43} + (62 \beta + 238) q^{45} + (134 \beta - 68) q^{47} + 49 q^{49} + (98 \beta - 502) q^{51} + ( - 172 \beta + 162) q^{53} + ( - 33 \beta - 77) q^{55} + ( - 58 \beta + 142) q^{57} + ( - 115 \beta + 229) q^{59} + (184 \beta + 98) q^{61} + ( - 35 \beta - 28) q^{63} + (112 \beta + 224) q^{65} + (31 \beta - 529) q^{67} + (61 \beta - 351) q^{69} + ( - 45 \beta + 607) q^{71} + ( - 42 \beta - 26) q^{73} + (52 \beta - 564) q^{75} + 77 q^{77} + (50 \beta - 446) q^{79} + (200 \beta - 255) q^{81} + ( - 42 \beta + 764) q^{83} + ( - 122 \beta - 770) q^{85} + (58 \beta - 558) q^{87} + ( - 119 \beta - 121) q^{89} + (14 \beta - 308) q^{91} + ( - 13 \beta - 33) q^{93} + ( - 238 \beta - 630) q^{95} + (55 \beta + 85) q^{97} + ( - 55 \beta - 44) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 5 q^{3} - 17 q^{5} + 14 q^{7} - 13 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 5 q^{3} - 17 q^{5} + 14 q^{7} - 13 q^{9} + 22 q^{11} - 86 q^{13} + 43 q^{15} - 66 q^{17} + 136 q^{19} + 35 q^{21} - 17 q^{23} + 151 q^{25} - 25 q^{27} + 124 q^{29} + 71 q^{31} + 55 q^{33} - 119 q^{35} - 135 q^{37} - 272 q^{39} - 618 q^{41} - 604 q^{43} + 538 q^{45} - 2 q^{47} + 98 q^{49} - 906 q^{51} + 152 q^{53} - 187 q^{55} + 226 q^{57} + 343 q^{59} + 380 q^{61} - 91 q^{63} + 560 q^{65} - 1027 q^{67} - 641 q^{69} + 1169 q^{71} - 94 q^{73} - 1076 q^{75} + 154 q^{77} - 842 q^{79} - 310 q^{81} + 1486 q^{83} - 1662 q^{85} - 1058 q^{87} - 361 q^{89} - 602 q^{91} - 79 q^{93} - 1498 q^{95} + 225 q^{97} - 143 q^{99}+O(q^{100}) \) 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
4.27492
−3.27492
0 −1.27492 0 −19.8248 0 7.00000 0 −25.3746 0
1.2 0 6.27492 0 2.82475 0 7.00000 0 12.3746 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(7\) \( -1 \)
\(11\) \( -1 \)

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 1232.4.a.o 2
4.b odd 2 1 154.4.a.h 2
12.b even 2 1 1386.4.a.s 2
28.d even 2 1 1078.4.a.o 2
44.c even 2 1 1694.4.a.h 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
154.4.a.h 2 4.b odd 2 1
1078.4.a.o 2 28.d even 2 1
1232.4.a.o 2 1.a even 1 1 trivial
1386.4.a.s 2 12.b even 2 1
1694.4.a.h 2 44.c even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1232))\):

\( T_{3}^{2} - 5T_{3} - 8 \) Copy content Toggle raw display
\( T_{5}^{2} + 17T_{5} - 56 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 5T - 8 \) Copy content Toggle raw display
$5$ \( T^{2} + 17T - 56 \) Copy content Toggle raw display
$7$ \( (T - 7)^{2} \) Copy content Toggle raw display
$11$ \( (T - 11)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 86T + 1792 \) Copy content Toggle raw display
$17$ \( T^{2} + 66T - 8544 \) Copy content Toggle raw display
$19$ \( T^{2} - 136T + 4396 \) Copy content Toggle raw display
$23$ \( T^{2} + 17T - 6212 \) Copy content Toggle raw display
$29$ \( T^{2} - 124T - 28988 \) Copy content Toggle raw display
$31$ \( T^{2} - 71T + 106 \) Copy content Toggle raw display
$37$ \( T^{2} + 135T + 438 \) Copy content Toggle raw display
$41$ \( T^{2} + 618T + 65328 \) Copy content Toggle raw display
$43$ \( T^{2} + 604T + 52672 \) Copy content Toggle raw display
$47$ \( T^{2} + 2T - 255872 \) Copy content Toggle raw display
$53$ \( T^{2} - 152T - 415796 \) Copy content Toggle raw display
$59$ \( T^{2} - 343T - 159044 \) Copy content Toggle raw display
$61$ \( T^{2} - 380T - 446348 \) Copy content Toggle raw display
$67$ \( T^{2} + 1027 T + 249988 \) Copy content Toggle raw display
$71$ \( T^{2} - 1169 T + 312784 \) Copy content Toggle raw display
$73$ \( T^{2} + 94T - 22928 \) Copy content Toggle raw display
$79$ \( T^{2} + 842T + 141616 \) Copy content Toggle raw display
$83$ \( T^{2} - 1486 T + 526912 \) Copy content Toggle raw display
$89$ \( T^{2} + 361T - 169214 \) Copy content Toggle raw display
$97$ \( T^{2} - 225T - 30450 \) Copy content Toggle raw display
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