Properties

Label 240.12.a.j
Level $240$
Weight $12$
Character orbit 240.a
Self dual yes
Analytic conductor $184.402$
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] = [240,12,Mod(1,240)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(240, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("240.1");
 
S:= CuspForms(chi, 12);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 240 = 2^{4} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 240.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: \(184.402363334\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{1609}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 402 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 15)
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 \(\beta = 4\sqrt{1609}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 243 q^{3} - 3125 q^{5} + ( - 319 \beta + 5432) q^{7} + 59049 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 243 q^{3} - 3125 q^{5} + ( - 319 \beta + 5432) q^{7} + 59049 q^{9} + (2596 \beta + 180896) q^{11} + (479 \beta - 1066866) q^{13} + 759375 q^{15} + ( - 16885 \beta - 3902294) q^{17} + ( - 52091 \beta + 7781112) q^{19} + (77517 \beta - 1319976) q^{21} + (173007 \beta + 18725124) q^{23} + 9765625 q^{25} - 14348907 q^{27} + ( - 772448 \beta - 35160334) q^{29} + (278147 \beta - 149292436) q^{31} + ( - 630828 \beta - 43957728) q^{33} + (996875 \beta - 16975000) q^{35} + (2256759 \beta + 118000478) q^{37} + ( - 116397 \beta + 259248438) q^{39} + (5581978 \beta - 232471294) q^{41} + (10266664 \beta + 121104300) q^{43} - 184528125 q^{45} + ( - 4951583 \beta + 2187898460) q^{47} + ( - 3465616 \beta + 671915065) q^{49} + (4103055 \beta + 948257442) q^{51} + ( - 7407982 \beta - 1094770694) q^{53} + ( - 8112500 \beta - 565300000) q^{55} + (12658113 \beta - 1890810216) q^{57} + ( - 36574724 \beta + 2740192928) q^{59} + ( - 3506734 \beta + 7278951990) q^{61} + ( - 18836631 \beta + 320754168) q^{63} + ( - 1496875 \beta + 3333956250) q^{65} + ( - 12642520 \beta + 7959194444) q^{67} + ( - 42040701 \beta - 4550205132) q^{69} + (73932560 \beta - 560280512) q^{71} + (17882198 \beta - 12260787174) q^{73} - 2373046875 q^{75} + ( - 43604352 \beta - 20336597184) q^{77} + ( - 22619515 \beta + 39621527780) q^{79} + 3486784401 q^{81} + ( - 345808512 \beta - 4622613348) q^{83} + (52765625 \beta + 12194668750) q^{85} + (187704864 \beta + 8543961162) q^{87} + ( - 78078714 \beta + 11058660618) q^{89} + (342932182 \beta - 9728925056) q^{91} + ( - 67589721 \beta + 36278061948) q^{93} + (162784375 \beta - 24315975000) q^{95} + (31846320 \beta - 80181836734) q^{97} + (153291204 \beta + 10681727904) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 486 q^{3} - 6250 q^{5} + 10864 q^{7} + 118098 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 486 q^{3} - 6250 q^{5} + 10864 q^{7} + 118098 q^{9} + 361792 q^{11} - 2133732 q^{13} + 1518750 q^{15} - 7804588 q^{17} + 15562224 q^{19} - 2639952 q^{21} + 37450248 q^{23} + 19531250 q^{25} - 28697814 q^{27} - 70320668 q^{29} - 298584872 q^{31} - 87915456 q^{33} - 33950000 q^{35} + 236000956 q^{37} + 518496876 q^{39} - 464942588 q^{41} + 242208600 q^{43} - 369056250 q^{45} + 4375796920 q^{47} + 1343830130 q^{49} + 1896514884 q^{51} - 2189541388 q^{53} - 1130600000 q^{55} - 3781620432 q^{57} + 5480385856 q^{59} + 14557903980 q^{61} + 641508336 q^{63} + 6667912500 q^{65} + 15918388888 q^{67} - 9100410264 q^{69} - 1120561024 q^{71} - 24521574348 q^{73} - 4746093750 q^{75} - 40673194368 q^{77} + 79243055560 q^{79} + 6973568802 q^{81} - 9245226696 q^{83} + 24389337500 q^{85} + 17087922324 q^{87} + 22117321236 q^{89} - 19457850112 q^{91} + 72556123896 q^{93} - 48631950000 q^{95} - 160363673468 q^{97} + 21363455808 q^{99}+O(q^{100}) \) 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
20.5562
−19.5562
0 −243.000 0 −3125.00 0 −45751.3 0 59049.0 0
1.2 0 −243.000 0 −3125.00 0 56615.3 0 59049.0 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 240.12.a.j 2
4.b odd 2 1 15.12.a.b 2
12.b even 2 1 45.12.a.e 2
20.d odd 2 1 75.12.a.d 2
20.e even 4 2 75.12.b.c 4
60.h even 2 1 225.12.a.g 2
60.l odd 4 2 225.12.b.h 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.12.a.b 2 4.b odd 2 1
45.12.a.e 2 12.b even 2 1
75.12.a.d 2 20.d odd 2 1
75.12.b.c 4 20.e even 4 2
225.12.a.g 2 60.h even 2 1
225.12.b.h 4 60.l odd 4 2
240.12.a.j 2 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{2} - 10864T_{7} - 2590228560 \) acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(240))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( (T + 243)^{2} \) Copy content Toggle raw display
$5$ \( (T + 3125)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 10864 T - 2590228560 \) Copy content Toggle raw display
$11$ \( T^{2} - 361792 T - 140771013888 \) Copy content Toggle raw display
$13$ \( T^{2} + 2133732 T + 1132296332852 \) Copy content Toggle raw display
$17$ \( T^{2} + 7804588 T + 7888201038036 \) Copy content Toggle raw display
$19$ \( T^{2} - 15562224 T - 9309926445520 \) Copy content Toggle raw display
$23$ \( T^{2} + \cdots - 419924260414080 \) Copy content Toggle raw display
$29$ \( T^{2} + 70320668 T - 14\!\cdots\!20 \) Copy content Toggle raw display
$31$ \( T^{2} + 298584872 T + 20\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{2} - 236000956 T - 11\!\cdots\!80 \) Copy content Toggle raw display
$41$ \( T^{2} + 464942588 T - 74\!\cdots\!60 \) Copy content Toggle raw display
$43$ \( T^{2} - 242208600 T - 26\!\cdots\!24 \) Copy content Toggle raw display
$47$ \( T^{2} - 4375796920 T + 41\!\cdots\!84 \) Copy content Toggle raw display
$53$ \( T^{2} + 2189541388 T - 21\!\cdots\!20 \) Copy content Toggle raw display
$59$ \( T^{2} - 5480385856 T - 26\!\cdots\!60 \) Copy content Toggle raw display
$61$ \( T^{2} - 14557903980 T + 52\!\cdots\!36 \) Copy content Toggle raw display
$67$ \( T^{2} - 15918388888 T + 59\!\cdots\!36 \) Copy content Toggle raw display
$71$ \( T^{2} + 1120561024 T - 14\!\cdots\!56 \) Copy content Toggle raw display
$73$ \( T^{2} + 24521574348 T + 14\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{2} - 79243055560 T + 15\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{2} + 9245226696 T - 30\!\cdots\!32 \) Copy content Toggle raw display
$89$ \( T^{2} - 22117321236 T - 34\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{2} + 160363673468 T + 64\!\cdots\!56 \) Copy content Toggle raw display
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