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Properties

Label 240.3.c.e
Level $240$
Weight $3$
Character orbit 240.c
Analytic conductor $6.540$
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] = [240,3,Mod(209,240)]
 
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, 1, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("240.209");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 240 = 2^{4} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 240.c (of order \(2\), degree \(1\), not 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: \(6.53952634465\)
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: \( x^{12} + 34x^{10} + 305x^{8} + 616x^{6} + 305x^{4} + 34x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{15}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 120)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + \beta_{7} q^{5} - \beta_{9} q^{7} + ( - \beta_{4} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + \beta_{7} q^{5} - \beta_{9} q^{7} + ( - \beta_{4} + 1) q^{9} + (\beta_{4} - \beta_{3}) q^{11} + ( - \beta_{11} - \beta_{10} + \cdots - \beta_1) q^{13}+ \cdots + ( - 4 \beta_{11} + 4 \beta_{10} + \cdots + 46) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 8 q^{9} - 16 q^{15} + 4 q^{21} + 36 q^{25} + 48 q^{31} + 128 q^{39} - 68 q^{45} - 252 q^{49} - 48 q^{51} + 48 q^{55} + 144 q^{61} + 268 q^{69} - 304 q^{75} - 432 q^{79} - 188 q^{81} + 336 q^{85} + 560 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 34x^{10} + 305x^{8} + 616x^{6} + 305x^{4} + 34x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{11} - \nu^{10} + 35 \nu^{9} - 35 \nu^{8} + 340 \nu^{7} - 340 \nu^{6} + 956 \nu^{5} - 956 \nu^{4} + \cdots - 647 ) / 216 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - \nu^{11} - \nu^{10} - 35 \nu^{9} - 35 \nu^{8} - 340 \nu^{7} - 340 \nu^{6} - 956 \nu^{5} - 956 \nu^{4} + \cdots - 647 ) / 72 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{11} - 4 \nu^{10} + 25 \nu^{9} - 138 \nu^{8} + 4 \nu^{7} - 1288 \nu^{6} - 1964 \nu^{5} - 3072 \nu^{4} + \cdots - 282 ) / 36 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 5 \nu^{11} - 4 \nu^{10} + 173 \nu^{9} - 138 \nu^{8} + 1628 \nu^{7} - 1288 \nu^{6} + 4028 \nu^{5} + \cdots - 282 ) / 36 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -35\nu^{10} - 1189\nu^{8} - 10640\nu^{6} - 21220\nu^{4} - 9755\nu^{2} - 577 ) / 27 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 3 \nu^{11} + 64 \nu^{10} + 99 \nu^{9} + 2172 \nu^{8} + 816 \nu^{7} + 19384 \nu^{6} + 1032 \nu^{5} + \cdots + 1038 ) / 18 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 137 \nu^{11} - 26 \nu^{10} - 4649 \nu^{9} - 880 \nu^{8} - 41480 \nu^{7} - 7796 \nu^{6} - 81680 \nu^{5} + \cdots - 280 ) / 36 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 137 \nu^{11} - 26 \nu^{10} + 4649 \nu^{9} - 880 \nu^{8} + 41480 \nu^{7} - 7796 \nu^{6} + 81680 \nu^{5} + \cdots - 280 ) / 36 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -655\nu^{11} - 22205\nu^{9} - 197572\nu^{7} - 383900\nu^{5} - 162043\nu^{3} - 7337\nu ) / 108 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 713 \nu^{11} - 167 \nu^{10} - 24191 \nu^{9} - 5665 \nu^{8} - 215732 \nu^{7} - 50492 \nu^{6} + \cdots - 2461 ) / 72 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 733 \nu^{11} + 73 \nu^{10} - 24883 \nu^{9} + 2471 \nu^{8} - 222244 \nu^{7} + 21892 \nu^{6} + \cdots + 491 ) / 72 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( - 2 \beta_{11} - 4 \beta_{10} + 6 \beta_{9} - 3 \beta_{8} + 3 \beta_{7} - \beta_{6} + 3 \beta_{5} + \cdots + 1 ) / 48 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{11} - \beta_{10} - 4\beta_{6} - 9\beta_{5} + 6\beta_{4} + 4\beta_{3} - 12\beta_{2} - 36\beta _1 - 140 ) / 24 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 64 \beta_{11} + 86 \beta_{10} - 114 \beta_{9} + 105 \beta_{8} - 105 \beta_{7} + 11 \beta_{6} + \cdots - 11 ) / 48 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 6\beta_{8} + 6\beta_{7} + 18\beta_{6} + 45\beta_{5} - 18\beta_{4} - 18\beta_{3} + 36\beta_{2} + 108\beta _1 + 382 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 1466 \beta_{11} - 1744 \beta_{10} + 1950 \beta_{9} - 2649 \beta_{8} + 2649 \beta_{7} - 139 \beta_{6} + \cdots + 139 ) / 48 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 305 \beta_{11} + 305 \beta_{10} - 1296 \beta_{8} - 1296 \beta_{7} - 2452 \beta_{6} - 6345 \beta_{5} + \cdots - 42620 ) / 24 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 31516 \beta_{11} + 35630 \beta_{10} - 35322 \beta_{9} + 59883 \beta_{8} - 59883 \beta_{7} + 2057 \beta_{6} + \cdots - 2057 ) / 48 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 405 \beta_{11} - 405 \beta_{10} + 1380 \beta_{8} + 1380 \beta_{7} + 2205 \beta_{6} + 5787 \beta_{5} + \cdots + 34642 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 663278 \beta_{11} - 731260 \beta_{10} + 674682 \beta_{9} - 1291845 \beta_{8} + 1291845 \beta_{7} + \cdots + 33991 ) / 48 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 237761 \beta_{11} + 237761 \beta_{10} - 752976 \beta_{8} - 752976 \beta_{7} - 1116724 \beta_{6} + \cdots - 16638620 ) / 24 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 13822672 \beta_{11} + 15043898 \beta_{10} - 13332606 \beta_{9} + 27258279 \beta_{8} - 27258279 \beta_{7} + \cdots - 610613 ) / 48 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/240\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(97\) \(161\) \(181\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
209.1
− 0.220185i
0.220185i
− 0.304307i
0.304307i
− 0.723561i
0.723561i
− 1.38205i
1.38205i
− 3.28615i
3.28615i
− 4.54164i
4.54164i
0 −2.72256 − 1.26002i 0 −0.689011 + 4.95230i 0 − 0.735748i 0 5.82469 + 6.86097i 0
209.2 0 −2.72256 + 1.26002i 0 −0.689011 − 4.95230i 0 0.735748i 0 5.82469 − 6.86097i 0
209.3 0 −2.49147 − 1.67109i 0 4.19906 − 2.71439i 0 12.7692i 0 3.41489 + 8.32698i 0
209.4 0 −2.49147 + 1.67109i 0 4.19906 + 2.71439i 0 − 12.7692i 0 3.41489 − 8.32698i 0
209.5 0 −0.938195 − 2.84952i 0 −4.88807 + 1.05205i 0 − 6.81219i 0 −7.23958 + 5.34682i 0
209.6 0 −0.938195 + 2.84952i 0 −4.88807 − 1.05205i 0 6.81219i 0 −7.23958 − 5.34682i 0
209.7 0 0.938195 − 2.84952i 0 4.88807 − 1.05205i 0 − 6.81219i 0 −7.23958 − 5.34682i 0
209.8 0 0.938195 + 2.84952i 0 4.88807 + 1.05205i 0 6.81219i 0 −7.23958 + 5.34682i 0
209.9 0 2.49147 − 1.67109i 0 −4.19906 + 2.71439i 0 12.7692i 0 3.41489 − 8.32698i 0
209.10 0 2.49147 + 1.67109i 0 −4.19906 − 2.71439i 0 − 12.7692i 0 3.41489 + 8.32698i 0
209.11 0 2.72256 − 1.26002i 0 0.689011 − 4.95230i 0 − 0.735748i 0 5.82469 − 6.86097i 0
209.12 0 2.72256 + 1.26002i 0 0.689011 + 4.95230i 0 0.735748i 0 5.82469 + 6.86097i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 209.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
15.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 240.3.c.e 12
3.b odd 2 1 inner 240.3.c.e 12
4.b odd 2 1 120.3.c.a ✓ 12
5.b even 2 1 inner 240.3.c.e 12
5.c odd 4 2 1200.3.l.y 12
8.b even 2 1 960.3.c.j 12
8.d odd 2 1 960.3.c.k 12
12.b even 2 1 120.3.c.a ✓ 12
15.d odd 2 1 inner 240.3.c.e 12
15.e even 4 2 1200.3.l.y 12
20.d odd 2 1 120.3.c.a ✓ 12
20.e even 4 2 600.3.l.g 12
24.f even 2 1 960.3.c.k 12
24.h odd 2 1 960.3.c.j 12
40.e odd 2 1 960.3.c.k 12
40.f even 2 1 960.3.c.j 12
60.h even 2 1 120.3.c.a ✓ 12
60.l odd 4 2 600.3.l.g 12
120.i odd 2 1 960.3.c.j 12
120.m even 2 1 960.3.c.k 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.3.c.a ✓ 12 4.b odd 2 1
120.3.c.a ✓ 12 12.b even 2 1
120.3.c.a ✓ 12 20.d odd 2 1
120.3.c.a ✓ 12 60.h even 2 1
240.3.c.e 12 1.a even 1 1 trivial
240.3.c.e 12 3.b odd 2 1 inner
240.3.c.e 12 5.b even 2 1 inner
240.3.c.e 12 15.d odd 2 1 inner
600.3.l.g 12 20.e even 4 2
600.3.l.g 12 60.l odd 4 2
960.3.c.j 12 8.b even 2 1
960.3.c.j 12 24.h odd 2 1
960.3.c.j 12 40.f even 2 1
960.3.c.j 12 120.i odd 2 1
960.3.c.k 12 8.d odd 2 1
960.3.c.k 12 24.f even 2 1
960.3.c.k 12 40.e odd 2 1
960.3.c.k 12 120.m even 2 1
1200.3.l.y 12 5.c odd 4 2
1200.3.l.y 12 15.e even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(240, [\chi])\):

\( T_{7}^{6} + 210T_{7}^{4} + 7680T_{7}^{2} + 4096 \) Copy content Toggle raw display
\( T_{17}^{6} - 264T_{17}^{4} + 13968T_{17}^{2} - 165888 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} - 4 T^{10} + \cdots + 531441 \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots + 244140625 \) Copy content Toggle raw display
$7$ \( (T^{6} + 210 T^{4} + \cdots + 4096)^{2} \) Copy content Toggle raw display
$11$ \( (T^{6} + 336 T^{4} + \cdots + 1083392)^{2} \) Copy content Toggle raw display
$13$ \( (T^{6} + 768 T^{4} + \cdots + 6553600)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} - 264 T^{4} + \cdots - 165888)^{2} \) Copy content Toggle raw display
$19$ \( (T^{3} - 780 T - 5648)^{4} \) Copy content Toggle raw display
$23$ \( (T^{6} - 1050 T^{4} + \cdots - 13148192)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} + 4416 T^{4} + \cdots + 807698432)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} - 12 T^{2} + \cdots + 31104)^{4} \) Copy content Toggle raw display
$37$ \( (T^{6} + 2592 T^{4} + \cdots + 237899776)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} + 4356 T^{4} + \cdots + 772087808)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} + 4050 T^{4} + \cdots + 1224440064)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} - 5754 T^{4} + \cdots - 2393766432)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} - 11400 T^{4} + \cdots - 21525635072)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} + 11856 T^{4} + \cdots + 1313998848)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} - 36 T^{2} + \cdots + 1984)^{4} \) Copy content Toggle raw display
$67$ \( (T^{6} + 5106 T^{4} + \cdots + 1763584)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} + 19200 T^{4} + \cdots + 124856041472)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} + 26400 T^{4} + \cdots + 238331428864)^{2} \) Copy content Toggle raw display
$79$ \( (T^{3} + 108 T^{2} + \cdots - 234400)^{4} \) Copy content Toggle raw display
$83$ \( (T^{6} - 14586 T^{4} + \cdots - 387755552)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} + 27648 T^{4} + \cdots + 278628139008)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} + 13344 T^{4} + \cdots + 16777216)^{2} \) Copy content Toggle raw display
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