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Properties

Label 240.7.c.e
Level $240$
Weight $7$
Character orbit 240.c
Analytic conductor $55.213$
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] = [240,7,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, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("240.209");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 240 = 2^{4} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 7 \)
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: \(55.2129800688\)
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: \( x^{12} + 1880 x^{10} + 1266870 x^{8} + 399545800 x^{6} + 62009694600 x^{4} + 4432082624000 x^{2} + 109931031040000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{22}\cdot 3^{8}\cdot 5^{9} \)
Twist minimal: no (minimal twist has level 60)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} - \beta_{3} q^{5} + (\beta_{5} - \beta_1) q^{7} + (\beta_{6} + 59) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} - \beta_{3} q^{5} + (\beta_{5} - \beta_1) q^{7} + (\beta_{6} + 59) q^{9} + \beta_{11} q^{11} + (\beta_{10} + \beta_{5} + \cdots - 5 \beta_1) q^{13}+ \cdots + (393 \beta_{11} - 150 \beta_{10} + \cdots - 136096) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 712 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 712 q^{9} - 2480 q^{15} + 192 q^{19} + 5348 q^{21} + 18660 q^{25} - 40848 q^{31} + 45312 q^{39} + 45340 q^{45} - 242940 q^{49} + 40720 q^{51} + 24240 q^{55} - 99312 q^{61} + 108460 q^{69} - 126640 q^{75} - 626544 q^{79} - 798268 q^{81} - 732720 q^{85} - 1996032 q^{91} - 1632080 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 1880 x^{10} + 1266870 x^{8} + 399545800 x^{6} + 62009694600 x^{4} + 4432082624000 x^{2} + 109931031040000 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 2161174211789 \nu^{11} + 480484922547160 \nu^{10} + \cdots + 23\!\cdots\!00 ) / 44\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 2161174211789 \nu^{11} - 384387938037728 \nu^{10} + \cdots - 19\!\cdots\!00 ) / 44\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 14002169117329 \nu^{11} + 782772258266396 \nu^{10} + \cdots + 48\!\cdots\!00 ) / 26\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 171785352648929 \nu^{11} + \cdots - 24\!\cdots\!00 ) / 66\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 27240080124843 \nu^{11} + 96096984509432 \nu^{10} + \cdots + 47\!\cdots\!00 ) / 89\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 881775295611 \nu^{11} - 15699416590600 \nu^{10} + \cdots - 90\!\cdots\!00 ) / 19\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 660899180472969 \nu^{11} + \cdots + 36\!\cdots\!00 ) / 66\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 982770994181987 \nu^{11} + \cdots + 18\!\cdots\!00 ) / 66\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 10\!\cdots\!99 \nu^{11} + \cdots + 60\!\cdots\!00 ) / 66\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 110435887939899 \nu^{11} - 120121230636790 \nu^{10} + \cdots - 59\!\cdots\!00 ) / 27\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 24017870888963 \nu^{11} + \cdots + 81\!\cdots\!00 \nu ) / 58\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 2 \beta_{11} + \beta_{9} - \beta_{7} - 10 \beta_{6} - 90 \beta_{5} + 57 \beta_{4} + 283 \beta_{3} + \cdots + 4 ) / 3600 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 16 \beta_{11} - 21 \beta_{10} + 28 \beta_{9} + 54 \beta_{8} - 16 \beta_{7} + 86 \beta_{6} + \cdots - 225632 ) / 720 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 1948 \beta_{11} - 1665 \beta_{10} - 569 \beta_{9} + 569 \beta_{7} + 5690 \beta_{6} + 9900 \beta_{5} + \cdots - 2276 ) / 1440 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 3650 \beta_{11} + 2715 \beta_{10} - 9701 \beta_{9} - 11481 \beta_{8} + 3650 \beta_{7} + \cdots + 30027100 ) / 180 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 421258 \beta_{11} + 461625 \beta_{10} + 122204 \beta_{9} - 122204 \beta_{7} - 1222040 \beta_{6} + \cdots + 488816 ) / 360 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 1260784 \beta_{11} - 578199 \beta_{10} + 3786940 \beta_{9} + 3682554 \beta_{8} - 1260784 \beta_{7} + \cdots - 8380472768 ) / 72 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 140785604 \beta_{11} - 170671515 \beta_{10} - 41417707 \beta_{9} + 41417707 \beta_{7} + \cdots - 165670828 ) / 144 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 89987290 \beta_{11} + 30408210 \beta_{10} - 281128441 \beta_{9} - 251957571 \beta_{8} + \cdots + 548126853980 ) / 6 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 29579633162 \beta_{11} + 37371854700 \beta_{10} + 8793634981 \beta_{9} - 8793634981 \beta_{7} + \cdots + 35174539924 ) / 36 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 460452788480 \beta_{11} - 133110669705 \beta_{10} + 1458258641900 \beta_{9} + 1264027192830 \beta_{8} + \cdots - 27\!\cdots\!60 ) / 36 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 49974254127980 \beta_{11} - 64228670133525 \beta_{10} - 14944476215965 \beta_{9} + \cdots - 59777904863860 ) / 72 \) 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
− 18.2420i
18.2420i
11.1131i
− 11.1131i
− 29.1428i
29.1428i
13.3561i
− 13.3561i
− 7.15535i
7.15535i
− 18.5699i
18.5699i
0 −24.4963 − 11.3548i 0 124.726 − 8.26448i 0 577.271i 0 471.136 + 556.302i 0
209.2 0 −24.4963 + 11.3548i 0 124.726 + 8.26448i 0 − 577.271i 0 471.136 − 556.302i 0
209.3 0 −22.8848 − 14.3278i 0 −93.9389 − 82.4651i 0 − 279.744i 0 318.429 + 655.777i 0
209.4 0 −22.8848 + 14.3278i 0 −93.9389 + 82.4651i 0 279.744i 0 318.429 − 655.777i 0
209.5 0 −7.66273 − 25.8898i 0 −37.2664 + 119.316i 0 − 46.7246i 0 −611.565 + 396.774i 0
209.6 0 −7.66273 + 25.8898i 0 −37.2664 − 119.316i 0 46.7246i 0 −611.565 − 396.774i 0
209.7 0 7.66273 − 25.8898i 0 37.2664 − 119.316i 0 − 46.7246i 0 −611.565 − 396.774i 0
209.8 0 7.66273 + 25.8898i 0 37.2664 + 119.316i 0 46.7246i 0 −611.565 + 396.774i 0
209.9 0 22.8848 − 14.3278i 0 93.9389 + 82.4651i 0 − 279.744i 0 318.429 − 655.777i 0
209.10 0 22.8848 + 14.3278i 0 93.9389 − 82.4651i 0 279.744i 0 318.429 + 655.777i 0
209.11 0 24.4963 − 11.3548i 0 −124.726 + 8.26448i 0 577.271i 0 471.136 − 556.302i 0
209.12 0 24.4963 + 11.3548i 0 −124.726 − 8.26448i 0 − 577.271i 0 471.136 + 556.302i 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 240.7.c.e 12
3.b odd 2 1 inner 240.7.c.e 12
4.b odd 2 1 60.7.b.a ✓ 12
5.b even 2 1 inner 240.7.c.e 12
12.b even 2 1 60.7.b.a ✓ 12
15.d odd 2 1 inner 240.7.c.e 12
20.d odd 2 1 60.7.b.a ✓ 12
20.e even 4 2 300.7.g.i 12
60.h even 2 1 60.7.b.a ✓ 12
60.l odd 4 2 300.7.g.i 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.7.b.a ✓ 12 4.b odd 2 1
60.7.b.a ✓ 12 12.b even 2 1
60.7.b.a ✓ 12 20.d odd 2 1
60.7.b.a ✓ 12 60.h even 2 1
240.7.c.e 12 1.a even 1 1 trivial
240.7.c.e 12 3.b odd 2 1 inner
240.7.c.e 12 5.b even 2 1 inner
240.7.c.e 12 15.d odd 2 1 inner
300.7.g.i 12 20.e even 4 2
300.7.g.i 12 60.l odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{7}^{\mathrm{new}}(240, [\chi])\):

\( T_{7}^{6} + 413682T_{7}^{4} + 26976849408T_{7}^{2} + 56934147702784 \) Copy content Toggle raw display
\( T_{17}^{6} - 44984520T_{17}^{4} + 337924580874000T_{17}^{2} - 372126567799205120000 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} + \cdots + 15\!\cdots\!21 \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots + 14\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( (T^{6} + \cdots + 56934147702784)^{2} \) Copy content Toggle raw display
$11$ \( (T^{6} + \cdots + 46\!\cdots\!00)^{2} \) Copy content Toggle raw display
$13$ \( (T^{6} + \cdots + 15\!\cdots\!04)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} + \cdots - 37\!\cdots\!00)^{2} \) Copy content Toggle raw display
$19$ \( (T^{3} - 48 T^{2} + \cdots - 543444769296)^{4} \) Copy content Toggle raw display
$23$ \( (T^{6} + \cdots - 37\!\cdots\!00)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} + \cdots + 64\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} + \cdots + 5778576379264)^{4} \) Copy content Toggle raw display
$37$ \( (T^{6} + \cdots + 17\!\cdots\!96)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} + \cdots + 62\!\cdots\!00)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} + \cdots + 35\!\cdots\!64)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} + \cdots - 10\!\cdots\!00)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + \cdots - 55\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} + \cdots + 56\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} + \cdots + 21\!\cdots\!76)^{4} \) Copy content Toggle raw display
$67$ \( (T^{6} + \cdots + 42\!\cdots\!84)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots + 80\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} + \cdots + 41\!\cdots\!96)^{2} \) Copy content Toggle raw display
$79$ \( (T^{3} + \cdots + 74\!\cdots\!28)^{4} \) Copy content Toggle raw display
$83$ \( (T^{6} + \cdots - 18\!\cdots\!00)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} + \cdots + 35\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} + \cdots + 40\!\cdots\!24)^{2} \) Copy content Toggle raw display
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