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Properties

Label 240.3.bj.b
Level $240$
Weight $3$
Character orbit 240.bj
Analytic conductor $6.540$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [240,3,Mod(47,240)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(240, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 0, 2, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("240.47");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 240 = 2^{4} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 240.bj (of order \(4\), degree \(2\), 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: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 66x^{14} + 1495x^{12} + 15218x^{10} + 73233x^{8} + 159560x^{6} + 136096x^{4} + 27456x^{2} + 1600 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{14}\cdot 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} + \beta_{6} q^{5} + \beta_{4} q^{7} + ( - \beta_{14} - \beta_{9} + \beta_{7}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} + \beta_{6} q^{5} + \beta_{4} q^{7} + ( - \beta_{14} - \beta_{9} + \beta_{7}) q^{9} + (\beta_{15} + \beta_{13} + \cdots + \beta_{5}) q^{11}+ \cdots + (8 \beta_{15} - 5 \beta_{13} + \cdots + \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 8 q^{13} - 40 q^{21} - 24 q^{25} - 64 q^{33} - 136 q^{37} + 152 q^{45} + 328 q^{57} + 320 q^{61} + 256 q^{73} - 640 q^{81} - 424 q^{85} - 528 q^{93} - 800 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 66x^{14} + 1495x^{12} + 15218x^{10} + 73233x^{8} + 159560x^{6} + 136096x^{4} + 27456x^{2} + 1600 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 23461987 \nu^{15} - 5283914 \nu^{14} - 1545813794 \nu^{13} - 347782468 \nu^{12} + \cdots - 46723897280 ) / 5274322560 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 23461987 \nu^{15} + 5283914 \nu^{14} - 1545813794 \nu^{13} + 347782468 \nu^{12} + \cdots + 46723897280 ) / 1758107520 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2923127 \nu^{15} + 192548740 \nu^{13} + 4345209501 \nu^{11} + 43923430952 \nu^{9} + \cdots + 36738697744 \nu ) / 97672640 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 70860311 \nu^{15} - 60525730 \nu^{14} + 4668108613 \nu^{13} - 3987710120 \nu^{12} + \cdots - 768599160640 ) / 1318580640 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 70860311 \nu^{15} - 60525730 \nu^{14} - 4668108613 \nu^{13} - 3987710120 \nu^{12} + \cdots - 768599160640 ) / 1318580640 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 291520843 \nu^{15} + 127344054 \nu^{14} - 19203753938 \nu^{13} + 8387722548 \nu^{12} + \cdots + 1609189493760 ) / 5274322560 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 291520843 \nu^{15} + 127344054 \nu^{14} + 19203753938 \nu^{13} + 8387722548 \nu^{12} + \cdots + 1609189493760 ) / 5274322560 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 163784257 \nu^{15} + 107730008 \nu^{14} + 10791491438 \nu^{13} + 7094722936 \nu^{12} + \cdots + 1331756725760 ) / 2637161280 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 408718161 \nu^{15} + 175794958 \nu^{14} - 26922784086 \nu^{13} + 11581003076 \nu^{12} + \cdots + 2203358264320 ) / 5274322560 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 408003275 \nu^{15} - 83952326 \nu^{14} - 26877087898 \nu^{13} - 5530957372 \nu^{12} + \cdots - 1063109864000 ) / 5274322560 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 408718161 \nu^{15} - 175794958 \nu^{14} - 26922784086 \nu^{13} - 11581003076 \nu^{12} + \cdots - 2203358264320 ) / 5274322560 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 175059751 \nu^{15} + 109646400 \nu^{14} - 11531634506 \nu^{13} + 7221678240 \nu^{12} + \cdots + 1372479960960 ) / 1318580640 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 192969820 \nu^{15} + 111748777 \nu^{14} + 12710906843 \nu^{13} + 7361180354 \nu^{12} + \cdots + 1411115574880 ) / 1318580640 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 391594487 \nu^{15} + 25795882672 \nu^{13} + 582181545925 \nu^{11} + 5885843803836 \nu^{9} + \cdots + 4983351040112 \nu ) / 2637161280 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 408003275 \nu^{15} + 83952326 \nu^{14} - 26877087898 \nu^{13} + 5530957372 \nu^{12} + \cdots + 1063109864000 ) / 1758107520 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{15} - 3\beta_{11} + 3\beta_{10} - 3\beta_{9} + 3\beta_{7} - 3\beta_{6} - 12\beta_{3} - \beta_{2} - 3\beta_1 ) / 12 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - \beta_{15} - 6 \beta_{13} + 6 \beta_{12} + 3 \beta_{11} - 9 \beta_{10} - 9 \beta_{9} - 6 \beta_{8} + \cdots - 96 ) / 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 3 \beta_{15} - 3 \beta_{14} - 3 \beta_{13} + 4 \beta_{11} - 9 \beta_{10} + 4 \beta_{9} + \cdots + 15 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - \beta_{15} + 174 \beta_{13} - 210 \beta_{12} - 147 \beta_{11} + 351 \beta_{10} + 357 \beta_{9} + \cdots + 1944 ) / 12 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 443 \beta_{15} + 930 \beta_{14} + 750 \beta_{13} - 171 \beta_{11} + 1329 \beta_{10} - 171 \beta_{9} + \cdots - 2469 \beta_1 ) / 12 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 72 \beta_{15} - 868 \beta_{13} + 1114 \beta_{12} + 963 \beta_{11} - 1952 \beta_{10} - 2077 \beta_{9} + \cdots - 8728 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 12463 \beta_{15} - 34104 \beta_{14} - 26292 \beta_{13} - 471 \beta_{11} - 37389 \beta_{10} + \cdots + 72837 \beta_1 ) / 12 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 19511 \beta_{15} + 161946 \beta_{13} - 211770 \beta_{12} - 201771 \beta_{11} + 382425 \beta_{10} + \cdots + 1574904 ) / 12 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 62739 \beta_{15} + 191271 \beta_{14} + 145767 \beta_{13} + 13336 \beta_{11} + 188217 \beta_{10} + \cdots - 376323 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 711757 \beta_{15} - 5134074 \beta_{13} + 6751446 \beta_{12} + 6719151 \beta_{11} - 12403419 \beta_{10} + \cdots - 49398216 ) / 12 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 11775839 \beta_{15} - 37559214 \beta_{14} - 28530810 \beta_{13} - 3319041 \beta_{11} + \cdots + 71599929 \beta_1 ) / 12 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 4026864 \beta_{15} + 27366056 \beta_{13} - 36048722 \beta_{12} - 36571871 \beta_{11} + \cdots + 262411032 ) / 2 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 374397091 \beta_{15} + 1216986420 \beta_{14} + 923552760 \beta_{13} + 115687323 \beta_{11} + \cdots - 2291565753 \beta_1 ) / 12 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 796137515 \beta_{15} - 5271562518 \beta_{13} + 6947799846 \beta_{12} + 7107791175 \beta_{11} + \cdots - 50493666696 ) / 12 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 1998243487 \beta_{15} - 6547175363 \beta_{14} - 4967065607 \beta_{13} - 638747984 \beta_{11} + \cdots + 12268409959 \beta_1 ) / 2 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(31\) \(97\) \(161\) \(181\)
\(\chi(n)\) \(-1\) \(\beta_{3}\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
47.1
− 0.358209i
0.355001i
− 2.35500i
− 5.67442i
− 1.64179i
− 3.21402i
1.21402i
3.67442i
0.358209i
− 0.355001i
2.35500i
5.67442i
1.64179i
3.21402i
− 1.21402i
− 3.67442i
0 −2.67546 − 1.35716i 0 0.326908 − 4.98930i 0 2.54476 − 2.54476i 0 5.31621 + 7.26209i 0
47.2 0 −2.50689 + 1.64787i 0 4.83664 + 1.26762i 0 −6.12570 + 6.12570i 0 3.56902 − 8.26209i 0
47.3 0 −1.64787 + 2.50689i 0 −4.83664 − 1.26762i 0 6.12570 − 6.12570i 0 −3.56902 − 8.26209i 0
47.4 0 −1.35716 − 2.67546i 0 −0.326908 + 4.98930i 0 2.54476 − 2.54476i 0 −5.31621 + 7.26209i 0
47.5 0 1.35716 + 2.67546i 0 −0.326908 + 4.98930i 0 −2.54476 + 2.54476i 0 −5.31621 + 7.26209i 0
47.6 0 1.64787 − 2.50689i 0 −4.83664 − 1.26762i 0 −6.12570 + 6.12570i 0 −3.56902 − 8.26209i 0
47.7 0 2.50689 − 1.64787i 0 4.83664 + 1.26762i 0 6.12570 − 6.12570i 0 3.56902 − 8.26209i 0
47.8 0 2.67546 + 1.35716i 0 0.326908 − 4.98930i 0 −2.54476 + 2.54476i 0 5.31621 + 7.26209i 0
143.1 0 −2.67546 + 1.35716i 0 0.326908 + 4.98930i 0 2.54476 + 2.54476i 0 5.31621 − 7.26209i 0
143.2 0 −2.50689 − 1.64787i 0 4.83664 − 1.26762i 0 −6.12570 − 6.12570i 0 3.56902 + 8.26209i 0
143.3 0 −1.64787 − 2.50689i 0 −4.83664 + 1.26762i 0 6.12570 + 6.12570i 0 −3.56902 + 8.26209i 0
143.4 0 −1.35716 + 2.67546i 0 −0.326908 − 4.98930i 0 2.54476 + 2.54476i 0 −5.31621 − 7.26209i 0
143.5 0 1.35716 − 2.67546i 0 −0.326908 − 4.98930i 0 −2.54476 − 2.54476i 0 −5.31621 − 7.26209i 0
143.6 0 1.64787 + 2.50689i 0 −4.83664 + 1.26762i 0 −6.12570 − 6.12570i 0 −3.56902 + 8.26209i 0
143.7 0 2.50689 + 1.64787i 0 4.83664 − 1.26762i 0 6.12570 + 6.12570i 0 3.56902 + 8.26209i 0
143.8 0 2.67546 − 1.35716i 0 0.326908 + 4.98930i 0 −2.54476 − 2.54476i 0 5.31621 − 7.26209i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 47.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
5.c odd 4 1 inner
12.b even 2 1 inner
15.e even 4 1 inner
20.e even 4 1 inner
60.l odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 240.3.bj.b ✓ 16
3.b odd 2 1 inner 240.3.bj.b ✓ 16
4.b odd 2 1 inner 240.3.bj.b ✓ 16
5.c odd 4 1 inner 240.3.bj.b ✓ 16
12.b even 2 1 inner 240.3.bj.b ✓ 16
15.e even 4 1 inner 240.3.bj.b ✓ 16
20.e even 4 1 inner 240.3.bj.b ✓ 16
60.l odd 4 1 inner 240.3.bj.b ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
240.3.bj.b ✓ 16 1.a even 1 1 trivial
240.3.bj.b ✓ 16 3.b odd 2 1 inner
240.3.bj.b ✓ 16 4.b odd 2 1 inner
240.3.bj.b ✓ 16 5.c odd 4 1 inner
240.3.bj.b ✓ 16 12.b even 2 1 inner
240.3.bj.b ✓ 16 15.e even 4 1 inner
240.3.bj.b ✓ 16 20.e even 4 1 inner
240.3.bj.b ✓ 16 60.l odd 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{8} + 5800T_{7}^{4} + 944784 \) acting on \(S_{3}^{\mathrm{new}}(240, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} + 160 T^{12} + \cdots + 43046721 \) Copy content Toggle raw display
$5$ \( (T^{8} + 6 T^{6} + \cdots + 390625)^{2} \) Copy content Toggle raw display
$7$ \( (T^{8} + 5800 T^{4} + 944784)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} - 394 T^{2} + 27000)^{4} \) Copy content Toggle raw display
$13$ \( (T^{4} - 2 T^{3} + \cdots + 14400)^{4} \) Copy content Toggle raw display
$17$ \( (T^{8} + 128084 T^{4} + 625000000)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} - 1076 T^{2} + 172800)^{4} \) Copy content Toggle raw display
$23$ \( (T^{8} + 845284 T^{4} + 298598400)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} - 2110 T^{2} + 625000)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} + 2556 T^{2} + 1555200)^{4} \) Copy content Toggle raw display
$37$ \( (T^{4} + 34 T^{3} + \cdots + 883600)^{4} \) Copy content Toggle raw display
$41$ \( (T^{4} + 3776 T^{2} + 3317760)^{4} \) Copy content Toggle raw display
$43$ \( (T^{8} + 1909540 T^{4} + 910665586944)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} + \cdots + 7290000000000)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + \cdots + 126247696000000)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 16534 T^{2} + 43200000)^{4} \) Copy content Toggle raw display
$61$ \( (T^{2} - 40 T - 564)^{8} \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots + 77\!\cdots\!84)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} - 19564 T^{2} + 90828000)^{4} \) Copy content Toggle raw display
$73$ \( (T^{4} - 64 T^{3} + \cdots + 900)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} - 21776 T^{2} + 74401200)^{4} \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots + 97\!\cdots\!00)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} - 2956 T^{2} + 718240)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} + 200 T^{3} + \cdots + 49702500)^{4} \) Copy content Toggle raw display
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