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Properties

Label 31.9.b.b
Level $31$
Weight $9$
Character orbit 31.b
Analytic conductor $12.629$
Analytic rank $0$
Dimension $18$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [31,9,Mod(30,31)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(31, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("31.30");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 31 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 31.b (of order \(2\), degree \(1\), 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: \(12.6287369119\)
Analytic rank: \(0\)
Dimension: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} + 95810 x^{16} + 3843218640 x^{14} + 83743705221696 x^{12} + \cdots + 57\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{20}\cdot 3^{8}\cdot 7^{2} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{2} q^{2} + \beta_1 q^{3} + ( - \beta_{3} + \beta_{2} + 140) q^{4} + ( - \beta_{4} - 8 \beta_{2} + 17) q^{5} + \beta_{6} q^{6} + ( - \beta_{7} - \beta_{4} + 37 \beta_{2} - 308) q^{7} + (\beta_{7} + \beta_{5} - 2 \beta_{4} + \cdots - 529) q^{8}+ \cdots + (\beta_{11} + \beta_{8} - 2 \beta_{4} + \cdots - 4086) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{2} + \beta_1 q^{3} + ( - \beta_{3} + \beta_{2} + 140) q^{4} + ( - \beta_{4} - 8 \beta_{2} + 17) q^{5} + \beta_{6} q^{6} + ( - \beta_{7} - \beta_{4} + 37 \beta_{2} - 308) q^{7} + (\beta_{7} + \beta_{5} - 2 \beta_{4} + \cdots - 529) q^{8}+ \cdots + ( - 72 \beta_{17} + \cdots + 685491 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q - 2 q^{2} + 2522 q^{4} + 292 q^{5} - 5472 q^{7} - 9754 q^{8} - 73522 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q - 2 q^{2} + 2522 q^{4} + 292 q^{5} - 5472 q^{7} - 9754 q^{8} - 73522 q^{9} + 54308 q^{10} - 264784 q^{14} + 202418 q^{16} - 56414 q^{18} + 377156 q^{19} + 190140 q^{20} - 1140034 q^{25} - 870008 q^{28} - 1173842 q^{31} - 2397306 q^{32} + 6930852 q^{33} + 4750064 q^{35} - 19467210 q^{36} + 3135428 q^{38} - 681276 q^{39} + 17586220 q^{40} - 1205840 q^{41} + 9516676 q^{45} - 4150064 q^{47} + 10788990 q^{49} + 40810482 q^{50} + 17963112 q^{51} - 149792048 q^{56} - 39211724 q^{59} - 58107422 q^{62} + 3318904 q^{63} + 166431826 q^{64} + 137200020 q^{66} - 138448012 q^{67} - 52165728 q^{69} - 140786384 q^{70} + 2474212 q^{71} + 10731434 q^{72} + 418883012 q^{76} - 338212356 q^{78} + 324459812 q^{80} - 10577842 q^{81} + 85367816 q^{82} - 305000388 q^{87} - 383483092 q^{90} + 99725664 q^{93} + 346935656 q^{94} + 293130824 q^{95} - 125184144 q^{97} + 172809978 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} + 95810 x^{16} + 3843218640 x^{14} + 83743705221696 x^{12} + \cdots + 57\!\cdots\!00 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 24\!\cdots\!57 \nu^{16} + \cdots + 90\!\cdots\!00 ) / 14\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 40\!\cdots\!97 \nu^{16} + \cdots + 14\!\cdots\!00 ) / 14\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 18\!\cdots\!83 \nu^{16} + \cdots + 51\!\cdots\!00 ) / 46\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 16\!\cdots\!77 \nu^{16} + \cdots - 47\!\cdots\!00 ) / 35\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 24\!\cdots\!57 \nu^{17} + \cdots - 90\!\cdots\!00 \nu ) / 14\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 11\!\cdots\!53 \nu^{16} + \cdots + 44\!\cdots\!00 ) / 14\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 83\!\cdots\!07 \nu^{16} + \cdots + 14\!\cdots\!00 ) / 63\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 29\!\cdots\!23 \nu^{16} + \cdots - 93\!\cdots\!00 ) / 14\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 24\!\cdots\!67 \nu^{17} + \cdots + 67\!\cdots\!00 \nu ) / 32\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 22\!\cdots\!03 \nu^{16} + \cdots - 52\!\cdots\!00 ) / 90\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 23\!\cdots\!99 \nu^{17} + \cdots - 32\!\cdots\!00 \nu ) / 28\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 62\!\cdots\!77 \nu^{17} + \cdots + 57\!\cdots\!00 \nu ) / 64\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 25\!\cdots\!07 \nu^{17} + \cdots + 15\!\cdots\!00 \nu ) / 16\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 62\!\cdots\!53 \nu^{17} + \cdots + 37\!\cdots\!00 \nu ) / 32\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 20\!\cdots\!35 \nu^{17} + \cdots - 25\!\cdots\!00 \nu ) / 32\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 92\!\cdots\!33 \nu^{17} + \cdots + 30\!\cdots\!00 \nu ) / 10\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{11} + \beta_{8} - 2\beta_{4} + 6\beta_{3} + 11\beta_{2} - 10647 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{16} - \beta_{15} - 5\beta_{14} + 3\beta_{13} + 6\beta_{12} - 15\beta_{10} - 3\beta_{6} - 15590\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 14548 \beta_{11} - 4912 \beta_{9} - 20894 \beta_{8} - 5466 \beta_{7} + 114 \beta_{5} + \cdots + 165926730 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 5352 \beta_{17} + 21008 \beta_{16} + 68834 \beta_{15} + 114154 \beta_{14} - 84090 \beta_{13} + \cdots + 272118556 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 216144830 \beta_{11} + 137047082 \beta_{9} + 408688816 \beta_{8} + 202497504 \beta_{7} + \cdots - 2897585114694 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 215060466 \beta_{17} - 396125854 \beta_{16} - 2515760062 \beta_{15} - 1991239652 \beta_{14} + \cdots - 4973379593228 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 3268474480456 \beta_{11} - 2974561528936 \beta_{9} - 7946089323944 \beta_{8} - 5913277551048 \beta_{7} + \cdots + 52\!\cdots\!88 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 6352184729256 \beta_{17} + 7507182145736 \beta_{16} + 73087161052592 \beta_{15} + \cdots + 92\!\cdots\!44 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 49\!\cdots\!68 \beta_{11} + \cdots - 99\!\cdots\!96 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 16\!\cdots\!84 \beta_{17} + \cdots - 17\!\cdots\!84 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 73\!\cdots\!56 \beta_{11} + \cdots + 18\!\cdots\!28 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 40\!\cdots\!92 \beta_{17} + \cdots + 33\!\cdots\!72 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 10\!\cdots\!68 \beta_{11} + \cdots - 36\!\cdots\!92 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 96\!\cdots\!48 \beta_{17} + \cdots - 65\!\cdots\!20 \beta_1 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 13\!\cdots\!04 \beta_{11} + \cdots + 70\!\cdots\!48 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 22\!\cdots\!56 \beta_{17} + \cdots + 12\!\cdots\!48 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/31\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
30.1
− 130.105i
130.105i
− 8.18525i
8.18525i
− 132.514i
132.514i
− 95.9907i
95.9907i
− 69.6427i
69.6427i
− 144.959i
144.959i
− 57.7052i
57.7052i
− 72.4854i
72.4854i
− 132.643i
132.643i
−30.4926 − 130.105i 673.798 −79.8895 3967.24i 2200.79 −12739.8 −10366.3 2436.04
30.2 −30.4926 130.105i 673.798 −79.8895 − 3967.24i 2200.79 −12739.8 −10366.3 2436.04
30.3 −22.8491 − 8.18525i 266.081 −387.783 187.026i 501.787 −230.341 6494.00 8860.48
30.4 −22.8491 8.18525i 266.081 −387.783 − 187.026i 501.787 −230.341 6494.00 8860.48
30.5 −15.5457 − 132.514i −14.3300 −674.689 2060.03i −3905.33 4202.48 −10999.0 10488.5
30.6 −15.5457 132.514i −14.3300 −674.689 − 2060.03i −3905.33 4202.48 −10999.0 10488.5
30.7 −12.5546 − 95.9907i −98.3829 691.979 1205.12i 2150.13 4449.12 −2653.22 −8687.50
30.8 −12.5546 95.9907i −98.3829 691.979 − 1205.12i 2150.13 4449.12 −2653.22 −8687.50
30.9 2.03935 − 69.6427i −251.841 333.045 − 142.026i −2494.40 −1035.67 1710.89 679.195
30.10 2.03935 69.6427i −251.841 333.045 142.026i −2494.40 −1035.67 1710.89 679.195
30.11 11.2145 − 144.959i −130.234 −460.481 − 1625.64i 3027.74 −4331.44 −14452.1 −5164.08
30.12 11.2145 144.959i −130.234 −460.481 1625.64i 3027.74 −4331.44 −14452.1 −5164.08
30.13 17.1502 − 57.7052i 38.1295 728.004 − 989.656i 1299.93 −3736.52 3231.11 12485.4
30.14 17.1502 57.7052i 38.1295 728.004 989.656i 1299.93 −3736.52 3231.11 12485.4
30.15 20.6668 − 72.4854i 171.115 −709.853 − 1498.04i −2680.44 −1754.30 1306.87 −14670.4
30.16 20.6668 72.4854i 171.115 −709.853 1498.04i −2680.44 −1754.30 1306.87 −14670.4
30.17 29.3712 − 132.643i 606.665 705.668 − 3895.88i −2836.18 10299.4 −11033.2 20726.3
30.18 29.3712 132.643i 606.665 705.668 3895.88i −2836.18 10299.4 −11033.2 20726.3
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 30.18
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
31.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 31.9.b.b ✓ 18
31.b odd 2 1 inner 31.9.b.b ✓ 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
31.9.b.b ✓ 18 1.a even 1 1 trivial
31.9.b.b ✓ 18 31.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{9} + T_{2}^{8} - 1782 T_{2}^{7} + 14 T_{2}^{6} + 1027232 T_{2}^{5} - 870960 T_{2}^{4} + \cdots - 32375116800 \) acting on \(S_{9}^{\mathrm{new}}(31, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{9} + T^{8} + \cdots - 32375116800)^{2} \) Copy content Toggle raw display
$3$ \( T^{18} + \cdots + 57\!\cdots\!00 \) Copy content Toggle raw display
$5$ \( (T^{9} + \cdots + 80\!\cdots\!00)^{2} \) Copy content Toggle raw display
$7$ \( (T^{9} + \cdots - 69\!\cdots\!00)^{2} \) Copy content Toggle raw display
$11$ \( T^{18} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( T^{18} + \cdots + 45\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( T^{18} + \cdots + 59\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( (T^{9} + \cdots + 44\!\cdots\!72)^{2} \) Copy content Toggle raw display
$23$ \( T^{18} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{18} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{18} + \cdots + 23\!\cdots\!61 \) Copy content Toggle raw display
$37$ \( T^{18} + \cdots + 83\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( (T^{9} + \cdots - 13\!\cdots\!76)^{2} \) Copy content Toggle raw display
$43$ \( T^{18} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( (T^{9} + \cdots - 48\!\cdots\!00)^{2} \) Copy content Toggle raw display
$53$ \( T^{18} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( (T^{9} + \cdots + 23\!\cdots\!72)^{2} \) Copy content Toggle raw display
$61$ \( T^{18} + \cdots + 42\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( (T^{9} + \cdots - 20\!\cdots\!00)^{2} \) Copy content Toggle raw display
$71$ \( (T^{9} + \cdots + 19\!\cdots\!56)^{2} \) Copy content Toggle raw display
$73$ \( T^{18} + \cdots + 62\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{18} + \cdots + 94\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{18} + \cdots + 34\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{18} + \cdots + 26\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( (T^{9} + \cdots - 75\!\cdots\!00)^{2} \) Copy content Toggle raw display
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