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Properties

Label 684.2.bo.c
Level $684$
Weight $2$
Character orbit 684.bo
Analytic conductor $5.462$
Analytic rank $0$
Dimension $12$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [684,2,Mod(73,684)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(684, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([0, 0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("684.73");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 684 = 2^{2} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 684.bo (of order \(9\), degree \(6\), 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: \(5.46176749826\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(2\) over \(\Q(\zeta_{9})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 6 x^{11} - 3 x^{10} + 70 x^{9} - 15 x^{8} - 426 x^{7} + 64 x^{6} + 1659 x^{5} + 267 x^{4} + \cdots + 4161 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 76)
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_{11} - \beta_{10} - \beta_{7} + \cdots - 1) q^{5} + ( - \beta_{9} - \beta_{7} + \cdots + \beta_1) q^{7} + ( - \beta_{11} - 2 \beta_{9} + \cdots + 1) q^{11} + (\beta_{11} + \beta_{9} - \beta_{8} + \cdots - 1) q^{13}+ \cdots + (2 \beta_{11} - 4 \beta_{10} + \cdots - 2) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 3 q^{7} - 3 q^{11} - 9 q^{13} + 3 q^{17} - 12 q^{19} + 12 q^{23} - 18 q^{25} - 27 q^{29} + 6 q^{31} - 33 q^{35} - 12 q^{37} - 3 q^{41} + 27 q^{43} + 15 q^{47} + 9 q^{49} + 21 q^{53} - 27 q^{55} + 48 q^{59}+ \cdots + 12 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 6 x^{11} - 3 x^{10} + 70 x^{9} - 15 x^{8} - 426 x^{7} + 64 x^{6} + 1659 x^{5} + 267 x^{4} + \cdots + 4161 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 1861 \nu^{11} + 59764 \nu^{10} - 261706 \nu^{9} - 459237 \nu^{8} + 3512934 \nu^{7} + \cdots - 104999089 ) / 8740601 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 3014 \nu^{11} - 25876 \nu^{10} + 250822 \nu^{9} + 84243 \nu^{8} - 2839567 \nu^{7} + \cdots + 91337490 ) / 8740601 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 7410 \nu^{11} + 92642 \nu^{10} - 242771 \nu^{9} - 590557 \nu^{8} + 2422014 \nu^{7} + \cdots - 39192874 ) / 8740601 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 7410 \nu^{11} + 11132 \nu^{10} - 276099 \nu^{9} + 170744 \nu^{8} + 2370458 \nu^{7} + \cdots - 26703616 ) / 8740601 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 8405 \nu^{11} + 8018 \nu^{10} - 346679 \nu^{9} + 455105 \nu^{8} + 2627431 \nu^{7} + \cdots - 11328251 ) / 8740601 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 15815 \nu^{11} - 89341 \nu^{10} - 80323 \nu^{9} + 1088115 \nu^{8} - 105905 \nu^{7} + \cdots + 1553197 ) / 8740601 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 16036 \nu^{11} - 88198 \nu^{10} + 124775 \nu^{9} + 357074 \nu^{8} - 2014841 \nu^{7} + \cdots + 33629265 ) / 8740601 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 1506 \nu^{11} + 8283 \nu^{10} + 6301 \nu^{9} - 90477 \nu^{8} - 1422 \nu^{7} + 485184 \nu^{6} + \cdots - 745423 ) / 514153 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 32367 \nu^{11} - 175660 \nu^{10} + 8466 \nu^{9} + 1222749 \nu^{8} - 536869 \nu^{7} + \cdots + 20539212 ) / 8740601 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 382 \nu^{11} - 2101 \nu^{10} + 162 \nu^{9} + 12670 \nu^{8} + 605 \nu^{7} - 42146 \nu^{6} + \cdots - 476163 ) / 80189 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{10} - \beta_{9} + \beta_{8} - \beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} + \beta_{2} + \beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{11} - 3\beta_{10} - \beta_{9} + \beta_{8} - 2\beta_{7} + \beta_{6} + 2\beta_{5} - 3\beta_{4} + 3\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{11} - 9 \beta_{10} - 7 \beta_{9} + 6 \beta_{8} - 12 \beta_{7} + 8 \beta_{6} - 13 \beta_{4} + \cdots + 5 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 8 \beta_{11} - 28 \beta_{10} - 22 \beta_{9} + 10 \beta_{8} - 34 \beta_{7} + 11 \beta_{6} + 12 \beta_{5} + \cdots + 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 11 \beta_{11} - 62 \beta_{10} - 71 \beta_{9} + 26 \beta_{8} - 108 \beta_{7} + 26 \beta_{6} + 25 \beta_{5} + \cdots - 6 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 26 \beta_{11} - 152 \beta_{10} - 261 \beta_{9} + 42 \beta_{8} - 327 \beta_{7} - 12 \beta_{6} + 87 \beta_{5} + \cdots - 5 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 12 \beta_{11} - 288 \beta_{10} - 804 \beta_{9} + 76 \beta_{8} - 866 \beta_{7} - 236 \beta_{6} + \cdots + 18 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 236 \beta_{11} - 480 \beta_{10} - 2610 \beta_{9} + 88 \beta_{8} - 2462 \beta_{7} - 1334 \beta_{6} + \cdots + 384 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 1334 \beta_{11} - 495 \beta_{10} - 7824 \beta_{9} + 204 \beta_{8} - 6291 \beta_{7} - 5298 \beta_{6} + \cdots + 1889 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 5298 \beta_{11} + 678 \beta_{10} - 23140 \beta_{9} + 726 \beta_{8} - 16756 \beta_{7} - 18648 \beta_{6} + \cdots + 7815 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/684\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(343\) \(533\)
\(\chi(n)\) \(-\beta_{4} + \beta_{6}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
73.1
2.25236 + 0.642788i
−1.25236 + 0.642788i
2.25236 − 0.642788i
−1.25236 − 0.642788i
−1.75227 − 0.342020i
2.75227 − 0.342020i
2.26253 − 0.984808i
−1.26253 − 0.984808i
2.26253 + 0.984808i
−1.26253 + 0.984808i
−1.75227 + 0.342020i
2.75227 + 0.342020i
0 0 0 −0.485063 + 2.75093i 0 1.68285 + 2.91479i 0 0 0
73.2 0 0 0 0.658711 − 3.73574i 0 −0.0695116 − 0.120398i 0 0 0
253.1 0 0 0 −0.485063 − 2.75093i 0 1.68285 − 2.91479i 0 0 0
253.2 0 0 0 0.658711 + 3.73574i 0 −0.0695116 + 0.120398i 0 0 0
289.1 0 0 0 −0.216181 + 0.181398i 0 −0.579936 + 1.00448i 0 0 0
289.2 0 0 0 0.982226 − 0.824185i 0 1.67233 − 2.89656i 0 0 0
397.1 0 0 0 −3.00735 + 1.09458i 0 0.278396 + 0.482195i 0 0 0
397.2 0 0 0 2.06765 − 0.752564i 0 −1.48413 − 2.57059i 0 0 0
541.1 0 0 0 −3.00735 − 1.09458i 0 0.278396 − 0.482195i 0 0 0
541.2 0 0 0 2.06765 + 0.752564i 0 −1.48413 + 2.57059i 0 0 0
613.1 0 0 0 −0.216181 − 0.181398i 0 −0.579936 − 1.00448i 0 0 0
613.2 0 0 0 0.982226 + 0.824185i 0 1.67233 + 2.89656i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 73.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.e even 9 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 684.2.bo.c 12
3.b odd 2 1 76.2.i.a ✓ 12
12.b even 2 1 304.2.u.e 12
19.e even 9 1 inner 684.2.bo.c 12
57.j even 18 1 1444.2.a.g 6
57.j even 18 2 1444.2.e.h 12
57.l odd 18 1 76.2.i.a ✓ 12
57.l odd 18 1 1444.2.a.h 6
57.l odd 18 2 1444.2.e.g 12
228.u odd 18 1 5776.2.a.by 6
228.v even 18 1 304.2.u.e 12
228.v even 18 1 5776.2.a.bw 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.2.i.a ✓ 12 3.b odd 2 1
76.2.i.a ✓ 12 57.l odd 18 1
304.2.u.e 12 12.b even 2 1
304.2.u.e 12 228.v even 18 1
684.2.bo.c 12 1.a even 1 1 trivial
684.2.bo.c 12 19.e even 9 1 inner
1444.2.a.g 6 57.j even 18 1
1444.2.a.h 6 57.l odd 18 1
1444.2.e.g 12 57.l odd 18 2
1444.2.e.h 12 57.j even 18 2
5776.2.a.bw 6 228.v even 18 1
5776.2.a.by 6 228.u odd 18 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{12} + 9 T_{5}^{10} + 19 T_{5}^{9} - 72 T_{5}^{8} - 36 T_{5}^{7} + 64 T_{5}^{6} - 1251 T_{5}^{5} + \cdots + 729 \) acting on \(S_{2}^{\mathrm{new}}(684, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} + 9 T^{10} + \cdots + 729 \) Copy content Toggle raw display
$7$ \( T^{12} - 3 T^{11} + \cdots + 9 \) Copy content Toggle raw display
$11$ \( T^{12} + 3 T^{11} + \cdots + 729 \) Copy content Toggle raw display
$13$ \( T^{12} + 9 T^{11} + \cdots + 130321 \) Copy content Toggle raw display
$17$ \( T^{12} - 3 T^{11} + \cdots + 8982009 \) Copy content Toggle raw display
$19$ \( T^{12} + 12 T^{11} + \cdots + 47045881 \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 151585344 \) Copy content Toggle raw display
$29$ \( T^{12} + 27 T^{11} + \cdots + 263169 \) Copy content Toggle raw display
$31$ \( T^{12} - 6 T^{11} + \cdots + 130321 \) Copy content Toggle raw display
$37$ \( (T^{6} + 6 T^{5} + \cdots + 11096)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 360278361 \) Copy content Toggle raw display
$43$ \( T^{12} - 27 T^{11} + \cdots + 1203409 \) Copy content Toggle raw display
$47$ \( T^{12} - 15 T^{11} + \cdots + 729 \) Copy content Toggle raw display
$53$ \( T^{12} - 21 T^{11} + \cdots + 95004009 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 12621848409 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 1418049649 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 6080256576 \) Copy content Toggle raw display
$71$ \( T^{12} + 45 T^{10} + \cdots + 16842816 \) Copy content Toggle raw display
$73$ \( T^{12} - 30 T^{11} + \cdots + 36864 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 1548343801 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 11939714361 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 7695324729 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 16983563041 \) Copy content Toggle raw display
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