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Properties

Label 340.2.o.a
Level $340$
Weight $2$
Character orbit 340.o
Analytic conductor $2.715$
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] = [340,2,Mod(21,340)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(340, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 0, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("340.21");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 340 = 2^{2} \cdot 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 340.o (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: \(2.71491366872\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 6 x^{10} - 16 x^{9} + 9 x^{8} - 72 x^{7} + 114 x^{6} - 144 x^{5} + 391 x^{4} - 484 x^{3} + \cdots + 121 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{10} + \beta_{7} + \beta_{6} + 1) q^{3} + \beta_{3} q^{5} + ( - \beta_{11} + \beta_{8} + \cdots + \beta_{2}) q^{7} + (\beta_{11} + \beta_{10} + \cdots - \beta_{3}) q^{9} + ( - \beta_{11} + 2 \beta_{9} + \cdots - 2) q^{11}+ \cdots + ( - 3 \beta_{11} - 4 \beta_{10} + \cdots - 5) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 4 q^{3} - 12 q^{11} + 16 q^{13} - 4 q^{17} + 16 q^{21} + 12 q^{23} - 20 q^{27} + 4 q^{29} - 8 q^{31} - 48 q^{33} - 8 q^{35} - 20 q^{37} + 36 q^{39} + 8 q^{41} + 8 q^{45} - 8 q^{47} + 24 q^{51} + 16 q^{55}+ \cdots - 40 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 6 x^{10} - 16 x^{9} + 9 x^{8} - 72 x^{7} + 114 x^{6} - 144 x^{5} + 391 x^{4} - 484 x^{3} + \cdots + 121 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 237029484 \nu^{11} + 452648926 \nu^{10} + 1781832384 \nu^{9} + 7020348 \nu^{8} + \cdots + 229900141235 ) / 91818516077 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 6748724960 \nu^{11} - 7586462510 \nu^{10} - 49427698226 \nu^{9} + 46585030124 \nu^{8} + \cdots - 730333528452 ) / 1010003676847 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 1681562281 \nu^{11} + 1206883623 \nu^{10} + 10971218980 \nu^{9} - 18940257967 \nu^{8} + \cdots - 291050856503 ) / 59411980991 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 1877462551 \nu^{11} + 702587919 \nu^{10} + 11535649657 \nu^{9} - 25605083005 \nu^{8} + \cdots - 392384744796 ) / 59411980991 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 32219944720 \nu^{11} + 37654083852 \nu^{10} + 222519008952 \nu^{9} - 272711613374 \nu^{8} + \cdots - 2812980959800 ) / 1010003676847 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 2354972 \nu^{11} + 1834921 \nu^{10} + 15435730 \nu^{9} - 25839746 \nu^{8} + 945970 \nu^{7} + \cdots - 385736208 ) / 49800487 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 73257346909 \nu^{11} + 56161613079 \nu^{10} + 490247008135 \nu^{9} - 801822345473 \nu^{8} + \cdots - 12607579566516 ) / 1010003676847 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 84496178552 \nu^{11} - 52566459528 \nu^{10} - 527061304371 \nu^{9} + \cdots + 17037121470248 ) / 1010003676847 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 85003027341 \nu^{11} + 45952103197 \nu^{10} + 532895178107 \nu^{9} - 1072415229638 \nu^{8} + \cdots - 17011911372159 ) / 1010003676847 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 86757664220 \nu^{11} - 64707732221 \nu^{10} - 564718779799 \nu^{9} + \cdots + 13315455160535 ) / 1010003676847 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 127836924130 \nu^{11} + 85006522887 \nu^{10} + 831492598779 \nu^{9} - 1466105836771 \nu^{8} + \cdots - 22714181280351 ) / 1010003676847 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{11} - \beta_{10} - 2\beta_{9} - \beta_{8} - 2\beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{11} + 2\beta_{10} + \beta_{9} + \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} + \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{11} + \beta_{10} + 4 \beta_{9} + \beta_{8} - 6 \beta_{6} + 4 \beta_{5} - 3 \beta_{4} - \beta_{3} + \cdots + 8 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 3 \beta_{11} - 7 \beta_{10} - 6 \beta_{9} + \beta_{8} - 4 \beta_{7} + 4 \beta_{6} - 6 \beta_{5} + \cdots - 1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 3 \beta_{11} + 29 \beta_{10} - 24 \beta_{9} - 27 \beta_{8} + 18 \beta_{7} + 22 \beta_{6} - 8 \beta_{5} + \cdots + 10 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 7 \beta_{11} + 15 \beta_{10} + 44 \beta_{9} + 11 \beta_{8} + 12 \beta_{7} - 34 \beta_{6} + 31 \beta_{5} + \cdots - 1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 13 \beta_{11} - 101 \beta_{10} + 74 \beta_{9} + 137 \beta_{8} - 98 \beta_{7} + 86 \beta_{6} + \cdots + 28 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 138 \beta_{11} - 16 \beta_{10} - 369 \beta_{9} - 212 \beta_{8} - 19 \beta_{7} + 56 \beta_{6} - 196 \beta_{5} + \cdots + 83 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 329 \beta_{11} + 599 \beta_{10} + 572 \beta_{9} - 17 \beta_{8} + 666 \beta_{7} - 314 \beta_{6} + \cdots - 514 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 668 \beta_{11} + 66 \beta_{10} + 1756 \beta_{9} + 1072 \beta_{8} - 120 \beta_{7} + 187 \beta_{6} + \cdots + 346 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 4875 \beta_{11} - 4321 \beta_{10} - 8194 \beta_{9} - 2885 \beta_{8} - 4208 \beta_{7} - 36 \beta_{6} + \cdots + 1514 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/340\mathbb{Z}\right)^\times\).

\(n\) \(137\) \(171\) \(241\)
\(\chi(n)\) \(1\) \(1\) \(\beta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
21.1
0.150127 + 1.01316i
−0.609374 − 2.33025i
−0.411629 − 1.88205i
0.671074 − 0.0762761i
−1.47591 + 1.40653i
1.67572 − 0.131109i
0.150127 − 1.01316i
−0.609374 + 2.33025i
−0.411629 + 1.88205i
0.671074 + 0.0762761i
−1.47591 − 1.40653i
1.67572 + 0.131109i
0 −1.16724 + 1.16724i 0 −0.707107 + 0.707107i 0 −2.24258 − 2.24258i 0 0.275124i 0
21.2 0 −1.11911 + 1.11911i 0 0.707107 − 0.707107i 0 −0.260019 − 0.260019i 0 0.495178i 0
21.3 0 −0.0789919 + 0.0789919i 0 −0.707107 + 0.707107i 0 1.97356 + 1.97356i 0 2.98752i 0
21.4 0 0.849725 − 0.849725i 0 0.707107 − 0.707107i 0 2.54612 + 2.54612i 0 1.55594i 0
21.5 0 1.26939 − 1.26939i 0 0.707107 − 0.707107i 0 −3.70031 − 3.70031i 0 − 0.222687i 0
21.6 0 2.24623 − 2.24623i 0 −0.707107 + 0.707107i 0 1.68323 + 1.68323i 0 − 7.09107i 0
81.1 0 −1.16724 − 1.16724i 0 −0.707107 − 0.707107i 0 −2.24258 + 2.24258i 0 − 0.275124i 0
81.2 0 −1.11911 − 1.11911i 0 0.707107 + 0.707107i 0 −0.260019 + 0.260019i 0 − 0.495178i 0
81.3 0 −0.0789919 − 0.0789919i 0 −0.707107 − 0.707107i 0 1.97356 − 1.97356i 0 − 2.98752i 0
81.4 0 0.849725 + 0.849725i 0 0.707107 + 0.707107i 0 2.54612 − 2.54612i 0 − 1.55594i 0
81.5 0 1.26939 + 1.26939i 0 0.707107 + 0.707107i 0 −3.70031 + 3.70031i 0 0.222687i 0
81.6 0 2.24623 + 2.24623i 0 −0.707107 − 0.707107i 0 1.68323 − 1.68323i 0 7.09107i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 21.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.c even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 340.2.o.a ✓ 12
3.b odd 2 1 3060.2.be.b 12
4.b odd 2 1 1360.2.bt.c 12
5.b even 2 1 1700.2.o.d 12
5.c odd 4 1 1700.2.m.c 12
5.c odd 4 1 1700.2.m.f 12
17.c even 4 1 inner 340.2.o.a ✓ 12
17.d even 8 1 5780.2.a.m 6
17.d even 8 1 5780.2.a.n 6
17.d even 8 2 5780.2.c.h 12
51.f odd 4 1 3060.2.be.b 12
68.f odd 4 1 1360.2.bt.c 12
85.f odd 4 1 1700.2.m.c 12
85.i odd 4 1 1700.2.m.f 12
85.j even 4 1 1700.2.o.d 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
340.2.o.a ✓ 12 1.a even 1 1 trivial
340.2.o.a ✓ 12 17.c even 4 1 inner
1360.2.bt.c 12 4.b odd 2 1
1360.2.bt.c 12 68.f odd 4 1
1700.2.m.c 12 5.c odd 4 1
1700.2.m.c 12 85.f odd 4 1
1700.2.m.f 12 5.c odd 4 1
1700.2.m.f 12 85.i odd 4 1
1700.2.o.d 12 5.b even 2 1
1700.2.o.d 12 85.j even 4 1
3060.2.be.b 12 3.b odd 2 1
3060.2.be.b 12 51.f odd 4 1
5780.2.a.m 6 17.d even 8 1
5780.2.a.n 6 17.d even 8 1
5780.2.c.h 12 17.d even 8 2

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(340, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} - 4 T^{11} + \cdots + 4 \) Copy content Toggle raw display
$5$ \( (T^{4} + 1)^{3} \) Copy content Toggle raw display
$7$ \( T^{12} - 44 T^{9} + \cdots + 21316 \) Copy content Toggle raw display
$11$ \( T^{12} + 12 T^{11} + \cdots + 24964 \) Copy content Toggle raw display
$13$ \( (T^{6} - 8 T^{5} - 2 T^{4} + \cdots + 68)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} + 4 T^{11} + \cdots + 24137569 \) Copy content Toggle raw display
$19$ \( T^{12} + 104 T^{10} + \cdots + 1183744 \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 220997956 \) Copy content Toggle raw display
$29$ \( T^{12} - 4 T^{11} + \cdots + 33856 \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 1726734916 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 17587003456 \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 6066540544 \) Copy content Toggle raw display
$43$ \( T^{12} + 116 T^{10} + \cdots + 1296 \) Copy content Toggle raw display
$47$ \( (T^{6} + 4 T^{5} + \cdots + 292)^{2} \) Copy content Toggle raw display
$53$ \( T^{12} + 144 T^{10} + \cdots + 135424 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 154157056 \) Copy content Toggle raw display
$61$ \( T^{12} + 8 T^{11} + \cdots + 295936 \) Copy content Toggle raw display
$67$ \( (T^{6} - 16 T^{5} + \cdots + 928964)^{2} \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 1942870084 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 520569856 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 445969924 \) Copy content Toggle raw display
$83$ \( T^{12} + 356 T^{10} + \cdots + 1336336 \) Copy content Toggle raw display
$89$ \( (T^{6} + 28 T^{5} + \cdots + 32996)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} + 44 T^{11} + \cdots + 20720704 \) Copy content Toggle raw display
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