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Properties

Label 342.7.d.a
Level $342$
Weight $7$
Character orbit 342.d
Analytic conductor $78.678$
Analytic rank $0$
Dimension $10$
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] = [342,7,Mod(37,342)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(342, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("342.37");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 342 = 2 \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 342.d (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: \(78.6784965980\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} + 5050x^{8} + 7354489x^{6} + 2475755792x^{4} + 232626987584x^{2} + 2900002611200 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{29}]\)
Coefficient ring index: \( 2^{11} \)
Twist minimal: no (minimal twist has level 38)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{6} q^{2} - 32 q^{4} + (\beta_1 + 11) q^{5} + (\beta_{3} + \beta_{2} + 2 \beta_1 - 23) q^{7} + 32 \beta_{6} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{6} q^{2} - 32 q^{4} + (\beta_1 + 11) q^{5} + (\beta_{3} + \beta_{2} + 2 \beta_1 - 23) q^{7} + 32 \beta_{6} q^{8} + ( - \beta_{9} - 6 \beta_{6} + 10 \beta_{5}) q^{10} + ( - \beta_{4} + 4 \beta_{3} + \cdots - 364) q^{11}+ \cdots + (34 \beta_{9} - 1132 \beta_{8} + \cdots + 2175 \beta_{5}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 320 q^{4} + 112 q^{5} - 224 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 320 q^{4} + 112 q^{5} - 224 q^{7} - 3644 q^{11} + 10240 q^{16} + 10420 q^{17} - 17230 q^{19} - 3584 q^{20} - 37712 q^{23} - 52078 q^{25} + 7104 q^{26} + 7168 q^{28} + 161720 q^{35} - 25152 q^{38} + 6308 q^{43} + 116608 q^{44} - 322220 q^{47} - 235770 q^{49} - 377880 q^{55} + 445920 q^{58} + 426304 q^{61} - 59424 q^{62} - 327680 q^{64} - 333440 q^{68} - 786076 q^{73} + 293280 q^{74} + 551360 q^{76} - 2303716 q^{77} + 114688 q^{80} - 455136 q^{82} + 101500 q^{83} - 1261380 q^{85} + 1206784 q^{92} - 3106292 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} + 5050x^{8} + 7354489x^{6} + 2475755792x^{4} + 232626987584x^{2} + 2900002611200 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 1003849 \nu^{8} - 4834130050 \nu^{6} - 6581721953521 \nu^{4} + \cdots - 10\!\cdots\!20 ) / 486640529977344 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 3036701 \nu^{8} - 14052901482 \nu^{6} - 16755790470373 \nu^{4} + \cdots + 32\!\cdots\!16 ) / 11\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 80563009 \nu^{8} - 413005498610 \nu^{6} - 584839838800809 \nu^{4} + \cdots - 57\!\cdots\!20 ) / 27\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 94415663 \nu^{8} + 484279228654 \nu^{6} + 700898949271623 \nu^{4} + \cdots + 29\!\cdots\!84 ) / 27\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 128881 \nu^{9} - 758922410 \nu^{7} - 1584249446409 \nu^{5} + \cdots + 48\!\cdots\!36 \nu ) / 25\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 128881 \nu^{9} + 758922410 \nu^{7} + 1584249446409 \nu^{5} + \cdots + 20\!\cdots\!24 \nu ) / 12\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 2748818487 \nu^{9} + 13702415612990 \nu^{7} + \cdots + 12\!\cdots\!28 \nu ) / 57\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 83926954043 \nu^{9} - 412231451711590 \nu^{7} + \cdots - 60\!\cdots\!32 \nu ) / 65\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 97505079905 \nu^{9} - 489205973118514 \nu^{7} + \cdots - 14\!\cdots\!28 \nu ) / 52\!\cdots\!52 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{6} + 2\beta_{5} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -3\beta_{4} - 3\beta_{3} + \beta_{2} - 2\beta _1 - 1009 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3\beta_{9} + 6\beta_{8} + 29\beta_{7} - 2556\beta_{6} - 4269\beta_{5} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 7515\beta_{4} + 7854\beta_{3} - 1885\beta_{2} + 3743\beta _1 + 2156385 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -1113\beta_{9} - 23766\beta_{8} - 73943\beta_{7} + 7288856\beta_{6} + 9801079\beta_{5} ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -17515695\beta_{4} - 19389633\beta_{3} + 4436769\beta_{2} - 7253208\beta _1 - 4953548237 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -19152537\beta_{9} + 85880598\beta_{8} + 174468057\beta_{7} - 19310070620\beta_{6} - 22676527321\beta_{5} ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 40590401331\beta_{4} + 47391878700\beta_{3} - 10844673557\beta_{2} + 13578744841\beta _1 + 11468377737353 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 96438196491 \beta_{9} - 273620269878 \beta_{8} - 408662294491 \beta_{7} + \cdots + 52544592576699 \beta_{5} ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/342\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(325\)
\(\chi(n)\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1
3.82791i
48.7374i
− 16.6843i
11.5377i
− 47.4187i
− 3.82791i
− 48.7374i
16.6843i
− 11.5377i
47.4187i
− 5.65685i 0 −32.0000 −146.942 0 −183.624 181.019i 0 831.227i
37.2 − 5.65685i 0 −32.0000 −39.9185 0 −25.1565 181.019i 0 225.813i
37.3 − 5.65685i 0 −32.0000 9.64927 0 −472.908 181.019i 0 − 54.5845i
37.4 − 5.65685i 0 −32.0000 88.1981 0 443.109 181.019i 0 − 498.924i
37.5 − 5.65685i 0 −32.0000 145.013 0 126.579 181.019i 0 − 820.316i
37.6 5.65685i 0 −32.0000 −146.942 0 −183.624 − 181.019i 0 − 831.227i
37.7 5.65685i 0 −32.0000 −39.9185 0 −25.1565 − 181.019i 0 − 225.813i
37.8 5.65685i 0 −32.0000 9.64927 0 −472.908 − 181.019i 0 54.5845i
37.9 5.65685i 0 −32.0000 88.1981 0 443.109 − 181.019i 0 498.924i
37.10 5.65685i 0 −32.0000 145.013 0 126.579 − 181.019i 0 820.316i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 37.10
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 342.7.d.a 10
3.b odd 2 1 38.7.b.a ✓ 10
12.b even 2 1 304.7.e.e 10
19.b odd 2 1 inner 342.7.d.a 10
57.d even 2 1 38.7.b.a ✓ 10
228.b odd 2 1 304.7.e.e 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.7.b.a ✓ 10 3.b odd 2 1
38.7.b.a ✓ 10 57.d even 2 1
304.7.e.e 10 12.b even 2 1
304.7.e.e 10 228.b odd 2 1
342.7.d.a 10 1.a even 1 1 trivial
342.7.d.a 10 19.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{5} - 56T_{5}^{4} - 24475T_{5}^{3} + 1262450T_{5}^{2} + 65160000T_{5} - 723900000 \) acting on \(S_{7}^{\mathrm{new}}(342, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 32)^{5} \) Copy content Toggle raw display
$3$ \( T^{10} \) Copy content Toggle raw display
$5$ \( (T^{5} - 56 T^{4} + \cdots - 723900000)^{2} \) Copy content Toggle raw display
$7$ \( (T^{5} + 112 T^{4} + \cdots + 122526279250)^{2} \) Copy content Toggle raw display
$11$ \( (T^{5} + \cdots + 73\!\cdots\!80)^{2} \) Copy content Toggle raw display
$13$ \( T^{10} + \cdots + 25\!\cdots\!28 \) Copy content Toggle raw display
$17$ \( (T^{5} + \cdots - 12\!\cdots\!70)^{2} \) Copy content Toggle raw display
$19$ \( T^{10} + \cdots + 23\!\cdots\!01 \) Copy content Toggle raw display
$23$ \( (T^{5} + \cdots - 41\!\cdots\!00)^{2} \) Copy content Toggle raw display
$29$ \( T^{10} + \cdots + 43\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{10} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{10} + \cdots + 78\!\cdots\!88 \) Copy content Toggle raw display
$41$ \( T^{10} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{5} + \cdots + 26\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( (T^{5} + \cdots + 52\!\cdots\!40)^{2} \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots + 55\!\cdots\!72 \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots + 63\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{5} + \cdots + 12\!\cdots\!80)^{2} \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 12\!\cdots\!88 \) Copy content Toggle raw display
$71$ \( T^{10} + \cdots + 28\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{5} + \cdots + 13\!\cdots\!50)^{2} \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( (T^{5} + \cdots - 54\!\cdots\!40)^{2} \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots + 99\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots + 71\!\cdots\!08 \) Copy content Toggle raw display
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