Properties

Label 342.8.a.i
Level $342$
Weight $8$
Character orbit 342.a
Self dual yes
Analytic conductor $106.836$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [342,8,Mod(1,342)]
 
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, 0]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("342.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 342 = 2 \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 342.a (trivial)

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: yes
Analytic conductor: \(106.835678716\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17953}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 4488 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 38)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(\beta = \frac{1}{2}(1 + \sqrt{17953})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 8 q^{2} + 64 q^{4} + ( - 3 \beta + 36) q^{5} + (14 \beta - 181) q^{7} + 512 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + 8 q^{2} + 64 q^{4} + ( - 3 \beta + 36) q^{5} + (14 \beta - 181) q^{7} + 512 q^{8} + ( - 24 \beta + 288) q^{10} + (\beta - 1362) q^{11} + ( - 23 \beta - 7045) q^{13} + (112 \beta - 1448) q^{14} + 4096 q^{16} + ( - 122 \beta + 19341) q^{17} + 6859 q^{19} + ( - 192 \beta + 2304) q^{20} + (8 \beta - 10896) q^{22} + ( - 377 \beta + 37137) q^{23} + ( - 207 \beta - 36437) q^{25} + ( - 184 \beta - 56360) q^{26} + (896 \beta - 11584) q^{28} + (1929 \beta - 80871) q^{29} + (4 \beta - 129736) q^{31} + 32768 q^{32} + ( - 976 \beta + 154728) q^{34} + (1005 \beta - 195012) q^{35} + ( - 3412 \beta - 262378) q^{37} + 54872 q^{38} + ( - 1536 \beta + 18432) q^{40} + (910 \beta - 503280) q^{41} + (7393 \beta + 139412) q^{43} + (64 \beta - 87168) q^{44} + ( - 3016 \beta + 297096) q^{46} + ( - 731 \beta + 699120) q^{47} + ( - 4872 \beta + 88866) q^{49} + ( - 1656 \beta - 291496) q^{50} + ( - 1472 \beta - 450880) q^{52} + (20567 \beta + 682701) q^{53} + (4119 \beta - 62496) q^{55} + (7168 \beta - 92672) q^{56} + (15432 \beta - 646968) q^{58} + (22285 \beta - 1361619) q^{59} + ( - 21231 \beta - 1977358) q^{61} + (32 \beta - 1037888) q^{62} + 262144 q^{64} + (20376 \beta + 56052) q^{65} + ( - 53243 \beta - 40657) q^{67} + ( - 7808 \beta + 1237824) q^{68} + (8040 \beta - 1560096) q^{70} + ( - 26944 \beta - 2087898) q^{71} + (30704 \beta + 434897) q^{73} + ( - 27296 \beta - 2099024) q^{74} + 438976 q^{76} + ( - 19235 \beta + 309354) q^{77} + ( - 13342 \beta - 3440194) q^{79} + ( - 12288 \beta + 147456) q^{80} + (7280 \beta - 4026240) q^{82} + (23654 \beta + 1220838) q^{83} + ( - 62049 \beta + 2338884) q^{85} + (59144 \beta + 1115296) q^{86} + (512 \beta - 697344) q^{88} + ( - 50588 \beta - 8690568) q^{89} + ( - 94789 \beta - 169991) q^{91} + ( - 24128 \beta + 2376768) q^{92} + ( - 5848 \beta + 5592960) q^{94} + ( - 20577 \beta + 246924) q^{95} + ( - 154542 \beta + 3253238) q^{97} + ( - 38976 \beta + 710928) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 16 q^{2} + 128 q^{4} + 69 q^{5} - 348 q^{7} + 1024 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 16 q^{2} + 128 q^{4} + 69 q^{5} - 348 q^{7} + 1024 q^{8} + 552 q^{10} - 2723 q^{11} - 14113 q^{13} - 2784 q^{14} + 8192 q^{16} + 38560 q^{17} + 13718 q^{19} + 4416 q^{20} - 21784 q^{22} + 73897 q^{23} - 73081 q^{25} - 112904 q^{26} - 22272 q^{28} - 159813 q^{29} - 259468 q^{31} + 65536 q^{32} + 308480 q^{34} - 389019 q^{35} - 528168 q^{37} + 109744 q^{38} + 35328 q^{40} - 1005650 q^{41} + 286217 q^{43} - 174272 q^{44} + 591176 q^{46} + 1397509 q^{47} + 172860 q^{49} - 584648 q^{50} - 903232 q^{52} + 1385969 q^{53} - 120873 q^{55} - 178176 q^{56} - 1278504 q^{58} - 2700953 q^{59} - 3975947 q^{61} - 2075744 q^{62} + 524288 q^{64} + 132480 q^{65} - 134557 q^{67} + 2467840 q^{68} - 3112152 q^{70} - 4202740 q^{71} + 900498 q^{73} - 4225344 q^{74} + 877952 q^{76} + 599473 q^{77} - 6893730 q^{79} + 282624 q^{80} - 8045200 q^{82} + 2465330 q^{83} + 4615719 q^{85} + 2289736 q^{86} - 1394176 q^{88} - 17431724 q^{89} - 434771 q^{91} + 4729408 q^{92} + 11180072 q^{94} + 473271 q^{95} + 6351934 q^{97} + 1382880 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
67.4944
−66.4944
8.00000 0 64.0000 −166.483 0 763.922 512.000 0 −1331.87
1.2 8.00000 0 64.0000 235.483 0 −1111.92 512.000 0 1883.87
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(19\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 342.8.a.i 2
3.b odd 2 1 38.8.a.c 2
12.b even 2 1 304.8.a.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.8.a.c 2 3.b odd 2 1
304.8.a.b 2 12.b even 2 1
342.8.a.i 2 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} - 69T_{5} - 39204 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(342))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 8)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 69T - 39204 \) Copy content Toggle raw display
$7$ \( T^{2} + 348T - 849421 \) Copy content Toggle raw display
$11$ \( T^{2} + 2723 T + 1849194 \) Copy content Toggle raw display
$13$ \( T^{2} + 14113 T + 47419908 \) Copy content Toggle raw display
$17$ \( T^{2} - 38560 T + 304915287 \) Copy content Toggle raw display
$19$ \( (T - 6859)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 73897 T + 727281168 \) Copy content Toggle raw display
$29$ \( T^{2} + 159813 T - 10315913526 \) Copy content Toggle raw display
$31$ \( T^{2} + 259468 T + 16830838944 \) Copy content Toggle raw display
$37$ \( T^{2} + 528168 T + 17489301548 \) Copy content Toggle raw display
$41$ \( T^{2} + 1005650 T + 249116260800 \) Copy content Toggle raw display
$43$ \( T^{2} - 286217 T - 224831764452 \) Copy content Toggle raw display
$47$ \( T^{2} - 1397509 T + 485859505512 \) Copy content Toggle raw display
$53$ \( T^{2} - 1385969 T - 1418308915764 \) Copy content Toggle raw display
$59$ \( T^{2} + 2700953 T - 405173436054 \) Copy content Toggle raw display
$61$ \( T^{2} + 3975947 T + 1928935887694 \) Copy content Toggle raw display
$67$ \( T^{2} + 134557 T - 12718841223612 \) Copy content Toggle raw display
$71$ \( T^{2} + 4202740 T + 1157380019748 \) Copy content Toggle raw display
$73$ \( T^{2} - 900498 T - 4028508966511 \) Copy content Toggle raw display
$79$ \( T^{2} + 6893730 T + 11081929595552 \) Copy content Toggle raw display
$83$ \( T^{2} - 2465330 T - 991765457112 \) Copy content Toggle raw display
$89$ \( T^{2} + 17431724 T + 64480164517536 \) Copy content Toggle raw display
$97$ \( T^{2} - 6351934 T - 97107139603184 \) Copy content Toggle raw display
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