Properties

Label 1682.4.a.j
Level $1682$
Weight $4$
Character orbit 1682.a
Self dual yes
Analytic conductor $99.241$
Analytic rank $0$
Dimension $4$
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] = [1682,4,Mod(1,1682)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1682, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1682.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1682 = 2 \cdot 29^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1682.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: \(99.2412126297\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 100x^{2} - 135x + 1548 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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 a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 q^{2} + ( - \beta_1 + 1) q^{3} + 4 q^{4} + ( - \beta_{2} - \beta_1 - 2) q^{5} + ( - 2 \beta_1 + 2) q^{6} + (\beta_{2} + 10) q^{7} + 8 q^{8} + (\beta_{3} + \beta_{2} + 24) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 q^{2} + ( - \beta_1 + 1) q^{3} + 4 q^{4} + ( - \beta_{2} - \beta_1 - 2) q^{5} + ( - 2 \beta_1 + 2) q^{6} + (\beta_{2} + 10) q^{7} + 8 q^{8} + (\beta_{3} + \beta_{2} + 24) q^{9} + ( - 2 \beta_{2} - 2 \beta_1 - 4) q^{10} + ( - 2 \beta_{3} - \beta_1 - 3) q^{11} + ( - 4 \beta_1 + 4) q^{12} + (\beta_{3} - \beta_{2} + 3 \beta_1 + 24) q^{13} + (2 \beta_{2} + 20) q^{14} + (3 \beta_{3} - 4 \beta_{2} + 2 \beta_1 + 28) q^{15} + 16 q^{16} + ( - 2 \beta_{2} - 9 \beta_1 - 22) q^{17} + (2 \beta_{3} + 2 \beta_{2} + 48) q^{18} + (\beta_{3} + 3 \beta_{2} - 4 \beta_1 + 29) q^{19} + ( - 4 \beta_{2} - 4 \beta_1 - 8) q^{20} + ( - 2 \beta_{3} + 5 \beta_{2} - 9 \beta_1 + 30) q^{21} + ( - 4 \beta_{3} - 2 \beta_1 - 6) q^{22} + ( - \beta_{3} - 4 \beta_{2} - 7 \beta_1 + 70) q^{23} + ( - 8 \beta_1 + 8) q^{24} + (3 \beta_{3} + 4 \beta_{2} - 14 \beta_1 + 111) q^{25} + (2 \beta_{3} - 2 \beta_{2} + 6 \beta_1 + 48) q^{26} + ( - 3 \beta_{3} - 6 \beta_{2} - 13 \beta_1 - 5) q^{27} + (4 \beta_{2} + 40) q^{28} + (6 \beta_{3} - 8 \beta_{2} + 4 \beta_1 + 56) q^{30} + ( - 6 \beta_{3} - 6 \beta_{2} + 19 \beta_1 - 88) q^{31} + 32 q^{32} + (3 \beta_{3} + 23 \beta_{2} + 38 \beta_1 + 91) q^{33} + ( - 4 \beta_{2} - 18 \beta_1 - 44) q^{34} + ( - 15 \beta_{2} + 9 \beta_1 - 222) q^{35} + (4 \beta_{3} + 4 \beta_{2} + 96) q^{36} + ( - 3 \beta_{3} + 7 \beta_{2} + 3 \beta_1 - 86) q^{37} + (2 \beta_{3} + 6 \beta_{2} - 8 \beta_1 + 58) q^{38} + ( - 2 \beta_{3} - 19 \beta_{2} - 45 \beta_1 - 168) q^{39} + ( - 8 \beta_{2} - 8 \beta_1 - 16) q^{40} + (4 \beta_{3} - 2 \beta_{2} - 19 \beta_1 + 125) q^{41} + ( - 4 \beta_{3} + 10 \beta_{2} - 18 \beta_1 + 60) q^{42} + (7 \beta_{2} + 30 \beta_1 + 25) q^{43} + ( - 8 \beta_{3} - 4 \beta_1 - 12) q^{44} + (3 \beta_{3} - 28 \beta_{2} - 58 \beta_1 - 164) q^{45} + ( - 2 \beta_{3} - 8 \beta_{2} - 14 \beta_1 + 140) q^{46} + ( - 8 \beta_{3} + 11 \beta_{2} - 14 \beta_1 - 50) q^{47} + ( - 16 \beta_1 + 16) q^{48} + ( - 2 \beta_{3} + 27 \beta_{2} - 18 \beta_1 - 21) q^{49} + (6 \beta_{3} + 8 \beta_{2} - 28 \beta_1 + 222) q^{50} + (13 \beta_{3} - \beta_{2} + 29 \beta_1 + 388) q^{51} + (4 \beta_{3} - 4 \beta_{2} + 12 \beta_1 + 96) q^{52} + ( - 2 \beta_{3} - 20 \beta_{2} - 10 \beta_1 - 10) q^{53} + ( - 6 \beta_{3} - 12 \beta_{2} - 26 \beta_1 - 10) q^{54} + ( - 3 \beta_{3} - 2 \beta_{2} + 112 \beta_1 - 136) q^{55} + (8 \beta_{2} + 80) q^{56} + ( - 3 \beta_{3} + 8 \beta_{2} - 39 \beta_1 + 267) q^{57} + ( - 15 \beta_{3} - 10 \beta_{2} - 2 \beta_1 + 268) q^{59} + (12 \beta_{3} - 16 \beta_{2} + 8 \beta_1 + 112) q^{60} + (23 \beta_{2} - 11 \beta_1 - 378) q^{61} + ( - 12 \beta_{3} - 12 \beta_{2} + 38 \beta_1 - 176) q^{62} + (\beta_{3} + 29 \beta_{2} + 18 \beta_1 + 354) q^{63} + 64 q^{64} + ( - 6 \beta_{3} - 9 \beta_{2} - 105 \beta_1 + 150) q^{65} + (6 \beta_{3} + 46 \beta_{2} + 76 \beta_1 + 182) q^{66} + ( - 16 \beta_{3} - 16 \beta_{2} - 28 \beta_1 - 144) q^{67} + ( - 8 \beta_{2} - 36 \beta_1 - 88) q^{68} + (16 \beta_{3} - 2 \beta_{2} - 50 \beta_1 + 362) q^{69} + ( - 30 \beta_{2} + 18 \beta_1 - 444) q^{70} + (5 \beta_{3} - 42 \beta_{2} - 59 \beta_1 - 162) q^{71} + (8 \beta_{3} + 8 \beta_{2} + 192) q^{72} + ( - 7 \beta_{3} + 14 \beta_{2} - 33 \beta_1 + 258) q^{73} + ( - 6 \beta_{3} + 14 \beta_{2} + 6 \beta_1 - 172) q^{74} + (3 \beta_{3} + \beta_{2} - 144 \beta_1 + 825) q^{75} + (4 \beta_{3} + 12 \beta_{2} - 16 \beta_1 + 116) q^{76} + ( - 8 \beta_{3} + 25 \beta_{2} - 81 \beta_1 + 206) q^{77} + ( - 4 \beta_{3} - 38 \beta_{2} - 90 \beta_1 - 336) q^{78} + (4 \beta_{3} + 18 \beta_{2} - 87 \beta_1 + 260) q^{79} + ( - 16 \beta_{2} - 16 \beta_1 - 32) q^{80} + (\beta_{3} - 11 \beta_{2} + 63 \beta_1 - 57) q^{81} + (8 \beta_{3} - 4 \beta_{2} - 38 \beta_1 + 250) q^{82} + ( - 3 \beta_{3} - 48 \beta_{2} + 15 \beta_1 - 39) q^{83} + ( - 8 \beta_{3} + 20 \beta_{2} - 36 \beta_1 + 120) q^{84} + (27 \beta_{3} + 5 \beta_{2} + 11 \beta_1 + 718) q^{85} + (14 \beta_{2} + 60 \beta_1 + 50) q^{86} + ( - 16 \beta_{3} - 8 \beta_1 - 24) q^{88} + (20 \beta_{3} + 10 \beta_{2} + 10 \beta_1 + 599) q^{89} + (6 \beta_{3} - 56 \beta_{2} - 116 \beta_1 - 328) q^{90} + (11 \beta_{3} - 17 \beta_{2} + 81 \beta_1 - 150) q^{91} + ( - 4 \beta_{3} - 16 \beta_{2} - 28 \beta_1 + 280) q^{92} + ( - \beta_{3} + 17 \beta_{2} + 165 \beta_1 - 1026) q^{93} + ( - 16 \beta_{3} + 22 \beta_{2} - 28 \beta_1 - 100) q^{94} + (15 \beta_{3} - 55 \beta_{2} - 13 \beta_1 - 458) q^{95} + ( - 32 \beta_1 + 32) q^{96} + (19 \beta_{3} + 8 \beta_{2} - 54 \beta_1 - 39) q^{97} + ( - 4 \beta_{3} + 54 \beta_{2} - 36 \beta_1 - 42) q^{98} + ( - 33 \beta_{3} + 44 \beta_{2} - 130 \beta_1 - 1334) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 8 q^{2} + 4 q^{3} + 16 q^{4} - 7 q^{5} + 8 q^{6} + 39 q^{7} + 32 q^{8} + 96 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 8 q^{2} + 4 q^{3} + 16 q^{4} - 7 q^{5} + 8 q^{6} + 39 q^{7} + 32 q^{8} + 96 q^{9} - 14 q^{10} - 14 q^{11} + 16 q^{12} + 98 q^{13} + 78 q^{14} + 119 q^{15} + 64 q^{16} - 86 q^{17} + 192 q^{18} + 114 q^{19} - 28 q^{20} + 113 q^{21} - 28 q^{22} + 283 q^{23} + 32 q^{24} + 443 q^{25} + 196 q^{26} - 17 q^{27} + 156 q^{28} + 238 q^{30} - 352 q^{31} + 128 q^{32} + 344 q^{33} - 172 q^{34} - 873 q^{35} + 384 q^{36} - 354 q^{37} + 228 q^{38} - 655 q^{39} - 56 q^{40} + 506 q^{41} + 226 q^{42} + 93 q^{43} - 56 q^{44} - 625 q^{45} + 566 q^{46} - 219 q^{47} + 64 q^{48} - 113 q^{49} + 886 q^{50} + 1566 q^{51} + 392 q^{52} - 22 q^{53} - 34 q^{54} - 545 q^{55} + 312 q^{56} + 1057 q^{57} + 1067 q^{59} + 476 q^{60} - 1535 q^{61} - 704 q^{62} + 1388 q^{63} + 256 q^{64} + 603 q^{65} + 688 q^{66} - 576 q^{67} - 344 q^{68} + 1466 q^{69} - 1746 q^{70} - 601 q^{71} + 768 q^{72} + 1011 q^{73} - 708 q^{74} + 3302 q^{75} + 456 q^{76} + 791 q^{77} - 1310 q^{78} + 1026 q^{79} - 112 q^{80} - 216 q^{81} + 1012 q^{82} - 111 q^{83} + 452 q^{84} + 2894 q^{85} + 186 q^{86} - 112 q^{88} + 2406 q^{89} - 1250 q^{90} - 572 q^{91} + 1132 q^{92} - 4122 q^{93} - 438 q^{94} - 1762 q^{95} + 128 q^{96} - 145 q^{97} - 226 q^{98} - 5413 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 100x^{2} - 135x + 1548 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} - 6\nu^{2} - 58\nu + 198 ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{3} + 9\nu^{2} + 52\nu - 348 ) / 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta_{2} + 2\beta _1 + 50 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 6\beta_{3} + 9\beta_{2} + 70\beta _1 + 102 \) 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
9.89066
3.50085
−6.34023
−7.05127
2.00000 −8.89066 4.00000 −13.5392 −17.7813 11.6486 8.00000 52.0438 −27.0785
1.2 2.00000 −2.50085 4.00000 6.39202 −5.00170 −1.89287 8.00000 −20.7458 12.7840
1.3 2.00000 7.34023 4.00000 −18.8845 14.6805 33.2247 8.00000 26.8790 −37.7690
1.4 2.00000 8.05127 4.00000 19.0317 16.1025 −3.98045 8.00000 37.8230 38.0634
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(29\) \(-1\)

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 1682.4.a.j yes 4
29.b even 2 1 1682.4.a.g 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1682.4.a.g 4 29.b even 2 1
1682.4.a.j yes 4 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} - 4T_{3}^{3} - 94T_{3}^{2} + 331T_{3} + 1314 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1682))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 2)^{4} \) Copy content Toggle raw display
$3$ \( T^{4} - 4 T^{3} - 94 T^{2} + \cdots + 1314 \) Copy content Toggle raw display
$5$ \( T^{4} + 7 T^{3} - 447 T^{2} + \cdots + 31104 \) Copy content Toggle raw display
$7$ \( T^{4} - 39 T^{3} + 131 T^{2} + \cdots + 2916 \) Copy content Toggle raw display
$11$ \( T^{4} + 14 T^{3} - 6110 T^{2} + \cdots + 8827542 \) Copy content Toggle raw display
$13$ \( T^{4} - 98 T^{3} - 105 T^{2} + \cdots - 1829088 \) Copy content Toggle raw display
$17$ \( T^{4} + 86 T^{3} - 5752 T^{2} + \cdots - 2685942 \) Copy content Toggle raw display
$19$ \( T^{4} - 114 T^{3} - 1454 T^{2} + \cdots - 4967384 \) Copy content Toggle raw display
$23$ \( T^{4} - 283 T^{3} + \cdots - 25030296 \) Copy content Toggle raw display
$29$ \( T^{4} \) Copy content Toggle raw display
$31$ \( T^{4} + 352 T^{3} + \cdots - 1161813984 \) Copy content Toggle raw display
$37$ \( T^{4} + 354 T^{3} + \cdots - 255228624 \) Copy content Toggle raw display
$41$ \( T^{4} - 506 T^{3} + \cdots - 90592182 \) Copy content Toggle raw display
$43$ \( T^{4} - 93 T^{3} - 92746 T^{2} + \cdots + 11676096 \) Copy content Toggle raw display
$47$ \( T^{4} + 219 T^{3} + \cdots + 1883605644 \) Copy content Toggle raw display
$53$ \( T^{4} + 22 T^{3} + \cdots + 5304263616 \) Copy content Toggle raw display
$59$ \( T^{4} - 1067 T^{3} + \cdots - 246032616 \) Copy content Toggle raw display
$61$ \( T^{4} + 1535 T^{3} + \cdots - 43523877984 \) Copy content Toggle raw display
$67$ \( T^{4} + 576 T^{3} + \cdots - 866678784 \) Copy content Toggle raw display
$71$ \( T^{4} + 601 T^{3} + \cdots - 19289176992 \) Copy content Toggle raw display
$73$ \( T^{4} - 1011 T^{3} + \cdots - 4444444536 \) Copy content Toggle raw display
$79$ \( T^{4} - 1026 T^{3} + \cdots + 152742025204 \) Copy content Toggle raw display
$83$ \( T^{4} + 111 T^{3} + \cdots + 17589089088 \) Copy content Toggle raw display
$89$ \( T^{4} - 2406 T^{3} + \cdots - 60143265909 \) Copy content Toggle raw display
$97$ \( T^{4} + 145 T^{3} + \cdots + 4303874709 \) Copy content Toggle raw display
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