Properties

Label 74.8.a.b
Level $74$
Weight $8$
Character orbit 74.a
Self dual yes
Analytic conductor $23.116$
Analytic rank $1$
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] = [74,8,Mod(1,74)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(74, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("74.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 74 = 2 \cdot 37 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 74.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: \(23.1164918858\)
Analytic rank: \(1\)
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: \( x^{4} - 2x^{3} - 2177x^{2} - 14018x + 634476 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
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.

\(f(q)\) \(=\) \( q + 8 q^{2} + (\beta_{2} - 10) q^{3} + 64 q^{4} + ( - \beta_{3} - 4 \beta_{2} + \cdots - 89) q^{5}+ \cdots + (6 \beta_{3} - 65 \beta_{2} + \cdots - 292) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 8 q^{2} + (\beta_{2} - 10) q^{3} + 64 q^{4} + ( - \beta_{3} - 4 \beta_{2} + \cdots - 89) q^{5}+ \cdots + (40917 \beta_{3} - 125449 \beta_{2} + \cdots + 4301113) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 32 q^{2} - 41 q^{3} + 256 q^{4} - 363 q^{5} - 328 q^{6} - 774 q^{7} + 2048 q^{8} - 1079 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 32 q^{2} - 41 q^{3} + 256 q^{4} - 363 q^{5} - 328 q^{6} - 774 q^{7} + 2048 q^{8} - 1079 q^{9} - 2904 q^{10} - 309 q^{11} - 2624 q^{12} - 20827 q^{13} - 6192 q^{14} - 22940 q^{15} + 16384 q^{16} - 48756 q^{17} - 8632 q^{18} - 69068 q^{19} - 23232 q^{20} - 640 q^{21} - 2472 q^{22} - 50237 q^{23} - 20992 q^{24} - 3581 q^{25} - 166616 q^{26} - 368414 q^{27} - 49536 q^{28} - 205195 q^{29} - 183520 q^{30} - 172283 q^{31} + 131072 q^{32} - 205234 q^{33} - 390048 q^{34} - 584964 q^{35} - 69056 q^{36} - 202612 q^{37} - 552544 q^{38} + 329055 q^{39} - 185856 q^{40} - 1018945 q^{41} - 5120 q^{42} + 1263046 q^{43} - 19776 q^{44} + 1279606 q^{45} - 401896 q^{46} - 420930 q^{47} - 167936 q^{48} - 482790 q^{49} - 28648 q^{50} + 728262 q^{51} - 1332928 q^{52} + 2051230 q^{53} - 2947312 q^{54} - 1442891 q^{55} - 396288 q^{56} - 926198 q^{57} - 1641560 q^{58} + 357914 q^{59} - 1468160 q^{60} - 2507513 q^{61} - 1378264 q^{62} + 2879054 q^{63} + 1048576 q^{64} + 3097954 q^{65} - 1641872 q^{66} + 586879 q^{67} - 3120384 q^{68} - 252895 q^{69} - 4679712 q^{70} - 130272 q^{71} - 552448 q^{72} + 3517417 q^{73} - 1620896 q^{74} + 8154290 q^{75} - 4420352 q^{76} + 8777590 q^{77} + 2632440 q^{78} + 3790171 q^{79} - 1486848 q^{80} + 15888376 q^{81} - 8151560 q^{82} + 12973460 q^{83} - 40960 q^{84} + 14870322 q^{85} + 10104368 q^{86} + 23063695 q^{87} - 158208 q^{88} + 18852848 q^{89} + 10236848 q^{90} - 2046622 q^{91} - 3215168 q^{92} + 4005314 q^{93} - 3367440 q^{94} + 30367150 q^{95} - 1343488 q^{96} + 14580104 q^{97} - 3862320 q^{98} + 16949258 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 2x^{3} - 2177x^{2} - 14018x + 634476 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 60\nu^{2} - 2591\nu - 78200 ) / 1378 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -15\nu^{3} + 478\nu^{2} + 18195\nu - 323508 ) / 1378 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + 15\beta_{2} + 15\beta _1 + 1086 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -60\beta_{3} + 478\beta_{2} + 1691\beta _1 + 13040 \) 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
14.6748
−26.8431
−33.6696
47.8378
8.00000 −82.6714 64.0000 114.337 −661.371 130.721 512.000 4647.56 914.695
1.2 8.00000 1.06074 64.0000 156.529 8.48592 −1336.24 512.000 −2185.87 1252.23
1.3 8.00000 18.2200 64.0000 −129.247 145.760 −460.861 512.000 −1855.03 −1033.98
1.4 8.00000 22.3907 64.0000 −504.619 179.126 892.385 512.000 −1685.66 −4036.95
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(37\) \( +1 \)

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 74.8.a.b 4
4.b odd 2 1 592.8.a.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
74.8.a.b 4 1.a even 1 1 trivial
592.8.a.a 4 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 41T_{3}^{3} - 2994T_{3}^{2} + 36855T_{3} - 35775 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(74))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 8)^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 41 T^{3} + \cdots - 35775 \) Copy content Toggle raw display
$5$ \( T^{4} + \cdots + 1167253500 \) Copy content Toggle raw display
$7$ \( T^{4} + \cdots + 71837713524 \) Copy content Toggle raw display
$11$ \( T^{4} + \cdots + 70652437172795 \) Copy content Toggle raw display
$13$ \( T^{4} + \cdots - 38\!\cdots\!96 \) Copy content Toggle raw display
$17$ \( T^{4} + \cdots - 18\!\cdots\!92 \) Copy content Toggle raw display
$19$ \( T^{4} + \cdots - 13\!\cdots\!40 \) Copy content Toggle raw display
$23$ \( T^{4} + \cdots + 30\!\cdots\!64 \) Copy content Toggle raw display
$29$ \( T^{4} + \cdots - 14\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{4} + \cdots + 19\!\cdots\!52 \) Copy content Toggle raw display
$37$ \( (T + 50653)^{4} \) Copy content Toggle raw display
$41$ \( T^{4} + \cdots - 96\!\cdots\!85 \) Copy content Toggle raw display
$43$ \( T^{4} + \cdots - 50\!\cdots\!44 \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots + 82\!\cdots\!76 \) Copy content Toggle raw display
$53$ \( T^{4} + \cdots - 27\!\cdots\!68 \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots + 12\!\cdots\!32 \) Copy content Toggle raw display
$61$ \( T^{4} + \cdots - 71\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots - 18\!\cdots\!48 \) Copy content Toggle raw display
$71$ \( T^{4} + \cdots - 15\!\cdots\!20 \) Copy content Toggle raw display
$73$ \( T^{4} + \cdots - 27\!\cdots\!53 \) Copy content Toggle raw display
$79$ \( T^{4} + \cdots - 15\!\cdots\!56 \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots + 47\!\cdots\!88 \) Copy content Toggle raw display
$89$ \( T^{4} + \cdots - 18\!\cdots\!60 \) Copy content Toggle raw display
$97$ \( T^{4} + \cdots + 35\!\cdots\!80 \) Copy content Toggle raw display
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