Properties

Label 74.8.a.a
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: \( x^{4} - 2x^{3} - 405x^{2} - 2998x - 4396 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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.

\(f(q)\) \(=\) \( q - 8 q^{2} + (\beta_{3} + 2 \beta_1 - 14) q^{3} + 64 q^{4} + (4 \beta_{3} - \beta_{2} - 6 \beta_1 + 32) q^{5} + ( - 8 \beta_{3} - 16 \beta_1 + 112) q^{6} + (4 \beta_{2} + 17 \beta_1 - 426) q^{7} - 512 q^{8} + ( - 65 \beta_{3} - 6 \beta_{2} + \cdots + 1108) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 8 q^{2} + (\beta_{3} + 2 \beta_1 - 14) q^{3} + 64 q^{4} + (4 \beta_{3} - \beta_{2} - 6 \beta_1 + 32) q^{5} + ( - 8 \beta_{3} - 16 \beta_1 + 112) q^{6} + (4 \beta_{2} + 17 \beta_1 - 426) q^{7} - 512 q^{8} + ( - 65 \beta_{3} - 6 \beta_{2} + \cdots + 1108) q^{9}+ \cdots + (173397 \beta_{3} - 13527 \beta_{2} + \cdots - 4397460) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 32 q^{2} - 53 q^{3} + 256 q^{4} + 111 q^{5} + 424 q^{6} - 1666 q^{7} - 2048 q^{8} + 4609 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 32 q^{2} - 53 q^{3} + 256 q^{4} + 111 q^{5} + 424 q^{6} - 1666 q^{7} - 2048 q^{8} + 4609 q^{9} - 888 q^{10} - 4593 q^{11} - 3392 q^{12} + 7847 q^{13} + 13328 q^{14} + 18900 q^{15} + 16384 q^{16} + 23172 q^{17} - 36872 q^{18} + 23696 q^{19} + 7104 q^{20} + 69416 q^{21} + 36744 q^{22} + 24105 q^{23} + 27136 q^{24} - 138149 q^{25} - 62776 q^{26} - 433646 q^{27} - 106624 q^{28} - 140949 q^{29} - 151200 q^{30} - 664609 q^{31} - 131072 q^{32} - 240450 q^{33} - 185376 q^{34} - 248544 q^{35} + 294976 q^{36} + 202612 q^{37} - 189568 q^{38} - 2288827 q^{39} - 56832 q^{40} - 709737 q^{41} - 555328 q^{42} - 128962 q^{43} - 293952 q^{44} - 1755342 q^{45} - 192840 q^{46} - 445842 q^{47} - 217088 q^{48} - 1602774 q^{49} + 1105192 q^{50} - 2883630 q^{51} + 502208 q^{52} - 975870 q^{53} + 3469168 q^{54} - 644145 q^{55} + 852992 q^{56} + 3494630 q^{57} + 1127592 q^{58} - 1812858 q^{59} + 1209600 q^{60} - 2955031 q^{61} + 5316872 q^{62} - 3362482 q^{63} + 1048576 q^{64} + 666 q^{65} + 1923600 q^{66} + 2737235 q^{67} + 1483008 q^{68} - 1781673 q^{69} + 1988352 q^{70} + 4958184 q^{71} - 2359808 q^{72} - 931591 q^{73} - 1620896 q^{74} + 4945810 q^{75} + 1516544 q^{76} + 4352514 q^{77} + 18310616 q^{78} + 5813561 q^{79} + 454656 q^{80} + 16394896 q^{81} + 5677896 q^{82} + 2120460 q^{83} + 4442624 q^{84} - 4845402 q^{85} + 1031696 q^{86} + 7965333 q^{87} + 2351616 q^{88} + 8833716 q^{89} + 14042736 q^{90} - 18886274 q^{91} + 1542720 q^{92} + 3024182 q^{93} + 3566736 q^{94} - 3151794 q^{95} + 1736704 q^{96} - 22666876 q^{97} + 12822192 q^{98} - 17931894 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 2x^{3} - 405x^{2} - 2998x - 4396 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{3} + 16\nu^{2} + 241\nu - 516 ) / 10 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} - 6\nu^{2} - 371\nu - 1454 ) / 10 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta_{2} + 13\beta _1 + 197 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 16\beta_{3} + 6\beta_{2} + 449\beta _1 + 2636 \) 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
−1.99165
−13.3612
−6.83090
24.1837
−8.00000 −92.6631 64.0000 −162.307 741.305 −829.706 −512.000 6399.45 1298.46
1.2 −8.00000 −36.0598 64.0000 −19.7362 288.479 −50.9272 −512.000 −886.690 157.889
1.3 −8.00000 20.4940 64.0000 375.302 −163.952 −980.897 −512.000 −1766.99 −3002.41
1.4 −8.00000 55.2289 64.0000 −82.2583 −441.831 195.531 −512.000 863.231 658.066
\(n\): 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.a 4
4.b odd 2 1 592.8.a.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
74.8.a.a 4 1.a even 1 1 trivial
592.8.a.b 4 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 53T_{3}^{3} - 5274T_{3}^{2} - 107325T_{3} + 3782025 \) 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
$3$ \( T^{4} + 53 T^{3} + \cdots + 3782025 \) Copy content Toggle raw display
$5$ \( T^{4} - 111 T^{3} + \cdots - 98892000 \) Copy content Toggle raw display
$7$ \( T^{4} + \cdots - 8104255332 \) Copy content Toggle raw display
$11$ \( T^{4} + \cdots - 169823359884525 \) Copy content Toggle raw display
$13$ \( T^{4} + \cdots - 32\!\cdots\!32 \) Copy content Toggle raw display
$17$ \( T^{4} + \cdots + 92\!\cdots\!60 \) Copy content Toggle raw display
$19$ \( T^{4} + \cdots + 31\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{4} + \cdots + 46\!\cdots\!92 \) Copy content Toggle raw display
$29$ \( T^{4} + \cdots - 12\!\cdots\!60 \) Copy content Toggle raw display
$31$ \( T^{4} + \cdots + 25\!\cdots\!48 \) Copy content Toggle raw display
$37$ \( (T - 50653)^{4} \) Copy content Toggle raw display
$41$ \( T^{4} + \cdots + 83\!\cdots\!95 \) Copy content Toggle raw display
$43$ \( T^{4} + \cdots + 23\!\cdots\!44 \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots + 76\!\cdots\!12 \) Copy content Toggle raw display
$53$ \( T^{4} + \cdots - 14\!\cdots\!28 \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots - 78\!\cdots\!72 \) Copy content Toggle raw display
$61$ \( T^{4} + \cdots - 31\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots - 21\!\cdots\!48 \) Copy content Toggle raw display
$71$ \( T^{4} + \cdots + 68\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{4} + \cdots + 42\!\cdots\!11 \) Copy content Toggle raw display
$79$ \( T^{4} + \cdots + 82\!\cdots\!88 \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots + 23\!\cdots\!20 \) Copy content Toggle raw display
$89$ \( T^{4} + \cdots - 17\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{4} + \cdots + 16\!\cdots\!20 \) Copy content Toggle raw display
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