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Properties

Label 74.4.b.a
Level $74$
Weight $4$
Character orbit 74.b
Analytic conductor $4.366$
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] = [74,4,Mod(73,74)]
 
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([1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("74.73");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 74 = 2 \cdot 37 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 74.b (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: \(4.36614134042\)
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: \( x^{10} + 212x^{8} + 15052x^{6} + 392769x^{4} + 2690496x^{2} + 2985984 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{7}\cdot 3^{2} \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q - \beta_{5} q^{2} + ( - \beta_{2} + 1) q^{3} - 4 q^{4} + (\beta_{7} + 2 \beta_{5}) q^{5} + ( - \beta_{5} + \beta_1) q^{6} + (\beta_{3} + \beta_{2}) q^{7} + 4 \beta_{5} q^{8} + ( - \beta_{6} - \beta_{4} + \beta_{3} + \cdots + 16) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{5} q^{2} + ( - \beta_{2} + 1) q^{3} - 4 q^{4} + (\beta_{7} + 2 \beta_{5}) q^{5} + ( - \beta_{5} + \beta_1) q^{6} + (\beta_{3} + \beta_{2}) q^{7} + 4 \beta_{5} q^{8} + ( - \beta_{6} - \beta_{4} + \beta_{3} + \cdots + 16) q^{9}+ \cdots + ( - 48 \beta_{6} - 7 \beta_{4} + \cdots + 52) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 14 q^{3} - 40 q^{4} - 4 q^{7} + 172 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 14 q^{3} - 40 q^{4} - 4 q^{7} + 172 q^{9} + 76 q^{10} - 50 q^{11} - 56 q^{12} + 160 q^{16} - 312 q^{21} - 700 q^{25} + 492 q^{26} + 848 q^{27} + 16 q^{28} - 240 q^{30} - 508 q^{33} - 568 q^{34} - 688 q^{36} + 82 q^{37} + 336 q^{38} - 304 q^{40} - 1194 q^{41} + 200 q^{44} + 60 q^{46} + 464 q^{47} + 224 q^{48} + 2382 q^{49} - 692 q^{53} + 1108 q^{58} - 1700 q^{62} + 2300 q^{63} - 640 q^{64} + 604 q^{65} + 1114 q^{67} + 1880 q^{70} - 1460 q^{71} + 2082 q^{73} + 968 q^{74} - 5160 q^{75} - 6096 q^{77} + 1004 q^{78} + 4978 q^{81} - 1364 q^{83} + 1248 q^{84} + 104 q^{85} + 1400 q^{86} - 2600 q^{90} + 5084 q^{95} + 508 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} + 212x^{8} + 15052x^{6} + 392769x^{4} + 2690496x^{2} + 2985984 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 49\nu^{8} + 5060\nu^{6} + 73924\nu^{4} - 1042479\nu^{2} + 4950720 ) / 5557896 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 247\nu^{8} + 31808\nu^{6} + 744424\nu^{4} - 20315457\nu^{2} - 277495956 ) / 8336844 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -203\nu^{8} - 43018\nu^{6} - 2996984\nu^{4} - 69411501\nu^{2} - 170771814 ) / 4168422 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 955\nu^{9} + 174236\nu^{7} + 11460100\nu^{5} + 332514171\nu^{3} + 3169891584\nu ) / 1600674048 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 653\nu^{8} + 117844\nu^{6} + 6738392\nu^{4} + 126844389\nu^{2} + 414195120 ) / 8336844 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 4877\nu^{9} + 1100164\nu^{7} + 81458780\nu^{5} + 2086850637\nu^{3} + 9887785344\nu ) / 533558016 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 6325\nu^{9} + 1283300\nu^{7} + 85626172\nu^{5} + 2049826869\nu^{3} + 13178567808\nu ) / 400168512 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 11011\nu^{9} + 2019260\nu^{7} + 118682980\nu^{5} + 2281437891\nu^{3} + 4155048576\nu ) / 533558016 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} + \beta_{4} - \beta_{3} - 42 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -4\beta_{9} + 10\beta_{8} - 8\beta_{7} - 4\beta_{5} - 71\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -91\beta_{6} - 94\beta_{4} + 70\beta_{3} + 54\beta_{2} + 2952 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 358\beta_{9} - 1078\beta_{8} + 866\beta_{7} + 2908\beta_{5} + 5453\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 7759\beta_{6} + 7936\beta_{4} - 5197\beta_{3} - 7632\beta_{2} - 226551 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -31636\beta_{9} + 99040\beta_{8} - 74552\beta_{7} - 387340\beta_{5} - 433313\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -642673\beta_{6} - 656425\beta_{4} + 409789\beta_{3} + 820080\beta_{2} + 17946726 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 2868556\beta_{9} - 8615170\beta_{8} + 5995088\beta_{7} + 40517284\beta_{5} + 35021375\beta_1 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/74\mathbb{Z}\right)^\times\).

\(n\) \(39\)
\(\chi(n)\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
73.1
9.15606i
2.87017i
1.17028i
− 6.35485i
− 8.84166i
− 9.15606i
− 2.87017i
− 1.17028i
6.35485i
8.84166i
− 2.00000i −8.15606 −4.00000 6.18414i 16.3121i 23.1710 8.00000i 39.5214 12.3683
73.2 − 2.00000i −1.87017 −4.00000 − 8.73820i 3.74034i −6.27788 8.00000i −23.5025 −17.4764
73.3 − 2.00000i −0.170276 −4.00000 19.7476i 0.340553i −28.6201 8.00000i −26.9710 39.4952
73.4 − 2.00000i 7.35485 −4.00000 16.2134i − 14.7097i 31.9123 8.00000i 27.0938 32.4267
73.5 − 2.00000i 9.84166 −4.00000 − 14.4069i − 19.6833i −22.1853 8.00000i 69.8583 −28.8138
73.6 2.00000i −8.15606 −4.00000 − 6.18414i − 16.3121i 23.1710 − 8.00000i 39.5214 12.3683
73.7 2.00000i −1.87017 −4.00000 8.73820i − 3.74034i −6.27788 − 8.00000i −23.5025 −17.4764
73.8 2.00000i −0.170276 −4.00000 − 19.7476i − 0.340553i −28.6201 − 8.00000i −26.9710 39.4952
73.9 2.00000i 7.35485 −4.00000 − 16.2134i 14.7097i 31.9123 − 8.00000i 27.0938 32.4267
73.10 2.00000i 9.84166 −4.00000 14.4069i 19.6833i −22.1853 − 8.00000i 69.8583 −28.8138
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 73.10
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 74.4.b.a ✓ 10
3.b odd 2 1 666.4.c.d 10
4.b odd 2 1 592.4.g.d 10
37.b even 2 1 inner 74.4.b.a ✓ 10
111.d odd 2 1 666.4.c.d 10
148.b odd 2 1 592.4.g.d 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
74.4.b.a ✓ 10 1.a even 1 1 trivial
74.4.b.a ✓ 10 37.b even 2 1 inner
592.4.g.d 10 4.b odd 2 1
592.4.g.d 10 148.b odd 2 1
666.4.c.d 10 3.b odd 2 1
666.4.c.d 10 111.d odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(74, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 4)^{5} \) Copy content Toggle raw display
$3$ \( (T^{5} - 7 T^{4} + \cdots + 188)^{2} \) Copy content Toggle raw display
$5$ \( T^{10} + \cdots + 62132541696 \) Copy content Toggle raw display
$7$ \( (T^{5} + 2 T^{4} + \cdots + 2947488)^{2} \) Copy content Toggle raw display
$11$ \( (T^{5} + 25 T^{4} + \cdots - 33993108)^{2} \) Copy content Toggle raw display
$13$ \( T^{10} + \cdots + 11\!\cdots\!56 \) Copy content Toggle raw display
$17$ \( T^{10} + \cdots + 60\!\cdots\!64 \) Copy content Toggle raw display
$19$ \( T^{10} + \cdots + 718725483611136 \) Copy content Toggle raw display
$23$ \( T^{10} + \cdots + 27\!\cdots\!56 \) Copy content Toggle raw display
$29$ \( T^{10} + \cdots + 12\!\cdots\!56 \) Copy content Toggle raw display
$31$ \( T^{10} + \cdots + 11\!\cdots\!76 \) Copy content Toggle raw display
$37$ \( T^{10} + \cdots + 33\!\cdots\!93 \) Copy content Toggle raw display
$41$ \( (T^{5} + 597 T^{4} + \cdots - 671986566)^{2} \) Copy content Toggle raw display
$43$ \( T^{10} + \cdots + 41\!\cdots\!04 \) Copy content Toggle raw display
$47$ \( (T^{5} - 232 T^{4} + \cdots + 69952577472)^{2} \) Copy content Toggle raw display
$53$ \( (T^{5} + 346 T^{4} + \cdots + 494344579464)^{2} \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots + 42\!\cdots\!04 \) Copy content Toggle raw display
$61$ \( T^{10} + \cdots + 31\!\cdots\!04 \) Copy content Toggle raw display
$67$ \( (T^{5} + \cdots + 16621395102016)^{2} \) Copy content Toggle raw display
$71$ \( (T^{5} + 730 T^{4} + \cdots - 731949946272)^{2} \) Copy content Toggle raw display
$73$ \( (T^{5} + \cdots - 21354004243006)^{2} \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots + 39\!\cdots\!76 \) Copy content Toggle raw display
$83$ \( (T^{5} + \cdots + 465096834945216)^{2} \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots + 50\!\cdots\!84 \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots + 41\!\cdots\!76 \) Copy content Toggle raw display
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