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Properties

Label 74.6.b.a
Level $74$
Weight $6$
Character orbit 74.b
Analytic conductor $11.868$
Analytic rank $0$
Dimension $14$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [74,6,Mod(73,74)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("74.73"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(74, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 74 = 2 \cdot 37 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 74.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(11.8684026662\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} + 2080 x^{12} + 1660392 x^{10} + 661933561 x^{8} + 142107556840 x^{6} + 16073178612240 x^{4} + \cdots + 98\!\cdots\!24 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{19} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{9} q^{2} + (\beta_{2} + 1) q^{3} - 16 q^{4} + ( - \beta_{11} + \beta_{9}) q^{5} + ( - \beta_{9} - 2 \beta_1) q^{6} + (\beta_{3} + \beta_{2} - 22) q^{7} + 16 \beta_{9} q^{8} + ( - \beta_{4} - \beta_{3} - 3 \beta_{2} + 55) q^{9}+ \cdots + ( - 99 \beta_{7} + 60 \beta_{6} + \cdots + 21176) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q + 14 q^{3} - 224 q^{4} - 312 q^{7} + 772 q^{9} + 136 q^{10} + 262 q^{11} - 224 q^{12} + 3584 q^{16} + 4908 q^{21} + 6280 q^{25} - 3960 q^{26} - 14596 q^{27} + 4992 q^{28} - 8160 q^{30} + 8336 q^{33}+ \cdots + 295324 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} + 2080 x^{12} + 1660392 x^{10} + 661933561 x^{8} + 142107556840 x^{6} + 16073178612240 x^{4} + \cdots + 98\!\cdots\!24 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 732692153 \nu^{12} + 1388013518416 \nu^{10} + 958944732453480 \nu^{8} + \cdots + 38\!\cdots\!16 ) / 70\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 37964590870309 \nu^{12} + \cdots + 20\!\cdots\!56 ) / 21\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 95\!\cdots\!01 \nu^{12} + \cdots - 50\!\cdots\!04 ) / 13\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 961480202784347 \nu^{12} + \cdots + 50\!\cdots\!08 ) / 91\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 100577138548913 \nu^{12} + \cdots + 52\!\cdots\!72 ) / 54\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 32\!\cdots\!41 \nu^{12} + \cdots + 17\!\cdots\!44 ) / 13\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 15\!\cdots\!19 \nu^{13} + \cdots - 38\!\cdots\!56 \nu ) / 14\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 32625886415201 \nu^{13} + \cdots + 17\!\cdots\!20 \nu ) / 14\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 21\!\cdots\!67 \nu^{13} + \cdots - 10\!\cdots\!60 \nu ) / 37\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 33\!\cdots\!11 \nu^{13} + \cdots + 17\!\cdots\!00 \nu ) / 11\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 28\!\cdots\!09 \nu^{13} + \cdots - 15\!\cdots\!96 \nu ) / 47\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 46\!\cdots\!77 \nu^{13} + \cdots - 24\!\cdots\!08 \nu ) / 56\!\cdots\!80 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} + \beta_{3} + 5\beta_{2} - 297 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -17\beta_{13} + 46\beta_{12} - 13\beta_{11} - 22\beta_{10} + 738\beta_{9} + 21\beta_{8} - 484\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 87\beta_{7} + 48\beta_{6} - 60\beta_{5} - 724\beta_{4} - 808\beta_{3} - 5711\beta_{2} + 143541 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 17687 \beta_{13} - 46234 \beta_{12} + 22825 \beta_{11} + 18604 \beta_{10} - 857850 \beta_{9} + \cdots + 296890 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 109326 \beta_{7} - 44457 \beta_{6} + 37329 \beta_{5} + 516220 \beta_{4} + 633334 \beta_{3} + \cdots - 89023824 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 14892599 \beta_{13} + 38235712 \beta_{12} - 26237719 \beta_{11} - 13894318 \beta_{10} + \cdots - 204754786 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 105428565 \beta_{7} + 34106985 \beta_{6} - 17103417 \beta_{5} - 376529455 \beta_{4} - 498788020 \beta_{3} + \cdots + 62247660021 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 11959842629 \beta_{13} - 30294526030 \beta_{12} + 25011535537 \beta_{11} + 10326494062 \beta_{10} + \cdots + 149489344456 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 92185196307 \beta_{7} - 25476957948 \beta_{6} + 6305333208 \beta_{5} + 280905227428 \beta_{4} + \cdots - 45947506772709 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 9472517496767 \beta_{13} + 23761804188250 \beta_{12} - 21814880693473 \beta_{11} + \cdots - 112420204206730 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 76864301517342 \beta_{7} + 19164897212217 \beta_{6} - 1341457270593 \beta_{5} - 213208001006452 \beta_{4} + \cdots + 34\!\cdots\!00 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 74\!\cdots\!15 \beta_{13} + \cdots + 85\!\cdots\!78 \beta_1 ) / 2 \) 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.

Copy content comment:embeddings in the coefficient field
 
Copy content 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
− 27.9988i
− 13.6641i
− 13.5752i
4.09028i
13.6225i
15.7052i
21.8202i
27.9988i
13.6641i
13.5752i
− 4.09028i
− 13.6225i
− 15.7052i
− 21.8202i
− 4.00000i −26.9988 −16.0000 47.3009i 107.995i −174.456 64.0000i 485.936 189.204
73.2 − 4.00000i −12.6641 −16.0000 34.3650i 50.6564i 159.077 64.0000i −82.6204 137.460
73.3 − 4.00000i −12.5752 −16.0000 − 57.3659i 50.3006i −82.9360 64.0000i −84.8653 −229.464
73.4 − 4.00000i 5.09028 −16.0000 35.2058i − 20.3611i 28.3294 64.0000i −217.089 140.823
73.5 − 4.00000i 14.6225 −16.0000 − 90.2169i − 58.4898i 121.556 64.0000i −29.1837 −360.868
73.6 − 4.00000i 16.7052 −16.0000 − 3.61371i − 66.8208i −250.319 64.0000i 36.0633 −14.4548
73.7 − 4.00000i 22.8202 −16.0000 51.3247i − 91.2806i 42.7478 64.0000i 277.759 205.299
73.8 4.00000i −26.9988 −16.0000 − 47.3009i − 107.995i −174.456 − 64.0000i 485.936 189.204
73.9 4.00000i −12.6641 −16.0000 − 34.3650i − 50.6564i 159.077 − 64.0000i −82.6204 137.460
73.10 4.00000i −12.5752 −16.0000 57.3659i − 50.3006i −82.9360 − 64.0000i −84.8653 −229.464
73.11 4.00000i 5.09028 −16.0000 − 35.2058i 20.3611i 28.3294 − 64.0000i −217.089 140.823
73.12 4.00000i 14.6225 −16.0000 90.2169i 58.4898i 121.556 − 64.0000i −29.1837 −360.868
73.13 4.00000i 16.7052 −16.0000 3.61371i 66.8208i −250.319 − 64.0000i 36.0633 −14.4548
73.14 4.00000i 22.8202 −16.0000 − 51.3247i 91.2806i 42.7478 − 64.0000i 277.759 205.299
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 73.14
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 74.6.b.a ✓ 14
3.b odd 2 1 666.6.c.a 14
4.b odd 2 1 592.6.g.a 14
37.b even 2 1 inner 74.6.b.a ✓ 14
111.d odd 2 1 666.6.c.a 14
148.b odd 2 1 592.6.g.a 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
74.6.b.a ✓ 14 1.a even 1 1 trivial
74.6.b.a ✓ 14 37.b even 2 1 inner
592.6.g.a 14 4.b odd 2 1
592.6.g.a 14 148.b odd 2 1
666.6.c.a 14 3.b odd 2 1
666.6.c.a 14 111.d odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(74, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 16)^{7} \) Copy content Toggle raw display
$3$ \( (T^{7} - 7 T^{6} + \cdots + 122001300)^{2} \) Copy content Toggle raw display
$5$ \( T^{14} + \cdots + 30\!\cdots\!64 \) Copy content Toggle raw display
$7$ \( (T^{7} + \cdots + 84811816418400)^{2} \) Copy content Toggle raw display
$11$ \( (T^{7} + \cdots - 11\!\cdots\!64)^{2} \) Copy content Toggle raw display
$13$ \( T^{14} + \cdots + 14\!\cdots\!24 \) Copy content Toggle raw display
$17$ \( T^{14} + \cdots + 59\!\cdots\!16 \) Copy content Toggle raw display
$19$ \( T^{14} + \cdots + 25\!\cdots\!24 \) Copy content Toggle raw display
$23$ \( T^{14} + \cdots + 13\!\cdots\!36 \) Copy content Toggle raw display
$29$ \( T^{14} + \cdots + 81\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{14} + \cdots + 21\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{14} + \cdots + 77\!\cdots\!93 \) Copy content Toggle raw display
$41$ \( (T^{7} + \cdots + 44\!\cdots\!10)^{2} \) Copy content Toggle raw display
$43$ \( T^{14} + \cdots + 15\!\cdots\!96 \) Copy content Toggle raw display
$47$ \( (T^{7} + \cdots - 48\!\cdots\!32)^{2} \) Copy content Toggle raw display
$53$ \( (T^{7} + \cdots - 35\!\cdots\!28)^{2} \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots + 20\!\cdots\!56 \) Copy content Toggle raw display
$61$ \( T^{14} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( (T^{7} + \cdots + 13\!\cdots\!76)^{2} \) Copy content Toggle raw display
$71$ \( (T^{7} + \cdots + 39\!\cdots\!44)^{2} \) Copy content Toggle raw display
$73$ \( (T^{7} + \cdots - 29\!\cdots\!66)^{2} \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( (T^{7} + \cdots - 37\!\cdots\!16)^{2} \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots + 35\!\cdots\!04 \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
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