Properties

Label 76.2.d.a
Level $76$
Weight $2$
Character orbit 76.d
Analytic conductor $0.607$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 76 = 2^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 76.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.606863055362\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.14453810176.1
Defining polynomial: \( x^{8} + 3x^{6} + 6x^{4} + 12x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + \beta_{4} q^{3} + (\beta_{2} - 1) q^{4} + ( - \beta_{6} + \beta_{2} - 1) q^{5} + (\beta_{6} - \beta_{5} - \beta_{2}) q^{6} + ( - \beta_{6} + \beta_{5}) q^{7} + (\beta_{3} - \beta_1) q^{8} + (\beta_{6} - \beta_{2} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + \beta_{4} q^{3} + (\beta_{2} - 1) q^{4} + ( - \beta_{6} + \beta_{2} - 1) q^{5} + (\beta_{6} - \beta_{5} - \beta_{2}) q^{6} + ( - \beta_{6} + \beta_{5}) q^{7} + (\beta_{3} - \beta_1) q^{8} + (\beta_{6} - \beta_{2} + 1) q^{9} + ( - \beta_{7} + \beta_{3} - \beta_1) q^{10} + ( - \beta_{5} - \beta_{2}) q^{11} + (\beta_{7} - 2 \beta_{4} - \beta_{3}) q^{12} + (\beta_{7} - \beta_{4} - 2 \beta_{3} + \beta_1) q^{13} + ( - \beta_{7} + 2 \beta_{4}) q^{14} + (\beta_{7} - 2 \beta_{4} - \beta_1) q^{15} + (\beta_{6} + \beta_{5} - \beta_{2}) q^{16} - q^{17} + (\beta_{7} - \beta_{3} + \beta_1) q^{18} + ( - \beta_{7} + \beta_{5} + \beta_{4} + \beta_{2} + \beta_1) q^{19} + (\beta_{6} + \beta_{5} + 3) q^{20} + ( - 2 \beta_{7} + \beta_{4} + 2 \beta_{3} - 4 \beta_1) q^{21} + ( - 2 \beta_{4} - \beta_{3}) q^{22} + (\beta_{6} + \beta_{2}) q^{23} + ( - 3 \beta_{6} + \beta_{5} + \beta_{2} - 2) q^{24} + (\beta_{6} - \beta_{2}) q^{25} + ( - 3 \beta_{6} - \beta_{5} + \beta_{2} - 2) q^{26} + ( - \beta_{7} - \beta_{4} + \beta_1) q^{27} + (2 \beta_{6} - 2 \beta_{5} - \beta_{2} + 3) q^{28} + (\beta_{4} + 2 \beta_{3} + 2 \beta_1) q^{29} + ( - 2 \beta_{6} + 2 \beta_{5} - 2) q^{30} + (\beta_{7} - \beta_1) q^{31} + (\beta_{7} + 2 \beta_{4} - \beta_{3}) q^{32} + (\beta_{7} + 3 \beta_1) q^{33} - \beta_1 q^{34} + (2 \beta_{6} - 3 \beta_{5} - \beta_{2}) q^{35} + ( - \beta_{6} - \beta_{5} - 3) q^{36} + ( - \beta_{7} - 3 \beta_1) q^{37} + (\beta_{6} - \beta_{5} + 2 \beta_{4} + \beta_{3} + \beta_{2} + 2) q^{38} + (3 \beta_{6} - 2 \beta_{5} + \beta_{2}) q^{39} + (\beta_{7} + 2 \beta_{4} + 3 \beta_1) q^{40} + (\beta_{7} + 3 \beta_1) q^{41} + (3 \beta_{6} + \beta_{5} - 3 \beta_{2} + 8) q^{42} + ( - 2 \beta_{6} - \beta_{5} - 3 \beta_{2}) q^{43} + ( - 3 \beta_{6} + \beta_{5} + 2 \beta_{2} + 1) q^{44} + ( - \beta_{6} + \beta_{2} - 5) q^{45} + (\beta_{7} + \beta_{3}) q^{46} + (3 \beta_{5} + 3 \beta_{2}) q^{47} + ( - 3 \beta_{7} + 2 \beta_{4} + \beta_{3} - 2 \beta_1) q^{48} + ( - 4 \beta_{6} + 4 \beta_{2} - 4) q^{49} + (\beta_{7} - \beta_{3}) q^{50} - \beta_{4} q^{51} + ( - 3 \beta_{7} - 2 \beta_{4} + \beta_{3} - 2 \beta_1) q^{52} + (2 \beta_{7} - \beta_{4} - 2 \beta_{3} + 4 \beta_1) q^{53} + ( - \beta_{6} + \beta_{5} + 3 \beta_{2} + 2) q^{54} + ( - 2 \beta_{6} + \beta_{5} - \beta_{2}) q^{55} + (2 \beta_{7} - 4 \beta_{4} - \beta_{3} + 3 \beta_1) q^{56} + ( - \beta_{7} + 3 \beta_{6} - 3 \beta_{2} - 3 \beta_1 + 4) q^{57} + (3 \beta_{6} + \beta_{5} + \beta_{2} - 4) q^{58} + ( - 3 \beta_{7} + \beta_{4} + 3 \beta_1) q^{59} + ( - 2 \beta_{7} + 4 \beta_{4} - 2 \beta_1) q^{60} + (3 \beta_{6} - 3 \beta_{2} + 7) q^{61} + ( - 2 \beta_{2} - 2) q^{62} + ( - 2 \beta_{6} + 3 \beta_{5} + \beta_{2}) q^{63} + (\beta_{6} - 3 \beta_{5} - 3 \beta_{2} - 2) q^{64} + ( - 2 \beta_{7} - 6 \beta_1) q^{65} + (2 \beta_{2} - 6) q^{66} + (2 \beta_{7} + \beta_{4} - 2 \beta_1) q^{67} + ( - \beta_{2} + 1) q^{68} + (\beta_{7} - \beta_{4} - 2 \beta_{3} + \beta_1) q^{69} + (2 \beta_{7} - 6 \beta_{4} - \beta_{3}) q^{70} + (2 \beta_{7} + 2 \beta_{4} - 2 \beta_1) q^{71} + ( - \beta_{7} - 2 \beta_{4} - 3 \beta_1) q^{72} + ( - 2 \beta_{6} + 2 \beta_{2} + 3) q^{73} + ( - 2 \beta_{2} + 6) q^{74} + ( - \beta_{7} + \beta_{4} + \beta_1) q^{75} + (\beta_{7} + 3 \beta_{6} - \beta_{5} - 2 \beta_{4} + \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 1) q^{76} + (3 \beta_{6} - 3 \beta_{2} + 5) q^{77} + (3 \beta_{7} - 4 \beta_{4} + \beta_{3}) q^{78} + (3 \beta_{7} - 4 \beta_{4} - 3 \beta_1) q^{79} + (2 \beta_{6} - 2 \beta_{5} - 6) q^{80} + ( - 2 \beta_{6} + 2 \beta_{2} - 7) q^{81} + (2 \beta_{2} - 6) q^{82} + ( - 4 \beta_{6} + 2 \beta_{5} - 2 \beta_{2}) q^{83} + (3 \beta_{7} + 2 \beta_{4} - 3 \beta_{3} + 8 \beta_1) q^{84} + (\beta_{6} - \beta_{2} + 1) q^{85} + ( - 2 \beta_{7} - 2 \beta_{4} - 3 \beta_{3}) q^{86} + ( - \beta_{6} - 2 \beta_{5} - 3 \beta_{2}) q^{87} + ( - 3 \beta_{7} + 2 \beta_{4} + 2 \beta_{3} + \beta_1) q^{88} + ( - 3 \beta_{7} + 2 \beta_{4} + 4 \beta_{3} - 5 \beta_1) q^{89} + ( - \beta_{7} + \beta_{3} - 5 \beta_1) q^{90} + ( - \beta_{7} + 7 \beta_{4} + \beta_1) q^{91} + (\beta_{6} + \beta_{5} - \beta_{2} - 4) q^{92} + ( - 2 \beta_{6} + 2 \beta_{2}) q^{93} + (6 \beta_{4} + 3 \beta_{3}) q^{94} + (2 \beta_{6} - \beta_{5} - 4 \beta_{4} + \beta_{2}) q^{95} + (3 \beta_{6} - \beta_{5} - \beta_{2} + 10) q^{96} + ( - 2 \beta_{4} - 4 \beta_{3} - 4 \beta_1) q^{97} + ( - 4 \beta_{7} + 4 \beta_{3} - 4 \beta_1) q^{98} + (2 \beta_{6} - \beta_{5} + \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 6 q^{4} - 4 q^{5} - 2 q^{6} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 6 q^{4} - 4 q^{5} - 2 q^{6} + 4 q^{9} - 6 q^{16} - 8 q^{17} + 20 q^{20} - 10 q^{24} - 4 q^{25} - 6 q^{26} + 22 q^{28} - 16 q^{30} - 20 q^{36} + 18 q^{38} + 50 q^{42} + 16 q^{44} - 36 q^{45} - 16 q^{49} + 22 q^{54} + 20 q^{57} - 38 q^{58} + 44 q^{61} - 20 q^{62} - 18 q^{64} - 44 q^{66} + 6 q^{68} + 32 q^{73} + 44 q^{74} - 16 q^{76} + 28 q^{77} - 48 q^{80} - 48 q^{81} - 44 q^{82} + 4 q^{85} - 38 q^{92} + 8 q^{93} + 74 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 3x^{6} + 6x^{4} + 12x^{2} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} + \nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{7} + \nu^{5} - 2\nu^{3} - 4\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{6} + \nu^{4} - 2\nu^{2} - 4 ) / 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{6} + 3\nu^{4} + 6\nu^{2} + 8 ) / 4 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{7} + 3\nu^{5} + 6\nu^{3} + 8\nu ) / 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} - \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{6} + \beta_{5} - \beta_{2} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{7} + 2\beta_{4} - \beta_{3} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{6} - 3\beta_{5} - 3\beta_{2} - 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( \beta_{7} - 6\beta_{4} - 3\beta_{3} - 2\beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/76\mathbb{Z}\right)^\times\).

\(n\) \(21\) \(39\)
\(\chi(n)\) \(-1\) \(-1\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
75.1
−1.06789 0.927153i
−1.06789 + 0.927153i
−0.331077 1.37491i
−0.331077 + 1.37491i
0.331077 1.37491i
0.331077 + 1.37491i
1.06789 0.927153i
1.06789 + 0.927153i
−1.06789 0.927153i 1.19935 0.280776 + 1.98019i 1.56155 −1.28078 1.11198i 0.868210i 1.53610 2.37495i −1.56155 −1.66757 1.44780i
75.2 −1.06789 + 0.927153i 1.19935 0.280776 1.98019i 1.56155 −1.28078 + 1.11198i 0.868210i 1.53610 + 2.37495i −1.56155 −1.66757 + 1.44780i
75.3 −0.331077 1.37491i −2.35829 −1.78078 + 0.910404i −2.56155 0.780776 + 3.24245i 4.15286i 1.84130 + 2.14700i 2.56155 0.848071 + 3.52191i
75.4 −0.331077 + 1.37491i −2.35829 −1.78078 0.910404i −2.56155 0.780776 3.24245i 4.15286i 1.84130 2.14700i 2.56155 0.848071 3.52191i
75.5 0.331077 1.37491i 2.35829 −1.78078 0.910404i −2.56155 0.780776 3.24245i 4.15286i −1.84130 + 2.14700i 2.56155 −0.848071 + 3.52191i
75.6 0.331077 + 1.37491i 2.35829 −1.78078 + 0.910404i −2.56155 0.780776 + 3.24245i 4.15286i −1.84130 2.14700i 2.56155 −0.848071 3.52191i
75.7 1.06789 0.927153i −1.19935 0.280776 1.98019i 1.56155 −1.28078 + 1.11198i 0.868210i −1.53610 2.37495i −1.56155 1.66757 1.44780i
75.8 1.06789 + 0.927153i −1.19935 0.280776 + 1.98019i 1.56155 −1.28078 1.11198i 0.868210i −1.53610 + 2.37495i −1.56155 1.66757 + 1.44780i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 75.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
19.b odd 2 1 inner
76.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 76.2.d.a 8
3.b odd 2 1 684.2.f.b 8
4.b odd 2 1 inner 76.2.d.a 8
8.b even 2 1 1216.2.h.d 8
8.d odd 2 1 1216.2.h.d 8
12.b even 2 1 684.2.f.b 8
19.b odd 2 1 inner 76.2.d.a 8
57.d even 2 1 684.2.f.b 8
76.d even 2 1 inner 76.2.d.a 8
152.b even 2 1 1216.2.h.d 8
152.g odd 2 1 1216.2.h.d 8
228.b odd 2 1 684.2.f.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.2.d.a 8 1.a even 1 1 trivial
76.2.d.a 8 4.b odd 2 1 inner
76.2.d.a 8 19.b odd 2 1 inner
76.2.d.a 8 76.d even 2 1 inner
684.2.f.b 8 3.b odd 2 1
684.2.f.b 8 12.b even 2 1
684.2.f.b 8 57.d even 2 1
684.2.f.b 8 228.b odd 2 1
1216.2.h.d 8 8.b even 2 1
1216.2.h.d 8 8.d odd 2 1
1216.2.h.d 8 152.b even 2 1
1216.2.h.d 8 152.g odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(76, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 3 T^{6} + 6 T^{4} + 12 T^{2} + \cdots + 16 \) Copy content Toggle raw display
$3$ \( (T^{4} - 7 T^{2} + 8)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} + T - 4)^{4} \) Copy content Toggle raw display
$7$ \( (T^{4} + 18 T^{2} + 13)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + 15 T^{2} + 52)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 41 T^{2} + 416)^{2} \) Copy content Toggle raw display
$17$ \( (T + 1)^{8} \) Copy content Toggle raw display
$19$ \( T^{8} - 16 T^{6} + 718 T^{4} + \cdots + 130321 \) Copy content Toggle raw display
$23$ \( (T^{4} + 19 T^{2} + 52)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 73 T^{2} + 104)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} - 20 T^{2} + 32)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 44 T^{2} + 416)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 44 T^{2} + 416)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 123 T^{2} + 208)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 135 T^{2} + 4212)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 97 T^{2} + 104)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} - 175 T^{2} + 5408)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 11 T - 8)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} - 95 T^{2} + 8)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} - 124 T^{2} + 512)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 8 T - 1)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} - 244 T^{2} + 11552)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + 236 T^{2} + 13312)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 232 T^{2} + 6656)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 292 T^{2} + 1664)^{2} \) Copy content Toggle raw display
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