Properties

Label 74.2.c.c
Level $74$
Weight $2$
Character orbit 74.c
Analytic conductor $0.591$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 74 = 2 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 74.c (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.590892974957\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.4406832.1
Defining polynomial: \( x^{6} - x^{5} + 6x^{4} + 7x^{3} + 24x^{2} + 5x + 1 \) 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: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - x^{5} + 6x^{4} + 7x^{3} + 24x^{2} + 5x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -5\nu^{5} + 30\nu^{4} - 31\nu^{3} + 120\nu^{2} + 25\nu + 595 ) / 149 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 7\nu^{5} - 42\nu^{4} + 103\nu^{3} - 168\nu^{2} - 35\nu - 386 ) / 149 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 30\nu^{5} - 31\nu^{4} + 186\nu^{3} + 174\nu^{2} + 744\nu + 6 ) / 149 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 88\nu^{5} - 81\nu^{4} + 486\nu^{3} + 719\nu^{2} + 1944\nu + 405 ) / 149 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 125\nu^{5} - 154\nu^{4} + 775\nu^{3} + 725\nu^{2} + 2951\nu + 25 ) / 149 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{5} - \beta_{4} - \beta_{3} - \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} - 4\beta_{3} + \beta _1 - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 5\beta_{2} + 7\beta _1 - 15 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -9\beta_{5} + 3\beta_{4} + 28\beta_{3} + 3\beta_{2} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -55\beta_{5} + 31\beta_{4} + 139\beta_{3} - 55\beta _1 + 139 ) / 2 \) Copy content Toggle raw display

Character values

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

\(n\) \(39\)
\(\chi(n)\) \(\beta_{3}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
47.1
−0.105378 0.182520i
1.43310 + 2.48220i
−0.827721 1.43366i
−0.105378 + 0.182520i
1.43310 2.48220i
−0.827721 + 1.43366i
−0.500000 + 0.866025i −1.26704 2.19457i −0.500000 0.866025i −1.97779 3.42563i 2.53407 1.26704 + 2.19457i 1.00000 −1.71076 + 2.96312i 3.95558
47.2 −0.500000 + 0.866025i −0.258652 0.447998i −0.500000 0.866025i 2.10755 + 3.65038i 0.517304 0.258652 + 0.447998i 1.00000 1.36620 2.36632i −4.21509
47.3 −0.500000 + 0.866025i 1.52569 + 2.64257i −0.500000 0.866025i −0.629755 1.09077i −3.05137 −1.52569 2.64257i 1.00000 −3.15544 + 5.46539i 1.25951
63.1 −0.500000 0.866025i −1.26704 + 2.19457i −0.500000 + 0.866025i −1.97779 + 3.42563i 2.53407 1.26704 2.19457i 1.00000 −1.71076 2.96312i 3.95558
63.2 −0.500000 0.866025i −0.258652 + 0.447998i −0.500000 + 0.866025i 2.10755 3.65038i 0.517304 0.258652 0.447998i 1.00000 1.36620 + 2.36632i −4.21509
63.3 −0.500000 0.866025i 1.52569 2.64257i −0.500000 + 0.866025i −0.629755 + 1.09077i −3.05137 −1.52569 + 2.64257i 1.00000 −3.15544 5.46539i 1.25951
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 63.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 74.2.c.c 6
3.b odd 2 1 666.2.f.j 6
4.b odd 2 1 592.2.i.e 6
37.c even 3 1 inner 74.2.c.c 6
37.c even 3 1 2738.2.a.o 3
37.e even 6 1 2738.2.a.n 3
111.i odd 6 1 666.2.f.j 6
148.i odd 6 1 592.2.i.e 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
74.2.c.c 6 1.a even 1 1 trivial
74.2.c.c 6 37.c even 3 1 inner
592.2.i.e 6 4.b odd 2 1
592.2.i.e 6 148.i odd 6 1
666.2.f.j 6 3.b odd 2 1
666.2.f.j 6 111.i odd 6 1
2738.2.a.n 3 37.e even 6 1
2738.2.a.o 3 37.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{6} + 8T_{3}^{4} + 8T_{3}^{3} + 64T_{3}^{2} + 32T_{3} + 16 \) acting on \(S_{2}^{\mathrm{new}}(74, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + T + 1)^{3} \) Copy content Toggle raw display
$3$ \( T^{6} + 8 T^{4} + 8 T^{3} + 64 T^{2} + \cdots + 16 \) Copy content Toggle raw display
$5$ \( T^{6} + T^{5} + 18 T^{4} + 25 T^{3} + \cdots + 441 \) Copy content Toggle raw display
$7$ \( T^{6} + 8 T^{4} - 8 T^{3} + 64 T^{2} + \cdots + 16 \) Copy content Toggle raw display
$11$ \( (T^{3} - 4 T^{2} - 16 T + 48)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 2 T + 4)^{3} \) Copy content Toggle raw display
$17$ \( T^{6} - 3 T^{5} + 38 T^{4} + \cdots + 3969 \) Copy content Toggle raw display
$19$ \( T^{6} + 8 T^{5} + 72 T^{4} + \cdots + 12544 \) Copy content Toggle raw display
$23$ \( (T^{3} - 8 T^{2} + 4 T + 12)^{2} \) Copy content Toggle raw display
$29$ \( (T^{3} + 3 T^{2} - 5 T - 3)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} - 4 T^{2} - 24 T - 16)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} + 11 T^{5} + 134 T^{4} + \cdots + 50653 \) Copy content Toggle raw display
$41$ \( T^{6} - 7 T^{5} + 62 T^{4} + 85 T^{3} + \cdots + 9 \) Copy content Toggle raw display
$43$ \( (T^{3} - 8 T^{2} - 20 T + 148)^{2} \) Copy content Toggle raw display
$47$ \( (T^{3} + 16 T^{2} + 56 T + 48)^{2} \) Copy content Toggle raw display
$53$ \( T^{6} + 22 T^{5} + 344 T^{4} + \cdots + 46656 \) Copy content Toggle raw display
$59$ \( T^{6} + 4 T^{5} + 176 T^{4} + \cdots + 451584 \) Copy content Toggle raw display
$61$ \( T^{6} + 9 T^{5} + 62 T^{4} + 157 T^{3} + \cdots + 49 \) Copy content Toggle raw display
$67$ \( T^{6} - 16 T^{5} + 192 T^{4} + \cdots + 256 \) Copy content Toggle raw display
$71$ \( T^{6} - 12 T^{5} + 104 T^{4} + \cdots + 1296 \) Copy content Toggle raw display
$73$ \( (T^{3} - 2 T^{2} - 20 T - 8)^{2} \) Copy content Toggle raw display
$79$ \( T^{6} - 4 T^{5} + 88 T^{4} + \cdots + 20736 \) Copy content Toggle raw display
$83$ \( T^{6} + 4 T^{5} + 80 T^{4} + \cdots + 17424 \) Copy content Toggle raw display
$89$ \( T^{6} + 9 T^{5} + 230 T^{4} + \cdots + 301401 \) Copy content Toggle raw display
$97$ \( (T^{3} - 13 T^{2} + 27 T - 7)^{2} \) Copy content Toggle raw display
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