Properties

Label 74.2.b.a
Level $74$
Weight $2$
Character orbit 74.b
Analytic conductor $0.591$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 74 = 2 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 74.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.590892974957\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{21})\)
Defining polynomial: \( x^{4} + 11x^{2} + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 11x^{2} + 25 \) : Copy content Toggle raw display

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

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
73.1
2.79129i
1.79129i
2.79129i
1.79129i
1.00000i −2.79129 −1.00000 3.79129i 2.79129i −2.00000 1.00000i 4.79129 −3.79129
73.2 1.00000i 1.79129 −1.00000 0.791288i 1.79129i −2.00000 1.00000i 0.208712 0.791288
73.3 1.00000i −2.79129 −1.00000 3.79129i 2.79129i −2.00000 1.00000i 4.79129 −3.79129
73.4 1.00000i 1.79129 −1.00000 0.791288i 1.79129i −2.00000 1.00000i 0.208712 0.791288
\(n\): e.g. 2-40 or 990-1000
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 74.2.b.a 4
3.b odd 2 1 666.2.c.b 4
4.b odd 2 1 592.2.g.c 4
5.b even 2 1 1850.2.d.e 4
5.c odd 4 1 1850.2.c.g 4
5.c odd 4 1 1850.2.c.h 4
8.b even 2 1 2368.2.g.j 4
8.d odd 2 1 2368.2.g.h 4
12.b even 2 1 5328.2.h.m 4
37.b even 2 1 inner 74.2.b.a 4
37.d odd 4 1 2738.2.a.h 2
37.d odd 4 1 2738.2.a.k 2
111.d odd 2 1 666.2.c.b 4
148.b odd 2 1 592.2.g.c 4
185.d even 2 1 1850.2.d.e 4
185.h odd 4 1 1850.2.c.g 4
185.h odd 4 1 1850.2.c.h 4
296.e even 2 1 2368.2.g.j 4
296.h odd 2 1 2368.2.g.h 4
444.g even 2 1 5328.2.h.m 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
74.2.b.a 4 1.a even 1 1 trivial
74.2.b.a 4 37.b even 2 1 inner
592.2.g.c 4 4.b odd 2 1
592.2.g.c 4 148.b odd 2 1
666.2.c.b 4 3.b odd 2 1
666.2.c.b 4 111.d odd 2 1
1850.2.c.g 4 5.c odd 4 1
1850.2.c.g 4 185.h odd 4 1
1850.2.c.h 4 5.c odd 4 1
1850.2.c.h 4 185.h odd 4 1
1850.2.d.e 4 5.b even 2 1
1850.2.d.e 4 185.d even 2 1
2368.2.g.h 4 8.d odd 2 1
2368.2.g.h 4 296.h odd 2 1
2368.2.g.j 4 8.b even 2 1
2368.2.g.j 4 296.e even 2 1
2738.2.a.h 2 37.d odd 4 1
2738.2.a.k 2 37.d odd 4 1
5328.2.h.m 4 12.b even 2 1
5328.2.h.m 4 444.g even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(74, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$3$ \( (T^{2} + T - 5)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} + 15T^{2} + 9 \) Copy content Toggle raw display
$7$ \( (T + 2)^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} - 3 T - 3)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 15T^{2} + 9 \) Copy content Toggle raw display
$17$ \( T^{4} + 60T^{2} + 144 \) Copy content Toggle raw display
$19$ \( T^{4} + 60T^{2} + 144 \) Copy content Toggle raw display
$23$ \( T^{4} + 15T^{2} + 9 \) Copy content Toggle raw display
$29$ \( T^{4} + 15T^{2} + 9 \) Copy content Toggle raw display
$31$ \( T^{4} + 99T^{2} + 2025 \) Copy content Toggle raw display
$37$ \( (T^{2} - 8 T + 37)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 15 T + 51)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} + 6 T - 12)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} - 6 T - 12)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 60T^{2} + 144 \) Copy content Toggle raw display
$61$ \( T^{4} + 231 T^{2} + 11025 \) Copy content Toggle raw display
$67$ \( (T^{2} + T - 47)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 84)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 5 T - 41)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} + 231 T^{2} + 11025 \) Copy content Toggle raw display
$83$ \( (T^{2} - 12 T - 48)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 204T^{2} + 3600 \) Copy content Toggle raw display
show more
show less