Properties

Label 12.4.b.a
Level $12$
Weight $4$
Character orbit 12.b
Analytic conductor $0.708$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 12 = 2^{2} \cdot 3 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 12.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.708022920069\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{3}, \sqrt{-5})\)
Defining polynomial: \( x^{4} + x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{4} \)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{3} - \beta_1) q^{3} + (\beta_{3} + \beta_{2} - 2) q^{4} + (\beta_{3} - \beta_{2} - 2 \beta_1) q^{5} + (\beta_{3} - 3 \beta_{2} + \beta_1 - 6) q^{6} + (\beta_{3} + \beta_{2}) q^{7} + ( - 4 \beta_{3} + 4 \beta_{2} - 4 \beta_1) q^{8} + ( - 3 \beta_{3} + 3 \beta_{2} + 6 \beta_1 - 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + ( - \beta_{3} - \beta_1) q^{3} + (\beta_{3} + \beta_{2} - 2) q^{4} + (\beta_{3} - \beta_{2} - 2 \beta_1) q^{5} + (\beta_{3} - 3 \beta_{2} + \beta_1 - 6) q^{6} + (\beta_{3} + \beta_{2}) q^{7} + ( - 4 \beta_{3} + 4 \beta_{2} - 4 \beta_1) q^{8} + ( - 3 \beta_{3} + 3 \beta_{2} + 6 \beta_1 - 3) q^{9} + ( - 2 \beta_{3} - 2 \beta_{2} + 20) q^{10} + (5 \beta_{3} - 5 \beta_{2} + 10 \beta_1) q^{11} + (5 \beta_{3} - 3 \beta_{2} - 4 \beta_1 + 30) q^{12} - 10 q^{13} + ( - 4 \beta_{3} + 4 \beta_{2} - 2 \beta_1) q^{14} + ( - \beta_{3} + 9 \beta_{2} - 10 \beta_1) q^{15} + ( - 4 \beta_{3} - 4 \beta_{2} - 56) q^{16} + (4 \beta_{3} - 4 \beta_{2} - 8 \beta_1) q^{17} + (6 \beta_{3} + 6 \beta_{2} - 3 \beta_1 - 60) q^{18} + ( - 9 \beta_{3} - 9 \beta_{2}) q^{19} + (8 \beta_{3} - 8 \beta_{2} + 24 \beta_1) q^{20} + (3 \beta_{3} - 3 \beta_{2} - 6 \beta_1 + 30) q^{21} + (10 \beta_{3} + 10 \beta_{2} + 60) q^{22} + ( - 14 \beta_{3} + 14 \beta_{2} - 28 \beta_1) q^{23} + ( - 8 \beta_{3} + 28 \beta_1 + 72) q^{24} + 45 q^{25} - 10 \beta_1 q^{26} + (6 \beta_{3} - 27 \beta_{2} + 33 \beta_1) q^{27} + ( - 2 \beta_{3} - 2 \beta_{2} - 60) q^{28} + ( - 17 \beta_{3} + 17 \beta_{2} + 34 \beta_1) q^{29} + ( - 26 \beta_{3} + 6 \beta_{2} - 8 \beta_1 - 60) q^{30} + (29 \beta_{3} + 29 \beta_{2}) q^{31} + (16 \beta_{3} - 16 \beta_{2} - 48 \beta_1) q^{32} + (15 \beta_{3} - 15 \beta_{2} - 30 \beta_1 - 120) q^{33} + ( - 8 \beta_{3} - 8 \beta_{2} + 80) q^{34} + (10 \beta_{3} - 10 \beta_{2} + 20 \beta_1) q^{35} + ( - 27 \beta_{3} + 21 \beta_{2} - 72 \beta_1 + 6) q^{36} - 130 q^{37} + (36 \beta_{3} - 36 \beta_{2} + 18 \beta_1) q^{38} + (10 \beta_{3} + 10 \beta_1) q^{39} + (24 \beta_{3} + 24 \beta_{2} + 80) q^{40} + ( - 14 \beta_{3} + 14 \beta_{2} + 28 \beta_1) q^{41} + ( - 6 \beta_{3} - 6 \beta_{2} + 30 \beta_1 + 60) q^{42} + ( - 29 \beta_{3} - 29 \beta_{2}) q^{43} + ( - 40 \beta_{3} + 40 \beta_{2} + 40 \beta_1) q^{44} + ( - 3 \beta_{3} + 3 \beta_{2} + 6 \beta_1 + 240) q^{45} + ( - 28 \beta_{3} - 28 \beta_{2} - 168) q^{46} + ( - 28 \beta_{3} + 28 \beta_{2} - 56 \beta_1) q^{47} + (44 \beta_{3} + 12 \beta_{2} + 80 \beta_1 - 120) q^{48} + 283 q^{49} + 45 \beta_1 q^{50} + ( - 4 \beta_{3} + 36 \beta_{2} - 40 \beta_1) q^{51} + ( - 10 \beta_{3} - 10 \beta_{2} + 20) q^{52} + (61 \beta_{3} - 61 \beta_{2} - 122 \beta_1) q^{53} + (75 \beta_{3} - 9 \beta_{2} + 21 \beta_1 + 198) q^{54} + ( - 40 \beta_{3} - 40 \beta_{2}) q^{55} + (8 \beta_{3} - 8 \beta_{2} - 56 \beta_1) q^{56} + ( - 27 \beta_{3} + 27 \beta_{2} + 54 \beta_1 - 270) q^{57} + (34 \beta_{3} + 34 \beta_{2} - 340) q^{58} + (25 \beta_{3} - 25 \beta_{2} + 50 \beta_1) q^{59} + (32 \beta_{3} - 48 \beta_{2} - 40 \beta_1 - 240) q^{60} - 442 q^{61} + ( - 116 \beta_{3} + 116 \beta_{2} - 58 \beta_1) q^{62} + ( - 33 \beta_{3} + 27 \beta_{2} - 60 \beta_1) q^{63} + ( - 48 \beta_{3} - 48 \beta_{2} + 352) q^{64} + ( - 10 \beta_{3} + 10 \beta_{2} + 20 \beta_1) q^{65} + ( - 30 \beta_{3} - 30 \beta_{2} - 120 \beta_1 + 300) q^{66} + (95 \beta_{3} + 95 \beta_{2}) q^{67} + (32 \beta_{3} - 32 \beta_{2} + 96 \beta_1) q^{68} + ( - 42 \beta_{3} + 42 \beta_{2} + 84 \beta_1 + 336) q^{69} + (20 \beta_{3} + 20 \beta_{2} + 120) q^{70} + (150 \beta_{3} - 150 \beta_{2} + 300 \beta_1) q^{71} + ( - 60 \beta_{3} - 84 \beta_{2} + 12 \beta_1 - 240) q^{72} + 410 q^{73} - 130 \beta_1 q^{74} + ( - 45 \beta_{3} - 45 \beta_1) q^{75} + (18 \beta_{3} + 18 \beta_{2} + 540) q^{76} + ( - 30 \beta_{3} + 30 \beta_{2} + 60 \beta_1) q^{77} + ( - 10 \beta_{3} + 30 \beta_{2} - 10 \beta_1 + 60) q^{78} + ( - 11 \beta_{3} - 11 \beta_{2}) q^{79} + ( - 96 \beta_{3} + 96 \beta_{2} + 32 \beta_1) q^{80} + (18 \beta_{3} - 18 \beta_{2} - 36 \beta_1 - 711) q^{81} + (28 \beta_{3} + 28 \beta_{2} - 280) q^{82} + ( - 181 \beta_{3} + 181 \beta_{2} - 362 \beta_1) q^{83} + (54 \beta_{3} + 6 \beta_{2} + 72 \beta_1 - 60) q^{84} - 320 q^{85} + (116 \beta_{3} - 116 \beta_{2} + 58 \beta_1) q^{86} + (17 \beta_{3} - 153 \beta_{2} + 170 \beta_1) q^{87} + (40 \beta_{3} + 40 \beta_{2} - 720) q^{88} + (94 \beta_{3} - 94 \beta_{2} - 188 \beta_1) q^{89} + (6 \beta_{3} + 6 \beta_{2} + 240 \beta_1 - 60) q^{90} + ( - 10 \beta_{3} - 10 \beta_{2}) q^{91} + (112 \beta_{3} - 112 \beta_{2} - 112 \beta_1) q^{92} + (87 \beta_{3} - 87 \beta_{2} - 174 \beta_1 + 870) q^{93} + ( - 56 \beta_{3} - 56 \beta_{2} - 336) q^{94} + ( - 90 \beta_{3} + 90 \beta_{2} - 180 \beta_1) q^{95} + ( - 32 \beta_{3} + 192 \beta_{2} - 176 \beta_1 + 96) q^{96} + 770 q^{97} + 283 \beta_1 q^{98} + (105 \beta_{3} + 135 \beta_{2} - 30 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{4} - 24 q^{6} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 8 q^{4} - 24 q^{6} - 12 q^{9} + 80 q^{10} + 120 q^{12} - 40 q^{13} - 224 q^{16} - 240 q^{18} + 120 q^{21} + 240 q^{22} + 288 q^{24} + 180 q^{25} - 240 q^{28} - 240 q^{30} - 480 q^{33} + 320 q^{34} + 24 q^{36} - 520 q^{37} + 320 q^{40} + 240 q^{42} + 960 q^{45} - 672 q^{46} - 480 q^{48} + 1132 q^{49} + 80 q^{52} + 792 q^{54} - 1080 q^{57} - 1360 q^{58} - 960 q^{60} - 1768 q^{61} + 1408 q^{64} + 1200 q^{66} + 1344 q^{69} + 480 q^{70} - 960 q^{72} + 1640 q^{73} + 2160 q^{76} + 240 q^{78} - 2844 q^{81} - 1120 q^{82} - 240 q^{84} - 1280 q^{85} - 2880 q^{88} - 240 q^{90} + 3480 q^{93} - 1344 q^{94} + 384 q^{96} + 3080 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/12\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(7\)
\(\chi(n)\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1
−0.866025 1.11803i
−0.866025 + 1.11803i
0.866025 1.11803i
0.866025 + 1.11803i
−1.73205 2.23607i 3.46410 3.87298i −2.00000 + 7.74597i 8.94427i −14.6603 1.03776i 7.74597i 20.7846 8.94427i −3.00000 26.8328i 20.0000 15.4919i
11.2 −1.73205 + 2.23607i 3.46410 + 3.87298i −2.00000 7.74597i 8.94427i −14.6603 + 1.03776i 7.74597i 20.7846 + 8.94427i −3.00000 + 26.8328i 20.0000 + 15.4919i
11.3 1.73205 2.23607i −3.46410 + 3.87298i −2.00000 7.74597i 8.94427i 2.66025 + 14.4542i 7.74597i −20.7846 8.94427i −3.00000 26.8328i 20.0000 + 15.4919i
11.4 1.73205 + 2.23607i −3.46410 3.87298i −2.00000 + 7.74597i 8.94427i 2.66025 14.4542i 7.74597i −20.7846 + 8.94427i −3.00000 + 26.8328i 20.0000 15.4919i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 12.4.b.a 4
3.b odd 2 1 inner 12.4.b.a 4
4.b odd 2 1 inner 12.4.b.a 4
8.b even 2 1 192.4.c.b 4
8.d odd 2 1 192.4.c.b 4
12.b even 2 1 inner 12.4.b.a 4
16.e even 4 2 768.4.f.c 8
16.f odd 4 2 768.4.f.c 8
24.f even 2 1 192.4.c.b 4
24.h odd 2 1 192.4.c.b 4
48.i odd 4 2 768.4.f.c 8
48.k even 4 2 768.4.f.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
12.4.b.a 4 1.a even 1 1 trivial
12.4.b.a 4 3.b odd 2 1 inner
12.4.b.a 4 4.b odd 2 1 inner
12.4.b.a 4 12.b even 2 1 inner
192.4.c.b 4 8.b even 2 1
192.4.c.b 4 8.d odd 2 1
192.4.c.b 4 24.f even 2 1
192.4.c.b 4 24.h odd 2 1
768.4.f.c 8 16.e even 4 2
768.4.f.c 8 16.f odd 4 2
768.4.f.c 8 48.i odd 4 2
768.4.f.c 8 48.k even 4 2

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(12, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 4T^{2} + 64 \) Copy content Toggle raw display
$3$ \( T^{4} + 6T^{2} + 729 \) Copy content Toggle raw display
$5$ \( (T^{2} + 80)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + 60)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} - 1200)^{2} \) Copy content Toggle raw display
$13$ \( (T + 10)^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} + 1280)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 4860)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} - 9408)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 23120)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 50460)^{2} \) Copy content Toggle raw display
$37$ \( (T + 130)^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} + 15680)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 50460)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} - 37632)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 297680)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} - 30000)^{2} \) Copy content Toggle raw display
$61$ \( (T + 442)^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} + 541500)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 1080000)^{2} \) Copy content Toggle raw display
$73$ \( (T - 410)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} + 7260)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} - 1572528)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 706880)^{2} \) Copy content Toggle raw display
$97$ \( (T - 770)^{4} \) Copy content Toggle raw display
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