Properties

Label 448.3.s.d
Level $448$
Weight $3$
Character orbit 448.s
Analytic conductor $12.207$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 448 = 2^{6} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 448.s (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(12.2071158433\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
Defining polynomial: \( x^{4} + 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 14)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_{3} + \beta_1 + 2) q^{3} + ( - 2 \beta_{3} + 2 \beta_{2} - \beta_1 + 1) q^{5} + ( - 2 \beta_{3} + 3 \beta_{2} + 2 \beta_1 + 3) q^{7} + (2 \beta_{3} + 2 \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{3} + \beta_1 + 2) q^{3} + ( - 2 \beta_{3} + 2 \beta_{2} - \beta_1 + 1) q^{5} + ( - 2 \beta_{3} + 3 \beta_{2} + 2 \beta_1 + 3) q^{7} + (2 \beta_{3} + 2 \beta_{2}) q^{9} + ( - \beta_{3} + 2 \beta_{2} + 9 \beta_1) q^{11} + (2 \beta_{2} - 12 \beta_1 - 6) q^{13} + ( - 2 \beta_{3} + \beta_{2} - 9) q^{15} + ( - 2 \beta_{3} - 5 \beta_1 - 10) q^{17} + (\beta_{3} - \beta_{2} + \beta_1 - 1) q^{19} + ( - 4 \beta_{3} + 6 \beta_{2} + 11 \beta_1 - 8) q^{21} + ( - 3 \beta_{3} - 3 \beta_{2} + 15 \beta_1 + 15) q^{23} + ( - 4 \beta_{3} + 8 \beta_{2} - 2 \beta_1) q^{25} + ( - 3 \beta_{2} + 6 \beta_1 + 3) q^{27} + ( - 4 \beta_{3} + 2 \beta_{2} - 12) q^{29} + (15 \beta_{3} - 7 \beta_1 - 14) q^{31} + ( - 12 \beta_{3} + 12 \beta_{2} + 15 \beta_1 - 15) q^{33} + ( - 7 \beta_{3} + 7 \beta_{2} - 35 \beta_1 - 7) q^{35} + (8 \beta_{3} + 8 \beta_{2} + 31 \beta_1 + 31) q^{37} + (4 \beta_{3} - 8 \beta_{2} - 6 \beta_1) q^{39} + (10 \beta_{2} - 4 \beta_1 - 2) q^{41} + ( - 4 \beta_{3} + 2 \beta_{2} + 2) q^{43} + (6 \beta_{3} - 24 \beta_1 - 48) q^{45} + ( - \beta_{3} + \beta_{2} - 29 \beta_1 + 29) q^{47} + ( - 16 \beta_{3} + 10 \beta_{2} - 40 \beta_1 - 25) q^{49} + ( - 7 \beta_{3} - 7 \beta_{2} - 27 \beta_1 - 27) q^{51} + ( - 4 \beta_{3} + 8 \beta_{2} - 39 \beta_1) q^{53} + ( - 15 \beta_{2} - 6 \beta_1 - 3) q^{55} + 3 q^{57} + (25 \beta_{3} + 13 \beta_1 + 26) q^{59} + ( - 32 \beta_{3} + 32 \beta_{2} - 7 \beta_1 + 7) q^{61} + ( - 2 \beta_{3} + 10 \beta_{2} - 12 \beta_1 - 60) q^{63} + (14 \beta_{3} + 14 \beta_{2} - 42 \beta_1 - 42) q^{65} + (15 \beta_{3} - 30 \beta_{2} - 29 \beta_1) q^{67} + (6 \beta_{2} - 6 \beta_1 - 3) q^{69} + ( - 20 \beta_{3} + 10 \beta_{2} - 6) q^{71} + (16 \beta_{3} + 53 \beta_1 + 106) q^{73} + ( - 10 \beta_{3} + 10 \beta_{2} + 22 \beta_1 - 22) q^{75} + ( - 14 \beta_{3} - 14 \beta_{2} - 21 \beta_1 - 42) q^{77} + (5 \beta_{3} + 5 \beta_{2} + 55 \beta_1 + 55) q^{79} + (18 \beta_{3} - 36 \beta_{2} - 9 \beta_1) q^{81} + ( - 4 \beta_{2} + 136 \beta_1 + 68) q^{83} + (16 \beta_{3} - 8 \beta_{2} + 9) q^{85} + ( - 18 \beta_{3} - 24 \beta_1 - 48) q^{87} + ( - 24 \beta_{3} + 24 \beta_{2} + 63 \beta_1 - 63) q^{89} + (20 \beta_{3} + 12 \beta_{2} - 48 \beta_1 - 30) q^{91} + (8 \beta_{3} + 8 \beta_{2} + 69 \beta_1 + 69) q^{93} + (3 \beta_{3} - 6 \beta_{2} + 15 \beta_1) q^{95} + ( - 26 \beta_{2} + 44 \beta_1 + 22) q^{97} + ( - 36 \beta_{3} + 18 \beta_{2} - 36) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 6 q^{3} + 6 q^{5} + 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 6 q^{3} + 6 q^{5} + 8 q^{7} - 18 q^{11} - 36 q^{15} - 30 q^{17} - 6 q^{19} - 54 q^{21} + 30 q^{23} + 4 q^{25} - 48 q^{29} - 42 q^{31} - 90 q^{33} + 42 q^{35} + 62 q^{37} + 12 q^{39} + 8 q^{43} - 144 q^{45} + 174 q^{47} - 20 q^{49} - 54 q^{51} + 78 q^{53} + 12 q^{57} + 78 q^{59} + 42 q^{61} - 216 q^{63} - 84 q^{65} + 58 q^{67} - 24 q^{71} + 318 q^{73} - 132 q^{75} - 126 q^{77} + 110 q^{79} + 18 q^{81} + 36 q^{85} - 144 q^{87} - 378 q^{89} - 24 q^{91} + 138 q^{93} - 30 q^{95} - 144 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/448\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(129\) \(197\)
\(\chi(n)\) \(1\) \(1 + \beta_{1}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
129.1
−0.707107 1.22474i
0.707107 + 1.22474i
−0.707107 + 1.22474i
0.707107 1.22474i
0 −0.621320 0.358719i 0 5.74264 3.31552i 0 6.24264 3.16693i 0 −4.24264 7.34847i 0
129.2 0 3.62132 + 2.09077i 0 −2.74264 + 1.58346i 0 −2.24264 + 6.63103i 0 4.24264 + 7.34847i 0
257.1 0 −0.621320 + 0.358719i 0 5.74264 + 3.31552i 0 6.24264 + 3.16693i 0 −4.24264 + 7.34847i 0
257.2 0 3.62132 2.09077i 0 −2.74264 1.58346i 0 −2.24264 6.63103i 0 4.24264 7.34847i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 448.3.s.d 4
4.b odd 2 1 448.3.s.c 4
7.d odd 6 1 inner 448.3.s.d 4
8.b even 2 1 14.3.d.a 4
8.d odd 2 1 112.3.s.b 4
24.f even 2 1 1008.3.cg.l 4
24.h odd 2 1 126.3.n.c 4
28.f even 6 1 448.3.s.c 4
40.f even 2 1 350.3.k.a 4
40.i odd 4 2 350.3.i.a 8
56.e even 2 1 784.3.s.c 4
56.h odd 2 1 98.3.d.a 4
56.j odd 6 1 14.3.d.a 4
56.j odd 6 1 98.3.b.b 4
56.k odd 6 1 784.3.c.e 4
56.k odd 6 1 784.3.s.c 4
56.m even 6 1 112.3.s.b 4
56.m even 6 1 784.3.c.e 4
56.p even 6 1 98.3.b.b 4
56.p even 6 1 98.3.d.a 4
168.i even 2 1 882.3.n.b 4
168.s odd 6 1 882.3.c.f 4
168.s odd 6 1 882.3.n.b 4
168.ba even 6 1 126.3.n.c 4
168.ba even 6 1 882.3.c.f 4
168.be odd 6 1 1008.3.cg.l 4
280.bk odd 6 1 350.3.k.a 4
280.bv even 12 2 350.3.i.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.3.d.a 4 8.b even 2 1
14.3.d.a 4 56.j odd 6 1
98.3.b.b 4 56.j odd 6 1
98.3.b.b 4 56.p even 6 1
98.3.d.a 4 56.h odd 2 1
98.3.d.a 4 56.p even 6 1
112.3.s.b 4 8.d odd 2 1
112.3.s.b 4 56.m even 6 1
126.3.n.c 4 24.h odd 2 1
126.3.n.c 4 168.ba even 6 1
350.3.i.a 8 40.i odd 4 2
350.3.i.a 8 280.bv even 12 2
350.3.k.a 4 40.f even 2 1
350.3.k.a 4 280.bk odd 6 1
448.3.s.c 4 4.b odd 2 1
448.3.s.c 4 28.f even 6 1
448.3.s.d 4 1.a even 1 1 trivial
448.3.s.d 4 7.d odd 6 1 inner
784.3.c.e 4 56.k odd 6 1
784.3.c.e 4 56.m even 6 1
784.3.s.c 4 56.e even 2 1
784.3.s.c 4 56.k odd 6 1
882.3.c.f 4 168.s odd 6 1
882.3.c.f 4 168.ba even 6 1
882.3.n.b 4 168.i even 2 1
882.3.n.b 4 168.s odd 6 1
1008.3.cg.l 4 24.f even 2 1
1008.3.cg.l 4 168.be odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} - 6T_{3}^{3} + 9T_{3}^{2} + 18T_{3} + 9 \) acting on \(S_{3}^{\mathrm{new}}(448, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} - 6 T^{3} + 9 T^{2} + 18 T + 9 \) Copy content Toggle raw display
$5$ \( T^{4} - 6 T^{3} - 9 T^{2} + 126 T + 441 \) Copy content Toggle raw display
$7$ \( T^{4} - 8 T^{3} + 42 T^{2} + \cdots + 2401 \) Copy content Toggle raw display
$11$ \( T^{4} + 18 T^{3} + 261 T^{2} + \cdots + 3969 \) Copy content Toggle raw display
$13$ \( T^{4} + 264T^{2} + 7056 \) Copy content Toggle raw display
$17$ \( T^{4} + 30 T^{3} + 351 T^{2} + \cdots + 2601 \) Copy content Toggle raw display
$19$ \( T^{4} + 6 T^{3} + 9 T^{2} - 18 T + 9 \) Copy content Toggle raw display
$23$ \( T^{4} - 30 T^{3} + 837 T^{2} + \cdots + 3969 \) Copy content Toggle raw display
$29$ \( (T^{2} + 24 T + 72)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + 42 T^{3} - 615 T^{2} + \cdots + 1447209 \) Copy content Toggle raw display
$37$ \( T^{4} - 62 T^{3} + 4035 T^{2} + \cdots + 36481 \) Copy content Toggle raw display
$41$ \( T^{4} + 1224 T^{2} + 345744 \) Copy content Toggle raw display
$43$ \( (T^{2} - 4 T - 68)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} - 174 T^{3} + 12609 T^{2} + \cdots + 6335289 \) Copy content Toggle raw display
$53$ \( T^{4} - 78 T^{3} + 4851 T^{2} + \cdots + 1520289 \) Copy content Toggle raw display
$59$ \( T^{4} - 78 T^{3} - 1215 T^{2} + \cdots + 10517049 \) Copy content Toggle raw display
$61$ \( T^{4} - 42 T^{3} - 5409 T^{2} + \cdots + 35964009 \) Copy content Toggle raw display
$67$ \( T^{4} - 58 T^{3} + 6573 T^{2} + \cdots + 10297681 \) Copy content Toggle raw display
$71$ \( (T^{2} + 12 T - 1764)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} - 318 T^{3} + \cdots + 47485881 \) Copy content Toggle raw display
$79$ \( T^{4} - 110 T^{3} + 9525 T^{2} + \cdots + 6630625 \) Copy content Toggle raw display
$83$ \( T^{4} + 27936 T^{2} + \cdots + 189778176 \) Copy content Toggle raw display
$89$ \( T^{4} + 378 T^{3} + \cdots + 71419401 \) Copy content Toggle raw display
$97$ \( T^{4} + 11016 T^{2} + \cdots + 6780816 \) Copy content Toggle raw display
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