Properties

Label 672.2.p.a
Level 672
Weight 2
Character orbit 672.p
Analytic conductor 5.366
Analytic rank 0
Dimension 16
CM No
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 672 = 2^{5} \cdot 3 \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 672.p (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(5.36594701583\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{17} \)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{4} q^{3} + \beta_{2} q^{5} + \beta_{7} q^{7} - q^{9} +O(q^{10})\) \( q + \beta_{4} q^{3} + \beta_{2} q^{5} + \beta_{7} q^{7} - q^{9} + \beta_{3} q^{11} + ( \beta_{2} + \beta_{7} - \beta_{8} - \beta_{10} - \beta_{12} ) q^{13} -\beta_{1} q^{15} -\beta_{15} q^{17} + ( 2 \beta_{4} + \beta_{9} - \beta_{13} ) q^{19} + \beta_{6} q^{21} + ( \beta_{5} + \beta_{6} + \beta_{8} ) q^{23} + ( 1 + \beta_{3} + \beta_{11} ) q^{25} -\beta_{4} q^{27} + ( -\beta_{1} - \beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} - \beta_{10} ) q^{29} + ( \beta_{2} + 2 \beta_{6} + \beta_{7} - \beta_{8} - \beta_{10} + \beta_{12} ) q^{31} -\beta_{9} q^{33} + ( -2 - \beta_{9} - \beta_{11} - \beta_{15} ) q^{35} + ( \beta_{1} + \beta_{5} + \beta_{7} + \beta_{10} ) q^{37} + ( -\beta_{1} - \beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} - \beta_{10} ) q^{39} + ( 4 \beta_{4} - 2 \beta_{13} + \beta_{15} ) q^{41} + ( \beta_{3} + \beta_{14} ) q^{43} -\beta_{2} q^{45} + ( -2 \beta_{8} - 2 \beta_{12} ) q^{47} + ( -1 + 2 \beta_{4} + \beta_{9} - \beta_{11} + \beta_{13} + \beta_{14} ) q^{49} -\beta_{14} q^{51} + ( -3 \beta_{1} - \beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} - \beta_{10} ) q^{53} + ( 3 \beta_{2} - 2 \beta_{6} + \beta_{8} - \beta_{12} ) q^{55} + ( -2 + \beta_{3} - \beta_{11} ) q^{57} + 2 \beta_{13} q^{59} + ( \beta_{2} + \beta_{7} + \beta_{8} - \beta_{10} + \beta_{12} ) q^{61} -\beta_{7} q^{63} -2 \beta_{14} q^{65} + ( 4 - \beta_{3} + 2 \beta_{11} - \beta_{14} ) q^{67} + ( -\beta_{6} - \beta_{7} + \beta_{10} - \beta_{12} ) q^{69} + ( -\beta_{5} + \beta_{6} + \beta_{8} ) q^{71} -2 \beta_{15} q^{73} + ( \beta_{4} - \beta_{9} - \beta_{13} ) q^{75} + ( -\beta_{1} - \beta_{2} + \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} + \beta_{10} + 2 \beta_{12} ) q^{77} + ( 2 \beta_{1} - 2 \beta_{6} + \beta_{7} - 2 \beta_{8} + \beta_{10} ) q^{79} + q^{81} + ( -2 \beta_{9} - 2 \beta_{13} + 2 \beta_{15} ) q^{83} + ( \beta_{1} + \beta_{5} - 2 \beta_{6} + 3 \beta_{7} - 2 \beta_{8} + 3 \beta_{10} ) q^{85} + ( -\beta_{2} - \beta_{7} + \beta_{8} + \beta_{10} + \beta_{12} ) q^{87} + ( -4 \beta_{4} + 2 \beta_{13} + \beta_{15} ) q^{89} + ( 4 - \beta_{3} - 2 \beta_{4} + \beta_{9} + \beta_{13} + \beta_{14} - 2 \beta_{15} ) q^{91} + ( -\beta_{1} + \beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} - \beta_{10} ) q^{93} + ( -2 \beta_{6} - 2 \beta_{8} ) q^{95} + ( -4 \beta_{4} + 2 \beta_{9} - 2 \beta_{13} + 2 \beta_{15} ) q^{97} -\beta_{3} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q - 16q^{9} + O(q^{10}) \) \( 16q - 16q^{9} + 8q^{11} + 16q^{25} - 24q^{35} + 8q^{43} - 8q^{49} - 16q^{57} + 40q^{67} + 16q^{81} + 56q^{91} - 8q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} + x^{14} - 4 x^{12} - 4 x^{10} + 16 x^{8} - 16 x^{6} - 64 x^{4} + 64 x^{2} + 256\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{14} + \nu^{12} + 16 \nu^{6} - 16 \nu^{4} \)\()/64\)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{11} + \nu^{9} + 2 \nu^{7} + 32 \nu \)\()/16\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{14} - \nu^{12} + 16 \nu^{6} + 48 \nu^{4} \)\()/64\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{15} + \nu^{13} + 2 \nu^{11} - 8 \nu^{9} - 8 \nu^{7} + 32 \nu^{5} - 64 \nu \)\()/128\)
\(\beta_{5}\)\(=\)\((\)\( -\nu^{14} - \nu^{12} + 8 \nu^{10} + 8 \nu^{8} - 16 \nu^{6} + 16 \nu^{4} - 128 \)\()/64\)
\(\beta_{6}\)\(=\)\((\)\( \nu^{13} - \nu^{11} + 4 \nu^{10} - 2 \nu^{9} - 4 \nu^{8} - 8 \nu^{6} + 16 \nu^{5} + 32 \nu^{4} - 48 \nu^{3} + 32 \nu^{2} - 32 \nu - 128 \)\()/64\)
\(\beta_{7}\)\(=\)\((\)\( \nu^{15} + \nu^{13} - 4 \nu^{12} - 4 \nu^{11} + 4 \nu^{10} + 4 \nu^{9} - 8 \nu^{8} + 8 \nu^{7} - 16 \nu^{6} - 64 \nu^{5} - 32 \nu^{3} + 128 \nu^{2} + 128 \nu - 128 \)\()/128\)
\(\beta_{8}\)\(=\)\((\)\( -\nu^{13} + \nu^{11} + 4 \nu^{10} + 2 \nu^{9} - 4 \nu^{8} - 8 \nu^{6} - 16 \nu^{5} + 32 \nu^{4} + 48 \nu^{3} + 32 \nu^{2} + 32 \nu - 128 \)\()/64\)
\(\beta_{9}\)\(=\)\((\)\( \nu^{15} + \nu^{13} + 32 \nu^{3} \)\()/64\)
\(\beta_{10}\)\(=\)\((\)\( -\nu^{15} - \nu^{13} - 4 \nu^{12} + 4 \nu^{11} + 4 \nu^{10} - 4 \nu^{9} - 8 \nu^{8} - 8 \nu^{7} - 16 \nu^{6} + 64 \nu^{5} + 32 \nu^{3} + 128 \nu^{2} - 128 \nu - 128 \)\()/128\)
\(\beta_{11}\)\(=\)\((\)\( \nu^{14} - 3 \nu^{12} - 4 \nu^{10} + 16 \nu^{8} - 48 \nu^{4} + 128 \nu^{2} + 128 \)\()/64\)
\(\beta_{12}\)\(=\)\((\)\( \nu^{15} - 7 \nu^{11} - 4 \nu^{10} + 2 \nu^{9} + 4 \nu^{8} + 24 \nu^{7} + 8 \nu^{6} - 32 \nu^{5} - 32 \nu^{4} - 16 \nu^{3} - 32 \nu^{2} + 96 \nu + 128 \)\()/64\)
\(\beta_{13}\)\(=\)\((\)\( -\nu^{15} + 3 \nu^{13} + 4 \nu^{11} - 8 \nu^{9} + 8 \nu^{7} + 48 \nu^{5} - 64 \nu^{3} - 64 \nu \)\()/64\)
\(\beta_{14}\)\(=\)\((\)\( -3 \nu^{14} + \nu^{12} + 12 \nu^{10} - 8 \nu^{8} - 32 \nu^{6} + 80 \nu^{4} + 128 \nu^{2} - 256 \)\()/64\)
\(\beta_{15}\)\(=\)\((\)\( -\nu^{15} + \nu^{13} + 4 \nu^{11} - 2 \nu^{9} - 4 \nu^{7} + 32 \nu^{5} - 128 \nu \)\()/32\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{15} + \beta_{13} - \beta_{12} - \beta_{6} + \beta_{2}\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{14} + \beta_{11} - \beta_{5} + \beta_{1}\)\()/4\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{15} - \beta_{13} + \beta_{12} + 2 \beta_{9} + 2 \beta_{8} - \beta_{6} + 4 \beta_{4} + \beta_{2}\)\()/4\)
\(\nu^{4}\)\(=\)\((\)\(-\beta_{14} + \beta_{11} - 2 \beta_{10} + 4 \beta_{8} - 2 \beta_{7} + 4 \beta_{6} - \beta_{5} + 2 \beta_{3} - 3 \beta_{1} + 4\)\()/4\)
\(\nu^{5}\)\(=\)\((\)\(-\beta_{15} + \beta_{13} - \beta_{12} + 4 \beta_{10} + 2 \beta_{9} - 2 \beta_{8} - 4 \beta_{7} + \beta_{6} - 4 \beta_{4} + 3 \beta_{2}\)\()/4\)
\(\nu^{6}\)\(=\)\((\)\(\beta_{14} - \beta_{11} + 2 \beta_{10} - 4 \beta_{8} + 2 \beta_{7} - 4 \beta_{6} + \beta_{5} + 6 \beta_{3} + 11 \beta_{1} - 4\)\()/4\)
\(\nu^{7}\)\(=\)\((\)\(\beta_{15} + 7 \beta_{13} + 9 \beta_{12} + 4 \beta_{10} - 2 \beta_{9} + 10 \beta_{8} - 4 \beta_{7} - \beta_{6} - 12 \beta_{4} - 3 \beta_{2}\)\()/4\)
\(\nu^{8}\)\(=\)\((\)\(-\beta_{14} + 9 \beta_{11} - 10 \beta_{10} + 4 \beta_{8} - 10 \beta_{7} + 4 \beta_{6} + 7 \beta_{5} + 2 \beta_{3} - 3 \beta_{1} - 12\)\()/4\)
\(\nu^{9}\)\(=\)\((\)\(15 \beta_{15} - 7 \beta_{13} - 9 \beta_{12} - 4 \beta_{10} + 2 \beta_{9} - 10 \beta_{8} + 4 \beta_{7} + \beta_{6} - 52 \beta_{4} + 19 \beta_{2}\)\()/4\)
\(\nu^{10}\)\(=\)\((\)\(\beta_{14} - 9 \beta_{11} + 10 \beta_{10} - 4 \beta_{8} + 10 \beta_{7} - 4 \beta_{6} + 25 \beta_{5} - 2 \beta_{3} + 35 \beta_{1} + 76\)\()/4\)
\(\nu^{11}\)\(=\)\((\)\(-15 \beta_{15} + 39 \beta_{13} - 23 \beta_{12} + 4 \beta_{10} - 2 \beta_{9} + 10 \beta_{8} - 4 \beta_{7} - 33 \beta_{6} - 76 \beta_{4} - 19 \beta_{2}\)\()/4\)
\(\nu^{12}\)\(=\)\((\)\(31 \beta_{14} + 9 \beta_{11} - 42 \beta_{10} + 4 \beta_{8} - 42 \beta_{7} + 4 \beta_{6} - 25 \beta_{5} - 30 \beta_{3} + 29 \beta_{1} - 12\)\()/4\)
\(\nu^{13}\)\(=\)\((\)\(47 \beta_{15} - 7 \beta_{13} - 9 \beta_{12} - 68 \beta_{10} + 66 \beta_{9} - 10 \beta_{8} + 68 \beta_{7} + \beta_{6} + 76 \beta_{4} + 51 \beta_{2}\)\()/4\)
\(\nu^{14}\)\(=\)\((\)\(-63 \beta_{14} + 23 \beta_{11} - 22 \beta_{10} + 124 \beta_{8} - 22 \beta_{7} + 124 \beta_{6} - 7 \beta_{5} - 34 \beta_{3} + 3 \beta_{1} + 140\)\()/4\)
\(\nu^{15}\)\(=\)\((\)\(-79 \beta_{15} + 39 \beta_{13} - 23 \beta_{12} + 68 \beta_{10} + 126 \beta_{9} - 54 \beta_{8} - 68 \beta_{7} + 31 \beta_{6} - 204 \beta_{4} - 83 \beta_{2}\)\()/4\)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/672\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(421\) \(449\) \(577\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
559.1
−1.40199 + 0.185533i
−0.310478 + 1.37971i
0.474920 1.33209i
1.20933 0.733159i
−1.20933 0.733159i
−0.474920 1.33209i
0.310478 + 1.37971i
1.40199 + 0.185533i
−1.40199 0.185533i
−0.310478 1.37971i
0.474920 + 1.33209i
1.20933 + 0.733159i
−1.20933 + 0.733159i
−0.474920 + 1.33209i
0.310478 1.37971i
1.40199 0.185533i
0 1.00000i 0 −3.84444 0 1.62140 + 2.09071i 0 −1.00000 0
559.2 0 1.00000i 0 −2.33443 0 0.490487 2.59989i 0 −1.00000 0
559.3 0 1.00000i 0 −1.58069 0 −2.37995 + 1.15578i 0 −1.00000 0
559.4 0 1.00000i 0 −1.12786 0 2.11337 1.59175i 0 −1.00000 0
559.5 0 1.00000i 0 1.12786 0 −2.11337 + 1.59175i 0 −1.00000 0
559.6 0 1.00000i 0 1.58069 0 2.37995 1.15578i 0 −1.00000 0
559.7 0 1.00000i 0 2.33443 0 −0.490487 + 2.59989i 0 −1.00000 0
559.8 0 1.00000i 0 3.84444 0 −1.62140 2.09071i 0 −1.00000 0
559.9 0 1.00000i 0 −3.84444 0 1.62140 2.09071i 0 −1.00000 0
559.10 0 1.00000i 0 −2.33443 0 0.490487 + 2.59989i 0 −1.00000 0
559.11 0 1.00000i 0 −1.58069 0 −2.37995 1.15578i 0 −1.00000 0
559.12 0 1.00000i 0 −1.12786 0 2.11337 + 1.59175i 0 −1.00000 0
559.13 0 1.00000i 0 1.12786 0 −2.11337 1.59175i 0 −1.00000 0
559.14 0 1.00000i 0 1.58069 0 2.37995 + 1.15578i 0 −1.00000 0
559.15 0 1.00000i 0 2.33443 0 −0.490487 2.59989i 0 −1.00000 0
559.16 0 1.00000i 0 3.84444 0 −1.62140 + 2.09071i 0 −1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 559.16
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
7.b Odd 1 yes
8.d Odd 1 yes
56.e Even 1 yes

Hecke kernels

There are no other newforms in \(S_{2}^{\mathrm{new}}(672, [\chi])\).