Properties

Label 354.2.c.b
Level 354
Weight 2
Character orbit 354.c
Analytic conductor 2.827
Analytic rank 0
Dimension 10
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 354 = 2 \cdot 3 \cdot 59 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 354.c (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(2.82670423155\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: 10.0.41542366334681088.1
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{10} - x^{9} - x^{8} - 2 x^{7} + 2 x^{6} + 14 x^{5} + 6 x^{4} - 18 x^{3} - 27 x^{2} - 81 x + 243\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -\nu^{9} - 2 \nu^{8} - 23 \nu^{7} - 49 \nu^{6} + 31 \nu^{5} - 47 \nu^{4} + 141 \nu^{3} + 27 \nu^{2} - 1296 \nu - 405 \)\()/648\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{8} - 7 \nu^{7} - 4 \nu^{6} + 13 \nu^{5} - 4 \nu^{4} - 7 \nu^{3} + 12 \nu^{2} + 9 \nu + 27 \)\()/108\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{9} + 8 \nu^{8} + 17 \nu^{7} - 11 \nu^{6} + 11 \nu^{5} - 49 \nu^{4} + 213 \nu^{3} + 225 \nu^{2} - 54 \nu - 81 \)\()/648\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{9} + 8 \nu^{8} - 19 \nu^{7} + 25 \nu^{6} + 47 \nu^{5} + 23 \nu^{4} - 183 \nu^{3} + 45 \nu^{2} - 594 \nu + 567 \)\()/648\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{9} + 8 \nu^{8} - \nu^{7} + 7 \nu^{6} + 29 \nu^{5} - 13 \nu^{4} + 15 \nu^{3} - 189 \nu^{2} + 243 \)\()/324\)
\(\beta_{6}\)\(=\)\((\)\( -\nu^{9} + \nu^{8} + \nu^{7} + 2 \nu^{6} - 2 \nu^{5} - 14 \nu^{4} - 6 \nu^{3} + 18 \nu^{2} + 27 \nu + 81 \)\()/81\)
\(\beta_{7}\)\(=\)\((\)\( -2 \nu^{9} - 4 \nu^{8} - \nu^{7} + 19 \nu^{6} + 17 \nu^{5} - 22 \nu^{4} - 33 \nu^{3} - 45 \nu^{2} + 27 \nu + 324 \)\()/162\)
\(\beta_{8}\)\(=\)\((\)\( -\nu^{9} - 2 \nu^{8} + 4 \nu^{7} + 5 \nu^{6} + 4 \nu^{5} - 20 \nu^{4} - 48 \nu^{3} + 81 \nu + 162 \)\()/81\)
\(\beta_{9}\)\(=\)\((\)\( 8 \nu^{9} + 7 \nu^{8} - 5 \nu^{7} - 22 \nu^{6} - 59 \nu^{5} - 2 \nu^{4} + 159 \nu^{3} + 90 \nu^{2} + 81 \nu - 729 \)\()/324\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{4} + \beta_{2} - \beta_{1}\)\()/3\)
\(\nu^{2}\)\(=\)\((\)\(-3 \beta_{5} + 2 \beta_{4} + 3 \beta_{3} + \beta_{2} - \beta_{1}\)\()/3\)
\(\nu^{3}\)\(=\)\(-\beta_{8} + \beta_{7} - \beta_{4} + \beta_{3} + 1\)
\(\nu^{4}\)\(=\)\((\)\(-6 \beta_{9} - 6 \beta_{8} - 3 \beta_{7} - 3 \beta_{6} - 3 \beta_{5} + \beta_{4} - \beta_{2} - 5 \beta_{1} + 6\)\()/3\)
\(\nu^{5}\)\(=\)\((\)\(-6 \beta_{9} + 3 \beta_{8} - 12 \beta_{6} + 3 \beta_{5} + \beta_{4} + 15 \beta_{3} + 8 \beta_{2} - 2 \beta_{1} - 12\)\()/3\)
\(\nu^{6}\)\(=\)\(2 \beta_{9} - 3 \beta_{8} + 7 \beta_{7} + \beta_{6} - \beta_{5} + 5 \beta_{4} + 2 \beta_{3} - 3 \beta_{2} - 3 \beta_{1} - 9\)
\(\nu^{7}\)\(=\)\((\)\(-12 \beta_{9} + 15 \beta_{8} - 18 \beta_{7} - 15 \beta_{6} + 9 \beta_{5} + 2 \beta_{4} + 27 \beta_{3} - 23 \beta_{2} - \beta_{1} - 6\)\()/3\)
\(\nu^{8}\)\(=\)\((\)\(-6 \beta_{9} - 15 \beta_{8} - 33 \beta_{7} + 51 \beta_{6} + 36 \beta_{5} + 29 \beta_{4} + 3 \beta_{3} - 2 \beta_{2} - 16 \beta_{1} - 30\)\()/3\)
\(\nu^{9}\)\(=\)\(30 \beta_{9} + 26 \beta_{8} + 5 \beta_{7} - 45 \beta_{6} + 7 \beta_{5} + 24 \beta_{4} + 16 \beta_{3} - 2 \beta_{1} + 25\)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/354\mathbb{Z}\right)^\times\).

\(n\) \(61\) \(119\)
\(\chi(n)\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
353.1
1.65530 0.509894i
1.65530 + 0.509894i
1.43134 0.975334i
1.43134 + 0.975334i
−0.0738304 1.73048i
−0.0738304 + 1.73048i
−0.869925 1.49774i
−0.869925 + 1.49774i
−1.64288 0.548592i
−1.64288 + 0.548592i
1.00000 −1.65530 0.509894i 1.00000 2.90260i −1.65530 0.509894i −3.06916 1.00000 2.48002 + 1.68805i 2.90260i
353.2 1.00000 −1.65530 + 0.509894i 1.00000 2.90260i −1.65530 + 0.509894i −3.06916 1.00000 2.48002 1.68805i 2.90260i
353.3 1.00000 −1.43134 0.975334i 1.00000 0.0999120i −1.43134 0.975334i 2.48626 1.00000 1.09745 + 2.79206i 0.0999120i
353.4 1.00000 −1.43134 + 0.975334i 1.00000 0.0999120i −1.43134 + 0.975334i 2.48626 1.00000 1.09745 2.79206i 0.0999120i
353.5 1.00000 0.0738304 1.73048i 1.00000 2.30520i 0.0738304 1.73048i 3.25240 1.00000 −2.98910 0.255524i 2.30520i
353.6 1.00000 0.0738304 + 1.73048i 1.00000 2.30520i 0.0738304 + 1.73048i 3.25240 1.00000 −2.98910 + 0.255524i 2.30520i
353.7 1.00000 0.869925 1.49774i 1.00000 1.66014i 0.869925 1.49774i −3.57946 1.00000 −1.48646 2.60585i 1.66014i
353.8 1.00000 0.869925 + 1.49774i 1.00000 1.66014i 0.869925 + 1.49774i −3.57946 1.00000 −1.48646 + 2.60585i 1.66014i
353.9 1.00000 1.64288 0.548592i 1.00000 2.54852i 1.64288 0.548592i −0.0900537 1.00000 2.39809 1.80254i 2.54852i
353.10 1.00000 1.64288 + 0.548592i 1.00000 2.54852i 1.64288 + 0.548592i −0.0900537 1.00000 2.39809 + 1.80254i 2.54852i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 353.10
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
177.d Even 1 yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \( T_{11}^{5} + 2 T_{11}^{4} - 25 T_{11}^{3} - 54 T_{11}^{2} + 54 T_{11} + 108 \) acting on \(S_{2}^{\mathrm{new}}(354, [\chi])\).