Properties

Label 3450.2.d.y
Level $3450$
Weight $2$
Character orbit 3450.d
Analytic conductor $27.548$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 3450 = 2 \cdot 3 \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3450.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(27.5483886973\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{6})\)
Defining polynomial: \(x^{4} + 9\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{2} \)\(/3\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} + 3 \nu \)\()/3\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{3} + 3 \nu \)\()/3\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + \beta_{2}\)\()/2\)
\(\nu^{2}\)\(=\)\(3 \beta_{1}\)
\(\nu^{3}\)\(=\)\((\)\(-3 \beta_{3} + 3 \beta_{2}\)\()/2\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3450\mathbb{Z}\right)^\times\).

\(n\) \(277\) \(1151\) \(1201\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2899.1
−1.22474 1.22474i
1.22474 + 1.22474i
1.22474 1.22474i
−1.22474 + 1.22474i
1.00000i 1.00000i −1.00000 0 1.00000 1.44949i 1.00000i −1.00000 0
2899.2 1.00000i 1.00000i −1.00000 0 1.00000 3.44949i 1.00000i −1.00000 0
2899.3 1.00000i 1.00000i −1.00000 0 1.00000 3.44949i 1.00000i −1.00000 0
2899.4 1.00000i 1.00000i −1.00000 0 1.00000 1.44949i 1.00000i −1.00000 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
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3450.2.d.y 4
5.b even 2 1 inner 3450.2.d.y 4
5.c odd 4 1 3450.2.a.bf 2
5.c odd 4 1 3450.2.a.bl yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3450.2.a.bf 2 5.c odd 4 1
3450.2.a.bl yes 2 5.c odd 4 1
3450.2.d.y 4 1.a even 1 1 trivial
3450.2.d.y 4 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3450, [\chi])\):

\( T_{7}^{4} + 14 T_{7}^{2} + 25 \)
\( T_{11} - 2 \)
\( T_{13}^{4} + 56 T_{13}^{2} + 400 \)
\( T_{17}^{4} + 30 T_{17}^{2} + 9 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T^{2} )^{2} \)
$3$ \( ( 1 + T^{2} )^{2} \)
$5$ \( T^{4} \)
$7$ \( 25 + 14 T^{2} + T^{4} \)
$11$ \( ( -2 + T )^{4} \)
$13$ \( 400 + 56 T^{2} + T^{4} \)
$17$ \( 9 + 30 T^{2} + T^{4} \)
$19$ \( ( -20 + 4 T + T^{2} )^{2} \)
$23$ \( ( 1 + T^{2} )^{2} \)
$29$ \( ( 5 + T )^{4} \)
$31$ \( ( -2 + T )^{4} \)
$37$ \( 2809 + 110 T^{2} + T^{4} \)
$41$ \( ( -24 + T^{2} )^{2} \)
$43$ \( 144 + 120 T^{2} + T^{4} \)
$47$ \( 529 + 50 T^{2} + T^{4} \)
$53$ \( 64 + 80 T^{2} + T^{4} \)
$59$ \( ( 10 + T )^{4} \)
$61$ \( ( -24 + T^{2} )^{2} \)
$67$ \( ( 4 + T^{2} )^{2} \)
$71$ \( ( -95 - 2 T + T^{2} )^{2} \)
$73$ \( 625 + 146 T^{2} + T^{4} \)
$79$ \( ( 10 + T )^{4} \)
$83$ \( 361 + 62 T^{2} + T^{4} \)
$89$ \( ( -45 - 6 T + T^{2} )^{2} \)
$97$ \( 44944 + 440 T^{2} + T^{4} \)
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