Properties

Label 3450.2.d.w
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: \(\Q(i, \sqrt{73})\)
Defining polynomial: \(x^{4} + 37 x^{2} + 324\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 37 x^{2} + 324\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} + 19 \nu \)\()/18\)
\(\beta_{3}\)\(=\)\( \nu^{2} + 19 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} - 19\)
\(\nu^{3}\)\(=\)\(18 \beta_{2} - 19 \beta_{1}\)

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
3.77200i
4.77200i
3.77200i
4.77200i
1.00000i 1.00000i −1.00000 0 −1.00000 3.00000i 1.00000i −1.00000 0
2899.2 1.00000i 1.00000i −1.00000 0 −1.00000 3.00000i 1.00000i −1.00000 0
2899.3 1.00000i 1.00000i −1.00000 0 −1.00000 3.00000i 1.00000i −1.00000 0
2899.4 1.00000i 1.00000i −1.00000 0 −1.00000 3.00000i 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.w 4
5.b even 2 1 inner 3450.2.d.w 4
5.c odd 4 1 3450.2.a.bh 2
5.c odd 4 1 3450.2.a.bj yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3450.2.a.bh 2 5.c odd 4 1
3450.2.a.bj yes 2 5.c odd 4 1
3450.2.d.w 4 1.a even 1 1 trivial
3450.2.d.w 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 \(S_{2}^{\mathrm{new}}(3450, [\chi])\):

\( T_{7}^{2} + 9 \)
\( T_{11}^{2} - 3 T_{11} - 16 \)
\( T_{13}^{4} + 37 T_{13}^{2} + 324 \)
\( T_{17}^{4} + 61 T_{17}^{2} + 36 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T^{2} )^{2} \)
$3$ \( ( 1 + T^{2} )^{2} \)
$5$ \( T^{4} \)
$7$ \( ( 9 + T^{2} )^{2} \)
$11$ \( ( -16 - 3 T + T^{2} )^{2} \)
$13$ \( 324 + 37 T^{2} + T^{4} \)
$17$ \( 36 + 61 T^{2} + T^{4} \)
$19$ \( ( -6 + 7 T + T^{2} )^{2} \)
$23$ \( ( 1 + T^{2} )^{2} \)
$29$ \( ( 3 + T )^{4} \)
$31$ \( ( -72 + 2 T + T^{2} )^{2} \)
$37$ \( 144 + 49 T^{2} + T^{4} \)
$41$ \( ( -16 + 3 T + T^{2} )^{2} \)
$43$ \( 36 + 61 T^{2} + T^{4} \)
$47$ \( 4 + 77 T^{2} + T^{4} \)
$53$ \( ( 16 + T^{2} )^{2} \)
$59$ \( ( 6 + T )^{4} \)
$61$ \( ( -72 + 2 T + T^{2} )^{2} \)
$67$ \( 5184 + 148 T^{2} + T^{4} \)
$71$ \( ( -18 + T + T^{2} )^{2} \)
$73$ \( 4761 + 154 T^{2} + T^{4} \)
$79$ \( ( 24 + 13 T + T^{2} )^{2} \)
$83$ \( ( 1 + T^{2} )^{2} \)
$89$ \( ( 138 + 25 T + T^{2} )^{2} \)
$97$ \( 64 + 308 T^{2} + T^{4} \)
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