Properties

Label 36.33.d.a
Level 36
Weight 33
Character orbit 36.d
Self dual yes
Analytic conductor 233.520
Analytic rank 0
Dimension 1
CM discriminant -4
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 36 = 2^{2} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 33 \)
Character orbit: \([\chi]\) \(=\) 36.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(233.519958512\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 4)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

\(f(q)\) \(=\) \( q - 65536q^{2} + 4294967296q^{4} + 196496109694q^{5} - 281474976710656q^{8} + O(q^{10}) \) \( q - 65536q^{2} + 4294967296q^{4} + 196496109694q^{5} - 281474976710656q^{8} - 12877569044905984q^{10} + 1330087744899070082q^{13} + 18446744073709551616q^{16} - 1427124567881986562q^{17} + 843944364926958567424q^{20} + 15327656759489517883011q^{25} - 87168630449705456893952q^{26} - 462878764200641031680642q^{29} - 1208925819614629174706176q^{32} + 93528035680713871327232q^{34} + 13364603395730595798238082q^{37} - 55308737899853156674699264q^{40} + 117479780930606773712920318q^{41} + 1104427674243920646305299201q^{49} - 1004513313389905043981008896q^{50} + 5712683365151896823002038272q^{52} + 6730923570418671225390264958q^{53} + 30335222690653210652222554112q^{58} - 71388302423745245269146679678q^{61} + 79228162514264337593543950336q^{64} + 261357067424332763791265574908q^{65} - 6129453346371264271301476352q^{68} + 606523506722898114294021899522q^{73} - 875862648142600326233330941952q^{74} + 3624713447004776475833090965504q^{80} - 7699154923068245522049945960448q^{82} - 280424425637541180733385932028q^{85} - 17408498582555430732193248126722q^{89} + 84097663609016849910038850800642q^{97} - 72379772059249583476264088436736q^{98} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/36\mathbb{Z}\right)^\times\).

\(n\) \(19\) \(29\)
\(\chi(n)\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
0
−65536.0 0 4.29497e9 1.96496e11 0 0 −2.81475e14 0 −1.28776e16
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 36.33.d.a 1
3.b odd 2 1 4.33.b.a 1
4.b odd 2 1 CM 36.33.d.a 1
12.b even 2 1 4.33.b.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4.33.b.a 1 3.b odd 2 1
4.33.b.a 1 12.b even 2 1
36.33.d.a 1 1.a even 1 1 trivial
36.33.d.a 1 4.b odd 2 1 CM

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} - 196496109694 \) acting on \(S_{33}^{\mathrm{new}}(36, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 65536 T \)
$3$ 1
$5$ \( 1 - 196496109694 T + \)\(23\!\cdots\!25\)\( T^{2} \)
$7$ \( ( 1 - 33232930569601 T )( 1 + 33232930569601 T ) \)
$11$ \( ( 1 - 45949729863572161 T )( 1 + 45949729863572161 T ) \)
$13$ \( 1 - 1330087744899070082 T + \)\(44\!\cdots\!81\)\( T^{2} \)
$17$ \( 1 + 1427124567881986562 T + \)\(23\!\cdots\!61\)\( T^{2} \)
$19$ \( ( 1 - \)\(28\!\cdots\!81\)\( T )( 1 + \)\(28\!\cdots\!81\)\( T ) \)
$23$ \( ( 1 - \)\(61\!\cdots\!61\)\( T )( 1 + \)\(61\!\cdots\!61\)\( T ) \)
$29$ \( 1 + \)\(46\!\cdots\!42\)\( T + \)\(62\!\cdots\!41\)\( T^{2} \)
$31$ \( ( 1 - \)\(72\!\cdots\!81\)\( T )( 1 + \)\(72\!\cdots\!81\)\( T ) \)
$37$ \( 1 - \)\(13\!\cdots\!82\)\( T + \)\(15\!\cdots\!81\)\( T^{2} \)
$41$ \( 1 - \)\(11\!\cdots\!18\)\( T + \)\(40\!\cdots\!81\)\( T^{2} \)
$43$ \( ( 1 - \)\(13\!\cdots\!01\)\( T )( 1 + \)\(13\!\cdots\!01\)\( T ) \)
$47$ \( ( 1 - \)\(56\!\cdots\!21\)\( T )( 1 + \)\(56\!\cdots\!21\)\( T ) \)
$53$ \( 1 - \)\(67\!\cdots\!58\)\( T + \)\(15\!\cdots\!41\)\( T^{2} \)
$59$ \( ( 1 - \)\(21\!\cdots\!41\)\( T )( 1 + \)\(21\!\cdots\!41\)\( T ) \)
$61$ \( 1 + \)\(71\!\cdots\!78\)\( T + \)\(13\!\cdots\!21\)\( T^{2} \)
$67$ \( ( 1 - \)\(16\!\cdots\!81\)\( T )( 1 + \)\(16\!\cdots\!81\)\( T ) \)
$71$ \( ( 1 - \)\(41\!\cdots\!21\)\( T )( 1 + \)\(41\!\cdots\!21\)\( T ) \)
$73$ \( 1 - \)\(60\!\cdots\!22\)\( T + \)\(42\!\cdots\!21\)\( T^{2} \)
$79$ \( ( 1 - \)\(23\!\cdots\!21\)\( T )( 1 + \)\(23\!\cdots\!21\)\( T ) \)
$83$ \( ( 1 - \)\(50\!\cdots\!81\)\( T )( 1 + \)\(50\!\cdots\!81\)\( T ) \)
$89$ \( 1 + \)\(17\!\cdots\!22\)\( T + \)\(24\!\cdots\!21\)\( T^{2} \)
$97$ \( 1 - \)\(84\!\cdots\!42\)\( T + \)\(37\!\cdots\!41\)\( T^{2} \)
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