Properties

Label 9.9.b.a
Level 9
Weight 9
Character orbit 9.b
Analytic conductor 3.666
Analytic rank 0
Dimension 2
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 9 = 3^{2} \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 9.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.66640749055\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 3\sqrt{-2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} + 238 q^{4} + 233 \beta q^{5} + 1652 q^{7} + 494 \beta q^{8} +O(q^{10})\) \( q + \beta q^{2} + 238 q^{4} + 233 \beta q^{5} + 1652 q^{7} + 494 \beta q^{8} -4194 q^{10} -5036 \beta q^{11} -26272 q^{13} + 1652 \beta q^{14} + 52036 q^{16} + 3579 \beta q^{17} + 46640 q^{19} + 55454 \beta q^{20} + 90648 q^{22} -77332 \beta q^{23} -586577 q^{25} -26272 \beta q^{26} + 393176 q^{28} -144953 \beta q^{29} -196444 q^{31} + 178500 \beta q^{32} -64422 q^{34} + 384916 \beta q^{35} + 2819414 q^{37} + 46640 \beta q^{38} -2071836 q^{40} -166507 \beta q^{41} -2213464 q^{43} -1198568 \beta q^{44} + 1391976 q^{46} + 384892 \beta q^{47} -3035697 q^{49} -586577 \beta q^{50} -6252736 q^{52} + 1222983 \beta q^{53} + 21120984 q^{55} + 816088 \beta q^{56} + 2609154 q^{58} + 2768264 \beta q^{59} -17405302 q^{61} -196444 \beta q^{62} + 10108216 q^{64} -6121376 \beta q^{65} -14322664 q^{67} + 851802 \beta q^{68} -6928488 q^{70} -3687708 \beta q^{71} -8906992 q^{73} + 2819414 \beta q^{74} + 11100320 q^{76} -8319472 \beta q^{77} + 32758844 q^{79} + 12124388 \beta q^{80} + 2997126 q^{82} + 20055628 \beta q^{83} -15010326 q^{85} -2213464 \beta q^{86} + 44780112 q^{88} -13891371 \beta q^{89} -43401344 q^{91} -18405016 \beta q^{92} -6928056 q^{94} + 10867120 \beta q^{95} -24451744 q^{97} -3035697 \beta q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 476q^{4} + 3304q^{7} + O(q^{10}) \) \( 2q + 476q^{4} + 3304q^{7} - 8388q^{10} - 52544q^{13} + 104072q^{16} + 93280q^{19} + 181296q^{22} - 1173154q^{25} + 786352q^{28} - 392888q^{31} - 128844q^{34} + 5638828q^{37} - 4143672q^{40} - 4426928q^{43} + 2783952q^{46} - 6071394q^{49} - 12505472q^{52} + 42241968q^{55} + 5218308q^{58} - 34810604q^{61} + 20216432q^{64} - 28645328q^{67} - 13856976q^{70} - 17813984q^{73} + 22200640q^{76} + 65517688q^{79} + 5994252q^{82} - 30020652q^{85} + 89560224q^{88} - 86802688q^{91} - 13856112q^{94} - 48903488q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/9\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
8.1
1.41421i
1.41421i
4.24264i 0 238.000 988.535i 0 1652.00 2095.86i 0 −4194.00
8.2 4.24264i 0 238.000 988.535i 0 1652.00 2095.86i 0 −4194.00
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 9.9.b.a 2
3.b odd 2 1 inner 9.9.b.a 2
4.b odd 2 1 144.9.e.a 2
5.b even 2 1 225.9.c.a 2
5.c odd 4 2 225.9.d.a 4
9.c even 3 2 81.9.d.b 4
9.d odd 6 2 81.9.d.b 4
12.b even 2 1 144.9.e.a 2
15.d odd 2 1 225.9.c.a 2
15.e even 4 2 225.9.d.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9.9.b.a 2 1.a even 1 1 trivial
9.9.b.a 2 3.b odd 2 1 inner
81.9.d.b 4 9.c even 3 2
81.9.d.b 4 9.d odd 6 2
144.9.e.a 2 4.b odd 2 1
144.9.e.a 2 12.b even 2 1
225.9.c.a 2 5.b even 2 1
225.9.c.a 2 15.d odd 2 1
225.9.d.a 4 5.c odd 4 2
225.9.d.a 4 15.e even 4 2

Hecke kernels

This newform subspace is the entire newspace \(S_{9}^{\mathrm{new}}(9, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 494 T^{2} + 65536 T^{4} \)
$3$ 1
$5$ \( 1 + 195952 T^{2} + 152587890625 T^{4} \)
$7$ \( ( 1 - 1652 T + 5764801 T^{2} )^{2} \)
$11$ \( 1 + 27785566 T^{2} + 45949729863572161 T^{4} \)
$13$ \( ( 1 + 26272 T + 815730721 T^{2} )^{2} \)
$17$ \( 1 - 13720948544 T^{2} + 48661191875666868481 T^{4} \)
$19$ \( ( 1 - 46640 T + 16983563041 T^{2} )^{2} \)
$23$ \( 1 - 48977682530 T^{2} + \)\(61\!\cdots\!61\)\( T^{4} \)
$29$ \( 1 - 622288126160 T^{2} + \)\(25\!\cdots\!21\)\( T^{4} \)
$31$ \( ( 1 + 196444 T + 852891037441 T^{2} )^{2} \)
$37$ \( ( 1 - 2819414 T + 3512479453921 T^{2} )^{2} \)
$41$ \( 1 - 15470807999360 T^{2} + \)\(63\!\cdots\!41\)\( T^{4} \)
$43$ \( ( 1 + 2213464 T + 11688200277601 T^{2} )^{2} \)
$47$ \( 1 - 44956019993570 T^{2} + \)\(56\!\cdots\!21\)\( T^{4} \)
$53$ \( 1 - 97597007293520 T^{2} + \)\(38\!\cdots\!21\)\( T^{4} \)
$59$ \( 1 - 155721734882114 T^{2} + \)\(21\!\cdots\!41\)\( T^{4} \)
$61$ \( ( 1 + 17405302 T + 191707312997281 T^{2} )^{2} \)
$67$ \( ( 1 + 14322664 T + 406067677556641 T^{2} )^{2} \)
$71$ \( 1 - 1046721637212770 T^{2} + \)\(41\!\cdots\!21\)\( T^{4} \)
$73$ \( ( 1 + 8906992 T + 806460091894081 T^{2} )^{2} \)
$79$ \( ( 1 - 32758844 T + 1517108809906561 T^{2} )^{2} \)
$83$ \( 1 + 2735523396260830 T^{2} + \)\(50\!\cdots\!81\)\( T^{4} \)
$89$ \( 1 - 4399714222730624 T^{2} + \)\(15\!\cdots\!61\)\( T^{4} \)
$97$ \( ( 1 + 24451744 T + 7837433594376961 T^{2} )^{2} \)
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