Properties

Label 225.12.b.d
Level $225$
Weight $12$
Character orbit 225.b
Analytic conductor $172.877$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 225.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(172.877215626\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 1)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 12 \beta q^{2} + 1472 q^{4} - 8372 \beta q^{7} + 42240 \beta q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + 12 \beta q^{2} + 1472 q^{4} - 8372 \beta q^{7} + 42240 \beta q^{8} - 534612 q^{11} + 288869 \beta q^{13} + 401856 q^{14} + 987136 q^{16} + 3452967 \beta q^{17} - 10661420 q^{19} - 6415344 \beta q^{22} + 9321636 \beta q^{23} - 13865712 q^{26} - 12323584 \beta q^{28} + 128406630 q^{29} - 52843168 q^{31} + 98353152 \beta q^{32} - 165742416 q^{34} - 91106657 \beta q^{37} - 127937040 \beta q^{38} - 308120442 q^{41} + 8562854 \beta q^{43} - 786948864 q^{44} - 447438528 q^{46} - 1343674248 \beta q^{47} + 1696965207 q^{49} + 425215168 \beta q^{52} - 798027849 \beta q^{53} + 1414533120 q^{56} + 1540879560 \beta q^{58} - 5189203740 q^{59} + 6956478662 q^{61} - 634118016 \beta q^{62} - 2699296768 q^{64} - 7740913442 \beta q^{67} + 5082767424 \beta q^{68} - 9791485272 q^{71} - 731895661 \beta q^{73} + 4373119536 q^{74} - 15693610240 q^{76} + 4475771664 \beta q^{77} - 38116845680 q^{79} - 3697445304 \beta q^{82} - 14667549834 \beta q^{83} - 411016992 q^{86} - 22582010880 \beta q^{88} - 24992917110 q^{89} + 9673645072 q^{91} + 13721448192 \beta q^{92} + 64496363904 q^{94} + 37506784273 \beta q^{97} + 20363582484 \beta q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2944 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2944 q^{4} - 1069224 q^{11} + 803712 q^{14} + 1974272 q^{16} - 21322840 q^{19} - 27731424 q^{26} + 256813260 q^{29} - 105686336 q^{31} - 331484832 q^{34} - 616240884 q^{41} - 1573897728 q^{44} - 894877056 q^{46} + 3393930414 q^{49} + 2829066240 q^{56} - 10378407480 q^{59} + 13912957324 q^{61} - 5398593536 q^{64} - 19582970544 q^{71} + 8746239072 q^{74} - 31387220480 q^{76} - 76233691360 q^{79} - 822033984 q^{86} - 49985834220 q^{89} + 19347290144 q^{91} + 128992727808 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/225\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
199.1
1.00000i
1.00000i
24.0000i 0 1472.00 0 0 16744.0i 84480.0i 0 0
199.2 24.0000i 0 1472.00 0 0 16744.0i 84480.0i 0 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 225.12.b.d 2
3.b odd 2 1 25.12.b.b 2
5.b even 2 1 inner 225.12.b.d 2
5.c odd 4 1 9.12.a.b 1
5.c odd 4 1 225.12.a.b 1
15.d odd 2 1 25.12.b.b 2
15.e even 4 1 1.12.a.a 1
15.e even 4 1 25.12.a.b 1
20.e even 4 1 144.12.a.d 1
45.k odd 12 2 81.12.c.b 2
45.l even 12 2 81.12.c.d 2
60.l odd 4 1 16.12.a.a 1
105.k odd 4 1 49.12.a.a 1
105.w odd 12 2 49.12.c.c 2
105.x even 12 2 49.12.c.b 2
120.q odd 4 1 64.12.a.f 1
120.w even 4 1 64.12.a.b 1
165.l odd 4 1 121.12.a.b 1
195.s even 4 1 169.12.a.a 1
240.z odd 4 1 256.12.b.c 2
240.bb even 4 1 256.12.b.e 2
240.bd odd 4 1 256.12.b.c 2
240.bf even 4 1 256.12.b.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.12.a.a 1 15.e even 4 1
9.12.a.b 1 5.c odd 4 1
16.12.a.a 1 60.l odd 4 1
25.12.a.b 1 15.e even 4 1
25.12.b.b 2 3.b odd 2 1
25.12.b.b 2 15.d odd 2 1
49.12.a.a 1 105.k odd 4 1
49.12.c.b 2 105.x even 12 2
49.12.c.c 2 105.w odd 12 2
64.12.a.b 1 120.w even 4 1
64.12.a.f 1 120.q odd 4 1
81.12.c.b 2 45.k odd 12 2
81.12.c.d 2 45.l even 12 2
121.12.a.b 1 165.l odd 4 1
144.12.a.d 1 20.e even 4 1
169.12.a.a 1 195.s even 4 1
225.12.a.b 1 5.c odd 4 1
225.12.b.d 2 1.a even 1 1 trivial
225.12.b.d 2 5.b even 2 1 inner
256.12.b.c 2 240.z odd 4 1
256.12.b.c 2 240.bd odd 4 1
256.12.b.e 2 240.bb even 4 1
256.12.b.e 2 240.bf even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{12}^{\mathrm{new}}(225, [\chi])\):

\( T_{2}^{2} + 576 \) Copy content Toggle raw display
\( T_{11} + 534612 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 576 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 280361536 \) Copy content Toggle raw display
$11$ \( (T + 534612)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 333781196644 \) Copy content Toggle raw display
$17$ \( T^{2} + 47691924412356 \) Copy content Toggle raw display
$19$ \( (T + 10661420)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + \cdots + 347571590865984 \) Copy content Toggle raw display
$29$ \( (T - 128406630)^{2} \) Copy content Toggle raw display
$31$ \( (T + 52843168)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 33\!\cdots\!96 \) Copy content Toggle raw display
$41$ \( (T + 308120442)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + \cdots + 293289874501264 \) Copy content Toggle raw display
$47$ \( T^{2} + 72\!\cdots\!16 \) Copy content Toggle raw display
$53$ \( T^{2} + 25\!\cdots\!04 \) Copy content Toggle raw display
$59$ \( (T + 5189203740)^{2} \) Copy content Toggle raw display
$61$ \( (T - 6956478662)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 23\!\cdots\!56 \) Copy content Toggle raw display
$71$ \( (T + 9791485272)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 21\!\cdots\!84 \) Copy content Toggle raw display
$79$ \( (T + 38116845680)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 86\!\cdots\!24 \) Copy content Toggle raw display
$89$ \( (T + 24992917110)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 56\!\cdots\!16 \) Copy content Toggle raw display
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