Properties

Label 225.12.b.f
Level 225
Weight 12
Character orbit 225.b
Analytic conductor 172.877
Analytic rank 0
Dimension 4
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: \(4\)
Coefficient field: \(\Q(i, \sqrt{151})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{2}\cdot 5^{2} \)
Twist minimal: no (minimal twist has level 5)
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.

\(f(q)\) \(=\) \( q + ( \beta_{1} - \beta_{3} ) q^{2} + ( -3488 + 2 \beta_{2} ) q^{4} + ( 2895 \beta_{1} + 176 \beta_{3} ) q^{7} + ( -12312 \beta_{1} + 1640 \beta_{3} ) q^{8} +O(q^{10})\) \( q + ( \beta_{1} - \beta_{3} ) q^{2} + ( -3488 + 2 \beta_{2} ) q^{4} + ( 2895 \beta_{1} + 176 \beta_{3} ) q^{7} + ( -12312 \beta_{1} + 1640 \beta_{3} ) q^{8} + ( 309088 + 880 \beta_{2} ) q^{11} + ( -170713 \beta_{1} + 4288 \beta_{3} ) q^{13} + ( 667236 + 2719 \beta_{2} ) q^{14} + ( 3002816 - 9856 \beta_{2} ) q^{16} + ( -65897 \beta_{1} - 42176 \beta_{3} ) q^{17} + ( -2662660 - 9152 \beta_{2} ) q^{19} + ( -4474592 \beta_{1} - 221088 \beta_{3} ) q^{22} + ( 2947197 \beta_{1} + 11152 \beta_{3} ) q^{23} + ( 40380868 - 175001 \beta_{2} ) q^{26} + ( -8184288 \beta_{1} - 34888 \beta_{3} ) q^{28} + ( 47070190 + 76608 \beta_{2} ) q^{29} + ( 122271732 - 240240 \beta_{2} ) q^{31} + ( 31365056 \beta_{1} - 629696 \beta_{3} ) q^{32} + ( -222679036 - 23721 \beta_{2} ) q^{34} + ( 1050161 \beta_{1} + 6344576 \beta_{3} ) q^{37} + ( 47087612 \beta_{1} + 1747460 \beta_{3} ) q^{38} + ( 372871658 + 755040 \beta_{2} ) q^{41} + ( -31497505 \beta_{1} + 4636368 \beta_{3} ) q^{43} + ( -121362944 - 2451264 \beta_{2} ) q^{44} + ( -234097428 + 2936045 \beta_{2} ) q^{46} + ( 70103077 \beta_{1} + 6835024 \beta_{3} ) q^{47} + ( 970838707 - 1019040 \beta_{2} ) q^{49} + ( 642066080 \beta_{1} - 49099144 \beta_{3} ) q^{52} + ( 56916029 \beta_{1} - 62251328 \beta_{3} ) q^{53} + ( 1995276960 - 2580888 \beta_{2} ) q^{56} + ( -369370898 \beta_{1} - 39409390 \beta_{3} ) q^{58} + ( 3658757780 + 5860576 \beta_{2} ) q^{59} + ( -758212838 - 1785600 \beta_{2} ) q^{61} + ( 1428216372 \beta_{1} - 146295732 \beta_{3} ) q^{62} + ( -409765888 + 11809664 \beta_{2} ) q^{64} + ( 786714507 \beta_{1} - 30563824 \beta_{3} ) q^{67} + ( -228688736 \beta_{1} + 133930488 \beta_{3} ) q^{68} + ( -16469235772 + 1826800 \beta_{2} ) q^{71} + ( 1499142443 \beta_{1} - 113205952 \beta_{3} ) q^{73} + ( 34384099036 - 5294415 \beta_{2} ) q^{74} + ( -662696320 + 26596856 \beta_{2} ) q^{76} + ( 1736737440 \beta_{1} + 309159488 \beta_{3} ) q^{77} + ( 1651411560 + 19187872 \beta_{2} ) q^{79} + ( -3731525782 \beta_{1} - 297367658 \beta_{3} ) q^{82} + ( 664955121 \beta_{1} - 366939408 \beta_{3} ) q^{83} + ( 28353046948 - 36133873 \beta_{2} ) q^{86} + ( 4039743744 \beta_{1} - 576551680 \beta_{3} ) q^{88} + ( -6337385430 - 48509376 \beta_{2} ) q^{89} + ( 45318929532 + 17631728 \beta_{2} ) q^{91} + ( -10158578592 \beta_{1} + 550541224 \beta_{3} ) q^{92} + ( 30144882764 + 63268053 \beta_{2} ) q^{94} + ( -154035187 \beta_{1} + 1515290176 \beta_{3} ) q^{97} + ( 6510340147 \beta_{1} - 1072742707 \beta_{3} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 13952q^{4} + O(q^{10}) \) \( 4q - 13952q^{4} + 1236352q^{11} + 2668944q^{14} + 12011264q^{16} - 10650640q^{19} + 161523472q^{26} + 188280760q^{29} + 489086928q^{31} - 890716144q^{34} + 1491486632q^{41} - 485451776q^{44} - 936389712q^{46} + 3883354828q^{49} + 7981107840q^{56} + 14635031120q^{59} - 3032851352q^{61} - 1639063552q^{64} - 65876943088q^{71} + 137536396144q^{74} - 2650785280q^{76} + 6605646240q^{79} + 113412187792q^{86} - 25349541720q^{89} + 181275718128q^{91} + 120579531056q^{94} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 75 x^{2} + 1444\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 5 \nu^{3} - 185 \nu \)\()/19\)
\(\beta_{2}\)\(=\)\((\)\( -30 \nu^{3} + 3390 \nu \)\()/19\)
\(\beta_{3}\)\(=\)\( 12 \nu^{2} - 450 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} + 6 \beta_{1}\)\()/120\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} + 450\)\()/12\)
\(\nu^{3}\)\(=\)\((\)\(37 \beta_{2} + 678 \beta_{1}\)\()/120\)

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
−6.14410 0.500000i
6.14410 + 0.500000i
6.14410 0.500000i
−6.14410 + 0.500000i
83.7292i 0 −4962.58 0 0 15973.7i 244036.i 0 0
199.2 63.7292i 0 −2013.42 0 0 41926.3i 2204.06i 0 0
199.3 63.7292i 0 −2013.42 0 0 41926.3i 2204.06i 0 0
199.4 83.7292i 0 −4962.58 0 0 15973.7i 244036.i 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.f 4
3.b odd 2 1 25.12.b.c 4
5.b even 2 1 inner 225.12.b.f 4
5.c odd 4 1 45.12.a.d 2
5.c odd 4 1 225.12.a.h 2
15.d odd 2 1 25.12.b.c 4
15.e even 4 1 5.12.a.b 2
15.e even 4 1 25.12.a.c 2
60.l odd 4 1 80.12.a.j 2
105.k odd 4 1 245.12.a.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.12.a.b 2 15.e even 4 1
25.12.a.c 2 15.e even 4 1
25.12.b.c 4 3.b odd 2 1
25.12.b.c 4 15.d odd 2 1
45.12.a.d 2 5.c odd 4 1
80.12.a.j 2 60.l odd 4 1
225.12.a.h 2 5.c odd 4 1
225.12.b.f 4 1.a even 1 1 trivial
225.12.b.f 4 5.b even 2 1 inner
245.12.a.b 2 105.k odd 4 1

Hecke kernels

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

\( T_{2}^{4} + 11072 T_{2}^{2} + 28472896 \)
\( T_{11}^{2} - 618176 T_{11} - 325428448256 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + 2880 T^{2} + 8287808 T^{4} + 12079595520 T^{6} + 17592186044416 T^{8} \)
$3$ 1
$5$ 1
$7$ \( 1 - 5896330900 T^{2} + 15946824262997918598 T^{4} - \)\(23\!\cdots\!00\)\( T^{6} + \)\(15\!\cdots\!01\)\( T^{8} \)
$11$ \( ( 1 - 618176 T + 245194892966 T^{2} - 176372827291625536 T^{3} + \)\(81\!\cdots\!21\)\( T^{4} )^{2} \)
$13$ \( 1 - 1140153047180 T^{2} + \)\(55\!\cdots\!38\)\( T^{4} - \)\(36\!\cdots\!20\)\( T^{6} + \)\(10\!\cdots\!61\)\( T^{8} \)
$17$ \( 1 - 116879825889660 T^{2} + \)\(57\!\cdots\!78\)\( T^{4} - \)\(13\!\cdots\!40\)\( T^{6} + \)\(13\!\cdots\!21\)\( T^{8} \)
$19$ \( ( 1 + 5325320 T + 194538827137638 T^{2} + \)\(62\!\cdots\!80\)\( T^{3} + \)\(13\!\cdots\!61\)\( T^{4} )^{2} \)
$23$ \( 1 - 2072692881139220 T^{2} + \)\(28\!\cdots\!58\)\( T^{4} - \)\(18\!\cdots\!80\)\( T^{6} + \)\(82\!\cdots\!41\)\( T^{8} \)
$29$ \( ( 1 - 94140380 T + 23426350431097358 T^{2} - \)\(11\!\cdots\!20\)\( T^{3} + \)\(14\!\cdots\!41\)\( T^{4} )^{2} \)
$31$ \( ( 1 - 244543464 T + 34393316207729486 T^{2} - \)\(62\!\cdots\!84\)\( T^{3} + \)\(64\!\cdots\!61\)\( T^{4} )^{2} \)
$37$ \( 1 - 273812295186452780 T^{2} + \)\(81\!\cdots\!38\)\( T^{4} - \)\(86\!\cdots\!20\)\( T^{6} + \)\(10\!\cdots\!61\)\( T^{8} \)
$41$ \( ( 1 - 745743316 T + 929792912462405846 T^{2} - \)\(41\!\cdots\!56\)\( T^{3} + \)\(30\!\cdots\!81\)\( T^{4} )^{2} \)
$43$ \( 1 - 3285052879347844100 T^{2} + \)\(43\!\cdots\!98\)\( T^{4} - \)\(28\!\cdots\!00\)\( T^{6} + \)\(74\!\cdots\!01\)\( T^{8} \)
$47$ \( 1 - 8397835342270441140 T^{2} + \)\(29\!\cdots\!18\)\( T^{4} - \)\(51\!\cdots\!60\)\( T^{6} + \)\(37\!\cdots\!81\)\( T^{8} \)
$53$ \( 1 + 5703220206102687060 T^{2} + \)\(15\!\cdots\!18\)\( T^{4} + \)\(48\!\cdots\!40\)\( T^{6} + \)\(73\!\cdots\!81\)\( T^{8} \)
$59$ \( ( 1 - 7317515560 T + 55027608950440780118 T^{2} - \)\(22\!\cdots\!40\)\( T^{3} + \)\(90\!\cdots\!81\)\( T^{4} )^{2} \)
$61$ \( ( 1 + 1516425676 T + 85869525433683691566 T^{2} + \)\(65\!\cdots\!36\)\( T^{3} + \)\(18\!\cdots\!21\)\( T^{4} )^{2} \)
$67$ \( 1 - \)\(35\!\cdots\!60\)\( T^{2} + \)\(60\!\cdots\!78\)\( T^{4} - \)\(52\!\cdots\!40\)\( T^{6} + \)\(22\!\cdots\!21\)\( T^{8} \)
$71$ \( ( 1 + 32938471544 T + \)\(73\!\cdots\!26\)\( T^{2} + \)\(76\!\cdots\!24\)\( T^{3} + \)\(53\!\cdots\!41\)\( T^{4} )^{2} \)
$73$ \( 1 - \)\(66\!\cdots\!20\)\( T^{2} + \)\(24\!\cdots\!58\)\( T^{4} - \)\(65\!\cdots\!80\)\( T^{6} + \)\(96\!\cdots\!41\)\( T^{8} \)
$79$ \( ( 1 - 3302823120 T + \)\(12\!\cdots\!58\)\( T^{2} - \)\(24\!\cdots\!80\)\( T^{3} + \)\(55\!\cdots\!41\)\( T^{4} )^{2} \)
$83$ \( 1 - \)\(35\!\cdots\!60\)\( T^{2} + \)\(64\!\cdots\!78\)\( T^{4} - \)\(59\!\cdots\!40\)\( T^{6} + \)\(27\!\cdots\!21\)\( T^{8} \)
$89$ \( ( 1 + 12674770860 T + \)\(43\!\cdots\!78\)\( T^{2} + \)\(35\!\cdots\!40\)\( T^{3} + \)\(77\!\cdots\!21\)\( T^{4} )^{2} \)
$97$ \( 1 - \)\(36\!\cdots\!40\)\( T^{2} + \)\(10\!\cdots\!18\)\( T^{4} - \)\(18\!\cdots\!60\)\( T^{6} + \)\(26\!\cdots\!81\)\( T^{8} \)
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