Properties

Label 225.4.f.c
Level $225$
Weight $4$
Character orbit 225.f
Analytic conductor $13.275$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 225.f (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(13.2754297513\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \(x^{12} - 16 x^{10} - 14 x^{8} - 512 x^{6} + 3889 x^{4} + 126224 x^{2} + 506944\)
Coefficient ring: \(\Z[a_1, \ldots, a_{29}]\)
Coefficient ring index: \( 2^{11}\cdot 3^{4} \)
Twist minimal: no (minimal twist has level 45)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{6} q^{2} + ( -6 \beta_{1} + \beta_{5} ) q^{4} + ( -2 - 2 \beta_{1} + \beta_{4} ) q^{7} + ( -3 \beta_{7} - 7 \beta_{8} - 3 \beta_{9} - \beta_{10} + \beta_{11} ) q^{8} +O(q^{10})\) \( q -\beta_{6} q^{2} + ( -6 \beta_{1} + \beta_{5} ) q^{4} + ( -2 - 2 \beta_{1} + \beta_{4} ) q^{7} + ( -3 \beta_{7} - 7 \beta_{8} - 3 \beta_{9} - \beta_{10} + \beta_{11} ) q^{8} + ( -7 \beta_{6} - 7 \beta_{8} + 3 \beta_{9} + \beta_{11} ) q^{11} + ( 9 - 9 \beta_{1} + 2 \beta_{2} - 3 \beta_{3} - 2 \beta_{5} ) q^{13} + ( -3 \beta_{6} - 7 \beta_{7} + 3 \beta_{8} - \beta_{10} ) q^{14} + ( -54 + 9 \beta_{2} - 5 \beta_{3} + 5 \beta_{4} ) q^{16} + ( -8 \beta_{6} - 12 \beta_{7} + 12 \beta_{9} + \beta_{10} + \beta_{11} ) q^{17} + ( -20 \beta_{1} + 5 \beta_{3} + 5 \beta_{4} + 4 \beta_{5} ) q^{19} + ( -88 - 88 \beta_{1} + 6 \beta_{2} - \beta_{4} + 6 \beta_{5} ) q^{22} + ( -2 \beta_{7} + 8 \beta_{8} - 2 \beta_{9} + 6 \beta_{10} - 6 \beta_{11} ) q^{23} + ( -6 \beta_{6} - 6 \beta_{8} + 33 \beta_{9} - 7 \beta_{11} ) q^{26} + ( 48 - 48 \beta_{1} + 6 \beta_{2} - \beta_{3} - 6 \beta_{5} ) q^{28} + ( -25 \beta_{6} - 4 \beta_{7} + 25 \beta_{8} + 5 \beta_{10} ) q^{29} + ( -104 - 4 \beta_{2} + 5 \beta_{3} - 5 \beta_{4} ) q^{31} + ( 29 \beta_{6} - 38 \beta_{7} + 38 \beta_{9} - 6 \beta_{10} - 6 \beta_{11} ) q^{32} + ( -56 \beta_{1} - 10 \beta_{3} - 10 \beta_{4} - 12 \beta_{5} ) q^{34} + ( -69 - 69 \beta_{1} - 8 \beta_{2} + 7 \beta_{4} - 8 \beta_{5} ) q^{37} + ( -47 \beta_{7} - 6 \beta_{8} - 47 \beta_{9} - 9 \beta_{10} + 9 \beta_{11} ) q^{38} + ( 38 \beta_{6} + 38 \beta_{8} + 29 \beta_{9} + 16 \beta_{11} ) q^{41} + ( 8 - 8 \beta_{1} - 16 \beta_{2} + 14 \beta_{3} + 16 \beta_{5} ) q^{43} + ( 91 \beta_{6} - 53 \beta_{7} - 91 \beta_{8} - 3 \beta_{10} ) q^{44} + ( 56 - 28 \beta_{2} + 10 \beta_{3} - 10 \beta_{4} ) q^{46} + ( -58 \beta_{6} - 23 \beta_{7} + 23 \beta_{9} + 9 \beta_{10} + 9 \beta_{11} ) q^{47} + ( -111 \beta_{1} + 16 \beta_{5} ) q^{49} + ( 26 + 26 \beta_{1} - 25 \beta_{2} - 23 \beta_{4} - 25 \beta_{5} ) q^{52} + ( 5 \beta_{7} + 34 \beta_{8} + 5 \beta_{9} - 10 \beta_{10} + 10 \beta_{11} ) q^{53} + ( 25 \beta_{6} + 25 \beta_{8} - 13 \beta_{9} - 5 \beta_{11} ) q^{56} + ( 322 - 322 \beta_{1} - 31 \beta_{2} + 6 \beta_{3} + 31 \beta_{5} ) q^{58} + ( -41 \beta_{6} + 31 \beta_{7} + 41 \beta_{8} - 17 \beta_{10} ) q^{59} + ( 8 - 48 \beta_{2} + 15 \beta_{3} - 15 \beta_{4} ) q^{61} + ( 118 \beta_{6} + 47 \beta_{7} - 47 \beta_{9} + 9 \beta_{10} + 9 \beta_{11} ) q^{62} + ( 78 \beta_{1} - 10 \beta_{3} - 10 \beta_{4} - 57 \beta_{5} ) q^{64} + ( -136 - 136 \beta_{1} + 44 \beta_{2} - 2 \beta_{4} + 44 \beta_{5} ) q^{67} + ( 10 \beta_{7} + 16 \beta_{8} + 10 \beta_{9} + 30 \beta_{10} - 30 \beta_{11} ) q^{68} + ( -16 \beta_{6} - 16 \beta_{8} + 4 \beta_{9} - 32 \beta_{11} ) q^{71} + ( -331 + 331 \beta_{1} + 18 \beta_{2} + 12 \beta_{3} - 18 \beta_{5} ) q^{73} + ( -38 \beta_{6} - \beta_{7} + 38 \beta_{8} + 9 \beta_{10} ) q^{74} + ( -40 + 104 \beta_{2} - 25 \beta_{3} + 25 \beta_{4} ) q^{76} + ( -32 \beta_{6} - 12 \beta_{7} + 12 \beta_{9} - 24 \beta_{10} - 24 \beta_{11} ) q^{77} + ( 672 \beta_{1} + 25 \beta_{3} + 25 \beta_{4} + 28 \beta_{5} ) q^{79} + ( 654 + 654 \beta_{1} - 35 \beta_{2} + 3 \beta_{4} - 35 \beta_{5} ) q^{82} + ( 153 \beta_{7} + 78 \beta_{8} + 153 \beta_{9} + \beta_{10} - \beta_{11} ) q^{83} + ( -82 \beta_{6} - 82 \beta_{8} - 194 \beta_{9} + 46 \beta_{11} ) q^{86} + ( -664 + 664 \beta_{1} + 102 \beta_{2} - 67 \beta_{3} - 102 \beta_{5} ) q^{88} + ( 36 \beta_{6} + 163 \beta_{7} - 36 \beta_{8} + 42 \beta_{10} ) q^{89} + ( 396 + 24 \beta_{2} - 25 \beta_{3} + 25 \beta_{4} ) q^{91} + ( -144 \beta_{6} + 170 \beta_{7} - 170 \beta_{9} - 10 \beta_{10} - 10 \beta_{11} ) q^{92} + ( -648 \beta_{1} - 5 \beta_{3} - 5 \beta_{4} + 48 \beta_{5} ) q^{94} + ( -231 - 231 \beta_{1} + 50 \beta_{2} + 78 \beta_{4} + 50 \beta_{5} ) q^{97} + ( -48 \beta_{7} - 255 \beta_{8} - 48 \beta_{9} - 16 \beta_{10} + 16 \beta_{11} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 24 q^{7} + O(q^{10}) \) \( 12 q - 24 q^{7} + 108 q^{13} - 648 q^{16} - 1056 q^{22} + 576 q^{28} - 1248 q^{31} - 828 q^{37} + 96 q^{43} + 672 q^{46} + 312 q^{52} + 3864 q^{58} + 96 q^{61} - 1632 q^{67} - 3972 q^{73} - 480 q^{76} + 7848 q^{82} - 7968 q^{88} + 4752 q^{91} - 2772 q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} - 16 x^{10} - 14 x^{8} - 512 x^{6} + 3889 x^{4} + 126224 x^{2} + 506944\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -3 \nu^{10} - 56 \nu^{8} + 634 \nu^{6} + 3248 \nu^{4} + 83197 \nu^{2} + 418024 \)\()/182400\)
\(\beta_{2}\)\(=\)\((\)\( -71 \nu^{10} + 2538 \nu^{8} - 4502 \nu^{6} - 356014 \nu^{4} - 1896111 \nu^{2} - 1578712 \)\()/3620640\)
\(\beta_{3}\)\(=\)\((\)\( 1029 \nu^{10} - 22972 \nu^{8} + 158738 \nu^{6} - 2560724 \nu^{4} + 22729229 \nu^{2} + 78379688 \)\()/9051600\)
\(\beta_{4}\)\(=\)\((\)\( 5723 \nu^{10} - 110024 \nu^{8} + 167406 \nu^{6} - 2434608 \nu^{4} + 44277523 \nu^{2} + 559571096 \)\()/18103200\)
\(\beta_{5}\)\(=\)\((\)\( 1265 \nu^{10} - 2936 \nu^{8} - 23478 \nu^{6} - 1362480 \nu^{4} - 16244903 \nu^{2} - 51726904 \)\()/2896512\)
\(\beta_{6}\)\(=\)\((\)\( -113621 \nu^{11} - 12179272 \nu^{9} + 76074438 \nu^{7} + 253832976 \nu^{5} + 12758183579 \nu^{3} + 93476563288 \nu \)\()/ 25778956800 \)
\(\beta_{7}\)\(=\)\((\)\( 6773 \nu^{11} - 146104 \nu^{9} + 577306 \nu^{7} - 5223568 \nu^{5} + 79774373 \nu^{3} + 554840616 \nu \)\()/ 226131200 \)
\(\beta_{8}\)\(=\)\((\)\( -1093919 \nu^{11} + 7804552 \nu^{9} - 15466318 \nu^{7} + 1380835984 \nu^{5} + 6018524881 \nu^{3} + 28260515592 \nu \)\()/ 25778956800 \)
\(\beta_{9}\)\(=\)\((\)\( -1095 \nu^{11} + 7908 \nu^{9} + 18534 \nu^{7} + 745404 \nu^{5} + 8928141 \nu^{3} - 11971272 \nu \)\()/21482464\)
\(\beta_{10}\)\(=\)\((\)\( 72101 \nu^{11} - 1867040 \nu^{9} + 15340242 \nu^{7} - 118083000 \nu^{5} + 659420461 \nu^{3} + 7004722280 \nu \)\()/ 1288947840 \)
\(\beta_{11}\)\(=\)\((\)\( 12569 \nu^{11} - 59505 \nu^{9} - 670717 \nu^{7} - 13483505 \nu^{5} - 70039446 \nu^{3} + 713598880 \nu \)\()/ 161118480 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(3 \beta_{11} + 3 \beta_{10} - \beta_{9} + 12 \beta_{8} + \beta_{7} - 6 \beta_{6}\)\()/24\)
\(\nu^{2}\)\(=\)\((\)\(-4 \beta_{5} - 3 \beta_{4} + 11 \beta_{3} - 20 \beta_{2} - 64 \beta_{1} + 64\)\()/24\)
\(\nu^{3}\)\(=\)\((\)\(9 \beta_{11} + 3 \beta_{10} + 53 \beta_{9} + 66 \beta_{8} + 137 \beta_{7} - 120 \beta_{6}\)\()/24\)
\(\nu^{4}\)\(=\)\((\)\(12 \beta_{5} - 35 \beta_{4} + 5 \beta_{3} - 268 \beta_{2} + 1136\)\()/24\)
\(\nu^{5}\)\(=\)\((\)\(-171 \beta_{11} + 177 \beta_{10} - 615 \beta_{9} + 1704 \beta_{8} + 1259 \beta_{7} - 1518 \beta_{6}\)\()/24\)
\(\nu^{6}\)\(=\)\((\)\(676 \beta_{5} - 1157 \beta_{4} + 909 \beta_{3} - 3212 \beta_{2} + 5824 \beta_{1} + 25216\)\()/24\)
\(\nu^{7}\)\(=\)\((\)\(-489 \beta_{11} + 6429 \beta_{10} - 2133 \beta_{9} + 21438 \beta_{8} + 12439 \beta_{7} - 24264 \beta_{6}\)\()/24\)
\(\nu^{8}\)\(=\)\((\)\(252 \beta_{5} - 4255 \beta_{4} + 5385 \beta_{3} - 14588 \beta_{2} - 20480 \beta_{1} + 129968\)\()/8\)
\(\nu^{9}\)\(=\)\((\)\(24939 \beta_{11} + 63663 \beta_{10} + 24551 \beta_{9} + 292920 \beta_{8} + 220021 \beta_{7} - 359538 \beta_{6}\)\()/24\)
\(\nu^{10}\)\(=\)\((\)\(30812 \beta_{5} - 127323 \beta_{4} + 201011 \beta_{3} - 706676 \beta_{2} - 856384 \beta_{1} + 4399744\)\()/24\)
\(\nu^{11}\)\(=\)\((\)\(96009 \beta_{11} + 680739 \beta_{10} - 305227 \beta_{9} + 4045218 \beta_{8} + 3762665 \beta_{7} - 4953432 \beta_{6}\)\()/24\)

Character values

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

\(n\) \(101\) \(127\)
\(\chi(n)\) \(-1\) \(\beta_{1}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
107.1
−3.78139 + 0.0336790i
−0.347140 2.27426i
2.02004 2.30794i
−2.02004 + 2.30794i
0.347140 + 2.27426i
3.78139 0.0336790i
−3.78139 0.0336790i
−0.347140 + 2.27426i
2.02004 + 2.30794i
−2.02004 2.30794i
0.347140 2.27426i
3.78139 + 0.0336790i
−3.74771 + 3.74771i 0 20.0907i 0 0 −1.80948 1.80948i 45.3126 + 45.3126i 0 0
107.2 −2.62140 + 2.62140i 0 5.74350i 0 0 10.8652 + 10.8652i −5.91519 5.91519i 0 0
107.3 −0.287902 + 0.287902i 0 7.83422i 0 0 −15.0557 15.0557i −4.55871 4.55871i 0 0
107.4 0.287902 0.287902i 0 7.83422i 0 0 −15.0557 15.0557i 4.55871 + 4.55871i 0 0
107.5 2.62140 2.62140i 0 5.74350i 0 0 10.8652 + 10.8652i 5.91519 + 5.91519i 0 0
107.6 3.74771 3.74771i 0 20.0907i 0 0 −1.80948 1.80948i −45.3126 45.3126i 0 0
143.1 −3.74771 3.74771i 0 20.0907i 0 0 −1.80948 + 1.80948i 45.3126 45.3126i 0 0
143.2 −2.62140 2.62140i 0 5.74350i 0 0 10.8652 10.8652i −5.91519 + 5.91519i 0 0
143.3 −0.287902 0.287902i 0 7.83422i 0 0 −15.0557 + 15.0557i −4.55871 + 4.55871i 0 0
143.4 0.287902 + 0.287902i 0 7.83422i 0 0 −15.0557 + 15.0557i 4.55871 4.55871i 0 0
143.5 2.62140 + 2.62140i 0 5.74350i 0 0 10.8652 10.8652i 5.91519 5.91519i 0 0
143.6 3.74771 + 3.74771i 0 20.0907i 0 0 −1.80948 + 1.80948i −45.3126 + 45.3126i 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 143.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.c odd 4 1 inner
15.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 225.4.f.c 12
3.b odd 2 1 inner 225.4.f.c 12
5.b even 2 1 45.4.f.a 12
5.c odd 4 1 45.4.f.a 12
5.c odd 4 1 inner 225.4.f.c 12
15.d odd 2 1 45.4.f.a 12
15.e even 4 1 45.4.f.a 12
15.e even 4 1 inner 225.4.f.c 12
20.d odd 2 1 720.4.w.d 12
20.e even 4 1 720.4.w.d 12
60.h even 2 1 720.4.w.d 12
60.l odd 4 1 720.4.w.d 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.4.f.a 12 5.b even 2 1
45.4.f.a 12 5.c odd 4 1
45.4.f.a 12 15.d odd 2 1
45.4.f.a 12 15.e even 4 1
225.4.f.c 12 1.a even 1 1 trivial
225.4.f.c 12 3.b odd 2 1 inner
225.4.f.c 12 5.c odd 4 1 inner
225.4.f.c 12 15.e even 4 1 inner
720.4.w.d 12 20.d odd 2 1
720.4.w.d 12 20.e even 4 1
720.4.w.d 12 60.h even 2 1
720.4.w.d 12 60.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(225, [\chi])\):

\( T_{2}^{12} + 978 T_{2}^{8} + 149073 T_{2}^{4} + 4096 \)
\( T_{7}^{6} + 12 T_{7}^{5} + 72 T_{7}^{4} - 2560 T_{7}^{3} + 97344 T_{7}^{2} + 369408 T_{7} + 700928 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 4096 + 149073 T^{4} + 978 T^{8} + T^{12} \)
$3$ \( T^{12} \)
$5$ \( T^{12} \)
$7$ \( ( 700928 + 369408 T + 97344 T^{2} - 2560 T^{3} + 72 T^{4} + 12 T^{5} + T^{6} )^{2} \)
$11$ \( ( 2390999552 + 5840832 T^{2} + 4344 T^{4} + T^{6} )^{2} \)
$13$ \( ( 40033352 + 17341224 T + 3755844 T^{2} + 113600 T^{3} + 1458 T^{4} - 54 T^{5} + T^{6} )^{2} \)
$17$ \( \)\(81\!\cdots\!96\)\( + 4537848559116288 T^{4} + 138301968 T^{8} + T^{12} \)
$19$ \( ( 604661760000 + 226022400 T^{2} + 26640 T^{4} + T^{6} )^{2} \)
$23$ \( \)\(44\!\cdots\!96\)\( + 54082669119602688 T^{4} + 946426368 T^{8} + T^{12} \)
$29$ \( ( -3276052522632 + 815112972 T^{2} - 58854 T^{4} + T^{6} )^{2} \)
$31$ \( ( 333184 + 19728 T + 312 T^{2} + T^{3} )^{4} \)
$37$ \( ( 431026413512 - 30656156424 T + 1090188324 T^{2} - 14597920 T^{3} + 85698 T^{4} + 414 T^{5} + T^{6} )^{2} \)
$41$ \( ( 1313254395137288 + 39690796812 T^{2} + 367686 T^{4} + T^{6} )^{2} \)
$43$ \( ( 12363096915968 + 421035245568 T + 7169347584 T^{2} + 9036800 T^{3} + 1152 T^{4} - 48 T^{5} + T^{6} )^{2} \)
$47$ \( \)\(11\!\cdots\!96\)\( + \)\(10\!\cdots\!88\)\( T^{4} + 22535002368 T^{8} + T^{12} \)
$53$ \( \)\(95\!\cdots\!76\)\( + 48541234921077547008 T^{4} + 14798523408 T^{8} + T^{12} \)
$59$ \( ( -1998258773488128 + 52376651712 T^{2} - 421176 T^{4} + T^{6} )^{2} \)
$61$ \( ( -70019072 - 341088 T - 24 T^{2} + T^{3} )^{4} \)
$67$ \( ( 25774917628854272 + 153730046214144 T + 458448159744 T^{2} - 325457920 T^{3} + 332928 T^{4} + 816 T^{5} + T^{6} )^{2} \)
$71$ \( ( 826210599600128 + 88146496512 T^{2} + 602976 T^{4} + T^{6} )^{2} \)
$73$ \( ( 3552805641982472 - 34127756459256 T + 163913239044 T^{2} + 888350720 T^{3} + 1972098 T^{4} + 1986 T^{5} + T^{6} )^{2} \)
$79$ \( ( 23809566408966144 + 446739648768 T^{2} + 1996512 T^{4} + T^{6} )^{2} \)
$83$ \( \)\(40\!\cdots\!76\)\( + \)\(30\!\cdots\!48\)\( T^{4} + 4125230408448 T^{8} + T^{12} \)
$89$ \( ( -17269984128833928 + 928594032012 T^{2} - 2356326 T^{4} + T^{6} )^{2} \)
$97$ \( ( 1300409140717475912 + 2088546858965544 T + 1677175223364 T^{2} - 182245120 T^{3} + 960498 T^{4} + 1386 T^{5} + T^{6} )^{2} \)
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