Properties

Label 225.4.f.d
Level $225$
Weight $4$
Character orbit 225.f
Analytic conductor $13.275$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

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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: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: 16.0.11007531417600000000.1
Defining polynomial: \(x^{16} - 7 x^{12} + 48 x^{8} - 7 x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{16}\cdot 3^{12} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{13} q^{2} + ( \beta_{9} + \beta_{15} ) q^{4} + ( 7 \beta_{3} + 2 \beta_{14} ) q^{7} + ( -2 \beta_{10} + 2 \beta_{12} ) q^{8} +O(q^{10})\) \( q + \beta_{13} q^{2} + ( \beta_{9} + \beta_{15} ) q^{4} + ( 7 \beta_{3} + 2 \beta_{14} ) q^{7} + ( -2 \beta_{10} + 2 \beta_{12} ) q^{8} + ( \beta_{4} + 7 \beta_{6} ) q^{11} -21 \beta_{2} q^{13} + ( -4 \beta_{5} + 17 \beta_{7} ) q^{14} + ( 26 + 6 \beta_{1} ) q^{16} + ( -7 \beta_{11} - 21 \beta_{13} ) q^{17} + ( -35 \beta_{9} + 14 \beta_{15} ) q^{19} + ( 63 \beta_{3} + 5 \beta_{14} ) q^{22} + ( -23 \beta_{10} - 31 \beta_{12} ) q^{23} -21 \beta_{6} q^{26} + ( -97 \beta_{2} + 9 \beta_{8} ) q^{28} + ( -17 \beta_{5} - 3 \beta_{7} ) q^{29} + ( 91 + 14 \beta_{1} ) q^{31} + ( -28 \beta_{11} + 40 \beta_{13} ) q^{32} + ( -189 \beta_{9} - 7 \beta_{15} ) q^{34} + ( 14 \beta_{3} - 12 \beta_{14} ) q^{37} + ( -28 \beta_{10} - 35 \beta_{12} ) q^{38} + ( 14 \beta_{4} + 14 \beta_{6} ) q^{41} + ( -77 \beta_{2} - 30 \beta_{8} ) q^{43} + ( -18 \beta_{5} + 32 \beta_{7} ) q^{44} + ( -279 - 77 \beta_{1} ) q^{46} + ( 7 \beta_{11} + 77 \beta_{13} ) q^{47} + ( -344 \beta_{9} - 84 \beta_{15} ) q^{49} + ( -21 \beta_{3} - 21 \beta_{14} ) q^{52} + ( -40 \beta_{10} + 4 \beta_{12} ) q^{53} + ( -14 \beta_{4} - 6 \beta_{6} ) q^{56} + ( 27 \beta_{2} + 31 \beta_{8} ) q^{58} + ( 7 \beta_{5} - 91 \beta_{7} ) q^{59} + ( 413 + 28 \beta_{1} ) q^{61} + ( -28 \beta_{11} + 161 \beta_{13} ) q^{62} + ( 152 \beta_{9} + 48 \beta_{15} ) q^{64} + ( -99 \beta_{3} + 14 \beta_{14} ) q^{67} + ( 70 \beta_{10} + 56 \beta_{12} ) q^{68} + ( 18 \beta_{4} - 94 \beta_{6} ) q^{71} + ( 154 \beta_{2} - 28 \beta_{8} ) q^{73} + ( 24 \beta_{5} - 46 \beta_{7} ) q^{74} + ( -595 + 21 \beta_{1} ) q^{76} + ( 93 \beta_{11} - 297 \beta_{13} ) q^{77} + ( 638 \beta_{9} + 28 \beta_{15} ) q^{79} + ( 126 \beta_{3} - 14 \beta_{14} ) q^{82} + ( 77 \beta_{10} - 231 \beta_{12} ) q^{83} + ( -60 \beta_{4} + 73 \beta_{6} ) q^{86} + ( 216 \beta_{2} + 28 \beta_{8} ) q^{88} + ( 70 \beta_{5} + 154 \beta_{7} ) q^{89} + ( 441 + 126 \beta_{1} ) q^{91} + ( -30 \beta_{11} - 416 \beta_{13} ) q^{92} + ( 693 \beta_{9} + 63 \beta_{15} ) q^{94} + ( -679 \beta_{3} + 56 \beta_{14} ) q^{97} + ( 168 \beta_{10} + 764 \beta_{12} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + O(q^{10}) \) \( 16 q + 416 q^{16} + 1456 q^{31} - 4464 q^{46} + 6608 q^{61} - 9520 q^{76} + 7056 q^{91} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - 7 x^{12} + 48 x^{8} - 7 x^{4} + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{12} + 161 \)\()/24\)
\(\beta_{2}\)\(=\)\((\)\( 3 \nu^{13} - 20 \nu^{9} + 136 \nu^{5} + 19 \nu \)\()/12\)
\(\beta_{3}\)\(=\)\((\)\( 15 \nu^{15} - 104 \nu^{11} + 712 \nu^{7} - 53 \nu^{3} \)\()/12\)
\(\beta_{4}\)\(=\)\((\)\( 3 \nu^{15} + 47 \nu^{13} - 336 \nu^{9} + 2256 \nu^{5} + 843 \nu^{3} - 329 \nu \)\()/48\)
\(\beta_{5}\)\(=\)\((\)\( -3 \nu^{15} + 47 \nu^{13} - 336 \nu^{9} + 2256 \nu^{5} - 843 \nu^{3} - 329 \nu \)\()/48\)
\(\beta_{6}\)\(=\)\((\)\( -3 \nu^{15} - 55 \nu^{13} + 384 \nu^{9} - 2640 \nu^{5} - 987 \nu^{3} + 385 \nu \)\()/48\)
\(\beta_{7}\)\(=\)\((\)\( 3 \nu^{15} - 55 \nu^{13} + 384 \nu^{9} - 2640 \nu^{5} + 987 \nu^{3} + 385 \nu \)\()/48\)
\(\beta_{8}\)\(=\)\((\)\( -13 \nu^{13} + 88 \nu^{9} - 608 \nu^{5} - 85 \nu \)\()/8\)
\(\beta_{9}\)\(=\)\((\)\( 7 \nu^{14} - 48 \nu^{10} + 330 \nu^{6} - \nu^{2} \)\()/18\)
\(\beta_{10}\)\(=\)\((\)\( 25 \nu^{14} + 19 \nu^{12} - 176 \nu^{10} - 128 \nu^{8} + 1216 \nu^{6} + 928 \nu^{4} - 351 \nu^{2} - 69 \)\()/48\)
\(\beta_{11}\)\(=\)\((\)\( -25 \nu^{14} + 19 \nu^{12} + 176 \nu^{10} - 128 \nu^{8} - 1216 \nu^{6} + 928 \nu^{4} + 351 \nu^{2} - 69 \)\()/48\)
\(\beta_{12}\)\(=\)\((\)\( 29 \nu^{14} + 23 \nu^{12} - 208 \nu^{10} - 160 \nu^{8} + 1424 \nu^{6} + 1088 \nu^{4} - 411 \nu^{2} - 81 \)\()/48\)
\(\beta_{13}\)\(=\)\((\)\( 29 \nu^{14} - 23 \nu^{12} - 208 \nu^{10} + 160 \nu^{8} + 1424 \nu^{6} - 1088 \nu^{4} - 411 \nu^{2} + 81 \)\()/48\)
\(\beta_{14}\)\(=\)\((\)\( 67 \nu^{15} - 464 \nu^{11} + 3184 \nu^{7} - 237 \nu^{3} \)\()/8\)
\(\beta_{15}\)\(=\)\((\)\( 21 \nu^{14} - 144 \nu^{10} + 984 \nu^{6} - 3 \nu^{2} \)\()/8\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-3 \beta_{8} + 2 \beta_{7} + 2 \beta_{6} + \beta_{5} + \beta_{4} - 9 \beta_{2}\)\()/36\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{15} + \beta_{13} + \beta_{12} + 2 \beta_{11} - 2 \beta_{10} + 9 \beta_{9}\)\()/12\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{7} - \beta_{6} + \beta_{5} - \beta_{4}\)\()/6\)
\(\nu^{4}\)\(=\)\((\)\(4 \beta_{13} - 4 \beta_{12} + 5 \beta_{11} + 5 \beta_{10} - 3 \beta_{1} + 21\)\()/12\)
\(\nu^{5}\)\(=\)\((\)\(-15 \beta_{8} - 7 \beta_{7} - 7 \beta_{6} - 8 \beta_{5} - 8 \beta_{4} - 99 \beta_{2}\)\()/36\)
\(\nu^{6}\)\(=\)\((\)\(-4 \beta_{15} + 27 \beta_{9}\)\()/3\)
\(\nu^{7}\)\(=\)\((\)\(13 \beta_{14} + 6 \beta_{7} - 6 \beta_{6} + 7 \beta_{5} - 7 \beta_{4} - 87 \beta_{3}\)\()/12\)
\(\nu^{8}\)\(=\)\((\)\(29 \beta_{13} - 29 \beta_{12} + 34 \beta_{11} + 34 \beta_{10} + 21 \beta_{1} - 141\)\()/12\)
\(\nu^{9}\)\(=\)\((\)\(-47 \beta_{7} - 47 \beta_{6} - 55 \beta_{5} - 55 \beta_{4}\)\()/18\)
\(\nu^{10}\)\(=\)\((\)\(-55 \beta_{15} - 76 \beta_{13} - 76 \beta_{12} - 89 \beta_{11} + 89 \beta_{10} + 369 \beta_{9}\)\()/12\)
\(\nu^{11}\)\(=\)\((\)\(89 \beta_{14} - 41 \beta_{7} + 41 \beta_{6} - 48 \beta_{5} + 48 \beta_{4} - 597 \beta_{3}\)\()/12\)
\(\nu^{12}\)\(=\)\(24 \beta_{1} - 161\)
\(\nu^{13}\)\(=\)\((\)\(699 \beta_{8} - 322 \beta_{7} - 322 \beta_{6} - 377 \beta_{5} - 377 \beta_{4} + 4689 \beta_{2}\)\()/36\)
\(\nu^{14}\)\(=\)\((\)\(377 \beta_{15} - 521 \beta_{13} - 521 \beta_{12} - 610 \beta_{11} + 610 \beta_{10} - 2529 \beta_{9}\)\()/12\)
\(\nu^{15}\)\(=\)\((\)\(-281 \beta_{7} + 281 \beta_{6} - 329 \beta_{5} + 329 \beta_{4}\)\()/6\)

Character values

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

\(n\) \(101\) \(127\)
\(\chi(n)\) \(-1\) \(-\beta_{9}\)

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
0.596975 0.159959i
−0.596975 + 0.159959i
0.418778 1.56290i
−0.418778 + 1.56290i
1.56290 0.418778i
−1.56290 + 0.418778i
0.159959 0.596975i
−0.159959 + 0.596975i
0.596975 + 0.159959i
−0.596975 0.159959i
0.418778 + 1.56290i
−0.418778 1.56290i
1.56290 + 0.418778i
−1.56290 0.418778i
0.159959 + 0.596975i
−0.159959 0.596975i
−2.80252 + 2.80252i 0 7.70820i 0 0 −25.0049 25.0049i −0.817763 0.817763i 0 0
107.2 −2.80252 + 2.80252i 0 7.70820i 0 0 25.0049 + 25.0049i −0.817763 0.817763i 0 0
107.3 −1.07047 + 1.07047i 0 5.70820i 0 0 −7.85846 7.85846i −14.6742 14.6742i 0 0
107.4 −1.07047 + 1.07047i 0 5.70820i 0 0 7.85846 + 7.85846i −14.6742 14.6742i 0 0
107.5 1.07047 1.07047i 0 5.70820i 0 0 −7.85846 7.85846i 14.6742 + 14.6742i 0 0
107.6 1.07047 1.07047i 0 5.70820i 0 0 7.85846 + 7.85846i 14.6742 + 14.6742i 0 0
107.7 2.80252 2.80252i 0 7.70820i 0 0 −25.0049 25.0049i 0.817763 + 0.817763i 0 0
107.8 2.80252 2.80252i 0 7.70820i 0 0 25.0049 + 25.0049i 0.817763 + 0.817763i 0 0
143.1 −2.80252 2.80252i 0 7.70820i 0 0 −25.0049 + 25.0049i −0.817763 + 0.817763i 0 0
143.2 −2.80252 2.80252i 0 7.70820i 0 0 25.0049 25.0049i −0.817763 + 0.817763i 0 0
143.3 −1.07047 1.07047i 0 5.70820i 0 0 −7.85846 + 7.85846i −14.6742 + 14.6742i 0 0
143.4 −1.07047 1.07047i 0 5.70820i 0 0 7.85846 7.85846i −14.6742 + 14.6742i 0 0
143.5 1.07047 + 1.07047i 0 5.70820i 0 0 −7.85846 + 7.85846i 14.6742 14.6742i 0 0
143.6 1.07047 + 1.07047i 0 5.70820i 0 0 7.85846 7.85846i 14.6742 14.6742i 0 0
143.7 2.80252 + 2.80252i 0 7.70820i 0 0 −25.0049 + 25.0049i 0.817763 0.817763i 0 0
143.8 2.80252 + 2.80252i 0 7.70820i 0 0 25.0049 25.0049i 0.817763 0.817763i 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 143.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 225.4.f.d 16
3.b odd 2 1 inner 225.4.f.d 16
5.b even 2 1 inner 225.4.f.d 16
5.c odd 4 2 inner 225.4.f.d 16
15.d odd 2 1 inner 225.4.f.d 16
15.e even 4 2 inner 225.4.f.d 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
225.4.f.d 16 1.a even 1 1 trivial
225.4.f.d 16 3.b odd 2 1 inner
225.4.f.d 16 5.b even 2 1 inner
225.4.f.d 16 5.c odd 4 2 inner
225.4.f.d 16 15.d odd 2 1 inner
225.4.f.d 16 15.e even 4 2 inner

Hecke kernels

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

\( T_{2}^{8} + 252 T_{2}^{4} + 1296 \)
\( T_{7}^{8} + 1578978 T_{7}^{4} + 23854493601 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1296 + 252 T^{4} + T^{8} )^{2} \)
$3$ \( T^{16} \)
$5$ \( T^{16} \)
$7$ \( ( 23854493601 + 1578978 T^{4} + T^{8} )^{2} \)
$11$ \( ( 2022084 + 2916 T^{2} + T^{4} )^{4} \)
$13$ \( ( 1750329 + T^{4} )^{4} \)
$17$ \( ( 973651921932816 + 90066312 T^{4} + T^{8} )^{2} \)
$19$ \( ( 57684025 + 20090 T^{2} + T^{4} )^{4} \)
$23$ \( ( 983810135798785296 + 2229304392 T^{4} + T^{8} )^{2} \)
$29$ \( ( 451902564 - 78516 T^{2} + T^{4} )^{4} \)
$31$ \( ( -539 - 182 T + T^{2} )^{8} \)
$37$ \( ( 126307670167593216 + 893687328 T^{4} + T^{8} )^{2} \)
$41$ \( ( 12446784 + 63504 T^{2} + T^{4} )^{4} \)
$43$ \( ( \)\(11\!\cdots\!61\)\( + 56090700738 T^{4} + T^{8} )^{2} \)
$47$ \( ( 6388130259801205776 + 7295371272 T^{4} + T^{8} )^{2} \)
$53$ \( ( 3451996098297593856 + 17103117312 T^{4} + T^{8} )^{2} \)
$59$ \( ( 7624433124 - 460404 T^{2} + T^{4} )^{4} \)
$61$ \( ( 135289 - 826 T + T^{2} )^{8} \)
$67$ \( ( 75017234240001 + 12465376578 T^{4} + T^{8} )^{2} \)
$71$ \( ( 501401664 + 564624 T^{2} + T^{4} )^{4} \)
$73$ \( ( 1448496168863498496 + 122891938848 T^{4} + T^{8} )^{2} \)
$79$ \( ( 138208471696 + 884648 T^{2} + T^{4} )^{4} \)
$83$ \( ( \)\(20\!\cdots\!96\)\( + 1318660873992 T^{4} + T^{8} )^{2} \)
$89$ \( ( 914104263744 - 2603664 T^{2} + T^{4} )^{4} \)
$97$ \( ( \)\(84\!\cdots\!61\)\( + 11211233284818 T^{4} + T^{8} )^{2} \)
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