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: \( x^{16} - 7x^{12} + 48x^{8} - 7x^{4} + 1 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q + \beta_{13} q^{2} + (\beta_{15} + \beta_{9}) q^{4} + (2 \beta_{14} + 7 \beta_{3}) q^{7} + (2 \beta_{12} - 2 \beta_{10}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{13} q^{2} + (\beta_{15} + \beta_{9}) q^{4} + (2 \beta_{14} + 7 \beta_{3}) q^{7} + (2 \beta_{12} - 2 \beta_{10}) q^{8} + (7 \beta_{6} + \beta_{4}) q^{11} - 21 \beta_{2} q^{13} + (17 \beta_{7} - 4 \beta_{5}) q^{14} + (6 \beta_1 + 26) q^{16} + ( - 21 \beta_{13} - 7 \beta_{11}) q^{17} + (14 \beta_{15} - 35 \beta_{9}) q^{19} + (5 \beta_{14} + 63 \beta_{3}) q^{22} + ( - 31 \beta_{12} - 23 \beta_{10}) q^{23} - 21 \beta_{6} q^{26} + (9 \beta_{8} - 97 \beta_{2}) q^{28} + ( - 3 \beta_{7} - 17 \beta_{5}) q^{29} + (14 \beta_1 + 91) q^{31} + (40 \beta_{13} - 28 \beta_{11}) q^{32} + ( - 7 \beta_{15} - 189 \beta_{9}) q^{34} + ( - 12 \beta_{14} + 14 \beta_{3}) q^{37} + ( - 35 \beta_{12} - 28 \beta_{10}) q^{38} + (14 \beta_{6} + 14 \beta_{4}) q^{41} + ( - 30 \beta_{8} - 77 \beta_{2}) q^{43} + (32 \beta_{7} - 18 \beta_{5}) q^{44} + ( - 77 \beta_1 - 279) q^{46} + (77 \beta_{13} + 7 \beta_{11}) q^{47} + ( - 84 \beta_{15} - 344 \beta_{9}) q^{49} + ( - 21 \beta_{14} - 21 \beta_{3}) q^{52} + (4 \beta_{12} - 40 \beta_{10}) q^{53} + ( - 6 \beta_{6} - 14 \beta_{4}) q^{56} + (31 \beta_{8} + 27 \beta_{2}) q^{58} + ( - 91 \beta_{7} + 7 \beta_{5}) q^{59} + (28 \beta_1 + 413) q^{61} + (161 \beta_{13} - 28 \beta_{11}) q^{62} + (48 \beta_{15} + 152 \beta_{9}) q^{64} + (14 \beta_{14} - 99 \beta_{3}) q^{67} + (56 \beta_{12} + 70 \beta_{10}) q^{68} + ( - 94 \beta_{6} + 18 \beta_{4}) q^{71} + ( - 28 \beta_{8} + 154 \beta_{2}) q^{73} + ( - 46 \beta_{7} + 24 \beta_{5}) q^{74} + (21 \beta_1 - 595) q^{76} + ( - 297 \beta_{13} + 93 \beta_{11}) q^{77} + (28 \beta_{15} + 638 \beta_{9}) q^{79} + ( - 14 \beta_{14} + 126 \beta_{3}) q^{82} + ( - 231 \beta_{12} + 77 \beta_{10}) q^{83} + (73 \beta_{6} - 60 \beta_{4}) q^{86} + (28 \beta_{8} + 216 \beta_{2}) q^{88} + (154 \beta_{7} + 70 \beta_{5}) q^{89} + (126 \beta_1 + 441) q^{91} + ( - 416 \beta_{13} - 30 \beta_{11}) q^{92} + (63 \beta_{15} + 693 \beta_{9}) q^{94} + (56 \beta_{14} - 679 \beta_{3}) q^{97} + (764 \beta_{12} + 168 \beta_{10}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 416 q^{16} + 1456 q^{31} - 4464 q^{46} + 6608 q^{61} - 9520 q^{76} + 7056 q^{91}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( ( \nu^{12} + 161 ) / 24 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 3\nu^{13} - 20\nu^{9} + 136\nu^{5} + 19\nu ) / 12 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 15\nu^{15} - 104\nu^{11} + 712\nu^{7} - 53\nu^{3} ) / 12 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 3\nu^{15} + 47\nu^{13} - 336\nu^{9} + 2256\nu^{5} + 843\nu^{3} - 329\nu ) / 48 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -3\nu^{15} + 47\nu^{13} - 336\nu^{9} + 2256\nu^{5} - 843\nu^{3} - 329\nu ) / 48 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -3\nu^{15} - 55\nu^{13} + 384\nu^{9} - 2640\nu^{5} - 987\nu^{3} + 385\nu ) / 48 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 3\nu^{15} - 55\nu^{13} + 384\nu^{9} - 2640\nu^{5} + 987\nu^{3} + 385\nu ) / 48 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -13\nu^{13} + 88\nu^{9} - 608\nu^{5} - 85\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 7\nu^{14} - 48\nu^{10} + 330\nu^{6} - \nu^{2} ) / 18 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 25\nu^{14} + 19\nu^{12} - 176\nu^{10} - 128\nu^{8} + 1216\nu^{6} + 928\nu^{4} - 351\nu^{2} - 69 ) / 48 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -25\nu^{14} + 19\nu^{12} + 176\nu^{10} - 128\nu^{8} - 1216\nu^{6} + 928\nu^{4} + 351\nu^{2} - 69 ) / 48 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 29\nu^{14} + 23\nu^{12} - 208\nu^{10} - 160\nu^{8} + 1424\nu^{6} + 1088\nu^{4} - 411\nu^{2} - 81 ) / 48 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 29\nu^{14} - 23\nu^{12} - 208\nu^{10} + 160\nu^{8} + 1424\nu^{6} - 1088\nu^{4} - 411\nu^{2} + 81 ) / 48 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 67\nu^{15} - 464\nu^{11} + 3184\nu^{7} - 237\nu^{3} ) / 8 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 21\nu^{14} - 144\nu^{10} + 984\nu^{6} - 3\nu^{2} ) / 8 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -3\beta_{8} + 2\beta_{7} + 2\beta_{6} + \beta_{5} + \beta_{4} - 9\beta_{2} ) / 36 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{15} + \beta_{13} + \beta_{12} + 2\beta_{11} - 2\beta_{10} + 9\beta_{9} ) / 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} ) / 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 4\beta_{13} - 4\beta_{12} + 5\beta_{11} + 5\beta_{10} - 3\beta _1 + 21 ) / 12 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -15\beta_{8} - 7\beta_{7} - 7\beta_{6} - 8\beta_{5} - 8\beta_{4} - 99\beta_{2} ) / 36 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -4\beta_{15} + 27\beta_{9} ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 13\beta_{14} + 6\beta_{7} - 6\beta_{6} + 7\beta_{5} - 7\beta_{4} - 87\beta_{3} ) / 12 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 29\beta_{13} - 29\beta_{12} + 34\beta_{11} + 34\beta_{10} + 21\beta _1 - 141 ) / 12 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( -47\beta_{7} - 47\beta_{6} - 55\beta_{5} - 55\beta_{4} ) / 18 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( -55\beta_{15} - 76\beta_{13} - 76\beta_{12} - 89\beta_{11} + 89\beta_{10} + 369\beta_{9} ) / 12 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 89\beta_{14} - 41\beta_{7} + 41\beta_{6} - 48\beta_{5} + 48\beta_{4} - 597\beta_{3} ) / 12 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 24\beta _1 - 161 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 699\beta_{8} - 322\beta_{7} - 322\beta_{6} - 377\beta_{5} - 377\beta_{4} + 4689\beta_{2} ) / 36 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 377\beta_{15} - 521\beta_{13} - 521\beta_{12} - 610\beta_{11} + 610\beta_{10} - 2529\beta_{9} ) / 12 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( -281\beta_{7} + 281\beta_{6} - 329\beta_{5} + 329\beta_{4} ) / 6 \) Copy content Toggle raw display

Character values

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

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

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 \(S_{4}^{\mathrm{new}}(225, [\chi])\):

\( T_{2}^{8} + 252T_{2}^{4} + 1296 \) Copy content Toggle raw display
\( T_{7}^{8} + 1578978T_{7}^{4} + 23854493601 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{8} + 252 T^{4} + 1296)^{2} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( (T^{8} + 1578978 T^{4} + \cdots + 23854493601)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + 2916 T^{2} + 2022084)^{4} \) Copy content Toggle raw display
$13$ \( (T^{4} + 1750329)^{4} \) Copy content Toggle raw display
$17$ \( (T^{8} + 90066312 T^{4} + \cdots + 973651921932816)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 20090 T^{2} + 57684025)^{4} \) Copy content Toggle raw display
$23$ \( (T^{8} + 2229304392 T^{4} + \cdots + 98\!\cdots\!96)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} - 78516 T^{2} + \cdots + 451902564)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} - 182 T - 539)^{8} \) Copy content Toggle raw display
$37$ \( (T^{8} + 893687328 T^{4} + \cdots + 12\!\cdots\!16)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 63504 T^{2} + 12446784)^{4} \) Copy content Toggle raw display
$43$ \( (T^{8} + 56090700738 T^{4} + \cdots + 11\!\cdots\!61)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} + 7295371272 T^{4} + \cdots + 63\!\cdots\!76)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + 17103117312 T^{4} + \cdots + 34\!\cdots\!56)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} - 460404 T^{2} + \cdots + 7624433124)^{4} \) Copy content Toggle raw display
$61$ \( (T^{2} - 826 T + 135289)^{8} \) Copy content Toggle raw display
$67$ \( (T^{8} + 12465376578 T^{4} + \cdots + 75017234240001)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 564624 T^{2} + \cdots + 501401664)^{4} \) Copy content Toggle raw display
$73$ \( (T^{8} + 122891938848 T^{4} + \cdots + 14\!\cdots\!96)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 884648 T^{2} + \cdots + 138208471696)^{4} \) Copy content Toggle raw display
$83$ \( (T^{8} + 1318660873992 T^{4} + \cdots + 20\!\cdots\!96)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} - 2603664 T^{2} + \cdots + 914104263744)^{4} \) Copy content Toggle raw display
$97$ \( (T^{8} + 11211233284818 T^{4} + \cdots + 84\!\cdots\!61)^{2} \) Copy content Toggle raw display
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