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: \( x^{12} - 16x^{10} - 14x^{8} - 512x^{6} + 3889x^{4} + 126224x^{2} + 506944 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q - \beta_{6} q^{2} + (\beta_{5} - 6 \beta_1) q^{4} + (\beta_{4} - 2 \beta_1 - 2) q^{7} + (\beta_{11} - \beta_{10} - 3 \beta_{9} - 7 \beta_{8} - 3 \beta_{7}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{6} q^{2} + (\beta_{5} - 6 \beta_1) q^{4} + (\beta_{4} - 2 \beta_1 - 2) q^{7} + (\beta_{11} - \beta_{10} - 3 \beta_{9} - 7 \beta_{8} - 3 \beta_{7}) q^{8} + (\beta_{11} + 3 \beta_{9} - 7 \beta_{8} - 7 \beta_{6}) q^{11} + ( - 2 \beta_{5} - 3 \beta_{3} + 2 \beta_{2} - 9 \beta_1 + 9) q^{13} + ( - \beta_{10} + 3 \beta_{8} - 7 \beta_{7} - 3 \beta_{6}) q^{14} + (5 \beta_{4} - 5 \beta_{3} + 9 \beta_{2} - 54) q^{16} + (\beta_{11} + \beta_{10} + 12 \beta_{9} - 12 \beta_{7} - 8 \beta_{6}) q^{17} + (4 \beta_{5} + 5 \beta_{4} + 5 \beta_{3} - 20 \beta_1) q^{19} + (6 \beta_{5} - \beta_{4} + 6 \beta_{2} - 88 \beta_1 - 88) q^{22} + ( - 6 \beta_{11} + 6 \beta_{10} - 2 \beta_{9} + 8 \beta_{8} - 2 \beta_{7}) q^{23} + ( - 7 \beta_{11} + 33 \beta_{9} - 6 \beta_{8} - 6 \beta_{6}) q^{26} + ( - 6 \beta_{5} - \beta_{3} + 6 \beta_{2} - 48 \beta_1 + 48) q^{28} + (5 \beta_{10} + 25 \beta_{8} - 4 \beta_{7} - 25 \beta_{6}) q^{29} + ( - 5 \beta_{4} + 5 \beta_{3} - 4 \beta_{2} - 104) q^{31} + ( - 6 \beta_{11} - 6 \beta_{10} + 38 \beta_{9} - 38 \beta_{7} + 29 \beta_{6}) q^{32} + ( - 12 \beta_{5} - 10 \beta_{4} - 10 \beta_{3} - 56 \beta_1) q^{34} + ( - 8 \beta_{5} + 7 \beta_{4} - 8 \beta_{2} - 69 \beta_1 - 69) q^{37} + (9 \beta_{11} - 9 \beta_{10} - 47 \beta_{9} - 6 \beta_{8} - 47 \beta_{7}) q^{38} + (16 \beta_{11} + 29 \beta_{9} + 38 \beta_{8} + 38 \beta_{6}) q^{41} + (16 \beta_{5} + 14 \beta_{3} - 16 \beta_{2} - 8 \beta_1 + 8) q^{43} + ( - 3 \beta_{10} - 91 \beta_{8} - 53 \beta_{7} + 91 \beta_{6}) q^{44} + ( - 10 \beta_{4} + 10 \beta_{3} - 28 \beta_{2} + 56) q^{46} + (9 \beta_{11} + 9 \beta_{10} + 23 \beta_{9} - 23 \beta_{7} - 58 \beta_{6}) q^{47} + (16 \beta_{5} - 111 \beta_1) q^{49} + ( - 25 \beta_{5} - 23 \beta_{4} - 25 \beta_{2} + 26 \beta_1 + 26) q^{52} + (10 \beta_{11} - 10 \beta_{10} + 5 \beta_{9} + 34 \beta_{8} + 5 \beta_{7}) q^{53} + ( - 5 \beta_{11} - 13 \beta_{9} + 25 \beta_{8} + 25 \beta_{6}) q^{56} + (31 \beta_{5} + 6 \beta_{3} - 31 \beta_{2} - 322 \beta_1 + 322) q^{58} + ( - 17 \beta_{10} + 41 \beta_{8} + 31 \beta_{7} - 41 \beta_{6}) q^{59} + ( - 15 \beta_{4} + 15 \beta_{3} - 48 \beta_{2} + 8) q^{61} + (9 \beta_{11} + 9 \beta_{10} - 47 \beta_{9} + 47 \beta_{7} + 118 \beta_{6}) q^{62} + ( - 57 \beta_{5} - 10 \beta_{4} - 10 \beta_{3} + 78 \beta_1) q^{64} + (44 \beta_{5} - 2 \beta_{4} + 44 \beta_{2} - 136 \beta_1 - 136) q^{67} + ( - 30 \beta_{11} + 30 \beta_{10} + 10 \beta_{9} + 16 \beta_{8} + 10 \beta_{7}) q^{68} + ( - 32 \beta_{11} + 4 \beta_{9} - 16 \beta_{8} - 16 \beta_{6}) q^{71} + ( - 18 \beta_{5} + 12 \beta_{3} + 18 \beta_{2} + 331 \beta_1 - 331) q^{73} + (9 \beta_{10} + 38 \beta_{8} - \beta_{7} - 38 \beta_{6}) q^{74} + (25 \beta_{4} - 25 \beta_{3} + 104 \beta_{2} - 40) q^{76} + ( - 24 \beta_{11} - 24 \beta_{10} + 12 \beta_{9} - 12 \beta_{7} - 32 \beta_{6}) q^{77} + (28 \beta_{5} + 25 \beta_{4} + 25 \beta_{3} + 672 \beta_1) q^{79} + ( - 35 \beta_{5} + 3 \beta_{4} - 35 \beta_{2} + 654 \beta_1 + 654) q^{82} + ( - \beta_{11} + \beta_{10} + 153 \beta_{9} + 78 \beta_{8} + 153 \beta_{7}) q^{83} + (46 \beta_{11} - 194 \beta_{9} - 82 \beta_{8} - 82 \beta_{6}) q^{86} + ( - 102 \beta_{5} - 67 \beta_{3} + 102 \beta_{2} + 664 \beta_1 - 664) q^{88} + (42 \beta_{10} - 36 \beta_{8} + 163 \beta_{7} + 36 \beta_{6}) q^{89} + (25 \beta_{4} - 25 \beta_{3} + 24 \beta_{2} + 396) q^{91} + ( - 10 \beta_{11} - 10 \beta_{10} - 170 \beta_{9} + 170 \beta_{7} - 144 \beta_{6}) q^{92} + (48 \beta_{5} - 5 \beta_{4} - 5 \beta_{3} - 648 \beta_1) q^{94} + (50 \beta_{5} + 78 \beta_{4} + 50 \beta_{2} - 231 \beta_1 - 231) q^{97} + (16 \beta_{11} - 16 \beta_{10} - 48 \beta_{9} - 255 \beta_{8} - 48 \beta_{7}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 24 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 24 q^{7} + 108 q^{13} - 648 q^{16} - 1056 q^{22} + 576 q^{28} - 1248 q^{31} - 828 q^{37} + 96 q^{43} + 672 q^{46} + 312 q^{52} + 3864 q^{58} + 96 q^{61} - 1632 q^{67} - 3972 q^{73} - 480 q^{76} + 7848 q^{82} - 7968 q^{88} + 4752 q^{91} - 2772 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( ( -3\nu^{10} - 56\nu^{8} + 634\nu^{6} + 3248\nu^{4} + 83197\nu^{2} + 418024 ) / 182400 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -71\nu^{10} + 2538\nu^{8} - 4502\nu^{6} - 356014\nu^{4} - 1896111\nu^{2} - 1578712 ) / 3620640 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 1029\nu^{10} - 22972\nu^{8} + 158738\nu^{6} - 2560724\nu^{4} + 22729229\nu^{2} + 78379688 ) / 9051600 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 5723\nu^{10} - 110024\nu^{8} + 167406\nu^{6} - 2434608\nu^{4} + 44277523\nu^{2} + 559571096 ) / 18103200 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 1265\nu^{10} - 2936\nu^{8} - 23478\nu^{6} - 1362480\nu^{4} - 16244903\nu^{2} - 51726904 ) / 2896512 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 113621 \nu^{11} - 12179272 \nu^{9} + 76074438 \nu^{7} + 253832976 \nu^{5} + 12758183579 \nu^{3} + 93476563288 \nu ) / 25778956800 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 6773\nu^{11} - 146104\nu^{9} + 577306\nu^{7} - 5223568\nu^{5} + 79774373\nu^{3} + 554840616\nu ) / 226131200 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 1093919 \nu^{11} + 7804552 \nu^{9} - 15466318 \nu^{7} + 1380835984 \nu^{5} + 6018524881 \nu^{3} + 28260515592 \nu ) / 25778956800 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -1095\nu^{11} + 7908\nu^{9} + 18534\nu^{7} + 745404\nu^{5} + 8928141\nu^{3} - 11971272\nu ) / 21482464 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 72101 \nu^{11} - 1867040 \nu^{9} + 15340242 \nu^{7} - 118083000 \nu^{5} + 659420461 \nu^{3} + 7004722280 \nu ) / 1288947840 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 12569\nu^{11} - 59505\nu^{9} - 670717\nu^{7} - 13483505\nu^{5} - 70039446\nu^{3} + 713598880\nu ) / 161118480 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( 3\beta_{11} + 3\beta_{10} - \beta_{9} + 12\beta_{8} + \beta_{7} - 6\beta_{6} ) / 24 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -4\beta_{5} - 3\beta_{4} + 11\beta_{3} - 20\beta_{2} - 64\beta _1 + 64 ) / 24 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 9\beta_{11} + 3\beta_{10} + 53\beta_{9} + 66\beta_{8} + 137\beta_{7} - 120\beta_{6} ) / 24 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 12\beta_{5} - 35\beta_{4} + 5\beta_{3} - 268\beta_{2} + 1136 ) / 24 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -171\beta_{11} + 177\beta_{10} - 615\beta_{9} + 1704\beta_{8} + 1259\beta_{7} - 1518\beta_{6} ) / 24 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 676\beta_{5} - 1157\beta_{4} + 909\beta_{3} - 3212\beta_{2} + 5824\beta _1 + 25216 ) / 24 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -489\beta_{11} + 6429\beta_{10} - 2133\beta_{9} + 21438\beta_{8} + 12439\beta_{7} - 24264\beta_{6} ) / 24 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 252\beta_{5} - 4255\beta_{4} + 5385\beta_{3} - 14588\beta_{2} - 20480\beta _1 + 129968 ) / 8 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 24939\beta_{11} + 63663\beta_{10} + 24551\beta_{9} + 292920\beta_{8} + 220021\beta_{7} - 359538\beta_{6} ) / 24 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 30812\beta_{5} - 127323\beta_{4} + 201011\beta_{3} - 706676\beta_{2} - 856384\beta _1 + 4399744 ) / 24 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 96009 \beta_{11} + 680739 \beta_{10} - 305227 \beta_{9} + 4045218 \beta_{8} + 3762665 \beta_{7} - 4953432 \beta_{6} ) / 24 \) Copy content Toggle raw display

Character values

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

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

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + 978 T^{8} + 149073 T^{4} + \cdots + 4096 \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( (T^{6} + 12 T^{5} + 72 T^{4} + \cdots + 700928)^{2} \) Copy content Toggle raw display
$11$ \( (T^{6} + 4344 T^{4} + \cdots + 2390999552)^{2} \) Copy content Toggle raw display
$13$ \( (T^{6} - 54 T^{5} + 1458 T^{4} + \cdots + 40033352)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} + 138301968 T^{8} + \cdots + 81\!\cdots\!96 \) Copy content Toggle raw display
$19$ \( (T^{6} + 26640 T^{4} + \cdots + 604661760000)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + 946426368 T^{8} + \cdots + 44\!\cdots\!96 \) Copy content Toggle raw display
$29$ \( (T^{6} - 58854 T^{4} + \cdots - 3276052522632)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} + 312 T^{2} + 19728 T + 333184)^{4} \) Copy content Toggle raw display
$37$ \( (T^{6} + 414 T^{5} + \cdots + 431026413512)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} + 367686 T^{4} + \cdots + 13\!\cdots\!88)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} - 48 T^{5} + \cdots + 12363096915968)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + 22535002368 T^{8} + \cdots + 11\!\cdots\!96 \) Copy content Toggle raw display
$53$ \( T^{12} + 14798523408 T^{8} + \cdots + 95\!\cdots\!76 \) Copy content Toggle raw display
$59$ \( (T^{6} - 421176 T^{4} + \cdots - 19\!\cdots\!28)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} - 24 T^{2} - 341088 T - 70019072)^{4} \) Copy content Toggle raw display
$67$ \( (T^{6} + 816 T^{5} + \cdots + 25\!\cdots\!72)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} + 602976 T^{4} + \cdots + 826210599600128)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} + 1986 T^{5} + \cdots + 35\!\cdots\!72)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} + 1996512 T^{4} + \cdots + 23\!\cdots\!44)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + 4125230408448 T^{8} + \cdots + 40\!\cdots\!76 \) Copy content Toggle raw display
$89$ \( (T^{6} - 2356326 T^{4} + \cdots - 17\!\cdots\!28)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} + 1386 T^{5} + \cdots + 13\!\cdots\!12)^{2} \) Copy content Toggle raw display
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