Properties

Label 225.4.e.c
Level $225$
Weight $4$
Character orbit 225.e
Analytic conductor $13.275$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(13.2754297513\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.15759792.1
Defining polynomial: \(x^{6} - 3 x^{5} + 16 x^{4} - 27 x^{3} + 52 x^{2} - 39 x + 9\)
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 45)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( \beta_{1} - \beta_{5} ) q^{2} + ( -1 - \beta_{2} + \beta_{5} ) q^{3} + ( 4 \beta_{3} - 3 \beta_{4} + 2 \beta_{5} ) q^{4} + ( 17 - 2 \beta_{1} - 4 \beta_{2} + 5 \beta_{3} - \beta_{4} ) q^{6} + ( -16 + \beta_{1} + 6 \beta_{2} - 16 \beta_{3} + 6 \beta_{4} - \beta_{5} ) q^{7} + ( 9 - 3 \beta_{1} - 3 \beta_{2} ) q^{8} + ( -2 + 4 \beta_{1} + 7 \beta_{2} - 22 \beta_{3} + 5 \beta_{4} - 2 \beta_{5} ) q^{9} +O(q^{10})\) \( q + ( \beta_{1} - \beta_{5} ) q^{2} + ( -1 - \beta_{2} + \beta_{5} ) q^{3} + ( 4 \beta_{3} - 3 \beta_{4} + 2 \beta_{5} ) q^{4} + ( 17 - 2 \beta_{1} - 4 \beta_{2} + 5 \beta_{3} - \beta_{4} ) q^{6} + ( -16 + \beta_{1} + 6 \beta_{2} - 16 \beta_{3} + 6 \beta_{4} - \beta_{5} ) q^{7} + ( 9 - 3 \beta_{1} - 3 \beta_{2} ) q^{8} + ( -2 + 4 \beta_{1} + 7 \beta_{2} - 22 \beta_{3} + 5 \beta_{4} - 2 \beta_{5} ) q^{9} + ( -3 + 14 \beta_{1} + 9 \beta_{2} - 3 \beta_{3} + 9 \beta_{4} - 14 \beta_{5} ) q^{11} + ( -9 + 11 \beta_{1} + 3 \beta_{2} + 4 \beta_{3} + 10 \beta_{4} - 17 \beta_{5} ) q^{12} + ( -17 \beta_{3} - 3 \beta_{4} + 14 \beta_{5} ) q^{13} + ( -18 \beta_{3} + 3 \beta_{4} + 24 \beta_{5} ) q^{14} + ( 11 + 2 \beta_{1} - 18 \beta_{2} + 11 \beta_{3} - 18 \beta_{4} - 2 \beta_{5} ) q^{16} + ( 57 + 2 \beta_{1} - 3 \beta_{2} ) q^{17} + ( 14 + 14 \beta_{1} - 4 \beta_{2} + 13 \beta_{3} - 5 \beta_{4} + 17 \beta_{5} ) q^{18} + ( -55 - 10 \beta_{1} - 9 \beta_{2} ) q^{19} + ( -51 - 36 \beta_{1} - 33 \beta_{3} - 3 \beta_{4} + 12 \beta_{5} ) q^{21} + ( 123 \beta_{3} - 33 \beta_{4} + 40 \beta_{5} ) q^{22} + ( -48 \beta_{3} - 36 \beta_{4} + 9 \beta_{5} ) q^{23} + ( 3 + 6 \beta_{1} + 3 \beta_{2} + 21 \beta_{3} - 6 \beta_{4} + 18 \beta_{5} ) q^{24} + ( 153 - 14 \beta_{1} - 39 \beta_{2} ) q^{26} + ( -42 - 45 \beta_{1} - 24 \beta_{2} - 27 \beta_{3} + 9 \beta_{4} + 24 \beta_{5} ) q^{27} + ( 175 - 19 \beta_{1} - 27 \beta_{2} ) q^{28} + ( 117 - 14 \beta_{1} - 30 \beta_{2} + 117 \beta_{3} - 30 \beta_{4} + 14 \beta_{5} ) q^{29} + ( -127 \beta_{3} - 33 \beta_{4} + 62 \beta_{5} ) q^{31} + ( 42 \beta_{3} - 49 \beta_{5} ) q^{32} + ( 115 - 58 \beta_{1} - 71 \beta_{2} - 8 \beta_{3} - 38 \beta_{4} + 18 \beta_{5} ) q^{33} + ( 39 + 56 \beta_{1} - 9 \beta_{2} + 39 \beta_{3} - 9 \beta_{4} - 56 \beta_{5} ) q^{34} + ( 191 - 46 \beta_{1} - 52 \beta_{2} + 28 \beta_{3} - 62 \beta_{4} + 26 \beta_{5} ) q^{36} + ( -143 + 24 \beta_{1} + 51 \beta_{2} ) q^{37} + ( -75 - 26 \beta_{1} + 21 \beta_{2} - 75 \beta_{3} + 21 \beta_{4} + 26 \beta_{5} ) q^{38} + ( -153 + 14 \beta_{1} + 39 \beta_{2} - 179 \beta_{3} + 79 \beta_{4} - 20 \beta_{5} ) q^{39} + ( 72 \beta_{3} - 69 \beta_{4} + 40 \beta_{5} ) q^{41} + ( -303 + 27 \beta_{1} + 75 \beta_{2} - 432 \beta_{3} + 108 \beta_{4} - 21 \beta_{5} ) q^{42} + ( -169 + 8 \beta_{1} - 87 \beta_{2} - 169 \beta_{3} - 87 \beta_{4} - 8 \beta_{5} ) q^{43} + ( 291 - 124 \beta_{1} - 15 \beta_{2} ) q^{44} + ( -72 - 6 \beta_{1} + 9 \beta_{2} ) q^{46} + ( 207 - 59 \beta_{1} - 15 \beta_{2} + 207 \beta_{3} - 15 \beta_{4} + 59 \beta_{5} ) q^{47} + ( 275 + 61 \beta_{1} + 17 \beta_{2} + 161 \beta_{3} + 41 \beta_{4} - 36 \beta_{5} ) q^{48} + ( 189 \beta_{3} - 111 \beta_{4} - 26 \beta_{5} ) q^{49} + ( -20 + 6 \beta_{1} - 50 \beta_{2} - 39 \beta_{3} + 9 \beta_{4} + 56 \beta_{5} ) q^{51} + ( -109 + 108 \beta_{1} - 21 \beta_{2} - 109 \beta_{3} - 21 \beta_{4} - 108 \beta_{5} ) q^{52} + ( 228 - 70 \beta_{1} - 24 \beta_{2} ) q^{53} + ( -87 + 60 \beta_{1} + 30 \beta_{2} - 420 \beta_{3} + 111 \beta_{4} - 72 \beta_{5} ) q^{54} + ( -237 + 48 \beta_{1} + 54 \beta_{2} - 237 \beta_{3} + 54 \beta_{4} - 48 \beta_{5} ) q^{56} + ( 86 + 18 \beta_{1} + 92 \beta_{2} + 75 \beta_{3} - 21 \beta_{4} - 26 \beta_{5} ) q^{57} + ( -18 \beta_{3} + 12 \beta_{4} - 175 \beta_{5} ) q^{58} + ( -36 \beta_{3} - 108 \beta_{4} - 82 \beta_{5} ) q^{59} + ( 448 - 20 \beta_{1} + 75 \beta_{2} + 448 \beta_{3} + 75 \beta_{4} + 20 \beta_{5} ) q^{61} + ( 579 - 30 \beta_{1} - 153 \beta_{2} ) q^{62} + ( -258 - 6 \beta_{1} + 120 \beta_{2} + 303 \beta_{3} - 21 \beta_{4} + 21 \beta_{5} ) q^{63} + ( -500 + 72 \beta_{1} + 3 \beta_{2} ) q^{64} + ( -315 + 236 \beta_{1} + 87 \beta_{2} - 341 \beta_{3} + 103 \beta_{4} - 302 \beta_{5} ) q^{66} + ( 637 \beta_{3} - 21 \beta_{4} - 41 \beta_{5} ) q^{67} + ( 261 \beta_{3} - 153 \beta_{4} + 80 \beta_{5} ) q^{68} + ( 72 + 6 \beta_{1} - 9 \beta_{2} + 399 \beta_{3} + 156 \beta_{4} - 78 \beta_{5} ) q^{69} + ( 81 - 76 \beta_{1} - 249 \beta_{2} ) q^{71} + ( -186 + 57 \beta_{1} + 30 \beta_{2} - 300 \beta_{3} + 78 \beta_{4} - 87 \beta_{5} ) q^{72} + ( 172 - 44 \beta_{1} - 192 \beta_{2} ) q^{73} + ( 33 - 242 \beta_{1} - 21 \beta_{2} + 33 \beta_{3} - 21 \beta_{4} + 242 \beta_{5} ) q^{74} + ( 23 \beta_{3} + 171 \beta_{4} - 36 \beta_{5} ) q^{76} + ( 237 \beta_{3} - 3 \beta_{4} + 240 \beta_{5} ) q^{77} + ( 128 + 78 \beta_{1} - 22 \beta_{2} - 27 \beta_{3} - 3 \beta_{4} + 220 \beta_{5} ) q^{78} + ( 154 - 112 \beta_{1} - 192 \beta_{2} + 154 \beta_{3} - 192 \beta_{4} + 112 \beta_{5} ) q^{79} + ( -300 + 60 \beta_{1} + 240 \beta_{2} - 87 \beta_{3} - 6 \beta_{4} + 78 \beta_{5} ) q^{81} + ( 135 - 221 \beta_{1} - 51 \beta_{2} ) q^{82} + ( -360 + 105 \beta_{1} + 354 \beta_{2} - 360 \beta_{3} + 354 \beta_{4} - 105 \beta_{5} ) q^{83} + ( -27 + 54 \beta_{1} - 75 \beta_{2} + 93 \beta_{3} - 30 \beta_{4} + 240 \beta_{5} ) q^{84} + ( 531 \beta_{3} - 111 \beta_{4} + 98 \beta_{5} ) q^{86} + ( 65 + 235 \beta_{1} - \beta_{2} + 83 \beta_{3} - 13 \beta_{4} - 60 \beta_{5} ) q^{87} + ( -429 + 234 \beta_{1} + 93 \beta_{2} - 429 \beta_{3} + 93 \beta_{4} - 234 \beta_{5} ) q^{88} + ( 549 + 168 \beta_{1} + 240 \beta_{2} ) q^{89} + ( -377 - 286 \beta_{1} + 135 \beta_{2} ) q^{91} + ( -501 - 141 \beta_{1} - 261 \beta_{2} - 501 \beta_{3} - 261 \beta_{4} + 141 \beta_{5} ) q^{92} + ( -579 + 30 \beta_{1} + 153 \beta_{2} - 465 \beta_{3} + 441 \beta_{4} - 96 \beta_{5} ) q^{93} + ( -633 \beta_{3} + 162 \beta_{4} - 340 \beta_{5} ) q^{94} + ( 588 - 56 \beta_{1} - 147 \beta_{2} + 791 \beta_{3} - 238 \beta_{4} + 56 \beta_{5} ) q^{96} + ( -100 - 78 \beta_{1} - 60 \beta_{2} - 100 \beta_{3} - 60 \beta_{4} + 78 \beta_{5} ) q^{97} + ( -867 - 248 \beta_{1} + 189 \beta_{2} ) q^{98} + ( 208 + 208 \beta_{1} + 172 \beta_{2} + 575 \beta_{3} + 53 \beta_{4} + 160 \beta_{5} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - q^{2} - 9 q^{3} - 11 q^{4} + 84 q^{6} - 43 q^{7} + 54 q^{8} + 57 q^{9} + O(q^{10}) \) \( 6 q - q^{2} - 9 q^{3} - 11 q^{4} + 84 q^{6} - 43 q^{7} + 54 q^{8} + 57 q^{9} - 14 q^{11} - 75 q^{12} + 40 q^{13} + 27 q^{14} + 13 q^{16} + 332 q^{17} - 3 q^{18} - 328 q^{19} - 144 q^{21} - 376 q^{22} + 171 q^{23} - 63 q^{24} + 868 q^{26} - 162 q^{27} + 1034 q^{28} + 335 q^{29} + 352 q^{31} - 77 q^{32} + 708 q^{33} + 52 q^{34} + 1086 q^{36} - 804 q^{37} - 178 q^{38} - 390 q^{39} - 187 q^{41} - 513 q^{42} - 602 q^{43} + 1964 q^{44} - 402 q^{46} + 665 q^{47} + 1074 q^{48} - 430 q^{49} - 180 q^{51} - 456 q^{52} + 1460 q^{53} + 639 q^{54} - 705 q^{56} + 486 q^{57} + 217 q^{58} + 298 q^{59} + 1439 q^{61} + 3228 q^{62} - 2205 q^{63} - 3138 q^{64} - 966 q^{66} - 1849 q^{67} - 710 q^{68} - 873 q^{69} + 140 q^{71} - 261 q^{72} + 736 q^{73} + 320 q^{74} - 204 q^{76} - 948 q^{77} + 432 q^{78} + 382 q^{79} - 1251 q^{81} + 1150 q^{82} - 831 q^{83} - 909 q^{84} - 1580 q^{86} - 258 q^{87} - 1428 q^{88} + 3438 q^{89} - 1420 q^{91} - 1623 q^{92} - 2178 q^{93} + 2077 q^{94} + 1155 q^{96} - 282 q^{97} - 4328 q^{98} - 762 q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - 3 x^{5} + 16 x^{4} - 27 x^{3} + 52 x^{2} - 39 x + 9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{2} - \nu + 4 \)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{4} + 2 \nu^{3} - 11 \nu^{2} + 10 \nu - 12 \)\()/3\)
\(\beta_{3}\)\(=\)\((\)\( 2 \nu^{5} - 5 \nu^{4} + 30 \nu^{3} - 40 \nu^{2} + 88 \nu - 39 \)\()/3\)
\(\beta_{4}\)\(=\)\((\)\( -7 \nu^{5} + 18 \nu^{4} - 103 \nu^{3} + 141 \nu^{2} - 289 \nu + 126 \)\()/3\)
\(\beta_{5}\)\(=\)\((\)\( -8 \nu^{5} + 20 \nu^{4} - 117 \nu^{3} + 157 \nu^{2} - 334 \nu + 147 \)\()/3\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-2 \beta_{5} + 2 \beta_{4} - \beta_{3} + \beta_{2} + \beta_{1} + 1\)\()/3\)
\(\nu^{2}\)\(=\)\((\)\(-2 \beta_{5} + 2 \beta_{4} - \beta_{3} + \beta_{2} + 4 \beta_{1} - 11\)\()/3\)
\(\nu^{3}\)\(=\)\((\)\(13 \beta_{5} - 10 \beta_{4} + 17 \beta_{3} - 5 \beta_{2} - 2 \beta_{1} - 8\)\()/3\)
\(\nu^{4}\)\(=\)\((\)\(28 \beta_{5} - 22 \beta_{4} + 35 \beta_{3} - 20 \beta_{2} - 38 \beta_{1} + 79\)\()/3\)
\(\nu^{5}\)\(=\)\((\)\(-77 \beta_{5} + 47 \beta_{4} - 139 \beta_{3} + \beta_{2} - 29 \beta_{1} + 112\)\()/3\)

Character values

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

\(n\) \(101\) \(127\)
\(\chi(n)\) \(\beta_{3}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
76.1
0.500000 2.88506i
0.500000 + 1.98116i
0.500000 + 0.0378788i
0.500000 + 2.88506i
0.500000 1.98116i
0.500000 0.0378788i
−2.28679 3.96084i −3.36330 + 3.96084i −6.45882 + 11.1870i 0 23.3794 + 4.26387i −10.0573 17.4197i 22.4912 −4.37646 26.6429i 0
76.2 −0.0874923 0.151541i −5.19394 + 0.151541i 3.98469 6.90169i 0 0.477395 + 0.773837i 4.23186 + 7.32979i −2.79440 26.9541 1.57419i 0
76.3 1.87428 + 3.24635i 4.05724 3.24635i −3.02587 + 5.24096i 0 18.1432 + 7.08665i −15.6746 27.1492i 7.30318 5.92239 26.3425i 0
151.1 −2.28679 + 3.96084i −3.36330 3.96084i −6.45882 11.1870i 0 23.3794 4.26387i −10.0573 + 17.4197i 22.4912 −4.37646 + 26.6429i 0
151.2 −0.0874923 + 0.151541i −5.19394 0.151541i 3.98469 + 6.90169i 0 0.477395 0.773837i 4.23186 7.32979i −2.79440 26.9541 + 1.57419i 0
151.3 1.87428 3.24635i 4.05724 + 3.24635i −3.02587 5.24096i 0 18.1432 7.08665i −15.6746 + 27.1492i 7.30318 5.92239 + 26.3425i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 151.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 225.4.e.c 6
5.b even 2 1 45.4.e.b 6
5.c odd 4 2 225.4.k.c 12
9.c even 3 1 inner 225.4.e.c 6
9.c even 3 1 2025.4.a.s 3
9.d odd 6 1 2025.4.a.q 3
15.d odd 2 1 135.4.e.b 6
45.h odd 6 1 135.4.e.b 6
45.h odd 6 1 405.4.a.j 3
45.j even 6 1 45.4.e.b 6
45.j even 6 1 405.4.a.h 3
45.k odd 12 2 225.4.k.c 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.4.e.b 6 5.b even 2 1
45.4.e.b 6 45.j even 6 1
135.4.e.b 6 15.d odd 2 1
135.4.e.b 6 45.h odd 6 1
225.4.e.c 6 1.a even 1 1 trivial
225.4.e.c 6 9.c even 3 1 inner
225.4.k.c 12 5.c odd 4 2
225.4.k.c 12 45.k odd 12 2
405.4.a.h 3 45.j even 6 1
405.4.a.j 3 45.h odd 6 1
2025.4.a.q 3 9.d odd 6 1
2025.4.a.s 3 9.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} + T_{2}^{5} + 18 T_{2}^{4} - 11 T_{2}^{3} + 292 T_{2}^{2} + 51 T_{2} + 9 \) acting on \(S_{4}^{\mathrm{new}}(225, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 9 + 51 T + 292 T^{2} - 11 T^{3} + 18 T^{4} + T^{5} + T^{6} \)
$3$ \( 19683 + 6561 T + 324 T^{2} - 81 T^{3} + 12 T^{4} + 9 T^{5} + T^{6} \)
$5$ \( T^{6} \)
$7$ \( 28483569 - 1040715 T + 267516 T^{2} + 19059 T^{3} + 1654 T^{4} + 43 T^{5} + T^{6} \)
$11$ \( 1896428304 - 122631168 T + 7320184 T^{2} - 126520 T^{3} + 3012 T^{4} + 14 T^{5} + T^{6} \)
$13$ \( 5679732496 - 184792528 T + 9026864 T^{2} - 52648 T^{3} + 4052 T^{4} - 40 T^{5} + T^{6} \)
$17$ \( ( -156324 + 8920 T - 166 T^{2} + T^{3} )^{2} \)
$19$ \( ( 57316 + 7292 T + 164 T^{2} + T^{3} )^{2} \)
$23$ \( 3746541681 - 295823097 T + 33824628 T^{2} + 704025 T^{3} + 34074 T^{4} - 171 T^{5} + T^{6} \)
$29$ \( 11463342489 + 2926248177 T + 782851006 T^{2} - 9370019 T^{3} + 84894 T^{4} - 335 T^{5} + T^{6} \)
$31$ \( 97238137683600 - 137382616080 T + 3665151504 T^{2} - 14817816 T^{3} + 137836 T^{4} - 352 T^{5} + T^{6} \)
$37$ \( ( -3335284 + 24708 T + 402 T^{2} + T^{3} )^{2} \)
$41$ \( 52247959475625 + 322359380175 T + 3340579834 T^{2} + 6116911 T^{3} + 79566 T^{4} + 187 T^{5} + T^{6} \)
$43$ \( 238533321586576 - 153765680944 T + 9396725384 T^{2} + 36882560 T^{3} + 352448 T^{4} + 602 T^{5} + T^{6} \)
$47$ \( 6187571675289 - 237009867723 T + 7424292766 T^{2} - 58386899 T^{3} + 346944 T^{4} - 665 T^{5} + T^{6} \)
$53$ \( ( -3250536 + 106300 T - 730 T^{2} + T^{3} )^{2} \)
$59$ \( 16224618881802816 - 45173097261024 T + 163730383744 T^{2} - 149067880 T^{3} + 443448 T^{4} - 298 T^{5} + T^{6} \)
$61$ \( 3092861048569009 - 32773423076579 T + 267254918066 T^{2} - 736785779 T^{3} + 1481414 T^{4} - 1439 T^{5} + T^{6} \)
$67$ \( 43587484566793281 + 228333683246643 T + 810102262338 T^{2} + 1604656455 T^{3} + 2325124 T^{4} + 1849 T^{5} + T^{6} \)
$71$ \( ( 223775052 - 685460 T - 70 T^{2} + T^{3} )^{2} \)
$73$ \( ( 134927744 - 372928 T - 368 T^{2} + T^{3} )^{2} \)
$79$ \( 19133137244437056 - 53658650075616 T + 203324256864 T^{2} - 128458200 T^{3} + 533848 T^{4} - 382 T^{5} + T^{6} \)
$83$ \( 913637410143880041 + 1109708868397833 T + 2142164521980 T^{2} + 946919079 T^{3} + 1851534 T^{4} + 831 T^{5} + T^{6} \)
$89$ \( ( 125506395 + 238491 T - 1719 T^{2} + T^{3} )^{2} \)
$97$ \( 285551326213696 + 1143471728352 T + 9344268672 T^{2} + 14714152 T^{3} + 147192 T^{4} + 282 T^{5} + T^{6} \)
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