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 $2$

Related objects

Downloads

Learn more

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: \( x^{6} - 3x^{5} + 16x^{4} - 27x^{3} + 52x^{2} - 39x + 9 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q + ( - \beta_{5} + \beta_1) q^{2} + (\beta_{5} - \beta_{2} - 1) q^{3} + (2 \beta_{5} - 3 \beta_{4} + 4 \beta_{3}) q^{4} + ( - \beta_{4} + 5 \beta_{3} - 4 \beta_{2} - 2 \beta_1 + 17) q^{6} + ( - \beta_{5} + 6 \beta_{4} - 16 \beta_{3} + 6 \beta_{2} + \beta_1 - 16) q^{7} + ( - 3 \beta_{2} - 3 \beta_1 + 9) q^{8} + ( - 2 \beta_{5} + 5 \beta_{4} - 22 \beta_{3} + 7 \beta_{2} + 4 \beta_1 - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{5} + \beta_1) q^{2} + (\beta_{5} - \beta_{2} - 1) q^{3} + (2 \beta_{5} - 3 \beta_{4} + 4 \beta_{3}) q^{4} + ( - \beta_{4} + 5 \beta_{3} - 4 \beta_{2} - 2 \beta_1 + 17) q^{6} + ( - \beta_{5} + 6 \beta_{4} - 16 \beta_{3} + 6 \beta_{2} + \beta_1 - 16) q^{7} + ( - 3 \beta_{2} - 3 \beta_1 + 9) q^{8} + ( - 2 \beta_{5} + 5 \beta_{4} - 22 \beta_{3} + 7 \beta_{2} + 4 \beta_1 - 2) q^{9} + ( - 14 \beta_{5} + 9 \beta_{4} - 3 \beta_{3} + 9 \beta_{2} + 14 \beta_1 - 3) q^{11} + ( - 17 \beta_{5} + 10 \beta_{4} + 4 \beta_{3} + 3 \beta_{2} + 11 \beta_1 - 9) q^{12} + (14 \beta_{5} - 3 \beta_{4} - 17 \beta_{3}) q^{13} + (24 \beta_{5} + 3 \beta_{4} - 18 \beta_{3}) q^{14} + ( - 2 \beta_{5} - 18 \beta_{4} + 11 \beta_{3} - 18 \beta_{2} + 2 \beta_1 + 11) q^{16} + ( - 3 \beta_{2} + 2 \beta_1 + 57) q^{17} + (17 \beta_{5} - 5 \beta_{4} + 13 \beta_{3} - 4 \beta_{2} + 14 \beta_1 + 14) q^{18} + ( - 9 \beta_{2} - 10 \beta_1 - 55) q^{19} + (12 \beta_{5} - 3 \beta_{4} - 33 \beta_{3} - 36 \beta_1 - 51) q^{21} + (40 \beta_{5} - 33 \beta_{4} + 123 \beta_{3}) q^{22} + (9 \beta_{5} - 36 \beta_{4} - 48 \beta_{3}) q^{23} + (18 \beta_{5} - 6 \beta_{4} + 21 \beta_{3} + 3 \beta_{2} + 6 \beta_1 + 3) q^{24} + ( - 39 \beta_{2} - 14 \beta_1 + 153) q^{26} + (24 \beta_{5} + 9 \beta_{4} - 27 \beta_{3} - 24 \beta_{2} - 45 \beta_1 - 42) q^{27} + ( - 27 \beta_{2} - 19 \beta_1 + 175) q^{28} + (14 \beta_{5} - 30 \beta_{4} + 117 \beta_{3} - 30 \beta_{2} - 14 \beta_1 + 117) q^{29} + (62 \beta_{5} - 33 \beta_{4} - 127 \beta_{3}) q^{31} + ( - 49 \beta_{5} + 42 \beta_{3}) q^{32} + (18 \beta_{5} - 38 \beta_{4} - 8 \beta_{3} - 71 \beta_{2} - 58 \beta_1 + 115) q^{33} + ( - 56 \beta_{5} - 9 \beta_{4} + 39 \beta_{3} - 9 \beta_{2} + 56 \beta_1 + 39) q^{34} + (26 \beta_{5} - 62 \beta_{4} + 28 \beta_{3} - 52 \beta_{2} - 46 \beta_1 + 191) q^{36} + (51 \beta_{2} + 24 \beta_1 - 143) q^{37} + (26 \beta_{5} + 21 \beta_{4} - 75 \beta_{3} + 21 \beta_{2} - 26 \beta_1 - 75) q^{38} + ( - 20 \beta_{5} + 79 \beta_{4} - 179 \beta_{3} + 39 \beta_{2} + \cdots - 153) q^{39}+ \cdots + (160 \beta_{5} + 53 \beta_{4} + 575 \beta_{3} + 172 \beta_{2} + 208 \beta_1 + 208) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - q^{2} - 9 q^{3} - 11 q^{4} + 84 q^{6} - 43 q^{7} + 54 q^{8} + 57 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - q^{2} - 9 q^{3} - 11 q^{4} + 84 q^{6} - 43 q^{7} + 54 q^{8} + 57 q^{9} - 14 q^{11} - 75 q^{12} + 40 q^{13} + 27 q^{14} + 13 q^{16} + 332 q^{17} - 3 q^{18} - 328 q^{19} - 144 q^{21} - 376 q^{22} + 171 q^{23} - 63 q^{24} + 868 q^{26} - 162 q^{27} + 1034 q^{28} + 335 q^{29} + 352 q^{31} - 77 q^{32} + 708 q^{33} + 52 q^{34} + 1086 q^{36} - 804 q^{37} - 178 q^{38} - 390 q^{39} - 187 q^{41} - 513 q^{42} - 602 q^{43} + 1964 q^{44} - 402 q^{46} + 665 q^{47} + 1074 q^{48} - 430 q^{49} - 180 q^{51} - 456 q^{52} + 1460 q^{53} + 639 q^{54} - 705 q^{56} + 486 q^{57} + 217 q^{58} + 298 q^{59} + 1439 q^{61} + 3228 q^{62} - 2205 q^{63} - 3138 q^{64} - 966 q^{66} - 1849 q^{67} - 710 q^{68} - 873 q^{69} + 140 q^{71} - 261 q^{72} + 736 q^{73} + 320 q^{74} - 204 q^{76} - 948 q^{77} + 432 q^{78} + 382 q^{79} - 1251 q^{81} + 1150 q^{82} - 831 q^{83} - 909 q^{84} - 1580 q^{86} - 258 q^{87} - 1428 q^{88} + 3438 q^{89} - 1420 q^{91} - 1623 q^{92} - 2178 q^{93} + 2077 q^{94} + 1155 q^{96} - 282 q^{97} - 4328 q^{98} - 762 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu^{2} - \nu + 4 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{4} + 2\nu^{3} - 11\nu^{2} + 10\nu - 12 ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2\nu^{5} - 5\nu^{4} + 30\nu^{3} - 40\nu^{2} + 88\nu - 39 ) / 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -7\nu^{5} + 18\nu^{4} - 103\nu^{3} + 141\nu^{2} - 289\nu + 126 ) / 3 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -8\nu^{5} + 20\nu^{4} - 117\nu^{3} + 157\nu^{2} - 334\nu + 147 ) / 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -2\beta_{5} + 2\beta_{4} - \beta_{3} + \beta_{2} + \beta _1 + 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{5} + 2\beta_{4} - \beta_{3} + \beta_{2} + 4\beta _1 - 11 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 13\beta_{5} - 10\beta_{4} + 17\beta_{3} - 5\beta_{2} - 2\beta _1 - 8 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 28\beta_{5} - 22\beta_{4} + 35\beta_{3} - 20\beta_{2} - 38\beta _1 + 79 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -77\beta_{5} + 47\beta_{4} - 139\beta_{3} + \beta_{2} - 29\beta _1 + 112 ) / 3 \) Copy content Toggle raw display

Character values

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

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

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 \( T_{2}^{6} + T_{2}^{5} + 18T_{2}^{4} - 11T_{2}^{3} + 292T_{2}^{2} + 51T_{2} + 9 \) acting on \(S_{4}^{\mathrm{new}}(225, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} + T^{5} + 18 T^{4} - 11 T^{3} + \cdots + 9 \) Copy content Toggle raw display
$3$ \( T^{6} + 9 T^{5} + 12 T^{4} + \cdots + 19683 \) Copy content Toggle raw display
$5$ \( T^{6} \) Copy content Toggle raw display
$7$ \( T^{6} + 43 T^{5} + 1654 T^{4} + \cdots + 28483569 \) Copy content Toggle raw display
$11$ \( T^{6} + 14 T^{5} + \cdots + 1896428304 \) Copy content Toggle raw display
$13$ \( T^{6} - 40 T^{5} + \cdots + 5679732496 \) Copy content Toggle raw display
$17$ \( (T^{3} - 166 T^{2} + 8920 T - 156324)^{2} \) Copy content Toggle raw display
$19$ \( (T^{3} + 164 T^{2} + 7292 T + 57316)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} - 171 T^{5} + \cdots + 3746541681 \) Copy content Toggle raw display
$29$ \( T^{6} - 335 T^{5} + \cdots + 11463342489 \) Copy content Toggle raw display
$31$ \( T^{6} - 352 T^{5} + \cdots + 97238137683600 \) Copy content Toggle raw display
$37$ \( (T^{3} + 402 T^{2} + 24708 T - 3335284)^{2} \) Copy content Toggle raw display
$41$ \( T^{6} + 187 T^{5} + \cdots + 52247959475625 \) Copy content Toggle raw display
$43$ \( T^{6} + \cdots + 238533321586576 \) Copy content Toggle raw display
$47$ \( T^{6} - 665 T^{5} + \cdots + 6187571675289 \) Copy content Toggle raw display
$53$ \( (T^{3} - 730 T^{2} + 106300 T - 3250536)^{2} \) Copy content Toggle raw display
$59$ \( T^{6} - 298 T^{5} + \cdots + 16\!\cdots\!16 \) Copy content Toggle raw display
$61$ \( T^{6} - 1439 T^{5} + \cdots + 30\!\cdots\!09 \) Copy content Toggle raw display
$67$ \( T^{6} + 1849 T^{5} + \cdots + 43\!\cdots\!81 \) Copy content Toggle raw display
$71$ \( (T^{3} - 70 T^{2} - 685460 T + 223775052)^{2} \) Copy content Toggle raw display
$73$ \( (T^{3} - 368 T^{2} - 372928 T + 134927744)^{2} \) Copy content Toggle raw display
$79$ \( T^{6} - 382 T^{5} + \cdots + 19\!\cdots\!56 \) Copy content Toggle raw display
$83$ \( T^{6} + 831 T^{5} + \cdots + 91\!\cdots\!41 \) Copy content Toggle raw display
$89$ \( (T^{3} - 1719 T^{2} + 238491 T + 125506395)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} + \cdots + 285551326213696 \) Copy content Toggle raw display
show more
show less