Properties

Label 225.4.e.b
Level $225$
Weight $4$
Character orbit 225.e
Analytic conductor $13.275$
Analytic rank $0$
Dimension $4$
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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-11})\)
Defining polynomial: \( x^{4} - x^{3} - 2x^{2} - 3x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 9)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{3} + \beta_1) q^{2} + ( - \beta_{3} - \beta_{2} - \beta_1 + 1) q^{3} + (3 \beta_{3} - 3 \beta_{2} + \beta_1 - 4) q^{4} + ( - 3 \beta_{3} + 3 \beta_{2} - 15 \beta_1 + 12) q^{6} + (3 \beta_{3} + 2 \beta_1) q^{7} + (\beta_{2} - 16) q^{8} + (3 \beta_{3} - 6 \beta_{2} + 21 \beta_1 - 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{3} + \beta_1) q^{2} + ( - \beta_{3} - \beta_{2} - \beta_1 + 1) q^{3} + (3 \beta_{3} - 3 \beta_{2} + \beta_1 - 4) q^{4} + ( - 3 \beta_{3} + 3 \beta_{2} - 15 \beta_1 + 12) q^{6} + (3 \beta_{3} + 2 \beta_1) q^{7} + (\beta_{2} - 16) q^{8} + (3 \beta_{3} - 6 \beta_{2} + 21 \beta_1 - 3) q^{9} + (8 \beta_{3} - 37 \beta_1) q^{11} + (2 \beta_{3} + 5 \beta_{2} - 22 \beta_1 + 52) q^{12} + (15 \beta_{3} - 15 \beta_{2} - 2 \beta_1 - 13) q^{13} + (8 \beta_{3} - 8 \beta_{2} + 26 \beta_1 - 34) q^{14} + (9 \beta_{3} - \beta_1) q^{16} + ( - 9 \beta_{2} - 54) q^{17} + (18 \beta_{3} - 27 \beta_{2} - 72) q^{18} + ( - 27 \beta_{2} - 52) q^{19} + ( - 7 \beta_{3} + 8 \beta_{2} - 46 \beta_1 + 34) q^{21} + ( - 21 \beta_{3} + 21 \beta_{2} + 27 \beta_1 - 6) q^{22} + (19 \beta_{3} - 19 \beta_{2} - 26 \beta_1 + 7) q^{23} + (15 \beta_{3} + 18 \beta_{2} + 9 \beta_1 - 24) q^{24} + ( - 28 \beta_{2} - 146) q^{26} + ( - 36 \beta_{3} + 18 \beta_{2} + 18 \beta_1 + 117) q^{27} + ( - 18 \beta_{2} - 92) q^{28} + ( - \beta_{3} + 26 \beta_1) q^{29} + (3 \beta_{3} - 3 \beta_{2} + 20 \beta_1 - 23) q^{31} + (9 \beta_{3} - 9 \beta_{2} + 207 \beta_1 - 216) q^{32} + (66 \beta_{3} - 21 \beta_{2} - 165 \beta_1 + 6) q^{33} + ( - 63 \beta_{3} - 117 \beta_1) q^{34} + ( - 15 \beta_{3} - 60 \beta_{2} - 141 \beta_1 - 12) q^{36} + ( - 54 \beta_{2} - 2) q^{37} + ( - 79 \beta_{3} - 241 \beta_1) q^{38} + (17 \beta_{3} + 11 \beta_{2} - 124 \beta_1 + 253) q^{39} + (98 \beta_{3} - 98 \beta_{2} + 17 \beta_1 - 115) q^{41} + ( - 18 \beta_{3} + 60 \beta_{2} - 12 \beta_1 + 162) q^{42} + ( - 6 \beta_{3} + 47 \beta_1) q^{43} + (79 \beta_{2} - 76) q^{44} + ( - 12 \beta_{2} - 138) q^{46} + (91 \beta_{3} + 154 \beta_1) q^{47} + ( - 7 \beta_{3} + 17 \beta_{2} - 145 \beta_1 + 88) q^{48} + (21 \beta_{3} - 21 \beta_{2} - 267 \beta_1 + 246) q^{49} + (63 \beta_{3} + 36 \beta_{2} + 117 \beta_1 + 18) q^{51} + ( - 54 \beta_{3} - 358 \beta_1) q^{52} + (162 \beta_{2} + 54) q^{53} + (81 \beta_{3} + 54 \beta_{2} - 27 \beta_1 + 324) q^{54} + ( - 46 \beta_{3} - 10 \beta_1) q^{56} + (79 \beta_{3} - 2 \beta_{2} + 241 \beta_1 + 164) q^{57} + (24 \beta_{3} - 24 \beta_{2} + 18 \beta_1 - 42) q^{58} + ( - 136 \beta_{3} + 136 \beta_{2} + 467 \beta_1 - 331) q^{59} + (105 \beta_{3} - 272 \beta_1) q^{61} + ( - 26 \beta_{2} - 70) q^{62} + (57 \beta_{3} - 78 \beta_{2} - 24 \beta_1 - 192) q^{63} + ( - 153 \beta_{2} - 440) q^{64} + ( - 48 \beta_{3} + 33 \beta_{2} + 222 \beta_1 - 330) q^{66} + ( - 66 \beta_{3} + 66 \beta_{2} - 461 \beta_1 + 527) q^{67} + ( - 171 \beta_{3} + 171 \beta_{2} - 261 \beta_1 + 432) q^{68} + (45 \beta_{3} - 33 \beta_{2} - 204 \beta_1 + 297) q^{69} + (144 \beta_{2} + 756) q^{71} + ( - 27 \beta_{3} + 99 \beta_{2} - 333 \beta_1) q^{72} + ( - 243 \beta_{2} + 106) q^{73} + ( - 56 \beta_{3} - 380 \beta_1) q^{74} + ( - 183 \beta_{3} + 183 \beta_{2} - 673 \beta_1 + 856) q^{76} + ( - 71 \beta_{3} + 71 \beta_{2} + 118 \beta_1 - 47) q^{77} + (174 \beta_{3} + 90 \beta_{2} + 342 \beta_1 + 78) q^{78} + ( - 309 \beta_{3} + 556 \beta_1) q^{79} + ( - 135 \beta_{3} - 135 \beta_{2} + 351 \beta_1 - 351) q^{81} + ( - 213 \beta_{2} - 1014) q^{82} + ( - 107 \beta_{3} + 460 \beta_1) q^{83} + (110 \beta_{3} + 56 \beta_{2} + 218 \beta_1 + 52) q^{84} + (35 \beta_{3} - 35 \beta_{2} - \beta_1 - 34) q^{86} + ( - 51 \beta_{3} + 24 \beta_{2} + 42 \beta_1 + 42) q^{87} + ( - 165 \beta_{3} + 693 \beta_1) q^{88} + (72 \beta_{2} - 162) q^{89} + ( - 69 \beta_{2} - 425) q^{91} + (2 \beta_{3} - 430 \beta_1) q^{92} + ( - 17 \beta_{3} + 43 \beta_{2} + 16 \beta_1 + 71) q^{93} + (336 \beta_{3} - 336 \beta_{2} + 882 \beta_1 - 1218) q^{94} + ( - 198 \beta_{3} + 423 \beta_{2} + 342 \beta_1 + 360) q^{96} + (102 \beta_{3} + 317 \beta_1) q^{97} + (225 \beta_{2} + 324) q^{98} + (279 \beta_{3} - 81 \beta_{2} - 1080 \beta_1 + 504) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 3 q^{2} + 3 q^{3} - 5 q^{4} + 9 q^{6} + 7 q^{7} - 66 q^{8} + 45 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 3 q^{2} + 3 q^{3} - 5 q^{4} + 9 q^{6} + 7 q^{7} - 66 q^{8} + 45 q^{9} - 66 q^{11} + 156 q^{12} - 11 q^{13} - 60 q^{14} + 7 q^{16} - 198 q^{17} - 216 q^{18} - 154 q^{19} + 21 q^{21} - 33 q^{22} + 33 q^{23} - 99 q^{24} - 528 q^{26} + 432 q^{27} - 332 q^{28} + 51 q^{29} - 43 q^{31} - 423 q^{32} - 198 q^{33} - 297 q^{34} - 225 q^{36} + 100 q^{37} - 561 q^{38} + 759 q^{39} - 132 q^{41} + 486 q^{42} + 88 q^{43} - 462 q^{44} - 528 q^{46} + 399 q^{47} + 21 q^{48} + 513 q^{49} + 297 q^{51} - 770 q^{52} - 108 q^{53} + 1215 q^{54} - 66 q^{56} + 1221 q^{57} - 60 q^{58} - 798 q^{59} - 439 q^{61} - 228 q^{62} - 603 q^{63} - 1454 q^{64} - 990 q^{66} + 988 q^{67} + 693 q^{68} + 891 q^{69} + 2736 q^{71} - 891 q^{72} + 910 q^{73} - 816 q^{74} + 1529 q^{76} - 165 q^{77} + 990 q^{78} + 803 q^{79} - 567 q^{81} - 3630 q^{82} + 813 q^{83} + 642 q^{84} - 33 q^{86} + 153 q^{87} + 1221 q^{88} - 792 q^{89} - 1562 q^{91} - 858 q^{92} + 213 q^{93} - 2100 q^{94} + 1080 q^{96} + 736 q^{97} + 846 q^{98} + 297 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( ( \nu^{3} + 2\nu^{2} - 2\nu - 3 ) / 6 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{3} + \nu^{2} + 5\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2\nu^{3} + \nu^{2} + 2\nu - 9 ) / 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} - 2\beta _1 + 2 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{3} + 2\beta_{2} + 8\beta _1 + 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 4\beta_{3} - 2\beta_{2} - 2\beta _1 + 11 ) / 3 \) Copy content Toggle raw display

Character values

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

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

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
−1.18614 1.26217i
1.68614 + 0.396143i
−1.18614 + 1.26217i
1.68614 0.396143i
−0.686141 1.18843i 5.05842 + 1.18843i 3.05842 5.29734i 0 −2.05842 6.82701i −2.55842 4.43132i −19.3723 24.1753 + 12.0232i 0
76.2 2.18614 + 3.78651i −3.55842 3.78651i −5.55842 + 9.62747i 0 6.55842 21.7518i 6.05842 + 10.4935i −13.6277 −1.67527 + 26.9480i 0
151.1 −0.686141 + 1.18843i 5.05842 1.18843i 3.05842 + 5.29734i 0 −2.05842 + 6.82701i −2.55842 + 4.43132i −19.3723 24.1753 12.0232i 0
151.2 2.18614 3.78651i −3.55842 + 3.78651i −5.55842 9.62747i 0 6.55842 + 21.7518i 6.05842 10.4935i −13.6277 −1.67527 26.9480i 0
\(n\): e.g. 2-40 or 990-1000
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.b 4
5.b even 2 1 9.4.c.a 4
5.c odd 4 2 225.4.k.b 8
9.c even 3 1 inner 225.4.e.b 4
9.c even 3 1 2025.4.a.g 2
9.d odd 6 1 2025.4.a.n 2
15.d odd 2 1 27.4.c.a 4
20.d odd 2 1 144.4.i.c 4
45.h odd 6 1 27.4.c.a 4
45.h odd 6 1 81.4.a.a 2
45.j even 6 1 9.4.c.a 4
45.j even 6 1 81.4.a.d 2
45.k odd 12 2 225.4.k.b 8
60.h even 2 1 432.4.i.c 4
180.n even 6 1 432.4.i.c 4
180.n even 6 1 1296.4.a.i 2
180.p odd 6 1 144.4.i.c 4
180.p odd 6 1 1296.4.a.u 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9.4.c.a 4 5.b even 2 1
9.4.c.a 4 45.j even 6 1
27.4.c.a 4 15.d odd 2 1
27.4.c.a 4 45.h odd 6 1
81.4.a.a 2 45.h odd 6 1
81.4.a.d 2 45.j even 6 1
144.4.i.c 4 20.d odd 2 1
144.4.i.c 4 180.p odd 6 1
225.4.e.b 4 1.a even 1 1 trivial
225.4.e.b 4 9.c even 3 1 inner
225.4.k.b 8 5.c odd 4 2
225.4.k.b 8 45.k odd 12 2
432.4.i.c 4 60.h even 2 1
432.4.i.c 4 180.n even 6 1
1296.4.a.i 2 180.n even 6 1
1296.4.a.u 2 180.p odd 6 1
2025.4.a.g 2 9.c even 3 1
2025.4.a.n 2 9.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} - 3T_{2}^{3} + 15T_{2}^{2} + 18T_{2} + 36 \) acting on \(S_{4}^{\mathrm{new}}(225, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 3 T^{3} + 15 T^{2} + 18 T + 36 \) Copy content Toggle raw display
$3$ \( T^{4} - 3 T^{3} - 18 T^{2} - 81 T + 729 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} - 7 T^{3} + 111 T^{2} + \cdots + 3844 \) Copy content Toggle raw display
$11$ \( T^{4} + 66 T^{3} + 3795 T^{2} + \cdots + 314721 \) Copy content Toggle raw display
$13$ \( T^{4} + 11 T^{3} + 1947 T^{2} + \cdots + 3334276 \) Copy content Toggle raw display
$17$ \( (T^{2} + 99 T + 1782)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 77 T - 4532)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} - 33 T^{3} + 3795 T^{2} + \cdots + 7322436 \) Copy content Toggle raw display
$29$ \( T^{4} - 51 T^{3} + 1959 T^{2} + \cdots + 412164 \) Copy content Toggle raw display
$31$ \( T^{4} + 43 T^{3} + 1461 T^{2} + \cdots + 150544 \) Copy content Toggle raw display
$37$ \( (T^{2} - 50 T - 23432)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + 132 T^{3} + \cdots + 5606565129 \) Copy content Toggle raw display
$43$ \( T^{4} - 88 T^{3} + 6105 T^{2} + \cdots + 2686321 \) Copy content Toggle raw display
$47$ \( T^{4} - 399 T^{3} + \cdots + 813276324 \) Copy content Toggle raw display
$53$ \( (T^{2} + 54 T - 215784)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 798 T^{3} + \cdots + 43678881 \) Copy content Toggle raw display
$61$ \( T^{4} + 439 T^{3} + \cdots + 1829786176 \) Copy content Toggle raw display
$67$ \( T^{4} - 988 T^{3} + \cdots + 43305193801 \) Copy content Toggle raw display
$71$ \( (T^{2} - 1368 T + 296784)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 455 T - 435398)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} - 803 T^{3} + \cdots + 392522298256 \) Copy content Toggle raw display
$83$ \( T^{4} - 813 T^{3} + \cdots + 5010940944 \) Copy content Toggle raw display
$89$ \( (T^{2} + 396 T - 3564)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} - 736 T^{3} + \cdots + 2459267281 \) Copy content Toggle raw display
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