Properties

Label 225.4.e.d
Level $225$
Weight $4$
Character orbit 225.e
Analytic conductor $13.275$
Analytic rank $0$
Dimension $14$
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: \(14\)
Relative dimension: \(7\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
Defining polynomial: \( x^{14} - 2 x^{13} + 48 x^{12} - 60 x^{11} + 1605 x^{10} - 1800 x^{9} + 23232 x^{8} - 2346 x^{7} + 209529 x^{6} - 55412 x^{5} + 765088 x^{4} + 276096 x^{3} + 1572480 x^{2} + \cdots + 82944 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{6} \)
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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{2} + \beta_1) q^{2} + (\beta_{10} - \beta_{7}) q^{3} + ( - \beta_{11} + \beta_{10} - \beta_{5} + 5 \beta_{4} - \beta_1 - 5) q^{4} + ( - \beta_{11} - \beta_{9} + \beta_{6} - \beta_{5} + 4 \beta_{4} - \beta_{2} + 2 \beta_1 - 4) q^{6} + ( - \beta_{13} - \beta_{11} + \beta_{10} - \beta_{9} + \beta_{6} + 3 \beta_{4} + 1) q^{7} + (\beta_{12} + \beta_{11} + \beta_{10} + 2 \beta_{9} + \beta_{8} - \beta_{5} - \beta_{4} + \beta_{3} + 5 \beta_{2} + \cdots + 1) q^{8}+ \cdots + (\beta_{13} - \beta_{11} + \beta_{8} - \beta_{7} + \beta_{6} + 2 \beta_{4} + \beta_{3} + 2 \beta_{2} - 2 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} + \beta_1) q^{2} + (\beta_{10} - \beta_{7}) q^{3} + ( - \beta_{11} + \beta_{10} - \beta_{5} + 5 \beta_{4} - \beta_1 - 5) q^{4} + ( - \beta_{11} - \beta_{9} + \beta_{6} - \beta_{5} + 4 \beta_{4} - \beta_{2} + 2 \beta_1 - 4) q^{6} + ( - \beta_{13} - \beta_{11} + \beta_{10} - \beta_{9} + \beta_{6} + 3 \beta_{4} + 1) q^{7} + (\beta_{12} + \beta_{11} + \beta_{10} + 2 \beta_{9} + \beta_{8} - \beta_{5} - \beta_{4} + \beta_{3} + 5 \beta_{2} + \cdots + 1) q^{8}+ \cdots + (5 \beta_{13} + 48 \beta_{12} + 33 \beta_{11} + 73 \beta_{10} + 17 \beta_{9} + \cdots + 301) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 2 q^{2} + 5 q^{3} - 36 q^{4} - 31 q^{6} + 22 q^{7} + 36 q^{8} + 17 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q - 2 q^{2} + 5 q^{3} - 36 q^{4} - 31 q^{6} + 22 q^{7} + 36 q^{8} + 17 q^{9} + 23 q^{11} - 287 q^{12} + 96 q^{13} - 21 q^{14} - 324 q^{16} + 322 q^{17} + 89 q^{18} + 558 q^{19} + 180 q^{21} + 311 q^{22} - 96 q^{23} + 48 q^{24} + 716 q^{26} + 470 q^{27} - 674 q^{28} - 296 q^{29} - 244 q^{31} + 314 q^{32} + 211 q^{33} - 125 q^{34} - 2399 q^{36} - 808 q^{37} - 305 q^{38} + 634 q^{39} - 47 q^{41} - 1941 q^{42} + 525 q^{43} - 110 q^{44} + 1434 q^{46} - 164 q^{47} - 2051 q^{48} - 1225 q^{49} + 1517 q^{51} + 1682 q^{52} + 1012 q^{53} - 4066 q^{54} - 981 q^{56} - 337 q^{57} + 1183 q^{58} - 85 q^{59} - 828 q^{61} - 1572 q^{62} + 828 q^{63} + 4472 q^{64} + 4930 q^{66} + 1093 q^{67} - 2473 q^{68} - 822 q^{69} - 656 q^{71} + 4626 q^{72} - 4170 q^{73} - 1316 q^{74} - 2789 q^{76} - 24 q^{77} + 5314 q^{78} - 2110 q^{79} - 2167 q^{81} + 124 q^{82} - 1290 q^{83} + 5775 q^{84} - 2569 q^{86} - 3604 q^{87} + 2271 q^{88} + 6096 q^{89} + 6676 q^{91} - 2763 q^{92} + 696 q^{93} + 517 q^{94} - 593 q^{96} + 1787 q^{97} + 2558 q^{98} + 2320 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 2 x^{13} + 48 x^{12} - 60 x^{11} + 1605 x^{10} - 1800 x^{9} + 23232 x^{8} - 2346 x^{7} + 209529 x^{6} - 55412 x^{5} + 765088 x^{4} + 276096 x^{3} + 1572480 x^{2} + \cdots + 82944 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 20061126179 \nu^{13} + 306116042626 \nu^{12} - 1952416155384 \nu^{11} + 13978725434868 \nu^{10} - 67214635202751 \nu^{9} + \cdots + 81\!\cdots\!72 ) / 34\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 102404815617 \nu^{13} + 5266027591948 \nu^{12} - 8326481101332 \nu^{11} + 268246879929564 \nu^{10} + \cdots + 63\!\cdots\!56 ) / 68\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 2939472898789 \nu^{13} + 5818762419041 \nu^{12} - 140176351013994 \nu^{11} + 170511125461188 \nu^{10} + \cdots + 93\!\cdots\!52 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 376371909409 \nu^{13} + 779547299216 \nu^{12} - 17633627499780 \nu^{11} + 23331293206188 \nu^{10} + \cdots - 63\!\cdots\!28 ) / 81\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 2181076654577 \nu^{13} + 4927484464048 \nu^{12} - 102520174110372 \nu^{11} + 135970911492204 \nu^{10} + \cdots - 22\!\cdots\!24 ) / 40\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 11125627233907 \nu^{13} + 27262271328908 \nu^{12} - 529205531611572 \nu^{11} + 877253848876644 \nu^{10} + \cdots + 11\!\cdots\!76 ) / 20\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 11877650203069 \nu^{13} - 84950703264836 \nu^{12} + 708621055239924 \nu^{11} + \cdots - 58\!\cdots\!92 ) / 20\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 10816153989491 \nu^{13} - 13376805115054 \nu^{12} + 497974097019636 \nu^{11} - 253186185774972 \nu^{10} + \cdots - 56\!\cdots\!88 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 4781648360579 \nu^{13} - 8884383807886 \nu^{12} + 231881123650104 \nu^{11} - 261650594664228 \nu^{10} + \cdots + 41\!\cdots\!28 ) / 40\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 6918798848131 \nu^{13} + 13573388130414 \nu^{12} - 327948078905676 \nu^{11} + 388609132067052 \nu^{10} + \cdots + 26\!\cdots\!08 ) / 34\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 2499552103127 \nu^{13} + 3909850857688 \nu^{12} - 117122204938752 \nu^{11} + 101188546669524 \nu^{10} + \cdots + 88\!\cdots\!96 ) / 10\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 47810889109763 \nu^{13} + 95353418254572 \nu^{12} + \cdots + 18\!\cdots\!84 ) / 68\!\cdots\!00 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{11} - \beta_{10} + \beta_{7} - 13\beta_{4} - \beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{12} - \beta_{11} - \beta_{10} - 2\beta_{9} - \beta_{8} + \beta_{5} + \beta_{4} - \beta_{3} - 21\beta_{2} - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 3 \beta_{13} - 3 \beta_{12} - 30 \beta_{11} + 24 \beta_{10} + 3 \beta_{8} + 9 \beta_{7} - 27 \beta_{5} + 294 \beta_{4} - 25 \beta _1 - 291 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 30 \beta_{13} + 66 \beta_{12} + 80 \beta_{11} + 4 \beta_{10} + 78 \beta_{9} + 26 \beta_{7} - 78 \beta_{6} + 36 \beta_{5} - 62 \beta_{4} - 36 \beta_{3} + 529 \beta_{2} - 529 \beta _1 + 24 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 130 \beta_{12} + 130 \beta_{11} + 382 \beta_{10} + 8 \beta_{9} - 122 \beta_{8} - 1029 \beta_{7} + 851 \beta_{5} - 106 \beta_{4} - 146 \beta_{3} + 585 \beta_{2} + 7591 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 819 \beta_{13} - 771 \beta_{12} - 1515 \beta_{11} + 687 \beta_{10} + 819 \beta_{8} - 939 \beta_{7} + 2490 \beta_{6} - 2658 \beta_{5} + 3033 \beta_{4} + 1914 \beta_{3} + 14197 \beta _1 - 3600 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 4053 \beta_{13} + 840 \beta_{12} + 19675 \beta_{11} - 30877 \beta_{10} - 480 \beta_{9} + 18262 \beta_{7} + 480 \beta_{6} - 4365 \beta_{5} - 202939 \beta_{4} + 4365 \beta_{3} - 13585 \beta_{2} + \cdots - 3045 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 34738 \beta_{12} - 34738 \beta_{11} - 29248 \beta_{10} - 74966 \beta_{9} - 22408 \beta_{8} + 9672 \beta_{7} + 48166 \beta_{5} + 14278 \beta_{4} - 19768 \beta_{3} - 391569 \beta_{2} + \cdots + 132320 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 125772 \beta_{13} - 166692 \beta_{12} - 681957 \beta_{11} + 522429 \beta_{10} + 125772 \beta_{8} + 385404 \beta_{7} - 18264 \beta_{6} - 547401 \beta_{5} + 5722041 \beta_{4} + 32136 \beta_{3} + \cdots - 5637885 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 622773 \beta_{13} + 1551186 \beta_{12} + 2507858 \beta_{11} + 304264 \beta_{10} + 2204082 \beta_{9} + 579293 \beta_{7} - 2204082 \beta_{6} + 1029213 \beta_{5} - 5182736 \beta_{4} + \cdots + 376317 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 3988855 \beta_{12} + 3988855 \beta_{11} + 11399653 \beta_{10} + 566912 \beta_{9} - 3783527 \beta_{8} - 26396937 \beta_{7} + 19434914 \beta_{5} - 2321479 \beta_{4} + \cdots + 161007274 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 17562210 \beta_{13} - 14227458 \beta_{12} - 44588544 \beta_{11} + 17387382 \beta_{10} + 17562210 \beta_{8} - 25965810 \beta_{7} + 64059678 \beta_{6} - 74597040 \beta_{5} + \cdots - 170775582 \) Copy content Toggle raw display

Character values

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

\(n\) \(101\) \(127\)
\(\chi(n)\) \(-1 + \beta_{4}\) \(1\)

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
2.65775 4.60336i
2.13089 3.69081i
1.09722 1.90044i
0.112625 0.195072i
−0.785104 + 1.35984i
−1.52087 + 2.63422i
−2.69252 + 4.66357i
2.65775 + 4.60336i
2.13089 + 3.69081i
1.09722 + 1.90044i
0.112625 + 0.195072i
−0.785104 1.35984i
−1.52087 2.63422i
−2.69252 4.66357i
−2.65775 4.60336i 5.19389 + 0.153351i −10.1273 + 17.5410i 0 −13.0981 24.3169i −6.71686 11.6339i 65.1396 26.9530 + 1.59298i 0
76.2 −2.13089 3.69081i −4.39640 2.76977i −5.08138 + 8.80120i 0 −0.854448 + 22.1284i 15.3820 + 26.6423i 9.21718 11.6567 + 24.3541i 0
76.3 −1.09722 1.90044i −0.206141 + 5.19206i 1.59221 2.75778i 0 10.0934 5.30509i 1.38302 + 2.39547i −24.5436 −26.9150 2.14059i 0
76.4 −0.112625 0.195072i −2.06755 4.76710i 3.97463 6.88426i 0 −0.697071 + 0.940215i −15.5970 27.0148i −3.59257 −18.4505 + 19.7124i 0
76.5 0.785104 + 1.35984i 3.89934 3.43441i 2.76722 4.79297i 0 7.73164 + 2.60611i 17.1199 + 29.6525i 21.2519 3.40971 26.7838i 0
76.6 1.52087 + 2.63422i −4.01867 + 3.29398i −0.626094 + 1.08443i 0 −14.7890 5.57636i −6.85611 11.8751i 20.5251 5.29939 26.4748i 0
76.7 2.69252 + 4.66357i 4.09553 + 3.19791i −10.4993 + 18.1853i 0 −3.88642 + 27.7102i 6.28510 + 10.8861i −69.9976 6.54673 + 26.1943i 0
151.1 −2.65775 + 4.60336i 5.19389 0.153351i −10.1273 17.5410i 0 −13.0981 + 24.3169i −6.71686 + 11.6339i 65.1396 26.9530 1.59298i 0
151.2 −2.13089 + 3.69081i −4.39640 + 2.76977i −5.08138 8.80120i 0 −0.854448 22.1284i 15.3820 26.6423i 9.21718 11.6567 24.3541i 0
151.3 −1.09722 + 1.90044i −0.206141 5.19206i 1.59221 + 2.75778i 0 10.0934 + 5.30509i 1.38302 2.39547i −24.5436 −26.9150 + 2.14059i 0
151.4 −0.112625 + 0.195072i −2.06755 + 4.76710i 3.97463 + 6.88426i 0 −0.697071 0.940215i −15.5970 + 27.0148i −3.59257 −18.4505 19.7124i 0
151.5 0.785104 1.35984i 3.89934 + 3.43441i 2.76722 + 4.79297i 0 7.73164 2.60611i 17.1199 29.6525i 21.2519 3.40971 + 26.7838i 0
151.6 1.52087 2.63422i −4.01867 3.29398i −0.626094 1.08443i 0 −14.7890 + 5.57636i −6.85611 + 11.8751i 20.5251 5.29939 + 26.4748i 0
151.7 2.69252 4.66357i 4.09553 3.19791i −10.4993 18.1853i 0 −3.88642 27.7102i 6.28510 10.8861i −69.9976 6.54673 26.1943i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 151.7
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.d 14
5.b even 2 1 45.4.e.c 14
5.c odd 4 2 225.4.k.d 28
9.c even 3 1 inner 225.4.e.d 14
9.c even 3 1 2025.4.a.bb 7
9.d odd 6 1 2025.4.a.ba 7
15.d odd 2 1 135.4.e.c 14
45.h odd 6 1 135.4.e.c 14
45.h odd 6 1 405.4.a.n 7
45.j even 6 1 45.4.e.c 14
45.j even 6 1 405.4.a.m 7
45.k odd 12 2 225.4.k.d 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.4.e.c 14 5.b even 2 1
45.4.e.c 14 45.j even 6 1
135.4.e.c 14 15.d odd 2 1
135.4.e.c 14 45.h odd 6 1
225.4.e.d 14 1.a even 1 1 trivial
225.4.e.d 14 9.c even 3 1 inner
225.4.k.d 28 5.c odd 4 2
225.4.k.d 28 45.k odd 12 2
405.4.a.m 7 45.j even 6 1
405.4.a.n 7 45.h odd 6 1
2025.4.a.ba 7 9.d odd 6 1
2025.4.a.bb 7 9.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{14} + 2 T_{2}^{13} + 48 T_{2}^{12} + 60 T_{2}^{11} + 1605 T_{2}^{10} + 1800 T_{2}^{9} + 23232 T_{2}^{8} + 2346 T_{2}^{7} + 209529 T_{2}^{6} + 55412 T_{2}^{5} + 765088 T_{2}^{4} - 276096 T_{2}^{3} + 1572480 T_{2}^{2} + \cdots + 82944 \) acting on \(S_{4}^{\mathrm{new}}(225, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} + 2 T^{13} + 48 T^{12} + \cdots + 82944 \) Copy content Toggle raw display
$3$ \( T^{14} - 5 T^{13} + \cdots + 10460353203 \) Copy content Toggle raw display
$5$ \( T^{14} \) Copy content Toggle raw display
$7$ \( T^{14} - 22 T^{13} + \cdots + 44\!\cdots\!36 \) Copy content Toggle raw display
$11$ \( T^{14} - 23 T^{13} + \cdots + 40\!\cdots\!24 \) Copy content Toggle raw display
$13$ \( T^{14} - 96 T^{13} + \cdots + 25\!\cdots\!76 \) Copy content Toggle raw display
$17$ \( (T^{7} - 161 T^{6} + \cdots + 60588009792)^{2} \) Copy content Toggle raw display
$19$ \( (T^{7} - 279 T^{6} + \cdots - 1377989598400)^{2} \) Copy content Toggle raw display
$23$ \( T^{14} + 96 T^{13} + \cdots + 16\!\cdots\!44 \) Copy content Toggle raw display
$29$ \( T^{14} + 296 T^{13} + \cdots + 14\!\cdots\!76 \) Copy content Toggle raw display
$31$ \( T^{14} + 244 T^{13} + \cdots + 86\!\cdots\!16 \) Copy content Toggle raw display
$37$ \( (T^{7} + 404 T^{6} + \cdots + 83646911884544)^{2} \) Copy content Toggle raw display
$41$ \( T^{14} + 47 T^{13} + \cdots + 23\!\cdots\!21 \) Copy content Toggle raw display
$43$ \( T^{14} - 525 T^{13} + \cdots + 17\!\cdots\!56 \) Copy content Toggle raw display
$47$ \( T^{14} + 164 T^{13} + \cdots + 42\!\cdots\!84 \) Copy content Toggle raw display
$53$ \( (T^{7} - 506 T^{6} + \cdots + 11\!\cdots\!92)^{2} \) Copy content Toggle raw display
$59$ \( T^{14} + 85 T^{13} + \cdots + 35\!\cdots\!64 \) Copy content Toggle raw display
$61$ \( T^{14} + 828 T^{13} + \cdots + 14\!\cdots\!16 \) Copy content Toggle raw display
$67$ \( T^{14} - 1093 T^{13} + \cdots + 85\!\cdots\!09 \) Copy content Toggle raw display
$71$ \( (T^{7} + 328 T^{6} + \cdots - 32\!\cdots\!28)^{2} \) Copy content Toggle raw display
$73$ \( (T^{7} + 2085 T^{6} + \cdots - 82\!\cdots\!12)^{2} \) Copy content Toggle raw display
$79$ \( T^{14} + 2110 T^{13} + \cdots + 26\!\cdots\!64 \) Copy content Toggle raw display
$83$ \( T^{14} + 1290 T^{13} + \cdots + 43\!\cdots\!84 \) Copy content Toggle raw display
$89$ \( (T^{7} - 3048 T^{6} + \cdots - 32\!\cdots\!50)^{2} \) Copy content Toggle raw display
$97$ \( T^{14} - 1787 T^{13} + \cdots + 22\!\cdots\!76 \) Copy content Toggle raw display
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