Properties

Label 414.8.a
Level $414$
Weight $8$
Character orbit 414.a
Rep. character $\chi_{414}(1,\cdot)$
Character field $\Q$
Dimension $66$
Newform subspaces $16$
Sturm bound $576$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 414 = 2 \cdot 3^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 414.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 16 \)
Sturm bound: \(576\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_0(414))\).

Total New Old
Modular forms 512 66 446
Cusp forms 496 66 430
Eisenstein series 16 0 16

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(2\)\(3\)\(23\)FrickeDim
\(+\)\(+\)\(+\)$+$\(8\)
\(+\)\(+\)\(-\)$-$\(6\)
\(+\)\(-\)\(+\)$-$\(9\)
\(+\)\(-\)\(-\)$+$\(10\)
\(-\)\(+\)\(+\)$-$\(6\)
\(-\)\(+\)\(-\)$+$\(8\)
\(-\)\(-\)\(+\)$+$\(10\)
\(-\)\(-\)\(-\)$-$\(9\)
Plus space\(+\)\(36\)
Minus space\(-\)\(30\)

Trace form

\( 66 q + 4224 q^{4} - 110 q^{5} + 1908 q^{7} + O(q^{10}) \) \( 66 q + 4224 q^{4} - 110 q^{5} + 1908 q^{7} - 3536 q^{10} + 12178 q^{11} + 24388 q^{13} - 3616 q^{14} + 270336 q^{16} + 1016 q^{17} - 10598 q^{19} - 7040 q^{20} - 115312 q^{22} + 1198706 q^{25} - 176640 q^{26} + 122112 q^{28} - 366864 q^{29} - 108640 q^{31} + 39520 q^{34} - 7168 q^{35} - 451986 q^{37} + 1521040 q^{38} - 226304 q^{40} - 1101520 q^{41} + 2578558 q^{43} + 779392 q^{44} + 194672 q^{46} - 685976 q^{47} + 7133682 q^{49} + 1612000 q^{50} + 1560832 q^{52} - 518934 q^{53} - 952016 q^{55} - 231424 q^{56} - 2042080 q^{58} - 466904 q^{59} + 978802 q^{61} - 6971584 q^{62} + 17301504 q^{64} + 862980 q^{65} + 3635586 q^{67} + 65024 q^{68} + 679424 q^{70} + 1585392 q^{71} - 11336128 q^{73} + 1847504 q^{74} - 678272 q^{76} - 8674248 q^{77} - 4767292 q^{79} - 450560 q^{80} + 3408864 q^{82} + 9174278 q^{83} + 2514860 q^{85} + 3084144 q^{86} - 7379968 q^{88} - 33276156 q^{89} - 9153592 q^{91} + 6620928 q^{94} - 1388648 q^{95} - 41678104 q^{97} - 10223488 q^{98} + O(q^{100}) \)

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_0(414))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 3 23
414.8.a.a 414.a 1.a $1$ $129.327$ \(\Q\) None \(-8\) \(0\) \(230\) \(106\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-8q^{2}+2^{6}q^{4}+230q^{5}+106q^{7}+\cdots\)
414.8.a.b 414.a 1.a $2$ $129.327$ \(\Q(\sqrt{85}) \) None \(16\) \(0\) \(110\) \(74\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+8q^{2}+2^{6}q^{4}+(55-3\beta )q^{5}+(37+\cdots)q^{7}+\cdots\)
414.8.a.c 414.a 1.a $3$ $129.327$ \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(-24\) \(0\) \(-92\) \(-858\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-8q^{2}+2^{6}q^{4}+(-31-\beta _{2})q^{5}-286q^{7}+\cdots\)
414.8.a.d 414.a 1.a $3$ $129.327$ 3.3.285765.1 None \(-24\) \(0\) \(570\) \(-1382\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-8q^{2}+2^{6}q^{4}+(186+13\beta _{1}+\beta _{2})q^{5}+\cdots\)
414.8.a.e 414.a 1.a $3$ $129.327$ \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(24\) \(0\) \(-390\) \(-470\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+8q^{2}+2^{6}q^{4}+(-131+5\beta _{1}+8\beta _{2})q^{5}+\cdots\)
414.8.a.f 414.a 1.a $3$ $129.327$ \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(24\) \(0\) \(-160\) \(-364\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+8q^{2}+2^{6}q^{4}+(-55-5\beta _{1})q^{5}+\cdots\)
414.8.a.g 414.a 1.a $3$ $129.327$ 3.3.3351293.1 None \(24\) \(0\) \(162\) \(-296\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+8q^{2}+2^{6}q^{4}+(55+5\beta _{1}-2\beta _{2})q^{5}+\cdots\)
414.8.a.h 414.a 1.a $4$ $129.327$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(-32\) \(0\) \(-342\) \(-30\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-8q^{2}+2^{6}q^{4}+(-85-\beta _{1}+\beta _{2}+\cdots)q^{5}+\cdots\)
414.8.a.i 414.a 1.a $4$ $129.327$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(-32\) \(0\) \(-270\) \(2022\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-8q^{2}+2^{6}q^{4}+(-68+\beta _{1}-\beta _{2}+\cdots)q^{5}+\cdots\)
414.8.a.j 414.a 1.a $4$ $129.327$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(-32\) \(0\) \(-180\) \(534\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-8q^{2}+2^{6}q^{4}+(-45+\beta _{1}-\beta _{2}+\cdots)q^{5}+\cdots\)
414.8.a.k 414.a 1.a $4$ $129.327$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(32\) \(0\) \(90\) \(-222\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+8q^{2}+2^{6}q^{4}+(23-\beta _{1})q^{5}+(-56+\cdots)q^{7}+\cdots\)
414.8.a.l 414.a 1.a $4$ $129.327$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(32\) \(0\) \(162\) \(1218\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+8q^{2}+2^{6}q^{4}+(40+\beta _{1})q^{5}+(304+\cdots)q^{7}+\cdots\)
414.8.a.m 414.a 1.a $6$ $129.327$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(-48\) \(0\) \(250\) \(-292\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-8q^{2}+2^{6}q^{4}+(42-\beta _{1})q^{5}+(-7^{2}+\cdots)q^{7}+\cdots\)
414.8.a.n 414.a 1.a $6$ $129.327$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(48\) \(0\) \(-250\) \(-292\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+8q^{2}+2^{6}q^{4}+(-42+\beta _{1})q^{5}+(-7^{2}+\cdots)q^{7}+\cdots\)
414.8.a.o 414.a 1.a $8$ $129.327$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(-64\) \(0\) \(0\) \(1080\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-8q^{2}+2^{6}q^{4}-\beta _{1}q^{5}+(135+\beta _{2}+\cdots)q^{7}+\cdots\)
414.8.a.p 414.a 1.a $8$ $129.327$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(64\) \(0\) \(0\) \(1080\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+8q^{2}+2^{6}q^{4}+\beta _{1}q^{5}+(135+\beta _{2}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_0(414))\) into lower level spaces

\( S_{8}^{\mathrm{old}}(\Gamma_0(414)) \cong \) \(S_{8}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 6}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 8}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(18))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(23))\)\(^{\oplus 6}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(46))\)\(^{\oplus 3}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(69))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(138))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(207))\)\(^{\oplus 2}\)