Properties

Label 234.8.a
Level $234$
Weight $8$
Character orbit 234.a
Rep. character $\chi_{234}(1,\cdot)$
Character field $\Q$
Dimension $35$
Newform subspaces $17$
Sturm bound $336$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 234 = 2 \cdot 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 234.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 17 \)
Sturm bound: \(336\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_0(234))\).

Total New Old
Modular forms 302 35 267
Cusp forms 286 35 251
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.

\(2\)\(3\)\(13\)FrickeDim
\(+\)\(+\)\(+\)\(+\)\(3\)
\(+\)\(+\)\(-\)\(-\)\(4\)
\(+\)\(-\)\(+\)\(-\)\(5\)
\(+\)\(-\)\(-\)\(+\)\(6\)
\(-\)\(+\)\(+\)\(-\)\(3\)
\(-\)\(+\)\(-\)\(+\)\(4\)
\(-\)\(-\)\(+\)\(+\)\(6\)
\(-\)\(-\)\(-\)\(-\)\(4\)
Plus space\(+\)\(19\)
Minus space\(-\)\(16\)

Trace form

\( 35 q - 8 q^{2} + 2240 q^{4} - 538 q^{5} + 560 q^{7} - 512 q^{8} + O(q^{10}) \) \( 35 q - 8 q^{2} + 2240 q^{4} - 538 q^{5} + 560 q^{7} - 512 q^{8} - 416 q^{10} + 5424 q^{11} + 2197 q^{13} - 19696 q^{14} + 143360 q^{16} + 70708 q^{17} - 11052 q^{19} - 34432 q^{20} - 125152 q^{22} + 120772 q^{23} + 779707 q^{25} - 52728 q^{26} + 35840 q^{28} + 209830 q^{29} - 685720 q^{31} - 32768 q^{32} + 220944 q^{34} - 942850 q^{35} + 641442 q^{37} + 433824 q^{38} - 26624 q^{40} - 1059050 q^{41} + 1041926 q^{43} + 347136 q^{44} - 463712 q^{46} + 120792 q^{47} + 2760681 q^{49} + 301576 q^{50} + 140608 q^{52} - 638966 q^{53} + 2518936 q^{55} - 1260544 q^{56} - 1636592 q^{58} + 6833196 q^{59} - 3254170 q^{61} - 3896544 q^{62} + 9175040 q^{64} + 685464 q^{65} + 8934344 q^{67} + 4525312 q^{68} + 437728 q^{70} + 6644512 q^{71} - 1738214 q^{73} + 2979552 q^{74} - 707328 q^{76} - 7778712 q^{77} + 5703364 q^{79} - 2203648 q^{80} - 3256944 q^{82} + 10107288 q^{83} + 18345792 q^{85} + 3421760 q^{86} - 8009728 q^{88} + 501358 q^{89} - 2693522 q^{91} + 7729408 q^{92} + 3835664 q^{94} + 3102940 q^{95} - 8541962 q^{97} + 15183672 q^{98} + O(q^{100}) \)

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_0(234))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 3 13
234.8.a.a 234.a 1.a $1$ $73.098$ \(\Q\) None 26.8.a.b \(-8\) \(0\) \(-321\) \(-181\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-8q^{2}+2^{6}q^{4}-321q^{5}-181q^{7}+\cdots\)
234.8.a.b 234.a 1.a $1$ $73.098$ \(\Q\) None 26.8.a.c \(-8\) \(0\) \(245\) \(-587\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-8q^{2}+2^{6}q^{4}+245q^{5}-587q^{7}+\cdots\)
234.8.a.c 234.a 1.a $1$ $73.098$ \(\Q\) None 78.8.a.b \(-8\) \(0\) \(466\) \(328\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-8q^{2}+2^{6}q^{4}+466q^{5}+328q^{7}+\cdots\)
234.8.a.d 234.a 1.a $1$ $73.098$ \(\Q\) None 26.8.a.a \(8\) \(0\) \(-385\) \(-293\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+8q^{2}+2^{6}q^{4}-385q^{5}-293q^{7}+\cdots\)
234.8.a.e 234.a 1.a $1$ $73.098$ \(\Q\) None 78.8.a.a \(8\) \(0\) \(6\) \(-316\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+8q^{2}+2^{6}q^{4}+6q^{5}-316q^{7}+2^{9}q^{8}+\cdots\)
234.8.a.f 234.a 1.a $2$ $73.098$ \(\Q(\sqrt{94}) \) None 78.8.a.h \(-16\) \(0\) \(-440\) \(1304\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-8q^{2}+2^{6}q^{4}+(-220+5\beta )q^{5}+\cdots\)
234.8.a.g 234.a 1.a $2$ $73.098$ \(\Q(\sqrt{2305}) \) None 26.8.a.e \(-16\) \(0\) \(-215\) \(705\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-8q^{2}+2^{6}q^{4}+(-105-5\beta )q^{5}+\cdots\)
234.8.a.h 234.a 1.a $2$ $73.098$ \(\Q(\sqrt{10}) \) None 78.8.a.g \(-16\) \(0\) \(-80\) \(-856\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-8q^{2}+2^{6}q^{4}+(-40+11\beta )q^{5}+\cdots\)
234.8.a.i 234.a 1.a $2$ $73.098$ \(\Q(\sqrt{589}) \) None 78.8.a.f \(-16\) \(0\) \(76\) \(912\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-8q^{2}+2^{6}q^{4}+(38+4\beta )q^{5}+(456+\cdots)q^{7}+\cdots\)
234.8.a.j 234.a 1.a $2$ $73.098$ \(\Q(\sqrt{114}) \) None 78.8.a.c \(16\) \(0\) \(-400\) \(2408\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+8q^{2}+2^{6}q^{4}+(-200+5\beta )q^{5}+\cdots\)
234.8.a.k 234.a 1.a $2$ $73.098$ \(\Q(\sqrt{2454}) \) None 78.8.a.e \(16\) \(0\) \(104\) \(248\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+8q^{2}+2^{6}q^{4}+(52+\beta )q^{5}+(124+\cdots)q^{7}+\cdots\)
234.8.a.l 234.a 1.a $2$ $73.098$ \(\Q(\sqrt{105}) \) None 26.8.a.d \(16\) \(0\) \(146\) \(-1780\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+8q^{2}+2^{6}q^{4}+(73+4\beta )q^{5}+(-890+\cdots)q^{7}+\cdots\)
234.8.a.m 234.a 1.a $2$ $73.098$ \(\Q(\sqrt{235}) \) None 78.8.a.d \(16\) \(0\) \(260\) \(-1104\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+8q^{2}+2^{6}q^{4}+(130+2\beta )q^{5}+(-552+\cdots)q^{7}+\cdots\)
234.8.a.n 234.a 1.a $3$ $73.098$ \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None 234.8.a.n \(-24\) \(0\) \(138\) \(78\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-8q^{2}+2^{6}q^{4}+(46-\beta _{1})q^{5}+(26+\cdots)q^{7}+\cdots\)
234.8.a.o 234.a 1.a $3$ $73.098$ \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None 234.8.a.n \(24\) \(0\) \(-138\) \(78\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+8q^{2}+2^{6}q^{4}+(-46+\beta _{1})q^{5}+(26+\cdots)q^{7}+\cdots\)
234.8.a.p 234.a 1.a $4$ $73.098$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None 234.8.a.p \(-32\) \(0\) \(-112\) \(-192\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-8q^{2}+2^{6}q^{4}+(-28+\beta _{2})q^{5}+(-48+\cdots)q^{7}+\cdots\)
234.8.a.q 234.a 1.a $4$ $73.098$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None 234.8.a.p \(32\) \(0\) \(112\) \(-192\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+8q^{2}+2^{6}q^{4}+(28-\beta _{2})q^{5}+(-48+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_0(234))\) into lower level spaces

\( S_{8}^{\mathrm{old}}(\Gamma_0(234)) \simeq \) \(S_{8}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 6}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 8}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(13))\)\(^{\oplus 6}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(18))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(26))\)\(^{\oplus 3}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(39))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(78))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(117))\)\(^{\oplus 2}\)