Properties

Label 338.10.a
Level $338$
Weight $10$
Character orbit 338.a
Rep. character $\chi_{338}(1,\cdot)$
Character field $\Q$
Dimension $116$
Newform subspaces $18$
Sturm bound $455$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 338 = 2 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 338.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 18 \)
Sturm bound: \(455\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(3\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{10}(\Gamma_0(338))\).

Total New Old
Modular forms 423 116 307
Cusp forms 395 116 279
Eisenstein series 28 0 28

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

\(2\)\(13\)FrickeDim
\(+\)\(+\)\(+\)\(28\)
\(+\)\(-\)\(-\)\(30\)
\(-\)\(+\)\(-\)\(31\)
\(-\)\(-\)\(+\)\(27\)
Plus space\(+\)\(55\)
Minus space\(-\)\(61\)

Trace form

\( 116 q + 6 q^{3} + 29696 q^{4} + 2428 q^{5} - 2496 q^{6} - 1180 q^{7} + 742654 q^{9} + O(q^{10}) \) \( 116 q + 6 q^{3} + 29696 q^{4} + 2428 q^{5} - 2496 q^{6} - 1180 q^{7} + 742654 q^{9} + 37344 q^{10} + 50144 q^{11} + 1536 q^{12} - 112192 q^{14} + 26864 q^{15} + 7602176 q^{16} - 1133024 q^{17} + 292736 q^{18} - 1003748 q^{19} + 621568 q^{20} + 1318796 q^{21} - 2584992 q^{22} - 4348236 q^{23} - 638976 q^{24} + 43709058 q^{25} + 3565020 q^{27} - 302080 q^{28} + 13295158 q^{29} + 1154368 q^{30} - 12553764 q^{31} + 9944252 q^{33} + 19081216 q^{34} - 2528052 q^{35} + 190119424 q^{36} + 14144908 q^{37} - 5710944 q^{38} + 9560064 q^{40} + 62310264 q^{41} - 15161408 q^{42} + 12363346 q^{43} + 12836864 q^{44} - 16074512 q^{45} + 36739136 q^{46} - 11195932 q^{47} + 393216 q^{48} + 632511124 q^{49} - 19040128 q^{50} + 206075264 q^{51} - 136724670 q^{53} - 177181632 q^{54} + 125124600 q^{55} - 28721152 q^{56} - 183224560 q^{57} + 99567680 q^{58} + 54241756 q^{59} + 6877184 q^{60} + 25019766 q^{61} + 67342592 q^{62} - 842734212 q^{63} + 1946157056 q^{64} + 262039424 q^{66} - 312664320 q^{67} - 290054144 q^{68} + 357645940 q^{69} + 332467776 q^{70} + 296235812 q^{71} + 74940416 q^{72} + 557073280 q^{73} - 123161376 q^{74} + 118852002 q^{75} - 256959488 q^{76} + 1033459644 q^{77} - 1178812176 q^{79} + 159121408 q^{80} + 5741961652 q^{81} - 313225280 q^{82} + 1193508480 q^{83} + 337611776 q^{84} + 325342444 q^{85} + 1737090112 q^{86} - 716486260 q^{87} - 661757952 q^{88} - 1435566888 q^{89} + 2336204256 q^{90} - 1113148416 q^{92} + 1752294000 q^{93} + 1059316992 q^{94} + 1569077040 q^{95} - 163577856 q^{96} + 313419948 q^{97} - 926367744 q^{98} + 6184680436 q^{99} + O(q^{100}) \)

Decomposition of \(S_{10}^{\mathrm{new}}(\Gamma_0(338))\) 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 13
338.10.a.a 338.a 1.a $1$ $174.082$ \(\Q\) None 2.10.a.a \(-16\) \(-156\) \(-870\) \(952\) $+$ $+$ $\mathrm{SU}(2)$ \(q-2^{4}q^{2}-156q^{3}+2^{8}q^{4}-870q^{5}+\cdots\)
338.10.a.b 338.a 1.a $1$ $174.082$ \(\Q\) None 26.10.a.c \(-16\) \(75\) \(1979\) \(10115\) $+$ $+$ $\mathrm{SU}(2)$ \(q-2^{4}q^{2}+75q^{3}+2^{8}q^{4}+1979q^{5}+\cdots\)
338.10.a.c 338.a 1.a $1$ $174.082$ \(\Q\) None 26.10.a.a \(16\) \(-273\) \(-1015\) \(-3955\) $-$ $+$ $\mathrm{SU}(2)$ \(q+2^{4}q^{2}-273q^{3}+2^{8}q^{4}-1015q^{5}+\cdots\)
338.10.a.d 338.a 1.a $1$ $174.082$ \(\Q\) None 26.10.a.b \(16\) \(192\) \(1310\) \(5810\) $-$ $+$ $\mathrm{SU}(2)$ \(q+2^{4}q^{2}+192q^{3}+2^{8}q^{4}+1310q^{5}+\cdots\)
338.10.a.e 338.a 1.a $3$ $174.082$ 3.3.2119705.1 None 26.10.a.e \(-48\) \(156\) \(1272\) \(-17058\) $+$ $+$ $\mathrm{SU}(2)$ \(q-2^{4}q^{2}+(52-\beta _{1}-\beta _{2})q^{3}+2^{8}q^{4}+\cdots\)
338.10.a.f 338.a 1.a $3$ $174.082$ \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None 26.10.a.d \(48\) \(0\) \(-248\) \(2956\) $-$ $+$ $\mathrm{SU}(2)$ \(q+2^{4}q^{2}-\beta _{2}q^{3}+2^{8}q^{4}+(-79+11\beta _{1}+\cdots)q^{5}+\cdots\)
338.10.a.g 338.a 1.a $5$ $174.082$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None 26.10.b.a \(-80\) \(81\) \(-1213\) \(8715\) $+$ $-$ $\mathrm{SU}(2)$ \(q-2^{4}q^{2}+(2^{4}+\beta _{1})q^{3}+2^{8}q^{4}+(-3^{5}+\cdots)q^{5}+\cdots\)
338.10.a.h 338.a 1.a $5$ $174.082$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None 26.10.c.a \(-80\) \(81\) \(-914\) \(-3323\) $+$ $+$ $\mathrm{SU}(2)$ \(q-2^{4}q^{2}+(2^{4}+\beta _{1})q^{3}+2^{8}q^{4}+(-183+\cdots)q^{5}+\cdots\)
338.10.a.i 338.a 1.a $5$ $174.082$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None 26.10.c.a \(80\) \(81\) \(914\) \(3323\) $-$ $+$ $\mathrm{SU}(2)$ \(q+2^{4}q^{2}+(2^{4}+\beta _{1})q^{3}+2^{8}q^{4}+(183+\cdots)q^{5}+\cdots\)
338.10.a.j 338.a 1.a $5$ $174.082$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None 26.10.b.a \(80\) \(81\) \(1213\) \(-8715\) $-$ $-$ $\mathrm{SU}(2)$ \(q+2^{4}q^{2}+(2^{4}+\beta _{1})q^{3}+2^{8}q^{4}+(3^{5}+\cdots)q^{5}+\cdots\)
338.10.a.k 338.a 1.a $6$ $174.082$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None 26.10.c.b \(-96\) \(81\) \(-1693\) \(-473\) $+$ $+$ $\mathrm{SU}(2)$ \(q-2^{4}q^{2}+(14-\beta _{1})q^{3}+2^{8}q^{4}+(-281+\cdots)q^{5}+\cdots\)
338.10.a.l 338.a 1.a $6$ $174.082$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None 26.10.c.b \(96\) \(81\) \(1693\) \(473\) $-$ $+$ $\mathrm{SU}(2)$ \(q+2^{4}q^{2}+(14-\beta _{1})q^{3}+2^{8}q^{4}+(281+\cdots)q^{5}+\cdots\)
338.10.a.m 338.a 1.a $10$ $174.082$ \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None 26.10.e.a \(-160\) \(-162\) \(3088\) \(11376\) $+$ $-$ $\mathrm{SU}(2)$ \(q-2^{4}q^{2}+(-2^{4}-\beta _{1})q^{3}+2^{8}q^{4}+\cdots\)
338.10.a.n 338.a 1.a $10$ $174.082$ \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None 26.10.e.a \(160\) \(-162\) \(-3088\) \(-11376\) $-$ $-$ $\mathrm{SU}(2)$ \(q+2^{4}q^{2}+(-2^{4}-\beta _{1})q^{3}+2^{8}q^{4}+\cdots\)
338.10.a.o 338.a 1.a $12$ $174.082$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None 338.10.a.o \(-192\) \(-399\) \(562\) \(-3161\) $+$ $+$ $\mathrm{SU}(2)$ \(q-2^{4}q^{2}+(-33+\beta _{1}-\beta _{2})q^{3}+2^{8}q^{4}+\cdots\)
338.10.a.p 338.a 1.a $12$ $174.082$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None 338.10.a.o \(192\) \(-399\) \(-562\) \(3161\) $-$ $-$ $\mathrm{SU}(2)$ \(q+2^{4}q^{2}+(-33+\beta _{1}-\beta _{2})q^{3}+2^{8}q^{4}+\cdots\)
338.10.a.q 338.a 1.a $15$ $174.082$ \(\mathbb{Q}[x]/(x^{15} - \cdots)\) None 338.10.a.q \(-240\) \(324\) \(-2164\) \(-4227\) $+$ $-$ $\mathrm{SU}(2)$ \(q-2^{4}q^{2}+(22+\beta _{1})q^{3}+2^{8}q^{4}+(-145+\cdots)q^{5}+\cdots\)
338.10.a.r 338.a 1.a $15$ $174.082$ \(\mathbb{Q}[x]/(x^{15} - \cdots)\) None 338.10.a.q \(240\) \(324\) \(2164\) \(4227\) $-$ $+$ $\mathrm{SU}(2)$ \(q+2^{4}q^{2}+(22+\beta _{1})q^{3}+2^{8}q^{4}+(145+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{10}^{\mathrm{old}}(\Gamma_0(338))\) into lower level spaces

\( S_{10}^{\mathrm{old}}(\Gamma_0(338)) \simeq \) \(S_{10}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 3}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(13))\)\(^{\oplus 4}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(26))\)\(^{\oplus 2}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(169))\)\(^{\oplus 2}\)