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Properties

Label 50.11.c
Level $50$
Weight $11$
Character orbit 50.c
Rep. character $\chi_{50}(7,\cdot)$
Character field $\Q(\zeta_{4})$
Dimension $30$
Newform subspaces $7$
Sturm bound $82$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 50 = 2 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 50.c (of order \(4\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 5 \)
Character field: \(\Q(i)\)
Newform subspaces: \( 7 \)
Sturm bound: \(82\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{11}(50, [\chi])\).

Total New Old
Modular forms 162 30 132
Cusp forms 138 30 108
Eisenstein series 24 0 24

Trace form

\( 30 q + 32 q^{2} + 124 q^{3} + 24320 q^{6} - 10604 q^{7} - 16384 q^{8} + O(q^{10}) \) \( 30 q + 32 q^{2} + 124 q^{3} + 24320 q^{6} - 10604 q^{7} - 16384 q^{8} - 903040 q^{11} + 63488 q^{12} + 988434 q^{13} - 7864320 q^{16} + 52906 q^{17} + 7111648 q^{18} - 32791240 q^{21} + 13506304 q^{22} + 26934364 q^{23} + 33741120 q^{26} + 53985640 q^{27} + 5429248 q^{28} - 20948040 q^{31} - 8388608 q^{32} + 100984528 q^{33} + 231511040 q^{36} - 78377634 q^{37} - 146994560 q^{38} + 504257360 q^{41} + 72606464 q^{42} - 291961716 q^{43} - 202983680 q^{46} - 877618884 q^{47} - 32505856 q^{48} + 1559723560 q^{51} + 506078208 q^{52} - 59690386 q^{53} - 538050560 q^{56} + 395251120 q^{57} - 468378880 q^{58} + 4790965360 q^{61} + 3714249344 q^{62} + 5999298844 q^{63} - 6050536960 q^{66} - 2285641604 q^{67} - 27087872 q^{68} - 10038076040 q^{71} + 3641163776 q^{72} + 4474810454 q^{73} - 959180800 q^{76} - 4559799088 q^{77} - 14413532544 q^{78} - 11290034870 q^{81} + 13760829184 q^{82} + 7763146644 q^{83} + 472427520 q^{86} - 44300553440 q^{87} - 6915227648 q^{88} + 34742508360 q^{91} + 13790394368 q^{92} - 22411006592 q^{93} - 6375342080 q^{96} - 30947240394 q^{97} - 23335064288 q^{98} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{11}^{\mathrm{new}}(50, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
50.11.c.a 50.c 5.c $2$ $31.768$ \(\Q(\sqrt{-1}) \) None 50.11.c.a \(-32\) \(-234\) \(0\) \(-16086\) $\mathrm{SU}(2)[C_{4}]$ \(q+(-2^{4}-2^{4}i)q^{2}+(-117+117i)q^{3}+\cdots\)
50.11.c.b 50.c 5.c $2$ $31.768$ \(\Q(\sqrt{-1}) \) None 10.11.c.b \(-32\) \(-114\) \(0\) \(-13906\) $\mathrm{SU}(2)[C_{4}]$ \(q+(-2^{4}-2^{4}i)q^{2}+(-57+57i)q^{3}+\cdots\)
50.11.c.c 50.c 5.c $2$ $31.768$ \(\Q(\sqrt{-1}) \) None 10.11.c.a \(-32\) \(366\) \(0\) \(16814\) $\mathrm{SU}(2)[C_{4}]$ \(q+(-2^{4}-2^{4}i)q^{2}+(183-183i)q^{3}+\cdots\)
50.11.c.d 50.c 5.c $2$ $31.768$ \(\Q(\sqrt{-1}) \) None 50.11.c.a \(32\) \(234\) \(0\) \(16086\) $\mathrm{SU}(2)[C_{4}]$ \(q+(2^{4}+2^{4}i)q^{2}+(117-117i)q^{3}+\cdots\)
50.11.c.e 50.c 5.c $6$ $31.768$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None 10.11.c.c \(96\) \(-128\) \(0\) \(-13512\) $\mathrm{SU}(2)[C_{4}]$ \(q+(2^{4}+2^{4}\beta _{1})q^{2}+(-21+21\beta _{1}+\beta _{3}+\cdots)q^{3}+\cdots\)
50.11.c.f 50.c 5.c $8$ $31.768$ 8.0.\(\cdots\).1 None 50.11.c.f \(-128\) \(-336\) \(0\) \(-8544\) $\mathrm{SU}(2)[C_{4}]$ \(q+(-2^{4}+2^{4}\beta _{1})q^{2}+(-42-42\beta _{1}+\cdots)q^{3}+\cdots\)
50.11.c.g 50.c 5.c $8$ $31.768$ 8.0.\(\cdots\).1 None 50.11.c.f \(128\) \(336\) \(0\) \(8544\) $\mathrm{SU}(2)[C_{4}]$ \(q+(2^{4}-2^{4}\beta _{1})q^{2}+(42+42\beta _{1}+\beta _{3}+\cdots)q^{3}+\cdots\)

Decomposition of \(S_{11}^{\mathrm{old}}(50, [\chi])\) into lower level spaces

\( S_{11}^{\mathrm{old}}(50, [\chi]) \simeq \) \(S_{11}^{\mathrm{new}}(5, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{11}^{\mathrm{new}}(10, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{11}^{\mathrm{new}}(25, [\chi])\)\(^{\oplus 2}\)