Properties

Label 3850.2.a
Level $3850$
Weight $2$
Character orbit 3850.a
Rep. character $\chi_{3850}(1,\cdot)$
Character field $\Q$
Dimension $94$
Newform subspaces $53$
Sturm bound $1440$
Trace bound $19$

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Defining parameters

Level: \( N \) \(=\) \( 3850 = 2 \cdot 5^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3850.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 53 \)
Sturm bound: \(1440\)
Trace bound: \(19\)
Distinguishing \(T_p\): \(3\), \(13\), \(17\), \(19\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(3850))\).

Total New Old
Modular forms 744 94 650
Cusp forms 697 94 603
Eisenstein series 47 0 47

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

\(2\)\(5\)\(7\)\(11\)FrickeDim.
\(+\)\(+\)\(+\)\(+\)\(+\)\(8\)
\(+\)\(+\)\(+\)\(-\)\(-\)\(5\)
\(+\)\(+\)\(-\)\(+\)\(-\)\(5\)
\(+\)\(+\)\(-\)\(-\)\(+\)\(6\)
\(+\)\(-\)\(+\)\(+\)\(-\)\(5\)
\(+\)\(-\)\(+\)\(-\)\(+\)\(5\)
\(+\)\(-\)\(-\)\(+\)\(+\)\(7\)
\(+\)\(-\)\(-\)\(-\)\(-\)\(7\)
\(-\)\(+\)\(+\)\(+\)\(-\)\(6\)
\(-\)\(+\)\(+\)\(-\)\(+\)\(4\)
\(-\)\(+\)\(-\)\(+\)\(+\)\(3\)
\(-\)\(+\)\(-\)\(-\)\(-\)\(9\)
\(-\)\(-\)\(+\)\(+\)\(+\)\(6\)
\(-\)\(-\)\(+\)\(-\)\(-\)\(8\)
\(-\)\(-\)\(-\)\(+\)\(-\)\(8\)
\(-\)\(-\)\(-\)\(-\)\(+\)\(2\)
Plus space\(+\)\(41\)
Minus space\(-\)\(53\)

Trace form

\( 94q - 2q^{2} - 4q^{3} + 94q^{4} - 2q^{8} + 86q^{9} + O(q^{10}) \) \( 94q - 2q^{2} - 4q^{3} + 94q^{4} - 2q^{8} + 86q^{9} - 2q^{11} - 4q^{12} - 12q^{13} - 4q^{14} + 94q^{16} + 4q^{17} - 18q^{18} + 8q^{19} + 2q^{22} - 8q^{23} + 32q^{26} - 16q^{27} + 28q^{29} - 8q^{31} - 2q^{32} - 4q^{33} + 20q^{34} + 86q^{36} - 12q^{37} + 12q^{38} - 40q^{39} + 12q^{41} + 4q^{42} + 8q^{43} - 2q^{44} + 16q^{46} - 24q^{47} - 4q^{48} + 94q^{49} - 48q^{51} - 12q^{52} + 28q^{53} + 24q^{54} - 4q^{56} - 8q^{57} + 4q^{58} - 36q^{59} + 12q^{61} + 40q^{62} + 8q^{63} + 94q^{64} + 24q^{67} + 4q^{68} + 64q^{69} + 72q^{71} - 18q^{72} - 12q^{73} + 28q^{74} + 8q^{76} + 4q^{77} + 40q^{78} - 32q^{79} + 142q^{81} - 20q^{82} + 32q^{83} - 8q^{86} - 16q^{87} + 2q^{88} + 84q^{89} + 4q^{91} - 8q^{92} + 24q^{93} + 40q^{94} - 28q^{97} - 2q^{98} + 54q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(3850))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 5 7 11
3850.2.a.a \(1\) \(30.742\) \(\Q\) None \(-1\) \(-1\) \(0\) \(-1\) \(+\) \(+\) \(+\) \(+\) \(q-q^{2}-q^{3}+q^{4}+q^{6}-q^{7}-q^{8}+\cdots\)
3850.2.a.b \(1\) \(30.742\) \(\Q\) None \(-1\) \(0\) \(0\) \(-1\) \(+\) \(-\) \(+\) \(+\) \(q-q^{2}+q^{4}-q^{7}-q^{8}-3q^{9}-q^{11}+\cdots\)
3850.2.a.c \(1\) \(30.742\) \(\Q\) None \(-1\) \(0\) \(0\) \(-1\) \(+\) \(-\) \(+\) \(+\) \(q-q^{2}+q^{4}-q^{7}-q^{8}-3q^{9}-q^{11}+\cdots\)
3850.2.a.d \(1\) \(30.742\) \(\Q\) None \(-1\) \(0\) \(0\) \(-1\) \(+\) \(-\) \(+\) \(+\) \(q-q^{2}+q^{4}-q^{7}-q^{8}-3q^{9}-q^{11}+\cdots\)
3850.2.a.e \(1\) \(30.742\) \(\Q\) None \(-1\) \(0\) \(0\) \(-1\) \(+\) \(-\) \(+\) \(-\) \(q-q^{2}+q^{4}-q^{7}-q^{8}-3q^{9}+q^{11}+\cdots\)
3850.2.a.f \(1\) \(30.742\) \(\Q\) None \(-1\) \(0\) \(0\) \(1\) \(+\) \(+\) \(-\) \(+\) \(q-q^{2}+q^{4}+q^{7}-q^{8}-3q^{9}-q^{11}+\cdots\)
3850.2.a.g \(1\) \(30.742\) \(\Q\) None \(-1\) \(0\) \(0\) \(1\) \(+\) \(-\) \(-\) \(+\) \(q-q^{2}+q^{4}+q^{7}-q^{8}-3q^{9}-q^{11}+\cdots\)
3850.2.a.h \(1\) \(30.742\) \(\Q\) None \(-1\) \(1\) \(0\) \(-1\) \(+\) \(+\) \(+\) \(+\) \(q-q^{2}+q^{3}+q^{4}-q^{6}-q^{7}-q^{8}+\cdots\)
3850.2.a.i \(1\) \(30.742\) \(\Q\) None \(-1\) \(1\) \(0\) \(-1\) \(+\) \(-\) \(+\) \(-\) \(q-q^{2}+q^{3}+q^{4}-q^{6}-q^{7}-q^{8}+\cdots\)
3850.2.a.j \(1\) \(30.742\) \(\Q\) None \(-1\) \(2\) \(0\) \(-1\) \(+\) \(+\) \(+\) \(+\) \(q-q^{2}+2q^{3}+q^{4}-2q^{6}-q^{7}-q^{8}+\cdots\)
3850.2.a.k \(1\) \(30.742\) \(\Q\) None \(-1\) \(2\) \(0\) \(-1\) \(+\) \(+\) \(+\) \(+\) \(q-q^{2}+2q^{3}+q^{4}-2q^{6}-q^{7}-q^{8}+\cdots\)
3850.2.a.l \(1\) \(30.742\) \(\Q\) None \(-1\) \(2\) \(0\) \(-1\) \(+\) \(+\) \(+\) \(-\) \(q-q^{2}+2q^{3}+q^{4}-2q^{6}-q^{7}-q^{8}+\cdots\)
3850.2.a.m \(1\) \(30.742\) \(\Q\) None \(1\) \(-2\) \(0\) \(-1\) \(-\) \(+\) \(+\) \(+\) \(q+q^{2}-2q^{3}+q^{4}-2q^{6}-q^{7}+q^{8}+\cdots\)
3850.2.a.n \(1\) \(30.742\) \(\Q\) None \(1\) \(-2\) \(0\) \(1\) \(-\) \(-\) \(-\) \(-\) \(q+q^{2}-2q^{3}+q^{4}-2q^{6}+q^{7}+q^{8}+\cdots\)
3850.2.a.o \(1\) \(30.742\) \(\Q\) None \(1\) \(-2\) \(0\) \(1\) \(-\) \(+\) \(-\) \(-\) \(q+q^{2}-2q^{3}+q^{4}-2q^{6}+q^{7}+q^{8}+\cdots\)
3850.2.a.p \(1\) \(30.742\) \(\Q\) None \(1\) \(-1\) \(0\) \(1\) \(-\) \(-\) \(-\) \(+\) \(q+q^{2}-q^{3}+q^{4}-q^{6}+q^{7}+q^{8}+\cdots\)
3850.2.a.q \(1\) \(30.742\) \(\Q\) None \(1\) \(-1\) \(0\) \(1\) \(-\) \(+\) \(-\) \(-\) \(q+q^{2}-q^{3}+q^{4}-q^{6}+q^{7}+q^{8}+\cdots\)
3850.2.a.r \(1\) \(30.742\) \(\Q\) None \(1\) \(0\) \(0\) \(-1\) \(-\) \(+\) \(+\) \(+\) \(q+q^{2}+q^{4}-q^{7}+q^{8}-3q^{9}-q^{11}+\cdots\)
3850.2.a.s \(1\) \(30.742\) \(\Q\) None \(1\) \(0\) \(0\) \(-1\) \(-\) \(+\) \(+\) \(+\) \(q+q^{2}+q^{4}-q^{7}+q^{8}-3q^{9}-q^{11}+\cdots\)
3850.2.a.t \(1\) \(30.742\) \(\Q\) None \(1\) \(0\) \(0\) \(1\) \(-\) \(+\) \(-\) \(+\) \(q+q^{2}+q^{4}+q^{7}+q^{8}-3q^{9}-q^{11}+\cdots\)
3850.2.a.u \(1\) \(30.742\) \(\Q\) None \(1\) \(0\) \(0\) \(1\) \(-\) \(+\) \(-\) \(+\) \(q+q^{2}+q^{4}+q^{7}+q^{8}-3q^{9}-q^{11}+\cdots\)
3850.2.a.v \(1\) \(30.742\) \(\Q\) None \(1\) \(0\) \(0\) \(1\) \(-\) \(-\) \(-\) \(+\) \(q+q^{2}+q^{4}+q^{7}+q^{8}-3q^{9}-q^{11}+\cdots\)
3850.2.a.w \(1\) \(30.742\) \(\Q\) None \(1\) \(0\) \(0\) \(1\) \(-\) \(+\) \(-\) \(+\) \(q+q^{2}+q^{4}+q^{7}+q^{8}-3q^{9}-q^{11}+\cdots\)
3850.2.a.x \(1\) \(30.742\) \(\Q\) None \(1\) \(0\) \(0\) \(1\) \(-\) \(-\) \(-\) \(+\) \(q+q^{2}+q^{4}+q^{7}+q^{8}-3q^{9}-q^{11}+\cdots\)
3850.2.a.y \(1\) \(30.742\) \(\Q\) None \(1\) \(0\) \(0\) \(1\) \(-\) \(-\) \(-\) \(-\) \(q+q^{2}+q^{4}+q^{7}+q^{8}-3q^{9}+q^{11}+\cdots\)
3850.2.a.z \(1\) \(30.742\) \(\Q\) None \(1\) \(1\) \(0\) \(1\) \(-\) \(-\) \(-\) \(+\) \(q+q^{2}+q^{3}+q^{4}+q^{6}+q^{7}+q^{8}+\cdots\)
3850.2.a.ba \(1\) \(30.742\) \(\Q\) None \(1\) \(2\) \(0\) \(-1\) \(-\) \(+\) \(+\) \(+\) \(q+q^{2}+2q^{3}+q^{4}+2q^{6}-q^{7}+q^{8}+\cdots\)
3850.2.a.bb \(1\) \(30.742\) \(\Q\) None \(1\) \(2\) \(0\) \(1\) \(-\) \(+\) \(-\) \(-\) \(q+q^{2}+2q^{3}+q^{4}+2q^{6}+q^{7}+q^{8}+\cdots\)
3850.2.a.bc \(2\) \(30.742\) \(\Q(\sqrt{33}) \) None \(-2\) \(-4\) \(0\) \(-2\) \(+\) \(+\) \(+\) \(+\) \(q-q^{2}-2q^{3}+q^{4}+2q^{6}-q^{7}-q^{8}+\cdots\)
3850.2.a.bd \(2\) \(30.742\) \(\Q(\sqrt{3}) \) None \(-2\) \(-2\) \(0\) \(-2\) \(+\) \(+\) \(+\) \(-\) \(q-q^{2}+(-1+\beta )q^{3}+q^{4}+(1-\beta )q^{6}+\cdots\)
3850.2.a.be \(2\) \(30.742\) \(\Q(\sqrt{6}) \) None \(-2\) \(0\) \(0\) \(-2\) \(+\) \(-\) \(+\) \(+\) \(q-q^{2}+\beta q^{3}+q^{4}-\beta q^{6}-q^{7}-q^{8}+\cdots\)
3850.2.a.bf \(2\) \(30.742\) \(\Q(\sqrt{10}) \) None \(-2\) \(0\) \(0\) \(-2\) \(+\) \(+\) \(+\) \(+\) \(q-q^{2}+\beta q^{3}+q^{4}-\beta q^{6}-q^{7}-q^{8}+\cdots\)
3850.2.a.bg \(2\) \(30.742\) \(\Q(\sqrt{2}) \) None \(-2\) \(0\) \(0\) \(2\) \(+\) \(+\) \(-\) \(+\) \(q-q^{2}+\beta q^{3}+q^{4}-\beta q^{6}+q^{7}-q^{8}+\cdots\)
3850.2.a.bh \(2\) \(30.742\) \(\Q(\sqrt{7}) \) None \(-2\) \(0\) \(0\) \(2\) \(+\) \(-\) \(-\) \(+\) \(q-q^{2}+\beta q^{3}+q^{4}-\beta q^{6}+q^{7}-q^{8}+\cdots\)
3850.2.a.bi \(2\) \(30.742\) \(\Q(\sqrt{2}) \) None \(-2\) \(0\) \(0\) \(2\) \(+\) \(+\) \(-\) \(+\) \(q-q^{2}+\beta q^{3}+q^{4}-\beta q^{6}+q^{7}-q^{8}+\cdots\)
3850.2.a.bj \(2\) \(30.742\) \(\Q(\sqrt{5}) \) None \(-2\) \(2\) \(0\) \(-2\) \(+\) \(+\) \(+\) \(-\) \(q-q^{2}+(1+\beta )q^{3}+q^{4}+(-1-\beta )q^{6}+\cdots\)
3850.2.a.bk \(2\) \(30.742\) \(\Q(\sqrt{3}) \) None \(-2\) \(2\) \(0\) \(2\) \(+\) \(-\) \(-\) \(-\) \(q-q^{2}+(1+\beta )q^{3}+q^{4}+(-1-\beta )q^{6}+\cdots\)
3850.2.a.bl \(2\) \(30.742\) \(\Q(\sqrt{3}) \) None \(2\) \(-2\) \(0\) \(-2\) \(-\) \(+\) \(+\) \(-\) \(q+q^{2}+(-1+\beta )q^{3}+q^{4}+(-1+\beta )q^{6}+\cdots\)
3850.2.a.bm \(2\) \(30.742\) \(\Q(\sqrt{3}) \) None \(2\) \(-2\) \(0\) \(-2\) \(-\) \(+\) \(+\) \(-\) \(q+q^{2}+(-1+\beta )q^{3}+q^{4}+(-1+\beta )q^{6}+\cdots\)
3850.2.a.bn \(2\) \(30.742\) \(\Q(\sqrt{2}) \) None \(2\) \(0\) \(0\) \(-2\) \(-\) \(-\) \(+\) \(+\) \(q+q^{2}+\beta q^{3}+q^{4}+\beta q^{6}-q^{7}+q^{8}+\cdots\)
3850.2.a.bo \(2\) \(30.742\) \(\Q(\sqrt{7}) \) None \(2\) \(0\) \(0\) \(-2\) \(-\) \(+\) \(+\) \(+\) \(q+q^{2}+\beta q^{3}+q^{4}+\beta q^{6}-q^{7}+q^{8}+\cdots\)
3850.2.a.bp \(2\) \(30.742\) \(\Q(\sqrt{6}) \) None \(2\) \(0\) \(0\) \(2\) \(-\) \(-\) \(-\) \(+\) \(q+q^{2}+\beta q^{3}+q^{4}+\beta q^{6}+q^{7}+q^{8}+\cdots\)
3850.2.a.bq \(2\) \(30.742\) \(\Q(\sqrt{10}) \) None \(2\) \(0\) \(0\) \(2\) \(-\) \(-\) \(-\) \(+\) \(q+q^{2}+\beta q^{3}+q^{4}+\beta q^{6}+q^{7}+q^{8}+\cdots\)
3850.2.a.br \(3\) \(30.742\) 3.3.940.1 None \(-3\) \(-3\) \(0\) \(-3\) \(+\) \(-\) \(+\) \(-\) \(q-q^{2}+(-1+\beta _{1})q^{3}+q^{4}+(1-\beta _{1}+\cdots)q^{6}+\cdots\)
3850.2.a.bs \(3\) \(30.742\) 3.3.316.1 None \(-3\) \(-2\) \(0\) \(3\) \(+\) \(+\) \(-\) \(-\) \(q-q^{2}+(-1+\beta _{1})q^{3}+q^{4}+(1-\beta _{1}+\cdots)q^{6}+\cdots\)
3850.2.a.bt \(3\) \(30.742\) 3.3.316.1 None \(-3\) \(-2\) \(0\) \(3\) \(+\) \(+\) \(-\) \(-\) \(q-q^{2}+(-1-\beta _{2})q^{3}+q^{4}+(1+\beta _{2})q^{6}+\cdots\)
3850.2.a.bu \(3\) \(30.742\) 3.3.892.1 None \(3\) \(0\) \(0\) \(3\) \(-\) \(+\) \(-\) \(-\) \(q+q^{2}-\beta _{1}q^{3}+q^{4}-\beta _{1}q^{6}+q^{7}+\cdots\)
3850.2.a.bv \(3\) \(30.742\) 3.3.316.1 None \(3\) \(2\) \(0\) \(-3\) \(-\) \(-\) \(+\) \(-\) \(q+q^{2}+(1-\beta _{1})q^{3}+q^{4}+(1-\beta _{1})q^{6}+\cdots\)
3850.2.a.bw \(3\) \(30.742\) 3.3.940.1 None \(3\) \(3\) \(0\) \(3\) \(-\) \(+\) \(-\) \(-\) \(q+q^{2}+(1-\beta _{1})q^{3}+q^{4}+(1-\beta _{1})q^{6}+\cdots\)
3850.2.a.bx \(4\) \(30.742\) 4.4.10304.1 None \(-4\) \(0\) \(0\) \(4\) \(+\) \(-\) \(-\) \(+\) \(q-q^{2}-\beta _{2}q^{3}+q^{4}+\beta _{2}q^{6}+q^{7}+\cdots\)
3850.2.a.by \(4\) \(30.742\) 4.4.10304.1 None \(4\) \(0\) \(0\) \(-4\) \(-\) \(-\) \(+\) \(+\) \(q+q^{2}+\beta _{2}q^{3}+q^{4}+\beta _{2}q^{6}-q^{7}+\cdots\)
3850.2.a.bz \(5\) \(30.742\) 5.5.117688.1 None \(-5\) \(0\) \(0\) \(5\) \(+\) \(-\) \(-\) \(-\) \(q-q^{2}-\beta _{2}q^{3}+q^{4}+\beta _{2}q^{6}+q^{7}+\cdots\)
3850.2.a.ca \(5\) \(30.742\) 5.5.117688.1 None \(5\) \(0\) \(0\) \(-5\) \(-\) \(-\) \(+\) \(-\) \(q+q^{2}+\beta _{2}q^{3}+q^{4}+\beta _{2}q^{6}-q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(3850))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(3850)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(35))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(55))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(70))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(77))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(110))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(154))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(175))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(275))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(350))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(385))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(550))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(770))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1925))\)\(^{\oplus 2}\)