Properties

Label 3696.2.a
Level $3696$
Weight $2$
Character orbit 3696.a
Rep. character $\chi_{3696}(1,\cdot)$
Character field $\Q$
Dimension $60$
Newform subspaces $42$
Sturm bound $1536$
Trace bound $17$

Related objects

Downloads

Learn more

Defining parameters

Level: \( N \) \(=\) \( 3696 = 2^{4} \cdot 3 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3696.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 42 \)
Sturm bound: \(1536\)
Trace bound: \(17\)
Distinguishing \(T_p\): \(5\), \(13\), \(17\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(3696))\).

Total New Old
Modular forms 792 60 732
Cusp forms 745 60 685
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.

\(2\)\(3\)\(7\)\(11\)FrickeDim.
\(+\)\(+\)\(+\)\(+\)\(+\)\(4\)
\(+\)\(+\)\(+\)\(-\)\(-\)\(4\)
\(+\)\(+\)\(-\)\(+\)\(-\)\(4\)
\(+\)\(+\)\(-\)\(-\)\(+\)\(4\)
\(+\)\(-\)\(+\)\(+\)\(-\)\(4\)
\(+\)\(-\)\(+\)\(-\)\(+\)\(4\)
\(+\)\(-\)\(-\)\(+\)\(+\)\(4\)
\(+\)\(-\)\(-\)\(-\)\(-\)\(4\)
\(-\)\(+\)\(+\)\(+\)\(-\)\(5\)
\(-\)\(+\)\(+\)\(-\)\(+\)\(2\)
\(-\)\(+\)\(-\)\(+\)\(+\)\(2\)
\(-\)\(+\)\(-\)\(-\)\(-\)\(5\)
\(-\)\(-\)\(+\)\(+\)\(+\)\(2\)
\(-\)\(-\)\(+\)\(-\)\(-\)\(5\)
\(-\)\(-\)\(-\)\(+\)\(-\)\(5\)
\(-\)\(-\)\(-\)\(-\)\(+\)\(2\)
Plus space\(+\)\(24\)
Minus space\(-\)\(36\)

Trace form

\( 60 q - 8 q^{5} + 60 q^{9} + O(q^{10}) \) \( 60 q - 8 q^{5} + 60 q^{9} - 8 q^{13} - 8 q^{15} + 8 q^{17} - 16 q^{19} + 68 q^{25} - 8 q^{29} - 16 q^{31} - 8 q^{37} + 8 q^{41} - 8 q^{45} + 60 q^{49} - 16 q^{51} - 8 q^{53} + 24 q^{61} + 48 q^{65} + 24 q^{67} + 32 q^{69} + 48 q^{71} + 40 q^{73} - 16 q^{79} + 60 q^{81} + 48 q^{83} + 16 q^{85} + 24 q^{89} + 24 q^{91} + 48 q^{95} - 8 q^{97} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(3696))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 3 7 11
3696.2.a.a \(1\) \(29.513\) \(\Q\) None \(0\) \(-1\) \(-4\) \(1\) \(+\) \(+\) \(-\) \(-\) \(q-q^{3}-4q^{5}+q^{7}+q^{9}+q^{11}-2q^{13}+\cdots\)
3696.2.a.b \(1\) \(29.513\) \(\Q\) None \(0\) \(-1\) \(-3\) \(-1\) \(-\) \(+\) \(+\) \(+\) \(q-q^{3}-3q^{5}-q^{7}+q^{9}-q^{11}-7q^{13}+\cdots\)
3696.2.a.c \(1\) \(29.513\) \(\Q\) None \(0\) \(-1\) \(-3\) \(-1\) \(+\) \(+\) \(+\) \(+\) \(q-q^{3}-3q^{5}-q^{7}+q^{9}-q^{11}-3q^{13}+\cdots\)
3696.2.a.d \(1\) \(29.513\) \(\Q\) None \(0\) \(-1\) \(-3\) \(1\) \(-\) \(+\) \(-\) \(-\) \(q-q^{3}-3q^{5}+q^{7}+q^{9}+q^{11}+3q^{13}+\cdots\)
3696.2.a.e \(1\) \(29.513\) \(\Q\) None \(0\) \(-1\) \(-2\) \(1\) \(+\) \(+\) \(-\) \(-\) \(q-q^{3}-2q^{5}+q^{7}+q^{9}+q^{11}-2q^{13}+\cdots\)
3696.2.a.f \(1\) \(29.513\) \(\Q\) None \(0\) \(-1\) \(-1\) \(-1\) \(+\) \(+\) \(+\) \(-\) \(q-q^{3}-q^{5}-q^{7}+q^{9}+q^{11}-5q^{13}+\cdots\)
3696.2.a.g \(1\) \(29.513\) \(\Q\) None \(0\) \(-1\) \(-1\) \(1\) \(+\) \(+\) \(-\) \(+\) \(q-q^{3}-q^{5}+q^{7}+q^{9}-q^{11}+q^{13}+\cdots\)
3696.2.a.h \(1\) \(29.513\) \(\Q\) None \(0\) \(-1\) \(-1\) \(1\) \(-\) \(+\) \(-\) \(+\) \(q-q^{3}-q^{5}+q^{7}+q^{9}-q^{11}+q^{13}+\cdots\)
3696.2.a.i \(1\) \(29.513\) \(\Q\) None \(0\) \(-1\) \(0\) \(-1\) \(-\) \(+\) \(+\) \(-\) \(q-q^{3}-q^{7}+q^{9}+q^{11}+2q^{13}-2q^{19}+\cdots\)
3696.2.a.j \(1\) \(29.513\) \(\Q\) None \(0\) \(-1\) \(0\) \(1\) \(-\) \(+\) \(-\) \(-\) \(q-q^{3}+q^{7}+q^{9}+q^{11}+6q^{13}+4q^{17}+\cdots\)
3696.2.a.k \(1\) \(29.513\) \(\Q\) None \(0\) \(-1\) \(1\) \(1\) \(+\) \(+\) \(-\) \(-\) \(q-q^{3}+q^{5}+q^{7}+q^{9}+q^{11}-5q^{13}+\cdots\)
3696.2.a.l \(1\) \(29.513\) \(\Q\) None \(0\) \(-1\) \(1\) \(1\) \(+\) \(+\) \(-\) \(-\) \(q-q^{3}+q^{5}+q^{7}+q^{9}+q^{11}+3q^{13}+\cdots\)
3696.2.a.m \(1\) \(29.513\) \(\Q\) None \(0\) \(-1\) \(2\) \(1\) \(-\) \(+\) \(-\) \(+\) \(q-q^{3}+2q^{5}+q^{7}+q^{9}-q^{11}-2q^{13}+\cdots\)
3696.2.a.n \(1\) \(29.513\) \(\Q\) None \(0\) \(-1\) \(2\) \(1\) \(+\) \(+\) \(-\) \(+\) \(q-q^{3}+2q^{5}+q^{7}+q^{9}-q^{11}+6q^{13}+\cdots\)
3696.2.a.o \(1\) \(29.513\) \(\Q\) None \(0\) \(-1\) \(3\) \(-1\) \(-\) \(+\) \(+\) \(-\) \(q-q^{3}+3q^{5}-q^{7}+q^{9}+q^{11}-q^{13}+\cdots\)
3696.2.a.p \(1\) \(29.513\) \(\Q\) None \(0\) \(1\) \(-4\) \(-1\) \(-\) \(-\) \(+\) \(-\) \(q+q^{3}-4q^{5}-q^{7}+q^{9}+q^{11}-6q^{13}+\cdots\)
3696.2.a.q \(1\) \(29.513\) \(\Q\) None \(0\) \(1\) \(-3\) \(-1\) \(+\) \(-\) \(+\) \(+\) \(q+q^{3}-3q^{5}-q^{7}+q^{9}-q^{11}-q^{13}+\cdots\)
3696.2.a.r \(1\) \(29.513\) \(\Q\) None \(0\) \(1\) \(-3\) \(1\) \(-\) \(-\) \(-\) \(-\) \(q+q^{3}-3q^{5}+q^{7}+q^{9}+q^{11}+q^{13}+\cdots\)
3696.2.a.s \(1\) \(29.513\) \(\Q\) None \(0\) \(1\) \(-2\) \(-1\) \(-\) \(-\) \(+\) \(+\) \(q+q^{3}-2q^{5}-q^{7}+q^{9}-q^{11}+2q^{13}+\cdots\)
3696.2.a.t \(1\) \(29.513\) \(\Q\) None \(0\) \(1\) \(-2\) \(-1\) \(-\) \(-\) \(+\) \(-\) \(q+q^{3}-2q^{5}-q^{7}+q^{9}+q^{11}+6q^{13}+\cdots\)
3696.2.a.u \(1\) \(29.513\) \(\Q\) None \(0\) \(1\) \(-1\) \(-1\) \(-\) \(-\) \(+\) \(-\) \(q+q^{3}-q^{5}-q^{7}+q^{9}+q^{11}-3q^{13}+\cdots\)
3696.2.a.v \(1\) \(29.513\) \(\Q\) None \(0\) \(1\) \(-1\) \(-1\) \(+\) \(-\) \(+\) \(-\) \(q+q^{3}-q^{5}-q^{7}+q^{9}+q^{11}-3q^{13}+\cdots\)
3696.2.a.w \(1\) \(29.513\) \(\Q\) None \(0\) \(1\) \(-1\) \(1\) \(-\) \(-\) \(-\) \(+\) \(q+q^{3}-q^{5}+q^{7}+q^{9}-q^{11}-q^{13}+\cdots\)
3696.2.a.x \(1\) \(29.513\) \(\Q\) None \(0\) \(1\) \(0\) \(-1\) \(+\) \(-\) \(+\) \(-\) \(q+q^{3}-q^{7}+q^{9}+q^{11}+2q^{13}-8q^{17}+\cdots\)
3696.2.a.y \(1\) \(29.513\) \(\Q\) None \(0\) \(1\) \(0\) \(1\) \(-\) \(-\) \(-\) \(-\) \(q+q^{3}+q^{7}+q^{9}+q^{11}-2q^{13}-4q^{17}+\cdots\)
3696.2.a.z \(1\) \(29.513\) \(\Q\) None \(0\) \(1\) \(1\) \(-1\) \(-\) \(-\) \(+\) \(+\) \(q+q^{3}+q^{5}-q^{7}+q^{9}-q^{11}-q^{13}+\cdots\)
3696.2.a.ba \(1\) \(29.513\) \(\Q\) None \(0\) \(1\) \(2\) \(-1\) \(+\) \(-\) \(+\) \(+\) \(q+q^{3}+2q^{5}-q^{7}+q^{9}-q^{11}-6q^{13}+\cdots\)
3696.2.a.bb \(1\) \(29.513\) \(\Q\) None \(0\) \(1\) \(2\) \(1\) \(-\) \(-\) \(-\) \(+\) \(q+q^{3}+2q^{5}+q^{7}+q^{9}-q^{11}+2q^{13}+\cdots\)
3696.2.a.bc \(2\) \(29.513\) \(\Q(\sqrt{3}) \) None \(0\) \(-2\) \(0\) \(-2\) \(-\) \(+\) \(+\) \(+\) \(q-q^{3}+\beta q^{5}-q^{7}+q^{9}-q^{11}+2q^{13}+\cdots\)
3696.2.a.bd \(2\) \(29.513\) \(\Q(\sqrt{41}) \) None \(0\) \(-2\) \(1\) \(2\) \(+\) \(+\) \(-\) \(+\) \(q-q^{3}+\beta q^{5}+q^{7}+q^{9}-q^{11}-\beta q^{13}+\cdots\)
3696.2.a.be \(2\) \(29.513\) \(\Q(\sqrt{5}) \) None \(0\) \(-2\) \(2\) \(-2\) \(-\) \(+\) \(+\) \(+\) \(q-q^{3}+q^{5}-q^{7}+q^{9}-q^{11}+(-1+\cdots)q^{13}+\cdots\)
3696.2.a.bf \(2\) \(29.513\) \(\Q(\sqrt{33}) \) None \(0\) \(2\) \(-3\) \(-2\) \(+\) \(-\) \(+\) \(-\) \(q+q^{3}+(-1-\beta )q^{5}-q^{7}+q^{9}+q^{11}+\cdots\)
3696.2.a.bg \(2\) \(29.513\) \(\Q(\sqrt{17}) \) None \(0\) \(2\) \(-3\) \(2\) \(+\) \(-\) \(-\) \(+\) \(q+q^{3}+(-1-\beta )q^{5}+q^{7}+q^{9}-q^{11}+\cdots\)
3696.2.a.bh \(2\) \(29.513\) \(\Q(\sqrt{17}) \) None \(0\) \(2\) \(-3\) \(2\) \(+\) \(-\) \(-\) \(+\) \(q+q^{3}+(-1-\beta )q^{5}+q^{7}+q^{9}-q^{11}+\cdots\)
3696.2.a.bi \(2\) \(29.513\) \(\Q(\sqrt{17}) \) None \(0\) \(2\) \(1\) \(2\) \(+\) \(-\) \(-\) \(-\) \(q+q^{3}+\beta q^{5}+q^{7}+q^{9}+q^{11}+(2+\cdots)q^{13}+\cdots\)
3696.2.a.bj \(2\) \(29.513\) \(\Q(\sqrt{17}) \) None \(0\) \(2\) \(3\) \(-2\) \(+\) \(-\) \(+\) \(+\) \(q+q^{3}+(1+\beta )q^{5}-q^{7}+q^{9}-q^{11}+\cdots\)
3696.2.a.bk \(2\) \(29.513\) \(\Q(\sqrt{17}) \) None \(0\) \(2\) \(3\) \(2\) \(+\) \(-\) \(-\) \(-\) \(q+q^{3}+(1+\beta )q^{5}+q^{7}+q^{9}+q^{11}+\cdots\)
3696.2.a.bl \(2\) \(29.513\) \(\Q(\sqrt{21}) \) None \(0\) \(2\) \(6\) \(-2\) \(-\) \(-\) \(+\) \(-\) \(q+q^{3}+3q^{5}-q^{7}+q^{9}+q^{11}+q^{13}+\cdots\)
3696.2.a.bm \(3\) \(29.513\) 3.3.568.1 None \(0\) \(-3\) \(1\) \(-3\) \(+\) \(+\) \(+\) \(+\) \(q-q^{3}-\beta _{2}q^{5}-q^{7}+q^{9}-q^{11}+\beta _{1}q^{13}+\cdots\)
3696.2.a.bn \(3\) \(29.513\) 3.3.961.1 None \(0\) \(-3\) \(1\) \(-3\) \(+\) \(+\) \(+\) \(-\) \(q-q^{3}+\beta _{1}q^{5}-q^{7}+q^{9}+q^{11}+(1+\cdots)q^{13}+\cdots\)
3696.2.a.bo \(3\) \(29.513\) 3.3.229.1 None \(0\) \(-3\) \(4\) \(3\) \(-\) \(+\) \(-\) \(-\) \(q-q^{3}+(1-\beta _{1})q^{5}+q^{7}+q^{9}+q^{11}+\cdots\)
3696.2.a.bp \(3\) \(29.513\) 3.3.837.1 None \(0\) \(3\) \(0\) \(3\) \(-\) \(-\) \(-\) \(+\) \(q+q^{3}-\beta _{2}q^{5}+q^{7}+q^{9}-q^{11}-\beta _{2}q^{13}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(3696))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(3696)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 20}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 16}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(33))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(42))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(44))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(48))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(56))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(66))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(77))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(84))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(88))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(112))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(132))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(154))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(168))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(176))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(231))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(264))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(308))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(336))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(462))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(528))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(616))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(924))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1232))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1848))\)\(^{\oplus 2}\)