Properties

Label 400.8.a
Level $400$
Weight $8$
Character orbit 400.a
Rep. character $\chi_{400}(1,\cdot)$
Character field $\Q$
Dimension $65$
Newform subspaces $38$
Sturm bound $480$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 400 = 2^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 400.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 38 \)
Sturm bound: \(480\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_0(400))\).

Total New Old
Modular forms 438 68 370
Cusp forms 402 65 337
Eisenstein series 36 3 33

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

\(2\)\(5\)FrickeDim
\(+\)\(+\)\(+\)\(16\)
\(+\)\(-\)\(-\)\(17\)
\(-\)\(+\)\(-\)\(15\)
\(-\)\(-\)\(+\)\(17\)
Plus space\(+\)\(33\)
Minus space\(-\)\(32\)

Trace form

\( 65 q + 26 q^{3} + 662 q^{7} + 46317 q^{9} + O(q^{10}) \) \( 65 q + 26 q^{3} + 662 q^{7} + 46317 q^{9} - 3388 q^{11} + 3278 q^{13} - 11258 q^{17} - 60448 q^{19} + 3356 q^{21} + 7282 q^{23} + 12188 q^{27} - 68378 q^{29} + 8340 q^{31} - 166544 q^{33} - 104322 q^{37} + 268748 q^{39} - 133454 q^{41} - 142462 q^{43} - 1589618 q^{47} + 6831653 q^{49} + 1514416 q^{51} - 392442 q^{53} + 725144 q^{57} - 5391876 q^{59} - 5518 q^{61} + 4296854 q^{63} + 4606486 q^{67} + 2445284 q^{69} - 3556636 q^{71} + 1901070 q^{73} + 7780240 q^{77} + 9605360 q^{79} + 27940077 q^{81} - 3397918 q^{83} - 19863444 q^{87} + 10608630 q^{89} - 18655020 q^{91} + 9069672 q^{93} + 8946158 q^{97} - 46333544 q^{99} + O(q^{100}) \)

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_0(400))\) 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 5
400.8.a.a 400.a 1.a $1$ $124.954$ \(\Q\) None 50.8.a.d \(0\) \(-87\) \(0\) \(-1366\) $-$ $+$ $\mathrm{SU}(2)$ \(q-87q^{3}-1366q^{7}+5382q^{9}+1083q^{11}+\cdots\)
400.8.a.b 400.a 1.a $1$ $124.954$ \(\Q\) None 8.8.a.a \(0\) \(-84\) \(0\) \(-456\) $+$ $+$ $\mathrm{SU}(2)$ \(q-84q^{3}-456q^{7}+4869q^{9}+2524q^{11}+\cdots\)
400.8.a.c 400.a 1.a $1$ $124.954$ \(\Q\) None 200.8.a.a \(0\) \(-69\) \(0\) \(174\) $+$ $+$ $\mathrm{SU}(2)$ \(q-69q^{3}+174q^{7}+2574q^{9}-7111q^{11}+\cdots\)
400.8.a.d 400.a 1.a $1$ $124.954$ \(\Q\) None 50.8.a.c \(0\) \(-57\) \(0\) \(1174\) $-$ $-$ $\mathrm{SU}(2)$ \(q-57q^{3}+1174q^{7}+1062q^{9}+7563q^{11}+\cdots\)
400.8.a.e 400.a 1.a $1$ $124.954$ \(\Q\) None 5.8.a.a \(0\) \(-48\) \(0\) \(-1644\) $-$ $+$ $\mathrm{SU}(2)$ \(q-48q^{3}-1644q^{7}+117q^{9}-172q^{11}+\cdots\)
400.8.a.f 400.a 1.a $1$ $124.954$ \(\Q\) None 50.8.a.a \(0\) \(-43\) \(0\) \(-974\) $-$ $-$ $\mathrm{SU}(2)$ \(q-43q^{3}-974q^{7}-338q^{9}-87q^{11}+\cdots\)
400.8.a.g 400.a 1.a $1$ $124.954$ \(\Q\) None 40.8.a.a \(0\) \(-36\) \(0\) \(776\) $+$ $+$ $\mathrm{SU}(2)$ \(q-6^{2}q^{3}+776q^{7}-891q^{9}+124q^{11}+\cdots\)
400.8.a.h 400.a 1.a $1$ $124.954$ \(\Q\) None 40.8.c.a \(0\) \(-34\) \(0\) \(-106\) $+$ $-$ $\mathrm{SU}(2)$ \(q-34q^{3}-106q^{7}-1031q^{9}+1324q^{11}+\cdots\)
400.8.a.i 400.a 1.a $1$ $124.954$ \(\Q\) None 200.8.a.d \(0\) \(-9\) \(0\) \(694\) $+$ $-$ $\mathrm{SU}(2)$ \(q-9q^{3}+694q^{7}-2106q^{9}-4901q^{11}+\cdots\)
400.8.a.j 400.a 1.a $1$ $124.954$ \(\Q\) None 20.8.a.a \(0\) \(-6\) \(0\) \(-706\) $-$ $+$ $\mathrm{SU}(2)$ \(q-6q^{3}-706q^{7}-2151q^{9}+3840q^{11}+\cdots\)
400.8.a.k 400.a 1.a $1$ $124.954$ \(\Q\) None 200.8.a.d \(0\) \(9\) \(0\) \(-694\) $+$ $+$ $\mathrm{SU}(2)$ \(q+9q^{3}-694q^{7}-2106q^{9}-4901q^{11}+\cdots\)
400.8.a.l 400.a 1.a $1$ $124.954$ \(\Q\) None 2.8.a.a \(0\) \(12\) \(0\) \(1016\) $-$ $+$ $\mathrm{SU}(2)$ \(q+12q^{3}+1016q^{7}-2043q^{9}-1092q^{11}+\cdots\)
400.8.a.m 400.a 1.a $1$ $124.954$ \(\Q\) None 10.8.a.a \(0\) \(28\) \(0\) \(104\) $-$ $+$ $\mathrm{SU}(2)$ \(q+28q^{3}+104q^{7}-1403q^{9}+5148q^{11}+\cdots\)
400.8.a.n 400.a 1.a $1$ $124.954$ \(\Q\) None 40.8.c.a \(0\) \(34\) \(0\) \(106\) $+$ $-$ $\mathrm{SU}(2)$ \(q+34q^{3}+106q^{7}-1031q^{9}+1324q^{11}+\cdots\)
400.8.a.o 400.a 1.a $1$ $124.954$ \(\Q\) None 50.8.a.a \(0\) \(43\) \(0\) \(974\) $-$ $+$ $\mathrm{SU}(2)$ \(q+43q^{3}+974q^{7}-338q^{9}-87q^{11}+\cdots\)
400.8.a.p 400.a 1.a $1$ $124.954$ \(\Q\) None 8.8.a.b \(0\) \(44\) \(0\) \(-1224\) $+$ $+$ $\mathrm{SU}(2)$ \(q+44q^{3}-1224q^{7}-251q^{9}+3164q^{11}+\cdots\)
400.8.a.q 400.a 1.a $1$ $124.954$ \(\Q\) None 50.8.a.c \(0\) \(57\) \(0\) \(-1174\) $-$ $+$ $\mathrm{SU}(2)$ \(q+57q^{3}-1174q^{7}+1062q^{9}+7563q^{11}+\cdots\)
400.8.a.r 400.a 1.a $1$ $124.954$ \(\Q\) None 200.8.a.a \(0\) \(69\) \(0\) \(-174\) $+$ $-$ $\mathrm{SU}(2)$ \(q+69q^{3}-174q^{7}+2574q^{9}-7111q^{11}+\cdots\)
400.8.a.s 400.a 1.a $1$ $124.954$ \(\Q\) None 50.8.a.d \(0\) \(87\) \(0\) \(1366\) $-$ $-$ $\mathrm{SU}(2)$ \(q+87q^{3}+1366q^{7}+5382q^{9}+1083q^{11}+\cdots\)
400.8.a.t 400.a 1.a $2$ $124.954$ \(\Q(\sqrt{31}) \) None 10.8.b.a \(0\) \(-56\) \(0\) \(-408\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(-28+\beta )q^{3}+(-204-7\beta )q^{7}+\cdots\)
400.8.a.u 400.a 1.a $2$ $124.954$ \(\Q(\sqrt{2521}) \) None 100.8.a.b \(0\) \(-40\) \(0\) \(-280\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(-20-\beta )q^{3}+(-140-6\beta )q^{7}+\cdots\)
400.8.a.v 400.a 1.a $2$ $124.954$ \(\Q(\sqrt{649}) \) None 25.8.a.c \(0\) \(-40\) \(0\) \(600\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(-20-\beta )q^{3}+(300+42\beta )q^{7}+(-1138+\cdots)q^{9}+\cdots\)
400.8.a.w 400.a 1.a $2$ $124.954$ \(\Q(\sqrt{1129}) \) None 20.8.a.b \(0\) \(-20\) \(0\) \(1660\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(-10-\beta )q^{3}+(830+9\beta )q^{7}+(2429+\cdots)q^{9}+\cdots\)
400.8.a.x 400.a 1.a $2$ $124.954$ \(\Q(\sqrt{3889}) \) None 200.8.a.l \(0\) \(-4\) \(0\) \(-1144\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(-2-\beta )q^{3}+(-572-2\beta )q^{7}+(1706+\cdots)q^{9}+\cdots\)
400.8.a.y 400.a 1.a $2$ $124.954$ \(\Q(\sqrt{29}) \) None 5.8.b.a \(0\) \(0\) \(0\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q-3\beta q^{3}-39\beta q^{7}-1143q^{9}+6828q^{11}+\cdots\)
400.8.a.z 400.a 1.a $2$ $124.954$ \(\Q(\sqrt{46}) \) None 40.8.a.b \(0\) \(4\) \(0\) \(844\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(2+\beta )q^{3}+(422+17\beta )q^{7}+(761+\cdots)q^{9}+\cdots\)
400.8.a.ba 400.a 1.a $2$ $124.954$ \(\Q(\sqrt{3889}) \) None 200.8.a.l \(0\) \(4\) \(0\) \(1144\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(2+\beta )q^{3}+(572+2\beta )q^{7}+(1706+\cdots)q^{9}+\cdots\)
400.8.a.bb 400.a 1.a $2$ $124.954$ \(\Q(\sqrt{19}) \) None 5.8.a.b \(0\) \(20\) \(0\) \(-100\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(10+\beta )q^{3}+(-50-7\beta )q^{7}+(2777+\cdots)q^{9}+\cdots\)
400.8.a.bc 400.a 1.a $2$ $124.954$ \(\Q(\sqrt{6}) \) None 40.8.a.c \(0\) \(36\) \(0\) \(-404\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(18+\beta )q^{3}+(-202-63\beta )q^{7}+(-1479+\cdots)q^{9}+\cdots\)
400.8.a.bd 400.a 1.a $2$ $124.954$ \(\Q(\sqrt{649}) \) None 25.8.a.c \(0\) \(40\) \(0\) \(-600\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(20-\beta )q^{3}+(-300+42\beta )q^{7}+(-1138+\cdots)q^{9}+\cdots\)
400.8.a.be 400.a 1.a $2$ $124.954$ \(\Q(\sqrt{2521}) \) None 100.8.a.b \(0\) \(40\) \(0\) \(280\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(20-\beta )q^{3}+(140-6\beta )q^{7}+(734+\cdots)q^{9}+\cdots\)
400.8.a.bf 400.a 1.a $2$ $124.954$ \(\Q(\sqrt{31}) \) None 10.8.b.a \(0\) \(56\) \(0\) \(408\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(28+\beta )q^{3}+(204-7\beta )q^{7}+(1697+\cdots)q^{9}+\cdots\)
400.8.a.bg 400.a 1.a $2$ $124.954$ \(\Q(\sqrt{601}) \) None 40.8.a.d \(0\) \(76\) \(0\) \(796\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(38-\beta )q^{3}+(398-7\beta )q^{7}+(1661+\cdots)q^{9}+\cdots\)
400.8.a.bh 400.a 1.a $3$ $124.954$ 3.3.40101.1 None 200.8.a.o \(0\) \(-97\) \(0\) \(1630\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(-2^{5}-\beta _{1})q^{3}+(543+2\beta _{1}+\beta _{2})q^{7}+\cdots\)
400.8.a.bi 400.a 1.a $3$ $124.954$ 3.3.40101.1 None 200.8.a.o \(0\) \(97\) \(0\) \(-1630\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(2^{5}+\beta _{1})q^{3}+(-543-2\beta _{1}-\beta _{2})q^{7}+\cdots\)
400.8.a.bj 400.a 1.a $4$ $124.954$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None 40.8.c.b \(0\) \(-32\) \(0\) \(-1632\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(-8-\beta _{1})q^{3}+(-408+3\beta _{1}-2\beta _{2}+\cdots)q^{7}+\cdots\)
400.8.a.bk 400.a 1.a $4$ $124.954$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None 20.8.c.a \(0\) \(0\) \(0\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{3}+(6\beta _{1}+\beta _{2})q^{7}+(509+\beta _{3})q^{9}+\cdots\)
400.8.a.bl 400.a 1.a $4$ $124.954$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None 40.8.c.b \(0\) \(32\) \(0\) \(1632\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(8+\beta _{1})q^{3}+(408-3\beta _{1}+2\beta _{2}-\beta _{3})q^{7}+\cdots\)

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_0(400))\) into lower level spaces

\( S_{8}^{\mathrm{old}}(\Gamma_0(400)) \simeq \) \(S_{8}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 12}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(4))\)\(^{\oplus 9}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 10}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 6}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 8}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 3}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 6}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(25))\)\(^{\oplus 5}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(80))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(100))\)\(^{\oplus 3}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(200))\)\(^{\oplus 2}\)