Properties

Label 676.6.d.a
Level $676$
Weight $6$
Character orbit 676.d
Analytic conductor $108.419$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 676 = 2^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 676.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(108.419462194\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 4)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -12 q^{3} + 54 i q^{5} + 88 i q^{7} -99 q^{9} +O(q^{10})\) \( q -12 q^{3} + 54 i q^{5} + 88 i q^{7} -99 q^{9} -540 i q^{11} -648 i q^{15} -594 q^{17} + 836 i q^{19} -1056 i q^{21} + 4104 q^{23} + 209 q^{25} + 4104 q^{27} -594 q^{29} + 4256 i q^{31} + 6480 i q^{33} -4752 q^{35} + 298 i q^{37} + 17226 i q^{41} + 12100 q^{43} -5346 i q^{45} + 1296 i q^{47} + 9063 q^{49} + 7128 q^{51} + 19494 q^{53} + 29160 q^{55} -10032 i q^{57} + 7668 i q^{59} -34738 q^{61} -8712 i q^{63} + 21812 i q^{67} -49248 q^{69} -46872 i q^{71} -67562 i q^{73} -2508 q^{75} + 47520 q^{77} -76912 q^{79} -25191 q^{81} + 67716 i q^{83} -32076 i q^{85} + 7128 q^{87} -29754 i q^{89} -51072 i q^{93} -45144 q^{95} -122398 i q^{97} + 53460 i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 24 q^{3} - 198 q^{9} + O(q^{10}) \) \( 2 q - 24 q^{3} - 198 q^{9} - 1188 q^{17} + 8208 q^{23} + 418 q^{25} + 8208 q^{27} - 1188 q^{29} - 9504 q^{35} + 24200 q^{43} + 18126 q^{49} + 14256 q^{51} + 38988 q^{53} + 58320 q^{55} - 69476 q^{61} - 98496 q^{69} - 5016 q^{75} + 95040 q^{77} - 153824 q^{79} - 50382 q^{81} + 14256 q^{87} - 90288 q^{95} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/676\mathbb{Z}\right)^\times\).

\(n\) \(339\) \(509\)
\(\chi(n)\) \(1\) \(-1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
337.1
1.00000i
1.00000i
0 −12.0000 0 54.0000i 0 88.0000i 0 −99.0000 0
337.2 0 −12.0000 0 54.0000i 0 88.0000i 0 −99.0000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 676.6.d.a 2
13.b even 2 1 inner 676.6.d.a 2
13.d odd 4 1 4.6.a.a 1
13.d odd 4 1 676.6.a.a 1
39.f even 4 1 36.6.a.a 1
52.f even 4 1 16.6.a.b 1
65.f even 4 1 100.6.c.b 2
65.g odd 4 1 100.6.a.b 1
65.k even 4 1 100.6.c.b 2
91.i even 4 1 196.6.a.e 1
91.z odd 12 2 196.6.e.g 2
91.bb even 12 2 196.6.e.d 2
104.j odd 4 1 64.6.a.f 1
104.m even 4 1 64.6.a.b 1
117.y odd 12 2 324.6.e.a 2
117.z even 12 2 324.6.e.d 2
143.g even 4 1 484.6.a.a 1
156.l odd 4 1 144.6.a.c 1
195.j odd 4 1 900.6.d.a 2
195.n even 4 1 900.6.a.h 1
195.u odd 4 1 900.6.d.a 2
208.l even 4 1 256.6.b.c 2
208.m odd 4 1 256.6.b.g 2
208.r odd 4 1 256.6.b.g 2
208.s even 4 1 256.6.b.c 2
260.l odd 4 1 400.6.c.f 2
260.s odd 4 1 400.6.c.f 2
260.u even 4 1 400.6.a.d 1
312.w odd 4 1 576.6.a.bd 1
312.y even 4 1 576.6.a.bc 1
364.p odd 4 1 784.6.a.d 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4.6.a.a 1 13.d odd 4 1
16.6.a.b 1 52.f even 4 1
36.6.a.a 1 39.f even 4 1
64.6.a.b 1 104.m even 4 1
64.6.a.f 1 104.j odd 4 1
100.6.a.b 1 65.g odd 4 1
100.6.c.b 2 65.f even 4 1
100.6.c.b 2 65.k even 4 1
144.6.a.c 1 156.l odd 4 1
196.6.a.e 1 91.i even 4 1
196.6.e.d 2 91.bb even 12 2
196.6.e.g 2 91.z odd 12 2
256.6.b.c 2 208.l even 4 1
256.6.b.c 2 208.s even 4 1
256.6.b.g 2 208.m odd 4 1
256.6.b.g 2 208.r odd 4 1
324.6.e.a 2 117.y odd 12 2
324.6.e.d 2 117.z even 12 2
400.6.a.d 1 260.u even 4 1
400.6.c.f 2 260.l odd 4 1
400.6.c.f 2 260.s odd 4 1
484.6.a.a 1 143.g even 4 1
576.6.a.bc 1 312.y even 4 1
576.6.a.bd 1 312.w odd 4 1
676.6.a.a 1 13.d odd 4 1
676.6.d.a 2 1.a even 1 1 trivial
676.6.d.a 2 13.b even 2 1 inner
784.6.a.d 1 364.p odd 4 1
900.6.a.h 1 195.n even 4 1
900.6.d.a 2 195.j odd 4 1
900.6.d.a 2 195.u odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} + 12 \) acting on \(S_{6}^{\mathrm{new}}(676, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( ( 12 + T )^{2} \)
$5$ \( 2916 + T^{2} \)
$7$ \( 7744 + T^{2} \)
$11$ \( 291600 + T^{2} \)
$13$ \( T^{2} \)
$17$ \( ( 594 + T )^{2} \)
$19$ \( 698896 + T^{2} \)
$23$ \( ( -4104 + T )^{2} \)
$29$ \( ( 594 + T )^{2} \)
$31$ \( 18113536 + T^{2} \)
$37$ \( 88804 + T^{2} \)
$41$ \( 296735076 + T^{2} \)
$43$ \( ( -12100 + T )^{2} \)
$47$ \( 1679616 + T^{2} \)
$53$ \( ( -19494 + T )^{2} \)
$59$ \( 58798224 + T^{2} \)
$61$ \( ( 34738 + T )^{2} \)
$67$ \( 475763344 + T^{2} \)
$71$ \( 2196984384 + T^{2} \)
$73$ \( 4564623844 + T^{2} \)
$79$ \( ( 76912 + T )^{2} \)
$83$ \( 4585456656 + T^{2} \)
$89$ \( 885300516 + T^{2} \)
$97$ \( 14981270404 + T^{2} \)
show more
show less