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 $2$

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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: \( x^{2} + 1 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q - 12 q^{3} + 27 \beta q^{5} + 44 \beta q^{7} - 99 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 12 q^{3} + 27 \beta q^{5} + 44 \beta q^{7} - 99 q^{9} - 270 \beta q^{11} - 324 \beta q^{15} - 594 q^{17} + 418 \beta q^{19} - 528 \beta q^{21} + 4104 q^{23} + 209 q^{25} + 4104 q^{27} - 594 q^{29} + 2128 \beta q^{31} + 3240 \beta q^{33} - 4752 q^{35} + 149 \beta q^{37} + 8613 \beta q^{41} + 12100 q^{43} - 2673 \beta q^{45} + 648 \beta q^{47} + 9063 q^{49} + 7128 q^{51} + 19494 q^{53} + 29160 q^{55} - 5016 \beta q^{57} + 3834 \beta q^{59} - 34738 q^{61} - 4356 \beta q^{63} + 10906 \beta q^{67} - 49248 q^{69} - 23436 \beta q^{71} - 33781 \beta q^{73} - 2508 q^{75} + 47520 q^{77} - 76912 q^{79} - 25191 q^{81} + 33858 \beta q^{83} - 16038 \beta q^{85} + 7128 q^{87} - 14877 \beta q^{89} - 25536 \beta q^{93} - 45144 q^{95} - 61199 \beta q^{97} + 26730 \beta q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 24 q^{3} - 198 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 24 q^{3} - 198 q^{9} - 1188 q^{17} + 8208 q^{23} + 418 q^{25} + 8208 q^{27} - 1188 q^{29} - 9504 q^{35} + 24200 q^{43} + 18126 q^{49} + 14256 q^{51} + 38988 q^{53} + 58320 q^{55} - 69476 q^{61} - 98496 q^{69} - 5016 q^{75} + 95040 q^{77} - 153824 q^{79} - 50382 q^{81} + 14256 q^{87} - 90288 q^{95}+O(q^{100}) \) Copy content Toggle raw display

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 \(S_{6}^{\mathrm{new}}(676, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

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