Properties

Label 8.8.b.a
Level $8$
Weight $8$
Character orbit 8.b
Analytic conductor $2.499$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 8 = 2^{3} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 8.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.49908020387\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Defining polynomial: \( x^{6} - 3x^{5} - 10x^{4} - 24x^{3} - 320x^{2} - 3072x + 32768 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{15} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 + 1) q^{2} + \beta_{4} q^{3} + (\beta_{3} + \beta_1 + 19) q^{4} + ( - \beta_{5} - \beta_{4} + \beta_{3} + 2 \beta_1) q^{5} + (2 \beta_{5} + 6 \beta_{4} - \beta_{2} + 2 \beta_1 + 44) q^{6} + ( - 2 \beta_{5} + 2 \beta_{4} - 6 \beta_{3} + 2 \beta_{2} + 16 \beta_1 - 112) q^{7} + (4 \beta_{5} - 20 \beta_{4} + 2 \beta_{3} + 2 \beta_{2} + 10 \beta_1 + 250) q^{8} + ( - 4 \beta_{5} + 4 \beta_{4} - 12 \beta_{3} - 4 \beta_{2} - 112 \beta_1 - 481) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 + 1) q^{2} + \beta_{4} q^{3} + (\beta_{3} + \beta_1 + 19) q^{4} + ( - \beta_{5} - \beta_{4} + \beta_{3} + 2 \beta_1) q^{5} + (2 \beta_{5} + 6 \beta_{4} - \beta_{2} + 2 \beta_1 + 44) q^{6} + ( - 2 \beta_{5} + 2 \beta_{4} - 6 \beta_{3} + 2 \beta_{2} + 16 \beta_1 - 112) q^{7} + (4 \beta_{5} - 20 \beta_{4} + 2 \beta_{3} + 2 \beta_{2} + 10 \beta_1 + 250) q^{8} + ( - 4 \beta_{5} + 4 \beta_{4} - 12 \beta_{3} - 4 \beta_{2} - 112 \beta_1 - 481) q^{9} + (4 \beta_{5} - 52 \beta_{4} + 8 \beta_{3} - 10 \beta_{2} - 12 \beta_1 - 280) q^{10} + ( - 8 \beta_{5} - 11 \beta_{4} + 8 \beta_{3} + 16 \beta_{2} - 208 \beta_1) q^{11} + (8 \beta_{5} + 88 \beta_{4} - 6 \beta_{3} + 20 \beta_{2} + 74 \beta_1 - 682) q^{12} + (5 \beta_{5} + 37 \beta_{4} - 5 \beta_{3} - 32 \beta_{2} + 438 \beta_1) q^{13} + ( - 16 \beta_{5} + 80 \beta_{4} + 16 \beta_{3} - 40 \beta_{2} + \cdots + 2008) q^{14}+ \cdots + (7488 \beta_{5} + 54531 \beta_{4} - 7488 \beta_{3} + \cdots - 289152 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 6 q^{2} + 116 q^{4} + 268 q^{6} - 688 q^{7} + 1512 q^{8} - 2918 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 6 q^{2} + 116 q^{4} + 268 q^{6} - 688 q^{7} + 1512 q^{8} - 2918 q^{9} - 1656 q^{10} - 4088 q^{12} + 12048 q^{14} + 17872 q^{15} + 35344 q^{16} + 1452 q^{17} - 89062 q^{18} - 114768 q^{20} + 152860 q^{22} - 1296 q^{23} + 282512 q^{24} - 39314 q^{25} - 316968 q^{26} - 480800 q^{28} + 821648 q^{30} - 89280 q^{31} + 817056 q^{32} + 53880 q^{33} - 1009108 q^{34} - 1253556 q^{36} + 974124 q^{38} - 328208 q^{39} + 954464 q^{40} + 521244 q^{41} - 1093088 q^{42} - 1096344 q^{44} + 929840 q^{46} + 1566432 q^{47} + 853920 q^{48} - 511050 q^{49} - 148626 q^{50} + 823952 q^{52} - 1077064 q^{54} - 3270256 q^{55} - 2468928 q^{56} - 1889896 q^{57} + 3130744 q^{58} + 5715168 q^{60} - 7055808 q^{62} + 5776816 q^{63} - 4792768 q^{64} + 1416480 q^{65} + 7926264 q^{66} + 6608040 q^{68} - 7406912 q^{70} - 7597104 q^{71} - 11363944 q^{72} + 2089564 q^{73} + 7744200 q^{74} + 9241288 q^{76} - 9471184 q^{78} + 16015904 q^{79} - 12600384 q^{80} - 723058 q^{81} + 10715932 q^{82} + 4220608 q^{84} - 5639076 q^{86} - 37453776 q^{87} + 1541200 q^{88} + 2169084 q^{89} - 121864 q^{90} + 669600 q^{92} + 15503712 q^{94} + 48537936 q^{95} + 21402176 q^{96} - 1088308 q^{97} - 14983242 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 3x^{5} - 10x^{4} - 24x^{3} - 320x^{2} - 3072x + 32768 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{5} + 3\nu^{4} + 10\nu^{3} + 24\nu^{2} + 320\nu + 2560 ) / 512 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{5} - 3\nu^{4} - 10\nu^{3} - 24\nu^{2} + 7872\nu - 6656 ) / 256 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -5\nu^{5} - 49\nu^{4} + 242\nu^{3} + 760\nu^{2} + 3136\nu + 26624 ) / 512 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 3\nu^{5} + 7\nu^{4} + 50\nu^{3} - 104\nu^{2} - 1088\nu - 15872 ) / 256 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 15\nu^{5} + 51\nu^{4} - 182\nu^{3} + 3032\nu^{2} - 2496\nu - 90112 ) / 512 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + 2\beta _1 + 16 ) / 32 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 4\beta_{5} - 4\beta_{4} + 4\beta_{3} - \beta_{2} + 14\beta _1 + 152 ) / 32 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -4\beta_{5} + 68\beta_{4} + 28\beta_{3} - \beta_{2} + 206\beta _1 + 1000 ) / 32 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 28\beta_{5} + 164\beta_{4} - 132\beta_{3} + 11\beta_{2} + 2086\beta _1 + 11816 ) / 32 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 140\beta_{5} + 1076\beta_{4} - 20\beta_{3} + 319\beta_{2} - 7090\beta _1 + 136136 ) / 32 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/8\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(7\)
\(\chi(n)\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1
−4.85268 + 2.90715i
−4.85268 2.90715i
0.776001 + 5.60338i
0.776001 5.60338i
5.57668 + 0.949035i
5.57668 0.949035i
−9.70536 5.81430i 40.2163i 60.3879 + 112.860i 324.492i 233.829 390.313i −956.960 70.1132 1446.46i 569.651 1886.69 3149.31i
5.2 −9.70536 + 5.81430i 40.2163i 60.3879 112.860i 324.492i 233.829 + 390.313i −956.960 70.1132 + 1446.46i 569.651 1886.69 + 3149.31i
5.3 1.55200 11.2068i 21.9408i −123.183 34.7858i 184.916i −245.885 34.0522i 1051.96 −581.015 + 1326.49i 1705.60 −2072.30 286.989i
5.4 1.55200 + 11.2068i 21.9408i −123.183 + 34.7858i 184.916i −245.885 + 34.0522i 1051.96 −581.015 1326.49i 1705.60 −2072.30 + 286.989i
5.5 11.1534 1.89807i 76.9497i 120.795 42.3397i 338.443i 146.056 + 858.247i −438.996 1266.90 701.506i −3734.25 −642.387 3774.77i
5.6 11.1534 + 1.89807i 76.9497i 120.795 + 42.3397i 338.443i 146.056 858.247i −438.996 1266.90 + 701.506i −3734.25 −642.387 + 3774.77i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.6
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8.8.b.a 6
3.b odd 2 1 72.8.d.b 6
4.b odd 2 1 32.8.b.a 6
8.b even 2 1 inner 8.8.b.a 6
8.d odd 2 1 32.8.b.a 6
12.b even 2 1 288.8.d.b 6
16.e even 4 2 256.8.a.r 6
16.f odd 4 2 256.8.a.q 6
24.f even 2 1 288.8.d.b 6
24.h odd 2 1 72.8.d.b 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.8.b.a 6 1.a even 1 1 trivial
8.8.b.a 6 8.b even 2 1 inner
32.8.b.a 6 4.b odd 2 1
32.8.b.a 6 8.d odd 2 1
72.8.d.b 6 3.b odd 2 1
72.8.d.b 6 24.h odd 2 1
256.8.a.q 6 16.f odd 4 2
256.8.a.r 6 16.e even 4 2
288.8.d.b 6 12.b even 2 1
288.8.d.b 6 24.f even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{8}^{\mathrm{new}}(8, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} - 6 T^{5} - 40 T^{4} + \cdots + 2097152 \) Copy content Toggle raw display
$3$ \( T^{6} + 8020 T^{4} + \cdots + 4610229696 \) Copy content Toggle raw display
$5$ \( T^{6} + \cdots + 412405245440000 \) Copy content Toggle raw display
$7$ \( (T^{3} + 344 T^{2} - 1048384 T - 441929216)^{2} \) Copy content Toggle raw display
$11$ \( T^{6} + 52294004 T^{4} + \cdots + 21\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( T^{6} + 171080144 T^{4} + \cdots + 24\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( (T^{3} - 726 T^{2} + \cdots - 9112197964104)^{2} \) Copy content Toggle raw display
$19$ \( T^{6} + 3360814100 T^{4} + \cdots + 47\!\cdots\!04 \) Copy content Toggle raw display
$23$ \( (T^{3} + 648 T^{2} + \cdots + 2134822184448)^{2} \) Copy content Toggle raw display
$29$ \( T^{6} + 55662621776 T^{4} + \cdots + 42\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{3} + 44640 T^{2} + \cdots + 18\!\cdots\!28)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} + 490654094672 T^{4} + \cdots + 61\!\cdots\!64 \) Copy content Toggle raw display
$41$ \( (T^{3} - 260622 T^{2} + \cdots + 17\!\cdots\!00)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} + 124911737588 T^{4} + \cdots + 77\!\cdots\!56 \) Copy content Toggle raw display
$47$ \( (T^{3} - 783216 T^{2} + \cdots + 15\!\cdots\!96)^{2} \) Copy content Toggle raw display
$53$ \( T^{6} + 3916631783120 T^{4} + \cdots + 12\!\cdots\!36 \) Copy content Toggle raw display
$59$ \( T^{6} + 6619585104052 T^{4} + \cdots + 55\!\cdots\!04 \) Copy content Toggle raw display
$61$ \( T^{6} + 4505952081744 T^{4} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{6} + 1291377394260 T^{4} + \cdots + 75\!\cdots\!24 \) Copy content Toggle raw display
$71$ \( (T^{3} + 3798552 T^{2} + \cdots + 38\!\cdots\!92)^{2} \) Copy content Toggle raw display
$73$ \( (T^{3} - 1044782 T^{2} + \cdots + 21\!\cdots\!72)^{2} \) Copy content Toggle raw display
$79$ \( (T^{3} - 8007952 T^{2} + \cdots + 49\!\cdots\!40)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + 37884069033748 T^{4} + \cdots + 63\!\cdots\!16 \) Copy content Toggle raw display
$89$ \( (T^{3} - 1084542 T^{2} + \cdots + 11\!\cdots\!20)^{2} \) Copy content Toggle raw display
$97$ \( (T^{3} + 544154 T^{2} + \cdots - 16\!\cdots\!24)^{2} \) Copy content Toggle raw display
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