LMFDB - Newform orbit 136.4.s.a

Newspace parameters

Level:	$ N $	$=$	$ 136 = 2^{3} \cdot 17 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	136.s (of order $16$, degree $8$, minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$8.02425976078$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\Q(\zeta_{16})$
Defining polynomial:	$ x^{8} + 1 $ x^8 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{16}]$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/136\mathbb{Z}\right)^\times$.

$n$	$69$	$103$	$105$
$\chi(n)$	$-1$	$-1$	$\zeta_{16}$

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} $

3.1

0.923880	+	0.382683i
−0.923880	+	0.382683i
0.382683	+	0.923880i
−0.382683	−	0.923880i
0.923880	−	0.382683i
−0.923880	−	0.382683i
−0.382683	+	0.923880i
0.382683	−	0.923880i

2.61313

−

1.08239i

6.61374

1.31555i

5.65685

−

5.65685i

18.7065

−

3.72095i

8.65914

−

20.9050i

17.0661

7.06900i

11.1

−2.61313

−

1.08239i

−1.54267

−

7.75551i

5.65685

5.65685i

−4.36332

21.9359i

−8.65914

−

20.9050i

−32.8234

13.5959i

27.1

−1.08239

2.61313i

−3.92868

−

2.62506i

−5.65685

−

5.65685i

11.1120

−

7.42479i

20.9050

−

8.65914i

−1.78887

−

4.31871i

75.1

1.08239

−

2.61313i

−5.14239

7.69613i

−5.65685

−

5.65685i

14.5449

21.7679i

−20.9050

8.65914i

−22.4538

−

54.2082i

91.1

2.61313

1.08239i

6.61374

−

1.31555i

5.65685

5.65685i

18.7065

3.72095i

8.65914

20.9050i

17.0661

−

7.06900i

99.1

−2.61313

1.08239i

−1.54267

7.75551i

5.65685

−

5.65685i

−4.36332

−

21.9359i

−8.65914

20.9050i

−32.8234

−

13.5959i

107.1

1.08239

2.61313i

−5.14239

−

7.69613i

−5.65685

5.65685i

14.5449

−

21.7679i

−20.9050

−

8.65914i

−22.4538

54.2082i

131.1

−1.08239

−

2.61313i

−3.92868

2.62506i

−5.65685

5.65685i

11.1120

7.42479i

20.9050

8.65914i

−1.78887

4.31871i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
8.d	odd	2	1	CM by $\Q(\sqrt{-2}) $
17.e	odd	16	1	inner
136.s	even	16	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	136.4.s.a	✓	8
8.d	odd	2	1	CM	136.4.s.a	✓	8
17.e	odd	16	1	inner	136.4.s.a	✓	8
136.s	even	16	1	inner	136.4.s.a	✓	8

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
136.4.s.a	✓	8	1.a	even	1	1	trivial
136.4.s.a	✓	8	8.d	odd	2	1	CM
136.4.s.a	✓	8	17.e	odd	16	1	inner
136.4.s.a	✓	8	136.s	even	16	1	inner

This newform subspace can be constructed as the kernel of the linear operator $ T_{3}^{8} + 8T_{3}^{7} + 72T_{3}^{6} - 480T_{3}^{5} - 4174T_{3}^{4} - 36832T_{3}^{3} + 45332T_{3}^{2} + 1253240T_{3} + 5438402 $ T3^8 + 8*T3^7 + 72*T3^6 - 480*T3^5 - 4174*T3^4 - 36832*T3^3 + 45332*T3^2 + 1253240*T3 + 5438402 acting on $S_{4}^{\mathrm{new}}(136, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{8} + 4096 $ T^8 + 4096
$3$	$ T^{8} + 8 T^{7} + 72 T^{6} + \cdots + 5438402 $ T^8 + 8T^7 + 72T^6 - 480T^5 - 4174T^4 - 36832T^3 + 45332T^2 + 1253240*T + 5438402
$5$	$ T^{8} $ T^8
$7$	$ T^{8} $ T^8
$11$	$ T^{8} - 6476 T^{6} + \cdots + 221297408642 $ T^8 - 6476T^6 + 266200T^5 + 19046166T^4 - 663679192T^3 + 16110661164T^2 - 31877460648T + 221297408642
$13$	$ T^{8} $ T^8
$17$	$ T^{8} + \cdots + 582622237229761 $ T^8 + 13680T^6 + 93571200T^4 + 330201943920*T^2 + 582622237229761
$19$	$ T^{8} + 38160 T^{6} + \cdots + 37949591784976 $ T^8 + 38160T^6 + 728092800T^4 - 235077963840*T^2 + 37949591784976
$23$	$ T^{8} $ T^8
$29$	$ T^{8} $ T^8
$31$	$ T^{8} $ T^8
$37$	$ T^{8} $ T^8
$41$	$ T^{8} - 160 T^{7} + \cdots + 11\!\cdots\!02 $ T^8 - 160T^7 - 28960T^6 - 4857760T^5 + 10730058082T^4 - 18044605191696T^3 + 7658082631848500T^2 - 1204385713455385768*T + 114350423123470996802
$43$	$ (T^{4} - 104 T^{3} + 80322 T^{2} + \cdots + 774053858)^{2} $ (T^4 - 104T^3 + 80322T^2 + 15502324*T + 774053858)^2
$47$	$ T^{8} $ T^8
$53$	$ T^{8} $ T^8
$59$	$ (T^{4} + 460 T^{3} + 410758 T^{2} + \cdots + 233236802)^{2} $ (T^4 + 460T^3 + 410758T^2 - 18271908*T + 233236802)^2
$61$	$ T^{8} $ T^8
$67$	$ (T^{4} + 1094692 T^{2} + \cdots + 121606351778)^{2} $ (T^4 + 1094692*T^2 + 121606351778)^2
$71$	$ T^{8} $ T^8
$73$	$ T^{8} + 3312 T^{7} + \cdots + 71\!\cdots\!82 $ T^8 + 3312T^7 + 6196752T^6 + 7519032440T^5 + 6259365182306T^4 + 2849495810064192T^3 + 712423431833963052T^2 + 101096484791183440120*T + 7162896661218455662082
$79$	$ T^{8} $ T^8
$83$	$ (T^{4} + 1736 T^{3} + \cdots + 12454523138)^{2} $ (T^4 + 1736T^3 + 827922T^2 - 60920836*T + 12454523138)^2
$89$	$ T^{8} + 11402977910412 T^{4} + \cdots + 31\!\cdots\!44 $ T^8 + 11402977910412*T^4 + 31067835242991551691747844
$97$	$ T^{8} + 3507988 T^{6} + \cdots + 18\!\cdots\!22 $ T^8 + 3507988T^6 + 131424912T^5 + 4496501769414T^4 - 5644899612142712T^3 + 2851913581902246244T^2 + 6682086919855102926680T + 1836695297646940033321922
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Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 136 = 2^{3} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	136.s (of order \(16\), degree \(8\), minimal)

\(f(q)\)	\(=\)	\( q + ( - 2 \zeta_{16}^{5} + 2 \zeta_{16}) q^{2} + ( - 5 \zeta_{16}^{7} - \zeta_{16}^{5} + \zeta_{16}^{4} + 5 \zeta_{16}^{2} - \zeta_{16} - 1) q^{3} - 8 \zeta_{16}^{6} q^{4} + ( - 10 \zeta_{16}^{7} + 4 \zeta_{16}^{5} - 10 \zeta_{16}^{4} + 10 \zeta_{16}^{3} - 4 \zeta_{16}^{2} + \cdots + 10) q^{6} + \cdots + ( - 13 \zeta_{16}^{6} + 13 \zeta_{16}^{4} - 10 \zeta_{16}^{2} + 27 \zeta_{16} - 10) q^{9} +O(q^{10}) \) q + (-2z^5 + 2z) * q^2 + (-5z^7 - z^5 + z^4 + 5z^2 - z - 1) * q^3 - 8z^6 q^4 + (-10z^7 + 4z^5 - 10z^4 + 10z^3 - 4z^2 + 10) q^6 + (-16z^7 - 16z^3) * q^8 + (-13z^6 + 13z^4 - 10z^2 + 27z - 10) * q^9	\( q + ( - 2 \zeta_{16}^{5} + 2 \zeta_{16}) q^{2} + ( - 5 \zeta_{16}^{7} - \zeta_{16}^{5} + \zeta_{16}^{4} + 5 \zeta_{16}^{2} - \zeta_{16} - 1) q^{3} - 8 \zeta_{16}^{6} q^{4} + ( - 10 \zeta_{16}^{7} + 4 \zeta_{16}^{5} - 10 \zeta_{16}^{4} + 10 \zeta_{16}^{3} - 4 \zeta_{16}^{2} + \cdots + 10) q^{6} + \cdots + (717 \zeta_{16}^{7} - 1340 \zeta_{16}^{6} + 728 \zeta_{16}^{5} + 728 \zeta_{16}^{4} + \cdots - 192) q^{99} +O(q^{100}) \) q + (-2z^5 + 2z) * q^2 + (-5z^7 - z^5 + z^4 + 5z^2 - z - 1) * q^3 - 8z^6 q^4 + (-10z^7 + 4z^5 - 10z^4 + 10z^3 - 4z^2 + 10) q^6 + (-16z^7 - 16z^3) * q^8 + (-13z^6 + 13z^4 - 10z^2 + 27z - 10) * q^9 + (25z^6 - 25z^5 + 9z^4 + 9z^3 - 25z^2 + 25z) * q^11 + (8z^7 + 8z^6 - 40z^5 - 8z^3 + 8z^2 + 40) q^12 - 64z^4 q^16 + (45z^7 + 38z^5 + 38z) q^17 + (-6z^7 - 54z^6 + 46z^5 - 46z^3 + 54z^2 + 6z) * q^18 + (90z^7 - 53z^5 + 53z) q^19 + (100z^7 - 100z^6 + 18z^5 + 18z^4 + 18z + 18) q^22 + (-80z^6 - 80z^5 + 32z^3 - 80z^2 + 80z - 32) q^24 + 125z^5 q^25 + (73z^7 + 68z^6 - 68z^5 - 73z^4 + 62z^3 - 27z^2 - 27z + 62) q^27 + (-128z^5 - 128z) * q^32 + (-116z^7 - 5z^6 + 211z^5 - 268z^4 + 211z^3 - 5z^2 - 116z) q^33 + (90z^4 + 152z^2 - 90) * q^34 + (-216z^7 + 80z^6 - 104z^4 + 104z^2 - 80) * q^36 + (-212z^6 + 180z^4 - 180) * q^38 + (-20z^7 - 20z^4 - 20z^3 - 261z^2 + 261z + 20) q^41 + (171z^6 - 171z^4 + 26z^2 + 26) q^43 + (-200z^7 + 200z^4 - 200z^3 + 72z^2 + 72z - 200) q^44 + (-320z^6 + 64z^5 + 64z^4 - 320z^3 - 64z + 64) q^48 - 343z^3 q^49 + (250z^6 + 250z^2) * q^50 + (145z^7 + 149z^6 + 235z^4 + 145z^3 - 301z + 235) q^51 + (190z^7 - 82z^6 - 270z^5 + 270z^4 + 82z^3 - 190z^2 - 22z - 22) q^54 + (-355z^7 + 450z^6 + 106z^5 - 175z^4 + 175z^3 - 106z^2 - 450z + 355) q^57 + (308z^6 + 308z^4 + 115z^2 - 115) q^59 - 512z^2 q^64 + (654z^6 - 536z^5 + 190z^4 - 20z^3 + 190z^2 - 536z + 654) * q^66 + (352z^7 + 387z^5 + 387z^3 + 352z) * q^67 + (-304z^7 + 360z^5 + 304z^3) q^68 + (-48z^7 - 48z^5 - 432z^4 + 368z^3 - 368z + 432) q^72 + (-414z^5 + 414z^4 - 215z^3 - 215z^2 + 414z - 414) q^73 + (625z^7 - 125z^6 - 125z^5 + 625z^4 + 125z^2 - 125z) * q^75 + (-424z^7 + 720z^5 - 424z^3) q^76 + (-351z^7 + 351z^5 - 329z^4 - 270z^3 - 191z^2 - 270z - 329) * q^81 + (522z^7 - 522z^6 - 80z^5 - 80z^4 - 522z^3 + 522z^2) * q^82 + (-434z^6 + 241z^4 + 241z^2 - 434) q^83 + (290z^7 - 394z^5 + 394z^3 - 290z) * q^86 + (-144z^7 - 144z^6 + 800z^5 - 800z^4 + 144z^3 + 144z^2) * q^88 + (-983z^7 - 983z^5 + 470z^3 - 470z) * q^89 + (-640z^7 + 256z^6 - 640z^4 - 640z^3 + 256z - 640) q^96 + (-18z^7 - 18z^6 - 955z^4 - 18z^3 + 18z^2 + 955z) * q^97 + (-686z^4 - 686) q^98 + (717z^7 - 1340z^6 + 728z^5 + 728z^4 - 1340z^3 + 717z^2 + 192z - 192) q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{3} + 80 q^{6} - 80 q^{9}+O(q^{10}) \) 8 * q - 8 * q^3 + 80 * q^6 - 80 * q^9	\( 8 q - 8 q^{3} + 80 q^{6} - 80 q^{9} + 320 q^{12} + 144 q^{22} - 256 q^{24} + 496 q^{27} - 720 q^{34} - 640 q^{36} - 1440 q^{38} + 160 q^{41} + 208 q^{43} - 1600 q^{44} + 512 q^{48} + 1880 q^{51} - 176 q^{54} + 2840 q^{57} - 920 q^{59} + 5232 q^{66} + 3456 q^{72} - 3312 q^{73} - 2632 q^{81} - 3472 q^{83} - 5120 q^{96} - 5488 q^{98} - 1536 q^{99}+O(q^{100}) \) 8 * q - 8 * q^3 + 80 * q^6 - 80 * q^9 + 320 * q^12 + 144 * q^22 - 256 * q^24 + 496 * q^27 - 720 * q^34 - 640 * q^36 - 1440 * q^38 + 160 * q^41 + 208 * q^43 - 1600 * q^44 + 512 * q^48 + 1880 * q^51 - 176 * q^54 + 2840 * q^57 - 920 * q^59 + 5232 * q^66 + 3456 * q^72 - 3312 * q^73 - 2632 * q^81 - 3472 * q^83 - 5120 * q^96 - 5488 * q^98 - 1536 * q^99

Introduction

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Newspace parameters

Newform invariants

$q$-expansion

Character values

Embeddings

Inner twists

Twists

Hecke kernels

Hecke characteristic polynomials

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	136.4.s.a

Level	$136$
Weight	$4$
Character orbit	136.s
Analytic conductor	$8.024$
Analytic rank	$0$
Dimension	$8$
CM discriminant	-8
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(8.02425976078\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Coefficient field:	\(\Q(\zeta_{16})\)
Defining polynomial:	\( x^{8} + 1 \) x^8 + 1
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{16}]$

\(n\)	\(69\)	\(103\)	\(105\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(\zeta_{16}\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 131.1
Significant digits:
Format:

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
8.d	odd	2	1	CM by \(\Q(\sqrt{-2}) \)
17.e	odd	16	1	inner
136.s	even	16	1	inner

$p$	$F_p(T)$
$2$	\( T^{8} + 4096 \) T^8 + 4096
$3$	\( T^{8} + 8 T^{7} + 72 T^{6} + \cdots + 5438402 \) T^8 + 8T^7 + 72T^6 - 480T^5 - 4174T^4 - 36832T^3 + 45332T^2 + 1253240*T + 5438402
$5$	\( T^{8} \) T^8
$7$	\( T^{8} \) T^8
$11$	\( T^{8} - 6476 T^{6} + \cdots + 221297408642 \) T^8 - 6476T^6 + 266200T^5 + 19046166T^4 - 663679192T^3 + 16110661164T^2 - 31877460648T + 221297408642
$13$	\( T^{8} \) T^8
$17$	\( T^{8} + \cdots + 582622237229761 \) T^8 + 13680T^6 + 93571200T^4 + 330201943920*T^2 + 582622237229761
$19$	\( T^{8} + 38160 T^{6} + \cdots + 37949591784976 \) T^8 + 38160T^6 + 728092800T^4 - 235077963840*T^2 + 37949591784976
$23$	\( T^{8} \) T^8
$29$	\( T^{8} \) T^8
$31$	\( T^{8} \) T^8
$37$	\( T^{8} \) T^8
$41$	\( T^{8} - 160 T^{7} + \cdots + 11\!\cdots\!02 \) T^8 - 160T^7 - 28960T^6 - 4857760T^5 + 10730058082T^4 - 18044605191696T^3 + 7658082631848500T^2 - 1204385713455385768*T + 114350423123470996802
$43$	\( (T^{4} - 104 T^{3} + 80322 T^{2} + \cdots + 774053858)^{2} \) (T^4 - 104T^3 + 80322T^2 + 15502324*T + 774053858)^2
$47$	\( T^{8} \) T^8
$53$	\( T^{8} \) T^8
$59$	\( (T^{4} + 460 T^{3} + 410758 T^{2} + \cdots + 233236802)^{2} \) (T^4 + 460T^3 + 410758T^2 - 18271908*T + 233236802)^2
$61$	\( T^{8} \) T^8
$67$	\( (T^{4} + 1094692 T^{2} + \cdots + 121606351778)^{2} \) (T^4 + 1094692*T^2 + 121606351778)^2
$71$	\( T^{8} \) T^8
$73$	\( T^{8} + 3312 T^{7} + \cdots + 71\!\cdots\!82 \) T^8 + 3312T^7 + 6196752T^6 + 7519032440T^5 + 6259365182306T^4 + 2849495810064192T^3 + 712423431833963052T^2 + 101096484791183440120*T + 7162896661218455662082
$79$	\( T^{8} \) T^8
$83$	\( (T^{4} + 1736 T^{3} + \cdots + 12454523138)^{2} \) (T^4 + 1736T^3 + 827922T^2 - 60920836*T + 12454523138)^2
$89$	\( T^{8} + 11402977910412 T^{4} + \cdots + 31\!\cdots\!44 \) T^8 + 11402977910412*T^4 + 31067835242991551691747844
$97$	\( T^{8} + 3507988 T^{6} + \cdots + 18\!\cdots\!22 \) T^8 + 3507988T^6 + 131424912T^5 + 4496501769414T^4 - 5644899612142712T^3 + 2851913581902246244T^2 + 6682086919855102926680T + 1836695297646940033321922
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