Newform orbit 234.8.a.j

• Label: 234.8.a.j
• Level: $234$
• Weight: $8$
• Character orbit: 234.a
• Self dual: yes
• Analytic conductor: $73.098$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	$N$	$=$	$234 = 2 \cdot 3^{2} \cdot 13$
Weight:	$k$	$=$	$8$
Character orbit:	$[\chi]$	$=$	234.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	$73.0980959633$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{114})$
Defining polynomial:	$x^{2} - 114$
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$2\cdot 3$
Twist minimal:	no (minimal twist has level 78)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the $q$-expansion are expressed in terms of $\beta = 6\sqrt{114}$. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	q + 8 * q^2 + 64 * q^4 + (5*b - 200) * q^5 + (-9*b + 1204) * q^7 + 512 * q^8 + (40*b - 1600) * q^10 + (98*b + 662) * q^11 - 2197 * q^13 + (-72*b + 9632) * q^14 + 4096 * q^16 + (-106*b + 8406) * q^17 + (73*b + 1524) * q^19 + (320*b - 12800) * q^20 + (784*b + 5296) * q^22 + (720*b - 7412) * q^23 + (-2000*b + 64475) * q^25 - 17576 * q^26 + (-576*b + 77056) * q^28 + (-2660*b - 3050) * q^29 + (2307*b + 19528) * q^31 + 32768 * q^32 + (-848*b + 67248) * q^34 + (7820*b - 425480) * q^35 + (2182*b + 228822) * q^37 + (584*b + 12192) * q^38 + (2560*b - 102400) * q^40 + (-1523*b - 368256) * q^41 + (7418*b + 372916) * q^43 + (6272*b + 42368) * q^44 + (5760*b - 59296) * q^46 + (-14100*b + 317834) * q^47 + (-21672*b + 958497) * q^49 + (-16000*b + 515800) * q^50 - 140608 * q^52 + (-1300*b + 474978) * q^53 + (-16290*b + 1878560) * q^55 + (-4608*b + 616448) * q^56 + (-21280*b - 24400) * q^58 + (5024*b + 812722) * q^59 + (7852*b + 977338) * q^61 + (18456*b + 156224) * q^62 + 262144 * q^64 + (-10985*b + 439400) * q^65 + (11995*b + 1771576) * q^67 + (-6784*b + 537984) * q^68 + (62560*b - 3403840) * q^70 + (-62378*b + 1677674) * q^71 + (-57266*b + 1496534) * q^73 + (17456*b + 1830576) * q^74 + (4672*b + 97536) * q^76 + (112034*b - 2822680) * q^77 + (38308*b - 2298136) * q^79 + (20480*b - 819200) * q^80 + (-12184*b - 2946048) * q^82 + (924*b + 9924270) * q^83 + (63230*b - 3856320) * q^85 + (59344*b + 2983328) * q^86 + (50176*b + 338944) * q^88 + (3059*b - 3377508) * q^89 + (19773*b - 2645188) * q^91 + (46080*b - 474368) * q^92 + (-112800*b + 2542672) * q^94 + (-6980*b + 1193160) * q^95 + (38442*b + 10288862) * q^97 + (-173376*b + 7667976) * q^98
$\operatorname{Tr}(f)(q)$	$=$	2 * q + 16 * q^2 + 128 * q^4 - 400 * q^5 + 2408 * q^7 + 1024 * q^8 - 3200 * q^10 + 1324 * q^11 - 4394 * q^13 + 19264 * q^14 + 8192 * q^16 + 16812 * q^17 + 3048 * q^19 - 25600 * q^20 + 10592 * q^22 - 14824 * q^23 + 128950 * q^25 - 35152 * q^26 + 154112 * q^28 - 6100 * q^29 + 39056 * q^31 + 65536 * q^32 + 134496 * q^34 - 850960 * q^35 + 457644 * q^37 + 24384 * q^38 - 204800 * q^40 - 736512 * q^41 + 745832 * q^43 + 84736 * q^44 - 118592 * q^46 + 635668 * q^47 + 1916994 * q^49 + 1031600 * q^50 - 281216 * q^52 + 949956 * q^53 + 3757120 * q^55 + 1232896 * q^56 - 48800 * q^58 + 1625444 * q^59 + 1954676 * q^61 + 312448 * q^62 + 524288 * q^64 + 878800 * q^65 + 3543152 * q^67 + 1075968 * q^68 - 6807680 * q^70 + 3355348 * q^71 + 2993068 * q^73 + 3661152 * q^74 + 195072 * q^76 - 5645360 * q^77 - 4596272 * q^79 - 1638400 * q^80 - 5892096 * q^82 + 19848540 * q^83 - 7712640 * q^85 + 5966656 * q^86 + 677888 * q^88 - 6755016 * q^89 - 5290376 * q^91 - 948736 * q^92 + 5085344 * q^94 + 2386320 * q^95 + 20577724 * q^97 + 15335952 * q^98

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  

$\iota_m(\nu)$

$a_{2}$

$a_{3}$

$a_{4}$

$a_{5}$

$a_{6}$

$a_{7}$

$a_{8}$

$a_{9}$

$a_{10}$

1.1

−10.6771
10.6771

8.00000

0

64.0000

−520.312

0

1780.56

512.000

0

−4162.50

1.2

8.00000

0

64.0000

120.312

0

627.438

512.000

0

962.499

Atkin-Lehner signs

$p$	Sign
$2$	$-1$
$3$	$-1$
$13$	$+1$

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	234.8.a.j		2
3.b	odd	2	1		78.8.a.c	✓	2
12.b	even	2	1		624.8.a.k		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
78.8.a.c	✓	2	3.b	odd	2	1
234.8.a.j		2	1.a	even	1	1	trivial
624.8.a.k		2	12.b	even	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{5}^{2} + 400T_{5} - 62600$ acting on $S_{8}^{\mathrm{new}}(\Gamma_0(234))$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$(T - 8)^{2}$
$3$	$T^{2}$
$5$	$T^{2} + 400T - 62600$
$7$	$T^{2} - 2408 T + 1117192$
$11$	$T^{2} - 1324 T - 38976572$