Newform orbit 338.8.a.e

• Label: 338.8.a.e
• Level: $338$
• Weight: $8$
• Character orbit: 338.a
• Self dual: yes
• Analytic conductor: $105.586$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	$N$	$=$	$338 = 2 \cdot 13^{2}$
Weight:	$k$	$=$	$8$
Character orbit:	$[\chi]$	$=$	338.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	$105.586138614$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{2305})$
Defining polynomial:	$x^{2} - x - 576$
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 26)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the $q$-expansion are expressed in terms of $\beta = \frac{1}{2}(1 + \sqrt{2305})$. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	q - 8 * q^2 + (-b + 44) * q^3 + 64 * q^4 + (5*b - 110) * q^5 + (8*b - 352) * q^6 + (-49*b - 328) * q^7 - 512 * q^8 + (-87*b + 325) * q^9 + (-40*b + 880) * q^10 + (-190*b - 212) * q^11 + (-64*b + 2816) * q^12 + (392*b + 2624) * q^14 + (325*b - 7720) * q^15 + 4096 * q^16 + (-1157*b + 3890) * q^17 + (696*b - 2600) * q^18 + (-1738*b - 1900) * q^19 + (320*b - 7040) * q^20 + (-1779*b + 13792) * q^21 + (1520*b + 1696) * q^22 + (-2088*b - 19224) * q^23 + (512*b - 22528) * q^24 + (-1075*b - 51625) * q^25 + (-1879*b - 31816) * q^27 + (-3136*b - 20992) * q^28 + (2952*b - 149610) * q^29 + (-2600*b + 61760) * q^30 + (4716*b + 132592) * q^31 - 32768 * q^32 + (-7958*b + 100112) * q^33 + (9256*b - 31120) * q^34 + (3505*b - 105040) * q^35 + (-5568*b + 20800) * q^36 + (-2407*b + 179570) * q^37 + (13904*b + 15200) * q^38 + (-2560*b + 56320) * q^40 + (15002*b - 112346) * q^41 + (14232*b - 110336) * q^42 + (5593*b + 425380) * q^43 + (-12160*b - 13568) * q^44 + (10760*b - 286310) * q^45 + (16704*b + 153792) * q^46 + (-3901*b - 887840) * q^47 + (-4096*b + 180224) * q^48 + (34545*b + 667017) * q^49 + (8600*b + 413000) * q^50 + (-53641*b + 837592) * q^51 + (-17918*b + 817406) * q^53 + (15032*b + 254528) * q^54 + (18890*b - 523880) * q^55 + (25088*b + 167936) * q^56 + (-72834*b + 917488) * q^57 + (-23616*b + 1196880) * q^58 + (-33578*b + 980060) * q^59 + (20800*b - 494080) * q^60 + (-79118*b - 263882) * q^61 + (-37728*b - 1060736) * q^62 + (16874*b + 2348888) * q^63 + 262144 * q^64 + (63664*b - 800896) * q^66 + (-102558*b + 1600324) * q^67 + (-74048*b + 248960) * q^68 + (-70560*b + 356832) * q^69 + (-28040*b + 840320) * q^70 + (-42755*b + 131480) * q^71 + (44544*b - 166400) * q^72 + (62280*b - 4327418) * q^73 + (19256*b - 1436560) * q^74 + (5400*b - 1652300) * q^75 + (-111232*b - 121600) * q^76 + (82018*b + 5432096) * q^77 + (-174168*b - 685360) * q^79 + (20480*b - 450560) * q^80 + (141288*b - 1028375) * q^81 + (-120016*b + 898768) * q^82 + (251496*b + 1662612) * q^83 + (-113856*b + 882688) * q^84 + (140935*b - 3760060) * q^85 + (-44744*b - 3403040) * q^86 + (276546*b - 8283192) * q^87 + (97280*b + 108544) * q^88 + (-194272*b + 1767190) * q^89 + (-86080*b + 2290480) * q^90 + (-133632*b - 1230336) * q^92 + (70196*b + 3117632) * q^93 + (31208*b + 7102720) * q^94 + (172990*b - 4796440) * q^95 + (32768*b - 1441792) * q^96 + (228332*b - 11833570) * q^97 + (-276360*b - 5336136) * q^98 + (-26776*b + 9452380) * q^99
$\operatorname{Tr}(f)(q)$	$=$	2 * q - 16 * q^2 + 87 * q^3 + 128 * q^4 - 215 * q^5 - 696 * q^6 - 705 * q^7 - 1024 * q^8 + 563 * q^9 + 1720 * q^10 - 614 * q^11 + 5568 * q^12 + 5640 * q^14 - 15115 * q^15 + 8192 * q^16 + 6623 * q^17 - 4504 * q^18 - 5538 * q^19 - 13760 * q^20 + 25805 * q^21 + 4912 * q^22 - 40536 * q^23 - 44544 * q^24 - 104325 * q^25 - 65511 * q^27 - 45120 * q^28 - 296268 * q^29 + 120920 * q^30 + 269900 * q^31 - 65536 * q^32 + 192266 * q^33 - 52984 * q^34 - 206575 * q^35 + 36032 * q^36 + 356733 * q^37 + 44304 * q^38 + 110080 * q^40 - 209690 * q^41 - 206440 * q^42 + 856353 * q^43 - 39296 * q^44 - 561860 * q^45 + 324288 * q^46 - 1779581 * q^47 + 356352 * q^48 + 1368579 * q^49 + 834600 * q^50 + 1621543 * q^51 + 1616894 * q^53 + 524088 * q^54 - 1028870 * q^55 + 360960 * q^56 + 1762142 * q^57 + 2370144 * q^58 + 1926542 * q^59 - 967360 * q^60 - 606882 * q^61 - 2159200 * q^62 + 4714650 * q^63 + 524288 * q^64 - 1538128 * q^66 + 3098090 * q^67 + 423872 * q^68 + 643104 * q^69 + 1652600 * q^70 + 220205 * q^71 - 288256 * q^72 - 8592556 * q^73 - 2853864 * q^74 - 3299200 * q^75 - 354432 * q^76 + 10946210 * q^77 - 1544888 * q^79 - 880640 * q^80 - 1915462 * q^81 + 1677520 * q^82 + 3576720 * q^83 + 1651520 * q^84 - 7379185 * q^85 - 6850824 * q^86 - 16289838 * q^87 + 314368 * q^88 + 3340108 * q^89 + 4494880 * q^90 - 2594304 * q^92 + 6305460 * q^93 + 14236648 * q^94 - 9419890 * q^95 - 2850816 * q^96 - 23438808 * q^97 - 10948632 * q^98 + 18877984 * q^99

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

24.5052
−23.5052

−8.00000

19.4948

64.0000

12.5260

−155.958

−1528.76

−512.000

−1806.95

−100.208

1.2

−8.00000

67.5052

64.0000

−227.526

−540.042

823.755

−512.000

2369.95

1820.21

Atkin-Lehner signs

$p$	Sign
$2$	$+1$
$13$	$+1$

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	338.8.a.e		2
13.b	even	2	1		26.8.a.e	✓	2
13.d	odd	4	2		338.8.b.f		4
39.d	odd	2	1		234.8.a.g		2
52.b	odd	2	1		208.8.a.f		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
26.8.a.e	✓	2	13.b	even	2	1
208.8.a.f		2	52.b	odd	2	1
234.8.a.g		2	39.d	odd	2	1
338.8.a.e		2	1.a	even	1	1	trivial
338.8.b.f		4	13.d	odd	4	2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{8}^{\mathrm{new}}(\Gamma_0(338))$:

$T_{3}^{2} - 87T_{3} + 1316$

$T_{5}^{2} + 215T_{5} - 2850$

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$(T + 8)^{2}$
$3$	$T^{2} - 87T + 1316$
$5$	$T^{2} + 215T - 2850$
$7$	$T^{2} + 705 T - 1259320$
$11$	$T^{2} + 614 T - 20708376$