LMFDB - Newform orbit 325.6.a.f

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [325,6,Mod(1,325)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(325, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 6, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("325.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$N$	$=$	$325 = 5^{2} \cdot 13$
Weight:	$k$	$=$	$6$
Character orbit:	$[\chi]$	$=$	325.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$52.1247414392$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{6} - 2x^{5} - 161x^{4} + 328x^{3} + 6584x^{2} - 10688x - 47440$ x^6 - 2x^5 - 161x^4 + 328x^3 + 6584x^2 - 10688*x - 47440
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$2^{4}$
Twist minimal:	no (minimal twist has level 65)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label

\iota_m(\nu)

a_{2}

a_{3}

a_{4}

a_{5}

a_{6}

a_{7}

a_{8}

a_{9}

a_{10}

1.1

9.62003
7.62628
4.15349
−2.16876
−7.05260
−10.1784

−9.62003

−10.7175

60.5450

103.103

−18.4289

−274.604

−128.135

1.2

−7.62628

29.7832

26.1601

−227.135

−171.199

44.5366

644.042

1.3

−4.15349

−29.2293

−14.7485

121.404

−42.5171

194.170

611.349

1.4

2.16876

2.56930

−27.2965

5.57220

75.5966

−128.600

−236.399

1.5

7.05260

−22.8190

17.7392

−160.933

−141.347

−100.576

277.708

1.6

10.1784

10.4132

71.6007

105.990

125.896

403.073

−134.565

Atkin-Lehner signs

$p$	Sign
$5$	$+1$
$13$	$-1$

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	325.6.a.f		6
5.b	even	2	1		65.6.a.e	✓	6
5.c	odd	4	2		325.6.b.f		12
15.d	odd	2	1		585.6.a.k		6
20.d	odd	2	1		1040.6.a.r		6
65.d	even	2	1		845.6.a.g		6

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
65.6.a.e	✓	6	5.b	even	2	1
325.6.a.f		6	1.a	even	1	1	trivial
325.6.b.f		12	5.c	odd	4	2
585.6.a.k		6	15.d	odd	2	1
845.6.a.g		6	65.d	even	2	1
1040.6.a.r		6	20.d	odd	2	1

This newform subspace can be constructed as the kernel of the linear operator $T_{2}^{6} + 2T_{2}^{5} - 161T_{2}^{4} - 328T_{2}^{3} + 6584T_{2}^{2} + 10688T_{2} - 47440$ T2^6 + 2*T2^5 - 161*T2^4 - 328*T2^3 + 6584*T2^2 + 10688*T2 - 47440 acting on $S_{6}^{\mathrm{new}}(\Gamma_0(325))$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{6} + 2 T^{5} + \cdots - 47440$ T^6 + 2T^5 - 161T^4 - 328T^3 + 6584T^2 + 10688*T - 47440
$3$	$T^{6} + 20 T^{5} + \cdots - 5696136$ T^6 + 20T^5 - 1046T^4 - 20080T^3 + 150636T^2 + 1979280*T - 5696136
$5$	$T^{6}$ T^6
$7$	$T^{6} + \cdots + 180452981952$ T^6 + 172T^5 - 21708T^4 - 3597472T^3 + 91505904T^2 + 12546409664*T + 180452981952
$11$	$T^{6} + \cdots - 674919089487832$ T^6 - 800T^5 - 206738T^4 + 237595936T^3 - 6483260692T^2 - 10499695593152*T - 674919089487832
$13$	$(T - 169)^{6}$ (T - 169)^6
$17$	$T^{6} + \cdots + 12\!\cdots\!24$ T^6 + 4396T^5 + 4869172T^4 - 960709280T^3 - 2439928389904T^2 + 380252212894912*T + 127189810361849024
$19$	$T^{6} + \cdots + 58\!\cdots\!40$ T^6 - 5304T^5 + 2913118T^4 + 24207093504T^3 - 38879087374420T^2 + 7572388849871456*T + 5808654868911896040
$23$	$T^{6} + \cdots + 84\!\cdots\!48$ T^6 - 1140T^5 - 11627686T^4 + 9352643952T^3 + 24049262758444T^2 - 29513582574123024*T + 8400583456909960248
$29$	$T^{6} + \cdots - 24\!\cdots\!20$ T^6 + 3664T^5 - 52851648T^4 - 163663782400T^3 + 607799257638912T^2 + 1030966528660209664*T - 2447523844877627064320
$31$	$T^{6} + \cdots - 47\!\cdots\!84$ T^6 - 16664T^5 + 51216702T^4 + 543432097728T^3 - 4268297726426196T^2 + 9598590691238595168*T - 4769294212190703192984
$37$	$T^{6} + \cdots - 39\!\cdots\!12$ T^6 + 4488T^5 - 147307480T^4 - 505285377408T^3 + 4745029149611712T^2 + 7408416581173799424*T - 39020514602213745050112
$41$	$T^{6} + \cdots - 78\!\cdots\!32$ T^6 - 12436T^5 - 256586188T^4 + 2359387215712T^3 + 24108587736637168T^2 - 100511082931917954880*T - 785926703272282800589632
$43$	$T^{6} + \cdots - 34\!\cdots\!28$ T^6 + 1516T^5 - 763665350T^4 + 91100020912T^3 + 147669336867385004T^2 - 387776188391137912080*T - 3476049892877075020269128
$47$	$T^{6} + \cdots + 16\!\cdots\!00$ T^6 + 212T^5 - 507148364T^4 - 1058589710176T^3 + 63348445299267824T^2 + 243211301818430862144*T + 161322090496477727147200
$53$	$T^{6} + \cdots - 51\!\cdots\!36$ T^6 - 15612T^5 - 1206561196T^4 + 28782836945184T^3 - 140551415066137104T^2 - 179651622420045508032*T - 51383605166899636412736
$59$	$T^{6} + \cdots + 29\!\cdots\!80$ T^6 + 11896T^5 - 1595764818T^4 - 32111780588928T^3 + 265931057997507564T^2 + 8148149100822143132064*T + 29510181073952662770371880
$61$	$T^{6} + \cdots - 11\!\cdots\!68$ T^6 - 57232T^5 + 168842912T^4 + 22516772324352T^3 - 8568183035110400T^2 - 1648284295655640842240*T - 1137697860559129609043968
$67$	$T^{6} + \cdots - 31\!\cdots\!72$ T^6 - 25428T^5 - 3096671524T^4 + 68303341554976T^3 + 1520691410235169008T^2 - 4890342881168593097536*T - 31245851153354757778663872
$71$	$T^{6} + \cdots - 14\!\cdots\!12$ T^6 - 214104T^5 + 16334031214T^4 - 486962868474240T^3 + 1722407988696542828T^2 + 143212398667900270177760*T - 1466115382321133772846583512
$73$	$T^{6} + \cdots - 20\!\cdots\!08$ T^6 + 60808T^5 - 4754925912T^4 - 163610851054464T^3 + 7164863488481828544T^2 + 69032902098390133971456*T - 2040742962559722855865655808
$79$	$T^{6} + \cdots + 22\!\cdots\!80$ T^6 - 36024T^5 - 6846795288T^4 + 323321950242432T^3 + 4310204272652567232T^2 - 318004582684501394155008*T + 2251311065394611275520110080
$83$	$T^{6} + \cdots - 22\!\cdots\!12$ T^6 + 34684T^5 - 15785810284T^4 - 585489998596384T^3 + 48951160346967223792T^2 + 1280407626992062127816128*T - 2223018364367521107568878912
$89$	$T^{6} + \cdots - 21\!\cdots\!00$ T^6 + 52028T^5 - 26124810276T^4 - 1026375912831456T^3 + 172232026863871220208T^2 + 4581489073610645943261120*T - 211932928456723516735558065600
$97$	$T^{6} + \cdots + 10\!\cdots\!20$ T^6 + 129316T^5 - 24831820292T^4 - 1936192290937376T^3 + 140634712155264785648T^2 - 2280925248148682270381504*T + 10341173985646490456610374720
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Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

$f(q)$	$=$	$q - \beta_1 q^{2} + (\beta_{2} - 3) q^{3} + (\beta_{3} + \beta_{2} - \beta_1 + 23) q^{4} + (\beta_{5} - 2 \beta_{4} + \beta_{3} + \cdots - 10) q^{6} + (\beta_{5} - \beta_{4} - 6 \beta_1 - 27) q^{7} + ( - \beta_{5} - 5 \beta_{4} + \cdots + 29) q^{8}+ \cdots + ( - 1219 \beta_{5} - 3106 \beta_{4} + \cdots + 46690) q^{99}+O(q^{100})$ q - b1 * q^2 + (b2 - 3) * q^3 + (b3 + b2 - b1 + 23) * q^4 + (b5 - 2b4 + b3 - 3b2 - 10) * q^6 + (b5 - b4 - 6b1 - 27) q^7 + (-b5 - 5b4 + 2b3 + 2b2 - 15b1 + 29) * q^8 + (-4b5 - 3b4 - 2b2 + 18b1 + 167) * q^9 + (b5 - 4b4 - 8b3 + b2 + 28b1 + 124) q^11 + (-3b5 - 11b4 + b3 + 13b2 - 12b1 + 148) * q^12 + 169 * q^13 + (-12b4 + 6b3 + 30b2 + 10b1 + 358) * q^14 + (10b5 - 10b4 + 9b3 + 9b2 - 47b1 + 69) q^16 + (-2b5 - 19b4 + 24b3 - 12b2 - 6b1 - 734) q^17 + (12b5 + 38b4 + 8b3 - 52b2 - 99b1 - 1012) q^18 + (-5b5 + 20b4 + 24b3 + 63b2 - 124b1 + 948) q^19 + (6b5 + 5b4 + 8b3 - 92b2 - 198b1 + 212) q^21 + (23b5 + 4b4 - 15b3 - 15b2 + 50b1 - 1364) q^22 + (-2b5 - 43b4 + 40b3 + 37b2 - 6b1 + 205) q^23 + (7b5 + 43b4 + 49b3 + 113b2 - 186b1 + 890) q^24 - 169b1 q^26 + (-10b5 + 100b4 + 16b3 + 276b2 + 336b1 - 150) q^27 + (10b5 - 70b4 + 68b3 + 20b2 - 366b1 + 38) q^28 + (54b5 + 9b4 - 32b3 - 14b2 - 94b1 - 602) q^29 + (-43b5 - 53b4 - 16b3 - b2 - 302b1 + 2892) * q^31 + (23b5 - 5b4 - 8b3 + 232b2 + 47b1 + 1703) q^32 + (-32b5 + 172b4 - 8b3 + 274b2 - 120b1 + 2324) q^33 + (-18b5 - 66b4 + 78b3 + 258b2 + 306b1 + 220) q^34 + (-40b5 + 132b4 - 153b3 + 239b2 + 201b1 + 329) q^36 + (-6b5 - 108b4 - 104b3 - 74b2 + 640b1 - 984) q^37 + (-15b5 - 108b4 + 127b3 - 209b2 - 1686b1 + 5516) q^38 + (169b2 - 507) q^39 + (-46b5 + 187b4 - 168b3 + 108b2 + 6b1 + 2122) q^41 + (-130b5 + 110b4 + 62b3 + 562b2 - 360b1 + 11740) q^42 + (-48b5 - 48b4 - 368b3 - 539b2 + 176b1 - 475) q^43 + (-71b5 - 19b4 + 83b3 + 239b2 + 610b1 - 5954) q^44 + (47b5 - 260b4 + 223b3 + 367b2 - 1100b1 - 286) q^46 + (47b5 + 229b4 - 224b3 - 392b2 - 338b1 - 69) q^47 + (11b5 - 5b4 + 67b3 - 605b2 - 2088b1 + 3276) q^48 + (-88b5 + 52b4 + 32b3 + 368b2 - 424b1 - 4347) q^49 + (54b5 + 224b4 + 128b3 - 526b2 - 1800b1 - 4204) q^51 + (169b3 + 169b2 - 169b1 + 3887) q^52 + (-18b5 + 471b4 - 56b3 + 556b2 - 962b1 + 3114) q^53 + (64b5 - 300b4 - 420b3 - 2060b2 - 552b1 - 22676) q^54 + (4b5 - 100b4 + 434b3 + 338b2 - 2254b1 + 8266) q^56 + (-128b5 - 936b4 + 24b3 + 442b2 + 2304b1 + 13888) q^57 + (-76b5 - 398b4 - 172b3 + 800b2 + 596b1 + 6704) q^58 + (89b5 + 46b4 - 536b3 + 505b2 - 96b1 - 1812) q^59 + (130b5 - 67b4 - 80b3 + 622b2 - 1270b1 + 10126) q^61 + (223b5 + 374b4 + 685b3 - 23b2 - 2398b1 + 16632) q^62 + (323b5 - 21b4 + 144b3 - 996b2 - 966b1 - 34777) q^63 + (-108b5 - 360b4 - 175b3 - 655b2 - 941b1 - 6521) q^64 + (10b5 + 140b4 - 166b3 - 2622b2 - 2754b1 + 1692) q^66 + (-45b5 - 27b4 + 16b3 - 938b2 + 3886b1 + 2645) q^67 + (334b5 - 94b4 - 480b3 - 144b2 - 1858b1 + 3166) q^68 + (-142b5 + 413b4 + 272b3 + 718b2 - 2838b1 + 13300) q^69 + (209b5 - 575b4 - 80b3 - 35b2 - 466b1 + 35758) q^71 + (-23b5 - 571b4 - 586b3 - 1562b2 + 5823b1 + 19347) q^72 + (-64b5 - 8b4 - 632b3 + 1474b2 - 2352b1 - 8838) q^73 + (362b5 + 304b4 - 258b3 - 66b2 + 3992b1 - 31824) q^74 + (-57b5 - 669b4 + 1201b3 + 1429b2 - 4982b1 + 63262) q^76 + (218b5 + 745b4 - 488b3 - 1352b2 - 2910b1 + 1312) q^77 + (169b5 - 338b4 + 169b3 - 507b2 - 1690) * q^78 + (240b5 + 828b4 - 912b3 - 1518b2 + 2232b1 + 4674) q^79 + (-468b5 - 1313b4 - 896b3 + 522b2 + 7494b1 + 56251) q^81 + (162b5 + 1122b4 - 462b3 - 3234b2 + 1426b1 - 1420) q^82 + (-851b5 + 263b4 + 208b3 + 112b2 - 2678b1 - 4567) q^83 + (286b5 + 50b4 + 746b3 - 1094b2 - 7260b1 + 2964) q^84 + (389b5 + 2566b4 - 331b3 - 415b2 + 10156b1 + 1214) q^86 + (150b5 - 170b4 - 176b3 - 4188b2 - 6204b1 + 2062) q^87 + (-483b5 - 183b4 + 469b3 - 1499b2 + 3368b1 + 5292) q^88 + (-164b5 + 608b4 + 1648b3 - 2432b2 + 2192b1 - 10158) q^89 + (169b5 - 169b4 - 1014b1 - 4563) q^91 + (411b5 - 1017b4 + 1039b3 + 2539b2 - 6700b1 + 50148) q^92 + (306b5 + 492b4 + 792b3 + 5670b2 + 3216b1 - 3702) q^93 + (-496b5 + 1444b4 - 1158b3 - 462b2 + 4870b1 + 24650) q^94 + (-975b5 - 487b4 - 109b3 + 771b2 + 942b1 + 91586) q^96 + (280b5 + 590b4 + 288b3 + 4920b2 + 2844b1 - 20954) q^97 + (376b5 + 152b4 + 936b3 - 2376b2 + 3147b1 + 17024) q^98 + (-1219b5 - 3106b4 + 1160b3 + 1323b2 + 13104b1 + 46690) q^99
$\operatorname{Tr}(f)(q)$	$=$	$6 q - 2 q^{2} - 20 q^{3} + 134 q^{4} - 52 q^{6} - 172 q^{7} + 138 q^{8} + 1034 q^{9} + 800 q^{11} + 832 q^{12} + 1014 q^{13} + 2108 q^{14} + 322 q^{16} - 4396 q^{17} - 6142 q^{18} + 5304 q^{19} + 1072 q^{21}+ \cdots + 301264 q^{99}+O(q^{100})$ 6 * q - 2 * q^2 - 20 * q^3 + 134 * q^4 - 52 * q^6 - 172 * q^7 + 138 * q^8 + 1034 * q^9 + 800 * q^11 + 832 * q^12 + 1014 * q^13 + 2108 * q^14 + 322 * q^16 - 4396 * q^17 - 6142 * q^18 + 5304 * q^19 + 1072 * q^21 - 8008 * q^22 + 1140 * q^23 + 4756 * q^24 - 338 * q^26 - 800 * q^27 - 524 * q^28 - 3664 * q^29 + 16664 * q^31 + 9894 * q^32 + 13092 * q^33 + 1380 * q^34 + 1818 * q^36 - 4488 * q^37 + 30112 * q^38 - 3380 * q^39 + 12436 * q^41 + 68336 * q^42 - 1516 * q^43 - 35124 * q^44 - 4556 * q^46 - 212 * q^47 + 16712 * q^48 - 27842 * q^49 - 27664 * q^51 + 22646 * q^52 + 15612 * q^53 - 132912 * q^54 + 44420 * q^56 + 86796 * q^57 + 39664 * q^58 - 11896 * q^59 + 57232 * q^61 + 95488 * q^62 - 207956 * q^63 - 39914 * q^64 + 9908 * q^66 + 25428 * q^67 + 16236 * q^68 + 72404 * q^69 + 214104 * q^71 + 130806 * q^72 - 60808 * q^73 - 182104 * q^74 + 366636 * q^76 + 5192 * q^77 - 8788 * q^78 + 36024 * q^79 + 350514 * q^81 + 1124 * q^82 - 34684 * q^83 + 6024 * q^84 + 29204 * q^86 + 8640 * q^87 + 40520 * q^88 - 52028 * q^89 - 29068 * q^91 + 283232 * q^92 - 26508 * q^93 + 157572 * q^94 + 547908 * q^96 - 129316 * q^97 + 113942 * q^98 + 301264 * q^99

$\beta_{1}$	$=$	$\nu$ v
$\beta_{2}$	$=$	$( -\nu^{5} - 2\nu^{4} + 125\nu^{3} + 212\nu^{2} - 2956\nu - 4800 ) / 240$ (-v^5 - 2v^4 + 125v^3 + 212v^2 - 2956v - 4800) / 240
$\beta_{3}$	$=$	$( \nu^{5} + 2\nu^{4} - 125\nu^{3} + 28\nu^{2} + 3196\nu - 8400 ) / 240$ (v^5 + 2v^4 - 125v^3 + 28v^2 + 3196v - 8400) / 240
$\beta_{4}$	$=$	$( -\nu^{4} + 10\nu^{3} + 125\nu^{2} - 808\nu - 2260 ) / 60$ (-v^4 + 10v^3 + 125v^2 - 808*v - 2260) / 60
$\beta_{5}$	$=$	$( \nu^{4} + 2\nu^{3} - 101\nu^{2} - 116\nu + 1288 ) / 12$ (v^4 + 2v^3 - 101v^2 - 116*v + 1288) / 12

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