LMFDB - Newform orbit 416.2.w.d

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [416,2,Mod(225,416)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(416, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("416.225");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 416 = 2^{5} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	416.w (of order $6$, degree $2$, minimal)

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:	no
Analytic conductor:	$3.32177672409$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$6$ over $\Q(\zeta_{6})$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 12x^{10} + 108x^{8} - 430x^{6} + 1284x^{4} - 36x^{2} + 1 $ x^12 - 12x^10 + 108x^8 - 430x^6 + 1284x^4 - 36*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{9}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $q$-expansion are expressed in terms of a basis $1,\beta_1,\ldots,\beta_{11}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q - \beta_{9} q^{3} + (\beta_{6} - \beta_{5} - 2 \beta_{4}) q^{5} - \beta_{3} q^{7} + ( - \beta_{7} + 2 \beta_{4}) q^{9}+O(q^{10}) $ q - b9 * q^3 + (b6 - b5 - 2b4) q^5 - b3 * q^7 + (-b7 + 2b4) q^9	$ q - \beta_{9} q^{3} + (\beta_{6} - \beta_{5} - 2 \beta_{4}) q^{5} - \beta_{3} q^{7} + ( - \beta_{7} + 2 \beta_{4}) q^{9} + ( - \beta_{10} + \beta_{8}) q^{11} + (\beta_{6} + \beta_{4} + \beta_{2} + 2) q^{13} + ( - \beta_{11} - \beta_{10} + \cdots - \beta_{8}) q^{15}+ \cdots + (2 \beta_{11} + 2 \beta_{10} + \cdots - 2 \beta_1) q^{99}+O(q^{100}) $ q - b9 * q^3 + (b6 - b5 - 2b4) q^5 - b3 * q^7 + (-b7 + 2b4) q^9 + (-b10 + b8) * q^11 + (b6 + b4 + b2 + 2) * q^13 + (-b11 - b10 + b9 - b8) * q^15 + b4 * q^17 + (-b9 - b8) * q^19 + (2b7 - 2b6 + 2b5 - 2b4 - b2 - 3) * q^21 + (-b11 + b9 + 2b1) q^23 + (-2b6 - 2b5 - b2 - 1) * q^25 + (-b10 + b8 - b3) * q^27 + (-b7 - b6 + 2b5 + b2 - 2) q^29 + (b11 + b10 - 2b9 - b3 - b1) q^31 + (b7 + 2b5 - b4 - 2b2 - 4) * q^33 + (2b9 - 2b8) * q^35 + (b7 + b6 + b2) * q^37 + (-b10 - 3b9 + 2b8 + b3 - b1) * q^39 + (-2b6 - b4 + 1) q^41 + (2b10 - 2b9 - b3) * q^43 + (3b5 + b4 - 1) q^45 + (b11 - b10 - 2b9 + 2b8 + b3 - b1) * q^47 + (-b7 + 2b6 - 4b5 + 2b4 + b2 + 6) q^49 + b8 * q^51 + (-b6 - b5 - 4) * q^53 + 2b9 q^55 + (-2b7 + 10b4 + b2 + 5) * q^57 + (2b9 + 2b8 + b3) * q^59 + (-b7 + 2b6 - b5 - 4b4 + 1) * q^61 + (2b11 + 2b10 + 2b9 - 6b8) * q^63 + (b7 + 4b6 - 4b5 - 6b4 - b2) q^65 + (-2b11 + 2b10 + b9 - 4b8) q^67 + (b7 + 8b6 - 4b5 - 5b4 + 4) q^69 + (b9 + b8 + b1) * q^71 + (-2b7 - 4b6 + 4b5 + b2 - 4) q^73 + (-2b11 - b10 + 4b9 + b8 + 2b3 + 4b1) * q^75 + (-2b6 - 2b5 + b2 + 11) * q^77 + (2b10 + 2b8 + 2b3) q^79 + (-2b6 + 4b5 + 3b4 - 1) q^81 + (2b11 - 2b10 + 2b8 + 2b3 - 2b1) q^83 + (b5 + b4 + 1) * q^85 + (2b11 - b9 + b8 - b1) q^87 + (-b7 + 4b6 - 3b4 - b2 + 3) * q^89 + (2b10 + 2b9 - 2b8 - b3 - 2b1) * q^91 + (-6b6 + 4b4 - 4) * q^93 + (-2b11 - 2b10 + 3b9 - b8 + b3 + b1) q^95 + (b7 - 5b4 - 2b2 - 10) * q^97 + (2b11 + 2b10 + 4b9 - 4b8 - 2b3 - 2b1) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 12 q^{9}+O(q^{10}) $ 12 * q - 12 * q^9	$ 12 q - 12 q^{9} + 12 q^{13} - 6 q^{17} - 12 q^{25} - 6 q^{29} - 30 q^{33} - 6 q^{37} + 30 q^{41} + 24 q^{49} - 48 q^{53} + 18 q^{61} - 12 q^{65} + 6 q^{69} + 132 q^{77} + 6 q^{81} + 12 q^{85} + 30 q^{89} - 36 q^{93} - 90 q^{97}+O(q^{100}) $ 12 * q - 12 * q^9 + 12 * q^13 - 6 * q^17 - 12 * q^25 - 6 * q^29 - 30 * q^33 - 6 * q^37 + 30 * q^41 + 24 * q^49 - 48 * q^53 + 18 * q^61 - 12 * q^65 + 6 * q^69 + 132 * q^77 + 6 * q^81 + 12 * q^85 + 30 * q^89 - 36 * q^93 - 90 * q^97

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{12} - 12x^{10} + 108x^{8} - 430x^{6} + 1284x^{4} - 36x^{2} + 1 $ x^12 - 12*x^10 + 108*x^8 - 430*x^6 + 1284*x^4 - 36*x^2 + 1 :

$\beta_{1}$	$=$	$ ( -\nu^{11} + 9\nu^{9} - 81\nu^{7} + 107\nu^{5} - 3\nu^{3} - 5733\nu ) / 240 $ (-v^11 + 9v^9 - 81v^7 + 107v^5 - 3v^3 - 5733*v) / 240
$\beta_{2}$	$=$	$ ( \nu^{10} - 9\nu^{8} + 61\nu^{6} - 107\nu^{4} + 3\nu^{2} + 913 ) / 240 $ (v^10 - 9v^8 + 61v^6 - 107v^4 + 3v^2 + 913) / 240
$\beta_{3}$	$=$	$ ( -\nu^{11} + 9\nu^{9} - 71\nu^{7} + 107\nu^{5} - 3\nu^{3} - 3443\nu ) / 120 $ (-v^11 + 9v^9 - 71v^7 + 107v^5 - 3v^3 - 3443*v) / 120
$\beta_{4}$	$=$	$ ( 27\nu^{10} - 323\nu^{8} + 2907\nu^{6} - 11529\nu^{4} + 34561\nu^{2} - 969 ) / 960 $ (27v^10 - 323v^8 + 2907v^6 - 11529v^4 + 34561*v^2 - 969) / 960
$\beta_{5}$	$=$	$ ( -17\nu^{10} + 203\nu^{8} - 1847\nu^{6} + 7339\nu^{4} - 22201\nu^{2} - 251 ) / 360 $ (-17v^10 + 203v^8 - 1847v^6 + 7339v^4 - 22201*v^2 - 251) / 360
$\beta_{6}$	$=$	$ ( 71\nu^{10} - 839\nu^{8} + 7511\nu^{6} - 29677\nu^{4} + 88813\nu^{2} - 4957 ) / 1440 $ (71v^10 - 839v^8 + 7511v^6 - 29677v^4 + 88813*v^2 - 4957) / 1440
$\beta_{7}$	$=$	$ ( -101\nu^{10} + 1229\nu^{8} - 11061\nu^{6} + 44407\nu^{4} - 131503\nu^{2} + 3687 ) / 960 $ (-101v^10 + 1229v^8 - 11061v^6 + 44407v^4 - 131503*v^2 + 3687) / 960
$\beta_{8}$	$=$	$ ( -409\nu^{11} + 4921\nu^{9} - 44329\nu^{7} + 177203\nu^{5} - 530387\nu^{3} + 29603\nu ) / 1440 $ (-409v^11 + 4921v^9 - 44329v^7 + 177203v^5 - 530387v^3 + 29603v) / 1440
$\beta_{9}$	$=$	$ ( 421\nu^{11} - 5029\nu^{9} + 45181\nu^{7} - 178487\nu^{5} + 530423\nu^{3} + 14593\nu ) / 1440 $ (421v^11 - 5029v^9 + 45181v^7 - 178487v^5 + 530423v^3 + 14593v) / 1440
$\beta_{10}$	$=$	$ ( 749\nu^{11} - 8981\nu^{9} + 80789\nu^{7} - 321103\nu^{5} + 957127\nu^{3} - 12103\nu ) / 1440 $ (749v^11 - 8981v^9 + 80789v^7 - 321103v^5 + 957127v^3 - 12103v) / 1440
$\beta_{11}$	$=$	$ ( 161\nu^{11} - 1929\nu^{9} + 17361\nu^{7} - 69067\nu^{5} + 206403\nu^{3} - 5787\nu ) / 240 $ (161v^11 - 1929v^9 + 17361v^7 - 69067v^5 + 206403v^3 - 5787v) / 240

Display $\nu^j$ in terms of $\beta_i$

$\nu$	$=$	$ ( \beta_{9} + \beta_{8} + \beta_{3} ) / 2 $ (b9 + b8 + b3) / 2
$\nu^{2}$	$=$	$ ( \beta_{7} - \beta_{6} + 2\beta_{5} + 9\beta_{4} - \beta_{2} + 7 ) / 2 $ (b7 - b6 + 2b5 + 9b4 - b2 + 7) / 2
$\nu^{3}$	$=$	$ ( -\beta_{11} - 12\beta_{10} + 24\beta_{9} + 12\beta_{3} + \beta_1 ) / 4 $ (-b11 - 12b10 + 24b9 + 12*b3 + b1) / 4
$\nu^{4}$	$=$	$ ( 5\beta_{7} - 14\beta_{6} + 7\beta_{5} + 55\beta_{4} - 7 ) / 2 $ (5b7 - 14b6 + 7b5 + 55b4 - 7) / 2
$\nu^{5}$	$=$	$ ( -11\beta_{11} - 70\beta_{10} + 74\beta_{9} - 78\beta_{8} ) / 4 $ (-11b11 - 70b10 + 74b9 - 78b8) / 4
$\nu^{6}$	$=$	$ -24\beta_{6} - 24\beta_{5} + 12\beta_{2} - 145 $ -24b6 - 24b5 + 12*b2 - 145
$\nu^{7}$	$=$	$ ( -229\beta_{9} - 229\beta_{8} - 205\beta_{3} - 48\beta_1 ) / 2 $ (-229b9 - 229b8 - 205b3 - 48b1) / 2
$\nu^{8}$	$=$	$ ( -109\beta_{7} + 325\beta_{6} - 650\beta_{5} - 2085\beta_{4} + 109\beta_{2} - 1435 ) / 2 $ (-109b7 + 325b6 - 650b5 - 2085b4 + 109*b2 - 1435) / 2
$\nu^{9}$	$=$	$ ( 757\beta_{11} + 2412\beta_{10} - 5688\beta_{9} + 432\beta_{8} - 2412\beta_{3} - 757\beta_1 ) / 4 $ (757b11 + 2412b10 - 5688b9 + 432b8 - 2412b3 - 757b1) / 4
$\nu^{10}$	$=$	$ ( -449\beta_{7} + 4358\beta_{6} - 2179\beta_{5} - 12907\beta_{4} + 2179 ) / 2 $ (-449b7 + 4358b6 - 2179b5 - 12907b4 + 2179) / 2
$\nu^{11}$	$=$	$ ( 5639\beta_{11} + 14254\beta_{10} - 17714\beta_{9} + 21174\beta_{8} ) / 4 $ (5639b11 + 14254b10 - 17714b9 + 21174b8) / 4

Display $\beta_i$ in terms of $\nu^j$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/416\mathbb{Z}\right)^\times$.

$n$	$261$	$287$	$353$
$\chi(n)$	$1$	$1$	$-\beta_{4}$

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} $

225.1

0.145015	−	0.0837246i
2.19011	−	1.26446i
2.04509	−	1.18073i
−2.04509	+	1.18073i
−2.19011	+	1.26446i
−0.145015	+	0.0837246i
0.145015	+	0.0837246i
2.19011	+	1.26446i
2.04509	+	1.18073i
−2.04509	−	1.18073i
−2.19011	−	1.26446i
−0.145015	−	0.0837246i

−1.48362

2.56970i

−

1.23519i

4.16083

−

2.40226i

−2.90226

−

5.02685i

225.2

−1.13286

1.96217i

3.99777i

−0.981633

0.566746i

−1.06675

−

1.84766i

225.3

−0.515266

0.892467i

0.701519i

−2.54439

1.46900i

0.969002

1.67836i

225.4

0.515266

−

0.892467i

0.701519i

2.54439

−

1.46900i

0.969002

1.67836i

225.5

1.13286

−

1.96217i

3.99777i

0.981633

−

0.566746i

−1.06675

−

1.84766i

225.6

1.48362

−

2.56970i

−

1.23519i

−4.16083

2.40226i

−2.90226

−

5.02685i

257.1

−1.48362

−

2.56970i

1.23519i

4.16083

2.40226i

−2.90226

5.02685i

257.2

−1.13286

−

1.96217i

−

3.99777i

−0.981633

−

0.566746i

−1.06675

1.84766i

257.3

−0.515266

−

0.892467i

−

0.701519i

−2.54439

−

1.46900i

0.969002

−

1.67836i

257.4

0.515266

0.892467i

−

0.701519i

2.54439

1.46900i

0.969002

−

1.67836i

257.5

1.13286

1.96217i

−

3.99777i

0.981633

0.566746i

−1.06675

1.84766i

257.6

1.48362

2.56970i

1.23519i

−4.16083

−

2.40226i

−2.90226

5.02685i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
13.e	even	6	1	inner
52.i	odd	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	416.2.w.d	✓	12
4.b	odd	2	1	inner	416.2.w.d	✓	12
8.b	even	2	1		832.2.w.j		12
8.d	odd	2	1		832.2.w.j		12
13.e	even	6	1	inner	416.2.w.d	✓	12
13.f	odd	12	1		5408.2.a.bm		6
13.f	odd	12	1		5408.2.a.bn		6
52.i	odd	6	1	inner	416.2.w.d	✓	12
52.l	even	12	1		5408.2.a.bm		6
52.l	even	12	1		5408.2.a.bn		6
104.p	odd	6	1		832.2.w.j		12
104.s	even	6	1		832.2.w.j		12

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
416.2.w.d	✓	12	1.a	even	1	1	trivial
416.2.w.d	✓	12	4.b	odd	2	1	inner
416.2.w.d	✓	12	13.e	even	6	1	inner
416.2.w.d	✓	12	52.i	odd	6	1	inner
832.2.w.j		12	8.b	even	2	1
832.2.w.j		12	8.d	odd	2	1
832.2.w.j		12	104.p	odd	6	1
832.2.w.j		12	104.s	even	6	1
5408.2.a.bm		6	13.f	odd	12	1
5408.2.a.bm		6	52.l	even	12	1
5408.2.a.bn		6	13.f	odd	12	1
5408.2.a.bn		6	52.l	even	12	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{3}^{12} + 15T_{3}^{10} + 165T_{3}^{8} + 804T_{3}^{6} + 2880T_{3}^{4} + 2880T_{3}^{2} + 2304 $ T3^12 + 15*T3^10 + 165*T3^8 + 804*T3^6 + 2880*T3^4 + 2880*T3^2 + 2304 acting on $S_{2}^{\mathrm{new}}(416, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{12} $ T^12
$3$	$ T^{12} + 15 T^{10} + \cdots + 2304 $ T^12 + 15T^10 + 165T^8 + 804T^6 + 2880T^4 + 2880*T^2 + 2304
$5$	$ (T^{6} + 18 T^{4} + \cdots + 12)^{2} $ (T^6 + 18T^4 + 33T^2 + 12)^2
$7$	$ T^{12} - 33 T^{10} + \cdots + 65536 $ T^12 - 33T^10 + 849T^8 - 7408T^6 + 49152T^4 - 61440*T^2 + 65536
$11$	$ T^{12} - 33 T^{10} + \cdots + 65536 $ T^12 - 33T^10 + 849T^8 - 7408T^6 + 49152T^4 - 61440*T^2 + 65536
$13$	$ (T^{6} - 6 T^{5} + \cdots + 2197)^{2} $ (T^6 - 6T^5 - 6T^4 + 106T^3 - 78T^2 - 1014*T + 2197)^2
$17$	$ (T^{2} + T + 1)^{6} $ (T^2 + T + 1)^6
$19$	$ T^{12} - 45 T^{10} + \cdots + 1679616 $ T^12 - 45T^10 + 1485T^8 - 21708T^6 + 233280T^4 - 699840*T^2 + 1679616
$23$	$ T^{12} + \cdots + 2127792384 $ T^12 + 135T^10 + 13413T^8 + 557364T^6 + 16928064T^4 + 221967936*T^2 + 2127792384
$29$	$ (T^{6} + 3 T^{5} + \cdots + 961)^{2} $ (T^6 + 3T^5 + 30T^4 - T^3 + 534T^2 + 651T + 961)^2
$31$	$ (T^{6} + 96 T^{4} + \cdots + 576)^{2} $ (T^6 + 96T^4 + 1152T^2 + 576)^2
$37$	$ (T^{6} + 3 T^{5} + \cdots + 5043)^{2} $ (T^6 + 3T^5 - 36T^4 - 117T^3 + 1644T^2 - 4797*T + 5043)^2
$41$	$ (T^{6} - 15 T^{5} + \cdots + 6627)^{2} $ (T^6 - 15T^5 + 72T^4 + 45T^3 - 696T^2 - 423*T + 6627)^2
$43$	$ T^{12} + \cdots + 150994944 $ T^12 + 99T^10 + 7113T^8 + 241536T^6 + 6008832T^4 + 33030144*T^2 + 150994944
$47$	$ (T^{6} + 108 T^{4} + \cdots + 43264)^{2} $ (T^6 + 108T^4 + 3792T^2 + 43264)^2
$53$	$ (T^{3} + 12 T^{2} + 27 T - 4)^{4} $ (T^3 + 12T^2 + 27T - 4)^4
$59$	$ T^{12} + \cdots + 10485760000 $ T^12 - 153T^10 + 16305T^8 - 882112T^6 + 34799616T^4 - 727449600*T^2 + 10485760000
$61$	$ (T^{6} - 9 T^{5} + \cdots + 56169)^{2} $ (T^6 - 9T^5 + 102T^4 - 285T^3 + 2574T^2 - 4977*T + 56169)^2
$67$	$ T^{12} + \cdots + 385571451136 $ T^12 - 285T^10 + 56685T^8 - 5752012T^6 + 425242560T^4 - 15237965760*T^2 + 385571451136
$71$	$ T^{12} - 69 T^{10} + \cdots + 160000 $ T^12 - 69T^10 + 4365T^8 - 26524T^6 + 129216T^4 - 158400*T^2 + 160000
$73$	$ (T^{6} + 234 T^{4} + \cdots + 288300)^{2} $ (T^6 + 234T^4 + 16089T^2 + 288300)^2
$79$	$ (T^{6} - 396 T^{4} + \cdots - 786432)^{2} $ (T^6 - 396T^4 + 43008T^2 - 786432)^2
$83$	$ (T^{6} + 228 T^{4} + \cdots + 16384)^{2} $ (T^6 + 228T^4 + 6912T^2 + 16384)^2
$89$	$ (T^{6} - 15 T^{5} + \cdots + 2157312)^{2} $ (T^6 - 15T^5 - 105T^4 + 2700T^3 + 19680T^2 - 457920*T + 2157312)^2
$97$	$ (T^{6} + 45 T^{5} + \cdots + 62208)^{2} $ (T^6 + 45T^5 + 855T^4 + 8100T^3 + 38880T^2 + 77760*T + 62208)^2
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Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 416 = 2^{5} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	416.w (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\( q - \beta_{9} q^{3} + (\beta_{6} - \beta_{5} - 2 \beta_{4}) q^{5} - \beta_{3} q^{7} + ( - \beta_{7} + 2 \beta_{4}) q^{9}+O(q^{10}) \) q - b9 * q^3 + (b6 - b5 - 2b4) q^5 - b3 * q^7 + (-b7 + 2b4) q^9	\( q - \beta_{9} q^{3} + (\beta_{6} - \beta_{5} - 2 \beta_{4}) q^{5} - \beta_{3} q^{7} + ( - \beta_{7} + 2 \beta_{4}) q^{9} + ( - \beta_{10} + \beta_{8}) q^{11} + (\beta_{6} + \beta_{4} + \beta_{2} + 2) q^{13} + ( - \beta_{11} - \beta_{10} + \cdots - \beta_{8}) q^{15}+ \cdots + (2 \beta_{11} + 2 \beta_{10} + \cdots - 2 \beta_1) q^{99}+O(q^{100}) \) q - b9 * q^3 + (b6 - b5 - 2b4) q^5 - b3 * q^7 + (-b7 + 2b4) q^9 + (-b10 + b8) * q^11 + (b6 + b4 + b2 + 2) * q^13 + (-b11 - b10 + b9 - b8) * q^15 + b4 * q^17 + (-b9 - b8) * q^19 + (2b7 - 2b6 + 2b5 - 2b4 - b2 - 3) * q^21 + (-b11 + b9 + 2b1) q^23 + (-2b6 - 2b5 - b2 - 1) * q^25 + (-b10 + b8 - b3) * q^27 + (-b7 - b6 + 2b5 + b2 - 2) q^29 + (b11 + b10 - 2b9 - b3 - b1) q^31 + (b7 + 2b5 - b4 - 2b2 - 4) * q^33 + (2b9 - 2b8) * q^35 + (b7 + b6 + b2) * q^37 + (-b10 - 3b9 + 2b8 + b3 - b1) * q^39 + (-2b6 - b4 + 1) q^41 + (2b10 - 2b9 - b3) * q^43 + (3b5 + b4 - 1) q^45 + (b11 - b10 - 2b9 + 2b8 + b3 - b1) * q^47 + (-b7 + 2b6 - 4b5 + 2b4 + b2 + 6) q^49 + b8 * q^51 + (-b6 - b5 - 4) * q^53 + 2b9 q^55 + (-2b7 + 10b4 + b2 + 5) * q^57 + (2b9 + 2b8 + b3) * q^59 + (-b7 + 2b6 - b5 - 4b4 + 1) * q^61 + (2b11 + 2b10 + 2b9 - 6b8) * q^63 + (b7 + 4b6 - 4b5 - 6b4 - b2) q^65 + (-2b11 + 2b10 + b9 - 4b8) q^67 + (b7 + 8b6 - 4b5 - 5b4 + 4) q^69 + (b9 + b8 + b1) * q^71 + (-2b7 - 4b6 + 4b5 + b2 - 4) q^73 + (-2b11 - b10 + 4b9 + b8 + 2b3 + 4b1) * q^75 + (-2b6 - 2b5 + b2 + 11) * q^77 + (2b10 + 2b8 + 2b3) q^79 + (-2b6 + 4b5 + 3b4 - 1) q^81 + (2b11 - 2b10 + 2b8 + 2b3 - 2b1) q^83 + (b5 + b4 + 1) * q^85 + (2b11 - b9 + b8 - b1) q^87 + (-b7 + 4b6 - 3b4 - b2 + 3) * q^89 + (2b10 + 2b9 - 2b8 - b3 - 2b1) * q^91 + (-6b6 + 4b4 - 4) * q^93 + (-2b11 - 2b10 + 3b9 - b8 + b3 + b1) q^95 + (b7 - 5b4 - 2b2 - 10) * q^97 + (2b11 + 2b10 + 4b9 - 4b8 - 2b3 - 2b1) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 12 q^{9}+O(q^{10}) \) 12 * q - 12 * q^9	\( 12 q - 12 q^{9} + 12 q^{13} - 6 q^{17} - 12 q^{25} - 6 q^{29} - 30 q^{33} - 6 q^{37} + 30 q^{41} + 24 q^{49} - 48 q^{53} + 18 q^{61} - 12 q^{65} + 6 q^{69} + 132 q^{77} + 6 q^{81} + 12 q^{85} + 30 q^{89} - 36 q^{93} - 90 q^{97}+O(q^{100}) \) 12 * q - 12 * q^9 + 12 * q^13 - 6 * q^17 - 12 * q^25 - 6 * q^29 - 30 * q^33 - 6 * q^37 + 30 * q^41 + 24 * q^49 - 48 * q^53 + 18 * q^61 - 12 * q^65 + 6 * q^69 + 132 * q^77 + 6 * q^81 + 12 * q^85 + 30 * q^89 - 36 * q^93 - 90 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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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	416.2.w.d

Level	$416$
Weight	$2$
Character orbit	416.w
Analytic conductor	$3.322$
Analytic rank	$0$
Dimension	$12$
CM	no
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(3.32177672409\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Relative dimension:	\(6\) over \(\Q(\zeta_{6})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{12} - 12x^{10} + 108x^{8} - 430x^{6} + 1284x^{4} - 36x^{2} + 1 \) x^12 - 12x^10 + 108x^8 - 430x^6 + 1284x^4 - 36*x^2 + 1
Coefficient ring:	\(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index:	\( 2^{8} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

\(\beta_{1}\)	\(=\)	\( ( -\nu^{11} + 9\nu^{9} - 81\nu^{7} + 107\nu^{5} - 3\nu^{3} - 5733\nu ) / 240 \) (-v^11 + 9v^9 - 81v^7 + 107v^5 - 3v^3 - 5733*v) / 240
\(\beta_{2}\)	\(=\)	\( ( \nu^{10} - 9\nu^{8} + 61\nu^{6} - 107\nu^{4} + 3\nu^{2} + 913 ) / 240 \) (v^10 - 9v^8 + 61v^6 - 107v^4 + 3v^2 + 913) / 240
\(\beta_{3}\)	\(=\)	\( ( -\nu^{11} + 9\nu^{9} - 71\nu^{7} + 107\nu^{5} - 3\nu^{3} - 3443\nu ) / 120 \) (-v^11 + 9v^9 - 71v^7 + 107v^5 - 3v^3 - 3443*v) / 120
\(\beta_{4}\)	\(=\)	\( ( 27\nu^{10} - 323\nu^{8} + 2907\nu^{6} - 11529\nu^{4} + 34561\nu^{2} - 969 ) / 960 \) (27v^10 - 323v^8 + 2907v^6 - 11529v^4 + 34561*v^2 - 969) / 960
\(\beta_{5}\)	\(=\)	\( ( -17\nu^{10} + 203\nu^{8} - 1847\nu^{6} + 7339\nu^{4} - 22201\nu^{2} - 251 ) / 360 \) (-17v^10 + 203v^8 - 1847v^6 + 7339v^4 - 22201*v^2 - 251) / 360
\(\beta_{6}\)	\(=\)	\( ( 71\nu^{10} - 839\nu^{8} + 7511\nu^{6} - 29677\nu^{4} + 88813\nu^{2} - 4957 ) / 1440 \) (71v^10 - 839v^8 + 7511v^6 - 29677v^4 + 88813*v^2 - 4957) / 1440
\(\beta_{7}\)	\(=\)	\( ( -101\nu^{10} + 1229\nu^{8} - 11061\nu^{6} + 44407\nu^{4} - 131503\nu^{2} + 3687 ) / 960 \) (-101v^10 + 1229v^8 - 11061v^6 + 44407v^4 - 131503*v^2 + 3687) / 960
\(\beta_{8}\)	\(=\)	\( ( -409\nu^{11} + 4921\nu^{9} - 44329\nu^{7} + 177203\nu^{5} - 530387\nu^{3} + 29603\nu ) / 1440 \) (-409v^11 + 4921v^9 - 44329v^7 + 177203v^5 - 530387v^3 + 29603v) / 1440
\(\beta_{9}\)	\(=\)	\( ( 421\nu^{11} - 5029\nu^{9} + 45181\nu^{7} - 178487\nu^{5} + 530423\nu^{3} + 14593\nu ) / 1440 \) (421v^11 - 5029v^9 + 45181v^7 - 178487v^5 + 530423v^3 + 14593v) / 1440
\(\beta_{10}\)	\(=\)	\( ( 749\nu^{11} - 8981\nu^{9} + 80789\nu^{7} - 321103\nu^{5} + 957127\nu^{3} - 12103\nu ) / 1440 \) (749v^11 - 8981v^9 + 80789v^7 - 321103v^5 + 957127v^3 - 12103v) / 1440
\(\beta_{11}\)	\(=\)	\( ( 161\nu^{11} - 1929\nu^{9} + 17361\nu^{7} - 69067\nu^{5} + 206403\nu^{3} - 5787\nu ) / 240 \) (161v^11 - 1929v^9 + 17361v^7 - 69067v^5 + 206403v^3 - 5787v) / 240

\(\nu\)	\(=\)	\( ( \beta_{9} + \beta_{8} + \beta_{3} ) / 2 \) (b9 + b8 + b3) / 2
\(\nu^{2}\)	\(=\)	\( ( \beta_{7} - \beta_{6} + 2\beta_{5} + 9\beta_{4} - \beta_{2} + 7 ) / 2 \) (b7 - b6 + 2b5 + 9b4 - b2 + 7) / 2
\(\nu^{3}\)	\(=\)	\( ( -\beta_{11} - 12\beta_{10} + 24\beta_{9} + 12\beta_{3} + \beta_1 ) / 4 \) (-b11 - 12b10 + 24b9 + 12*b3 + b1) / 4
\(\nu^{4}\)	\(=\)	\( ( 5\beta_{7} - 14\beta_{6} + 7\beta_{5} + 55\beta_{4} - 7 ) / 2 \) (5b7 - 14b6 + 7b5 + 55b4 - 7) / 2
\(\nu^{5}\)	\(=\)	\( ( -11\beta_{11} - 70\beta_{10} + 74\beta_{9} - 78\beta_{8} ) / 4 \) (-11b11 - 70b10 + 74b9 - 78b8) / 4
\(\nu^{6}\)	\(=\)	\( -24\beta_{6} - 24\beta_{5} + 12\beta_{2} - 145 \) -24b6 - 24b5 + 12*b2 - 145
\(\nu^{7}\)	\(=\)	\( ( -229\beta_{9} - 229\beta_{8} - 205\beta_{3} - 48\beta_1 ) / 2 \) (-229b9 - 229b8 - 205b3 - 48b1) / 2
\(\nu^{8}\)	\(=\)	\( ( -109\beta_{7} + 325\beta_{6} - 650\beta_{5} - 2085\beta_{4} + 109\beta_{2} - 1435 ) / 2 \) (-109b7 + 325b6 - 650b5 - 2085b4 + 109*b2 - 1435) / 2
\(\nu^{9}\)	\(=\)	\( ( 757\beta_{11} + 2412\beta_{10} - 5688\beta_{9} + 432\beta_{8} - 2412\beta_{3} - 757\beta_1 ) / 4 \) (757b11 + 2412b10 - 5688b9 + 432b8 - 2412b3 - 757b1) / 4
\(\nu^{10}\)	\(=\)	\( ( -449\beta_{7} + 4358\beta_{6} - 2179\beta_{5} - 12907\beta_{4} + 2179 ) / 2 \) (-449b7 + 4358b6 - 2179b5 - 12907b4 + 2179) / 2
\(\nu^{11}\)	\(=\)	\( ( 5639\beta_{11} + 14254\beta_{10} - 17714\beta_{9} + 21174\beta_{8} ) / 4 \) (5639b11 + 14254b10 - 17714b9 + 21174b8) / 4

\(n\)	\(261\)	\(287\)	\(353\)
\(\chi(n)\)	\(1\)	\(1\)	\(-\beta_{4}\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 225.6
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{12} \) T^12
$3$	\( T^{12} + 15 T^{10} + \cdots + 2304 \) T^12 + 15T^10 + 165T^8 + 804T^6 + 2880T^4 + 2880*T^2 + 2304
$5$	\( (T^{6} + 18 T^{4} + \cdots + 12)^{2} \) (T^6 + 18T^4 + 33T^2 + 12)^2
$7$	\( T^{12} - 33 T^{10} + \cdots + 65536 \) T^12 - 33T^10 + 849T^8 - 7408T^6 + 49152T^4 - 61440*T^2 + 65536
$11$	\( T^{12} - 33 T^{10} + \cdots + 65536 \) T^12 - 33T^10 + 849T^8 - 7408T^6 + 49152T^4 - 61440*T^2 + 65536
$13$	\( (T^{6} - 6 T^{5} + \cdots + 2197)^{2} \) (T^6 - 6T^5 - 6T^4 + 106T^3 - 78T^2 - 1014*T + 2197)^2
$17$	\( (T^{2} + T + 1)^{6} \) (T^2 + T + 1)^6
$19$	\( T^{12} - 45 T^{10} + \cdots + 1679616 \) T^12 - 45T^10 + 1485T^8 - 21708T^6 + 233280T^4 - 699840*T^2 + 1679616
$23$	\( T^{12} + \cdots + 2127792384 \) T^12 + 135T^10 + 13413T^8 + 557364T^6 + 16928064T^4 + 221967936*T^2 + 2127792384
$29$	\( (T^{6} + 3 T^{5} + \cdots + 961)^{2} \) (T^6 + 3T^5 + 30T^4 - T^3 + 534T^2 + 651T + 961)^2
$31$	\( (T^{6} + 96 T^{4} + \cdots + 576)^{2} \) (T^6 + 96T^4 + 1152T^2 + 576)^2
$37$	\( (T^{6} + 3 T^{5} + \cdots + 5043)^{2} \) (T^6 + 3T^5 - 36T^4 - 117T^3 + 1644T^2 - 4797*T + 5043)^2
$41$	\( (T^{6} - 15 T^{5} + \cdots + 6627)^{2} \) (T^6 - 15T^5 + 72T^4 + 45T^3 - 696T^2 - 423*T + 6627)^2
$43$	\( T^{12} + \cdots + 150994944 \) T^12 + 99T^10 + 7113T^8 + 241536T^6 + 6008832T^4 + 33030144*T^2 + 150994944
$47$	\( (T^{6} + 108 T^{4} + \cdots + 43264)^{2} \) (T^6 + 108T^4 + 3792T^2 + 43264)^2
$53$	\( (T^{3} + 12 T^{2} + 27 T - 4)^{4} \) (T^3 + 12T^2 + 27T - 4)^4
$59$	\( T^{12} + \cdots + 10485760000 \) T^12 - 153T^10 + 16305T^8 - 882112T^6 + 34799616T^4 - 727449600*T^2 + 10485760000
$61$	\( (T^{6} - 9 T^{5} + \cdots + 56169)^{2} \) (T^6 - 9T^5 + 102T^4 - 285T^3 + 2574T^2 - 4977*T + 56169)^2
$67$	\( T^{12} + \cdots + 385571451136 \) T^12 - 285T^10 + 56685T^8 - 5752012T^6 + 425242560T^4 - 15237965760*T^2 + 385571451136
$71$	\( T^{12} - 69 T^{10} + \cdots + 160000 \) T^12 - 69T^10 + 4365T^8 - 26524T^6 + 129216T^4 - 158400*T^2 + 160000
$73$	\( (T^{6} + 234 T^{4} + \cdots + 288300)^{2} \) (T^6 + 234T^4 + 16089T^2 + 288300)^2
$79$	\( (T^{6} - 396 T^{4} + \cdots - 786432)^{2} \) (T^6 - 396T^4 + 43008T^2 - 786432)^2
$83$	\( (T^{6} + 228 T^{4} + \cdots + 16384)^{2} \) (T^6 + 228T^4 + 6912T^2 + 16384)^2
$89$	\( (T^{6} - 15 T^{5} + \cdots + 2157312)^{2} \) (T^6 - 15T^5 - 105T^4 + 2700T^3 + 19680T^2 - 457920*T + 2157312)^2
$97$	\( (T^{6} + 45 T^{5} + \cdots + 62208)^{2} \) (T^6 + 45T^5 + 855T^4 + 8100T^3 + 38880T^2 + 77760*T + 62208)^2
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