LMFDB - Newform orbit 207.4.g.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [207,4,Mod(68,207)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(207, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([5, 3]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("207.68");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 207 = 3^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	207.g (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:	$12.2133953712$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$6$ over $\Q(\zeta_{6})$
Coefficient field:	12.0.57352136505929721.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 17x^{9} + 73x^{6} - 3672x^{3} + 46656 $ x^12 - 17x^9 + 73x^6 - 3672*x^3 + 46656
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 3 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{6}]$

$q$-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_{11} + 2 \beta_{10} + \cdots - \beta_{4}) q^{2}+ \cdots + ( - 10 \beta_{10} - 5 \beta_{9} + \cdots - 8 \beta_{4}) q^{9}+O(q^{10}) $ q + (-b11 + 2b10 + b9 + b8 + b6 - b4) q^2 + (2b11 + b10 - b9 + 2b7 - b6) * q^3 + (-3b11 + 3b10 + 4b9 + 2b8 - b6 + 3b5 + 8b2) * q^4 + (5b10 - 5b9 + 10b6 - b5 - b4 - 6b2 - 5b1 + 6) q^6 + (-8b11 + 8b10 + 8b9 - 8b7 - 8b6 + 8b5 + b3 + 46b2 - b1 - 23) q^8 + (-10b10 - 5b9 - 5b6 - 8b4) * q^9	$ q + ( - \beta_{11} + 2 \beta_{10} + \cdots - \beta_{4}) q^{2}+ \cdots + (343 \beta_{11} - 343 \beta_{10} + \cdots - 343 \beta_{5}) q^{98}+O(q^{100}) $ q + (-b11 + 2b10 + b9 + b8 + b6 - b4) q^2 + (2b11 + b10 - b9 + 2b7 - b6) * q^3 + (-3b11 + 3b10 + 4b9 + 2b8 - b6 + 3b5 + 8b2) * q^4 + (5b10 - 5b9 + 10b6 - b5 - b4 - 6b2 - 5b1 + 6) q^6 + (-8b11 + 8b10 + 8b9 - 8b7 - 8b6 + 8b5 + b3 + 46b2 - b1 - 23) q^8 + (-10b10 - 5b9 - 5b6 - 8b4) * q^9 + (6b10 - 30b9 - 24b8 + 16b7 + 12b6 - b5 - b4 + 13b3 + 13b2 - 13b1 + 21) * q^12 + (4b10 + 39b9 + 43b8 - 12b7 + 8b6 - 12b5 - 12b4) q^13 + (-24b11 - b10 + 22b9 - 29b8 - 45b7 - 68b6 + 45b5 + 24b4 + 64b2 - 64) * q^16 + (7b11 + 52b10 + 26b9 + 78b8 + 26b6 - 31b3 + 2b2) q^18 + (46b3 + 23b2) * q^23 + (-47b11 + 16b10 - 31b9 - 135b8 + 43b7 - 31b6 - 8b4 + 40b3 + 40b2 - 40b1 + 48) * q^24 + (-125b2 + 125) q^25 + (-32b10 + 51b9 + 134b8 - 19b7 + 51b6 - 19b4 - 110b3 - 110b2 + 110b1 + 55) q^26 + (-141b2 + 22b1 + 141) * q^27 + (-94b10 - 188b9 - 127b8 - 28b7 - 33b6 - 28b5) * q^29 + (-6b11 + 173b10 + 2b9 - 6b7 + 2b6 - 6b4) * q^31 + (-71b11 - 120b10 - 41b9 - 64b8 - 135b7 - 169b6 + 64b5 + 135b4 + 8b3 + 184b2 - 360) * q^32 + (29b11 + 33b10 - 153b9 + 29b7 - 33b6 + 64b5 - 85b3 + 29b2) * q^36 + (5b10 + 10b9 - 5b6 - 98b5 + 50b3 - 187b2) * q^39 + (-64b11 + 39b10 - 103b9 + 181b8 + 142b6 - 64b5) * q^41 + (-69b11 - 23b10 + 253b9 - 69b7 + 23b6 - 69b5) * q^46 + (-134b11 - 16b10 + 16b9 - 134b7 + 59b6 - 134b5 - 134b4) q^47 + (-128b11 + 209b10 + 274b9 - 192b8 - 65b6 + 37b5 - 8b4 + 225b3 + 393b2 - 121b1 + 102) * q^48 - 343b2 q^49 + (125b10 + 125b8 + 125b7 + 250b6 - 125b5 - 125b4) * q^50 + (165b11 + 110b10 + 55b9 + 715b8 + 55b6 - 165b4 - 306b3 - 307b2 + 153b1 + 154) q^52 + (141b10 + 132b9 + 273b8 + 97b7 + 282b6 - 163b5 - 163b4) q^54 + (183b11 + 20b10 - 61b9 + 183b7 - 61b6 + 183b4 + 99b3 - 224b2 - 198b1 - 99) q^58 + (170b3 - 113b2 + 396) * q^59 + (-181b11 - 165b10 - 149b9 - 314b8 - 16b6 + 181b5 + 181b4 + 163b3 + 748b2 - 163b1 - 374) * q^62 + (168b11 - 544b10 - 136b9 - 360b8 - 192b7 - 536b6 + 168b5 + 360b4 + 45b3 + 45b1 - 990) * q^64 + (-161b10 - 322b9 - 483b8 - 92b7 - 322b6) q^69 + (-46b11 - 219b10 - 392b9 - 611b8 + 173b6 + 46b5 + 46b4) q^71 + (56b11 + 245b10 - 527b9 + 56b7 + 148b6 + 360b5 + 161b4 + 264b2 - 248b1 - 264) q^72 + (-48b11 - 621b10 - 16b9 - 637b8 - 605b6 - 48b5 - 48b4) q^73 + (250b11 + 250b10 + 125b9 + 375b8 + 125b6) q^75 + (220b11 - 443b10 + 269b9 + 220b7 + 443b6 - 287b5 - 573b2 + 211b1 + 573) * q^78 + (304b11 + 106b10 + 53b9 + 159b8 + 53b6) q^81 + (309b11 + 103b10 + 100b9 + 309b7 - 103b6 + 309b5 - 117b3 - 117b1 + 629) * q^82 + (239b10 + 478b9 - 239b6 + 226b5 - 338b3 - 338b2 + 338b1 + 129) q^87 + (-161b11 - 529b10 + 529b9 - 161b7 - 184b6 - 161b5 - 161b4 - 184b2 + 368b1 + 184) q^92 + (386b10 + 193b9 + 193b6 - 332b4 + 453b2 - 356b1 - 453) * q^93 + (-59b10 + 777b9 + 718b8 + 177b7 - 118b6 + 177b5 + 177b4 - 225b3 + 397b2 + 450b1 + 225) * q^94 + (-720b11 + 372b10 + 894b9 - 624b8 - 376b7 - 66b6 + 64b5 + 19b4 + 320b3 + 1265b2 + 265b1 - 561) q^96 + (343b11 - 343b10 - 343b9 + 343b7 + 343b6 - 343b5) * q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 48 q^{4} + 51 q^{6}+O(q^{10}) $ 12 * q + 48 * q^4 + 51 * q^6	$ 12 q + 48 q^{4} + 51 q^{6} + 330 q^{12} - 384 q^{16} + 105 q^{18} + 816 q^{24} + 750 q^{25} + 780 q^{27} - 3240 q^{32} + 429 q^{36} - 1272 q^{39} + 3270 q^{48} - 2058 q^{49} + 465 q^{52} - 2235 q^{58} + 3564 q^{59} - 12150 q^{64} - 840 q^{72} + 2805 q^{78} + 8250 q^{82} - 480 q^{87} - 1650 q^{93} + 4407 q^{94} - 897 q^{96}+O(q^{100}) $ 12 * q + 48 * q^4 + 51 * q^6 + 330 * q^12 - 384 * q^16 + 105 * q^18 + 816 * q^24 + 750 * q^25 + 780 * q^27 - 3240 * q^32 + 429 * q^36 - 1272 * q^39 + 3270 * q^48 - 2058 * q^49 + 465 * q^52 - 2235 * q^58 + 3564 * q^59 - 12150 * q^64 - 840 * q^72 + 2805 * q^78 + 8250 * q^82 - 480 * q^87 - 1650 * q^93 + 4407 * q^94 - 897 * q^96

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{12} - 17x^{9} + 73x^{6} - 3672x^{3} + 46656 $ x^12 - 17*x^9 + 73*x^6 - 3672*x^3 + 46656 :

$\beta_{1}$	$=$	$ ( -29\nu^{9} - 803\nu^{6} - 2117\nu^{3} + 106488 ) / 78840 $ (-29v^9 - 803v^6 - 2117*v^3 + 106488) / 78840
$\beta_{2}$	$=$	$ ( 17\nu^{9} - 73\nu^{6} + 1241\nu^{3} - 46656 ) / 15768 $ (17v^9 - 73v^6 + 1241*v^3 - 46656) / 15768
$\beta_{3}$	$=$	$ ( 17\nu^{9} - 73\nu^{6} - 1387\nu^{3} - 46656 ) / 13140 $ (17v^9 - 73v^6 - 1387*v^3 - 46656) / 13140
$\beta_{4}$	$=$	$ ( \nu^{10} - 5059\nu ) / 2190 $ (v^10 - 5059*v) / 2190
$\beta_{5}$	$=$	$ ( 323\nu^{10} - 1387\nu^{7} + 39347\nu^{4} - 886464\nu ) / 473040 $ (323v^10 - 1387v^7 + 39347v^4 - 886464v) / 473040
$\beta_{6}$	$=$	$ ( - 323 \nu^{11} + 174 \nu^{10} + 1387 \nu^{8} + 4818 \nu^{7} - 39347 \nu^{5} + 12702 \nu^{4} + \cdots - 638928 \nu ) / 2838240 $ (-323v^11 + 174v^10 + 1387v^8 + 4818v^7 - 39347v^5 + 12702v^4 + 886464v^2 - 638928v) / 2838240
$\beta_{7}$	$=$	$ ( -\nu^{11} + 17\nu^{8} - 73\nu^{5} + 3672\nu^{2} ) / 7776 $ (-v^11 + 17v^8 - 73v^5 + 3672*v^2) / 7776
$\beta_{8}$	$=$	$ ( -\nu^{11} - 6\nu^{10} + 5059\nu^{2} + 17214\nu ) / 13140 $ (-v^11 - 6v^10 + 5059v^2 + 17214*v) / 13140
$\beta_{9}$	$=$	$ ( - 323 \nu^{11} + 1122 \nu^{10} + 1387 \nu^{8} - 4818 \nu^{7} - 39347 \nu^{5} - 12702 \nu^{4} + \cdots - 3079296 \nu ) / 2838240 $ (-323v^11 + 1122v^10 + 1387v^8 - 4818v^7 - 39347v^5 - 12702v^4 + 886464v^2 - 3079296v) / 2838240
$\beta_{10}$	$=$	$ ( 323\nu^{11} + 1296\nu^{10} - 1387\nu^{8} + 39347\nu^{5} - 886464\nu^{2} - 3718224\nu ) / 2838240 $ (323v^11 + 1296v^10 - 1387v^8 + 39347v^5 - 886464v^2 - 3718224v) / 2838240
$\beta_{11}$	$=$	$ ( -\nu^{11} + 2431\nu^{2} ) / 2628 $ (-v^11 + 2431*v^2) / 2628

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

$\nu$	$=$	$ ( 2\beta_{10} + \beta_{9} + \beta_{6} - 3\beta_{4} ) / 3 $ (2b10 + b9 + b6 - 3b4) / 3
$\nu^{2}$	$=$	$ ( -3\beta_{11} + 10\beta_{10} + 5\beta_{9} + 15\beta_{8} + 5\beta_{6} ) / 3 $ (-3b11 + 10b10 + 5b9 + 15b8 + 5*b6) / 3
$\nu^{3}$	$=$	$ -5\beta_{3} + 6\beta_{2} $ -5b3 + 6b2
$\nu^{4}$	$=$	$ ( -19\beta_{10} - 38\beta_{9} + 19\beta_{6} + 33\beta_{5} ) / 3 $ (-19b10 - 38b9 + 19b6 + 33b5) / 3
$\nu^{5}$	$=$	$ ( 57\beta_{11} + 85\beta_{10} - 85\beta_{9} + 57\beta_{7} - 85\beta_{6} ) / 3 $ (57b11 + 85b10 - 85b9 + 57b7 - 85*b6) / 3
$\nu^{6}$	$=$	$ -29\beta_{2} - 85\beta _1 + 29 $ -29b2 - 85b1 + 29
$\nu^{7}$	$=$	$ ( 539\beta_{10} - 539\beta_{9} + 1078\beta_{6} - 87\beta_{5} - 87\beta_{4} ) / 3 $ (539b10 - 539b9 + 1078b6 - 87b5 - 87*b4) / 3
$\nu^{8}$	$=$	$ ( -365\beta_{10} - 730\beta_{9} - 1095\beta_{8} + 1617\beta_{7} - 730\beta_{6} ) / 3 $ (-365b10 - 730b9 - 1095b8 + 1617b7 - 730*b6) / 3
$\nu^{9}$	$=$	$ 365\beta_{3} + 365\beta_{2} - 365\beta _1 + 2869 $ 365b3 + 365b2 - 365*b1 + 2869
$\nu^{10}$	$=$	$ ( 10118\beta_{10} + 5059\beta_{9} + 5059\beta_{6} - 8607\beta_{4} ) / 3 $ (10118b10 + 5059b9 + 5059b6 - 8607b4) / 3
$\nu^{11}$	$=$	$ ( -15177\beta_{11} + 24310\beta_{10} + 12155\beta_{9} + 36465\beta_{8} + 12155\beta_{6} ) / 3 $ (-15177b11 + 24310b10 + 12155b9 + 36465b8 + 12155*b6) / 3

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/207\mathbb{Z}\right)^\times$.

$n$	$28$	$47$
$\chi(n)$	$-1$	$\beta_{2}$

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} $

68.1

0.350289	+	2.42431i
−1.15833	+	2.15830i
−1.28998	−	2.08229i
−2.27466	−	0.908798i
1.92437	−	1.51552i
2.44831	−	0.0760102i
0.350289	−	2.42431i
−1.15833	−	2.15830i
−1.28998	+	2.08229i
−2.27466	+	0.908798i
1.92437	+	1.51552i
2.44831	+	0.0760102i

−3.27982

1.89361i

5.07027

1.13683i

3.17148

−

5.49317i

−18.7823

5.87250i

−

6.27554i

24.4152

11.5280i

68.2

−3.01424

1.74027i

−0.239114

−

5.19065i

2.05710

−

3.56301i

9.75390

15.2297i

−

13.5247i

−26.8856

2.48232i

68.3

−1.83739

1.06082i

−4.37568

2.80240i

−1.74933

3.02993i

5.06699

−

9.79090i

−

24.3960i

11.2931

−

24.5248i

68.4

−1.51161

0.872729i

−1.55061

−

4.95940i

−2.47669

4.28975i

6.67213

6.14341i

−

22.6096i

−22.1912

15.3802i

68.5

4.79143

−

2.76633i

−3.51966

3.82257i

11.3052

−

19.5812i

−6.28969

28.0521i

−

80.8346i

−2.22404

−

26.9082i

68.6

4.85163

−

2.80109i

4.61479

2.38824i

11.6922

−

20.2515i

29.0790

−

1.33956i

−

86.1865i

15.5926

22.0425i

137.1