LMFDB - Newform orbit 1900.2.s.c

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1900,2,Mod(49,1900)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1900, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 3, 4]))

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

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

chi := DirichletCharacter("1900.49");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1900 = 2^{2} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1900.s (of order $6$, degree $2$, not 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:	$15.1715763840$
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} - 9x^{10} + 59x^{8} - 180x^{6} + 403x^{4} - 198x^{2} + 81 $ x^12 - 9x^10 + 59x^8 - 180x^6 + 403x^4 - 198*x^2 + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 3^{2} $
Twist minimal:	no (minimal twist has level 380)
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_{11} + \beta_{6}) q^{3} + (\beta_{11} + \beta_{10}) q^{7} + (\beta_{5} - \beta_{3} + \cdots - 2 \beta_1) q^{9}+O(q^{10}) $ q + (b11 + b6) * q^3 + (b11 + b10) * q^7 + (b5 - b3 + 3b2 - 2b1) * q^9	$ q + (\beta_{11} + \beta_{6}) q^{3} + (\beta_{11} + \beta_{10}) q^{7} + (\beta_{5} - \beta_{3} + \cdots - 2 \beta_1) q^{9}+ \cdots + (5 \beta_{8} + \beta_{5} + \cdots - 2 \beta_1) q^{99}+O(q^{100}) $ q + (b11 + b6) * q^3 + (b11 + b10) * q^7 + (b5 - b3 + 3b2 - 2b1) * q^9 + (b5 + 2) * q^11 - b7 * q^13 + (2b9 - b6 + b4) q^17 + (b5 + b2 + b1) * q^19 + (b3 + 6b2 - b1 - 6) q^21 + (-b9 - 5b7 + b4) q^23 + (4b11 - 3b10 + b9 + b6 + 2b4) q^27 + (b8 + 2b5 - 2b3 + 3b2 - 4b1) * q^29 + (b8 - 2b3 - b1 + 4) q^31 + (6b10 - 2b9 - 6b7 + b6 - b4) q^33 + (b11 + b10) * q^37 + b5 * q^39 + (3b8 - 3b5 + 3b2 - 3) q^41 + (-b11 - 3b10 + 2b9 + 3b7 - 2b6 + b4) * q^43 + (2b9 + 3b7 - b6 - 2b4) q^47 + (-b8 - 2b5 + 2b3 + b1) * q^49 + (-2b8 - b5 + b3 - 9b2 + 2b1) q^51 + (b9 - 4b7 - b4) q^53 + (-3b11 + 9b10 - 3b9 - 9b7 - b6 - 2b4) q^57 + (-b8 + b5 + 2b3 + 3b2 - 2b1 - 3) q^59 + (-2b8 - 3b2) * q^61 - 5b6 q^63 + (2b9 - 2b6 - 2b4) q^67 + (-b8 + 7b5 + 2b3 + b1 - 3) * q^69 + (2b8 - 2b5 - b3 - 3b2 + b1 + 3) q^71 + (-7b10 + 2b9 + 7b7 - b6 + b4) q^73 + (b11 + 8b10 - b9 - b6 - 2b4) * q^77 + (2b8 - 2b5 - 10b2 + 10) q^79 + (-5b8 + 5b5 + 6b2 - 6) q^81 + (2b11 + 10b10 + b9 + b6 + 2b4) q^83 + (9b11 + b9 + b6 + 2b4) * q^87 + (4b8 + 2b2) * q^89 + (b8 + b2) * q^91 + (8b11 - 3b10 + 2b9 + 3b7 + 7b6 + b4) q^93 + (5b11 - 4b10 + 4b9 + 4b7 + 3b6 + 2b4) * q^97 + (5b8 + b5 - b3 + 3b2 - 2b1) q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 16 q^{9}+O(q^{10}) $ 12 * q + 16 * q^9	$ 12 q + 16 q^{9} + 20 q^{11} - 32 q^{21} + 12 q^{29} + 44 q^{31} - 4 q^{39} - 12 q^{41} + 12 q^{49} - 48 q^{51} - 12 q^{59} - 14 q^{61} - 60 q^{69} + 18 q^{71} + 64 q^{79} - 46 q^{81} + 4 q^{89} + 4 q^{91} + 6 q^{99}+O(q^{100}) $ 12 * q + 16 * q^9 + 20 * q^11 - 32 * q^21 + 12 * q^29 + 44 * q^31 - 4 * q^39 - 12 * q^41 + 12 * q^49 - 48 * q^51 - 12 * q^59 - 14 * q^61 - 60 * q^69 + 18 * q^71 + 64 * q^79 - 46 * q^81 + 4 * q^89 + 4 * q^91 + 6 * q^99

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{12} - 9x^{10} + 59x^{8} - 180x^{6} + 403x^{4} - 198x^{2} + 81 $ x^12 - 9*x^10 + 59*x^8 - 180*x^6 + 403*x^4 - 198*x^2 + 81 :

$\beta_{1}$	$=$	$ ( 117\nu^{10} - 767\nu^{8} + 7472\nu^{6} - 27234\nu^{4} + 90554\nu^{2} - 60864 ) / 21995 $ (117v^10 - 767v^8 + 7472v^6 - 27234v^4 + 90554*v^2 - 60864) / 21995
$\beta_{2}$	$=$	$ ( -1298\nu^{10} + 10953\nu^{8} - 71803\nu^{6} + 202311\nu^{4} - 490451\nu^{2} + 240966 ) / 197955 $ (-1298v^10 + 10953v^8 - 71803v^6 + 202311v^4 - 490451*v^2 + 240966) / 197955
$\beta_{3}$	$=$	$ ( 461\nu^{10} - 5466\nu^{8} + 28501\nu^{6} - 98847\nu^{4} + 142112\nu^{2} - 186237 ) / 65985 $ (461v^10 - 5466v^8 + 28501v^6 - 98847v^4 + 142112*v^2 - 186237) / 65985
$\beta_{4}$	$=$	$ ( 569\nu^{11} - 6174\nu^{9} + 40474\nu^{7} - 169668\nu^{5} + 474413\nu^{3} - 729693\nu ) / 197955 $ (569v^11 - 6174v^9 + 40474v^7 - 169668v^5 + 474413v^3 - 729693v) / 197955
$\beta_{5}$	$=$	$ ( -288\nu^{10} + 1888\nu^{8} - 9933\nu^{6} + 12896\nu^{4} - 6336\nu^{2} - 68439 ) / 21995 $ (-288v^10 + 1888v^8 - 9933v^6 + 12896v^4 - 6336*v^2 - 68439) / 21995
$\beta_{6}$	$=$	$ ( 81\nu^{11} - 531\nu^{9} + 3481\nu^{7} - 3627\nu^{5} + 1782\nu^{3} + 98293\nu ) / 21995 $ (81v^11 - 531v^9 + 3481v^7 - 3627v^5 + 1782v^3 + 98293v) / 21995
$\beta_{7}$	$=$	$ ( 288\nu^{11} - 1888\nu^{9} + 9933\nu^{7} - 12896\nu^{5} + 6336\nu^{3} + 90434\nu ) / 65985 $ (288v^11 - 1888v^9 + 9933v^7 - 12896v^5 + 6336v^3 + 90434v) / 65985
$\beta_{8}$	$=$	$ ( 1138\nu^{10} - 12348\nu^{8} + 80948\nu^{6} - 273351\nu^{4} + 552916\nu^{2} - 271656 ) / 65985 $ (1138v^10 - 12348v^8 + 80948v^6 - 273351v^4 + 552916*v^2 - 271656) / 65985
$\beta_{9}$	$=$	$ ( 2027\nu^{11} - 15732\nu^{9} + 103132\nu^{7} - 234954\nu^{5} + 506489\nu^{3} + 445716\nu ) / 197955 $ (2027v^11 - 15732v^9 + 103132v^7 - 234954v^5 + 506489v^3 + 445716v) / 197955
$\beta_{10}$	$=$	$ ( 88\nu^{11} - 783\nu^{9} + 4868\nu^{7} - 13716\nu^{5} + 26581\nu^{3} - 2916\nu ) / 7155 $ (88v^11 - 783v^9 + 4868v^7 - 13716v^5 + 26581v^3 - 2916v) / 7155
$\beta_{11}$	$=$	$ ( -1298\nu^{11} + 10953\nu^{9} - 71803\nu^{7} + 202311\nu^{5} - 450860\nu^{3} + 43011\nu ) / 39591 $ (-1298v^11 + 10953v^9 - 71803v^7 + 202311v^5 - 450860v^3 + 43011v) / 39591

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

$\nu$	$=$	$ ( -\beta_{9} + 2\beta_{6} + \beta_{4} ) / 3 $ (-b9 + 2*b6 + b4) / 3
$\nu^{2}$	$=$	$ ( -2\beta_{8} + 2\beta_{5} + \beta_{3} - 9\beta_{2} - \beta _1 + 9 ) / 3 $ (-2b8 + 2b5 + b3 - 9*b2 - b1 + 9) / 3
$\nu^{3}$	$=$	$ ( 3\beta_{11} + 5\beta_{9} + 5\beta_{6} + 10\beta_{4} ) / 3 $ (3b11 + 5b9 + 5b6 + 10b4) / 3
$\nu^{4}$	$=$	$ -\beta_{8} + 2\beta_{5} - 2\beta_{3} - 12\beta_{2} - 4\beta_1 $ -b8 + 2b5 - 2b3 - 12b2 - 4b1
$\nu^{5}$	$=$	$ ( 18\beta_{11} + 9\beta_{10} + 46\beta_{9} - 9\beta_{7} - 5\beta_{6} + 23\beta_{4} ) / 3 $ (18b11 + 9b10 + 46b9 - 9b7 - 5b6 + 23b4) / 3
$\nu^{6}$	$=$	$ ( 32\beta_{8} - 5\beta_{5} - 64\beta_{3} - 32\beta _1 - 153 ) / 3 $ (32b8 - 5b5 - 64b3 - 32b1 - 153) / 3
$\nu^{7}$	$=$	$ ( 106\beta_{9} - 81\beta_{7} - 116\beta_{6} - 106\beta_{4} ) / 3 $ (106b9 - 81b7 - 116b6 - 106b4) / 3
$\nu^{8}$	$=$	$ 51\beta_{8} - 51\beta_{5} - 55\beta_{3} + 222\beta_{2} + 55\beta _1 - 222 $ 51b8 - 51b5 - 55b3 + 222b2 + 55*b1 - 222
$\nu^{9}$	$=$	$ ( -495\beta_{11} - 531\beta_{10} - 493\beta_{9} - 493\beta_{6} - 986\beta_{4} ) / 3 $ (-495b11 - 531b10 - 493b9 - 493b6 - 986*b4) / 3
$\nu^{10}$	$=$	$ ( -191\beta_{8} - 835\beta_{5} + 835\beta_{3} + 2952\beta_{2} + 1670\beta_1 ) / 3 $ (-191b8 - 835b5 + 835b3 + 2952b2 + 1670*b1) / 3
$\nu^{11}$	$=$	$ ( -2505\beta_{11} - 3078\beta_{10} - 4624\beta_{9} + 3078\beta_{7} - 193\beta_{6} - 2312\beta_{4} ) / 3 $ (-2505b11 - 3078b10 - 4624b9 + 3078b7 - 193b6 - 2312b4) / 3

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1900\mathbb{Z}\right)^\times$.

$n$	$77$	$401$	$951$
$\chi(n)$	$-1$	$-\beta_{2}$	$1$

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} $

49.1

−0.617942	+	0.356769i
−1.65604	+	0.956115i
−1.90412	+	1.09935i
1.90412	−	1.09935i
1.65604	−	0.956115i
0.617942	−	0.356769i
−0.617942	−	0.356769i
−1.65604	−	0.956115i
−1.90412	−	1.09935i
1.90412	+	1.09935i
1.65604	+	0.956115i
0.617942	+	0.356769i

−2.77509

−

1.60220i

−

2.20440i

3.63409

6.29444i

49.2

−2.22469

−

1.28442i

−

3.56885i

1.79949

3.11682i

49.3

−0.315621

−

0.182224i

0.635552i

−1.43359

−

2.48305i

49.4

0.315621

0.182224i

−

0.635552i

−1.43359

−

2.48305i

49.5

2.22469

1.28442i

3.56885i

1.79949

3.11682i

49.6

2.77509

1.60220i

2.20440i

3.63409

6.29444i

349.1

−2.77509

1.60220i

2.20440i

3.63409

−

6.29444i

349.2

−2.22469

1.28442i

3.56885i

1.79949

−

3.11682i

349.3

−0.315621

0.182224i

−

0.635552i

−1.43359

2.48305i

349.4

0.315621

−

0.182224i

0.635552i

−1.43359

2.48305i

349.5

2.22469

−

1.28442i

−

3.56885i

1.79949

−

3.11682i

349.6

2.77509

−

1.60220i

−

2.20440i

3.63409

−

6.29444i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.b	even	2	1	inner
19.c	even	3	1	inner
95.i	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1900.2.s.c		12
5.b	even	2	1	inner	1900.2.s.c		12
5.c	odd	4	1		380.2.i.b	✓	6
5.c	odd	4	1		1900.2.i.c		6
15.e	even	4	1		3420.2.t.v		6
19.c	even	3	1	inner	1900.2.s.c		12
20.e	even	4	1		1520.2.q.i		6
95.i	even	6	1	inner	1900.2.s.c		12
95.l	even	12	1		7220.2.a.o		3
95.m	odd	12	1		380.2.i.b	✓	6
95.m	odd	12	1		1900.2.i.c		6
95.m	odd	12	1		7220.2.a.n		3
285.v	even	12	1		3420.2.t.v		6
380.v	even	12	1		1520.2.q.i		6

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
380.2.i.b	✓	6	5.c	odd	4	1
380.2.i.b	✓	6	95.m	odd	12	1
1520.2.q.i		6	20.e	even	4	1
1520.2.q.i		6	380.v	even	12	1
1900.2.i.c		6	5.c	odd	4	1
1900.2.i.c		6	95.m	odd	12	1
1900.2.s.c		12	1.a	even	1	1	trivial
1900.2.s.c		12	5.b	even	2	1	inner
1900.2.s.c		12	19.c	even	3	1	inner
1900.2.s.c		12	95.i	even	6	1	inner
3420.2.t.v		6	15.e	even	4	1
3420.2.t.v		6	285.v	even	12	1
7220.2.a.n		3	95.m	odd	12	1
7220.2.a.o		3	95.l	even	12	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{3}^{12} - 17T_{3}^{10} + 219T_{3}^{8} - 1172T_{3}^{6} + 4747T_{3}^{4} - 630T_{3}^{2} + 81 $ T3^12 - 17*T3^10 + 219*T3^8 - 1172*T3^6 + 4747*T3^4 - 630*T3^2 + 81 acting on $S_{2}^{\mathrm{new}}(1900, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{12} $ T^12
$3$	$ T^{12} - 17 T^{10} + \cdots + 81 $ T^12 - 17T^10 + 219T^8 - 1172T^6 + 4747T^4 - 630*T^2 + 81
$5$	$ T^{12} $ T^12
$7$	$ (T^{6} + 18 T^{4} + \cdots + 25)^{2} $ (T^6 + 18T^4 + 69T^2 + 25)^2
$11$	$ (T^{3} - 5 T^{2} + 9)^{4} $ (T^3 - 5*T^2 + 9)^4
$13$	$ (T^{4} - T^{2} + 1)^{3} $ (T^4 - T^2 + 1)^3
$17$	$ T^{12} - 81 T^{10} + \cdots + 43046721 $ T^12 - 81T^10 + 4779T^8 - 131220T^6 + 2644083T^4 - 11691702*T^2 + 43046721
$19$	$ (T^{6} - 18 T^{4} + \cdots + 6859)^{2} $ (T^6 - 18T^4 - 7T^3 - 342*T^2 + 6859)^2
$23$	$ T^{12} - 118 T^{10} + \cdots + 4100625 $ T^12 - 118T^10 + 11143T^8 - 324108T^6 + 7495011T^4 - 5631525*T^2 + 4100625
$29$	$ (T^{6} - 6 T^{5} + \cdots + 263169)^{2} $ (T^6 - 6T^5 + 117T^4 - 540T^3 + 9639T^2 - 41553*T + 263169)^2
$31$	$ (T^{3} - 11 T^{2} + \cdots + 71)^{4} $ (T^3 - 11T^2 + 14T + 71)^4
$37$	$ (T^{6} + 18 T^{4} + \cdots + 25)^{2} $ (T^6 + 18T^4 + 69T^2 + 25)^2
$41$	$ (T^{6} + 6 T^{5} + \cdots + 18225)^{2} $ (T^6 + 6T^5 + 99T^4 - 108T^3 + 4779T^2 + 8505*T + 18225)^2
$43$	$ T^{12} + \cdots + 121550625 $ T^12 - 149T^10 + 19407T^8 - 394256T^6 + 6163711T^4 - 30803850*T^2 + 121550625
$47$	$ T^{12} + \cdots + 69257922561 $ T^12 - 198T^10 + 26487T^8 - 1991628T^6 + 109614627T^4 - 3346720173*T^2 + 69257922561
$53$	$ T^{12} - 109 T^{10} + \cdots + 6561 $ T^12 - 109T^10 + 10747T^8 - 123444T^6 + 1277127T^4 - 91854*T^2 + 6561
$59$	$ (T^{6} + 6 T^{5} + \cdots + 263169)^{2} $ (T^6 + 6T^5 + 117T^4 + 540T^3 + 9639T^2 + 41553*T + 263169)^2
$61$	$ (T^{6} + 7 T^{5} + \cdots + 3969)^{2} $ (T^6 + 7T^5 + 66T^4 + 7T^3 + 730T^2 + 1071*T + 3969)^2
$67$	$ T^{12} + \cdots + 796594176 $ T^12 - 200T^10 + 32880T^8 - 1367552T^6 + 45049600T^4 - 200954880*T^2 + 796594176
$71$	$ (T^{6} - 9 T^{5} + \cdots + 729)^{2} $ (T^6 - 9T^5 + 99T^4 + 108T^3 + 567T^2 - 486*T + 729)^2
$73$	$ T^{12} - 186 T^{10} + \cdots + 390625 $ T^12 - 186T^10 + 30735T^8 - 716896T^6 + 14791071T^4 - 2413125*T^2 + 390625
$79$	$ (T^{6} - 32 T^{5} + \cdots + 732736)^{2} $ (T^6 - 32T^5 + 716T^4 - 8144T^3 + 67472T^2 - 263648*T + 732736)^2
$83$	$ (T^{6} + 373 T^{4} + \cdots + 731025)^{2} $ (T^6 + 373T^4 + 33426T^2 + 731025)^2
$89$	$ (T^{6} - 2 T^{5} + \cdots + 5184)^{2} $ (T^6 - 2T^5 + 136T^4 + 120T^3 + 17568T^2 - 9504*T + 5184)^2
$97$	$ T^{12} + \cdots + 8653650625 $ T^12 - 309T^10 + 79935T^8 - 4617664T^6 + 212933391T^4 - 1446166650*T^2 + 8653650625
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Classical	Maass
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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:	\( N \)	\(=\)	\( 1900 = 2^{2} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1900.s (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\( q + (\beta_{11} + \beta_{6}) q^{3} + (\beta_{11} + \beta_{10}) q^{7} + (\beta_{5} - \beta_{3} + \cdots - 2 \beta_1) q^{9}+O(q^{10}) \) q + (b11 + b6) * q^3 + (b11 + b10) * q^7 + (b5 - b3 + 3b2 - 2b1) * q^9	\( q + (\beta_{11} + \beta_{6}) q^{3} + (\beta_{11} + \beta_{10}) q^{7} + (\beta_{5} - \beta_{3} + \cdots - 2 \beta_1) q^{9}+ \cdots + (5 \beta_{8} + \beta_{5} + \cdots - 2 \beta_1) q^{99}+O(q^{100}) \) q + (b11 + b6) * q^3 + (b11 + b10) * q^7 + (b5 - b3 + 3b2 - 2b1) * q^9 + (b5 + 2) * q^11 - b7 * q^13 + (2b9 - b6 + b4) q^17 + (b5 + b2 + b1) * q^19 + (b3 + 6b2 - b1 - 6) q^21 + (-b9 - 5b7 + b4) q^23 + (4b11 - 3b10 + b9 + b6 + 2b4) q^27 + (b8 + 2b5 - 2b3 + 3b2 - 4b1) * q^29 + (b8 - 2b3 - b1 + 4) q^31 + (6b10 - 2b9 - 6b7 + b6 - b4) q^33 + (b11 + b10) * q^37 + b5 * q^39 + (3b8 - 3b5 + 3b2 - 3) q^41 + (-b11 - 3b10 + 2b9 + 3b7 - 2b6 + b4) * q^43 + (2b9 + 3b7 - b6 - 2b4) q^47 + (-b8 - 2b5 + 2b3 + b1) * q^49 + (-2b8 - b5 + b3 - 9b2 + 2b1) q^51 + (b9 - 4b7 - b4) q^53 + (-3b11 + 9b10 - 3b9 - 9b7 - b6 - 2b4) q^57 + (-b8 + b5 + 2b3 + 3b2 - 2b1 - 3) q^59 + (-2b8 - 3b2) * q^61 - 5b6 q^63 + (2b9 - 2b6 - 2b4) q^67 + (-b8 + 7b5 + 2b3 + b1 - 3) * q^69 + (2b8 - 2b5 - b3 - 3b2 + b1 + 3) q^71 + (-7b10 + 2b9 + 7b7 - b6 + b4) q^73 + (b11 + 8b10 - b9 - b6 - 2b4) * q^77 + (2b8 - 2b5 - 10b2 + 10) q^79 + (-5b8 + 5b5 + 6b2 - 6) q^81 + (2b11 + 10b10 + b9 + b6 + 2b4) q^83 + (9b11 + b9 + b6 + 2b4) * q^87 + (4b8 + 2b2) * q^89 + (b8 + b2) * q^91 + (8b11 - 3b10 + 2b9 + 3b7 + 7b6 + b4) q^93 + (5b11 - 4b10 + 4b9 + 4b7 + 3b6 + 2b4) * q^97 + (5b8 + b5 - b3 + 3b2 - 2b1) q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 16 q^{9}+O(q^{10}) \) 12 * q + 16 * q^9	\( 12 q + 16 q^{9} + 20 q^{11} - 32 q^{21} + 12 q^{29} + 44 q^{31} - 4 q^{39} - 12 q^{41} + 12 q^{49} - 48 q^{51} - 12 q^{59} - 14 q^{61} - 60 q^{69} + 18 q^{71} + 64 q^{79} - 46 q^{81} + 4 q^{89} + 4 q^{91} + 6 q^{99}+O(q^{100}) \) 12 * q + 16 * q^9 + 20 * q^11 - 32 * q^21 + 12 * q^29 + 44 * q^31 - 4 * q^39 - 12 * q^41 + 12 * q^49 - 48 * q^51 - 12 * q^59 - 14 * q^61 - 60 * q^69 + 18 * q^71 + 64 * q^79 - 46 * q^81 + 4 * q^89 + 4 * q^91 + 6 * q^99

Introduction

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Newspace parameters

Newform invariants

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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	1900.2.s.c

Level	$1900$
Weight	$2$
Character orbit	1900.s
Analytic conductor	$15.172$
Analytic rank	$0$
Dimension	$12$
CM	no
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(15.1715763840\)
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:	\( x^{12} - 9x^{10} + 59x^{8} - 180x^{6} + 403x^{4} - 198x^{2} + 81 \) x^12 - 9x^10 + 59x^8 - 180x^6 + 403x^4 - 198*x^2 + 81
Coefficient ring:	\(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index:	\( 3^{2} \)
Twist minimal:	no (minimal twist has level 380)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

\(\beta_{1}\)	\(=\)	\( ( 117\nu^{10} - 767\nu^{8} + 7472\nu^{6} - 27234\nu^{4} + 90554\nu^{2} - 60864 ) / 21995 \) (117v^10 - 767v^8 + 7472v^6 - 27234v^4 + 90554*v^2 - 60864) / 21995
\(\beta_{2}\)	\(=\)	\( ( -1298\nu^{10} + 10953\nu^{8} - 71803\nu^{6} + 202311\nu^{4} - 490451\nu^{2} + 240966 ) / 197955 \) (-1298v^10 + 10953v^8 - 71803v^6 + 202311v^4 - 490451*v^2 + 240966) / 197955
\(\beta_{3}\)	\(=\)	\( ( 461\nu^{10} - 5466\nu^{8} + 28501\nu^{6} - 98847\nu^{4} + 142112\nu^{2} - 186237 ) / 65985 \) (461v^10 - 5466v^8 + 28501v^6 - 98847v^4 + 142112*v^2 - 186237) / 65985
\(\beta_{4}\)	\(=\)	\( ( 569\nu^{11} - 6174\nu^{9} + 40474\nu^{7} - 169668\nu^{5} + 474413\nu^{3} - 729693\nu ) / 197955 \) (569v^11 - 6174v^9 + 40474v^7 - 169668v^5 + 474413v^3 - 729693v) / 197955
\(\beta_{5}\)	\(=\)	\( ( -288\nu^{10} + 1888\nu^{8} - 9933\nu^{6} + 12896\nu^{4} - 6336\nu^{2} - 68439 ) / 21995 \) (-288v^10 + 1888v^8 - 9933v^6 + 12896v^4 - 6336*v^2 - 68439) / 21995
\(\beta_{6}\)	\(=\)	\( ( 81\nu^{11} - 531\nu^{9} + 3481\nu^{7} - 3627\nu^{5} + 1782\nu^{3} + 98293\nu ) / 21995 \) (81v^11 - 531v^9 + 3481v^7 - 3627v^5 + 1782v^3 + 98293v) / 21995
\(\beta_{7}\)	\(=\)	\( ( 288\nu^{11} - 1888\nu^{9} + 9933\nu^{7} - 12896\nu^{5} + 6336\nu^{3} + 90434\nu ) / 65985 \) (288v^11 - 1888v^9 + 9933v^7 - 12896v^5 + 6336v^3 + 90434v) / 65985
\(\beta_{8}\)	\(=\)	\( ( 1138\nu^{10} - 12348\nu^{8} + 80948\nu^{6} - 273351\nu^{4} + 552916\nu^{2} - 271656 ) / 65985 \) (1138v^10 - 12348v^8 + 80948v^6 - 273351v^4 + 552916*v^2 - 271656) / 65985
\(\beta_{9}\)	\(=\)	\( ( 2027\nu^{11} - 15732\nu^{9} + 103132\nu^{7} - 234954\nu^{5} + 506489\nu^{3} + 445716\nu ) / 197955 \) (2027v^11 - 15732v^9 + 103132v^7 - 234954v^5 + 506489v^3 + 445716v) / 197955
\(\beta_{10}\)	\(=\)	\( ( 88\nu^{11} - 783\nu^{9} + 4868\nu^{7} - 13716\nu^{5} + 26581\nu^{3} - 2916\nu ) / 7155 \) (88v^11 - 783v^9 + 4868v^7 - 13716v^5 + 26581v^3 - 2916v) / 7155
\(\beta_{11}\)	\(=\)	\( ( -1298\nu^{11} + 10953\nu^{9} - 71803\nu^{7} + 202311\nu^{5} - 450860\nu^{3} + 43011\nu ) / 39591 \) (-1298v^11 + 10953v^9 - 71803v^7 + 202311v^5 - 450860v^3 + 43011v) / 39591

\(\nu\)	\(=\)	\( ( -\beta_{9} + 2\beta_{6} + \beta_{4} ) / 3 \) (-b9 + 2*b6 + b4) / 3
\(\nu^{2}\)	\(=\)	\( ( -2\beta_{8} + 2\beta_{5} + \beta_{3} - 9\beta_{2} - \beta _1 + 9 ) / 3 \) (-2b8 + 2b5 + b3 - 9*b2 - b1 + 9) / 3
\(\nu^{3}\)	\(=\)	\( ( 3\beta_{11} + 5\beta_{9} + 5\beta_{6} + 10\beta_{4} ) / 3 \) (3b11 + 5b9 + 5b6 + 10b4) / 3
\(\nu^{4}\)	\(=\)	\( -\beta_{8} + 2\beta_{5} - 2\beta_{3} - 12\beta_{2} - 4\beta_1 \) -b8 + 2b5 - 2b3 - 12b2 - 4b1
\(\nu^{5}\)	\(=\)	\( ( 18\beta_{11} + 9\beta_{10} + 46\beta_{9} - 9\beta_{7} - 5\beta_{6} + 23\beta_{4} ) / 3 \) (18b11 + 9b10 + 46b9 - 9b7 - 5b6 + 23b4) / 3
\(\nu^{6}\)	\(=\)	\( ( 32\beta_{8} - 5\beta_{5} - 64\beta_{3} - 32\beta _1 - 153 ) / 3 \) (32b8 - 5b5 - 64b3 - 32b1 - 153) / 3
\(\nu^{7}\)	\(=\)	\( ( 106\beta_{9} - 81\beta_{7} - 116\beta_{6} - 106\beta_{4} ) / 3 \) (106b9 - 81b7 - 116b6 - 106b4) / 3
\(\nu^{8}\)	\(=\)	\( 51\beta_{8} - 51\beta_{5} - 55\beta_{3} + 222\beta_{2} + 55\beta _1 - 222 \) 51b8 - 51b5 - 55b3 + 222b2 + 55*b1 - 222
\(\nu^{9}\)	\(=\)	\( ( -495\beta_{11} - 531\beta_{10} - 493\beta_{9} - 493\beta_{6} - 986\beta_{4} ) / 3 \) (-495b11 - 531b10 - 493b9 - 493b6 - 986*b4) / 3
\(\nu^{10}\)	\(=\)	\( ( -191\beta_{8} - 835\beta_{5} + 835\beta_{3} + 2952\beta_{2} + 1670\beta_1 ) / 3 \) (-191b8 - 835b5 + 835b3 + 2952b2 + 1670*b1) / 3
\(\nu^{11}\)	\(=\)	\( ( -2505\beta_{11} - 3078\beta_{10} - 4624\beta_{9} + 3078\beta_{7} - 193\beta_{6} - 2312\beta_{4} ) / 3 \) (-2505b11 - 3078b10 - 4624b9 + 3078b7 - 193b6 - 2312b4) / 3

\(n\)	\(77\)	\(401\)	\(951\)
\(\chi(n)\)	\(-1\)	\(-\beta_{2}\)	\(1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 49.6
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{12} \) T^12
$3$	\( T^{12} - 17 T^{10} + \cdots + 81 \) T^12 - 17T^10 + 219T^8 - 1172T^6 + 4747T^4 - 630*T^2 + 81
$5$	\( T^{12} \) T^12
$7$	\( (T^{6} + 18 T^{4} + \cdots + 25)^{2} \) (T^6 + 18T^4 + 69T^2 + 25)^2
$11$	\( (T^{3} - 5 T^{2} + 9)^{4} \) (T^3 - 5*T^2 + 9)^4
$13$	\( (T^{4} - T^{2} + 1)^{3} \) (T^4 - T^2 + 1)^3
$17$	\( T^{12} - 81 T^{10} + \cdots + 43046721 \) T^12 - 81T^10 + 4779T^8 - 131220T^6 + 2644083T^4 - 11691702*T^2 + 43046721
$19$	\( (T^{6} - 18 T^{4} + \cdots + 6859)^{2} \) (T^6 - 18T^4 - 7T^3 - 342*T^2 + 6859)^2
$23$	\( T^{12} - 118 T^{10} + \cdots + 4100625 \) T^12 - 118T^10 + 11143T^8 - 324108T^6 + 7495011T^4 - 5631525*T^2 + 4100625
$29$	\( (T^{6} - 6 T^{5} + \cdots + 263169)^{2} \) (T^6 - 6T^5 + 117T^4 - 540T^3 + 9639T^2 - 41553*T + 263169)^2
$31$	\( (T^{3} - 11 T^{2} + \cdots + 71)^{4} \) (T^3 - 11T^2 + 14T + 71)^4
$37$	\( (T^{6} + 18 T^{4} + \cdots + 25)^{2} \) (T^6 + 18T^4 + 69T^2 + 25)^2
$41$	\( (T^{6} + 6 T^{5} + \cdots + 18225)^{2} \) (T^6 + 6T^5 + 99T^4 - 108T^3 + 4779T^2 + 8505*T + 18225)^2
$43$	\( T^{12} + \cdots + 121550625 \) T^12 - 149T^10 + 19407T^8 - 394256T^6 + 6163711T^4 - 30803850*T^2 + 121550625
$47$	\( T^{12} + \cdots + 69257922561 \) T^12 - 198T^10 + 26487T^8 - 1991628T^6 + 109614627T^4 - 3346720173*T^2 + 69257922561
$53$	\( T^{12} - 109 T^{10} + \cdots + 6561 \) T^12 - 109T^10 + 10747T^8 - 123444T^6 + 1277127T^4 - 91854*T^2 + 6561
$59$	\( (T^{6} + 6 T^{5} + \cdots + 263169)^{2} \) (T^6 + 6T^5 + 117T^4 + 540T^3 + 9639T^2 + 41553*T + 263169)^2
$61$	\( (T^{6} + 7 T^{5} + \cdots + 3969)^{2} \) (T^6 + 7T^5 + 66T^4 + 7T^3 + 730T^2 + 1071*T + 3969)^2
$67$	\( T^{12} + \cdots + 796594176 \) T^12 - 200T^10 + 32880T^8 - 1367552T^6 + 45049600T^4 - 200954880*T^2 + 796594176
$71$	\( (T^{6} - 9 T^{5} + \cdots + 729)^{2} \) (T^6 - 9T^5 + 99T^4 + 108T^3 + 567T^2 - 486*T + 729)^2
$73$	\( T^{12} - 186 T^{10} + \cdots + 390625 \) T^12 - 186T^10 + 30735T^8 - 716896T^6 + 14791071T^4 - 2413125*T^2 + 390625
$79$	\( (T^{6} - 32 T^{5} + \cdots + 732736)^{2} \) (T^6 - 32T^5 + 716T^4 - 8144T^3 + 67472T^2 - 263648*T + 732736)^2
$83$	\( (T^{6} + 373 T^{4} + \cdots + 731025)^{2} \) (T^6 + 373T^4 + 33426T^2 + 731025)^2
$89$	\( (T^{6} - 2 T^{5} + \cdots + 5184)^{2} \) (T^6 - 2T^5 + 136T^4 + 120T^3 + 17568T^2 - 9504*T + 5184)^2
$97$	\( T^{12} + \cdots + 8653650625 \) T^12 - 309T^10 + 79935T^8 - 4617664T^6 + 212933391T^4 - 1446166650*T^2 + 8653650625
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